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.
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.
The computational cost of describing a general quantum system fully coupled by anharmonic interactions scales exponentially with the system size. Thus, an efficient basis representation of wave functions is essential, and when it comes to the large-amplitude motion of high-dimensional systems, the dynamic bases of Gaussian functions are often employed. The time dependence of such bases is determined from the variational principle or from classical dynamics; the former is challenging in implementation due to singular matrices, while the latter may not cover the configuration space relevant to quantum dynamics. Here we describe a method using Quantum Trajectory-guided Adaptable Gaussian (QTAG) bases "tuned"-including the basis position, phase, and width-to the wave function evolution, thanks to the continuity of the probability density in the course of the quantum trajectory dynamics. Thus, an efficient basis in configuration space is generated, bypassing the variational equations on the parameters of the Gaussians. We also propose a time propagator with basis transformation by projections which lends efficiency and stability to the QTAG dynamics, as demonstrated on standard tests and the ammonia inversion model.
In this paper, we present a variational treatment of the linear dependence for a non-orthogonal time-dependent basis set in solving the Schrödinger equation. The method is based on (i) the definition of a linearly independent working space and (ii) a variational construction of the propagator over finite time steps. The second point allows the method to properly account for changes in the dimensionality of the working space along the time evolution. In particular, the time evolution is represented by a semi-unitary transformation. Tests are carried out on a quartic double-well potential with Gaussian basis functions whose centers evolve according to classical equations of motion. We show that the resulting dynamics converges to the exact one and is unitary by construction.
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.
Bohm theory is a formulation of Quantum Mechanics that characterises the state of a quantum system according to both the wave function, as in the conventional formulation, and the coordinates (positions) of all the particles that evolve in time drawing quantum continuous trajectories. Furthermore, a statistical ensemble of all the possible trajectories, raising from the impossibility to know the initial position of all the particles, establishes the exact correspondence with the traditional Quantum Mechanics. From a computational point of view, Bohm theory has found many applications in Chemical Physics especially to develop new methodologies for solving the Schrödinger equation and to address semi-classical approximations of Quantum Mechanics. From a theoretical point of view, the most appealing feature of Bohm theory is its capability to supply a conceptual map between the quantum formalism and our representation of what a chemical system is. Chemical systems are composed of molecules, but the same idea of molecule requires a specific arrangement in the space of particles, i.e., the nuclei of the atoms. The statistical description of conventional Quantum Mechanics on the basis of wave function alone is insufficient to establish a clear correspondence with such a picture of molecules. Indeed, chemists employ usually Classical Mechanics in order to overcome this drawback of the standard quantum theory. On the other hand, if the particles position is included in the quantum formalism, as Bohm theory does, the map can be defined in a self-consistent way. In other words, Bohm theory appears to be the suitable quantum framework to represent molecules and their motion. The chemical representation of molecular systems finds a natural correspondence with a single Bohm trajectory, since it is always implicitly assumed that molecular components have specific spatial position independently of our knowledge about it. Consequently, we develop a quantum method whose fundamental assumption is that a single Bohm trajectory, i.e., a quantum molecular trajectory, describes the molecular systems and the molecular motion correctly. First of all, we examine the correspondence between a single Bohm trajectory and the conventional Quantum Mechanics, without using the ensemble of trajectories. We verify that such a correspondence exists through numerical simulations and we prove formally that the statistical properties of a single Bohm trajectory explain the probabilistic description of Quantum Mechanics. Once the consistency of this original approach has been established, we investigate the predicted properties. For instance, we take into account the constants of motion (such as the energy) corresponding to the time evolution of the coordinates and the behaviour of simple chemical systems, e.g., the vibrational motion of single molecules interacting with a resonant field. In this way, unexpected features of the molecular motion are found. Secondly, we tackle the challenge of describing many components systems (like the chemical systems in ordinary conditions). As a matter of fact, the computation of the Bohm trajectory and of the wave function is extremely demanding. However, the statistical properties of the Bohm trajectory allow the derivation of stochastic theories for examining the dynamics of open quantum systems, i.e., few molecules (or few degrees of freedom) interacting with their environment (the other molecules). One of the developed stochastic methods correlates the dynamics of the reduced density matrix, for the degrees of freedom of interest, to the evolution of the corresponding Bohm coordinates. In other words, the Bohm equation, determining the set of all the particles velocities according to the full wave function, is replaced with a stochastic one that approximates the velocity of a subset of coordinates according to the reduced density matrix. In such a way, the quantum fluctuations induced by the environment are taken into account. The advantage of this method concerns its capability of describing quantum systems, including open quantum systems, in terms of a quantum trajectory. This could allow the understanding of the molecular motion during a spectroscopical experiment. The possibility of investigating reactive systems, such as conformational changes, is particularly interesting. As a matter of fact, chemical reactions can be completely characterised only through the particles motion and we define the suit- able quantum methodology providing a self-consistent description of the molecular motion.
A hybrid method is proposed to propagate system-bath quantum dynamics that use both basis functions and coupled quantum trajectories. In it, the bath is represented with an ensemble of Bohmian trajectories while the system degrees of freedom are accounted through reduced density matrices. By retaining the Hilbert space structure for the system, the method is able to capture interference processes that are challenging to describe in Bohmian dynamics due to singularities that these processes introduce in the quantum potential. By adopting quantum trajectories to represent the bath, the method beats the exponential scaling of the computational cost with the bath size. This combination makes the method suitable for large-scale ground and excited state fully quantum molecular dynamics simulations. Equations of motion for the quantum trajectories and reduced density matrices are derived from the Schrödinger equation and a computational algorithm to solve these equations is proposed. Through computations in two-dimensional model systems, the method is shown to offer an accurate description of subsystem observables and of quantum decoherence, which is difficult to obtain when the quantum nature of the bath is ignored. The scaling of the method is demonstrated using a model with 21 degrees of freedom. The limit of independent trajectories is recovered when the mass of bath degrees of freedom is much larger than the one of the system, in agreement with mixed quantum-classical descriptions.
A hybrid method is proposed to propagate system-bath quantum dynamics that use both basis functions and coupled quantum trajectories. In it, the bath is represented with an ensemble of Bohmian trajectories while the system degrees of freedom are accounted through reduced density matrices. By retaining the Hilbert space structure for the system, the method is able to capture interference processes that are challenging to describe in Bohmian dynamics due to singularities that these processes introduce in the quantum potential. By adopting quantum trajectories to represent the bath, the method beats the exponential scaling of the computational cost with the bath size. This combination makes the method suitable for large-scale ground and excited state fully quantum molecular dynamics simulations. Equations of motion for the quantum trajectories and reduced density matrices are derived from the Schrodinger equation and a computational algorithm to solve these equations is proposed. Through computations in two-dimensional model systems, the method is shown to offer an accurate description of subsystem observables and of quantum decoherence, which is difficult to obtain when the quantum nature of the bath is ignored. The scaling of the method is demonstrated using a model with 21 degrees of freedom. The limit of independent trajectories is recovered when the mass of bath degrees of freedom is much larger than the one of the system, in agreement with mixed quantum-classical descriptions.