Page 261 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

244 R. CARBÓ AND E. BESALÚ
as an ideal framework to construct a really general perturbation theory scheme. Next
discussion will try to prove this.

Let us write a perturbed hamiltonian by a set of k independent perturbation
operators using the following expression involving a NSS:

where the vectors s and L of the NSS are omitted, assuming that s=1 and all the possible
forms of vector p have to be generated. In equation (44) the first parameter vector value
gives the unperturbed hamiltonian H(0), thus the convention must hold, and any
other vector index p structure generates a set of perturbation operators The
final parameter vector K contains the maximal order of the perturbation, which can be
different for every operator. The symbol is an element of the scalar set of
perturbation parameters. Both H(p) and can be considered products of perturbation
operators and the attached parameters.
That is:

and

The adequate technique here is to substitute the usual Rayleigh-Schrödinger
scalar perturbation order by a vector perturbation order n.

The perturbed energies and wavefunctions for the i-th system state can be
expressed in a similar way as in scalar perturbation theory:

and

being the expressions (47) and (48) the generalization of equations (34) and (35)
respectively.
Substituting equations (44), (47) and (48) into the perturbed Schrödinger
secular equation produces the n-th order equation:

which when n=0 yields the unperturbed Schrödinger equation.