Herman-Kluk semiclassical dynamics in action-angle representation: new approaches to mapping quantum degrees of freedom

A general approach to mapping a discrete quantum mechanical problem by a continuous Hamiltonian is presented. The method is based on the representation of the quantum number by a continuous action variable that extends from -infinity to infinity. The projection of this Hilbert space onto the set of integer quantum numbers reduces the Hamiltonian to a discrete matrix of interest. The theory allows the application of the semiclassical methods to discrete quantum mechanical problems and, in particular, to problems where quantum Hamiltonians are coupled to continuous degrees of freedom. The Herman Kluk semiclassical propagation is used to calculate the nonadiabatic dynamics for a model avoided crossing system. The results demonstrate several advantages of the new theory compared to the existing mapping approaches.     

Incorporation of quantum dots into a silica matrix using a compatible precursor

A method is developed for one-stage incorporation of cadmium-sulfide quantum dots synthesized in the presence of mercaptosuccinic acid into a silica matrix formed from a precursor containing ethylene-glycol residues—tetrakis(2-hydroxyethyl) orthosilica. This precursor has not previously been applied for this purpose. It is more compatible with diverse substances than tetraethoxysilane, which is traditionally used. Moreover, it is advantageous in its unlimited solubility in water; release of ethylene glycol, which does not precipitate quantum dots, rather than alcohol upon hydrolysis; and the feasibility of performing the sol-gel process at any pH value in a range of 2–10 without the addition of acid or alkali and without heating. When the precursor (50 wt %) is added to a dispersion of quantum dots, the system is transformed into a gel in as little as a few minutes. The synthesized hybrid materials are optically transparent. Therewith, the quantum dots incorporated into the silica matrix exhibit luminescence, with their spectral characteristics remaining almost unchanged.     

Incorporation of nonadiabatic transition into wave-packet dynamics

Nonadiabatic wave-packet dynamics is factorized into purely adiabatic propagation and instantaneous localized nonadiabatic transition. A general formula is derived for the quantum-mechanical local nonadiabatic operator which is implemented within the framework of the R-matrix method. The operator can be used for incorporating the nonadiabatic transition in semiclassical wave-packet dynamics.     

Two more approaches for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics

We show two more approaches for generating trajectory-based dynamics in the phase space formulation of quantum mechanics: "equilibrium continuity dynamics" (ECD) in the spirit of the phase space continuity equation in classical mechanics, and "equilibrium Hamiltonian dynamics" (EHD) in the spirit of the Hamilton equations of motion in classical mechanics. Both ECD and EHD can recover exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Both ECD and EHD conserve the quasi-probability within the infinitesimal volume dx(t)dp(t) around the phase point (x(t), p(t)) along the trajectory. Numerical tests of both approaches in the Wigner phase space have been made for two strongly anharmonic model problems and a double well system, for each potential auto-correlation functions of both linear and nonlinear operators have been calculated. The results suggest EHD and ECD are two additional potential useful approaches for describing quantum effects for complex systems in condense phase.     

Monte Carlo and event-driven dynamics of Brownian particles with orientational degrees of freedom

Recently, a simple scaling argument was introduced that allows us to map, with some precautions, Brownian and Monte Carlo dynamics for spherical particles. Here, we extend the scaling to study systems that have orientational degrees of freedom and carefully asses its validity over a wide region of temperature and density. Our work allows us to devise a Brownian Monte Carlo algorithm that produces, to a good approximation, physically meaningful trajectories with a minimum programming effort, although at the expense of some sampling efficiency.     

An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. I. Theories

We have reformulated and generalized our recent work [J. Liu and W. H. Miller, J. Chem. Phys. 126, 234110 (2007)] into an approach for generating a family of trajectory-based dynamics methods in the phase space formulation of quantum mechanics. The approach (equilibrium Liouville dynamics) is in the spirit of Liouville's theorem in classical mechanics. The trajectory-based dynamics is able to conserve the quantum canonical distribution for the thermal equilibrium system and approaches classical dynamics in the classical (? → 0), high temperature (β → 0), and harmonic limits. Equilibrium Liouville dynamics provides the framework for the development of novel theoretical∕computational tools for studying quantum dynamical effects in large∕complex molecular systems.     

Multiple time scale molecular dynamics for fluids with orientational degrees of freedom. I. Microcanonical ensemble

We propose a new approach to eliminate the resonance instabilities inherent in multiple time step molecular dynamics simulations. The approach is developed within the microcanonical ensemble on the basis of an energy-constrained technique in the presence of orientational degrees of freedom. While the single and standard multiscale methods are restricted to small time steps of 5 and 8 fs, respectively, it is shown in simulations of water that the algorithms we have derived postpone the appearance of the instabilities to larger steps of about 16 fs. Such steps are close to the upper theoretical limit of 20 fs peculiar to the microcanonical ensemble and can be used without affecting static and dynamical properties.     

Transition rate prefactors for systems of many degrees of freedom

When a minimum on the potential energy surface is surrounded by multiple saddle points with similar energy barriers, the transition pathways with greater prefactors are more important than those that have similar energy barriers but smaller prefactors. In this paper, we present a theoretical formulation for the prefactors, computing the probabilities for transition paths from a minimum to its surrounding saddle points. We apply this formulation to a system of 2 degrees of freedom and a system of 14 degrees of freedom. The first is Brownian motion in a two-dimensional potential whose global anharmonicities play a dominant role in determining the transition rates. The second is a Lennard-Jones (LJ) cluster of seven particles in two dimensions. Low lying transition states of the LJ cluster, which can be reached directly from a minimum without passing through another minimum, are identified without any presumption of their characteristics nor of the product states they lead to. The probabilities are computed for paths going from an equilibrium ensemble of states near a given minimum to the surrounding transition states. These probabilities are directly related to the prefactors in the rate formula. This determination of the rate prefactors includes all anharmonicities, near or far from transition states, which are pertinent in the very sophisticated energy landscape of LJ clusters and in many other complex systems.     

An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. II. Thermal correlation functions

We show the exact expression of the quantum mechanical time correlation function in the phase space formulation of quantum mechanics. The trajectory-based dynamics that conserves the quantum canonical distribution-equilibrium Liouville dynamics (ELD) proposed in Paper I is then used to approximately evaluate the exact expression. It gives exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Various methods have been presented for the implementation of ELD. Numerical tests of the ELD approach in the Wigner or Husimi phase space have been made for a harmonic oscillator and two strongly anharmonic model problems, for each potential autocorrelation functions of both linear and nonlinear operators have been calculated. It suggests ELD can be a potentially useful approach for describing quantum effects for complex systems in condense phase.     

Multistate multimode vibronic dynamics: entanglement of electronic and vibrational degrees of freedom in the benzene radical cation

An earlier theoretical treatment of multimode and multistate vibronic coupling in the benzene radical cation [Koppel et al., J. Chem. Phys. 117, 2657 (2002)] is extended to investigate also the behavior of the nuclear degrees of freedom and to include additional electronic states. The five lowest doublet electronic states are considered which have been shown earlier to be all interconnected through a series of conical intersections of their potential-energy surfaces. In the most extensive calculations, they are all included simultaneously in the quantum dynamical calculations performed, which represent a system of unprecedented complexity treated in this way. The results are compared with various reduced-dimensionality treatments (i.e., employing reduced vibrational and electronic function spaces). The different temporal behavior of the various electronic populations is emphasized and traced to the different locations of the various seams of conical intersections: due to the coherent oscillations of the time-dependent wave packet this leads to an oscillatory behavior in some cases and to monotonous behavior in others. A seemingly irreversible behavior of the system dynamics in this strictly microscopic treatment is confirmed. The importance of this benchmark system to highlight complex, entangled multimode, and multistate vibronic dynamics is pointed out.     

Multiple time scale molecular dynamics for fluids with orientational degrees of freedom. II. Canonical and isokinetic ensembles

We have developed several multiple time stepping techniques to overcome the limitations on efficiency of molecular dynamics simulations of complex fluids. They include the modified canonical and isokinetic schemes, as well as the extended isokinetic Nosé-Hoover chain approach. The latter generalizes the method of Minary, Tuckerman, and Martyna for translational motion [Phys. Rev. Lett. 93, 150201 (2004)] to systems with both translational and orientational degrees of freedom. Although the microcanonical integrators are restricted to relatively small outer time steps of order of 16 fs, we show on the basis of molecular dynamics simulations of ambient water that in the canonical and isokinetic thermostats the size of these steps can be increased to 50 and 75 fs, respectively (at the same inner time step of 4 fs). Within the generalized isokinetic Nosé-Hoover chain algorithm we have derived, huge outer time steps of order of 500 fs can be used without losing numerical stability and affecting equilibrium properties.     

An alternative to the effective number of degrees of freedom

 In order to calculate the expanded uncertainty when one or more of the components of the standard uncertainty is based on a small number of degrees of freedom, the ISO Guide recommends utilising the 'effective number of degrees of freedom'. Calculating and utilising the 'effective number of degrees of freedom' to obtain a confidence interval both depend upon approximations, which, however, are adequate for determining an expanded uncertainty. This paper puts forward an alternative and simpler approximation which is also adequate but which is much easier to apply than the multi-step process given in the ISO Guide. Received: 2 July 1998 · Accepted: 12 August 1998     

Individual degrees of freedom and the solvation properties of water

Using molecular dynamics simulations in conjunction with home-developed Split Integration Symplectic Method we effectively decouple individual degrees of freedom of water molecules and connect them to corresponding thermostats. In this way, we facilitate elucidation of structural, dynamical, spectral, and hydration properties of bulk water at any given combination of rotational, translational, and vibrational temperatures. Elevated rotational temperature of the water medium is found to severely hinder hydration of polar molecules, to affect hydration of ionic species in a nonmonotonous way and to somewhat improve hydration of nonpolar species. As proteins consist of charged, polar, and nonpolar amino-acid residues, the developed methodology is also applied to critically evaluate the hypothesis that the overall decrease in protein hydration and the change in the subtle balance between hydration of various types of amino-acid residues provide a plausible physical mechanism through which microwaves enhance aberrant protein folding and aggregation.     

Incorporation dynamics of molecular guests into two-dimensional supramolecular host networks at the liquid-solid interface

The objective of this work is to study both the dynamics and mechanisms of guest incorporation into the pores of 2D supramolecular host networks at the liquid-solid interface. This was accomplished by adding molecular guests to prefabricated self-assembled porous monolayers and the simultaneous acquisition of scanning tunneling microscopy (STM) topographs. The incorporation of the same guest molecule (coronene) into two different host networks was compared, where the pores of the networks either featured a perfect geometric match with the guest (for trimesic acid host networks) or were substantially larger than the guest species (for benzenetribenzoic acid host networks). Even the moderate temporal resolution of standard STM experiments in combination with a novel injection system was sufficient to reveal clear differences in the incorporation dynamics in the two different host networks. Further experiments were aimed at identifying a possible solvent influence. The interpretation of the results is aided by molecular mechanics (MM) and molecular dynamics (MD) simulations.     

Incorporation of rotational effects into vibrational distributions in chemical reactions

Many models of chemical reactions yield only vib-translational distributions. A method by which such distributions are modified to account for a final assumed or calculated rotational distribution is presented. Two examples demonstrate the reliability of the method.     

Geometry optimization of metal complexes using natural internal coordinates: Representation of skeletal degrees of freedom

The geometry optimization using natural internal coordinates was applied for transition metal complexes. The original definitions were extended here for the skeletal degrees of freedom which are related to the translational and rotational displacements of the ηⁿ-bonded ligands. We suggest definitions for skeletal coordinates of ηⁿ-bonded small unsaturated rings and chains. The performance of geometry optimizations using the suggested coordinates were tested on various conformers of 14 complexes. Consideration was given to alternative representations of the skeletal internal coordinates, and the performance of optimization is compared. Using the skeletal internal coordinates presented here, most transition metal complexes were optimized between 10 and 20 geometry optimization cycles in spite of the usually poor starting geometry and crude approximation for the Hessian. We also optimized the geometry of some complexes in Cartesian coordinates using the Hessian from a parametrized redundant force field. We found that it took between two and three times as many iterations to reach convergence in Cartesian coordinates than using natural internal coordinates. © 1997 by John Wiley & Sons, Inc.     

Direct observation of the substitution effects on the hydrogen bridge dynamics in selected Schiff bases--a comparative molecular dynamics study

We have studied substituent effects on the properties of the intramolecular hydrogen bond of some ortho-hydroxy Schiff bases using density functional theory (DFT) based first-principle molecular dynamics (FPMD) and path integral molecular dynamics. The studied compounds possess a strong intramolecular hydrogen bond (r((O???N)) ≤ 2.6 A?), which can be tuned by substitution to either (i) enhance the basicity of the acceptor moiety by induction effects or (ii) decrease the hydrogen bond length through steric repulsion. DFT calculations and FPMD were employed to investigate structural and dynamical properties of the selected molecules, while quantum effects on the structural properties were assessed using path integral FPMD. The simulations were performed in vacuo and in the solid state to study the influence of the environment on the hydrogen bond and spectroscopic properties. We give computational support to the suggestion that induction effects are less effective to tune the intramolecular hydrogen bond properties of the discussed ortho-hydroxy Schiff bases than the steric or the environmental effects.