Page 262 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

APPLICATIONS OF NESTED SUMMATION SYMBOLS TO QUANTUM CHEMISTRY 245
Thus, the n-th order energy correction for the i-th system’s state can be
written as:

provided that the orthogonality condition:

holds between the unperturbed state wavefunction and their perturbation corrections up
to any order.

The wavefunction corrections can be obtained similarly through a resolvent
operator technique which will be discussed below. The n-th wavefunction correction for
the i-th state of the perturbed system can be written in the same manner as it is customary
when developing some scalar perturbation theory scheme: by means of a linear
combination of the unperturbed state wavefunctions, excluding the i-th unperturbed state.
That is:

Using expression (52) into equation (49), after some straightforward
manipulation, one can obtain the equivalent rule in order to construct the n-th order
wavefunction correction:

where a set of Resolvent Operators for the i-th state are easily defined as follows:

with the weighted projector sum Z i (0) defined in turn as:

being the set of projectors over the unperturbed states:

In this context equations (50) and (53) can be considered forming a
completely general perturbation theory for nondegenerate systems, although a recent
development permits to extend the formalism to degenerate states [1e].