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.
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.
The quantum nature of nuclei often affects molecular structure and properties, which are associated with the motion of protons and other light nuclei at low temperature. However, incorporation of the nuclear quantum effects into theoretical studies of large molecular systems is an outstanding challenge in theoretical chemistry. In this paper, the de Broglie-Bohm formulation of the time-dependent Schrödinger equation is used as a framework for the development of approximate quantum corrections to the dynamics of a trajectory ensemble, representing a time-dependent wave function. Specifically, the trajectory-centered local Least Square Fit (L-LSF) to the trajectory attributes is introduced, formally analyzed and illustrated on model potentials, typically used to benchmark approximate and semiclassical dynamics methods. The L-LSF method improves the dynamics accuracy, compared to the globally defined approximations, developed previously, and is shown to converge to the exact (generally impractical for large systems) limit of quantum dynamics.
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.
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.
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.
A method of molecular dynamics with quantum corrections, practical for studies of large molecular systems, is reviewed. The approach is based on the Bohmian formulation of the time-dependent Schrödinger equation in which a wavefunction is represented by an ensemble of interdependent trajectories. The quantum effects come from the quantum potential acting on trajectories on par with the usual classical potential. The quantum potential is determined from the evolving nuclear wavefunction, i.e. from the quantum trajectory (QT) ensemble itself. For practical and conceptual reasons the quantum potential and corresponding quantum nuclear effect are computed only for the selected light nuclei. For studies of reactive chemical processes, the classical potential is computed on-the-fly using the density functional tight binding method of electronic structure. A massively parallel implementation, based on the message passing interface allows for efficient simulations of ensembles of thousands of trajectories describing systems of up to 200 atoms. As a biochemical application, the approximate QT approach is used to model the tunnelling-dominated proton transfer in soybean-lipoxygenase-1. A materials science application is represented by a study of the nuclear quantum effect on adsorption of hydrogen and deuterium on a C37H15 molecule, which is a model ‘flake’ of graphene.
Introduction We now move to considering the dynamics of a system of nuclei and electrons. Of course, both electrons and nuclei are subject to the laws of quantum mechanics, but since nuclei are 2000 to 200,000 times heavier than electrons, we expect that classical mechanics will be a much better approximation for the motion of nuclei than for the motion of electrons. This means that we expect a level of approximation to be valid, where some of the degrees of freedom (d.o.f.) of a system behave essentially classically and others behave essentially quantum-mechanically. The system then is of a mixed quantum/classical nature. Most often the quantum subsystem consists of system of electrons in a dynamical field of classical nuclei, but the quantum subsystem may also be a selection of generalized nuclear coordinates (e.g., corresponding to high-frequency vibrations) while other generalized coordinates are supposed to behave classically, or describe the motion of a proton in a classical environment. So, in this chapter we consider the dynamics of a quantum system in a non-stationary potential. In Section 5.2 we consider the time-dependent potential as externally given, without taking notice of the fact that the sources of the time-dependent potential are moving nuclei, which are quantum particles themselves, feeling the interaction with the quantum d.o.f. Thus we consider the time evolution of the quantum system, which now involves mixing-in of excited states, but we completely ignore the back reaction of the quantum system onto the d.o.f. that cause the time-dependent potential, i.e., the moving nuclei.