Exchange energy, uniform electron density

The essence of the local density approximation is the assumption that the exchange-correlation energy of a non-uniform electron density can be approximated as a sum of contributions from small volume elements each characterized by uniform electron density (exc(p)(r)). Since the exchange-correlation energy of uniform electron gas is available with arbitrary accuracy, the LDA exchange-correlation functional takes the following simple form ... 

For uniform electron densities, ExciPaiPp] should reduce to the formulas for the exchange-correlation energy of a uniform free-electron gas (the LSDA) ... 

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... 

The local density approximation (LDA) replaces the exchange and correlation potential, which is non-local, by a local potential. This latter is constructed from an approximate expression of the exchange-correlation energy xc( ) of an homogeneous electron gas. At each point r of the system, the uniform electron density n in xc( ) is functionally replaced by the actual density n f). The Hamiltonian and the Poisson s equation are then solved in a self-consistent manner. There have been various applications of these methods to oxides ... 

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 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. ... 

---