The Many-body Perturbation Theory

It should perhaps be stated at this point that the use of diagrams in the many-body perturbation theory is not obligatory. The whole of the theoretical apparatus can be set up in entirely algebraic terms. However, the diagrams are both more physical and easier to handle than the algebraic expressions and it is well worth the effort required to familiarize oneself with the diagrammatic rules and conventions. 

The linked diagram expansion has, indeed, been derived by many authors and we shall, therefore, content ourselves with a brief outline of the Goldstone derivation here referring the interested reader elsewhere for full details.8-5 

The many-body perturbation theory is developed in terms of some set of single particle states, f Pi which are eigenfunctions of some single-particle operator, /, 

Use of the interaction representation in time-dependent perturbation theory and an adiabatic switching, ( a 0 of the perturbation yields the evolution operator 

Analysis of the products of field operators in these equations leads to a representation of the wave function and of the level shift in terms of diagrams of the type first introduced by Feynman. These diagrams provide a simple pictorial description of electron correlation effects in terms of the particle-hole formalism. 

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Ei=i N F(i), perturbation theory (see Appendix D for an introduetion to time-independent perturbation theory) is used to determine the Ci amplitudes for the CSFs. The MPPT proeedure is also referred to as the many-body perturbation theory (MBPT) method. The two names arose beeause two different sehools of physies and ehemistry developed them for somewhat different applieations. Later, workers realized that they were identieal in their working equations when the UHF H is employed as the unperturbed Hamiltonian. In this text, we will therefore refer to this approaeh as MPPT/MBPT. 

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. 

In the many-body perturbation theory the non-relativistic Hamiltonian H is partitioned in the following way ... 

At the correlated level the many-body perturbation theory is applied, the localized version of which (LMBPT) has already proven to be useful in the study of molecular electronic structure. The LMBPT is a double perturbation theory, and the perturbational correction are calculated as ... 

On the convergence of the many-body perturbation theory second-order energy component for negative ions using systematically constructed basis sets of primitive Gaussian-type functions... 

Using the F ion as a prototype, the convergence of the many-body perturbation theory second-order energy component for negative ions is studied when a systematic procedure for the construction of even-tempered btisis sets of primitive Gaussian type functions is employed. Calculations are reported for sequences of even-tempered basis sets originally developed for neutral atoms and for basis sets containing supplementary diffuse functions. 

The many-body perturbation theory [39] [40] [41] was used to model the electronic structure of the atomic systems studied in this work. The theory developed with respect to a Hartree-Fock reference function constructed from canonical orbitals is employed. This formulation is numerically equivalent to the M ler-Plesset theory[42] [43]. 

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. 

The ab initio calculated energies were obtained at the SCF level, followed by the evaluation of the second-order electronic correlation contribution with the many-body perturbation theory [SCF+MBPT(2)]. These calculations were performed on HF/3-21G(d) optimized geometries and include the zero-point vibrational energy corrections. 

The paper is organized as follows. A complete description of the theoretical methods used is given in Section 2 considering, in the different subsections, the Density Functional Theory (DFT) (Section 2.1), the A self-consistent (A-SCF) approach (Section 2.1.1), and the Many-Body perturbation theory (Section 2.2) through the GW (Section 2.2.1) and the Bethe-Salpeter (Section 2.2.2) methods. 

Next the results from the relativistic random-phase approximation (RRPA) and the many-body perturbation theory (MBPT), also shown in Table 5.1, will be discussed. Because both calculations include basically the same electron-electron interactions, rather good agreement exists, and it is sufficient to concentrate only on the RRPA model. 

In terms of basic physical effects included, the calculations of Drake and of Persson et al. are equivalent up to all terms of order a3 (assuming that the Many Body Perturbation Theory expansion has converged sufficiently well), and also terms of order a4Z6 and aAZb. Any difference between the two calculations should therefore scale as a4Z4, at least through the intermediate range of Z. 

Since the Dirac equation is written for one electron, the real problem of ah initio methods for a many-electron system is an accurate treatment of the instantaneous electron-electron interaction, called electron correlation. The latter is of the order of magnitude of relativistic effects and may contribute to a very large extent to the binding energy and other properties. The DCB Hamiltonian (Equation 3) accounts for the correlation effects in the first order via the Vy term. Some higher order of magnitude correlation effects are taken into account by the configuration interaction (Cl), the many-body perturbation theory (MBPT) and by the presently most accurate coupled cluster (CC) technique. 

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... 

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... 

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