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Classification of Basis Functions by Excitation Level
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 doublyexcited configurations. Since the Hamiltonian
contains only twobody 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.

(tex2html_deferredtex2html_deferred4.tex2html_deferred14H =
[
]
) 
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 HartreeFock method (HartreeFock guarantees
that offdiagonal 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 offdiagonal 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 HartreeFock 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
nonzero 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 oneelectron properties (dipole moment, etc.)
Next: Energy Contributions of the
Up: Reducing the Size of
Previous: Symmetry Restrictions on the
C. David Sherrill
20000418