Properties of multipole wave functions and of Wannier functions are studied jointly, because both are generated by unitary transformations of Bloch waves and because their symmetry and localization depend on the analytic behavior and phase normalization of Bloch waves. Fourier theory relates the amplitude and convergence of the tails of Wannier functions to the singularities of Bloch waves in $\overrightarrow{\mathrm{k}}$ space, specifically to the contribution of singularities to integrals over $\overrightarrow{\mathrm{k}}$. These singularities may include branch cut surfaces bounded by curves of degeneracy. We discuss the convergence of the coefficients of a linear-combination-of-atomic-orbitals expansion in orbitals introduced recently by Kohn, which are Fourier coefficients of the eigenvectors ${\overrightarrow{\mathrm{U}}}^{\left(\mu \right)}\left(\overrightarrow{\mathrm{k}}\right)$ of the Hamiltonian in Kohn's basis. The phases of Bloch waves are fixed by a constraint on the Wannier functions. The symmetry of Bloch waves determines then both the species and the center of symmetry of the Wannier functions. An application to the conduction band of copper is presented. Contributions of neighborhoods of loci of degeneracy in the Brillouin zone to the multipole wave functions are singled out and the rate of convergence of their Wannier series is examined. The series converges faster than the corresponding Wannier series of Bloch waves.

• Received 24 October 1977

DOI:https://doi.org/10.1103/PhysRevB.18.4104