Self-energy, operator

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]. 

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... 

Here, an effective one-electron operator matrix has Fock and energy-dependent, self-energy terms. Prom this matrix expression, one may abstract one-electron equations in terms of the generalized Fock and energy-dependent, self-energy operators ... 

In Equation (12), the self-energy operator Z(r, rEbik) is, in general, non-local and depends on energy. Therefore, to solve the Schrodinger equation, a series of approximations have to be introduced. 

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 ... 

As mentioned in Section 2, the CPs of solids have to be calculated on the quasi-particle scheme. In order to calculate the quasi-particle states, non-local and energy-dependent self-energy in Equation (13) must be evaluated in a real system. In practice, the exact self-energy for real systems are impossible to compute, and we always resort to approximate forms. A more realistic but relatively simple approximation to the selfenergy is the GWA proposed by Hedin [7]. In the GW A, the self-energy operator in Equation (12) is... 

From the self-energy operator in Equation (38), the self-energy value in GWA is calculated as... 

Now the Fock operator is supplemented by the self-energy operator E(E). This operator depends on an energy parameter E and is nonlocal. All... 

Perturbative expressions for the self-energy operator can achieve this goal for large, closed-shell molecules. In this review, we will concentrate on an approximation developed for this purpose, the partial third-order, or P3, approximation. P3 calculations have been carried out for a variety of molecules. A tabulation of these calculations is given in Table 5.1. 

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. 

Fig. 3.15. Exact second order self-energy operator...

It is not difficult to present an exact formula containing all corrections produced by the Uehling potential in Fig. 2.2 (compare with the respective expression for the self-energy operator above)... 

The second order perturbation theory term with two one-loop self-energy operators does not generate any logarithm squared contribution for the state with nonzero angular momentum since the respective nonrelativistic wave function vanishes at the origin. Only the two-loop vertex in Fig. 3.24 produces a logarithm squared term in this case. The respective perturbation potential determined by the second term in the low-momentum expansion of the two-loop Dirac form factor [111] has the form... 

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 ... 

As follows from Eq. (12), the Green s function does not have a singular part, but a self-energy operator, , does have it [42]... 

Now that the self-energy operators are known, the zeroth Keldysh functions can be easily obtained from Eqs. (68) and (69) ... 

Canonical Hartree-Fock orbitals have been the usual choice among quantum chemists. Effects of electron correlation and orbital relaxation in final states are described by the self-energy operator, X(E). The latter operator can be written as a sum of energy-independent and energy-dependent parts according to... 

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