Many-body perturbation theory, applications

The main advantage suggested by the use of the localized many-body perturbation theory (LMBPT) is that the local effects can be separated from the non-local ones. The summations in the corrections at a given order can be truncated. As to the practical applicability of the localized representation, a localization (separation) method, satisfying a double requirement is highly desired. Well-localized (separated) orbitals with small off-diagonal Lagrangianmultipliers are required (Kapuy etal., 1983). 

K. F. Freed, Tests and applications of complete model space quasidegenerate many-body perturbation theory for molecules, in Many-Body Methods in Quantum Chemistry (U. Kaldor, ed.), Springer, Berlin, 1989, p. 1. 

Ei=i n F(i), perturbation theory (see Appendix D for an introduction to time-independent perturbation theory) is used to determine the Cj amplitudes for the CSFs. The MPPT procedure is also referred to as the many-body perturbation theory (MBPT) method. The two names arose because two different schools of physics and chemistry developed them for somewhat different applications. Later, workers realized that they were identical in their working equations when the UHF H is employed as the unperturbed Hamiltonian. In this text, we will therefore refer to this approach as MPPT/MBPT. 

Although HF theory is useful in its own right for many kinds of investigations, there are some applications for which the neglect of electron correlation or the assumption that the error is constant (and so will cancel) is not warranted. Post-Hartree-Fock methods seek to improve the description of the electron-electron interactions using HF theory as a reference point. Improvements to HF theory can be made in a variety of ways, including the method of configuration interaction (Cl) and by use of many-body perturbation theory (MBPT). It is beyond the scope of this text to treat Cl and MBPT methods in any but the most cursory manner. However, both methods can be introduced from aspects of the theory already discussed. 

We now consider the use of perturbation theory for the case where the complete operator A is the Hamiltonian, H. Mpller and Plesset (1934) proposed choices for A and V with this goal in mind, and the application of their prescription is now typically referred to by the acronym MPn where n is the order at which the perturbation theory is truncated, e.g., MP2, MP3, etc. Some workers in the field prefer the acronym MBPTn, to emphasize the more general nature of many-body perturbation theory (Bartlett 1981). 

There have been attempts to generalize the many-body perturbation theory to cover the relativistic regime in a rigorous and systematic manner [238-241]. Unfortunately, practical applications, so far, are only to simple atoms or ions. 

Since his appointment at the University of Waterloo, Paldus has fully devoted himself to theoretical and methodological aspects of atomic and molecular electronic structure, while keeping in close contact with actual applications of these methods in computational quantum chemistry. His contributions include the examination of stability conditions and symmetry breaking in the independent particle models,109 many-body perturbation theory and Green s function approaches to the many-electron correlation problem,110 the development of graphical methods for the time-independent many-fermion problem,111 and the development of various algebraic approaches and an exploration of convergence properties of perturbative methods. His most important... 

S. Wilson, K. Jankowski, and J. Paldus, Int.]. Quantum Chem., 28,525 (1985). Applicability of Non-Degenerate Many-Body Perturbation Theory to Quasi-Degenerate Electronic States. II. A Two-State Model. 

An interesting application of many-body perturbation theory using a discrete orbital basis has been reported by Robb.162 In calculations on BH, comparison was made between the results and those of Houlden et a/.152 using Cl. Most of the pair-pair interaction energy can be recovered by this method. 

Wormer PES, Hettema H (1992) Many-body perturbation theory of frequency-dependent polarizabilities andVan derWaals coefficients Application to H20-H20 and A r Nil,. J Chem Phys 97 5592-5606... 

The pioneering work on the application of the many-body perturbation theory to atomic and molecular systems was performed by Kelly.5-17-21 He applied the method to atoms using numerical solutions of the Hartree-Fock equations. Many other calculations on atomic systems were subsequently... 

This article is divided into seven parts. The many-body perturbation theory is discussed in the next section. The algebraic approximation is discussed in some detail in section 3 since this approximation is fundamental to most molecular applications. In the fourth section, the truncation of the many-body perturbation series is discussed, and, since other approaches to the many-electron correlation problem may be regarded as different ways of truncating the many-body perturbation expansion, we briefly discuss the relation to other approaches. Computational aspects of many-body perturbative calculations are considered in section 5. In section 6, some typical applications to molecules are given. In the final section, some other aspects of the many-body perturbation theory of molecules are briefly discussed and possible directions for future investigations are outlined. 

Quasi-degeneracy Effects.—The convergence properties of the non-degenerate formulation of the many-body perturbation theory deteriorate when quasidegeneracy is present in the reference spectrum. In view of its simplicity, however, there is considerable interest in exploring the range of applicability of the nondegenerate formalism. 

General Remarks.—In this section, the computational aspects of the application of the many-body perturbation theory to molecular correlation energies are discussed. 

The type of correlated method that has enjoyed the most widespread application to H-bonded systems is many-body perturbation theory, also commonly referred to as Mpller-Plesset (MP) perturbation theory This approach considers the true Hamiltonian as a sum of its Hartree-Fock part plus an operator corresponding to electron correlation. In other words, the unperturbed Hamiltonian consists of the interaction of the electrons with the nuclei, plus their kinetic energy, to which is added the Hartree-Fock potential the interaction of each electron with the time-averaged field generated by the others. The perturbation thus becomes the difference between the correct interelectronic repulsion operator, with its instantaneous correlation between electrons, and the latter Hartree-Fock potential. In this formalism, the Hartree-Fock energy is equed to the sum of the zeroth and first-order perturbation energy corrections. 

Dr Steven Wilson Many body perturbation theory and its application to the molecular structure problem... 

This review continues four earlier biennial reviews published in this series " in 2000, 2002, 2004 and 2006. In turn, these reviews built on my report, written almost three decades ago, for a previous Specialist Periodical Reports series (Theoretical Chemistry) and published in 1981. This earlier review was one of the first to survey the application of many-body methods, and, in particular, many-body perturbation theory to the molecular structure problem. 

In section 4, we turn our attention to the many applications of second order many-body perturbation theory in chemical modelling. We provide a brief S5mopsis of the applications of MP2 theory during the reporting period. 

This, being the fifth in a series of biennial reports to the series Chemical Modelling-Applications and Theory, it brings to ten years the period for which detailed reviews of the literature of many-body perturbation theory and its application to the molecular structure problem have been presented. (The first report, published in 2000, updated a previous report written some twenty years earlier for another series. For the first and second reports only applications of many-body perturbation theory to the problem of molecular electronic structure were considered, but with the third and subsequent reports the word electronic was dropped to indicate a broader remit). 

Published in 2002, the second report covered the period June 1999 to May 2001 and provided an opportunity to review the wide range of applications to which many-body perturbation theory in its simplest form, that is Moller-Plesset perturbation theory through second order, was being put at the turn of the millennium. The main development considered in this second report were ... 

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