Rayleigh-Schrodinger perturbation theory formal development

E. Schrodinger, Ann. Phys. 80 (1926), 437. The quantal formalism substantially follows the classical method developed by Lord Rayleigh (Theory of Sound [1894]) and is commonly referred to as Rayleigh-Schrodinger perturbation theory. ... 

The renewal of interest in Brillouin-Wigner perturbation theory for many-body systems seen in recent years, is driven by the need to develop a robust multi-reference theory. Multi-reference formalisms are an important prerequisite for theoretical descriptions of dissociative phenomena and of many electronically excited states. Brillouin-Wigner perturbation theory is seen as a remedy to a problem which plagues multi-reference Rayleigh-Schrodinger perturbation theory the so-called intruder state problem. 

Brillouin-Wigner perturbation theory can be developed for both the single reference function case and the multireference function case using a common formalism. This contrasts with the situation for Rayleigh-Schrodinger perturbation theory. We shall, therefore, consider the single reference and multireference formalisms together. 

This use of the unperturbed population to calculate the change in revenue is a crade example of a certain level of estimation (called first order ) in Rayleigh-Schrodinger perturbation theory. We will now proceed to develop the theory more formally in the context of wavefunctions and energies. The above example has been presented to encourage the reader to anticipate that there is a lot of simple good sense in the results of perturbation theory even though the mathematical development is rather cumbersome and unintuitive. 

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. 

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