Exchange and correlation

Let us now revisit the Hamilton operator for a many-electron system as given in Equation (2.5). It contains the kinetic energies of the electrons, their Coulomb potentials in the fields of the nuclei and, also, the electron-electron interactions which we have not further specified yet. Fortunately, the success of one-electron theories in chemistry is due to the fact that an explicit consideration of the electron-electron interactions may often be ignored for a qualitatively appropriate description, simply because these interactions are implicitly contained in some parameters for example, one might want to recall that the spatial extent of the atomic orbitals (see Section 2.2) depends on the amount of interelectronic screening. Nonetheless, the explicit inclusion of electron-electron interactions is needed for accurate calculations and, just as important, also for imderstanding. Here is a qualitative description. 

Electron-electron interactions fall into two classes, depending on the type of electron that is involved. We all know that each electron carries the same (elementary) charge —e and is further characterized by its nonclassical angular momentum, the so-called spin, which can be thought of, in a classical picture, as indicating the electron s rotation around its own axis, say, clockwise or anticlockwise. Thus, electrons come either as spin-up (, or just a) or as spin-down (J., j6) electrons. 

16) This classical idea is highly questionable spin is a function of the wave function alone, and the particle possesses only a position in space [96], 

17) It is common practice to explain Hund s rules by referring to the relative energetic positioning of atomic (or molecular) states, not (one-electron) orbitals, and because the states represent the many-electron wave functions, they already incorporate the 

Sketch (a) shows the correct filling, and the incorrect one is given by the smaller grey sketch (b). Here we have abstained, for reasons of simplicity, from distinguishing between a spin-up (a) and a spin-down (jS) electron. Now, explicitly taking the spin into account, the above filling (a) is more 

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Kohn W and Sham L J 1965 Self-consistent equations including exchange and correlation effects Phys. Rev A 140 1133-8... 

More advanced teclmiques take into account quasiparticle corrections to the DFT-LDA eigenvalues. Quasiparticles are a way of conceptualizing the elementary excitations in electronic systems. They can be detennined in band stmcture calculations that properly include the effects of exchange and correlation. In the... 

Flere we distinguish between nuclear coordinates R and electronic coordinates r is the single-particle kinetic energy operator, and Vp is the total pseudopotential operator for the interaction between the valence electrons and the combined nucleus + frozen core electrons. The electron-electron and micleus-micleus Coulomb interactions are easily recognized, and the remaining tenu electronic exchange and correlation... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

Gunnarsson O and B I Lundqvist 1976. Exchange and Correlation in Atoms, Molecules, and Solids by the Spin-density-functional Formalism. Physical Review B13.-4274-4298. 

Kohn W and L J Sham 1965. Self-consistent Equations Including Exchange and Correlation Effects. Physical Review A140 1133-1138. 

A variety of functionals have been defined, generally distinguished by the way that they treat the exchange and correlation components ... 

Local exchange and correlation functionals involve only the values of the electron spin densities. Slater and Xa are well-known local exchange functionals, and the local spin density treatment of Vosko, Wilk and Nusair (VWN) is a widely-used local correlation functional. 

E is usually divided into separate parts, referred to as the exchange and correlation parts, but actually corresponding to same-spin and mixed-spin interactions, respectively ... 

Self-Consistent Equations Including Exchange and Correlation Effects W. Kohn and L. J. Sham Physical Review 140 (1965) All33... 

The first term on the right-hand side is a contribution from external fields, usually zero. The second term is the contribution from the kinetic energy and the nuclear attraction. The third term is the Coulomb repulsion between the electrons, and the final term is a composite exchange and correlation term. 

As mentioned above, a KS-LCAO calculation adds one additional step to each iteration of a standard HF-LCAO calculation a quadrature to calculate the exchange and correlation functionals. The accuracy of such calculations therefore depends on the number of grid points used, and this has a memory resource implication. The Kohn-Sham equations are very similar to the HF-LCAO ones and most cases converge readily. 

It is customary to separate xc into two parts, a pure exchange and a correlation part although it is not clear that this is a valid assumption (cf. the above discussion of the definition of exchange and correlation). Each of these energies is often written in terms of the energy per particle (energy density), and c-... 

The total density is the sum of die a and /3 contributions, p = Pa + Pp, and for a closed-shell singlet these are identical (p, = pp). Functionals for the exchange and correlation energies may be formulated in terms of separate spin-densities however, they are often given instead as functions of the spin polarization C, (normalized difference between p and pp), and the radius of the effective volume containing one electron, rs-... 

Theoretical calculations were performed with the linear muffin tin orbital (LMTO) method and the local density approximation for exchange and correlation. This method was used in combination with supercell models containing up to 16 atoms to calculate the DOS. The LMTO calculations are run self consistently and the DOS obtained are combined with the matrix elements for the transitions from initial to final states as described in detail elsewhere (Botton et al., 1996a) according to the method described by Vvedensky (1992). A comparison is also made between spectra calculated for some of the B2 compounds using the Korringa-Kohn-Rostoker (KKR) method. 

Calculations were done with a full-potential version of the LMTO method with nonoverlapping spheres. The contributions from the interstitial region were accounted for by expanding the products of Hankel functions in a series of atom-ce- -ered Hankels of three different kinetic energies. The corrected tetrahedron method was used for Brillouin zone integration. Electronic exchange and correlation contributions to the total energy were obtained from the local-density functional calculated by Ceperley and Alder " and parametrized by Vosko, Wilk, and Nusair. ... 

The local density approximation is highly successful and has been used in density functional calculations for many years now. There were several difficulties in implementing better approximations, but in 1991 Perdew et al. successfully parametrised a potential known as the generalised gradient approximation (GGA) which expresses the exchange and correlation potential as a function of both the local density and its gradient ... 

E [n] includes all many-body contributions to the total energy, in particular the exchange and correlation energies. 

The first term is the familiar one-electron operator, the second term represents the Coulomb potential, and the third term is called exchange-correlation potential. HF and DFT differ only in this last term. In HF theory there is only a nonlocal exchange term, while in DFT the term is local and supposed to cover both exchange and correlation. It arises as a functional derivative with respect to the density ... 

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