Orthogonal Orbitals and Generalized Wannier Functions

The invariance properties of a one-electron Hamiltonian $H=T+V$ with respect to the transformations of a space group $G_H$ are used to show how the eigenfunctions of $H$ can be expanded in terms of equivalent local orbitals. These orbitals are built from a suitable set of eigenfunctions and are shown to be orthogonal to each other. They are associated with the points $M$ of a lattice $\mathcal{L}$ which is invariant with respect to $G_H$ and can be obtained from each other by space transformations. Group theory is used to write explicitly the unitary relations connecting the set of eigenfunctions of $H$ and the corresponding orbitals. In crystals, it is shown that the eigenfunctions belonging to an energy band can often be described by means of one set of orbitals, provided that certain simple conditions are fulfilled. These conditions depend on the properties of the levels which correspond to the points $\Sigma$ of maximum symmetry in the reciprocal space. These requirements determine also the nature of the lattice $\mathcal{L}$ and the chemical bonding in the band.