In this contribution, we expose a new mixed quantum-classical reaction dynamics method whosenovelty is that it is entirely based on trajectories. Our approach rests on the concept of quantumtrajectory, which has been long been considered a philosophical and interpretative tool for quantummechanics [1]. Recently, quantum trajectories have been “rediscovered” with the goal of developinga totally new and unconventional computational method for doing quantum reaction dynamics [2,3].The most recent formulation, due to Bill Poirier and coworkers [4-6], sums up as a system ofcoupled ordinary differential equations (ODEs), that can be propagated in an extremely stable andaccurate manner. Potential advantages of this method are easily seen: (i) accurate propagation canbe extended to very large internuclear distances at low temperature; (ii) the propagation of onequantum degree of freedom together with several—possibly many—classical degrees of freedomsimply amounts to solving a “cheap” classical-like ODE system.Here we elaborate on our companion communication “Quantum trajectory capture at low and ultralowtemperature”. The above quantum trajectory approach is introduced and its physical as well ascomputational advantages are detailed, with special emphasis on the capture process.
In this contribution, we expose a new mixed quantum-classical reaction dynamics method whosenovelty is that it is entirely based on trajectories. Our approach rests on the concept of quantumtrajectory, which has been long been considered a philosophical and interpretative tool for quantummechanics [1]. Recently, quantum trajectories have been “rediscovered” with the goal of developinga totally new and unconventional computational method for doing quantum reaction dynamics [2,3].The most recent formulation, due to Bill Poirier and coworkers [4-6], sums up as a system ofcoupled ordinary differential equations (ODEs), that can be propagated in an extremely stable andaccurate manner. Potential advantages of this method are easily seen: (i) accurate propagation canbe extended to very large internuclear distances at low temperature; (ii) the propagation of onequantum degree of freedom together with several—possibly many—classical degrees of freedomsimply amounts to solving a “cheap” classical-like ODE system.Here we elaborate on our companion communication “Quantum trajectory capture at low and ultralowtemperature”. The above quantum trajectory approach is introduced and its physical as well ascomputational advantages are detailed, with special emphasis on the capture process.
The classical capture model for chemical reactions assumes that all trajectories with enough energyto cross the (centrifugal) barrier in the effective potential actually do so, and then proceed to formthe products. However, wave effects (not accounted for in a classical approach, by definition)become increasingly important with decreasing temperature and may dominate the collisionalbehavior at ultra-low temperatures.In this contribution we propose a new approach that describes the capture process by evolvingquantum trajectories [1,2] in the reactant channel up to a given limit, the capture distance, on theinside of the centrifugal barrier. The quantum trajectories take full account of quantum effects alongthe reaction path direction in the entrance channel. In particular, tunneling and quantum reflection,which can change the reaction rate by orders of magnitude, are accurately computed. In addition,quantum trajectories give valuable physical insight into the quantum capture process all along theapproach of the reactants. They also provide a technical means to determine the optimum capturedistance.Moreover, quantum trajectories along the reactant approach can easily be coupled with classicaltrajectories that drive the rest of the degrees of freedom of the system [3]. Apart from the captureprocess, which is guided by a quantum trajectory, all other coordinates are treated classically [3].This mixed quantum-classical capture approach is thus highly “classical-like” [3] and is aimed atlarge system reactions with quantum effects, still out of reach for current quantum scattering codes.Preliminary results will be presented at the conference.The authors are very grateful to Bill Poirier for his ongoing encouragement.[1] B. Poirier, “Bohmian mechanics without pilot waves”, Chem. Phys. 370, 4 (2010)[2] J. Schiff and B. Poirier, “Quantum mechanics without wavefunctions”, J. Chem. Phys. 136,
While direct reactions with activation barriers are now routinely calculated, reactions going through a longlivedintermediate complex are much more difficult to study[1, 2]. Complex-forming reactions are often barrierlessand are thus relevant to the field of cold and utra-cold chemistry [2]. With decreasing temperature, wave effectsbecome increasingly important and may dominate the collisional behavior at ultralow temperatures. Capture theories(close-coupling expansion with boundary conditions applied in the reactant channel) are often used to studycomplex-forming reactions [3, 4].The Langevin capture model is often used to describe barrierless reactive collisions. At very low temperature,quantum effects may alter this simple capture image and dramatically affect the reaction probability. In this talk,we use the trajectory-ensemble reformulation of quantum mechanics without wavefunctions recently proposed byPoirier and coworkers[5, 6] to compute adiabatic-channel capture probabilities and cross-sections for the reactionLi + CaH(v = 0; j = 0)!LiH + Ca at low and ultra-low temperatures. The captured quantum trajectory takes fullaccount of tunneling and quantum reflection along the radial collision coordinate. Our approach turns out to bevery fast and accurate, down to extremely low temperatures.
The Langevin capture model is often used to describe barrierless reactive collisions. At very low temperatures, quantum effects may alter this simple capture image and dramatically affect the reaction probability. In this paper, we use the trajectory-ensemble reformulation of quantum mechanics, as recently proposed by one of the authors (Poirier) to compute adiabatic-channel capture probabilities and cross-sections for the highly exothermic reaction Li + CaH(v = 0, j = 0) → LiH + Ca, at low and ultra-low temperatures. Each captured quantum trajectory takes full account of tunneling and quantum reflection along the radial collision coordinate. Our approach is found to be very fast and accurate, down to extremely low temperatures. Moreover, it provides an intuitive and practical procedure for determining the capture distance (i.e., where the capture probability is evaluated), which would otherwise be arbitrary.
The method, described in the paper, incorporates ideas of the Bohmian formulation of quantum mechanics into a trajectory approach designed for systems comprised of light (quantum) and heavy (nearly classical) particles. The goal is to treat all degrees of freedom on the same footing but, for practical reasons, to include the quantum-mechanical effects into dynamics for just the light particles. The total wavefunction, taken as a product of slow and fast components representing the heavy and light particles respectively is represented in terms of trajectories. The fast component depends on the trajectory-guided configurations of the heavy particles. The initial conditions for the trajectories are defined by the full-dimensional initial wavefunction according to Bohm's prescription of associating the trajectory momentum with the gradient of the wavefunction phase. The time-dependent Schrödinger equation is solved for the quantum degrees of freedom using an ensemble of the quantum trajectories (or other time-propagation method) for each guiding trajectory of the classical subspace. Neglect of the quantum force acting on the heavy particles allows decoupling of the quantum calculations associated with different guiding trajectories. The approach allows reconstruction of the full-dimensional wavefunction and conserves the total wavefunction energy as illustrated for the vibrationally non-adiabatic model.
In this chapter we formulate quantum trajectory dynamics in imaginary time which seamlessly connects to Bohmian dynamics in real time. Approximate determination of the quantum potential and mixed representation of wavefunctions with nodes make this trajectory approach stable and scalable to high dimensions, while including the T&F Cat # K11224, Chapter 23, Page 362, 15-9-2010 leading quantum effects into wavefunction evolution. Reaction probabilities, energy eigenstates and reaction rate constants can be obtained by propagating an ensemble of quantum trajectories in real and/or imaginary time.
Chemical dynamics, in principle, should be understood by solving the time-dependent Schrödinger equation for a molecular system, describing motion of the nuclei and electrons. However, the computational efforts to solve this partial second-order differential equation scales exponentially with the system size, which prevents us from getting exact numerical solutions for systems larger than 4-5 atoms. Thus, approximations simplifying the picture are necessary. The so-called Born-Oppenheimer approximation, separating motion of the electrons and nuclei is the central one: solution to the electronic Schrödinger equation defines the potential energy surface on which the nuclear motion unfolds, and there are standard quantum chemistry software packages for solving the electronic Schrödinger equation. For the nuclear Schrödinger equation, however, there are no widely applicable quantum-mechanical approaches, and most simulations are performed using classical Newtonian mechanics which is often adequate due to large nuclear masses. However, the nuclear quantum effects are significant for chemical processes involving light nuclei at low energies, and including these effects into simulation, even approximately, is highly desirable. In this dissertation, an approximate methodology of including quantum-mechanical effects within the quantum trajectory or the de Broglie-Bohm formulation of the Schrödinger equations is developed. Use of the trajectory framework makes the approach scalable to hundreds of degrees of freedom. The methodology is applied to study high-dimensional systems (solid He4 and others) relevant to chemistry.