  In materials with strong electron-phonon (e-ph) interactions, charge carriers
can distort the surrounding lattice and become trapped, forming self-localized
(small) polarons. We recently developed an ab initio approach based on
canonical transformations to efficiently compute the formation and energetics
of small polarons[1]. A different approach based on a Landau-Pekar energy
functional has been proposed in the recent literature [2,3]. In this work, we
analyze and compare these two methods in detail. We show that the small polaron
energy is identical in the two formalisms when using the same polaron wave
function. We also show that our canonical transformation formalism can predict
polaron band structures and can properly treat zero- and finite-temperature
lattice vibration effects, although at present using a fixed polaron wave
function. Conversely, the energy functional approach can compute the polaron
wave function, but as we show here it neglects lattice vibrations and cannot
address polaron self-localization and thermal band narrowing. Taken together,
this work relates two different methods developed recently to study polarons
from first-principles, highlighting their merits and shortcomings and
discussing them both in a unified formalism.
