Rayleigh-Schrodinger perturbation theory, multi-reference

In this section, we shall follow the convention that. .. are eigenfunctions of 

Let m be the dimension of the model space. Furthermore, let there be m well-defined eigenfunctions of the full Hamiltonian, Jf, 

The functions P are termed the model functions. It is clear that 

It is assumed that these model functions are Unearly independent and that they span the entire model space. There is, therefore, a one-to-one correspondence between the m model functions and m exact solutions of the Schrddinger equation. 

Lindgren defines a state-independent wave operator, i , which transforms all the model functions into the corresponding exact functions 

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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. 

Multi-reference Brillouin-Wigner theory overcomes the intruder state problem because the energy is contained in the denominator factors. Calculations are therefore state-specific , that is, they are performed for one state at a time. This is in contrast to multi-reference Rayleigh-Schrodinger perturbation theory, which is applied to a manifold of states simultaneously. Multi-reference Brillouin-Wigner perturbation theory is applied to a single state. Wenzel and Steiner [77] write ... 

Figure 1.2. In multi-reference Rayleigh-Schrodinger perturbation theory, states from outside the reference space, IP, which assume an energy below that of any state among the reference set for —1 < A < 0 are termed backdoor intruder states. Unlike the intruder states corresponding to 0 < A < +1, which often have a physical origin, backdoor intruder states are frequently unphysical.

Whereas the multi-reference Rayleigh-Schrodinger perturbation theory approximates a manifold of states simultaneously, the multi-reference Brillouin-Wigner perturbation theory approach is applied to a single state - it is said to be state-specific . The multi-reference Brillouin-Wigner perturbation theory avoids the intruder state problem. If a particular Brillouin-Wigner-based formulation is not a valid many-body method, then a posteriori correction can be applied. This correction is designed to restore the extensivity of the method. This extensivity may be restored approximately... 

MR-BWPT multi-reference Brillouin-Wigner perturbation theory MR-RSPT multi-reference Rayleigh-Schrodinger perturbation theory... 

In spite of this progress, problems remain and the description of electron correlation in molecules will remain an active field of research in the years ahead. The most outstanding problem is the development of robust theoretical apparatus for handling multi-reference treatments. Methods based on Rayleigh-Schrodinger perturbation theory suffer from the so-called intruder state problem. In recent years, it has been recognized that Brillouin-Wigner perturbation theory shows promise as a robust technique for the multi-reference problem which avoids the intruder state problem. 

The relationship between single-reference Brillouin-Wigner perturbation theory and its Rayleigh-Schrodinger counterpart is well known, but for completeness we include a brief account of the single-reference case in Section 4.4.1 before turning to the multi-reference case in Section 4.4.2. 

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