Small-polaron theory of phonon-assisted defect tunneling with quadratic defect-lattice coupling

The usual polaronlike treatment of phonon-assisted tunneling of lattice defects is a linear-coupling theory in which the defect-lattice interaction is linear in lattice displacements. We extend the treatment to include coupling terms quadratic in lattice displacements and momenta. Both the linear and the quadratic coupling are treated to all orders in the interaction so that the resulting theory is a union of small-polaron theory with Lifshitz perturbed phonon theory. Our formal results are presented in diagramatic series form and in terms of integral equations. Approximations suitable for high-and low-temperature ranges are discussed.