Relationship between path-integral and scaling theories of small polarons

The detailed correspondence between the Emin-Holstein scaling theory and Feynman path-integral method for small-polaron formation is presented. In particular it is shown that the first-cumulant approximation to the path integral using a harmonic trial action in the adiabatic limit is identical to the use of a Gaussian trial wave function in the Emin-Holstein one-parameter scaling theory for the ground-state energy with polaron radius. This is then generalized to a simple two-parameter scaling theory involving both the polaron radius as well as nonadiabaticity of the electron-phonon interaction.