Kohn-Sham equations solving

To. solve the Kohn-Sham equations a self-consistent approach is taken. An initial guess of the density is fed into Equation (3.47) from which a set of orbitals can be derived, leading to an improved value for the density, which is then used in the second iteration, and so on until convergence is achieved. 

In addition to the energy terms for the exchange-correlation contribution (which enables the total energy to be determined) it is necessary to have corresponding terms for the potential, Vxc[p(i )]/ which are used to solve the Kohn-Sham equations. These are obtained as the appropriate first derivatives using Equation (3.52). 

To solve the Kohn-Sham equations a number of different approaches and strategies have been proposed. One important way in which these can differ is in the choice of basis set for expanding the Kohn-Sham orbitals. In most (but not all) DPT programs for calculating the properties of molecular systems (rather than for solid-state materials) the Kohn-Sham orbitals are expressed as a linear combination of atomic-centred basis functions ... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

In this equation Exc is the exchange correlation functional [46], is the partial charge of an atom in the classical region, Z, is the nuclear charge of an atom in the quantum region, is the distance between an electron and quantum atom q, r, is the distance between an electron and a classical atom c is the distance between two quantum nuclei, and r is the coordinate of a second electron. Once the Kohn-Sham equations have been solved, the various energy terms of the DF-MM method are evaluated as... 

Since surface charges depend on the electrostatic potential (Eq. 4.20), Eqs. 4.20-4.22 are solved in an iterative way leading to self-consistent surface charges. At the end of this procedure, surface charges and the electrostatic potential satisfy the boundary condition specified in Eq. 4.21. In practical applications, this self-consistent procedure for calculating reaction field potential is coupled to self-consistent procedure which governs solving the Kohn-Sham equations. A special case for infinite dielectric constant outside the cavity... 

Going back to Eq. (8), the problem now is that one must minimize Ev with respect to the y/i, instead of directly minimizing it with respect to p. This can, however, be achieved by solving Eq. (9), called the Kohn-Sham equation ... 

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... 

In Equations 10.1, 10.4, and 10.5, the standard implicit sum on the index (x is assumed. The relativistic Kohn-Sham equations are obtained by minimization of the Equation 10.4 with respect to the orbitals. In contrast to the nonrelativistic case, the variational procedure gives rise to an infinite set of coupled equations (see the summation restrictions in Equation 10.3) that have to be solved in a self-consistent manner ... 

Garza, J., Vargas, R. and Vela, A. 1988. Numerical self-consistent-field method to solve the Kohn-Sham equations in confined many-electron atoms. Phys. Rev. E. 58 3949-54. 

Most practical electronic structure calculations using density functional theory [1] involve solving the Kohn-Sham equations [2], The only unknown quantity in a Kohn-Sham spin-density functional calculation is the exchange-correlation energy (and its functional derivative) [2]... 

It has been shown by Levy and Ferdew [100] that it is possible to solve the Kohn-Sham equations by means of a density-constrained variation of the... 

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,... 

Thus, it is clearly equivalent to solving the Kohn-Sham equations or to dealing with the constrained variational principle for T. ... 

The computational procedure in Eq. (82) can also be written from a dual perspective, in which the Kohn- ham potential is the fundamental descriptor [60]. In this perspective, one solves the Kohn-Sham equations,... 

If you have a vague sense that there is something circular about our discussion of the Kohn-Sham equations you are exactly right. To solve the Kohn-Sham equations, we need to define the Hartree potential, and to define the Hartree potential we need to know the electron density. But to find the electron density, we must know the single-electron wave functions, and to know these wave functions we must solve the Kohn-Sham equations. To break this circle, the problem is usually treated in an iterative way as outlined in the following algorithm 1 2 3... 

Solve the Kohn-Sham equations defined using the trial electron density to find the single-particle wave functions, i i,(r). 

Compare the calculated electron density, nKs(r), with the electron density used in solving the Kohn-Sham equations, (r). If the two densities are the same, then this is the ground-state electron density, and it can be used to compute the total energy. If the two densities are different, then the trial electron density must be updated in some way. Once this is done, the process begins again from step 2. 

This approximation uses only the local density to define the approximate exchange-correlation functional, so it is called the local density approximation (LDA). The LDA gives us a way to completely define the Kohn-Sham equations, but it is crucial to remember that the results from these equations do not exactly solve the true Schrodinger equation because we are not using the true exchange-correlation functional. 

We now have all the pieces in place to perform an HF calculation—a basis set in which the individual spin orbitals are expanded, the equations that the spin orbitals must satisfy, and a prescription for forming the final wave function once the spin orbitals are known. But there is one crucial complication left to deal with one that also appeared when we discussed the Kohn-Sham equations in Section 1.4. To find the spin orbitals we must solve the singleelectron equations. To define the Hartree potential in the single-electron equations, we must know the electron density. But to know the electron density, we must define the electron wave function, which is found using the individual spin orbitals To break this circle, an HF calculation is an iterative procedure that can be outlined as follows ... 

Dispersion interactions are, roughly speaking, associated with interacting electrons that are well separated spatially. DFT also has a systematic difficulty that results from an unphysical interaction of an electron with itself. To understand the origin of the self-interaction error, it is useful to look at the Kohn-Sham equations. In the KS formulation, energy is calculated by solving a series of one-electron equations of the form... 

The Kohn-Sham equations are approximations because the exact functional needed to transform the electron density function p into the energy is unknown. They are approximations to an exact description because the equations (as distinct from methods of solving them) involve no approximations, with the ominous caveat that the form of the p-to-E functional Exc is left unspecified. 

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