Essentials of Computational Chemistry

7.3 CONFIGURATION INTERACTION 213
where εi is the MO eigenvalue. Thus, all matrix elements between the HF determinants and
singly excited determinants are zero, since to be singly excited, r must not be equal to i.
This result is known as Brillouin’s theorem (Brillouin 1934).
It is not the case that arbitrary matrix elements between other determinants differing
by only one occupied orbital are equal to zero. Nevertheless, the Condon–Slater rules and
Brillouin’s theorem ensure that the CI matrix in a broad sense is reasonably sparse, as
illustrated in Figure 7.4. With that in mind, let us return to the question of which excitations
to include in a ‘non-full’ CI. What if we only keep single excitations? In that case, we
see from Figure 7.4 that the CI matrix will be block diagonal. One ‘block’ will be the HF
energy, H11, and the other will be the singles/singles region. Since a block diagonal matrix
can be fully diagonalized block by block, and since the HF result is already a block by itself,
Ψ i a
Ψ ij ab
Ψ ijk abc
Ψ HF
Ψ i a
Ψ HF E HF 0
0
d
e
n
s
e
0
dense
sparse
very
sparse
Ψ ij ab Ψ ijk abc
dense
sparse
sparse
extremely
sparse
0
very sparse
extremely sparse
extremely sparse
Figure 7.4 Structure of the CI matrix as blocked by classes of determinants. The HF block is the (1,1)
position, the matrix elements between the HF and singly excited determinants are zero by Brillouin’s
theorem, and between the HF and triply excited determinants are zero by the Condon–Slater rules.
In a system of reasonable size, remaining regions of the matrix become increasingly sparse, but the
number of determinants in each block grows to be extremely large. Thus, the (1,1) eigenvalue is most
affected by the doubles, then by the singles, then by the triples, etc