Mixed Quantum-classical Method Research Articles

Numerical tests of coherence-corrected surface hopping methods using a donor-bridge-acceptor model system.

Surface hopping (SH) is a popular mixed quantum-classical method for modeling nonadiabatic excited state processes in molecules and condensed phase materials. The method is simple, efficient, and easy to implement, but the use of classical and independent nuclear trajectories introduces an overcoherence in the electronic density matrix which, if ignored, often leads to spurious results, such as overestimated reaction rates. Several methods have been proposed to incorporate decoherence into SH simulations, but a lack of insightful benchmarks makes their relative accuracy unknown. Herein, we run numerical simulations of common coherence-corrected SH methods including Truhlar's decay-of-mixing (DOM) and Subotnik's augmented SH using a Donor-bridge-Acceptor (DbA) model system. Numerical simulations are carried out in the superexchange regime, where charge transfer proceeds from a donor to an acceptor as a result of donor-bridge and bridge-acceptor couplings. The computed donor-to-acceptor reaction rates are compared to the reference Marcus theory results. For the DbA model under consideration, augmented SH recovers Marcus theory with quantitative accuracy, whereas DOM is only qualitatively accurate depending on whether predefined parameters in the decoherence rate are chosen wisely. We propose a general method for parameterizing the decoherence rate in the DOM method, which improves the method's reaction rates and presumably increases its transferability. Overall, the decoherence method of choice must be chosen with great care and this work provides insight using an exactly solvable model.

Mixed quantum-classical simulation of the hydride transfer reaction catalyzed by dihydrofolate reductase based on a mapped system-harmonic bath model.

The hydride transfer reaction catalyzed by dihydrofolate reductase is studied using a recently developed mixed quantum-classical method to investigate the nuclear quantum effects on the reaction. Molecular dynamics simulation is first performed based on a two-state empirical valence bond potential to map the atomistic model to an effective double-well potential coupled to a harmonic bath. In the mixed quantum-classical simulation, the hydride degree of freedom is quantized, and the effective harmonic oscillator modes are treated classically. It is shown that the hydride transfer reaction rate using the mapped effective double-well/harmonic-bath model is dominated by the contribution from the ground vibrational state. Further comparison with the adiabatic reaction rate constant based on the Kramers theory confirms that the reaction is primarily vibrationally adiabatic, which agrees well with the high transmission coefficients found in previous theoretical studies. The calculated kinetic isotope effect is also consistent with the experimental and recent theoretical results.

A comparative study of different methods for calculating electronic transition rates.

We present a comprehensive comparison of the following mixed quantum-classical methods for calculating electronic transition rates: (1) nonequilibrium Fermi's golden rule, (2) mixed quantum-classical Liouville method, (3) mean-field (Ehrenfest) mixed quantum-classical method, and (4) fewest switches surface-hopping method (in diabatic and adiabatic representations). The comparison is performed on the Garg-Onuchic-Ambegaokar benchmark charge-transfer model, over a broad range of temperatures and electronic coupling strengths, with different nonequilibrium initial states, in the normal and inverted regimes. Under weak to moderate electronic coupling, the nonequilibrium Fermi's golden rule rates are found to be in good agreement with the rates obtained via the mixed quantum-classical Liouville method that coincides with the fully quantum-mechanically exact results for the model system under study. Our results suggest that the nonequilibrium Fermi's golden rule can serve as an inexpensive yet accurate alternative to Ehrenfest and the fewest switches surface-hopping methods.

Accurate Long-Time Mixed Quantum-Classical Liouville Dynamics via the Transfer Tensor Method.

In this Letter, we combine the recently introduced transfer tensor method with the mixed quantum-classical Liouville method. The resulting protocol provides an accurate, general, flexible and robust new route for simulating the reduced dynamics of the quantum subsystem for arbitrarily long times, starting with computationally feasible short-time mixed quantum-classical Liouville dynamical maps. The accuracy and feasibility of the methodology are demonstrated on a spin-boson benchmark model.

Laser-induced electron localization in H₂⁺: mixed quantum-classical dynamics based on the exact time-dependent potential energy surface.

We study the exact nuclear time-dependent potential energy surface (TDPES) for laser-induced electron localization with a view to eventually developing a mixed quantum-classical dynamics method for strong-field processes. The TDPES is defined within the framework of the exact factorization [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett., 2010, 105, 123002] and contains the exact effect of the couplings to the electronic subsystem and to any external fields within a scalar potential. We compare its features with those of the quasistatic potential energy surfaces (QSPES) often used to analyse strong-field processes. We show that the gauge-independent component of the TDPES has a mean-field-like character very close to the density-weighted average of the QSPESs. Oscillations in this component are smoothened out by the gauge-dependent component, and both components are needed to yield the correct force on the nuclei. Once the localization begins to set in, the gradient of the exact TDPES tracks one QSPES and then switches to the other, similar to the description provided by surface-hopping between QSPESs. We show that evolving an ensemble of classical nuclear trajectories on the exact TDPES accurately reproduces the exact dynamics. This study suggests that the mixed quantum-classical dynamics scheme based on evolving multiple classical nuclear trajectories on the exact TDPES will be a novel and useful method to simulate strong field processes.

State-dependent molecular dynamics.

This paper proposes a new mixed quantum mechanics (QM)—molecular mechanics (MM) approach, where MM is replaced by quantum Hamilton mechanics (QHM), which inherits the modeling capability of MM, while preserving the state-dependent nature of QM. QHM, a single mechanics playing the roles of QM and MM simultaneously, will be employed here to derive the three-dimensional quantum dynamics of diatomic molecules. The resulting state-dependent molecular dynamics including vibration, rotation and spin are shown to completely agree with the QM description and well match the experimental vibration-rotation spectrum. QHM can be incorporated into the framework of a mixed quantum-classical Bohmian method to enable a trajectory interpretation of orbital-spin interaction and spin entanglement in molecular dynamics.

Interplay of radiative and nonradiative transitions in surface hopping with radiation-molecule interactions.

We implemented a method for the treatment of field induced transitions in trajectory surface hopping simulations, in the framework of the local diabatization scheme, especially suited for on-the-fly dynamics. The method is applied to a simple one-dimensional model with an avoided crossing and compared with quantum wavepacket dynamics. The results show the importance of introducing a proper decoherence correction to surface hopping, in order to obtain meaningful results. Also the energy conservation policy of standard surface hopping must be revised: in fact, the quantum wavepacket energetics is well reproduced if energy absorption/emission is allowed for in the hops determined by radiation-molecule coupling. To our knowledge, this is the first time the issues of decoherence and energy conservation have been analyzed in depth to devise a mixed quantum-classical method for dynamics with molecule-field interactions.

Surface hopping simulation of vibrational predissociation of methanol dimer

The mixed quantum-classical surface hopping method is applied to the vibrational predissociation of methanol dimer, and the results are compared to more exact quantum calculations. Utilizing the vibrational SCF basis, the predissociation problem is cast into a curve crossing problem between dissociative and quasibound surfaces with different vibrational character. The varied features of the dissociative surfaces, arising from the large amplitude OH torsion, generate rich predissociation dynamics. The fewest switches surface hopping algorithm of Tully [J. Chem. Phys. 93, 1061 (1990)] is applied to both diabatic and adiabatic representations. The comparison affords new insight into the criterion for selecting the suitable representation. The adiabatic method's difficulty with low energy trajectories is highlighted. In the normal crossing case, the diabatic calculations yield good results, albeit showing its limitation in situations where tunneling is important. The quadratic scaling of the rates on coupling strength is confirmed. An interesting resonance behavior is identified and is dealt with using a simple decoherence scheme. For low lying dissociative surfaces that do not cross the quasibound surface, the diabatic method tends to overestimate the predissociation rate whereas the adiabatic method is qualitatively correct. Analysis reveals the major culprits involve Rabi-like oscillation, treatment of classically forbidden hops, and overcoherence. Improvements of the surface hopping results are achieved by adopting a few changes to the original surface hopping algorithms.

Stereodynamics of chemical reactions: quasi-classical, quantum and mixed quantum-classical theories

Abstract In this review, some benchmark works by Han and coworkers on the stereodynamics of typical chemical reactions, triatomic reactions H + D2, Cl + H2 and O + H2 and polyatomic reaction Cl+CH4/CD4, are presented by using the quasi-classical, quantum and mixed quantum-classical methods. The product alignment and orientation in these A+BC model reactions are discussed in detail. We have also compared our theoretical results with experimental measurements and demonstrated that our theoretical results are in good agreement with the experimental results. Quasi-classical trajectory (QCT) method ignores some quantum effects like the tunneling effect and zero-point energy. The quantum method will be very time-consuming. Moreover, the mixed quantum-classical method can take into account some quantum effects and hence is expected to be applicable to large systems and widely used in chemical stereodynamics studies.

Subpicosecond Dynamics of Metal-to-Ligand Charge-Transfer Excited States in Solvated [Ru(bpy)3]2+ Complexes

Subpicosecond dynamics of metal-to-ligand charge-transfer (MLCT) excited states of [Ru(bpy)3]2+ (bpy = 2,2′-bipyridine) in solution is investigated through a mixed time-dependent quantum-classical dynamics computational method that can be applied to a wide class of solvated transition-metal complexes. Both solute and solvent systems are treated at the same atomistic quantum dynamics level of theory. We study the processes of induced polarization, depolarization, and interligand electron-transfer (ILET) kinetics within the complex as well as solvation dynamics effects on the MLCT state relaxation. Liquid acetonitrile and water are considered as solvent media. It is found that the characteristic time scale for ILET is in the range of hundreds of femtoseconds for both solvent cases and polarization relaxation is even faster.

Mixed quantum-classical approach to multiphoton dissociation of the hydrogen molecular ion

We present a mixed quantum-classical propagation method for the time-dependent dynamics of nuclear as well as electronic degrees of freedom for para-H2+ exposed to short intense laser pulses of 800 nm wavelength. Depending on the initial vibrational state, the angular distributions of photofragments show characteristic shapes in very good agreement with our full-dimensional nuclear dynamics quantum calculations. The results can be understood in terms of two-dimensional adiabatic Floquet surfaces, which depend on the internuclear separation and rotation angle, demonstrating that adiabatic light-dressed surfaces are also a useful concept for full-dimensional nuclear dynamics. Using kinetic energy release spectra, we are also able to extract the contributions of different photon channels in the quantum-classical case.

Mixed time slicing in path integral simulations

A simple and efficient scheme is presented for using different time slices for different degrees of freedom in path integral calculations. This method bridges the gap between full quantization and the standard mixed quantum-classical (MQC) scheme and, therefore, still provides quantum mechanical effects in the less-quantized variables. Underlying the algorithm is the notion that time slices (beads) may be "collapsed" in a manner that preserves quantization in the less quantum mechanical degrees of freedom. The method is shown to be analogous to multiple-time step integration techniques in classical molecular dynamics. The algorithm and its associated error are demonstrated on model systems containing coupled high- and low-frequency modes; results indicate that convergence of quantum mechanical observables can be achieved with disparate bead numbers in the different modes. Cost estimates indicate that this procedure, much like the MQC method, is most efficient for only a relatively few quantum mechanical degrees of freedom, such as proton transfer. In this regime, however, the cost of a fully quantum mechanical simulation is determined by the quantization of the least quantum mechanical degrees of freedom.

Finite temperature application of the corrected propagator method to reactive dynamics in a condensed-phase environment.

The recently proposed mixed quantum-classical method is extended to applications at finite temperatures. The method is designed to treat complex systems consisting of a low-dimensional quantum part (the primary system) coupled to a dissipative bath described classically. The method is based on a formalism showing how to systematically correct the approximate zeroth-order evolution rule. The corrections are defined in terms of the total quantum Hamiltonian and are taken to the classical limit by introducing the frozen Gaussian approximation for the bath degrees of freedom. The evolution of the primary system is governed by the corrected propagator yielding the exact quantum dynamics. The method has been tested on a standard model system describing proton transfer in a condensed-phase environment: a symmetric double-well potential bilinearly coupled to a bath of harmonic oscillators. Flux correlation functions and thermal rate constants have been calculated at two different temperatures for a range of coupling strengths. The results have been compared to the fully quantum simulations of Topaler and Makri [J. Chem. Phys. 101, 7500 (1994)] with the real path integral method.

Mixed quantum-classical dynamics with time-dependent external fields: A time-dependent density-functional-theory approach

A mixed quantum-classical method aimed at the study of nonadiabatic dynamics in the presence of external electromagnetic fields is developed within the framework of time-dependent density functional theory. To this end, we use a trajectory-based description of the quantum nature of the nuclear degrees of freedom according to Tully's fewest switches trajectories surface hopping, where both the nonadiabatic coupling elements between the different potential energy surfaces, and the coupling with the external field are given as functionals of the ground-state electron density or, equivalently, of the corresponding Kohn-Sham orbitals. The method is applied to the study of the photodissociation dynamics of some simple molecules in gas phase.

Mixed Quantum-Classical Reaction Path Dynamics of HCl Elimination from Chloroethane

The dynamics of four-centered HCl elimination from chloroethane are studied using a mixed quantum-classical method based on a reaction path Hamiltonian. Both the structural details of the reaction and the partitioning of the exit-channel potential energy to the products are analyzed. The minimum energy path was calculated at the B3LYP/6-311++G(2d,2p) level of theory, which was followed by energy-partitioning dynamics computations. Selective vibrational excitation of the HCl product was observed, leading to a vibrational state distribution in good agreement with experiment. Differences between HCl elimination from C(2)H(5)Cl and HF elimination from C(2)H(5)F, particularly in the ethylene fragment, were observed and are discussed.

Modeling vibrational resonance in linear hydrocarbon chain with a mixed quantum-classical method

The quantum dynamics of a vibrational excitation in a linear hydrocarbon model system is studied with a new mixed quantum-classical method. The method is suited to treat many-body systems consisting of a low dimensional quantum primary part coupled to a classical bath. The dynamics of the primary part is governed by the quantum corrected propagator, with the corrections defined in terms of matrix elements of zeroth order propagators. The corrections are taken to the classical limit by introducing the frozen Gaussian approximation for the bath degrees of freedom. The ability of the method to describe dynamics of multidimensional systems has been tested. The results obtained by the method have been compared to previous quantum simulations performed with the quasiadiabatic path integral method.

Isotope Effects on the Vibrational Relaxation and Multidimensional Infrared Spectra of the Hydrogen Stretch in a Hydrogen-Bonded Complex Dissolved in a Polar Liquid

Isotope effects on rate processes and spectra are often used to elucidate the nature of the interactions underlying molecular structure and dynamics. In this paper, we present a computational study of the effect of substituting hydrogen by deuterium in a solvated hydrogen-bonded complex on the rates of the various processes involved in the vibrational relaxation of the hydrogen/deuterium stretch and on the corresponding 1D and 2D infrared spectra. The vibrational relaxation is simulated via the mixed quantum-classical Liouville method, where the proton/deuteron is treated quantum mechanically while the other particles are treated in a classical-like manner. We find that the vibrational relaxation pathway and the rates of the various steps in it are similar for the deuterium and hydrogen stretches. However, we also find that isotope substitution modifies the 1D and 2D spectra of the stretch in a qualitative manner. The differences between the spectra are explained in terms of the narrowing and broadening of the fundamental and overtone transition frequency ranges, respectively, and the smaller transition dipole moments in the case of the deuterium stretch. Our results demonstrate that isotope substitution may have a rather dramatic effect on the infrared spectra of a vibrational mode strongly coupled to its environment even though the rate and pathway of the underlying vibrational relaxation may not be overly sensitive to it.

Mixed Quantum-Classical Reaction Path Dynamics of C2H5F → C2H4 + HF

A mixed quantum-classical method for calculating product energy partitioning based on a reaction path Hamiltonian is presented and applied to HF elimination from fluoroethane. The goal is to describe the effect of the potential energy release on the product energies using a simple model of quantized transverse vibrational modes coupled to a classical reaction path via the path curvature. Calculations of the minimum energy path were done at the B3LYP/6-311++G(2d,2p) and MP2/6-311++G** levels of theory, followed by energy-partitioning dynamics calculations. The results for the final HF vibrational state distribution were found to be in good qualitative agreement with both experimental studies and quasiclassical trajectory simulations.

Tunneling dynamics with a mixed quantum-classical method: Quantum corrected propagator combined with frozen Gaussian wave packets

The recently developed mixed quantum-classical propagation method is extended to treat tunneling effects in multidimensional systems. Formulated for systems consisting of a quantum primary part and a classical bath of heavier particles, the method employs a frozen Gaussian description for the bath degrees of freedom, while the dynamics of the quantum subsystem is governed by a corrected propagator. The corrections are defined in terms of matrix elements of zeroth-order propagators. The method is applied to a model system of a double-well potential bilinearly coupled to a harmonic oscillator. The extension of the method, which includes nondiagonal elements of the correction propagator, enables an accurate treatment of tunneling in an antisymmetric double-well potential.

The effect of protonation on the photodissociation processes in formamide – An ab initio surface hopping dynamics study

Photodeactivation processes of protonated formamide were investigated using the multireference configuration interaction calculations with singles and doubles (MR-CISD) method while dynamics were simulated by employing the mixed quantum-classical direct trajectory method with surface hopping based on multi-configurational self-consistent wave functions. The dynamics calculations from the first excited singlet state in O-protonated formamide resembled those found for the second valence excited state of the parent molecule. Two deactivation processes were found: the C–N (major) and C–O (minor) dissociations with very short lifetimes (33 fs). Similarly, the major process for photodecomposition in the first excited state of N-protonated formamide resembles that for the parent formamide, and involves C–N dissociation with a lifetime around 390 fs. However, 55% of trajectories remained undissociated and undeactivated until 1000 fs, indicating the existence of other deactivation processes on a longer time scale.