Non-local energy-dependent

For the actual quasi-particle excitations E one has to solve the Dyson equation (2.1) or p.2) with a non-local energy-dependent self-energy 2. sham and Kohn have suggested LDA approximations for which is also a functional of density. To this they split 2 into a local and non-local component... 

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]. 

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. 

Molecular quantum potential and non-local interaction depend on molecular size and the nature of intramolecular cohesion. Macromolecular assemblies such as polymers, biopolymers, liquids, glasses, crystals and quasicrystals are different forms of condensed matter with characteristic quanmm potentials. The one property they have in common is non-local long-range interaction, albeit of different intensity. Without enquiring into the mechanism of their formation, various forms of condensed matter are considered to have well-defined electronic potential energies that depend on the nuclear framework. A regular array of nuclei in a structure such as diamond maximizes cohesive interaction between nuclei and electrons, precisely balanced by the quantum potential, almost as in an atom. 

There are complicating issues in defmmg pseudopotentials, e.g. the pseudopotential in equation Al.3.78 is state dependent, orbitally dependent and the energy and spatial separations between valence and core electrons are sometimes not transparent. These are not insunnoimtable issues. The state dependence is usually weak and can be ignored. The orbital dependence requires different potentials for different angular momentum components. This can be incorporated via non-local operators. The distinction between valence and core states can be addressed by incorporating the core level in question as part of the valence shell. For... 

What does this mean We have replaced the non-local and therefore fairly complicated exchange term of Hartree-Fock theory as given in equation (3-3) by a simple approximate expression which depends only on the local values of the electron density. Thus, this expression represents a density functional for the exchange energy. As noted above, this formula was originally explicitly derived as an approximation to the HF scheme, without any reference to density functional theory. To improve the quality of this approximation an adjustable, semiempirical parameter a was introduced into the pre-factor Cx which leads to the Xa or Hartree-Fock-Slater (HFS) method which enjoyed a significant amount of popularity among physicists, but never had much impact in chemistry,... 

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... 

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. 

The kinetic energy can be rewritten as a sum of the Weiszacker term [10] (a local component which depends only on the density) plus a non-local one ... 

In comparing Eq. (13) to the Kohn-Sham equations Eq. (3) one concludes that E(.r, x E), since it is derived from exact many-electron theory [22], is the exact Coulomb (direct) plus exchange-correlation potential. It is non-local and also energy-dependent. In view of this it is hard to see how the various forms of constructed local exchange correlation potentials that are in use today can ever capture the full details of the correlation problem. 

The superconducting properties induced in the normal metal manifest themselves in many different ways, including energy-dependent transport properties and a modification of the local density of states. For instance, the conductance of a normal conductor connected to a superconducting electrode shows a striking re-entrant behavior [4]. At non-zero temperature and/or bias, the conductance of the normal metal is enhanced as compared to the normal-state. At zero temperature and zero bias, the expected conductance coincides with the normal-state value. The conductance has therefore a non-monotonous behavior. 

---