Kohn-Sham equations self-consistent solution

We have skipped over a whole series of important details in this process (How close do the two electron densities have to be before we consider them to be the same What is a good way to update the trial electron density How should we define the initial density ), but you should be able to see how this iterative method can lead to a solution of the Kohn-Sham equations that is self-consistent. 

QR Method. The first relativistic method is the so-called quasi-relativistic (QR) method. It has been developed by Snijders, Ziegler and co-workers (13). In this approach, a Pauli Hamiltonian is included into the self-consistent solution of the Kohn-Sham equations of DFT. The Pauli operator is in a DFT framework given by... 

The calculation of the ground-state energy of the Wigner electron crystal necessitates the self-consistent solution of the Slater-Kohn-Sham equations for the Bloch orbitals of a single fully occupied energy band, since there is one electron per unit cell and one is concerned with the spin-polarized state [45], This was accomplished by standard computational routines for energy band-... 

Using Kohn-Sham theorem one can solve equation (4) by obtaining the self-consistent solution of the following Hamiltonian ... 

A self-consistent scalar-relativistic (SR) version of the LCGTO-DF method has also been developed recently." "The SR variant employs a unitary second-order Douglas-Kroll-Hess (DKH) "" transformation for decoupling large and small components of the full four-component spinor solutions to the Dirac-Kohn-Sham equation. The approximate DKH transformation, very appropriate and efficient for molecular calculations, has been implemented this variant utilizes nuclear potential-based projectors and leaves the electron-electron interaction untransformed. 

It is necessary to determine the set of eigenfimctions that minimize the Kohn-Sham energy fimctional E[ ipi ]. These are given by self-consistent solutions of the Kohn-Sham equations ... 

There are a set of Schrbdinger-like independent-particle equations which must be solved subject to the condition that the effective potential V r) = Ve (r) + Vn(r) + Vf r) and the density n(r,a) are consistent An actual calculation utilizes a numerical procedure that successively changes VJ(.and nto approach a self-consistent solution. The computationally intensive step in Figure 8.5 is solve KS (that is Kohn-Sham) equation for a given potential Veg-. This step is considered as a black box that uniquely solves the equation for a given input the potential to determine an output electronic density u ° P (r). Except for the exact solution, the input and output potentials and densities do not agree. To arrive at the solution one defines a new potential operationally which can start a new cycle... 

Fig. 6.1 Flow diagram for obtaining a self-consistent solution of the Kohn-Sham equation...

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. 

Initially, this tight-binding scheme has been designed to model materials with tetrahedral local order. Porezag et aL [76] have proposed a TB scheme based on the DPT theory (DFTB). In this approach, the pseudoatomic orbitals Slater-type orbitals and optimized to be solutions of the self-consistent modified atomic Kohn-Sham equations ... 

In the case of hybrid functionals, still another mode of implementation has become popular. This alternative, which also avoids solution of Eq. (91), is to calculate the derivative of the hybrid functional with respect to the singleparticle orbitals, and not with respect to the density as in (91). The resulting single-particle equation is of Hartree-Fock form, with a nonlocal potential, and with a weight factor in front of the Fock term. Strictly speaking, the orbital derivative is not what the HK theorem demands, but rather a Hartree-Fock like procedure, but in practice it is a convenient and successful approach. This scheme, in which self-consistency is obtained with respect to the singleparticle orbitals, can be considered an evolution of the Hartree-Fock Kohn-Sham method [6], and is how hybrids are commonly implemented. Recently, it has also been used for Meta-GGAs [2]. For occupied orbitals, results obtained from orbital selfconsistency differ little from those obtained from the OEP. 

The Kohn-Sham-Dirac equation (28) has to be solved self consistently, since the crystal potential and the XC-field depend via the (magnetization) density on its solutions. For a local orbital method it is advantageous to use a strictly local language for all relevant quantities, so that computationally expensive transformations between different numerical representations are avoided during the self consistency cycle. In the (R)FPLO method, the density n(r) and the magnetization density m(r) = m r)z are represented as lattice sums... 

In 1965 Kohn and Sham used the variational principle of Hohenberg and Kohn to derive a system of one-electron equations which, like the Hartree approach, can be self-consistently solved. For this case, however the electron densities obtained from the orbitals (called Kohn-Sham orbitals) are an exact solution to the many-body problem (for a complete basis) given the density functional. Hence the task of determining the electronic energy is changed from calculating the full many-body wavefunction to determining the best approximation to the density functional. 

W. Kohn and L. H. Sham, Phys. Rev. A, 140, 1133 (1965). Self-Consistent Equations Including Exchange and Correlation Effects. M. Levy, Proc. Natl. Acad. Sci. U.S.A., 76, 6062 (1979). Universal Variational Functionals of Electron Densities, First-Order Density Matrices, and Natural Spin-Orbitals and Solution of the v-Representability Problem. R. G. Parr and W. Yang, Eds., Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1988. J. Labanowski and J. Andzelm, Eds., Density Functional Methods in Chemistry, Springer Verlag, Heidelberg, 1991. L.J. Bartolotti and K. Flurchick, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1995, Vol. 7, pp. 187-216. An Introduction to Density Functional Theory. 

---