Many-body self energy

Some of the main issues of MBPT can be illustrated by considering the second-order many-body self-energy, which in the HF potential is defined by... 

The error in this term was estimated by various methods [40]. In one, the self-energy was scaled, E (e ) —> AE (e ), with A chosen to reproduce experimental removal energies. Although the values of A 0.8 thus obtained are quite different from unity, E pnc was found to vary by only a few tenths of a percent. This behavior could be traced to accidental cancellations between numerator and denominator contributions in the implicit sum over states (see next section) in the present approach. The cancellations are fortunate, however, because they imply that corrections to the many-body self-energy beyond second order play a reduced role in the PNC effect. 

The pseudopotential is derived from an all-electron SIC-LDA atomic potential. The relaxation correction takes into account the relaxation of the electronic system upon the excitation of an electron [44]- The authors speculate that ... the ability of the SIRC potential to produce considerably better band structures than DFT-LDA may reflect an extra nonlocality in the SIRC pseudopotential, related to the nonlocality or orbital dependence in the SIC all-electron potential. In addition, it may mimic some of the energy and the non-local space dependence of the self-energy operator occurring in the GW approximation of the electronic many body problem [45]. 

The electrostatic energy is calculated using the distributed multipolar expansion introduced by Stone [39,40], with the expansion carried out through octopoles. The expansion centers are taken to be the atom centers and the bond midpoints. So, for water, there are five expansion points (three at the atom centers and two at the O-H bond midpoints), while in benzene there are 24 expansion points. The induction or polarization term is represented by the interaction of the induced dipole on one fragment with the static multipolar field on another fragment, expressed in terms of the distributed localized molecular orbital (LMO) dipole polarizabilities. That is, the number of polarizability points is equal to the number of bonds and lone pairs in the molecule. One can opt to include inner shells as well, but this is usually not useful. The induced dipoles are iterated to self-consistency, so some many body effects are included. 

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]. 

The self-energy determines the many-body accm acy for /rk), (k), thus G E). Following Hedin [7] and Hedin and Lundquist [8], E Ef is expanded in terms of a screened potential W, rather than the bare Coulomb potential v (atomic units are used throughout) ... 

Fig. 4.2. Charge distribution and surface potential in a jellium model, (a) Distribution of the positive charge (a uniform background abruptly drops to zero at the boundary) and the negative charge density, determined by a self-consistent field calculation. (b) Potential energy as seen by an electron. By including all the many-body effects, including the exchange potential and the correlation potential, the classical image potential provides an adequate approximation. (After Bardeen, 1936 see Herring, 1992.)...

The BW form of PT is formally very simple. However, the operators in it depend on the exact energy of the state studied. This requires a self-consistency procedure and limits its application to one energy level at a time. The Rayleigh-Schrodinger (RS) PT does not have these shortcomings, and is, therefore, a more suitable basis for many-body calculations of many-electron systems than the BW form of the theory, it is applicable to a group of levels simultaneously. 

Neglecting off-diagonal elements of the self-energy matrix in the canonical Hartree-Fock basis in (1.15) constitutes the quasiparticle approximation. With this approximation, the calculation of EADEs is simplified, for each KT result may be improved with many-body corrections that reside in a diagonal element of the self-energy matrix. 

In the early 1990s, Brenner and coworkers [163] developed interaction potentials for model explosives that include realistic chemical reaction steps (i.e., endothermic bond rupture and exothermic product formation) and many-body effects. This potential, called the Reactive Empirical Bond Order (REBO) potential, has been used in molecular dynamics simulations by numerous groups to explore atomic-level details of self-sustained reaction waves propagating through a crystal [163-171], The potential is based on ideas first proposed by Abell [172] and implemented for covalent solids by Tersoff [173]. It introduces many-body effects through modification of the pair-additive attractive term by an empirical bond-order function whose value is dependent on the local atomic environment. The form that has been used in the detonation simulations assumes that the total energy of a system of N atoms is ... 

This operator is consistent with the leading terms in quasiparticle self-energies implied by many-body theory [275, 407]. 

The asymptotic structure of the exchange potential vx(r) was derived via the relationship between density functional theory and many-body perturbation theory as established by Sham26. The integral equation relating vxc(r) to the nonlocal exchange-correlation component Exc(r, rf ) of the self-energy (r, r7 >) is... 

In this section I should like to outline a general approach to the renormalization (infinite order treatment) of the self-energy Z i(E) and describe how this scheme has been partly implemented in the actual calculations to be presented in Sect. 6. The general, formal theory of many-body systems and renormalization is well known (see e.g.27,70) and references therein). However, in actual applications each physical system has its own charac-... 

Complete multiconfiguration-self consistent-field (CMC-SCF) technique designates the method where a given occupied molecular orbital of the set is excited to all unoccupied molecular orbitals. If an occupied orbital is excited to one or more, but not all, of the unoccupied orbitals, the technique is described as incomplete MC-SCF (IMC-SCF). The reader is referred to refs. 13 and 14 for details of the derivation. The CMC-SCF formalism differs from most many body techniques presented to date insofar as the Hartree-Fock energy is not assumed to be the zero order energy. 

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