Rayleigh-Schrodinger approach

The idea of the perturbational approach is very simple. We know everything about a non-perturbed problem. Then we slightly perturb the system and everything changes. If the perturbation is small, it seems there is a good chance that there 

Assuming we used the basis functions that satisfy Postulate V (on symmetry). 

Perturbational theory is notorious for quite clumsy equations. Unfortunately, there is no way round if we want to explain how to calculate things. However, in practise only a few of these equations will be used - they will be hi hghted in frames. 

Let us begin our story. We would like to solve the Schrodinger equation 

We assume that form an orthonormal set, which is natural. We are in- 

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Rayleigh-Schrodinger many-body perturbation theory — RSPT). In this approach, the total Hamiltonian of the system is divided or partitioned into two parts a zeroth-order part, Hq (which has... 

In the above formula, Q is the nuclear coordinate, p, and I/r are the ground state and excited electronic terms. Here Kv is provided through the traditional Rayleigh-Schrodinger perturbation formula and K0 have an electrostatic meaning. This expression will be called traditional approach, which has, in principle, quantum correctness, but requires some amendments when different particular approaches of electronic structure calculation are employed (see the Bersuker s work in this volume). In the traditional formalism the vibronic constants P0 dH/dQ Pr) can be tackled with the electric field integrals at nuclei, while the K0 is ultimately related with electric field gradients. Computationally, these are easy to evaluate but the literally use of equations (1) and (2) definitions does not recover the total curvature computed by the ab initio method at hand. 

Another approach to the problem of computing the electron correlation energy is the M0ller54-Plesset55 (MP) perturbation theory (which is philosophically akin to the many-body perturbation theory of solid-state physics). The mechanics are the conventional Rayleigh-Schrodinger perturbation theory One introduces a generalized electronic Hamiltonian Hi, where... 

We now apply standard Rayleigh-Schrodinger perturbation theory, using the Hartree-Fock determinant as the zero-order state, and expand the perturbed states in the set of excited determinants. This approach gives rise to Moller-Plesset perturbation theory [1]. To first order, we recover the Hartree-Fock energy hf and, to second order, we obtain the second-order Moller-Plesset (MP2) energy ... 

Since both these models represent extended systems, the exploitation of the shell-model or Cl-type variational methods was a priori excluded. This placed emphasis on the development of perturbative approaches for this type of problem. It was soon realized that the most efficient methodological approach must be based on a mathematical formalism that was originally developed in quantum field theory. Moreover, in view of the extended character of the studied systems, it was absolutely essential that the method employed yields energies that are linearly proportional to the particle number N in the system or, in today s parlance, that it must be size extensive, so that the limiting procedure when N->-oo makes sense. In terms of MBPT, this imphes that only the connected or finked energy terms be present in the perturbation series, a requirement that automatically leads to the Rayleigh-Schrodinger PT. 

Formalism of the so-called response theory is another, quite universal language for the description of the more general approach to Rayleigh-Schrodinger perturbation theory suggested above, in which the summation over excited states is effectively replaced with solving a large system of linear equa-... 

Such second-order EPs have been used (Doll and Reinhardt, 1972 Purvis and Ohrn, 1974) to compute atomic and molecular ionization potentials, electron affinities, and even electron-atom shape resonance positions and lifetimes with some success. Based upon the experience gained to date, however, we cannot expect the accuracy of this approach to be better than 0.5 eV, even for systems that are described reasonably well by a singleconfiguration reference function. Often, this numerical accuracy is not satisfactory and hence the above formalism must be advanced to higher order (or replaced by another development that does not depend upon the Rayleigh-Schrodinger order concept). An example of such a second-order EP calculation is given in Problem 6.1. 

The form of perturbation theory developed in this section is called Rayleigh-Schrodinger perturbation theory, other approaches exist.)... 

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