In this work we present a method for performing LCAO (linear combination of atomic orbitals) band structure calculations (tight binding) in crystalline solids. In the first part of the article we apply group theoretical methods to the establishment of a least-squares scheme for the calculation of the matrix of the crystal potential: This scheme is based on a well-defined choice of independent parameters for the Bloch vector-dependent matrix elements and on the considerations of the symmetries between these independent parameters and their Fourier coefficients. In the second part of this work we deal with the representation of the matrices of the identity operator and of the operator of the kinetic energy by linear combinations in terms of two center integrals: We express these linear combinations by a closed formula, which can be easily programmed on a computer, and we mention a method by which the two center integrals can be evaluated numerically fast and accurately. Finally we apply our theory to the derivation of numerical results: We determine the electronic states and the high-momentum components of Compton rates in the alkali metal rubidium and we compare the results obtained with those of augmented plane-wave (APW) calculations.