First-order many-body theory

The calculation describes the differential cross sections and 2 P electron impact coherence parameters quite well. For the 2 P differential cross section it is contrasted with a variant of the distorted-wave Born approximation, first-order many-body theory, where the distorted waves are both calculated in the initial-state Hartree—Fock potential. 

Kucharski SA, Bartlett RJ (1986) First-order many-body perturbation theory and its relationship to various coupled-cluster approaches. Adv Quantum Chem 18 281-344... 

For systems containing light elements, however, Eq. (2) currently offers the most accurate method for calculating of Ep from first principles and for the estimation of E g,.,. for the calibration of non-relativistic many-body theory. For diatomic molecules high precision accurate numerical methods are available for the calculation of Epp, and, furthermore, Egg,., is about an order of magnitude larger than Ep for light elements. 

The Hartree-Fock ground state of the F anion is described by orbitals of s Emd of p symmetry. In the first part of this study, attention was restricted to the convergence of the second order many-body perturbation theory component of the correlation energy for stematically constructed even-tempered basis sets of primitive Gaussian-typ>e functions of s and p symmetry. 

Foundations to the CC methods were laid by Coester and Kuemmel,1 Cizek,2 Hubbard,3 Sinanoglu,4 and Primas,5 while Cizek2 first presented the CC equations in explicit form. Also Hubbard3 called attention to the equivalence of CC methods and infinite-order many-body perturbation theory (MBPT) methods. From this latter viewpoint, the CC method is a device to sum to infinity certain classes of MBPT diagrams or all possible MBPT diagrams when the full set of coupled-cluster equations is solved. The latter possibility would require solving a series of coupled equations involving up to IV-fold excitations for N electrons. Practical applications require the truncation of the cluster operators to low N values. 

The first purpose of the present paper is to set the approximate theory which was developed in a series of papers3-5 , into a more general context of the many body theory. The thinking behind the rather drastic simplifications of Refs.3-51 is exposed in Sections 2 and 3. Our further aim here is to present our previous results, scattered in literature, in an unified text, in order to emphasize their logical interdependence. This is done in Sections 3 and 4. The Section 4 contains some previously unpublished calculations added here for the sake of completeness. 

We can now interpret the role of the components of the extended state y A)) in the following way According to the orthonormality relation (28), the extended states have always the p-norm 1. Since the first (cind physical) component contributes in zeroth order only for ph index pairs, the other components (the extensions) have to take over in the remaining cases. The second component of the extended state of Eq. (47) or Eq. (48) is a kind of symmetric partner of the first component (which is also present in the polarisation propagator of traditional many-body theory). On its own, it leads to a negative contribution to the p-product because the metric p introduces... 

The second-order many-body perturbation theory Goldstone energy diagrams are shown in Figure 3. The first of these is the direct term and the second the... 

These results are equal to the total first-order pair correlation energy, obtained in Exercise 5.3, for the dimer using localized orbitals. The total first-order pair correlation energy is identical to the second-order many-body perturbation result for the correlation energy (see Chapter 6). The above results are a reflection of the fact that many-body perturbation theory is invariant to unitary transformations of degenerate orbitals. 

On the other hand, the orbital-dependent treatment of correlation represents a much more serious challenge than that of exchange The systematic derivation of such functionals via standard many-body theory leads to rather complicated expressions. Their rigorous application within the OPM not only requires the evaluation of Coulomb matrix elements between the complete set of KS states, but, in principle, also relies on the knowledge of higher order response functions. In practical calculations, these first-principles functionals necessarily turn out to be rather inefficient, even if they are only treated perturbatively. In addition, the potential resulting from a large class of such functionals is non-physical for finite systems. Both problems are related to the presence of unoccupied states in the functionals which seems inevitable as soon as some variant of standard many-body theory is used for the derivation. 

Accuracy of the SLG approximation can be improved by perturbation theory. Second quantization provides us a powerful tool in developing a many-body theory suitable to derive interbond delocalization and correlation effects. The first question concerns the partitioning of the Hamiltonian to a zeroth-order part and perturbation. LFsing a straightforward generalization of the Moller-Plesset (1934) partitioning, the zeroth-order Hamiltonian is chosen as the sum of the effective intrabond Hamiltonians ... 

For second-order many body or M0ller-Plesset perturbation theory (MBPT2 or MP2 see M0ller-Plesset Perturbation Theory), the first-order wavefunction, I l, can be written as... 

In Bohmian mechanics, the way the full problem is tackled in order to obtain operational formulas can determine dramatically the final solution due to the context-dependence of this theory. More specifically, developing a Bohmian description within the many-body framework and then focusing on a particle is not equivalent to directly starting from the reduced density matrix or from the one-particle TD-DFT equation. Being well aware of the severe computational problems coming from the first and second approaches, we are still tempted to claim that those are the most natural ways to deal with a many-body problem in a Bohmian context. 

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]. 

This work (actually very difficult to read, and using a very heavy formalism) had the effect of a bomb in Brussels. Prigogine associated himself with Robert Brout (who was at that time a postdoc in Bmssels) in order to understand, deepen, and develop Van Hove s ideas. The first result of this collaboration was a basic paper (1956, MSN. 12) on the general theory of weakly coupled classical many-body systems. Although still influenced by Van Hove s paper, this work by Brout and Prigogine is a generalization of the latter, as well as a simpler and more transparent presentation. 

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. 

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