Exchange self-energy

Godby R W, Schluter M and Sham L J 1988 Self-energy operators and exchange-correlation potentials in semiconductors Phys. Rev. B 37 10159-75... 

First, the self-energy operator is replaced by a local exchange-correlation potential, which is given by the functional derivative of the exchange-correlation energy with respect to the electron density ... 

One obvious drawback of the LDA is that, when we replace unknown exchange-correlation energy by the known form of the exchange-correlation for a homogeneous electron gas in Equation (17), we have a problem in that cancelation of self-Coulomb... 

The XC energy represents the correction to the Coulomb energy for the self-energy of an electron in a many-electron system. The latter is due to both the direct self-energy of the electron as well as the redistribution of electronic density around each electron because of the Pauli exclusion principle and the Coulomb interaction. As an example, we now discuss the case of Fermi hole and the exchange energy in Hartree-Fock (HF) theory [16]. For brevity, we restrict ourselves to closed-shell cases. 

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]. 

Given an expression for the self-energy operator, equations (2) and (4) must he solved self-consistently. E(E) is also called the exchange-correlation potential, it is manifestly non-local and energy dependent. 

Each of these properties can be expressed equivalently in terms of x(r, r co). In the current work, x(, r co) contains exchange and correlation contributions to it also contains terms that (in effect) remove the self-energy (35-37) from the final result for (Vge). 

For the details and derivation of the physical interpretation we refer the reader to the original literature14,15. Since the Coulomb self-energy component of the KS electron-interaction energy functional and its derivative, the Hartree potential, are known functionals of the density, we provide in Section HA the expressions governing the interpretation of the KS exchange-correlation energy... 

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... 

It is easy to see that the self-energy operator - average electron-electron interaction -can be considered also as a linear superoperator in the space of the matrices it depends on. Indeed, from the point of view of the 2M-dimensional space of spin-orbitals E[/5] acts as a 2M x 2M matrix, so that E[p] is a 2M x 2M matrix constructed after another 2M x 2M matrix p. On the other hand, it easy to see from the definition of the Coulomb and exchange operators in eq. (1.145) that the result of calculating each of them (and thus of the sum of them) taking a sum of two functions p (x x ) + p2(x x ) and/or a product of this function by a number Ap x x ) as its argument yields respectively a sum of the results of the actions of S and the product in the same number as the result of action of S ... 

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