Applications of Many-body Perturbation Theory

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. 

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). 

As in my three most recent reviews, "" I have attempted to provide a snapshot of the many applications of many-body perturbation theory in its simplest form, i.e. Moller-Plesset theory designated MP2, during the period under review by performing a literature search for publications with the string MP2 in the title only. During the period covered by the present report, a total of 80 papers were discovered satisfying this criterion. 

This report covers developments in the theory and application of many-body perturbation theory to molecular systems during the period June 2003 through to May 2005. It thus continues three earlier reviews in this series1 3 which, in turn, built on my report, written about twenty-five years ago, for a previous Specialist Periodical Reports series published in 1981.4... 

Applications of many-body perturbation theory to the molecular electronic structure problem have been published during the reporting period in an ever increasing range of scientific areas. In particular, in its second order form, many-body perturbation theory continues to be the most widely used ab initio quantum chemical method for describing the effects of electron correlation. A review of the numerous applications reported during the period under consideration is given in Section 4. 

In my previous report3,1 surveyed progress that is being made in extending the range of applicability of many-body perturbation theory to larger systems making it an alternative to the widely employed density functional theory. 

This report describes progress made on the application of many-body perturbation theory to the problem of molecular structure in recent years. However, since it is almost two decades since the last report51 on this subject in Specialist Periodical Reports, the opportunity will be taken to provide some historical perspective on the developments during the past two decades. 

The applications of many-body perturbation theory in contemporary research in the molecular sciences are manifold and it is certainly not possible to describe more than a mere fraction of the enormous number of publications which have exploited this approach to the molecular structure problem over recent years. Calculations based on second order many-body perturbation theory or MP2 theory are particularly prevalent offering unique advantages in terms of efficiency and accuracy over many other theoretical and computational approaches. Here, we shall briefly describe the use of graphical user interfaces and then concentrate on two recent applications of the many-body perturbation theory which have established new levels of precision. 

This chapter covers developments in the theory and the applications of many-body perturbation theory to the molecular electronic stracture problem during the period June 1999 through to May 2001. It thus provides a snapshot of both theoretical developments and application areas at the turn of the century. The emphasis in this review is on applications, of which there are an ever growing number, particularly using finite- and low-order theory. 

Schulman, J. M. and Kaufman, D. N. (1970). Application of many-body perturbation theory to the hydrogen molecule. J. Chem. Phys., 53, 477-484. 

The application of many-body perturbation theory to molecules involves the direct application of the Rayleigh-Schrodinger formalism with specific choices of reference Hamiltonian. The most familiar of these is that first presented by Mpller and Plesset... 

Computational and Experimental Chemistry Developments and Applications provides an eclectic survey of contemporary problems in theoretical chemistry and applied chemistry. The problems addressed in its pages vary from the prediction of a novel spiro quantum chemistry edifice of carbon-based structures to applications of many body perturbation theory in helium-like ions, and also from the elucidation of a novel d5mamic elasticity theory applied to carbon, to the description of equalization principles in chemistry. The book is divided in to two main parts. Part I entitled Exotic Carbon Allotropes is in four chapters and describes the theoretical work of Bucknum et al. applied to the emuneration of novel carbon patterns and their properties. Part II entitled, New Developments in Computational and Experimental Chemistry comprises the last eight chapters of the book and provides an interesting survey of contemporary problems in theoretical chemistry and applied chemistry. 

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). 

In previous reports to this series, the increasing use of many-body perturbation theory in molecular electronic structure studies was measured by interrogating the Institute for Scientific Information (ISI) databases. In particular, I determined the number of incidences of the string MP2 in titles and/or ke5rwords and/or abstracts. This acronym is frequently associated with the simplest form of many-body perturbation theory. This assessment of the use of second order many-body perturbation theory will undoubtedly miss many routine applications but should serve to convey both the extent and the breadth of contemporary application areas. 

This report has continued our biennial survey of Many-body Perturbation Theory and Its Application to the Molecular Structure Problem covering the reporting period assigned to this volume June 2003 to May 2005. 

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. 

Coupled electron pair and cluster expansions. - The linked diagram theorem of many-body perturbation theory and the connected cluster structure of the exact wave function was first established by Hubbard211 in 1958 and exploited in the context of the nuclear correlation problem by Coester212 and by Coester and Kummel.213 Cizek214-216 described the first systematic application to molecular systems and Paldus et al.217 described the first ab initio application. The analysis of the coupled cluster equations in terms of the many-body perturbation theory for closed-shell molecular systems is well understood and has been described by a number of authors.9-11,67,69,218-221 In 1992, Paldus221 summarized the situtation for open-shell systems one must nonetheless admit... 

Many-body methods, based on the linked-cluster expansion (LCE), were first developed by Brueckner [1] and Goldstone [2] in the 1950s for nuclear physics problems. Perturbation-theory applications to atomic and molecular systems (in a numerical, one-center frame) were pioneered by Kelly [3] in the early 1960s. Basis sets were later introduced, first in second-order [4] and then in third-order [5]. The 1970s saw a proliferation of molecular applications with basis sets, under the names of many-body perturbation theory (MBPT) [6] or the Moller-Plesset method [7]. Nowadays, many-body methods offer some of the most powerful tools in the quantum chemistry arsenal, in particular the coupled-cluster (CC) method, and are available in many widely used quantum chemistry program packages. 

Application of the above-described method to different poly-mers " has yielded reasonably good agreement with experiment only (see the next section) if one employs a good basis set and substitutes into the Green matrix elements (8.22) not the HF one-electron eneigies, but rather the quasi-particle energies, which contain also correlation contributions at least in the second order of many-body perturbation theory (see Section 5.3). 

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. 

---