A direct numerical method for the energy band problem: Preliminary results for Li

A numerical method of calculating energy bands is presented which makes it possible to circumvent the difficulties associated with integration of the relevant matrix elements. The method employs a Diophantine numerical integration procedure, previously applied to molecular systems. Preliminary results indicate that the convergence of crystal energy bands is even more rapid than the convergence obtained in previous molecular calculations. The main advantage of our method is that integration is done directly on matrix elements of the hamiltonian without separately evaluating and storing many basis integrals. No multicenter integrals need be done and no approximations are made. Another advantage of the direct evaluation of matrix elements is that there is no loss of significant figures as experienced in going from basis integrals to the final matrix elements.