Kohn-Sham effective

The potential V/A(r) at point r is a Lagrange multiplier corresponding to the fixed-density constraint at that point. As A varies between 0 and 1, V A(r) must be adjusted such that the electron density remains fixed at n(r). At A = 1, 1/A coincides with the actual external potential Ki(r) while at A = 0, it coincides with the Kohn-Sham effective potential,... 

The analytic form of the first two terms in the Kohn-Sham effective potential (Vrff [p](r)) is known. They represent the external potential (vext which is the nuclear attraction potential in most cases) and Coulomb repulsion between electrons. The second term is an explicit functional of electron density. The last term, however, represents the quantum many-body effects and has a traditional name of exchange-correlation potential. vxc is the functional derivative of the component of the total energy functional called conventionally exchange-correlation energy (Exc[p]) ... 

The effective potential VKohn-Sham effective potential for the isolated subsystem A at the electron density p = pA ... 

The KSCED effective potential in Eq. 31 can be simply related with the Kohn-Sham effective potential [pa + Pb, A) f°r the whole system ... 

The matrix K relates the changes in the Coulomb and the exchange-correlation parts of the Kohn-Sham effective potential (truncated to the linear terms) and the... 

Here [ Svo r )/5vefr r)) y is the inverse of the transpose of the potential-potential response function, Svo r)/dveff r ))ji, r eff is the Kohn-Sham effective potential, and (0 ) is the lowest unoccupied... 

J. Chen, J. B. Krieger, R. O. Esquivel, M. J. Stott, G. J. lafrate. Kohn-Sham effective potentials for spin-polarized atomic systems. Phys. Rev. A, 54(3) (1996) 1910-1921. 

Fig. 5.5. The Kohn-Sham effective potential vks r) for two widely separated open shell atoms. Whereas the classical contributions to uks do not show any pathological behaviour, the exchange-correlation potential takes a positive value Ib — I a around the atom B...

Writing the Euler-Lagrange equations in terms of the single-particle wave functions (tpi) the variation principle finally leads to the effective singleelectron equation, well-known as the Kohn-Sham (KS) equation ... 

It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

Since the Fock operator is a effective one-electron operator, equation (1-29) describes a system of N electrons which do not interact among themselves but experience an effective potential VHF. In other words, the Slater determinant is the exact wave function of N noninteracting particles moving in the field of the effective potential VHF.5 It will not take long before we will meet again the idea of non-interacting systems in the discussion of the Kohn-Sham approach to density functional theory. 

In order to distinguish these orbitals from their Hartree-Fock counterparts, they are usually termed Kohn-Sham orbitals, or briefly KS orbitals. The connection of this artificial system to the one we are really interested in is now established by choosing the effective potential Vs such that the density resulting from the summation of the moduli of the squared orbitals tpj exactly equals the ground state density of our real target system of interacting electrons,... 

Gritsenko, O. V., Baerends, E. J., 1997, Electron Correlation Effects on the Shape of the Kohn-Sham Molecular Orbitals , Theor. Chem. Acc., 96, 44. 

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