Page 252 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

APPLICATIONS OF NESTED SUMMATION SYMBOLS TO QUANTUM CHEMISTRY 235
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.