Total Energies from Many-Body Theory

4 Total Energies from Many-Body Theory 

MBPT-based schemes can be meant as an alternative for those situations in which known DFT models are inaccurate, but whose complexity makes the implementation of QMC difficult. In this section, after a brief summary of the theoretical foundations, we will review some of the recent applications which, so far, have been restricted to model systems but in which LDA/GGAs clearly show their limitations. Finally, we will present a simplified many-body theory amenable for its implementation in a DFT-fashion. 

MBPT provides several ways to obtain each of the different contributions to the ground-state energy Perhaps the best known, owing to its role in the construction of xc energy functionals, is the expression based on the adiabatic-connection-fluctuation-dissipation (ACFD) theorem [75,76] 

xa (i ) is the causal density response function at imaginary frequencies of a system in which the electrons interact through a modified Coulomb potential w (r) = A/r, and whose ground state density is equal to the actual one. Xa (i ) is related to the polarisation function P (ia ) through the equality  

The same information can be also extracted from the self-energy operator and the Green s function. Namely, using the adiabatic-connection that led to (5.22)1° 

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In this chapter, after a brief introduction to MBPT and Hedin s GW approximation, we will summarise some peculiar aspects of the Kohn-Sham xc energy functional, showing that some of them can be illuminated using MBPT. Then, we will discuss how to obtain ground-state total energies from GW. Finally, we will present a way to combine techniques from many-body and density functional theories within a generalised version of Kohn-Sham (KS) DFT. 

The atomic interactions of the system are derived from a many-body empirical potential, the attractive part of which is expressed within the SMA of the TB theory ", while the repulsive term is a pair-potential of Bom-Mayer type. Accordingly, the total energy of the system is written 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. 

As a second model potential we shall briefly discuss the PES for the water dimer. Analytical potentials developed from ab initio calculations have been available since the mid seventies, when Clementi and collaborators proposed their MCY potential [49], More recent calculations by dementi s group led to the development of the NCC surface, which also included many-body induction effects (see below) [50]. Both potentials were fitted to the total energy and therefore their individual energy components are not faithfully represented. For the purposes of the present discussion we will focus on another ab initio potential, which was designed primarily with the interaction energy components in mind by Millot and Stone [51]. This PES was obtained by applying the same philosophy as in the case of ArCC>2, i.e., both the template and calibration originate from the quantum chemical calculations, and are rooted in the perturbation theory of intermolecular forces. 

A fully relativistic treatment of more than one particle would have to start from a full QED treatment of the system (Chapter 1), and perform a perturbation expansion in terms of the radiation frequency. There is no universally accepted way of doing this, and a full relativistic many-body equation has not yet been developed. For many-particle systems it is assumed that each electron can be described by a Dirac operator (ca n -I- P me and the many-electron operator is a sum of such terms, in analogy with the kinetic energy in non-relativistic theory. Furthermore, potential energy operators are added to form a total operator equivalent to the Hamiltonian operator in non-relativistic theory. Since this approach gives results that agree with experiments, the assumptions appear justified. 

The most important equation, derived in this work, is the extended Born-Handy formula, valid in the adiabatic limit as well as in the case of break down of the B-0 approximation. Since due to the many-body formulation the extended Born-Handy formula can be expressed in the CPHF compatible form, the extended CPHF equations, describing the non-adiabatic systems, will immediately follow from the presented theory. We shall call them COM CPHF equations. Whereas in the adiabatic limit the extended Bom-Handy formula represents only small corrections to the system total energy, in non-adiabatic systems it plays three important roles (1) removes the electron degeneracies, (2) is responsible for the symmetry breaking, and (3) forms the molecular and crystalline stmcture. 

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