Kohn-Sham density functional theory procedures

Kohn and Sham later proved that Slater s intuitively motivated suggestion can be justified theoretically and procedures which combine the orbital-based Hartree kinetic functional with density-based exchange-correlation functionals are now called Kohn-Sham density functional theories. They are shown as family 3 in Figure 1. 

The most uniformly successful family of methods begins with the simplest possible n-electron wavefunction satisfying the Pauli antisymmetry principle - a Slater determinant [2] of one-electron functions % r.to) called spinorbitals. Each spinorbital is a product of a molecular orbital xpt(r) and a spinfunction a(to) or P(co). The V /.(r) are found by the self-consistent-field (SCF) procedure introduced [3] into quantum chemistry by Hartree. The Hartree-Fock (HF) [4] and Kohn-Sham density functional (KS) [5,6] theories are both of this type, as are their many simplified variants [7-16],... 

We may ask now, whether the same procedure may be applied to density-functional theory, just by replacing the Fock operator by the corresponding Kohn-Sham operator. To this end we have to look at the minimization of the total energy with respect to the density of a multi-determinantal wavefunction 4. We write the density as ... 

Theoretical considerations leading to a density functional theory (DFT) formulation of the reaction field (RF) approach to solvent effects are discussed. The first model is based upon isolelectronic processes that take place at the nucleus of the host system. The energy variations are derived from the nuclear transition state (ZTS) model. The solvation energy is expressed in terms of the electrostatic potential at the nucleus of a pseudo atom having a fractional nuclear charge. This procedure avoids the introduction of arbitrary ionic radii in the calculation of insertion energy, since all integrations involved are performed over [O.ooJ The quality of the approximations made are discussed within the frame of the Kohn-Sham formulation of density functional theory. 

We also discuss the generalization of density-functional theory to n-partical states, nDFT, and the possible extension of the local density approximation , nLDA. We will see there that the difficulty of describing the state of a system properly in terms of n-particle states presents no formal difficultie since DFT is directed only at the determination of the particle density rather than individual-particle wave functions. The extent to which practical applications of nDFT within a generalized Kohn-Sham scheme will provide a viable procedure is commented upon below. 

Instead of supposing there to be a single Kohn-Sham potential, one can think of it as a vector in Fock space. For each sheet ft = N of the latter, there is a component vKS(r,N) and a corresponding set of Kohn-Sham equations. Density functional theory and Kohn-Sham theory hold separately on each sheet. Ensemble-average properties are then composed of weighted contributions from each sheet, computable sheet by sheet via the techniques of DFT and the KS equations. Nevertheless, though completely valid, this procedure would yield for the reactivity indices f(r), s(r), and S the results already obtained directly from Eqs. (28). We are left without proper definitions of chemical-reactivity indices for systems with discrete spectra at T = 0 [43]. 

We have now come to the discussion of the central equations which form the basis of almost any practical application of density functional theory the Kohn-Sham equations. Kohn and Sham [8] introduced an auxiliary noninteracting system of particles with the property that it yields the same ground state density as the real interacting system. In order to put the Kohn-Sham procedure on a rigorous basis we introduce the functional... 

We shall see that the method of Kohn and Sham in density functional theory actually provides a sound theoretical base for this method which has been used Over the years simply as a numerical convenience. The density functional method uses a set of fictional molecular orbitals which do not themselves have any physical interpretation and whose only property is to generate an electron density which is exact. The whole of the experimental calibration procedure is thrown into the generation of a potential (the exchange/correlation potential) which can, in principle, be universal that is, not dependent on the particular molecule under study. The huge number of parameters required in earlier semi-empirical methods (some for every atom) is replaced by choice of a form for this potential and a few universal parameters (up to a dozen). 

Kohn and Sham took E c as a functional of the density p. An alternative procedure, the optimized effective potential (OEP) method, takes as a functional of the occupied KS orbitals, in the hope that this will make it easier to develop accurate functionals. The OEP method leads to equations that are hard to solve. Kreiger, Li and lafrate (KLI) developed an accurate approximation to the OEP equations, thereby making them easier to deal with, and the KLI method has given good results in DF calculations on atoms [J. B. Krieger, Y. li, and G. I. lafrate in E. K U. Gross and R. M. Dreizler (eds.). Density Functional Theory, Plenum, 1995, pp. 191-216]. 

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