Exchange-correlation energy, density

We have used the basis set of the Linear-Muffin-Tin-Orbital (LMTO) method in the atomic sphere approximation (ASA). The LMTO-ASA is based on the work of Andersen and co-workers and the combined technique allows us to treat all phases on equal footing. To treat itinerant magnetism we have employed the Vosko-Wilk-Nusair parametrization for the exchange-correlation energy density and potential. In conjunction with this we have treated the alloying effects for random and partially ordered phases with a multisublattice generalization of the coherent potential approximation (CPA). 

The exchange-correlation energy density can be split into two parts exchange component Ex n) and correlation component e Cn). The explicit expression for the exchange component is known from Hartree-Fock theory but the correlation component is known only numerically. Several parametrisations exist for the exchange-correlation energy and potential of a homogeneous gas system which can be used for the LDA calculations within DFT. 

Wesolowski, T. A., Parisel, O., Ellinger, Y., Weber, J., 1997, Comparative Study of Benzene---X (X = 02, N2, CO) Complexes Using Density Functional Theory The Importance of an Accurate Exchange-Correlation Energy Density at High Reduced Density Gradients , J. Phys. Chem. A, 101, 7818. 

In Eq. [47], epc ( ) and exc (n) are the exchange-correlation energy densities for the nonpolarized (paramagnetic) and fully polarized (ferromagnetic) homogeneous electron gas. The form of both exc(n) and exc(n) has been conveniently parameterized by von Barth and Hedin. Other interpolations have also been proposed24,33 for eKC(n, J ). The results for the homogeneous electron gas can be used to construct an LSDA... 

Because of different scaling properties, the exchange-correlation energy density functional (XCEDF) can be further decomposed into separate exchange and correlation components, ... 

Now, whereas the long range behaviour of the exchange-correlation energy density should follow [10] ... 

The term non-local is used sometimes in die literature in association with gradient-dependent (GGA) functionals. This nomenclature is not applied in this work. The LDA-, GGA-, and meta-GGA functionals are referred to as semi-local as they do not account for any long-range non-locality of the exchange-correlation energy density excseim local(r) = exc(p(r), Vp(r), V2p(r), r(r)) whereas... 

Fournier, R. Andzelm, J. Salahub, D. R., to be published in Xa theory, eq. (5) can be further simplified by using the linear relationship between the exchange-correlation energy density, and the exchange-correlation potential, In general LSDF theory, this simple relationship does not apply. 

Integrals involving the exchange-correlation potential r,c or the exchange-correlation energy density cannot be evaluated analytically so that further sets of auxiliary functions are introduced. (In practice and 6,<, behave similarly so that a common set is used to fit both functions.) The exchange-correlation basis (XCB) also consists of Hermite Gaussians... 

The volume derivatives of the potential parameters appearing in (7.46) may be found in Sect.4.5. To include the surface term of (7.31), the potential v in (4.44,48) should be replaced by the exchange-correlation energy density exc as in (7.35), i.e. we imply that the electrostatic potential is zero at S. Hence, we have the relations... 

The conventional exchange-correlation energy density is generated by the choice [3,4]... 

Fig. 5 Exact and fitted (using Eq. (21)) deviation of the RPA conventional exchange-correlation energy density from RPA-LDA, as a function of position z for the Airy gas model with force...

To see if there is a second-order gradient expansion for the conventional exchange-correlation energy density in RPA, we have fitted B in... 

In the LDA framework, the total electronic energy includes the exchange-correlation energy density functional of the form [43]... 

The quest for various approximations for the exchange-correlation energy density/(p) has spanned decades in quantum chemistry and was recently reviewed [92]. Here, we will present the red line of its implementation, as it will be further used for the current applications. The benchmark density functional stands as the Slater exchange approximation, derived within the so-called Xa theory [179] ... 

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