We use a one-dimensional tight binding model with an impurity site characterized by electron-vibration coupling, to describe electron transfer and localization at zero temperature, aiming to examine the process of polaron formation in this system. In particular we focus on comparing a semiclassical approach that describes nuclear motion in this many vibronic-states system on the Ehrenfest dynamics level to a numerically exact fully quantum calculation based on the Bonca-Trugman method [J. Bonča and S. A. Trugman, Phys. Rev. Lett. 75, 2566 (1995)]. In both approaches, thermal relaxation in the nuclear subspace is implemented in equivalent approximate ways: In the Ehrenfest calculation the uncoupled (to the electronic subsystem) motion of the classical (harmonic) oscillator is simply damped as would be implied by coupling to a Markovian zero temperature bath. In the quantum calculation, thermal relaxation is implemented by augmenting the Liouville equation for the oscillator density matrix with kinetic terms that account for the same relaxation. In both cases we calculate the probability to trap the electron by forming a polaron and the probability that it escapes to infinity. Comparing these calculations, we find that while both result in similar long time yields for these processes, the Ehrenfest-dynamics based calculation fails to account for the correct time scale for the polaron formation. This failure results, as usual, from the fact that at the early stage of polaron formation the classical nuclear dynamics takes place on an unphysical average potential surface that reflects the distributed electronic population in the system, while the quantum calculation accounts fully for correlations between the electronic and vibrational subsystems.
We use a one-dimensional tight binding model with an impurity site characterized by electron-vibration coupling, to describe electron transfer and localization at zero temperature, aiming to examine the process of polaron formation in this system. In particular we focus on comparing a semiclassical approach that describes nuclear motion in this many vibronic-states system on the Ehrenfest dynamics level to a numerically exact fully quantum calculation based on the Bonca-Trugman method [J. Bonča and S. A. Trugman, Phys. Rev. Lett. 75, 2566 (1995)]. In both approaches, thermal relaxation in the nuclear subspace is implemented in equivalent approximate ways: In the Ehrenfest calculation the uncoupled (to the electronic subsystem) motion of the classical (harmonic) oscillator is simply damped as would be implied by coupling to a Markovian zero temperature bath. In the quantum calculation, thermal relaxation is implemented by augmenting the Liouville equation for the oscillator density matrix with kinetic terms that account for the same relaxation. In both cases we calculate the probability to trap the electron by forming a polaron and the probability that it escapes to infinity. Comparing these calculations, we find that while both result in similar long time yields for these processes, the Ehrenfest-dynamics based calculation fails to account for the correct time scale for the polaron formation. This failure results, as usual, from the fact that at the early stage of polaron formation the classical nuclear dynamics takes place on an unphysical average potential surface that reflects the distributed electronic population in the system, while the quantum calculation accounts fully for correlations between the electronic and vibrational subsystems.
We consider the quantum evolution and relaxation of an electron or hole which is coupled to a set of bath modes. In most applications the bath modes would be the vibronic coordinates but the model considered applies to any type of dynamic boson environment. The method is developed specifically for the problem of dynamic polaron formation in small nonperiodic systems. It can describe a broad group of experimental situations, including in particular electron localization in organics and polymeric materials and devices. The immediate bath is allowed to dissipate energy to a secondary bath. The bath obeys classical dynamics which puts some restriction on the range of validity of this approach. Using the density matrix formalism on a tight binding model consisting of a linear chain coupled to vibronic coordinates, we demonstrate in real time how the interaction with a dissipative bath makes the initial quantum distribution reach a steady-state population. This calculation is based on the Ehrenfest dynamics approximation. As an example we consider coupling at a single impurity site and find that for given parameters (bath coupling, site energy, and relaxation rate), the particle becomes dynamically localized in space on a particular time scale. This localized particle can be called a polaron. We define a population formation time in the same way as done in the experimental measurement. This formation time is studied as a function of the coupling strength, bandwidth, and energy dissipation rate. Energy dissipation plays a crucial role in the spatial localization process. The formation time shortens as the electron-vibration coupling increases, and as the intersite tunneling increases, but lengthens with impurity trap depth. Polaron formation is suppressed for sufficiently wide electronic bands.
We present results for the time evolution of a one-dimensional system consisting of an electron, described by a tight-binding Hamiltonian and a harmonic latttice, coupled by a deformation-type potential. We solve numerically the nonlinear system of equations of motion for this model in order to study the effects of varying the electronic effective mass for several initial conditions and coupling strengths. Different types of localized and extended states are formed with features that are absent from the traditional polaronic states and depend very strongly on the initial electronic configuration and effective mass in a very often unexpected manner. We find that, in general, an increase of the initial electronic energy decreases the ability of the system to form localized states. However, a large effective mass favors localized polaron formation for initially localized electrons, but this is not always the case for initially extended electronic states. In the latter case, increasing the effective mass of an electron initially close to the bottom of the band makes localization more difficult, while for an initially highly excited electron, localized polaron formation is possible only when the electronic effective mass and the atomic masses of the lattice become of the same order. Finally, for a small parameter range, we find an impressive recurrence, a periodic and a complete exchange between the electronic and vibrational degrees of freedom of a small part of the initial electronic energy.
Self-trapping has been traditionally studied on the assumption that quasiparticles interact with harmonic phonons and that this interaction is linear in the displacement of the phonon. To complement recent semiclassical studies of anharmonicity and nonlinearity in this context, we present below a fully quantum-mechanical analysis of a two-site system, where the oscillator is described by a tunably anharmonic potential, with a square well with infinite walls and the harmonic potential as its extreme limits, and wherein the interaction is nonlinear in the oscillator displacement. We find that even highly anharmonic polarons behave similar to their harmonic counterparts in that self-trapping is preserved for long times in the limit of strong coupling, and that the polaronic tunneling time scale depends exponentially on the polaron binding energy. Further, in agreement, with earlier results related to harmonic polarons, the semiclassical approximation agrees with the full quantum result in the massive oscillator limit of small oscillator frequency and strong quasiparticle-oscillator coupling.
The formation of a polaron quasiparticle from a bare electron is studied in the framework of the Holstein model of electron-phonon coupling. Using Schrödinger's formalism, we calculate the time evolution of the distribution of the electron and phonon density, lattice deformation, and the electron-phonon (el-ph) correlation functions in real space. The quantum dynamical nature of the phonons is preserved. The polaron formation time is related to the dephasing time of the continuum of unbound phonon excited states in the spectral function, which depends on the phonon frequency and the el-ph coupling strength. As the el-ph coupling increases, qualitative changes in polaron formation occur when the one-phonon polaron bound state forms. In the adiabatic regime, we find that a potential barrier between the quasi-free and heavy polaron states exists in both 2D and 3D, which is crucial for polaron formation dynamics. We compare to recent experiments.
Electronic energy flow in an isolated molecular system involves coupling between the electronic and nuclear subsystems, and the coupled system evolves to a statistical mixture of pure states. In semiclassical theories, nuclear motion is treated using classical mechanics, and electronic motion is treated as an open quantal system coupled to a "bath" of nuclear coordinates. We have previously shown how this can be simulated by a time-dependent Schrödinger equation with coherent switching and decay of mixing, where the decay of mixing terms model the dissipative effect of the environment on the electronic subdynamics (i.e., on the reduced dynamics of the electronic subsystem). In the present paper we reformulate the problem as a Liouville-von Neumann equation of motion (i.e., we propagate the reduced density matrix of the electronic subsystem), and we introduce the assumption of first-order linear decay. We specifically examine the cases of equal relaxation times for both longitudinal (i.e., population) decay and transverse decay (i.e., dephasing) and of longitudinal relaxation only, yielding the linear decay of mixing (LDM) and the population-driven decay of mixing (PDDM) schemes, respectively. Because we do not generally know the basis in which coherence decays, that is, the pointer basis, we judge the semiclassical methods in part by their ability to give good results in both the adiabatic and diabatic bases. The accuracy in the prediction of physical observables is shown to be robust not only with respect to basis but also with respect to the way in which demixing is incorporated into the master equation for the density matrix. The success of the PDDM scheme is particularly interesting because it incorporates the least amount of decoherence (i.e., the PDDM scheme is the most similar of the methods discussed to the fully coherent semiclassical Ehrenfest method). For both the new and previous decay of mixing schemes, four kinds of decoherent state switching algorithms are analyzed and compared to one another: natural switching (NS), self-consistent switching (SCS), coherent switching (CS), and globally coherent switching (GCS). The CS formulations are examples of a non-Markovian method, in which the system retains some memory of its history, whereas the GCS, SCS, and NS schemes are Markovian (time local). These methods are tested against accurate quantum mechanical results using 17 multidimensional atom-diatom test cases. The test cases include avoided crossings, conical interactions, and systems with noncrossing diabatic potential energy surfaces. The CS switching algorithm, in which the state populations are controlled by a coherent stochastic algorithm for each complete passage through a strong interaction region, but successive strong-interaction regions are not mutually coherent, is shown to be the most accurate of the switching algorithms tested for the LDM and PDDM methods as well as for the previous decay of mixing methods, which are reformulated here as Liouville-von Neumann equations with nonlinear decay of mixing (NLDM). We also demonstrate that one variant of the PDDM method with CS performs almost equally well in the adiabatic and diabatic representations, which is a difficult objective for semiclassical methods. Thus decay of mixing methods provides powerful mixed quantum-classical methods for modeling non-Born-Oppenheimer polyatomic dynamics including photochemistry, charge-transfer, and other electronically nonadiabatic processes.
We calculate the real space and time formation of a polaron quasiparticle from a bare electron. The time-dependent Schr\"odinger equation for the Holstein model of electron-phonon coupling is numerically integrated in a large Hilbert space to obtain the time evolution of the electron and phonon densities and the electron-phonon (el-ph) correlation functions. The quantum dynamical nature of the phonons is preserved. As the el-ph coupling increases, qualitative changes in polaron formation occur when the one-phonon polaron bound state forms. A potential barrier between the quasifree and heavy polaron states exists in dimensions $D\ensuremath{\geqslant}2$, consistent with earlier adiabatic theory. We compare to recent experiments.