computational tours de force are, although of limited direct application, important
for evaluating the efficacy, by comparison, of other methods.
The simplest implementation of CI is analogous to the Roothaan–Hall imple-
mentation of the HF method: Eqs.5.168lead to a CI matrix, as the HF equations
(Eqs.5.164¼5.52) lead to an HF matrix (Fock matrix;Section 5.2.3.6). Do not
confuse a matrix with a determinant (Section 4.3.3)! We saw that the Fock matrixF
can be calculated from thec’s andf’s of Eq.5.164¼5.52(starting with a “guess”
of thec’s), and thatF(after transformation to an orthogonalized matrixF^0 and
diagonalization) gives eigenvalueseand eigenvectorsc, i.e.Fleads to the energy
levels and the wavefunctions (cf) of the component MOsc; all this was shown in
detail in Section 5.2.3.6.5. Similarly, a CI matrix can be calculated in which the
determinantsDplay the role that the basis functionsfplay in the Fock matrix,
since theD’s in Eqs.5.168are analogous to thef’s in Eq.5.164¼5.52. TheD’s are
composed of spin orbitalscaandcb, and the spin factors can be integrated out,
reducing the elements of the CI matrix to expressions involving the basis functions
and the coefficients of the spatial component MOsc. The CI matrix can thus be
calculated from the MOs resulting from an HF calculation. Orthogonalization and
diagonalization of the CI matrix gives the energies and the wavefunctions of the
ground stateC 1 and, fromideterminants,i#1 excited states. A full CI matrix
would give the energies and wavefunctions of the ground state and all the excited
states obtainable from the basis set being used. Full CI with an infinitely large basis
set would give the exact energies of all the electronic states; more realistically, full
CI with a large basis set gives good energies for the ground and many excited states.
Full CI is out of the question for any but small molecules, and the expansion of
Eq.5.169must usually be limited by including only the most important terms.
Which terms can be neglected depends partly on the purpose of the calculation.
For example, in calculating the ground state energy quadruply excited states are,
unexpectedly, much more important thantriply and singly excited ones, but the
latter are usually included too because theyaffecttheelectrondistributionofthe
ground state, and in calculating excitedstateenergiessingleexcitationsare
important. A CI calculation in which all theD’s involve only single excitations
is called CIS (CI singles); such a calculation yields the energies and wavefunc-
tions of excited states and often gives a reasonable account of electronic spectra.
Another common kind of CI calculation isCIsinglesanddoubles(CISD,which
actually indirectly includes triply andquadruply excited states). Various mathe-
matical devices have been developed to make CI calculations recover a good
deal of the correlation energy despite thenecessity of (judicious) truncation of the
CI expansion. Perhaps the currently mostwidely-usedimplementationsofCI
aremulticonfigurational SCF(MCSCF) and its variantcomplete active space
SCF(CASSCF), and the coupled-cluster (CC) and relatedquadratic CI(QCI)
methods.
The CI strict analogue of the iterative refinement of the coefficients that we saw
in HF calculations (Section 5.2.3.6.5) would refine just the weighting factors of the
determinants (thec’s of Eqs. (5.168), but in the MCSCF version of CI the spatial
MOswithinthe determinants are also optimized (by optimizing thec’s of the
5.4 Post-Hartree–Fock Calculations: Electron Correlation 273