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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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