DERIVATIVES OF EIGENVALUES FROM AB-INITIO HARTREE-FOCK CRYSTAL ORBITAL CALCULATIONS WITH RESPECT TO THE QUASI-MOMENTUM AND THE ITERATIVE SOLUTION OF THE INVERSE DYSON EQUATION IN THE CORRELATION-PROBLEM

We derive an analytical formula for the calculation of densities of st ates from ab initio Hartree-Fock Crystal Orbital (HF-CO) results, usin g an exact expression for the derivatives of the eigenvalues with resp ect to the quasimomentum k following from first order perturbation the ory. The result is completely equivalent to that reported previously b y Delhalle. We show that derivatives of the GO-coefficients with respe ct to k cannot be derived from first order perturbation theory, becaus e one coefficient in the wavefunction is not defined. Further, due to the arbitrary phase factors at each CO, the form of such derivatives i s not unique, but depends on the actual phase. Second derivatives of e igenvalues, and thus effective masses, are also obtained in this way, because for this purpose the unknown coefficient in the first order wa vefunction is not necessary. In principle, perturbation theory can als o yield expressions for higher order derivatives. We develop a CO form alism based on real quantities only and show that with this approach w ell defined phases are obtained. There are no more artificial numerica l discontinuities in the phases and in this way the matrices introduce d by Ladik to avoid complex calculus can be related directly to a basi s set transformation. Further we discuss the use of phase factors for the construction of Wannier functions in correlation calculations on p olymers, as well as the properties for the iterative solution of the i nverse Dyson equation. Finally we describe the exploitation of helical symmetry without rotating two-electron integrals but instead with rot ations on density matrices in an Appendix.