Lennard-Jones-Brillouin-Wigner

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... 

Explicitly, the first few terms in the Lennard-Jones Brillouin Wigner perturbation series take the form... 

The Lennard-Jones Brillouin Wigner perturbation expansion is a simple geometric series. However, it contains the unknown exact energy within the denominators. This expansion is, therefore, not a simple power series in the perturbation. 

Rayleigh-Schrodinger Perturbation Theory.—In Rayleigh-Schrodinger perturbation theory the unknown energy in the denominators of the Lennard-Jones Brillouin Wigner expansion is avoided. This enables a size-consistent theory to be derived. 

This should be compared with the energy coefficients in the Lennard-Jones, Brillouin, Wigner expansion (23) which can be re-written in the form... 

The Lennard-Jones, Brillouin, Wigner perturbation expansion is not a simple power series in A since each depends on the exact energy, S. Each energy coefficient in the Rayleigh-Schrodinger perturbation expansion consists of a principal term of the form... 

The Rayleigh-Schrodinger peiturbation theory can be derived from the Lennard-Jones-Brillouin-Wigner perturbation theory by expanding the energy-dependent denominators which occur in the latter... 

J. E. Lennard-Jones [37]. (Indeed, some authors [38,39] refer to the method as Lennard-Jones-Brillouin-Wigner perturbation theory .) Subsequently, L. Brillouin published his famous paper in 1932 [40] whilst E.P. Wigner s paper appeared some 2 years later [41]. In this section, we provide a brief synopsis of each of these important papers. 

As we have noted already, some authors, such as Dalgarno [38] and Wilson [39], term what is nowadays called Brillouin-Wigner perturbation theory Lennard-Jones-Brillouin-Wigner perturbation theory in recognition of the seminal contribution of Sir John Lennard-Jones. 

The perturbation theory of Lennard-Jones, Brillouin, and Wigner is not size consistent. 

The perturbation expansion of Lennard-Jones, Brillouin and Wigner does not lead to expressions which are directly proportional to the number of electrons in the system being studied. 

Research over the past two decades has demonstrated that, after being abandoned for almost half a century in favor of Rayleigh-SchrodingCT-based many-body formalism, the Brillouin-Wigner approach has much to offer in studies of the quantum many-body problem. This short review should convince the reader that the formalism of Brillouin and Wigner, and of Lennard-Jones, has much to contfibute to modern many-body theory. 

See Chap. 1 of reference [51] for a brief discussion of the contribution of Lennard-Jones to Brillouin-Wigner theory. 

The seminal paper on what is nowadays called Brillouin-Wigner perturbation theory by Lennard-Jones was published in the Proceedings of the Royal Society of London in 1931. It was communicated by R.H. Fowler and received on 1st September, 1930. Here we reproduce Lennard-Jones introduction ... 

Wigner s contribution to Brillouin-Wigner perturbation theory appeared some 2 years after Brillouin s paper in 1935. It was published in Mathematischer und Naturwissenschaftlicher Anzeiger der Ungarischen Akademie der Wi -senschaften. Wigner cites the earlier work of Brillouin but not that of Lennard-Jones. The first part of Wigner s paper is in Hungarian ... 

The papers by Lennard-Jones, by BriUouin and by Wigner represent the genesis of what is now termed Brillouin-Wigner perturbation theory . 

This perspective on Brillouin-Wigner perturbation theory was mentioned by Lennard-Jones in his seminal 1930 paper [37]. It was also described in the review byDalgarno [38] published in 1961. 

This advantage of the Brillouin-Wigner perturbation theory was recognized in the original papers by Lennard-Jones [37] and of Brillouin [40]. It is also described in the review by Dalgarno [38]. 

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