The energy band interpolation is a well known problem in solid state theory. It is often disconnected from the consideration of wavefunctions, in particular in the very popular k.p and tight-binding approaches. However, the local wavefunction is also important since the matrix elements of different operators in real space constitute the basic ingredient to evaluate many physical observables. This is particularly true when one tries to evaluate interactions (in particular, short range interactions) between quansi-particles. In the empirical tight binding method, the single-electron wavefunctions are developed on a basis of orbitals whose spatial form is not taken into account in the construction of the Hamiltonian. This work adresses this theoretical problem that remained open since the pioneering work of Slater and Koster (1954). We take as a starting point a basis of atomic orbitals (in practice, Slater orbitals) with adjustable screening coefficients. We calculate the overlap matrix between the different functions of the basis and obtain a new, orthogonal basis using the Löwdin orthogonalization procedure. The projections of the Hamiltonian eigenstates on the new basis give the electronic Bloch functions, which are then used to compute the momentum matrix elements in real space, that are iteratively compared with their k-space counterpart. This method allows a self-consistent fitting of the screening coefficients of starting atomic orbitals, therefore complements the tight-binding theory with a description of local wavefunctions. Our results compare satisfactorily with results of ab initio and empirical pseudo-potential calculations. Beyond semiconductor nanostructures, this work is a fundamental step toward modeling many-body effects from post-processing wave-functions within the Slater and Koster theory. A first test of this approach was made by computing the fine structure of excitons in bulk GaAs. We have calculated the dispersion of exciton fine structure, treating on equal footing all the ingredients of the problem, from details of the single particle dispersions (valence band warping, electron and hole spin splittings) to direct and exchange Coulomb interactions. Our results are in very good agreement with experimental values of exciton binding energy and longitudinal-transverse splitting.
