On the Role of the Nonlocal Hartree-Fock (HF) Exchange in Narrow-Band Materials

Abstract

The influence of the nonlocal Hartree-Fock (HF) exchange in narrow-band materials with finite band gaps is analyzed. The mean-field Hamiltonian in the crystal orbital (CO) basis contains two k-dependent matrix elements responsible for the dispersion of the one-electron levels: the classical tight-binding integrals (kinetic energy of the electrons) that decay exponentially as well as the (non)local exchange. The asymptotic behavior of the two-electron potential is determined by the fall-off of the intercell bond-order matrices; their range exceeds significantly the spatial extension of the k-dependent one-electron integrals. The analytic structure of the HF dispersions of narrow-band systems (weak intercell interactions) is largely influenced by the magnitude of the Fermi-correlation beyond the direct neighbors. The associated ε(k) curves differ strongly from idealized tight-binding relations. The bands are broadened and show enhanced energy gradients in certain domains of k-space. Nonlinearities in the ε(k) relations are a direct consequence of finite neighbor's approximations adopted for the evaluation of the lattice sums. The analytic structures of such HF bands are intermediate between idealized tight-binding relations of covalent solids, on one side, and HF dispersions of metals in (nearly-)free electron-gas approximations, on the other, that show divergent ε(k) gradients at the Fermi level. The exchange influence in insulating narrow-band materials is restricted to the filled one-particle space; this is demonstrated by a perturbational analysis. The crucial importance of reliable numerical integration procedures for the determination of the intercell bond-order matrices is pointed out. Standard techniques may lead to artificial periodicities pretending unphysical decay properties of the electronic exchange. Dispersion patterns of a simple one-orbital model are analyzed as a function of the mutual strength of the kinetic hopping integrals and the HF exchange as well as the spatial extension of the k-dependent two-electron potential. The validity of the theoretical expectations deduced from simple model calculations is studied for two complex polymers. Important one-electron properties of one-dimensional (1 D) porphyrinato nickel(II) derivatives (2 and 3) are investigated by means of semiempirical SCF (self-consistent-field) HF INDO (intermediate neglect of differential overlap) CO calculations. The lattice spacings of 2 and 3 differ by 0.31 Å. This geometrical distinction allows for an inversion of the relative importance of the k-dependent one- and two-electron contributions to the mean-field operator. The exchange influence on the width of the HF dispersion in 3 exceeds the one-electron part by nearly one order of magnitude. It is shown that band structure properties of narrow-band systems are neither properly described by one-electron models of the Wolfsberg-Helmholtz-type nor by bare (unscreened) HF dispersions. The width of a mean-field band calculated within a nonlocal exchange approximation has to be corrected for quasi-particle (QP) interactions beyond the HF scheme (i.e., long-range and short-range correlations and relaxations) as well as for electron (optical) phonon interactions. The phononic coupling leads to a narrowing of the band width via Franck-Condon-like vibrational overlaps; this part is independent of the theoretical details of the electronic structure investigation. Important physical consequences of the exchange-control in narrow-band solids are shortly discussed.