Incorporation of Quantum Effects for Selected Degrees of Freedom into the Trajectory-Based Dynamics Using Spatial Domains

The approach of defining quantum corrections on nuclear dynamics of molecular systems incorporated approximately into selected degrees of freedom, is described. The approach is based on the Madelung-de-Broglie-Bohm formulation of time-dependent quantum mechanics which represents a wavefunction in terms of an ensemble of trajectories. The trajectories follow classical laws of motion except that the quantum potential, dependent on the wavefunction amplitude and its derivatives, is added to the external, classical potential. In this framework the quantum potential, determined approximately for practical reasons, is included only into the “quantum” degrees of freedom describing light particles such as protons, while neglecting with the quantum force for the heavy, nearly classical nuclei. The entire system comprised of light and heavy particles is described by a single wavefunction of full dimensionality. The coordinate space of heavy particles is divided into spatial domains or subspaces. The quantum force acting on the light particles is determined for each domain of similar configurations of the heavy nuclei. This approach effectively introduces parametric dependence of the reduced dimensionality quantum force, on classical degrees of freedom. This strategy improves accuracy of the quantum force and does not restrict interaction between the domains. The concept is illustrated for two-dimensional scattering systems, where the quantum force is required to reproduce vibrational energy of the quantum degree of freedom

Simplified Calculation of the Stability Matrix for Semiclassical Propagation

We present a simple method of calculation of the stability (monodromy) matrix that enters the widely used semiclassical propagator of Herman and Kluk and almost all other semiclassical propagators. The method is based on the unitarity of classical propagation and does not involve any approximations. The number of auxiliary differential equations per trajectory scales linearly rather than quadratically with the system size. Just the first derivatives of the potential surface are needed. The method is illustrated on the collinear H3 system

Semiclassical Dynamics with Quantum Trajectories: Formulation and Comparison with the Semiclassical Initial Value Representation Propagator

We present a time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential (AQP). One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare AQP method to the widely used semiclassical propagator of Herman and Kluk by computing energy-resolved transmission probabilities for the Eckart barrier from the wave packet time-correlation functions. We find the results obtained with the Herman–Kluk propagator to be essentially equivalent to those of AQP method with a one-Gaussian density fit for several barrier widths

Bohmian Dynamics on Subspaces Using Linearized Quantum Force

In the de Broglie–Bohm formulation of quantum mechanics the time-dependent Schrödinger equation is solved in terms of quantum trajectories evolving under the influence of quantum and classical potentials. For a practical implementation that scales favorably with system size and is accurate for semiclassical systems, we use approximate quantum potentials. Recently, we have shown that optimization of the nonclassical component of the momentum operator in terms of fitting functions leads to the energy-conserving approximate quantum potential. In particular, linear fitting functions give the exact time evolution of a Gaussian wave packet in a locally quadratic potential and can describe the dominant quantum-mechanical effects in the semiclassical scattering problems of nuclear dynamics. In this paper we formulate the Bohmian dynamics on subspaces and define the energy-conserving approximate quantum potential in terms of optimized nonclassical momentum, extended to include the domain boundary functions. This generalization allows a better description of the non-Gaussian wave packets and general potentials in terms of simple fitting functions. The optimization is performed independently for each domain and each dimension. For linear fitting functions optimal parameters are expressed in terms of the first and second moments of the trajectory distribution. Examples are given for one-dimensional anharmonic systems and for the collinear hydrogen exchange reaction

Semiclassical Approach to the Hydrogen-exchange Reaction- Reactive and Transition-state Dynamics

Scattering matrix elements and symmetric transition-state resonances for the collinear H 2 + H → H + H 2 reaction are obtained using a time-dependent approach. The correlation function between reactant channel wavepackets and product channel wavepackets is used to determine the S-matrix elements. In a similar fashion, autocorrelation functions are used to extract the positions and widths of transition-state resonances. The time propagation of the wavepackets is performed by the improved semiclassical frozen Gaussian method of Herman and Kluk, which is an initial value, uniformly converged method. The agreement between the quantum and semiclassical results is far better than that obtained previously for this system by other semiclassical methods

Correlation Function Formulation for the State Selected Total Reaction Probability

A correlation function formulation for the state-selected total reaction probability, Nα(E), is suggested. A wave packet, correlating with a specific set of internal reactant quantum numbers, α, is propagated forward in time until bifurcation is complete at which time the nonreactive portion of the amplitude is discarded. The autocorrelation function of the remaining amplitude is then computed and Fourier transformed to obtain a reactivity spectrum. Dividing by the corresponding spectrum of the original, unfiltered, wave packet normalizes the reactivity spectrum, yielding the total reaction probability from the internal state, α. The procedure requires negligible storage and just one time-energy Fourier transform for each initial reactant state, independent of the number of open channels of products. The method is illustrated numerically for the one-dimensional Eckart barrier, using both quantum-mechanical and semiclassical propagation methods. Summing over internal states of reactants gives the cumulative reaction probability, N(E). The relation to the trace formula [W. H. Miller, S. D. Schwartz, J. W. Tromp, J. Chem. Phys. 79, 4889 (1983)], N(E)=12(2πℏ)2 tr(F̄δ(H−E)F̄δ(H−E)), is established, and a new variant of the trace formula is presented

Cumulative Reaction Probability in Terms of Reactant-Product Wave Packet Correlation Functions

We present new expressions for the cumulative reaction probability (N(E)), cast in terms of time-correlation functions of reactant and product wave packets. The derivation begins with a standard trace expression for the cumulative reaction probability, expressed in terms of the reactive scattering matrix elements in an asymptotic internal basis. By combining the property of invariance of the trace with a wave packet correlation function formulation of reactive scattering, we obtain an expression for N(E) in terms of the correlation matrices of incoming and outgoing wave packets which are arbitrary in the internal coordinates. This formulation, like other recent formulations of N(E), allows calculation of the quantum dynamics just in the interaction region of the potential, and removes the need for knowledge of the asymptotic eigenstates. However, unlike earlier formulations, the present formulation is fully compatible with both exact and approximate methods of wave packet propagation. We illustrate this by calculating N(E) for the collinear hydrogen exchange reaction, both quantally and semiclassically. These results indicate that the use of wave packet cross-correlation functions, as opposed to a coordinate basis and flux operators, regularizes the semiclassical calculation, suggesting that the semiclassical implementation described here may be applied fruitfully to systems with more degrees of freedom

Semiclassical Nonadiabatic Dynamics with Quantum Trajectories

Dynamics based on quantum trajectories with approximate quantum potential is generalized to nonadiabatic systems and its semiclassical properties are discussed. The formulation uses the mixed polar-coordinate space representation of a wave function. The polar part describes the overall time evolution of the wave-function components semiclassically using the single-surface approximate quantum potential. The coordinate part represents a complex“population” amplitude, which in case of localized coupling can be solved for quantum mechanically in an efficient manner. In the high-energy regime this is accomplished by using a small basis determined by the coupling between surfaces. An illustration is given for a typical curve-crossing problem. The energy-resolved probabilities obtained from the time evolution of two wave packets for a wide range of energies are in excellent agreement with exact results for energies above the threshold of the diabatic reaction, including the case of total nonadiabatic transition

Semiclassical Application of the Mo/ller Operators in Reactive Scattering

Mo/ller operators in the formulation of reaction probabilities in terms of wave packet correlation functions allow us to define the wave packets in the interaction region rather than in the asymptotic region of the potential surface. We combine Mo/ller operators with the semiclassical propagator of Herman and Kluk. This does not involve further approximations and can be used with any initial value representation (IVR) semiclassical propagator. Time propagation in asymptotic regions of the potential due to Mo/ller operators reduces the oscillations of the propagator integrand and improves convergence of the results with respect to the number of trajectories. The effectiveness of Mo/ller operators for semiclassical reaction probability calculation is demonstrated for the collinear hydrogen exchange reaction. Full convergence is achieved and the number of classical trajectories is reduced by a factor of 10 compared to the calculation without Mo/ller operators

Semiclassical Calculation of Cumulative Reaction Probabilities

Calculation of chemical reaction rates lies at the very core of theoretical chemistry. The essential dynamical quantity which determines the reaction rate is the energy-dependent cumulative reaction probability, N(E), whose Boltzmann average gives the thermal rate constant, k(T). Converged quantum mechanical calculations of N(E) remain a challenge even for three- and four-atom systems, and a longstanding goal of theoreticians has been to calculate N(E) accurately and efficiently using semiclassical methods. In this article we present a variety of methods for achieving this goal, by combining semiclassical initial value propagation methods with a reactant–product wavepacket correlation function approach to reactive scattering. The correlation function approach, originally developed for transitions between asymptotic internal states of reactants and products, is here reformulated using wavepackets in an arbitrary basis, so that N(E) can be calculated entirely from trajectory dynamics in the vicinity of the transition state. This is analogous to the approaches pioneered by Miller for the quantum calculation ofN(E), and leads to a reduction in the number of trajectories and the propagation time. Numerical examples are presented for both one-dimensional test problems and for the collinear hydrogen exchange reaction