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PhD Thesis
Optimization of Densities in
Hartree-Fock and Density-functional Theory
Atomic Orbital Based Response Theory
and
Benchmarking for Radicals
Lea Thøgersen
Department of Chemistry
University of Aarhus
2005

"Experiments are the only means of knowledge at our disposal.
The rest is poetry, imagination."
Max Planck

Contents
Preface .........................................................................................................................v
List of Publications ....................................................................................................vii
Part 1 Improving Self-consistent Field Convergence.................................................1
1.1 Introduction .....................................................................................................................1
1.2 The Self-consistent Field Method....................................................................................2
1.3 A Survey of Methods for Improving SCF Convergence .................................................5
1.3.1 Energy Minimization.............................................................................................6
1.3.2 Damping and Extrapolation...................................................................................7
1.3.3 Level Shifting......................................................................................................11
1.4 Development of SCF Optimization Algorithms ............................................................12
1.4.1 Dynamically Level Shifted Roothaan-Hall .........................................................13
1.4.1.1 RH Step with Control of Density Change..............................................13
1.4.1.2 The Trust Region RH Level Shift ..........................................................15
1.4.1.3 DIIS and Dynamically Level Shifted RH ..............................................16
1.4.1.4 Line Search TRRH.................................................................................18
1.4.1.5 Optimal Level Shift without MO Information.......................................19
1.4.1.6 The Trace Purification Scheme..............................................................23
1.4.2 Density Subspace Minimization..........................................................................25
1.4.2.1 The Trust Region DSM Parameterization..............................................25
1.4.2.2 The Trust Region DSM Energy Function ..............................................26
1.4.2.3 The Trust Region DSM Minimization ...................................................27
1.4.2.4 Line Search TRDSM..............................................................................29
1.4.2.5 The Missing Term..................................................................................30
1.4.3 Energy Minimization Exploiting the Density Subspace .....................................32
1.4.3.1 The Augmented RH Energy model........................................................33
1.4.3.2 The Augmented RH Optimization .........................................................34
1.4.3.3 Applications ...........................................................................................36
1.5 The Quality of the Energy Models for HF and DFT .....................................................37
1.5.1 The Quality of the TRRH Energy Model............................................................39
1.5.2 The Quality of the TRDSM Energy Model.........................................................42
1.6 Convergence for Problems with Several Stationary Points...........................................44
1.6.1 Walking Away from Unstable Stationary Points ................................................46
1.6.1.1 Theory....................................................................................................46
1.6.1.2 Examples................................................................................................47
i

1.7 Scaling .......................................................................................................................... 48
1.7.1 Scaling of TRRH ................................................................................................ 49
1.7.2 Scaling of TRDSM ............................................................................................. 51
1.8 Applications.................................................................................................................. 51
1.8.1 Calculations on Small Molecules ....................................................................... 52
1.8.2 Calculations on Metal Complexes...................................................................... 54
1.9 Conclusion .................................................................................................................... 56
Part 2 Atomic Orbital Based Response Theory........................................................ 59
2.1 Introduction................................................................................................................... 59
2.2 AO Based Response Equations in Second Quantization .............................................. 60
2.2.1 The Parameterization.......................................................................................... 60
2.2.2 The Linear Response Function ........................................................................... 62
2.2.3 The Time Development of the Reference State.................................................. 63
2.2.4 The First-order Equation .................................................................................... 64
2.2.5 Pairing................................................................................................................. 66
2.3 Solving the Response Equations................................................................................... 68
2.3.1 Preconditioning................................................................................................... 69
2.3.2 Projections .......................................................................................................... 70
2.4 The Excited State Gradient ........................................................................................... 71
2.4.1 Construction of the Lagrangian .......................................................................... 71
2.4.2 The Lagrange Multipliers ................................................................................... 72
2.4.3 The Geometrical Gradient .................................................................................. 73
2.4.4 The First-order Excited State Properties............................................................. 74
2.5 Test Calculations........................................................................................................... 75
2.6 Conclusion .................................................................................................................... 76
Part 3 Benchmarking for Radicals............................................................................ 77
3.1 Introduction................................................................................................................... 77
3.2 Computational Methods................................................................................................ 77
3.3 Numerical Results......................................................................................................... 79
3.3.1 Convergence of CC and CI Hierarchies ............................................................. 79
3.3.2 The Potential Curve for CN................................................................................ 80
3.3.3 Spectroscopic Constants and Atomization Energy for CN................................. 81
3.3.4 The Vertical Electron Affinity of CN................................................................. 82
3.3.5 The Equilibrium Geometry of CCH ................................................................... 83
3.4 Conclusion .................................................................................................................... 84
ii

Summary....................................................................................................................87
Dansk Resumé ...........................................................................................................89
Appendix A................................................................................................................91
Appendix B................................................................................................................93
Acknowledgements....................................................................................................95
References..................................................................................................................97
iii

Preface
The present PhD thesis is the outcome of four years of PhD studies at the Faculty of Science,
University of Aarhus, Denmark.
The thesis is divided into three distinct parts which can be read independently. Part 1 deals with the
optimization of the one-electron density in Hartree Fock and density functional theory, and Part 2
deals with atomic orbital based response theory for Hartree Fock and density functional theory. Part
2 thus naturally follows after Part 1. In Part 3 benchmark results from FCI calculations on the
radicals CN and CCH are given.
The work presented in Part 1 has resulted in papers I - III as listed in the following List of
Publications and the work presented in Part 3 has resulted in papers V – VI. The work presented in
Part 2 was initialized in the fall 2004 and will result in paper IV. The development of improved
optimization algorithms for self-consistent field calculations is the subject on which I have spent the
most of my time, and Part 1 therefore makes up the larger part of this thesis.
The work has been carried out under the supervision of and in collaboration with Dr. Jeppe Olsen
and Professor Poul Jørgensen at the University of Aarhus. Some work was carried out during visits
at The Royal Institute of Technology in Stockholm, Sweden, the University of Trieste, Italy and the
University of Oslo, Norway. The following people have also contributed to the work presented in
this thesis (see List of Publications): Paweł Sałek (The Royal Institute of Technology in
Stockholm), Sonia Coriani (University of Trieste), Trygve Helgaker (University of Oslo), Stinne
Høst (University of Aarhus), Danny Yeager (Texas A&M University), Andreas Köhn (University of
Aarhus), Jürgen Gauss (University of Mainz), Péter Szalay (Eötvös Loránd University) and Mihály
Kállay (University of Mainz).
The outline of the thesis is as follows: Part 1 is based on the published papers I – II and the
unpublished paper III, but can be read independently of the papers. Certain discussions in the papers
I - II are left out of the thesis and only referred to, as they might as well be read in the papers. Other
discussions not published in the papers are presented in this thesis, including the latest
developments of the algorithms. Part 2 is simply paper IV in preparation. Part 3 is based on the
published papers V – VI and is basically a short version of paper V combined with selected results
from paper VI. Also this part can be read independently of the papers.
v

List of Publications
This thesis includes the following papers. Number I, II, V and VI have already been published and
are attached this thesis, whereas III and IV are in preparation.
Part 1
I. The Trust-region Self-consistent Field Method: Towards a Black Box optimization in Hartree-
Fock and Kohn-Sham Theories,
L. Thøgersen, J. Olsen, D. Yeager, P. Jørgensen, P. Sałek, and T. Helgaker,
J. Chem. Phys. 121, 16 (2004)
II. The Trust-region Self-consistent Field Method in Kohn-Sham Density-functional Theory,
L. Thøgersen, J. Olsen, A. Köhn, P. Jørgensen, P. Sałek, and T. Helgaker,
J. Chem. Phys. 123, 074103 (2005)
III. Augmented Roothaan-Hall for converging Densities in Hartree-Fock and Density-functional
Theory,
S. Høst, L. Thøgersen, P. Jørgensen and J. Olsen
Part 2
IV. Atomic Orbital Based Response Theory,
L. Thøgersen, P. Jørgensen, J. Olsen and S. Coriani
Part 3
V. A Coupled Cluster and Full Configuration Interaction Study of CN and CN - ,
L. Thøgersen and J. Olsen,
Chem. Phys. Lett. 393, 36 (2004)
VI. Equilibrium Geometry of the Ethynyl (CCH) Radical,
P. G. Szalay, L. Thøgersen, J. Olsen, M. Kállay and J. Gauss,
J. Phys. Chem. A 108, 3030 (2004).
vii

Part 1
Improving Self-consistent Field Convergence
1.1 Introduction
The Hartree-Fock (HF) self-consistent field (SCF) method has been around in an orbital formulation
since 1951, where it was introduced by Roothaan 1 and Hall 2 , but today it is as significant as ever.
Even though numerous higher correlated methods with superior accuracy have been developed
since then, most of them still use the Hartree-Fock wave function as the reference function, and are
thus still dependent on a functioning Hartree-Fock optimization. When Kohn and Sham 3 recognized
in 1965 that the Roothaan-Hall SCF scheme had a lot to offer the density optimization in density
functional theory (DFT), the DFT methods entered the chemical scene. Now it was in theory also
possible to obtain results at the exact level from SCF calculations; if only the correct functional
could be found. The developments in computer hardware and linear scaling SCF algorithms over
the last decade have made it possible to carry out ab initio quantum chemical calculations on biomolecules
with hundreds of amino acids and on large molecules relevant for nano-science.
Quantum chemical calculations are thus evolving to become a widespread tool for use in several
scientific branches. It is therefore important that the algorithms work as black-boxes, such that the
user outside quantum chemistry does not have to be concerned with the details of the calculations.
Since no scientific results neither from the higher correlated calculations nor from the large-scale
calculations can be achieved if the SCF optimization does not converge, it is necessary to take an
interest in developing a sound, stable optimization scheme that can handle the complexity in the
problems of the future.
This part of my thesis is a contribution to the quest for a black-box SCF optimization algorithm with
optimal convergence properties. In Section 1.2, the basic Hartree-Fock/Kohn-Sham theory and
notation of this part of the thesis is stated, and in Section 1.3 the efforts through the years to
1

Part 1
Improving Self-consistent Field Convergence
improve the Roothaan-Hall SCF scheme are reviewed. Our contributions to the development of
stable and physical sound SCF optimization schemes are presented in Section 1.4, and in Section
1.5 we study the quality of the schemes when applied for HF and DFT. Optimization of problems
with several stationary points is discussed in Section 1.6, in Section 1.7 the scaling of the algorithms
is accounted for, and Section 1.8 contains some convergence examples for HF and DFT calculations
using the algorithms presented in Section 1.4. Finally, Section 1.9 contains concluding remarks;
reviewing the results of this part of the thesis.
1.2 The Self-consistent Field Method
In the following we consider a closed-shell system with N/2 electron pairs. The basic theory of the
Hartree-Fock (HF) and the Kohn-Sham (KS) density optimizations will be described
simultaneously, and the differences will be noted as they appear. Since we are interested in
extending the algorithms presented to large scale calculations, a formulation without reference to
the delocalized molecular orbitals (MOs) is essential, and thus the focus will be on the density in the
atomic orbital (AO) basis rather than the MOs themselves. All through the thesis, SCF will be used
as a general term for HF and KS-DFT methods since they have the SCF optimization scheme in
common. The orbital index convention used in this thesis is i, j, k, l for occupied MOs, a, b, c, d for
virtual MOs, p, q for MOs in general, and Greek letters µ, ν, ρ, σ for AOs.
For closed-shell restricted Hartree-Fock or DFT, the electronic energy is given by
E = 2TrhD + Tr DG( D) + h + E ( D ), (1.1)
SCF nuc XC
where h is the one-electron Hamiltonian matrix in the AO basis, h nuc is the nuclear-nuclear repulsion
contribution, and D is the (scaled) one-electron density matrix in the AO basis, D = ½D AO , which
satisfies the symmetry, trace, and idempotency conditions,
D
T
Tr DS =
= D
N
2
DSD = D ,
(1.2)
of a valid one-electron density matrix. S is the AO overlap matrix. The elements of G(D) are given
by
∑
∑
G ( D ) = 2 g D −γ g D , (1.3)
µν µνρσ ρσ µσρν ρσ
ρσ
ρσ
where g µνρσ are the two-electron AO integrals. The first term in Eq. (1.3) represents the Coulomb
contribution, and the second term is the contribution from exact exchange, with γ = 1 in Hartree-
Fock theory, γ = 0 in pure DFT, and γ ≠ 0 in hybrid DFT. The exchange-correlation energy E XC (D)
in Eq. (1.1) is a nonlinear and non-quadratic functional of the electronic density. This term is only
2

The Self-consistent Field Method
present in the energy expression for the DFT level of theory - the Hartree-Fock energy is expressed
only by the first three terms of Eq. (1.1). The form of E XC depends on the DFT functional chosen for
the calculation.
The first derivative of the electronic energy with respect to the density is found as
where
(1) ∂ESCF
( D)
ESCF
( D) = = 2 F( D)
, (1.4)
∂D
1
2
(1)
XC
FD ( ) = h+ GD ( ) + E ( D )
(1.5)
is the Kohn-Sham matrix in DFT and, if the last term is excluded, the Fock matrix in Hartree-Fock
(1)
theory. From now on F(D) is simply referred to as the Fock matrix. E XC ( D ) is the first derivative
of the term E XC expanded in the density.
The Fock matrix is by design an effective one-electron Hamiltonian which is itself dependent on the
eigenfunctions. Optimizing the electronic energy is thus a nonlinear problem and an iterative
scheme must be applied. In 1951 Roothaan and Hall suggested an iterative procedure 1,2 in which a
set of molecular orbitals (MOs) are constructed in each step through a diagonalization of the current
Fock matrix, which in the AO formulation is written as
FC = SCε , (1.6)
where S is the AO overlap matrix, ε is a diagonal matrix containing the orbital energies, and the
eigenvectors C contain the MO coefficients. The MOs, φ p , are linear combinations of a finite set of
one-electron basis functions, χ µ , with C µp as expansion coefficients
ϕ
p
= ∑ χ C . (1.7)
µ
µ µ p
For the closed shell case the MOs can be divided into an occupied (φ occ ) and a virtual (φ virt ) part,
where the occupied MOs each contain two electrons and the virtual orbitals are empty. If the aufbau
ordering rule is applied, the occupied MOs are chosen as those with the lowest eigenvalues.
A new trial density D can then be constructed from the occupied orbitals as
occ
T
occ
D = C C . (1.8)
From this density a new Fock matrix can be evaluated from Eq. (1.5) and diagonalizing it according
to Eq. (1.6) establishes the iterative procedure. The iterative cycle stops when self-consistency is
obtained, that is, when the new density, energy or molecular orbitals do not change within some
convergence threshold compared to the previous ones.
3

Part 1
Improving Self-consistent Field Convergence
In an iterative scheme it is necessary to have a start guess. For the SCF case it should be a one
electron density which fulfils Eq. (1.2), created directly or from a start guess of the molecular
orbitals as in Eq. (1.8). Different approaches are used; a simple and easily applicable possibility is
to obtain the starting orbitals by diagonalization of the one-electron Hamiltonian (H1-core). This is
the start guess most widely used in this thesis since it is always available. Another popular
possibility is to create a semi-empirical start guess where the orbitals resulting from a semiempirical
calculation (e.g. Hückel) on the molecule are fitted to the current basis.
n = n+1
no
D 0
F(D n
)
F(D n
) D n+1
D n+1
≈ D n
yes
The steps of the self-consistent field (SCF) scheme are summarized
from the density point of view in Fig. 1.1: From a density matrix start
guess a Fock matrix is constructed. From this Fock matrix a new density
matrix can be found and so an iteration procedure is established which
continues until self consistency. The step creating a new density from a
Fock matrix will be referred to as the Roothaan-Hall (RH) step
throughout this thesis, regardless if it is a diagonalization of the Fock
matrix or some alternative scheme.
The purpose of an SCF optimization is typically to find the global
D conv
minimum. Since the HF/KS equations are nonlinear, several stationary
Fig. 1.1 Flow diagram of
points might exist, and depending on the start guess and the
the SCF scheme.
optimization procedure, the converged result can be representing a local
minimum as well as a global or even a saddle point. By evaluating the lowest Hessian eigenvalue it
can be realized whether the stationary point is a minimum or a saddle point, but no simple test can
reveal whether a minimum is global or not. The use of the term “convergence” in this thesis will
simply refer to the iterative development from the start guess to a self-consistent density with a
gradient below the convergence threshold. The issues connected with problems where several
stationary points can be found are discussed in Section 1.6.
Since Roothaan and Hall suggested the iterative diagonalization procedure as a means to solve the
Hartree-Fock equations and Kohn and Sham suggested using the same scheme for optimizing the
electron density for density functional theory 3 , the SCF methods have been used extensively in
quantum chemistry. Unfortunately, it turned out that the simple fixed point scheme sketched in Fig.
1.1 converges only in simple cases. Already around 1960 it was recognized that the method
sometimes fails to converge and that divergent behavior in some cases is intrinsic 4,5 .
4

A Survey of Methods for Improving SCF Convergence
1.3 A Survey of Methods for Improving SCF Convergence
Numerous suggestions have been made to improve upon the convergence of Roothaan and Hall’s
original scheme or to replace it with an alternative scheme. The suggestions can be crudely divided
into three different categories; energy minimization, damping/extrapolation, and level shifting.
Furthermore the different suggestions in these categories have been combined in various ways. The
two latter categories are modifications to the Roothaan-Hall scheme, whereas energy minimization
is a means of avoiding the iterative diagonalization scheme and instead use some optimization
scheme on an energy function.
To my knowledge these categories embrace all convergence improvements suggested over the
years, except for the method of fractionally occupying orbitals around the Fermi level 6 which does
not fit in any of the categories. As mentioned, the start guess has a great impact on the optimization,
and a poor start guess with the wrong electron configuration can use many iterations changing to a
more optimal electron configuration and in some cases the proper electron configuration is never
found and the calculation diverges. In the methods using fractional occupations, a number of
orbitals around the Fermi level are allowed to have non-integral occupation. The non-integral
occupations are determined from the Fermi-Dirac distribution which is a function of the
temperature. The non-integral occupations are updated in each iteration, and corrected such that the
total number of electrons is constant. During the optimization either the temperature is decreased to
T = 0K or the number of orbitals allowed to have non-integral occupation is decreased, to have only
integer occupations at the end of the optimization. It is thus possible to optimize the electron
configuration in an effective manner in the beginning of the SCF optimization, and when the proper
configuration has been found, the rest of the optimization has a better chance of convergence since
the start guess in a way has been improved.
In the following, the focus will be on the efforts to improve the convergence behavior of the SCF
scheme through optimization algorithm development in the three categories listed above. Other
efforts bear as much significance and should also be acknowledged, in particular should be
mentioned the generalizations of many well-functioning schemes to the unrestricted level of theory
which has its own challenges. Also the quest for construction of an improved start guess is
important. It is obvious that with an improved start guess, less is demanded from the optimization
method and thus some convergence problems inherent in the methods could be avoided. In the last
decade the effort in SCF scheme development has for a large part been put in decreasing the scaling
of the methods to allow calculations on larger molecules. Scaling is a very important subject and it
should not be ignored. Section 1.7 will therefore discuss the scaling of the algorithms presented in
5

Part 1
Improving Self-consistent Field Convergence
this thesis. Despite the importance of these three SCF related subjects, the rest of this section will be
almost solely on efforts to improve convergence through optimization algorithm development.
1.3.1 Energy Minimization
One of the problems in the simple Roothaan-Hall procedure is the lack of guarantees for energy
decrease in the iterative steps. This was pointed out by McWeeny, and he thus introduced a steepest
descent procedure 7,8 as an energy minimization alternative to Roothaan and Hall’s repeated
diagonalizations. Steepest descent optimizations have the benefit that a decrease in energy can be
guaranteed for each step. McWeeny’s scheme suffers, however, from a slow convergence rate 5 as
often seen for steepest descent methods. Fletcher and Reeves proposed the conjugate gradient
optimization method 9 instead, which often is more efficient than steepest descent and is guaranteed
to converge in a number of steps equal to the dimension of the problem.
A decade later Hilliers and Saunders suggested an improvement to the McWeeny scheme called
energy-weighted steepest descent 10 , in which the coordinates in the orbital space are energyweighted.
In 1976 this work was generalized by Seeger and Pople. They realized that another
problem in the simple Roothaan procedure is the possibility for discontinuous changes in the
orbitals which do not necessarily lower the energy. To ensure energy descent it is necessary to be
able to follow such changes continuously, and methods like the steepest descent have the possibility
to do so. Their procedure proceeds in small steps, where the new occupied trial orbitals are selected
based on a criterion of overlap with the previous set. This technique ensures stability and avoids
switching of orbital occupation. The step is found by a univariate search 11 in the energy, on a path
that passes through the point corresponding to the next iteration step of the classical procedure.
Their scheme can therefore also be seen as a polynomial interpolation along a path joining
successive SCF cycles. Half a decade later, Camp and King followed the same strategy of a
univariant cubic fit technique 12 , but with a different parameterization. Stanton also suggested a
similar approach 13 , but whereas the Seeger-Pople approach requires the evaluation of the Fock
matrix at interior points on the interpolative path, Stanton’s scheme uses a cubic interpolation,
where only the end point properties are needed, making it a less expensive method.
Another way of improving the convergence properties is to evaluate the gradient and Hessian of the
electronic energy analytically with respect to some variational parameter, and then optimize the
energy through Newton-Raphson steps resulting in a quadratically convergent 14 scheme, at least in
the region close to the optimized state where a second order approximation is reasonable. These
methods are computationally very expensive since a four index transformation is required to obtain
the Hessian information. In 1981 Bacskay proposed a quadratically convergent SCF (QC-SCF)
method 15 which escapes the four index transformation while requiring four or five micro iterations
6

A Survey of Methods for Improving SCF Convergence
per step (in non-problematic cases), each of which is about as expensive computationally as
building a Fock matrix. His method was inspired from single excitation configuration interaction
(SX-CI) and multi-configurational SCF (MC-SCF). A possible divergence of the scheme can be
overcome by moderating the orbital update step by the augmented Hessian method 16 or trust radius
techniques 17 . Even though it is still quite expensive, the method is also used today for cases with
convergence problems, since a decrease in energy can be ensured step by step and it has quadratic
convergence properties near the optimized state.
Around 1995, the interest for linear scaling SCF methods took on, since the development in
computer hardware had made calculations on large molecules possible. With newly developed
algorithms the evaluation of the Fock matrix, with the formal scaling of N 4 arising from the fourindex
integrals, could now routinely be decreased to a near-linear scaling. The diagonalization with
a N 3 scaling in standard Roothaan-Hall was now the bottle neck. Inspiration was found in tight
binding theory 18-20 , where a number of linear scaling approaches had been suggested earlier 21 . To
obtain linear scaling of the RH step it is necessary to avoid the diagonalization and to ensure
sparsity in the matrices. This is a problem since the convenient canonical MO basis is inherently
delocalized. Some of the well known schemes were reformulated in localized MOs 22 , while others
developed strict AO formulations 20,23-25 . Most of the suggested linear scaling methods did not arise
so much to improve convergence as to improve the scaling, and will therefore not be discussed in
further detail.
Very recently Francisco, Martínez and Martínez introduced their globally convergent trust region
methods for SCF 26 , where the standard fixed-point Roothaan-Hall step is replaced by a trust region
optimization of a model energy function. This algorithm has very nice features since it can be
proved to be globally convergent, and the step sizes are controlled dynamically through a trust
region update scheme. The convergence rate seems rather random though; sometimes perfect and
sometimes hopeless, but only small test examples have been published, so time will show.
1.3.2 Damping and Extrapolation
In his SCF study of atoms, Hartree noted convergence difficulties and suggested a so-called
damping scheme 27 as a modification to the iterative procedure. Instead of using the newly
constructed density D n+1 , which corresponds to a full step, a linear combination of the new density
matrix with the previous one is constructed
damp
Dn+ 1
= Dn + λ( Dn+ 1 − Dn ) = λDn+
1 + ( 1 −λ)
D n , (1.9)
7

Part 1
Improving Self-consistent Field Convergence
where λ – the damping factor - is a scalar chosen between zero and one. The iterative sequence is
then continued with D damp as the new density. Hartree found that this scheme could force
convergence in problematic cases.
To get an idea of the effect of the damping factor, we consider a block-diagonal Fock matrix in the
MO basis
F
MO
⎛ εo
Fov
⎞
= ⎜ ⎟ , (1.10)
⎝Fvo
εv
⎠
where ‘o’ denotes occupied, ‘v’ virtual and [ε o ] ij = δ ij ε i and [ε v ] ab = δ ab ε a . The change in electronic
energy from the first order variation of the occupied orbitals through first-order perturbation theory
is then given as
virtual occupied 2
( 1)
−Fai
SCF
4
a i
εa
− εi
∆ E =
∑ ∑ . (1.11)
( )
If this first order term is negative and sufficiently small such that the higher order contributions are
insignificant, then a decrease in the electronic energy is seen. If the MOs obey the aufbau principle,
then all ε i < ε a and it is clear that the term is negative as desired. The Hartree damping of Eq. (1.9)
roughly corresponds to multiplying the numerator of Eq. (1.11) by the factor λ, which is positive
and less than one
virtual occupied 2
( 1)
−λFai
SCF
4
a i
εa
− εi
∆ E =
∑ ∑ , (1.12)
( )
thus giving the opportunity to obtain a negative first order change of arbitrarily small magnitude,
making the higher order terms insignificant. Though this would seem promising, the aufbau
principle is seldom obeyed all through the optimization.
If λ could be freely chosen, the damping technique would lead to an extrapolation scheme in the
densities. Since SCF generates an iterative sequence where each step only depends upon the
preceding, it was natural to apply the mathematical extrapolation methods (e.g. the Aitken
extrapolation 28 procedures) on SCF to improve in particular the convergence rate close to the
minimum. When the individual MO expansion coefficients are chosen as the extrapolated
parameters, as Winter and Dunning Jr. 29 suggested, unphysical result may be obtained, though they
can be corrected at the end of the calculation. Nielsen used instead the density matrix as the
extrapolated parameter 30 and an eigenvalue extrapolation instead of the Aitken method. This led to a
scheme more similar to Hartree damping, but with λ found within the eigenvalue extrapolation
scheme.
8

A Survey of Methods for Improving SCF Convergence
Different approaches have been taken to dynamically find the damping factor λ. Zerner and
Hehenberger 31 found it based on an extrapolation of the Mulliken gross population. Karlström 32
expressed the electronic energy in the damped density E(D damp ) and used the first derivative with
respect to λ, to choose in each iteration the λ that minimized the electronic energy.
None of these schemes were very successful solving the convergence problems. They all had some
particular problematic cases they could handle better than the predecessors, but in general they did
not catch on. Pulay then suggested in the early 1980s to use the norm of a linear combination of
error vectors e i from the individual iterations, where the vanishing of the error vector is a necessary
and sufficient condition for SCF convergence. The norm is then optimized with respect to the
coefficients c i
n
e ( c)
= ∑ ciei
, (1.13)
where n is the number of previous iterations, and the coefficients are restricted to add up to 1
n
i=
1
i=
1
∑ ci
= 1. (1.14)
The resulting coefficients are used to construct a favorable linear combination of the previous Fock
matrices
n
F = ∑ ciF i , (1.15)
i=
1
which is diagonalized to obtain a new density, and so the iterative procedure is reestablished. This
was the first density subspace minimization scheme that deliberately exploited the information
obtained in the previous iterations and he named the approach DIIS 33 for “Direct Inversion in the
Iterative Subspace”. For the special case of two matrices, the DIIS density corresponds to the
damped density of Eq. (1.9), but with no restrictions on λ. A decade later the DIIS algorithm was a
standard option in most ab initio programs and had effectively solved a number of the convergence
problems. The orbital rotation gradient was typically used as the error vector for wave function
optimizations, and Sellers pointed out 34 that the DIIS algorithm exploits the second-order
information contained in a set of gradients to obtain quadratic convergence behavior. Some
numerical problems were seen though, where numerical instabilities appeared because of linear
dependencies in the space of error vectors. Sellers introduced the C2-DIIS method 34 , which is
similar to DIIS except the restriction is on the squares of the coefficients
n
2
∑ ci
= 1 , (1.16)
i=
1
9

Part 1
Improving Self-consistent Field Convergence
with a renormalization at the end. This gives an eigenvalue problem to be solved instead of the set
of linear equations in normal DIIS, and thus singularities are more easily handled. However, one of
the examples (Pd 2 in the Hyla-Kripsin basis set 35 ) given in ref. 34 , where DIIS supposedly diverges,
converges for our plain DIIS implementation to 10 -7 in the energy in 14 iterations.
Even though DIIS is successful, examples of divergence with no relation to numerical instabilities
have been encountered over the years. In the year 2000 Cancès and Le Bris presented a damping
algorithm named the Optimal damping Algorithm 36 (ODA) that ensures a decrease in energy at each
iteration and converges toward a solution to the HF equations. In ODA the damping factor λ is
found based on the minimum of the Hartree-Fock energy for the damped density in Eq. (1.9)
E
damp
( Dn+
1
, λ) = E ( Dn ) + 2λTrF( Dn )( Dn+
−Dn
)
HF HF 1
2
+ λ Tr ( D −D ) G( D − D ) + h ,
n+ 1 n n+
1 n nuc
(1.17)
much like Karlström did it in 1979. The damping factor is thus optimized in each iteration, hence
the name of the algorithm.
Recently Kudin, Scuseria, and Cancès proposed a method in which the gradient-norm minimization
in DIIS is replace by a minimization of an approximation to the true energy function and they
named it the energy DIIS (EDIIS) method 37 . Where the ODA used the energy expression of Eq.
(1.17) to find the optimal λ, EDIIS uses an approximation of the Hartree-Fock energy for the
averaged density
n
EDIIS 1
n
D = ∑ ciD i , (1.18)
i=
1
( , ) = ∑ i SCF ( i ) −
2 ∑ i j Tr( ( i − j ) ⋅( i − j ))
i= 1 i, j=
1
n
E Dc c E D c c F F D D , (1.19)
where the sum of the coefficients c i is still restricted to 1. They combine the scheme with DIIS, such
that the EDIIS optimized coefficients are used to construct the averaged Fock matrix if all
coefficients fall between 0 and 1. If not, the coefficients from the DIIS scheme are used instead. The
EDIIS scheme introduces some Hessian information not found in DIIS and thus improves
convergence in cases where the start guess has a Hessian structure far from the optimized one. For
non-problematic cases and near the optimized state EDIIS has a slower convergence rate than DIIS,
but it has been demonstrated that EDIIS can converge cases where DIIS diverges.
Recently, we suggested another subspace minimization algorithm along the same line as EDIIS, but
with a smaller idempotency error in the energy model and the same orbital rotation gradient in the
subspace as the SCF energy (the EDIIS energy model actually has a different gradient). We named
it TRDSM 38 for trust region density subspace minimization since a trust region optimization is
10

A Survey of Methods for Improving SCF Convergence
carried out of the energy model in the subspace of previous densities. In the second paper on
TRDSM 39 , a comparison with the EDIIS and DIIS models can be found stating explicitly that the
EDIIS energy model does not have the correct gradient and is wrong for other reasons as well at the
DFT level of theory.
Many of the energy minimization techniques can be combined with a damping or extrapolation
scheme to improve the convergence. Typically, DIIS has been the choice 24,40,41 , but TRDSM could
be used just as well.
1.3.3 Level Shifting
In 1973 Saunders and Hillier introduced the level shift concept 42 . They suggested adding a positive
scalar µ to the diagonal of the virtual-virtual block of the Fock matrix in the MO basis, Eq. (1.10),
before diagonalizing
MO
MO
( µ ( ) )
F + I− D C = Cε , (1.20)
where I is the identity matrix and D MO is the scaled one-electron density matrix in the MO basis
with 1 in the diagonal of the occupied-occupied block and zeros for the rest.
To compare level shifting with the damping scheme of Hartree 27 , consider the first order variation in
the energy change as in Eq. (1.11); the level shift µ then corresponds to adding a positive constant to
the denominator
virtual occupied 2
( 1)
−Fai
SCF
4
a i a i
∆ E =
∑ ∑ . (1.21)
( ε − ε + µ )
The level shift thus has, as the damping factor, the possibility to decrease the magnitude of the term.
The problems with respect to the aufbau principle mentioned in connection with the damping can be
overcome with the level shift. The level shift can separate the occupied orbitals from the virtuals
and thereby ensure a positive denominator and an overall decrease in energy. As the level shift is
increased towards infinity, the obtained decrease in energy will correspond to that of the steepest
descent method as explained in Section 1.4.1.4, and thus the convergence will be slow. This
connection between a large gap between the occupied and the virtual orbitals (HOMO-LUMO gap)
and slow convergence was exploited by Bhattacharyya in 1978 to accelerate convergence for cases
with large HOMO-LUMO gaps. His “reverse level shift” technique 43 uses a negative level shift
instead of a positive, thus decreasing the gap and accelerating the convergence.
In 1977, Carbó, Hernández and Sanz claimed unconditional convergence for an SCF process with a
properly used level shift 44 , and two decades later, Cancès and Le Bris 45 made a formal proof that for
11

Part 1
Improving Self-consistent Field Convergence
any initial guess D 0 , there exists a level shift µ 0 > 0 such that for level shift parameters µ > µ 0 , the
energy decreases at each step and converges towards a stationary value.
The level shift technique is still routinely used for cases where the DIIS scheme has problems. The
level shifts are typically found on a trial and error basis. Recently, we advocated the use of a level
shift to control the changes introduced in the Roothaan-Hall step 38 , and we suggested a way of
optimizing the level shift at each iteration based on physical arguments and without guesswork. The
algorithm is based on the trust region philosophy in which a model energy function is optimized,
but restricted with respect to the step length. We thus named the algorithm trust region Roothaan-
Hall (TRRH), even though it is not a true trust region optimization scheme like e.g. the energy
minimization of Francisco, Martínez, and Martínez 26 or our TRDSM scheme 38 .
Level shifting can be combined with a damping or extrapolation scheme. When the TRRH approach
is combined with the subspace minimization method TRDSM it seems to outperform DIIS in
stability and to have a better or similar convergence rate, as will be illustrated in the following
sections. Combining level shifting with DIIS can occasionally be a benefit, but typically DIIS and
level-shifting does not work well together, and in Section 1.4.1.3 we will try to justify this.
1.4 Development of SCF Optimization Algorithms
The SCF scheme as it typically looks today is sketched in Fig. 1.2. Compared to Fig. 1.1, the step
is inserted, illustrating a density subspace minimization, where
some function f is minimized with respect to the coefficients c i
which expand the previous densities D i . The function f could
be the gradient norm as in DIIS or some energy model
D 0
F(D n
)
n
approximating the SCF energy in the subspace of the previous
D = ∑ciDi,minf
( c)
densities as in EDIIS and TRDSM. In the Roothaan-Hall step
i=
1
, the averaged Fock matrix F found from the optimization in
n
n = n+1 F =
is then used instead of the most recent Fock matrix F(D n ) to
∑ciF( Di)
i=
1
find a new trial density D n+1 . In general, the averaged density
matrix D is not idempotent and therefore does not represent a
valid density matrix; moreover, since the Kohn-Sham matrix
F D n+1
(unlike the Fock matrix) is nonlinear in the density matrix, the
averaged Kohn-Sham matrix F is different from FD. ( ) For
these reasons, the averaged Fock matrix F cannot be
no
D n+1
≈ D n
yes
D conv
associated uniquely with a valid Fock matrix. Usually, this
Fig. 1.2 Flow diagram of the SCF
does not matter much since the subsequent diagonalization of scheme including the density
the Fock matrix nevertheless produces a valid density matrix subspace minimization step.
12

Development of SCF Optimization Algorithms
according to Eq. (1.8). The complications arising from the use of the averaged Fock matrix is
disregarded in the following, noting that the errors introduced by this approach may easily be
corrected for, if necessary.
The rest of this part of the thesis will focus on the work we have done over the last couple of years
to improve SCF convergence. We have made developments in all of the three categories of the
previous section. The density subspace minimization scheme TRDSM and the level shift scheme in
TRRH, both briefly described in the previous section, make up a total scheme we have named
TRSCF, where each SCF iteration contains a TRDSM and a TRRH step. The first subsection will
go into further detail on TRRH and will thus be concerned with our modifications to step in Fig.
1.2. The second subsection will likewise go into further detail on TRDSM and will describe the
scheme we apply in step . In the third subsection, a recently developed energy minimization
procedure will be presented. The procedure merges step and integrating a subspace
minimization in the optimization of a new trial density.
This section will primarily take the Hartree-Fock point of view, acknowledging that with small
adjustments and the word Fock replaced by Kohn-Sham, it would describe the DFT situation as
well. In Section 1.5 the differences appearing when the algorithms are applied to the HF and DFT
cases, respectively, will be discussed.
1.4.1 Dynamically Level Shifted Roothaan-Hall
The problems inherent to the RH diagonalization method are the discontinuous changes in the
density and the lack of guarantees for energy decrease. To overcome these problems, we introduced
in 2004 a means to restrict the RH step to the trust region of the RH energy model, with the purpose
of both controlling the changes in the density and ensuring an energy decrease. Since then, the same
ideas have been put forward by Francisco et. al. 26 as well, suggesting a trust region optimization of
a RH energy model.
In this section, our trust region Roothaan-Hall scheme and related subjects are discussed. In
particular, we present two different schemes for dynamic level shifting and an alternative to
diagonalization.
1.4.1.1 RH Step with Control of Density Change
The solution of the traditional Roothaan–Hall eigenvalue problem Eq. (1.6) may be regarded as the
minimization of the sum of the energies of the occupied MOs 8,46
RH
subject to MO orthonormality constraints
E
∑
( D) = 2 ε = 2TrF D (1.22)
i
i
0
13

Part 1
Improving Self-consistent Field Convergence
T
occ occ = N
C SC I , (1.23)
where F 0 is typically obtained as a weighted sum of the previous Fock matrices such as F in Eq.
(1.15). Since Eq. (1.22) represents a crude model of the true Hartree-Fock energy (with the same
first-order term, but different zero- and second-order terms), it has a rather small trust radius. A
global minimization of E RH (D), as accomplished by the solution of the Roothaan–Hall eigenvalue
problem Eq. (1.6), may therefore easily lead to steps that are longer than the trust radius and hence
unreliable. To avoid such steps, we shall impose on the optimization of Eq. (1.22) the constraint that
the new density matrix D does not differ much from the old D 0 , that is, the S-norm of the density
difference should be equal to a small number ∆
2
2
D− D0 S
= Tr ( D−D0 ) S( D− D0 ) S = − 2Tr D0SDS + N = ∆, (1.24)
where N is the number of electrons – see Eq. (1.2) – and the S-norm used throughout this thesis is
defined as
2
S
A = Tr ASAS (1.25)
for symmetric A. The optimization of Eq. (1.22) subject to the constraints Eq. (1.23) and Eq. (1.24)
may be carried out by introducing the Lagrangian
1
T
L = 2TrFD 0 −2µ
( TrDSDS 0 − ( N −∆)
) −2Trη( CoccSCocc
−I N ) , (1.26)
2
where µ is the undetermined multiplier associated with the constraint Eq. (1.24), whereas the
symmetric matrix η contains the multipliers associated with the MO orthonormality constraints.
Differentiating this Lagrangian with respect to the MO coefficients and setting the result equal to
zero, we arrive at the level-shifted Roothaan–Hall equations:
( F − µ SD S) C ( µ ) = SC ( µ ) λ ( µ ). (1.27)
0 0 occ occ
Since the density matrix, Eq. (1.8), is invariant to unitary transformations among the occupied MOs
in C occ ( µ ), we may transform this eigenvalue problem to the canonical basis:
( F − µ SD S) C ( µ ) = SC ( µ ) ε ( µ ) , (1.28)
0 0 occ occ
where the diagonal matrix ε(µ) contains the orbital energies. Note that, since D 0 S projects onto the
part of C occ that is occupied in D 0 (see ref. 46 ), the level-shift parameter µ shifts only the energies of
the occupied MOs. Therefore, the role of µ is to modify the difference between the energies of the
occupied and virtual MOs - in particular, the HOMO–LUMO gap.
Clearly, the success of the trust region Roothaan–Hall (TRRH) method will depend on our ability to
make a judicious choice of the level-shift parameter µ in Eq. (1.28). In our standard TRRH
implementation, we determine µ by requiring that D(µ) does not differ much from D 0 in the sense of
2
14

Development of SCF Optimization Algorithms
Eq. (1.24), thereby ensuring a continuous and controlled development of the density matrix from the
initial guess to the converged one.
1.4.1.2 The Trust Region RH Level Shift
The constraint on the change in the AO density Eq. (1.24) refers to a change which may arise not
only from small changes in many MOs but also from large changes in a few MOs or even in a
single MO. To obtain a high level of control, we shall require that the changes in the individual
new
MOs are all small. Expanding the MOs ϕ i , obtained by diagonalization of Eq. (1.28), in the old
MOs, we obtain
occ
virt
new old new old old new old
i = j i j + a i a
j
a
∑ ∑ , (1.29)
ϕ ϕ ϕ ϕ ϕ ϕ ϕ
where the first summation is over the occupied MOs and the second over the virtual MOs. The
new
squared norm of the projection of ϕ i onto the MO space associated with D 0 is therefore
orb old new
i j i
j
2
a = ∑ ϕ ϕ . (1.30)
To ensure small individual MO changes in each iteration (to within a unitary transformation of the
occupied MOs), we shall therefore require
orb orb orb
min
min i
i
min
a = a ≥ A , (1.31)
orb
where Amin
is close to one (0.98 or 0.975 in practice). This way of controlling the changes in the
density was also used by Seeger and Pople in their steepest descent method 11 .
To illustrate how this scheme is used in practice, detailed
information from the TRRH step in iteration 7 of a HF/6-31G and
an LDA/6-31G calculation on the zinc complex depicted in Fig.
1.3 is displayed in Fig. 1.4 and Fig. 1.5, respectively. In the upper
orb orb
panels is illustrated how a search for amin
= Amin
determines the
optimal level shift µ for the TRRH step. The TRRH energy model
is more accurate for HF than for DFT (see Section 1.5.1), and
consequently larger changes can be handled in the TRRH step for Fig. 1.3 Zn 2+ in complex with
orb
ethylenediamine-N,N'-disuccinic
HF than for DFT. A
min
is thus set to 0.975 for HF and 0.98 for
acid (EDDS).
DFT. In the lower panels is seen that the chosen level shifts avoid
an increase in the energy which would have been the case if the Roothaan-Hall step was not level
shifted (µ = 0). Notice also that an even lower energy would have been obtained by reducing the
level shift, but then the restrictions on the overlap should be loosened, and this would result in
15

Part 1
Improving Self-consistent Field Convergence
energy increase in other iterations. In short, the identification of µ from the overlap requirement
a
orb
min
orb
min
= A appears to be a good and secure way to control the step sizes in the optimization.
orb
a min
1.0
0.8
orb
A min = 0.975
orb
a min
1.0
0.8
orb
A min = 0.98
0.6
0.6
0.4
0.2
0.0
A
0 2 4 6 8 10
µ
0.4
0.2
0.0
A
0 2 4 6 8 10
µ
40.0
20.0
RH
∆E HF
40.0
20.0
RH
∆E LDA
∆E / a.u.
0.0
-20.0
-40.0
RH
∆E
0 2 4 6 8 10
µ
Fig. 1.4 HF/6-31G, iteration 7. (A) The overlap
orb
RH
a
min
and (B) the changes in the HF energy ∆ E HF
RH
and in the RH energy model ∆ E as a function of
the level shift µ.
B
∆E / a.u.
0.0
-20.0
-40.0
∆E RH
0 2 4 6 8 10
µ
Fig. 1.5 LDA/6-31G, iteration 7. (A) The overlap
orb
a
min
and (B) the changes in the LDA energy
RH
RH
∆ E LDA
and in the RH energy model ∆ E as a
function of the level shift µ.
B
1.4.1.3 DIIS and Dynamically Level Shifted RH
For accelerating the SCF convergence, DIIS is a simple and in general very successful scheme. We
would expect to get an even better performance and improve the stability of the scheme if DIIS was
combined with a dynamically level shifted RH step like TRRH instead of the standard RH with no
control of the step. To investigate how a combination of DIIS and TRRH performs, we carried out a
number of DIIS-TRRH optimizations. A typical example is seen in Fig. 1.7 and an extraordinary
example is seen in Fig. 1.8.
Fig. 1.6 Cd 2+ complexed with an
imidazole ring.
16

Development of SCF Optimization Algorithms
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
1.E-08
DIIS
DIIS-TRRH
TRSCF
0 5 10 15 20 25
Iteration
Fig. 1.7 LDA/STO-3G calculations with a H1-core
start guess on the cadmium complex in Fig. 1.6.
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
TRSCF
DIIS-TRRH
DIIS
0 5 10 15 20 25 30
Iteration
Fig. 1.8 LDA/STO-3G calculations with a Hückel
start guess on the zinc complex in Fig. 1.3.
Somewhat surprisingly the calculations rarely converge with the DIIS-TRRH method. To
understand this behavior, we note that, in the global region, the TRRH method typically produces
gradients that do not change much, even though large changes may occur in the energy. In such
cases, the DIIS method may stall, not being able to identify a good combination of density matrices.
This behavior is illustrated in Table 1-1, where the gradient norm and Kohn–Sham energy of the
first six iterations of the cadmium complex calculations in Fig. 1.7 are listed.
Table 1-1. The Gradient norm ||g||=||4(SDF-FDS)|| in the first six
iterations of the cadmium complex calculations of Fig. 1.7.
DIIS DIIS-TRRH TRSCF
It. E KS ||g|| E KS ||g|| E KS ||g||
1 -5597.0 7.8 -5597.0 7.8 -5597.0 7.8
2 -5502.3 14.9 -5598.4 7.2 -5598.3 7.1
3 -5602.1 9.7 -5600.3 8.5 -5603.7 9.3
4 -5628.5 2.1 -5599.9 7.7 -5611.1 9.1
5 -5627.4 3.5 -5599.9 7.8 -5616.8 7.7
6 -5628.8 0.8 -5600.2 8.1 -5622.7 7.5
conv no conv conv
The TRSCF and DIIS-TRRH gradients stay almost the same during these iterations, stalling the
DIIS-TRRH optimization but not the TRSCF optimization, whose energy decreases in each
iteration. In the pure DIIS optimization, by contrast, the gradient changes significantly from
iteration to iteration; at the same time, the energy decreases at each iteration except the second and
fifth, where also the gradient norms increase. Eventually, DIIS enters the local region with its rapid
rate of convergence although we note a sudden, large increase in the energy in iterations 10 and 11.
However, these changes are accompanied with large increases in the gradient norm, allowing DIIS
to recover safely.
17

Part 1
Improving Self-consistent Field Convergence
In the example Fig. 1.8 standard DIIS diverges. TRSCF converges, but a minimum level shift of 0.1
is used all through the calculation. When DIIS is combined with TRRH in this case, also using a
minimum level shift of 0.1, it converges as well as TRSCF. Table 1-2 contains the gradient norm
and Kohn-Sham energy of the first six iterations of the calculations in Fig. 1.8.
Table 1-2. The gradient norm ||g||=||4(SDF-FDS)|| in the first six
iterations of the zinc complex calculations of Fig. 1.8.
DIIS DIIS-TRRH TRSCF
It. E KS ||g|| E KS ||g|| E KS ||g||
1 -2826.95 11.6 -2826.95 11.6 -2826.95 11.6
2 -2745.49 24.0 -2830.11 3.3 -2830.06 3.4
3 -2809.38 13.6 -2831.04 1.6 -2831.11 1.5
4 -2819.16 9.7 -2831.44 0.8 -2831.42 1.1
5 -2776.74 15.4 -2831.34 1.5 -2831.40 1.5
6 -2826.55 7.0 -2831.41 1.5 -2831.47 0.9
no conv conv conv
In this case the gradient norms for the TRSCF calculation change significantly and a decrease in
gradient relates directly to a decrease in the energy, where in the first example there were no direct
connection between the gradient norm and the energy. The DIIS-TRRH calculation follows the
same gradient behavior as TRSCF, just as in the first example, and they both converge. The DIIS
gradient norm changes, but does not decrease as in the first example. There is still the connection
between small gradients and low energies though, so why DIIS cannot find the proper directions in
this case is not evident.
In our experience DIIS should not be used in connection with a dynamic level shift scheme like
TRRH, since for all but the simplest cases DIIS-TRRH diverged if DIIS converged. We
encountered, however, the example in Fig. 1.8 where DIIS does not converge and DIIS-TRRH does,
but it was the exception.
1.4.1.4 Line Search TRRH
In view of the relative crudeness of the E RH (D) model, a more robust approach for choosing the
level shift µ than the one presented in Section 1.4.1.2 consists of performing a line search along the
RH
path defined by µ to obtain the minimum of the energy E SCF ( D ( µ )). Strictly speaking, this
optimization is not a line search but rather a univariate search. A univariate search has previously
been used by Seeger and Pople 11 to stabilize convergence of the RH procedure.
For µ → ∞ Eq. (1.28) becomes equivalent to solving the eigenvalue equation
0 0
0 occ = occ
SD SC SC η , (1.32)
18

Development of SCF Optimization Algorithms
where η has eigenvalues 1 for the set of orbitals that are occupied in D 0 and eigenvalues 0 for the
set of virtual orbitals. Eq. (1.32) thus effectively divides the molecular orbitals into a set that is
occupied and a set that is unoccupied. If D 0 is idempotent, it can be reconstructed from the occupied
0
set of eigenvectors C occ . If D 0 is not idempotent, a purification of D 0 is obtained
( ) T
occ
idem 0 0
0
= occ
D C C . (1.33)
Since F 0 is the gradient of E(D 0 ), the step from Eq. (1.28) corresponding to a large µ is in the
steepest descent direction, and will therefore give a decrease in the Hartree-Fock energy compared
to the energy at D 0 . Thus a µ exists for which the energy decreases and a line search can then find
the µ leading to the largest decrease in the energy. Using the same example as in Section 1.4.1.2,
Fig. 1.9 and Fig. 1.10 illustrate how the optimal µ is chosen for the line search TRRH (TRRH-LS)
algorithm. A simple search in the energy change for the RH step is carried out, where the energy
change is found as
( ) SCF ( )
RH
idem
∆ E ( µ ) = E D( µ ) − E D , (1.34)
SCF SCF
0
and the µ leading to the largest decrease in energy is chosen as marked on the figures.
40.0
20.0
RH
∆E HF
40.0
20.0
∆E / a.u.
0.0
-20.0
-40.0
RH
∆E
0 2 4 µ 6 8 10
Fig. 1.9 HF/6-31G, iteration 7. The changes in the
RH
HF energy ∆ E HF
and in the RH energy model
RH
∆ E as a function of the level shift µ.
∆E / a.u.
0.0
-20.0
-40.0
RH
∆E LDA
∆E RH
0 2 4 µ 6 8 10
Fig. 1.10 LDA/6-31G, iteration 7. The changes in
RH
the LDA energy ∆ E LDA
and in the RH energy
RH
model ∆ E as a function of the level shift µ.
The TRRH-LS algorithm thus ensures an energy decrease in the RH step, but is of course much
more expensive than the standard method, requiring the repeated construction of the Fock matrix for
a single RH step. However, the first derivative dE
SCF dµ can be evaluated from the Fock matrix,
RH
and a cubic spline interpolation can thus be made from only two points on the ∆ E SCF
curve.
1.4.1.5 Optimal Level Shift without MO Information
As seen from Eq. (1.29) the individual MOs are used to find a suitable level shift in the TRRH
scheme. We are very much aware that this is the most import point to improve on in our scheme. To
obtain this MO information, the cubically scaling diagonalization of the Fock matrix is necessary,
19

Part 1
Improving Self-consistent Field Convergence
and furthermore the MO coefficient matrices C are inherently non-sparse. Several linear or nearlinear
scaling alternatives to diagonalization have been suggested in the literature 18-20 . These
methods could be reformulated with a dynamical level shift scheme like ours if the scheme could do
without the MO information, but it is not an easy task to find a good dynamic level shift scheme
with a high level of control without the knowledge of the developments in the individual MOs. The
search used to find the level shift in TRRH-LS is directly applicable since it is not dependent on the
MO information; the problem is only the number of Fock evaluations. The Fock evaluation is still
expensive even though algorithms which make the evaluation of the Fock matrix cheaper are
continually developed.
This section describes a very recently developed approach to find the optimal level shift in the
TRRH step without the use of individual MOs or knowledge of the HOMO-LUMO gap. So far it
has proven to be the most successful level shift scheme we have studied. The scheme is build on the
assumption that the TRRH step is taken in connection with a TRDSM step (or some other density
subspace minimization method). In this case it can be exploited that TRDSM is a very good energy
model (see Section 1.4.2.2) and can be trusted with the responsibility to find the best direction as
long as not too much new information is introduced to the density subspace in each step.
A new density, found by diagonalization of a level shifted Fock matrix or by some alternative, can
be split in a part D ⊥
that can be described in the previous densities and a part D with new
information orthogonal to the existing subspace
D can be expanded in the previous densities as
⊥
D( µ ) = D + D . (1.35)
n
D = ∑ωiDi
, (1.36)
i=
1
where n is the number of previously stored densities D i and the expansion coefficients ω i are
dependent on µ and determined in a least-squares manner
n
−1
ω i ( µ ) = ∑ ⎡⎣M ⎤⎦
Tr D jSD( µ ) S, Mij = Tr DiSD jS . (1.37)
j=
1
ij
⊥
It is obvious that when µ → ∞ then D → 0 since the new density then approaches the initial
density D 0 , see Eq. (1.32) and (1.33), which belongs to the set of previous densities. Thus, there is a
⊥
connection between D and µ which we can exploit. If the ratio d orth ⊥ 2
of the square norm D
S
2
relative to D
S
is small, only small changes to the density subspace are introduced;
20

Development of SCF Optimization Algorithms
d
orth
⊥ 2
S
2
S
D
⊥ ⊥
Tr D SD S
= = < δ , (1.38)
D Tr DSDS
⊥
where δ is some small number and D can be found as D ⊥ = D−
D . To illustrate how this is used
in a dynamic level shift scheme, the examples from the previous sections are again seen in Fig. 1.11
and Fig. 1.12.
In the rest of the thesis the level shift scheme described in Section 1.4.1.2 will be referred to as the
C-shift scheme since it involves the eigenvectors C from the diagonalization of the Fock matrix,
and the level shift scheme described in this section will be referred to as the d orth -shift scheme. If
nothing is mentioned about the level shift scheme, the C-shift is implied.
1.0
0.8
A
1.0
0.8
A
d orth
0.6
0.4
d orth
0.6
0.4
0.2
δ = 0.08
0.2
δ = 0.03
0.0
0 2 4 6 8 10
µ
0.0
0 2 4 6 8 10
µ
40.0
20.0
RH
∆E HF
B
40.0
20.0
RH
∆E LDA
B
∆E / a.u.
0.0
-20.0
-40.0
RH
∆E
0 2 4 µ 6 8 10
Fig. 1.11 HF/6-31G iteration 7. (A) The ratio d orth
RH
and (B) the changes in the HF energy ∆ E HF
and in
RH
the RH energy model ∆ E as a function of the
level shift µ.
∆E / a.u.
0.0
-20.0
-40.0
RH
∆E
0 2 4 µ 6 8 10
Fig. 1.12 LDA/6-31G iteration 7. (A) The ratio d orth
RH
and (B) the changes in the LDA energy ∆ E LDA
and
RH
in the RH energy model ∆ E as a function of the
level shift µ.
The upper panels now display the search made in d orth , and it is clearly seen that d orth → 0 for µ → ∞
as expected, and increases for µ → 0. As for the C-shift scheme we can allow larger changes in the
HF method than in DFT, and thus δ is set to 0.08 for HF and 0.03 for DFT. In the lower panels are
seen that this level shift avoids an increase in the energy just as the C-shift scheme, but the level
shift chosen here is closer to the optimal line search level shift, and thus leads to a larger decrease in
the energy than was the case for the C-shift scheme.
21

Part 1
Improving Self-consistent Field Convergence
In the C-shift scheme seen in Eq. (1.31) the changes introduced are controlled compared to the
previous density, whereas in the d orth -shift scheme the changes are controlled compared to the
subspace of all the previous densities. This scheme is thus less restrictive than the C-shift scheme,
but it seems that the C-shift scheme is too restrictive, ignoring the stability gained from the
subspace information. To compare the overall effect of the two level shift schemes on the SCF
convergence, calculations are given in Fig. 1.13 and Fig. 1.14, for HF and LDA, respectively. The
HF calculations are on CrC with bond distance 2.00Å in the STO-3G basis and the LDA
calculations are on the zinc complex seen in Fig. 1.3 in the 6-31G basis, both cases for which DIIS
diverges. The starting orbitals have been obtained by diagonalization of the one-electron
Hamiltonian (H1-core start guess).
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
TRSCF
d orth -shift
DIIS
TRSCF
C-shift
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
TRSCF
d orth -shift
DIIS
TRSCF
C-shift
1.E-08
0 4 8 12 16
Iteration
Fig. 1.13 SCF convergence for HF/STO-3G calculations
on CrC.
1.E-08
0 8 16 24 32
Iteration
Fig. 1.14 SCF convergence for LDA/6-31G calculations
on the zinc complex in Fig. 1.3.
The only difference in the “TRSCF/d orth -shift” and the “TRSCF/C-shift” optimizations is the way
the level shift is found in the TRRH step. Since DIIS diverges, the examples display the stability of
the TRSCF algorithm, and the ability of the two level shifting schemes to handle problematic cases.
In all examples studied so far, both problematic and simple, the d orth -shift has proven as good as or
better than the C-shift. The cost of the level shift search process is similar in the two schemes; the
matrix M in Eq. (1.37) is updated in each iteration as a part of TRDSM and is then reused for the
d orth -shift scheme in TRRH.
In Table 1-3 The SCF energy change in each iteration is divided in the part of the change obtained
from the RH and DSM step, respectively, and it is seen how the RH step is now allowed to accept
larger changes in the density, but still in a controlled manner, thus leading to larger decreases in the
energy and improved convergence.
22

Development of SCF Optimization Algorithms
Table 1-3. The SCF energy change for each RH and DSM step
in the TRSCF calculations in Fig. 1.13.
C-shift
d orth -shift
It.
RH
DSM
RH DSM
∆ E HF ∆ E HF
∆ E HF ∆ E HF
2 -1.1768 0.0000 -1.3976 0.0000
3 -1.8964 -3.8998 -4.1319 -4.5865
4 -1.6764 -1.9603 -1.8021 -1.0448
5 -0.3655 -1.7543 -0.2103 -0.1200
6 -0.1881 -0.1624 -0.0111 -0.0463
7 -0.0932 -0.1505 -0.0036 -0.0037
8 0.0065 -0.0212 -0.0001 -0.0008
9 -0.0039 -0.0154
10 0.0002 -0.0009
1.4.1.6 The Trace Purification Scheme
The dynamic level shift scheme described in the previous section has no reference to the MO basis.
This opens the possibility to replace the diagonalizations in the TRRH step with some alternative
scheme without affecting the overall result.
There have been many suggestions as to how the diagonalization can be replaced by a linear scaling
algorithm 47 . The trace purification (TP) scheme 19,48 , however, is a simple and useful approach and it
has thus been implemented in our SCF program in a local version of DALTON 38,49 . The trace
purification scheme was originally formulated for tight binding theory by Palser and
Manolopoulos 19 and later improved by Niklasson 48 , and is linear scaling when formulated in an
orthogonal basis. The scheme uses the trace and idempotency properties of the density to iteratively
find the new density from a suitable start guess constructed from the Fock matrix.
Since the SCF optimization is formulated in the non-orthogonal AO basis to avoid the delocalized
MO basis, it is necessary to transform the matrices to an orthogonal basis. This is done by a
Cholesky decomposition 50 of the AO overlap matrix S
T
S = LL , (1.39)
where L then is used to transform the Fock matrix to an orthogonal basis
orth -1 −T
F = L FL . (1.40)
The density resulting from the trace purification scheme will also be in the orthogonal basis and
should be transformed back as
−T orth -1
D = L D L . (1.41)
Since the AO overlap matrix does not change during the optimization, the Cholesky decomposition
and the inversion of L can be done once and for all in the beginning of the calculation.
23

Part 1
Improving Self-consistent Field Convergence
F orth
R
λ min
Estimate and
for F orth
λ max
0
orth
( λ
max
I
−
F
)
=
( λ
−
λ
)
max
min
1
x n +1 = 2x n - x n
2
n = n + 1
Tr Rn > N
yes
R
n+ 1 =
R
2
n
no
2
n+ 1 = 2 n − n
R R R
x n +1
no
Tr Rn N ε
+ 1 − <
yes
D orth = R n+1
Fig. 1.15 Flow diagram for the trace purification (TP)
scheme. N is the number of electrons.
0
x n +1 = x n
2
0 x n
1
Fig. 1.16 The purifying polynomials used in
the trace purification scheme. The orange line
is the McWeeny purification polynomial
x n+1 = 3x n 2 – 2x n 3 .
The trace purification is carried out by the Niklasson model with second order purification
polynomials, and is schematized in Fig. 1.15. The initial density guess R 0 is obtained by
normalizing the Fock matrix such that it only has eigenvalues between 0 and 1. To do this, the
bounds for the Fock eigenvalues, λ min and λ max , must be found. They can be estimated using
Gerschgorin’s theorem or the Lanczos algorithm for eigenvalues 51 with only a small extra
computational cost. R is then iteratively purified, and the purification function applied in each
iteration is chosen based on the trace of the matrix R, always keeping the direction towards the
correct trace condition. The purification functions are sketched in Fig. 1.16 including the McWenny
purification function 8 . One of the functions used in the scheme has a stationary point for x = 1 and
the other has a stationary point for x = 0; depending of the function chosen we thus go towards a
larger or smaller trace. When R fulfils the trace and/or idempotency conditions Eq. (1.2) of the one
electron density within some threshold ε, the new density D orth = R has been found and the density
to use in the next TRSCF iteration can be evaluated from Eq. (1.41).
The number of purification iterations required to obtain a new density depends on the threshold ε.
For the test calculations carried out so far, the threshold has been an error of 10 -7 in the trace, and
the number of iterations ranges from 30 to 70 for a single RH step, with the typical number being
closer to 30 than 70. Still, it is less expensive than the diagonalization as soon as more than a couple
24

Development of SCF Optimization Algorithms
of thousand basis functions are needed. The scaling of the TRRH step in general and the trace
purification scheme in particular is illustrated and discussed in Section 1.7.1.
1.4.2 Density Subspace Minimization
The DIIS scheme seems to have been the overall most successful of all the suggestions on how to
improve SCF convergence described in Section 1.3. DIIS was the first scheme to take advantage of
the information contained in the densities and Fock matrices of the previous iterations, and this
made the difference.
This is also exploited in the EDIIS scheme by Kudin et. al. 37 in which an energy model is optimized
with respect to the linear combination of previous densities. The density subspace minimization
presented in this section is an improvement to EDIIS with a smaller idempotency error in the
density, the correct gradient compared to SCF, and thus better convergence properties in both the
local and global region of the optimization.
1.4.2.1 The Trust Region DSM Parameterization
After a sequence of Roothaan-Hall iterations, we have determined a set of density matrices D i and a
corresponding set of Fock matrices F i = F(D i ). An improved density D and Fock matrix F should
now be found as a linear combination of the previous n + 1 stored matrices. Taking D 0 as the
reference density matrix, the improved density matrix can be written
n
= 0 +∑ ci
i=
0
D D D , (1.42)
which, ideally, should satisfy the symmetry, trace and idempotency conditions Eq. (1.2) of a valid
one-electron density matrix. Whereas the symmetry condition is trivially satisfied for any such
linear combination, the trace condition holds only for combinations that satisfy the constraint
n
i=
0
i
∑ ci
= 0 , (1.43)
leading to a set of n + 1 constrained parameters c i with 0 ≤ i ≤ n. Alternatively, an unconstrained set
of n parameters c i with 1 ≤ i ≤ n can be used, with c 0 defined so that the trace condition is fulfilled:
c
0
n
=−∑ c . (1.44)
i=
1
i
In terms of these independent parameters, the density matrix D becomes
where we have introduced the notation
D = D0 + D + , (1.45)
25

Part 1
Improving Self-consistent Field Convergence
D
+
=
n
∑
i=
1
c D
i0
D = D −D
i0 i 0 .
i
(1.46)
Unlike the symmetry and trace conditions in Eq. (1.2), the idempotency condition is in general not
fulfilled for linear combinations of D i . Still, for any averaged density matrix D in Eq. (1.45) that
does not fulfill the idempotency condition, we may generate a purified density matrix with a smaller
idempotency error by the transformation 8
D = 3DSD−2DSDSD. (1.47)
Introducing the idempotency correction
Dδ = D − D, (1.48)
we may then write the purified averaged density matrix in the form
D = D + D + D . (1.49)
0 + δ
1.4.2.2 The Trust Region DSM Energy Function
Having established a useful parameterization of the averaged density matrix Eq. (1.45) and having
considered its purification Eq. (1.47), let us now consider how to determine the best set of
coefficients c i . Expanding the energy in the purified averaged density matrix, Eq. (1.49), around the
reference density matrix D 0 , we obtain to second order
T
( ) ( ) ( ) (1) 1
T
D = D + D+ + D E + ( D+ + D ) E (2) ( D+
+ D )
E E δ δ δ . (1.50)
SCF(2) SCF 0 0 2
0
To evaluate the terms containing
(1)
E
0
and
(2)
E
0
we make the identifications
(1)
0
= 2 0
2 2
0 + = 2 + + +
E F (1.51)
( )
( )
E D F O D , (1.52)
which follow from Eq. (1.4) and from the second-order Taylor expansion of about D 0 . The
n
notation Eq. (1.46) has now been generalized to the Fock matrix F+ = ∑ c
i=
1 iF i0
. Ignoring the
terms quadratic in D δ in Eq. (1.50) and quadratic in D + in Eq. (1.52), we then obtain the DSM
energy
DSM
E () = ESCF ( 0 ) + 2Tr + 0 + Tr + + + 2Tr δ 0 + 2Tr δ +
(1)
E0
c D DF DF DF DF. (1.53)
Finally, for a more compact notation, we introduce the weighted Fock matrix
n
0 + 0 ci
i0
i=
1
and find that the DSM energy may be written in the form
F = F + F = F +∑ F , (1.54)
26

Development of SCF Optimization Algorithms
DSM
( ) ( )
where the first term is quadratic in the expansion coefficients c i
E c = E D + 2TrDδ
F, (1.55)
( ) SCF 0 0
E D = E ( D) + 2TrDF + + TrDF, + +
(1.56)
and the second, idempotency-correction term is quartic in these coefficients:
( )
2TrDδ F = Tr 6DSD −4DSDSD −2D F . (1.57)
The derivatives of E DSM (c) are straightforwardly obtained by inserting the expansions of F and D ,
using the independent parameter representation. The expressions are given in Error! Reference
source not found..
The energy function E DSM (c) in Eq. (1.55) provides an excellent approximation to the exact SCF
energy E SCF (c) about D 0 , with an error quadratic in D δ (see Section 1.5.2). The EDIIS energy model
corresponds to the first term E( D ) in Eq. (1.55) and has thus an error linear in D δ .
1.4.2.3 The Trust Region DSM Minimization
The DSM energy, Eq. (1.55), is minimized with respect to the independent parameters c i with 1 ≤ i
≤ n. The vector containing the parameters is initialized to zero c (0) = 0 such that D = D 0 , where D 0
is chosen as the density matrix with the lowest energy E SCF (D i ), usually the one from the latest
TRRH step. The minimization is then carried out by the trust region method 52 , taking a number of
steps from the initial parameters c (0) to the final optimized parameters c* as illustrated in Fig. 1.17.
c (0) = 0 c*
c (1) c (2) c (3) ....
Fig. 1.17 Steps in the trust region minimization of the DSM energy.
We thus consider in each step the second-order Taylor expansion of the DSM energy in Eq. (1.55).
Introducing the step vector
( i+
1) ( i)
∆c = c −c , (1.58)
we obtain
E
i
( )
DSM ( ) T 1 T
(2)
+ = E0
+ +
2
c ∆c ∆c g ∆c H∆c , (1.59)
where the energy, gradient, and Hessian at the expansion point are given by
E
DSM 2 DSM
DSM ( i)
∂E ( c) ∂ E ( c)
= E ( c ), g = , H =
∂c
i
∂c
0 2
c= c
c=
c
() () i
. (1.60)
27

Part 1
Improving Self-consistent Field Convergence
DSM ( i)
We then introduce a trust region of radius h for E ( c + )
(2)
∆c and require that steps are always
taken inside or to the boundary of this region. To determine a step to the boundary, we restrict the
step to have the length h in the S metric norm M
n
2 2
S
= ∑ ∆cM i ij∆ cj
= h
ij=
1
∆c . (1.61)
In the unconstrained formulation defined by Eq. (1.44), the metric M of Eq. (1.37), is found as
M = Tr DSDS−Tr DSDS− Tr DSDS+ Tr DSDS, i, j ≠ 0 , (1.62)
ij i j i 0 0 j
0 0
Introducing the undetermined multiplier ν for the step-size constraint, we arrive at the following
Lagrangian for minimization on the boundary of the trust region:
L E h . (1.63)
T T T 2
( ∆c,
ν ) = + ∆c g+ 1 ∆c H∆c − 1 ν ( ∆c M∆c − )
0 2 2
Differentiating this Lagrangian and setting the derivatives equal to zero, we obtain the equations
∂L
= g+ H∆c− ν M∆c = 0
∂∆c
(1.64)
∂ L 1 T 2
2 ( ∆c M∆c − h ) 0 .
∂ν
(1.65)
The optimization of the Lagrangian thus corresponds to the solution of the following set of linear
equations:
H− M ∆c =−g
(1.66)
( ν )
where the multiplier ν is iteratively adjusted until the step is to the boundary of the trust region Eq.
(1.65). The step length restriction may be lifted by setting ν = 0 as needed for steps inside the trust
region.
To illustrate how the level shift parameter ν in Eq. (1.66) is determined, we consider in Fig. 1.18
and Fig. 1.19 the third and fourth DSM step respectively, in iteration five of the HF/STO-3G
calculation on CrC seen in Fig. 1.13. The step length ||∆c|| S is plotted as a function of ν. The plots
consist of branches between asymptotes where ν makes the matrix on the left hand side of Eq.
(1.66) singular. This happens whenever ν equals one of the Hessian eigenvalues. The lowest
eigenvalue ω 1 of the Hessian H is found, and the level shift parameter is chosen in the interval -∞ <
ν < min(0,ω 1 ). The proper value is found where the step length function crosses the line
DSM
representing the trust radius h, as marked in Fig. 1.18. If the step that minimizes E
(2)
is inside the
trust region, ν = 0 is chosen as is the case in Fig. 1.19. The trust region is updated during the
iterative procedure and therefore h is different in the two steps.
28

Development of SCF Optimization Algorithms
3
3
2
2
1
h = 0.34
1
h = 0.44
0
-5 -2.5 0 2.5 5 7.5
ν
Fig. 1.18 The step length as a function of the
multiplier ν in the third DSM step.
0
-5 -2.5 0 ν 2.5 5 7.5
Fig. 1.19 The step length as a function of the
multiplier ν in the fourth DSM step.
Each of the trust region steps require the construction of the gradient g and the Hessian H in the
density subspace, and the solution of the level shifted Newton equations Eq. (1.66). Since E DSM is a
local model of the true energy function E SCF , it resembles E SCF only in a small region about the
initial point c (0) . The DSM iterations are therefore terminated if the total step length after p iterations
||c (p) – c (0) || S exceeds some preset value k. If a minimum of E DSM is found inside the trust region ||c (p)
– c (0) || S < k, then the step ||c* - c (0) || S to the minimum is taken and the iterations are terminated. This
is the typical situation.
When the trust region minimization has terminated, an improved density matrix D can be
constructed. However, to avoid the expensive calculation of the Fock matrix from D we use instead
the averaged density matrix from eq. (1.45) and exploit that the Fock matrix is linear in the density
for Hartree-Fock such that F( D ) is simply the averaged Fock matrix of Eq. (1.54). For DFT this is
an approximation, but typically insignificant improvements are obtained by evaluating the correct
Kohn-Sham matrix. The improved Fock matrix and density matrix then enters the TRRH step as F 0
and D 0 , respectively.
By construction E DSM (c) is lowered at each iteration of the trust region minimization. Since E DSM is
a local model to the true energy E SCF , the lowering of E DSM will also lead to a lowering of E SCF
provided the total step is sufficiently short and thus stays in the local region.
1.4.2.4 Line Search TRDSM
As in the TRRH step, the averaged density matrix D may also be determined by a line search and
we denote this line search algorithm TRDSM-LS. Here, the line search is made in the direction
defined by the first step c (1) of the TRDSM algorithm—that is, the step at the expansion point D 0 .
As in the TRRH step, such a line search is guaranteed to reduce the energy. The first step is scaled
by a parameter α,
29

Part 1
Improving Self-consistent Field Convergence
tot
(1)
∆c = α ⋅ c (1.67)
DSM
and a search is made in ∆ E SCF
to find the step ∆c tot that leads to the largest decrease in energy.
E SCF (α) is found by evaluating the averaged density of Eq. (1.45) for the coefficients (c 0 + ∆c tot ),
purifying it as in Eq. (1.32)–(1.33) and inserting it in the energy expression of Eq. (1.1). Then
DSM
∆ E SCF ( α)
can be found as DSM
∆ E ( α ) = E ( α ) − E ( D ). (1.68)
SCF
SCF SCF 0
Fig. 1.20 and Fig. 1.21 illustrate the search in α, again for iteration seven of the HF and LDA
calculations on the zinc complex in Fig. 1.3. For α = 0, no step is taken and hence no energy
decrease is seen. For the marked choice of α, the optimal step length is obtained.
0
-5
-10
-15
-20
-25
-30
-35
0 4 8 12 16 20
α
Fig. 1.20 Decrease in HF energy as a function of
the step length α.
0
-5
-10
-15
-20
-25
0 4 8 12 16 20
α
Fig. 1.21 Decrease in LDA energy as a function of
the step length α.
1.4.2.5 The Missing Term
In the construction of the TRDSM energy model Eq. (1.55), the term of second order in the
idempotency correction D δ was neglected from Eq. (1.50), since this term required a new Fock
evaluation F(D δ ), which would increase the expenses of the scheme considerably. This section will
be concerned with this neglected term and how a part of it can be described without the evaluation
of a new Fock matrix, leading to an improved energy model for TRDSM at no considerable extra
cost. The actual effect of this improvement to the energy model will then be discussed through a
case study. This section will only be concerned with Hartree-Fock theory and examples, but it might
equally well be done for DFT even though the improvement should be less significant since for
DFT, also terms of order ||D + || 3 are neglected. These are of the same size as the neglected term
quadratic in D δ . In Section 1.5.2 these errors are discussed.
Since the only neglect in the DSM energy model Eq. (1.55) for Hartree-Fock is the term quadratic
in D δ , and since the only term quadratic in the density is TrDG(D), the HF energy for the density D
can be written as
30

Development of SCF Optimization Algorithms
( D) = ( D) + D F+
D G( D )
E HF E 2Tr δ Tr δ δ , (1.69)
where E ( D ) is seen in Eq. (1.56). Even though a new Fock matrix h + G(D δ ) should be evaluated
to describe the last term exactly, a part of the term can be described in the subspace of the previous
densities.
As exploited in the level-shift scheme Section 1.4.1.5, a density or density difference, in this case
D δ , can be divided in a part that can be described in the subspace of the previous densities D
δ
and
an unknown part orthogonal to the space
D
δ
D
⊥
δ
δ =
δ
+
⊥
δ
D D D
is expanded in the previous densities D i as
. (1.70)
D
δ
n
= ∑ωiD
i=
0
i
, (1.71)
where the expansion coefficients ω i are determined in a least-squares manner
ω
n
i =
−1
⎡⎣
⎤⎦
Tr
ij
j=
0
j δ , Mij = Tr i j
∑ M D SD S D SD S . (1.72)
Inserting Eq. (1.70) for D δ in Eq. (1.69), an improved DSM energy model can be written
DSM
( c) = ( D) + D F+ ( D −D ) G( D )
Eimp E 2Tr δ Tr 2 δ δ δ
where only previous density and Fock matrices enter. The relation
, (1.73)
Tr AG( B) = Tr BG( A )
(1.74)
⊥ ⊥
for symmetric matrices A and B is used and the term ( )
Tr Dδ G D
δ
is neglected. A second order
Taylor expansion of the improved DSM energy can then be made as in Eq. (1.59) and a trust region
minimization carried out.
To study the improvement to the energy function, two TRSCF calculations are carried out on the
cadmium complex seen in Fig. 1.6 in the STO-3G basis and with a H1-core start guess. The
convergence profiles of the calculations are displayed in Fig. 1.22, the one denoted “Improved
TRDSM” is a TRSCF calculation just as the one denoted “TRSCF” with the only difference that the
improved energy model in Eq. (1.73) is used for TRDSM instead of the one in Eq. (1.55). To
illustrate the impact of the improvement in a single TRDSM step, a line search like the one in Fig.
1.20 is made in iteration 7 of the same TRSCF calculation as in Fig. 1.22. Apart from displaying the
change in SCF energy as a function of the step length α, also the DSM energy of Eq. (1.55) and the
improved DSM energy of Eq. (1.73) are evaluated for the different choices of α, and their energy
changes found as well.
31

Part 1
Improving Self-consistent Field Convergence
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
1.E-08
TRSCF
Improved TRDSM
0 5 10 15 20
Iteration
Fig. 1.22 Convergence for the cadmium complex in
Fig. 1.6, both for TRSCF with no improvements,
and for TRSCF where E is used in TRDSM.
DSM
imp
∆E / E h
1.0
0.0
-1.0
-2.0
-3.0
-4.0
DSM
∆E
DSM
∆E HF
DSM
∆E imp
0 2 4 6 8 10 12
α
Fig. 1.23 TRDSM line search for iteration 7 in the
TRSCF optimization Fig. 1.22. For different α in
DSM
Eq. (1.67), the changes in E
HF
, E DSM DSM
and E
imp
compared to E HF (D 0 ) are found.
It is seen in Fig. 1.23 that the improved DSM energy describes the HF energy better than the
standard DSM energy does, just as expected. As the step moves away from the expansion point, the
part of the energy which cannot be described in the old densities grows and both the DSM energy
models become poor.
The improvements presented in this section add complexity to the TRDSM algorithm, even though
the computational cost is not significant. As seen in Fig. 1.22 and Fig. 1.23, the improvements to the
TRSCF calculation are minor. The overall gain does not justify the extra complexity added to the
TRDSM algorithm.
1.4.3 Energy Minimization Exploiting the Density Subspace
Section 1.3.1 describes how different approaches have been taken to avoid the diagonalization in
the Roothaan-Hall step. Replacing the standard diagonalization of the Fock matrix can be done for
the purpose of improving either the convergence properties or the scaling of the algorithm or for
both reasons. With the purpose of improving both, a newly developed scheme is presented in this
section, in which an energy minimization replaces the standard diagonalization in the SCF
optimization.
When the RH energy model is minimized, the density subspace information used with great success
in TRDSM is ignored. The novel idea is thus to exploit the valuable information saved in the
density subspace of the previous densities to construct an improved RH energy model and minimize
this model instead of the RH model. This makes the TRDSM step redundant since a density
subspace minimization now is included in the RH energy model minimization.
The Hessian update methods 40,53 , in which an approximate Hessian is updated in each iteration and
an approximate Newton step is taken, exploit some of the same ideas, but they are all based on
32

Development of SCF Optimization Algorithms
approximate second order energy expansions in the orbital rotation parameters and therefore do not
include the third and higher order terms included in the RH energy.
In the following subsections the improved RH energy model and its minimization will be described.
The SCF convergence of a test case is then displayed, in which the new energy minimization
approach is compared to standard DIIS and the TRSCF schemes. As the scheme has not yet been
extended to DFT, this section will only consider HF theory and calculations.
1.4.3.1 The Augmented RH Energy model
If the Hartree-Fock energy, Eq. (1.1), is expanded through second order around some reference
density D 0
E ( D) = E ( D ) + 2TrF( D )( D− D ) + Tr( D−D ) G( D−D ) , (1.75)
HF HF 0 0 0 0 0
the first two terms are recognized as E RH (D) from Eq. (1.22) plus the terms of zeroth order E HF (D 0 )
and - E RH (D 0 )
( ) ( ) ( )
RH
RH
E ( D) = E ( D) + E ( D ) − E ( D ) + Tr D−D G D−D . (1.76)
HF HF 0 0 0 0
In a standard RH step, the energy function to minimize is the RH energy, neglecting the last term
which contains the Hessian information, because it is too expensive to evaluate. Since Hessian
information is very valuable to an optimization, the scheme presented in this section will replace the
diagonalization in the RH step by an energy minimization of an augmented RH (ARH) energy
model, where as much Hessian information as possible is included without directly evaluating new
Fock matrices. This is done by exploiting the information contained in the density and Fock
matrices of the previous iterations.
As previously exploited, a density or density difference, in this case ∆ = D – D 0 , can be split in a
part that can be described in the subspace of the n + 1 previous densities ∆ and an unknown part
orthogonal to the space
⊥
∆
∆ is expanded in the previous densities D i as
D− D = ∆ = ∆ + ∆
0
n
i=
0
⊥
. (1.77)
∆ = ∑ωiDi
, (1.78)
where n is the number of previously stored densities and the expansion coefficients ω i are
determined in a least-squares manner
ω
n
i =
−1
⎡⎣
⎤⎦
Tr
ij
j=
0
j , Mij = Tr i j
∑ M D S∆S D SD S . (1.79)
33

Part 1
Improving Self-consistent Field Convergence
⊥ ⊥
Inserting Eq. (1.77) in the last term of Eq. (1.76) and neglecting the term Tr ∆ G ( ∆ ) , the
augmented Roothaan-Hall energy model can be written as
( ) ( ) ( )
ARH ( ) RH ( ) ( ) RH
E D = E D + EHF D0 − E ( D0 ) + Tr 2∆−∆ G ∆ , (1.80)
where G ( ∆ ) is evaluated as a linear combination of previous Fock matrices
n
( ) ∑ωi ( i ) ∑ωi ( i )
G ∆ = G D = ( F D − h ). (1.81)
i= 1 i=
1
The energy model E ARH has no intrinsic restrictions with respect to how different the densities
spanning the subspace are allowed to be, and this is one of the benefits compared to the TRSCF
scheme. For the TRDSM energy model, the purification implicit in the DSM energy makes no sense
if the densities are too different, in particular if they have different electron configurations. In ARH,
configuration shifts can be handled without problems, and whereas old, obsolete densities pollute
the DSM energy model, they simply disappear from the ARH energy model, since their weights ω i
diminish.
We expect a faster convergence rate for ARH compared to TRSCF, mainly because the RH and
DSM steps are merged to an energy model with correct gradient (not just in the subspace) and an
approximate Hessian, which is improved in each iteration using the information from the previous
density and Fock matrices.
1.4.3.2 The Augmented RH Optimization
The density for which the ARH energy model should be optimized can be expanded in the antisymmetric
matrix X
n
D ( X () () () ()
) = exp 1
( − XS ) D i 0 exp ( SX ) = D i ⎡ i 0
+
0 , ⎤ + ⎡⎡ i
2 0
, ⎤ , ⎤
⎣ D X ⎦ ⎣ ⎦
+
⎣
D X X ⎦
, (1.82)
() i
S S S
where D
0
is the reference density from which the step X is taken. Optimizing the ARH energy is
thus a nonlinear problem and an iterative scheme should be applied.
A Newton-Raphson (NR) optimization of the ARH energy is therefore carried out, and the steps are
ARH
found minimizing a second order approximation of the ARH energy E
(2)
by the preconditioned
conjugate gradient (PCG) method. The second order approximation of the ARH energy, where the
constant terms are excluded, can be written as
34

Development of SCF Optimization Algorithms
E
where
() i
() i
( X)
= 2Tr F0 ⎡
0
, ⎤ + Tr ⎡
0
⎡
0
, ⎤ , ⎤
⎣
D X
⎦
F
⎣⎣ D X
⎦
X
⎦
ARH
(2) S S S
() i
(1) (2)
( D0
D0
) ∑( ωi
ωi
) G( Di
)
+ 2Tr − +
i, j=
1
n
i=
1
n
n
() i
(0) (1) () (0)
0 S ⎣ 0 S ⎦
i= 1
S
i=
1
i
∑( ωi ωi ) ( i ) ⎡
⎤ ∑ωi
( i )
+ 2Tr ⎡ , ⎤ Tr ⎡ , ⎤
⎣
D X
⎦
+ G D +
⎣
D X
⎦
, X G D
n
∑
( ) ⎤DG i ( Dj
)
(0) (1) (2) (1) (1)
j i i i j
− Tr ⎡
⎣
2 ω ω + ω + ω ω
⎦
,
(1.83)
ω
ω
ω
n
(0) −1
( )
i = ∑ ⎡⎣ ⎤⎦
Tr
ij
j=
1
i
( j 0 )
M D SD S
i
( j
⎡ ⎤ )
n
(1) −1
( )
i = ∑ ⎡⎣ ⎤⎦ Tr
0
,
ij ⎣ ⎦S
j=
1
M D S D X S
( ⎡
i
j
⎡ ⎤ ⎤
0
)
n
(2) 1 −1
( )
i =
2 ∑ ⎡⎣ ⎤⎦ij
⎣⎣ ⎦S ⎦
j=
1
S
M Tr D S D , X , X S .
(1.84)
If the summations are put in the most favorable way, the number of matrix multiplications is limited
and independent of subspace size. Only the update of the metric M takes a number of matrix
multiplications linearly in the subspace size.
ARH
∂E (2)
∂X
From the derivative , the problem to be solved by PCG is set up for the current reference
() i
density D
0
where i denotes the Newton-Raphson step number. Through the whole NR
optimization D 0 and F 0 are the density and Fock matrices from the previous SCF iteration. The NR
step X found by PCG is used to evaluate a new density from Eq. (1.82) and if the new density is
similar to the previous one, the Newton-Raphson optimization has converged, if not, the density is
() i
used as reference density D in the next step.
0
The final density matrix resulting from the NR optimization is then used to evaluate a new Fock
matrix, and so the SCF iterative procedure is established. The SCF scheme for the described
algorithm is illustrated in Fig. 1.24.
35

Part 1
Improving Self-consistent Field Convergence
( 0 )
D 0
( 0 )
( )
F D n
ARH
min E(2) ( X ) ( i
D )
n
by PCG
( i 1
D
) ( X)
n +
i = i + 1
n = n + 1
no
( i+
1) ( i)
no
n ≈ Dn
yes
( 0 ) ( i+
1)
n+ 1 = D n
D ( 0 ) ( 0 )
n+ 1 ≈ D n
D
D
yes
D conv
1.4.3.3 Applications
Fig. 1.24 Flow diagram of the SCF optimization with
the diagonalization of the Fock matrix replaced by a
minimization of the ARH energy. The light blue box
embraces the Newton-Raphson optimization of E ARH .
SCF calculations have been carried out using the ARH scheme. In Fig. 1.25 the convergence of
HF/STO-3G calculations on CrC with 2.00Å bond distance are displayed. Results are given for the
augmented RH scheme, DIIS and TRSCF with the C-shift and d orth -shift schemes, respectively. For
the first iterations in the ARH optimization a limit is put on the ||X|| S norm to avoid changes in the
densities which go beyond the region that is well described by the energy model.
The ARH scheme is clearly superior for this test case, even with the convergence improvements for
TRSCF obtained with the d orth -shift scheme; ARH is almost an iteration in front of ‘TRSCF/d orth -
shift’ in the local region. The standard DIIS approach does not converge at all for this case.
36

The Quality of the Energy Models for HF and DFT
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
1.E-08
1.E-10
DIIS
TRSCF C-shift std.
TRSCF dnew
orth -shift
ARH
1 3 5 7 9
Iteration
Fig. 1.25 HF/STO-3G calculations on CrC using
different approaches.
0.8
0.6
0.4
0.2
0.0
1 3 5 7 9
Iteration
Fig. 1.26 Details from the ARH optimization in
Fig. 1.25: The part of the density change which can
be described in the subspace of the previous
densities.
To illustrate how information gradually is obtained from the previous densities in ARH, the part of
the density change ∆D = D n+1 - D n in each iteration that can be described in the previous densities
∆D is found as in Eq. (1.78)-(1.79), and the ratio
∆D
∆D
is depicted in Fig. 1.26. It is
seen how the description of ∆D improves during the first five iterations until a significant part of the
Hessian is described, then a qualified step is taken to another region, and the new density is
therefore not well described in the previous densities. This step is followed by a significant decrease
in SCF energy of two orders of magnitude. The same pattern is repeated after two additional
iterations.
Even though only preliminary results are given in this section, the ARH energy minimization seems
promising, taking the best of the RH and DSM energy models, and improving the convergence
compared to TRSCF, which already saw better or as good convergence rates as DIIS. It could be
expected that this scheme has the ability to converge in fewest SCF iterations overall. The future
success of ARH is dependent on the development of effective ways of solving the nonlinear
equations in X, e.g. by setting up a good preconditioner.
1.5 The Quality of the Energy Models for HF and DFT
Having considered the theory behind the TRRH and TRDSM steps in Section 1.4.1 and 1.4.2
without being concerned with the approximations introduced in the energy functions, this section
takes a closer look at the errors in the energy models compared to the SCF energy. The SCF
optimization of Hartree-Fock and Kohn-Sham-DFT energies is similar; the only difference lies in
the energy expressions to be optimized. The approximations in the energy models will thus also
differ in HF and DFT, and while Section 1.2 described the HF and DFT theory in a generic manner,
this section will focus on the differences, ignoring the general elements already stated in Section
1.2.
S
S
37

Part 1
Improving Self-consistent Field Convergence
To make the differences in the HF and DFT energy expressions clear, we will now study them
separately:
E
= 2TrhD + Tr DG ( D ) + h , (1.85)
HF HF nuc
E = 2TrhD + Tr DG ( D) + h + E ( D ), (1.86)
DFT DFT nuc XC
where
[ G HF ( D ) ] = 2 gµνρσ Dρσ − gµσρν Dρσ
, (1.87)
µν
∑
ρσ
∑
ρσ
[ G DFT ( D )]
= 2 gµνρσ Dρσ −γ gµσρν Dρσ
. (1.88)
µν
∑
ρσ
∑
ρσ
The second term in Eq. (1.87) and Eq. (1.88) is the contribution from exact exchange, with γ = 0 in
pure DFT (LDA), and γ ≠ 0 in hybrid DFT. The exchange-correlation energy E XC (D) in Eq. (1.86) is
a functional of the electronic density. In the local-density approximation (LDA), the exchangecorrelation
energy is local in the density, whereas in the generalized gradient approximation (GGA),
it is also local in the squared density gradient, and may thus be expressed as
EXC ( D) = ∫ f ( ρ( x), ζ( x)
) dx. (1.89)
Here the electron density ρ(x) and its squared gradient norm ζ(x) are given by
T
ρ( x) = χ ( xDχ ) ( x),
ζ( x) =∇ρ( x) ⋅∇ρ( x),
(1.90)
where χ(x) is a column vector containing the AOs. Note that the exchange-correlation energy
density f(ρ(x), ζ(x)) in Eq. (1.89) is a nonlinear (and non-quadratic) function of ρ(x) and ζ(x). In the
following is relied on an expansion of E XC (D) around some reference density matrix D 0
E
T
T
XC XC 0 0 XC 2 0 XC 0
(1) (2)
( D) = E ( D ) +
1
( D− D ) E + ( D−D ) E ( D− D ) + , (1.91)
( n)
where the derivatives E
XC
have been evaluated at D = D 0 and where for convenience a vectormatrix
notation for D, E
(1)
XC
, and E (2)
XC
is used. The precise form of E XC depends on the DFT
functional chosen for the calculation.
It is often more problematic to obtain convergence for DFT than HF, mainly for two reasons: The
HOMO-LUMO gap ∆ε ai is smaller for DFT than for HF, and a determinant with a well separated
occupied and virtual part has better convergence properties than one with a lot of close lying
states 54,55 . Also, since the exchange-correlation is nonlinear and non-quadratic in the density, the
higher order terms in the density not present in Hartree-Fock theory introduces some extra
approximations to the SCF scheme for DFT. In this section these differences and their consequences
for the convergence properties will be discussed for the TRSCF algorithm. It is here assumed that if
the energy models employed in TRSCF were of the same quality for HF and DFT, that is, had errors
38

The Quality of the Energy Models for HF and DFT
of the same order compared to the true SCF energy, then the convergence properties would also be
of the same quality.
The study is mainly performed in the MO basis with a block diagonal Fock matrix as in Eq. (1.10)
and the reference density matrix
MO
D
0
2δ
ij
MO ⎛ 0 ⎞
D0
= ⎜ ⎟
⎝ 0 0 ⎠
. (1.92)
It is also exploited that any valid density matrix D may be expressed in terms of a valid reference
density matrix D 0 as
MO
MO
D ( K)
= exp( −K) D exp( K ) , (1.93)
and can thus be expanded in orders of K through the BCH-expansion 46
MO MO MO 1 MO 3
=
0
+ ⎡
0
⎤ + ⎡⎡
2 0
⎤ ⎤ +
0
D ( K) D ⎣D , K⎦ ⎣⎣D , K⎦, K⎦
O ( K ). (1.94)
The anti-symmetric rotation matrix may be written in the form
⎛ 0 −κ
⎞
K = ⎜ ⎟ , (1.95)
⎝κ 0 ⎠
where κ holds the orbital rotation parameters. The diagonal block matrices representing rotations
among the occupied MOs and among the virtual MOs are zero since the density matrix in Eq. (1.8)
is invariant to such rotations.
In the following subsections the RH energy model Eq. (1.22) and the DSM energy model Eq. (1.55)
are analyzed separately with respect to differences for HF and DFT.
1.5.1 The Quality of the TRRH Energy Model
To compare the RH energy model to the SCF energy, both are expanded about a reference density
matrix D 0 (neglecting the possible difference between F 0 and F(D 0 ) noted in Section 1.4)
E
T
RH RH
E ( D) = E ( D0) + 2Tr F( D0)
( D−D 0 ), (1.96)
( D) = E ( D ) + 2TrF( D )( D− D ) + Tr( D−D ) G( D−D
)
(1)
+ E ( D) − E ( D ) −Tr ( D−D ) E ( D ),
(1.97)
SCF SCF 0 0 0 0 0
XC XC 0 0 XC 0
where the last three terms of Eq. (1.97) only are present in DFT theory. These expansions have the
same first-order term 2TrF(D 0 )(D - D 0 ) and thus the same first derivative with respect to the orbital
rotation parameters κ ai of Eq. (1.95)
RH
(1) ∂E
( κ )
⎡ ⎤
⎣
E
RH ⎦
= = −4F
ai , (1.98)
ai ∂κ
ai
κ=
0
39

Part 1
Improving Self-consistent Field Convergence
(1) ∂ESCF
( κ )
⎡ ⎤
⎣
E
SCF ⎦
= = −4F
ai . (1.99)
ai ∂κ ai κ=
0
The expressions are found replacing D in Eqs. (1.96) and (1.97) with D MO in Eq. (1.94) and
differentiating with respect to κ ai .
All higher order terms in κ arising from 2TrF(D 0 )(D - D 0 ) are consequently also shared for the SCF
and RH energies whereas terms of second and higher order arising from the last term(s) in Eq. 1.94
are neglected in the RH energy model. To study the differences, the second order derivatives in κ
are found in the same way as the first derivatives
2 RH
(2) ∂ E ( κ)
⎡ ⎤
⎣
E
RH ⎦
= = 4δ ij δ ab ( ε a −ε
i )
(1.100)
aibj ∂κ
∂κ
2
ai
bj
κ=
0
(2) ∂ ESCF
( κ)
⎡ ⎤
⎣
E
SCF ⎦
= = 4δδ ij ab ( ε a − ε i ) + W aibj , (1.101)
aibj ∂κ
∂κ
ai
bj
κ=
0
where
HF
16 4( )
W = g − g + g
(1.102)
aibj
aibj abij ajib
( )
DFT
Waibj
= 16gaibj − 4 γ gabij + gajib
+ ⎡ ( ) ⎤ ⎣
E κ ⎦
. (1.103)
(2)
XC
aibj
(2)
E XC ( κ ) is the second derivative of the term E XC expanded in the orbital rotation parameters κ. The
error in the RH energy model can then be said to depend partly on the size of W and partly on the
size of the third and higher order contributions from the nonlinear terms in Eq. (1.97) which are not
included in Eq. (1.96). This general consideration goes for DFT as well as HF, but with different
impact. As seen in Eq. (1.102) and (1.103), the definition of W differs in the two approaches and
even differs depending on which DFT functional is chosen. Furthermore, since the size of the
HOMO-LUMO gap ∆ε ai = ε a - ε i is typically smaller in DFT, the term 4δ ij δ ab (ε a – ε i ) will have
different weights in Eq. (1.101) depending on the method. Also the size of the third and higher
order contributions in Eq. (1.97) would be expected to differ for HF and DFT, since for DFT both
the terms Tr(D - D 0 )G(D - D 0 ) and E XC (D) contribute whereas HF only contains the Tr(D - D 0 )G(D
- D 0 ) term. In the beginning of the optimization, where large steps are taken, the size of the third
and higher order contributions is the potential source of error. Near convergence this should be less
of an issue, and in this region the size of the lowest Hessian eigenvalues should be the decisive error
source.
HF and LDA calculations have been carried out and the part of the SCF energy change arising from
RH
the RH step ∆ E SCF
has been found as well as the change in the RH energy model ∆E RH in each
iteration.
40

The Quality of the Energy Models for HF and DFT
4.0
2.0
0.0
-2.0
HF
LDA
0 5 10 15 20
Iteration
Fig. 1.27 Calculations on the cadmium complex in
Fig. 1.6 in the STO-3G basis set.
3.0
2.0
1.0
0.0
-1.0
-2.0
HF
LDA
0 5 10 15 20 25
Iteration
Fig. 1.28 Calculations on the zinc complex in Fig.
1.3 in the 6-31G basis set.
The change in the RH energy model is found as
idem
( n )
RH
E 2Tr + 1 0
∆ = F D −D , (1.104)
idem
where D
0
is the reference density matrix, typically a D from the previous TRDSM step purified
as in Eqs. (1.32)-(1.33), and D n+1 is the new density found from diagonalization of the Fock matrix.
In the C-shift scheme the criterion Eq. (1.31) ensures that the occupied and virtual orbitals do not
mix, and thus the Hessian, Eq. (1.100), is positive and the RH energy decreases. The SCF energy
change is found as
RH
idem
SCF SCF n+
1 SCF 0
∆ E = E ( D ) − E ( D ). (1.105)
The ratio between Eq. (1.104) and Eq. (1.105) contains information of the quality of the RH energy
model. If the errors are negligible, the ratio is close to 1. If the ratio is larger than one, the RH
energy model exaggerates the energy decrease, and if it is between 0 and 1 it underestimates the
energy decrease. If it is negative, the SCF energy increases even though the RH energy model
predicts an energy decrease.
RH RH
For two test cases the ∆E ∆ E SCF
ratio is displayed in Fig. 1.27 and Fig. 1.28, respectively. It is
clearly seen that generally, the RH energy model is better for HF than for DFT, in particular,
negative values are seen for the LDA ratios. The errors in the RH energy model for the LDA
calculations get worse as convergence is approached, so it would be expected that the significant
source of error is the neglected term W in the Hessian rather than the higher order terms. Since
locally the lowest Hessian eigenvalue should be the one controlling the optimization, this theory is
inspected evaluating the lowest Hessian eigenvalue for both the RH energy model and for SCF
according to Eq. (1.100) and Eq. (1.101), respectively, at convergence of the two test cases. The
results are compared in Table 1-4.
41

Part 1
Improving Self-consistent Field Convergence
Table 1-4 The lowest Hessian eigenvalues for the RH energy
model and SCF energy at convergence of the calculations in Fig.
1.27 and Fig. 1.28. The deviation is found as
( ⎡ (2) ⎤ ⎡ (2) ⎤ )
(2)
RH SCF
100% ⎡ ⎤
⎣
E
⎦
−
⎣
E
⎦
⋅
⎣
E
SCF ⎦
.
(2)
SCF
(2)
RH
min min min
cadmium complex zinc complex
HF LDA HF LDA
⎡
⎣
E ⎤
⎦ min
0.557 0.017 1.000 0.290
⎡ ⎤
⎣
E
⎦ min
1.112 0.014 1.621 0.281
Deviation 100% -21% 62% -2%
As expected, the lowest Hessian eigenvalue for the RH energy model, that is the HOMO-LUMO
gap, is much smaller for LDA than for HF, but surprisingly it is seen that the Hessian prediction in
the RH energy model for LDA is much better than the one for HF. Of course this is only the lowest
eigenvalue, and we have not studied the corresponding eigenvector. We know for sure that the size
of the orbital rotation parameters κ ai decreases during the optimization and should be very small at
convergence, where only small adjustments to the density are made. It is thus difficult to imagine
that terms of third and higher order in κ should be the reason for the larger errors in the DSM
energy model for LDA compared to HF.
This is a matter we will investigate further in the future since it is not understood at the moment.
The importance of the higher order terms should be examined directly to understand how they affect
the errors, and the Hessian should be studied more carefully introducing information about the
direction of the eigenvalues. However, it can still be concluded from Fig. 1.27 and Fig. 1.28 that the
RH energy model is poorer for LDA than for HF optimizations.
1.5.2 The Quality of the TRDSM Energy Model
The TRDSM energy model of Section 1.4.2.2 is formulated in a general manner and is as applicable
to DFT theory as to HF theory. Still, the model will be poorer for DFT than for HF because of the
general exchange-correlation term appearing in the DFT energy.
For the DSM energy model there are in general four possible sources of errors:
1. The purified density D still has an idempotency error.
2. The term
1 T [2]
2 δ 0 δ
D E D in E( D ) , Eq. (1.50), is neglected.
3. E( D ) , Eq. (1.50), is truncated after second order.
4.
( 2 )
0 +
E D in Eq. (1.50) is approximated by 2 F + .
42

The Quality of the Energy Models for HF and DFT
Let us take a closer look at the errors one by one. In ref. 39 a general order analysis of the purified
density D used in the parameterization of the DSM energy is given, and the results are summarized
in Table 1-5.
Table 1-5. Comparison of the properties of the unpurified density D and the purified
density D . c is the density expansion coefficients and κ is the orbital rotation parameters
that change D 0 to another density in the subspace D i .
D
Differences D+ = D− D0 = ( c κ )
O
2
Dδ = D
− D = O ( c κ )
Idempotency error
2
4
DSD − D = O ( c κ ) DSD − D
= O ( c 2 κ )
Trace error Tr DS − N / 2 = 0
2 4
Tr DS − N / 2 = O ( c κ )
In the D column, the order of the idempotency correction D δ and the idempotency error for D are
found. These are the same for DFT and HF; the idempotency error is of order c 2 ||κ|| 4 , and since D δ
is of the order c||κ|| 2 , the error connected to the neglect of the term second order in D δ , will be of
order c 2 ||κ|| 4 as well.
The third possible source of errors is the truncation of the energy E( D ) after second order in the
density. Since the Hartree-Fock energy is quadratic in the density, this truncation leads to no errors
for HF, but for DFT there will be an error of order ||D + || 3 and from the first column in Table 1-5 it is
seen that it can be written as an error of order c 3 ||κ|| 3 , since D + is of the order c||κ||. Also since the
(3)
HF energy is quadratic in the density, no third derivative E
0
exists and thus the Taylor expansion
( 2 )
used to find E0 D+ = 2F + is terminated for HF, but for DFT terms of order ||D + || 2 are neglected.
( 2 )
Since E0 D + is multiplied by D + in the energy function Eq. (1.50), this gives an error for DFT of
the order ||D + || 3 or as before c 3 ||κ|| 3 . The sizes of the introduced errors are summarized in Table 1-6.
Table 1-6. Comparison of the errors introduced in the DSM energy model for
HF and DFT respectively.
D
1 Idempotency error DSD − D
2 Neglected term
3 Truncation of ( )
4 Approximation of
( )
error in HF
error in DFT
( 2 4
O c κ )
2 4
O ( c κ )
1 T [2]
D
2 δ
E0
D
2 4
2 4
δ O ( c κ ) O ( c κ )
E D 0 3 3
O ( c κ )
2
E0 D +
0 3 3
O ( c κ )
Depending on the sizes of c and ||κ|| respectively, the error for DFT will be of same or lower order
than the one for HF. To inspect whether or not the DSM energy is a poorer model for DFT than for
HF, a number of calculations have been carried out, and the sizes of ||D δ || and ||D + || for the DSM
step in each iteration are examined. Since D δ is of the order c||κ|| 2 and D + is of the order c||κ||, the
43

Part 1
Improving Self-consistent Field Convergence
size of ||D δ || 2 and ||D + || 3 will indicate whether the error in the energy model is controlled by the
( c 2 4
3 3
O κ ) or the ( c κ )
O error. The test cases showed similar behavior and results from HF
and LDA calculations on the cadmium complex in Fig. 1.6 with a STO-3G basis and a H1-core start
guess are displayed in Fig. 1.29 and Fig. 1.30.
1.E+00
1.E-02
1.E-04
1.E-06
1.E-08
1.E-10
4
||D+||^4D + S
||D+||^3
3
D + S
2||Ddelta||^2 2
2 Dδ
S
dEDSM
E − E
HF
DSM
2 5 8 11 14 17 20
Iteration
1.E+01
1.E-01
1.E-03
1.E-05
1.E-07
1.E-09
1.E-11
4
||D+||^4D + S
3
||D+||^3D + S
||Ddelta||^2 2
D δ S
dEDSM
E − E
LDA
DSM
2 5 8 11 14 17 20 23
Iteration
Fig. 1.29 HF/STO-3G calculation. The size of
different density norms compared to the actual
error in the DSM energy model.
Fig. 1.30 LDA/STO-3G calculation. The size of
different density norms compared to the actual
error in the DSM energy model.
DSM
The SCF energy at the end of a DSM step ESCF
is found by purifying the resulting D by Eq. (1.32)
–(1.33) and evaluating the SCF energy, Eq. (1.1), for this density. The DSM energy, Eq. (1.55), is
DSM DSM
also evaluated and the error of the DSM energy model is then found as the size ESCF
− E .
For the HF calculation this error is expected to be of the size ||D δ || 2 , and it is seen in Fig. 1.29 that
this is actually the case; if ||D δ || 2 is multiplied by 2, there is a remarkable fit. Also it is seen that if
the error in the DSM energy for HF should be expressed in the density differences D + , it would be
the density differences to the third rather than the fourth order. For the DFT calculation the
interesting point was to see whether or not ||D + || 3 is the controlling error. In Fig. 1.30 is seen that
even though there is not an obvious fit as for HF, ||D δ || 2 seems to be the dominant error here as well.
Still, if the error should be expressed in the density differences D + , it would be the density
differences to the third rather than the fourth order as expected for DFT.
In conclusion it seems that the dominating error in the DSM energy both for HF and DFT is ||D δ || 2 ,
that is, the idempotency correction squared. In comparison it should be mentioned that the EDIIS
model 37 by Kudin, Scuseria, and Cancès corresponds to E( D ) in Eq. (1.55) and thus has an error of
the order ||D δ || compared to the SCF energy.
1.6 Convergence for Problems with Several Stationary Points
The HF equation is a nonlinear equation and, therefore, it presents in principle several solutions.
Several minima might exist, and even though it is typically preferred to find the global minimum,
44

Convergence for Problems with Several Stationary Points
no optimization method can make that a guarantee. Furthermore, it cannot be tested if the minimum
found is a local or the global minimum without knowledge of the whole surface. Depending on the
start guess and the optimization approach, an optimization can converge to different stationary
points. Further, it is necessary to decide in which subspace of orbital rotations the desired solution
should be found, since a solution representing a stable stationary point in one subspace is not
necessarily stable in another.
Orbital rotations can be divided in real and complex rotations and each of those can be further
divided in singlet and triplet rotations. Each of those can then again be divided in rotations within
the different point group symmetries. Generally, we do not consider the complex rotations, and we
only optimize in the real space. Further, when optimizing a closed shell wave function, only the
total-symmetric part of the singlet rotations is considered. A stationary point in the subspace of real,
total-symmetric, singlet rotations can be shown through elementary arguments to be a stationary
point for all types of rotations. However, a stationary point can both be a maximum, a saddle point
or a minimum. A way to realize if the stationary point also is a minimum is to evaluate the Hessian
eigenvalues. This is done within the subspace in which the solution should be stable. If a negative
Hessian eigenvalue is found in the subspace of singlet rotations, the stationary point is said to have
a singlet instability and if a negative Hessian eigenvalue is found in the subspace of triplet rotations,
it is said to have a triplet instability 54,56 . Triplet instabilities are connected to breaking the symmetry
between α and β orbitals. If a triplet instability is found, a minimum with a lower energy than the
current stationary point can be found, if the α and β parts are allowed to differ, typically leading to
2
a solution which is not an eigenfunction of Ŝ . Hence, the lower minimum could be found by an
unrestricted HF (UHF) optimization. A singlet instability found in the total-symmetric subspace
indicates that the current stationary point is a saddle point and a minimum with lower energy exists
within the subspace. If a singlet instability is found outside the total-symmetric subspace, orbitals of
different symmetries should be mixed to decrease the energy further, changing the symmetry of the
resulting wave function.
The aufbau ordering rule assumes that occupying the orbitals of lowest energy also leads to the
lowest Hartree-Fock energy. This cannot be proven to always apply for restricted HF as it can for
UHF 57 . Thus it is a risk when the aufbau ordering is forced upon an optimization, that a lower
energy with the aufbau ordering broken could exist. However in a study by Dardenne et. al. 58 , in
which different ordering schemes were tested, they found in all cases that the minimum was an
aufbau solution. The aufbau ordering was broken only for saddle points. In our schemes we always
apply the aufbau ordering rule, but if the RH step is level shifted to the end of the optimization, it
can force the convergence to a non-aufbau solution.
45

Part 1
Improving Self-consistent Field Convergence
1.6.1 Walking Away from Unstable Stationary Points
As concluded in the previous section, the Hessian eigenvalues should be tested to make sure the
optimized state is stable. This is expensive, so it is only done when it is expected that the problem
has several stationary points. Depending on the desired solution, only the relevant part of the
Hessian is checked. So far we have only considered singlet instabilities, but currently tests for triplet
instabilities are implemented as well.
The check for singlet instabilities is made on the converged wave function, finding the lowest
Hessian eigenvalue of the Hessian in the real, singlet subspace. If the lowest Hessian eigenvalue
turns out to be positive, we are sure to have a solution which is stable with respect to singlet
rotations, but if it is negative we are in a saddle point, and a minimum with a lower energy exists
within the subspace. We have in our SCF program implemented the possibility to test the singlet
Hessian and in case of a negative lowest Hessian eigenvalue follow the corresponding direction
downhill and away from the saddle point. The scheme and some examples of its use will be
described in the following.
1.6.1.1 Theory
When the SCF optimization has converged, the set of optimized orbitals described by their
expansion coefficients C opt are used to evaluate the lowest Hessian eigenvalues and the
corresponding eigenvectors by an iterative subspace method. If the lowest Hessian eigenvalue ε min is
found positive, then it is clear that the optimization has converged to a minimum. If on the other
hand the eigenvalue is negative, we know for sure that a lower stationary point exists.
We would then like to take a step downhill in the direction x corresponding to the negative
eigenvalue ε min
( 2 )
SCF
E x = εminx. (1.106)
This can be accomplished making a unitary transformation of the optimized expansion coefficients
C opt with x as the orbital rotation parameters to define the direction X dir of the step
X
dir
T
ai
⎡ 0 −x
⎤
= ⎢ ⎥ . (1.107)
⎣ xai
0 ⎦
The step length is controlled by a parameter α
Uα
= exp ( −α
X dir )
(1.108)
C′ ( α ) = C U . (1.109)
opt opt α
A line search is then carried out for α > 0 to find the lowest SCF energy in the direction X dir . This is
of course expensive since every point in the line search requires an evaluation of the Fock matrix
46

Convergence for Problems with Several Stationary Points
with respect to the new coefficients C opt ′ . When the SCF energy minimum in the direction X dir is
found, the corresponding coefficients should be the initial orbitals for a new SCF optimization,
hopefully now optimizing further downhill to a minimum. In problematic cases, e.g. with a very flat
saddle point close to the minimum, we have found it convenient to continue the optimization with
the line search scheme TRSCF-LS (the combination of TRRH-LS and TRDSM-LS described in
Sections 1.4.1.4 and 1.4.2.4) to ensure a continued decrease in the energy.
1.6.1.2 Examples
In Fig. 1.31 and Fig. 1.32 two examples of problems with several stationary points are given.
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
1.E-08
TRSCF
d orth -shift
TRSCF C-shift
Line search
0 20 40 60
Iteration
Fig. 1.31 HF calculations on the rhodium complex.
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
1.E-08
(1)
Line search
(2)
(3)
TRSCF
0 40 80 120
Iteration
Fig. 1.32 HF/STO-3G calculation on CrC.
The first example is a HF optimization on the rhodium complex seen in Fig. 1.33 in the
AhlrichsVDZ basis 59 combined with STO-3G on rhodium. For this example DIIS diverges, but the
TRSCF scheme with C-shift converges nicely in 38 iterations. However, when the Hessian is
inspected it is found that the lowest eigenvalue is negative, and a search in α is carried out in the
direction corresponding to the negative eigenvalue. This is
illustrated with the orange line in the picture. Since each
evaluation of a step-length α necessitates an evaluation of the
Rh Cl
Fock matrix, it is fair to display each line search step as an
iteration on the SCF iteration scale. When a minimum is found in
this direction, the corresponding orbitals are used as a start guess
for a new TRSCF optimization, and it is seen that it now Fig. 1.33 Rhodium complex.
converges nicely to a new and lower stationary point which is
found to be a minimum. When the d orth -shift scheme is applied in the TRRH steps instead of the C-
shift scheme, it turns out that convergence to the minimum is obtained with no problems, as seen
from Fig. 1.31, illustrating how the stationary point found from an SCF optimization not only
depends on the start guess, but also on the optimization procedure.
47

Part 1
Improving Self-consistent Field Convergence
The second example is a HF/STO-3G optimization of CrC with a bond distance on 2.00Å. The
example is also used in Fig. 1.13 and Fig. 1.25, but without discussing the stability of the converged
state. Also in this case DIIS diverges whereas TRSCF converges nicely in 12-13 iterations to a
stationary point which is found to have singlet instabilities. As for the first example, a line search is
carried out in the downhill direction and a new TRSCF optimization is started from the resulting
orbitals. This time the second optimization has more problems than was the case for the rhodium
example, but finally it converges to a minimum. Whereas in the rhodium case, only one plateau
corresponding to the saddle point could be seen, in this case three plateaus can be found, marked by
numbers on the figure. The first is the saddle point that TRSCF converges to, at E SCF =
− 1068.77014939 and with a lowest Hessian eigenvalue of -0.624. The second and third stationary
points are recognized as saddle points by TRSCF itself and it manages to move away. If a DIIS
optimization is carried out with a Hückel start guess, it converges to the second stationary point,
which has E SCF = -1069.21761813 and a lowest Hessian eigenvalue of -0.038, again demonstrating
that depending on the optimization procedure and start guess, different stationary points can be
found. It is thus necessary to check the Hessian of the result to know for sure that a minimum is
found, and in this case the final minimum has E SCF = -1069.30090709 and a lowest Hessian
eigenvalue of 0.043. CrC is well known for being a molecule with a complicated electronic energy
surface and has been the object for several theoretical studies 60 .
The scheme testing for singlet instabilities and walking away from unstable stationary points could
be integrated more efficiently in the optimization than is done here. It can be seen from Fig. 1.31
and Fig. 1.32 that the optimizations are completely converged before the Hessian check is made,
spending many iterations improving the unwanted result. The check could be made in an earlier
stage, saving a number of iterations. Also the steps taken in the line search could be optimized such
that fewer steps were necessary to find the minimum. Anyhow, it is convenient to have the
possibility to continue an optimization until a minimum is found.
1.7 Scaling
As mentioned in the introduction, it is now possible to apply ab-initio quantum chemical methods,
in particular HF and DFT, to large molecular systems of interest for biology and nano-science. This
is due to both the developments in integral screening and algorithms for the Fock matrix builder and
to approaches avoiding diagonalization and exploiting sparsity in the matrices. Since the TRSCF
scheme has properties which would be of great advantage for SCF calculations on large and
complex molecules, it is crucial that the scheme can be formulated in a linear or near-linear scaling
manner. We have not been concerned with the build of the Fock matrix, and any state-of-the-art,
linear or near-linear scaling approach could be used as the Fock builder for our scheme. The steps to
48

Scaling
consider are thus the Roothaan-Hall step TRRH, which evaluates a new density matrix, and the
density subspace minimization TRDSM, which improves convergence. In the following subsections
the scaling of these steps will be discussed.
1.7.1 Scaling of TRRH
The TRRH scheme with C-shift described in Section 1.4.1.2 requires the diagonalization of a level
shifted Fock matrix and the knowledge of the occupied molecular orbital coefficients. The
diagonalization scales as well as a matrix multiplication as N 3 , where N is the dimension of the
problem, in this case the number of basis functions. However, a diagonalization is ineffective and
cannot be nearly as well optimized as a matrix multiplication, and thus the scaling factor is much
larger for the diagonalization than for the matrix multiplication. Also, the matrix multiplication can
exploit sparsity and obtain a scaling linearly in the number of non-zero elements whereas sparsity is
not as easily exploited in diagonalizations. Furthermore, the molecular orbitals described by the
eigenvectors from the diagonalization of the Fock matrix are inherently delocalized and thus there is
no sparsity to exploit.
To obtain a linear scaling TRRH step it is thus necessary to avoid completely the diagonalizations
and any reference to the MO basis. This can be done in our SCF program – a local version of
DALTON 38,49 - by combining the d orth -shift scheme described in Section 1.4.1.5 with the trace
purification (TP) described in Section 1.4.1.6.
The trace purification scheme replaces the diagonalization of the level shifted Fock matrix and
makes it possible to exploit sparsity in the matrices. A sparse blocked matrix storage scheme has
been implemented for this purpose. In this scheme the columns and rows in the matrices are
permuted such that close lying atoms are collected in blocks, making it possible to exploit the
locality in the basis functions. Based on some drop tolerance for the size of matrix elements, pure
zero blocks can be found and neglected, both saving storage and computing time. A library has been
developed for the purpose of handling the matrix operations for this type of matrices and controlling
the truncation error arising from the neglect of elements 49 .
Calculations have been carried out on glycine chains of different length in the 4-31G basis set on a
3.4GHz Xeon/Nocona Machine with EM64T architecture and MKL BLAS+LAPACK library.
Timings have been made in the third iteration of the SCF optimization, measuring how much time
(CPU) is spent in the TRRH step in the case of full matrices and diagonalizations of the level
shifted Fock matrix (Diag./full) and in the case of sparse blocked matrices and the TP scheme
(TP/sparse). The results are seen in Fig. 1.34. Both in the full and sparse case the d orth -shift scheme
is applied.
49

Part 1
Improving Self-consistent Field Convergence
60
Time / min.
50
40
30
20
10
Diag./full
TP/sparse
0
400 1050 1700 2350 3000
Number of basis functions
Fig. 1.34 Timings of a TRRH step in case of
diagonalizations of full matrices (Diag./full) and in
case of trace purification of sparse blocked matrices
(TP/sparse).
The crossover is already around 1500 basis functions, and it is clear how the diagonalization
scheme quickly will become too time consuming if the number of basis functions is increased
further. Of course, this is a linear molecule as seen from Fig. 1.35, and the cross over will be later
for more three-dimensional molecules. The TP method does not have an exact linear scaling
because of the transformation to the orthogonal basis which gives rise to a quadratic term, but the
scaling factor on the quadratic term is very small. It should be noted that the dynamic level shift
scheme typically takes 5-10 diagonalizations or trace purifications to find the optimal level shift in
the first couple of iterations, and as the timings are from the third iteration, then not just one, but
several diagonalizations or purifications are included in the timings in Fig. 1.34. Currently a full
trace purification optimization (30-70 purification iterations) is carried out for each level shift tested
to find the optimal level shift. It is straightforward to optimize this process such that the purification
is not converged as hard for the level shifts tested and rejected, as for the final optimal level shift.
Fig. 1.35 Glycine chain.
To conclude, the scaling of the TRRH scheme with C-shift is dominated by the diagonalization, and
sparsity cannot be exploited. Still with a good Fock builder it can run effectively up to a couple of
thousand basis functions, but at some point the diagonalizations get too time consuming. For larger
systems the purification scheme with the d orth -shift scheme can be used with blocked sparse matrices
resulting in a near-linear scaling.
50

Applications
1.7.2 Scaling of TRDSM
For the density subspace minimization, a set of linear equations, Eq. (1.66), are solved in each DSM
step, but only in the dimension of the subspace which is much smaller than the number of basis
functions. It is therefore of no significance compared to the matrix additions and multiplications
needed to set up the DSM gradient g and Hessian H for the linear equations. For TRDSM it will
thus only be the number of matrix multiplication that determines the scaling. Nothing has to be
changed to exploit sparsity in the matrices, and linear scaling is automatically obtained from the
point where the number of non-zero elements in the matrices is linear scaling. For full matrices the
scaling is formally N 3 , where N is the number of basis functions, but as mentioned in the previous
subsection this is not a problem as it is for the diagonalization, since matrix multiplications can be
carried out with close to peak performance on computers. However, the number of matrix
multiplications should be kept at a minimum as it affects the scaling factor.
The number of matrix multiplications is dependent on the dimension of the subspace as the number
of gradient and Hessian elements grows with the size of the subspace, but even though the Hessian
is set up explicitly, the number of matrix multiplications only scales linearly with the dimension of
the subspace. The expressions for the DSM gradient and Hessian are found in 0, and it is seen that if
only the matrices FD i , SD i , FDiS and DSD i are evaluated, then all the terms for a Hessian
element can be expressed as the trace of two known matrices or their transpose. As the operation
TrAB scales quadratically instead of cubically, the overall scaling of TRDSM will be nN 3 for full
matrices, where n is the dimension of the subspace and N the dimension of the problem. For sparse
matrices both the matrix multiplications and TrAB scale linearly, but since n 2 TrABs are evaluated,
the overall scaling is n 2 N. However, the trace operations have a very small prefactor.
In the TRSCF scheme with C-shift the diagonalizations are thus the dominating operations, but
since both the TRRH and TRDSM step can be carried out without any reference to the MO basis
and with matrix multiplications as the most expensive operations, the TRSCF scheme is near-linear
scaling and has what it takes to be applied to really large molecular systems. It is still a work in
progress to get all the parts working together, so unfortunately no large scale TRSCF calculations
will appear in this thesis, and no benchmarks in which sparsity in the matrices is exploited for
TRDSM can be presented, but the whole framework is in place.
1.8 Applications
In this section, numerical examples are given to illustrate the convergence characteristics of the
TRSCF and ARH calculations. Comparisons are made with DIIS, the TRSCF-LS method, and the
globally convergent trust-region minimization method (GTR) of Francisco et. al. 26 .
51

Part 1
Improving Self-consistent Field Convergence
In Section 1.8.1 a set of small molecules used by Francisco et. al. to illustrate the convergence
characteristics of GTR is considered. Next in Section 1.8.2 the convergence of calculations on three
metal complexes is discussed for the DIIS, TRSCF and TRSCF-LS methods.
1.8.1 Calculations on Small Molecules
As an alternative to the RH diagonalization, Francisco et. al. have developed an energy
minimization method (GTR), where an energy model is minimized by a trust-region minimization.
They have proven that it is a globally convergent algorithm, that is, no matter the starting point; the
iterative steps will converge towards a stationary point. The best results are obtained when they
combine GTR with DIIS and thereby let DIIS accelerate the convergence. To examine the
convergence characteristics of TRSCF and ARH compared to GTR, calculations have been carried
out with the attempt to reproduce the conditions given in the paper by Francisco et. al.. Thus HF
calculations have been carried out with a maximum number of 10 previous density matrices for the
density subspace minimizations and convergence is obtained when the difference between two
consecutive energies is smaller than 10 -9 E h . The results are given in Table 1-7; the numbers found
with our SCF program are on a white background, whereas results copied from the GTR paper are
on a grey background.
Table 1-7 Number of iterations in HF calculations performed by each algorithm in some test problems. The
geometry of the molecules and the results in grey are taken from the paper by Francisco et. al. 26 , and
GTR+DIIS is their globally convergent trust-region algorithm with DIIS acceleration.
Algorithm
Molecule Basis Start guess DIIS TRSCF
C-shift
TRSCF
d orth -shift
ARH DIIS GTR
+DIIS
H 2 O STO-3G H1-core 7 7 7 6 5 5
6-31G H1-core 10 9 8 8 8 8
NH 3 STO-3G H1-core 7 8 7 6 7 7
6-31G H1-core 9 9 8 8 7 7
CO STO-3G H1-core 12 9 9 9 11 10
Hückel 8 8 8 - 7 7
CO(Dist) * STO-3G H1-core 39(a) 9 8 8 117(b) 10
Hückel 35 10 8 - 85 15
6-31G H1-core 24(a) 13 10 9 27(b) 115
Hückel 21(a) 10 10 - 36(b) 59
Cr 2 STO-3G H1-core 34(a) 14(a) 10(a) 12(a) 13 38
CrC STO-3G H1-core 29(a) 13(a) 11(a) 10(a) (X) 29
* Distorted geometry – double bond length compared to CO
(a) Negative Hessian eigenvalue.
(b) Converged to a higher energy than some of the other algorithms
(X) No convergence in 5001 iterations.
Let us first consider the results obtained from our SCF program. Comparing the TRSCF results
(both C-shift and d orth -shift) to the DIIS results, it is clear that the TRSCF method not only is an
52

Applications
improvement when DIIS cannot converge, but also for small simple examples, the convergence of
TRSCF is as good as or better than for DIIS. Also it is observed that in five instances DIIS converge
to a stationary point which is not a minimum, while that only happens in two instances for TRSCF.
This suggests that the TRSCF algorithm does not have a high tendency to converge to saddle points
compared to DIIS. Comparing the results obtained for TRSCF with the C-shift and the d orth -shift
schemes, only minor differences are seen for these small examples, but in all cases the d orth -shift
scheme presents a faster or similar convergence rate compared to the C-shift scheme. With the
ARH method the convergence is further improved compared to the TRSCF/d orth -shift scheme. It is
only a matter of saving a single iteration in some of the examples, but the tendency is clear. As the
algorithm is still in the implementation phase, no numbers can currently be obtained with the
Hückel start guess.
Comparing now the results from our SCF program with the results from the GTR paper, the obvious
peculiarity is the discrepancies between the DIIS results obtained by Francisco et. al. and by us. A
plain DIIS optimization should be completely reproducible, but there is a difference of two out of
seven iterations. These differences cannot be explained and make it more difficult to compare our
results with theirs. Furthermore it seems that they have not tested the Hessian eigenvalues at the
end; only if they for some other start guess or optimization method found a lower energy, it is noted
in their table, and thus we cannot know for sure if the given number of iterations corresponds to
convergence to a minimum. For Cr 2 and CrC it is very difficult to find the minimum, and several
saddle points exist where convergence can be obtained (see Section 1.6). It is thus an open question
whether the GTR+DIIS calculations for Cr 2 and CrC actually converge to a minimum or to a saddle
point as for the TRSCF methods.
In the examples where GTR+DIIS gives an improvement compared to their DIIS results, TRSCF
and ARH also give significant improvements to our DIIS results. For the distorted CO example,
TRSCF and ARH show better convergence than GTR+DIIS even if the results could be compared
directly. For all examples TRSCF and ARH converge in 7-14 iterations, whereas GTR+DIIS use
between five and 115. However, as discussed in Section 1.4.1.3, DIIS does not perform well when
the gradient and energy are not correlated as is often the case in the global region when using
TRRH, and could very well be the case for GTR as well. TRRH should be combined with a density
subspace minimization method in the energy (e.g. TRDSM), and the same probably applies for
GTR. We would thus suggest an implementation of TRDSM in connection with GTR.
In conclusion it has been illustrated that the TRSCF and ARH methods have very nice convergence
properties with improvements compared to DIIS in general and to GTR+DIIS as well, in case of
more problematic examples.
53

Part 1
Improving Self-consistent Field Convergence
1.8.2 Calculations on Metal Complexes
In reference 39 and throughout this part of the thesis, three molecules including transition metals
have been used for examples, namely the molecules in Fig. 1.3, Fig. 1.6 and Fig. 1.33. In this
section HF and LDA calculations on these metal complexes are given both for DIIS, TRSCF and
TRSCF-LS. For all calculations a H1-core start guess has been employed and a maximum of 10
matrices are used to define the subspace in the density subspace minimization. This is different
from the examples given in ref. 39, where the subspace dimension never was larger than eight.
Furthermore for the TRSCF calculations in ref. 39 the C-shift scheme was applied whereas in the
calculations reported here, the d orth -scheme has been applied.
TRSCF-LS is the TRSCF line search method in which the TRRH-LS and TRDSM-LS steps
described in Sections 1.4.1.4 and 1.4.2.4 are combined to set up an expensive, but highly robust
method, in which the lowest SCF energy is identified by a line search at each step. The convergence
results of the optimizations are seen in Fig. 1.36. For the cadmium complex a STO-3G basis set has
been applied, for the rhodium complex the AhlrichsVDZ basis set 59 has been applied except for the
rhodium which is described in the STO-3G basis and for the zinc complex the 6-31G basis set has
been applied.
The convergence of the TRSCF and TRSCF-LS methods is comparable for all cases in Fig. 1.36,
and in general the TRSCF calculations converge in fewer iterations than the TRSCF-LS calculations
do. As mentioned the line search method TRSCF-LS is much more expensive than TRSCF, and the
only reason for applying it instead of TRSCF is for very difficult examples, where convergence
cannot be obtained in any other way.
The convergence behavior of the DIIS method is somewhat more erratic than that of the TRSCF
methods since it makes no use of Hessian information and therefore cannot predict reliably what
directions will reduce the total energy. The HF calculation on the rhodium complex and the LDA
calculation on the zinc complex both diverge for the DIIS method. In general the erratic behavior is
in particular seen in the global region whereas in the local region, it converges as well as the
TRSCF method.
54

Applications
HF
LDA
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
A
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
A
1.E-08
0 5 10 15 20
Iteration
1.E-08
0 5 10 15 20
Iteration
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
B
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
B
1.E-08
0 10 20 30 40
Iteration
1.E-08
0 10 20 30 40
Iteration
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
C
Error in energy / E h
1.E+02
1.E+00
1.E-02
1.E-04
1.E-06
C
1.E-08
0 10 20 30 40
Iteration
1.E-08
0 10 20 30 40
Iteration
DIIS TRSCF TRSCF-LS
Fig. 1.36 Convergence of HF and LDA calculations on (A) the cadmium complex from Fig. 1.6,
(B) the rhodium complex from Fig. 1.33, and (C) the zinc complex from Fig. 1.3.
For the examples presented both in this and the previous subsection, the TRSCF convergence is as
good as or better than DIIS, and for problems where DIIS diverges, convergence is obtained with
the TRSCF methods. It thus seems that TRSCF has the properties of a good black-box optimization
algorithm.
55

Part 1
Improving Self-consistent Field Convergence
1.9 Conclusion
In this part of the thesis the trust region SCF (TRSCF) algorithm is presented as a means to improve
SCF convergence compared to methods typically used today e.g. DIIS. In the TRSCF method, both
the Roothaan-Hall (RH) step and the density-subspace minimization (DSM) steps are replaced by
optimizations of local energy models of the Hartree-Fock/Kohn-Sham energy E SCF . These local
models have the same gradient as the energy E SCF , but an approximate Hessian. Restricting the steps
of the TRSCF algorithm to the trust region of these local models, that is, to the region where the
local models approximate E SCF well, smooth and fast convergence may be obtained.
The developments through the years in SCF optimization algorithms are reviewed, and it is found
that the fundamental schemes used in TRSCF to improve convergence have been around for several
years; DIIS is actually a subspace minimization in the gradient norm, and level shifts have been
used to improve or force convergence since 1973. Anyhow, the level shifts have previously been
found on a trial and error basis as a constant parameter, whereas we advocate a dynamic level shift
scheme in which the level shift is used to control the density change in the RH step. As such the
level shift is optimized in each iteration to allow the density to change to the trust radius of the RH
energy model, hence the name trust region Roothaan-Hall (TRRH) for our RH scheme. Also, the
density subspace minimization has been improved compared to previous methods. An accurate
energy model is constructed in the iterative subspace, where only minor approximations are made
compared to the SCF energy. The trust region minimization of this energy model thus corresponds
well to a minimization of E SCF in the iterative subspace, thus resulting in an energy decrease in each
trust region DSM (TRDSM) step. The TRRH and TRDSM steps in combination make up a
successful scheme with a high convergence rate without compromising the control of the density
changes in each step.
Compared to ref. 38 and 39 , an alternative level shift scheme (d orth -shift) for the TRRH step is
presented which does not control the density change through the overlap of the individual orbitals,
but instead controls the amount of new information added to the density subspace. Thus the d orth -
shift scheme does not contain any reference to the MO basis and can be used in connection with
alternatives to diagonalization. Also, it is found that the d orth -shift scheme leads to a faster
convergence since the former level shift scheme is too restrictive, ignoring the well known changes
contained in the density subspace.
For TRDSM, an improvement of the energy model is developed, in which a part of the term
neglected in the DSM energy model compared to the SCF energy is recovered. However, the effects
of the improvement are found rather small compared to the extra complexity added to the algorithm.
56

Conclusion
An energy minimization algorithm is presented as well, replacing the standard RH-diagonalization
in the SCF optimization. The novel idea is to exploit the valuable information saved in the density
subspace of the previous densities to construct an improved RH energy model (augmented
Roothaan-Hall - ARH) and minimize this model instead of the RH model. This makes the TRDSM
step redundant since a density subspace minimization now is included in the minimization of the
RH energy model. We expect a faster convergence rate for ARH compared to TRSCF, mainly
because the RH and DSM steps are merged to an energy model with correct gradient (not just in the
subspace) and an approximate Hessian, which is improved in each iteration using the information
from the previous density and Fock matrices. The preliminary results from the ARH energy
minimization seems promising, with convergence improvements compared to TRSCF, which
already had better or as good convergence rates as DIIS.
The errors introduced in the TRRH and TRDSM energy models compared to the SCF energy are
studied. Since the DFT and HF energy expressions differ, the errors in the energy models are
potentially different for the two methods. It is found that the DSM energy model has the same error
of the order ||D δ || 2 for both HF and DFT, where D δ is the idempotency correction we impose on the
averaged density. For the RH energy model it is found by inspecting test cases that the errors are
larger for LDA than for HF, especially when convergence is approached. The error can be divided
into two sources, namely the error in the RH Hessian compared to the SCF Hessian, and the size of
the third and higher order contributions from the nonlinear terms in the SCF energy, which are not
included in the RH energy model. By further tests it seems that the Hessian is better described in
LDA than in HF, and since the errors are larger for LDA in particular close to convergence, it seems
unlikely that the third and higher order terms are causing the difference. The question why larger
errors are seen for LDA than for HF is thus still unanswered and it will be further investigated.
The stability of stationary points is discussed and a method to test and walk away from unstable
stationary points is described, and examples are given, where it has been applied. It is
acknowledged that such a method is very valuable since otherwise a minimum could not have been
found for the examples given.
The scaling of TRSCF is also considered. An alternative to diagonalization has been implemented
in our SCF program, where instead of diagonalizing the Fock matrix, the trace purification scheme
by Palser and Manolopoulos 19 and later Niklasson 48 is used. The purification scheme in combination
with the d orth -shift scheme make the TRRH step near-linearly scaling. The trace purification scheme
is linear scaling in an orthogonal basis, but since the optimization scheme is formulated in the nonorthogonal
AO basis, the transformation to an orthogonal basis has an N 2 scaling with a small
prefactor. Timings for the TRRH step with diagonalizations and with purifications are given, and it
57

Part 1
Improving Self-consistent Field Convergence
is seen that the trace purification scheme is a major improvement compared to diagonalization when
more that a couple of thousand basis functions are needed. The TRDSM step is based on matrix
multiplications and additions, so by construction it will be linearly scaling when sparsity in the
matrices is exploited.
As illustrated in the examples throughout this part of the thesis and in the applications section,
significant improvements to SCF convergence have been obtained. For both the TRSCF and ARH
examples presented, the convergence is as good as or better than DIIS, and for problems where
DIIS diverges, convergence is obtained with the TRSCF and ARH methods. The globally
convergent trust region method by Francisco et. al. 26 is found to be better only for the simplest
examples whereas for the rest, the TRSCF and ARH methods are found superior. The future success
of the TRSCF method depends on a well optimized implementation of the diagonalization
alternative combined with the dynamic level shift scheme, and sparsity being exploited in an
efficient manner such that it can compete with the linear scaling SCF programs used today. The
future success of the ARH method depends on finding efficient ways of solving the nonlinear
equations corresponding to the minimization of the energy model. For this purpose different
preconditioners will be tested.
To conclude, there are still some adjustments that should be done to improve the algorithms, but the
framework is in place. The SCF optimization algorithms presented in this thesis, each make up a
black-box optimization scheme for HF and DFT as there is one scheme without any user-adjustment
that lead to fast and stable convergence for both simple and problematic systems studied so far. We
are thus convinced that TRSCF and ARH are build to handle the optimization problems of the
future.
58

Part 2
Atomic Orbital Based Response Theory
2.1 Introduction
The first part of this thesis was concerned with the optimization of the one electron density matrix
for Hartree-Fock (HF) and density-functional theory (DFT). From such an optimized density,
information about excited states and how the system reacts to a perturbation (e.g. an external
electric field) may be obtained using response theory. Response theory and the derivation of
molecular properties will be the subject of this part of the thesis.
Response theory provides a rigorous approach for calculating molecular properties. As for the SCF
optimization algorithms, the theory has usually been formulated in the molecular orbital (MO) basis
which is inherently delocal, making the implicated matrices non-sparse. A reformulation in the local
atomic orbital (AO) basis is thus necessary to obtain linear scaling algorithms and permit
calculations of properties for large systems. Such a reformulation, in which an exponential
parameterization of the density matrix is employed, is given in a paper by Larsen et al. 61 .
The AO formulation of the response functions has a number of advantages compared to the MO
formulation, besides locality. The response equations and molecular property expressions are
simpler in the AO basis as the involved matrices (e.g. the Fock and property matrices) enter the
equations in the basis they are evaluated in originally. No transformation between bases is necessary
in the AO formulation as it is in the MO formulation. The AO formulation is particular convenient
for perturbation dependent basis sets. In the MO formulation a set of perturbation dependent
orthonormal molecular orbitals must be introduced. These orbitals have no physical content and
thus add artificial complexity to the problem. To exemplify the benefits of the AO formulation, the
expression for the excited state geometrical gradient is derived in Section 2.4.
59

Part 2
Atomic Orbital Based Response Theory
In the conventional MO formulation, number operators are redundant and can be eliminated.
However, in the AO basis the number operators are not redundant and must be included. Because of
this, the proof of pairing in the solutions of the response equations cannot be directly taken from the
MO basis to the AO basis. It is thus necessary to study the impact of the included number operators
on the solver for the AO response equations. This has been done in Section 2.2, using the method of
second quantization to formulate the AO based response equations. Implementation issues
connected to solving the AO response equations are discussed in Section 2.3. In Section 2.5 a
couple of simple examples are given, where the AO response solver is used to find ground and
excited state properties. In Section 2.6 the results of this part of the thesis are summarized.
2.2 AO Based Response Equations in Second Quantization
In this section the linear response equations are derived for Hartree-Fock theory, but with minor
technical changes they apply to DFT as well. The quadratic and higher response equations could
equally well be derived in this formulation; however, this is not necessary to arrive at the basic
conclusions.
2.2.1 The Parameterization
Consider a set of atomic orbitals (χ µ ) with the real and symmetric metric S. The creation and
annihilation operators for the atomic orbitals fulfil the anticommutation relation
†
µ , ν + νµ
⎣⎡a a ⎦ ⎤ = S . (2.1)
We will consider the following exponential operator
Tˆ
= exp ( iκˆ
), (2.2)
where ˆκ is a Hermitian one-electron operator
To examine the action of
ˆ κ = ∑ κ
(2.3)
µν
†
µν aµ aν
†
κ = κ .
(2.4)
exp( iκ ˆ)
, we consider the transformed creation operators
a = exp( iˆ) a exp( −iˆ
κ)
. (2.5)
† †
µ κ µ
It is seen that the transformed operators satisfy the same anticommutation relations as the
untransformed operators
⎡⎣a
a
⎤⎦ ⎡⎣ iˆ a iˆ iˆ a iˆ
⎤⎦
† †
µ , ν = exp( κ) µ exp( − κ),exp( κ) ν exp( − κ)
+ +
= exp( iˆ
κ) ⎡⎣a , a exp( − iˆ) = S .
†
µ ν ⎤⎦
κ
+
νµ
(2.6)
60

AO Based Response Equations in Second Quantization
The exponential operators of Eq. (2.2) are therefore the manifold of operators that conserves the
general metric S. In the special case where S = 1, the exponential operator reduces to the standard
exponential operator occurring in the second quantization formalism of the molecular orbital based
method. 46
Using the Baker-Champbell-Hausdorff expansion 46 and the anticommutation relation of Eq. (2.1),
we get
a
a i ˆ a ˆ ˆ a
† † † 1
†
µ = µ + ⎡⎣κ, µ ⎤⎦− ⎡ , ,
2 ⎣κ ⎡⎣κ
µ ⎤⎤ ⎦⎦ +
2
µ ∑ νµ ν 2 ∑ νµ ν
ν
ν
† † 1
†
= a + i ( κS ) a − ( κS ) a + . (2.7)
=
∑
ν
†
exp ( iκS
) a .
νµ
ν
To further investigate the properties of the above exponential transformation, we next consider the
transformation of a single determinant state 0 with exp( iκ ˆ)
0 = exp( iκˆ
) 0 . (2.8)
The properties of 0 may be obtained by comparing the expectation values of transformed
creation-annihilation operators
∆ = 0
a a 0
= 0 exp( −iˆ κ) a exp( iˆ κ) exp( −iˆ κ) a exp( iˆ
κ) 0
(2.9)
† †
µν µ ν µ ν
with the expectation values of the untransformed operators
†
µν aµ aν
∆ = 0 0 . (2.10)
To rewrite Eq. (2.9) in terms of Eq. (2.10) we use Eq. (2.7) to write the transformed creation- and
annihilation-operators in terms of the untransformed operators
∑
∑
exp( − iˆ
κ) a exp( iˆ) = exp( −i ) a
† †
µ κ
κS ρµ ρ
ρ
exp( − iˆ
κ) a exp( iˆ
κ) = exp( iSκ) a .
ν νρ ρ
ρ
T
T
( i ) exp ( i )
(2.11)
Substituting these expressions into Eq. (2.9) gives
∆ = exp - Sκ ∆ κ S . (2.12)
In Appendix B, it is shown that if 0 is a single determinant wave function, then ∆ fulfils Eqs.
(B-7), corresponding to the symmetry, trace, and idempotency condition for the one-electron
density. We will now show that if ∆ fulfils these equations then so does ∆ . The Hermiticity of ∆
follows from the Hermiticity of S and κ and will not be shown explicitly here. The trace relation is
shown as follows
61

Part 2
Atomic Orbital Based Response Theory
Tr ∆S = Tr ∆exp( iκ SS ) exp( −iSκ ) SS
−1 T −1 T −1
−1
T T −1
= Tr ∆exp( iκ S) exp( −iκ SS )
= Tr ∆S ,
(2.13)
where we have used the relation
−1 −1
B exp( A) B = exp( B AB ) . (2.14)
The same relation may be used to show the idempotency relation
−
( i ) ( i ) ( i ) ( i )
T T T −1 T
( iSκ ) ∆ ( iκ S ) ( iκ S) S ∆ ( iκ S )
T −1 T
( iSκ ) ∆S ∆exp
( iκ S )
T
T
( iSκ ) ∆ ( iκ S ) ∆
−1 T T 1 T T
∆S ∆
= exp − Sκ ∆exp κ S S exp − Sκ ∆exp
κ S
= exp − exp exp − exp
= exp −
= exp − exp = .
(2.15)
We can therefore conclude that ∆ fulfils Eqs. (B-7) and exp( iκ ˆ) 0 is therefore a legitimate
normalized single-determinant wave function. It can be shown that all matrices fulfilling Eqs. (B-7)
can be obtained from an appropriate choice of κ, so the transformation of Eq. (2.8) is a complete
parameterization.
2.2.2 The Linear Response Function
We will now use the parameterization of Eq. (2.8) for an arbitrary single-determinant wave function
to describe a Hartree-Fock wave function in an external, time-dependent field. The parameters in κ
will become time-dependent and we will in the following develop equations for obtaining these
parameters. The time-dependent Hamiltonian can be written as
H = H0 + Vt
, (2.16)
where H 0 is the Hamiltonian for the unperturbed system, and V t is a first-order perturbation. The
perturbation will be turned on adiabatically, and V t can be expressed as
∞
−∞
Vt
= ∫ dωVω
exp( ( − iω + ε ) t)
, (2.17)
where ε is a positive infinitesimal that ensures V t → 0 as t → -∞. The perturbation is required to be
Hermitian, so we have the relation
†
ω
V
= V . (2.18)
−ω
To determine the linear response function, we begin by considering the time dependence of the
expectation value 0
A 0 of a one-electron operator A. We need only expand the wave function
0 of Eq. (2.8) to first order in the external perturbation to obtain the linear response:
(1) (2)
t t
ˆ κ = ˆ κ + ˆ κ +. (2.19)
62

AO Based Response Equations in Second Quantization
(0)
ˆt κ
The zero-order contribution, , vanishes as the unperturbed wave function 0 is assumed to be
optimized for the zero-order Hamiltonian, so the Brillouin-conditions in the AO basis hold
∂
∂
κ µν
†
µ ν
0 H0 0 = i 0 ⎡⎣H0, a a ⎤⎦
0 = 0. (2.20)
Substitution of the expansion of ˆκ into Eq. (2.8) gives to first order:
(1)
0 A 0 = 0 A 0 −i 0 ⎡ ˆ κt
, A⎤
⎣ ⎦
0 . (2.21)
Since the response functions are defined in the frequency rather than the time domain, we formulate
the wave function corrections in the frequency space. By analogy with Eq. (2.17), we write
∞
−∞
Inserting Eq. (2.22) into Eq. (2.21) we obtain
(1) (1)
κt = ∫ dωκω
exp( ( − iω + ε ) t)
. (2.22)
∞
(1)
0 A 0 = 0 A 0 −i dω 0 ⎡ ˆ κω
, A⎤
⎣ ⎦
0 exp (( − iω + ε)
t)
. (2.23)
∫
-∞
Comparing Eq. (2.23) with the formal expansion of an expectation value in terms of a response
function
∞
-∞
0 A 0 = 0 A 0 + d ω A; V exp (( − iω + ε)
t)
, (2.24)
we may identify the linear response function as
∫
ω
ω
(1)
ω
AV ; ω 0 ˆ
ω
=−i ⎡κ
, A⎤
⎣ ⎦
0
. (2.25)
2.2.3 The Time Development of the Reference State
Before the explicit time-dependent equations are set up for determining the time-dependent
parameters of κ, it is convenient to rewrite ˆκ , Eq. (2.3), as
† † †
∑( µν µ ν ∗
µν ν µ ) ∑ µµ µ µ , (2.26)
ˆ κ = κ a a + κ a a + κ a a
µ > ν µ
which follows from the Hermiticity of ˆκ . The operators of ˆκ may be collected in a vector (here in
row form):
where the three classes of operators are defined as
† †
( )
Λ = Q D Q , (2.27)
Q
D
Q
† †
m aµ aν
= , µ > ν
† †
m = aµ aµ
m
†
ν µ
= a a , µ > ν .
(2.28)
63

Part 2
Atomic Orbital Based Response Theory
The parameters of κ may similarly be arranged in a vector
such that
⎛ ⎞ >
() i
κ µν µ ν
⎜ ⎟
() i () i
= ⎜ κµµ
⎟
⎜ () i
κ µ ν ,
µν
∗ ⎟ >
α (2.29)
⎝
⎠
ˆ() i ()
κ = ∑ αm
i Λm
. (2.30)
m
Here the index m on Λ runs over all three classes of operators listed in Eq. (2.28).
The single excitation operators a †
µ aν have by Eq. (2.27)-(2.28) been divided into a set of atomic
orbital excitations, corresponding to µ > ν and a set of atomic orbital deexcitations, corresponding to
µ < ν. As the atomic orbital excitations and deexcitation have the same formal properties, this
division does not have any physical content. However, the division will prove important when the
paired structure of the response equations is investigated in Section 2.2.5. Note that it is not possible
to exclude the number operators a †
µ aµ in the atomic orbital representation, whereas they are
redundant in the standard molecular orbital formulation.
In the presence of the time-dependent perturbation, we introduce the time transformed operator
basis
⎛ Q
⎞
† ⎜ ⎟
Λ
= ⎜ D
⎟ , (2.31)
⎜ † ⎟
⎝Q
⎠
where
and similarly for
†
Q m and D m .
Q = exp( iˆ
κ) Q exp( −iˆ
κ)
(2.32)
m
The time evolution of 0 may now be determined using Ehrenfest’s theorem for the transformed
†
operators of Λ in Eq. (2.31):
d † ∂
0 0 0
† 0 0
†
Λ −
⎛
Λ
⎞
= − ⎡ Λ , 0 + ⎤ 0
dt
2.2.4 The First-order Equation
m
⎜
i H V
∂t
⎟
⎣ t
⎝ ⎠
⎦ . (2.33)
We now expand Eq. (2.33) in orders of the external perturbation, restricting ourselves to terms that
are linear in the amplitudes. Inserting Eq. (2.19) into Eq. (2.33) and collecting the terms linear in the
perturbation, we obtain the first-order time-dependent equation
64

AO Based Response Equations in Second Quantization
† (1) † † (1)
κt
=− ⎡ t ⎤ +
0 ˆ κt
i 0 ⎡ , ⎤ 0 i 0 , V 0 0 ⎡ , ⎡H
, ⎤⎤
⎣ Λ
⎦ ⎣ Λ ⎦ ⎣ Λ ⎣ ⎦⎦
0 . (2.34)
To solve the time-dependent equation Eq. (2.34), we insert the frequency expansion of the wave
function correction of Eq. (2.22) and of the external perturbation Eq. (2.17)
∞
−∞
∞
∫−∞
∫
(1) (1)
( − i + t)( ⎡Λ
† ˆω
⎤ − ⎡Λ
† ⎡H0
ˆω
⎤⎤ )
dωexp ( ω ε) ω 0
⎣
, κ
⎦
0 0
⎣
,
⎣
, κ
⎦⎦
0
†
( i t)( i ⎡Λ
Vω
⎤ )
= dωexp ( − ω + ε) − 0 ⎣ , ⎦ 0 .
The first-order response equation is then found as
† (1) † (1)
ˆ
†
ω H0
ˆω i Vω
(2.35)
ω 0 ⎡ , κ ⎤ 0 0 ⎡ , ⎡ , κ ⎤⎤
⎣
Λ
⎦
−
⎣
Λ
⎣ ⎦⎦
0 = − 0 ⎡⎣ Λ , ⎤⎦
0 . (2.36)
The equation may be written in terms of the matrices
and the vector
E
= 0 ⎡⎣ Λ ,[ H0
, Λ ] ⎤⎦ 0 , (2.37)
[2] †
Smn = 0 ⎡⎣ Λm , Λn
⎤⎦ 0 , (2.38)
[2] †
mn m n
[1] †
ω = Λ
m
m
⎡⎣V
⎤⎦ 0 ⎡⎣ , Vω
⎤⎦ 0 . (2.39)
Using Eqs. (2.37)-(2.39) and (2.29)-(2.30), we now write the first-order response equations, Eq.
(2.36), in the form
( ω )
[2] − [2] (1) = i [1]
ω
E S α V , (2.40)
where E [2] and S [2] may be viewed as generalized electronic Hessian and overlap matrices 61,62 . The
[2] [2]
matrix elements E mn and S mn (Eq. (2.37) and (2.38)) can be expressed as matrix multiplications
and additions of the density, Fock and overlap matrices. 61
The linear response function is obtained by inserting the first-order correction as obtained in Eq.
(2.40) in the expression for the linear response function Eq. (2.25). Renaming the perturbation
operator V ω to B and introducing
we obtain
A
B
[1]
m =− ⎡ ⎣ Λm
[1] †
m = ⎡Λm
0 , A⎤
⎦ 0
(2.41)
0 ⎣ , B⎤
⎦ 0
−
( ) 1
[1] [2] [2] [1]
AB ; ω
=−A E −ωS B . (2.42)
The linear response function may thus be calculated by solving one set of linear equations at each
frequency. To be more explicit, denoting the solution vector to the linear response equation
B ω
−
( ω ) 1
[2] [2] [1]
N ( ) = E − S B , (2.43)
65

Part 2
Atomic Orbital Based Response Theory
the linear response function in Eq. (2.42) can be obtained as
[1]
B
AB ; ω
=−A N ( ω)
. (2.44)
2.2.5 Pairing
The excitation energies are identified as the poles of the linear response function of Eq. (2.42) and
are therefore solutions to the generalized eigenvalue problem
[2] [2]
E X = ωS X. (2.45)
In the MO formulation of response theory, it has been shown that the excitation energies are
paired 63 , so that if ω i is an eigenvalue for Eq. (2.45) then so is -ω i . It is important to understand how
pairing appears in the AO basis, in particular since this structural feature is exploited when the
equations are solved iteratively as is necessary for large problems. This is further discussed in
Section 2.3. Since the proof of the pairing given in the MO formulation cannot be directly
transferred to the AO formulation due to the presence of the diagonal operators D m , this section
gives the proof in the AO formulation.
The structure of E [2] and S [2] in the AO formulation is analyzed for the purpose of examining the
pairing structure. Dividing Λ into the tree classes of Eq. (2.28), the matrix E [2] may be written as
†
⎛ 0 ⎡⎣Q, ⎡⎣H0, Q ⎤⎤ ⎦⎦ 0 0 [ Q, [ H0, D]
] 0 0 [ Q, [ H0, Q]
] 0 ⎞
[2]
⎜
⎟
†
E = ⎜ 0 ⎣⎡D, ⎣⎡H0, Q ⎦⎦ ⎤⎤ 0 0 [ D, [ H0, D]
] 0 0 [ D, [ H0, Q]
] 0 ⎟. (2.46)
⎜ † † † †
⎟
⎝ 0 ⎣⎡Q , ⎣⎡H0, Q ⎦⎦ ⎤⎤ 0 0 ⎣⎡Q ,[ H0, D] ⎦⎤ 0 0 ⎣⎡Q ,[ H0, Q]
⎦⎤
0 ⎠
If we assume for simplicity that all orbitals and integrals for the unperturbed system are real, the
†
elements of for example the block 0 ⎡⎣Q ,[ H0
, Q ] ⎤⎦
0 are trivially rewritten as
† †
∗
0 ⎡⎣Qm, [ H0, Qn ] ⎤⎦ 0 = 0 ⎡⎣Qm, [ H0, Qn
] ⎤⎦
0
(2.47)
†
= 0 ⎡⎣Qm, ⎡⎣H0
, Qn
⎤⎤ ⎦⎦ 0 .
The nine blocks in Eq. (2.46) can then all be written in terms of the following four matrices
and we obtain
†
mn m 0 n
A = 0 ⎡⎣Q , ⎡⎣H , Q ⎤⎤ ⎦⎦ 0 ,
Bmn = 0 ⎡⎣Qm , ⎡⎣H0
, Qn
⎤⎤ ⎦⎦ 0 ,
(2.48)
Fmn = 0 ⎡⎣Qm , ⎡⎣H0
, Dn
⎤⎤ ⎦⎦ 0 ,
Gmn = 0 ⎡⎣Dm , ⎡⎣H0
, Dn
⎤⎤ ⎦⎦ 0 ,
⎛ A F B ⎞
[2] ⎜ T T
E = F G F
⎟
. (2.49)
⎜
⎟
⎝ B F A ⎠
66

AO Based Response Equations in Second Quantization
The matrix S [2] may in a similar way be written as
⎛ Σ Ω ∆ ⎞
[2] T T
S =
⎜
Ω 0 -Ω
⎟
⎜
- - -
⎟
⎝ ∆ Ω Σ ⎠
, (2.50)
where
†
mn ⎡Qm Qn
Σ = 0 ⎣ , ⎤⎦
0 ,
∆ mn = 0 ⎡⎣Qm , Qn
⎤⎦
0 ,
Ω = 0 [ Q , D ] 0 .
mn m n
(2.51)
Note that the block containing two diagonal operators vanishes as
† † † †
[ Dm
Dn
] = ⎡⎣aµ aµ aνaν ⎤⎦ = Sµν aµ aν − Sνµ aν aµ
= . (2.52)
0 , 0 0 , 0 0 0 0 0 0
To illustrate how the pairing is obtained in the AO formulation, we assume that the vector
⎛ Z ⎞
X =
⎜
U
⎟
⎜ ⎟
⎝Y
⎠
(2.53)
is an eigenvector for Eq. (2.45) with eigenvalue ω
⎛ A F B ⎞⎛ Z⎞ ⎛ Σ Ω ∆ ⎞⎛ Z ⎞
⎜ T T ⎟⎜ ⎟ T T
F G F U = ω
⎜
Ω 0 -Ω ⎟⎜
U
⎟
. (2.54)
⎟⎜ ⎟⎜ ⎜ ⎟⎜ ⎟ ⎜
- - -
⎟⎜ ⎟
⎝ B F A ⎠⎝Y⎠ ⎝ ∆ Ω Σ ⎠⎝Y
⎠
Multiplying the blocks of Eq. (2.54) gives three sets of equations
AZ + FU + BY = ω ( ΣZ + ΩU + ∆Y )
( )
T T T T
F Z+ GU+ F Y = ω Ω Z −Ω Y
BZ + FU + AY = ω ( −∆Z −ΩU −ΣY
).
(2.55)
We will now prove that the paired vector
X
P
⎛Y
⎞
=
⎜
U
⎟
⎜ ⎟
⎝ Z ⎠
(2.56)
is an eigenvector for Eq. (2.45) with eigenvalue –ω
⎛ A F B ⎞⎛Y⎞ ⎛ Σ Ω ∆ ⎞⎛Y
⎞
⎜ T T ⎟⎜ ⎟ T T
F G F U =−ω
⎜
Ω 0 -Ω ⎟⎜
U
⎟
. (2.57)
⎟⎜ ⎟⎜ ⎜ ⎟⎜ ⎟ ⎜
- - -
⎟⎜ ⎟
⎝ B F A ⎠⎝ Z⎠ ⎝ ∆ Ω Σ ⎠⎝ Z ⎠
Multiplying the blocks of Eq. (2.57) leads to the three sets of equations
67

Part 2
Atomic Orbital Based Response Theory
AY + FU + BZ = − ω ( ΣY + ΩU + ∆Z )
( )
T T T T
F Y+ GU+ F Z = −ω
Ω Y −Ω Z
BY + FU + AZ = −ω
( −∆Y −ΩU −ΣZ
),
(2.58)
which are identical to Eqs. (2.55). It is thus concluded that if X is an eigenvector of Eq. (2.45) with
eigenvalue ω, then X P is also an eigenvector with eigenvalue –ω.
2.3 Solving the Response Equations
For large systems, the response equations
( ω )
[2] [2] [1]
E − S N B ( ω ) = B (2.59)
are best solved using iterative algorithms. These algorithms rely on the ability to set up linear
transformations. Expressions for E [2] b and S [2] b, where b is a trial vector, have previously been
derived. 61 [2]
σ = E b (2.60)
[2]
ρ = S b. (2.61)
In each iteration, the response equations are set up and solved in a reduced space. For a reduced
space consisting of k trial vectors, the equations can be written as
where the reduced matrices are found as
( ω )
[2] [2] RED [1]
RED
−
RED
=
RED
E S X B , (2.62)
[2] T [2] T
RED ⎦ i j i j
ij
⎡
⎣
E ⎤ = b E b = b σ
[2] T [2] T
RED ⎦ i j i j
ij
⎡
⎣
S ⎤ = b S b = b ρ
[1] T [1]
RED ⎦
bi
B .
i
⎡
⎣
B ⎤ =
(2.63)
Normally when this type of iterative procedure is used, the reduced space is extended with one new
trial vector in each iteration. However, due to the pairing described in the previous section, the
linear transformations of E [2] and S [2] on a trial vector, here exemplified by E [2] b,
⎛ A F B ⎞⎛ Z⎞ ⎛ AZ+ FU+
BY ⎞
[2] ⎜ T T ⎟⎜ ⎟ ⎜ T T
E b = F G F U = F Z+ GU+ F Y
⎟
= σ , (2.64)
⎟⎜ ⎜ ⎟⎜ ⎟ ⎜
+ +
⎟
⎝ B F A ⎠⎝Y⎠ ⎝ BZ FU AY ⎠
may be obtained directly for the paired trial vector as well
⎛ A F B ⎞⎛Y⎞ ⎛ AY+ FU+
BZ ⎞
[2] P ⎜ T T ⎟⎜ ⎟ ⎜ T T ⎟ P
E b = F G F U = F Y+ GU+ F Z = σ . (2.65)
⎟⎜ ⎜ ⎟⎜ ⎟ ⎜
+ +
⎟
⎝ B F A ⎠⎝ Z⎠ ⎝ BY FU AZ ⎠
68

Solving the Response Equations
The reduced space is therefore extended with both vectors without additional cost. Furthermore,
when a trial vector and its paired counterpart are simultaneously added to the reduced space, the
paired structure of the response equations is preserved. With this structure preserved, the
eigenvalues in the reduced space will also be real and paired, and the lowest eigenvalue will
monotonically decrease towards the converged value as the reduced space is increased. 64
The solution vector in the reduced space X RED , can be expanded in the basis of trial vectors to
express the solution vector in the full space
k
B
. (2.66)
N
= ∑
i=
1
RED
( X i bi
)
The residual can then be found as
k
( ω )
R = E − S N
−B
k
∑
[2] [2] B [1]
= X ( σ −ωρ ) −B
i=
1
RED [1]
i i i
.
(2.67)
If the norm of the residual is smaller than some specified tolerance, the iterative procedure is ended
and the converged solution vector has been found
B
B
N ( ω ) = N . (2.68)
If the residual is too large, a new trial vector may be generated from the residual, preferably with a
preconditioner A to speed up the convergence
k+ 1 =
−1
b A R . (2.69)
The reduced space is then extended with b k+1 and bk
+ 2 = b
k + 1
and Eq. (2.62) is set up and solved
again, establishing the iterative procedure.
2.3.1 Preconditioning
As mentioned above, the residual found in each iteration should be preconditioned to obtain an
effective solver. As a consequence of the strict AO formulation, the electronic Hessian has no
diagonal dominance as was the case in the MO basis. This makes preconditioning a challenge. So
far, this problem has not been solved in our SCF response solver. Instead, a transformation is made
to the MO basis, where the preconditioning is carried out in the usual way using the orbital
eigenvalue differences,
k
P
MO
T
⎣⎡b + 1 ⎦⎤ = ⎣⎡C RkC ⎦⎤
( εa −εi
), (2.70)
k ai ai
69

Part 2
Atomic Orbital Based Response Theory
where C is the MO expansion coefficients and ε the orbital energies of the reference state. The
index a refers to virtual orbitals and i refers to occupied orbitals. The resulting vector is then back
transformed to the AO basis
MO
k + 1 =
k + 1
T
b Cb C . (2.71)
An AO alternative to this preconditioner should of course be found, since the reference to the MO
basis in this preconditioner introduces dense matrix intermediates. Moreover, at least one
diagonalization should be carried out at the end of the optimization of the reference state to obtain
the information on the MOs.
2.3.2 Projections
In the MO basis, the orbital rotations within the occupied and virtual spaces are redundant. The
response equations in the MO formulation are thus simply set up in the non-redundant occupiedvirtual
space to avoid linear dependencies. In the AO basis no such separation exists and the
equations are set up in the full space. To avoid redundancies in the AO formulation, projections
onto the non-redundant space should be made. In the exponential parameterization of the density
matrix used in our AO formulation of the response functions, the projector 23
where
P = P⊗ Q+ Q⊗P
T T
( X) = ∑ µν ρσ X ρσ = ( PXQ + QXP )
P P (2.72)
µν , ,
µν
ρσ
P = DS
Q = 1−DS,
(2.73)
projects onto the non-redundant parameter space. It can be shown that all new trial vectors b and
linear transformations σ and ρ should be projected onto the non-redundant space in the following
manner
b
σ
ρ
= P b
k+ 1 k+
1
T
k+ 1=
P σk+
1
T
k+ 1=
P ρk+
1
,
,
.
(2.74)
When solving the response equations as described in the beginning of this section, the vectors
projected as in Eq. (2.74) are used.
70

The Excited State Gradient
2.4 The Excited State Gradient
In this section the expression for the geometrical gradient of the singlet excited state is derived, to
illustrate how expressions for properties can straightforwardly be derived in the AO response
framework.
As for the derivations in Section 2.2 we assume that the wave function of the ground state is
optimized at the point of the potential surface, x 0 , where the excited state gradient is evaluated. The
variational condition is thus fulfilled at that point
FDS − SDF = 0, (2.75)
and the ground-state energy at x 0 is further obtained as
E
0
= 2TrhD + TrDG ( D ) + h , (2.76)
nuc
where h is the one-electron Hamiltonian matrix in the AO basis, h nuc is the nuclear-nuclear
repulsion, G holds the two-electron AO integrals and the Fock matrix F is given by h + G(D).
As mentioned previously, the excitation energy corresponding to the excitation from the ground
state 0 to the excited state f can be found from the poles of the linear response function for the
optimized ground state, 62 i.e. as the eigenvalue of the linear response generalized eigenvalue
equation as Eq. (2.45)
where ω f is the electronic excitation energy
and b f is the normalized eigenvector. 61,62
( ω f )
[2] [2] f
0
The excitation energy can then be obtained from Eq. (2.77) as
E − S b = , (2.77)
f
0
ω f = E − E
(2.78)
f
f † [2]
assuming that the eigenvectors b f satisfy the normalization condition
f
ω = b E b , (2.79)
f † [2] f
b S b = 1. (2.80)
Since we are interested in the molecular gradient for the excited state, f , the energy of the excited
state should be defined at arbitrary points on the potential surface.
2.4.1 Construction of the Lagrangian
The analytic expression for the excited state gradient is found using the Lagrangian technique 65 . We
construct the Lagrangian for the excited state energy E f = E 0 + ω f , using a matrix-vector notation,
( 1) ( )
f 0 f † [2] f f † [2] f
†
L = E + b E b −ω
b S b − −X FDS−SDF . (2.81)
71

Part 2
Atomic Orbital Based Response Theory
The variational condition on the ground state, Eq. (2.75), and the orthonormality constraint
condition on the eigenvectors, Eq. (2.80), are included, and they are multiplied by the Lagrange
multipliers ω and X , respectively.
We then require the Lagrangian to be variational in all parameters
∂L f
= SDF − FDS = 0
(2.82)
∂X
f
∂L
f † [2] f
= b S b − 1=
0
(2.83)
∂ω
f
∂L
[2] f [2] f
= E b − ωS b = 0
(2.84)
f †
∂b
f
∂L
f † [2] f † [2]
= b E − ωb S = 0
(2.85)
f
∂b
f 0 f † [2] f f † [2] f
∂L
∂E
∂b E b ∂b S b ∂( FDS −SDF
)
n
= + −ω
− X n
= 0
∂X ∂X ∂X ∂X ∑
, (2.86)
∂X
m m m m n
m
where X m are the orbital rotation parameters. Due to the 2n + 1 rule, and since the gradient is a firstorder
property, we only need to solve the above equations through zero order. Eqs. (2.82)-(2.85) are
thus already taken care of, and it is seen that the multiplier ω is determined as the eigenvalue of the
linear response equations, i.e. it corresponds to the excitation energy. It is then only necessary to
determine the Lagrange multipliers X such that Eq. (2.86) is also fulfilled.
2.4.2 The Lagrange Multipliers
To evaluate the terms in Eq. (2.86), the asymmetric Baker-Campbell-Hausdorff (BCH) expansion 46
of the exponentially parameterized density is applied
DX ( ) = exp( − XSD ) exp( SX) = D+ [ DX , ] S
+ , (2.87)
where
[ AB , ] S
= ASB−BSA. (2.88)
Since the derivatives are evaluated at the expansion point, only terms of first order in X are nonzero.
The last term in Eq. (2.86) is found to be equal to 61
[2]
[ , ] [ , ] ([ , ] ) ([ , ] )
E X = F X D S− S X D F+ G X D DS−SDG X D . (2.89)
S S S S
We can thus find X by solving the set of linear equations
E
[2]
0 f † [2] f f † [2] f
∂E
∂b E b ∂b S b
X = + −ω
∂X ∂X ∂X
From the matrix expressions for b f† E [2] b f and b f† S [2] b f 61
. (2.90)
72

The Excited State Gradient
( )
b E b F ⎡ b D b ⎤ G b D D b (2.91)
f † [2] f f f † f f †
=−Tr ⎣
⎡⎣ , ⎤⎦ , −Tr ⎡ , ⎤ ⎡ , ⎤
S ⎦S
⎣ ⎦S ⎣ ⎦S
f † [2] f f † f
b S b = Tr b S⎡⎣D,
b ⎤⎦
S (2.92)
and the relations for the two-electron integrals
T
S
T
( ) = ( )
G A G A (2.93)
Tr AG ( B ) = Tr BG ( A ) , (2.94)
the terms on the right hand side of Eq. (2.90) are found as
where
0
∂E
∂X
= 0 , (2.95)
f † [2] f
f f †
A
2 ω⎡
, ⎤
⎣
SDS ⎡b b ⎤ S
∂X S ⎦
, (2.96)
∂b S b
− ω
= − ⎣ ⎦
f † [2]
∂b E b
∂X
f
= ADS −SDA
, (2.97)
( ) ( ) ( ⎡ , ⎡ , ⎤ ⎤ )
f † f f f f f † f f †
A = Sb Fb S−Sb F − Fb S− Sb F b S+ G
⎣
b ⎣D b ⎦
( ⎡ ⎤ ) ( ⎡ ⎤ )
+ 2 ⎡
, − , ⎤
⎣
Sb G b D G b D b S
⎦
f † f f f †
⎣ ⎦S
⎣ ⎦S
S
S
⎦
S
(2.98)
and
[ ] A 1 1 †
M = M−
M (2.99)
2 2
[ ] S 1 1 †
M = M + M . (2.100)
2 2
Eq. (2.95) is straight forward since the variational condition Eq. (2.75) is fulfilled at the expansion
point.
2.4.3 The Geometrical Gradient
The excited state geometrical gradient should be expressed in terms of the first derivatives of the
one and two electron integral matrices h x , G x , S x and the density, Fock and overlap matrices at the
expansion point x 0 . The notation A x denotes the geometrical first derivative of A. In ref. 66 it was
found that the first derivative of the density D x (X) is given by the first derivative of the reference
density matrix D x which, from the idempotency condition for D, is found to be
x
x
D =−DS D. (2.101)
The first-order geometrical derivative is given by
f f 0 f † [2] f f † [2] f
dE dL dE
∂b E b ∂b S b ∂( FDS −SDF
)
= = + −ω
−X . (2.102)
dx dx dx ∂x ∂x ∂x
73

Part 2
Atomic Orbital Based Response Theory
The first term is simply the geometrical gradient of the ground state. In ref. 66 this was shown to be
E
0 x = 2Tr x + Tr x ( ) + Tr
x + hnuc
x
Dh DG D D F . (2.103)
The other terms are found as the derivative of the matrix expressions in Eq. (2.91) and (2.92)
f † [2]
∂b E b
∂x
f † [2]
f
f
( ( ))
=− Tr F + G D ⎡ , , ⎤ −Tr ⎡ , , ⎤
⎣
b D b
⎦
F
⎣
b D b
⎦
x x f f † f x f †
⎡⎣ ⎤⎦ ⎡ ⎤
S S ⎣ ⎦S
−Tr F⎡⎡⎣ , ⎤⎦ x
, ⎤ Tr ⎡⎡ ⎣ , ⎤⎦
, ⎤
⎣
b D b
⎦
F
⎣
b D b
⎦
f f † f f †
−
S S
S
†
( ⎡ ⎤ ) ⎡ ⎤
S
S
f x f † f †
( ⎡ ⎤ )( ⎡ ⎤ ⎡ ⎤
x )
x f f
− Tr G ⎣b , D⎦ ⎣D,
b ⎦
− 2Tr G ⎣b , D⎦ ⎣D , b ⎦ + ⎣D,
b ⎦
∂b S b
f † x f
− ω
= −ωTr b S ⎡ , ⎤
∂x
⎣D b ⎦ S
S
S S S
( ⎡ ⎤ ⎡ ⎤
x
⎡ ⎤ )
f † x f f f x
⎣ ⎦S ⎣ ⎦S ⎣ ⎦S
− ω Tr b S D , b S+ D, b S+
D,
b S
S
x
S
(2.104)
(2.105)
∂( FDS −SDF
)
x x x x A
− X = − 2X⎡
+ ( ) + + ⎤
∂x
⎣F DS G D DS FD S FDS ⎦ , (2.106)
where F x = h x + G x (D). Collecting the various terms we obtain
f
∂E
∂x
f f † x f † x f
( D ⎡
⎤
⎣
⎡⎣b D⎤⎦ b [ ]
S ⎦
D X
S ) h ⎡ ⎤
S
( ⎡ ⎤
S
)
S
⎣D b ⎦ G ⎣b D⎦
f f †
x x
( D ⎡⎡
⎣b D⎤⎦
b ⎤ [ D X]
) G D hnuc
= Tr 2 − , , − , −Tr , ,
+ Tr −
⎣
, ,
⎦
− , ( ) +
S
S
S
x f f † x f † f †
f
DG( ⎡
⎤
⎣
⎡⎣b D⎤⎦ b ) ( x
S ⎦
⎡
S S
) (
S
)
S ⎣D b ⎤⎦ ⎡⎣Db ⎤⎦ G ⎡⎣b D⎤⎦
x
x
DG( [ DX]
) ( ⎡ ⎤ [ ] x
S ⎣D X⎦
DX
S
S
) F
f x
( ⎡
f † † †
⎡⎣
b D ⎤ , ⎤ ⎡
f f
,
x
, ⎤ ⎡
f f
, , ⎤
⎣ ⎦ b
S ⎦
+
S S
x )
S ⎣⎣ ⎡b D⎦⎤ b
⎦
+
S ⎣⎣ ⎡b D⎦⎤
b
⎦
F
S
f † f x f f x
Tr b S( ⎡b , D ⎤ S ⎡b , D⎤ x
S ⎡b , D⎤
S )
−Tr , , − 2Tr , + , ,
−Tr , − Tr , + ,
− Tr ,
+ ω f ⎣ ⎦ + ⎣ ⎦ + ⎣ ⎦
f † x f
+ ω f Tr b S ⎡⎣b , D⎤⎦
S,
G b D b , ( [ , ] )
where ( ⎡
f
f †
, , ⎤
⎣
⎡⎣
⎤⎦S
⎦ )
S
G x x f
(D), ( ⎡ , ⎤ )
S
S S S
f
G D X , ( ⎡ , ⎤ )
(2.107)
G S ⎣ b D ⎦ and F can be evaluated, whereas
S
G ⎣ b D ⎦ , h x and nuc
x
h have to be evaluated for each geometrical perturbation.
S
Note that no two-electron integrals are represented explicitly, in order to obtain the best
performance – e.g. for linear scaling codes - no reference should be made to four-index integrals.
2.4.4 The First-order Excited State Properties
The expression for the first-order one-electron excited state properties for perturbation independent
basis sets is obtained from the expression for the excited state gradient by omitting all two-electron
derivative terms, as well as all terms involving the derivative of the overlap matrix
74

Test Calculations
( ⎡
†
⎡⎣
⎤⎦
⎤ [ ] )
x 2Tr x Tr f , , f , x x
= −
⎣
S ⎦
− +
S
S nuc
f h f Dh b D b D X h h . (2.108)
The first and last terms in Eq. (2.108) correspond to the ground state first order property as seen
from Eq. (2.103).
2.5 Test Calculations
To illustrate the possibilities of an AO response solver in connection with our SCF optimization
program, test calculations have been carried out on problematic cases from the first part of the
thesis. The lowest excitation energy and the average polarizability, both static and in a field with ω
= 0.03a.u., have been found for the zinc complex in Fig. 1.3 and the rhodium complex in Fig. 1.33.
The levels of theory chosen are those where DIIS could not optimize the reference state, namely
LDA/6-31G for the zinc complex and HF/AhlrichsVDZ with STO-3G on the rhodium for the
rhodium complex.
Table 2-1 Ground state properties obtained with our AO response solver. All numbers are in a.u.
The average polarizability Excitation
static ω = 0.03 energy
Rhodium complex HF/AhrichsVDZ 170.598 173.349 0.0938
Zinc complex LDA/6-31G 161.406 162.517 0.0713
The basis sets applied in the test calculations are not satisfactory for serious polarizability
calculations, and the numbers only demonstrate the perspectives of the AO response solver in
combination with the SCF optimization algorithms described in Part 1. When the solver is fully
implemented in the AO basis, we will be able to obtain molecular properties for large complex
molecules in a routine manner.
The implementation of the excited state gradient is a work in progress. So far we have implemented
calculation of first-order one-electron properties of the excited state for perturbation independent
basis sets as described in Section 2.4.4. The excited state dipole moment of the Rhodium complex
from above has been found as
Rh
Cl
µ = 5.960a.u.
Again it should be noted that the basis set is insufficient for this type of calculation. This is only to
demonstrate that it can be done.
75

Part 2
Atomic Orbital Based Response Theory
2.6 Conclusion
The atomic orbital (AO) based response equations have been derived using the second quantization
framework. In particular, the proof of pairing is considered. Since the diagonal elements in κ are not
redundant in the AO basis, the proof given in the MO basis cannot be directly applied. However, it
is shown that there is also pairing in the AO basis.
An AO response solver has been implemented similar to the solver in the MO basis with a few
exceptions. The lack of diagonal dominance in the electronic Hessian in the AO basis makes
preconditioning a difficult task. Optimally, the AO solver should be implemented in a linear scaling
manner with only matrix multiplications and additions, and without reference to the MO basis.
However, currently a transformation is made to the MO basis where the preconditioning is carried
out followed by a transformation back to the AO basis. The redundant orbital rotations, which are
simply left out of the MO equations, are removed in the AO formulation using projection operators.
The response equations and molecular property expressions are simpler in the AO formulation than
in the MO formulation. To demonstrate how expressions for properties can easily be derived in the
AO response framework, the expression for the geometrical gradient of the singlet excited state has
been derived.
To illustrate the possibilities of the AO optimization methods presented in Part 1, joined with the
AO response solver presented in this part of the thesis, test calculations are given for cases where
DIIS diverged when optimizing the reference state. The averaged polarizability and the lowest
excitation energy are given as well as the excited state dipole for one of the examples.
The derivation and implementation of the various molecular properties is straightforward in the AO
formulation compared to the MO formulation as exemplified by the excited state geometrical
gradient. Especially the derivation of higher derivatives of molecular properties is simplified, and it
will thus be natural to expand our response program in this direction. However, before calculations
of molecular properties of large and complex molecules can be carried out in a truly linear scaling
framework, the problems related to preconditioning of the AO solver must be solved.
76

Part 3
Benchmarking for Radicals
3.1 Introduction
To corroborate the reliability of ab initio quantum chemical predictions of molecular properties, it is
important to investigate and describe strengths and weaknesses of the many-electron models
through systematic benchmark studies on different kinds of molecules.
Regarding open-shell molecules, benchmarks have been reported comparing open- and closed-shell
molecules examining the accuracy of molecular properties computed by various many-electron
models. In a study of the atomization energies of 11 small molecules 67 no significant difference in
the performance for closed- and open-shell molecules was found for the CCSDT model. However,
in another study 68 it was found that even though the CCSD(T) model performs convincingly for
closed-shell molecules, the performance for open-shell molecules is less impressive.
In this part of the thesis full configuration interaction (FCI) benchmarks of molecular properties for
the small open-shell molecules CN and CCH are presented. In the FCI model, all Slater
determinants arising from distributing the electrons in the given one-electron basis with correct
symmetry and spin-projection are included. Errors due to truncation of the many-electron basis are
thus eliminated in an FCI calculation and it provides important benchmarks for other many-electron
models. For open-shell molecules, the number of FCI benchmarks is limited and the work presented
in this part of the thesis is an attempt to improve on this situation. We thus hope our results will
serve as valuable benchmarks for further analysis of open-shell methods.
3.2 Computational Methods
All calculations have been carried out with the quantum chemical program package LUCIA 69 , using
integrals and Hartree-Fock (HF) orbitals obtained from the DALTON 70 program. The calculations
77

Part 3
Benchmarking for Radicals
are based on a ROHF reference wave function, but no spin-adaption is imposed in the CI and CC
calculations.
All FCI calculations have been carried out in the Dunnings cc-pVDZ 71 basis set. Since the number
of determinants in the FCI model increases exponentially with the number of basis functions and
electrons, it is currently not feasible to do the FCI calculations on CN and CCH in the cc-pVTZ
basis. As the cc-pVDZ basis does not provide accurate geometries and energetics, 46 we will also
obtain the equilibrium geometry, harmonic frequency, and dissociation energy for CN using the ccpVTZ
71 basis set in coupled cluster calculations, including up to quadruple excitations. In addition,
FCI and CC calculations up to quadruples level have been carried out on CN and CN - in the basis
set aug-cc-pVDZ without the diffuse d-functions (aug´-cc-pVDZ) to obtain the vertical electron
affinity of CN.
We investigate two ways of defining the excitation-level in CC. The typical approach is to let the
excitation level identify the allowed number of orbital excitations, denoted CC(orb). If instead the
excitation level is taken to identify the spin-orbital excitation level, selected excitations, which
involve spin-flipping and other internal excitations, are excluded from the calculation for open-shell
molecules. This scheme will be referred to as CC(spin-orb). The difference between the two
definitions of the excitation level is illustrated in Fig. 3.1. The CI calculations will all be carried out
with orbital excitations.
Double
orbital
excitation
Triple
Spin-orbital
excitation
Fig. 3.1 An excitation which would be
included in a CCSD(orb) calculation, but
not in a CCSD(spin-orb) calculation.
In the following SD, SDT, SDTQ, SDTQ5, SDTQ56 and SDTQ567 denote excitation-spaces which
include up to 2, 3, 4, 5, 6 and 7 excitations from the occupied spin-orbitals respectively.
78

Numerical Results
3.3 Numerical Results
First, the convergence of the CC and CI hierarchies for the open shell molecule CN is studied. Next,
the potential curve for CN is obtained from CCSD, CCSDT, CCSDTQ, and FCI calculations at
various inter-nuclear distances. In Section 3.3.3, the equilibrium geometries, harmonic frequencies,
and dissociation energies obtained for CN are presented and in Section 3.3.4 the vertical electron
affinity for CN is found. Finally, in Section 3.3.5 a minor benchmark study is presented where the
equilibrium geometry of the intergalactic radical CCH is determined at the FCI level.
3.3.1 Convergence of CC and CI Hierarchies
The convergence of the CC and CI hierarchies are studied. For CN calculations have been carried
out at the experimental equilibrium distance 72 r exp = 1.1718Å at the levels CCSD through
CCSDTQ56. Both the orbital excitation and spin-orbital excitation approaches are considered. In
addition, calculations have been carried out at the levels CISD through CISDTQ567 and in FCI. In
all calculations the cc-pVDZ basis-set is used. The results are seen in Fig. 3.2.
1.E-01
1.E-02
CI
E dev / E h
1.E-03
1.E-04
1.E-05
CC(spinorb)
CC(orb)
1.E-06
SD
SDT
SDTQ
SDTQ5
SDTQ56
SDTQ567
Fig. 3.2 E dev for CC with spin-orbital and orbital
excitation levels and for CI with orbital excitation
levels. E dev = E – E FCI .
The first thing to note is the similarity of the two CC curves. Clearly the spin-orbital excitation
restriction does not affect the accuracy in a significant way, the deviation energies are in all cases
smaller for CC(orb), but the difference is negligible.
Comparing the CI curve with the CC curves, two trends are obvious; the smooth convergence of the
CC hierarchy compared to the CI hierarchy and the faster convergence of the CC hierarchy. The CC
energy obtained using up to n-fold excitations is roughly as accurate as the CI energy using up to
n+1-fold excitations. Both phenomena are explained by the inclusion of disconnected clusters in the
CC wave function. At a given level of CC theory, the CC wave function includes all the CI
configurations at the same level of CI theory plus some higher excitations arising from disconnected
clusters. Consequently, it covers the dynamical correlation better than CI and is thus at the given
79

Part 3
Benchmarking for Radicals
level closer to the FCI solution. Describing the convergence pattern of the CI and CC hierarchies
through orders of Møller-Plesset perturbation theory (MPPT), 73 the form of the curves can be
predicted. Because also disconnected products of excitations are included in the ansatz of CC, the
order of its error grows continually in the order of MPPT. Going from uneven to even excitation
levels, both methods have an increase in the order of error in energy of two orders of MPPT, thus,
the graphs are parallel. Going from even to uneven excitation levels, the CC error increases one
order, whereas the CI error remains unchanged, giving a greater slope for the CC curve. This
explains the parallel behavior going from uneven to even excitation levels and the smoother
convergence of the CC hierarchy compared to the CI hierarchy. The stepwise convergence
predicted by MPPT, which should be significant for CI and noticeably for CC, is not apparent
though. The reason could be that CN is not strictly mono-configurational.
The convergence patterns for CI and CC are very similar to the convergence patterns previously
reported for N 2 . 74 Therefore, it does not seem that the open-shell nature of CN leads to slow
convergence of the CI and CC hierarchies compared to closed shell cases.
3.3.2 The Potential Curve for CN
The potential curve for CN was determined from single-point calculations at the FCI level with
basis set cc-pVDZ. Close to equilibrium the energies were converged to 10 -9 E h making the
determination of accurate spectroscopic constants possible. The result is displayed in Fig. 3.3.
E FCI / E h
-92.15
-92.20
-92.25
-92.30
-92.35
-92.40
-92.45
-92.50
0.5 1.5 R / Å 2.5 3.5
Fig. 3.3 The potential curve for CN found from FCI
cc-pVDZ calculations.
E dev / E h
0.03
0.02
0.01
0.00
CCSD
CCSDT
CCSDTQ
0.9 1.2 R / Å 1.5 1.8
Fig. 3.4 E dev for the CC potential curves. E dev (R) =
E(R) – E FCI (R).
The potential curve was also created with the methods CCSD(orb), CCSDT(orb) and CCSDTQ(orb)
in the basis set cc-pVDZ. Since the weight of the reference HF- determinant decreases as the internuclear
distance increases, we examine the HF-coefficients from the FCI calculations and discover
that it is irrelevant to make single-reference CC calculations beyond R = 1.8Å, since the weight of
the reference has already dropped to 0.57 at that point. Fig. 3.4 displays the differences of the CC
80

Numerical Results
potential curves compared to the FCI curve. At a given inter-nuclear distance, the FCI energy has
been subtracted from the CC energy.
The decreasing weight of the reference ground state with increasing atomic distance is reflected in
the quality of the CC wave functions. The correlation in the wave function compensates partially for
the lack of a single dominant configuration; the higher the correlation level, the better the
compensation. This is illustrated by the slopes of the curves in Fig. 3.4. Furthermore, it should be
noticed how the deviation energy is nearly linear in R, with a slightly positive curvature around the
equilibrium geometry.
3.3.3 Spectroscopic Constants and Atomization Energy for CN
The equilibrium geometry and harmonic frequency for CN were found from single-point
calculations using quartic interpolation. The atomization energy was found at the experimental
equilibrium distance. The results are displayed in Table 3-1.
Table 3-1 Equilibrium geometry, harmonic frequency, and atomization energy for CN.
R eq / Å ω e / cm -1 D e / kJ/mol
CCSD(spin-orb) cc-pVDZ 1.1855 2114 629.2
CCSD(orb) cc-pVDZ 1.1860 2111 631.6
CCSDT(spin-orb) cc-pVDZ 1.1944 2046 662.9
CCSDT(orb) cc-pVDZ 1.1946 2043 663.0
CCSDTQ(spin-orb) cc-pVDZ 1.1964 2026 666.4
CCSDTQ(orb) cc-pVDZ 1.1964 2025 666.5
FCI cc-pVDZ 1.1969 2020 667.0
CCSD(spin-orb) cc-pVTZ 1.1688 2136 674.2
CCSDT(spin-orb) cc-pVTZ 1.1783 2067 714.4
CCSDTQ(spin-orb) cc-pVTZ 1.1804 2045 718.5
Experimental 72 1.1718 2069 ---
As mentioned in Section 3.2, it is not feasible to carry out FCI calculations at the cc-pVTZ level.
Still, the convergence of the CC hierarchy can be estimated by examining the changes in the
constants. Since the difference in accuracy between the models CC(orb) and CC(spin-orb) is
negligible compared to the deviation from FCI, only the CC(spin-orb) results are discussed from
now on and only the CC(spin-orb) numbers are found at the cc-pVTZ level.
The deviation curves for the coupled cluster energies (see Fig. 3.4) are increasing functions, and
thus the coupled cluster equilibrium bond lengths are shorter than the one found from FCI.
Furthermore, the positive curvature of the deviation-curves around the equilibrium leads to coupled
cluster frequencies that are higher than the FCI frequency.
81

Part 3
Benchmarking for Radicals
As expected, the cc-pVDZ basis set does not provide accurate geometries and frequencies, and the
cc-pVTZ numbers are clearly more in the range of the experimental data than the cc-pVDZ
numbers.
CCSD displays its insufficiency for prediction of equilibrium properties by differing from the FCI
values by 0.01Å in the geometry, 90 cm -1 in the frequency, and 35 kJ/mol in the atomization energy.
The errors in R eq and ω e are reduced by a factor of four going to the CCSDT level and a factor of
five going from the CCSDT to the CCSDTQ level. The error in the atomization energy is reduced
by a factor of nine going to the CCSDT level and a factor of eight going from the CCSDT to the
CCSDTQ level, but while the equilibrium geometry on the CCSDTQ level is only 0.0005Å from
the FCI value, the harmonic frequency is still about 5 cm -1 too high.
Both the equilibrium geometry and the harmonic frequency are apparently better approximated by
the CCSDT method than the CCSDTQ. This is due to a favorable cancellation in errors for CCSDT
calculations in small basis sets. By extrapolation to the larger aug-cc-pVQZ basis, 67,75 we get an
equilibrium distance of 1.1759Å and a harmonic frequency of 2060cm -1 at the CCSDTQ level.
3.3.4 The Vertical Electron Affinity of CN
Calculations on CN - and CN were carried out in the aug´-cc-pVDZ basis at the experimental
equilibrium geometry for CN. The FCI calculation on CN - is one of the largest FCI calculations
carried out so far containing about 20 billion Slater determinants. The vertical electron affinity (EA)
was found and is displayed in Table 3-2. Again only CC(spin-orb) calculations have been carried
out because of the rather small difference in performance of CC(spin-orb) and CC(orb).
Table 3-2 The vertical electron affinity of CN.
EA / E h EA - EA FCI
CCSD(spin-orb) aug’-cc-pVDZ 0.13025 0.00063
CCSDT(spin-orb) aug’-cc-pVDZ 0.12977 0.00014
CCSDTQ(spin-orb) aug’-cc-pVDZ 0.12966 0.00003
FCI aug’-cc-pVDZ 0.12962 ---
The convergence is remarkable; already at the CCSD level we are down to an error of 0.5% of the
FCI value, on the CCSDT level it is 0.1% and on the CCSDTQ level 0.02%. The reason for the
excellent convergence is found in a cancellation of errors that influence the result. The deviations of
the individual energies are always roughly an order of magnitude larger than the deviation of the
affinity, 75 but the errors cancel when the CN and CN - energies are subtracted. That the convergence
is from above is also noteworthy. This is because the CC hierarchy converges faster for CN - than for
82

Numerical Results
CN. This seems surprising since CN - contains one more electron than CN, but it could be explained
by CN - being more one-configurational than CN.
3.3.5 The Equilibrium Geometry of CCH
The equilibrium geometry of CCH found from FCI/cc-pVDZ calculations is used in ref. 76 to
calibrate coupled cluster calculations in larger basis sets. The FCI correction is assumed to be
independent of basis set.
To optimize for the two variables R(CC) and R(CH), the CCH radical is assumed linear and the CC
and CH bonds are then distorted in step-lengths of δ = 0.01Å from an initial geometry making a grid
of single-point calculations around the equilibrium geometry with R(CC) on the one axis and R(CH)
on the other. The initial geometry is taken from a CCSDT cc-pVDZ study 76 , the geometry being
R CCSDT (CC) = 1.23448Å and R CCSDT (CH) = 1.07924Å. The resulting potential energy surface is seen
in Fig. 3.5.
-76.4020
-76.4024
E FCI / E h
-76.4028
-76.4032
-76.4036
1.09924
1.08924
1.07924
1.06924
R (C-H)/Å
1.21448
1.22448
1.23448
R (C-C)/Å
1.24448
1.25448
1.05924
Fig. 3.5 The potential energy surface of CCH.
From finite-difference expressions with the error being of the order δ 4 , the gradient and Hessian are
found for the initial geometry and a Newton step is taken giving an improved guess for the
equilibrium geometry. The FCI equilibrium geometry is thus found as
FCI
CCSDT −1
R = R −H G, (3.1)
where G is the gradient, H the Hessian, and R CCSDT the CCSDT geometry.
The equilibrium geometry at the FCI level is found to be
83

Part 3
Benchmarking for Radicals
R FCI (CC) = 1.2367Å and R FCI (CH) = 1.0802Å.
The error in the resulting geometry is a sum of the error from the finite difference approximations
and the error from the Newton step. The gradient and Hessian carry an error of O(δ 4 ) where δ =
0.01Å, this is an error in the order of 10 -8 Å. The Newton step has an error of O((H -1 G) 2 ), in this
case H -1 G is of the size 10 -3 Å and so the error is in the order of 10 -6 Å. The error in total is thus in
the order of 10 -6 Å.
The gradient for the FCI equilibrium geometry has been found as above, making single-point
calculations at the FCI geometry and at geometries distorted in steps of 0.01Å from the FCI
geometry. The same finite-difference expressions as before are used. The gradient is found to be
⎡
FCI 1.8593 10
E
⎢
;3.0661 10
⎣
Å
⎤
Å⎥⎦
G −5 h
−5
= − ⋅ ⋅ h , (3.2)
thus verifying the correctness of the FCI geometry.
Since the geometry was determined at the CCSDT level to be R CCSDT (CC) = 1.23448Å and
R CCSDT (CH) = 1.07924Å, the error due to truncation of the many-electron basis in CCSDT is in the
order of 10 -3 Å. This is similar to the results obtained for CN. This also suggests that the quadruples
correction to the equilibrium geometry is in the order of 0.001-0.002Å.
3.4 Conclusion
Full configuration interaction (FCI) and coupled cluster (CC) calculations have been carried out on
CN using the cc-pVDZ and cc-pVTZ basis sets. The equilibrium bond distance, harmonic
frequency, atomization energy, and vertical electron affinity have been evaluated on the various
levels of theory.
As expected, the cc-pVDZ basis set does not provide accurate geometries and frequencies and
CCSD is insufficient for prediction of equilibrium properties. Apparently, the CCSDT method is a
better approximation than CCSDTQ for obtaining the equilibrium geometry and the harmonic
frequency. This is due to a favorable cancellation of errors for CCSDT calculations in small basis
sets. Also the vertical electron affinities are affected by cancellation of errors, and already at the
CCSD level, the error is less than 1mE h compared to the FCI value.
The convergence patterns for the CI and CC hierarchies are studied for CN and it is found similar to
the convergence patterns previously reported for N 2 . 74 Thus, it does not seem that the open-shell
nature of CN leads to slow convergence of the CI and CC hierarchies compared to closed shell
cases.
E
84

Conclusion
For a number of the CC calculations, the excitation levels have been defined by spin-orbital
excitations instead of orbital excitations. Certain internal excitations are thereby omitted, but it is
seen that this does not affect the accuracy in any significant way. For a given excitation level, the
energies obtained in the orbital formalism are in all cases closer to the FCI energy than the ones
obtained in the spin-orbital formalism. However, the difference is negligible.
The equilibrium geometry of CCH has been found at the FCI level in the cc-pVDZ basis set to be
R FCI (CC) = 1.2367Å and R FCI (CH) = 1.0802Å. The correction found to the initial CCSDT geometry
is in the order of 10 -3 Å. The FCI correction to the CCSDT equilibrium geometry of CN was of the
same order.
85

Summary
The developments in computer hardware and linear scaling algorithms over the last decade have
made it possible to carry out ab-initio quantum chemical calculations on bio-molecules with
hundreds of amino acids and on large molecules relevant for nano-science. Quantum chemical
calculations are thus evolving to become a widespread tool for use in several scientific branches. It
is therefore important that the algorithms work as black-boxes, such that the user outside quantum
chemistry does not have to be concerned with the details of the calculations. In particular Hartree
Fock (HF) and density functional theory (DFT) methods are employed for calculations on large
systems as they represent good compromises between relatively low computational costs and
reasonable accuracy of the results. The HF and DFT methods have been a fundamental part of
quantum chemistry for many years, and calculations on molecules of ever increasing size and
complexity are made possible due to increasing computer resources. The conventional algorithms
used for optimization of the one-electron density in HF and DFT are therefore continually tried on
their stability and general performance and occasionally they break down. In these cases the
calculation takes more time to complete than acceptable or no result can be obtained at all.
We have improved on this situation. In the first part of this thesis, algorithms are presented which
improve the optimization in HF and DFT significantly. The optimization has become more effective
and where the optimization broke down using conventional algorithms, it now converges without
problems. Furthermore, the presented algorithms have no problem-specific parameters and can thus
be used as black-boxes.
When the one-electron density has been optimized, molecular properties such as polarizabilities and
excitation energies can be calculated. Response theory is often used for this purpose. In the second
part of this thesis an atomic orbital (AO) based formulation of response theory is presented which
allows linear scaling calculations of molecular properties. Furthermore, the derivation of
expressions for molecular properties is simpler in the AO formulation than in the molecular orbital
formulation typically used. To illustrate the benefits, the expression for the geometrical derivative
of the excited state is derived in the AO formulation.
To confirm the reliability of quantum chemical predictions of molecular properties, it is important
to investigate and describe strengths and weaknesses of the quantum chemical models employed.
The full configuration interaction (FCI) model is exact within a certain basis set of atomic orbitals.
It is thus of great value to be able to compare results from approximate models with FCI results. In
the third part of this thesis FCI results are presented for two open-shell molecules, namely CN and
CCH. The FCI results are compared with results from approximate models used today for
calculations where an accuracy comparable to the experimental is needed.
87

Dansk Resumé
Udviklingen i det seneste årti indenfor computerhardware og lineært skalerende algoritmer har gjort
det muligt at udføre ab-initio kvantekemiske beregninger på bio-molekyler med hundredvis af
aminosyrer og på store molekyler relevant for nanoteknologi. Kvantekemiske beregninger udvikler
sig derfor til at være et bredt anvendt værktøj til brug for adskillige naturvidenskabelige grene. Det
er derfor vigtigt at algoritmerne fungerer som såkaldte black-boxes, således at brugere uden for
kvantekemi ikke behøver bekymre sig om detaljerne i beregningen. Især Hartree Fock (HF) og
density functional theory (DFT) metoderne er benyttet til beregninger på store systemer, da de
repræsenterer et godt kompromis mellem fornuftig nøjagtighed af resultaterne og relativ kort
beregningstid. HF og DFT er metoder, som har været anvendt i kvantekemien igennem mange år,
og da stadig større computer ressourcer er til rådighed bliver de brugt til at udføre beregninger på
stadigt større og mere komplekse molekyler. De algoritmer som benyttes i dag til optimering af den
en-elektroniske densitet i HF og DFT bliver derfor til stadighed testet på deres stabilitet og
effektivitet og til tider bryder de sammen. I disse tilfælde tager beregningen enten uacceptabelt lang
tid eller opgiver at levere et resultat.
Vi har forbedret denne situation. I den første del af afhandlingen præsenteres algoritmer, som
signifikant forbedrer optimeringen i HF og DFT. Optimeringen er blevet mere effektiv, og tilfælde
hvor optimeringen før brød sammen kan nu udføres uproblematisk. De præsenterede algoritmer har
desuden ingen problem-specifikke parametre og kan derfor betragtes som black-boxes.
Når den en-elektroniske densitet er optimeret, kan molekylære egenskaber såsom polarisabiliteter
og eksitationsenergier beregnes. Til det formål benyttes ofte responsteori. I anden del af
afhandlingen præsenteres en atomorbitalformulering af responsteori, som muliggør en lineær
skalering af egenskabsberegningerne. Desuden er udviklingen af udtryk for molekylære egenskaber
blevet simplere i atomorbitalformuleringen sammenlignet med molekylorbitalformuleringen som
ellers typisk benyttes. For at illustrere fordelene er udtrykket for den eksiterede tilstands
geometriske gradient udviklet i atomorbitalformuleringen.
For at bekræfte troværdigheden af kvantekemiske forudsigelser af molekylære egenskaber, er det
vigtigt at undersøge og beskrive styrker og svagheder ved de kvantekemiske modeller som
anvendes. Full configuration interaction (FCI) er en eksakt model inden for et bestemt sæt af
atomorbital basisfunktioner. Det er derfor værdifuldt at kunne sammenligne resultater fra
approksimative modeller med FCI resultater. I tredje del af afhandlingen er FCI resultater
præsenteret for to åben-skal molekyler, CN og CCH. Disse resultater er sammenlignet med
resultater fra approksimative modeller, som i dag bruges til at levere kvantekemiske beregninger
med en nøjagtighed, som i visse tilfælde overgår den eksperimentelle.
89

Appendix A
The Derivatives of the DSM Energy
The first and second derivatives of the DSM energy model with respect to c is found recalling that
and
DSM
( ) ( )
( ) ( ) 2Tr
E c = E D + 2TrFD δ , (A-1)
E D = E D0 + DF + 0 + TrDF, + +
(A-2)
n
D = c ( D −D ), (A-3)
+
∑
i=
1
The two terms in Eq. (A-1) is evaluated one by one:
and
∂E
∂c
( D )
x
i
i
0
D δ = 3DSD −2DSDSD −D. (A-4)
= Tr DF − Tr DF + Tr DF + Tr DF−Tr DF −Tr
DF (A-5)
x 0 0 x x x
0 0
∂
∂F
∂D
2TrFDδ
= 2Tr Dδ
+ 2TrF
∂c ∂c ∂c
x x x
∂Dδ
= 2TrFD
x δ + 2Tr F ,
∂c
x
δ
(A-6)
where
∂D
∂
δ
c x
= 3DSD + 3D SD −2DSDSD −2DSD SD −2D SDSD −D . (A-7)
The second derivative is found in the same manner
∂
where
2
E
∂c
x
( D )
∂c
y
x x x x x x
= 2TrDF + TrDF + TrDF −TrDF −TrDF −TrDF −TrDF, (A-8)
0 0 x y y x 0 x x 0 y 0 0 y
2
2
∂
∂ δ ∂ δ ∂ δ
2Tr δ = 2Tr D x + 2Tr D y + 2Tr
D
x y y x x y
FD F F F , (A-9)
∂c ∂c ∂c ∂c ∂c ∂c
2
∂ D
∂c
∂c
x
δ
y
= 3D SD + 3D SD −2DSD SD −2D SDSD −2DSD SD
y x x y y x y x x y
−2DSDSD−2DSDSD −2 DSDSD.
y x x y x y
(A-10)
91

Appendix B
The Density Matrix in the Atomic Orbital Basis
In this appendix we will briefly review the density matrix in the atomic orbital basis and derive the
most important relations. For convenience consider a single-determinant wave function with n
molecular orbitals occupied. The expectation value of a one-electron operator may then be written
as a sum over occupied spin-orbitals
0 hˆ
0
n
= ∑ h . (B-1)
i=
1
ii
Explicitly introducing the MO-AO transformation matrix C allow us to write the expectation value
as
0 hˆ
0
=
n
i=
1
ii
N n
⎛
∗
∑ hµν ∑Cµ iCν
i
µν , = 1 i=
1
N
h
⎞
= ⎜ ⎟
⎝ ⎠
=
∑
∑
h
D
µν µν
µν , = 1
,
(B-2)
where N is the number of AO basis functions and we have introduced D as
D
n
µν C ∗
µ iCνi
i=
1
= ∑ . (B-3)
It is of interest to study the relation between D and the expectation values ∆ of Eq. (2.10). To
accomplish this we consider the second quantization expression for 0 h ˆ 0 in the nonorthogonal
atomic orbital basis. According to ref. 46 one obtains
N
0 hˆ
0 =
0 0
µν , = 1
N
µν , = 1
N
h
1 1 †
aµ a
µν ν
= ∆
=
− −
∑ ( S hS )
−1 −1
∑ ( S hS )
∑
µν
−1 −1
( S ∆S )
µν
µν µν
µν , = 1
.
(B-4)
By comparing Eqs. (B-4) and (B-2) we have the identification
−1 −1
D = S ∆S . (B-5)
93

Thus, the density element D µν is only identical to the matrix element ∆ µν in an orthonormal basis.
Although it could be argued that it would be appropriate to call ∆ the one-electron density matrix in
the AO-basis, we will be consistent with the standard literature and call D the density matrix in the
AO basis, and ∆ the matrix of expectation values of creation-annihilation operators. From the
properties of the one-electron density matrix
D
†
= D
Tr DS = N
DSD = D ,
elec.
(B-6)
one straightforwardly obtains the following relations for ∆
∆
Tr ∆S
−1
†
−1
= ∆
= N
∆S ∆ = ∆.
elec.
(B-7)
Although Eqs. (B-6) and Eqs. (B-7) are formally equivalent, the equations for the standard AO
density matrix D are somewhat simpler to use as they contain the metric S whereas the equations for
∆ involves the inverted metric S -1 . It should be noted that Eqs. (B-7) are necessary and sufficient
conditions, so all three equations are fulfilled if and only if 0 is a normalized single-determinant
wave function.
94

Acknowledgements
A number of people have made my four years of PhD study a pleasant and interesting experience,
and I could not have done it without them. First of all I would like to thank Jeppe Olsen and Poul
Jørgensen for guidance and support through the years; they are a fantastic team. I am grateful to the
whole theoretical chemistry group for nice lunch breaks and cake-meetings, and I would like to
thank in particular Ove Christiansen for his career advices and Andreas Hesselman for sharing some
of his latest work with me. And Stinne, how I managed to get through the days before Stinne joined
the group is a mystery. It quickly turned out that we have much the same attitude towards life and
we have shared many a wholehearted opinion of the life as such and our work situation in
particular.
I would like to thank Pawel Salek for being good company during development and debugging of
Fortran90 code of the finest quality and for being willing to help with any problems that I might
have. A special thanks goes to Sonia Coriani and her husband Asger Halkier who took very good
care of me during my visits in Trieste (even though I still havn’t tasted her mum’s lasagna).
For a number of conferences, winter schools and summer schools a group of mainly Scandinavian
people made my trips an extra pleasant experience. They were always ready for some boozing and
all sorts of crazy ideas. In particular should be mentioned Patzke-guy; a gentleman disguised as a
theoretician, Pekka; the lizard king, Ulf; the sweet Swede, crazy Mikael, Ola, Tommy and all the
others. It has been some really fine hours spent with you guys, and I hope to see you all again,
maybe for a salmari or two – no miksi ei.
I also had the pleasure to spend a summer school with some of the students from the Copenhagen
group: Marianne, Anders, Jacob and Thorsten. Anders and Jacob got connected to the Aarhus group
at some point and have always been up for a nice chat and disgusting body noises to cheer up a grey
day at work.
I would like to thank Birgit Schiøtt for nice colleagueship in connection with teaching and for
coffee and talks in her office. I look forward to our collaboration on my next project.
I am grateful to the girl-gang; Louise, Trine, Cindie, and Rikke for keeping the connection to Århus
and for gossip, lunch dates and girl nights.
I would also like to thank my parents for raising me as a good girl who always did her homework,
otherwise I would never have gotten this far, and last but not least a great thanks goes to Kristoffer
for putting up with me and being considerate and caring when needed.
95

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99

Part 1
The Trust-region Self-consistent Field Method:
Towards a Black Box optimization in Hartree-Fock and Kohn-Sham Theories,
L. Thøgersen, J. Olsen, D. Yeager, P. Jørgensen, P. Sałek, and T. Helgaker,
J. Chem. Phys. 121, 16 (2004)

JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 1 1 JULY 2004
The trust-region self-consistent field method: Towards a black-box
optimization in Hartree–Fock and Kohn–Sham theories
Lea Thøgersen, Jeppe Olsen, Danny Yeager, a) and Poul Jørgensen
Department of Chemistry, University of Århus, DK-8000 Århus C, Denmark
Paweł Sałek
Laboratory of Theoretical Chemistry, The Royal Institute of Technology,
Teknikringen 30, Stockholm SE-10044, Sweden
Trygve Helgaker
Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Norway
Received 17 February 2004; accepted 5 April 2004
The trust-region self-consistent field TRSCF method is presented for optimizing the total energy
E SCF of Hartree–Fock theory and Kohn–Sham density-functional theory. In the TRSCF method,
both the Fock/Kohn–Sham matrix diagonalization step to obtain a new density matrix and the step
to determine the optimal density matrix in the subspace of the density matrices of the preceding
diagonalization steps have been improved. The improvements follow from the recognition that local
models to E SCF may be introduced by carrying out a Taylor expansion of the energy about the
current density matrix. At the point of expansion, the local models have the same gradient as E SCF
but only an approximate Hessian. The local models are therefore valid only in a restricted region—
the trust region—and steps can only be taken with confidence within this region. By restricting the
steps of the TRSCF model to be inside the trust region, a monotonic and significant reduction of the
total energy is ensured in each iteration of the TRSCF method. Examples are given where the
TRSCF method converges monotonically and smoothly, but where the standard DIIS method
diverges. © 2004 American Institute of Physics. DOI: 10.1063/1.1755673
I. INTRODUCTION
The steady progress in computer technology and
quantum-chemical methodology has widened the range of
users of quantum-chemical software packages to include a
vast number of practicing, experimental chemists. Routinely,
such users perform Hartree–Fock HF calculations and
Kohn–Sham KS density-functional theory DFT calculations
for molecules of a size and complexity that, a decade
ago, were beyond reach even for the most advanced research
codes. This development calls for further advances in the
automatization of the self-consistent field SCF procedure
used to optimize the HF and DFT energies, so as to ensure
that convergence may be reached in a routine manner even
for very complex molecules.
In the original formulation, the SCF procedure consists
of a sequence of Roothaan–Hall RH iterations. 1,2 At each
iteration, a Fock/KS matrix is first constructed from the current
approximation to the one-electron density matrix and
then diagonalized to yield an improved set of orbitals and
orbital energies and thus an improved density matrix. In the
subsequent iteration, this improved density matrix is then
used to construct a new Fock/KS matrix, thereby establishing
the iteration procedure. However, such a sequence of RH
a On leave. Permanent address: Department of Chemistry, Texas A&M University,
P.O. Box 30012, College Station, Texas 77842-3012.
iterations converges only in simple cases. To improve upon
the convergence, each RH iteration may be extended to include,
in addition to the diagonalization step, also a step
where the best density matrix is generated in the subspace of
the density matrices of the current and preceding RH iterations.
In the next RH iteration, this averaged density matrix
rather than the pure density matrix obtained in the last diagonalization
is used to construct the new Fock/KS matrix.
In this paper, we make improvements both to the RH
diagonalization step and to the density-subspace optimization
step of the SCF scheme. Our approach follows from the
recognition that, in both steps, we may construct local models
to the SCF energy function E SCF by a Taylor expansion of
the energy about the current density matrix. However, since,
at the point of expansion, these models have an exact gradient
but only an approximate Hessian, they are valid only in a
restricted region about the current approximation to the density
matrix—the trust region. Therefore, when these local
models are used in the course of the SCF optimization, it is
essential they are used only to generate steps within their
trust region. Only in this manner can it be ensured that the
SCF energy is systematically and sufficiently lowered at each
iteration.
In the RH diagonalization part of the SCF optimization,
the improvements are obtained by introducing an energy
function E RH that corresponds to the sum of the occupied
0021-9606/2004/121(1)/16/12/$22.00 16
© 2004 American Institute of Physics
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J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Self-consistent field method
17
orbital energies. 3 An unconstrained minimization of E RH results
in the same solution i.e., density matrix as obtained by
a diagonalization of the Fock/KS matrix. However, since, at
the point of expansion, the RH energy function E RH has only
the gradient in common with the true SCF energy E SCF ,a
global minimization of E RH may lead to steps that are too
long to be trusted. We therefore introduce a trust region
where E RH is a good approximation to E SCF . If a global
minimization of E RH leads to a step outside the trust region,
then the step to the minimum on the boundary of the trust
region for E RH is taken instead. This step is found by a
level-shifting technique, where the occupied molecular orbital
energies effectively are shifted by some constant to increase
the gap between the occupied and virtual molecular
orbitals. Level shifting has previously been used to improve
the convergence of the simple RH sequence of iterations. An
essential feature of our implementation is to adjust the level
shift in such a manner that the step is to the boundary of the
trust region, recognizing that only in this manner does a lowering
of E RH result in a lowering of E SCF . For this reason,
the resulting method is called the trust-region RH TRRH
method.
The optimization of the density matrix in the subspace of
the density matrices of the preceding RH iterations has a
long history. Early on, it was recognized that a simple averaging
of the density matrices of the last few RH iterations
significantly improves the convergence of the RH scheme.
This simple density-matrix averaging technique was later rationalized
and systematized in the direct inversion in iterative
subspace DIIS method of Pulay. 4 In the DIIS method,
an improved density matrix is obtained as a linear combination
of the previous density matrices by minimizing the norm
of the corresponding linear combination of gradients. The
DIIS method significantly speeds up the local convergence
and convergence can often be obtained to ground states of
rather complex molecules with a small gap between energies
of the highest occupied molecular orbital HOMO and the
lowest unoccupied molecular orbital LUMO and with a
large number of close-lying electronic states.
Several attempts have been made to modify the DIIS
algorithm so as to improve upon its global convergence behavior.
Recently, Kudin, Scuseria, and Cances proposed the
energy DIIS EDIIS method, where the DIIS gradient-norm
minimization is replaced by a minimization of an approximate
energy function. 5 In EDIIS, the variational parameters,
which are the linear expansion coefficients of the density
matrices from the previous RH iterations, may only take on
values that give densities in the convex set—that is, densities
with occupation numbers between 0 and 1. As the EDIIS
method is based on the minimization of an approximate energy
function, it may have some advantages in the global
region. However, it is worrying that a convex solution often
cannot be obtained and that the observed local convergence
of the EDIIS method is slower than in the standard DIIS
method.
In the DIIS and EDIIS methods, an improved density
matrix is obtained as a sum of the density matrices from the
preceding RH diagonalization steps. Consequently, the averaged
density matrix is not idempotent as required in HF and
KS theories. The deviation from idempotency may be reduced
using a purified density matrix as the one suggested by
McWeeny. 6 This has been done for the SCF energy minimization
by several workers including Nunes and Vanderbilt 7
and Daniels and Scuseria 8 and for the calculation of geometrical
derivatives by Ochsenfeld and co-workers. 9 It may
also be done for the EDIIS energy function. The energy function
then has the same gradient as E SCF , but also contains
terms which cannot be obtained from the densities and
Fock/KS matrices of the previous RH iterations. Neglecting
these terms, we arrive at the density-subspace minimization
DSM algorithm proposed in this paper. At the point of expansion,
the DSM energy function E DSM thus has the same
gradient as the true energy function E SCF but only an approximate
Hessian. Again, a trust region may be introduced
and only steps within this region are taken, ensuring that any
lowering of E DSM also corresponds to a lowering of E SCF .
The resulting method is called the trust-region DSM
TRDSM method.
In the next section, we first describe the standard optimization
of the SCF energy function in a density-matrix formulation.
The TRRH method is then discussed in Sec. II A
and the TRDSM method in Sec. II B. In Sec. III, we give
some numerical examples to demonstrate the performance of
the resulting trust-region SCF TRSCF method. The last
section contains some concluding remarks.
II. THEORY
For a closed-shell system with N/2 electron pairs, the
Hartree–Fock HF energy excluding the nuclear–nuclear repulsion
energy is given by 3
E SCF D2 TrhDTr DGD,
1
where D is the one-electron density matrix in the atomicorbital
AO basis, h is the one-electron Hamiltonian matrix
and GD is defined as
G D
2g g D , 2
where g is a two-electron integral in the AO basis. For
the energy in Eq. 1 to be a valid approximation to the true
HF energy, the density matrix D must satisfy the symmetry,
trace, and idempotency conditions:
D T D,
3
Tr DS N 2 ,
DSDD.
5
Similar conditions apply in the Kohn–Sham KS theory, but
the energy function of Eq. 1 must then be modified by
including the exchange-correlation term and by scaling or
complete removal of the exchange term from Eq. 2.
The traditional approach to the optimization of the HF
energy is an iterative one. From the current approximation to
the density matrix D n in iteration n, a Fock matrix is built
FD n hGD n
6
4
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18 J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Thøgersen et al.
and, following the Roothaan–Hall RH procedure, the Fock
matrix is diagonalized
FD n C occ SC occ ,
7
where S is the overlap matrix in the AO basis, to give a set of
occupied molecular orbitals MOs, from which a new approximation
to the density matrix is obtained as
D n1 C occ C T occ . 8
The iteration procedure is established using D n1 as the current
density in Eq. 6. The final solution to the minimization
problem is obtained when the D n and D n1 are the same.
This self-consistent field SCF procedure may also be used
in KS theory, the only difference being the addition of the
exchange-correlation potential and the scaling of the exchange
contribution in the Fock matrix to yield the KS matrix.
The pure RH iterations presented above often do not
converge. A powerful method for handling this divergence is
not to construct the Fock matrix from the density matrix D n
but rather from an average of all previous density matrices:
n
D¯n c i D i .
i1
The averaged density matrix D¯n is then used in place of the
pure density matrix D n in Eq. 6 to obtain the Fock matrix
F(D¯n) as
n
FD¯n c i FD i
10
i1
and the iteration procedure is established. In the course of the
TRSCF iterations, the following matrices are set up in the
order indicated: D 1 , F(D 1 ), D 2 , F(D 2 ), D¯2 , F(D¯2), D 3 ,
F(D 3 ), D¯3 , F(D¯3),.... Among these, D 1 , F(D 1 ), D 2 ,
F(D 2 ), D 3 , F(D 3 ), . . . are saved during the iteration procedure.
In the following, we describe improvements to the SCF
diagonalization and density-subspace optimization steps. In
Sec. II A, we describe how the trust-region RH TRRH
method is used to generate new density matrices by a modification
of the traditional RH method Eqs. 7 and 8. Next,
in Sec. II B, we introduce the trust-region density-subspace
minimization TRDSM method for calculating the averaged
density matrix of Eq. 9. In the following, we use the indices
i, j,k,l for occupied MOs and the indices a,b,c,d for the
virtual MOs.
A. The trust-region Roothaan–Hall method
As discussed in Ref. 3, the traditional RH method may
be viewed as a minimization of the sum of the orbital energies
of the occupied MOs
9
E RH 2
i
i 2TrFD¯D, 11
subject to orthonormality constraints on the occupied MOs
i :
i j ij .
12
Whereas D¯ is the current approximation to the HF/KS density
matrix, usually obtained as a linear combination of the
previous densities according to Eq. 9, the density matrix D
to be optimized in Eq. 11 is related to the occupied MOs
resulting from the diagonalization of F(D¯) as
DC occ C T occ . 13
To see this, consider the constrained minimization of E RH in
Eq. 11 expressed in terms of the Lagrangian
L2 TrFD¯D2 T TrCocc SC occ I N/2 , 14
where the multipliers ij ensure orthonormality among the
occupied MOs. Minimization of this Lagrangian leads to the
standard RH equations:
FD¯Cocc SC occ .
15
However, since E RH of Eq. 11 is only a crude model of the
true energy E SCF the gradient is correct at D¯ assuming D¯ is
idempotent, a global minimization of E RH according to Eq.
15 may easily lead to steps that are too long to be trusted as
they are outside the region where E RH is a good approximation
to E SCF . Steps outside the trust region may often not
lead to a reduction of the total energy E SCF .
1. The level-shifted Roothaan–Hall equations
To avoid too long steps, an additional constraint is imposed
on the optimization of Eq. 11, namely, that the new
density matrix D in Eq. 13 does not differ too much from
the old matrix D¯. This condition is conveniently expressed in
terms of the overlap between the density matrices in the S
metric norm
DD¯ S Tr DSD¯Sa N 2 Tr D¯SD¯S,
16
where Tr D¯SD¯S N/2 since D¯ is not necessarily idempotent.
Note that, for D equal to an idempotent D¯, a is equal to
one. For a sufficiently close to one, a step will therefore be
taken in the local region. In practice, we define sufficiently
close to one by the parameter a min 0.975.
Introducing an undetermined multiplier associated
with this new constraint, we obtain the following Lagrangian:
L2 TrFD¯D2Tr SD¯SDa N 2 Tr D¯SD¯S
2 TrC T occ SC occ I N/2 . 17
Differentiating this Lagrangian with respect to the MO coefficients
and setting the result equal to zero, we arrive at the
level-shifted RH equations
FD¯SD¯SC occ SC occ .
18
To interpret the level-shift term, we note that D¯S projects out
the component of C occ that is occupied in D¯ assuming idempotent
D¯), see Ref. 3. The level shift therefore works only on
the occupied part of F(D¯), shifting all the occupied orbital
energies and increasing the gap between the occupied and
virtual MOs, in particular the HOMO-LUMO gap.
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J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Self-consistent field method
19
where i HOMO () and a LUMO () are the HOMO and LUMO
orbital energies, respectively; in Fig. 1b, we have plotted
the overlap between the old and new density matrices as
given by
DD¯ S
a
,
DDS D¯D¯ S
22
where D()D() S is equal to N/2. For sufficiently large
, the HOMO-LUMO gap Eq. 21 is linear in . This linearity
of ai () for large arises from the dependence of
the orbital energies on in Eq. 19, where is effectively
subtracted from the occupied orbital energies. The MOs C¯occ
occupied in D¯ satisfy the generalized eigenvalue equations
SD¯SC¯occ SC¯occ ,
23
and become identical to the MOs C occ () obtained from Eq.
19 when tends to infinity. The corresponding density is
denoted
T
DC¯occ C¯occ
24
FIG. 1. For the fourth iteration of the rhodium calculation described in Sec.
III we have displayed as a function of the level-shift parameter ; a the
HOMO-LUMO gap ai , where min is the smallest accepted level-shift,
b the overlap a between the old and new density matrices, where opt is
the optimal level-shift, and c the change in the model energy E RH and the
actual energy E RH SCF .
Since the SCF energy E SCF is invariant with respect to
an orthogonal transformation between the MOs, Eq. 18
may be transformed to the canonical basis:
FD¯SD¯SC occ SC occ ,
where the diagonal matrix contains the orbital energies.
2. Choice of the RH level-shift parameter
19
The density matrix generated from the restricted RH solution
Eq. 19 depends on the level-shift parameter :
DC occ C T occ . 20
To see how is determined, we consider the determination
of in the fourth iteration of the rhodium-complex calculation
described in Sec. III. In Fig. 1a, we have plotted the
HOMO-LUMO gap as a function of ,
ai a LUMO i HOMO ,
21
and represents a purified D¯. In the linear regime of ai (),
there is a continuous development of the occupied MOs from
those occupied in D¯. As decreases and we enter the nonlinear
regime at min , the MOs in Eq. 20 no longer correspond
to those in Eq. 23. Comparing plot a and b in Fig.
1, we note that the region a()a min in Fig. 1b corresponds
roughly to the region min in Fig. 1a.
As we insist on a controlled, continuous development of
the MOs from those occupied in D¯, the level-shift parameter
should be restricted to the linear regime min . To determine
the optimal level-shift parameter opt , we therefore
begin by establishing the onset of linearity min by linear
extrapolation by means of two Fock/KS matrix diagonalizations,
giving the two ai values marked by crosses and the
linearly interpolated min value marked with an arrow. Next,
since, in the linear interval, a small corresponds to a large
step, we investigate whether min is acceptable by checking
if a( min )a min . If this step is too long, we backtrack by
increasing using inexact line search until an acceptable
value opt is found such that a( opt )a min , requiring a few
additional Fock/KS matrix diagonalizations. In Fig. 1b, the
accepted opt is marked with an arrow.
For a better understanding of this step, consider the Hessian
of the E RH energy function:
A RH ai,bj ij ab a i . 25
By restricting the level-shift parameter to min where
LUMO a () HOMO i ()0, we ensure that the effective Hessian
is positive definite and that the model energy function
E RH is reduced. We note that the Hessian of the true energy
function E SCF is given by the more complicated expression
A SCF ai,bj ij ab a i 4g aibj g abij g ajib . 26
Often, the orbital energy difference dominates the Hessian.
In such cases, we expect the above step to reduce the SCF
energy E SCF as well as the model function E RH . In any case,
when a sufficiently large level shift is added in Eq. 19, the
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20 J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Thøgersen et al.
Hessian structure of Eq. 25 becomes similar to that of the
true energy function E SCF in Eq. 26. The steps generated
from E RH with such level shifts will therefore have essentially
the same direction as the ones generated from E SCF .
By construction, the E RH energy function is lowered
when is chosen according to the above prescription
E RH 2TrFD¯DD¯0.
27
Since E RH is only a local model of the true energy function
E SCF , the associated change in the true energy
E RH SCF E SCF DE SCF D¯
28
may be either negative or positive, depending on how well
E RH represents E SCF for the chosen step. However, for sufficiently
small steps, E RH SCF 0, since the model function then
represents the true energy well.
Let us consider the relationship between the true lowering
E RH
SCF and the lowering predicted by the model function
E RH . Introducing the presumably small differential density
matrix
DD¯
29
and using the identity Tr AG(B)Tr BG(A) valid for symmetric
matrices A and B, we find that the change in the true
energy Eq. 28 may be written in the form
E RH SCF 2TrhDD¯TrD¯
GD¯Tr D¯GD¯
2 Trh2 TrGD¯Tr G,
30
which shows that the changes in the true energy and in the
model energy are related as
E RH SCF E RH Tr G.
31
If the last term which is second order in is negligible, the
energy lowering predicted by the local model E RH becomes
equal to E RH SCF . However, since the correction term is positive
strictly positive in the absence of exchange, its presence
in Eq. 31 shows that, for sufficiently large steps, a
lowering of the model function may not lead to a lowering of
the total energy. To avoid such steps, it would be useful to
provide an alternative prediction of E RH
SCF that is less expensive
than the calculation of Tr G itself. Section II A 3 is
concerned with this problem.
To demonstrate the efficiency of the chosen level shift
opt in the global region of a SCF optimization, we have for
the fourth iteration of the rhodium-complex calculation plotted
in Fig. 1c, E RH
SCF and E RH as a function of . The
energy gain E RH SCF is about optimal for the level shift opt .
Increasing gives a smaller energy gain while decreasing
gives a slight increase in the energy gain and from 4.5,
RH is actually positive. Note also that for opt , E RH
E SCF
RH
and E SCF start to differ indicating that the importance of
Tr G increases. The step representing a RH iteration
where 0 is far too long to be trusted and results in a
significant increase of the total energy.
3. Prediction of the energy close to the minimum
To develop a better prediction of E RH
SCF than E RH ,we
note that the only part, that cannot easily be evaluated from
known Fock-matrices, is the second-order contribution to Eq.
31 from that part of that does not belong to the linear
space spanned by the previous density matrices D i . To see
this, we decompose the current density matrix D into two
parts
DD D ,
32
where D belongs to the linear space spanned by the previous
density matrices and D belongs to its orthogonal complement.
We then expand D in the following manner:
n
D
i1
c i D i ,
33
where the expansion coefficients c i () are determined in a
least-squares manner
n
c i M 1 ij Tr D j SDS, M ij Tr D i SD j S.
j1
34
The change in the SCF energy associated with the change of
density matrix from D¯ to D may be expressed as
E RH SCF E SCF D E SCF D¯2 TrD FD
Tr D GD .
35
Ignoring the small term quadratic in D , we may now predict
the change in the SCF energy at little cost from the
expression
E P SCF E SCF D E SCF D¯2 TrD FD , 36
using only the density matrices and Fock/KS matrices of the
previous iterations. In particular in the later parts of the iteration
sequence, where the space spanned by the densities
of the preceding RH iterations is large, an accurate estimate
of E RH
SCF may be obtained from this formula. In the following,
we shall see how we may use this prediction to determine
the level shift when min 0 and a(0)a min .
P
To illustrate how E SCF is used to find the level-shift
parameter, consider as an example the determination of the
level-shift parameter in the ninth iteration of the rhodiumcomplex
calculation of Sec. III. The plot of the HOMO-
LUMO gap in Fig. 2a shows that the allowed level-shift
interval is 0. In Fig. 2b, we have plotted the overlap
a() as a function of . Since a(0)a min , we should,
according to the discussion in Sec. II A 2, use opt 0 to
determine the step. In short, considerations based on the
HOMO-LUMO gap and on the overlap with the averaged
density matrix indicate that the next density matrix should be
determined from the standard, unshifted RH equations.
However, from the nine density matrices of the previous
P
RH iterations, we can use E SCF () to predict the change in
E RH SCF () more accurately than with E RH (). Indeed, from
P
Fig. 2c, we see that E SCF () provides a good global representation
of E RH SCF (), with a minimum close to the minimum
of E RH SCF (). By contrast, the local model E RH ()
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J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Self-consistent field method
21
n
D¯ c i D i .
i1
37
Ideally, this averaged density should also fulfill the conditions
Eqs. 3–5. The symmetry condition Eq. 3 is trivially
satisfied since the averaged density Eq. 37 is a linear
combination of symmetric density matrices. The trace condition
Eq. 4 is also easily taken care of by imposing the
restriction
n
i1
c i 1
38
on the expansion coefficients
n
Tr D¯S c i Tr D i S N
i1
2 .
39
By contrast, the idempotency condition Eq. 5 cannot be
imposed on the averaged density matrix. However, the idempotency
may be significantly improved if, instead of working
with D¯, we work with the purified density matrix 6
D˜ 3D¯SD¯2D¯SD¯SD¯,
40
as proposed by Nunes and Vanderbilt. 7 The electronic energy
may be expressed in terms of the purified average density
matrix as
ED˜ 2 TrhD˜ Tr D˜ GD˜ .
41
FIG. 2. For the ninth iteration of the rhodium calculation described in Sec.
III we have displayed as a function of the level-shift parameter ; a the
HOMO-LUMO gap ai , where min 0, b the overlap a between the old
and new density matrices, where a min is the smallest accepted overlap and
c the change in the model energy E RH , the actual energy E RH
SCF and the
P
P
predicted energy E SCF . opt is found at the minimum of E SCF ().
gives a minimum at 0. Clearly, 0 should be avoided
in the calculation since it would lead to an increase in the
SCF energy. Instead, the value of the level-shift parameter
P
that corresponds to the minimum of E SCF denoted by opt )
is chosen for the calculation of the next density matrix.
This procedure may be summarized as follows. If min
0 and a(0)a min , then we calculate the predicted energies
P
P
P
E SCF (0) and E SCF () with 0. If E SCF (0)
P
E SCF (), then we use D0. Otherwise, we estimate the
P
minimum opt of E SCF () by an inexact line search and
use the density matrix D( opt ) at this minimum.
B. Density-subspace minimization
1. The DSM energy function
Let us assume that we have carried out n RH iterations
and that we have kept all previous density matrices D i and
the corresponding Fock matrices F i . We would now like to
construct an optimal density as a linear combination of the
densities from these iterations according to Eq. 9,
We note that the purified density is correct to first order in
the expansion coefficients c i and that E(D˜ ) thus contains
errors through second order in c i . To determine the best
average density matrix Eq. 37, we shall minimize Eq. 41
with respect to the expansion coefficients c i subject to the
condition Eq. 38.
One problem we encounter when minimizing Eq. 41 is
that new Fock matrices F(D˜ ) need to be evaluated. To avoid
this problem, we shall use an approximate form of Eq. 41.
Since the purified density matrix D˜ is close to the original
density matrix D¯, we can write it as
D˜ D¯,
42
where is the correction term. Inserting Eq. 42 into Eq.
41, we obtain
E2 TrhD¯Tr D¯GD¯2 Trh
2 TrGD¯Tr G.
43
Since is small, we may ignore the term quadratic in and
arrive at the density-subspace minimization DSM energy
function
E DSM c2 TrhD¯Tr D¯GD¯2 Trh2 TrGD¯
ED¯2 TrFD¯D˜ D¯.
44
Since is first order in the expansion coefficients c i , the
DSM energy differs from the true energy to second and
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22 J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Thøgersen et al.
higher orders in c i . The first contribution to the DSM energy
function may for example be evaluated using the energy expression
of the EDIIS algorithm, 5
Lc,,E 0 c T g 1 2 c T Hcc T 1
1 2 c T Mch 2 ,
52
ED¯
i
c i E SCF D i 1 2
ij
c i c j TrF i F j D i D j .
45
Using Eq. 40, we find that the second contribution may be
evaluated as
where 1 is a column vector with elements equal to 1. Differentiating
this Lagrangian and setting the derivatives equal to
zero, we obtain the equations
L
c gHcMc10,
53
2TrFD¯D˜ D¯2
ij
c i c j Tr F i D j
L
cT 10,
54
6
ijk
c i c j c k Tr F i D j SD k
L
1 2 c T Mch 2 0.
55
4
ijkl
c i c j c k c l Tr F i D j SD k SD l .
46
All contributions to the DSM energy function are therefore
easily calculated from the previous density and Fock/KS
matrices.
2. The trust-region DSM minimization
We minimize the DSM energy functional by the trustregion
method. 12 We thus consider the second-order Taylor
expansion of the DSM energy in Eq. 44 about c 0 . Introducing
the step vector
ccc 0 ,
we obtain
47
E DSM (2) cE 0 c T g 1 2 c T Hc,
48
where the energy, gradient, and Hessian at the expansion
point are given by
E 0 Ec 0 ,
g Ec
c
cc 0
, H 2 Ec
c 2 cc 0
. 49
As starting point c 0 , we choose the density matrix with the
lowest energy E SCF (D i ), usually from the last RH iteration.
The trace condition Eq. 38 imply
n
i1
c i 0.
50
We also introduce a trust region of radius h for E DSM (2) (c)
and require that steps are always taken inside or to the
boundary of this region. To determine a step to the boundary,
we restrict the step to have the length h in the S metric norm
of Eq. 34,
c S 2
ij
c i M ij c j h 2 . 51
Introducing the undetermined multipliers and for the
trace and step-size constraints, we arrive at the following
Lagrangian for minimization on the boundary of the trust
region:
The optimization of the Lagrangian thus corresponds to the
solution of the following set of linear equations:
HM
1 T
1
0
c
g 0 ,
56
where the multiplier is iteratively adjusted until the step is
to the boundary of the trust region Eq. 55. The step-length
restriction may be lifted by setting 0, as needed for steps
inside the trust region.
To understand the behavior of the step-length function,
we consider first the generalized eigenvalue problem
H 1
v 1 T 0 M 0
0 T
v , 57
where 0 is a column vector with zero elements, is a small
positive constant, and the eigenvector is normalized such that
v T v 2 1.
58
We first note that, for a finite , v0. Next, carrying out
block multiplications in Eq. 57, we obtain
Hv1Mv,
1 T v,
59
60
which upon elimination of from the first equation yields
the relation
Hv1 T v1 2 Mv.
61
Since (1 T v)1 is finite, we conclude that, as tends to zero,
the eigenvalue tends to either plus or minus infinity
1/2 . Next, substituting these values of into Eq. 60,
we find that v tends to the zero vector with elements proportional
to 1/2 and that , because of the normalization Eq.
58, tends to 1. In short, the eigenvalue problem Eq. 57
with 0 has two eigenvalues , whose eigenvectors
have zero elements except for the last element, which is
equal to 1. Finally, invoking the Hylleraas–Undheim interlace
theorem, 10,11 we conclude that the remaining n1 finite
eigenvalues of Eq. 57 bisects the n eigenvalues of the reduced
eigenvalue problem
HvMv.
62
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J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Self-consistent field method
23
Let us now consider the step length c() S as a function
of . In the diagonal representation of the augmented
matrix in the linear equations Eq. 57, we may write these
equations in the following uncoupled form:
h i m i i i , i1,2,3,...,n1. 63
Here, the h i and m i are the diagonal elements of the Hessian
and metric matrices, respectively, of the generalized eigenvalue
problem Eq. 57, whereas the i and i , respectively,
are the corresponding elements of the solution and gradient
vectors of Eq. 56. Since the last element of the gradient
vector in Eq. 56 is zero, the gradient vector has no contributions
from the eigenvectors with infinite eigenvalues
1 n1 0, 1 n1 64
assuming that the eigenvalues are sorted in increasing order
1 2 ¯ n1 . In the diagonal representation, therefore,
we may write the step norm in the form
c S
i2
n
m i i
2
h i m i 2 .
65
From this expression, we note that the step function consists
of n branches separated by n1 asymptotes at the finite
eigenvalues i . Moreover, it increases monotonically from
zero to infinity as increases from minus infinity and approaches
the lowest finite eigenvalue 2 . Therefore, there is
always one and only one 2 that gives rise to a
step of length h. As shown by Fletcher, 12 this value of
corresponds to the global minimum on the boundary of the
trust region.
In practice, we cannot easily determine the eigenvalues
i of the augmented eigenvalue problem Eq. 57. Instead,
we determine the eigenvalues i of the reduced problem Eq.
62 and restrict our search of to the smaller monotonic
interval 1 . Since 1 2 , it is possible that no
solution exists in this reduced interval. Mostly, however, this
restriction is mild since the two eigenvalues are usually
close. If no solution is found, we choose instead the slightly
shorter step obtained with 1 .
To illustrate how the level-shift parameter in Eq. 56
is determined, we consider the first Fig. 3a and third Fig.
3b DSM step in the eighth iteration of the rhodiumcomplex
calculation in Sec. III. We have plotted the steplength
function c() S as a function of . The plots consist
of a series of branches between asymptotes where
makes the matrix on the left-hand side of Eq. 56 singular.
The lowest eigenvalue 1 is marked with a vertical dashed
line in Figs. 3a and 3b. For minimization, the level-shift
parameter is chosen in the interval min( 1 ,0),
where 1 is the lowest eigenvalue of Eq. 62. The proper
value is found where the step-length function crosses the line
representing the trust radius h, as marked with a cross in Fig.
3a. If the step that minimizes E DSM (2) is inside the trust region,
0 is chosen as marked with a cross in Fig. 3b.
The trust region is updated during the iterative procedure.
FIG. 3. The step-length function c() S is plotted as a function of for
the first a and third b DSM step in the eighth iteration of the rhodium
calculation described in Sec. III. The trust radius h is represented by a
horizontal line. The proper value is marked with a cross.
3. Global optimization of the DSM function
The optimization of the E DSM energy is carried out in the
usual manner, requiring several trust-region steps, each of
which involves the construction of the gradient g and the
Hessian H, and the solution of the modified level-shifted
Newton equations Eq. 56. After p iterations, the density is
calculated from the coefficients
p
c p c (0) c i .
66
i1
However, since E DSM itself is a rather crude model of the
true energy function E SCF , it resembles E SCF only in a small
region about the initial point c (0) . The DSM iterations are
therefore terminated when the total step length c p c (0)
exceeds some preset value k. If a minimum of E DSM is found
inside the trust region c p c (0) k, then the step to the
minimum is taken and the iterations are terminated. This is
often the case.
Occasionally, the iterations start where the lowest eigenvalue
of the Hessian in Eq. 62 is negative. In the course of
the iterations, the Hessian can become positive definite and a
minimum is reached. In a few cases, however, a negative
Hessian eigenvalue may persist, changing little from iteration
to iteration. In our experience, a step along the eigenvector
corresponding to the negative eigenvalue cannot be
trusted. This direction is therefore projected out from the step
and the DSM function is minimized in the orthogonal subspace.
As an illustration, consider the first DSM step of the
tenth SCF iteration of the rhodium-complex calculation in
Sec. III. In Fig. 4, we have, for comparison, plotted the steplength
functions with the negative component kept and projected
out. The level shifts resulting from the two situations
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24 J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Thøgersen et al.
FIG. 4. The step-length function c() S is plotted as a function of with
the direction corresponding to the negative Hessian eigenvalue kept — and
projected out - --, respectively. The values resulting from the two
situations are marked with crosses.
are marked with crosses in Fig. 4. The level shift used in the
DSM optimization is, in this particular case, 0.
When the trust-region minimization is terminated, a new
RH iteration is initiated by constructing a new density and
associated Fock matrix
n
n
D¯ c i D i ,
i1
F¯ c i FD i ,
67
i1
where we have used the fact that the Fock matrix is linear in
the density. By construction E DSM (c) is lowered at each iteration
of the trust-region minimization. The total energy
lowering at the pth iteration is given by
E DSM E DSM c p E DSM c (0) .
68
Since E DSM is a local model to the true energy E SCF , the
lowering of E DSM will also lead to a lowering of E SCF provided
the total step is sufficiently short to be in the local
region.
4. Relationship to the DIIS method
The optimal density has previously been determined using
the DIIS scheme of Pulay. 4 In the DIIS method, the improved
density matrix is obtained as a linear combination of
the previous density matrices where the expansion coefficients
are determined by minimizing the norm of the error
vector, using the gradients of the previous iterations as error
vectors. To highlight the difference between TRDSM and
DIIS, we give below an alternative derivation of the DIIS
algorithm.
In an SCF calculation, the electronic gradient with the
averaged density matrix D¯ in Eq. 37 may be expressed in
the form, 3
gD¯4D¯SFD¯FD¯SD¯.
69
To determine the best linear combination of densities D i ,we
minimize the norm of the squared gradient
gD¯ 2 16 TrD¯SFD¯FD¯SD¯2 .
70
Inserting the expansion Eq. 37, we obtain a quartic polynomial
in c i ,
FIG. 5. The convergence of calculations on the rhodium complex using
AhlrichsVDZ basis Ref. 16 combined with STO-3G for Rh. The error in
the total energy is given for the TRSCF, the standard DIIS, and the QRHF
method as a function of the iteration number. Furthermore results are given
where DIIS is applied after nine TRSCF iterations.
gD¯ 2 16 Tr
i
c i gD i
i, j
c i c j D i SFD j D i
2
FD j D i SD i . 71
To simplify this expression, we neglect all cubic and quartic
terms
gD¯ 2 app c i c j gD i gD j . 72
i, j
Optimization of Eq. 72 subject to the constraint Eq. 38
gives the DIIS expression of the expansion coefficients in
Eq. 37.
III. APPLICATIONS
In this section, we examine the convergence characteristics
of the TRSCF algorithm. First, we consider a rhodiumcomplex
optimization as an example of a difficult case; next,
as a simpler case, we consider a calculation on H 2 O with the
OH bond lengths stretched to double length. For comparison,
we also give the convergence characteristics of the DIIS
algorithm 4 and the quadratically convergent restricted step
Hartree–Fock QRHF method. 13,14 All calculations are carried
out using a local version of the DALTON program
package. 17
A. The rhodium complex calculation
In Fig. 5, we have plotted the error in the energy at each
iteration of TRSCF, DIIS, and QRHF optimizations of the
rhodium complex with the geometry specified in Table I using
the AhlrichsVDZ basis 16 combined with STO-3G on Rh.
The starting orbitals have been obtained from diagonalizing
the one-electron Hamiltonian.
Clearly, the QRHF and DIIS methods do not work in this
case. In particular, the DIIS method is unable to handle the
global part of the optimization, where the initially indefinite
Hessian changes its structure and becomes positive definite.
Since the DIIS method relies solely on gradient information,
it does not see the negative eigenvalues and produces steps
that may or may not be in the right direction, leading to
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J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Self-consistent field method
25
TABLE I. Geometry of the rhodium complex.
x y z
Cl 2.783200 0.000000 0.000000
C 0.000000 1.750000 0.000000
C 0.000000 1.750000 0.000000
C 2.510000 1.247077 0.000000
C 2.510000 1.247077 0.000000
C 3.960000 1.247077 0.000000
C 3.960005 1.247074 0.000000
C 4.685005 0.008663 0.000000
C 6.585566 1.381712 0.000000
C 7.224161 0.912908 0.000000
H 1.802335 2.074803 0.000000
H 1.965500 2.190178 0.000000
H 4.323007 2.273792 0.000000
H 4.504500 2.190178 0.000000
H 6.215281 1.889842 0.889165
H 6.215281 1.889842 0.889165
H 7.169607 1.539271 0.889165
H 7.169607 1.539271 0.889165
H 7.674455 1.397244 0.000000
H 8.164527 0.363696 0.000000
N 1.790000 0.000000 0.000000
N 6.124978 0.017359 0.000000
O 0.122018 3.144673 0.000000
O 0.122018 3.144673 0.000000
Rh 0.0000000 0.000000 0.000000
divergence. Moreover, in this DIIS calculation, no level
shifts have been applied in the RH part of the optimization,
again leading to steps in the wrong direction. In short, the
DIIS method cannot be used for optimizations as complex as
the rhodium calculation. However, if the DIIS method is
started after the SCF local region has been reached by the
TRSCF algorithm, then the DIIS algorithm converges nicely
since the Hessian has the correct structure. In Fig. 5, we have
also plotted the errors in a calculation where the DIIS
method is started after nine TRSCF iterations. It then converges
in roughly the same manner as the pure TRSCF
method.
In the QRHF calculation, the total energy reduces slowly
and monotonically during the iteration procedure. However,
the resulting energy lowering is much too slow to be of any
practical value. Thus, after 14 iterations, the energy has decreased
by only 37 E h , which is insignificant compared with
the 237 E h needed for convergence.
To understand the difference between the QRHF and
TRSCF optimizations, let us recall the main features of the
two methods. Since the QRHF method is based on a local
quadratic model of E SCF , the QRHF orbital rotations are
correct to first order. However, no global information about
E SCF is available and only small steps can be trusted in the
optimization. When QRHF steps are taken to the boundary of
the trust region, level-shifted Newton equations are solved
with the Hessian of Eq. 26. By contrast, in the TRSCF
method, the RH optimization is based on the local energy
function E RH , which has the same gradient as E SCF but a
slightly different Hessian—compare Eqs. 25 and 26.
More important, E RH shares some global features with E SCF .
In the RH diagonalization step, a global optimization is carried
out for E RH . When an RH step is taken to the boundary
of the trust region of E RH , a level-shifted Fock eigenvalue
equation is solved where the level-shift parameter effectively
introduces a shift in the Hessian of E RH Eq. 25. The similarity
of the Hessians of E SCF and E RH makes the directions
of the steps taken by the QRHF and RH methods very similar
for sufficiently large level shifts, the essential difference
being the global character of the RH steps and the local
character of the QRHF steps. It is this local character of the
QRHF steps that prevents the QRHF method from being
efficient for systems as difficult as the rhodium complex.
Let us now consider the individual TRSCF iterations as
listed in Table II. The optimization begins with orbitals that
diagonalize the one-electron Hamiltonian, giving a start energy
of 5 466.530 208 964 75 E h . In Table II, the SCF energy
lowering E SCF is divided into two contributions, one
from the RH step and one from the DSM step. Recalling
from Eq. 24 that D() n is the purified D¯n ,
E DSM SCFn1
E SCF D n E SCF D n 73
becomes a realistic measure of the energy change in the
DSM part of the iteration. Similarly,
RH
E SCFn1
E SCF D n1 E SCF D n
74
becomes a realistic measure of the change in the RH part.
Clearly, the sum of Eqs. 73 and 74 is equal to the total
change E SCF . These exact energy changes should be compared
with the energy changes in the local models E RH and
E DSM given in Eqs. 27 and 68, respectively, also listed
in the table. Note that, to obtain E SCF D(), we must carry
out an additional energy calculation, which is here done only
for the purpose of this analysis.
For the DSM method, we have also indicated in Table II
how the trust-region optimization was terminated (exit DSM ):
M indicates that a minimum was determined in the full
space; PM indicates that a minimum was obtained in the
reduced space with the direction corresponding to the negative
Hessian eigenvalue projected out; and L indicates that
the iterations were terminated because the maximum step
length k was reached. For the RH steps, we have also listed
the level-shift parameter opt and the corresponding overlap
a( opt ) of Eq. 22.
The TRSCF iterations converge linearly, with a reduction
in the error of about a factor 2–4 at each iteration.
Moreover, the energy lowerings of the local models E RH
and E DSM are in good agreement with the actual SCF energy
changes, in the local as well as in the global part of the
optimization. Both the predicted and the actual energy
changes are negative in all iterations. In the global region,
E RH
SCF is usually significantly larger than E DSM SCF , whereas,
in the local region, they have similar sizes.
Except for three iterations in the global part of the SCF
optimization, the DSM trust-region method finds a minimum
within the step-length limit k. In the intermediate region, we
encounter components of the step vector that cannot be
trusted and have been projected out as described in Sec.
II B 3. The DSM iterations then reach a minimum in the orthogonal
subspace.
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26 J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Thøgersen et al.
TABLE II. Convergence details for the TRSCF calculation on the rhodium complex using AhlrichsVDZ basis combined with STO-3G on Rh. Energies given
in atomic units.
DSM
It. E SCF E SCF
E DSM RH
E SCF
E RH RH
opt
a( RH opt )
Exit DSM
1 18.94647615033 0.00000000009 0.00000000000 18.94647615024 19.21320649447 17.47 0.99382
2 45.45858825211 8.95768890498 7.10309975657 36.50089934714 38.75977508968 14.44 0.98630 M
3 59.81037380731 12.93651600370 8.85502694483 46.87385780361 51.53623100635 11.68 0.97940 M
4 63.34486220663 24.25263285599 21.63716388564 39.09222935064 48.71127100240 7.28 0.97288 L
5 30.22875461345 12.81783382045 12.23686585427 17.41092079300 21.38161936631 2.63 0.97384 L
6 11.56061105704 5.64904464510 4.74940263974 5.91156641194 7.60366893231 0.90 0.97552 L
7 4.61334906659 1.90220393646 1.51155035145 2.71114513013 3.30373325651 0.24 0.97792 M
8 2.16270415323 0.44637212140 0.44849600108 1.71633203184 1.49977814394 0.07 0.97876 M
9 0.60805181167 0.29078332276 0.21298647367 0.31726848890 0.60770324492 1.30 0.99823 M
10 0.16667264229 0.00294157325 0.00194422453 0.16373106904 0.22325882198 0.70 0.99934 PM
11 0.05893002647 0.00782290321 0.00662821837 0.05110712327 0.03977595787 0.00 0.99955 PM
12 0.01821537974 0.00935849099 0.00823957093 0.00885688875 0.00980424864 0.00 0.99989 PM
13 0.00829012952 0.00417695835 0.00382848541 0.00411317118 0.00413942925 0.00 0.99995 PM
14 0.00336772651 0.00246626574 0.00222734467 0.00090146077 0.00176102559 0.00 0.99998 PM
15 0.00144190516 0.00106346997 0.00091468267 0.00037843519 0.00066804948 0.00 1.00000 PM
16 0.00049317801 0.00040627140 0.00039284830 0.00008690661 0.00013209160 0.00 1.00000 PM
17 0.00005633666 0.00003203569 0.00002863768 0.00002430097 0.00003124073 0.00 1.00000 PM
18 0.00001495119 0.00000990523 0.00000917530 0.00000504595 0.00000926762 0.00 1.00000 PM
19 0.00000549749 0.00000312992 0.00000277915 0.00000236757 0.00000276315 0.00 1.00000 M
20 0.00000196603 0.00000126150 0.00000121565 0.00000070454 0.00000067573 0.00 1.00000 M
21 0.00000038264 0.00000022841 0.00000020736 0.00000015423 0.00000016335 0.00 1.00000 M
22 0.00000008720 0.00000004496 0.00000004404 0.00000004225 0.00000004536 0.00 1.00000 M
23 0.00000002788 0.00000001171 0.00000001049 0.00000001617 0.00000001603 0.00 1.00000 M
24 0.00000001286 0.00000000813 0.00000000800 0.00000000472 0.00000000514 0.00 1.00000 M
25 0.00000000294 0.00000000131 0.00000000127 0.00000000163 0.00000000186 0.00 1.00000 M
26 0.00000000119 0.00000000073 0.00000000072 0.00000000045 0.00000000056 0.00 1.00000 M
27 0.00000000035 0.00000000019 0.00000000019 0.00000000016 0.00000000022 0.00 1.00000 M
In the beginning of the SCF optimization, large level
shifts are applied in the RH diagonalization to ensure a continuous
development of the MOs. Thus, in the first few iterations,
the overlap constant a( opt ) is significantly larger than
the minimum accepted overlap of 0.975. However, the levelshift
parameter decreases during the subsequent SCF iterations
until, in the local region, no level shift is required and
conventional RH iterations are carried out. To summarize,
the TRSCF method gives a monotonic and significant energy
lowering both in the RH and in the DSM part of the optimization.
B. The water calculation
To demonstrate the performance of the TRSCF method
in a simpler case, we consider optimizations of H 2 O with the
OH bonds stretched to twice the equilibrium value 195.10
pm. In Figs. 6a and 6b, we have plotted the errors in the
energy during TRSCF, DIIS, and QRHF optimizations in the
cc-pVDZ basis. 15 In Fig. 6a, the initial guess of the orbitals
are the Hückel orbitals as implemented in the DALTON program.
With these initial orbitals, the TRSCF and DIIS methods
converge in a very similar manner to within a threshold
of 10 10 in ten iterations. In this case, therefore, gradient
information is sufficient for convergence. Although the
QRHF method outperforms the TRSCF and DIIS methods in
terms of iterations, this is of no practical value since, in each
QRHF step, about the same number of new Fock matrices
are needed to solve the Newton equations as is required to
find the optimized Hartree–Fock wave function with the
TRSCF and DIIS methods.
FIG. 6. The convergence of calculations on water with stretched bonds
using the cc-pVDZ basis and a aHückel start guess and b a one-electron
Hamiltonian start guess. The error in the total energy is given for the
TRSCF, the standard DIIS and the QRHF method as a function of the
iteration number.
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J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Self-consistent field method
27
In Fig. 6b, we have plotted the error of the energy in
H 2 O optimizations starting with the orbitals that diagonalize
the one-electron Hamiltonian. In this case, convergence to
10 10 is reached in 13 iterations with the TRSCF method and
in 18 iterations with the DIIS method. The main reason for
the better performance of the TRSCF algorithm is that, in the
global region, it gives a significant energy lowering in each
step, whereas the DIIS algorithm shows a much less systematic
behavior.
IV. CONCLUSION
A conventional SCF optimization consists of a sequence
of iterations, each of which begins with a Roothaan–Hall
RH diagonalization step, where a Fock/KS matrix is diagonalized
to obtain an improved density matrix, followed by an
averaging step, where the optimal density matrix is determined
in the subspace of the density matrices of the previous
RH diagonalization steps. In this paper, we have introduced a
trust-region SCF TRSCF algorithm, where improvements
have been made to both the diagonalization and the averaging
steps. In both steps, local energy model functions are
constructed which have the same gradient as the true energy
function E SCF but approximate Hessians. Recognizing the
locality of these energy functions, trust regions are introduced
as regions where they represent a good approximation
to E SCF and only steps inside these trust regions are allowed.
For the density-subspace minimization step, an energy
function is constructed and minimized with respect to the
coefficients of the linear combination of the previous density
matrices. Its functional form is based on a purified averaged
density matrix that is idempotent to first order. The advantages
of this model compared to EDIIS is the built-in density
purification, which helps to avoid problems arising from
non-idempotency. In addition, information about the Hessian
is extracted and used, leading to a monotonic and stable convergence.
The RH diagonalization step corresponds to a minimization
of an energy function E RH that represents the sum of the
orbital energies of the occupied MOs. Since this very simple
energy function is a local model function for E SCF , large
steps cannot be trusted. To generate steps to the boundary of
the trust region, level-shifted RH equations are solved where
the level shifts are determined in a systematic and general
manner, leading to a decrease in the model energy at each
iteration. If sufficiently small steps are taken, a similar decrease
is obtained in the SCF energy.
In the TRSCF algorithm a few diagonalizations are required
in each SCF iteration to obtain solutions for the levelshifted
RH equations in order to determine the optimal density
matrix. The number of diagonalizations may be reduced
in the local SCF region solving RH equations with zero level
shift with little consequence for the convergence. In the local
SCF region one may also safely use the DIIS algorithm if
desired.
The advantages of the TRSCF algorithm are demonstrated
by calculations on a rhodium complex and on a water
molecule with stretched bonds. In the rhodium-complex optimization,
the TRSCF algorithm converges monotonically
and fast, with a significant decrease in the energy in both the
RH part and DSM part at each iteration. By contrast, convergence
is not obtained with the DIIS method for this complex.
For the simpler water molecule, the TRSCF and DIIS methods
behave in a more similar manner, the TRSCF method
converging slightly faster than the DIIS method when the
initial orbitals are obtained by diagonalizing the one-electron
Hamiltonian. With the Hückel guess, the water convergence
is essentially obtained in the same number of steps for
the TRSCF and DIIS methods. In short, it appears that the
TRSCF algorithm, and its use of local energy model functions
to obtain significant reductions in E SCF in each iteration,
constitutes a significant step towards a black-box optimization
of SCF wave functions.
ACKNOWLEDGMENTS
This work has been supported by the Danish Natural
Research Council Grant No. 21-02-0467 and the Carlsbergfondet.
We also acknowledge support from the Danish Center
for Scientific Computing DCSC. D.Y. acknowledges
support from the Robert A. Welch Foundation, Grant No.
A-770.
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Theory Wiley, Chichester, 2000.
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www.kjemi.uio.no/software/dalton.
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Part 1
The Trust-region Self-consistent Field Method in Kohn-Sham Density-functional Theory,
L. Thøgersen, J. Olsen, A. Köhn, P. Jørgensen, P. Sałek, and T. Helgaker,
J. Chem. Phys. 123, 074103 (2005)

THE JOURNAL OF CHEMICAL PHYSICS 123, 074103 2005
The trust-region self-consistent field method
in Kohn–Sham density-functional theory
Lea Thøgersen, a Jeppe Olsen, Andreas Köhn, and Poul Jørgensen
Department of Chemistry, University of Århus, DK-8000 Århus C, Denmark
Paweł Sałek
Laboratory of Theoretical Chemistry, The Royal Institute of Technology, Roslagstullbacken 15,
Stockholm, S-10691 Sweden
Trygve Helgaker
Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Norway
Received 20 May 2005; accepted 7 June 2005; published online 22 August 2005
The trust-region self-consistent field TRSCF method is extended to the optimization of the Kohn–
Sham energy. In the TRSCF method, both the Roothaan–Hall step and the density-subspace
minimization step are replaced by trust-region optimizations of local approximations to the Kohn–
Sham energy, leading to a controlled, monotonic convergence towards the optimized energy.
Previously the TRSCF method has been developed for optimization of the Hartree–Fock energy,
which is a simple quadratic function in the density matrix. However, since the Kohn–Sham energy
is a nonquadratic function of the density matrix, the local energy functions must be generalized for
use with the Kohn–Sham model. Such a generalization, which contains the Hartree–Fock model as
a special case, is presented here. For comparison, a rederivation of the popular direct inversion in
the iterative subspace DIIS algorithm is performed, demonstrating that the DIIS method may be
viewed as a quasi-Newton method, explaining its fast local convergence. In the global region the
convergence behavior of DIIS is less predictable. The related energy DIIS technique is also
discussed and shown to be inappropriate for the optimization of the Kohn–Sham energy. © 2005
American Institute of Physics. DOI: 10.1063/1.1989311
I. INTRODUCTION
Computational methods rigorously based on the laws of
quantum mechanics are becoming an evermore important
component of scientific and technological progress in many
branches of natural science, including biochemistry and materials
science. Quantum-chemical codes, in particular, are
today routinely used to perform calculations on molecules
containing hundreds of atoms. Furthermore, with the advent
of density-functional theory DFT methods, molecules with
more complex electronic structure and larger parts of potential
surfaces may be calculated than with the Hartree–Fock
method. Most of these calculations are performed by nonspecialists,
not trained in quantum chemistry or in numerical
simulations. An important challenge is thus to develop
quantum-chemical techniques that allow the user to focus on
the physical and chemical interpretations of the results of the
calculations by eliminating or at least minimizing the need to
understand the details of the numerical algorithms.
A central numerical task of the Hartree–Fock wavefunction
theory and Kohn–Sham DFT is the minimization of
the electronic energy function with respect to the density
matrix of a single-determinant reference wave function. In
its original formulation, the self-consistent field SCF
method for optimizing Hartree–Fock and Kohn–Sham energies
E SCF consists of a sequence of Roothaan–Hall
a Electronic mail: lea@chem.au.dk
iterations. 1,2 At each iteration, the Fock/Kohn–Sham matrix
is first constructed from the current approximate atomicorbital
AO density matrix; next, an improved AO density
matrix is generated from the molecular orbitals MOs obtained
by diagonalization of this Fock/Kohn–Sham matrix.
Unfortunately, this simple SCF scheme converges only in
simple cases. To improve upon its convergence, the optimization
is modified by constructing the Fock/Kohn–Sham matrix
not directly from the AO density matrix of the last diagonalization
but rather from an averaged density matrix,
calculated in the subspace of the density matrices of the current
and previous iterations. In practice, the averaged AO
density matrix is calculated by the direct inversion in iterative
subspace DIIS method of Pulay, 3 nowadays implemented
in most electronic-structure programs. In the DIIS
method, the averaged density matrix is a linear combination
of density matrices, where the expansion coefficients are obtained
by minimizing the norm of the corresponding linear
combination of the gradients.
Over the years, several attempts have been made to improve
upon the DIIS method. In particular, Kudin et al. have
proposed the energy DIIS EDIIS method, 4 where the
gradient-norm minimization is replaced by a minimization of
an approximation to the true energy function E SCF , where the
expansion coefficients of the averaged density matrix are
used as variational parameters. For the special case of two
density matrices such an approach was first developed by
Karlström. 5
0021-9606/2005/1237/074103/17/$22.50
123, 074103-1
© 2005 American Institute of Physics
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074103-2 Thøgersen et al. J. Chem. Phys. 123, 074103 2005
Recently, we introduced the trust-region self-consistent
field TRSCF method 6 for SCF density-matrix optimizations.
In the TRSCF method, the diagonalization step trustregion
Roothaan–Hall TRRH and the density-optimization
step trust-region density-subspace minimization TRDSM
are realized as minimizations of local energy model functions
of E SCF . The local energy functions are expanded about
the current AO density matrix and have the same gradients as
the true energy E SCF but approximate Hessians. In the course
of the SCF optimization, each step is restricted to be within
the trust region of the current model, that is, within the region
where the model accurately represents the true energy
function. In TRDSM the steplength is controlled through a
standard trust-region optimization 7 and in TRRH the
steplength is controlled through a level shift. 8 In this manner,
a reliable and systematic energy lowering of E SCF is ensured
at each iteration.
In the first implementation of the TRSCF method, the
focus was on the optimization of the Hartree–Fock energy. In
this paper, the focus is on the optimization of the Kohn–
Sham energy. In the Kohn–Sham theory, the energy difference
between the highest occupied MO and lowest unoccupied
MO the HOMO-LUMO gap is usually much smaller
than that in the Hartree–Fock theory, making the optimization
more difficult. Here, we investigate the consequences of
this smaller HOMO-LUMO gap for the global and local convergence
characteristics for the Roothaan–Hall optimization
step. In the Hartree–Fock theory, the energy function is quadratic
in the density matrix, whereas, in the Kohn–Sham
theory, it becomes a nonquadratic function because of the
exchange-correlation contribution to the energy. In our previous
implementation of the TRSCF method, the model
function used to determine the averaged density matrix was
specially designed for the Hartree–Fock theory, assuming
that the energy depends quadratically on the density matrix.
For the Kohn–Sham theory, the model function must be generalized.
Such a generalization is presented here.
In this paper, the DIIS algorithm is also rederived to
understand better when it can safely be applied. In particular,
we find that the DIIS method may be viewed as a quasi-
Newton method in the local region, explaining its fast local
convergence. The convergence characteristics of the DIIS
method in the global region are less predictable.
Recently, and along the same lines as our TRRH method,
Francisco et al. introduced their globally convergent trustregion
methods for SCF, 9 where the standard fixed-point
Roothaan–Hall step is replaced by a trust-region optimization
of a model energy function. Any acceleration scheme,
such as DIIS, EDIIS, and the TRDSM method, can then be
combined with this method.
After an introduction to the SCF problem in Sec. II, we
examine the Roothaan–Hall scheme in Sec. III. In particular,
we identify the model energy function that is effectively being
optimized in the diagonalization step, demonstrating how
convergence can be improved upon by level shifting. In Sec.
IV, we consider the density-matrix averaging step. We establish
the model energy function of the weights of the density
matrices and perform an order analysis of the resulting
scheme, demonstrating that it represents a balanced approximation;
next, we compare our local energy function with the
EDIIS function, showing that the latter misses a term that is
necessary for calculating the correct gradient. After a brief
discussion of configuration shifts in Sec. V, we present in
Sec. VI a rederivation of the DIIS algorithm, establishing its
equivalence with the quasi-Newton method in the local region.
Section VII contains some convergence examples for
the DFT calculations, using the TRSCF algorithm and some
of its alternatives. Finally, Sec. VIII contains some concluding
remarks.
II. THE KOHN–SHAM ENERGY AND THE
ROOTHAAN–HALL METHOD
For a closed-shell system with N/2 electron pairs, the
Kohn–Sham energy excluding the nuclear-nuclear repulsion
contribution is given by 10
E KS D =2TrhD +TrDGD + E XC D.
Here D is the scaled one-electron density matrix in the AO
basis, D= 1 2 DAO ; h is the one-electron Hamiltonian matrix in
this basis; and the elements of GD are given by
G D =2
g D − g D ,
where g are the two-electron AO integrals. The first term
in Eq. 2 represents the Coulomb contribution and the second
term the contribution from exact exchange, with =1 in
the Hartree–Fock theory, =0 in the pure DFT, and 0 in
the hybrid DFT. The exchange-correlation energy E XC D in
Eq. 1 is a functional of the electron density. In the localdensity
approximation LDA, the exchange-correlation energy
is local in the density, whereas, in the generalized gradient
approximation GGA, it is also local in the squared
density gradient, that is, it may be expressed as
E XC D = fx,xdx.
Here the electron density x and its squared gradient norm
x are given by
x = T xDx,
x = x · x,
1
2
3
4a
4b
where x is a column vector containing the AOs. We note
that the exchange-correlation energy density fx,x in
Eq. 3 is a nonlinear and nonquadratic function of x and
x. In the following, we shall therefore rely on an expansion
of E XC D around some reference density matrix D 0 ,
E XC D = E XC D 0 + D − D 0 T 1
E XC
+ 1 2 D − D 0 T E 2 XC D − D 0 + ¯ , 5
where the derivatives E n XC
have been evaluated at D=D 0 and
where, for convenience, we have used a vector-matrix notation
for D, E 1 XC
, and E 2 XC
.
The first derivative of E KS D with respect to the density
matrix D is then given by
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074103-3 Trust-region self-consistent-field J. Chem. Phys. 123, 074103 2005
E 1 KS D = E KSD
=2FD, 6
D
where we have introduced the Fock/Kohn–Sham matrix,
FD = h + GD + 1 2 E XC
1 D.
We note, that for the energy in Eq. 1 to be a valid Kohn–
Sham energy, the density matrix D must satisfy the symmetry,
trace, and idempotency conditions,
D T = D,
Tr DS = N 2 ,
DSD = D,
7
8a
8b
8c
where S is the AO overlap matrix. Therefore, we cannot
carry out a free minimization of the total energy in Eq. 1,
but must restrict ourselves to those changes in the density
matrix that comply with these requirements.
The Kohn–Sham energy E KS is traditionally optimized
self-consistently by fixed-point iterations. From the current
approximation D 0 to the density matrix, the Kohn–Sham matrix
FD 0 is calculated from Eq. 7, followed by the solution
of the Roothaan–Hall generalized eigenvalue
equations: 1,2
FD 0 C occ = SC occ ,
where C occ is the set of occupied MOs and is a diagonal
matrix containing the associated eigenvalues orbital energies.
An improved density matrix is next calculated from the
occupied MOs as
D = C occ C T occ , 10
and the Roothaan–Hall fixed-point iteration is established by
constructing the Kohn–Sham matrix FD from this density
matrix, followed by diagonalization according to Eq. 9.
Note that, since
C occ UU T C T
occ = C occ C T occ , 11
where U is unitary, the Kohn–Sham density matrix in Eq.
10 and hence the energy are invariant to unitary transformations
among the occupied MOs.
The naive Roothaan–Hall fixed-point iteration outlined
above converges only in simple cases. To improve upon this
scheme, the new Kohn–Sham matrix is usually not calculated
directly from the density matrix obtained by diagonalization
of the previous Kohn–Sham matrix, but rather from
the density matrix obtained by diagonalizing some linear
combinations of the current and n previous Kohn–Sham matrices,
n
F¯ = F0 + c i FD i .
12
i=0
Typically, the coefficients c i are obtained by the DIIS method
as the weights of an improved density matrix,
9
n
D¯ = D 0 + c i D i .
i=0
13
Upon diagonalization of F¯ according to Eq. 9, the new
density matrix is obtained from Eq. 10, thereby establishing
the iterations. In general, the averaged density matrix in
Eq. 13 is not idempotent and therefore does not represent a
valid density matrix; moreover, since the Kohn–Sham matrix
unlike the Fock matrix is nonlinear in the density matrix,
the averaged Kohn–Sham matrix in Eq. 12 is different from
FD¯ . For these reasons, we cannot associate the averaged
Kohn–Sham matrix in Eq. 12 uniquely with a valid Kohn–
Sham matrix. Usually, this does not matter much since the
subsequent diagonalization of the Kohn–Sham matrix nevertheless
produces a valid density matrix according to Eq. 10.
In the following, we shall disregard the complications arising
from the use of the averaged Kohn–Sham matrix in Eq. 12,
noting that the errors introduced by this approach may easily
be corrected for, if necessary.
In the remainder of this paper, we discuss the TRSCF
method, which differs from the traditional SCF scheme by
the consistent use of trust-region techniques for optimization
control, both in the Roothaan–Hall diagonalization step in
Eq. 9 and in the construction of the averaged density matrix
in Eq. 13. In particular, the traditional Roothaan–Hall eigenvalue
problem is replaced by a level-shifted eigenvalue
problem, where the level shift is determined from trustregion
considerations, resulting in the TRRH step. Similarly,
the averaged density matrix is determined by a TRDSM
technique rather than by the traditional DIIS method. As we
shall see, the combined use of the TRRH and TRDSM
schemes in the TRSCF method leads to a highly efficient and
robust SCF scheme, characterized, in its most robust implementation,
by a monotonic convergence towards the optimized
Kohn–Sham energy.
III. TRUST-REGION ROOTHAAN–HALL OPTIMIZATION
A. The trust-region Roothaan–Hall method
We begin by noting that the solution of the traditional
Roothaan–Hall eigenvalue problem in Eq. 9 may be regarded
as the minimization of the sum of the energies of the
occupied MOs, 11
E RH D =2 i =2TrF 0 D,
14
i
subject to MO orthonormality constraints,
C T occ SC occ = I N/2 , 15
where F 0 is typically obtained as a weighted sum of the
Kohn–Sham matrices such as F¯ in Eq. 12. Since Eq. 14
represents a crude model of the true Kohn–Sham energy
with the same first-order term but different zero- and
second-order terms as discussed in Sec. III B, it has a rather
small trust radius. A global minimization of E RH D, asaccomplished
by the solution of the Roothaan–Hall eigenvalue
problem in Eq. 9, may therefore easily lead to steps that are
longer than the trust radius and hence unreliable. To avoid
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074103-4 Thøgersen et al. J. Chem. Phys. 123, 074103 2005
such steps, we shall impose on the optimization of Eq. 14
the constraint that the new density matrix D does not differ
much from the old matrix D 0 , that is, the S norm of the
density difference should be equal to a small number ,
D − D 0 2 S =TrD − D 0 SD − D 0 S =−2TrD 0 SDS + N
= . 16
The optimization of Eq. 14 subject to the constraints in
Eqs. 15 and 16 may be carried out by introducing the
Lagrangian
L =2TrF 0 D −2Tr DSD 0 S − 1 2N −
−2TrC T occ SC occ − I N/2 , 17
where is the undetermined multiplier associated with the
constraint in Eq. 16, whereas the symmetric matrix contains
the multipliers associated with the MO orthonormality
constraints. Differentiating this Lagrangian with respect to
the MO coefficients and setting the result equal to zero, we
arrive at the level-shifted Roothaan–Hall equations,
F 0 − SD 0 SC˜ occ = SC˜ occ.
18
Since the density matrix in Eq. 10 is invariant to unitary
transformations among the occupied MOs in C˜ occ, we
may transform this eigenvalue problem to the canonical basis,
F 0 − SD 0 SC occ = SC occ ,
19
where the diagonal matrix contains the orbital energies.
Note that, since D 0 S projects onto the part of C occ that is
occupied in D 0 see Ref. 11, the level-shift parameter
shifts only the energies of the occupied MOs. Therefore, the
role of is to modify the difference between the energies of
the occupied and virtual MOs, in particular, the HOMO-
LUMO gap.
Clearly, the success of the TRRH method will depend on
our ability to make a judicious choice of the level-shift parameter
in Eq. 19. In our standard TRRH implementation,
we determine by requiring that D does not differ
much from D 0 in the sense of Eq. 16, thereby ensuring a
continous and controlled development of the density matrix
from the initial guess to the converged one. In the following
sections we discuss how is determined in this standard
implementation.
In view of the relative crudeness of the E RH D model, a
more robust approach consists of performing a line search
along the path defined by to obtain the minimum of the
Kohn–Sham energy E KS D. Strictly speaking, this optimization
is not a line search but rather a one-parameter optimization.
One-parameter optimizations have previously
been used by Seeger and Pople 12 to stabilize convergence of
the RH procedure.
For → Eq. 19 becomes equivalent to solving the
eigenvalue equation,
0
SD 0 SC occ = SC 0 occ , 20
where has eigenvalue 1 for the set of orbitals that are
occupied in D 0 and eigenvalue 0 for the set of virtual orbitals.
Equation 20 thus effectively divide the molecular orbitals
into a set that is occupied and a set that is unoccupied,
where the density D 0 is obtained from the occupied set,
D 0 = C 0 occ C 0 occ T . 21
Since F 0 is the gradient of E KS at D 0 , the step from Eq. 19
for large is in the steepest-descent direction and will therefore
give a decrease in the Kohn–Sham energy compared to
the energy at D 0 . However, this TRRH line-search TRRH-
LS algorithm is more expensive than the standard method,
requiring the repeated construction of the Kohn–Sham matrix
at each SCF iteration.
B. Comparison of the Roothaan–Hall and Kohn–Sham
energy functions
To understand better our strategy for determining the
level-shift parameter in the Kohn–Sham energy optimizations,
we here examine the Roothaan–Hall model energy of
Eq. 14 in more detail, comparing it with the true Kohn–
Sham energy of Eq. 1. Expanding the Kohn–Sham and
Roothaan–Hall energies about the reference density matrix
D 0 and neglecting the differences between F 0 and FD 0
noted in Sec. II, we obtain
E KS D = E KS D 0 +2TrFD 0 D − D 0
+TrD − D 0 GD − D 0 + E XC D − E XC D 0
−TrD − D 0 E 1 XC D 0 ,
22
E RH D = E RH D 0 +2TrFD 0 D − D 0 .
23
These expansions have the same first-order term 2 Tr FD 0
D−D 0 but different zero- and second-order terms. In an
orthonormal MO basis, we may express any valid density
matrix D in terms of the reference density matrix D 0 as
DK = exp− KD 0 expK,
24
where the antisymmetric rotation matrix may be written in
the form
K = 0 − T
. 25
0
The diagonal block matrices representing rotations among
the occupied MOs and among the virtual MOs are zero since
the density matrix in Eq. 10 is invariant to such rotations
see Eq. 11. In terms of K, the first-order Roothaan–Hall
and Kohn–Sham energies may be written as
2TrFD 0 D − D 0 =2TrFD 0
exp− KD 0 expK − D 0 26
and thus share a series of higher-order terms in K. If these
shared higher-order terms are larger than the higher-order
terms that occur only in the Kohn–Sham energy in Eq. 22,
then the energy changes predicted by the Roothaan–Hall
function in Eq. 23 will be a good approximation to the
changes in the Kohn–Sham energy, even for large
rotations K.
Let us now compare the derivatives of the Roothaan–
Hall and Kohn–Sham energies with respect to the orbital-
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074103-5 Trust-region self-consistent-field J. Chem. Phys. 123, 074103 2005
rotation parameters ai in this paper, i, j, k, and l denote the
occupied indices and a, b, c, and d denote the virtual indices.
As already established, the two energy functions have
the same gradients,
E 1 KS ai =
E 1 RH ai =
E KS
=−4F ai ,
ai
=0
ERH
ai
=−4F ai . =0
27a
27b
The Hessians are most conveniently expressed in a basis
where the occupied-occupied and virtual-virtual blocks of
the Kohn–Sham matrix are diagonal,
F ab = ab a ,
28a
F ij = ij i .
28b
Since, at convergence where F is fully diagonal, the diagonal
elements a and i become the orbital energies, we shall refer
to these as the pseudo-orbital energies or sometimes just the
orbital energies. In this basis, the Hessians of the two energy
functions become
E 2 KS aibj =
2 E KS
=4 ij ab a − i + M aibj ,
ai bj=0
29a
E 2 RH aibj = 2 E RH
=4 ij ab a − i , 29b
ai bj=0
where
M aibj =16g aibj −4g abij + g ajib + E 2 XC D aibj . 30
Clearly, the Roothaan–Hall Hessian in Eq. 29b is positive
definite whenever the energies of the occupied orbitals are
lower than the energies of the virtual orbitals, that is, whenever
the HOMO-LUMO gap is positive. Furthermore, if the
differences a − i in the Hessians are large compared to M aibj
in Eq. 30, then E 2 RH
is a good approximation to E 2 KS
.
C. Quadratically convergent trust-region optimization
To minimize the Roothaan–Hall energy in Eq. 14, consider
the second-order expansion in the orbital-rotation parameters
,
E RH 2 = E RH + T E 1 RH + 1 2 T E 2 RH .
31
The unconstrained Newton step is obtained by setting the
gradient equal to zero,
E 2
RH
= E RH
1 + E 2 RH =0.
32
Solution of these equations yields the Newton step, with its
fast second-order convergence in the local region. In the global
region, far away from the true minimum, it is not reasonable
to accept large steps since the expansion in Eq. 31 is
only a valid approximation to E RH D for h, where h is
the trust radius. Furthermore, if E 2 RH
is indefinite, the Newton
step in Eq. 32 may not reduce the energy. Therefore, if the
Hessian is not positive definite or if the Newton step is too
large, we solve instead a modified set of equations, where we
minimize Eq. 31 subject to the constraint =h. To accomplish
this, we introduce an undetermined multiplier
and set up the Lagrangian
L, = E RH 2 + 1 2 T − h 2 ,
33
whose stationary points are determined from the equation
L,
= E 1
RH + E 2 RH + =0,
34
leading to the level-shifted Newton step,
=−E 2 RH + I −1 E 1 RH .
35
The multiplier is chosen such that =h and such that the
energy change predicted by E RH 2 is negative. Consider the
first- and second-order changes of the Roothaan–Hall energy,
E RH 1 − E RH = T E 1 RH =− T E 2 RH + I, 36a
E RH 2 − E RH = T E 1 RH + 1 2 T E 2 RH
=− 1 2 T E 2 RH + I − 1 2 T . 36b
2
If E RH
is positive definite, both corrections are negative for
2
0; if E RH
is indefinite, they are negative for − 1 ,
where 1 is the lowest negative eigenvalue i.e., the HOMO-
LUMO gap. In general, therefore, we choose such that
max0,− 1 . As discussed in Ref. 6, it is always possible
to find a level-shift parameter that satisfies this requirement.
D. The quadratically convergent SCF method
It is possible to optimize the Hartree–Fock and Kohn–
Sham energies in Eq. 1 directly, without invoking the
Roothaan–Hall energy function in Eq. 14. In the secondorder
trust-region Newton method, the optimization then
consists of a sequence of level-shifted Newton iterations. At
each iteration, the linear equation in Eq. 35 is solved, replacing
E RH
1 2 1
and E RH
by E KS
and E 2 KS
, respectively. The
resulting optimization scheme is known as the quadratically
convergent SCF QC-SCF method. 13,14 The method is quadratically
convergent in the local region and has a dynamic
update of the trust region as discussed by Fletcher. 7
E. The level-shift parameter in the TRRH method
1. The global region
A TRRH diagonalization step determined with =0 in
Eq. 19 corresponds to the global minimum of E RH D.
Therefore, when we impose the constraint in Eq. 16 on the
difference between the old and new density matrices, then
the step-size control is applied to a global optimization of
E RH D. By contrast, in the quadratically convergent trustregion
optimization of E RH in Eq. 35, step-size control
is applied to a local model of E RH , that is, to the optimization
of the second-order Taylor expansion of the energy
E RH 2 in Eq. 31 inside the trust region.
In the quadratically convergent trust-region method, we
direct the step towards the minimum by choosing the level-
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074103-6 Thøgersen et al. J. Chem. Phys. 123, 074103 2005
shift parameter in Eq. 35 such that the lowest diagonal
element of the Hessian LUMO a − HOMO i + becomes positive.
Alternatively, in the Kohn–Sham diagonalization step in Eq.
19, we may ensure positive definiteness by monitoring the
dependence of the pseudo-orbital energies on the levelshift
parameter in Eq. 19, adjusting it such that the
HOMO-LUMO gap,
ai = LUMO a − HOMO i ,
37
becomes positive. The configuration that defines the HOMO-
LUMO gap is identified from the eigenvalues of Eq. 20 that
are equal to one. Insisting on a smooth development of the
MOs from those that are occupied in D 0 to those that are
obtained by diagonalizing Eq. 19, we restrict to the interval
min , where min is the smallest value for
which the HOMO-LUMO gap is positive. In addition, the
step must be constrained such that Eq. 16 is fulfilled. In
passing, we note that the reference density matrix D 0 may
not always be idempotent, for example, it may be D¯ of Eq.
13, in which case its eigenvalues are not exactly 1. In such
cases, the matrix
D¯ 0 idem = C 0 occ C 0 occ T 38
constructed from the eigenvectors of Eq. 20 with D 0 replaced
by D¯ represents a purification of D¯ .
The constraint on the change in the AO density in Eq.
16 refers to a change which may arise not only from small
changes in many MOs but also from large changes in a few
MOs or even in a single MO. In the TRRH algorithm, we
shall require that the changes in the individual MOs are all
small. Expanding the MO new i , obtained by diagonalization
of Eq. 19, in the old MOs, we obtain
new i = j
old j new i old j + old a new i old a ,
a
39
where the first summation is over the occupied MOs and the
second over the virtual MOs. The squared norm of the projection
of new i onto the MO space associated with D 0 is
therefore
a orb i = old j new i 2 .
40
j
To ensure small individual MO changes at each iteration to
within a unitary transformation of the occupied MOs, we
shall therefore require
orb = min
a min
i
a orb i A orb min ,
41
where A orb min is close to 1. This constraint also ensures that the
HOMO-LUMO gap in Eq. 37 stays positive.
The Hessians of E RH and E KS in Eq. 29 both contain
the orbital-energy difference term, while the Hessian of E KS
also contains the terms M aibj of Eq. 30. When is large
compared to the M aibj terms, the step generated by the levelshifted
diagonalization in Eq. 19 is then of the same quality
as that generated by a quadratically convergent trust-region
optimization of E KS . However, since the step-size control in
Eq. 22 is imposed on the global optimization, the quality of
the step may be further improved relative to that obtained in
a QC-SCF optimization of the Kohn–Sham energy. When the
level shift is determined in the global region such that
a orb min A orb min we see often not just this one orbital but many for
which a orb i A orb min . In this way a large number of orbitals
change significantly.
2. The local region
To investigate the local convergence of the TRRH algorithm
in Eq. 19, we first note that, in the local region near
convergence, the gradient in Eq. 6 and thus the blocks F ov
and F vo between the occupied and virtual orbitals in the
Kohn–Sham matrix in the representation of Eq. 28,
F = o
F ov
F vo v
, 42
are small, see Eq. 27. Writing the unitary transformation of
F generated by K in Eq. 25 as
expKF exp− K = o
F ov
F vo v
+ − T F vo − T v
o F ov
+ T − F ov o T
+ O 2 , 43
− v F vo
we find that, to first order, the block diagonalization of the
Kohn–Sham matrix may be accomplished by solving the following
set of linear equations:
F vo + o − v = 0.
44
Since these equations are identical to the Newton equation in
Eq. 32, we conclude that, in the local region where the
higher-order terms in may be neglected, the block diagonalization
of the Kohn–Sham matrix is equivalent to the solution
of the equation
=−E 2 RH −1 E 1 RH .
45
Let these equations determine the step of iteration n and
expand the Kohn–Sham gradient at iteration n+1 about iteration
point n,
1
E KSn+1
1
= E KSn
1
= E KSn
2
+ E KSn n + O 2
2
− E KSn
2
E RHn
Using Eqs. 27 and 29, we then obtain
1
E KSn+1
1
= E KSn
2
− E RHn
1
−1 E RHn + O 2 . 46
2
+ M n E RHn −1 1
E KSn
2
=−M n E RHn −1 1
E KSn , 47
having neglected terms proportional to O 2 . Therefore, if
2
M n E RHn
−1 has eigenvalues larger than 1, a simple TRRH
sequence will diverge. This is particularly a problem in the
Kohn–Sham theory, where the HOMO-LUMO gap the lowest
eigenvalue of E 2 RH
often is small compared to the contribution
from M. To improve upon the local convergence,
we may increase the HOMO-LUMO gap by level shifting,
thereby reducing the magnitude of the eigenvalues of M n
2
E RHn
−1 . We note that, when the simple TRRH sequence
diverges, the TRSCF algorithm may still converge as TRRH
mainly serves to provide a new density and TRDSM then
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074103-7 Trust-region self-consistent-field J. Chem. Phys. 123, 074103 2005
FIG. 1. a The HOMO-LUMO gap ai , b the minimum overlap a orb min of
the new occupied orbitals with the previous set of occupied orbitals, and c
the changes in the model energy E RH —- and the Kohn–Sham energy
E RH KS ---. All as a function of the level-shift parameter in the TRRH step
of the seventh iteration of the zinc complex calculation seen in Fig. 5.
optimizes the combination of the various densities.
F. Examples of the trust-region
Roothaan–Hall algorithm
To illustrate how the TRRH algorithm is employed in the
different parts of a Kohn–Sham energy optimization, we here
consider how the level-shift parameter is determined in two
iterations of the zinc complex calculation depicted in Sec.
VII, Fig. 5. We first consider iteration 7, which is in the
global region of the optimization, and then proceed to iteration
22, as an example of a step in the local region.
In Figs. 1a and 1b, we have plotted the HOMOorb
LUMO gap ai of Eq. 37 and the overlap parameter a min
of Eq. 41, respectively, as functions of the level-shift parameter
. The corresponding changes in the Kohn–Sham
energy E RH KS dash line and in the Roothaan–Hall model
energy E RH full line of Eqs. 22 and 23 are plotted
in Fig. 1c. We note that the change in the Kohn–Sham
energy has been calculated as
E RH KS = E KS D − E KS D¯ 0 idem ,
48
where D and D¯ 0 idem are the density matrices calculated
from the solutions to the eigenvalue problems in Eqs. 19
and 20, respectively.
FIG. 2. a The HOMO-LUMO gap ai , b the minimum overlap a orb min of
the new occupied orbitals with the previous set of occupied orbitals, and c
the changes in the model energy E RH —- and the Kohn–Sham energy
E RH KS ---. All as a function of the level-shift parameter in the TRRH step
of the 22nd iteration of the zinc complex calculation seen in Fig. 5.
In Fig. 1a, we see that, in iteration 7, ai is linear
for 2.2, as the density matrix changes smoothly with
decreasing from that of Eq. 20 to that obtained by applying
the Aufbau principle to the solution of Eq. 19. For
2.2, the occupied and virtual orbitals defined by the previous
density interchange. The value of =5.078 used in this
iteration was chosen from the requirement a orb min =A orb min =0.98
in Eq. 41, restricting the new orbital component to 0.02.
Figure 1c shows that an even lower energy would have
been obtained by reducing the level shift to about 2.4, but it
would be very difficult to identify this optimal value of
without constructing additional Kohn–Sham matrices, since
the Roothaan–Hall model energy is not accurate for small .
In short, the identification of from the overlap requirement
a orb min =A orb min appears to be a good and secure way to control the
step sizes in the optimization.
Figures 2a–2c are equivalent to Figs. 1a–1c, but
for iteration 22 in the local part of the optimizaton. Notably,
the linear regime of ai in Fig. 2a now extends to
include =0, which corresponds to an unconstrained
Roothaan–Hall step. Also, since a orb min =1.0000 for =0, we
can no longer determine the level shift from the overlap criterion
a orb min =A orb min . From Fig. 2c, we see that E RH KS dash
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074103-8 Thøgersen et al. J. Chem. Phys. 123, 074103 2005
TABLE I. Convergence details for the TRRH steps in the TRSCF calculation
on the zinc complex in Fig. 5. Energies given in a.u.
Iteration RH a orb min RH
RH
E KS
line takes on its minimum value at =1.3; for smaller ,
the energy increases, giving a total increase of 6.0·10 −5 E h
for =0.
The TRRH energy increase in the local part of the SCF
optimization is particularly prominent for the DFT calculations.
In the Hartree–Fock calculations, the TRRH model
energy describes the SCF energy equally well in the local
and global regions of the optimization. To avoid the increase
in energy, we could add a constant minimum level shift, but
this may in some cases slow down the convergence. Typically,
the increase in the Kohn–Sham energy in the TRRH
steps in the local region of the optimization is compensated
by a larger energy decrease in the TRDSM step, ensuring
an overall decrease in the Kohn–Sham energy in the iteration.
In Table I, we have listed the values of several parameters
characterizing the TRRH steps in the TRSCF iterations
of the zinc complex calculation. In the first 17 iterations, the
constraint a orb min =A orb min is active and determines the level-shift
parameter. Note that, in the global region, E RH is a reasonable
good approximation to E RH KS . After iteration 17, the
local region of the Kohn–Sham energy optimization is approached
and E RH is no longer a good approximation to
E RH KS . In this region, the Kohn–Sham energy increases and it
is the TRDSM algorithm that ensures the calculations convergence
see Sec. VII, Table IV.
IV. TRUST-REGION DENSITY-SUBSPACE
MINIMIZATION IN DFT
E RH
2 22.57 0.994 −8.366 865 −8.411 913
3 26.71 0.980 −20.122 850 −20.895 267
4 30.54 0.980 −31.041 569 −35.286 269
5 19.21 0.980 −27.278 985 −31.363 274
6 10.31 0.980 −15.101 958 −18.277 717
7 5.07 0.980 −10.675 155 −13.082 691
8 2.96 0.980 −6.749 189 −7.197 438
9 2.18 0.981 −3.181 254 −4.589 630
10 4.68 0.980 0.394 694 −3.712 621
11 1.40 0.980 −1.676 644 −2.885 580
12 1.40 0.980 −1.743 634 −1.775 556
13 0.93 0.980 −0.402 427 −0.843 260
14 0.78 0.980 −0.376 675 −0.622 386
15 0.54 0.981 −0.211 002 −0.227 722
16 0.15 0.982 0.029 066 −0.199 268
17 0.07 0.980 0.010 452 −0.068 243
18 0.00 0.991 0.043 376 −0.037 071
19 0.00 0.997 0.012 644 −0.009 493
20 0.00 0.999 0.001 104 −0.000 931
21 0.00 0.999 0.000 352 −0.000 249
22 0.00 0.999 0.000 059 −0.000 049
23 0.00 0.999 0.000 010 −0.000 006
24 0.00 1.000 0.000 000 −0.000 000
After a sequence of the Roothaan–Hall iterations, we
have determined a set of the density matrices D i and a corresponding
set of the Kohn–Sham matrices F i =FD i . The
question then arises as to how to make the best use of the
information contained in these collected density and Kohn–
Sham matrices.
A. Parametrization of the DSM density matrix
Taking D 0 as the reference density matrix, we write the
improved density matrix as a linear combination of the current
and previous density matrices,
n
D¯ = D 0 + c i D i ,
49
i=0
which, ideally, should satisfy the symmetry, trace, and idempotency
conditions in Eq. 8 of a valid Kohn–Sham density
matrix. Whereas the symmetry condition in Eq. 8a is trivially
satisfied for any such linear combination, the trace condition
in Eq. 8b holds only for combinations that satisfy the
restriction
n
c i =0,
50
i=0
leading to a set of n+1 constrained parameters c i with 0
in. Alternatively, an unconstrained set of n parameters c i
with 1in can be used, with c 0 defined so that the trace
condition is fulfilled,
n
c 0 =− c i .
51
i=1
In terms of these independent parameters, the density matrix
D¯ becomes
D¯ = D 0 + D + ,
where we have introduced the notations
n
D + = c i D i0 ,
i=1
D i0 = D i − D 0 .
52
53a
53b
Unlike the symmetry and trace conditions in Eqs. 8a
and 8b, the idempotency condition in Eq. 8c is in general
not fulfilled for linear combinations of D i . Still, for any averaged
density matrix D¯ in Eq. 52 that does not fulfill the
idempotency condition, we may generate a purified density
matrix with a smaller idempotency error by the
transformation, 15
D˜ =3D¯ SD¯ −2D¯ SD¯ SD¯ .
54
The purification of the density matrix has previously been
used in connection with minimization of energy
functions. 16–19
Introducing the idempotency correction,
D = D˜ − D¯ ,
55
we may then write the purified averaged density matrix in
the form
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074103-9 Trust-region self-consistent-field J. Chem. Phys. 123, 074103 2005
D˜ = D 0 + D + + D .
56
In the following, we shall analyze the relative magnitudes of
the terms D + and D entering Eq. 56.
B. Order analysis of the purified averaged
density matrix
For simplicity, we shall work in the orthonormal MO
basis that diagonalizes the reference density matrix,
D 0 = I 0
57
0 0,
and consider the case with only one additional density matrix
D 1 . According to Eq. 24, an antisymmetric matrix K of the
form in Eq. 25 exists such that
D 1 = exp− KD 0 expK
= D 0 + − T −
T
− T + O 3 , 58
giving rise to the following averaged density matrix:
D¯ = D 0 + cD 10 = D 0 + − cT − c
T
− c c T + Oc 3 .
The idempotency error of D¯ is given by
59
D¯ D¯ − D¯ = c 2 − c T 0
0 T + Oc 4 , 60
showing that D¯ is idempotent only to first order in . To
reduce the idempotency error, we subject D¯ to the purification
in Eq. 54, obtaining
D˜ =3D¯ 2 −2D¯ 3 = D 0 + T − c2 T − c T
− c c 2 + Oc 3 .
Finally, comparing Eqs. 59 and 61, we obtain
D˜ = D¯ + Oc 2 ,
61
62
demonstrating that the impure and purified average density
matrices differ by terms proportional to c 2 . Since the
McWeeny purification in Eq. 54 converges quadratically,
we conclude that the idempotency error of Eq. 62 is proportional
to c 2 4 .
In a more general analysis, we would not assume an
orthonormal basis and we would also include several density
matrices D i =exp−K i D 0 expK i . The essential result is
then that we may write Eq. 56 as
n
D˜ = D 0 +
i=1
n
c i D i0 + O c i D i0 2,
i=1
63
where we have used the fact that D i0 is proportional to i .
We conclude that while D + is linear in c i and D i0 , the idempotency
correction D to D¯ is linear in c i but quadratic in D i0 .
The conclusions to be derived from this analysis are summarized
in Table II.
TABLE II. Comparison of the properties of the unpurified density D¯ and the
purified density D˜ .
C. Construction of the DSM energy function
Having established a useful parametrization of the averaged
density matrix in Eq. 52 and having considered its
purification in Eq. 54, let us now consider how to determine
the best set of coefficients c i . Expanding the energy for
the purified averaged density matrix in Eq. 56 around the
reference density matrix D 0 , we obtain to second order
ED˜ = ED 0 + D + + D T E 0
1
+ 1 2 D + + D T E 0 2 D + + D . 64
To evaluate the terms containing E 0 1 and E 0 2 , we make the
identifications,
E 0 1 =2F 0 ,
E 0 2 D + =2F + + OD + 2 ,
65
66
which follow from Eq. 6 and from the second-order Taylor
expansion of E 1 0
about D 0 , and where we have generalized
the notation in Eq. 53a to the Kohn–Sham matrix F +
= n
i=1
c i F i0 . Ignoring the terms quadratic in D in Eq. 64
and quadratic in D + in Eq. 66, we then obtain for the DSM
energy,
E DSM c = ED 0 +2TrD + F 0 +TrD + F +
+2TrD F 0 +2TrD F + .
67
Finally, for a more compact notation, we introduce the
weighted Kohn–Sham matrix,
n
F¯ = F0 + F + = F 0 + c i F i0 ,
68
i=1
and find that the DSM energy may be written in the form
E DSM c = ED¯ +2TrD F¯ ,
69
where the first term is quadratic in the expansion coefficients
c i ,
ED¯ = ED 0 +2TrD + F 0 +TrD + F + ,
70
and the second, idempotency-correction term is quartic in
these coefficients:
2TrD F¯ =Tr6D¯ SD¯ −4D¯ SD¯ SD¯ −2D¯ F¯ .
D¯
Differences D¯ −D 0 =Oc D˜ −D¯ =Oc 2
Idempotency error D¯ SD¯ −D¯ =Oc 2 D˜ SD˜ −D˜ =Oc 2 4
Trace error Tr D¯ S− N 2=0 TrD˜ S− N 2=Oc 2 4
71
The derivatives of E DSM (c) are straightforwardly obtained
by inserting the expansions of F¯ and D¯ , using the independent
parameter representation.
D˜
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074103-10 Thøgersen et al. J. Chem. Phys. 123, 074103 2005
TABLE III. Convergence details for the TRDSM steps in the TRSCF calculation
on the zinc complex in Fig. 5. Energies given in a.u.
Iteration D S
2
D + S
2
DSM
E KS
E DSM
3 1.612 753 6.129 310 −48.255 717 −49.742 656
4 1.488 082 12.140 844 −105.996 850 −111.554 301
5 0.206 716 1.594 214 −43.136 482 −41.110 879
6 1.504 099 3.162 679 −26.390 457 −26.511 025
7 0.096 714 1.468 925 −14.755 377 −14.499 582
8 0.110 282 1.525 848 −7.711 220 −7.278 600
9 0.086 759 1.569 113 −5.289 340 −5.165 696
10 0.423 825 1.614 867 −2.684 359 −3.500 173
11 0.196 628 1.002 744 −1.053 899 −1.126 867
12 0.111 409 0.867 238 −1.054 903 −0.936 180
13 0.093 520 0.729 574 −0.658 907 −0.621 180
14 0.054 596 0.324 338 −0.293 889 −0.238 992
15 0.045 721 0.201 434 −0.213 251 −0.170 060
16 0.026 474 0.242 928 −0.104 012 −0.096 482
17 0.011 746 0.071 203 −0.100 694 −0.093 602
18 0.001 512 0.022 758 −0.043 180 −0.042 748
19 0.000 687 0.040 675 −0.057 441 −0.056 819
20 0.000 122 0.011 897 −0.016 501 −0.016 416
21 0.000 025 0.001 164 −0.001 471 −0.001 453
22 0.000 001 0.000 308 −0.000 428 −0.000 427
23 0.000 000 0.000 050 −0.000 076 −0.000 076
24 0.000 000 0.000 009 −0.000 012 −0.000 012
25 0.000 000 0.000 000 −0.000 000 −0.000 000
D. Optimization of the DSM energy
The energy function E DSM c in Eq. 69 provides an
excellent approximation to the exact Kohn–Sham energy
E KS c about D 0 , with an error cubic in D + . It can be optimized
by the trust-region method, as described in Ref. 6,
yielding an improved density matrix D˜ , from which the
Kohn–Sham matrix of the next TRRH iteration is constructed.
However, to avoid the expensive calculation of the
Kohn–Sham matrix from D˜ , we use instead in our TRDSM
implementation the averaged Kohn–Sham matrix in Eq. 68.
As in the TRRH step in Sec. III A, the averaged density
matrix D¯ may also be determined by a line search. Here, the
line search is made in the direction defined by the first step
of the TRDSM algorithm, that is, the step at the expansion
point D 0 . As in the TRRH step, such a line search is guaranteed
to reduce the Kohn–Sham energy. We denote this line
search algorithm TRDSM-LS.
In the DSM scheme, we assume that the idempotency
correction D =D˜ −D¯ is small relative to D + =D¯ −D 0 , both
when discarding the terms quadratic in D in Eq. 64 and
when constructing the Kohn–Sham matrix from D¯ rather
than from D˜ in the subsequent Roothaan–Hall iteration. As is
seen from Eq. 63, this assumption holds if the old density
matrices D i are similar to D 0 . Formally, therefore, we should
include in the TRDSM only density matrices that are similar
to D 0 . In particular, if the orbital occupations change in the
course of the Roothaan–Hall iterations, we should discard all
density matrices that represent the old occupations.
To demonstrate the validity of the assumption, that D is
small compared to D + , we have in Table III listed D S
2
FIG. 3. The ratio between the norms of the idempotency correction to the
density D S 2 =D˜ −D¯ S 2 and the density change D + S 2 =D¯ −D 0 S 2 in the
TRDSM steps of the zinc complex calculation seen in Fig. 5.
=TrD SD S and D + 2 S =TrD + SD + S at each iteration of the
zinc complex calculation of Sec. VII. From Fig. 3, where the
ratio D 2 S /D + 2 S is plotted, we see that, apart from iteration
6, this ratio is always smaller than 0.3 and that it rapidly
converges to zero in the local region. The neglect of the
terms that are quadratic in D in the TRDSM method is thus
well justified. In Table III, we have also listed the model
energy change E DSM and the actual energy change E DSM KS ,
obtained as the difference between the Kohn–Sham energies
calculated from the idempotent D¯ obtained as in Eq. 38
and from D 0 : E DSM KS =E KS D¯ 0 idem −E KS D 0 . Clearly,
E DSM c is an extremely good representation of E KS c for
the step sizes taken by the TRDSM algorithm, as expected
since E DSM c and E KS c differ in terms that are cubic in D + .
E. Comparison of the DSM and EDIIS energies
Neglecting the idempotency correction in the DSM energy
in Eq. 69, we are left with ED¯ . In the Hartree–Fock
theory, this remaining term may be expressed in several
equivalent ways. First, it may be written as the energy of the
weighted density matrix,
E HF D¯ =2TrhD¯ +TrD¯ GD¯ ,
72
where the weighted density matrix is defined as note the
difference from Eq. 49
n
D¯ = d i D i ,
i=0
n
d i =1.
i=0
73
In their development of the EDIIS method, Kudin et al. 4
suggested the alternative form
n
E EDIIS D¯ = d i E SCF D i − 1 n
i=0
2 Tr d i d j F ij D ij , 74
i,j=0
where E SCF D may be the Hartree–Fock energy or the
Kohn–Sham energy. In the Hartree–Fock theory, Eqs. 70,
72, and 74 are equivalent since the Fock matrix is linear
in the density matrix. By contrast, in the DFT, where the
Kohn–Sham matrix contains terms that are nonlinear in the
density matrix, these expressions are not equivalent. Below,
we discuss some of the consequences of their nonequivalence
in the DFT.
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074103-11 Trust-region self-consistent-field J. Chem. Phys. 123, 074103 2005
Eliminating d 0 =1− n
i=1 d i from Eq. 74, we may express
the EDIIS energy in the independent representation of
Eqs. 52 and 53,
n
E EDIIS D¯ = E SCF D 0 + d i E SCF D i − E SCF D 0
i=1
n
−
i=1
n
d i Tr F i0 D i0 + d i d j Tr F j0 D j0
n
i,j=1
− 1 d i d j Tr F ij D ij .
2
i,j=1
75
Comparing this expression with ED¯ of Eq. 70, wefind
that they have the same values at the expansion point D 0 but
that their first derivatives differ since
ED¯
c k
=2TrF 0 D k0 , 76a
E EDIIS D¯
= E SCF D k − E SCF D 0 −TrF k0 D k0 . 76b
d k
In the Hartree–Fock theory, it is easy to see that Eqs. 76a
and 76b are identical.
The DSM gradient is
E DSM c ED¯
= +2 Tr D F¯
. 77
c k c k c k
Since E DSM is equal to E KS to first order, we have that
E DSM c
= E KS
. 78
c k c k
The EDIIS gradient at the expansion point is thus not equal
to the KS gradient as the last nonzero term in Eq. 77 the
term resulting from the idempotency correction is missing.
Further the correct gradient in the DSM can only be obtained
in the DFT if Eq. 76a and not Eq. 76b is used. It is thus
incorrect to use Eq. 76a in the DFT even though Eqs. 76a
and 76b are equivalent in Hartree–Fock.
V. CONFIGURATION SHIFT
IN THE TRSCF ALGORITHM
Since the TRSCF method has been designed for a
smooth and controlled convergence of the density matrix, it
does not allow for the abrupt changes in the orbitals associated
with configuration shifts. Nevertheless, it may sometimes
be advantageous to allow such shifts, as illustrated in
Fig. 4, where we compare two cadmium complex calculations
see Sec. VII for details. The “no-shift” optimization
proceeds carefully, allowing only small changes in the density
matrix at each iteration, whereas the “do-shift” optimization
is more daring, accepting abrupt configuration shifts
that reduce the total energy.
In Fig. 4a, we have plotted the error in the energy at
each iteration of the two optimizations. The first 13 iterations
are identical; the optimizations are in the global region and
orb
the level shift is determined from the requirement a min
FIG. 4. The TRSCF cadmium complex calculation described in Sec. VII. a
The convergence without abrupt configuration shift and with abrupt
configuration shift . b and c contain details of the TRRH step in
iteration 14; b the minimum overlap a orb
min for the new occupied orbitals
with the previous set of occupied orbitals and c the changes in the model
energy E RH — and the actual energy E RH KS ---. All as a function of the
level-shift parameter .
=A orb min =0.98. In iteration 14, the two optimizations differ. To
understand the reasons for these differences, we have in Fig.
4b plotted a orb min and in Fig. 4c E RH full line and
E RH KS dash line as functions of . For =0.25, there is an
abrupt shift in a orb min from 0.99 to 0.00, representing a configuration
shift where the LUMO for 0.25 becomes the
HOMO for 0.25. From Fig. 4c, we see that this shift
lowers the Kohn–Sham total energy. Because of the abrupt
change in a orb min at =0.25, we are unable to identify
a orb min =0.98. In the no-shift calculation, is chosen larger
than 0.25, whereas, in the do-shift calculation, the undamped
Roothaan–Hall step is taken with =0.
As the DSM energy model assumes small changes in the
density matrix, the density matrices of all previous iterations
are discarded in iteration 14 of the do-shift calculation, and a
rapid convergence to the optimized state is seen from that
point. In the no-shift calculation, an a orb min profile similar to
that of iteration 14 is obtained in the next few iterations. In
these iterations, the lowest Hessian eigenvalue is −0.95 a.u.
and the optimization proceeds towards a stationary point.
Finally, in iteration 22, the TRSCF algorithm identifies this
stationary point as a saddle point, moves out of this region,
and converges rapidly to the same minimum as the do-shift
optimization.
As this example illustrates, it is important to recognize
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074103-12 Thøgersen et al. J. Chem. Phys. 123, 074103 2005
and accept a favorable configuration shift. A configuration
shift may be recognized when an a orb min profile has an
abrupt change where on the right-hand side a orb min is close to 1
and on the left-hand side a orb min is close to 0. To maintain the
high degree of control characteristic of the TRSCF method,
the energy of the new configuration is checked before the
shift is accepted, at the cost of an additional Kohn–Sham
matrix build. As seen from Fig. 4a, this check is well worth
the effort, saving more than ten iterations, and thus it is made
an integrated part of our TRSCF implementation.
VI. THE DIIS METHOD VIEWED
AS A QUASI-NEWTON METHOD
Since its introduction by Pulay in 1980, the DIIS method
has been extensively and successfully used to accelerate the
convergence of SCF optimizations. We here present a rederivation
of the DIIS method to demonstrate that, in the iterative
subspace of density matrices, it is equivalent to a quasi-
Newton method. From this observation, we conclude that, in
the local region of the SCF optimization, the DIIS steps can
be used safely and will lead to fast convergence. The convergence
of the DIIS algorithm in the global region is also
discussed and is much more unpredictable.
We assume that, in the course of the SCF optimization,
we have determined a set of n+1 AO density matrices
D 0 ,D 1 ,D 2 ,...,D n and the associated Kohn–Sham or Fock
matrices FD 0 ,FD 1 ,FD 2 ,...,FD n . Since the electronic
gradient gD is given by 11
gD =4SDFD − FDDS,
79
we also have available the corresponding gradients
gD 0 ,gD 1 ,gD 2 ,...,gD n . We now wish to determine a
corrected density matrix,
n
D¯ = D 0 + c i D i0 , D i0 = D i − D 0 , 80
i=1
that minimizes the norm of the gradient gD¯ . For this purpose,
we parameterize the density matrix in terms of an antisymmetric
matrix X=−X T and the current density matrix
D 0 as 11
DX = exp− XSD 0 expSX.
81
With each old density matrix D i , we now associate an antisymmetric
matrix X i such that
D i = exp− X i SD 0 expSX i = D 0 + D 0 ,X i S + OX 2 i .
82
Introducing the averaged antisymmetric matrix,
n
X¯ = c i X i ,
i=1
we obtain
83
n
DX¯ = D 0 + c i D 0 ,X i S + OX¯ 2 ,
i=1
84
where we have used the S-commutator expansion of DX¯
analogeous to Eq. 82. Our task is hence to determine X¯ in
Eq. 83 such that DX¯ minimizes the gradient norm
gDX¯ . In passing, we note that, whereas D¯ is not in
general idempotent and therefore not a valid density matrix,
DX¯ is a valid, idempotent density matrix for all choices of
c i .
Expanding the gradient in Eq. 79 about the currentdensity
matrix D 0 , we obtain
gDX¯ = gD 0 + HD 0 X¯ + OX¯ 2 ,
85
where HD is the Jacobian matrix. Neglecting the higherorder
terms, our task is therefore to minimize the norm of the
gradient,
n
gc = gD 0 + c i HD 0 X i ,
86
i=1
with respect to the elements of c. For an estimate of
HD 0 X i , we truncate the expansion,
gD i = gD 0 + HD 0 X i + OX i 2 ,
and obtain the quasi-Newton condition,
gD i − gD 0 = HD 0 X i .
Inserting this condition into Eq. 86, we obtain
n
gc = gD 0 +
i=1
n
c i gD i − gD 0 = c i gD i ,
i=0
87
88
89
where we have introduced the parameter c 0 =1− n
i=1 c i . The
minimization of gc=gc may therefore be carried out as
a least-squares minimization of gc in Eq. 89 subject to the
constraint
n
c i =1.
90
i=0
If we consider gD i as an error vector for the density matrix
D i , this procedure becomes identical to the DIIS method.
From Eq. 86 we also see that DIIS may be viewed as a
minimization of the residual for the Newton equation in the
subspace of the density matrix differences D i −D 0 , i=1, n,
where the quasi-Newton condition is used to set up the subspace
equations. Since the quasi-Newton steps are reliable
only in the local region of the optimization, we conclude that
the DIIS method can be used safely only in this region, when
the electronic Hessian is positive definite.
The optimal combination of the density matrices is obtained
in the DIIS method, by carrying out a least-squares
minimization of the gradient norm subject to the constraint in
Eq. 90. However, since a small gradient norm in the global
region does not necessarily imply a low Kohn–Sham energy,
the DIIS convergence may be unpredictable. Furthermore,
we may encounter regions where the gradient norms are
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074103-13 Trust-region self-consistent-field J. Chem. Phys. 123, 074103 2005
ethylenediamine tetra-acetic acid EDTA. Next, in Sec.
VII B, we consider the calculations on five different systems.
All calculations have been carried out with a local version of
the DALTON program package. 20 Unless otherwise indicated,
the starting orbitals have been obtained by diagonalization of
the one-electron Hamiltonian.
FIG. 5. The convergence of different algorithms in a LDA/6-31G computation
with core Hamiltonian start guess for the zinc complex depicted in the
lower left corner. The algorithms being QC-SCF , DIIS , TRSCF
, and TRSCF-LS .
similar but the energies different. The DIIS method may then
diverge, not being able to identify the density matrix of lowest
energy, as illustrated in Sec. VII.
VII. APPLICATIONS
to
In this section, we give numerical examples to illustrate
the convergence characteristics of the Kohn–Sham TRSCF
calculations, comparing with the DIIS and QC-SCF calculations.
Comparisons are also made with the TRSCF-LS technique,
where the TRRH-LS and TRDSM-LS line-search
methods of Secs. III A and IV D are combined to set up an
expensive but highly robust method, in which the lowest
Kohn–Sham energy is identified by a line search at each step.
In Sec. VII A, we discuss the calculations on the zinc complex
in Fig. 5, where Zn 2+ is complexated with
ethylenediamine-N, N -disuccinic acid EDDS, an isomer
A. Calculations on the zinc complex
In Fig. 5, we have plotted the error in the Kohn–Sham
energy at each iteration of LDA/6-31G calculations on the
zinc complex. The standard TRSCF method performs
almost as well as the very smooth but much more expensive
TRSCF-LS method , giving a somewhat higher energy
between iterations 13 and 22. By contrast, the DIIS method
shows no sign of converging; after 100 iterations, the
Kohn–Sham gradient norm is still about 20. Whereas the
smooth TRSCF convergence arises because Hessian information
is used to ensure downhill TRRH and TRDSM steps
at each iteration, no such information is employed in the
DIIS method. Finally, the QC-SCF method converges
but exceedingly slow—even after 90 iterations it has not
reached the quadratically convergent local region! The difficulties
experienced with the QC-SCF method illustrate
clearly that the use of Hessian information by itself is no
guarantee of fast convergence.
More details about the TRSCF zinc complex calculation
are given in Tables I–V and in Figs. 1–3 and 6, partly discussed
in Secs. III F and IV D. In Table IV, we have listed
the changes in the Kohn–Sham energy generated separately
in the TRRH E RH KS and TRDSM E DSM KS steps at each
SCF iteration, and likewise the norms of the changes in the
TABLE IV. Convergence details for the TRSCF calculation on the zinc complex in Fig. 5. Energies given in a.u.
DSM
Iteration E KS E KS
RH
E KS
2
D¯ n−D n S DSM
D n+1 −D¯ n S
2
2 −8.366 865 0.000 000 −8.366 865 0.000 000 0.197 607
3 −68.378 567 −48.255 717 −20.122 850 6.129 310 1.141 536
4 −137.038 420 −105.996 850 −31.041 569 12.140 844 1.265 250
5 −70.415 468 −43.136 482 −27.278 985 1.594 214 1.031 844
6 −41.492 416 −26.390 457 −15.101 958 3.162 679 1.467 802
7 −25.430 533 −14.755 377 −10.675 155 1.468 925 1.364 944
8 −14.460 409 −7.711 220 −6.749 189 1.525 848 1.249 827
9 −8.470 594 −5.289 340 −3.181 254 1.569 113 1.040 337
10 −2.289 664 −2.684 359 0.394 694 1.614 867 0.817 844
11 −2.730 543 −1.053 899 −1.676 644 1.002 744 1.060 298
12 −2.798 537 −1.054 903 −1.743 634 0.867 238 0.632 009
13 −1.061 335 −0.658 907 −0.402 427 0.729 574 0.410 434
14 −0.670 565 −0.293 889 −0.376 675 0.324 338 0.351 715
15 −0.424 253 −0.213 251 −0.211 002 0.201 434 0.203 170
16 −0.074 945 −0.104 012 0.029 066 0.242 928 0.302 723
17 −0.090 241 −0.100 694 0.010 452 0.071 203 0.175 917
18 0.000 195 −0.043 180 0.043 376 0.022 758 0.126 709
19 −0.044 797 −0.057 441 0.012 644 0.047 885 0.032 787
20 −0.015 396 −0.016 501 0.001 104 0.011 897 0.002 976
21 −0.001 118 −0.001 471 0.000 352 0.001 164 0.000 668
22 −0.000 368 −0.000 428 0.000 059 0.000 308 0.000 111
23 −0.000 066 −0.000 076 0.000 010 0.000 050 0.000 019
24 −0.000 011 −0.000 012 0.000 000 0.000 009 0.000 001
25 −0.000 000 −0.000 000 0.000 000 0.000 000 0.000 000
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074103-14 Thøgersen et al. J. Chem. Phys. 123, 074103 2005
TABLE V. The density of each iteration compared to the optimized one.
Iteration D conv −D n S
2
a orb min conv,n
2 66.952 673 0.0965
3 65.174 713 0.0955
4 56.502 973 0.0927
5 51.210 143 0.1017
6 48.482 773 0.1411
7 42.682 641 0.1394
8 35.617 332 0.1992
9 26.551 913 0.3183
10 18.298 431 0.4094
11 14.152 342 0.4983
12 9.767 169 0.6927
13 6.184 621 0.6859
14 3.844 299 0.9187
15 2.240 436 0.9194
16 1.018 810 0.9771
17 0.200 374 0.9952
18 0.064 181 0.9984
19 0.043 906 0.9967
20 0.011 531 0.9996
21 0.001 092 0.9999
22 0.000 309 0.9999
23 0.000 053 0.9999
24 0.000 009 0.9999
25 0.000 000 0.9999
2
density matrix in the TRRH D n+1 −D¯ n S RH
and TRDSM
2
D¯ n−D n S DSM
steps. Remarkably, the TRDSM step consistently
reduces the energy more than the TRRH step. Indeed,
after iteration 15, each TRRH step increases rather
than decreases the energy. Apparently, in the local region, the
role of the TRRH step is reduced to that of improving that
variational space of the subsequent TRDSM step. From the
table, we also see that the largest changes in the density
matrix are generated by the TRDSM step rather than by the
TRRH step.
For the TRRH and TRDSM steps, we have at each iteration
determined the overlap a orb i in Eq. 40 of each generated
occupied orbital new i with the previous orbitals old j . In Fig.
6, the number of orbitals at each iteration with a orb i 0.98
i.e., with large rotations is illustrated in a bar chart. As we
require a orb i 0.98 in the Roothaan–Hall steps, the TRRH
FIG. 6. The number of occupied orbitals in the TRRH and TRDSM steps
with an overlap less than 0.98 to the previous set of occupied orbitals for
each step in the SCF iteration.
orb
bars simply represent the number of orbitals with a i
0.98. In the TRDSM step, however, no such restrictions
are imposed and a large number of orbitals with a orb i 0.98
are observed. Indeed, in the first few DSM steps, overlaps as
small as 0.76 occur, leading to far larger changes than those
accepted in the Roothaan–Hall step, emphasizing the important
role played by the TRDSM step in achieving orbital
reorganizations in a controlled manner.
In Table V, we have listed the norm of the difference
between the current-density matrix D n at each iteration and
the final converged density matrix D conv ; also, we have listed
a orb min conv,n, which is the smallest overlap in the sense of
Eq. 41 of the current occupied orbitals, with the converged
ones. Clearly, very large changes occur in the density matrix
and the orbitals in the course of the optimization, in particular,
during the first 17 iterations; in the remaining iterations,
only small adjustments are made. In spite of the large overall
changes made to the orbitals, they have been accomplished
in a controlled and reliable manner.
In Fig. 7, we have plotted the errors for the same LDA/
6-31G optimization as in Fig. 5, but with the starting orbitals
obtained from a Hückel calculation rather than from the diagonalization
of the one-electron Hamiltonian. Convergence
is now faster, with the TRSCF-LS and TRSCF methods
behaving in the same smooth manner as before. More
importantly, with this improved starting guess, the DIIS
method converges in almost the same number of iterations
as the TRSCF method, although less smoothly.
Finally, in Fig. 8, we have the same plot as in Fig. 7, but
in the STO-3G rather than 6-31G basis still with a Hückel
guess. Somewhat surprisingly, convergence is more difficult
in this smaller basis. Indeed, after 100 iterations, the DIIS
method has not yet converged, with a Kohn–Sham gradient
norm as large as 10. The standard TRSCF method
still converges, but now in a less smooth manner than the
TRSCF-LS method. As mentioned in Sec. III E 2, when
the HOMO-LUMO gap is particularly small, it may sometimes
be necessary to enforce a minimum TRRH level shift
to achieve convergence. Indeed, in the TRSCF optimization
in Fig. 8, we require 0.1 throughout the calculation.
B. Calculations on a variety of molecules
In Fig. 9, we have plotted the errors in the energy at each
SCF iteration, for a variety of molecules at the LDA level of
theory: the zinc complex from Fig. 5 in the 6-31G basis
set; the rhodium complex from Ref. 6 in the Ahlrichs-
VDZ basis 21 with STO-3G on the rhodium atom; a cadmium
complexed with an imidazole ring in the STO-3G basis;
the CH 3 CHO molecule in the cc-pVTZ basis, 22 and the
H 2 O molecule in the cc-pVTZ basis.
For the TRSCF-LS method, convergence is smooth for
all systems, as expected. Likewise, in the TRSCF calculations
with no restrictions enforced on the TRRH level-shift
parameter, convergence is still good although not as smooth
as in the TRSCF-LS calculations. The behavior of the DIIS
method is somewhat more erratic, in particular, in the global
region; in the local region, it converges as well as the TRSCF
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074103-15 Trust-region self-consistent-field J. Chem. Phys. 123, 074103 2005
FIG. 7. The convergence of different algorithms in a LDA/6-31G computation
with Hückel start guess for the zinc complex in Fig. 5. The algorithms
being DIIS , TRSCF , and TRSCF-LS .
method. These observations are in agreement with our discussion
in Sec. VI. The DIIS zinc complex calculation does
not converge as discussed above.
In Fig. 9, we have also included the results from the
DIIS-TRRH optimizations. These calculations differ from
the DIIS calculations in that we have used a level-shift parameter
in the Roothaan–Hall diagonalization step; alternatively,
DIIS-TRRH may be viewed as different from TRSCF
in that we have replaced the TRDSM steps by DIIS steps.
Somewhat surprisingly, only the water calculation converges
with the DIIS-TRRH method. To understand this behavior,
we note that, in the global region, the TRRH method typically
produces gradients that do not change much, even
though large changes may occur in the energy. In such cases,
the DIIS method may stall, not being able to identify a good
combination of density matrices.
This behavior is illustrated in Table VI, where we have
listed the gradient norm and Kohn–Sham energy of the first
six iterations of the cadmium complex calculation in Fig. 9.
The TRSCF and DIIS-TRRH gradients stay almost the same
during these iterations, stalling the DIIS-TRRH optimization
but not the TRSCF optimization, whose energy decreases in
each iteration. In the pure DIIS optimization, by contrast, the
gradient changes significantly from iteration to iteration; at
the same time, the energy decreases at each iteration except
the fifth, where also the gradient norm increases. Eventually,
DIIS enters the local region with its rapid rate of convergence
although we note, in the DIIS panel in Fig. 9, a sudden,
large increase in the energy for the cadmium complex
FIG. 9. The convergence in LDA calculations for a variety of molecules
using the TRSCF-LS, TRSCF, DIIS, and DIIS-TRRH approaches, respectively.
The molecules being a zinc complex , rhodium complex ,
cadmium complex , CH 3 CHO , and H 2 O .
calculation in iterations 10 and 11. However, these
changes are accompanied with large increases in the gradient
norm, allowing DIIS to recover safely.
VIII. CONCLUSIONS
FIG. 8. The convergence of different algorithms in a LDA/STO-3G computation
with Hückel start guess for the zinc complex in Fig. 5. The algorithms
being DIIS , TRSCF , and TRSCF-LS .
In this paper, the trust-region SCF TRSCF algorithm
introduced in Ref. 6 has been further developed to make it
applicable to the optimization of the Kohn–Sham energy. In
the TRSCF method, both the Roothaan–Hall step and the
density-subspace minimization DSM step are replaced by
optimizations of local energy models of the Hartree–Fock/
Kohn–Sham energy E SCF . These local models have the same
gradient as the true energy E SCF but an approximate Hessian.
Restricting the steps of the TRSCF algorithm to the trust
region of these local models, that is, to the region where the
local models approximate E SCF well, smooth and fast convergence
may be obtained to the optimized energy.
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074103-16 Thøgersen et al. J. Chem. Phys. 123, 074103 2005
TABLE VI. The gradient norm g=4SDF−FDS in the first six iterations of the cadmium complex calculations
seen in Fig. 9.
DIIS DIIS-TRRH TRSCF
Iteration
E KS g E KS g E KS g
1 −5597.0 7.8 −5597.0 7.8 −5597.0 7.8
2 −5502.3 14.9 −5598.4 7.2 −5598.3 7.1
3 −5602.1 9.7 −5600.3 8.5 −5603.7 9.3
4 −5628.5 2.1 −5599.9 7.7 −5611.1 9.1
5 −5627.4 3.5 −5599.9 7.8 −5616.8 7.7
6 −5628.8 0.8 −5600.2 8.1 −5622.7 7.5
conv no conv conv
In the previous implementation of the TRSCF algorithm,
the focus was on the optimization of the Hartree–Fock energy.
As the Kohn–Sham energy is nonquadratic in the density
matrix, the local DSM energy model has been generalized
and is now expanded about the current-density matrix
D 0 in the subspace of the density matrices D i of the previous
iterations. To satisfy the idempotency condition, the energy
model function is parametrized in terms of a purified averaged
density matrix. The local energy function is correct to
second order in D i −D 0 and can be set up solely in terms of
the density matrices and Kohn–Sham matrices of the previous
iterations. In the Hartree–Fock theory, the new local energy
model is identical to the one previously used in TRSCF
optimizations.
The EDIIS function is discussed in the context of the
proposed model. In the Hartree–Fock theory, the EDIIS function
is obtained from our proposed energy function by neglecting
terms that result from the purification of the density
matrix; the EDIIS function therefore does not reproduce the
Hartree–Fock gradient at the expansion point. In the DFT,
the EDIIS function is inappropriate for other reasons as well.
A rederivation of the original DIIS algorithm is also performed
to understand when it can safely be applied. In particular,
it is shown that the DIIS method may be viewed as a
quasi-Newton method, thus explaining its fast local convergence.
In the global region, its behavior is less predictable,
although we note that its gradient-norm minimization mechanism
usually allows it to recover safely from sudden, large
increases in the total energy brought on by the Roothaan–
Hall iterations.
The TRSCF scheme is tested both in a computationally
demanding, robust line-search implementation TRSCF-LS,
and in our standard implementation, where only the Fock/
Kohn–Sham matrices of previous iterations are used. Our
test calculations indicate not only that the TRSCF-LS
method is a highly stable and robust method, but also that the
standard TRSCF implementation converges rapidly in most
cases, with little degradation relative to the TRSCF-LS
scheme.
Relative to these schemes, the DIIS method is somewhat
more erratic since it makes no use of Hessian information
and therefore cannot predict reliably what directions will reduce
the total energy. For example, in situations where the
energy changes in the course of the iterations but the gradient
does not, the DIIS algorithm is unable to identify the density
matrix with the lowest energy and may diverge. Nevertheless,
the DIIS method handles most optimizations amazingly
well, which is particularly impressive in view of its very
simplicity; never has so few lines of code done so much
good for so many calculations. In general, however, it is
outperformed by the TRSCF method, which introduces Hessian
information at little extra cost, and is well founded in
the global as well as local regions of the optimization.
The current formulation of TRSCF requires a few diagonalizations
in each TRRH step, and to obtain linear scaling
these diagonalizations should be avoided. An even more efficient
algorithm may be obtained if the Roothaan–Hall and
DSM steps are integrated in such a manner that the information
from the previous density matrices are directly used in
the Roothaan–Hall optimization step. Work along these lines
is in progress.
ACKNOWLEDGMENTS
We thank Peter Taylor, Ditte Jørgensen, and Stephan
Sauer for providing some of the test examples. This work has
been supported by the Danish Natural Research Council. We
also acknowledge support from the Danish Center for Scientific
Computing DCSC.
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Part 3
A Coupled Cluster and Full Configuration Interaction Study of CN and CN - ,
L. Thøgersen and J. Olsen,
Chem. Phys. Lett. 393, 36 (2004)

Chemical Physics Letters 393 (2004) 36–43
www.elsevier.com/locate/cplett
A coupled cluster and full configuration interaction
study of CN and CN
Lea Thøgersen, Jeppe Olsen *
Department of Chemistry, Theoretical Chemistry, University of Aarhus, DK-8000 Aarhus, Denmark
Received 30 April 2004; in final form 27 May 2004
Abstract
Full configuration interaction (FCI) and coupled cluster (CC) calculations are carried out for the CN radical and CN using the
cc-pVDZ and an augmented cc-pVDZ basis set. In addition, CC calculations including up to quadruple excitations are carried out
using the cc-pVTZ basis. At the FCI level, the equilibrium distance is 1.1969 A, the harmonic frequency is 2020.1 cm 1 , the
electronic contribution to the atomization energy is 667 kJ/mol and the vertical electron affinity is 0.12962 E h . The contributions
from quadruple and quintuple excitations to the harmonic frequency are found to be 20 and 5 cm 1 , respectively. The quadruple
excitations give a contribution of 4 kJ/mol to the atomization energy and 0.00013 E h to the vertical electron affinity. None of the
calculations indicate that the convergence of the CC hierarchy is slower for open-shell than for closed-shell systems.
Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction
* Corresponding author. Fax: +45-861-961-99.
E-mail address: jeppe@chem.au.dk (J. Olsen).
The last decade has witnessed significant improvements
in the reliability of ab initio quantum chemical
predictions of spectroscopical and thermochemical data.
For closed shell molecules, equilibrium geometries [1],
harmonic frequencies [2] and reaction enthalpies [3,4]
may often be calculated with an accuracy that is equal to
or better than the experimental accuracy. Of central
importance for this development has been the developments
of hierarchies of basis sets [5], and CC methods
[6–8]. The coupled cluster (CC) method mostly used for
accurate calculations is the CCSD(T) method [9] which
augments the CC method including single and double
excitations (CCSD) [10] with a perturbative estimate of
triples contributions. For closed shell molecules, the
CCSD(T) method often exaggerates the contributions
from triple excitations [11]. As the signs of the triple and
quadruple corrections usually are identical, CCSD(T)
often gives results that are better than the CC method
including all single, double, and triple excitations
(CCSDT). The CCSD(T) method therefore often provides
results in surprisingly good agreement with the
much more expensive CC method including up to quadruple
excitations (CCSDTQ) [12]. Using triple-f basis
sets, the CCSD(T) method is especially accurate for
properties like internuclear distances and frequencies, as
the remaining basis-set errors and correlation errors
here usually are of opposite signs [1].
For open-shell molecules, CC methods with and
without spin-adaptation have been developed [7,13], and
the accuracy of CC calculations often matches the accuracy
obtained for closed shell molecules. In a study of
the atomization energies of 11 small molecules [2], Feller
and Sordo did not observe any systematic difference
between the accuracies obtained for closed- and openshell
molecules when the CCSDT method is used. The
performance of methods including perturbative estimates
of triple excitations as the CCSD(T) method is
less convincing for open-shell molecules. In a systematic
study of the performance of the CCSD(T) method for
the calculation of spectroscopical constants for 33 small
radicals [14], it was observed that the CCSD(T) method
did not provide constants that were significant more
accurate than those obtained with the CCSD method.
Several workers have suggested other methods combining
CCSD with the perturbative treatment of triple
0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.cplett.2004.06.001

L. Thøgersen, J. Olsen / Chemical Physics Letters 393 (2004) 36–43 37
excitations, but these alternative corrections do not
systematically perform better than the CCSD(T) method
[15].
The Schr€odinger equation within the Born–Oppenheimer
approximation may be solved in a given oneelectron
basis set using full configuration interaction
(FCI) calculations. In an FCI calculation, the wave
function includes all Slater determinants with correct
spin, symmetry and number of electrons. For a given
basis-set, FCI calculations eliminate the error due to
truncation of the many-electron basis, and provide
therefore important benchmarks for approximate orbital-based
methods. As the number of determinants in
the FCI expansion increase exponentially with the
number of basis functions and electrons, FCI calculations
may only be carried out for small molecules using
basis sets of double- or triple-f quality. For small closed
shell molecules, a number of FCI calculations have been
published [16,17], and these have given additional insight
into the accuracy of standard correlation methods.
For open-shell molecules, the number of FCI calculations
is more limited. Except for a recent FCI investigation
of the geometry of the CCH radical [18], no FCI
calculations have been published for open-shell molecules
with eight or more valence electrons using a correlation-consistent
basis-set [5]. The present study fills
this gab by providing an FCI benchmark for the openshell
molecule CN using the cc-pVDZ basis [5]. This
molecule is sufficiently small to allow FCI calculations
at numerous geometries, allowing the determination of
the FCI results for the equilibrium bond length, harmonic
frequency, and dissociation energy, as well as the
complete potential curve. We will furthermore study the
convergence of the CC energy as a function of the excitation-level
to see if an open-shell molecule exhibits the
same convergence pattern as previously determined for
closed-shell molecules [19–23]. The vertical electron affinity
will also be examined using CC and FCI calculations.
As the cc-pVDZ basis does not provide accurate
geometries or energetics [8], we will obtain the equilibrium
geometry, harmonic frequency and dissociation
energy using the cc-pVTZ basis set [5] and CC calculations
including up to quadruple excitations. We hope
that the data obtained here will assist in the analysis of
the accuracy of various open-shell perturbation and CC
methods, and especially the methods supplementing
CCSD with perturbative estimates of triple excitations.
2. Computational methods
The FCI and CC calculations were carried out using
the LUCIA
program [24]. The algorithms for performing
configuration interaction calculations are based on extensive
modifications of the algorithms originally published
in [25]. The CC code allows arbitrary excitation
levels out from a single closed shell or high-spin open
shell determinant. In contrast to the initial general CC
codes [19], the present codes [26] exhibit the same scaling
as the standard spin–orbital codes using explicitly coded
contractions. Another set of general CC codes with the
right scaling has been developed by Kallay and coworkers
[20,21], and a less efficient general CC code has
been developed by Hirata and Bartlett [22].
All calculations kept the lowest two sigma-orbitals,
corresponding to 1s(C) and 1s(N), doubly occupied. The
open-shell configuration interaction and CC calculations
used orbitals from restricted Hartree–Fock calculations.
No spin-adaptation was done in the open-shell
CC calculations. The integrals and HF-orbitals were
obtained using the DALTON
program [27].
In the following, the different spaces of determinants
or excitations are denoted SD, SDT, SDTQ, SDTQ5,
SDTQ56, SDTQ567 for the spaces including up to
2,3,4,5,6,7 excitations from the occupied spin–orbitals.
For open-shell molecules, an alternative way of classifying
excitations is to consider changes in orbital-occupations
instead of spin–orbital occupations [28]. All CI
calculations in the following are based on changes of
orbital-occupations, whereas we will discuss CC calculations
based on both divisions of excitations. Excitation
spaces based on changes of spin–orbital occupations will
be denoted (spin–orb), whereas the spaces based on
changes of orbital occupations will be denoted (orb).
Thus, the CCSD(spin–orb) excitation space contains all
single and double spin–orbital excitations.
Using the cc-pVDZ basis FCI, CI and CC calculations
were carried out. To examine the contributions
from quadruple excitations in a larger basis, CCSD,
CCSDT, and CCSDTQ calculations were performed
with the cc-pVTZ basis. For calculations of the electron
affinity, the aug-cc-pVDZ [29] basis set without diffuse
d-functions was used for CN and CN . The latter basis
is in the following called the aug 0 -cc-pVDZ basis.
3. Results
3.1. Convergence of CC and CI at the experimental
equilibrium geometry
At the experimental equilibrium distance (1.1718 A)
[30], the FCI wave function and energy was obtained
with an energy convergence threshold of 10 9 E h . The
FCI energy was obtained as )92.493262415 E h . At the
same internuclear distance, single reference CI and CC
energies were obtained with excitation levels from 2 to 7.
In Table 1, we give the deviations of the CI, CC(orb)
and CC(spin–orb) energies from the FCI energy. Fig. 1
is a single-logarithmic plot of these deviations.
The coupled-cluster energies using orbital-occupations
to define the excitation level are slightly below the

38 L. Thøgersen, J. Olsen / Chemical Physics Letters 393 (2004) 36–43
Table 1
Deviations of single reference CI- and CC-energies (E h ) from the FCI energy for CN
Largest exc. level E CI E FCI E CC ðorbÞ E FCI E CC ðspin–orbÞ E FCI
2 0.038240 0.015534 0.016517
3 0.022604 0.001563 0.001637
4 0.002391 0.000207 0.000230
5 0.000583 0.000019 0.000021
6 0.000031 0.000001 0.000002
7 0.000002 – –
0.1
0.01
Coupled Cluster(spin-orb)
Coupled cluster(orb)
Configuration Interaction
Deviation from FCI energy
0.001
0.0001
1e-05
1e-06
2 3 4 5 6
Excitation level
Fig. 1. The deviations (E h ) of CI and CC energies from the FCI energy as a function of excitation level for CN using the cc-pVDZ basis set.
energies using the smaller spaces based on spin–orbital
occupations. However, the differences between the two
choices are not significant compared to the deviations
from the FCI energy. Up to CCSDTQ5, the differences
between the two forms constitute at most 10% of the
deviation from the FCI energy. For the CCSDTQ56
expansions, the large difference between the two deviations
in Table 1 is caused by roundoff errors. Including
an additional digit, the CCSDTQ56 deviations are
0.0000015 and 0.0000013 E h for the spin–orbital and
orbital based divisions, respectively.
The CI-curves exhibit the behavior predicted by
perturbation theory [31]: the even-order excitations give
significantly larger reductions in the deviations than the
odd-order excitations. For CC expansions, perturbation
theory also predicts that adding even order excitations
give larger reductions in the deviations than adding odd
order excitations [8,31]. This is not observed in Fig. 1, as
the deviations of the CC energies nearly form straight
lines. Comparing the convergence of the CI and CC
hierarchies, it is observed that the CCSDT deviation is
slightly smaller than the CISDTQ error, and that the CC
energy obtained using up to n-fold excitations is as accurate
as the CI energy using up to n þ 1 fold excitations,
but less accurate than the CI energy using up to
n þ 2 fold excitations. To obtain an accuracy of 1 mE h
or less, one must include up to quadruple excitations for
the CC expansion, and up to quintuple excitations for
the CI expansion.
The convergence patterns for CI and CC discussed
above are very similar to the convergence patterns previously
reported for N 2 [23]. The similarity between the
convergences of CN and N 2 is more than qualitative. If
one combines the deviation curve for N 2 [23] with the
present deviation curve for CN in a single figure, the two
deviation curves are virtually identical. The deviations
of the CCSDT energies are thus 0.00156 E h and 0.00163
E h for CN and N 2 , respectively, and for a given excitation
level the deviations for CN and N 2 differ by at
most 10%.

L. Thøgersen, J. Olsen / Chemical Physics Letters 393 (2004) 36–43 39
From the above comparisons, it may be concluded,
that the open-shell nature of CN does not lead to slower
convergence of the CC hierarchy than previously observed
for N 2 . However, it should be noted, that the
convergence of the CC hierarchy for N 2 is rather slow
compared to the convergence observed for e.g., H 2 O [19]
and F 2 .
3.2. The potential curve for CN
FCI calculations were carried out at a number of
internuclear distances. To obtain accurate spectroscopic
constants, the CC energies were converged to 10 9 E h
Table 2
FCI energies (E h ) as a function of internuclear distance ( A) for CN
using the cc-pVDZ basis
R E R E
0.9 )92.169732 1.2118 )92.494065103
1.0 )92.384032 1.2169 )92.493765608
1.0918 )92.469313943 1.2369 )92.491837833
1.1318 )92.485677652 1.2518 )92.489691918
1.1518 )92.490432414 1.30 )92.479361
1.1569 )92.491327096 1.40 )92.447147
1.1718 )92.493262415 1.50 )92.408657
1.1769 )92.493704946 1.60 )92.370048
1.1869 )92.494267979 1.7577 )92.316388
1.1918 )92.494402963 2.05065 )92.255688
1.1919 )92.494404785 2.3436 )92.241450
1.1969 )92.494449358 2.9295 )92.240346
1.2019 )92.494404774 3.5154 )92.239697
1.2069 )92.494274026
for internuclear distances close to the experimental value.
For the remaining geometries the energy was converged
to 10 6 E h . The obtained FCI energies are listed
in Table 2. The graph of the FCI potential curve is given
in Fig. 2.
To associate the various internuclear distances with a
degree of bond-breaking it is useful to examine the coefficient
of the Hartree–Fock determinant in the FCI
wave-function. Around the equilibrium geometry, the
weight of the HF-determinant is about 0.92. Increasing
the internuclear distance leads to a steady lowering of
this weight and at 1.3 and 1.8 A, the weights are 0.79
and 0.57, respectively. From 1.8 to 2.0 A the weight
drops sharply so the weight at 2.0 A is 0.25 and at 2.5 A
less than 0.04. We may therefore say that the bond is
half broken at 1.8 A and broken at 2.5 A.
In addition to FCI calculations, CCSD(orb),
CCSDT(orb) and CCSDTQ(orb) calculations were
performed at the various internuclear distances up to 1.8
A. Although it is possible to converge the CC equations
for larger distances, we find this of less interest, due to
the breakdown of the single-reference approximation. In
Fig. 3, we plot the deviations of the CCSDT and
CCSDTQ energies from the FCI energy, and in Table 3,
we list the non-parallelity error (NPE), i.e., the difference
between the largest and smallest deviation from the
FCI energy.
At the equilibrium distance, both deviation curves in
Fig. 3 have a positive curvature. For internuclear distances
larger than the equilibrium distance, both the
CCSDT and CCSDTQ deviation curves are nearly
-92.15
-92.20
-92.25
FCI energy
-92.30
-92.35
-92.40
-92.45
-92.50
0.5 1 1.5 2 2.5 3 3.5 4
Internuclear distance
Fig. 2. The FCI potential curve for CN using the cc-pVDZ basis. The energies are in Hartrees and the inter-nuclear distances are in A.

40 L. Thøgersen, J. Olsen / Chemical Physics Letters 393 (2004) 36–43
0.007
0.006
CCSDT
CCSDTQ
Deviation from FCI energy
0.005
0.004
0.003
0.002
0.001
0
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
Internuclear distance
Fig. 3. The difference between the CCSDT and CCSDTQ energies and the FCI energy for CN using the cc-pVDZ basis. The energies are in Hartrees
and the inter-nuclear distances are in A.
Table 3
Non-parallelity error (NPE) (E h ) for CCSD, CCSDT, and CCSDTQ
Method
NPE
CCSD 0.042326
CCSDT 0.006355
CCSDTQ 0.001742
linear functions of the internuclear distance. Actually,
the slope of the CCSDT deviation is smaller for larger
internuclear distances than for the equilibrium distance.
The analogous CCSDT- and CCSDTQ-curves for the
nitrogen molecule exhibit maxima for an internuclear
distance around 1.5 A, (3 au) [23].
3.3. Spectroscopical constants for CN
Equilibrium geometries and harmonic frequencies
were obtained for the CCSD, CCSDT, CCSDTQ and
FCI methods using quartic interpolation of the energies.
The harmonic frequency for a given method was evaluated
at the equilibrium geometry of this method. In
Table 4 we list the obtained equilibrium distances and
frequencies. In addition, the table contains the CCSD,
CCSDT and CCSDTQ results for the cc-pVTZ basis.
We will first discuss the results obtained using the ccpVDZ
basis. The CC calculations using orbital-based
excitation spaces are slightly more accurate than those
using spin–orbital-based excitation spaces, but the differences
are small compared to the size of the deviations.
We will therefore, discuss only the spin–orbital based
Table 4
Equilibrium distance ( A) and harmonic frequency (cm 1 ) for CN
CCSD(orb) cc-pVDZ 1.1860 2111
CCSDT(orb) cc-pVDZ 1.1946 2043
CCSDTQ(orb) cc-pVDZ 1.1964 2025
CCSD(spin–orb) cc-pVDZ 1.1855 2114
CCSDT(spin–orb) cc-pVDZ 1.1944 2046
CCSDTQ(spin–orb) cc-pVDZ 1.1964 2026
FCI cc-pVDZ 1.1969 2020.1
CCSD(spin–orb) cc-pVTZ 1.1688 2136
CCSDT(spin–orb) cc-pVTZ 1.1783 2067
CCSDTQ(spin–orb) cc-pVTZ 1.1804 2045
Expt. 1.1718 2069
excitation spaces. Since the deviation curves for the CC
energies are increasing functions, the CC equilibrium
distances are necessarily shorter than the FCI equilibrium
distance. The causes of the errors of the harmonic
frequencies will be discussed in detail below. At the
CCSD level, the distance is 0.01 A shorter than the FCI
value and the harmonic frequency is about 90 cm 1
larger than the FCI value, stressing the inaccuracy of
this method for predicting equilibrium properties. The
errors are significantly reduced by the CCSDT method
with errors of 0.0025 A and 26 cm 1 for the equilibrium
distance and frequency, respectively. The errors are
further reduced by about a factor of five by using the
CCSDTQ instead of the CCSDT method. At the
CCSDTQ level, the equilibrium geometry is only 0.0005
R eq
x e

L. Thøgersen, J. Olsen / Chemical Physics Letters 393 (2004) 36–43 41
A smaller than the FCI value, but the frequency is 6
cm 1 too large. The deviations of the various CC
methods obtained here for CN are very similar to the
previously obtained deviations for N 2 . Thus, it has
previously been reported that the contribution from
connected quadruple excitations to the harmonic frequency
for this molecule [23,32] is 20 cm 1 .
It is currently not feasible to obtain FCI energies for
CN in the cc-pVTZ basis with an accuracy that is
sufficient to obtain the frequency with an accuracy of
1cm 1 or less. One can instead estimate the convergence
by examining the changes in the constants through the
CC hierarchy. It is seen from Table 4 that the changes
between the CCSDT and CCSDTQ results are very
similar in the cc-pVDZ and cc-pVTZ basis sets. In both
basis sets, the quadruple excitations increase the distance
by 0.0020 A and reduce the harmonic frequency
by about 20 cm 1 . This suggests that it may be feasible
to obtain the quadruple corrections to these constants in
rather small basis sets. It should be noted that although
the quadruple corrections to the properties are rather
constant, the quadruples corrections to the raw energies
are very different in the two basis sets.
The errors of the harmonic frequencies arise from
two sources. First of all, the positive curvatures of the
CC deviation curves around the equilibrium geometries
lead to CC frequencies that are larger than the FCI
frequency. Furthermore, as the third derivative of the
energies with respect to the distance in general is large
and negative, the somewhat shorter internuclear distances
obtained with the CC methods than with FCI
lead also to frequencies that are too large. These two
sources of errors may be analyzed in the cc-pVDZ basis
by evaluating the CC frequencies at the FCI equilibrium
geometry. For the orbital based methods one then obtains
the frequencies 2035, 2027 and 2022 cm 1 for the
CCSD, CCSDT and CCSDTQ methods. Whereas, the
CCSDT frequency evaluated at the optimized CCSDT
distance deviates from the FCI frequency by 23 cm 1 ,
the CCSDT frequency evaluated at the FCI geometry
thus deviates by only 7 cm 1 . Although the errors connected
with the positive curvatures of the deviation
curves are not vanishing, the major errors of the frequencies
seem to arise from the errors of the equilibrium
distances.
The experimental values for the equilibrium distance
and the harmonic frequency are 1.1718 A and 2069
cm 1 , respectively, [30]. Comparing the results obtained
using the cc-pVTZ basis to the experimental values, it is
observed that the CCSDT results are in better agreement
with experiment than the CCSDTQ results. A
better estimate of the importance of the quadruples
corrections may be obtained using CCSDT results for
large basis sets. Feller and Sordo [2] have calculated the
CCSDT spectroscopic constants for CN using the augcc-pVQZ
basis and obtained the equilibrium distance
1.1739 A and the harmonic frequency of 2082 cm 1 .
Adding our quadruples correction to these CCSDT results
gives an equilibrium geometry of 1.1759 A and a
harmonic frequency of 2060 cm 1 . To obtain spectroscopic
constants that are significantly more accurate
than the CCSDT results, other corrections, most important
core-correlation contributions, must be included
together with the quadruple excitations.
3.4. Atomization energy
It has previously been reported that quadruple and
even quintuple excitations may be important to obtain
atomization energies with high accuracy [3,4,12] In
Table 5, we list the atomization energies using the
CCSD, CCSDT, CCSDTQ, and FCI approaches with
the cc-pVDZ basis and the CCSD, CCSDT, and
CCSDTQ approaches with the cc-pVTZ basis set. All
molecular calculations were carried out at the experimental
equilibrium distance.
It is again noticed that there are no significant difference
between the results obtained using the CC(orb)
and CC(spin–orb) approaches. The two approaches
differ by only 0.1 kJ/mol at the CCSDT and CCSDTQ
levels.
The quadruple excitations change the atomization
energy by 4 kJ/mol with both the cc-pVDZ and the ccpVTZ
basis sets. These results are in agreement with
previous calculations of the contributions from connected
quadruple excitations [4]. From the difference
between CCSDTQ and the FCI atomization energy, it is
seen that the quintuple excitations contribute 0.5 kJ/mol
to the atomization energy. The above contribution from
quadruple and quintuple excitations are very similar to
the results previously reported for N 2 [3]. The contribution
from higher excitations to the atomization energy
of CN has previously been studied by Feller and Sordo
[2]. They obtained a significantly smaller contribution
from quadruple excitations, 0.3 kcal/mol or 1.2 kJ/mol.
There are several experimental measurements of the
atomization energies, and Feller and Sordo [2] quotes
Table 5
The electronic contribution to the dissociation energy (kJ/mol) for CN
CCSD(orb) cc-pVDZ 631.6
CCSDT(orb) cc-pVDZ 663.0
CCSDTQ(orb) cc-pVDZ 666.5
CCSD(spin–orb) cc-pVDZ 629.2
CCSDT(spin–orb) cc-pVDZ 662.9
CCSDTQ(spin–orb) cc-pVDZ 666.4
FCI cc-pVDZ 667.0
CCSD(spin–orb) cc-pVTZ 674.2
CCSDT(spin–orb) cc-pVTZ 714.4
CCSDTQ(spin–orb) cc-pVTZ 718.5
D e

42 L. Thøgersen, J. Olsen / Chemical Physics Letters 393 (2004) 36–43
values in the range 745–762 kJ/mol for the experimental
electronic contribution. Adding our estimate of quadruples
correction to the estimated CCSDT limit of 748
kJ/mol result of Feller and Sordo gives a value of 752 kJ/
mol for the electronic atomization energy for CN.
3.5. The vertical electron affinity
An FCI calculation for the CN anion using the aug 0 -
cc-pVDZ basis was carried out at the experimental
equilibrium geometry of the radical. The FCI calculation
contains about 20 billion Slater determinants and
sparsity of the CI-vectors was only used to reduce discstorage,
not computation time. This FCI calculation
represents one of the largest FCI calculations we hitherto
have carried out. The FCI energy for the anion was
obtained as )92.627391(2) E h . Combining this energy
with the FCI energy of )92.497766 E h for the radical in
the same basis set leads to an FCI value of 0.12962 E h
for the vertical electron affinity. CC expansions using
spin–orbital occupations for restrictions of excitations
were also carried out for the radical and the anion in the
aug 0 -cc-pVDZ basis and the resulting electron affinities
are given in Table 6.
As the differences between the CC calculations using
orbital and spin–orbital restrictions already have been
shown to be small, no orbital-restricted calculations
were carried out. Already at the CCSD level, the calculated
electron affinity differs from the FCI affinity by
less than 1 mE h , and at the CCSDT level the calculated
electron affinity differs from the FCI result by less than
0.1 mE h . The deviations of the CC energies from the
FCI energies for the radical and the anion are also listed
in Table 6. It is seen that the high accuracy of the CC
affinities is caused by cancellation of the errors of the
radical and anion – the deviation of the affinity is
roughly an order of magnitude smaller than the deviation
of the individual energies. It is also interesting to see
that the electron affinity converges from above – the CC
affinities are larger than the FCI affinity. As seen from
the other columns of Table 6, the CC expansion converges
slightly faster for the anion than for the radical.
The faster convergence of the anion may seem surprising
as the anion contains one more electron than the radical
but is probably caused by CN being slightly more
multiconfigurational than the anion. The electron affinity
of CN calculated using CC calculations in large
basis sets has been the subject of several recent studies
[33,34]. These studies also found small contributions to
the electron affinity from triple excitations.
4. Conclusion
Full configuration interaction calculations using the
cc-pVDZ basis and CC calculations using the cc-pVDZ
and cc-pVTZ basis sets have been carried out for the CN
radical at various geometries. Single reference configuration
interaction calculations were also carried out
using the cc-pVDZ basis at the experimental internuclear
distance. At the CCSDT level, the energies differ
from the FCI energy by 1.5 mE h , and at the CCSDTQ
level, the energies are 0.2 mE h from the FCI energy. The
CC energies converge toward the FCI energy in an approximately
linear fashion with a decrease in the deviation
by about a factor of 10 for each added excitation
level. This is in contrast to an analysis based on perturbation
theory, predicting that adding even orders
give larger decreases in the deviations than adding odd
orders. The observed convergence for CN in the ccpVDZ
basis is very similar to the convergence previously
reported for N 2 , indicating that the open-shell nature of
CN does not affect the convergence. A comparison of
the FCI and CC energies at various internuclear distances,
reveals that the deviations of the CC approaches
do not occur suddenly for large internuclear distances.
The deviations are instead nearly linear functions of the
internuclear distance.
At the FCI level, the equilibrium geometry and harmonic
frequency are obtained as 1.1969 A and 2020.1
cm 1 , respectively. The CCSDT and CCSDTQ frequencies
are 25 and 5 cm 1 above the FCI value, respectively.
The quadruple corrections to both the
equilibrium distance and the harmonic frequency were
found to be nearly identical in the cc-pVDZ and ccpVTZ
basis sets. The major errors of the CC frequencies
come from the errors of the distances where these are
evaluated.
For the electronic contribution to the atomization
energy, a value of 667.0 kJ/mol is obtained at the FCI
level using the cc-pVDZ basis set. The CCSDT and
CCSDTQ atomization energies are 4 and 0.5 kJ/mol
below the FCI atomization energy, respectively. The
quadruple contributions in the cc-pVDZ and cc-pVTZ
Table 6
The vertical electron affinity (E h ) of CN calculated in the aug 0 -cc-pVDZ basis
EA EA EA FCI E CN EFCI CN
E CN –EFCI
CN
CCSD(spin–orb) 0.13025 0.00063 0.01529 0.01466
CCSDT(spin–orb) 0.12977 0.00014 0.00154 0.00140
CCSDTQ(spin–orb) 0.12966 0.00003 0.00020 0.00016
FCI 0.12962

L. Thøgersen, J. Olsen / Chemical Physics Letters 393 (2004) 36–43 43
basis are determined as 3.5 and 4.1 kJ/mol, respectively,
indicating that a reliable estimate of quadruple contributions
may be obtained using rather small basis sets.
The FCI vertical electron affinity is obtained in the
aug 0 -cc-pVDZ basis as 0.12962 E h . Due to extensive
cancellations of errors, the FCI affinity is accurately
calculated at the CCSD and CCSDT levels with a contribution
from quadruple and higher excitations of
0.00014 E h . The CC hierarchy approaches the FCI affinity
from above, as the deviations for the anion are
slightly smaller than for the radical.
Acknowledgements
The work has been supported by the Danish Research
Council (Grant No. 9901973). The calculations
were carried out at the centre for supercomputing at
University of Aarhus (CSCAA). The support from the
Danish Centre for Supercomputing (DCSC) is gratefully
acknowledged.
References
[1] F. Pawlowski, P. Jørgensen, J. Olsen, F. Hegelund, T. Helgaker,
J. Gauss, K.L. Bak, J.F. Stanton, J. Chem. Phys. 116 (2002) 6482.
[2] D. Feller, J.A. Sordo, J. Chem. Phys. 113 (2000) 485.
[3] T. Helgaker, W. Klopper, A. Halkier, K.L. Bak, P. Jørgensen,
J. Olsen, in: J. Cioslowski, (Ed.), Understanding Chemical
Reactivity, vol. 22, Kluwer, Dordrecht, p. 1, 2001.
[4] A.D. Boese, M. Oren, O. Atasoylu, J.M.L. Martin, M. Kallay,
J. Gauss, J. Chem. Phys. 120 (2004) 4129.
[5] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007.
[6] R.J. Bartlett, in: D.R. Yarkony (Ed.), Modern Electronic Structure
Theory, Part I, 1047, World Scientific, Singapore, 1995.
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[8] T. Helgaker, P. Jørgensen, J. Olsen, Molecular Electronic-Structure
Theory, Wiley, 2000.
[9] K. Raghavachari, G.W. Trucks, J.A. Pople, M. Head-Gordon,
Chem. Phys. Lett. 157 (1989) 479.
[10] G.D. Purvis, R.J. Bartlett, J. Chem. Phys. 76 (1982) 1910.
[11] K.L. Bak, P. Jorgensen, J. Olsen, T. Helgaker, W. Klopper,
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[12] T.A. Ruden, T.U. Helgaker, P. Jørgensen, J. Olsen, Chem. Phys.
Lett. 371 (2003) 62.
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(2001) 9736.
[15] S.R. Gwaltney, M. Head-Gordon, J. Chem. Phys. 115 (2001)
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[16] J. Olsen, P. Jørgensen, H. Koch, A. Balkova, R.J. Bartlett,
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[17] H. Larsen, J. Olsen, P. Jørgensen, O. Christiansen, J. Chem. Phys.
113 (2000) 6677.
[18] P.G. Szalay, L.S. Thøgersen, J. Olsen, M. Kallay, J. Gauss,
J. Phys. Chem. A. 105 (2004) 3030.
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[20] M. Kallay, P.R. Surjan, J. Chem. Phys. 113 (2000) 1359.
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[22] S. Hirata, R.J. Bartlett, Chem. Phys. Lett. 321 (2000) 216.
[23] J.W. Krogh, J. Olsen, Chem. Phys. Lett. 344 (2001) 578.
[24] LUCIA, a general CI and CC code written by J. Olsen, University
of Aarhus with contributions from H. Larsen, M. F€ulscher.
[25] J. Olsen, B.O. Roos, P. Jørgensen, H.J.Aa. Jensen, J. Chem. Phys.
89 (1988) 2185.
[26] J. Olsen, unpublished.
[27] T. Helgaker et al DALTON, an ab initio electronic structure
program, Release 1.2. see http://www.kjemi.uio.no/software/dalton/dalton.html,
2001.
[28] X. Li, J. Paldus, J. Chem. Phys. 101 (1994) 8812.
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[32] S.A. Kucharski, J.D. Watts, R.J. Bartlett, Chem. Phys. Lett. 302
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[34] J.A. Sordo, J. Chem. Phys. 114 (2001) 1974.

Part 3
Equilibrium Geometry of the Ethynyl (CCH) Radical,
P. G. Szalay, L. Thøgersen, J. Olsen, M. Kállay and J. Gauss,
J. Phys. Chem. A 108, 3030 (2004).

3030 J. Phys. Chem. A 2004, 108, 3030-3034
Equilibrium Geometry of the Ethynyl (CCH) Radical †
Péter G. Szalay, ‡ Lea S. Thøgersen, § Jeppe Olsen, § Mihály Kállay, | and Ju1rgen Gauss* ,|
Department of Theoretical Chemistry, EötVös Loránd UniVersity, H-1518 Budapest, P.O. Box 32, Hungary,
Department of Chemistry, Aarhus UniVersity, DK-8000 Aarhus C, Denmark, and Institut für Physikalische
Chemie, UniVersität Mainz, D-55099 Mainz, Germany
ReceiVed: September 27, 2003; In Final Form: January 15, 2004
The equilibrium geometry of the ethynyl (CCH) radical has been obtained using the results of high-level
quantum chemical calculations and the available experimental data. In a purely quantum chemical approach,
the best theoretical estimates (1.208 Å for r CC and 1.061-1.063 Å for r CH ) have been obtained from CCSD-
(T), CCSDT, MR-AQCC, and full CI calculations with basis sets up to core-polarized pentuple-zeta quality.
In a mixed theoretical-experimental approach, empirical equilibrium geometrical parameters (1.207 Å for
r CC and 1.069 Å for r CH ) have been obtained from a least-squares fit to the experimental rotational constants
of four isotopomers of CCH which have been corrected for vibrational effects using computed vibrationinteraction
constants. These geometrical parameters lead to a consistent picture with remaining discrepancies
between theory and experiment of 0.001 Å for the CC and 0.006-0.008 Å for the CH distances, respectively.
The corresponding r s and r 0 geometries are shown not to be representative for the true equilibrium structure
of CCH.
I. Introduction
Considerable effort has been devoted to the determination
of the structure of the ethynyl (CCH) radical in its 2 Σ + electronic
ground state from the experimental 1 and the theoretical side. 2-7
Presently, experimental values for ground-state rotational constants
(B 0 ) for four isotopomers of CCH have been determined.
For CCH, a value of 43 674.528 94(115) MHz has been reported
by Müller et al. 8 in agreement with earlier measured values. 9-11
For 13 CCH and C 13 CH, values of 42 077.462(1) and 42 631.382-
(1) MHz have been obtained by McCarthy et al. 12 in excellent
agreement with a previous report of Bogey et al. 1,13 Finally,
for the deuterated form CCD, a value of 36 068.0310(96) MHz
has been reported by Bogey et al. 14
On the basis of the available experimental rotational constants,
Bogey et al. 1 determined a so-called substitution (r s ) structure.
However, the obtained bond distances are not in satisfactory
agreement with corresponding calculated equilibrium values; 2-7
in particular, the CH distance was unusually short (1.046 Å vs
calculated values of 1.062-1.070 Å). As has been already
pointed out by Bogey et al., 14 the observed discrepancy is
probably due to the large amplitude bending motion in CCH
which is not adequately accounted for in the substitution
approach 15 that provides the r s structure. Thus, determination
of the true equilibrium geometry is necessary to get a reliable
picture of the structure of the ethynyl radical.
Although the available rotational constants form a solid basis
for the experimental determination of the r 0 and r s geometry,
respectively, there is not enough experimental information
available to determine the equilibrium geometry. In particular,
the vibrational contributions to the rotational constants, which
in principle can be determined via the complete set of vibrationrotation
interaction constants, 16 cannot be obtained from the
available experimental data.
† Part of the special issue “Fritz Schaefer Festschrift”.
‡ Eötvös Loránd University.
§ Aarhus University.
| Universität Mainz.
As has been suggested long ago by Pulay et al. 17 and more
recently by others, 18,19 quantum chemical calculations can be
used to provide the lacking information. With computed
vibration-rotation interaction constants (R r ), it is possible to
correct experimental rotation constants for vibrational effects
and to obtain the corresponding equilibrium values
B e ) B 0 + 1 ∑ R r (1)
2 r
with the sum running over all vibrational degrees of freedom.
The accuracy of such a mixed experimental-theoretical (or
empirical) procedure for the determination of equilibrium
geometries has recently been investigated by Pawlowski et al. 20
for a set of 18 closed-shell molecules. It was concluded in this
study that errors in the determined empirical bond lengths are
below 0.001 Å, if the vibrational corrections to the rotational
constants are calculated at a sufficiently high level such as the
coupled-cluster singles and doubles (CCSD) level 21 augmented
by a perturbative treatment of triple excitations (CCSD(T)) 22
together with the cc-pVQZ set from Dunning’s correlationconsistent
basis-set hierarchy. 23 Although it is not clear whether
the same accuracy can be achieved for open-shell systems, this
combined experimental-theoretical procedure opens an interesting
possibility for the determination of a reliable equilibrium
geometry for CCH.
Alternatively, accurate equilibrium geometries can be obtained
via a purely theoretical approach. Such an approach can and
should take advantage of existing hierarchies of methods for
the treatment of electron correlation and establish basis-set
convergence by using basis-set sequences such as, for example,
the correlation-consistent sets developed by Dunning and coworkers.
23,24 As has been shown by Helgaker et al. 25 and more
recently also by Bak et al. 26 such a procedure can lead to an
accuracy of 0.002-0.003 Å in bond distances if CCSD(T)
calculations together with sufficiently large basis sets are carried
out. Again, this conclusion is mainly valid for closed-shell
10.1021/jp036885t CCC: $27.50 © 2004 American Chemical Society
Published on Web 02/17/2004

Equilibrium Geometry of Ethynyl Radical J. Phys. Chem. A, Vol. 108, No. 15, 2004 3031
molecules and needs to be checked for open-shell systems, for
which some further complications are expected. 27,28 Concerning
the use of multireference methods, a recent study on more than
60 electronic (closed- and open-shell) states of various diatomic
molecules found that approaches such as, for example, the
multireference-averaged quadratic coupled-cluster (MR-AQCC)
method, 29,30 provide bond distances with an accuracy close to
0.001 Å. As multireference methods together with a careful
selection of the reference space offer a well-balanced treatment
for both open- and closed-shell molecules, such calculations
should be considered useful complements to single-referencebased
CC calculations.
The aim of the present paper is to provide an accurate
equilibrium geometry for the electronic ground state of the
ethynyl radical by using both procedures outlined above. The
accuracy and reliability of the theoretically determined values
will be carefully investigated via benchmark calculations up to
the full configuration interaction (FCI) level. Calculated vibrational
corrections to the rotational constants are used to derive
equilibrium geometrical parameters from the available experimental
rotational constants. The accuracy achieved is judged
by a comparison of the results obtained with the two procedures.
II. Computational Methods
Theoretical determinations of the equilibrium geometry of
CCH have been carried out using various coupled-cluster (CC)
approaches and, to investigate possible multireference effects,
the multireference configuration interaction (MR-CI) and multireference-averaged
quadratic coupled-cluster (MR-AQCC)
methods.
Using the CC ansatz, calculations have been performed at
two levels beyond the coupled-cluster singles and doubles
(CCSD) 21 approximation, namely, at the CCSD(T) level which
includes connected triple excitations perturbatively on top of a
CCSD calculation 22,31 and at the CCSDT level 32-34 which
includes a full treatment of triple excitations. Both unrestricted
Hartree-Fock (UHF) and restricted open-shell Hartree-Fock
(ROHF) reference functions have been used in the CC calculations.
The MR-AQCC method can be considered an approximately
“extensive” version of the MR-CISD (multireference configuration
interaction with single and double excitations) method.
MR-AQCC and MR-CISD calculations have been carried out
with different reference (active) spaces. The n e factor in the
MR-AQCC calculations was chosen to be 9, that is, the core
electrons are not considered in the size-extensivity correction
(for details, see ref 30).
The hierarchy of correlation-consistent basis sets cc-pVXZ 23
and cc-pCVXZ 24 has been used with X ) D,T,Q, and 5.
Since the size of CCH renders FCI calculations with small
basis sets possible, FCI calculations (with a restricted openshell
HF reference) have been carried out for the geometry of
CCH employing the cc-pVDZ basis sets. These benchmark
results are used to calibrate the corresponding CC and MR-
AQCC results.
Geometry optimizations have been carried out with analytically
evaluated gradients in the case of the CCSD(T) 31,35-37 and
MR-AQCC calculations, 38,39 while in all other cases the
equilibrium geometry has been determined using purely numerical
methods.
The vibration-rotation interaction constants which are needed
to subtract the vibrational contribution from the experimental
rotational constants have been obtained at the UHF-CCSD(T)
and ROHF-CCSD(T) levels using cc-pVTZ, cc-pCVTZ, ccpVQZ,
and cc-pCVQZ basis sets 23,24 at the geometry optimized
at the same level. The required quantities (for the relevant
computational expressions, see, for example, ref 16) have been
determined using analytic derivative techniques, that is, the
harmonic force field was determined using either analytic
gradients (ROHF-CCSD(T)) 31 or analytic second derivatives
(UHF-CCSD(T)), 40,41 and the cubic force field has been
subsequently determined via numerical differentiation as described
in refs 19 and 42. In addition, to check the reliability
of the obtained force fields, UHF-CCSDT calculations of the
vibration-rotation interaction constants (within the frozen-core
approximation) have been carried out employing our recently
implemented general CC analytic second derivatives. 43
CC calculations have been performed with the Austin-Mainz
version of the ACES II program system. 44 The COLUMBUS
suite of programs 39,45 was used for the MR-AQCC and the
LUCIA code 46 for the FCI calculations. The CCSDT force field
calculations have been carried using the generalized CI/CC code
developed by one of us 47-49 which has been interfaced to the
ACES II program.
III. Results and Discussions
III.A. Choice of Reference Space in the Multireference
Treatments. The 2 Σ + ground state of CCH has a dominant
configuration of (1σ) 2 (2σ) 2 (3σ) 2 (4σ) 2 (1π) 4 5σ. An appropriate
reference space for the description of this electronic state within
a MR-AQCC treatment has to be selected in a careful manner.
In the present work, four different reference spaces have been
tested with respect to their performance for the equilibrium
geometry of CCH. In particular, the convergence of the
calculated geometrical parameters with increase of the reference
space is investigated.
The smallest reference space is of complete active space
(CAS) type and denoted by “5 × 5”, indicating that five
electrons are distributed within five orbitals, namely the openshell
5σ, the pairs of the π and π* orbitals (1π and 2π). The
next CAS reference space, denoted by “5 × 6”, considers in
addition the virtual 6σ orbital, while the largest CAS space (“5
× 8”) includes three virtual orbitals (6σ, 7σ, and 8σ). Finally,
to investigate the effect of including further “active” electrons,
the “5 × 6” space has been augmented by single and double
excitations involving the 3σ and/or 4σ orbital (in the following
denoted by “5 × 6 + 2d”). Note that in all considered cases,
the orbitals have been taken from MCSCF calculations using
the same space. All single and double excitations out of the
reference configurations have been included in the correlation
treatment within the MR-CISD and MR-AQCC calculations.
As the focus of these initial calculations is just the convergence
of the results with respect to the chosen reference space, the
calculations have been performed at the cc-pVDZ and cc-pVTZ
basis-set levels, respectively.
TABLE 1: Comparison of Geometrical Parameters (in Å)
for the 2 Σ + State of CCH with Respect to the Chosen
Reference Space in the MR-CISD and MR-AQCC
Treatments a 5 × 5 5 × 6 5 × 8 5 × 6 + 2d
r CC
MR-AQCC/cc-pVDZ (fc) 1.2369 1.2376 1.2379 1.2371
MR-CISD/cc-pVTZ (ae) 1.2093 1.2102 1.2102 1.2123
MR-AQCC/cc-pVTZ (ae) 1.2121 1.2129 1.2131 1.2126
r CH
MR-AQCC/cc-pVDZ (fc) 1.0794 1.0797 1.0807 1.0799
MR-CISD/cc-pVTZ (ae) 1.0546 1.0548 1.0552 1.0558
MR-AQCC/cc-pVTZ (ae) 1.0573 1.0575 1.0580 1.0580
a
fc ) frozen-core calculations, ae ) all-electron calculations.

3032 J. Phys. Chem. A, Vol. 108, No. 15, 2004 Szalay et al.
TABLE 2: Comparison of Geometrical Parameters (in Å) for the 2 Σ + State of CCH as Obtained at the CCSD(T), CCSDT, and
MR-AQCC Levels Using Different Basis Sets a
UHF-
CCSD(T)
ROHF-
CCSD(T)
r CC
UHF-
CCSDT
ROHF-
CCSDT
MR-
AQCC
UHF-
CCSD(T)
ROHF-
CCSD(T)
r CH
UHF-
CCSDT
ROHF-
CCSDT
MR-
AQCC
cc-pVDZ (fc) 1.2318 1.2353 1.2352 1.2354 1.2376 1.0797 1.0801 1.0801 1.0802 1.0797
cc-pVTZ (fc) 1.2120 1.2153 1.2150 1.2151 1.2173 1.0643 1.0646 1.0645 1.0645 1.0638
cc-pVQZ (fc) 1.2081 1.2113 1.2110 1.2110 1.2133 1.0642 1.0645 1.0644 1.0644 1.0635
cc-pV5Z (fc) 1.2072 1.2104 1.2098 1.2123 1.0639 1.0642 1.0642 1.0632
cc-pCVTZ (ae) 1.2087 1.2119 1.2132 1.0642 1.0645 1.0627
cc-pCVQZ (ae) 1.2052 1.2083 1.2096 1.0630 1.0632 1.0613
cc-pCV5Z (ae) 1.2043 1.2074 1.0626 1.0629
a
fc ) frozen-core calculations, ae ) all-electron calculations. b 5 × 6 reference space.
The corresponding results are compiled in Table 1. The most
significant observation is that there is a faster convergence of
the bond distance with increase of the reference space in the
MR-AQCC than in the MR-CISD calculations, as the MR-
AQCC results seem to be much less sensitive to the choice of
reference space. While the optimized bond distances obtained
with the two methods are very close when the largest reference
space (5 × 6 + 2d) is used, there are noticeable differences for
the smaller reference spaces. For these, the MR-AQCC results
are much closer to the “5 × 6 + 2d” values than the
corresponding MR-CISD results. In particular, the inclusion of
additional electrons in the reference space seems to be less
important when using the MR-AQCC ansatz. The results in
Table 1 thus indicate that the use of a “5 × 6” active space
seems to be a safe and economical choice for large-scale MR-
AQCC calculations on the 2 Σ + state of CCH. The remaining
error due to higher excitations is estimated to be about 0.001-
0.002 Å.
III.B. Comparison of MR-AQCC and CC Results. In Table
2 the CC and CH bond lengths obtained at CCSD(T), CCSDT,
and MR-AQCC levels using different basis sets are compared.
Focusing first on the coupled-cluster results, it is observed
that, independent of the chosen basis set, the CC distances
obtained at the UHF-CCSD(T) level are about 0.003 Å shorter
than the corresponding CCSDT values, while the corresponding
ROHF-CCSD(T) bond lengths are essentially identical to both
the UHF- and ROHF-CCSDT values. This unexpected difference
between the UHF and ROHF results is investigated in a
forthcoming article 28 where the failure of UHF-CCSD(T) is
traced back to a rapid change of the underlying UHF wave
function at certain bond distances. It will be shown in ref 28
that this breakdown of the UHF-CCSD(T) approach occurs for
the ethynyl radical at distances close to the equilibrium
geometry, and thus, the UHF-CCSD(T) results must be considered
unreliable. Interestingly, the full CCSDT approach seems
to be able to recover from these deficiencies of the underlying
UHF reference functions and provides results which are essentially
independent of the chosen reference functions.
For the CC distances the differences between ROHF-CCSD-
(T) and CCSDT are essentially negligible. When considering
in addition the MR-AQCC calculations (obtained with the “5
× 6” reference), we note that the MR-AQCC value for the CC
distance is even longer than the corresponding CCSDT value
(by about 0.002 Å). It is essentially impossible at this point to
decide whether the CCSDT or the MR-AQCC results should
be considered more accurate. 50 Good agreement of the ROHF-
CCSD(T) and CCSDT also suggests that ROHF-CCSD(T) can
be safely used with the larger basis sets where CCSDT is not
practical.
For the CH distance, all considered approaches yield essentially
the same result.
TABLE 3: Comparison of Geometrical Parameters (in Å)
for the 2 Σ + State of CCH at the CCSD(T), CCSDT, and
MR-AQCC Levels with Corresponding FCI Calculations a
III.C. Comparison with Full Configuration Interaction
Results. To judge the accuracy of MR-AQCC and CCSDT,
benchmark calculations at the FCI level using the cc-pVDZ basis
have been performed. The corresponding results are summarized
in Table 3. As these results show, the CH bond distances
obtained by any approach are in excellent agreement (differences
are less than 0.0005 Å), while for the CC bond distance the
FCI result falls between the corresponding CCSDT and MR-
AQCC values. This means that in comparison with FCI the
CCSDT value is about 0.001 Å too short, while MR-AQCC is
about 0.001 Å too long. Both methods thus exhibit errors which
are acceptable for our purpose.
III.D. Basis-Set Convergence. After discussing the issue of
electron correlation, we will now turn our interest to the basisset
effects. Results obtained with both the cc-pVXZ and ccpCVXZ
sequence of basis sets have been given in Table 2. In
the cc-pVXZ calculations, when employing the frozen-core
approximation, smooth convergence of the geometrical parameters
is observed. When going from cc-pVDZ to cc-pV5Z, both
bond distances are reduced, the CC distance by about 0.025 Å
and the CH distance by about 0.016 Å. The differences between
the cc-pVQZ and cc-pV5Z results are with 0.001 and 0.0003
Å already rather small so that the cc-pV5Z results can be
considered as nearly converged. However, the cc-pVXZ calculations
do not incorporate core-correlation effects. To consider
these properly, all-electron calculations using the core-valence
correlating cc-pCVXZ sets have been carried out. As for the
cc-pVXZ sequence, monotonic convergence is observed for the
geometrical parameters within this basis-set sequence and the
differences between quadruple- and pentuple-zeta results are
again small. From the results, it is further seen that core
correlation together with the additional consideration of core
polarization functions reduces the CC bond distance by about
0.003-0.004 Å, while the CH distance, as one might expect, is
less affected and shortened by only 0.001-0.002 Å.
Unfortunately, because of program limitations, it was not
possible to perform MR-AQCC calculations using the largest
cc-pCV5Z basis. However, the rather systematic difference
between the CCSD(T) and MR-AQCC results enables a
r CC
r CH
ROHF-CCSD(T) 1.2353 1.0801
UHF-CCSD(T) 1.2318 1.0797
UHF-CCSDT 1.2352 1.0801
ROHF-CCSDT 1.2354 1.0802
MR-AQCC b 1.2376 1.0797
FCI 1.2367 1.0802
a
All calculations with cc-pVDZ and core orbitals frozen in the
electron-correlation treatment. b 5 × 6 reference space.

Equilibrium Geometry of Ethynyl Radical J. Phys. Chem. A, Vol. 108, No. 15, 2004 3033
TABLE 4: Calculated Vibrational Corrections ∆B ) B e -
B 0 (in MHz) to the Rotational Constants of Different
Isotopomers of CCH from UHF- and ROHF-based CC
Calculations
CCSD(T)
cc-pVTZ
CCSD(T)
cc-pVQZ
CCSD(T)
cc-pCVTZ
CCSD(T)
cc-pCVQZ
CCSDT(fc)
cc-pVTZ a
UHF Reference Function
CCH 368.27 334.70 379.67 355.74 583.64
13
CCH 355.08 322.65 366.09 342.76 564.26
C 13 CH 366.25 333.16 377.31 353.24 580.54
CCD 168.07 151.12 175.33 167.52 258.47
ROHF Reference Function
CCH 531.16 479.58 568.24 495.37
13
CCH 513.21 463.24 549.12 478.15
C 13 CH 528.13 476.98 564.57 491.72
CCD 237.85 214.59 257.20 230.11
a
fc ) frozen-core calculation.
prediction of the corresponding value based on MR-AQCC/ccpCVQZ
and ROHF-CCSD(T)/cc-pCV5Z calculations. As the
use of the pentuple- instead of the quadruple-ζ set decreases
CC and CH bond distances by about 0.0009 and 0.0004 Å,
respectively, the estimated MR-AQCC/cc-pCV5Z values are
about 1.2087 and 1.0609 Å.
The influence of diffuse functions has been investigated at
the UHF-CCSD(T) level. It was found that the changes amounts
to less than 0.0003 Å when going from cc-pCVQZ to aug-ccpCVQZ.
III.E. Best Theoretical Estimates. On the basis of the
previous sections, we are now able to give a best theoretical
estimate for the equilibrium geometry of CCH. There are two
(almost) independent procedures: one uses the MR-AQCC data
while the other uses the CC data, respectively. At the MR-
AQCC level, the best directly calculated geometry has been
obtained with cc-pCVQZ basis set (r e (CC) ) 1.2096 Å and r e -
(CH) ) 1.0613 Å). This geometry should be “improved” by
the FCI correction obtained at the cc-pVDZ level, that is, by
-0.0009 and 0.0005 Å as well as corrected for the remaining
basis-set effect, that is, by -0.0009 Å and -0.0004 Å, for CC
and CH, respectively (see above). Assuming additivity of these
corrections, this leads to final values of 1.2078 and 1.0614 Å
for the CC and CH bond distance, respectively. A similar
extrapolation procedure starting from the ROHF-CCSD(T)/ccpCV5Z
results (1.2074 and 1.0628 Å) and employing corrections
due to full CCSDT (-0.0003 Å and -0.0001 Å) and FCI
(0.0013 and 0.0000 Å) leads to a final estimate of 1.2084 and
1.0627 Å for the two distances. The discrepancy of 0.001 to
0.002 Å between the values obtained with these two extrapolation
schemes is an indication for the accuracy of our theoretical
results.
It is noteworthy to mention that our best theoretical estimates
are in excellent agreement with recent recommendations for the
equilibrium geometry of CCH by Peterson and Dunning 7 based
on CCSD(T) calculations. The corresponding values are 1.2076
and 1.0619 Å.
III.F. Analysis of Experimental Rotational Constants.
After establishing a theoretical estimate for the equilibrium
geometry of CCH, we now focus on the analysis of the
experimental rotation constants using computed vibrational
corrections. These corrections to B, that is, ∆B ) B e - B 0 , have
been obtained at the UHF- and ROHF-CCSD(T) level using
the cc-pVXZ and cc-pCVXZ sets with X ) T and Q. The
calculated ∆B values are compiled in Table 4 and amount to
about 150-590 MHz, that is, about 0.5 to 1.5% of the values
of the corresponding rotational constants for the considered
isotopomer and thus are non-negligible. However, large discrepancies
are seen between the vibrational corrections computed
with UHF and ROHF reference functions. We thus
decided to check the reliability of the CCSD(T) force fields
via corresponding CCSDT calculations using the cc-pVTZ basis
set. As is seen from Table 4, the CCSDT calculations suggest
that the UHF-CCSD(T) force fields (as the corresponding
geometries) should be considered unreliable and that only the
ROHF-CCSD(T) approach yields vibrational corrections in good
agreement with the CCSDT approach. On the basis of these
calculations, we refrain from discussing the UHF-CCSD(T)
results any further and solely discuss the corresponding ROHF-
CCSD(T) results in the following.
For the least-squares fit of the geometrical parameters to the
rotational constants, the most recent B 0 values from refs 8, 12,
and 14, as given in the Introduction, have been used together
with the vibrational corrections compiled in Table 4. The
resulting empirical equilibrium geometries are summarized in
Table 5. According to the values reported there, an “empirical”
equilibrium geometry of r CC ) 1.207 Å and r CH ) 1.069 Å can
be given with 0.002 Å as a conservative error estimate 51 based
on the convergence of the results.
A comparison of the empirical equilibrium geometry with
our best theoretical estimates shows that the remaining discrepancies
are in the range of 0.001 to 0.002 Å for the CC and
0.006 to 0.008 Å for the CH distances. It appears that the
empirical value for the CC distance is slightly shorter and the
CH distance is longer than the corresponding theoretical values.
While these discrepancies can possibly be traced back to
remaining deficiencies in the theoretical treatment, another, and
maybe more likely, possibility is that these differences point to
so far unexplored limitations in the perturbational treatment of
the vibrational corrections (note that there is a low-lying Π state
which interacts with the electronic ground state through the
bending motion).
Nevertheless, the current study leads to a satisfactory agreement
between theory and experiment and thus provides a
consistent picture with respect to the equilibrium geometry.
Concerning previous efforts to determine the geometry of
CCH, we note that the r s (as well as the r 0 ) structures are rather
TABLE 5: Comparison of Geometrical Parameters (in Å) for the 2 Σ + State of CCH Obtained from Theory and Experiment
structure r CC r CH method ref
r e 1.2064 1.0678 from exptl B 0 with ∆B(ROHF-CCSD(T)/cc-pVTZ) this work
r e 1.2076 1.0657 from exptl B 0 with ∆B(ROHF-CCSD(T)/cc-pVQZ) this work
r e 1.2056 1.0689 from exptl B 0 with ∆B(ROHF-CCSD(T)/cc-pCVTZ) this work
r e 1.2075 1.0651 from exptl B 0 with ∆B(ROHF-CCSD(T)/cc-pCVQZ) this work
r e 1.2050 1.0703 from exptl B 0 with ∆B(UHF-CCSDT(fc)/cc-pVTZ) this work
r e 1.2078 1.0614 est from MR-AQCC this work
r e 1.2084 1.0627 est from CCSDT this work
r 0 1.2193 1.0457 from exptl B 0 this work
r s 1.21652 1.04653 from exptl B 0 1
r e 1.2076 1.0619 est from CCSD(T) 7

3034 J. Phys. Chem. A, Vol. 108, No. 15, 2004 Szalay et al.
different (compare Table 5). Both of them deviate by about
0.005 Å in the CC and by about 0.015 Å in the CH distance
from the equilibrium geometries obtained in this work. Apparently,
unlike often claimed, the substitution approach leading
to the r s structure is not able to eliminate vibrational effects in
the case of CCH, and thus, the r s and r 0 structure turn out to be
very similar. Our observation supports the speculation in ref 1
that the significantly too short CH distance is due to insufficient
account of vibrational effects, and in particular of the lowfrequency
bending motion, a well-known artifact of the substitution
approach to molecular structures.
IV. Conclusions
Equilibrium geometrical parameters for the 2 Σ + state of the
ethynyl radical have been obtained using two approaches. The
first purely theoretical procedure based on extensive CC, MR-
AQCC, and FCI calculations yields values of 1.208 Å for the
CC distance and 1.061-1.063 Å for the CH distance, while
the second approach based on the analysis of experimental
rotational constants using computed vibrational corrections
provides values of 1.207 and 1.069 Å. The observed differences
between the two approaches of 0.001-0.002 Å for CC and
0.006-0.008 Å for CH are somewhat larger than expected.
Among possible causes for this discrepancy, we consider
limitations in the perturbational treatment of the vibrational
corrections to the rotational constants. The r s and r 0 geometries
for CCH are, because of a missing or insufficient treatment of
these corrections, far away from the true equilibrium geometry.
Acknowledgment. The authors acknowledge fruitful discussions
with Professor J. F. Stanton (University of Texas,
Austin). This work has been supported by the Hungarian
Scientific Research Foundation (OTKA, Grants T032980 and
M042110), the Deutsche Forschungsgemeinschaft, the Fonds
der Chemischen Industrie, and the Danish Centre for Supercomputing
(DCSC). This research is part of an effort by a task
group of the International Union of Pure and Applied Chemistry
to determine structures, vibrational frequencies, and thermodynamic
functions of free radicals of importance in atmospheric
chemistry.
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(50) It should be mentioned here that our MR-AQCC results are in
excellent agreement with previous MR-CI calculations by Peterson and
Dunning. 7 Their best values at the MR-CI level (augmented by a Davidson
correction) using a full valence active space using a pV5Z basis for carbon
and a pVQZ basis for hydrogen of 1.2116 and 1.0643 Å are of comparable
quality as our MR-AQCC/pV5Z(fc) values of 1.2123 and 1.0632 Å.
(51) Note that the residuals in the least-squares fit were in all cases
smaller than 1.5 MHz.