Page 246 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

Applications of Nested Summation Symbols to Quantum Chemistry:
Formalism and Programming Techniques

R. CARBÓ and E. BESALÚ
Institute of Computational Chemistry, University of Girona
Albereda 5, 17071 Girona, Spain

1. Introduction

Our research on various Quantum Chemistry areas has been directed in a great
extend to the construction of general useful algorithms based on, as elementary as
possible, mathematical concepts [1,2]. We tried along this past period to obtain
computational procedures with sufficiently interesting features leading to a three fold
purpose. First, the results must be pedagogically adequate. Second, the algorithmic
structure must be susceptible of easy implementation to high level programming
languages. Third, the development must benefit the computational side of Chemistry as
Physics and be solidly grounded of Applied Mathematics principles.
In this sense, our intention was, such that the final working schemes can serve
to connect mathematical general formulae writing and computationally valid general
program structures. Thus, programming techniques can also be assisted by means of this
process, as well as Artificial Intelligence [1c] algorithms may use partially the results of
our outline in order to increase the performances of formulae generation and translation
programs.
With all this conditioning principles in mind, the present work tries to describe
in a first place the definition and properties of two fundamental symbols: Logical
Kronecker Deltas (LKD’s) and Nested Sums. The authors hope these symbol forms turn
to be as useful to the scientific community as they had been in the development of their
quest of a valid computational scheme based on PC machinery, whose main features had
been already explained by one of us, see for example reference [3].
Consequently, here are studied under the formulation of the Nested
Summation Symbols (NSS’s) symbolism some Quantum Chemical problems and topics.

2. Definition and Properties of the NSS

2.1 LOGICAL KRONECKER DELTA SYMBOL DEFINITION

Let us define a generalization of the Kronecker delta symbol and call it a
Logical Kronecker Delta (LKD). This symbol is written as and corresponds to a
function that can return two possible values: 1 if the logical argument L is true or 0
229
Y. Ellinger and M. Defrancéschi (eds.), Strategies and Applications in Quantum Chemistry. 229–248.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.