Electronic transitions with quantum trajectories

Effective Phase Space Representation of the Quantum Dynamics of Vibrational Predissociation of the ArBr2(B,ν =16···25) Complex

We perform trajectory-based simulations of the vibrational predissociation of the ArBr₂(B,ν=16···25) van der Waals triatomic complex, constrained to the T-shape geometry. To this aim, we employ a 2-fold mapping of the quantum dynamics into classical-like dynamics in an extended phase space. The effective phase space comprises two distinct sets of degrees of freedom, namely a collection of coupled harmonic oscillators and an ensemble of quantum trajectories. The time evolution of these variables represent bound and unbound motions of the quantum system, respectively. Quantum trajectories are propagated within the interacting trajectory representation. The comparison between the lifetimes of the predissociating complexes computed using the trajectory-based approach and the experimental results available for the target systems indicates that the present method is competitive with wavepacket propagation techniques. The competition between several simultaneous vibrational relaxation pathways was found to have a direct impact on the time scales of vibrational predissociation. Likewise, the analysis of the time evolution of the trajectories reveals the existence of regions in the effective phase space where transitions to vibrational states of higher energy are more likely to occur. The size and location of these regions influence the transient vibrational distributions and therefore the computed lifetimes. Furthermore, the mechanisms of energy redistribution along the dissociation coordinate are analyzed.

A walk through the approximations of ab initio multiple spawning

Full multiple spawning offers an in principle exact framework for excited-state dynamics, where nuclear wavefunctions in different electronic states are represented by a set of coupled trajectory basis functions that follow classical trajectories. The couplings between trajectory basis functions can be approximated to treat molecular systems, leading to the ab initio multiple spawning method which has been successfully employed to study the photochemistry and photophysics of several molecules. However, a detailed investigation of its approximations and their consequences is currently missing in the literature. In this work, we simulate the explicit photoexcitation and subsequent excited-state dynamics of a simple system, LiH, and we analyze (i) the effect of the ab initio multiple spawning approximations on different observables and (ii) the convergence of the ab initio multiple spawning results towards numerically exact quantum dynamics upon a progressive relaxation of these approximations. We show that, despite the crude character of the approximations underlying ab initio multiple spawning for this low-dimensional system, the qualitative excited-state dynamics is adequately captured, and affordable corrections can further be applied to ameliorate the coupling between trajectory basis functions.

Ultrafast dynamics induced by the interaction of molecules with electromagnetic fields: Several quantum, semiclassical, and classical approaches

Several strategies for simulating the ultrafast dynamics of molecules induced by interactions with electromagnetic fields are presented. After a brief overview of the theory of molecule-field interaction, we present several representative examples of quantum, semiclassical, and classical approaches to describe the ultrafast molecular dynamics, including the multiconfiguration time-dependent Hartree method, Bohmian dynamics, local control theory, semiclassical thawed Gaussian approximation, phase averaging, dephasing representation, molecular mechanics with proton transfer, and multipolar force fields. In addition to the general overview, some focus is given to the description of nuclear quantum effects and to the direct dynamics, in which the ab initio energies and forces acting on the nuclei are evaluated on the fly. Several practical applications, performed within the framework of the Swiss National Center of Competence in Research "Molecular Ultrafast Science and Technology," are presented: These include Bohmian dynamics description of the collision of H with H2, local control theory applied to the photoinduced ultrafast intramolecular proton transfer, semiclassical evaluation of vibrationally resolved electronic absorption, emission, photoelectron, and time-resolved stimulated emission spectra, infrared spectroscopy of H-bonding systems, and multipolar force fields applications in the condensed phase.

Quantum-Classical Nonadiabatic Dynamics: Coupled- vs Independent-Trajectory Methods

Trajectory-based mixed quantum-classical approaches to coupled electron-nuclear dynamics suffer from well-studied problems such as the lack of (or incorrect account for) decoherence in the trajectory surface hopping method and the inability of reproducing the spatial splitting of a nuclear wave packet in Ehrenfest-like dynamics. In the context of electronic nonadiabatic processes, these problems can result in wrong predictions for quantum populations and in unphysical outcomes for the nuclear dynamics. In this paper, we propose a solution to these issues by approximating the coupled electronic and nuclear equations within the framework of the exact factorization of the electron-nuclear wave function. We present a simple quantum-classical scheme based on coupled classical trajectories and test it against the full quantum mechanical solution from wave packet dynamics for some model situations which represent particularly challenging problems for the above-mentioned traditional methods.

Coupled-Trajectory Quantum-Classical Approach to Electronic Decoherence in Nonadiabatic Processes

We present a novel quantum-classical approach to nonadiabatic dynamics, deduced from the coupled electronic and nuclear equations in the framework of the exact factorization of the electron-nuclear wave function. The method is based on the quasiclassical interpretation of the nuclear wave function, whose phase is related to the classical momentum and whose density is represented in terms of classical trajectories. In this approximation, electronic decoherence is naturally induced as an effect of the coupling to the nuclei and correctly reproduces the expected quantum behavior. Moreover, the splitting of the nuclear wave packet is captured as a consequence of the correct approximation of the time-dependent potential of the theory. This new approach offers a clear improvement over Ehrenfest-like dynamics. The theoretical derivation presented in this Letter is supported by numerical results that are compared to quantum mechanical calculations.

The exact forces on classical nuclei in non-adiabatic charge transfer

The decomposition of electronic and nuclear motion presented in Abedi et al. [Phys. Rev. Lett. 105, 123002 (2010)] yields a time-dependent potential that drives the nuclear motion and fully accounts for the coupling to the electronic subsystem. Here, we show that propagation of an ensemble of independent classical nuclear trajectories on this exact potential yields dynamics that are essentially indistinguishable from the exact quantum dynamics for a model non-adiabatic charge transfer problem. We point out the importance of step and bump features in the exact potential that are critical in obtaining the correct splitting of the quasiclassical nuclear wave packet in space after it passes through an avoided crossing between two Born-Oppenheimer surfaces and analyze their structure. Finally, an analysis of the exact potentials in the context of trajectory surface hopping is presented, including preliminary investigations of velocity-adjustment and the force-induced decoherence effect.

Evaluation of the importance of spin-orbit couplings in the nonadiabatic quantum dynamics with quantum fidelity and with its efficient "on-the-fly" ab initio semiclassical approximation

We propose to measure the importance of spin-orbit couplings (SOCs) in the nonadiabatic molecular quantum dynamics rigorously with quantum fidelity. To make the criterion practical, quantum fidelity is estimated efficiently with the multiple-surface dephasing representation (MSDR). The MSDR is a semiclassical method that includes nuclear quantum effects through interference of mixed quantum-classical trajectories without the need for the Hessian of potential energy surfaces. Two variants of the MSDR are studied, in which the nuclei are propagated either with the fewest-switches surface hopping or with the locally mean field dynamics. The fidelity criterion and MSDR are first tested on one-dimensional model systems amenable to numerically exact quantum dynamics. Then, the MSDR is combined with "on-the-fly" computed electronic structure to measure the importance of SOCs and nonadiabatic couplings in the photoisomerization dynamics of CH(2)NH(2)(+) considering 20 electronic states and in the collision of F + H(2) considering six electronic states.

Measuring nonadiabaticity of molecular quantum dynamics with quantum fidelity and with its efficient semiclassical approximation

We propose to measure nonadiabaticity of molecular quantum dynamics rigorously with the quantum fidelity between the Born-Oppenheimer and fully nonadiabatic dynamics. It is shown that this measure of nonadiabaticity applies in situations where other criteria, such as the energy gap criterion or the extent of population transfer, fail. We further propose to estimate this quantum fidelity efficiently with a generalization of the dephasing representation to multiple surfaces. Two variants of the multiple-surface dephasing representation (MSDR) are introduced, in which the nuclei are propagated either with the fewest-switches surface hopping or with the locally mean field dynamics (LMFD). The LMFD can be interpreted as the Ehrenfest dynamics of an ensemble of nuclear trajectories, and has been used previously in the nonadiabatic semiclassical initial value representation. In addition to propagating an ensemble of classical trajectories, the MSDR requires evaluating nonadiabatic couplings and solving the Schrödinger (or more generally, the quantum Liouville-von Neumann) equation for a single discrete degree of freedom. The MSDR can be also used in the diabatic basis to measure the importance of the diabatic couplings. The method is tested on three model problems introduced by Tully and on a two-surface model of dissociation of NaI.

QUANTUM TRAJECTORY CALCULATIONS FOR BIPOLAR WAVEPACKET DYNAMICS IN ONE DIMENSION: SYNTHETIC SINGLE-WAVEPACKET PROPAGATION

In a previous paper [Park K, Poirier B, Parlant G, J Chem Phys129:194112, 2008], a synthetic quantum trajectory method (QTM) was successfully implemented for wave-packet dynamics in a one-dimensional (1D) symmetric Eckart barrier system, utilizing a "double-wavepacket" version of the bipolar decomposition, ψ = ψ+ + ψ- = (ψ1+ + ψ2+) + (ψ1- + ψ2-), to avoid a technical difficulty involving negligible initial ψ- density. In this paper, we develop a new synthetic algorithm which overcomes this difficulty directly, utilizing the original "single-wavepacket" version of the bipolar decomposition, ψ =ψ+ + ψ-, and also show that the initial propagation of ψ- is mainly governed by probability transfer from ψ+, rather than by the given initial conditions for ψ-. The new algorithm makes it possible to apply the synthetic bipolar QTM to asymptotically asymmetric as well as symmetric potential systems. Successful application results for both symmetric and asymmetric Eckart barrier systems in 1D are presented.

Supersymmetric approach to excited states

We present here a supersymmetric (SUSY) approach for determining excitation energies within the context of a quantum Monte Carlo scheme. By using the fact that SUSY quantum mechanics gives rises to a series of isospectral Hamiltonians, we show that Monte Carlo ground-state calculations in the SUSY partners can be used to reconstruct accurately both the spectrum and states of an arbitrary Schrodinger equation. Since the ground state of each partner potential is nodeless, we avoid any "node" problem typically associated with the Monte Carlo technique. Although we provide an example of using this approach to determine the tunneling states in a double-well potential, the method is applicable to any 1D potential problem. We conclude by discussing the extension to higher dimensions.

Description of bound reactive dynamics within the approximate quantum trajectory framework

The quantum trajectory framework incorporates quantum effects on dynamics through the quantum potential acting on a trajectory ensemble in addition to the classical potential. A global quadratic approximation to the quantum potential makes the method practical in many dimensions and captures dominant quantum effects in semiclassical systems. In this paper the approach is further developed to describe the "double well" dynamicsa prototype of the proton transfer reactionswhich exhibits the "hard" quantum effect of tunneling. Accurate description is achieved by combining the approximate quantum trajectory dynamics with the population amplitudes in the reactant and product wells. The quantum trajectory dynamics is defined by the asymptotic classical potentials. The population amplitudes represented in a small basis describe transfer between the wells. The method is exact if the reactant/product potentials are quadratic and the basis size is sufficiently large. In the semiclassical regime the trajectory dynamics is approximate and the basis size can be as small as two functions. The approach is fully compatible with the trajectory description of multidimensional systems capturing quantum tunneling along the reactive coordinate and zero-point energy flow among all degrees of freedom.

Reconciling semiclassical and Bohmian mechanics: IV. Multisurface dynamics

In previous articles (J. Chem. Phys. 2004, 121, 4501; 2006, 124, 034115; 2006, 124, 034116) a bipolar counter-propagating wave decomposition, Psi = Psi+ + Psi-, was presented for stationary states Psi of the one-dimensional Schrödinger equation, such that the components Psi+/- approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes or is wildly oscillatory. In this paper, the method is generalized for multisurface scattering applications and applied to several benchmark problems. A natural connection is established between intersurface transitions and (+ <--> -) transitions.

Quantum Trajectories from a Discrete−Variable Representation Method

A method for obtaining quantum trajectories from a discrete-variable representation computation of the quantum potential is presented. The method exploits the linearity of the Schrödinger equation, deals smoothly with the quantum potential singularities, and readily performs the time propagation up to fairly large total elapsed times. A one-dimensional test of the general n-dimensional formulation is included.

On the properties of a primitive semiclassical surface hopping propagator for nonadiabatic quantum dynamics

A previously developed nonadiabatic semiclassical surface hopping propagator [M. F. Herman J. Chem. Phys. 103, 8081 (1995)] is further studied. The propagator has been shown to satisfy the time-dependent Schrodinger equation (TDSE) through order h, and the O(h2) terms are treated as small errors, consistent with standard semiclassical analysis. Energy is conserved at each hopping point and the change in momentum accompanying each hop is parallel to the direction of the nonadiabatic coupling vector resulting in both transmission and reflection types of hops. Quantum mechanical analysis and numerical calculations presented in this paper show that the h2 terms involving the interstate coupling functions have significant effects on the quantum transition probabilities. Motivated by these data, the h2 terms are analyzed for the nonadiabatic semiclassical propagator. It is shown that the propagator can satisfy the TDSE for multidimensional systems by including another type of nonclassical trajectories that reflect on the same surfaces. This h2 analysis gives three conditions for these three types of trajectories so that their coefficients are uniquely determined. Besides the nonadiabatic semiclassical propagator, a numerically useful quantum propagator in the adiabatic representation is developed to describe nonadiabatic transitions.

Inelastic scattering matrix elements for the nonadiabatic collision B(P1∕22)+H2(Σg+1,j)↔B(P3∕22)+H2(Σg+1,j′)

Inelastic scattering matrix elements for the nonadiabatic collision B(2P1/2)+H2(1Sigmag+,j)<-->B(2P3/2)+H2(1Sigmag+,j') are calculated using the time dependent channel packet method (CPM). The calculation employs 1 2A', 2 2A', and 1 2A" adiabatic electronic potential energy surfaces determined by numerical computation at the multireference configuration-interaction level [M. H. Alexander, J. Chem. Phys. 99, 6041 (1993)]. The 1 2A' and 2 2A', adiabatic electronic potential energy surfaces are transformed to yield diabatic electronic potential energy surfaces that, when combined with the total B+H2 rotational kinetic energy, yield a set of effective potential energy surfaces [M. H. Alexander et al., J. Chem. Phys. 103, 7956 (1995)]. Within the framework of the CPM, the number of effective potential energy surfaces used for the scattering matrix calculation is then determined by the size of the angular momentum basis used as a representation. Twenty basis vectors are employed for these calculations, and the corresponding effective potential energy surfaces are identified in the asymptotic limit by the H2 rotor quantum numbers j=0, 2, 4, 6 and B electronic states 2Pja, ja=1/2, 3/2. Scattering matrix elements are obtained from the Fourier transform of the correlation function between channel packets evolving in time on these effective potential energy surfaces. For these calculations the H2 bond length is constrained to a constant value of req=1.402 a.u. and state to state scattering matrix elements corresponding to a total angular momentum of J=1/2 are discussed for j=0<-->j'=0,2,4 and 2P1/2<-->2P1/2, 2P3/2 over a range of total energy between 0.0 and 0.01 a.u.

Reconciling semiclassical and Bohmian mechanics. II. Scattering states for discontinuous potentials

In a previous paper [B. Poirier, J. Chem. Phys. 121, 4501 (2004)] a unique bipolar decomposition, psi = psi1 + psi2, was presented for stationary bound states Psi of the one-dimensional Schrodinger equation, such that the components psi1 and psi2 approach their semiclassical WKB analogs in the large action limit. Moreover, by applying the Madelung-Bohm ansatz to the components rather than to Psi itself, the resultant bipolar Bohmian mechanical formulation satisfies the correspondence principle. As a result, the bipolar quantum trajectories are classical-like and well behaved, even when psi has many nodes or is wildly oscillatory. In this paper, the previous decomposition scheme is modified in order to achieve the same desirable properties for stationary scattering states. Discontinuous potential systems are considered (hard wall, step potential, and square barrier/well), for which the bipolar quantum potential is found to be zero everywhere, except at the discontinuities. This approach leads to an exact numerical method for computing stationary scattering states of any desired boundary conditions, and reflection and transmission probabilities. The continuous potential case will be considered in a companion paper [C. Trahan and B. Poirier, J. Chem. Phys. 124, 034116 (2006), following paper].

Combining fixed- and moving-grid methods to study direct dissociation processes involving nonadiabatic transitions

We present a novel quantum-dynamics approach suitable for computing direct dissociation processes, including electronic transitions. This approach combines quantum trajectories in the Lagrangian reference frame with standard fixed-grid wave packets in order to overcome the limitations and difficulties of both techniques. As a model application, we consider the ultrafast photodissociation of H2 excited by a femtosecond extreme UV laser pulse.

Multidimensional quantum trajectories: Applications of the derivative propagation method

In a previous publication [J. Chem. Phys. 118, 9911 (2003)], the derivative propagation method (DPM) was introduced as a novel numerical scheme for solving the quantum hydrodynamic equations of motion (QHEM) and computing the time evolution of quantum mechanical wave packets. These equations are a set of coupled, nonlinear partial differential equations governing the time evolution of the real-valued functions C and S in the complex action, S=C(r,t) + iS(r,t)/Planck's over 2pi, where Psi(r,t)=exp(S). Past numerical solutions to the QHEM were obtained via ensemble trajectory propagation, where the required first- and second-order spatial derivatives were evaluated using fitting techniques such as moving least squares. In the DPM, however, equations of motion are developed for the derivatives themselves, and a truncated set of these are integrated along quantum trajectories concurrently with the original QHEM equations for C and S. Using the DPM quantum effects can be included at various orders of approximation; no spatial fitting is involved; there is no basis set expansion; and single, uncoupled quantum trajectories can be propagated (in parallel) rather than in correlated ensembles. In this study, the DPM is extended from previous one-dimensional (1D) results to calculate transmission probabilities for 2D and 3D wave packet evolution on coupled Eckart barrier/harmonic oscillator surfaces. In the 2D problem, the DPM results are compared to standard numerical integration of the time-dependent Schrodinger equation. Also in this study, the practicality of implementing the DPM for systems with many more degrees of freedom is discussed.

Modified quantum trajectory dynamics using a mixed wave function representation

Dynamics of quantum trajectories provides an efficient framework for description of various quantum effects in large systems, but it is unstable near the wave function density nodes where the quantum potential becomes singular. A mixed coordinate space/polar representation of the wave function is used to circumvent this problem. The resulting modified trajectory dynamics associated with the polar representation is nonsingular and smooth. The interference structure and the nodes of the wave function density are described, in principle, exactly in the coordinate representation. The approximate version of this approach is consistent with the semiclassical linearized quantum force method [S. Garashchuk and V. A. Rassolov, J. Chem. Phys. 120, 1181 (2004)]. This approach is exact for general wave functions with the density nodes in a locally quadratic potential.

Reconciling semiclassical and Bohmian mechanics. I. Stationary states

The semiclassical method is characterized by finite forces and smooth, well-behaved trajectories, but also by multivalued representational functions that are ill behaved at caustics. In contrast, quantum trajectory methods--based on Bohmian mechanics (quantum hydrodynamics)--are characterized by divergent forces and erratic trajectories near nodes, but also well-behaved, single-valued representational functions. In this paper, we unify these two approaches into a single method that captures the best features of both, and in addition, satisfies the correspondence principle. Stationary eigenstates in one degree of freedom are the primary focus, but more general applications are also anticipated.

A Bohmian total potential view to quantum effects. I. Methodology and simple model systems

The coherent-state wave packet dynamics of several model systems is analyzed in terms of Bohm's total potential. The quantum dynamics has been obtained by solving the time-dependent Schrodinger equation, and a method for obtaining the total potential from it, involving just matrix algebra, has been proposed. Contrary to what one may expect, it is shown that the time- and state-dependent features of the total potential admit a rationale, classical-like description of quantum effects, leading to a unified picture of them, which is not critically dependent, as for the key features, on the classical potential. An outstanding feature is found to be the relation of the state system's density amplitude and sharpness (in its dependence with position) with quantum effects. Sharp density profiles and low densities cause the total potential to strongly depart from the classical value, in both time regimes and position ranges, which provide a clearer, more deterministic view to quantum dynamics. Free motion as well as scattering processes by square and Eckart barriers have been analyzed by means of careful inspection of several time dependent snapshots. The result is an insightful picture of processes involving tunneling and antitunneling, including their dynamical variants, as well as resonances and quantization.

Bohmian dynamics on subspaces using linearized quantum force

In the de Broglie-Bohm formulation of quantum mechanics the time-dependent Schrodinger 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.

Energy conserving approximations to the quantum potential: Dynamics with linearized quantum force

Solution of the Schrodinger equation within the de Broglie-Bohm formulation is based on propagation of trajectories in the presence of a nonlocal quantum potential. We present a new strategy for defining approximate quantum potentials within a restricted trial function by performing the optimal fit to the log-derivatives of the wave function density. This procedure results in the energy-conserving dynamics for a closed system. For one particular form of the trial function leading to the linear quantum force, the optimization problem is solved analytically in terms of the first and second moments of the weighted trajectory distribution. This approach gives exact time-evolution of a correlated Gaussian wave function in a locally quadratic potential. The method is computationally cheap in many dimensions, conserves total energy and satisfies the criterion on the average quantum force. Expectation values are readily found by summing over trajectory weights. Efficient extraction of the phase-dependent quantities is discussed. We illustrate the efficiency and accuracy of the linear quantum force approximation by examining a one-dimensional scattering problem and by computing the wavepacket reaction probability for the hydrogen exchange reaction and the photodissociation spectrum of ICN in two dimensions.

A new method for solving the quantum hydrodynamic equations of motion: Application to two-dimensional reactive scattering

The de Broglie-Bohm hydrodynamic equations of motion are solved using a meshless method based on a moving least squares approach and an arbitrary Lagrangian-Eulerian frame of reference. A regridding algorithm adds and deletes computational points as needed in order to maintain a uniform interparticle spacing, and unitary time evolution is obtained by propagating the wave packet using averaged fields. The numerical instabilities associated with the formation of nodes in the reflected portion of the wave packet are avoided by adding artificial viscosity to the equations of motion. The methodology is applied to a two-dimensional model collinear reaction with an activation barrier. Reaction probabilities are computed as a function of both time and energy, and are in excellent agreement with those based on the quantum trajectory method.