Orbital Dependent Improvements of the Density Functional Formalism with Applications to Atoms, Molecules and Crystalline Silicon

Recently, it has been realized that significant improvements in the calculated electronic structure of single and multi-atom systems may be realized by incorporating orbital dependent improvements into the energy density -functional. These improvements correct for spurious electron self-interaction terms which are present in all of the existing approximations of the density-functional. The numerical successes of this theory are illustrated using examples from atoms, diatomic molecules and crystalline silicon. A proper variational procedure for orbital dependent theories is introduced and applied to a variety of electronic systems. It is demonstrated that, in addition to the usual Schrodinger-like equations, "localization equations" need to be satisfied which ensure that the resulting Lagrange multiplier matrix is well defined. By following this variational procedure and extending the theory so that fractional electron occupancies may be incorporated into the total energy functional, it is demonstrated that canonical orbitals may be retained in orbital dependent theories. The corresponding eigenvalues lead to good approximations to the ionization energies. The virial theorem for orbital dependent theories is discussed and used to monitor the importance of the localization equations in minimizing the energy functional. An all electron variational procedure for the direct calculation of Wannier functions is introduced. This method is used to carry out self-consistent field calculations on the silicon crystal in terms of local orbitals. These calculations are performed within the density-functional formalism with and without the self-interaction correction.