Page 75 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

60 C. VALDEMORO
the element of (relation 16), which appears between brackets, by its value in terms
of the electronic integrals (relation 8) one gets after some simple algebra

At first sight, this looks rather complicated, and in fact, evaluating this general rela-
tion is not trivial. On the other hand, the results are simple, closed form expressions.
The derivation of the algorithms for different values of p have been described in detail
previously [25,26,28]. Here we will just sketch the results obtained.
The main point about the SRH matrix is that its only non zero elements are those
where the set of the i indices and the set of the j indices are equal or differ at the
most in two indices.
The general form of the three different classes of the matrix elements are [28]:

• The set of i indices is equal to the set of j indices (although any ordering of the
indices is allowed).

Where the symbol has the same meaning as previously. In this relation,
denotes a permutation of the indices and A and B are coefficients.
• An index of the set i differs from that of an index of the set j, say
(although any ordering of the indices is allowed).

• Two indices of the set i differ from two indices of the set j, say and
(although any ordering of the indices is allowed).

A very convenient feature of this formalism is that the values of the A and B coef-
ficients only depend on three numbers: the number of electrons (N), the number of
orbitals of the basis (K), and the Spin quantum number (S).
The different orderings of the matrix element indices ( superscript in relations
19,20,21) give rise to different values of the A and B coefficients. Finally, the value of
these coefficients also depends on whether one or more of the matrix element indices
are repeated (appears twice).
For p = 1,2,3 the values of these coefficients have been explicitly obtained; and for
large p, a set of diagramatic rules have been reported [28] in order to determine the
coefficients in each case.
To calculate a p – SRH once the electronic integrals are known, is therefore a very
simple and rapid task.