Symmetrical Quasi-classical Research Articles

Dynamical free energy based model for quantum decision making

We propose a quantum-mechanical framework to describe the dynamics of human decision making, where the interactions between the open quantum system of decision makers and its surrounding environment at given temperature are relevantly taken into account. The temporal evolution of quantum system is described in terms of the time-dependent density matrix, from which we can evaluate those thermodynamic functions such as internal energy, (von Neumann) entropy and (Helmholtz) free energy. In this study we rely on the symmetrical quasi-classical (SQC) windowing approach to numerically solve the non-adiabatic quantum dynamics. We then consider the Hamiltonians to formulate the Prisoner’s Dilemma (PD) problem and calculate the population dynamics of the quantum states of two players’ decisions, where some quantum coherence or interference effects are observed. The evaluated free energy tends to decrease toward the thermodynamic equilibrium value, thus driving the dynamics of quantum system between the Nash equilibrium and Pareto optimum states. The calculated entropy often shows a tendency of temporal decrease, thus indicating the emergence of order in the quantum system. We have also applied the present scheme to the issue of the violation of the sure thing principle and found its occurrence through explicit numerical simulations for the PD problem. The present study may provide a theoretical basis to connect the dynamics of quantum decision making to the free energy principle in cognitive science.

Ab initio symmetric quasi-classical approach to investigate molecular Tully models.

We perform on-the-fly non-adiabatic molecular dynamics simulations using the symmetrical quasi-classical (SQC) approach with the recently suggested molecular Tully models: ethylene and fulvene. We attempt to provide benchmarks of the SQC methods using both the square and triangle windowing schemes as well as the recently proposed electronic zero-point-energy correction scheme (the so-called γ correction). We use the quasi-diabatic propagation scheme to directly interface the diabatic SQC methods with adiabatic electronic structure calculations. Our results showcase the drastic improvement of the accuracy by using the trajectory-adjusted γ-corrections, which outperform the widely used trajectory surface hopping method with decoherence corrections. These calculations provide useful and non-trivial tests to systematically investigate the numerical performance of various diabatic quantum dynamics approaches, going beyond simple diabatic model systems that have been used as the major workhorse in the quantum dynamics field. At the same time, these available benchmark studies will also likely foster the development of new quantum dynamics approaches based on these techniques.

Improving the Accuracy of Quasiclassical Mapping Hamiltonian Methods by Treating the Window Function Width as an Adjustable Parameter.

Mapping Hamiltonian methods for simulating electronically nonadiabatic molecular dynamics are based on representing the electronic population and coherence operators in terms of isomorphic mapping operators, which are given in terms of the auxiliary position and momentum operators. Adding a quasiclassical approximation then makes it possible to treat those auxiliary coordinates and momenta, as well as the nuclear coordinates and momenta, as classical-like phase-space variables. Within such quasiclassical mapping Hamiltonian methods, the initial sampling of the auxiliary coordinates and momenta and the calculation of expectation values of electronic observables at a later time are based on window functions whose functional form differ from one method to another. However, different methods also differ with respect to the way in which they treat the window width. More specifically, while the window width is treated as an adjustable parameter within the symmetrical quasiclassical (SQC) method, this has not been the case for methods based on the linearized semiclasscial (LSC) approximation. In the present study, we investigate the effect that turning the window width into an adjustable parameter within LSC-based methods has on their accuracy compared to SQC. The analysis is performed in the context of the spin-boson and Fenna-Matthews-Olson (FMO) complex benchmark models. We find that treating the window width in LSC-based methods as an adjustable parameter can make their accuracy comparable to that of the SQC method.

Simulating Absorption Spectra of Multiexcitonic Systems via Quasiclassical Mapping Hamiltonian Methods.

In this paper, we compare the ability of different quasiclassical mapping Hamiltonian methods to accurately simulate the absorption spectra of multiexcitonic molecular systems. Two distinctly different approaches for simulating the absorption spectra are considered: (1) a perturbative approach, which relies on the first-order perturbation theory with respect to the field-matter interaction; (2) a nonperturbative approach, which mimics the experimental measurement of the absorption spectra from the free-induction decay that follows a short laser pulse. The methods compared are several variations of the linearized semiclassical (LSC) method, the symmetrical quasiclassical (SQC) method, and the mean-field (Ehrenfest) method. The comparison is performed in the context of a biexcitonic model and a seven-excitonic model of the Fenna-Matthews-Olson (FMO) complex. The accuracy of the various methods is tested by comparing their predictions to the quantum-mechanically exact results obtained via the hierarchy of the equations of motion (HEOM) method, as well as to the results based on the Redfield quantum master equation. The results show that the LSC-based quasiclassical mapping Hamiltonian methods can yield the accurate and robust absorption spectra in the high-temperature and/or slow-bath limit, where the nuclear degrees of freedom can be treated as classical.

Benchmarking Quasiclassical Mapping Hamiltonian Methods for Simulating Electronically Nonadiabatic Molecular Dynamics.

Quasi-classical mapping Hamiltonian methods have recently emerged as a promising approach for simulating electronically nonadiabatic molecular dynamics. The classical-like dynamics of the overall system within these methods makes them computationally feasible, and they can be derived based on well-defined semiclassical approximations. However, the existence of a variety of different quasi-classical mapping Hamiltonian methods necessitates a systematic comparison of their respective advantages and limitations. Such a benchmark comparison is presented in this paper. The approaches compared include the Ehrenfest method, the symmetrical quasi-classical (SQC) method, and five variations of the linearized semiclassical (LSC) method, three of which employ a modified identity operator. The comparison is based on a number of popular nonadiabatic model systems; the spin-boson model, a Frenkel biexciton model, and Tully's scattering models 1 and 2. The relative accuracy of the different methods is tested by comparing with quantum-mechanically exact results for the dynamics of the electronic populations and coherences. We find that LSC with the modified identity operator typically performs better than the Ehrenfest and standard LSC approaches. In comparison to SQC, these modified methods appear to be slightly more accurate for condensed phase problems, but for scattering models there is little distinction between them.

Study of the exciton dynamics in perylene bisimide (PBI) aggregates with symmetrical quasiclassical dynamics based on the Meyer-Miller mapping Hamiltonian.

The exciton dynamics in one-dimensional stacked PBI (Perylene Bisimide) aggregates was studied with SQC-MM dynamics (Symmetrical Quasiclassical Dynamics based on the Meyer-Miller mapping Hamiltonian). Based on linear vibronic coupling models, one-dimensional PBI aggregates with different lengths were investigated. Based on an investigation of short PBI aggregates (10 sites and 80-140 modes) using both the SQC-MM and ML-MCTDH (multilayer multiconfigurational time-dependent Hartree) methods, we showed that SQC-MM dynamics give a reasonable description of the exciton dynamics for organic PBI aggregates composed of 101 sites and 808 modes. This allows us to employ SQC-MM dynamics in the explicit study of exciton diffusion for long one-dimensional PBI aggregates based on models with different site-site coupling strengths with/without static disorder by including all the involved electronic and vibrational degrees of freedom.

Trajectory-adjusted electronic zero point energy in classical Meyer-Miller vibronic dynamics: Symmetrical quasiclassical application to photodissociation.

An electronic zero-point energy (ZPE) adjustment protocol is presented within the context of the symmetrical quasiclassical (SQC) quantization of the electronic oscillator degrees of freedom (DOF) in classical Meyer-Miller (MM) vibronic dynamics for the molecular dynamics treatment of electronically nonadiabatic processes. The "adjustment" procedure maintains the same initial and final distributions of coordinates and momenta in the electronic oscillator DOF as previously given by the SQC windowing protocol but modifies the ZPE parameter in the MM Hamiltonian, on a per trajectory basis, so that the initial nuclear forces are precisely those corresponding to the initial electronic quantum state. Examples demonstrate that this slight modification to the standard SQC/MM approach significantly improves treatment of the multistate nonadiabatic dynamics following a Franck-Condon type vertical excitation onto a highly repulsive potential energy surface as is typical in the photodissociation context.

A symmetrical quasi-classical windowing model for the molecular dynamics treatment of non-adiabatic processes involving many electronic states.

In the previous work of Cotton and Miller [J. Chem. Phys. 145, 144108 (2016)], an improved symmetrical quasi-classical (SQC) windowing model for the molecular dynamics treatment of electronically non-adiabatic processes was developed in order to extend the original SQC approach to the regime of weak-coupling between the electronic states. The improved SQC model-based on triangular-shaped window functions-handled the weak-coupling limit as intended and, as a bonus, was shown to be universally superior to the original square/histogram SQC windowing model over all coupling regimes, but only for treating systems of two electronic states, as no higher-dimensional generalization was evident. This paper, therefore, provides a generalized version for treating an arbitrary number of electronic states. By construction, the benefits of the two-state triangle model-seamless treatment of weak-coupling and improved accuracy in all coupling regimes-carry over to the generalized version. Far more significant, however, is that the new model provides vastly improved windowing statistics in higher dimensions, enabling the SQC simulation of electronically non-adiabatic processes involving many more relevant electronic states than was previously practical. Capabilities are demonstrated with respect to a 24 pigment trimer model of the Fenna-Matthews-Olson light-harvesting complex, as well as treating similar 48- and 96-electronic state model problems, illustrating the scaling properties of the new method.

Initial sampling in symmetrical quasiclassical dynamics based on Li-Miller mapping Hamiltonian.

A symmetrical quasiclassical (SQC) dynamics approach based on the Li-Miller (LM) mapping Hamiltonian (SQC-LM) was employed to describe nonadiabatic dynamics. In principle, the different initial sampling procedures may be applied in the SQC-LM dynamics, and the results may be dependent on different initial sampling. We provided various initial sampling approaches and checked their influence. We selected two groups of models including site-exciton models for exciton dynamics and linear vibronic coupling models for conical intersections to test the performance of SQC-LM dynamics with the different initial sampling methods. The results were examined with respect to those of the accurate multiconfigurational time-dependent Hartree (MCTDH) quantum dynamics. For both the models, the SQC-LM method more-or-less gives a reasonable description of the population dynamics, while the influence of the initial sampling approaches on the final results is noticeable. It seems that the suitable initial sampling methods should be determined by the system under study. This indicates that the combination of the SQC-LM method with a suitable sampling approach may be a potential method in the description of nonadiabatic dynamics.

The symmetrical quasi-classical approach to electronically nonadiabatic dynamics applied to ultrafast exciton migration processes in semiconducting polymers.

In the last several years, a symmetrical quasi-classical (SQC) windowing model applied to the classical Meyer-Miller (MM) vibronic Hamiltonian has been shown to be a simple, efficient, general, and quite-accurate method for treating electronically nonadiabatic processes at the totally classical level. Here, the SQC/MM methodology is applied to ultrafast exciton dynamics in a Frenkel/site-exciton model of oligothiophene (OT) as a model of organic semiconductor polymers. In order to keep the electronic representation as compact and efficient as possible, the adiabatic version of the MM Hamiltonian was employed, with dynamical calculations carried out in the recently developed "kinematic momentum" representation, from which site/monomer-specific (diabatic) excitation probabilities were extracted using a new procedure developed in this work. The SQC/MM simulation results are seen to describe coherent exciton transport driven by planarization of a central torsion defect in the OT oligomer as well as to capture exciton self-trapping effects in good agreement with benchmark quantum calculations using the multi-layer multiconfiguration time-dependent Hartree approach. The SQC/MM calculations are also seen to significantly outperform the standard Ehrenfest approach, which shows serious discrepancies. These results are encouraging, not only because they illustrate a significant further application of the SQC/MM approach and its utility, but because they strongly suggest that classical mechanical simulations (with the potential for linear scaling efficiency) can be used to capture, quantitatively, important dynamical features of electronic excitation energy transfer in semiconducting polymers.

Mixed quantum-classical electrodynamics: Understanding spontaneous decay and zero-point energy

The dynamics of an electronic two-level system coupled to an electromagnetic field are simulated explicitly for one- and three-dimensional systems through semiclassical propagation of the Maxwell-Liouville equations. We consider three flavors of mixed quantum-classical dynamics: (i) the classical path approximation (CPA), (ii) Ehrenfest dynamics, and (iii) symmetrical quasiclassical (SQC) dynamics. Our findings are as follows: (i) The CPA fails to recover a consistent description of spontaneous emission, (ii) a consistent ``spontaneous'' emission can be obtained from Ehrenfest dynamics, provided that one starts in an electronic superposition state, and (iii) spontaneous emission is always obtained using SQC dynamics. Using the SQC and Ehrenfest frameworks, we further calculate the dynamics following an incoming pulse, but here we find very different responses: SQC and Ehrenfest dynamics deviate sometimes strongly in the calculated rate of decay of the transient excited state. Nevertheless, our work confirms the earlier observations by Miller [J. Chem. Phys. 69, 2188 (1978)] that Ehrenfest dynamics can effectively describe some aspects of spontaneous emission and highlights interesting possibilities for studying light-matter interactions with semiclassical mechanics.

Nonadiabatic Dynamics via the Symmetrical Quasi-Classical Method in the Presence of Anharmonicity.

The symmetrical quasi-classical (SQC) method recently proposed by Miller and Cotton allows one to simulate nonadiabatic dynamics based on an algorithm with classical-like scaling with respect to system size. This is made possible by casting the electronic degrees of freedom in terms of mapping variables that can be propagated in a classical-like manner. While SQC was shown to be rather accurate when applied to benchmark models with harmonic electronic potential energy surfaces, it was also found to become inaccurate and to suffer numerical instabilities when applied to anharmonic systems. In this paper, we propose an extended SQC (E-SQC) methodology for overcoming those discrepancies by describing the anharmonic nuclear modes, which are coupled to the electronic degrees of freedom, in terms of classical-like mapping variables. The accuracy of E-SQC relative to standard SQC is demonstrated on benchmark models with quartic and Morse potential energy surfaces.

Ultrafast Electron Transfer with Symmetrical Quasi-classical Dynamics based on Mapping Hamiltonian and Quantum Dynamics based on ML-MCTDH

Symmetrical quasi-classical (SQC) method based on mapping Hamiltonian is an efficient approach that is potentially useful to treat the nonadiabatic dynamics of very large systems. We try to evaluate the performance of this method in the ultrafast electron transfer processes involving a few of electronic states and a large number of vibrational modes. The multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) method was used to get the accurate dynamical results for benchmark. Although the population dynamics in the longtime limit show differences in the ML-MCTDH and SQC calculations, the SQC method gives acceptable results.

Vibrational Energy Relaxation: A Benchmark for Mixed Quantum-Classical Methods.

We investigate the ability of mixed quantum-classical methods to capture the dynamics of vibrational energy relaxation. Several methods, including surface hopping, and Ehrenfest and symmetrical quasiclassical (SQC) dynamics, are benchmarked for the exactly solvable model problem of a harmonic oscillator bilinearly coupled to a bath of harmonic oscillators. Results show that, very often, one can recover accurate vibrational relaxation rates and detailed balance using simple mixed quantum-classical approaches. A few anomalous results do appear, however, especially regarding Ehrenfest and SQC dynamics.

On the adiabatic representation of Meyer-Miller electronic-nuclear dynamics.

The Meyer-Miller (MM) classical vibronic (electronic + nuclear) Hamiltonian for electronically non-adiabatic dynamics-as used, for example, with the recently developed symmetrical quasiclassical (SQC) windowing model-can be written in either a diabatic or an adiabatic representation of the electronic degrees of freedom, the two being a canonical transformation of each other, thus giving the same dynamics. Although most recent applications of this SQC/MM approach have been carried out in the diabatic representation-because most of the benchmark model problems that have exact quantum results available for comparison are typically defined in a diabatic representation-it will typically be much more convenient to work in the adiabatic representation, e.g., when using Born-Oppenheimer potential energy surfaces (PESs) and derivative couplings that come from electronic structure calculations. The canonical equations of motion (EOMs) (i.e., Hamilton's equations) that come from the adiabatic MM Hamiltonian, however, in addition to the common first-derivative couplings, also involve second-derivative non-adiabatic coupling terms (as does the quantum Schrödinger equation), and the latter are considerably more difficult to calculate. This paper thus revisits the adiabatic version of the MM Hamiltonian and describes a modification of the classical adiabatic EOMs that are entirely equivalent to Hamilton's equations but that do not involve the second-derivative couplings. The second-derivative coupling terms have not been neglected; they simply do not appear in these modified adiabatic EOMs. This means that SQC/MM calculations can be carried out in the adiabatic representation, without approximation, needing only the PESs and the first-derivative coupling elements. The results of example SQC/MM calculations are presented, which illustrate this point, and also the fact that simply neglecting the second-derivative couplings in Hamilton's equations (and presumably also in the Schrödinger equation) can cause very significant errors.

A new symmetrical quasi-classical model for electronically non-adiabatic processes: Application to the case of weak non-adiabatic coupling

Previous work has shown how a symmetrical quasi-classical (SQC) windowing procedure can be used to quantize the initial and final electronic degrees of freedom in the Meyer-Miller (MM) classical vibronic (i.e, nuclear + electronic) Hamiltonian, and that the approach provides a very good description of electronically non-adiabatic processes within a standard classical molecular dynamics framework for a number of benchmark problems. This paper explores application of the SQC/MM approach to the case of very weak non-adiabatic coupling between the electronic states, showing (as anticipated) how the standard SQC/MM approach used to date fails in this limit, and then devises a new SQC windowing scheme to deal with it. Application of this new SQC model to a variety of realistic benchmark systems shows that the new model not only treats the weak coupling case extremely well, but it is also seen to describe the "normal" regime (of electronic transition probabilities ≳ 0.1) even more accurately than the previous "standard" model.

Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix.

It is pointed out that the classical phase space distribution in action-angle (a-a) variables obtained from a Wigner function depends on how the calculation is carried out: if one computes the standard Wigner function in Cartesian variables (p, x), and then replaces p and x by their expressions in terms of a-a variables, one obtains a different result than if the Wigner function is computed directly in terms of the a-a variables. Furthermore, the latter procedure gives a result more consistent with classical and semiclassical theory-e.g., by incorporating the Bohr-Sommerfeld quantization condition (quantum states defined by integer values of the action variable) as well as the Heisenberg correspondence principle for matrix elements of an operator between such states-and has also been shown to be more accurate when applied to electronically non-adiabatic applications as implemented within the recently developed symmetrical quasi-classical (SQC) Meyer-Miller (MM) approach. Moreover, use of the Wigner function (obtained directly) in a-a variables shows how our standard SQC/MM approach can be used to obtain off-diagonal elements of the electronic density matrix by processing in a different way the same set of trajectories already used (in the SQC/MM methodology) to obtain the diagonal elements.

The Symmetrical Quasi-Classical Model for Electronically Non-Adiabatic Processes Applied to Energy Transfer Dynamics in Site-Exciton Models of Light-Harvesting Complexes.

In a recent series of papers, it has been illustrated that a symmetrical quasi-classical (SQC) windowing model applied to the Meyer-Miller (MM) classical vibronic Hamiltonian provides an excellent description of a variety of electronically non-adiabatic benchmark model systems for which exact quantum results are available for comparison. In this paper, the SQC/MM approach is used to treat energy transfer dynamics in site-exciton models of light-harvesting complexes, and in particular, the well-known 7-state Fenna-Mathews-Olson (FMO) complex. Again, numerically "exact" results are available for comparison, here via the hierarchical equation of motion (HEOM) approach of Ishizaki and Fleming, and it is seen that the simple SQC/MM approach provides very reasonable agreement with the previous HEOM results. It is noted, however, that unlike most (if not all) simple approaches for treating these systems, because the SQC/MM approach presents a fully atomistic simulation based on classical trajectory simulation, it places no restrictions on the characteristics of the thermal baths coupled to each two-level site, e.g., bath spectral densities (SD) of any analytic functional form may be employed as well as discrete SD determined experimentally or from MD simulation (nor is there any restriction that the baths be harmonic), opening up the possibility of simulating more realistic variations on the basic site-exciton framework for describing the non-adiabatic dynamics of photosynthetic pigment complexes.

Ultrafast Nonadiabatic Dynamics of Singlet Fission: Quantum Dynamics with the Multilayer Multiconfigurational Time-Dependent Hartree (ML-MCTDH) Method

Singlet fission (SF) is supposed to potentially improve the efficiency of solar energy conversion in organic photovoltaic systems. The multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) method was employed to describe the singlet fission of the pentacene system with a three-state model. The ML-MCTDH result agrees well with the previous simulations using the Redfield theory, the hierarchical equation of motion (HEOM) and the symmetrical quasi-classical (SQC) theory. We carefully investigated the role of vibrational modes with different frequencies in singlet fission dynamics. Interestingly, we observed the important contribution of a few modes with frequency resonance to electronic transition. Such a finding can be understood by revisiting the superexchange mechanism within the framework of Fermi’s golden rule. As a numerically exact method, ML-MCTDH not only provides an accurate description of the microscopy insight of the SF dynamics but also provides benchmark results to examine the perfo...

Classical molecular dynamics simulation of electronically non-adiabatic processes.

Both classical and quantum mechanics (as well as hybrids thereof, i.e., semiclassical approaches) find widespread use in simulating dynamical processes in molecular systems. For large chemical systems, however, which involve potential energy surfaces (PES) of general/arbitrary form, it is usually the case that only classical molecular dynamics (MD) approaches are feasible, and their use is thus ubiquitous nowadays, at least for chemical processes involving dynamics on a single PES (i.e., within a single Born-Oppenheimer electronic state). This paper reviews recent developments in an approach which extends standard classical MD methods to the treatment of electronically non-adiabatic processes, i.e., those that involve transitions between different electronic states. The approach treats nuclear and electronic degrees of freedom (DOF) equivalently (i.e., by classical mechanics, thereby retaining the simplicity of standard MD), and provides "quantization" of the electronic states through a symmetrical quasi-classical (SQC) windowing model. The approach is seen to be capable of treating extreme regimes of strong and weak coupling between the electronic states, as well as accurately describing coherence effects in the electronic DOF (including the de-coherence of such effects caused by coupling to the nuclear DOF). A survey of recent applications is presented to illustrate the performance of the approach. Also described is a newly developed variation on the original SQC model (found universally superior to the original) and a general extension of the SQC model to obtain the full electronic density matrix (at no additional cost/complexity).