The Kohn-Sham Equations

The next key paper is that of Kohn and Sham. Here is the abstract, which is self-explanatory. 

Self-Consistent Equations Including Exchange and Correlation Effects W. Kohn and L. J. Sham Physical Review 140 (1965) All33 

The Kohn-Sham equations look like standard HF equations, except that the exchange term is replaced with an exchange-correlation potential whose form is unknown. 

In order to bring the notation back into line with other chapters, I will write the electronic energy for a molecular species as 

The first term on the right-hand side is a contribution from external fields, usually zero. The second term is the contribution from the kinetic energy and the nuclear attraction. The third term is the Coulomb repulsion between the electrons, and the final term is a composite exchange and correlation term. 

Since this Hamilton operator does not contain any electron-electron interactions it indeed describes a non-interacting system. Accordingly, its ground state wave function is represented by a Slater determinant (switching to 0S and (p rather than SD and % for the determinant and the spin orbitals, respectively, in order to underline that these new quantities are not related to the HF model) 

In order to distinguish these orbitals from their Hartree-Fock counterparts, they are usually termed Kohn-Sham orbitals, or briefly KS orbitals. The connection of this artificial system to the one we are really interested in is now established by choosing the effective potential Vs such that the density resulting from the summation of the moduli of the squared orbitals (p exactly equals the ground state density of our real target system of interacting electrons, 

At this point, we come back to our original problem finding a better way for the determination of the kinetic energy. The very clever idea of Kohn and Sham was to realize that if we are not able to accurately determine the kinetic energy through an explicit functional, 

Of course, the non-interacting kinetic energy is not equal to the hue kinetic energy of the interacting system, even if the systems share the same density, i. e., Ts A T.13 Kohn and Sham accounted for that by introducing the following separation of the functional F[p] 

So far so good, but before we are in business with this concept we need to find a prescription for how we can uniquely determine the orbitals in our non-interacting reference system. In other words, we ask how can we define Vs such that it really provides us with a Slater determinant which is characterized by exactly the same density as our real system To solve this problem, we write down the expression for the energy of our interacting, real 

In this section, we discuss first the basic formalism of density functional theory, and the Kohn-Sham equations whose solution yields the ground state density and energy of a system. We next describe a two-electron system, called Hooke s atom, which has the highly unusual feature of an analytic ground state, and is therefore of considerable pedagogical value. Lastly we discuss the notation and organization of this article. 

The problem of finding the ground-state properties of a system of N( 1) electrons is important in the study of atoms, molecules, clusters, surfaces, and solids. Since no exact solution exists in general, many approximate methods have been developed for approaching this problem. Each successful method has its own advantages and disadvantages. 

Wavefunction methods[l] have proved very successful in the study of small molecules. They have the important merit that their accuracy can be systematically improved by enlarging the size of the calculation. Unfortunately, since 

The density, on the other hand, is a function of only 3 spatial variables, r = x, y, z, so it is a much easier quantity to work with in practice. Furthermore, the groundbreaking work of Hohenberg and Kohn[2], and its subsequent extension in the constrained search formulation[3-5] proved that all quantities of interest could, in principle, be determined from knowledge of the density alone. 

The basic idea in density functional theory is to replace the Schrodinger equation for the interacting electronic system with a set of single-particle equations whose density is the same as that of the original system. These equations are the Kohn-Sham equations[6], and may be written 

DFT can be implemented in many ways. The minimization of an explicit energy functional, discussed up to this point, is not normally the most efficient among them. Much more widely used is the Kohn Sham approach. Interestingly, this approach owes its success and popularity partly to the fact that it does not exclusively work in terms of the particle (or charge) density, but brings a special kind of wave function (single-particle orbitals) back into the game. As 

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Once a solution of the Kohn-Sham equation is obtained, the total energy can be computed from... 

The wavevector is a good quantum number e.g., the orbitals of the Kohn-Sham equations [21] can be rigorously labelled by k and spin. In tln-ee dimensions, four quantum numbers are required to characterize an eigenstate. In spherically syimnetric atoms, the numbers correspond to n, /, m., s, the principal, angular momentum, azimuthal and spin quantum numbers, respectively. Bloch s theorem states that the equivalent... 

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. 

Density Eunctional Methods. The Kohn-Sham equations are... 

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... 

As mentioned above, a KS-LCAO calculation adds one additional step to each iteration of a standard HF-LCAO calculation a quadrature to calculate the exchange and correlation functionals. The accuracy of such calculations therefore depends on the number of grid points used, and this has a memory resource implication. The Kohn-Sham equations are very similar to the HF-LCAO ones and most cases converge readily. 

We may again chose a unitary transfonnation which makes tlie matrix of the Lagrange multiplier diagonal, producing a set of canonical Kohn-Sham (KS) orbitals. The resulting pseudo-eigenvalue equations are known as the Kohn-Sham equations. 

Morrison, R. C., Zhao, Q., 1995, Solution to the Kohn-Sham Equations Using Reference Densities from Accurate, Correlated Wave Functions for the Neutral Atoms Hehum Through Argon , Phys. Rev. A, 51, 1980. 

The Kohn-Sham equations have been applied to study gas-phase properties of many systems of biological interest. Most of such studies have been made for relatively small mol-... 

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... 

Although the Kohn-Sham equations form the quantum mechanical core of the DFT/SCRF methods, the final energetical results obtained by these methods also depend on other features of a particular DFT/SCRF implementation. It is important, therefore, to stress that the DFT/SCRF represents a whole family of methods. Each particular implementation may differ in the following details concerning ... 

Morrison, R. C., Q. Zhao, R. C. Morrison, and R. G. Parr. 1995. Solution of the Kohn-Sham equations using reference densities from accurate, correlated wave functions for the neutral atoms helium through argon. Phys. Rev. A51, 1980. 

For description of a time-dependent problem, the Kohn-Sham equations ... 

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 ... 

In the Kohn-Sham equation above, the Coulomb potential and the XC potential are obtained from their energy counterparts by taking the functional derivative of the latter with respect to the density. Thus... 

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... 

Our aim here is to apply the differential virial theorem to get an expression for the Kohn-Sham XC potential. To this end, we assume that a noninteracting system giving the same density as that of the interacting system exists. This system satisfies Equation 7.4, i.e., the Kohn-Sham equation. Since the total potential term of Kohn-Sham equation is the external potential for the noninteracting system, application of the differential virial relationship of Equation 7.41 to this system gives... 

Here the vector zs(r) is constructed from the kinetic energy tensor obtained by employing the solutions of the Kohn-Sham equation in Equation 7.42. Thus it is, in general, different from the vector z(r). A comparison of Equations 7.41 and 7.46 gives... 

The Kohn-Sham equations can be obtained from the minimalization of the noninteracting kinetic energy after expressing it with one-electron orbitals. Because Tl 0 is generally a linear combination of several Slater determinants, the form of the... 

Kohn-Sham equations is rather complicated for an arbitrarily selected set of weighting factors, and has to be derived separately for every different case of interest. For a spherically symmetric external potential and equal weighting factors, however, the Kohn-Sham equations have a very simple form, as shown in Ref. [72], In this case the noninteracting kinetic energy is given by... 

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. 

Within any sphere, the exchange-correlation potential in the Kohn-Sham equation is defined to be the conventional functional derivative of Ex c-... 

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