Page 248 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

APPLICATIONS OF NESTED SUMMATION SYMBOLS TO QUANTUM CHEMISTRY 231
c) The same can be said of the final parameter vector f, which may be a
product by a scalar of vector 1. As an example the notation:
displays the symbol which is constructed by n nested sums, whose
indices take the same values within each sum in the interval {l,m}.
d) When initial, final and increment values are implicit in the nested sum,
a simplified symbol such as may be also used.
e) When the vector dimension n is obvious, then the n subscript can be
omitted from the sum, as in: for instance.

2.4 MATHEMATICAL NSS PROPERTIES

Following from NSS’s definition, some properties of these operators arise and
have to be considered. Here are listed some of them:

a) NSS’s can be recognized as linear operators with respect to any general
expression placed at the right side of the symbol.

b) A product of two NSS’s of dimensions n and m is another NSS of
dimension or:

where the new index vectors are constructed using the direct sum of the
original vectors appearing in the product.
c) The symbol can be made by convention equivalent to the
unit operator.
d) The classical summation symbol is a particular case of the NSS one,
it can be written as:

e) Einstein's convention, by which a set of nested sums are omitted from
an expression, corresponds to obviate writing a NSS like

3. Computational implementation of a NSS: the GNDL algorithm

3.1 GENERAL CONSIDERATIONS
In standard high level language programming the dimension of the NSS: n,
signals the number of nested do loops which are necessary to reproduce the structure in
a computational environment. But the mathematical usefulness of this entity can be easily
recognized when the particular characteristic of this symbolic unit is analyzed: the
involved vector parameters could be chosen with arbitrary and variable dimensions. There
are many scientific and mathematical formulae which will benefit of this property, when
written in a paper or computationally implemented.