Page 251 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

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.

4.2 EXPLICIT 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