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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