Essentials of Computational Chemistry

212 7 INCLUDING ELECTRON CORRELATION IN MO THEORY
where i and j are occupied MOs in the HF ‘reference’ wave function, r and s are virtual
MOs in HF, and the additional CSFs appearing in the summations are generated by exciting
an electron from the occupied orbital(s) indicated by subscripts into the virtual orbital(s)
indicated by superscripts. Thus, the first summation on the r.h.s. of Eq. (7.10) includes all
possible single electronic excitations, the second includes all possible double excitations, etc.
If we assume that we do not have any problem with non-dynamical correlation, we may
assume that there is little need to reoptimize the MOs even if we do not plan to carry
out the expansion in Eq. (7.10) to its full CI limit. In that case, the problem is reduced to
determining the expansion coefficients for each excited CSF that is included. The energies
E of N different CI wave functions (i.e., corresponding to different variationally determined
sets of coefficients) can be determined from the N roots of the CI secular equation
H11
− E H12 ... H1N
H21 H22
− E ... H2N
. = 0 (7.11)
. . .. .
HN1 NN2 ... HNN − E
where
Hmn =〈m|H |n〉 (7.12)
H is the Hamiltonian operator and the numbering of the CSFs is arbitrary, but for convenience
we will take 1 = HF and then all singly excited determinants, all doubly excited,
etc. Solving the secular equation is equivalent to diagonalizing H, and permits determination
of the CI coefficients associated with each energy. While this is presented without derivation,
the formalism is entirely analogous to that used to develop Eq. (4.21).
To solve Eq. (7.11), we need to know how to evaluate matrix elements of the type defined
by Eq. (7.12). To simplify matters, we may note that the Hamiltonian operator is composed
only of one- and two-electron operators. Thus, if two CSFs differ in their occupied orbitals
by 3 or more orbitals, every possible integral over electronic coordinates hiding in the r.h.s.
of Eq. (7.12) will include a simple overlap between at least one pair of different, and hence
orthogonal, HF orbitals, and the matrix element will necessarily be zero. For the remaining
cases of CSFs differing by two, one, and zero orbitals, the so-called Condon–Slater rules,
which can be found in most quantum chemistry textbooks, detail how to evaluate Eq. (7.12)
in terms of integrals over the one- and two-electron operators in the Hamiltonian and the
HF MOs.
A somewhat special case is the matrix element between the HF determinant and a singly
excited CSF. The Condon–Slater rules applied to this situation dictate that
H1n =〈HF|H | r i 〉
(7.13)
=〈φr|F |φi〉
where F is the Fock operator and i and r are the occupied and virtual HF orbitals in the
single excitation. Since these orbitals are eigenfunctions of the Fock operator, we have
〈φr|F |φi〉 =εi〈φr|φi〉
(7.14)
= εiδir