Rigid muffin-tin approximation in plane-wave codes for fast modeling of phonon-mediated superconductors

Historically, the first practical method for calculating the electron-phonon (EP) coupling constant in metals was rigid muffin-tin approximation (RMTA), where the electronic potential within a sphere, called MT sphere, was assumed to follow ionic displacements rigidly. It was naturally implemented in many all-electron methods, such as linearized augmented-plane-wave and linear muffin-tin orbital methods, where atomic functions inside the MT sphere are evaluated explicitly. The method worked reasonably well for materials with localized d-orbitals and close packing. Subsequently, it was superseded by a much more accurate and universal linear-response method, well suited for fast pseudopotential-based plane-wave codes. While the method of choice today, it remains computationally demanding and cannot be applied for rapid scanning of hundreds or thousands candidate compounds, as is now common for computational design of new materials, specifically, new superconductors. To this end, RMTA would come handy, but so far it has not been formulated in the plane-wave representation. To assist future searches and add functionality to generally faster pseudopotential codes, current work presents a recipe for evaluation of EP coupling constant with RMTA in the plane-wave formalism.