A variational approach to nonlocal exciton–phonon coupling

Abstract

In this paper we apply variational energy band theory to a form of the Holstein Hamiltonian in which the influence of lattice vibrations (optical phonons) on both local site energies (local coupling) and transfers of electronic excitations between neighboring sites (nonlocal coupling) is taken into account. A flexible spanning set of orthonormal eigenfunctions of the joint exciton-phonon crystal momentum is used to arrive at a variational estimate (bound) of the ground state energy for every value of the joint crystal momentum, yielding a variational estimate of the lowest polaron energy band across the entire Brillouin zone, as well as the complete set of polaron Bloch functions associated with this band. The variation is implemented numerically, avoiding restrictive assumptions that have limited the scope of previous assaults on the same and similar problems. Polaron energy bands and the structure of the associated Bloch states are studied at general points in the three-dimensional parameter space of the model Hamiltonian (electronic tunneling, local coupling, nonlocal coupling), though our principal emphasis lay in under-studied area of nonlocal coupling and its interplay with electronic tunneling; a phase diagram summarizing the latter is presented. The common notion of a "self-trapping transition" is addressed and generalized.

Nonlocal electron-phonon coupling: Consequences for the nature of polaron states

“…At a certain coupling strength the free energies of the two minima become equal, marking the onset of the abrupt transition between the weakly dressed and strongly dressed states. As stressed elsewhere, 22 the discontinuities registered in such transitions might be the result of the changes in the polaron structure occurring too rapidly to be accurately accounted for by the variational entities used, and our method is not devoid of that. In order to illustrate the character of the dependence on g, in Fig.…”

“…It corresponds to the case of symmetric nonlocal coupling. 22 Furthermore, we define ⌫ m,mϮ1 ϵ⌫ 1 , U m,mϮ1 ϵU 1 , ⌬ m,mϮ1 ϵ⌬ 1 , Ṽ m,mϮ1 ϵṼ 1 , and R m ϪR mϮ1 ϭϯa. In Appendix B we derive the equalities…”

“…the free-energy minimization is a natural finite-temperature generalization of the ground-state variational principle of quantum mechanics ͑used by Zhao et al 22 in the study of zero-temperature polaron structure͒. Owing to the Jensen convexity of the function f (x)ϭe Ϫ␤x , the following theorem holds true ͑Bogoliubov's inequality͒: 26,27 For an arbitrary splitting of the Hamiltonian HϭH 0 ϩ(HϪH 0 ), under the condition that e Ϫ␤H and e Ϫ␤H 0 have finite traces for ␤Ͼ0, an upper bound to the free energy F is given by…”

“…For local coupling, a similar approach was followed by Yarkony and Silbey 21 and it is in this sense that our treatment constitutes a generalization of their work. A variational treatment of combined local and nonlocal coupling performed by Zhao et al 22 makes use of Toyozawa's Ansatz states, commonly employed in localcoupling theory. [23][24][25] However, the latter variational analysis is limited to the ground-state polaron structure, whereas our treatment also covers the regime of finite temperatures.…”

Polaron dynamics with a multitude of Davydov D2 trial states

“…Fortunately, the variational approach is still effective in dealing with polaron dynamics so long as a proper trial wave function is chosen. Previously, static properties of the Holstein polaron have been examined using a series of trial wave functions based upon phonon coherent states, such as the Toyozawa Ansatz [27,47,48], the GL Ansatz [27,28,48,49], and a delocalized form of the Davydov D 1 Ansatz [50]. By using these Ansätze, the ground state band and the self-trapping phenomenon were adequately investigated.…”