Potentials Kohn-Sham exchange

Gritsenko, O. V., R. van Leeuven, and E. J. Baerends. 1996. Molecular exchange-correlation Kohn-Sham potential and energy density from ab initio first- and second-order density matrices Examples for XH (X = Li, B, F). J. Chem. Phys. 104, 8535. 

Gritsenko O, Schipper PRT, Baerends EJ (1999) Approximation of the exchange-correlation Kohn-Sham potential with a statistical average of different orbital model potentials, Chem Phys Lett, 302 199-207... 

Gritsenko, O. V, Schipper, R. R. T., Baerends, E. J. (2000). Ensuring proper short-range and asymptotic behavior of the exchange-correlation kohn-sham potential by modeling with a statistical average of different orbital model potential. Int. J. Quantum Chem.l6,AffI-A 9. 

In addition to the energy terms for the exchange-correlation contribution (which enables the total energy to be determined) it is necessary to have corresponding terms for the potential, Vxc[p(i )]/ which are used to solve the Kohn-Sham equations. These are obtained as the appropriate first derivatives using Equation (3.52). 

The Kohn-Sham equations look like standard HF equations, except that the exchange term is replaced with an exchange-correlation potential whose form is unknown. 

Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in the corresponding time-dependent Kohn-Sham equations and which is rigorously defined as the functional derivative of the exchange-correlation action Axc[p] with respect to the time-dependent electron-density is approximated as the functional derivative of the standard, time-independent Exc with respect to the charge density at time t, i. e.,... 

Allen, M. J., Tozer, D. J., 2000, Kohn-Sham Calculations Using Hybrid Exchange-Correlation Functionals with Asymptotically Corrected Potentials , J. Chem. Phys., 113, 5185. 

However, one feature of the HF potential is that it is not a local potential. In the case of perfect data (i.e. zero experimental error), the fitted orbitals obtained are no longer Kohn-Sham orbitals, as they would have been if a local potential (for example, the local exchange approximation [27]) had been used. Since the fitted orbitals can be described as orbitals which minimise the HF energy and are constrained produce the real density , they are obviously quite closely related to the Kohn-Sham orbitals, which are orbitals which minimise the kinetic energy and produce the real density . In fact, Levy [16] has already considered these kind of orbitals within the context of hybrid density functional theories. 

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. 

Zhao, Q., R. C. Morrison, and R. G. Parr. 1994. From electron density to Kohn-Sham kinetic energies, orbital energies, exchange-correlation potentials, and exchange-correlation energies. Phys. Rev. A 50, 2138. 

J Exchange-Correlation Potential of Kohn-Sham Theory A Physical Perspective... 

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