Kohn-Sham gap

The problem of the cohesive energy, in principle, can be corrected with a better exchange-correlation functional. The band gap problem is more difficult because density functional formalism does not explicitly provide information on the excitation energies. The quasiparticle energies should, in general, be obtained from the one-particle Green s function.However, recently Sham and Schliiter have shown that there is a formal relationship between the minimum gap Eg of an insulator and the Kohn-Sham gap g obtained from a difference in eigenvalues ... 

E. J. Baerends, O. V. Gritsenko, and R. van Meet, Phys. Chem. Chem. Phys., 15, 16408-16425 (2013). The Kohn-Sham Gap, the Fundamental Gap and the Optical Gap The Physical Meaning of Occupied and Virtual Kohn-Sham Orbital Eneigies. 

Since all terms in E other than and are continuous functionals of n(r), the fundamental gap is the sum of the Kohn-Sham gap and the exchange orrelation discontinuity. Standard density fiinctionals (LDA and GGA) predict Axe = 0, and thus often underestimate the Wdamental gap. The fundamental and Kohn-Sham gaps are also illustrated in Fig. 1. 

To complete this presentation, we define a Kohn-Sham hardness (gap) by... 

All these functional derivatives are well defined and do not involve any actual derivative relative to the electron number. It is remarkable that the derivatives of the Kohn-Sham chemical potential /rs gives the so-called radical Fukui function [8] either in a frozen orbital approximation or by including the relaxation of the KS band structure. On the other hand, the derivative of the Kohn-Sham HOMO-FUMO gap (defined here as a positive quantity) is the so-called nonlinear Fukui function fir) [26,32,50] also called Fukui difference [51]. 

In spite of the absence of a typical chromophore, 1,2-dithiin is a bright reddish-orange color. Absorption maxima were found at 451 (2.75 eV), 279 (4.36 eV), and 248 nm (5.00 eV), and the colored band was assigned to a A excitation <1991JST(230)287>. The main reason for the colored absorption of 1,2-dithiin is the low HOMO-LUMO gap of the KS orbitals which amounts to only 3.6 eV (HOMO = highest occupied molecular orbital LUMO = lowest unoccupied molecular orbital KS = Kohn-Sham) <2000JMM177>. By comparison, saturated 1,2-dithiane is colorless (290 nm). 

In Kohn-Sham DFT based approaches, expressions that are of similar structure as Eqs. (9a) and (9b) are obtained, but in the form of contributions from all occupied Kohn-Sham MOs The excited-state wavefunctions are at the same time formally replaced by the unoccupied MOs, and the many-electron perturbation operators /T(M41, etc. by their one-electron counterparts //(M-41, etc. Orbital energies e and ea formally substitute the total energies of the states (see later). Thus, similar interpretations of NMR parameters can be worked out in which the highest occupied MO-lowest unoccupied MO gap (HLG) plays a highly important role. It must be emphasized, though, that there is no one-to-one correspondence between the excited states of the SOS equations and the unoccupied orbitals which enter the DFT expressions, nor between excitation energies and orbital energy differences, i.e., there are no one-determinantal wavefunctions in Kohn-Sham DFT perturbation theory which approximate the reference and excited states. 

We can also evaluate the lowest typical frequency for the electronic dynamics from the gap Egap, energy difference between the Lowest Unoccupied Molecular Orbital (LUMO) and the Highest Occupied Molecular Orbital (HOMO), of the Kohn-Sham non-interacting electron system, which determines the lowest curvature of the E c) Kohn-Sham energy functional ... 

We study the dielectric and energy loss properties of diamond via first-principles calculation of the (0,0)-element ( head element) of the frequency and wave-vector-dependent dielectric matrix eg.g CQ, The calculation uses all-electron Kohn-Sham states in the integral of the irreducihle polarizahility in the random phase approximation. We approximate the head element of the inverse matrix hy the inverse of the calculated head element, and integrate over frequencies and momenta to obtain the electronic energy loss of protons at low velocities. Numerical evaluation for diamond targets predicts that the band gap causes a strong nonlinear reduction of the electronic stopping power at ion velocities below 0.2 a.u. 

Many transition-metal oxides and fluorides are insulators which LSD incorrectly describes as metals[119]. For some of these materials, full-potential GGA calculations open up a small fundamental gap which LSD misses, and so correct the description. For others, GGA enhances the gap, and generally improves the energy bands[l 19]. Of course, the sizes of the band-structure gaps are not physically meaningful in LSD, GGA, or even with the exact Kohn-Sham potential for the neutral solid. To get a physically meaningful fundamental gap, one must take account... 

The conventional exchange-correlation functionals used in the Kohn-Sham equation contain electron correlations to some extent (see Sect. 4.5). Nevertheless, it has been recognized that the Kohn-Sham method is not able to reproduce accurate orbital energies. The cause for the inaccurate orbital energies has been investigated for many years in solid state physics from the viewpoint of the underestimation of band gaps. Let us consider the Kohn-Sham equation in Eq. (7.9). Perdew et al. 

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