Many-electron Perturbation Theory

We follow in this subsection the many-electron perturbation theory description given in [8]. Often, the Hartree-Fock approximation provides an accurate description of the system and the effects of the inclusion of correlations as, e.g., with the Cl or MCSCF methods, may be considered as important but small corrections. Accordingly, the correlation effects may be considered as a small perturbation and as such treated using the perturbation theory. This is the approach of [126] for the inclusion of correlation effects. 

For the sake of simplification we shall here consider the ground state but mention that the method in principle can be apphed for any state, i.e. also for an excited state. 

Our starting point is the Hartree-Fock equations (4.13) where the HF operator F, (4.14) is a single-electron operator, being a sum of the one-electron operator h, (4.8), local Coulomb J, (4.16) and nonlocal exchange K, (4.17) operators. 

Solving HF equations (4.13) gives not only the N occupied orbitals, but - in principle - a complete set of M (total number of AO basis functions) orbitals, since. F is a Hermitian operator. 

The operator F is a single-electron operator, which we formally wrote as F(i), where i numbers electrons. We define now first the AT-electron operator 

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As we turn up the interaction, EIMP becomes important. We must certainly modify our treatment of the K-shell electron and treat its excitation to all orders in the Born series. However, now we have to worry about the role that the other electrons play. Should we not introduce Pauli blocking operators in the Bom series Now we can make a hole in the L-shell, should we not allow this as a possible final state for the K-shell electron The answer to these questions is yes, if we wish to calculate exdusive cross sections, in perturbation theory. However it is not necessary to use many-electron perturbation theory, and we are only interested in an inclusive cross section. 

Bartlett R J and Purvis G D 1978 Many-body perturbation theory coupled-pair many-electron theory and the importance of quadruple excitations for the correlation problem int. J. Quantum Chem. 14 561-81... 

Another approach to electron correlation is Moller-Plesset perturbation theory. Qualitatively, Moller-Plesset perturbation theory adds higher excitations to Hartree-Fock theory as a non-iterative correction, drawing upon techniques from the area of mathematical physics known as many body perturbation theory. 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamilton operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of 1/2. Various combinations of these assumptions result in final expressions which differ by factors of 1 /2, 1/4 etc. from those given here. In the present book the MOs are always spin-MOs, and conversion of a restricted summation to an unrestricted is always noted explicitly. 

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. 

Many-body perturbation theory (MBPT) for periodic electron systems produces many terms. All but the first-order term (the exchange term) diverges for the electron gas and metallic systems. This behavior holds for both the total and self-energy. Partial summations of these MBPT terms must be made to obtain finite results. It is a well-known fact that the sum of the most divergent terms in a perturbation series, when convergent, leads often to remarkably accurate results [9-11]. 

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 ... 

From this, we may deduce that the relativistic correction to the correlation energy is dominated by the contribution from the s electron pair, and that the total relativistic effect involving the exchange of a single transverse Breit photon is obtained to sufficient accuracy for our present purposes at second-order in many-body perturbation theory. 

Paldus J, Li X (1999) Electron Correlation in Small Molecules Grafting Cl onto CC.203 1 -20 Pipek J, Bog4r F (1999) Many-Body Perturbation Theory with Localized Orbitals - Kapuy s Approach. 203 43 - 61... 

Next, we present some observations concerning the connection between the reconstruction process and the iterative solution of either CSE(p) or ICSE(p). The perturbative reconstruction functionals mentioned earlier each constitute a finite-order ladder-type approximation to the 3- and 4-RDMCs [46, 69] examples of the lowest-order corrections of this type are shown in Fig. 3. The hatched squares in these diagrams can be thought of as arising from the 2-RDM, which serves as an effective pair interaction for a form of many-body perturbation theory. Ordinarily, ladder-type perturbation expansions neglect three-electron (and higher) correlations, even when extended to infinite order in the effective pair interaction [46, 69], but iterative solution of the CSEs (or ICSEs) helps to... 

R. J. Bartlett, Many-body perturbation theory and coupled cluster theory for electron correlation in molecules. Ann. Rev. Phys. Chem. 32, 359 (1981). 

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]. 

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. 

In the previous chapter we have briefly discussed the use of the stationary many-body perturbation theory of the effective Hamiltonian, sketched in Chapter 3, to account for correlation effects. Here we shall continue such studies for electronic transitions on the example of the oxygen isoelectronic sequence. Having in mind (29.32) and the approximation Q(1) iiji the matrix element of the 1-transition operator Oei between the initial state... 

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. 

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