Time-dependent Self-consistent Field Approximation Research Articles

The Fate of Atomic Spin in Atomic Scattering off Surfaces.

We explore model electron dynamics of an atom scattering off a surface within the time-dependent complete active space self-consistent field (TD-CASSCF) approximation. We focus especially on the scattering of a hydrogen atom and its resulting spin dynamics starting from an initially spin-polarized state. Our results reveal competing electronic time scales that are governed by the electronic structure of the surface as well as the character of the atom. The time scales and nonadiabaticity of the dynamics are reported on by the final spin polarization of the scattered atom, which may be probed in future experiments.

Benchmark calculations for dissipative dynamics of a system coupled to an anharmonic bath with the multiconfiguration time-dependent Hartree method

In this paper, we present benchmark results for dissipative dynamics of a harmonic oscillator coupled to an anharmonic bath of Morse oscillators. The microscopic Hamiltonian has been chosen so that the anharmonicity can be adjusted as a free parameter, and its effect can be isolated. This leads to a temperature dependent spectral density of the bath, which is studied for ohmic and lorentzian cases. Also, we compare numerically exact multiconfiguration time-dependent Hartree results with approximate solutions using continuous configuration time-dependent self-consistent field and local coherent state approximation.

Generalized CC-TDSCF and LCSA: The system-energy representation

Typical (sub)system-bath quantum dynamical problems are often investigated by means of (approximate) reduced equations of motion. Wavepacket approaches to the dynamics of the whole system have gained momentum in recent years and there is hope that properly designed approximations to the wavefunction will allow one to correctly describe the subsystem evolution. The continuous-configuration time-dependent self-consistent field (CC-TDSCF) and local coherent-state approximation (LCSA) methods, for instance, use a simple Hartree product of bath single-particle-functions for each discrete variable representation (DVR) state introduced in the Hilbert space of the subsystem. Here we focus on the above two methods and replace the DVR states with the eigenstates of the subsystem Hamiltonian, i.e., we adopt an energy-local representation for the subsystem. We find that stable and semiquantitative results are obtained for a number of dissipative problems, at the same (small) computational cost of the original methods. Furthermore, we find that both methods give very similar results, thus suggesting that coherent-states are well suited to describe (local) bath states. As a whole, present results highlight the importance of the system basis-set in the selected-multiconfiguration expansion of the wavefunction. They suggest that accurate and yet computationally cheap methods may be simply obtained from CC-TDSCF/LCSA by letting the subsystem states be variationally optimized.

Variational determination of self-consistent interactions in atomic collisions

State-to-state transition amplitudes are obtained from a variational functional for two colliding atomic systems whose states are self-consistently coupled. Specifying classical center-of-mass trajectories, this approach leads to an extension of the time-dependent self-consistent field approximation which requires iterative solutions of equations for forward and backward motions in time. The variational procedure is described for trial wave functions and trial transition densities. It is briefly illustrated with a model of two interacting two-state atoms.

Quenching methods for chemical reaction dynamics within mixed quantum/classical approximation

We have derived the quenching methods, which were obtained previously based on physical intuitions (L. Wang and D. C. Clary, Chem. Phys. Lett., 1996, 262, 284; L. Wang, J. Chem. Phys., 1998, 108, 7538) from the time dependent Schrodinger equation using the quantum/classical multi-configuration time dependent self-consistent field (Q/C MCTDSCF) approximation. A theoretical ground for the quenching methods is therefore established through the present derivation. The present derivation shows that the quenching methods are just the Q/C MCTDSCF treatments. Moreover, implementations of the methods are given in detail. The relationships between the quenching methods and other relevant quantum/classical and quantum/semiclassical theories are also discussed.

Transition State Dynamics of Arn(ClHCl) (n = 0−5): Effects of Complex Formation on the Dynamics and Spectroscopy

The effects of intermolecular interactions on the transition state dynamics of the Cl + HCl reaction are investigated through time-dependent quantum simulations of the evolution of the ground state wave function for Arn(ClHCl-) on the Arn(ClHCl) potential when n = 0−5. A reduced dimensional approach has been employed in which the quantum/semiclassical time-dependent self-consistent field (Q/SC TDSCF) approximation is used to propagate the dynamics of the hydrogen and chlorine atoms, respectively, while the argon atoms are constrained to remain in the equilibrium configuration in Arn(ClHCl-). Analysis of the wave functions, obtained from these simulations, shows that as more argon atoms are introduced there is a change in the short time dynamics. When no argon atoms are present, the hydrogen atom rotates off of the Cl−Cl axis. As the number of argon atoms is increased, this motion becomes blocked, and when four or five argon atoms are present, the Cl−H−Cl subsystem is constrained to dissociate collinearly. This change in the dynamics of the hydrogen atom is reflected in an increase in the amount of energy that is transferred to the chlorine atoms as n is increased. It also results in qualitative changes in the calculated photoelectron spectra of Arn(ClHCl-).

Jastrow corrected time-dependent self-consistent field approximation

An improved approximation to the time-dependent Schrödinger equation is developed by correcting the time-dependent self-consistent field ansatz with a Jastrow prefactor defined via a set of variationally determined time-dependent parameters and a linearly independent set of prespecified spatial functions. The method is applicable in any number of dimensions, conserves norm and energy, is without parametric singularities, possesses an internal estimate of the accuracy, and has computational costs that scale algebraically with the number of degrees of freedom. The new formalism is applied to a two-dimensional double well potential to demonstrate the improved accuracy of the method. An extension of the method to electronically nonadiabatic problems is also presented.

Time-dependent studies of reaction dynamics: a test of mixed quantum/classical time-dependent self-consistent field approximations

Two approaches for applying the quantum/classical time-dependent self-consistent field approximation to collinear reaction dynamics are investigated for a generic heavy plus light–heavy (A+HB) reaction. In the first, a single quantum/classical trajectory is run, and in the second a small ensemble of quantum/classical trajectories are used. In the latter case, we derive a multiple trajectory extension to the quantum/classical TDSCF approximation starting from the Liouville–Von Neumann equation. In our formulation, the initial conditions for the classical degree of freedom are chosen from the phase space probability distribution given by the Wigner transform of the wave function that was used in the quantum simulation. Both approaches yield reaction probabilities that are in good agreement with the results of a two-dimensional quantum simulation, but the detailed dynamics are more accurately described by an ensemble average over several quantum/classical trajectories than by a single trajectory. The reasons for the relative successes of the two quantum/classical approaches will be discussed.

Comparison of self-consistent field treatments of kinetic couplings in calculations on polyatomic molecules

The accuracy of the vibrational self-consistent field (VSCF) and time-dependent self-consistent field (TDSCF) approximations for studies of vibrational dynamics of polyatomic systems is investigated for systems in which the most important off-diagonal coupling terms in the Hamiltonian are in the kinetic energy operator. To lowest order, these terms are zero in the VSCF treatment. This leads to large differences between the energies calculated using this approximation, compared to those obtained from a variational calculation. In contrast, the energy levels, obtained using the TDSCF approximation, are in good agreement with variational results. These questions are investigated using a collinear model of HCN in an internal coordinate representation.

A temperature-dependent Schrödinger equation based on a time-dependent self consistent field approximation

A temperature-dependent Schrödinger equation is proposed for the study of quantum dynamics in systems which interact with an external bath. The derivation is based on a time-dependent self consistent field (TDSCF) approximation for a system-bath wave function. As previously shown by Miller, this approximation leads to a quantum mechanical analog to the classical generalized Langevin equation. By replacing the time-evolution of the quantum mechanical bath observables with the corresponding classical trajectories, the fluctuating force and the nonlinear friction kernel in the Langevin-Schrödinger equation become temperature-dependent where the fluctuations intensity is proportional to T. Application of the new equation to the spin-boson model shows agreement with numerically exact path integral calculations for T>0. We relate the success of the TDSCF approximation in this case to the bath-induced noise which diminishes the importance of quantum mechanical system-bath correlations.

Discrete time-reversible propagation scheme for mixed quantum-classical dynamics

Abstract A time-reversible multiple time step quantum-classical propagation scheme within the time-dependent self-consistent field approximation (TDSCF) is presented. The equations of motion are obtained by transforming the time-dependent Schrodinger equation into a Hamilton-Jacobi-type formalism and describing the mixed quantum-classical time evolution by means of a Liouville operator, derived from a classical-type Hamiltonian, acting on a generalized phase space. The method is applied to a model system consisting of a harmonic oscillator non-linearly coupled to a bath of argon atoms and numerically tested with respect to norm and energy conservation and time-reversibility. The time-reversible multiple time step propagation scheme significantly contributes to the numerical stability in comparison to a non-reversible propagator with the same computational effort.

Finite-difference Heisenberg approach: an approximate treatment of many-particle systems, energy level separations and the time-dependent Schrödinger wavefunction

We extend our recently proposed finite-difference Heisenberg (FDH) approach to the quantum treatment of many-particle systems in the context of a time-dependent self-consistent field approximation. A simple method is then proposed for calculating the energy level separations directly from the matrix elements of the time-evolved operators using a time average and the velocity-dipole identity. Finally, we show that the FDH approach is not limited to the time evolution of operators, but that it also provides us, at no extra effort, with the time evolution of the Schrödinger wavefunction on a dynamic coordinate system. We illustrate our theory by applications to the multi-dimensional Hénon-Heiles model and 1D anharmonic oscillators with promising results in each case.

The validity of time-dependent self-consistent field approximation for non-stationary vibrational bound states of quantum clusters

The time-dependent self-consistent field (TDSCF) approximation is applied to the coupled-mode vibrational dynamics of highly anharmonic quantum clusters, and its validity and accuracy for such systems are tested against exact quantum calculations. The calculations are pursued for bound but non-stationary states of collinear models of (Ne)3 and B(H2)2. It is found that for a short timescale (t ⩽ 1 ps) the TDSCF approximation is in excellent accord with the exact results. This suggests that TDSCF should be an attractive quantitative approach for the short timescale vibrational dynamics of highly anharmonic quantum clusters.

Three-dimensional quantum wave packet study of the Ar–HCl photodissociation: A comparison between time-dependent self-consistent-field and exact treatments

The time-dependent self-consistent-field (TDSCF) approximation is used to study the photodissociation of the Ar–HCl cluster in a three-dimensional framework. The results are compared with numerically exact quantum calculations, and the properties and accuracy of the TDSCF approach are evaluated on this basis. The TDSCF approximation is used in Jacobi coordinates, and the total wave function is factorized into a wave packet for two coordinates associated with the H atom, and a wave packet for a single coordinate that describes the relative motion of the heavy particles. Quantitative agreement between the TDSCF and the exact results is found for most quantities calculated. The calculations show that photodissociation, and in particular the departure of the H atom is predominantly a direct process, but an appreciable amount of wave packet amplitude moving in excited state resonances is also found. This amplitude seems significantly larger than obtained in recent calculations by Schröder et al. [J. Chem. Phys. 100, 7239 (1994); Chem. Phys. Lett. 235, 316 (1995)]. The validity and computational efficiency of the TDSCF approach for realistic systems of this type is discussed.

Transition state dynamics of X + HX reactions using the time-dependent self-consistent field approximation

It has been demonstrated that the time-dependent self-consistent field approximation (TDSCF) provides an excellent approach for studying the three-dimensional quantum dynamics of ClHCl (J = 0) starting from the transition state (McCoy, A. B. et al. 1994, J. chem. Phys., 101, 1975). The question is now explored of whether this result is a consequence of the large difference between the masses of the Cl and H atoms or a fundamental property of the approximation. The quality of the results of TDSCF propagations on ClHCl is compared with that for a system in which the mass of Cl is reduced to equal that of deuterium. The lighter mass of the Cl atoms in the second system means that the wavefunction should be less well approximated by a product form than it was for ClHCl. This approximate separability of the wavefunction is the basis of TDSCF propagations. Therefore, over the 36 fs propagation times considered here, the TDSCF wavefunction for the system containing the light Cl atoms is not as accurate as that for ClHCl. Nevertheless, observable properties predicted by TDSCF propagations are in good agreement with those of numerically exact propagations for both systems.

Transition state dynamics of chemical reactions in clusters: A six-dimensional study of Ar(ClHCl)

Cluster effects on transition state dynamics are investigated through a time-dependent calculation of photodetachment spectrum of Ar(ClHCl)−. This system is studied by propagating a three-dimensional grid representation of the wave function in the H atom coordinates, coupled to three one-dimensional semiclassical Gaussians in the three lowest frequency normal coordinates of Ar(ClHCl)−. Over the 75 fs propagation times considered here, the motions of the three heavy atoms are very small. Therefore, we expect that the coupled quantum/semiclassical propagation will be in quantitative agreement with a full quantum treatment of this system. The couplings between the quantum and semiclassical degrees of freedom and between the three semiclassical degrees of freedom are introduced through the time-dependent self-consistent field (TDSCF) approximation. A computational bottleneck in applying the TDSCF approximation to such systems is that propagation of each of the modes requires the evaluation of the average of the potential over the wave functions in the three remaining modes. A solution to this problem for H transfer systems is presented and discussed. Comparisons between the dynamics of Ar(ClHCl) and ClHCl indicate that the presence of the Ar atom produces a hole in the wave function that is otherwise cylindrically symmetric. Possible observable experimental consequences of the introduction of the Ar atom to this system are discussed.

Time-dependent discrete variable representations for quantum wave packet propagation

We present an efficient method for exact wave function propagation with several degrees of freedom based on time-dependent discrete variable representations (TD-DVR) of the evolution operator. The key idea is to use basis sets that evolve in time according to appropriate reference Hamiltonians to construct TD-DVR grids. The initial finite basis representation is chosen to include the initial wavefunction and thus the evolution under the bare zeroth order Hamiltonian is described at each time by a single DVR point. For this reason TD-DVR grids offer optimal representations in time-dependent calculations, allowing significant reduction of grid size and large time steps while requiring numerical effort that (for systems with several degrees of freedom) scales almost linearly with the total grid size. The method is readily applicable to systems described by time-dependent Hamiltonians. TD-DVR grids based on the time-dependent self-consistent field approximation are shown to be very useful in the study of intramolecular or collision dynamics.

A quantitative approximation for the quantum dynamics of hydrogen transfer: Transition state dynamics and decay in ClHCl−

A general method for studying transition state spectroscopy and dynamics in hydrogen atom transfer reactions is presented. This approach is based on the time-dependent self-consistent field (TDSCF) approximation and is applied to a study of the ClHCl− photodetachment experiments of Metz et al. [Metz et al., J. Chem. Phys. 88, 1463 (1988)]. Comparison of results of exact time-dependent and TDSCF calculations are made for collinear and three-dimensional (J=0) approximations for the quantum dynamics. When ClHCl is constrained to be collinear, the TDSCF calculation overcorrelates the motions in the H atom displacement and ClCl extension coordinates. This results in relatively poor agreement with the exact result for many properties of the wave function. In contrast, when the system is propagated in the three vibrational coordinates of the system, the transition state dynamics are effectively over much more rapidly. Consequently, the TDSCF approximation yields results of very good quantitative accuracy over the time required for most of the wave function to decay off of the transition state. Comparison is also made between the wave function that results from the exact propagation and from TDSCF when the wave function in the ClCl stretch coordinate is approximated by a Gaussian wave packet. Here the magnitude of the overlap between the two TDSCF wave functions in the H atom coordinates, for quantum and semiclassical propagations of the wave function in the ClCl distance coordinate, is greater than 0.98 over the time of the propagations. These TDSCF calculations are repeated for a wave function that is approximated by a product of a two-dimensional wave function in the hydrogen atom coordinates and a one-dimensional wave function in the ClCl extension coordinate and even better quantitative agreement with the exact propagation is achieved. The success of this method for studying ClHCl gives us confidence that TDSCF will provide a general powerful tool for studies of hydrogen and proton transfer reactions in large systems.

Photodynamics of extended molecular systems. I. A theoretical approach for strongly coupled primary and secondary regions

A general time-dependent quantum mechanical approach to the interaction of visible and UV light with extended polyatomic systems is presented. It is treated as a two-step process: a photon absorption excites electronic transitions in the target system, and this is followed by a dynamical evolution of the system on the excited potential energy surface. The time evolution in a large polyatomic system is treated within a molecular time-correlation function (TCF) approach. For a general two-surface electronic excitation problem, an extension of these molecular TCFs from real to complex times is introduced to facilitate their computation. Time-dependent self-consistent field (TDSCF) equations for transition amplitudes are derived from a variational approach and are used for large polyatomic systems to factor the molecular TCFs into primary and secondary region TCFs. The primary and secondary regions are modeled by considering a general primary motion coupled to many harmonic degrees of freedom in the secondary region. In the strong coupling case, the Hamiltonian has a general dependence on the variables of the primary region, whereas it has linear and bilinear terms in the variables of the secondary region. A weak coupling limit is obtained by dropping the bilinear terms in the coupling. The complex time propagators for the secondary region dynamics are constructed analytically for both cases. The present treatment provides a quantal version of the generalized Langevin equations of molecular dynamics. It allows for the derivation of a fluctuation–dissipation theorem similar to the classical one, by introducing a displaced Boltzmann distribution and identifying dissipation and fluctuation forces within the TDSCF approximation.

Multiconfiguration time-dependent self-consistent field approximations in the numerical solution of quantum dynamical problems

The use of the time-dependent self-consistent field approximation (TDSCF) in the numerical solution of quantum curve crossing and tunneling dynamical problems is investigated. Particular emphasis is given to multiconfiguration TDSCF (MCTDSCF) approximations, which are shown to perform considerably better with only a small increase in computational effort. We investigate a number of simple models in which a “system” characterized by two electronic potential surfaces evolves while interacting with a “bath” mode described by an harmonic oscillator, and compare exact numerical solutions to one- and two-configuration TDSCF approximations. We also introduce and investigate a semiclassical approximation in which the “bath” mode is described by semiclassical wavepackets (one for each electronic state) and show that for all models investigated this scheme works very well in comparison with the fully quantum MCTDSCF approximation. This provides a potentially very useful method to simulate strongly quantum systems coupled to an essentially classical environment.