Kohn-Sham equations introduction

Introduction of a Basis The LCAO Ansatz in the Kohn-Sham Equations... 

Computational advances which have led to pseudopotential methods becoming efficient are Fast Fourier Transfoms and the approach of Car and Parrinello in which wavefunction coefficients are treated as d5mamical variables. Parts of the calculations on periodic systems are easier to calculate in real space and others in reciprocal space. Fourier transforms are needed to convert between the two spaces. The introduction of Fast Fourier Transforms substantially reduced the computational costs of this conversion. The key feature of the Car-Parrinello approach is the use of minimisation rather than diagonalisation of the Kohn-Sham equations to reach the ground state. 

The local nature of the effective potential in the one-electron Kohn-Sham equations affords efficient computational schemes. During the development of ADF, the remaining Coulomb problem, the two-electron-integral bottleneck, has been addressed by the introduction of auxiliary basis sets, so-called density fitting (Baerends et al. 1973). 

This introduction closes with a brief presentation of the Kohn-Sham [6] spin-density functional method, the most widely-used method of electronic-structure calculation in condensed-matter physics and one of the most widely-used methods in quantum chemistry. We seek the groimd-state total energy E and spin densities n (r), n (r) for a collection of N electrons interacting with one another and with an external potential w(r) (due to the nuclei in most practical cases). These are foimd by the selfconsistent solution of an auxiliary (fictitious) one-electron Schrodinger equation ... 

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. 

---