Molecular Crystal Model Research Articles

The hierarchy of Davydov's Ansätze: From guesswork to numerically "exact" many-body wave functions.

This Perspective presents an overview of the development of the hierarchy of Davydov's Ansätze and a few of their applications in many-body problems in computational chemical physics. Davydov's solitons originated in the investigation of vibrational energy transport in proteins in the 1970s. Momentum-space projection of these solitary waves turned up to be accurate variational ground-state wave functions for the extended Holstein molecular crystal model, lending unambiguous evidence to the absence of formal quantum phase transitions in Holstein systems. The multiple Davydov Ansätze have been proposed, with increasing Ansatz multiplicity, as incremental improvements of their single-Ansatz parents. For a given Hamiltonian, the time-dependent variational formalism is utilized to extract accurate dynamic and spectroscopic properties using Davydov's Ansätze as its trial states. A quantity proven to disappear for large multiplicities, the Ansatz relative deviation is introduced to quantify how closely the Schrödinger equation is obeyed. Three finite-temperature extensions to the time-dependent variation scheme are elaborated, i.e., the Monte Carlo importance sampling, the method of thermofield dynamics, and the method of displaced number states. To demonstrate the versatility of the methodology, this Perspective provides applications of Davydov's Ansätze to the generalized Holstein Hamiltonian, variants of the spin-boson model, and systems of cavity-assisted singlet fission, where accurate dynamic and spectroscopic properties of the many-body systems are given by the Davydov trial states.

The hierarchy of Davydov's Ansätze and its applications

AbstractThis review provides a bird's eye view over the development of the hierarchy of Davydov's Ansätze and its applications in a variety of problems in computational physical chemistry. Davydov's original solitons appeared in the 1970s as a candidate for vibrational energy carriers in proteins, thanks to their association with the Fröhlich Hamiltonian and the Holstein molecular crystal model. Momentum‐space projection of those solitary waves emerged to be great approximations to the ground‐state wave functions of the extended Holstein Hamiltonian, lending unambiguous evidence to the absence of formal quantum phase transitions in those systems. The multiple Davydov Ansätze are introduced, with increasing multiplicity, as incremental improvements of their corresponding single‐Ansatz parents. The time‐dependent variational formalism of Davydov's Ansätze is discussed in great detail, and the relative deviation of the Ansätze is constructed to quantify how faithfully they follow the Schrödinger equation, a quantity that is shown to vanish in the limit of large multiplicities. Three approaches to finite‐temperature variational dynamics of Davydov's Ansätze are demonstrated, namely, the Monte Carlo importance sampling, the method of thermo‐field dynamics, and the method of displaced number states. Applications of Davydov's Ansätze are given to variants of the spin‐boson model, the Landau–Zener transition, the Holstein Hamiltonian, energy transfer in light‐harvesting, and singlet fission in organic photovoltaics. As an example, simulation of multidimensional spectroscopic signals via Davydov's Ansätze is fully implemented for the finite‐temperature fission process in crystalline rubrene.This article is categorized under: Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics Software > Simulation Methods Theoretical and Physical Chemistry > Spectroscopy

Apoptosis of moving nonorthogonal basis functions in many-particle quantum dynamics

Due to the exponential increase of the numerical effort with the number of degrees of freedom, moving basis functions have a long history in quantum dynamics. In addition, spawning of new basis functions is routinely applied. Here we advocate the opposite process: the programmed removal of motional freedom of selected basis functions. This is a necessity for converged numerical results with respect to the size of a non-orthogonal basis, because generically two or more states approach each other too closely early on, rendering unstable the matrix inversion, required to make the equations of motion explicit. Applications to the sub-Ohmic spin-boson model as well as to polaron dynamics in a Holstein molecular crystal model demonstrate the power of the proposed methodology.

Dynamics of a one-dimensional Holstein polaron: The multiconfigurational Ehrenfest method.

We have extended the multiconfigurational Ehrenfest (MCE) approach to investigate the dynamics of a one-dimensional Holstein molecular crystal model. It has been shown that the extended MCE approach yields results in perfect agreement with benchmark calculations by the hierarchy equations of motion method. The accuracies of the MCE approach in describing the dynamical properties of the Holstein polaron over a wide range of exciton transfer integrals and exciton-phonon couplings are carefully examined by a detailed comparison with the fully variational multiple Davydov D2 ansatz. It is found that while the MCE approach and the multi-D2 ansatz produce almost exactly the same results for a small transfer integral, the results obtained by the multi-D2 ansatz start to deviate from those by the MCE approach at longer times for a large transfer integral. A large number of coherent state basis functions are required to characterize the delocalized features of the phonon wavefunction in the case of large transfer integral, which becomes computationally too demanding for the multi-D2 ansatz. The MCE approach, on the other hand, uses hundreds to thousands of trajectory guided basis functions and converges very well, thus providing an effective tool for accurate and efficient simulations of polaron dynamics.

Multi-polaron solutions, nonlocal effects and internal modes in a nonlinear chain

Multipolaron solutions were studied in the framework of the Holstein one-dimensional molecular crystal model. The study was performed in the continuous limit where the crystal model maps into the nonlinear Schrödinger equation for which a new periodic dnoidal solution was found for the multipolaron system. In addition, the stability of the multi-polaron solutions was examined, and it was found that cnoidal and dnoidal solutions stabilize in different ranges of the parameter space. Moreover, the model was studied under the influence of nonlocal effects and the polaronic dynamics was described in terms of internal solitonic modes.

Comparison between density functional theory and density functional tight binding approaches for finding the muon stopping site in organic molecular crystals.

Finding the possible stopping sites for muons inside a crystalline sample is a key problem of muon spectroscopy. In a previous study, we suggested a computational approach to this problem when dealing with muonium, the pseudoatom formed by a positive muon that has captured an electron, using density functional theory software in combination with a random structure searching approach that relies on a Poisson sphere distribution. In this work, we test this methodology further by applying it to muonium in three organic molecular crystal model systems: durene, bithiophene, and tetracyanoquinodimethane. Using the same sets of random structures, we compare the performance of density functional theory software CASTEP and the much faster lower level approximation of Density Functional Tight Binding provided by DFTB+ combined with the use of the 3ob-3-1 parameter set. We show the benefits and limitations of such an approach, and we propose the use of DFTB+ as a viable alternative to more cumbersome simulations for routine site-finding in organic materials. Finally, we introduce the Muon Spectroscopy Computational Project software suite, a library of Python tools meant to make these methods standardized and easy to use.

Dynamics of Coupled Electron-Boson Systems with the Multiple Davydov D1 Ansatz and the Generalized Coherent State.

The dynamics of a coupled electron-boson system is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz, and a second trial state based on a superposition of the time-dependent generalized coherent state (GCS Ansatz). The two Ansätze are applied to study population dynamics in the spin-boson model and the Holstein molecular crystal model, and a detailed comparison with numerically exact results obtained by the (multilayer) multiconfiguration time-dependent Hartree method and the hierarchy equations of motion approach is drawn. It is found that the two methodologies proposed here have significantly improved over that with the single D1 Ansatz, yielding quantitatively accurate results even in the critical cases of large energy biases and large transfer integrals. The two methodologies provide new effective tools for accurate, efficient simulation of many-body quantum dynamics thanks to a relatively small number of parameters which characterize the electron-nuclear wave functions. The wave-function-based approaches are capable of tracking explicitly detailed bosonic dynamics, which is absent by construct in approaches based on the reduced density matrix. The efficiency and flexibility of our methods are also advantages as compared with numerically exact approaches such as QUAPI and HEOM, especially at low temperatures and in the strong coupling regime.

Polaron dynamics with off-diagonal coupling: beyond the Ehrenfest approximation.

Treated traditionally by the Ehrenfest approximation, the dynamics of a one-dimensional molecular crystal model with off-diagonal exciton-phonon coupling is investigated in this work using the Dirac-Frenkel time-dependent variational principle with the multi-D2Ansatz. It is shown that the Ehrenfest method is equivalent to our variational method with the single D2Ansatz, and with the multi-D2Ansatz, the accuracy of our simulated dynamics is significantly enhanced in comparison with the semi-classical Ehrenfest dynamics. The multi-D2Ansatz is able to capture numerically accurate exciton momentum probability and help clarify the relation between the exciton momentum redistribution and the exciton energy relaxation. The results demonstrate that the exciton momentum distributions in the steady state are determined by a combination of the transfer integral and the off-diagonal coupling strength, independent of the excitonic initial conditions. We also probe the effect of the transfer integral and the off-diagonal coupling on exciton transport in both real and reciprocal space representations. Finally, the variational method with importance sampling is employed to investigate temperature effects on exciton transport using the multi-D2Ansatz, and it is demonstrated that the variational approach is valid in both low and high temperature regimes.

Fast, Accurate Simulation of Polaron Dynamics and Multidimensional Spectroscopy by Multiple Davydov Trial States.

By employing the Dirac-Frenkel time-dependent variational principle, we study the dynamical properties of the Holstein molecular crystal model with diagonal and off-diagonal exciton-phonon coupling. A linear combination of the Davydov D1 (D2) ansatz, referred to as the "multi-D1 ansatz" ("multi-D2 ansatz"), is used as the trial state with enhanced accuracy but without sacrificing efficiency. The time evolution of the exciton probability is found to be in perfect agreement with that of the hierarchy equations of motion, demonstrating the promise the multiple Davydov trial states hold as an efficient, robust description of dynamics of complex quantum systems. In addition to the linear absorption spectra computed for both diagonal and off-diagonal cases, for the first time, 2D spectra have been calculated for systems with off-diagonal exciton-phonon coupling by employing the multiple D2 ansatz to compute the nonlinear response function, testifying to the great potential of the multiple D2 ansatz for fast, accurate implementation of multidimensional spectroscopy. It is found that the signal exhibits a single peak for weak off-diagonal coupling, while a vibronic multipeak structure appears for strong off-diagonal coupling.

Dynamics of a One-Dimensional Holstein Polaron with the Hierarchical Equations of Motion Approach.

Dynamics of a one-dimensional Holstein molecular crystal model is investigated by making use of the hierarchical equations of motion (HEOM) introduced by Tanimura and Kubo [J. Phys. Soc. Jpn. 1989, 104, 101]. Our extended, numerically exact HEOM approach is capable of treating exciton-phonon coupling in a nonperturbative manner and is applicable to any temperature. It is revealed that strong exciton phonon coupling leads to excitonic localization, while a large exciton transfer integral facilitates exciton transport. Temperature effects on excitonic scattering have also been examined. A proof of concept, our work also serves as a benchmark for future comparisons with other numerical approaches to Holstein polaron dynamics.

Polaron dynamics with a multitude of Davydov D2 trial states.

We propose an extension to the Davydov D2 Ansatz in the dynamics study of the Holstein molecular crystal model with diagonal and off-diagonal exciton-phonon coupling using the Dirac-Frenkel time-dependent variational principle. The new trial state by the name of the "multi-D2 Ansatz" is a linear combination of Davydov D2 trial states, and its validity is carefully examined by quantifying how faithfully it follows the Schrödinger equation. Considerable improvements in accuracy have been demonstrated in comparison with the usual Davydov trial states, i.e., the single D1 and D2 Ansätze. With an increase in the number of the Davydov D2 trial states in the multi-D2 Ansatz, deviation from the exact Schrödinger dynamics is gradually diminished, leading to a numerically exact solution to the Schrödinger equation.

Analog superconducting quantum simulator for Holstein polarons

We propose an analog quantum simulator for the Holstein molecular-crystal model based on a superconducting circuit QED system in the dispersive regime. By varying the driving field on the superconducting resonators, one can readily access both the adiabatic and anti-adiabatic regimes of this model. Strong e-ph coupling required for small-polaron formation can also be reached. We show that small-polaron state of arbitrary quasimomentum can be generated by applying a microwave pulse to the resonators. We also show that significant squeezing in the resonator modes can be achieved in the polaron-crossover regime through a measurement-based scheme.

Polaronic discontinuities induced by off-diagonal coupling

In this paper, we study a form of the Holstein molecular crystal model in which the influence of lattice vibrations on the transfers of electronic excitations between neighboring sites (off-diagonal coupling) is taken into account. Using the Toyozawa Ansatz and the Lanczos algorithm, the Holstein Hamiltonian with two types of off-diagonal coupling is studied focusing on a number of analyticity issues in the ground state. For finite-sized lattices and antisymmetric coupling, a sequence of discontinuities are found in the polaron energy dispersion, the size of the ground-state phonon cloud, and the linearized von Neumann entropy used to quantify the quantum entanglement between the exciton and the phonons in the ground state. Such behavior is accompanied by a shift of the ground-state crystal momentum from zero to nonzero values as the coupling strength is increased. In the thermodynamic limit, all discontinuities associated with antisymmetric coupling vanish except the one corresponding to the initial departure of the ground-state wavevector from the Brillouin zone center. For the case of symmetric off-diagonal coupling, a smooth crossover is found to exist in all parameters regimes.

Charge Orderings and Phase Separations in Itinerant Fermion Systems at Half Filling

We analyse the ground state phase diagrams of the charge orderings in narrow band materials using two e ective models: (1) the spinless fermion model (t−W ) with repulsive intersite interaction (Wij > 0) and (2) the molecular crystal model with the coupling of electrons to intramolecular (crystal eld) vibrations. We present results for the case of half lled bands for d = 2 square lattice. The calculations are performed within the (broken symmetry) Hartree-Fock approximation. The study takes into consideration the e ects of frustrating next-nearest-neighbour hopping (t2) on the charge ordered states in these systems. We focus on the two cases: (i) homogeneous phases and phase separations involving checkerboard charge ordering with the nesting vector Q = (π, π) only and (ii) homogeneous phases and phase separations involving two types of charge ordering: (a) checkerboard charge ordering with the nesting vector Q = (π, π), and (b) collinear (CL) charge ordering with Q = (0, π) or Q = (π, 0).

Effects of Frustrating Hopping on Charge Ordered States in Itinerant Fermion Systems for Arbitrary Concentration in 2D Lattice

There is ongoing, intense, research in the eld of electron charge orderings (CO) and charge density waves phenomena, due to experimental discovery of such phases in numerous important compounds. The aim of this work is to extend recent advances in the eld by studying two simple e ective paradigmatic models used to describe CO in narrow band materials i.e. (i) a model of correlated electrons: the so-called t W model of spinless fermions with repulsive interaction W and (ii) the molecular crystal model with the coupling of electrons to intramolecular (crystal eld) vibrations in the static limit. The nite temperature phase diagrams are evaluated at arbitrary carriers concentration for several representative cases. Our calculations are performed within the (broken symmetry) HFA for d = 2 square lattice and arbitrary carriers concentration. In this contribution we focus on the e ects of next-nearest-neighbor hopping on the CO states in these systems and the problem of phase separations involving checkerboard CO with the nesting vector Q = (π, π). The results we show here are an extension of our previous work on the subject.

Generalized Holstein model for spin-dependent electron-transfer reactions

Some chemical reactions are described by electron transfer (ET) processes. The underlying mechanism could be modeled as a polaron motion in the molecular crystal---the Holstein model. By taking spin degrees of freedom into consideration, we generalize the Holstein model (molecular crystal model) to microscopically describe an ET chemical reaction. In our model, the electron spins in the radical pair simultaneously interact with a magnetic field and their nuclear-spin environments. By virtue of the perturbation approach, we obtain the chemical reaction rates for different initial states. It is discovered that the chemical reaction rate of the triplet state demonstrates its dependence on the direction of the magnetic field while the counterpart of the singlet state does not. This difference is attributed to the explicit dependence of the triplet state on the direction when the axis is rotated. Our model may provide a possible candidate for the microscopic origin of the avian compass.

Stationary soliton solutions for large adiabatic Holstein polaron in magnetic field in anisotropic solids

Large adiabatic polarons in anisotropic crystals in the presence of constant magnetic field have been studied within the Holstein molecular crystal model in the continuum approximation. It was shown that magnetic field directed along the symmetry axis induces transverse confinement which may stabilize large polarons. They represent localized (soliton-like) nonlinear structure uniformly propagating along the symmetry axis and rotating around it in the same time. Such objects exist in 3D lattice provided that coupling constant and magnetic field do not exceed certain critical values. In contrast with pure 1D systems existence of large polarons is possible in a quite wider region of the values of coupling constant which may attain considerably higher values than in the pure 1D media. Furthermore, polaron effective mass, depending on the intensity of the applied magnetic field, may be considerably lighter than that of the the pure 1D polarons for the same values of coupling constant. This is the most significant difference with respect to pure 1D systems in the absence of magnetic field and may have substantial impact on polaron transport properties.

Practical quantum mechanics-based fragment methods for predicting molecular crystal properties

Significant advances in fragment-based electronic structure methods have created a real alternative to force-field and density functional techniques in condensed-phase problems such as molecular crystals. This perspective article highlights some of the important challenges in modeling molecular crystals and discusses techniques for addressing them. First, we survey recent developments in fragment-based methods for molecular crystals. Second, we use examples from our own recent research on a fragment-based QM/MM method, the hybrid many-body interaction (HMBI) model, to analyze the physical requirements for a practical and effective molecular crystal model chemistry. We demonstrate that it is possible to predict molecular crystal lattice energies to within a couple kJ mol(-1) and lattice parameters to within a few percent in small-molecule crystals. Fragment methods provide a systematically improvable approach to making predictions in the condensed phase, which is critical to making robust predictions regarding the subtle energy differences found in molecular crystals.

Nature of the vibron self-trapped states in hydrogen-bonded macromolecular chains

We study the influence of temperature and the values of basic energy parameters on the character of vibron self-trapped states in quasi-one-dimensional hydrogen-bonded molecular chains. Investigations have been carried out within the one-dimensional Holstein molecular crystal model employing the variational extension of the Lang-Firsov unitary transformation. It was found that, in the low-temperature regime, only partially dressed small-polaron states may exist. With the rise of temperature, the system enters the metastability region, where partially dressed (light and mobile) and fully dressed (heavy and practically immobile) small-polaron states may exist simultaneously.

The effect of correlations on the non-ohmic behavior of the small-polaron hoppingconductivity in 1D and 3D disordered systems

According to percolation theory the investigation of charge transport in disordered systemsis equivalent to the study of the possibility of the passage of the carriers through a randomnetwork of impedances which interconnect the different lattice sites. When the site energiesare not the same, the energy of a site affects the incoming as well as the outgoingimpedances connected to the given site and this gives rise to correlations betweenneighboring impedances. This new condition characterizes the transport process andimposes the evaluation of the average number of sites accessible by a bond froma given site for all possible configurations of sites that satisfy the percolationcondition. The generalized molecular crystal model, appropriate for the study ofsmall-polaron hopping transport in disordered systems, and the Kubo formula permitthe evaluation of these impedances. Taking correlations into account, theoreticalpercolation considerations applicable to one-dimensional and three-dimensionaldisordered systems, lead to analytical expressions for the temperature and electric fielddependence of the DC conductivity at high (multi-phonon-assisted hopping) and low(few-phonon-assisted hopping) temperatures. The theoretical analysis reveals theeffect of correlations on the non-ohmic behavior of the small-polaron hoppingconductivity and permits the evaluation of the maximum hopping distance. Quantitativeestimates of this effect are presented comparing the theoretical results, includingcorrelations with those ignoring them, previously reported, applying them to recentexperimental data for a wide temperature range and from low up to moderate electricfields.