Kohn-Sham formalism

Table 4.1. Acronyms used in text for parametrizations for the exchange-correlation (Exc) functional. The acronyms for separate exchange (Ej and correlation (E ) components of A xc are specified when applicable. Throughout the text, density functional calculations following the Kohn-Sham formalism are referred to as DFT(XXX), where XXX stands either for the acronym of the approximate exchange-correlation functional or for the acronyms of the exchange and correlation functionals, separated by the / symbol.

A water dimer, which bears more relevance to biological systems, was recently investigated by many researchers. Results obtained using various implementations of the Kohn-Sham formalism were reported.87 109-116 119 122 124 125 128 Table 4.2 collects selected results. 

The Kohn-Sham theory made a dramatic impact in the field of ab initio molecular dynamics. In the 1985, Car and Parrinello38 introduced a new formalism to study dynamics of molecular systems in which the total energy functional defined as in the Kohn-Sham formalism proved to be instrumental for practical applications. In the Car-Parrinello method (CP), the equations of motion are based on a Lagrangian (Lcp) which includes fictitious degrees of freedom associated with the electronic state. It is defined as ... 

The SCRF approach became a standard tool167 for estimating solvent effects and was combined with various quantum chemical methods that range from semi-empirical161 to the post-Hartree-Fock ab initio ones. It can also be combined with the Kohn-Sham formalism where the Kohn-Sham Hamiltonian (Eq. 4.2) is used for the gas-phase Hamiltonian in Eq. 4.15. The effective Kohn-Sham Hamiltonian for the system embedded in the dielectric environment takes the following form ... 

Several techniques for calculating f p.p were recently combined with the Kohn-Sham formalism. They include ... 

Thermochemistry. Chen et al.168 combined the Kohn-Sham formalism with finite difference calculations of the reaction field potential. The effect of mobile ions into on the reaction field potential Poisson-Boltzman equation. The authors used the DFT(B88/P86)/SCRF method to study solvation energies, dipole moments of solvated molecules, and absolute pKa values for a variety of small organic molecules. The list of molecules studied with this approach was subsequently extended182. A simplified version, where the reaction field was calculated only at the end of the SCF cycle, was applied to study redox potentials of several iron-sulphur clusters181. 

Stanton et al. combined the Kohn-Sham formalism with molecular dynamics to study solvation enthalpies of several molecules189. Their implementation of Eq. 4.25 can be outlined as follows ... 

It is also possible to extend the Kohn-Sham formalism by defining an energy term Ts that includes the kinetic energy of the noninteracting system, and the total... 

The general relativistic Hohenberg-Kohn-Sham formalism, outlined above, contains the spin degrees of freedom in a complete form. Consequently, the spin and kinetic motion effects are not separable. Indeed, they are contained in the external potential term as one can see if such term is written using the orbital current... 

As in the case of the Schrodinger approach in which spin is introduced by giving a specific form to the wave function, the spin dependence in the Hohenberg-Kohn-Sham formalism in a nonrelativistic framework is introduced by imposing some form of restrictions to the functional. Namely, the total energy can be written as [3,5]... 

P. W. Ayers and M. Levy, Using the Kohn—Sham formalism in pair density-functional theories. Chem. Phys. Lett. 416, 211-216 (2005). 

A number of different methods have been proposed to introduce a self-interaction correction into the Kohn-Sham formalism (Perdew and Zunger 1981 KUmmel and Perdew 2003 Grafenstein, Kraka, and Cremer 2004). This correction is particularly useful in situations with odd numbers of electrons distributed over more than one atom, e.g., in transition-state structures (Patchkovskii and Ziegler 2002). Unfortunately, the correction introduces an additional level of self-consistency into the KS SCF process because it depends on the KS orbitals, and it tends to be difficult and time-consuming to converge the relevant equations. However, future developments in non-local correlation functionals may be able to correct for self-interaction error in a more efficient manner. 

Within the Kohn-Sham formalism of DFT, V is substituted by the Kohn-Sham potential Vks Expanding equation 7 a little, we note that spin-orbit effects are implicitly included in the ZORA Hamiltonian ... 

During the past 10 - 15 years, Kohn-Sham density functional theory has been a major factor in a dramatic expansion of the scope of computational chemistry and its capability for treating systems of practical importance [45-51]. Density functional methodology includes electronic correlation, so that the energies are more accurate than Hartree-Fock however the Kohn-Sham formalism is similar to the latter, as are therefore the demands upon computer resources. It is therefore feasible to treat relatively large systems at a reasonably high (post-Hartree-Fock) level. 

Current hybrid functionals do not improve this situation. Their non-local component (Hartree-Fock exchange) cannot give rise to any attraction. To describe quantitatively the long-range interactions, either a non-local approximation to Exc[p] must be applied within the Kohn-Sham framework or methods using other-than-Kohn-Sham formalism should be used. Some of such approaches will be discussed in the last section of this review. Here, we mention an especially promising combination of symmetry adapted perturbation theory with of the Kohn-Sham orbitals.125... 

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