  The interaction between electrons and lattice vibrations determines key
physical properties of materials, including their electrical and heat
transport, excited electron dynamics, phase transitions, and superconductivity.
We present a new ab initio method that employs atomic orbital (AO)
wavefunctions to compute the electron-phonon (e-ph) interactions in materials
and interpolate the e-ph coupling matrix elements to fine Brillouin zone grids.
We detail the numerical implementation of such AO-based e-ph calculations, and
benchmark them against direct density functional theory calculations and
Wannier function (WF) interpolation. The key advantages of AOs over WFs for
e-ph calculations are outlined. Since AOs are fixed basis functions associated
with the atoms, they circumvent the need to generate a material-specific
localized basis set with a trial-and-error approach, as is needed in WFs.
Therefore, AOs are ideal to compute e-ph interactions in chemically and
structurally complex materials for which WFs are challenging to generate, and
are also promising for high-throughput materials discovery. While our results
focus on AOs, the formalism we present generalizes e-ph calculations to
arbitrary localized basis sets, with WFs recovered as a special case.
