Many-body perturbation theory factorized

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 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... 

If one wishes to use RSPT to perform ab initio quantum-chemical calculations that yield size-consistent energies, then care must be taken in computing the factors that contribute to any given E For example, if were calculated as in Eq. (3.28), limitations of numerical precision might not give rise to the exact cancellation of size-inconsistent terms, which we know should occur. This would certainly be the case for an extended system (for which the size-inconsistent terms would dominate). In addition, it is unpleasant to have a formalism in which such improper terms arise in the first place. It is therefore natural to attempt to develop approaches to implementing RSPT in which the size-inconsistent factors are never even computed. Such an approach has been developed and is commonly referred to as many-body perturbation theory (MBPT). The method of implementing MBPT is discussed once we have completed the present treatment of RSPT. 

We know that the Rayleigh-Schrodinger perturbation theory series leads directly to the many-body perturbation theory by employing the linked diagram theorem. This theory uses factors of the form Eq—Ek) as denominators. Furthermore, this theory is fully extensive it scales linearly with electron number. The second term... 

The 6 (actually (-ifi) ) factor stems from the time integration in the trace. Thus, identifying the grand potential in eq. (2.32) with the grand potential from many-body perturbation theory-in eg.(2.31)... 

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