Many-body perturbation theory MBPT correlation

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. 

The reconstruction functionals may be understood as substantially renormalized many-body perturbation expansions. When exact lower RDMs are employed in the functionals, contributions from all orders of perturbation theory are contained in the reconstructed RDMs. As mentioned previously, the reconstruction exactly accounts for configurations in which at least one particle is statistically isolated from the others. Since we know the unconnected p-RDM exactly, all of the error arises from our imprecise knowledge of the connected p-RDM. The connected nature of the connected p-RDM will allow us to estimate the size of its error. For a Hamiltonian with no more than two-particle interactions, the connected p-RDM will have its first nonvanishing term in the (p — 1) order of many-body perturbation theory (MBPT) with a Hartree-Fock reference. This assertion may be understood by noticing that the minimum number of pairwise potentials V required to connectp particles completely is (p — 1). It follows from this that as the number of particles p in the reconstmcted RDM increases, the accuracy of the functional approximation improves. The reconstmction formula in Table I for the 2-RDM is equivalent to the Hartree-Fock approximation since it assumes that the two particles are statistically independent. Correlation corrections first appear in the 3-RDM functional, which with A = 0 is correct through first order of MBPT, and the 4-RDM functional with A = 0 is correct through second order of 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. 

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. 

The details of SAPT are beyond the scope of the present work. For our purposes it is enough to say that the fundamental components of the interaction energy are ordinarily expanded in terms of two perturbations the intermonomer interaction operator and the intramonomer electron correlation operator. Such a treatment provides us with fundamental components in the form of a double perturbation series, which should be judiciously limited to some low order, which produces a compromise between efficiency and accuracy. The most important corrections for two- and three-body terms in the interaction energy are described in Table 1. The SAPT corrections are directly related to the interaction energy evaluated by the supermolecular approach, Eq.(2), provided that many body perturbation theory (MBPT) is used [19,28]. Assignment of different perturbation and supermolecular energies is shown in Table 1. The power of this approach is its open-ended character. One can thoroughly analyse the role of individual corrections and evaluate them with carefully controlled effort and desired... 

Well-known procedures for the calculation of electron correlation energy involve using virtual Hartree-Fock orbitals to construct corresponding wavefunctions, since such methods computationally have a good convergence in many-body perturbation theory (MBPT). Although we know the virtual orbitals are not optimized in the SCF procedure. Alternatively, it is possible to transform the virtual orbitals to a number of functions. There are some techniques to do such transformation to natural orbitals, Brueckner orbitals and also the Davidson method. 

Constraints were then applied, such that the number of electrons in a orbitals was fixed at six and the number of electrons in n orbitals at four. The results of the two calculations are presented in Table I, where the effects on some of the properties of the nitrogen molecule are given. For comparison the corresponding SCF values are also presented. As can be seen from these results, the effects of the constraints on the CASSCF wavefunction are not negligible. They are, however, considerably smaller than the difference between the CASSCF and the SCF values. Better agreement with experiment can only be obtained by including dynamical correlation effects, for example, by means of a large multireference Cl calculation or a many-body perturbation theory (MBPT) calculation. ... 

In Chapter 2, the author describes state- and property-specific quantum chemistry including comments on certain aspects of the separation of electron correlation into its dynamical and nondynamical parts. The account contains also a historic appraisal of multiconfigurational schemes comparing Brueckner-Goldstone many-body perturbation theory, MBPT,... 

Many-body perturbation theory (MBPT) has been utilized as a convenient way of taking account of electron correlation beyond the Hartree-Fock (HF) approximation. In particular, its single-reference version is now fully established. M0ller-Plesset perturbation method [11], up to the fourth order, is provided as a standard tool in most... 

The ab initio HF treatment with correlation of large molecules is by no means a simple problem. On the other hand, without taking into account correlation effects only the ground state properties of a molecule in its equilibrium geometry can be calculated in a more or less reliable way. Further the standard method for the treatment of correlation, the configuration interaction (Cl) method, cannot be used well even for medium size systems, because it is not size consistent. Therefore, one has to apply either some form of many body perturbation theory (MBPT) or the coupled cluster (CC) approach, both in a certain approximation (both methods are size consistent). 

The V includes the effect of the finite nuclear size, while some finer effects, like QED, can be added to hpcB perturbatively. The DCB Hamiltonian in this form contains all effects through the second order in a, the fine-structure constant. Correlation effects are taken into account by configuration interaction (Cl), many-body perturbation theory (MBPT) and presently by the most accurate technique, CCSD (see the Chapter of U. Kaldor et al. in this issue). 

If effects from electron correlation on parity violating potentials shall be accounted for in a four-component framework, the situation becomes more complicated than in the Dirac Hartree-Fock case. This is related to the fact, that in four-component many body perturbation theory (MBPT) or in a four-component coupled cluster (CC) scheme the reduced density matrices on the respective computational level are required in order to determine the parity violating potentials. Since these densities were not available in analytic form, Thyssen, Laerdahl and Schwerdtfeger [153] used a finite field approach to compute parity violating potentials in a four-component framework on a correlated level. This amounts to adding the parity violating operator with different scaling factors A to the... 

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