Many-electron correlation problem perturbative approaches

Jiri Cizek s research program centers on the quantum theory of molecular electronic structure and related developments in quantum chemical methodology, coupled-cluster approaches to many-electron correlation problems,105 large-order perturbation theory,106 dynamical groups and exactly solvable models, lower bounds, and the use of symbolic computation language in physics and in chemistry. 

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... 

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. 

Of course, orbital models, such as the widely used Hartree-Fock approximation neglect the effects of electron correlation. One approximation which forms the basis of a computationally tractable approach to the electron correlation problem in atoms and molecules is the many-body perturbation... 

Analysis of Different Approaches to the Electron Correlation Problem in Molecules. - As well as forming the basis for electron correlation energy calculations the many-body perturbation theory has been shown to provide an invaluable tool for the analysis of other approaches to the correlation problem and can often serve to identify the strengths and weakness of a particular method. 

Finite basis set Hartree-Fock calculations yield not only an approximation for the occupied orbitals but also a representation of the spectrum which can be used in the treatment of correlation effects. In particular, the use of finite basis sets facilitates the effective evaluation of the sum-over-states which arise in the many-body perturbation theory of electron correlation effects in atoms and molecules. Basis sets have been developed for low order many-body perturbation theoretic treatments of the correlation problem which yield electron correlation energy components approaching the suh-milliHartree level of accuracy [20,21,22]. 

M0ller-Plesset second-order perturbation theory [78,162] is the most widely used approach to the electron correlation problem in contemporary ab initio molecular electronic structure studies [163-168], For systems which are well described by a single determinantal reference function, this theory - based on the use of Rayleigh-Schrodinger perturbation theory to describe electron correlation corrections to the Hartree-Fock independent electron model - affords an approach which combines accuracy with computational efficiency. The method, which is often designated mp2 , is based on the lowest order of the many-body perturbation theory expansion to take account of correlation effects. 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

A general approach to the intramonomer correlation problem is known as the many-electron (or many-body) SAPT method88,141 213-215. In this method the zeroth-order Hamiltonian H0 is decomposed as H0 = F + W, where F = FA + FB is the sum of the Fock operators, FA and FB, of monomer A and B, respectively, and W is the intramonomer correlation operator. The correlation operator can be written as W = WA + WB, where Wx = Hx — Fx, X = A or B. The total Hamiltonian can be now be represented as H = F + V + W. This partitioning of H defines a double perturbation expansion of the wave function and interaction energy. In the SRS theory the wave function is obtained by expanding the parametrized Schrodinger equation as a power series in and A,... 

Another approach to the problem of computing the electron correlation energy is the M0ller54-Plesset55 (MP) perturbation theory (which is philosophically akin to the many-body perturbation theory of solid-state physics). The mechanics are the conventional Rayleigh-Schrodinger perturbation theory One introduces a generalized electronic Hamiltonian Hi, where... 

Note that all of the above expressions are written in terms of single electron functions and no reference is made to many-electron functions. This is a fundamental characteristic of the many-body perturbation theoretic approach to the correlation problem. 

Many-body perturbation theory in its lowest order form, which is often designated MP2, continues to be the most widely used of the ab initio approaches to the molecular electronic structure problem which go beyond an independent particle model and take account of the effects of electron correlation. The main focus of the present review has been on some of the emerging fields in which MP2 calculations are being carried out. Obviously, within the limited space available it has not possible to cover all of the fields of application. Some selectivity has been necessary, but the choices made do provide a snapshot of the range of contemporary applications of chemical modelling using many-body perturbation theory. 

In order to have a more complete picture of the many-body problem for more general or complicated cases that DFT could help to treat, it is necessary to make a correspondence with the use of many-body perturbation theory. Under this wider classification of perturbation theory are included all the methods that treat electron correlation beyond the Hartree-Fock level, including configuration interaction, coupled cluster, etc. This perturbational approach has traditionally been known as second quantization, and its power for some applications can be seen when dealing with problems beyond the standard quantum mechanics. 

The theoretical description of any many body system is usually approached in two distinct stages. First, the solution of some independent particle model yielding a set of quasi-particles, or dressed particles, which are then used to formulate a systematic scheme for describing the corrections to the model. Perturbation theory, when developed with respect to a suitable reference model, affords the most systematic approach to the correlation problem which today, because it is non-iterative and, therefore, computationally very efficient, forms the basis of the most widely used approaches in contemporary electronic structure calculations, particularly when developed with respect to a Moller-Plesset zero order Hamiltonian. 

There are finally attempts to apply diagrammatic techniques of many-body perturbation theory S ), with the summation of certain diagrams to infinite order, to the correlation problem in atoms and molecules. A close relationship between this kind of approach and the independent electron-pair approximation has been demonstrated >. 

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