  The coupling between electrons and phonons plays important roles in physics,
chemistry and biology. However, the accurate calculation of the electron-phonon
coupling constants is computationally expensive as it involves solving the
Schrodinger equation for O(N) nuclear configurations, where N is the number of
nuclei. Herein we show that by considering the forces on the nuclei caused by
the addition or subtraction of an arbitrarily small electronic charge one may
calculate the electron-phonon coupling constants from O(1) solutions of the
Schrodinger equation. We show that Janak's theorem means that this procedure is
exact within the density functional formalism. We demonstrate that the O(1)
approach produces numerically accurate results by calculating the
electron-phonon coupling constants for a series of molecules ranging in size
from H_2 to C_60. We use our approach to introduce a computationally fast
approximation for the adiabatic ionisation potentials and electron affinities
which is shown to be accurate for large molecules. We also show that our
approach allows for the calculation isotope effects in O(0) time and discuss
the deuteration driven Mott transition in kappa-(BEDT-TTF)_2Cu[N(CN)_2]Br.

Materials Science;Strongly Correlated Electrons;Superconductivity;Atomic and Molecular Clusters;Chemical Physics;Computational Physics