Page 260 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

APPLICATIONS OF NESTED SUMMATION SYMBOLS TO QUANTUM CHEMISTRY 243

being the vector defined in equation (33) and where the terms
constitute the representation of the perturbation operator V within the characteristic basis
set of the unperturbed hamiltonian In equation (39) the primed summation symbols
are attached to sums performed over all index values except the i-th.

The n-th order correction for the energy takes the form [18]:

being defined in equations (33) and (36) respectively.
Equations (39) and (40) can be rewritten using the NSS formalism. The
corrections for the wavefunction take now the simple form:

and the corrections over the energies are expressed by equation (36).

In equation (41) the vectors 1 and L are n-dimensional and L components are
LKD’s of the type The operator R (j) is written as:
i

where is a projector-like operator defined in turn as:

Thus, one can see NSS as a useful device which permits to write in a compact
manner equations (39) and (40). Also it allows to easily obtain these formulae by means
of the NSS straightforward implementation, the GNDL algorithm.

5.4.2. General Rayleigh-Schrödinger perturbation theory
As it can be seen in equation (41), the NSS notation permits to write some
equations in an elegant and compact manner. This is due to the fact that NSS opens a new
door in order to obtain algebraic expressions. In this sense we propose that the use of NSS