Solving the Kohn-Sham Equations

The outline of the chapter is the following We start, in Sect. 6.2, by giving a technical overview on how to solve the Kohn-Sham equations. The next section is devoted to p seudo-potentials, an essential ingredient of many DFT calculations. In Sect. 6.4 we present our first test case, namely atoms, before we proceed to some plane-wave calculations in Sect. 6.5. The final example, methane calculated using a real-space implementation, is presented in Sect. 6.6. We will use atomic rniits throughout this chapter, except when explicitly stated otherwise. 

It is usually stated that the Kohn-Sham equations are simple to solve. By simple it is meant that for a given system, e.g., an atom, a molecule, or a solid, the computational effort to solve the Kohn-Sham equations is smaller than the one required by the traditional quantum chemistry methods, like Hartree-Fock (HF) or configuration interaction (Cl)h But it does not mean that it is easy or quick to write, or even to use, a DFT based computer program. Typically, such codes have several thousand lines (for example, the ABINIT [4] package - a plane-wave DFT code - recently reached 200,000 lines) and hundreds of input options. Even writing a suitable input file is often a matter of patience and experience. 

In spite of their differences, all codes try to solve the Kohn-Sham equations 

The notation fKs[tt] means that the Kohn-Sham potential, ukS) has a functional dependence on n, the electronic density, which is defined in terms of the Kohn-Sham wave-functions by 

The potential Uxc is defined as the sum of the external potential (normally the potential generated by the nuclei), the Hartree term and the exchange and correlation (xc) potential 

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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. 

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... 

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... 

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. 

Table III. Some Techniques for Solving the Kohn-Sham Equations ...

Ekardt [67] and Beck [68] used this approach and solved the Kohn-Sham equations self-consistently for this type of spherical jellium potential for Na clusters of various sizes. The results of such calculations are visualized for a 40-electron cluster of various metals or choices of rj values in Fig. 2, [62]. Going down in the periodic table from sodium to potassium, the potential becomes shallower and the levels are more loosely bound. The valence electrons in the coinage metal copper has a much higher electron density which will give a smaller r. value and more bound states. 

To solve the Kohn-Sham equations (13) one needs the ensemble exchange-correlation potential. In the following sections several approximate forms are discussed. 

In order to solve the Kohn Sham equations, an expansion of the one-electron wave functions on a basis set is performed. Both localized basis sets and plane wave ones are currently used. Localized basis sets have the advantage of their small size. However they are attached to the atomic positions, which yields non-zero Pulay forces in geometry optimization and molecular dynamics. Plane waves, on the other hand, provide a uniform sampling of space, whatever the specific conformation of the system they are independent of the atomic positions, but they require the use of pseudopotentials to mimick core electrons and a very large number of vectors is necessary in standard surface calculations. 

In order to solve the Kohn-Sham equations (Eqn. (2)) we used the molecular orbital-linear combination of atomic orbitals (MO-LCAO) approach. The molecular wave functions 0j are expanded the symmetry adapted orbitals Xj) which are also expanded in terms of the atomic orbitals... 

The purpose of this chapter will be to review the fundamentals of ab initio MD. We will consider here Density Functional Theory based ab initio MD, in particular in its Car-Parrinello version. We will start by introducing the basics of Density Functional Theory and the Kohn-Sham method, as the method chosen to perform electronic structure calculation. This will be followed by a rapid discussion on plane wave basis sets to solve the Kohn-Sham equations, including pseudopotentials for the core electrons. Then we will discuss the critical point of ab initio MD, i.e. coupling the electronic structure calculation to the ionic dynamics, using either the Born-Oppenheimer or the Car-Parrinello schemes. Finally, we will extend this presentation to the calculation of some electronic properties, in particular polarization through the modern theory of polarization in periodic systems. 

Eq. (9), and the forces on the nuclei can also be determined. More details on pseudopotential methods, on methods of solving the Kohn-Sham equations, and on the algorithms used can be found in the references described in Appendix A. 

In this case, the vectors G are members of the set of reciprocal lattice vectors and eqn (4.105) really amounts to a three-dimensional Fourier series. The task of solving the Kohn-Sham equations is then reduced to solving for the unknown coefficients ak+c- The only difficulty is that the potential that arises in the equation depends upon the solution. As a result, one resorts to iterative solution strategies. 

Here I will end my discussion on DFT, and instead move on to the problem of solving the Kohn-Sham equations. 

Thus, it is the same whether we solve the Kohn-Sham equations (85) or whether we deal with the constrained variational principle given by Eq. (102). 

One way to solve the Kohn-Sham equations Eq. (3) is to expand the molecular orbited wavefunctions in a set of symmetry adapted functions Xj(r) which is expanded as a linear combination of atomic orbitals Un/F/m with coefficients... 

Finally, we mention that further corrections may have to be added to the forces when other approximations in solving the Kohn-Sham equations are introduced. But we consider a discussion of these beyond the scope of this presentation. 

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