Page 255 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

238 R. CARBÓ AND E. BESALÚ

zero. This result encompass the well described zero-, one- and two-electron operator cases
[9], generalizing in this way the rules governing the calculation of operator matrix
elements between two Slater determinants. One can say that the general rule in order to
prevent automatic integral nullity is: "r-electron operators allow a maximal amount of r
spinorbital differences". This rule is connected to the Brillouin theorem [15].

The same expression can be used with the appropriate restrictions to obtain
matrix elements over Slater determinants made from non-orthogonal one-electron
functions. The logical Kronecker delta expression, appearing in equation (15) as defined
in (16)] must be substituted by a product of overlap integrals between the involved
spinorbitals.

5.2 CI WAVEFUNCTIONS

Using the approach already described for combination generation, one can
formulate in a short but completely general form the CI wavefunctions [16].

This kind of wavefunctions, in the complete CI framework, as Knowles and
Handy [16e] have proved feasible, for a system of m spin-orbitals and n electrons
can be written within the NSS formalism:

where the logical vector L is defined according to the combinations generation and the
terms D(j) are Slater determinants constructed as the one defined in equation (13). The
C(j) factors are the variational coefficients attached to each Slater determinant.

Also, an alternative formulation of equation (17) can be conceived if one
wants to distinguish between ground state, monoexcitations, biexcitations, ... and so on.
Such a possibility is symbolized in the following CI wavefunction expression for n
electrons, constructed as to include Slater determinants up to the p-th excited order.
One can initially start from n occupied spinorbitals and in unoccupied ones
Then, the CI wavefunction is written in this case as the linear combination:

where the index e, appearing in the first classical sum, signals the excitation order. That
is, for e=0 one has the ground state, for e=l the monoexcitations are obtained and so on.

In equation (18) the terms are n-electron Slater determinants formed
by the spin-orbitals numbered by means of the direct sum: of the vector index
parameters attached to the involved nested sums and to the occupied-unoccupied orbitals
respectively. That is: