Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200

226	J. WEBER ET AL.

CASE STUDIES OF ELECTROSTATIC PROPERTIES OF SMALL MOLECULES

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Acknowledgements

The authors are grateful to Professor D. Salahub for providing a copy of the deMon program. This work is part of Project 20-36131.92 of the Swiss National Science Foundation.

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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

2.1LOGICAL 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

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© 1996 Kluwer Academic Publishers. Printed in the Netherlands.

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.

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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

3.1GENERAL 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.

232	R. CARBÓ AND E. BESALÚ

NSS symbolism constitutes a link between mathematical formalism and program implementation techniques, because successive generation of j index vector elements can be programmed in a general but simple way under any high level language.

This can be achieved using a unique do or for loop statement construct, which is general and independent of the dimension of the involved nested sums. This kind of programming structure constitutes the GNDL algorithm.

NSS have not a direct translation to the usual high level languages. Present day compilers or standard language rules ignore such an interesting feature, see for example the practical final form of the standard Fortran 90 language [5]. Even high level language compilers have no capacity of processing more than a limited number of classical do loops in a nest, for example VAX Fortran and NDP Fortran compilers [6] have a l i m i t of 20 nested do loops. Thus, the GNDL structure is a good candidate to circumvent these limitations in any compiler.

It looks simple to introduce GNDL in the family of repetitive sentences found in high level languages. So we feel that a claim in this direction to language and compiler builders can be made here. Some immediate computational benefits in order to construct really general programs may be obtained.

3.2 A SCHEMATIC GNDL PROGRAM EXAMPLE

In order to show in a practical manner the computational implementation of a NSS, Program 1 represents a Fortran source code corresponding to the

NSS structure. The NSS implementation using a GNDL algorithm generates all the possible forms of vector j. According to this, Program 1 generates the indices of the n-dimensional N S S T h e dimension n and the initial, final and step index values collected in the vectors i, f, and s have not been specified and the question mark symbol stands for their possible values. These values depend on the concrete application given to the algorithm. Here it is assumed

that the step vector s has all its components positive definite.

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233

There, Application is a called procedure where the n nested loops converge and where their leading indices can be arbitrarily used in the desired internal application.

The j index values generation is sequential but the execution of Application can be performed into separate CPU’s, each one controlling the process attached to one of the forms of the vector j. In this manner, the full computation can be parallelized if desired.

In fact, this is a general algorithm, enabling to perform a parallel Application implementation if the nature of the problem asks for such a process and the available hardware allows to run it in this manner. A previous tentative description on GNDL, in a sequential programming framework, was initially made by Carbó and Bunge [4].

Various application examples have been constructed by the authors. Some

Fortran source codes on combinatorial analysis, product of an arbitrary number of matrices and determinant evaluation in a parallel transputer environment [7] have been tested and encouraging results obtained.

4. Mathematical application examples

As an illustration of the possible use of the described symbols, there will be presented first a set of possible purely mathematical application examples of NSS.

4.1 GENERATION OF VARIATIONS AND COMBINATIONS

A NSS can be used to generate variations and combinations of m elements belonging to an arbitrary set of mathematical objects. It is only necessary numbering in a canonical order, from 1 to m, all the elements in the set. This will produce a completely formal development which can be occasionally used for immediate implementation on any high level language. Although this direct translation will obviously lack of programming refinement in the first bulk program scheme, it may be considered a not too bad starting point in order to obtain a given optimized code.

Then, one can easily describe the expressions that stands for the generation of some combinatorial entities. It is required the implementation of the following NSS:

Depending on the definition of the logical vector L they are obtained different entities:

a)If L is obviated the NSS then represents the generation of all the possible variations with repetition, which can be formed making groups

of n elements out of the m element set.

b) When L is defined as

then they are reproduced

the m!/(m-n)! variations without repetition, which can be formed making groups of n objects taken from the m element set inside the nested sum. It is required the condition

c) If L is defined as

then the NSS creates the

m!/n!(m-n)! combinations related to a set of m elements, when they are

taken in groups of n out of the m element set.

234	R. CARBÓ AND E. BESALÚ

d)Combinations generation can be also performed by means of the implementation of the NSS obviating the logical vector L and defining

the parameter vectors i and f as	and
, respectively. This last choice	implies to rewrite

Program 1 in some special manner, where also the initial indices are modified, while the GNDL is executed.

4.2EXPLICIT EXPRESSION OF THE DETERMINANT OF AN ARBITRARY SQUARE MATRIX

	Using	the	NSS,	one	can reformulate	the	expression which gives	the
determinant	of an	arbitrary		(n×n)	square matrix A,	Det	| A | . A compact formula	of
Det | A | can	be written		in this	way as:

where the logical vector L is a function of the j vector indices and is defined as:

and the S(j) factor is a sign, which can be expressed by:

being P(j) the parity of the order of the values of the index of the vector j. This parity value can be expressed as:

Finally, the last term in equation (3) is a product of the elements of the

matrix:

Although this final determinant structure can easily lead to an immediate construction of sequential or parallel Fortran subroutines, there cannot be a claim such that this procedure will be better, from a computational point of view, than well established numerical ones, based on other grounds as Cholesky decomposition, see references [8] for more details. One can recall again the remarks already made at the beginning of section 3.1 above, and stress once more the formal nature of the programming immediate translation capabilities of NSS’s.

However, the previous determinant development form can be used as a very general interpretative formula, which can compete pedagogically and practically with other widespread alternatives, for example these usually employed in Quantum Chemistry, see

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reference [9] as a guide. In section 5 this determinant form is used, for example, to deal with Slater determinants.

One can easily see that, despite all criticisms which can arise from the programming technical side, the nested sum formalism permits to solve in a very elegant manner the following problem: Program in a chosen high level language a function procedure which can be used to compute the determinant of a general square matrix using the direct Laplace determinant definition [10].

4.3 TAYLOR SERIES EXPANSION OF A n-VARIABLE FUNCTION

The complete formula for the Taylor series expansion attached to a n-variable function f (x) in the neighbourhood of the point possess the following peculiar simple structure when using NSS’s:

The	terms are defined by means of the product:

Finally, is a short symbol expressing the m-th order partial derivative operators, acting first over the function f (x) and then, the resultant function, evaluated at the point The differential operators can be defined in the same manner as the terms present in equation (9), but using as second argument the nabla vector:

The expression (8) is very useful in the sense one can control the series truncation. This is so because the parameter in gives the order of the derivatives appearing in the expansion.

Although there are some general textbook approaches to equation (8), see reference [11] for example, we have not found the expression of the Taylor expansion in full as simple as it has been presented here. Moreover, many potential Taylor expansions are used in various physical and chemical applications; for instance in theoretical studies of molecular vibrational spectra [12] and other quantum chemical topics, see for example reference [13]. Then, the possibility to dispose of a compact and complete potential expression may appear useful.