Electron localization in spatially disordered systems

The homomorphic-cluster coherent-potential approximation is applied to obtain the average Green's function for spatially disordered systems in two and three dimensions, which is described by a tight-binding one-electron Hamiltonian with transfer energy between two $1s$ hydrogenic orbitals. The effective medium to be determined self-consistently is assumed to be an appropriate lattice, and a modified Hertz distribution is used for the random distribution of nearest-neighbor distance. With the use of the $L(E)$ criterion for localization, Anderson's transition is predicted to occur at a critical density ρ1/2aB=0.403 in two dimensions and ρ1/3aB=0.252 in three dimensions, where ρ is the number density of atoms and aB is the effective Bohr radius.