C 1 ¼c 11 D 1 þc 21 D 2 þc 31 D 3 þ(((þci 1 Di
C 2 ¼c 12 D 1 þc 22 D 2 þc 32 D 3 þ(((þci 2 Di
...
Ci¼c 1 iD 1 þc 2 iD 2 þc 3 iD 3 þ(((þciiDið 5 : 168 Þi.e. (cf. Eq.5.164¼5.52)
Ci¼Xis¼ 1csiDs s¼ 1 ; 2 ; 3 ;(((;iðtotal MOsÞð 5 : 169 ÞWhat is the physical meaning of all these total wavefunctionsC? In order of
correspondence to increasing energy (the expectation value of the integral of a
wavefunction over a Hamiltonian operator)C 1 is the wavefunction for the ground
electronic state andC 2 ,C 3 etc. represent the wavefunctions of excited electronic
states. The single-determinant HF wavefunction of Eq.5.163¼5.10(or the general
single-determinant wavefunction of Eq.5.12) is merely an approximation to theC 1
of Eqs.5.168. Each determinantD(or possibly a linear combination of a few
determinants for an open-shell species [ 96 ]), represents an idealized (in the sense of
contributing to therealelectron distribution) configuration, called aconfiguration
state functionorconfiguration function, CSF. A CSF is a linear combination of
determinants for equivalent states, states which differ only by whether anaor ab
electron was promoted. In many cases one determinant suffices for the Hartree–
Fock wavefunction, and then this determinant is the CSF. The CI wavefunctions of
Eqs.5.168or5.169, then, are linear combinations of CSFs. No single CSFfully
represents any particular electronic state. Each wavefunctionCiis the total wave-
function of one of the possible electronic states of the molecule, and the weighting
factorscin its expansion determine to what extent particular CSF’s (idealized
electronic states) contribute to anyCi. ForC 1 , representing the ground electronic
state, we expect the HF determinantD 1 to make the largest contribution to the
wavefunction.
If every possible idealized electronic state of the system, i.e. every possible
determinantD, were included in the expansions of Eqs.5.168, then the wavefunc-
tionsCwould befull CIwavefunctions. Full CI calculations are possible only for
very small molecules, because the promotion of electrons into virtual orbitals can
generate a huge number of states unless we have only a few electrons and orbitals.
Consider for example a full CI calculation on a very small system, H–H–H–H with
the 6–31G* basis set. We have eight basis functions and four electrons, giving eight
spatial MOs and 16 spin MOs, of which the lowest four are occupied. There are two
aelectrons to be promoted into six virtualaspin MOs, i.e. to be distributed among
eightaspin MOs, and likewise for thebelectrons andbspin orbitals. This can
be done in [8!/(8#2)!2!]^2 ¼784 ways. The number of configuration state functions
is about half this number of determinants (since some CSFs are composed of a
few determinants). CI calculations with more than five billion (sic) CSFs have been
performed on ethyne, C 2 H 2 [ 97 ]; rightly called benchmark calculations, such
272 5 Ab initio Calculations