Dirac exchange energy functional

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)... 

This proposition has been tested in the exact-exchange limit of the implied linear-response theory [329], The TDFT exchange response kernel disagrees qualitatively with the corresponding expression in Dirac s TDHF theory [79,289]. This can be taken as evidence that an exact local exchange potential does not exist in the form of a Frechet derivative of the exchange energy functional in TDFT theory. 

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 response kernel for the Hartree and the exchange energy functionals is /j, +fx = (u = (l/r12)(l — Pn), in agreement with Dirac [13] and with the second-quantized Hamiltonian. The response kernel fc is a linear operator such that... 

Table 1 Energies (in KeV) of single positive ions evaluated with (AH) a full relativistic kinetic energy functional without exchange [15] the c -order semi-relativistic functional (Eq. 46) without (1) and with (2) the relativistic exchange correction ((f-term), all using near-nuclear corrections, compared to Dirac-Fock (DF) values.

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 energy functional Etf[p] = Ttf[p] + ne />] + T[p] is known as Thomas-Fermi (TF) theory, inHiiHinv the Atp[p] exchange part (first derived by Block but commonly associated with the name of Dirac (constitutes the Thomas-Fermi-Dirac (TFD)... 

This began with a landmark paper by Slater in which he proposed that the effects of exchange in the wave function could be replaced by an exchange potential, proportional to p where p is the electron density function. This followed earlier work by Dirac, in which he showed how to add the exchange energy to the Thomas-Fermi theory of the atom. The exchange potential is also dependent on a factor, cv, which is allowed to vary somewhat from its value in a uniform electron gas. The method is called the Xa method and involves solving a series of one-electron wave equations in a self-consistent manner. 

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. 

The first generation is the local density approximation (LDA). This estimation involves the Dirac functional for exchange, which is nothing else than the functional proposed by Dirac [15] in 1927 for the so-called Thomas-Fermi-Dirac model of the atoms. For the correlation energy, some parameterizations have been proposed, and the formula can be considered as the limit of what can be obtained at this level of approximation [16-18], The Xa approximation falls into this category, since a known proportion of the exchange energy approximates the correlation. 

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