General Perturbation Algorithm

Let s consider the non-degenerate non-perturbed discrete (stationary) solved problem 

In these conditions, the perturbed eigen-states are generically written as a superposition of all non-perturbed eigen-energies  

With this specialization, the perturbed eigen-problem equivalently becomes  

Let s now analyze each order in perturbation, based on the above separate, however somehow iterative, equations. 

Order (0) The solution of this (unperturbed) problem is immediate  

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In this subsection we will combine the general ideas of the iterative perturbation algorithms by unitary transformations and the rotating wave transformation, to construct effective models. We first show that the preceding KAM iterative perturbation algorithms allow us to partition at a desired order operators in orthogonal Hilbert subspaces. Its relation with the standard adiabatic elimination is proved for the second order. We next apply this partitioning technique combined with RWT to construct effective dressed Hamiltonians from the Floquet Hamiltonian. This is illustrated in the next two Sections III.E and III.F for two-photon resonant processes in atoms and molecules. 

The second-order perturbation calculations following the CAS-SCF treatments previously described are intended to supply the deficiency of correlation, if any, included in the latter. To compute the energy corrections to be added to the variational zero-order values obtained by diagonalizing the active-space Hamiltonian matrices (see tables 3a and 3b of section 2.2), we have performed two different kinds of multireference perturbation Cl treatments, both of them being based on improvements of the general CIPSI algorithm (26] recently developed by the Ferrara-Pisa group[27,28,29,30]. 

Perturbation theory has been applied to anharmonic calculations of spectroscopy from ab initio potentials in a large number of studies [19-25,115-121]. In nearly all cases so far, second-order perturbation theory was employed. The representation of the anharmonic potential generally used in these studies is a polynomial in the normal modes, most often a quartic force field. A code implementing this vibrational method was recently incorporated by V. Barone in gaussian [24]. Calculations were carried out for relatively large molecules, such as pyrrole and furan [25], uracil and thiouracil [118], and azabenzenes [119]. We note that in addition to spectroscopy, the ab initio perturbation theoretic algorithms were also applied to the calculation of thermodynamic properties... 

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