has divergent terms when r gt 0 the initial condition has to be given avoiding the origin a wave function with angular momentum i is characterized by the behavior around the origin by where i 0 corresponds to the 3s state and i 1 to the 3p x state these values serve as an initial condition for numerical integration integration determines p a 2 and p 39 a 2 and they are immediately converted to r a 2 and r 39 a 2 this is done for even and odd solutions and all the necessary values g a 2 g 39 a 2 u a 2 and u 39 a 2 are determined to calculate u03c3 for given e then the dispersion relation is obtained through equation 5 and the wave function is constructed through equation 6 it is an energy value e that guarantees 3s or 3p wave functions in fact the expression in equation 11 just describes the wave function with angular momentum i and may accidentally correspond to 4s 4p 5s 5p etc functions with a different number of zeros the careful input of e is necessary fig1 shows the band structure of a si atomic chain as a function of lattice constant a the 3s and 3p x wave functions are calculated for a given pair of lattice constant a and energy e and g a 2 g 39 a 2 u a 2 and u 39 a 2 are numerically evaluated to give s in equation 5 allowed energy regions are for u03c3 gt 0 and forbidden energy regions are for u03c3 lt 0.