Page 258 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

APPLICATIONS OF NESTED SUMMATION SYMBOLS TO QUANTUM CHEMISTRY 241

where C(a,j[p]) and X(a) are products of the coefficients and the basis functions
respectively, appearing in the linear combinations (25) for every spinorbital. Now using
(26) in the spinorbital product appearing in the rightmost side of (24), one obtains using
a simplified NSS notation:

Finally the density function of (n-m)-th order can be expressed in terms of the
atomic spinorbitals as:

being the (n-m)-th order charge and bond order hypermatrices, defined as:

using the hypermatrix elements:

The equation (28) has the same structure as the well known LCAO form of
the first order density function [9]. Thus, it can be concluded that density functions of any
order exhibit the same formal structure. In this manner, it can be seen that NSS’s lead to
an interesting mnemotechnical rule.

5.4 PERTURBATION THEORY

In order to define the notation which we will use from now on, let us consider
the application of the perturbation theory to a system which has a perturbed hamiltonian
H composed by an unperturbed one, plus a perturbation operator

From here, the goal consists to find the eigenvalues and the eigenvectors of
the perturbed system, which we denote as the sets respectively. That is,
the target is focused into solving the eigenvalue problem: