Derivatives of eigenvalues from ab initio Hartree-Fock crystal orbital calculations with respect to the quasimomentum and the iterative solution of the inverse Dyson equation in the correlation problem

Abstract

We derive an analytical formula for the calculation of densities of states from ab initio Hartree-Fock Crystal Orbital (HF-CO) results, using an exact expression for the derivatives of the eigenvalues with respect to the quasimomentum k following from first order perturbation theory. The result is completely equivalent to that reported previously by Delhalle. We show that derivatives of the CO-coefficients with respect to k cannot be derived from first order perturbation theory, because one coefficient in the wavefunction is not defined. Further, due to the arbitrary phase factors at each CO, the form of such derivatives is not unique, but depends on the actual phase. Second derivatives of eigenvalues, and thus effective masses, are also obtained in this way, because for this purpose the unknown coefficient in the first order wavefunction is not necessary. In principle, perturbation theory can also yield expressions for higher order derivatives. We develop a CO formalism based on real quantities only and show that with this approach well defined phases are obtained. There are no more artificial numerical discontinuities in the phases and in this way the matrices introduced by Ladik to avoid complex calculus can be related directly to a basis set transformation. Further we discuss the use of phase factors for the construction of Wannier functions in correlation calculations on polymers, as well as the properties for the iterative solution of the inverse Dyson equation. Finally we describe the exploitation of helical symmetry without rotating two-electron integrals but instead with rotations on density matrices in an Appendix.