Exchange energy, uniform electron

Table III The ratio of approximate to exact exchange energy per electron of a uniform electron gas. For the approximations listed in Tables I and II but not listed here, this ratio is exactly 1. The Becke-Roussel exchange functional is a non-empirical meta-GGA based upon the hydrogen atom.

The exchange energy per electron ex (p) of the unpolarized uniform electron gas of density p r) depends on = p as sx p) = the correlation energy... 

Even with this approximation, we cannot assume that the relativistic exchange-correlation functional is the same as the nonrelativistic functional, though we might reasonably suppose that an expansion of the relativistic functional will yield the nonrelativistic functional with correction terms. This proves to be the case for the exchange energy of a relativistic uniform electron gas, for which we can write the exchange energy per electron as... 

Early attempts at deducing functionals for the kinetic and exchange energies considered a non-interacting uniform electron gas. For such a system it may be shown that T[p] and K[p] are given as... 

Here, exc(p(r)) is the exchange-correlation energy per particle of a uniform electron gas of density p( ). This energy per particle is weighted with the probability p(r) that there is in fact an electron at this position in space. Writing Exc in this way defines the local density approximation, LDA for short. The quantity exc(p(r)) can be further split into exchange and correlation contributions,... 

The exchange part, ex, which represents the exchange energy of an electron in a uniform electron gas of a particular density is, apart from the pre-factor, equal to the form found by Slater in his approximation of the Hartree-Fock exchange (Section 3.3) and was originally derived by Bloch and Dirac in the late 1920 s ... 

Table I The essentially-exact PW92 exchange-correlation energy per electron (in hartree) in a spin-unpolarized = 0) uniform electron gas of density parameter (in bohr), and the deviation (in hartree) of other approximations from PW92. (1 hartree = 27.21 eV = 627.5 kcal/moi.]...

B3LYP Model. A Hybrid Density Functional Model which improves on the Local Density Model by accounting explicitly for non-uniformity in electron distributions, and which also incorporates the Exchange Energy from the Hartree-Fock Model. The B3LYP model involves three adjustable parameters. 

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 distinction is probably best indicated by example. Following from Eq. (8.7) and the discussion in Section 8.1.2, the exchange energy for the uniform electron gas can be computed exactly, and is given by Eq. (8.23) with the constant a equal to. However, the Slater approach takes a value for a of 1, and the Xa model most typically uses j. All of these models have the same local dependence on the density, but only the first is typically referred to as LDA, while the other two are referred to by name as Slater (S) and Xa. ... 

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