Dirac exchange functional

The same investigations of the idealized uniform electron gas that identified the Dirac exchange functional, found that the correlation energy (per electron) could also be... 

The assumption is justified by the fact that the Slater-Dirac exchange functional can reproduce approximately 90 % of the HFx [3]. The condition for being SIF is... 

Following the success of the von Weizsacker approach in improving the Thomas-Fermi kinetic functional. Sham showed in 1971 that an analogous correction to the Dirac exchange functional can be derived, Kleinman later demonstrated that the Sham derivation was flawed and that his correction was too small by exactly 10/7, It is now agreed that the correct second-order alpha exchange functional is... 

The exchange part is given by the Dirac exchange-energy functional... 

One can improve upon the TF model by incorporating two-electron effects into P nlpl as the approximate, local Dirac exchange energy functional (cx is the Dirac exchange constant)... 

The kinetic and exchange energy functionals given by Eqs. (8) and (12), respectively, contain universal terms that just depend upon the one-particle density. In the case of the former, such term is p6/3, the Thomas-Fermi term [22,23] and for the latter, the set p(ri)(4+fc 3, where the first term p4 3 (for k = 0) is the Dirac exchange expression [24]. But in addition, in Eq. (8) we observe the presence of a factor, which we call Fis([p]jr) defined as ... 

For identical hydrons, the symmetry postulate of identical particles has to be fulfilled. For protons and tritons this means that the overall wave function must be antisymmetric under particle exchange and for deuterons it must be symmetric under particle exchange. Due to this correlation of spin and spatial state, the energy difference A between the lowest two spatial eigenstates can be treated as a pure spin Hamiltonian, similar to the Dirac exchange interaction of electronic spins. 

To keep the Dirac exchange [15] of the electron gas part complementary to Exc of Equation 5.7, the classic form of linear combinations is Fj 1 f , for hybrid functionals. Thus, the number of fitted... 

The combination of the Dirac-Kohn-Sham scheme with non-relativis-tic exchange-correlation functionals is sometimes termed the Dirac-Slater approach, since the first implementations for atoms [13] and molecules [14] used the Xa exchange functional. Because of the four-component (Dirac) structure, such methods are sometimes called fully relativistic although the electron interaction is treated without any relativistic corrections, and almost no results of relativistic density functional theory in its narrower sense [7] are included. For valence properties at least, the four-component structure of the effective one-particle equations is much more important than relativistic corrections to the functional itself. This is not really a surprise given the success of the Dirac-Coulomb operator in wave function based relativistic ab initio theory. Therefore a major part of the applications of relativistic density functional theory is done performed non-rela-tivistic functionals. 

Gas adsorption is a chemical interaction between the gas molecules and the semiconductor surface. This interaction is accompanied by charge exchange creating acceptor or donor like band gap level, whose occupation probability is given by the Fermi-Dirac distribution function (Kireev 1978). Its conduction behavior as acceptor or donor will depend on the type of the adsorbed molecule. 

Although the Thomas-Fermi method is an interesting theory representing the Hamiltonian operator as the functional only of the electron density, even qualitative discussions cannot be contemplated based on this method in actual electronic state calculations. Dirac considered that this problem may be attributed to the lack of exchange energy (see Sect. 2.4), which was proposed in the same year (Fock 1930), and proposed the first exchange functional of electron density p (Dirac 1930),... 

---