Localized Functions in Molecules and Crystals

The Wannier function in crystals is defined in terms of a differential equation and a variation procedure. This variation procedure is also used to define localized functions in molecules which can be used to build solutions of Schr\"odinger's equation. Illustrations of the procedure of building solutions of Schr\"odinger's equation from localized functions is given for examples taken from crystalline and molecular problems. Contact is made with Slater's work on the two-dimensional Mathieu problem. A numerical method of carrying out the variation procedure is discussed.