Towards a New Exchange-Correlation Density Functional for more Accurate Band Gap Predictions
Density-Functional Theory (DFT) offers a simplification to electronic structure problems by using the electron density instead of the wave-function. Unlike the wave-function which is a function of 3N variables (excluding spin) for an N -electron system, the density depends only on three variables, irrespective of the number of electrons in the system. While DFT, in principle, gives an accurate description of ground state properties, practical applications of DFT are based on approximations to the so-called exchange-correlation (xc) potential. The exchange-correlation potential describes the effects of the Pauli exclusion principle and the electron-electron Coulomb repulsion beyond a purely electrostatic interaction of the electrons. A common description of exchange-correlation functional is the so-called local density approximation (LDA) which locally substitutes the exchange-correlation energy density of an inhomogeneous system by that of an electron gas evaluated at that local density. While many ground state properties (such as lattice constants and bulk moduli) are well described in the LDA, the band gap is underestimated by as much as 50% in LDA compared to experiments. In this thesis, we focus on the development of an exchange-correlation functional with adjustable parameters which can give more accurate band gap energies. This functional is based on the xc potential derived in 1988 from a tight-binding approximation by Hanke and Sham (HS). Our contribution consists in expressing the HS potential in terms of the electron density and its gradient. This new expression for the xc functional was parameterized for the Si and Ge bulk systems and found to reduce the error in the LDA band gap prediction by an average of 22.3% for the systems (ZnO, MgO, ZnS, LiF, FeO and GaAs) that we tested it on.