Essentials of Computational Chemistry

216 7 INCLUDING ELECTRON CORRELATION IN MO THEORY
double excitations using perturbation theory (Head-Gordon et al. 1994; Section 7.4 presents
the basics of perturbation theory); this model is referred to as SF-CIS(D). Preliminary studies
on various challenging problems like homolytic bond dissociation energies and singlet–triplet
energy separations in biradicals have shown SF-CISD and SF-CIS(D) to be considerably
more robust than the corresponding non-spin-flip approaches (Krylov 2001; Krylov and
Sherrill 2002; Slipchenko and Krylov 2002).
7.3.2 Multireference
The formalism for multireference configuration interaction (MRCI) is quite similar to that
for single-reference CI, except that instead of the HF wave function serving as reference,
an MCSCF wave function is used. While it is computationally considerably more difficult
to construct the initial MCSCF wave function than a HF wave function, the significant
improvement of the virtual orbitals in the former case can make the CI itself more rapidly
convergent. Nevertheless, the number of matrix elements requiring evaluation in MRCI
calculations is enormous, and they are usually undertaken only for small systems. Typically,
MRCI is a useful method to study a large section of a PES, where significant changes in
bonding (and thus correlation energy) are taking place so a sophisticated method is needed
to accurately predict dynamical and non-dynamical correlation energies.
As with single-reference CI, most MRCI calculations truncate the CI expansion to include
only singles and doubles (MRCISD). An analog of Eq. (7.18) has been proposed to make
up for the non-size-extensivity this engenders (Bruna, Peyerimhoff, and Buenker, 1980).
MRCISD calculations with large basis sets can be better than similarly expensive full CI
calculations with smaller basis sets, illustrating that most of the correlation energy can be
captured by including only limited excitations, at least in those systems small enough to
permit thorough evaluation. Additional efficiencies can be gained by restricting the size of
the MCSCF reference to something smaller than a CAS reference and considering only the
reduced number of single and double excitations therefrom (Pitarch-Ruiz, Sanchez-Marin,
and Maynau 2002).
7.4 Perturbation Theory
7.4.1 General Principles
Often in pseudoeigenvalue equations, the nature of a particular operator makes it difficult
to work with. However, it is sometimes worthwhile to create a more tractable operator by
removing some particularly unpleasant portion of the original one. Using exact eigenfunctions
and eigenvalues of the simplified operator, it is possible to estimate the eigenfunctions
and eigenvalues of the more complete operator. Rayleigh–Schrödinger perturbation theory
provides a prescription for accomplishing this.
In the general case, we have some operator A that we can write as
A = A (0) + λV (7.19)