Small-polaron theory of phonon-assisted defect tunneling with quadratic defect-lattice coupling: An exact result

Two earlier papers by D. Tonks and B. G. Dick presented an extension of the usual polaronlike treatment of phonon-assisted tunneling of substitutional lattice defects to include defect-lattice coupling terms quadratic in atomic displacements and momenta. The formal results treating the defect-lattice interaction to all orders in perturbation theory were given in the form of a diagrammatic series. But the series was not summed exactly, it was approximated. Here, we exactly sum the diagrammatic series by solving an integral equation and use the solution to shed light on the one-phonon $F$ approximation made of the series in the earlier papers. The diagrammatic summation exactly includes both changes in vibrational normal modes and changes in atomic equilibrium positions with a change in defect site.