Kohn-Sham equations method determination

Since most modern DFT calculations use the Kohn-Sham method, it seems desirable to use the information available from a solution of the Kohn-Sham equations to determine the Fukui function. To this end, Cohen and coworkers [50-52] showed that one can relate the Fukui function to the frontier orbital densities of Kohn-Sham theory through... 

The main challenge, after having defined the Kohn-Sham equations, is to solve them in an accurate way where the main problem is to find an accurate description of the multi-centre Coulomb potential. The method used is, to a large extent, determined by the character of the atoms involved in the bonding. The electronic structure of alkali and noble metals is for example rather simple... 

Eq. (9), and the forces on the nuclei can also be determined. More details on pseudopotential methods, on methods of solving the Kohn-Sham equations, and on the algorithms used can be found in the references described in Appendix A. 

In our statement of the Kohn-Sham equations, nothing has been said about the way in which i/ (r) itself will be determined. Many methods share the common feature that the wave function is presumed to exist as a linear combination of some set of basis functions which might be written generically as... 

Unlike wavefimction methods, DFT does not attempt the daimting task of constructing (and finding a solution for) the many-body wavefimction. This is in contrast to the collection of techniques which one might collect under the familial umbrella of quantum chemical methods. Rather, it starts from the premise that it is the electron density that determines the ground-state energy of the system. This results in a formalism that is, in principle, exact and practice as well (via the one-particle Kohn-Sham equations [9]). 

The Kohn-Sham equations are exact but, of course, for practical calculations approximations have to be made and these will determine the accuracy, the speed, and the interpretability of approximate KS-DFT methods. 

In practical calculations making use of the Kohn-Sham method, the Kohn-Sham equation is used. This equation is a one-electron SCF equation applying the Slater determinant to the wavefunction of the Hartree method, similarly to the Hartree-Fock method. Therefore, in the same manner as the Hartree-Fock equation, this equation is derived to determine the lowest energy by means of the Lagrange multiplier method, subject to the normalization of the wavefunction (Parr and Yang 1994). As a consequence, it gives a similar Fock operator for the nonlinear equation. 

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

The Kohn-Sham equations have to be solved iteratively because the Coulomb term depends on the density (i.e., the orbitals to be determined). The main advantage of DFT methods is that they include some treatment of electron correlation at a computational cost equivalent to that of the HF method. The main disadvantage of DFT, however, is that there is no hierarchy of increasingly better functionals. The performance of a given functional must be... 

The procedure for determining the electron density and the energy of the system within the DFT method is similar to the approach used in the Hartree-Fock technique. The wavefunction is expressed as an antisymmetric determinant of occupied spin orbitals which are themselves expanded as a set of basis functions. The orbital expansion coefficients are the set of variable parameters with respect to which the DFT energy expression of equation 15 is optimized. The optimization procedure gives rise to the single particle Kohn-Sham equations which are similar, in many respects, to the Roothaan-Hall equations of Hartree-Fock theory. 

The conventional approach to calculate the polarizability of metal clusters is to solve the Kohn-Sham equations using suitable approximate forms for the exchange correlation functionals and a finite field method. We have recently carried out a systematic all electron DFT-based calculations for the polarizability and binding energy of sodium as well as lithium metal clusters [51,52]. It has been shown that the effect of electron correlation plays a significant role in determining the polarizability of metal clusters, although the effect is less pronounced for lithium clusters. Electron... 

Note that the Kohn-Sham Hamiltonian hKS [Eq. (4.1)] is a local operator, uniquely determined by electron density15. This is the main difference with respect to the Hartree-Fock equations which contain a nonlocal operator, namely the exchange part of the potential operator. In addition, the KS equations incorporate the correlation effects through Vxc whereas they are lacking in the Hartree-Fock SCF scheme. Nevertheless, though the latter model cannot be considered a special case of the KS equations, there are some similarities between the Hartree-Fock and the Kohn-Sham methods, as both lead to a set of one-electron equations allowing to describe an n-electron system. 

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