Exact Kohn-Sham potentials

As an example of these ideas, we plot in Figure 7.1 the exact Kohn-Sham potential for the neon atom. The potential has been obtained by applying the Zhao-Parr (ZP) method [12], which generates the exact Kohn-Sham potential for a given density to a highly accurate density of neon [13]. The corresponding... 

In this review we will give an overview of the properties (asymptotics, shell-structure, bond midpoint peaks) of exact Kohn-Sham potentials in atomic and molecular systems. Reproduction of these properties is a much more severe test for approximate density functionals than the reproduction of global quantities such as energies. Moreover, as the local properties of the exchange-correlation potential such as the atomic shell structure and the molecular bond midpoint peaks are closely related to the behavior of the exchange-correlation hole in these shell and bond midpoint regions, one might be able to construct... 

Are exact Kohn-Sham potentials equivalent to local functions 3... 

The first authors to realize that the OEP is the exact Kohn-Sham potential at the exchange-only level were apparently V. Sahni, J. Gruenebaum, and J.P. Perdew, Phys. Rev. B 26, 4371 (1982). 

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... 

A fictitious four-electron spherically symmetric system for which the exact Kohn-Sham potential would read vks = is considered in this work. The identification of the external potential Vgxtir) which would correspond to such simple form of the Kohn-Sham potential is of no concern for the present considerations. For the purposes of the present analyses, it is crucial that the two doubly occupied Kohn-Sham orbitals have the known exact analytic form of the hydrogenic wavefunctions Is and 2s. Note that the considered Kohn-Sham potential is neither the exact nor a reasonable approximant to the Kohn-Sham potential for a beryllium atom. The model bears some resemblance to a model used for a different purpose (the excited state of a two-electron system) in Ref. [32], Eq. (9). Although the analytical form of the dependence of b] on ha and is not obtained, the exact form of... 

Figure 27 Excitation energies for the He atom obtained with different approaches. KS marks the orbital eigenvalues for the exact Kohn-Sham potential, ALDA the results with the adiabatic local-density approximation, and for the TDOEP approaches different orbital-dependent functionals have been used. Both the ALDA and the TDOEP results have been obtained using the time-dependent density-functional theory...

Table 14 Fano parameters for different ionization resonances for some closed-shell atoms. Except for the calculations marked EXX, the Kohn-Sham single-particle eigenvalues have in all cases been shifted to match the ionization potential. Exact KS represents results with the exact Kohn-Sham potential, GGA those with a gradient-corrected density functional, EXX exact-exchange without correlation, EXX- -LDA the same but with the inclusion of correlation effects with a local-density approximation, and Exp. experimental results. The results are from ref. 91... 

Figure 1. Excitation energies from the ground-state of He, including the orbital energies of the exact Kohn-Sham potential, the time-dependent spin-split correction calculated within four different functional approximations, and experimental numbers. ...

Table 1. Singlet/triplet excitation energies in the helium and beryllium atoms, calculated from the exact Kohn-Sham potential by using approximate xc kernels (in millihartrees), and using the lowest 34 unoccupied orbitals of s and p symmetry for He, and the lowest 38 unoccupied orbitals of s, p, and d symmetry for Be. Exact values from Ref. [29] for He and from Ref. [30] for Be.

Equation (4.29), having the form of a one-particle equation, is fairly easy to solve mmierically. We stress, however, that the Kohn-Sham equation is not a mean-field approximation If we knew the exact Kohn-Sham potential, vks, we would obtain from (4.29) the exact Kohn-Sham orbitals, and from these the correct density of the system. 

This equation is a formally exact representation of the linear density response in the sense that, if we possessed the exact Kohn-Sham potential (so that we could extract /xc), a self-consistent solution of (4.69) would yield the response function, x, of the interacting system. 

Let us now discuss the correlation effects on the atomic shell structure. We plot in Fig. 7 some of the described potentials for the case of the beryllium atom. The exact exchange-correlation potential v c is calculated from an accurate Cl (Configuration Interaction) density using the procedure described in [20]. The potentials Vx, and u" , are calculated within the optimized potential model (OPM) [21,40,41] and are probably very close to their exact values which can be obtained from the solution for of the OPM integral equation [21,40,41] by insertion of the exact Kohn-Sham orbitals instead of the OPM... 

In the Flartree-Fock (FIF) method, the spurious self-interaction energy in the Flartree potential is exactly cancelled by the contributions to the energy from exchange. This would also occur in DFT if we knew the exact Kohn-Sham functional. In any approximate DFT functional, however, a systematic error arises due to incomplete cancellation of the self-interaction energy. 

The first term in brackets is the usual kinetic energy operator. The noninteracting reference system has the property that its one-determinantal wavefunction of the lowest N orbitals yields the exact density of the interacting system with external potential v(r) as a sum over densities of the occupied orbitals, that is, p(r) = Xl<)>,l2, and the corresponding exact energy E[p(r)]. The Kohn-Sham potential should account for all effects stemming from the electron-nuclear and electron-electron interactions. Not only does the Kohn-Sham potential contain the attractive potential v(r) of the nuclei and the classical Coulomb repulsion VCoul(r) within the electron density p(r), but it also accounts for all exchange and correlation effects, which have so to say been folded into a local potential vxc r) ... 

Nesbet, R.K. and Colle, R. (2000). Tests of the locality of exact Kohn-Sham exchange potentials, Phys. Rev. A 61, 012503. 

USING THE EXACT KOHN-SHAM EXCHANGE ENERGY DENSITY FUNCTIONAL AND POTENTIAL TO STUDY ERRORS INTRODUCED BY APPROXIMATE CORRELATION FUNCTIONALS... 

The relation between other than eHOMO eigenvalues of the exact Kohn-Sham orbitals and higher ionization potentials is currently an object of studies by Baerends and collaborators.95,96... 

There is a growing interest in determining the exact exchange, exchange-correlation and Kohn-Sham potential in the knowledge of the density[31-35]. The present author has also proposed a method that enables one to calculate these potentials if the density is known [37]. This method can be applied to ensemble states without any difficulty [16]. 

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