Kohn-Sham formulation of DFT

In the Kohn-Sham formulation of DFT the problem of finding the ground state energy of this system is exactly mapped onto one of finding the electron density which minimizes the total energy functional... 

A simple estimate of the computational difficulties involved with the customary quantum mechanical approach to the many-electron problem illustrates vividly the point [255]. Consider a real-space representation of ( ii 2, , at) on a mesh in which each coordinate is discretized by using 20 mesh points (which is not very much). For N electrons, becomes a variable of 3N coordinates (ignoring spin), and 20 values are required to describe on the mesh. The density n(r) is a function of three coordinates and requires only 20 values on the same mesh. Cl and the Kohn-Sham formulation of DFT (see below) additionally employ sets of single-particle orbitals. N such orbitals, used to build the density, require 20 values on the same mesh. (A Cl calculation employs in addition unoccupied orbitals and requires more values.) For = 10 electrons, the many-body wave function thus requires 20 °/20 10 times more storage space than the density and sets of single-particle orbitals 20 °/10x 20 10 times more. Clever use of symmetries can reduce these ratios, but the full many-body wave function remains inaccessible for real systems with more than a few electrons. 

The main task in the Kohn-Sham formulation of DFT is finding good approximations to the exchange-correlation functional contrast to wave function... 

Each one of the HF or Kohn-Sham MOs is expressed as a LCAO approach or, more generally, basis functions. Since the self-consistent field procedure in the Kohn-Sham formulation of DFT is very similar to that of the HF method, the choice of an appropriate basis set is equally important in DFT. 

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. 

TD-DFT is rooted in the Runge-Gross theorem [60] (which is not valid for the degenerate ground state), allowing the extension of the Hohemberg-Kohn-Sham formulation of the TD-DFT theory to the treatment of time-dependent phenomena ... 

Whereas the classic Kohn-Sham (KS) formulation of DFT is restricted to the time-independent case, the formalism of TD-DFT generalizes KS theory to include the case of a time-dependent, local external potential w(t) [27]. 

The original formulation of DFT by Kohn and Sham appeared to imply that an exact orbital theory could be expressed in terms of a local exchange-correlation potential[4] ( note added in proof , p. A1138). This conclusion would follow... 

The partitioning of F[p] into the three components T, J, and 4c in Eq. (16) is by no means unique. Different partitioning schemes give rise to different variants of DFT. In fact, the particular partitioning of Eq. (16), although very natural, is not the one that is normally used. The most popular variant of DFT is the Kohn-Sham formulation, to which we now turn our attention. 

Finally, there is an alternative and decidedly different way to incorporate electron correlation in quantum chemical calculations that is growing rapidly in importance DFT [100]. By using the Kohn-Sham formulation, DFT methods have been used extensively in quantum chemistry during the last decade and yield results that are superior to HF-SCF calculations at essentially the same cost. A further advantage seems to be that DFT appears to hold promise in the treatment of transition metal compounds, which is an area where standard methods (except elaborate MCSCF and MR-CI treatments) often fail catastrophically. Concerning the treatment of electron correlation, it should be noted that DFT methods — unlike the more traditional methods discussed so far—are semiempirical in nature and therefore only provide an implicit treatment. Correlation effects are incorporated in DFT (via an adequate parametrization) through the exchange-correlation functional and not explicitly treated in the usual sense. 

In practice, the Kohn-Sham formulation [5] of DFT is almost always used. This overcomes the major difficulty with finding a density-functional for the kinetic energy by introducing a set of orthonormal auxiliary functions (i.e. the Kohn-Sham orbitals), with occupation numbers / , which sum to the ground state density,... 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

Dispersion interactions are, roughly speaking, associated with interacting electrons that are well separated spatially. DFT also has a systematic difficulty that results from an unphysical interaction of an electron with itself. To understand the origin of the self-interaction error, it is useful to look at the Kohn-Sham equations. In the KS formulation, energy is calculated by solving a series of one-electron equations of the form... 

In the next section we shall recall the definitions of the chemical concepts relevant to this paper in the framework of DFT. In Section 3 we briefly review Strutinsky s averaging procedure and its formulation in the extended Kohn-Sham (EKS) scheme. The following section is devoted to the presentation and discussion of our results for the residual, shell-structure part of the ionization potential, electron affinity, electronegativity, and chemical hardness for the series of atoms from B to Ca. The last section will present some conclusions. 

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