An optimized LCAO method for crystals

A modification is proposed to the linear combination of atomic orbitals (LCAO) method for calculating electronic wave functions in crystals. Instead of using fixed atomic orbitals, atomic orbitals characterized by an internal parameter, such as scaling factor are used. This is optimized by applying a variational principle for Wannier functions, using as a trial function a localized orbital constructed from the atomic orbitals by a symmetric orthogonalization procedure. In this way, both the accuracy and range of applicability of the LCAO method for crystals are greatly increased. As shown for the example of the one-dimensional Mathieu problem, the optimum atomic orbital tends to expand as the interatomic distance decreases, and the optimized LCAO orbital tends to expand as the interatomic distance decreases, and the optimized LCAO method describes accurately the change in the energy bands and wave functions as the lattice parameter decreases and the electron states change from tight-binding to free-electron like. The application of the method to crystals containing defects and to amorphous materials is discussed.