Energy, exchange-correlation

1 Exchange-correlation energy definition, interpretation and exact properties 

The Thomas-Fermi approximation (34) for T[n is not very good. A more accurate scheme for treating the kinetic-energy functional of interacting electrons, T[n], is based on decomposing it into one part that represents the kinetic energy of noninteracting particles of density n, i.e., the quantity called above Ts[n], and one that represents the remainder, denoted Tc[n (the sub- 

Ts[n is not known exactly as a functional of n [and using the LDA to approximate it leads one back to the Thomas-Fermi approximation (34)], but it is easily expressed in terms of the single-particle orbitals fair) of a noninteracting system with density n, as 

26 Ts is defined as the expectation value of the kinetic-energy operator T with the Slater determinant arising from density n, i.e., Ts[n] = ( [n] T k[n]). Similarly, the full kinetic energy is defined as T[n = ( I,[n] Tj I,[n]. All consequences of antisymmetrization (i.e., exchange) are described by employing a determinantal wave function in defining Ts. Hence, Tc, the difference between Ts and T is a pure correlation effect. 

27This differs from the exchange energy used in Hartree-Fock theory only in the substitution of Hartree-Fock orbitals 4 fF(r) by Kohn-Sham orbitals 

The Thomas-Fermi approximation (42) for T ri is not very good. A more accurate scheme for treating the kinetic-energy functional of 

The exchange energy can be written explicitly in terms of the single-particle orbitals as 

For the correlation energy no general explicit expression is known, neither in terms of orbitals nor in terms of densities. A simple way to understand the origin of correlation is to recall that the Hartree energy is obtained in a variational calculation in which the many-body wave function is approximated as a product of single-particle orbitals. Use of an antisymmetrized product (a Slater determinant) produces the Hartree energy and the exchange 

---

Professor Axel Becke of Queens University, Belfast has been very actively involved in developing and improving exchange-correlation energy functionals. For a good recent overview, see ... 

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,... 

---