The use of localized Gaussian basis functions for large scale first principles density functional calculations with periodic boundary conditions (PBC) in 2 dimensions and 3 dimensions has been made possible by using a dual space approach. This new method is applied to the study of electronic properties of II-VI (II=Zn, Cd, Hg; VI=S, Se, Te, Po) and III-V (III=Al, Ga; V=As, N) semiconductors. Valence band offsets of heterojunctions are calculated including both bulk contributions and interfacial contributions. The results agree very well with available experimental data. The p(2X1) cation terminated surface reconstructions of CdTe and HgTe (100) are calculated using the local density approximation (LDA) with two-dimensional PBC and also using the ab initio Hartree-Fock (HF) method with a finite cluster. The LDA and HF results do not agree very well.