Dirac exchange

This result was rediscovered by Slater (1951) with a slightly different numerical coefficient of C. Authors often refer to a term Vx which is proportional to the one-third power of the electron density as a Slater-Dirac exchange potential. 

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

The electron-electron exchange term, Hex In equation (16) it is necessary to consider only He . As has been discussed, the energy difference between T and S states is equal to Je . With a minimal overlap integral due to a relatively large inter-radical separation. Hex can be given by the Dirac exchange operator [equation (18)],... 

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 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 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 ... 

The Thomas-Fermi kinetic energy density Ckp(r)5/3 derives directly from the first term on the RHS of Eq. (17), the Dirac exchange energy density —cxp(r)4/3 coming from the second term. Many-body perturbation theory on this state, in which electrons are fully delocalized, yields a precise result [36,37] for the correlation energy Ec in the high-density limit as A In rs + B, where for present purposes the correlation energy is defined as the difference between the true... 

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. 

Ex(lsd) is the local exchange energy, taken as the Dirac exchange... 

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... 

Note that the choice k = kp, regardless of the value of a, yields the LDA (Dirac) exchange formula (58) as the first term of x [p]- The remaining terms containing Vp, V p, and T, naturally arise as corrections to the LDA. 

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... 

For the Slater-Dirac exchange, the next relation is instead obtained using Eq. (14.18) ... 

Dirac 3 made an improvement to the TF model by introducing a formula for the exchange energy of a uniform electron gas. The Dirac exchange formula is... 

Unfortunately, adding the Dirac exchange formula to the TF model does not improve the quality of the calculated electron density. The TFD density suffers from the same undesirable characteristics as does the TF density. A major enhancement of these two overly simplified models was made through addition of an inhomogeneity the electron density correction to the kinetic energy density functional. This was first investigated by von Weizsacker, who derived a correction that depends upon the gradient of the density, namely. 

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