Kohn-Sham single-particle energies

The kinetic-energy discontinuity is thus simply the Kohn-Sham single-particle gap Aks, or HOMO-LUMO gap, whereas the exchange-correlation discontinuity Axe is a many-body elfect. The true fundamental gap A = (AT- -1)- - (7V—1) — 2 (N) is the discontinuity of the total ground-state energy functional, ... 

Politzer and Abu-Aw wad61 studied the calculated single-particle energies for 12 smaller molecules and compared them with experimental ionization potentials. They found that the Kohn-Sham orbital energies were at least 2 eV too high... 

In order to find the single-particle-like excitation energies it is necessary to investigate the structure of the Green function, given by the Dyson Eq. (14). Sham and Kohn here argued that... 

The electron density n(r) which minimizes the total energy is then given by the sum over single-particle Kohn-Sham states... 

Note that the summation in Eq. (344) extends over all single-particle transitions q a between occupied and unoccupied Kohn-Sham orbitals, including the continuum states. Up to this point, no approximations have been made. In order to actually calculate 2(o ), the eigenvalue problem (344) has to be truncated in one way or another. One possibility is to expand all quantities in Eq. (344) about one particular KS-orbital energy difference co... 

Expressing the single-particle kinetic energy Ts as an orbital functional (1.37) prevents direct minimization of the energy functional (1.38) with respect to n. Instead, one commonly employs a scheme suggested by Kohn and Sham [274], which starts by writing... 

In usual practice, all single-particle wave functions and energies are typically obtained by solving the single-particle Kohn-Sham equation of density-functional theory in the so-called local-density approximation (LDA) (see, e.g.. Ref. [48]). 

The ground-state energy of the non-interacting electrons in the field of the effective potential can be found from the eigenvalues of the single-particle Schrodinger equation (Kohn and Sham, 1965) ... 

A sometimes overlooked fact is that the Kohn-Sham equation is exact at this stage. It is much easier to solve than the coupled Schrodinger equations that would have to be solved for the original system, since it decouples into single particle equations. The only problem is that we have introduced the exchange-correlation energy, which is an unknown quantity, and which must be approximated. Fortunately, it will turn out to be relatively easy to find reasonably good local approximations for it. 

The Hartree-Fock and the Kohn-Sham Slater determinants are not identical, since they are composed of different single-particle orbitals, and thus the definition of exchange and correlation energy in DFT and in conventional quantum chemistry is slightly different [52]. 

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