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.
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.
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.
A classical limit of quantum dynamics can be defined by compensation of the quantum potential in the time-dependent Schrödinger equation. The quantum potential is a non-local quantity, defined in the trajectory-based form of the Schrödinger equation, due to Madelung, de Broglie, and Bohm, which formally generates the quantum-mechanical features in dynamics. Selective inclusion of the quantum potential for the degrees of freedom deemed "quantum," defines a hybrid quantum/classical dynamics, appropriate for molecular systems comprised of light and heavy nuclei. The wavefunction is associated with all of the nuclei, and the Ehrenfest, or mean-field, averaging of the force acting on the classical degrees of freedom, typical of the mixed quantum/classical methods, is avoided. The hybrid approach is used to examine evolution of light/heavy systems in the harmonic and double-well potentials, using conventional grid-based and approximate quantum-trajectory time propagation. The approximate quantum force is defined on spatial domains, which removes unphysical coupling of the wavefunction fragments corresponding to distinct classical channels or configurations. The quantum potential, associated with the quantum particle, generates forces acting on both quantum and classical particles to describe the backreaction.
We apply the MQCB (mixed quantum/classical mechanics based on Bohmian trajectories) [E. Gindensperger, C. Meier, and J. A. Beswick, J. Chem. Phys. 113, 9369 (2000)] to the case where the quantum degree of freedom contains both bound and continuum states. The quantum degree of freedom is treated by a wave packet that is propagated on a spatial grid with a Hamiltonian that depends parametrically on the classical degrees of freedom, while the classical degrees of freedom themselves are coupled to the wave function via quantum trajectories. The method is applied to a simple model of a light particle colliding with a heavy (classical) particle absorbed on a surface, which has been used by other authors to develop methods to combine quantum degrees of freedom including continuum states with classical mechanics. The results are compared to the Ehrenfest mean-field method as well as to full quantum results.
A novel time-dependent hybrid quantum/classical propagation scheme based on Bohmian quantum trajectories is presented. The quantum subsystem is described by a wave packet depending on the quantum variables x and, via the total potential energy of the system, parametrically on the classical trajectories X(t). The wave packet is used to calculate de Broglie–Bohm quantum trajectories x(t) which are used to calculate the force acting on the classical variables. Quantum corrections of the classical equation of motion are also included. The method is applied to a simple case of two coupled oscillators. Comparison between exact quantum and approximate results demonstrates that these MQCB (Mixed Quantum/Classical Bohmian) trajectories provide a good description of the energy exchange between the two oscillators.
Development of a general approach to construction of efficient high-dimensional bases is an outstanding challenge in quantum dynamics describing large amplitude motion of molecules and fragments. A number of approaches, proposed over the years, utilize Gaussian bases whose parameters are somehow-usually by propagating classical trajectories or by solving coupled variational equations-tailored to the shape of a wave function evolving in time. In this paper we define the time-dependent Gaussian bases through an ensemble of quantum or Bohmian trajectories, known to provide a very compact representation of a wave function due to conservation of the probability density associated with each trajectory. Though the exact numerical implementation of the quantum trajectory dynamics itself is, generally, impractical, the quantum trajectories can be obtained from the wave function expanded in a basis. The resulting trajectories are used to guide compact Gaussian bases, as illustrated on several model problems.
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.