Now we will discuss the importance of various excitation classes to
the CI wavefunction. As noted in equation (2.9), the CI
expansion is typically truncated according to excitation level; in the vast
majority of CI studies, the expansion is truncated (for computational
tractability) at doubly-excited configurations. Since the Hamiltonian
contains only two-body terms, only singles and doubles can interact
directly with the reference (for the sake of simplicity, we are assuming
only a single reference for now). This is a direct result of Slater's
Rules (cf. section 2.4).
The structure of the CI matrix
with respect to excitation level is given below (adapted from Szabo and
Ostlund [1], p. 235), where 
and 
represent blocks of singly, doubly, triply, and quadruply excited
determinants, respectively. The Hamiltonian matrix H is Hermitian;
if only real orbitals are used, as is usually the case, then the
Hamiltonian is also symmetric. Thus only the lower triangle of H is
shown below.
Note that the matrix elements 
are given as 0. This
is due to Brillouin's theorem, which is valid when our reference function
is obtained by the Hartree-Fock method (Hartree-Fock guarantees
that off-diagonal elements of the Fock matrix are zero, and it turns out
that the matrix element between two Slater determinants which differ by
one spin orbital is equal to an off-diagonal element of the Fock matrix).
Furthermore, the blocks 
which are not necessarily zero
may still be sparse; for example, the matrix element
, which belongs to the
block 
, will be nonzero only if a and b are
contained in the set 
and if r and s are contained
in the set 
.
Since only the doubles interact directly with the Hartree-Fock reference, we expect double excitations to make the largest contributions to the CI wavefunction, after the reference state. Indeed, this is what is observed. Even though singles, triples, etc. do not interact directly with the reference, they can still become part of the CI wavefunction (i.e. have non-zero coefficients) because they mix with the doubles, directly or indirectly. Although singles are much less important to the energy than doubles, they are generally included in CI treatments because of their relatively small number and because of their greater importance in describing one-electron properties (dipole moment, etc.)