Page 247 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

230 R. CARBÓ AND E. BESALÚ
otherwise. The Kronecker delta symbol is a particular case of a LKD, where in the logical
expression L there is involved an equivalence symbol.

2.2 DEFINITION OF THE NSS

The NSS concept corresponds to an operator attached to an arbitrary number
of nested sums. In other words, a NSS represents a set of summation symbols where the
number of them can be variable. In a general notation one can write a
where the meaning of this convention corresponds to perform all the sums involved in the
generation of all the possible values of the index vector j under the fulfillment of the set
of logical expressions collected in the components of the vector L. The elements of the
vector j have the following limits:

where the indices can be incremented or decremented respectively in steps of length .
The index n is the dimension of the NSS, that is: the number of summation symbols
embedded in the operator, and thus the dimension of the involved vectors j, i, f and s. The
set of all the vectors appearing as arguments of the NSS can be named parameters of the
NSS.

The logical vector L is of the type . The delta symbol corresponds to
a LKD. In this manner, the indices of the vector L are 0’s or 1’s. So, the convention of
a NSS stands for the generation of all the possible forms of the index vector j that are
attached to the logical vector

A NSS has a computational implementation we have called a GNDL [1,4].
The Fortran code of the algorithm implementing a GNDL can be found described in
Program 1 below. The GNDL algorithm constitutes the link between the mathematical
notation of the NSS and the computer codification of this operator.

2.3 SIMPLIFIED NESTED SUM NOTATION
Despite the general form adopted here to write a NSS, sometimes it is
superfluous to explicit all the involved parameters. When this circumstance does occur,
some parts of the general form can be dropped in an arbitrary manner. The most important
cases are:

a) If the logical vector L is not specified it will mean that all the possible
forms of the vector j must be generated with no restriction. In all the
remaining text this NSS form will be used.

b) When the step vector increment is irrelevant only the initial and final
vector index parameters need to be explicitly written. In this case, the
notation can be employed. Frequently, the step vector s is a n-
dimensional vector such as s=1.