PHYSICAL REVIEW B VOLUME 40, NUMBER 14 15 NOVEMBER 1989-I Unification of polaron and soliton theories of exciton transport David W. Brown and Zoran Ivic* Institute for Nonlinear Science, R-002, University of California, San Diego, La Jolla, California 92093 (Received 9 June 1989) We present a unified theory of polaron and soliton dynamics by combining time-dependent varia- tional methods recently applied to the theory of Davydov solitons with partial-dressing methods well known from polaron theory. We focus on the simplest partial-dressing assumption, applying a common dressing fraction to all phonon modes. Our fundamental result is a system of nonlinear evolution equations in which the tendency of a system to form Davydov solitons is balanced against its tendency to form small polarons. We subsequently apply time-independent variational methods to determine the optimal dressing fraction in a mean-field manner. The characterization of the par- tially dressed soliton states that results is complete with respect to the system parameter space. Consistent with prior works from polaron theory, we find a self-trapping transition that is only weakly modified by our inclusion of nonlinearities, and we reinterpret this transition in terms of our newly obtained soliton states. Applying our results to a central problem in bioenergetics, we obtain results markedly different from well-known results of Davydov's theory. I. INTRODUCTION The presence of foreign particles or excitations (elec- tronic or vibronic) in a deformable solid often causes lo- cal distortions of the host material which can profoundly affect the character and dynamics of would-be quasiparti- cles. Among the physical systems which present such challenging problems are polar crystals, where the prob- lem was first studied metal hydrides, wherein intersti- tial hydrogen isotopes can cause large volume dila- tions; ' organic molecular crystals such as anthracene and naphthalene, wherein the mobilities of photoinjected charge carriers exhibit novel temperature dependences; ' organic molecular crystals such as pyrene and a- perylene, wherein exciton spectra are profoundly affected by local distortions; ' and biological molecules such as DNA (Ref. 9) and the a-helix (Refs. 10 and ll) and bio- logical materials such as the molecular crystals acetani- lide (Refs. 12 and 13) and l-alanine (Ref. 14), wherein a number of phenomena have been related to conforma- tional excitations. The body of theory that has evolved to describe energy transport in deformable media necessarily embraces a number of different points of view, some of which appear, at different times and in different ways, to be in conflict. Central to most of the relevant literature is concept of the polaron. When a particle such as an exciton is created in or injected into a solid, the presence of the exciton in- duces a distortion in the surrounding medium. "Pola- ron" is generally given to mean the quasiparticle consist- ing of the original exciton together with the distortion it induces in the host medium. While conceptually simple, this definition is operationally inadequate since it con- tains no prescription by which to identify the relevant distortion. As immediate refinements there are the more useful concepts of the small polaron and the large pola- ron. "Small polaron" is generally given to mean a pola- ron which occupies a minimum number of host lattice sites, usually one. "Large polaron" is generally given to mean a polaron which occupies a number of host lattice sites well in excess of this minimum, usually a number large enough to justify a continuum approximation. Small-polaron theories are usually addressed to the non- adiabatic situation in which the exciton energy band- width is small relative to the typical phonon frequen- cy ' ' such treatments are largely based on the use of canonical transformations to identify sets of "small- polaron states" in which perturbation theory is often developed in orders of the ratio of the exciton bandwidth to the typical phonon frequency. On the other hand, large-polaron theories are usually addressed to the adia- batic limit in which the kinetic energy of lattice vibra- tions can be neglected in lowest order; ' ' such treat- ments are frequently based on the application of varia- tional methods. Small-polaron theory is, historically, a linear theory, while the theory of large polarons contains well-known nonlinearities. The nonlinearity implicit in the description of large polarons has given rise to trans- port theories based on the somewhat more specialized concept of envelope solitons. Roughly speaking, the greater part of perturbation theory is most appropriate in the small-polaron regime, where effective masses are large, and the greater part of soliton theory is most ap- propriate in the large-polaron regime, where effective masses are small. We put forward a theory based on recent advances in the theory of envelope solitons which allows a number of these sometimes confIicting perspectives to be unified in a single formulation. The theory we present covers the en- tire parameter range from the small-polaron regime to the large polaron regime, and recovers in appropriate limits small-polaron theory and Davydov's theory of en- velope solitons. In Sec. II we present the elements of the model on which our theory is based. In Sec. III we apply time-dependent variational methods to obtain equations of motion and special solutions in the small- and large- 40 9876 1989 The American Physical Society