Page 253 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

236 R. CARBÓ AND E. BESALÚ
5. Quantum chemical application examples

Several Quantum Chemical application examples of NSS’s follow. Some of
them had been chosen because they are related to the actual research in this field in our
Laboratory.

We do not pretend to give here an exhaustive account of all the possible
applications of NSS’s into Quantum Chemistry. Some areas, which for sure can be studied
from the nested summation point of view, like the Coupled Cluster Theory [14], are not
included here.
In fact, our interest in the present formulation, the use of NSS’s and LKD’s,
has been aroused when studying the integrals over Cartesian Exponential Type Orbitals
[la,b] and Generalized Perturbation Theory [ld,e]. The use of both symbols in this case
has been extensively studied in the above references, so we will not repeat here the
already published arguments. Instead we will show the interest of using nested sums in
a wide set of Quantum Chemical areas, which in some way or another had been included
in our research interests [lc].

5.1 SLATER DETERMINANTS
As it is shown in section 4.2, using NSS terminology, the general expression
for any determinant can be obtained. In this manner, this formulation can be transferred
into the Slater determinants [9], constructed by n spinorbitals associated to n electrons.
Adopting the following structure and notation for unnormalized Slater determinants:

where the logical vector L, defined in equation (4), is needed in order to obtain all the
variations without repetition of the values of the vector j indices. Here, a term constructed
by means of spinorbital products is present:

A similar definition of the symbol (12) can be taken into account, just using
the products of

The Slater determinant expression of equations (11) and (12) will be taken as
implicit in this paper from now on.
An operator, depending of an arbitrary number of electron coordinates, has
an easily expressible set of matrix elements, using two Slater determinants D(j) and D(k).
The term D(j) can be taken as a Slater determinant, formed by n functions
chosen from a set of in available spinorbitals, and ordered following the actual internal
values of the j index vectors. That is: