Trajectory-guided configuration interaction simulations of multidimensional quantum dynamics

We propose an approach to modelling multidimensional quantum systems which uses direct-dynamics trajectories to guide wavefunction propagation. First, trajectory simulations are used to generate a sample of dynamically relevant configurations on the potential energy surface (PES). Second, the sampled configurations are used to construct an n-mode representation of the PES using a greedy algorithm. Finally, the time-dependent Schr?dinger equation is solved using a configuration interaction expansion of the wavefunction, with individual basis functions derived directly from the 1-mode contributions to the n-mode PES. This approach is successfully demonstrated by application to a 20-dimensional benchmark problem describing tunnelling in the presence of coupled degrees of freedom.     

Influence of nuclear exchange on nonadiabatic electron processes in H(+)+H2 collisions

H(+)+H(2) collisions are studied by means of a semiclassical approach that explicitly accounts for nuclear rearrangement channels in nonadiabatic electron processes. A set of classical trajectories is used to describe the nuclear motion, while the electronic degrees of freedom are treated quantum mechanically in terms of a three-state expansion of the collision wavefunction. We describe electron capture and vibrational excitation, which can also involve nuclear exchange and dissociation, in the E = 2-1000 eV impact energy range. We compare dynamical results obtained with two parametrizations of the potential energy surface of H(3)(+) ground electronic state. Total cross sections for E > 10 eV agree with previous results using a vibronic close-coupling expansion, and with experimental data for E < 10 eV. Additionally, some prototypical features of both nuclear and electron dynamics at low E are discussed.     

Hybrid quantum/classical path integral approach for simulation of hydrogen transfer reactions in enzymes

A hybrid quantum/classical path integral Monte Carlo (QC-PIMC) method for calculating the quantum free energy barrier for hydrogen transfer reactions in condensed phases is presented. In this approach, the classical potential of mean force along a collective reaction coordinate is calculated using umbrella sampling techniques in conjunction with molecular dynamics trajectories propagated according to a mapping potential. The quantum contribution is determined for each configuration along the classical trajectory with path integral Monte Carlo calculations in which the beads move according to an effective mapping potential. This type of path integral calculation does not utilize the centroid constraint and can lead to more efficient sampling of the relevant region of conformational space than free-particle path integral sampling. The QC-PIMC method is computationally practical for large systems because the path integral sampling for the quantum nuclei is performed separately from the classical molecular dynamics sampling of the entire system. The utility of the QC-PIMC method is illustrated by an application to hydride transfer in the enzyme dihydrofolate reductase. A comparison of this method to the quantized classical path and grid-based methods for this system is presented.     

A coherent state approach to semiclassical nonadiabatic dynamics

A semiclassical (SC) approximation to the quantum mechanical propagator for nonadiabatic systems is derived. Our derivation starts with an exact path integral expression that uses canonical coherent states for the nuclear degrees of freedom and spin coherent states for the electronic degrees of freedom. A stationary path approximation (SPA) is then applied to the path integral to obtain the SC approximation. The SPA results in complex classical trajectories of both nuclear and electronic degrees of freedom and a double ended boundary condition. The root search problem is solved using the previously proposed "real trajectory local search" algorithm. The SC approximation is tested on three simple one dimensional two-state systems proposed by Tully [J. Chem. Phys. 93, 1061 (1990)], and the SC results are compared to Ehrenfest and surface hopping predictions. Excellent agreement with quantum results is reached when the SC trajectory is far away from caustics. We discuss the origin of caustics in this SC formalism and the strengths and weaknesses of this approach.     

On the influence of nuclear fluctuations on calculated NMR shieldings of benzene and ethylene: a Feynman path integral–ab initio investigation*

The absolute magnetic shieldings of benzene and ethylene have been theoretically studied under the conditions of thermal equilibrium, i.e., under explicit consideration of the nuclear degrees of freedom. For this purpose we have combined the Feynman path integral quantum Monte Carlo (PIMC) formalism with the gauge‐including atomic orbital (GIAO) approach in the Hartree–Fock (HF) approximation. The HF operator has been employed to derive the NMR parameters of the two hydrocarbons via an ensemble averaging over large sets of molecular configurations that are populated in thermal equilibrium. The nuclear fluctuations are responsible for a deshielding of the nuclei relative to the shieldings at the vibrationless minimum of the potential energy surface (PES). The influence of the nuclear degrees of freedom is largest for the isotropic part of the ¹³C shielding tensor. The theoretical results can be explained on the basis of simple geometrical considerations. The bond lengths in thermal equilibrium are larger than the bond lengths at the minimum of the PES. This length enhancement is the prerequisite for a deshielding of the nuclei in thermal equilibrium. The vibrational corrections of the nuclear magnetic resonance (NMR) parameters of benzene and ethylene are quantum driven; classical thermal degrees of freedom of the nuclei are of minor importance. Conceptual problems of theoretical studies of NMR parameters on the basis of a single molecular geometry are emphasized. The influence of the spatial uncertainty of the nuclei becomes decisive in molecules with light atoms. It is pointed out that the combination of the PIMC formalism with electronic Hamiltonians of state‐of‐the‐art quality renders possible accurate determinations of NMR parameters. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem 86: 280–296, 2002     

Quantum wave packet ab initio molecular dynamics: an approach to study quantum dynamics in large systems

A methodology to efficiently conduct simultaneous dynamics of electrons and nuclei is presented. The approach involves quantum wave packet dynamics using an accurate banded, sparse and Toeplitz representation for the discrete free propagator, in conjunction with ab initio molecular dynamics treatment of the electronic and classical nuclear degree of freedom. The latter may be achieved either by using atom-centered density-matrix propagation or by using Born-Oppenheimer dynamics. The two components of the methodology, namely, quantum dynamics and ab initio molecular dynamics, are harnessed together using a time-dependent self-consistent field-like coupling procedure. The quantum wave packet dynamics is made computationally robust by using adaptive grids to achieve optimized sampling. One notable feature of the approach is that important quantum dynamical effects including zero-point effects, tunneling, as well as over-barrier reflections are treated accurately. The electronic degrees of freedom are simultaneously handled at accurate levels of density functional theory, including hybrid or gradient corrected approximations. Benchmark calculations are provided for proton transfer systems and the dynamics results are compared with exact calculations to determine the accuracy of the approach.     

Simulating quantum dynamics in classical environments

Methods for simulating the dynamics of composite systems, where part of the system is treated quantum mechanically and its environment is treated classically, are discussed. Such quantum–classical systems arise in many physical contexts where certain degrees of freedom have an essential quantum character while the other degrees of freedom to which they are coupled may be treated classically to a good approximation. The dynamics of these composite systems are governed by a quantum–classical Liouville equation for either the density matrix or the dynamical variables which are operators in the Hilbert space of the quantum subsystem and functions of the classical phase space variables of the classical environment. Solutions of the evolution equations may be formulated in terms of surface-hopping dynamics involving ensembles of trajectory segments interspersed with quantum transitions. The surface-hopping schemes incorporate quantum coherence and account for energy exchanges between the quantum and classical degrees of freedom. Various simulation algorithms are discussed and illustrated with calculations on simple spin-boson models but the methods described here are applicable to realistic many-body environments.     

Semiclassical calculation of the vibrational echo

The infrared echo measurement probes the time scales of the molecular motions that couple to a vibrational transition. Computation of the echo observable within rigorous quantum mechanics is problematic for systems with many degrees of freedom, motivating the development of semiclassical approximations to the nonlinear optical response. We present a semiclassical approximation to the echo observable, based on the Herman-Kluk propagator. This calculation requires averaging over a quantity generated by two pairs of classical trajectories and associated stability matrices, connected by a pair of phase-space jumps. Quantum, classical, and semiclassical echo calculations are compared for a thermal ensemble of noninteracting anharmonic oscillators. The semiclassical approach uses input from classical mechanics to reproduce the significant features of a complete, quantum mechanical calculation of the nonlinear response.     

A semiclasical justification for the use of non-spreading wavepackets in dynamics calculations

A justification is given for the use of non-spreading or frozen gaussian packets in dynamics calculations. In this work an initial wavefunction or quantum density operator is expanded in a complete set of grussian wavepackets. It is demonstrated that the time evolution of this wavepacket expansion for the quantum wavefunction or density is correctly given within the approximations employed by the classical propagation of the avarage position and momentum of each gaussian packet, holding the shape of these individual gaussians fixed. The semiclassical approximation is employed for the quantum propagator and the stationary phase approximation for certain integrals is utilized in this derivation. This analysis demonstrates that the divergence of the classical trajectories associated with the individual gaussian packets accounts for the changes in shape of the quantum wavefunction or density, as has been suggested on intuitive grounds by Heller. The method should be exact for quadratic potentials and this is verified by explicitly applying it for the harmonic oscillator example.     

Elastic collisions of heavy ions at intermediate energies

The classical scattering cross section of two colliding nuclei is evaluated. Special attention is given to the influence of relativistic and magnetic field effects as well as retardation corrections on elastic trajectories of heavy ions. An analytical treatment is presented for a light particle impinging on a heavy target.     

Non-adiabatic molecular dynamics with quantum solvent effects

Three novel approaches extending quantum-classical non-adiabatic (NA) molecular dynamics (MD) to include quantum effects of solvent environments are described. In a standard NA-MD the solute subsystem is treated quantum mechanically, while the larger solvent part of a system is treated classically. The three novel approaches presented here are based on the Bohmian formulation of quantum mechanics, the stochastic Schrödinger equation for the evolution of open quantum systems and the quantized Hamilton dynamics generalization of classical mechanics. The approaches extend the standard NA-MD to incorporate the following quantum effects of the solvent. (1) Branching, i.e. the ability of solvent quantum wave packets to split and follow asymptotically diverging trajectories correlated with different quantum states of the solute. (2) Decoherence, i.e. loss of quantum interference within the solute subsystem induced by the diverging solvent trajectories. (3) Zero point energy that contributes to NA coupling and must be preserved during the energy exchange between solvent and solute degrees of freedom. The Bohmian quantum-classical mechanics, stochastic mean-field and quantized mean-field approximations incorporate the quantum solvent effects into the standard quantum-classical NA-MD in a straightforward and efficient way that can be easily applied to quantum dynamics of condensed phase chemical systems.     

Analysis of the nuclear-electronic orbital method for model hydrogen transfer systems

Fundamental issues associated with the application of the nuclear-electronic orbital (NEO) approach to hydrogen transfer systems are addressed. In the NEO approach, specified nuclei are treated quantum mechanically on the same level as the electrons, and mixed nuclear-electronic wavefunctions are calculated with molecular orbital methods. The positions of the nuclear basis function centers are optimized variationally. In the application of the NEO approach to hydrogen transfer systems, the hydrogen nuclei and all electrons are treated quantum mechanically. Within the NEO framework, the transferring hydrogen atom can be represented by two basis function centers to allow delocalization of the proton vibrational wavefunction. In this paper, the NEO approach is applied to the [He-H-He]+ and [He-H-He]++ model systems. Analyses of technical issues pertaining to flexibility of the basis set to describe both single and double well proton potential energy surfaces, linear dependency of the hydrogen basis functions, multiple minima in the basis function center optimization, convergence of the number of hydrogen basis function centers, and basis set superposition error are presented. The accuracy of the NEO approach is tested by comparison to grid calculations for these model systems.     

Alternative wavefunction ansatz for including explicit electron-proton correlation in the nuclear-electronic orbital approach

The nuclear-electronic orbital (NEO) approach treats specified nuclei quantum mechanically on the same level as the electrons with molecular orbital techniques. The explicitly correlated Hartree-Fock (NEO-XCHF) approach was developed to incorporate electron-nucleus dynamical correlation directly into the variational optimization of the nuclear-electronic wavefunction. In the original version of this approach, the Hartree-Fock wavefunction is multiplied by (1+G?), where G? is a geminal operator expressed as a sum of Gaussian type geminal functions that depend on the electron-proton distance. Herein, a new wavefunction ansatz is proposed to avoid the computation of five- and six-particle integrals and to simplify the computation of the lower dimensional integrals involving the geminal functions. In the new ansatz, denoted NEO-XCHF2, the Hartree-Fock wavefunction is multiplied by √(1+G?) rather than (1+G?). Although the NEO-XCHF2 ansatz eliminates the integrals that are quadratic in the geminal functions, it introduces terms in the kinetic energy integrals with no known analytical solution. A truncated expansion scheme is devised to approximate these problematic terms. An alternative hybrid approach, in which the kinetic energy terms are calculated with the original NEO-XCHF ansatz and the potential energy terms are calculated with the NEO-XCHF2 ansatz, is also implemented. Applications to a series of model systems with up to four electrons provide validation for the NEO-XCHF2 approach and the treatments of the kinetic energy terms.     

Six-dimensional quantum dynamics of H2 dissociative adsorption on the Pt(211) stepped surface

Results of experimental studies, and theoretical calculations utilizing classical trajectories, have shown that dissociation of H2 on the Pt(211) stepped surface is enhanced at low energies by a molecular trapping mechanism. Because quantum effects can play a large role at the low energies and long lifetimes that characterize molecular trapping, we have undertaken quantum dynamics calculations for this system, the first to treat all molecular degrees of freedom of a gas molecule reacting on a stepped metallic surface. The calculations show that molecular trapping persists in the quantum system, but only at much lower energies than experimentally seen, pointing to possible deficiencies in the potential energy surface. Classical and quasiclassical trajectory calculations on the same potential provide a reasonable picture of reaction overall, but many of the finer details are inaccurate, and certain classical reaction mechanisms are entirely invalid. We conclude that some skepticism should be shown toward any classical study for which long-lived trapping states play a role.