Exchange potential from Kohn-Sham equations

Given in Table 7.1 are the results [24] of the total energy of some atoms obtained by solving the Kohn-Sham equation self-consistently with the exchange potential Wx within the central field approximation. The energy is obtained from Equation 7.10... 

As is evident from the above, both the physics invoked to derive the potential of Equation 7.31 and the numerical results presented show that Wx gives an accurate exchange potential for the excited states. When the proposal was initially made, there was no mathematical proof of the existence of a Kohn-Sham equation for excited states. It is only during the past few years that DFT of excited states [34-37], akin to its ground-state counterpart, is being developed. 

The difficulty of this problem can be appreciated by noticing that in order to solve the Kohn-Sham equations exactly, one must have the exact exchange-correlation potential which, moreover, must be obtained from the exact exchange-correlation functional c[p( )] given by Eq. (160). As this functional is not known, the attempts to obtain a direct solution to the Kohn-Sham equations have had to rely on the use of approximate exchange-correlation functionals. This approximate direct method, however, does not satisfy the requirement of functional iV-representability,... 

In fact, the true form of the exchange-correlation functional whose existence is guaranteed by the Hohenberg-Kohn theorem is simply not known. Fortunately, there is one case where this functional can be derived exactly the uniform electron gas. In this situation, the electron density is constant at all points in space that is, n(r) = constant. This situation may appear to be of limited value in any real material since it is variations in electron density that define chemical bonds and generally make materials interesting. But the uniform electron gas provides a practical way to actually use the Kohn-Sham equations. To do this, we set the exchange-correlation potential at each position to be the known exchange-correlation potential from the uniform electron gas at the electron density observed at that position ... 

In comparing Eq. (13) to the Kohn-Sham equations Eq. (3) one concludes that E(.r, x E), since it is derived from exact many-electron theory [22], is the exact Coulomb (direct) plus exchange-correlation potential. It is non-local and also energy-dependent. In view of this it is hard to see how the various forms of constructed local exchange correlation potentials that are in use today can ever capture the full details of the correlation problem. 

The exchange-correlation potential is the source of both the strengths and the weaknesses of the DF approach. In HF theory, the analytical form of the term equivalent to Vxc, the exchange potential, arises directly during the derivation of the equations, but it depends upon the one-particle density matrix, making it expensive to calculate. In DF theory the analytical form of Vxc must be put into the calculations because it does not come from the derivation of the Kohn-Sham equations. Thus, it is possible to choose forms for Vxc that depend only upon the density and its derivatives and which are cheap to calculate (the so-called local and non-local density approximations). The Vxc factor can also be chosen to account for some of the correlation between the electrons, in contrast to HF methods for which additional calculations must be made. The drawback is that there does not appear to be any systematic way of improving the potential. Indeed, many such terms have been proposed. 

The above difficulty has been bypassed in actual applications of the Kohn-Sham equations by resorting to approximate exchange-correlation functionals. These functionals, however, as discussed in Sections 2.2 - 2.4, do not comply with the requirement of functional iV-representability. The calculation of the exact Kohn-Sham exchange-correlation potential is nonetheless feasible by means of the inverse method" provided that one has the exact ground-state one-particle density p(r). Although such densities can be obtained from experiment, the most accurate ones are obtained from highly accurate quantum mechanical calculations. 

This equation is usually solved self-consistently . An approximate charge is assumed to estimate the exchange-correlation potential and to determine the Hartree potential from equation A1.3,16. These approximate potentials are inserted in the Kohn-Sham equation and the total charge density is obtained from equation Al.3.14. The output charge density is used to construct new exchange-correlation and Hartree potentials. The process is repeated until the input and output charge densities or potentials are identical to within some prescribed tolerance. 

Appendix. First, the Kohn-Sham equations are converged for the metal slab without the solvent. Thereafter the DR procedure is applied simultaneously to the KS-DFT equations for the electron wave functions, and to the 3D-RISM/KH equations for the solvent site correlation functions. On each DR step for tpj k), the new wave functions of M states are orthonormalized by the Gram-Schmidt process. Then the exchange-correlation potential Vxc( ) is recomputed, the metal Hartree potential Vjj(r) is obtained from the Poisson equation by using the 3D-FFT, and the metal-solvent site potential is calculated. Next, several... 

Note that this correction has the problem that the Kohn-Sham equation is not invariant for the unitary transformation of occupied orbitals, even after the correction, differently from the Hartree-Fock equation. In the Hartree-Fock equation, the variations of the Coulomb self-interaction energy and its potential for the unitary transformations of occupied orbitals cancel out with those of the exchange self-interaction, while these are not compensated, even after the correction in the Kohn-Sham equation. Therefore, the effect of the self-interaction correction depends on the difference in occupied orbitals before and after the unitary transformation. For removing this difference, it is usual to localize the orbitals before the self-interaction correction (Johnson et al. 1994). Note, however, that there are various types of orbital localization methods, and the effect of the selfinteraction correction inevitably depends on them. Combining with the optimized effective potential (OEP) method (see Sect. 7.5) may be one of the most efficient ways to solve this problem. This combination enables us to consistently obtain localized potentials with no self-interaction error. 

In Chap. 4, the Kohn-Sham equation, which is the fundamental equation of DFT, and the Kohn-Sham method using this equation are described for the basic formalisms and application methods. This chapter first introduces the Thomas-Fermi method, which is conceptually the first DFT method. Then, the Hohenberg-Kohn theorem, which is the fundamental theorem of the Kohn-Sham method, is clarified in terms of its basics, problems, and solutions, including the constrained-search method. The Kohn-Sham method and its expansion to more general cases are explained on the basis of this theorem. This chapter also reviews the constrained-search-based method of exchange-correlation potentials from electron densities and... 

We are immediately confronted with the problem of how to find the unknown exchange-correlation energy Exc, which is replaced also by an unknown exchange-correlation potential in the form of a functional derivative Vxc = We obtain the Kohn-Sham equation (resembling the Fock equation) -jA -F ng (/>i = eifi, where wonder-potential t)g = t) -F Vcoyi -F i xc, fcoul stands for the sum of the usual Coulombic operators (as in the Hartree-Fock method) (built from the Kohn-Sham spinorbitals) and vxc is the potential to be found. 

Many of the questions raised here can, in principle, be addressed in the framework of Car-Parrinello and ab initio molecular dynamics where the parametrization of the potential is avoided by solving Hartree-Fock or Kohn-Sham equations at each time step. However, such simulations become computationally intractable for systems with a few hundred water molecules and simulation times of more than 100 ps. Recent controversy on the structural and dynamic properties of liquid water, apart from the limited accuracy of the modern exchange-correlation functionals, raised the issue of using the proper simulation protocols with the sufficient length of equilibration and production runs and the combination of pseudopotentials and basis sets yielding converged results. As such, they provide valuable benchmark data for validation of more approximate methods having much broader applicability for realistic and complex systems. 

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