On the network model of molecules and solids

Abstract

We consider the network model of molecules and solids, in which atoms are at fixed sites and electrons are allowed to move along one-dimensional bond paths between the atoms. We derive a difference equation which describes the model. There is a class of networks for which the shape of the constant energy surfaces is determined by the topology of the network alone, while the potential on the sites and along the bond paths determines the value of the energy attached to each surface. We discuss a necessary and sufficient condition for this to occur and examine in detail the energy band spectrum and dispersion curves. We also display a procedure for obtaining the secular equation for any periodic lattice. Explicit calculations of the dispersion curve for the cubic lattice are done in the WKB approximation and are compared with the exact results for a special potential.