Perturbation theory, Brillouin- Wigner

In this section, we shall first provide an elementary derivation of the Brillouin-Wigner perturbation theory and then present a comparable introduction to the Rayleigh-Schrodinger perturbation theory. The section concludes with a comparison of the advantages and disadvantages of the two theories. 

We seek to develop approximate solutions of the time-independent Schrodinger eigenvalue equation 

The exact eigenstate for the /U state is constructed using the completeness of the unperturbed basis, m), so that 

We have included the parameter A in eq. (1.13) which is set equal to unity in order to recover the perturbed problem. Equation (1.13) is the basic formula of the Brillouin-Wigner perturbation theory for a single-reference function. 

Erom eq. (1.13) we can develop a series expansion for the exact eigenfunction in powers of A with coefficients depending on the perturbed energy rather than the energy of the model Em, as would be the case in the more familiar Rayleigh-Schrodinger perturbation series. Iterating this basic formula, we find 

Let us go back to our problem we want to have Eq on the left-hand side of the last equation, while - for the time being - Eq occurs on the right-hand sides of both equations. To exit the situation we will treat Eq occurring on the right-hand side as a parameter manipulated in such a way as to obtain equality in both above equations. We may do it in two ways. One leads to Brillouin-Wigner perturbation theory, the other to Rayleigh-Schrodinger perturbation theory. 

Let us decide first at what n = M we terminate the series, i.e. to what order of perturbation theory the calculations will be carried out. Say, M = 3. Let us now take any reasonable value as a parameter of Eq. We insert this value into the right-hand side of eq. (10.80) for Eo and calculate the left-hand side, i.e. Eq. Then let us again insert the new Eq into the right-hand side and continue in this way until self-consistency, i.e. until (10.80) is satisfied. After Eq is known we go to eq. (10.79) and compute ij/Q (through a certain order, e.g., M). 

A unreasonable value will lead to numerical instabilities. Then we will learn that it was unreasonable to take it. 

Brillouin-Wigner perturbation theory has, as seen, the somewhat unpleasant feature that successive corrections to the wave function depend on the M assumed at the beginning. 

We may suspect - and this is true - that the Brillouin-Wigner perturbation theory is not size consistent. 

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Here we ignore any possible perturbation to the site energies at the ends of the chain, n = 1 and n = m.) We apply Brillouin-Wigner perturbation theory (Ohanian 1990), whereby the eigenvalue of a non-degenerate state can be expressed as... 

Lennard-Jones Brillouin Wigner Perturbation Theory.—Let us write the total hamiltonian operator as a sum of a zero-order operator and a perturbation... 

In Lennard-Jones Brillouin Wigner perturbation theory the wave operator is written as... 

Appendix 6. Brillouin-Wigner Perturbation Theory of the Quasi-species. Appendix 7. Renormalization of the Perturbation Theory Appendix 8. Statistical Convergence of Perturbation Theory Appendix 9. Variables, Mean Rate Constants, and Mean Selective Values for the Relaxed Error Threshold... 

APPENDIX 6. BRILLOUIN-WIGNER PERTURBATION THEORY OF THE QUASI-SPECIES... 

C. Huber, and R. J. Bartlett. Integral packages included are VMOL (J. Almlof and P. R. Taylor) VPROPS (P. R. Taylor) ABACUS (T. Helgaker, H. J. Aa Jensen, P. Jprgensen, J. Olsen, and P. R. Taylor). Brillouin-Wigner perturbation theory was implemented by J. Pittner. 

Multireference coupled cluster method based on the Brillouin-Wigner perturbation theory... 

Our interest in Brillouin-Wigner perturbation theory was stimulated by our finding [44] that this theoretical tool proved very useful in the scattering theory. The fundamental equation, known in the scattering theory as Lippmann-Schwinger equation, expresses the scattering operator as... 

Elimination of the size-extensivity error in the MRBWCC theory was suggested by Hubac and Wilson [68]. The Brillouin-Wigner perturbation theory has been known notoriously as a method not furnishing size extensivity. There was therefore every reason to believe that the size-extensivity error in MR BWCCSD originated from the use of the Brillouin-Wigner type resolvent (15) instead of the Rayleigh-Schrodinger... 

Chapter 18 - Multireference coupled cluster method based on the Brillouin-Wigner perturbation theory. Pages 465-481, Petr Carsky, Jin Pittner and Ivan Hubac... 

There has been a revival of interest in the Brillouin-Wigner perturbation theory since it is seen as a possible remedy to the intruder state problem. As... 

Wenzel and Steiner197 have pointed out "... the reference energy in Brillouin-Wigner perturbation theory is the fully dressed energy. .. This feature guarantees the existence of a natural gap and thereby rapid convergence of the perturbation... 

The use of Brillouin-Wigner perturbation theory in describing the many-body problem has been considered recently by Hubac and Wilson.198 These authors use the identity... 

Hubac and his co-workers222"231 have explored the use of Brillouin-Wigner perturbation theory in solving the coupled cluster equations. For the case of a single reference function, this approach is entirely equivalent to other formulations of the coupled cluster equations. However, for the multireference case, the Brillouin-Wigner coupled cluster theory shows some promise in that it appears to alleviate the intruder state problem. No doubt perturbative analysis will help to gain a deeper understanding of this approach. 

Brillouin-Wigner perturbation theory (p. 647) Brueckner function (p. 581)... 

Today, there remain a number of problems in molecular electronic structure theory. The most outstanding of these is undoubtedly the development of a robust theoretical apparatus for the accurate description of dissociative processes which usually demand the use of multi-reference functions. This requirement has recently kindled a renewal of interest in the Brillouin-Wigner perturbation theory and its application to such problems. This contribution describes the application of... 

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