The Exchange-Correlation Energy

In Ecjuation (3.47) we have written the external potential in the form appropriate to the interaction with M nuclei. , are the orbital energies and Vxc is known as the exchange-correlation functional, related to the exchange-correlation energy by ... 

L. iitortunately, this simple approach does not work well, but Becke has proposed a strategy which does seem to have much promise [Becke 1993a, b]. In his approach the exchange-correlation energy Exc is written in the following form ... 

Again the set of fitting functions may or may not be the same as the orbital and/or the density basis functions. Once the potential has been fitted, the exchange—correlation energy may be evaluated from integrals involving three functions, analogously to eq. 

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). 

To summarise, we have presented a way to improve an LMTO-ASA calculation of the electrostatic energy in a crystal. The method is stable and general in its formalism so that it should be applicable to a wide range of systems. In this talk we did not mention the exchange correlation energy. It is possible to make an expansion of the (xc(p(r)) in terms of the SSW s. Then the integral... 

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. 

One term in the above equation needs some additional comments, namely Vxc, the potential due to the exchange-correlation energy Exc. Since we do not know how this energy should be expressed, we of course also have no clue as to the explicit form of the corre-... 

The Exchange-Correlation Energy in the Kohn-Sham and Hartree-Fock... 

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,... 

A second major problem connected to the use of finite grids for the evaluation of the exchange-correlation energy is associated with the determination of derivatives of the energy, such as the gradients used in geometry optimizations. We use... 

Gritsenko, O. V., van Leuwen, R., Baerends, E. J., 1996, On the Optimal Mixing of the Exchange Energy and the Electron-Electron Interaction Part of the Exchange-Correlation Energy , Int. J. Quant. Chem. Quant. Chem. Symp.. 30, 1375. 

First, the self-energy operator is replaced by a local exchange-correlation potential, which is given by the functional derivative of the exchange-correlation energy with respect to the electron density ... 

The replacement of Equation (15) corresponds to the density functional method. But the exchange-correlation energy is generally unknown. Therefore, the unknown... 

Following Reference [27], we may write the variation of the exchange-correlation energy as ... 

In principle, the KS equations would lead to the exact electron density, provided the exact analytic formula of the exchange-correlation energy functional E was known. However, in practice, approximate expressions of Exc must be used, and the search of adequate functionals for this term is probably the greatest challenge of DFT8. The simplest model has been proposed by Kohn and Sham if the system is such that its electron density varies slowly, the local density approximation (LDA) may be introduced ... 

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