Publications
Preprints
Publications
2024

(2024) Journal of Chemical Physics. 161, 3, 036101. Abstract
The modified version of second and fourth order vibrational perturbation theory, whereby the Euclidean action for tunneling is computed on the inverted potential at a shifted energy that is ℏ^{2} dependent, is applied to a symmetric double well quartic potential. The mean energies of the doublets in each well are also computed using vibrational perturbation theory. Results show that the modified vibrational perturbation theory significantly improves the estimates of tunneling splitting energies both for the ground state and for excited state doublets.

(2024) Journal of Chemical Physics. 160, 18, 184110. Abstract
The combination of vibrational perturbation theory with the replacement of the harmonic oscillator quantization condition along the reaction coordinate with an imaginary action to be used in the uniform semiclassical approximation for the transmission probability has been shown in recent years to be a practical method for obtaining thermal reaction rates. To date, this theory has been developed systematically only up to second order in perturbation theory. Although it gives the correct leading order term in an ℏ^{2} expansion, its accuracy at lower temperatures, where tunneling becomes important, is not clear. In this paper, we develop the theory to fourth order in the action. This demands developing the quantum perturbation theory up to sixth order. Remarkably, we find that the fourth order theory gives the correct ℏ^{4} term in the expansion of the exact thermal rate. The relative magnitude of the fourth order correction as compared to the second order term objectively indicates the accuracy of the second order theory. We also extend the previous modified second order theory to the fourth order case, creating an ℏ^{2} modified potential for this purpose. The resulting theory is tested on the standard examples—symmetric and asymmetric Eckart potentials and a Gaussian potential. The modified fourth order theory is remarkably accurate for the asymmetric Eckart potential.

(2024) Journal of Physical Chemistry A. 128, 17, p. 34343448 Abstract
The uniform semiclassical expression for the energydependent transmission probability through a barrier has been a staple of reaction rate theory for almost 90 years. Yet, when using the classical Euclidean action, the transmission probability is identical to 1/2 when the energy equals the barrier height since the Euclidean action vanishes at this energy. This result is generally incorrect. It also leads to an inaccurate estimate of the leading order term in an ℏ^{2n} expansion of the thermal transmission coefficient. The central result of this paper is that adding an ℏ^{2} dependent correction to the uniform semiclassical expression, whether as a constant action or as a shift in the energy scale, not only corrects this inaccuracy but also leads to a theory that is more accurate than the previous one for almost any energy. Shifting the energy scale is a generalization of the vibrational perturbation theory 2 (VPT2) and is much more accurate than the “standard” VPT2 theory, especially when the potential is asymmetric. Shifting the action by a constant is a generalization of a result obtained by Yasumori and Fueki (YF) only for the Eckart barrier. The resulting modified VPT2 and YF semiclassical theories are applied to the symmetric and asymmetric Eckart barrier, a Gaussian barrier, and a tanh barrier. The onedimensional theories are also generalized to manydimensional systems. Their effect on the thermal instanton theory is discussed.

(2024) Journal of Chemical Physics. 160, 15, 150902. Abstract
Reaction rate theory has been at the center of physical chemistry for well over one hundred years. The evolution of the theory is not only of historical interest. Reliable and accurate computation of reaction rates remains a challenge to this very day, especially in view of the development of quantum chemistry methods, which predict the relevant force fields. It is still not possible to compute the numerically exact rate on the fly when the system has more than at most a few dozen anharmonic degrees of freedom, so one must consider various approximate methods, not only from the practical point of view of constructing numerical algorithms but also on conceptual and formal levels. In this Perspective, I present some of the recent analytical results concerning leading order terms in an ℏ^{2m} series expansion of the exact rate and their implications on various approximate theories. A second aspect has to do with the crossover temperature between tunneling and thermal activation. Using a uniform semiclassical transmission probability rather than the “primitive” semiclassical theory leads to the conclusion that there is no divergence problem associated with a “crossover temperature.” If one defines a semiclassical crossover temperature as the point at which the tunneling energy of the instanton equals the barrier height, then it is a factor of two higher than its previous estimate based on the “primitive” semiclassical approximation. In the low temperature tunneling regime, the uniform semiclassical theory as well as the “primitive” semiclassical theory were based on the classical Euclidean action of a periodic orbit on the inverted potential. The uniform semiclassical theory wrongly predicts that the “halfpoint,” which is the energy at which the transmission probability equals 1/2, for any barrier potential, is always the barrier energy. We describe here how augmenting the Euclidean action with constant terms of order ℏ^{2} can significantly improve the accuracy of the semiclassical theory and correct this deficiency. This also leads to a deep connection with and improvement of vibrational perturbation theory. The uniform semiclassical theory also enables an extension of the quantum version of Kramers’ turnover theory to temperatures below the “crossover temperature.” The implications of these recent advances on various approximate methods used to date are discussed at length, leading to the conclusion that reaction rate theory will continue to challenge us both on conceptual and practical levels for years to come.

(2024) Physical Review A. 109, 2, 022242. Abstract
The quantum version of Kramers turnover theory is generalized beyond the parabolic barrier approximation. The result is a uniform instantonbased quantum Kramers turnover theory that does not display any divergence at what is known as the crossover temperature. The theory is analyzed using a model of a particle trapped in a cubic potential. As the temperature is lowered, the maximum in the Kramers turnover curve moves to lower friction values. When the temperature is sufficiently low, the quantum rate at low friction becomes almost independent of the friction strength.
2023

(2023) Physical Chemistry Chemical Physics. 25, p. 3319833202 Abstract
In this Reply, we show that criticisms of perturbation theory for grazingincidence fastatom diffraction (GIFAD) are illfounded. We show explicitly that our formulation (W. Allison, S. MiretArtés and E. Pollak, Phys. Chem. Chem. Phys., 2022, 24, 15851) provides a similar precision in describing the observed phenomena as ab initio potentials. Since that is the main criterion to distinguish between methods, it seems reasonable to conclude that the perturbation approach using a Morsetype potential reproduces the essential aspects of the dynamics correctly. In addition we expand on the historical context and summarize the physical insights provided by our methods.

(2023) Journal of Chemical Physics. 159, 22, 224107. Abstract
Instantonbased rate theory is a powerful tool that is used to explore tunneling in manydimensional systems. Yet, it diverges at the socalled \u201ccrossover temperature.\u201d Using the uniform semiclassical transmission probability of Kemble [Phys. Rev. 48, 549 (1935)], we showed recently that in one dimension, one might derive a uniform semiclassical instanton rate theory, which has no divergence. In this paper, we generalize this uniform theory to manydimensional systems. The resulting theory uses the same input as in the previous instanton theory, yet does not suffer from the divergence. The application of the uniform theory to dissipative systems is considered and used to revise Wolynes\u2019 wellknown analytical expression for the rate [P. G. Wolynes, Phys. Rev. Lett. 47, 968 (1981)] so that it does not diverge at the \u201ccrossover temperature.\u201d

(2023) Journal of Physical Chemistry Letters. 14, 44, p. 98929899 Abstract
The instanton expression for the thermal transmission probability through a onedimensional barrier is derived by using the uniform semiclassical energydependent transmission coefficient of Kemble. The resulting theory does not diverge at the \u201ccrossover temperature\u201d but changes smoothly. The temperaturedependent energy of the instanton is the same as the barrier height when ℏβω^{\u2021} = π and not 2π as in the \u201cstandard\u201d instanton theory. The concept of a crossover temperature between tunneling and thermal activation, based on the divergence of the instanton rate, is obsolete. The theory is improved by assuring that at high energy when the energydependent transmission coefficient approaches unity the integrand decays exponentially as dictated by the Boltzmann factor and not as a Gaussian. This ensures that at sufficiently high temperatures the uniform theory reduces to the classical. Application to Eckart barriers demonstrates that the uniform theory provides a good estimate of the numerically exact result over the whole temperature range.

(2023) Physical Review A. 108, 3, 036201. Abstract
The central claim of Gavassino and Disconzi [Gavassino and Disconzi, Phys. Rev. A 107, 032209 (2023)2469992610.1103/PhysRevA.107.032209] that relativistic quantum tunneling is an entirely subluminal process is shown to be incorrect. The Hartman effect can lead to superluminal tunneling, but not to superluminal signaling.

(2023) ChemPhysChem. e202300272. Abstract
In this short review, we provide an update of recent developments in Kramers\u2019 theory of reaction rates. After a brief introduction stressing the importance of this theory initially developed for chemical reactions, we briefly present the main theoretical formalism starting from the generalized Langevin equation and continue by showing the main points of the modern Pollak, Grabert and Hänggi theory. Kramers\u2019 theory is then sketched for quantum and classical surface diffusion. As an illustration the surface diffusion of Na atoms on a Cu(110) surface is discussed showing escape rates, jump distributions and diffusion coefficients as a function of reduced friction. Finally, some very recent applications of turnover theory to different fields such as nanoparticle levitation, microcavity polariton dynamics and simulation of reaction in liquids are presented. We end with several open problems and future challenges faced up by Kramers turnover theory.

(2023) Physical Review A. 107, 2, 022203. Abstract
Nine decades after Wigner's formulation of quantum rate theory, his celebrated result was recently generalized to the asymmetric barrier using an exact firstorder expansion of the transmission probability in terms of ℏ2. This paper extends the firstorder quantum correction to secondorder correction of order ℏ4 for the thermally averaged transmission probability through an arbitrary barrier. The derivation employs a systematic expansion of the projection operator onto products and the thermal distribution which involves a Taylor expansion of the potential about the barrier up to eighth order. The resulting exact analytical expression is calibrated with numerical calculations of several model potentials and shows excellent agreement when the ℏ4 term is included. In comparison, the semiclassical transition state theory cannot reproduce the correct ℏ4 terms when the anharmonicity is treated on the level of VPT4 (vibrational perturbation theory  fourth order) and will potentially need a VPT6 expansion. Further analysis of the quartic barrier reveals suppressed transmission due to the dominant role of quantum reflection above the barrier. These results not only provide a conceptual framework but can also be applied to heavy atom tunneling and machine learning.

(2023) Physical review. A. 107, 1, 012204. Abstract
The development of computational resources has made it possible to determine upper bounds for atomic and molecular energies with high precision. Yet, error bounds to the computed energies have been available only as estimates. In this paper, the PollakMartinazzo lowerbound theory, in conjunction with correlated Gaussian basis sets, is elaborated and implemented to provide subpartspermillion convergence of the ground and excitedstate energies for the He, Li, and Be atoms. The quality of the lower bounds is comparable to that of the upper bounds obtained from the Ritz method. These results exemplify the power of lower bounds to provide tight estimates of atomic energies.
2022

(2022) The journal of physical chemistry letters. 13, 45, p. 1055810566 Abstract
Quantum tunneling is known to play an important role in the dynamics of systems with nonadiabatic couplings. However, until recently, the timedomain properties of nonadiabatic scattering have been severely underexplored. Using numerically exact quantum methods, we study the impact that nonadiabatic couplings have on the time it takes to tunnel through a barrier. We find that the Wigner phase time is the appropriate measure to use when determining the tunneling flight time also when considering nonadiabatic systems. The central result of the present study is that in an avoided crossing system in one dimension, the nonadiabatic couplings speed up the tunneling event, relative to the adiabatic case in which all nonadiabatic coupling is ignored. This has implications for both the study of quantum tunneling times and for the field of nonadiabatic scattering and chemistry.

(2022) The journal of physical chemistry. B. 126, 40, p. 79667974 Abstract
Singlemolecule experiments have now achieved a time resolution allowing observation of transition paths, the brief trajectory segments where the molecule undergoing an unfolding or folding transition enters the energetically or entropically unfavorable barrier region from the folded/unfolded side and exits to the unfolded/folded side, thereby completing the transition. This resolution, however, is yet insufficient to identify the precise entrance/exit events that mark the beginning and the end of a transition path: the nature of the diffusive dynamics is such that a molecular trajectory will recross the boundary between the barrier region and the folded/unfolded state, multiple times, at a time scale much shorter than that of the typical experimental resolution. Here we use theory and Brownian dynamics simulations to show that, as a result of such recrossings, the apparent transition path times are generally longer than the true ones. We quantify this effect using a simple model where the observed dynamics is a moving average of the true dynamics and discuss experimental implications of our results.

(2022) Physical chemistry chemical physics : PCCP. 24, 41, p. 2537325382 Abstract
Experimentally measured transition path time distributions are usually analyzed theoretically in terms of a diffusion equation over a free energy barrier. It is though well understood that the free energy profile separating the folded and unfolded states of a protein is characterized as a transition through many stable microstates which exist between the folded and unfolded states. Why is it then justified to model the transition path dynamics in terms of a diffusion equation, namely the Smoluchowski equation (SE)? In principle, van Kampen has shown that a nearest neighbor Markov chain of thermal jumps between neighboring microstates will lead in a continuum limit to the SE, such that the friction coefficient is proportional to the mean residence time in each microstate. However, the practical question of how many microstates are needed to justify modeling the transition path dynamics in terms of an SE has not been addressed. This is a central topic of this paper where we compare numerical results for transition paths based on the diffusion equation on the one hand and the nearest neighbor Markov jump model on the other. Comparison of the transition path time distributions shows that one needs at least a few dozen microstates to obtain reasonable agreement between the two approaches. Using the Markov nearest neighbor model one also obtains good agreement with the experimentally measured transition path time distributions for a DNA hairpin and PrP protein. As found previously when using the diffusion equation, the Markov chain model used here also reproduces the experimentally measured long time tail and confirms that the transition path barrier height is similar to 3k(B)T. This study indicates that in the future, when attempting to model experimentally measured transition path time distributions, one should perhaps prefer a nearest neighbor Markov model which is well defined also for rough energy landscapes. Such studies can also shed light on the minimal number of microstates needed to unravel the experimental data.

(2022) The journal of physical chemistry letters. 13, 30, p. 69666974 Abstract
Transition path flight times are studied for scattering on two electronic surfaces with a single crossing. These flight times reveal nontrivial quantum effects such as resonance lifetimes and nonclassical passage times and reveal that nonadiabatic effects often increase flight times. The flight times are computed using numerically exact time propagation and compared with results obtained from the Fewest Switches Surface Hopping (FSSH) method. Comparison of the two methods shows that the FSSH method is reliable for transition path times only when the scattering is classically allowed on the relevant adiabatic surfaces. However, where quantum effects such as tunneling and resonances dominate, the FSSH method is not adequate to accurately predict the correct times and transition probabilities. These results highlight limitations in methods which do not account for quantum interference effects, and suggest that measuring flight times is important for obtaining insights from the timedomain into quantum effects in nonadiabatic scattering.

(2022) The Journal of chemical physics. 157, 7, 074109. Abstract
Ninety years ago, Wigner derived the leading order expansion term in ℏ2 for the tunneling rate through a symmetric barrier. His derivation included two contributions: one came from the parabolic barrier, but a second term involved the fourthorder derivative of the potential at the barrier top. He left us with a challenge, which is answered in this paper, to derive the same but for an asymmetric barrier. A crucial element of the derivation is obtaining the ℏ2 expansion term for the projection operator, which appears in the fluxside expression for the rate. It is also reassuring that an analytical calculation of semiclassical transition state theory (TST) reproduces the anharmonic corrections to the leading order of ℏ2. The efficacy of the resulting expression is demonstrated for an Eckart barrier, leading to the conclusion that especially when considering heavy atom tunneling, one should use the expansion derived in this paper, rather than the parabolic barrier approximation. The rate expression derived here reveals how the classical TST limit is approached as a function of ℏ and, thus, provides critical insights to understand the validity of popular approximate theories, such as the classical Wigner, centroid molecular dynamics, and ring polymer molecular dynamics methods.

(2022) Physical chemistry chemical physics : PCCP. 24, 26, p. 1585115859 Abstract
Recent grazingincidence, fast atom diffraction (GIFAD) experiments have highlighted the well known observation that the distance between classical rainbow angles depends on the incident energy. The GIFAD experiments imply an incident vertical scattering angle, facilitating an analytic analysis using classical perturbation theory, which leads to the conclusion that the so called \u201cdynamic corrugation\u201d amplitude, as defined by Bocan et al., Phys. Rev. Lett., 2020 125, 096101 is, within firstorder perturbation theory, proportional to the tangent of the rainbow angle. Therefore it provides no further information about the interaction than is gleaned from the rainbow angle and its energy dependence. Perhaps more importantly, the resulting analytic theory reveals how the energy dependence of rainbow angles may be inverted into information on the force field governing the interaction of the incident projectile with the surface.

(2022) The Journal of chemical physics. 156, 24, 244101. Abstract
A coherent state phase space representation of operators, based on the Husimi distribution, is used to derive an exact expression for the symmetrized version of thermal correlation functions. In addition to the time and temperature independent phase space representation of the two operators whose correlation function is of interest, the integrand includes a nonnegative distribution function where only one imaginary time and one real time propagation are needed to compute it. The methodology is exemplified for the flux side correlation function used in rate theory. The coherent state representation necessitates the use of a smeared Gaussian flux operator whose coherent state phase space representation is identical to the classical flux expression. The resulting coherent state expression for the flux side correlation function has a number of advantages as compared to previous formulations. Since only one time propagation is needed, it is much easier to converge it with a semiclassical initial value representation. There is no need for forward\u2013backward approximations, and in principle, the computation may be implemented on the fly. It also provides a route for analytic semiclassical approximations for the thermal rate, as exemplified by a computation of the transmission factor through symmetric and asymmetric Eckart barriers using a thawed Gaussian approximation for both imaginary and real time propagations. As a byproduct, this example shows that one may obtain \u201cgood\u201d tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.

(2022) ACS Physical Chemistry Au. 2, 1, p. 2337 Abstract
A recently developed lower bound theory for Coulombic problems (E. Pollak, R. Martinazzo, J. Chem. Theory Comput. 2021, 17, 1535) is further developed and applied to the highly accurate calculation of the groundstate energy of two (He, Li^{+}, and H^{}) and three (Li) electron atoms. The method has been implemented with explicitly correlated manyparticle basis sets of Gaussian type, on the basis of the highly accurate (Ritz) upper bounds they can provide with relatively small numbers of functions. The use of explicitly correlated Gaussians is developed further for computing the variances, and the necessary modifications are here discussed. The computed lower bounds are of submilliHartree (parts per million relative) precision and for Li represent the best lower bounds ever obtained. Although not yet as accurate as the corresponding (Ritz) upper bounds, the computed bounds are orders of magnitude tighter than those obtained with other lower bound methods, thereby demonstrating that the proposed method is viable for lower bound calculations in quantum chemistry applications. Among several aspects, the optimization of the wave function is shown to play a key role for both the optimal solution of the lower bound problem and the internal check of the theory.
2021

(2021) Entropy. 23, 12, 1675. Abstract
In this work, our purpose is to show how the symmetry of identical particles can influence the time evolution of free particles in the nonrelativistic and relativistic domains. For this goal, we consider a system of either two distinguishable or indistinguishable (bosons and fermions) particles. Two classes of initial conditions have been studied: different initial locations with the same momenta, and the same locations with different momenta. The flight time distribution of particles arriving at a `screen' is calculated in each case. Fermions display broader distributions as compared with either distinguishable particles or bosons, leading to earlier and later arrivals for all the cases analyzed here. The symmetry of the wave function seems to speed up or slow down propagation of particles. Due to the cross terms, certain initial conditions lead to bimodality in the fermionic case. Within the nonrelativistic domain and when the shorttime survival probability is analyzed, if the cross term becomes important, one finds that the decay of the overlap of fermions is faster than for distinguishable particles which in turn is faster than for bosons. These results are of interest in the short time limit since they imply that the wellknown quantum Zeno effect would be stronger for bosons than for fermions.Fermions also arrive earlier than bosons when they are scattered by a delta barrier. Furthermore, the particle symmetry does not affect the mean tunneling flight time and it is given by the phase time for the distinguishable particle.

(2021) Scientific Reports. 11, 1, 23450. Abstract
Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The wellknown Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple\u2019s lower bounds with significantly improved results. We have recently formulated a SelfConsistent Lower Bound Theory (SCLBT), which improves upon Temple\u2019s results. In this paper, we further improve the SCLBT and compare its quality with Lehmann\u2019s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann\u2019s theory and is essential for the SCLBT method. Using two lattice Hamiltonians, we compared the improved SCLBT (iSCLBT) with its previous implementation as well as with Lehmann\u2019s lower bound theory. The novel iSCLBT exhibits a significant improvement over the previous version. Both Lehmann\u2019s theory and the SCLBT variants provide significantly better lower bounds than those obtained from Weinstein\u2019s and Temple\u2019s methods. Compared to each other, the Lehmann and iSCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for lowlying excited states as well. Compared to Lehmann\u2019s theory, one of the advantages of the iSCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used.

(2021) Physical chemistry chemical physics : PCCP. 23, 41, p. 2378723795 Abstract
Recent advances in experimental measurements of transition path time distributions have raised intriguing theoretical questions. The present interpretation of the experimental data indicates a small value of the fitted transition path barrier height as compared to the barrier height of the unfolded to folded transition. Secondly, as shown in this paper, it is essential to analyse the experimental data using absorbing boundary conditions at the end points used to determine the transition paths. Such an analysis reveals long time tails that have thus far eluded quantitative theoretical interpretation. Is this due to uncertainty in the experimental data or does it call for a rethinking of the theoretical interpretation? A detailed study of the transition path time distribution using a diffusive model leads to the following conclusions. a. The present experimental data is not accurate enough to discern between functional forms of bell shaped free energy barriers. b. Long time tails indicate the possible existence of a “trap” in the transition path region. c. The “trap” may be considered as a well in the free energy surface. d. The long time tail is quite sensitive to the form of the trap so that future measurements of the long time tail as a function of the location of the end points of the transition path may make it possible to not only determine the well depth but also to distinguish between different functional forms for the free energy surface. e. Introduction of a well along the transition path leads to good fits with the experimental data provided that the transition path barrier height is ∼3kBT, substantially higher than the estimates of ∼1kBT based on bell shaped functions. The results presented here negate the need of introducing multidimensional effects, free energy barrier asymmetry, subdiffusive memory kernels or systematic ruggedness to explain the experimentally measured data.

(2021) New Journal of Physics. 23, 6, 063044. Abstract
Different approaches for considering barrier crossing times are analyzed, with special emphasis on recent experiments which attempt to measure what is commonly referred to as the Larmor tunneling time. We show that that these experiments cannot reveal the Larmor time, due to the finite energy width of the incident particles. The Larmor time, which measures changes in spin polarization, is classified together with other measurements such as the Buttiker–Landauer oscillating barrier time as indirect measurements of interaction times of scattered particles. In contrast, we present a direct quantum mechanical measure of a barrier crossing time taken to be the difference between the mean flight time for a particle transmitted through a potential barrier incident on a screen and the time it would take to reach the same screen without the barrier. These metrics are asymptotic, in the sense that they infer a time from a measurement after the scattering event is over, whereas other measures like the dwell time are local. Some time measures are welldefined only for incident states which are monochromatic in energy, others are welldefined also for incident wavepackets whose incident energy width is finite. In this paper we compare the different approaches to conclude that only the flight time can be used to answer the provocative (but ultimately illposed) question: how much time does it take to tunnel through a barrier?

(2021) Physical review. A.. 103, 042215. Abstract
The oscillatingbarrier model was used by Büttiker and Landauer to determine a \u201ctraversal time for tunneling.\u201d The model sets a timescale but is not the physically measured flight time of a wave packet scattered on the oscillatingbarrier potential. In this paper we show that the flight time in the limit of a narrowinmomentum wave packet is given by the reflected phase time associated with the various branches of the scattered particle. This is but another example which establishes that tunneling flight times are a reflection of the Wigner phase times. As such, the oscillatingbarrier model does not add any new information about tunneling flight times which has not been elucidated previously using static barrier models.

(2021) Journal of Chemical Theory and Computation. 17, 3, p. 15351547 Abstract
As of the writing of this paper, lower bounds are not a staple of quantum chemistry computations and for good reason. All previous attempts at applying lower bound theory to Coulombic systems led to lower bounds whose quality was inferior to the Ritz upper bounds so that their added value was minimal. Even our recent improvements upon Temple\u2019s lower bound theory were limited to Lanczos basis sets and these are not available to atoms and molecules due to the Coulomb singularity. In the present paper, we overcome these problems by deriving a rather simple eigenvalue equation whose roots, under appropriate conditions, give lower bounds which are competitive with the Ritz upper bounds. The input for the theory is the Ritz eigenvalues and their variances; there is no need to compute the full matrix of the squared Hamiltonian. Along the way, we present a Cauchy\u2013Schwartz inequality which underlies many aspects of lower bound theory. We also show that within the matrix Hamiltonian theory used here, the methods of Lehmann and our recent selfconsistent lower bound theory (J. Chem. Phys.2020,115, 244110) are identical. Examples include implementation to the hydrogen and helium atoms.

(2021) Physical Review A. 103, 1, 012225. Abstract
Using the time parameter in the timedependent Schrödinger equation, we study the time of flight for a particle tunneling through a square barrier potential. Comparing the mean and variance of the energy and the flight time for transmitted and reflected particles, using both density and flux distributions, we find that, when accounting for momentum filtering, the suitably normalized transmitted and reflected distributions are identical in both the density and flux cases. In contrast to previous studies, we demonstrate that these results do not imply a vanishing tunneling time, but rather that the time it takes to tunnel through a square barrier is precisely given by the reflected phase time. For wide barriers, this becomes independent of the barrier width, as predicted independently by MacColl and Hartman. We show that these conclusions can be reached using a variety of arguments, including purely quantum mechanical ones. Analysis of the shapes of the distributions under consideration reveals that wavepacket reshaping is not an explanation for the MacCollHartman effect. The results presented here have direct implications for understanding recent experimental results in the study of the barrier crossing of rubidium atoms. The finite width of an incident wave packet significantly "masks"the tunneling time, and induces substantial asymmetry between the flight times of transmitted and reflected atoms.

(2021) Tunnelling in Molecules. Kastner J. & Kozuch S.(eds.). p. 399424 Abstract
How much time does it take for a particle to tunnel through a barrier? This question continues to baffle up to this very day, one of the reasons being the nonexistence of an energytime commutator analogous to the positionmomentum commutation relation. Instead of worrying about the definition of an operator one may consider time as a parameter in the timedependent Schrödinger equation and study the time evolution of a wavepacket appropriately scattered on a potential barrier. Using this approach, one may formally define a tunnelling flight time that may be measured in a timeofflight experiment. The resulting tunnelling flight time either vanishes or is very small. The implications of this observation with respect to recent hydrogen and helium atom attosecond photoionization experiments are discussed. The vanishing or small flight time does not contradict the finite time measured in Larmor clock experiments, where the tunnelling particle affects an external degree of freedom whose dynamics induced by the tunnelling may be interpreted in terms of a time scale. Flight times are also relevant to quantum reflection where coherences due to resonance scattering are not well accounted for by Wigner's phase time delay.
2020

(2020) RSC Advances. 10, 57, p. 3468134689 Abstract
The accurate determination of tunneling splittings in chemistry and physics is an ongoing challenge. However, the widely used variational methods only provide upper bounds for the energy levels, and thus do not give bounds on the gap between them. Here, we show how the selfconsistent lower bound theory developed previously can be applied to provide upper and lower bounds for tunneling splitting between symmetric and antisymmetric doublets in a symmetric doublewell potential. The tight bounds are due to the very high accuracy of the lower bounds obtained for the energy levels, using the selfconsistent lower bound theory. The accuracy of the lower bounds is comparable to that of the Ritz upper bounds. This is the first time that any theory gave upper and lower bounds to tunneling splittings.

(2020) New Journal of Physics. 22, 9, 093060. Abstract
Wavepacket tunneling, in the relativistic limit, is studied via solutions to the Dirac equation for a square barrier potential. Specifically, the arrival time distribution (the timedependent flux) is computed for wavepackets initiated far away from the barrier, and whose momentum is well below the threshold for abovebarrier transmission. The resulting distributions exhibit peaks at shorter times than those of photons with the same initial wavepacket transmitting through a vacuum. However, this apparent superluminality in time is accompanied by very low transmission probabilities. We discuss these observations, and related observations by other authors, in the context of published objections to the notion that tunneling can be superluminal in time. We find that many of these objections are not consistent with our observations, and conclude that postselected (for transmission) distributions of arrival times can be superluminal. However, the low probability of tunneling means a photon will most likely be seen first and therefore the superluminality does not imply superluminal signaling.

(2020) Proceedings of the National Academy of Sciences of the United States of America. 117, 28, p. 1618116186 Abstract
The Ritz upper bound to eigenvalues of Hermitian operators is essential for many applications in science. It is a staple of quantum chemistry and physics computations. The lower bound devised by Temple in 1928 [G. Temple, Proc. R. Soc. A Math. Phys. Eng. Sci. 119, 276293 (1928)] is not, since it converges too slowly. The need for a good lowerbound theorem and algorithm cannot be overstated, since an upper bound alone is not sufficient for determining differences between eigenvalues such as tunneling splittings and spectral features. In this paper, after 90 y, we derive a generalization and improvement of Temple's lower bound. Numerical examples based on implementation of the Lanczos tridiagonalization are provided for nontrivial lattice model Hamiltonians, exemplifying convergence over a range of 13 orders of magnitude. This lower bound is typically at least one order of magnitude better than Temple's result. Its rate of convergence is comparable to that of the Ritz upper bound. It is not limited to ground states. These results complement Ritz's upper bound and may turn the computation of lower bounds into a staple of eigenvalue and spectral problems in physics and chemistry.

(2020) Journal of Chemical Physics. 152, 24, 244110. Abstract
A rigorous practically applicable theory is presented for obtaining lower bounds to eigenvalues of Hermitian operators, whether the ground state or excited states. Algorithms are presented for computing "residual energies" whose magnitude is essential for the computation of the eigenvalues. Their practical application is possible due to the usage of the Lanczos method for creating a tridiagonal representation of the operator under study. The theory is selfconsistent, in the sense that a lower bound for one state may be used to improve the lower bounds for others, and this is then used selfconsistently until convergence. The theory is exemplified for a toy model of a quartic oscillator, where with only five states the relative error in the lower bound for the ground state is reduced to 6 . 10(6), which is the same as the relative error of the least upper bound obtained with the same basis functions. The lower bound method presented in this paper suggests that lower bounds may become a staple of eigenvalue computations.

(2020) Journal of Physical Chemistry A. 124, 16, p. 33003300 Abstract
The "Addition" by Goldman et al. to their paper "Correct Symmetry Treatment for X + X Reactions..." is in essence a correction of a serious misreading of our 1978 paper "Symmetry Numbers, Not Statistical Factors, Should Be Used in Absolute Rate Theory..." . This misreading led Goldman et al. to accuse us unjustly of major errors in rate theory. Goldman et al. misread us as recommending an additional factor of 2 in their definition of the rate constant. Naturally, error results. We saw neither the original paper by Goldman et al. nor the subsequent "Addition" before publication. Too bad; this misunderstanding could have easily been avoided.

(2020) Physical Review A. 101, 2, 022506. Abstract
Quantum reflection of thermal He atoms from various surfaces (glass slide, GaAs wafer, flat, and structured Cr) at grazing conditions is studied within the elastic closecoupling formalism. Comparison with the experimental results of Zhao et al. [Phys. Rev. Lett. 105, 133203 (2010)] is quite reasonable but the conclusions of the present theoretical analysis are different from those discussed in the experimental work. The universal linear behavior observed in the dependence of the reflection probability on the incident wavevector component perpendicular to the surface is only valid at small values of the component whereas, at larger values, deviation from the linearity is evident, approaching a quadratic dependence at higher values. The surface roughness seems to play no important role in this scattering. Moreover, the claim that one observes a transition from quantum to classical reflection seems to be imprecise.
2019

(2019) Journal of Chemical Theory and Computation. 15, 7, p. 40794087 Abstract
Ninety years ago Temple (Proc. R. Soc. (London) 1928, A119, 276) derived a lower bound for the groundstate energy. The bound was tested and invariably found to be poor as compared to the upper bound obtained through the Rayleigh Ritz procedure due to the fact that it is based also on the second moment of the Hamiltonian. In this paper we (a) improve upon Temple's lower bound estimate for the overlap squared of the true groundstate wave function with the approximate one and (b) describe in detail and generalize our recent improvement on the Temple lower bound based on utilization of higherorder basis functions derived by the Arnoldi algorithm. Both improvements combined lead to a lower bound on the groundstate energy whose accuracy is better than that of the Temple lower bound. This is exemplified by considering the groundstate energy of a quartic potential where one finds that the improvements lead to a lower bound whose quality is comparable to that of the upper bound. The applicability of the method to atoms and molecules is discussed.

(2019) Journal of Chemical Physics. 151, 2, 024703. Abstract
The recently improved Pollak, Grabert, and Hanggi (PGH) turnover theory for activated surface diffusion, including finite barrier effects, is extended and studied in the quantum domain. Analytic expressions are presented for the diffusion coefficient, escape rate, hopping distribution, and mean squared path length of particles initially trapped in one of the wells of a periodic potential, moving under the influence of a frictional and Gaussian random force. Tunneling is included by assuming incoherent quantum hopping at temperatures which are above the crossover temperature between deep tunneling and thermal activation. In the improved version of PGH theory as applied to activated surface diffusion, the potential governing the motion of the unstable mode remains periodic but with a scaled mass which increases with the friction strength. Application of the theory to a periodic cosine potential demonstrates that in the weak damping regime quantum diffusion is slower than classical diffusion due to above barrier quantum reflection which significantly shortens the mean squared path length as compared to the classical result. Finite barrier corrections increase this quantum suppression of diffusion or, equivalently, the inverse isotope effect, whereby the diffusion is faster for a heavier mass. Published under license by AIP Publishing.

(2019) Journal of Chemical Theory and Computation. 15, 3, p. 14981502 Abstract
The Arnoldi iterative method for determining eigenvalues is based on the observation that the effect of operating with the Hamiltonian on a vector may be expressed as a sum of parallel and perpendicular contributions. This identity is used here to improve the previous lowerbound estimate of the groundstate energy by Temple, derived 90 years ago [Temple. Proc. Roy. Soc. (London) 1928, A119, 276]. The significantly improved lower bound is exemplified by considering a quartic and a Morse potential. The lower bound is valid for any Hermitian operator whose discrete spectrum is bounded from below.

(2019) Physical Review A. 99, 1, 012108. Abstract
Time averaging of weak values using the quantum transition path time probability distribution enables us to establish a general uncertainty relation for the weak values of two not necessarily Hermitian operators. This new relation is a weak value analog of the Schrodinger strong value uncertainty relation. It leads to the conclusion that it is possible to determine with high accuracy the simultaneous mean weak values of noncommuting operators by judicious choice of the pre and postselected states even when the postselected state is not an eigenfunction of one of the respective operators. When the time fluctuations of the two weak values are proportional to each other there is no uncertainty limitation on their variances and, in principle, their means can be determined with arbitrary precision even though their corresponding operators do not commute. To exemplify these properties we consider specific weak value uncertainty relations for the timeenergy, coordinatemomentum, and coordinatekineticenergy pairs. In addition we analyze spin operators and the SternGerlach experiment in weak and strong inhomogeneous magnetic fields. This classic case leads to anomalous spin values when the field is weak. The weak value uncertainty relation implies that anomalous spin values are associated with large variances so that their measurement demands increased signal averaging. These examples establish the importance of considering the time dependence of weak values in scattering experiments.
2018

(2018) Physical Review A. 98, 6, 063604. Abstract
Quantum reflection is a universal property of atoms and molecules when scattered from surfaces in ultracold collisions. Recent experimental work has documented the quantum reflection and diffraction of He atoms, dimers, trimers, and neon atoms when reflected from a grating. Conditions for the observation of emerging beam resonances have been discussed and measured. In this paper, we provide a theoretical simulation of the quantum reflection from a grating for those systems. We confirm the universal dependence on the incident de Broglie wavelength with the threshold angles where the emerging beam resonances are observed. However, the angular dependence of the reflection efficiencies, that is, the ratio of scattered intensity into specific diffraction channels relative to the total intensity is found to be dependent on the details of the particle surface interaction.

(2018) Journal of Chemical Theory and Computation. 14, 10, p. 53105323 Abstract
The vibronic absorption spectrum of the electric dipole forbidden and vibronically allowed S_{1}(^{1} A_{2}) ← S_{0}(^{1} A_{1}) transition of formaldehyde is calculated by Gaussian wavepacket and semiclassical methods, along with numerically exact reference calculations, using the potential energy surface of Fu, Shepler, and Bowman ( J. Am. Chem. Soc. 2011, 133, 7957). Specifically, the variational multiconfigurational Gaussian (vMCG) approach and the HermanKluk semiclassical initial value representation (HKSCIVR) are compared to assess the accuracy and convergence of these methods, benchmarked against numerically exact timedependent wavepacket propagation (TDWP) on the reference potential energy surface. The vMCG calculation is shown to converge quite well with about 100 variationally evolving Gaussian functions and using a local cubic expansion instead of the conventional local harmonic approximation. By contrast, the HKSCIVR approach with ∼10^{5} trajectories reproduces the vibrationally structured spectral envelope correctly but yields a strongly broadened spectrum. The comparison of the computed absorption spectrum with experiment shows that the relevant vibronic progressions are reasonably reproduced by all computations, but deviations of the order of 10100 cm^{1} occur, underscoring that both electronic structure calculations and dynamical approaches remain challenging in the calculation of typical smallmolecule excitedstate spectra by trajectorybased methods.

(2018) Journal of Physical Chemistry Letters. 9, 20, p. 60666071 Abstract
Kramers's original paper on the diffusion model of chemical reactions was based on the consideration that only the barrier region determines the outcome of transmission over a barrier. Subsequently it became understood that Kramers's approach was identical to variational transition state theory (VTST) and as such used only thermodynamic information. Here, using Kramers's philosophy in conjunction with perturbation theory and the realization that the dynamics which is ratedetermining usually occurs in the vicinity of the transition state leads to a novel stochastic rate theory in which the momentum change induced by the medium is the stochastic variable. A first successful application of the theory is to the old and challenging problem of motion over a cusped barrier. This has implications for the study of transition path time distributions as well as the theory of tunneling via nonadiabatic coupling.

(2018) The Journal of chemical physics. 149, 16, 164114. Abstract
Scattering through a double slit potential is one of the most fundamental problems in quantum mechanics. It is well understood that due to the superposition of amplitudes, one observes a spatial interference pattern in the scattered wavefunction reflecting the superposition of amplitudes coming from both slits. However, the effect of the double slit on the mean time it takes to traverse the slit has not been considered previously. Using a transition path time formalism, we show that when a single Gaussian wavepacket is scattered through a double slit potential, one finds not only oscillations in the scattered density resulting from the spatial interference created by the splitting of the wavepacket but also an oscillatory pattern in the mean scattering time. Long times are associated with low values of a suitably defined momentum, and short times with higher values. The double slit thus serves as a momentum filtering device. We also find an interference pattern in the time averaged momentum weak value profile of the scattered particle implying that the double slit also acts as a weak momentum filter. These results not only demonstrate the value of considering transition path time distributions in their quantum mechanical context but also present a challenge to semiclassical approximationscan they account for temporal interference? Published by AIP Publishing.

(2018) Physical Review A. 98, 4, 042112. Abstract
Weak values have been shown to be helpful especially when considering them as the outcomes of weak measurements. In this paper we show that, in principle, the real and imaginary parts of the weak value of any operator may be elucidated from expectation values of suitably defined density, flux, and Hermitian commutator operators. Expectation values are the outcomes of strong (projective) measurements, implying that weak values are general properties of operators in association with pre and postselection and they need not be preferentially associated with weak measurements. They should be considered as an important measurable property which provides added information compared with the "standard" diagonal expectation value of an operator. As the first specific example we consider the determination of the real and imaginary parts of the weak value of the momentum operator employing projective timeofflight experiments. Then the results are analyzed from the point of view of Bohmian mechanics. Finally, we consider recent neutron interferometry experiments used to determine the weak values of the neutron spin.

(2018) New Journal of Physics. 20, 073016. Abstract
Using the quantum transition path time probability distribution we show that time averaging of weak values leads to unexpected results. We prove a weak value timeenergy uncertainty principle and timeenergy commutation relation. We also find that time averaging allows one to predict in advance the momentum of a particle at a post selected point in space with accuracy greater than the limit of h/2 as dictated by the uncertainty principle. This comes at a costit is not possible at the same time to predict when the particle will arrive at the post selected point. A specific example is provided for one dimensional scattering from a square barrier.

(2018) Physical Review A. 97, 4, 042102. Abstract
Quantum threshold reflection is a wellknown quantum phenomenon which prescribes that at threshold, except for special circumstances, a quantum particle scattering from any potential, even if attractive at long range, will be reflected with unit probability. In the past, this property had been associated with the socalled badlands region of the potential, where the semiclassical description of the scattering fails due to a rapid spatial variation of the de Broglie wavelength. This badlands region occurs far from the strong interaction region of the potential and has therefore been used to "explain" the quantum reflection phenomenon. In this paper we show that the badlands region of the interaction potential is immaterial. The extremely long wavelength of the scattered particle at threshold is much longer than the spatial extension of the badlands region, which therefore does not affect the scattering. For this purpose, we review and generalize the proof for the existence of quantum threshold reflection to stress that it is only a consequence of continuity and boundary conditions. The nonlocal character of the scattering implies that the whole interaction potential is involved in the phenomenon. We then provide a detailed numerical study of the threshold scattering of a particle by a Morse potential and an Eckart potential, especially in the time domain. We compare exact quantum computations with incoherent results obtained from a classical Wigner approximation. This study shows that close to threshold the timedependent amplitude of the scattered particle is negligible in the badlands region and is the same whether the potential has a reflecting wall as in the Morse potential or a steplike structure as in the Eckart smooth step potential. The mean flight time of the particle is not shortened due to a local reflection from the badlands region or due to the lower density of the wave function at short distances. This study should serve to definitely rule out the badlands region as a qualitative guide to the properties of quantum threshold reflection.

(2018) Journal of Physical Chemistry A. 122, 14, p. 35633571 Abstract
The time it takes a particle to tunnel through the asymmetric Eckart barrier potential is investigated using Gaussian wavepackets, where the barrier serves as a model for the potential along a chemical reaction coordinate. We have previously shown that the, in principle experimentally measurable, tunneling flight time, which determines the time taken by the transmitted particle to traverse the barrier, vanishes for symmetric potentials like the Eckart and square barrier [Petersen, J.; Pollak, E. J. Phys. Chem. Lett. 2017, 9, 4017]. Here we show that the same result is obtained for the asymmetric Eckart barrier potential, and therefore, the zero tunneling flight time seems to be a general result for onedimensional timeindependent potentials. The wavepacket dynamics is simulated using both an exact quantum mechanical method and a classical Wigner prescription. The excellent agreement between the two methods shows that quantum coherences are not important in pure onedimensional tunneling and reinforces the conclusion that the tunneling flight time vanishes.

(2018) Journal of Chemical Physics. 148, 7, 074111. Abstract
The quantum phenomenon of above barrier reflection is investigated from a timedependent perspective using Gaussian wavepackets. The transition path time distribution, which in principle is experimentally measurable, is used to study the mean flight times R and T associated with the reflection and the transmission over the barrier paying special attention to their dependence on the width of the barrier. Both flight times, and their difference Delta t, exhibit two distinct regimes depending on the ratio of the spatial width of the incident wavepacket and the length of the barrier. When the ratio is larger than unity, the reflection and transmission dynamics are coherent and dominated by the resonances above the barrier. The flight times (R/T) and the flight time difference Delta t oscillate as a function of the barrier width (almost in phase with the transmission probability). These oscillations reflect a momentum filtering effect related to the coherent superposition of the reflected and transmitted waves. For a ratio less than unity, the barrier reflection and transmission dynamics are incoherent and the oscillations are absent. The barrier width which separates the coherent and incoherent regimes is identified analytically. The oscillatory structure of the time difference Delta t as a function of the barrier width in the coherent regime is absent when considered in terms of the Wigner phase time delays for reflection and transmission. We conclude that the Wigner phase time does not correctly describe the temporal properties of above barrier reflection. We also find that the structure of the reflected and transmitted wavepackets depends on the coherence of the process. In the coherent regime, the wavepackets can have an overlapping peak structure, but the peaks are not fully resolved. In the incoherent regime, the wavepackets split in time into distinct separated Gaussian like waves, each one reflecting the number of times the wavepacket crosses the barrier region before exiting. A classical Wigner approximation, using classical trajectories which upon reaching an edge of the barrier are reflected or transmitted as if the edge was a step potential, is quantitative in the incoherent regime. The implications of the coherence observed on resonance reactive scattering are discussed.
2017

(2017) Journal of Physical Chemistry Letters. 8, 17, p. 40174022 Abstract
Attosecond ionization experiments have not resolved the question "What is the tunneling time?". Different definitions of tunneling time lead to different results. Second, a zero tunneling time for a material particle suggests that the nonrelativistic theory includes speeds greater than the speed of light. Chemical reactions, occurring via tunneling, should then not be considered in terms of a nonrelativistic quantum theory calling into question quantum dynamics computations on tunneling reactions. To answer these questions, we define a new experimentally measurable paradigm, the tunneling flight time, and show that it vanishes for scattering through an Eckart or a square barrier, irrespective of barrier length or height, generalizing the Hartman effect. We explain why this result does not lead to experimental measurement of speeds greater than the speed of light. We show that this tunneling is an incoherent process by comparing a classical Wigner theory with exact quantum mechanical computations.

(2017) Physical Review A. 95, 4, 042108. Abstract
The transitionpathtime distribution is formalized for quantum systems and applied to a number of examples. Using a symmetrized thermal density, transition times are studied for the free particle, a δfunction potential, a squarebarrier potential, and symmetricdoublewell dynamics at very low temperature. These studies exemplify extreme nonlocality for motion in δfunction potentials, vanishing tunneling times for the squarebarrier potential, and varying transit times in the symmetricdoublewell potential. In all cases, there are regions where the longer the distance traversed, the shorter the mean transit time is. For the thermal density correlation functions studied here, the Hartman effect exemplifies itself through the independence of the transit time on the barrier height. However, due to the thermal distribution, the transit time does depend on the barrier width, initially decreasing with increasing width but then increasing again.

(2017) Journal of Physical Chemistry Letters. 8, 5, p. 10091013 Abstract
The quantum reflection measured previously by Zhao et al. (Phys. Rev. A 2008, 78, 010902(R)) for the scattering of He atoms off of a microstructured grating is described and analyzed theoretically. Using the closecoupling formalism with a complex absorbing potential and describing the longrange interaction in terms of the Casimirvan der Waals potential, we find probabilities and diffraction patterns that are in fairly good agreement with the experimental results. The central outcomes of this study are twofold. First is the theoretical confirmation that, indeed, the phenomenon of quantum reflection may be detected not only through the elastic peak but also in terms of a quantum reflected diffraction pattern. Second, we demonstrate that the phenomenon of quantum reflection is the result of a coherent process where all of the potential regions are involved on an equal footing. It is a nonlocal property and cannot be related only to the longrange badlands region of the potential of interaction.

(2017) Physical Review Letters. 118, 7, 070401. Abstract
A quantum mechanical transition path time probability distribution is formulated and its properties are studied using a parabolic barrier potential model. The average transit time is well defined and readily calculated. It is smaller than the analogous classical mechanical average transit time, vanishing at the crossover temperature. It provides a direct route for determining tunneling times. The average time may be also used to define a coarse grained momentum of the system for the passage from one side of the barrier to the other. The product of the uncertainty in this coarse grained momentum with the uncertainty in the location of the particle is shown under certain conditions to be smaller than the. h/2 formal uncertainty limit. The model is generalized to include friction in the form of a bilinear interaction with a harmonic bath. Using an Ohmic friction model one finds that increasing the friction, increases the transition time. Only moderate values of the reduced friction coefficient are needed for the quantum transition time and coarse grained uncertainty to approach the classical limit which is smaller than. h/2 when the friction is not too small. These results show how one obtains classical dynamics from a pure quantum system without invoking any further assumptions, approximations, or postulates.

(2017) Journal of Physical Chemistry Letters. 8, 2, p. 352356 Abstract
The standard approaches to tunneling times are replaced by considering time correlation functions. A class of correlation functions that is always positive is identified and used to define quantum mechanical transition time probability distributions. The formalism is used to study the quantum dynamics of a thermal position correlation function of a parabolic barrier Hamiltonian. The transition time probability distribution between two points distributed symmetrically about the barrier top shifts to shorter times as the temperature is reduced and tunneling is increased. A study of the mean transition time as a function of the distance between the center of the initial and final densities shows that when the temperature is sufficiently low and tunneling dominates the dynamics, increasing the length of the path traversed decreases the mean transition time. The introduction of friction to the dynamics does not \u201cdestroy\u201d this phenomenon, except when the friction coefficient is very large.
2016

(2016) Journal of Physical Chemistry A. 120, 28, p. 54465456 Abstract
This year we celebrate the 80th anniversary of the LandauTeller model for energy exchange in a collinear collision of an atom with a harmonic diatomic molecule. Even after 80 years though, the analytic theory to date has not included in it the backinfluence of the oscillator's motion on the energy transfer between the approaching particle and the molecule. This is the topic of the present paper. The backinfluence can be obtained by employing classical secondorder perturbation theory. The secondorder theory is used in both a classical and semiclassical context. Classically, analytic expressions are derived for the final phase and action of the diatom, after the collision. The energy loss of the atom is shown to decrease linearly with the increasing energy of the oscillator. The magnitude of this decrease is a direct consequence of the backreaction of the oscillator on the translational motion. The qualitative result is universal, in the sense that it is not dependent on the details of the interaction of the atom with the oscillator. A numerical application to a model collision of an Ar atom with a Br_{2} diatom demonstrates the importance and accuracy of the secondorder perturbation theory. The same results are then used to derive a secondorder perturbation theory semiclassical expression for the quantum transition probability from initial vibrational state n_{i} to final vibrational state n_{f} of the oscillator. A comparison of the theory with exact quantum data is presented for a model collision of Br_{2} with a hydrogen molecule, where the hydrogen molecule is considered as a single approaching particle.

(2016) Journal of Physical Chemistry A. 120, 19, p. 31553164 Abstract
Kramers turnover theory as derived by Pollak, Grabert, and Hänggi (PGH) suffers from a few drawbacks. First, the energy loss in PGH theory is not a monotonic function of the friction. Second, the theory is not applicable to surface diffusion, because the effective potential for the system does not conserve the periodicity of the potential. Third, when the reduced barrier height is low, it is rather inaccurate. In this paper, we present a modification of PGH theory that alleviates these drawbacks. We also introduce a finite barrier correction term which takes into consideration that the energy interval of the escaping particle is bounded from below. The resulting theory is tested for motion on a cubic potential and relatively low reduced barriers.

(2016) Physical Chemistry Chemical Physics. 18, 41, p. 2887228882 Abstract
The recent experimental measurement of the transition path time distributions of proteins presents several challenges to theory. Firstly, why do the fits of the experimental data to a theoretical expression lead to barrier heights which are much lower than the free energies of activation of the observed transitions? Secondly, there is the theoretical question of determining the transition path time distribution, without invoking the Smoluchowski limit. In this paper, we derive an exact expression for a transition path time distribution which is valid for arbitrary memory friction using the normal mode transformation which underlies Kramers' rate theory. We then recall that for low barriers, there is a noticeable difference between the transition path time distribution obtained with absorbing boundary conditions and free boundary conditions. For the former, the transition times are shorter, since recrossings of the boundaries are disallowed. As a result, if one uses the distribution based on absorbing boundary conditions to fit the experimental data, one will find that the transition path barrier will be larger than the values found based on a theory with free boundary conditions. We then introduce the paradigm of a transition path barrier height, and show that one should always expect it to be much smaller than the activation energy.

(2016) Faraday Discussions. 195, p. 111138 Abstract
Kramers' turnover theory, based on the dynamics of the collective unstable normal mode (also known as PGH theory), is extended to the motion of a particle on a periodic potential interacting bilinearly with a dissipative harmonic bath. This is achieved by considering the small parameter of the problem to be the deviation of the collective bath mode from its value along the reaction coordinate, defined by the unstable normal mode. With this change, the effective potential along the unstable normal mode remains periodic, albeit with a renormalized mass, or equivalently a renormalized lattice length. Using second order classical perturbation theory, this not only enables the derivation of the hopping rates and the diffusion coefficient, but also the derivation of finite barrier corrections to the theory. The analytical results are tested against numerical simulation data for a simple cosine potential, ohmic friction, and different reduced barrier heights.
2015

(2015) Journal of Chemical Physics. 143, 22, 224114. Abstract
One of the challenges facing onthefly ab initio semiclassical time evolution is the large expense needed to converge the computation. In this paper, we suggest that a significant saving in computational effort may be achieved by employing a semiclassical initial value representation (SCIVR) of the quantum propagator based on the Heisenberg interaction representation. We formulate and test numerically a modification and simplification of the previous semiclassical interaction representation of Shao and Makri [J. Chem. Phys. 113, 3681 (2000)]. The formulation is based on the wavefunction form of the semiclassical propagation instead of the operator form, and so is simpler and cheaper to implement. The semiclassical interaction representation has the advantage that the phase and prefactor vary relatively slowly as compared to the "standard" SCIVR methods. This improves its convergence properties significantly. Using a onedimensional model system, the approximation is compared with HermanKluk's frozen Gaussian and Heller's thawed Gaussian approximations. The convergence properties of the interaction representation approach are shown to be favorable and indicate that the interaction representation is a viable way of incorporating onthefly force field information within a semiclassical framework.

(2015) The Journal of chemical physics. 143, 17, 179901. Abstract
We have discovered a numerical error in the plots of Fig. 6 in the original paper (panels (a) and (b)).1 The correct result is presented in Fig. 6. The corrected figure does not impact any of the conclusions of the paper.figureFIG. 6. The 2 dimensional (panel (a)) and 3 dimensional (panel (b)) quantum (diffractiveblack line) and classical Wigner (red circles) angular distributions associated with the scattering along the [100] surface direction.

(2015) Journal of Chemical Physics. 143, 10, 104104. Abstract
Originally, the challenge of solving Kramers' turnover theory was limited to Ohmic friction, or equivalently, motion of the escaping particle governed by a Langevin equation. Mel'nikov and Meshkov [J. Chem. Phys. 85, 1018 (1986)] (MM) presented a solution valid for Ohmic friction. The turnover theory was derived more generally and for memory friction by Pollak, Grabert, and Hänggi [J. Chem. Phys. 91, 4073 (1989)] (PGH). Mel'nikov proceeded to also provide finite barrier corrections to his theory [Phys. Rev. E 48, 3271 (1993)]. Finite barrier corrections were derived only recently within the framework of PGH theory [E. Pollak and R. Ianconescu, J. Chem. Phys. 140, 154108 (2014)]. A comprehensive comparison between MM and PGH theories including finite barrier corrections and using Ohmic friction showed that the two methods gave quantitatively similar results and were in quantitative agreement with numerical simulation data. In the present paper, we extend the study of the turnover theories to exponential memory friction. By comparing with numerical simulation, we find that PGH theory is rather accurate, even in the strong friction long memory time limit, while MM theory fails. However, inclusion of finite barrier corrections to PGH theory leads to failure in this limit. The long memory time invalidates the fundamental assumption that consecutive traversals of the well are independent of each other. Why PGH theory without finite barrier corrections remains accurate is a puzzle.

(2015) The Journal of chemical physics. 143, 6, 064706. Abstract
A second order classical perturbation theory is developed to calculate the sticking probability of a particle scattered from an uncorrugated thermal surface. An analytic expression for the temperature dependent energy loss of the particle to the surface is derived by employing a onedimensional generalized Langevin equation. The surface temperature reduces the energy loss, since the thermal surface transfers energy to the particle. Using a Gaussian energy loss kernel and the multiple collision theory of Fan and Manson [J. Chem. Phys. 130, 064703 (2009)], enables the determination of the fraction of particles trapped on the surface after subsequent momentum reversals of the colliding particle. This then leads to an estimate of the trapping probability. The theory is tested for the model scattering of Ar on a LiF(100) surface. Comparison with numerical simulations shows excellent agreement of the analytical theory with simulations, provided that the energy loss is determined by the second order perturbation theory.

(2015) Journal of Chemical Physics. 143, 1, 014705. Abstract
Inplane two and three dimensional diffraction patterns are computed for the vertical scattering of an Ar atom from a frozen LiF(100) surface. Suitable collimation of the incoming wavepacket serves to reveal the quantum mechanical diffraction. The interaction potential is based on a fit to an ab initio potential calculated using density functional theory with dispersion corrections. Due to the potential coupling found between the two horizontal surface directions, there are noticeable differences between the quantum angular distributions computed for two and three dimensional scattering. The quantum results are compared to analogous classical Wigner computations on the same surface and with the same conditions. The classical dynamics largely provides the envelope for the quantum diffractive scattering. The classical results also show that the corrugation along the [110] direction of the surface is smaller than along the [100] direction, in qualitative agreement with experimental observations of unimodal and bimodal scattering for the [110] and [100] directions, respectively.

(2015) Journal of Physical Chemistry C. 119, 26, p. 1453214541 Abstract
A secondorder semiclassical perturbation theory is developed and applied to the elastic scattering of an atom from a corrugated surface. Analytical expressions for the diffraction pattern in the momentum space are obtained based on a sine corrugation function and a Morse potential for the interaction of the particle with the surface. The theory is implemented for a model of the inplane scattering of Ar atoms from a LiF(100) surface. The resulting diffraction intensities are compared with secondorder perturbation theory classical distributions and closecoupling results for two incident energies of 300 and 700 meV. The previous firstorder perturbation theory predicts a symmetric diffraction pattern about the elastic peak, while the secondorder semiclassical perturbation theory accounts correctly for the asymmetry in the diffraction pattern.

(2015) Journal of Chemical Physics. 142, 17, 174102. Abstract
The semiclassical perturbation theory of Hubbard and Miller [J. Chem. Phys. 80, 5827 (1984)] is further developed to include the full multiphonon transitions in atomsurface scattering. A practically applicable expression is developed for the angular scattering distribution by utilising a discretized bath of oscillators, instead of the continuum limit. At sufficiently low surface temperature good agreement is found between the present multiphonon theory and the previous one, and twophonon theory derived in the continuum limit in our previous study [Daon, Pollak, and MiretArtés, J. Chem. Phys. 137, 201103 (2012)]. The theory is applied to the measured angular distributions of Ne, Ar, and Kr scattered from a Cu(111) surface. We find that the present multiphonon theory substantially improves the agreement between experiment and theory, especially at the higher surface temperatures. This provides evidence for the importance of multiphonon transitions in determining the angular distribution as the surface temperature is increased.
2014

(2014) Journal of Chemical Physics. 141, 23, 234509. Abstract
The dissipative harmonic oscillator is studied as a model for vibrational relaxation in a liquid environment. Continuum limit expressions are derived for the timedependent average energy, average width of the population, and the vibrational population itself. The effect of the magnitude of the solutesolvent interaction, expressed in terms of a friction coefficient, solvent temperature, and initial energy of the oscillator on the relaxation has been studied. These results shed light on the recent femtosecond stimulated Raman scattering probe of the 1570 cm^{1} C=C stretching mode of transStilbene in the first (S_{1}) excited electronic state. When the oscillator is initially cold with respect to the bath temperature, its average energy and width increase in time. When it is initially hot, the average energy and width decrease with time in qualitative agreement with the experimental observations.

(2014) Journal of Chemical Physics. 140, 15, 154108. Abstract
Kramers [Physica 7, 284 (1940)], in his seminal paper, derived expressions for the rate of crossing a barrier in the underdamped limit of weak friction and the moderate to strong friction limit. The challenge of obtaining a uniform expression for the rate, valid for all damping strengths is known as Kramers turnover theory. Two different solutions have been presented. Mel'nikov and Meshkov [J. Chem. Phys. 85, 1018 (1986)] (MM) considered the motion of the particle, treating the friction as a perturbation parameter. Pollak, Grabert, and H nggi [J. Chem. Phys. 91, 4073 (1989)] (PGH), considered the motion along the unstable mode which is separable from the bath in the barrier region. In practice, the two theories differ in the way an energy loss parameter is estimated. In this paper, we show that previous numerical attempts to resolve the quality of the two approaches were incomplete and that at least for a cubic potential with Ohmic friction, the quality of agreement of both expressions with numerical simulation is similar over a large range of friction strengths and temperatures. Mel'nikov [Phys. Rev. E 48, 3271 (1993)], in a later paper, improved his theory by introducing finite barrier corrections. In this paper we note that previous numerical tests of the finite barrier corrections were also incomplete. They did not employ the exact rate expression, but a harmonic approximation to it. The central part of this paper, is to include finite barrier corrections also within the PGH formalism. Tests on a cubic potential demonstrate that finite barrier corrections significantly improve the agreement of both MM and PGH theories when compared with numerical simulations. c 2014 AIP Publishing LLC.

(2014) Physical Review A. 89, 3, 032104. Abstract
Several approaches to the semiclassical dynamics of fermions have been proposed in the past. The main subject under discussion was the inclusion of the Pauli principle, i.e., the fact that two electrons with parallel spins must be in orthogonal states. In the past, this was sometimes achieved by adding repulsive Pauli potentials or by using antisymmetric trial states. In this article we show that (a) the use of semiclassical propagators based on classical trajectories is sufficient to account for the Pauli principle, but (b) a semiclassical wavefunction approach is not satisfactory.

(2014) Journal of Chemical Physics. 140, 1, 014104. Abstract
We propose a semiphenomenological Markovian Master equation for describing the quantum dynamics of atomsurface scattering. It embodies the Lindbladlike structure and can describe both damping and pumping of energy between the system and the bath. It preserves positivity and correctly accounts for the vanishing of the interaction of the particle with the surface when the particle is distant from the surface. As a numerical test, we apply it to a model of an Ar atom scattered from a LiF surface, allowing for interaction only in the vertical direction. At low temperatures, we find that the quantum mechanical average energy loss is smaller than the classical energy loss. The numerical results obtained from the space dependent friction master equation are compared with numerical simulations for a discretized bath, using the multiconfigurational time dependent Hartree methodology. The agreement between the two simulations is quantitative.

(2014) Journal of Chemical Physics. 140, 2, 024709. Abstract
A second order classical perturbation theory is developed and applied to elastic atom corrugated surface scattering. The resulting theory accounts for experimentally observed asymmetry in the final angular distributions. These include qualitative features, such as reduction of the asymmetry in the intensity of the rainbow peaks with increased incidence energy as well as the asymmetry in the location of the rainbow peaks with respect to the specular scattering angle. The theory is especially applicable to soft corrugated potentials. Expressions for the angular distribution are derived for the exponential repulsive and Morse potential models. The theory is implemented numerically to a simplified model of the scattering of an Ar atom from a LiF(100) surface.
2013

(2013) Journal of Chemical Physics. 139, 15, 154311. Abstract
Internal conversion is an inherently quantum mechanical process. To date, "ab initio" computation of internal conversion rates was limited to harmonic based approximations. These are questionable since the typical transition to the ground electronic state occurs at energies which are far from the harmonic limit. It is thus of interest to study the applicability of the Semiclassical Initial Value Representation (SCIVR) approach which is in principle amenable to "on the fly" studies even with "many" degrees of freedom. In this work we apply the HermanKlukSCIVR methodology to compute the internal conversion rates for formaldehyde for a variety of initial vibronic states. The SCIVR computation gives reasonable agreement with experiment, while the harmonic approximation typically gives rates that are too high.

(2013) Journal of Chemical Physics. 139, 1, 011101. Abstract
Transstilbene in nhexane is excited with excess vibrational energy in the range 07000 cm(1). In the excited electronic state, the Raman linewidth of the ethylenic C=C stretching mode at 1570 cm(1) is followed with similar to 100 fs time resolution. Upon excitation with substantial excess energy, the width of the peak is initially broad and then narrows within a few picoseconds, as observed previously by Iwata and Hamaguchi [Chem. Phys. Lett. 196, 462 (1992)]. This narrowing is understood as being caused by cooling of the initially hot molecule, by the surrounding solvent. In this Communication, we report that upon excitation without excess energy, the width is initially relatively narrow and then broadens on a picosecond time scale. The broadening is attributed to heating of the molecule by solvent collisions. It follows that the nascent population in the excited electronic state is cold as compared with the solvent. Such reduction of the initial vibrational energy may affect the rate for the subsequent photoreaction, especially in the absence of the solvent. (C) 2013 AIP Publishing LLC.

(2013) Journal of Chemical Physics. 139, 4, 044707. Abstract
A density functional theory with dispersion corrections is used to study the scattering of an Ar atom on the LiF(100) surface. On the fly classical trajectories are propagated to study the inplane angular and energy loss distributions of the scattered Ar atom. The computations are carried out for a frozen surface and a surface in which the crystal atoms are initially at T = 0 K. Two dimensional as well as three dimensional computations are presented. We find that the results agree qualitatively with measured experimental results. These computations show the impact of three dimensional effects on the scattering such as narrowing of the angular distance between rainbow peaks and inversion of asymmetry properties of the angular distribution. The computations also reaffirm the prediction that one should observe energy loss rainbows in the scattering of Ar from the LiF(100) surface. (C) 2013 AIP Publishing LLC.

(2013) Journal of Chemical Physics. 138, 16, 164116. Abstract
The Kramers turnover problem, that is, obtaining a uniform expression for the rate of escape of a particle over a barrier for any value of the external friction was solved in the 1980s. Two formulations were given, one by Melnikov and Meshkov (MM) [V. I. Melnikov and S. V. Meshkov, J. Chem. Phys. 85, 1018 (1986)10.1063/1.451844], which was based on a perturbation expansion for the motion of the particle in the presence of friction. The other, by Pollak, Grabert, and Hänggi (PGH) [E. Pollak, H. Grabert, and P. Hänggi, J. Chem. Phys. 91, 4073 (1989)10.1063/1.456837], valid also for memory friction, was based on a perturbation expansion for the motion along the collective unstable normal mode of the particle. Both theories did not take into account the temperature dependence of the average energy loss to the bath. Increasing the bath temperature will reduce the average energy loss. In this paper, we analyse this effect, using a novel perturbation theory. We find that within the MM approach, the thermal energy gained from the bath diverges, the average energy gain becomes infinite implying an essential failure of the theory. Within the PGH approach increasing the bath temperature reduces the average energy loss but only by a finite small amount of the order of the inverse of the reduced barrier height. Then, this does not seriously affect the theory. Analysis and application for a cubic potential and Ohmic friction are presented.
2012

(2012) Journal of Chemical Physics. 137, 20, 201103. Abstract
Inspired by the semiclassical perturbation theory of Hubbard and Miller [J. Chem. Phys. 80, 5827 (1984)10.1063/1.446609], we derive explicit expressions for the angular distribution of particles scattered from thermal surfaces. At very low surface temperature, the observed experimental background scattering is proportional to the spectral density of the phonons. The angular distribution is a sum of diffraction peaks and a broad background reflecting the spectral density. The theory is applied to measured angular distributions of Ne, Ar, and Kr scattered from a Cu(111) surface.

(2012) Journal Of Physical Chemistry B. 116, 43, p. 1296612971 Abstract
Experiments in recent years have shown that there is a large kinetic isotope effect in the rate of transfer of hydrogen or deuterium in enzymatic reactions of soybean lipoxygenase1. The kinetic isotope effect (KIE) is only weakly temperature dependent but varies significantly in the presence of mutants whose functional groups are located rather far from the reaction center. In this paper we suggest that variational transition state theory as applied to dissipative systems, above the crossover temperature between deep tunneling and thermal activation, may be used as a paradigm for understanding the dynamics of these reactions. We find that the theory fits the experimental data rather well. The effects of different mutants are readily interpreted in terms of the friction they exert on the reaction center. Increasing the distal functional group increases the friction and thus lowers the kinetic isotope effect.

(2012) Surface Science Reports. 67, 78, p. 161200 Abstract
The scattering of heavy atoms and molecules from surfaces is oftentimes dominated by classical mechanics. A large body of experiments have gathered data on the angular distributions of the scattered species, their energy loss distribution, sticking probability, dependence on surface temperature and more. For many years these phenomena have been considered theoretically in the framework of the "washboard model" in which the interaction of the incident particle with the surface is described in terms of hard wall potentials. Although this class of models has helped in elucidating some of the features it left open many questions such as: true potentials are clearly not hard wall potentials, it does not provide a realistic framework for phonon scattering, and it cannot explain the incident angle and incident energy dependence of rainbow scattering, nor can it provide a consistent theory for sticking. In recent years we have been developing a classical perturbation theory approach which has provided new insight into the dynamics of atomsurface scattering. The theory includes both surface corrugation as well as interaction with surface phonons in terms of harmonic baths which are linearly coupled to the system coordinates. This model has been successful in elucidating many new features of rainbow scattering in terms of frictions and bath fluctuations or noise. It has also given new insight into the origins of asymmetry in atomic scattering from surfaces. New phenomena deduced from the theory include friction induced rainbows, energy loss rainbows, a theory of superrainbows, and more. In this review we present the classical theory of atomsurface scattering as well as extensions and implications for semiclassical scattering and the further development of a quantum theory of surface scattering. Special emphasis is given to the inversion of scattering data into information on the particlesurface interactions.

(2012) Journal of Chemical Physics. 136, 20, 204707. Abstract
The semiclassical perturbation theory formalism of Hubbard and Miller [J. Chem. Phys. 78, 1801 (1983)] for atom surface scattering is used to explore the possibility of observation of heavy atom diffractive scattering. In the limit of vanishing the semiclassical theory is shown to reduce to the classical perturbation theory. The quantum diffraction pattern is sensitive to the characteristics of the beam of incoming particles. Necessary conditions for observation of quantum diffraction are derived for the angular width of the incoming beam. An analytic expression for the angular distribution as a function of the angular and momentum variance of the incoming beam is obtained. We show both analytically and through some numerical results that increasing the angular width of the incident beam leads to decoherence of the quantum diffraction peaks and one approaches the classical limit. However, the incoherence of the beam in the parallel direction does not destroy the diffraction pattern. We consider the specific example of Ar atoms scattered from a rigid LiF(100) surface.

(2012) Molecular Physics. 110, 910, p. 861873 Abstract
Three different methods that are based on the coherent control of a time evolved wavefunction are used to determine the eigenvalues of Hermitian matrices. These methods are of special interest for determining eigenvalues of very large matrices and they replace the standard matrix diagonalization by a minimization problem of a few optimal time or phase variables. Upon inversion, the optimal time or phase variables directly provide the energies of higher eigenstates spanned by the initial wavefunction, without having to compute the wavefunctions themselves. The methods are applied to determine the electronic energies of the He and C atoms as well as a model harmonic oscillator system. All three methods scale as N ^{2} for a matrix whose dimension is N and they use as input only the overlap of the time evolved initial wavefunction with itself.

(2012) Chemical Physics. 399, p. 135141 Abstract
Gaussian approximations to the Boltzmann operator have proven themselves in recent years as useful tools for the study of the thermodynamic properties of rare gas clusters. They are, however, not necessarily correct at very low temperatures. In this article we introduce a firstorder correction term to the frozen Gaussian imaginary time propagator and apply it to the argon trimer. Our findings show that the correction term provides objective access to the quality of the propagator's results and clearly defines the "best" Gaussian width parameter. The strength of the correction monitored as a function of the temperature indicates that the results of the Gaussian propagator become questionable below a certain temperature. The interesting thermodynamic transition from a bounded trimer to three body dissociation lies in the temperature range for which the Gaussian approximation is predicted to be accurate.

(2012) Journal of Chemical Physics. 136, 9, 094101. Abstract
A continuum limit frozen Gaussian approximation is formulated for the reduced thermal density matrix for dissipative systems. The imaginary time dynamics is obtained from a novel generalized Langevin equation for the system coordinates. The method is applied to study the thermal density in a double well potential in the presence of Ohmiclike friction. We find that the approximation describes correctly the delocalization of the density due to quantization of the vibrations in the well. It also accounts for the friction induced reduction of the tunneling density in the barrier region.

(2012) Journal Of PhysicsCondensed Matter. 24, 10, 104001. Abstract
It is shown that a straightforward measure of the temperature dependence of energy resolved atomsurface scattering spectra measured under classical conditions can be related to the strength of the surface corrugation. Using classical perturbation theory combined with a Langevin bath formalism for describing energy transfer, explicit expressions for the scattering probabilities are obtained for both twodimensional, inplane scattering and full threedimensional scattering. For strong surface corrugations results expressed as analytic closedform equations for the scattering probability are derived which demonstrate that the temperature dependence of the scattering probability weakens with increasing corrugation strength. The relationship to the inelastic rainbow is briefly discussed.
2011

(2011) Journal of Physical Chemistry A. 115, 25, p. 71897198 Abstract
A stochastic theory is formulated for the sticking probability of a projectile scattered from a surface. The theory is then explored by applying it to a generalized Langevin equation model of the scattering dynamics. The theory succeeds in describing the known features of trapping on surfaces. At low energies sticking will occur only if there is an attractive interaction between the projectile and the surface. The probability of sticking at low energies is greater the lower the temperature and the deeper the attractive well of the particle as it approaches the surface. The sticking probability in the absence of horizontal friction tends to be lower as the stiffness of the surface increases. However, in the presence of horizontal friction, increased stiffness may lead to an increase in the sticking coefficient. A cos^{2}(θ _{i}) scaling is found only in the absence of corrugation and horizontal friction. The theory is then applied successfully to describe experimentally measured sticking probabilities for the scattering of Xe on a Pt(111) surface.

(2011) Journal of Chemical Physics. 134, 23, 234305. Abstract
Internal conversion is an inherently quantum mechanical process. To date, on the fly computation of internal conversion rates is limited to harmonic approximations, which would seem to be especially unsuitable, given that the typical transition to the ground electronic state occurs at energies which are far from the harmonic limit. It is thus of interest to study the applicability of the semiclassial initial value representation (SCIVR) approach which is in principle amenable to on the fly studies even with many degrees of freedom. In this paper we study the applicability of the HermanKluk (HK) SCIVR to a model system with two coupled and anharmonic degrees of freedom. We find that (a) the HK SCIVR is a good approximation to the exact quantum dynamics; (b) computation of the first order correction to the HKSCIVR approximation corroborates the accuracy; (c) by studying a large parameter range, we find that the harmonic approximation is mostly unsatisfactory; and (d) for the specific model used, the coupling between the modes was found to be relatively unimportant. These results imply that the HKSCIVR methodology is a good candidate for on the fly studies of internal conversion processes of large molecules.

(2011) Journal of Chemical Physics. 134, 13, 134104. Abstract
The frozen Gaussian approximation to the quantum propagator may be a viable method for obtaining "on the fly" quantum dynamical information on systems with many degrees of freedom. However, it has two severe limitations, it rapidly loses normalization and one needs to know the Gaussian averaged potential, hence it is not a purely local theory in the force field. These limitations are in principle remedied by using the HermanKluk (HK) form for the semiclassical propagator. The HK propagator approximately conserves unitarity for relatively long times and depends only locally on the bare potential and its second derivatives. However, the HK propagator involves a much more expensive computation due to the need for evaluating the monodromy matrix elements. In this paper, we (a) derive a new formula for the normalization integral based on a prefactor free HK propagator which is amenable to "on the fly" computations; (b) show that a frozen Gaussian version of the normalization integral is not readily computable "on the fly"; (c) provide a new insight into how the HK prefactor leads to approximate unitarity; and (d) how one may construct a prefactor free approximation which combines the advantages of the frozen Gaussian and the HK propagators. The theoretical developments are backed by numerical examples on a Morse oscillator and a quartic double well potential.

(2011) Journal of Chemical Physics. 134, 1, 011103. Abstract
Typically one expects that when a heavy particle collides with a surface, the scattered angular distribution will follow classical mechanics. The heavy mass usually assures that the coherence length of the incident particle in the direction of the propagation of the particle (the parallel direction) will be much shorter than the characteristic lattice length of the surface, thus leading to a classical description. Recent work on molecular interferometry has shown that extreme collimation of the beam creates a perpendicular coherence length which is sufficiently long so as to observe interference of very heavy species passing through a grating. Here we show, using quantum mechanical simulations, that the same effect will lead to quantum diffraction of heavy particles colliding with a surface. The effect is robust with respect to the incident energy, the angle of incidence, and the mass of the particle.

(2011) Journal of Chemical Physics. 134, 2, 024319. Abstract
Exact timedependent wavepacket calculations of helium atom scattering from model symmetric, chiral, and hexagonal surfaces are presented and compared with their classical counterparts. Analysis of the momentum distribution of the scattered wavepacket provides a convenient method to obtain the resulting energy and angle resolved scattering distributions. The classical distributions are characterized by standard rainbow scattering from corrugated surfaces. It is shown that the classical results are closely related to their quantum counterparts and capture the qualitative features appearing therein. Both the quantum and classical distributions are capable of distinguishing between the structures of the three surfaces. (C) 2011 American Institute of Physics. [doi:10.1063/1.3519811]

(2011) Journal of Chemical Physics. 134, 4, 044107. Abstract
Semiclassical Gaussian approximations to the Boltzmann operator have become an important tool for the investigation of thermodynamic properties of clusters of atoms at low temperatures. Usually, numerically expensive thawed Gaussian variants are applied. In this article, we introduce a numerically much cheaper frozen Gaussian approximation to the imaginary time propagator with a width matrix especially suited for the dynamics of clusters. The quality of the results is comparable to that of thawed Gaussian methods based on the singleparticle ansatz. We apply the method to the argon trimer and investigate the dissociation process of the cluster. The results clearly show a classicallike transition from a bounded moiety to three free particles at a temperature T ≈ 20 K, whereas previous studies of the system were not able to resolve this transition. Quantum effects, i.e., differences with the purely classical case manifest themselves in the lowtemperature behavior of the mean energy and specific heat as well as in a slight shift of the transition temperature. We also discuss the influence of an artificial confinement of the atoms usually introduced to converge numerical computations. The results show that restrictive confinements often implemented in studies of clusters can influence the thermodynamic properties drastically. This finding may have implications on other studies of atomic clusters.
2010

(2010) Chemical Physics. 375, 23, p. 337347 Abstract
In this work, we extend to three dimensions our previous stochastic classical theory on surface rainbow scattering. The stochastic phonon bath is modeled in terms of linear coupling of the phonon modes to the motion of the scattered particle. We take into account the three polarizations of the phonons. Closed formulae are derived for the angular and energy loss distributions. They are readily implemented when assuming that the vertical interaction with the surface is described by a Morse potential. The hard wall limit of the theory is derived and applied to some model corrugated potentials. We find that rainbow structure of the scattered angular distribution reflects the underlying symmetries of the surface. We also distinguish between "normal rainbows" and "super rainbows". The latter occur when the two eigenvalues of the Hessian of the corrugation function vanish simultaneously.

(2010) Physical review letters. 105, 13, 136101. Abstract
Measurements of the atomicscale motion of H and D atoms on the Pt(111) surface, above the crossover temperature to deep tunneling, are presented. The results indicate that quantum effects are significant up to the highest temperature studied (250K). The motion is shown to correspond to nearest neighbor hopping diffusion on a well defined fcc (111) lattice. The measurements provide information on the adiabatic potential of both the adsorption site and the transition state and give strong empirical support for a dissipative transitionstate theory description of the quantum contribution to the motion.

(2010) Journal Of PhysicsCondensed Matter. 22, 30, 304004. Abstract
A classical perturbation theory is developed to study rotational rainbow scattering of molecules from uncorrugated frozen surfaces. Considering the interaction of the rigid rotor with the translational motion towards the surface to be weak allows for a perturbative treatment, in which the known zeroth order motion is that of a freely rotating molecule hitting a surface. Using perturbation theory leads to explicit expressions for the angular momentum deflection function with respect to the initial orientational angle of the rotor that are valid for any magnitude of the initial angular momentum. The rotational rainbows appear as peaks both in the final angular momentum and rotational energy distributions, as well as peaks in the angular distribution, although the surface is assumed to be uncorrugated. The derived analytic expressions are compared with numerical simulation data. Even when the rotational motion is significantly coupled to the translational motion, the predictions of the perturbative treatment remain qualitatively correct.

(2010) Physical Review E. 81, 3, 036704. Abstract
A useful approximation for the thermal operator exp (β Ĥ) is based on its representation in terms of either frozen or thawed Gaussian states. Such approximate representations are leadingorder terms in respective series representations of the thermal operator. A numerical study of the convergence properties of the frozen Gaussian series representation has been recently published. In this paper, we extend the previous study to include also the convergence properties of the more expensive thawed Gaussian series representation of the thermal operator. We consider three different formulations for the series representation and apply them to a quartic doublewell potential to find that the thawed Gaussian series representation converges faster than the frozen Gaussian one. Further analysis is presented as to the convergence properties and the numerical efficiency of three different thawed Gaussian series representation. The unsymmetrized form converges most rapidly, however, the lower order approximations of the symmetrized forms are more accurate. Comparison with a standard discretized pathintegral evaluation demonstrates that the Gaussian based perturbation series representation converges much faster.

(2010) Physical review letters. 104, 11, 116103. Abstract
The rainbow is due to extrema of the angular deflection function of light impinging on water drops. Generically, extrema of suitably defined deflection functions lead to rainbows. These include angular and rotational rainbows in surface scattering and more. Here we introduce the concept of an "energyloss deflection function" for scattering of particles from a periodic surface whose extrema lead to a new formthe "energyloss rainbow" which appears as multiple maxima in the final energy distribution of the scattered particle. Energyloss rainbows are caused by frictional phonon effects which induce structure in the energyloss distribution instead of "washing it out." We provide evidence that they have been observed in Ne scattering on selfassembled monolayers.
2009

(2009) Physical Review A. 80, 5, 052103. Abstract
A generalized timedependent perturbation theory is derived for superoperators. Instead of using the "standard" breakup of the Hamiltonian into a known zeroth order term and a correction, we use the approximate superpropagator to define the correction superoperator which is then used to obtain a series representation of the exact Liouville operator. The theory reduces to known limits and may be used for a perturbation expansion of classical Wigner and Husimi dynamics as well as for recent phasespacebased semiclassical approximations. The theory is demonstrated for a model quartic potential.

(2009) Physical Review B. 80, 16, Abstract
Inplane atom surface scattering perturbation theory within a generalized Langevin equation formalism is proposed to account for the asymmetry found in angular distributions of heavy rare gas atoms scattered by corrugated surfaces. We show that when the surface corrugation is represented in terms of the first two (sine) Fourier components, one finds an asymmetric angular distribution. This asymmetry reflects the ratchetlike form of the effective corrugation. Adding in higherorder terms can also increase the number of rainbow scattering angles. Three rainbows are found for a secondorder sine term in the corrugation, four symmetrically spaced rainbow angles are found when adding in a secondorder cosine term to the corrugation. Analytic expressions for the angular distribution are derived in terms of a Morse oscillator model. The theory accounts well for the asymmetry and predicts its disappearance as the incident scattering angle is increased. It also features a decrease in the distance between the rainbow angles as the angle of incidence is increased and as the incident energy is increased. The theory is successfully applied to the experimental results of Kondo et al. [Eur. Phys. J. D 38, 129 (2006)] for the scattering of Ar on LiF(100) and the results of Amirav et al. [J. Chem. Phys. 87, 1796 (1987)] for the scattering of Xe on Ge(100) and Ar and Kr on Ag(100).

(2009) Physical Review B. 80, 11, 115404. Abstract
The scattering of argon atoms from a hydrogen saturated tungsten (100) surface was measured almost two decades ago by Schweizer [Surf. Sci. 249, 335 (1991)]. Angular distributions with rainbow features were measured as a function of surface temperature, incident kinetic energy and incident angle. In this paper, we show that a recently formulated classical Wigner theory of atom surface scattering accounts well for the measured distributions and their properties. Parameters were fit to a corrugated Morse potential, with Ohmic friction. Ab initio quantum chemistry computations verify that the fitted Morse potential parameters are in qualitative agreement with computed ArW and ArHW potentials of interaction.

(2009) Physical Review A. 79, 6, 062507. Abstract
The final momentum distribution for the scattering of He from a corrugated surface representation of Cu(110) is obtained from semiclassical theory. We derive a formally exact expression for the distribution which involves the absolute value squared of a single overlap of the initial wave function with the final momentum state. This reduces the number of phasespace integrals appearing in the semiclassical expressions and therefore leads to a large reduction in the computational effort. In addition, other energydependent observables are directly accessible from the momentum distribution without the need for further simulations. Using this formalism, we compare the quality of results obtained using a classical Wigner approximation and the frozen Gaussian, HermanKluk, and thawed Gaussian semiclassical propagators. We find that the thawed Gaussian is not only the best approximation, but it also converges more rapidly than the other semiclassical methods. The frozen Gaussian HermanKluk propagator is superior to the frozen Gaussian propagator. In contrast, the classical Wigner approach is qualitatively wrong as it does not properly account for the interference which dominates the angular distribution.

(2009) Journal of Chemical Physics. 131, 4, 044116. Abstract
Thawed Gaussian wavepackets have been used in recent years to compute approximations to the thermal density matrix. From a numerical point of view, it is cheaper to employ frozen Gaussian wave packets. In this paper, we provide the formalism for the computation of thermal densities using frozen Gaussian wave packets. We show that the exact density may be given in terms of a series, in which the zeroth order term is the frozen Gaussian. A numerical test of the methodology is presented for deep tunneling in the quartic double well potential. In all cases, the series is observed to converge. The convergence of the diagonal density matrix element is much faster than that of the antidiagonal one, suggesting that the methodology should be especially useful for the computation of partition functions. As a by product of this study, we find that the density matrix in configuration space can have more than two saddle points at low temperatures. This has implications for the use of the quantum instanton theory.

(2009) Journal of Chemical Physics. 130, 19, 194710. Abstract
A classical Wigner inplane atom surface scattering perturbation theory within the generalized Langevin equation formalism is proposed and discussed with applications to the ArAg(111) system. The theory generalizes the wellknown formula of Brako as well as the "washboard model." Explicit expressions are derived for the joint angular and final momentum distributions, joint final energy, and angular distributions as well as average energy losses to the surface. The theory provides insight into the intertwining between the energy loss and angular dependence of the scattering. At low energies the energy loss in the horizontal direction is expected to be large, leading to a shift of the maximum of the angular distribution to subspecular angles, while at high energies the energy loss in the vertical direction dominates, leading to a superspecular maximum in the angular distribution. The same effect underlies the negative slope of the average final (relative) energy versus scattering angle at low energies which becomes positive at high energies. The theory also predicts that the full width at half maximum of the angular distribution varies as the square root of the temperature. We show how the theory provides insight into the experimental results for scattering of Ar from the Ag(111) surface.

(2009) Journal of Chemical Physics. 130, 4, 041103. Abstract
The anharmonic S_{0} → S_{1} vibronic absorption spectrum of the formaldehyde molecule is computed on the fly using semiclassical dynamics. This first example of an onthefly semiclassical computation of a vibronic spectrum was achieved using a unit prefactor modified frozen Gaussian semiclassical propagator for the excited state. A sample of 6000 trajectories sufficed for obtaining a converged spectrum, which is in reasonable agreement with experiment. Similar agreement is not obtained when using a harmonic approximation for the spectrum, demonstrating the need for a full anharmonic computation. This first example provides a resolution of ∼100 cm _{}1. Potential ways of improving the methodology and obtaining higher resolution and accuracy are discussed.
2008

(2008) Biophysical Journal. 95, 9, p. 42584265 Abstract
The potential energy profile for many complex reactions of proteins, such as folding or allosteric conformational change, involves many different scales of molecular motion along the reaction coordinate. Although it is natural to model the dynamics of motion along such rugged energy landscapes as diffusional (the Smoluchowski equation; SE), problems arise because the frictional forces generated by the molecular surround are typically not strong enough to justify the use of the SE. Here, we discuss the fundamental theory behind the SE and note that it may be justified through a master equation when reduced to its continuum limit. However, the SE cannot be used for rough energy landscapes, where the continuum limit is ill defined. Instead, we suggest that one should use a mean first passage time expression derived from a master equation, and show how this approach can be used to glean information about the underlying dynamics of barrier crossing. We note that the potential profile in the SE is that of the microbarriers between conformational substates, and that there is a temperaturedependent, effective friction associated with the long residence time in the microwells that populate the rough landscape. The number of recrossings of the overall barrier is temperaturedependent, governed by the microbarriers and not by the effective friction. We derive an explicit expression for the mean number of recrossings and its temperature dependence. Finally, we note that the mean first passage time can be used as a departure point for measuring the roughness of the landscape.

(2008) Journal of Chemical Physics. 128, 16, Abstract
We present a theoretical study of the S(0)> S(1) and S(0)

(2008) Physical Review E. 77, 2, 021107. Abstract
The time correlation functions for a Gaussian wavepacket preparation of the dissipative harmonic oscillator evolving from three initial conditions for the heat bath are calculated and compared with each other for Ohmic heat baths. The three initial distributions for the bath are the factorized, partially factorized, and unfactorized distributions. Explicit analytical formulas are derived and then used to study the effect of the three initial distributions on the subsequent dynamics. We find that the transient behavior does not depend sensitively on the initial condition as long as the initial Gaussian wave function of the system is centered at the equilibrium point. Differences become noticeable as the center of the wave packet is significantly shifted from the equilibrium point. These observations justify to some extent the prevalent use of factorized initial conditions for studying real time quantum dynamics in dissipative systems. The total energy in the system is also calculated for the three initial states and its relation to features in the decay is pointed out.

(2008) Journal Of Physical Chemistry B. 112, 2, p. 213218 Abstract
We present a computation of the classical momentum and velocity correlation functions of Br_{2} considered as an idealized molecular wire connecting dissipated lead atoms at each end of the dimer. It is demonstrated that coupling of the diatomic relative momentum to the leads may result in momenta that are not equal to the massweighted velocity. These differences show up in numerical simulations of both the average value and time correlations of the bond momentum and velocity. These observations are supported by analytical predictions for the average temperature of the diatomic. They imply that the "standard recipes" for modeling the system with a generalized Langevin equation are insufficient.

(2008) Journal of Chemical Physics. 129, 5, 054107. Abstract
The scattering of atoms from surfaces is studied within the classical Wigner formalism. A new analytical expression is derived for the angular distribution and its surface temperature dependence. The expression is valid in the limit of weak coupling between the vertical motion with respect to the surface and the horizontal motion of the atom along the periodic surface. The surface temperature dependence is obtained in the limit of weak coupling between the horizontal atomic motion and the surface phonons. The resulting expression, which takes into account the surface corrugation, leads to an almost symmetric double peaked angular distribution, with peaks at the rainbow angles. The analytic expression agrees with model numerical computations. It provides a good qualitative description for the experimentally measured angular distribution of Ne and Ar scattered from a Cu surface.

(2008) Journal of Chemical Physics. 129, 6, 064515. Abstract
A recently formulated continuum limit semiclassical initial value series representation (SCIVR) of the quantum dynamics of dissipative systems is applied to the study of vibrational relaxation of model harmonic and anharmonic oscillator systems. As is well known, the classical dynamics of dissipative systems may be described in terms of a generalized Langevin equation. The continuum limit SCIVR uses the Langevin trajectories as input, albeit with a quantum noise rather than a classical noise. Combining this development with the forwardbackward form of the prefactorfree propagator leads to a tractable scheme for computing quantum thermal correlation functions. Here we present the first implementation of this continuum limit SCIVR series method to study two model problems of vibrational relaxation. Simulations of the dissipative harmonic oscillator system over a wide range of parameters demonstrate that at most only the first two terms in the SCIVR series are needed for convergence of the correlation function. The methodology is then applied to the vibrational relaxation of a dissipative Morse oscillator. Here, too, the SCIVR series converges rapidly as the first two terms are sufficient to provide the quantum mechanical relaxation with an estimated accuracy on the order of a few percent. The results in this case are compared with computations obtained using the classical Wigner approximation for the relaxation dynamics.
2007

(2007) Journal of Chemical Physics. 127, 7, Abstract
In this paper, we consider a dissipative system in which the system is coupled linearly to a harmonic bath. In the continuum limit, the bath is defined via a spectral density and the classical system dynamics is given in terms of a generalized Langevin equation. Using the path integral formulation and factorized initial conditions, it is well known that one can integrate out the harmonic bath, leaving only a path integral over the system degrees of freedom. However, the semiclassical initial value representation treatment of dissipative systems has usually been limited to a discretized treatment of the bath in terms of a finite number of bath oscillators. In this paper, the continuum limit of the semiclassical initial value representation is derived for dissipative systems. As in the path integral, the action is modified with an added nonlocal term, which expresses the influence of the bath on the dynamics. The first order correction term to the semiclassical initial value approximation is also derived in the continuum limit. (c) 2007 American Institute of Physics.

(2007) Journal of Chemical Physics. 126, 16, Abstract
There have been quite a few attempts in recent years to provide an initial value coherent state representation for the imaginary time propagator exp(beta H). The most notable is the recent time evolving Gaussian approximation of Frantsuzov and Mandelshtam [J. Chem. Phys. 121, 9247 (2004)] which may be considered as an expansion of the imaginary time propagator in terms of coherent states whose momentum is zero. In this paper, a similar but different expression is developed in which exp(beta H) is represented in a series whose terms are weighted phase space averages of coherent states. Such a representation allows for the formulation of a new and simplified forwardbackward semiclassical initial value representation expression for thermal correlation functions. (c) 2007 American Institute of Physics.

(2007) Physical Review E. 75, 4, 041103. Abstract
Usually one finds that dissipation tends to make a quantum system more classical in nature. We study the effect of momentum dissipation on a quantum system. The momentum of the particle is coupled bilinearly to the momenta of a harmonic oscillator heat bath. For a harmonic oscillator system we find that the position and momentum variances for momentum coupling are, respectively, identical to momentum and position variances for spatial friction. This implies that momentum coupling leads to an increase in the fluctuations in position as the temperature is lowered, exactly the opposite of the classicallike localization of the oscillator, found with spatial friction. For a parabolic barrier, momentum coupling causes an increase in the unstable normal mode barrier frequency, as compared to the lowering of the barrier frequency in the presence of purely spatial coupling. This increase in the frequency leads to an enhancement of the thermal tunneling flux, which below the crossover temperature becomes exponentially large. The crossover temperature between tunneling and thermal activation increases with momentum friction so that quantum effects in the escape are relevant at higher temperatures.

(2007) Journal of Chemical Theory and Computation. 3, 2, p. 344349 Abstract
The RayleighRitz functional is used in conjunction with an approximate time evolution to improve ab initio estimates of groundstate energies. The improvement is due in part to the introduction of a novel variational "normalization function" for the approximate propagator. An additional variational parameter was introduced in the form of a constant shift energy of the Hamiltonian. The approximate propagator used was the frozen Gaussian propagator; however, the trajectories evolved on the coherentstate averaged Hamiltonian (Q representation). For Coulombic forces, this removes the singularity, easing the computation. An additional variational parameter was the width parameter used for the coherent states appearing in the frozen Gaussian propagator. Using an initial combination of nine Gaussian functions for He, with an initial energy of 2.5115 au, the variational method, with a very short time interval of integration, led to an improved energy of 2.81 +/ 0.04 au.

The semiclassical initial value series representation of the quantum propagator(2007) p. 259271 Abstract
One of the central open challenges of the 21st century is the computation of real time quantum dynamics for systems with "many" degrees of freedom. A promising approach for obtaining approximate real time quantum dynamics is through the use of the semiclassical initial value approximation for the exact quantum propagator. The main drawback of this class of approximations was its ad hoc nature, it was in many senses an uncontrolled approximation scheme. This drawback has been recently remedied by showing that the semiclassical initial value representation (SCIVR) propagator is just a leading order term in a formally exact series representation of the true quantum propagator. In this review we present the SCIVR series representation, its successes and future challenges in applications to "large" systems. In addition, a new interaction representation initial value series representation for the exact quantum propagator is formulated.

(2007) Journal of Chemical Physics. 126, 16, 164108. Abstract
A numerical solution for the quantum dynamics of the spin boson problem is obtained using the semiclassical initial value series representation approach to the quantum dynamics. The zeroth order term of the series is computed using the new forwardbackward representation for correlation functions presented in the preceding adjacent paper. This leads to a rapid convergence of the Monte Carlo sampling, as compared to previous attempts. The zeroth order results are already quite accurate. The first order term of the series is small, demonstrating the rapid convergence of the semiclassical initial value representation series. This is the first time that the first order term in the semiclassical initial value representation series has been converged for systems with the order of 50 degrees of freedom.
2006

(2006) Molecular Physics. 104, 1, p. 1121 Abstract
The second order perturbation theory expression for the time dependent populations and rates of photoinduced electron transfer reactions has been previously derived by R.D. Coalson, D.G. Evans and A. Nitzan (J. chem. Phys., 101, 436 (1994)) and by M. Cho and R.J. Silbey (J. chem. Phys., 103, 595 (1995)). Here, we adapt these expressions for the study and analysis of the excitation laser frequency dependence of the time dependent populations and rates. Our model consists of a molecule with three electronic states, each supporting a manifold of harmonic internal vibrations of the molecule. In contrast to previous expectations, we find that in the region of significant absorption, the photoinduced electron transfer rate is almost independent of the frequency and the temporal width of the excitation laser. This conclusion implies that control of the excitation rate through the excitation laser frequency is possible only if external noise destroys the coherence of the excitation process.

(2006) Physical Review E. 73, 4, 041105. Abstract
Stochastic acceleration, defined in terms of a stochastic equation of motion for the acceleration, is derived from a Hamiltonian model. A free particle is coupled bilinearly to a harmonic bath through the particle's momentum and coordinate. Under appropriate conditions, momentum coupling induces velocity diffusion which is not destroyed by the spatial coupling. Spatialmomentum coupling may induce spatial subdiffusion. The thermodynamic equilibrium theory presented in this paper does not violate the second law of thermodynamics, although the average velocity squared of the particle may increase in time without bound.

(2006) Journal of Chemical Physics. 125, 13, 133502. Abstract
Frantsuzov and Mandelshtam [J. Chem. Phys. 121, 9247 (2004)] have recently demonstrated that a time evolving Gaussian approximation (TEGA) to the imaginary time propagator exp (ΒH) is useful for numerical computations of anharmonically coupled systems with many degrees of freedom. In this paper we derive a new exact series representation for the imaginary time propagator whose leading order term is the TEGA. One can thus use the TEGA not only as an approximation but also to obtain the exact imaginary time propagator. We also show how the TEGA may be generalized to provide a family of TEGA's. Finally, we find that the equations of motion governing the evolution of the center and width of the Gaussian may be thought of as introducing a quantum friction term to the classical evolution equations.

(2006) Journal of Chemical Physics. 125, 16, 164104. Abstract
The forwardbackward (FB) approximation as applied to semiclassical initial value representations (SCIVR's) has enabled the practical application of the SCIVR methodology to systems with many degrees of freedom. However, to date a systematic representation of the exact quantum dynamics in terms of the FBSCIVR has proven elusive. In this paper, we provide a new derivation of a forwardbackward phase space SCIVR expression (FBPSSCIVR) derived previously by Thompson and Makri [Phys. Rev. E 59, R4729 (1999)]. This enables us to represent quantum correlation functions exactly in terms of a series whose leading order term is the FBPSSCIVR expression. Numerical examples for systems with over 50 degrees of freedom are presented for the spin boson problem. Comparison of the FBPSSCIVR with the numerically exact results of Wang [J. Chem. Phys. 113, 9948 (2000)] obtained using a multiconfigurational time dependent method shows that the leading order FBPSSCIVR term already provides an excellent approximation.
2005

(2005) Journal Of PhysicsCondensed Matter. 17, 49, p. S4133S4150 Abstract
An elementary process occurring on surfaces is diffusion. The dynamics is simplest when the concentration of adsorbates is sufficiently small that interaction between adsorbates can be ignored. But even for this tracer diffusion process, much remains to be uncovered. Here, we present the interplay between experimental measurement of tracer diffusion and its theoretical interpretation, which leads to good estimates of the interaction of the adparticle with the surface. We show how the results from three different experimental techniques  field ion microscopy, scanning tunnelling microscopy and quasielastic helium atom scattering  can be interpreted. Using the generalized Langevin equation as a model for the diffusion dynamics, we show how the turnover theory for activated diffusion may be used to describe the measured time evolution of the adparticle distribution on the surface. The different activation energy measured for hopping over single or double lattice lengths is shown to come from the added energy loss to the surface, as the particle moves over the longer path. We discuss some of the issues which are not yet clear; these include quantum effects, such as the quantum suppression of diffusion, vibrationally assisted diffusion, multidimensional effects and diffusion in the presence of external fields.

(2005) Chaos. 15, 2, 026116. Abstract
A brief history is presented, outlining the development of rate theory during the past century. Starting from Arrhenius [Z. Phys. Chem. 4, 226 (1889)], we follow especially the formulation of transition state theory by Wigner [Z. Phys. Chem. Abt. B 19, 203 (1932)] and Eyring [J. Chem. Phys. 3, 107 (1935)]. Transition state theory (TST) made it possible to obtain quick estimates for reaction rates for a broad variety of processes even during the days when sophisticated computers were not available. Arrhenius' suggestion that a transition state exists which is intermediate between reactants and products was central to the development of rate theory. Although Wigner gave an abstract definition of the transition state as a surface of minimal unidirectional flux, it took almost half of a century until the transition state was precisely defined by Pechukas [Dynamics of Molecular Collisions B, edited by W. H. Miller (Plenum, New York, 1976)], but even this only in the realm of classical mechanics. Eyring, considered by many to be the father of TST, never resolved the question as to the definition of the activation energy for which Arrhenius became famous. In 1978, Chandler [J. Chem. Phys. 68, 2959 (1978)] finally showed that especially when considering condensed phases, the activation energy is a free energy, it is the barrier height in the potential of mean force felt by the reacting system. Parallel to the development of rate theory in the chemistry community, Kramers published in 1940 [Physica (Amsterdam) 7, 284 (1940)] a seminal paper on the relation between Einstein's theory of Brownian motion [Einstein, Ann. Phys. 17, 549 (1905)] and rate theory. Kramers' paper provided a solution for the effect of friction on reaction rates but left us also with some challenges. He could not derive a uniform expression for the rate, valid for all values of the friction coefficient, known as the Kramers turnover problem. He also did not establish the connection between his approach and the TST developed by the chemistry community. For many years, Kramers' theory was considered as providing a dynamic correction to the thermodynamic TST. Both of these questions were resolved in the 1980s when Pollak [J. Chem. Phys. 85, 865 (1986)] showed that Kramers' expression in the moderate to strong friction regime could be derived from TST, provided that the bath, which is the source of the friction, is handled at the same level as the system which is observed. This then led to the Mel'nikovPollakGrabertHänggi [Mel'nikov and Meshkov, J. Chem. Phys. 85, 1018 (1986); Pollak, Grabert, and Hänggi, J. Chem. Phys. 91, 4073 (1989)] solution of the turnover problem posed by Kramers. Although classical rate theory reached a high level of maturity, its quantum analog leaves the theorist with serious challenges to this very day. As noted by Wigner [Trans. Faraday Soc. 34, 29 (1938)], TST is an inherently classical theory. A definite quantum TST has not been formulated to date although some very useful approximate quantum rate theories have been invented. The successes and challenges facing quantum rate theory are outlined. An open problem which is being investigated intensively is rate theory away from equilibrium. TST is no longer valid and cannot even serve as a conceptual guide for understanding the critical factors which determine rates away from equilibrium. The nonequilibrium quantum theory is even less well developed than the classical, and suffers from the fact that even today, we do not know how to solve the real time quantum dynamics for systems with "many" degrees of freedom.

(2005) Journal of Chemical Theory and Computation. 1, 3, p. 439443 Abstract
Short time information on the time evolution of wave packets is combined with the variational theorem to determine eigenvalues and eigenfunctions. As in the Filter Diagonalization Method the input that is needed is a correlation function and its time derivative. The method is iterative and convergent. The time interval needed is short, for example, the determination of tunneling splitting energies δEis obtained in a time interval which is substantially shorter than the Fourier time 2πp/δE. The method is applied to some model problems including determining the ground tunneling state in a quartic double well potential using numerically exact short time results obtained from the semiclassical initial value representation series of the exact propagator. This is another example in which tunneling is obtained using only coherent classical paths. Implications of the method for ab initio computation of molecular electronic energies is discussed.

(2005) New Journal of Physics. 7, 022. Abstract
A secondorder cumulant expansion is used to derive continuum limit expressions for the electronic absorption spectrum of a polyatomic molecule interacting with a bath, within the Condon approximation and weak fields. The small expansion parameter is the difference between the vibrational Hamiltonians in the ground and excited electronic states. The secondorder cumulant expansion is shown to be a good approximation for a reasonable model of a polyatomic molecule with 45 degrees of freedom. Friction tends to shift the maximum in the absorption peak to the blue. When the vibrational frequencies in the excited electronic state are lower than those in the ground electronic state, one finds a stochastic resonance feature. Friction first narrows the peak and then broadens it. This narrowing is absent when one shifts only the equilibrium positions in the excited state relative to the ground state.

(2005) Journal of Physical Chemistry A. 109, 1, p. 122132 Abstract
A correlation function formalism is applied to compute the twophoton absorption spectrum of benzene. Using harmonic Hamiltonians for the ground and excited electronic states, we find that the theory agrees qualitatively with the experimentally observed sparsity of the thermal twophoton absorption spectrum as compared with the singlephoton absorption spectrum. An expression for the average vibrational energy in the excited state is derived. We find that cooling of the nascent vibrational energy in the electronically excited state is not as extensive in the twophoton absorption process as compared to the singlephoton case.

(2005) Journal of Chemical Theory and Computation. 1, 3, p. 345352 Abstract
One of the central advantages of the Herman Kluk Semiclassical Initial Value Representation (SCIVR) of the quantum propagator is that through its prefactor it approximately conserves unitarity for relatively long times. Its main disadvantage is that the prefactor appearing in the SCIVR propagator is expensive to compute as the dimensionality of the problem increases. When using the SCIVR series method for computation of the numerically exact quantum dynamics, the expense becomes even larger, since each term in the series involves a product of propagators, each with its own prefactor. This expense can be eliminated if one uses prefactor free propagators; however, these do not conserve unitarity as well as the HK propagator. As a compromise, we suggest the use of a hybrid propagator, in which the system variables are treated with the HermanKluk prefactor, while the bath variables are treated as prefactor free. Numerical application to a quartic oscillator coupled bilinearly to five harmonic bath oscillators demonstrates the viability of the hybrid method. The results presented are also a first application of the SCIVR series method to a system with six degrees of freedom. Convergence to the numerically exact answer using Monte Carlo sampling is obtained with at most the first two terms in the SCIVR series.
2004

(2004) Physical review letters. 93, 14, p. 14040111404014 140401. Abstract
The deep quantum tunnelling was analyzed using coherent classical paths involving real time classical trajectories. It was observed that a (Gaussian) wave packet, localized on one side of a barrier with mean energy and an energy variance significantly smaller than the barrier height was scattered off the barrier. It was found that the classical path contribution became too small, when the barrier was not parabolic. The results show that the wave packet will always possess a tail whose energy is larger than the barrier light.

(2004) Journal of Physics A: Mathematical and General. 37, 41, p. 96699676 Abstract
A general expression for thawed semiclassical initialvalue representation propagators has been derived in the multidimensional form. The thawed Gaussian propagator of Heller and the coherentstateaveraged thawed Gaussian propagator of Baranger et al (2001 J. Phys. A: Math. Gen. 34 7227) are some examples of the more general class. The derivation is based on the demand that the correction operator associated with the semiclassical propagator includes only cubic and higherorder terms of the averaged potential.

(2004) Journal of Physical Chemistry A. 108, 39, p. 77787784 Abstract
We present a theoretical study of the effect of Dushinskii rotations on the vibrational population created in an excited electronic state through photoexcitation. Special attention is given to the effect of Dushinskii rotations on the possibility of cooling the vibrational population in the excited state, relative to the thermal distribution in the ground state. The absorption spectrum and corresponding average energy in the excited state are calculated using a closedform expression for the harmonic correlation function between the ground and excited electronic states, which includes the effects of Dushinskii rotations, equilibrium position shifts, and frequency shifts between the excited and groundelectronicstate normal modes. We investigate numerically the separate and joint effects of rotation, position shifts, and frequency shifts on the absorption spectrum and average vibrational energy in the excited electronic state. We find that, although the Dushinskii rotations generally diminish the cooling effect, the effect does not disappear and, in some cases, may also increase slightly.

(2004) Journal of Chemical Physics. 121, 8, p. 33843392 Abstract
A recently developed class of prefactor free semiclassical initial value representations (SCIVR) of the quantum propagator was discussed. A numerical study of the prefactor free SCIVR series for scattering through a double slit potential was also discussed. The SCIVR series was also computed using the optimized HermanKulk SCIVR. The results show that the sum of the zeroth order and the first order terms in the series is sufficient for an accurate determination of the diffraction pattern.

(2004) Journal of Chemical Physics. 120, 22, p. 1076810779 Abstract
An analytical theory which was based on a Hamiltonian equivalent of the generalized Langevin equation, for the line shape, temperaturedependent shift and broadening of the translational or Tmode peak is presented. The theory can be used to infer physical parameters of the adatomsurface interaction. For the line shape a firstorder perturbative solution of the normalmode coordinates was used. For the shift and broadening, a perturbative expansion in the instantaneous system frequency was employed.

(2004) Journal of Chemical Physics. 120, 20, p. 96309637 Abstract
In a previous paper [J. Chem. Phys. 119, 11864 (2003)], we derived a set of two coupled equations which describe electron transfer in the presence of dissipation at high temperature. Employing the low temperature extension of the FokkerPlanck operator, suggested by Haake and Reibold [Phys. Rev. A 32, 2462 (1985)] and Ankerhold [Europhys. Lett. 61, 301 (2003)], we show that one may extend the generalized Zusman equations in a similar manner to low temperature. Numerical simulation shows that addition of the temperaturedependent term which couples the coordinate and momentum causes an increase in the electron transfer rate as compared to the rate obtained from the previous high temperature equations. The increase in the rate comes from the increase in the equilibrium variances of the coordinate and momentum. The low temperature quantum theory allows for higher energy portions of phase space to contribute to the electron transfer rate where the rate is higher thus enhancing the overall rate. (C) 2004 American Institute of Physics.
2003

(2003) Journal of Chemical Physics. 119, 22, p. 1186411877 Abstract
An analytic study of the density matrix and Wigner representation equations for dissipative electron transfer is presented. An explicit expression is derived for the offdiagonal Green's function, which shows a very fast relaxation in time if the barrier to reaction is greater than the thermal energy. This fast relaxation invalidates previous attempts to derive coupled equations for the density in the large friction limit. The fast offdiagonal relaxation disallows an adiabatic elimination of the momentum even in the large friction limit. We then show, with the aid of the boundary layer method, how one can use the same analysis to derive a set of two coupled equations for the diagonal densities. These equations are a generalization to phase space of the large friction Zusman equations [Chem. Phys. 49, 295 (1980)]. Adiabatic elimination of the momentum from these generalized Zusman equations is correct in the large friction limit and naturally leads back to the Zusman equations. Numerical solution of the generalized Zusman equations is presented for symmetric electron transfer for weak and strong electronic coupling, moderate and high barriers, and a large range of damping. The numerical results provide new insight into the friction dependence of the rate in the weak damping regime and show that previous analytic expressions for the rate are only qualitative in nature. (C) 2003 American Institute of Physics.

(2003) Journal of Chemical Physics. 119, 21, p. 1105811063 Abstract
The SCIVR series representation of the quantum propagator were used to optimize the width parameter of the coherent state. Such optimization led to improved convergence of the series. It was shown that the SCIVR series can be computed using Monte Carlo methods.

(2003) Journal of Chemical Physics. 119, 20, p. 1094110952 Abstract
The kinetics of onedimensional particle diffusion over the wells of a periodic potential when the escape and trapping of particles is limited by energy relaxation due to coupling with a bath were analyzed. The energy relaxation mechanism was determined by the form of the energy relaxation operator. An analysis of the diffusion dynamics for different energy relaxation models was performed, focusing on the Gaussian and exponential forms. In addition, some generalizations, valid for any reasonable kernel and presence of quantum effects, were detailed.

(2003) Journal of Physical Chemistry A. 107, 37, p. 71127117 Abstract
A systematic method is developed to obtain increasingly accurate semiclassical initial value representation (IVR) approximations to the exact quantum propagator. The main result is a series of correction terms of increasing order in a "correction operator", which describes the difference between the exact evolution equation and the equation obeyed by the semiclassical propagator. Each term in the series involves only phase space integrals of classical trajectories and is therefore, in principle, amenable to numerical computation. The properties of the "correction operator" are studied for three different representations of the semiclassical propagator. For initial times, we find that the propagator suggested recently by Baranger et al. is superior to a thawed Gaussian propagator or the HermanKluk propagator.

(2003) Journal of Chemical Physics. 119, 5, p. 27802791 Abstract
A study was performed on Kramers' turnover theory for diffusion of Na atoms on a Cu(001) surface. The measurement was performed by helium scattering. Kramer's theory determined the hopping distribution in terms of effective frequency and the energy loss of the particle to the bath as it traverses from one barrier to the next.

(2003) Journal of Chemical Physics. 118, 10, p. 43574364 Abstract
The dynamics and dissipative tunneling in a symmetric quartic double well potential was studied with the aim of numerically investigating the WignerFokkerPlanck (WFP) equation. Numerical results were compared with the numerically exact path integral results of Stockburger and Mak for the position autocorrelation function in a symmetric quartic double well potential. Good agreement was obtained, indicating that the WFP equation manages to treat well both coherences as well as tunneling phenomena.

(2003) Physical review letters. 91, 19, Abstract
A new exact representation of the quantum propagator is derived in terms of semiclassical initial value representations. The resulting expression may be expanded in a series, of which the leading order term is the semiclassical one. Motion of a Gaussian wave packet on a symmetric double well potential is used to demonstrate numerical convergence of the series and the ability to compute each element in the series using Monte Carlo methods.

(2003) p. 136150 Abstract
The study of quantum stochastic processes presents severe difficulties, both on the theory level as well as on technical grounds. The numerically exact solution remains prohibitive even today. In this paper we review and present new results for three different methods used for the modelling of quantum stochastic processes. These include a mixed quantum classical approach, semiclassical initial value representations of the quantum propagator and the reduced density matrix approach as typified by the quantum WignerFokkerPlanck equation. A new semiclassical initial value representation that does away with cumbersome prefactors which depend on the monodromy matrix elements but is exact for a harmonic oscillator is presented and its properties analysed. A recently proposed systematic method for improving semiclassical initial value representations is reviewed. The generalization of the WignerFokkerPlanck equation to stochastic processes with memory is obtained by using a novel integral equation representation.
2002

(2002) Computer Physics Communications. 147, 3, p. 759769 Abstract
The denoising characteristics for the representation of experimental data in terms of the Hermite Distributed Approximating Functionals (HDAF's) are analyzed with respect to signals corrupted with Gaussian noise. The HDAF performance is compared to both the ideal window and running averages representations of the same data. We find that the HDAF filter combines the best features of both. That is, the HDAF filter provides approximately the same noise reduction and bandwidth as the ideal filter while at the same time remaining limited in range in both the physical and Fourier spaces.

(2002) Journal of Chemical Physics. 116, 14, p. 59255932 Abstract
An attempt was made to check the analytical properties of the semiclassical initial value representation (IVR) propagator. It was found that the IVR approximation is not exact at short times. However, the deviation does not seem to be too important for typical smooth potentials of interest.

(2002) Journal of Chemical Physics. 116, 14, p. 60886101 Abstract
An attempt is made to understand the origin and extent of photoinduced cooling of the vibrational population in the excited electronic state. A theory is given which suggests that vibrational cooling is observable in the naphthalene molecule. The pressure, frequency, and width dependence of the fluorescence decay provide unique insight into the excitation and decay dynamics of photoexcited polyatomic molecules.

(2002) Journal of Chemical Physics. 116, 7, p. 27182727 Abstract
A general formulation of mixed quantum classical rate theory (MQCLT) for dissipative systems was derived. The resulting theory was applied to the model system of a quartic double well potential studied by Topaler and Makri. It was shown that the MQCLT provides a substantial improvement over previous quantum transition state theory as well as centroid transition state theory computations.
2001

(2001) Journal of Physical Chemistry A. 105, 49, p. 1096110966 Abstract
An ab initio harmonic study is presented for the nascent vibrational energy distribution of roomtemperature benzene when photoexcited to the S_{1} state. The dependence on photoexcitation frequency and pulse width is investigated. We find, that even though the transition is symmetry disallowed, the HerzbergTeller mechanism by which the nuclear motion induces the transition, can lead to cooling of the molecule at the transition frequencies corresponding to a mode of E_{2g} symmetry. The extent of cooling decreases with increasing pulse width, but even with a pulse width of 90 cm^{1} one still finds significant cooling of the vibrational population. Cooling is also found for deuterated benzene. The energy deposited in the molecule is found to be very sensitive to the excitation frequency, provided that the pulse width is sufficiently narrow.

(2001) Journal of Chemical Physics. 115, 15, p. 68766880 Abstract
An attempt is made to illuminate the relationship between Quantum Transition State Theory (QTST) and Mixed Quantum CLassical rate Theory (MQCLT) on the one hand and the HansenAndersen approach on the other. It is shown that MQCLT gives the exact first couple of nonzero initial time derivatives of the fluxflux autocorrelation function, hence, it falls under the category of approximations considered by Hansel and Andersen.

(2001) Journal of Chemical Physics. 115, 4, p. 18671874 Abstract
The microcanonical energy dependent electron transfer rate constant was calculated both in the activated and in the activationless regime using the recently developed Short Time Inverse Laplace Transform (STILT) method. Results show that STILT is also a viable method for obtaining microcanonical electron transfer rates for anharmonic coupled systems.

(2001) Journal Of Physical Chemistry B. 105, 28, p. 65006506 Abstract
The nature of the nascent vibrational distribution in the excited donor state in photoinduced electron transfer is shown to have a profound effect on the electrontransfer rate. In polyatomic molecules, excitation at wavelengths in the vicinity of the ground state to ground state excitation frequency may lead to significant cooling of the excited vibrational state distribution. This cooling is shown to lead to a slowing down of the electrontransfer rate. A theory for photoinduced electron transfer is developed to include the nonequilibrium nature of the excited donor vibrational distribution. The rate expression is shown to be the standard Golden rule thermal rate expression but at an effective temperature which depends on the ground electronic state temperature and the photoexcitation frequency. A simple numerical model is presented to demonstrate the cooling and control of the electrontransfer rate by variation of the excitation frequency.

(2001) Chemical Physics. 268, 13, p. 295313 Abstract
Two formulations of quantum transition state theory (QTST) for dissipative systems, based on the symmetrized and Kubo form of the thermal flux operator are presented. Numerical results for a symmetric double well potential are compared with the numerically exact results of Topaler and Makri [J. Chem. Phys. 101 (1994) 7500] and with centroid transition state theory. The two forms give similar answers and are similar in accuracy to the centroid theory. QTST however, is found to always bounds the numerically exact result from above. QTST can be further improved, using a variational theory or by using the mixed quantum classical version of the theory.

(2001) Journal of Chemical Physics. 114, 22, p. 97419746 Abstract
Mixed quantum classical rate theory (MQCLT) was applied to the collinear hydrogen exchange reaction on the PK II potential energy surfaces and LiuSiegbahnTruhlarHorowitz (LSTH) potential energy surface. Thermal reaction rate was computed by combining classical trajectories with a numerically exact quantum Monte Carlo evaluation of the thermal flux operator. The MQCLT results were compared to centroid rate theory computations and quantum transition state theory (QTST). Computed rates were found to effect the results from above for temperatures ranging from T=200K to T=1000K.


(2001) Theoretical Chemistry Accounts. 105, 3, p. 173181 Abstract
Fourier transforms occur in a variety of chemical systems and processes. A few examples include obtaining spectral information from correlation functions, energy relaxation processes, spectral densities obtained from force autocorrelation functions, etc. In this article, a new functional transform, named the dual propagation inversion (DPI) is introduced. The DPI functional transform can be applied to a variety of problems in chemistry, such as Fourier transforms of time correlation functions, energy relaxation processes, rate theory, etc. The present illustrative application is to generating the frequency representation of a discrete, truncated timedomain signal. The DPI result is compared with the traditional Fourier transform applied to the same truncated time signal. For both noisefree and noisecorrupted timetruncated signals, the DPI spectrum is found to be more accurate, particularly as the signal is more severely truncated. In the DPI, the distributedapproximatingfunctional free propagator is used to propagate and denoise the signal simultaneously.
2000

An approximate short time Laplace transform inversion method(2000) Journal of Chemical Physics. 113, 11, p. 45334548 Abstract
The "standard" numerical methods used for inverting the Laplace transform are based on a regularization of an exact inversion formula. They are very sensitive to noise in the Laplace transformed function. In this article we suggest a different strategy. The inversion formula we use is an approximate one, but it is stable with respect to noise. The new approximate expression is obtained from a short time expansion of the Bromwich inversion formula. We show that this approximate result can be significantly improved when iterated, while remaining stable with respect to noise. The iterated method is exact for the class of functions of type E^{m}e^{aE}. The method is applied to a harmonic model of the stilbene molecule, to a truncated exponent series, and to the fluxflux correlation function for the parabolic barrier. These examples demonstrate the utility of the method for application to problems of interest in molecular dynamics.

(2000) Journal of Physical Chemistry A. 104, 9, p. 17991803 Abstract
The recently formulated quantum transition state theory (QTST) in which the quantum projection operator is approximated by its parabolic barrier limit and the symmetrized thermal flux is evaluated numerically exactly, is applied to the collinear hydrogen exchange reaction. The results are found to bound the exact results from above for temperatures ranging from T = 200 K to T = 1000 K. The QTST rate is almost exact at high temperature and is a factor of 3.7 greater than the exact rate at T = 200 K, where there is extensive tunneling. Contour plots of the quantum transition state theory reactive flux reveal that the theory accounts well for the "corner cutting" observed in the collinear hydrogen exchange reaction at low temperatures. These results demonstrate that one may estimate quantum rates of bimolecular reactions, using only thermodynamic information.

(2000) Journal of Chemical Physics. 112, 9, p. 39383941 Abstract
The room temperature photoinduced fluorescence decay of isolated transstilbene and transstilbene in the presence of 1 atm of Ar gas was measured as a function of the excitation laser frequency. Lifetimes were measured both to the blue and the red of the ground vibrational state of the ground electronic state (S0) to the ground vibrational state of the S1 state transition frequency omega(00). The lifetime was found to decrease on both sides of omega(00). The dependence of the decay rate on laser frequency in the presence of Ar gas is much weaker than for the isolated molecule. Both observations corroborate previous theoretical predictions of laser cooling of thermal transstilbene upon excitation at the omega(00) frequency. The experimental results are in good agreement with theoretical analysis. (C) 2000 American Institute of Physics. [S00219606(00)018092].

(2000) Annalen der Physik. 9, 9, p. 764775 Abstract
The bridge length dependence of the classical transfer rate from donor to acceptor is studied for symmetric bridged systems. The reaction rate is shown to be factorizable into an escape rate from the donor well and a transmission factor through the bridge. As expected for a diffusing particle this transmission factor is inversely proportional to the bridge length, for long bridges. The PollakGrabertHanggi turnover theory is shown to be applicable for all friction strengths and bridge lengths studied in this paper.
1999

(1999) Journal of Chemical Physics. 111, 16, p. 72447254 Abstract
The recently formulated mixed quantum classical rate theory (MQCLT) is implemented for a model system with two degrees of freedom. In MQCLT, one must compute the Wigner representation of the symmetrized thermal flux operator. This phase space flux distribution is then multiplied by the classical reaction probability to obtain the rate. The major computational difficulty is the multidimensional Fourier transform necessary for obtaining the Wigner distribution. The Fourier transform reintroduces a sign problem when attempting to estimate the MQCLT rate using Monte Carlo methods. Two different methods for overcoming the sign problem are explored in this paper. Numerical results are presented for a model problem of an Eckart barrier coupled bilinearly to a slow oscillator and compared with numerically exact results.

(1999) Surface Science. 437, 1, p. 198206 Abstract
A theory is presented for the diffusion coefficient and the hopping distribution of an adatom on a surface in the presence of external fields. Relatively simple expressions are derived for the probability of multiple hops in the exponential hopping limit. This limit is the one which is usually found in the diffusion of a metal atom on a metal surface. In this limit the barrier height (in units of k_{B}T) is large compared with the bias created by the field and the energy loss of the particle as it traverses from one barrier to the next. The hopping distribution is obtained for constant and time varying fields in the adiabatic limit. Typically, the presence of an external field will increase the probability of long hops. The magnitude of the field needed to appreciably increase the probability of multiple hops is 10^{8}10^{9}V m^{1}.

Numerical inversion of the Laplace transform(1999) Journal of Chemical Physics. 110, 23, p. 1117611186 Abstract
A generalization of Doetsch's formula [Math. Z. 42, 263 (1937)] is derived to develop a stable numerical inversion of the onesided Laplace transform (C) over cap (beta). The necessary input is only the values of C ( b) on the positive real axis. The method is applicable provided that the functions (C) over cap (beta) belong to the function space L(alpha)(2) defined by the condition that G(x) = e(x alpha)(C) over cap(e(x)), alpha>0 has to be square integrable. The inversion algorithm consists of two sequential Fourier transforms where the second Fourier integration requires a cutoff, whose magnitude depends on the accuracy of the data. For high accuracy data, the cutoff tends to infinity and the inversion is very accurate. The presence of noise in the signal causes a lowering of the cutoff and a lowering of the accuracy of the inverted data. The optimal cutoff value is shown to be one which leads to an inversion which remains consistent with the original data and its noise level. The method is demonstrated for some model problems: a harmonic partition function, resonant transmission through a barrier, noisy correlation functions, and noisy Monte Carlo generated data for tunneling coefficients obtained via the recently introduced quantum transition state theory (QTST). (C) 1999 American Institute of Physics. [S00219606(99)004213].

(1999) Journal of Chemical Physics. 110, 24, p. 1189011905 Abstract
A detailed theoretical study is presented for the vibrational population distribution of polyatomic molecules which results from electronic excitation from a thermal ground state. If the vibrational frequencies of the excited state are lower than the groundstate frequencies and if position shifts are not too large, then there exist excitation frequencies for which the excitedstate vibrational distribution will be cooled in comparison to the ground state. An analytic theory for the vibrational distribution in the excited state is obtained by noting that the fast dephasing of a polyatomic molecule after excitation allows for the development of a Gaussian approximation for the excitation process. We show that the equilibrium energy distribution of a polyatomic molecule as well as the nascent distribution after excitation are well approximated as Gaussian. The average energy in the excited state is then found to be a quadratic function of the excitation frequency. If cooling takes place, it will usually be maximal for an excitation frequency which is to the red of the ground electronic state to ground electronic state excitation frequency. Cooling is not necessarily a quantum effect, it may also be found in the classical limit, in which one ignores quantization of the vibrational levels. The generality of the Gaussian approximation opens the way for theoretical treatment of anharmonic polyatomic molecules, using quantum Monte Carlo techniques. (C) 1999 American Institute of Physics.

(1999) Journal of Chemical Physics. 110, 17, p. 82468253 Abstract
A new method is given for the computation of quantum mechanical microcanonical densities of states of large molecules. The method is based on the observation that for large molecules with many vibrational degrees of freedom, the complex time partition function dephases rapidly allowing for a good shorttime approximation. The shorttime approximation up to third order gives an Airy function expression for the thermal density of states at a given temperature T. The microcanonical density of states is then deduced from the thermal density. The input needed for the method is the first three moments of the Hamiltonian at a series of temperatures, which adequately cover the energy range of interest. These may be computed using quantum Monte Carlo methods. The method is tested for a harmonic model of transstilbene, a separable anharmonic model of cyclopropane, and a separable anharmonic model of a system with 50 degrees of freedom. The shorttime Airy method is found to give accurate estimates for the density of states, the integrated density of states, and RRKM microcanonical rate constants. (C) 1999 American Institute of Physics. [S00219606(99)306164].

(1999) Surface Science. 421, 12, p. 7388 Abstract
Kramers' theory is used to derive simple expressions for the hopping distribution in multidimensional activated surface diffusion. The expressions are tested against one and twodimensional numerically exact simulations. The present expressions are valid provided that the average energy loss of the particle as it goes from one barrier to the next is of the order of k_{B}T or more. The ratio of double hops to single hops is shown to obey an Arrheniuslike behavior, with a prefactor that is proportional to T. The added activation energy is proportional to the average energy loss of the diffusing particle. The magnitude of the energy loss depends on the coupling between modes: the stronger the coupling, the larger the energy loss and the smaller is the multiple hopping probability. The theory is used to analyze recent experiments on the diffusion of the Pt atom on a Pt(110)(1×2) missing row reconstructed surface.

(1999) Journal of Chemical Physics. 110, 1, p. 8087 Abstract
A recently formulated quantum transition state theory is applied to scattering over an Eckart barrier coupled bilinearly to a harmonic mode. Results are compared with the numerically exact and the centroid density method computations of McRae et al. [J. Chem. Phys. 97, 7392 (1992)]. We find that quantum transition state theory is of comparable accuracy to the centroid approximation for all parameter ranges studied.
1998

(1998) Europhysics Letters. 44, 4, p. 416422 Abstract
We investigate the motion of an overdamped Brownian particle in a periodic potential with weak thermal noise and a timeperiodic unbiased (i.e. 〈F(t)〉 = 0) external driving force F(t). By introducing appropriate "waitingperiods", where F(t) vanishes, an arbitrarily strong enhancement of diffusion in a symmetric potential is possible. In asymmetric periodic potentials (ratchets) the net flux of particles can be directed in both directions, even in the absence of thermal noise. For finite temperatures we observe and explain additional, purenoiseinduced flux reversal phenomena.

(1998) Chemical Physics. 235, 13, p. 131146 Abstract
Many unimolecular reactions are initiated by photoexcitation of a polyatomic molecule at room temperature from its S_{0} ground state to an electronically excited S_{1} state. This excitation will generally lead to a nonisothermal initial distribution of energy in the excited state. Collisions with a buffer gas at room temperature tend to reequilibrate the reacting molecule. The ensuing radiative and nonradiative decay will depend on the competition between the energy dependent unimolecular decay rate and the energy relaxation. In this paper we describe a Gaussian binary collision theory which includes all three aspects  radiative decay, nonradiative decay and relaxation. The Gaussian property is justified when the reacting species is large enough, i.e. it has a large enough number of degrees of freedom such that the equilibrium distribution of the molecule can be described by a Gaussian. Guided by experimental observation, we adapt a Gaussian transition probability, which is similar to Mel'nikov's, to describe the relaxation dynamics. An analytic solution for the Gaussian master equation is presented. We find that pressure induced decay which is faster than the initial decay rate is an experimental signature of an initial cold distribution of reactants. This signature was observed experimentally in the isomerization of transstilbene. Application to the decay dynamics of the transstilbene molecule shows that an initial temperature of 230 K for transstilbene in the excited S_{1} state suffices for good agreement between the theoretical and experimental survival probability measured at a gas temperature of 300 K.

(1998) Journal of Chemical Physics. 108, 23, p. 97119725 Abstract
The exact quantum expression for the thermal rate of reaction is the trace of a product of two operators. It may therefore be written exactly as a phase space integral over the Wigner phase space representations of the two operators. The two are a projection operator onto the product's space, which is difficult to compute, and the symmetrized thermal flux operator, which can be computed using Monte Carlo methods. A quantum transition state theory was presented recently, in which the exact projection operator was replaced by its parabolic barrier limit. Alternatively, the exact projection operator may be replaced by its classical limit. Both approximations give thermodynamic estimates for the quantum rates. In this paper, we derive a perturbation theory expansion for the projection operator about the parabolic barrier limit and the classical limit. The correction terms are then used to evaluate the leading order corrections to the rate estimates based on the parabolic barrier or classical limits of the projection operator. The expansion is applied to a symmetric and an asymmetric Eckart barrier. The first two terms in the expansion give excellent results for temperatures above the crossover between quantum tunneling and thermal activation. For deep tunneling and asymmetric systems, the use of variational transition state theory, the classical limit, and perturbation theory leads to significant improvement in the estimate of the tunneling rate. Multidimensional extensions are presented and discussed.


(1998) Journal of Chemical Physics. 108, 7, p. 27332743 Abstract
An old challenge in rate theory is the formulation of a quantum thermodynamic theory of rates which gives accurate estimates but does not demand any real time propagation. In this paper we attempt to answer the challenge by extending an idea suggested by Voth, Chandler and Miller [J. Phys. Chem. 93, 7009 (1989)]. A new quantum expression for the rate is derived by replacing the exact time dependent dynamics with the analytically known dynamics of a parabolic barrier and utilizing the symmetrized thermal flux operator. The new rate expression is exact for a parabolic barrier, and leads by derivation rather than by ansatz to a phase space integration of a Wigner thermal flux distribution function. The semiclassical limit is similar but not identical to Miller's semiclassical transition state theory. Numerical computations on the symmetric and asymmetric one dimensional Eckart barrier give results which are equal to or greater than the exact ones, as expected from a transition state theory. In contrast to other approaches, the present theory is a leading term in an expansion which may be used to systematically improve the results and assess their validity.

Quantum harmonic transition state theory  Application to isomerization of stilbene in liquid ethane(1998) Journal of Chemical Physics. 108, 7, p. 27562764 Abstract
A harmonic quantum transition state theory, suggested recently by Pollak and Gershinsky [in Lectures on Stochastic Dynamics, edited by W. Lutz and T. Poeschel, Lecture Notes in Physics (Springer Verlag, New York, 1997)], is applied for the first time to a realistic reacting system. The isomerization of transstilbene in the gas phase and in the presence of dense liquid ethane solvent is investigated. We find that the overall quantum effect at room temperature is rather small. The quantum correction to the classical reaction rate estimate is approximately 23% for gas phase stilbene at room temperature. The addition of the dense solvent lowers the correction down to 13%, thus making the reacting system even more "classical," justifying the extensive use of classical molecular dynamics in investigating this reaction.

(1998) Physical Review E. 58, 5, p. 54365448 Abstract
The exact quantum rate may be represented as a phase space trace of a product of two operators: of the symmetrized thermal flux operator and a projection operator onto the product space. A semiclassical analysis of the phase space representation of these two operators is presented and used to explain recent results found for a quantum thermodynamic rate theory. For low temperatures, the central object that is responsible for the oscillatory nature of the flux operator is a periodic orbit on the upside down potential surface whose period is [Formula Presented] The semiclassical analysis of the flux distribution explains why a variation of the dividing surface leads to improved thermodynamic rate estimates in asymmetric systems. The semiclassical limit (stationary phase limit) of the projection operator is shown to be identical to the classical projection operator. A semiclassical rate theory is obtained using the product of the semiclassical flux distribution and either the parabolic barrier or the classical projection operator and compared with the exact rate and approximate quantum thermodynamic estimates.
1997

(1997) Journal of Chemical Physics. 107, 24, p. 1053210538 Abstract
This paper presents a continuation of our previous theoretical studies on the rate of isomerization of transstilbene from the first excited electronic state based on the potential energy surface of Vachev et al. [J. Phys. Chem. 99, 5247 (1995)]. Harmonic RRKM computations and molecular dynamics and Monte Carlo based classical rates are presented for deuterated isotopes of stilbene as well as hexane clusters of stilbene of varying size. Good agreement with experiment is found for energy dependent rates of d_{12} vs h_{12} stilbene. However, we find that the rate for d_{2} stilbene is greater than for d_{10} stilbene in contradiction to the experimental observations. For the hexane clusters we find that addition of hexane molecules causes a systematic decrease in the rate, in agreement with experiment.

(1997) Journal of Chemical Physics. 107, 9, p. 35423549 Abstract
A simple expression is derived for the survival probability of a reactive chemical species which is initially prepared at a temperature which differs from its surrounding. The competition between relaxation of reactants back to the external equilibrium and the possibility of reaction may prevent the usual single exponential kinetics for the survival probability. The theory is accurate for activated reactions with moderate (V^{‡}/k_{B}T≥3) to high reduced barrier heights. It is especially relevant for multidimensional systems where the characteristic energy at which a molecule dissociates is greater than the barrier height.


(1997) Journal of Chemical Physics. 107, 1, p. 6469 Abstract
Analysis of the symmetrized thermal flux operator leads to explicit expressions for its eigenvalues and eigenfunctions. At any point in configuration space one finds two nonzero eigenvalues of opposite sign. The associated eigenfunctions are L^{2} integrable. The eigenfunctions and eigenvalues are expressed in terms of the thermal density matrix in the vicinity of the transition state. The positive eigenvalue of the thermal flux operator gives an upper bound to the rate and allows for a formulation of a quantum mechanical variational transition state theory. This new upper bound, though, is only a slight improvement over previous theories.

(1997) Journal of Chemical Physics. 107, 3, p. 812824 Abstract
Previous theoretical and experimental investigations of the transstilbene isomerization reaction in the excited S_{1} state indicated that the gas phase thermal rate at room temperature is much smaller than the thermal rate in the liquid phase. This was based on the observations that: (a) A combination of measured energydependent rates and RRKM calculations led to an isolated molecule thermal rate at T= 300 K of 2 × 10^{9} s^{1}; (b) An experiment of Balk and Fleming [J. Phys. Chem. 90, 3975 (1986)] in which stilbene vapor at 300 K excited at the S_{0} to S_{1} zero point to zero point electronic transition energy (0^{0}_{0}), gave a lifetime in the excited state of ∼780 ps. The liquid state lifetime in ethane is ∼30 ps. In this paper we present theoretical computations of the rate in the gas and liquid phases, based on a new potential model of Vachev et al. [J. Phys. Chem. 99, 5247 (1995)]. We find that: (a) RRKM rates are in agreement with measured energydependent rates; (b) The thermal rate derived from the new RRKM rates is the same as the thermal rate in liquid ethane; (c) The laser excitation experiment of Balk and Fleming leads to laser cooling of the excited state suggesting that their measured lifetime is longer than the lifetime in the liquid. The surrounding liquid heats up the molecule on a time scale which is faster than the isomerization lifetime. Experiments are suggested to verify this interpretation.

(1997) Journal of Chemical Physics. 106, 18, p. 76787699 Abstract
The turnover theory for activated rate processes, is extended to multidimensional systems. The theory derived in this paper accounts for the competition between intramolecular and intermolecular relaxation. The extent of chaotic motion of the system modes directly affects the rate of energy diffusion in the system. The more chaos, the faster the energy diffusion and the larger the rate. The dependence of the rate on the intramolecular coupling strength is well accounted for. The theory is applied to a model twodimensional system studied previously by Straub and Berne [J. Chem. Phys. 85, 2999 (1986)]. The theory, which is the multidimensional generalization of the onedimensional Pollak, Grabert, and Hänggi (PGH) turnover theory [J. Chem. Phys. 91, 4073 (1989)] accounts well for the rate even in the case of extreme anisotropic friction. The theory is cast in terms of the collective normal modes of the system and the bath and is thus applicable also to memory friction.

(1997) Advances in Chemical Physics: Chemical Reactions And Their Control On The Femtosecond Time Scale. 101, p. 141183 Abstract

(1997) Advances in Chemical Physics: Chemical Reactions And Their Control On The Femtosecond Time Scale. 101, p. 391408 Abstract
1996

(1996) Surface Science. 365, 1, p. 159167 Abstract
A comparison is presented between macroscopic constants relating to the diffusion dynamics of a Cu atom on a Cu surface and their microscopic estimates. Comparison of elastic constants, speed of sound and friction coefficient indicates that a generalized Langevin equation (GLE) description of the diffusion dynamics of the Cu adatom is quite reasonable. This serves as a further justification for the recent GLEbased analysis of the experimental measurement of hopping distributions of metal atoms on metal surfaces.

(1996) Journal of Chemical Physics. 105, 10, p. 43884390 Abstract
A theoretical investigation of the experimental measurements of the isomerization rate of transstilbene in liquids is presented. Monte Carlo and molecular dynamics simulations of the reaction indicate that the predominant solvent effect is in raising the isomerization barrier in the potential of mean force as the solvent density is increased. Dynamic friction effects are small. Good agreement is obtained between the numerical and experimental rates. (C) 1996 American Institute of Physics.

(1996) Surface Science. 355, 13, p. L366L370 Abstract
In a recent experiment, Senft and Ehrlich (Phys. Rev. Lett. 74 (1995) 294) reported the observation of long hops in the migration of Pd on W(211). In this Letter we consider a possible microscopic mechanism responsible for such hops. We demonstrate that (a) Kramer's theory provides a good framework for analysis of the experimental data; (b) that the measured temperature dependence may be reasonably well accounted for; (c) the experimental results do not unequivocally demonstrate a substantial amount of long jumps.

(1996) Physical review letters. 77, 13, p. 26622665 Abstract
We present a new method for the semiclassical quantization of classically integrable as well as nonintegrable systems. The method is based on the semiclassical approximation of the equilibrium density matrix, using classical trajectories on the upside down potential surface. Periodic orbits do not play any special role. Explicit results are given for the case of the classically chaotic potential kx^{2}y^{2}/2.

(1996) Journal of Chemical Physics. 104, 3, p. 11111119 Abstract
We develop an expression for the rate of energy relaxation of a nonlinear oscillator coupled to a linear, dissipative bath. This particular type of model has wide applicability to studies of relaxation rates of vibrational modes in chemical systems. The energy relaxation rate is estimated by relating the anharmonic oscillator to an effective harmonic reference system. The theoretical predictions compare favorably with simulation results for the energy relaxation of a Morse oscillator (i) coupled to an Ohmic bath and (ii) coupled to a bath with exponentially decaying friction. The dependence of the initial relaxation rate on the excitation energy of a Morse oscillator is qualitatively different for the two cases. When the oscillator is coupled to an Ohmic bath, the initial relaxation rate decreases as a function of the excitation energy. When exponentially decaying friction is employed, however, the initial relaxation rate is an increasing function of the excitation energy.

(1996) Journal of Chemical Physics. 105, 20, p. 90939103 Abstract
Variational transition state theory is used to compute the rate of nonadiabatic electron transfer for a model of two sets of shifted harmonic oscillators. The calculations provide new insight on the suitability of the energy gap as a reaction coordinate. The relationship to the standard generalized Langevin equation model of electron transfer is established, and provides a framework for the application of variational transition state theory in a realistic simulation of electron transfer in a microscopic (nonlinear) bath.

(1996) Journal of Chemical Physics. 104, 17, p. 65476559 Abstract
A numerical study of the effect of dissipation on the radiationless transition rate in the adiabatic and solventcontrolled limits is presented. For light particle reactions, the nonlinearity of the potential surface in the vicinity of the barrier top is important, and the potential may be approximated as a cusped double well potential, provided that the nonadiabatic coupling is small compared to the thermal energy. Three different theoretical approaches for calculation of the thermally activated rate are analyzed and compared with exact numerical results. We find that Variational Transition State Theory (VTST) with a planar dividing surface, as well as the approach of Calef and Wolynes (CW), provide a good description of the rate of symmetric reactions. A rate expression suggested by Dekker is found to be the least accurate. The CW approach is most accurate in the strong damping regime, while VTST is better in the weak damping regime. The accuracy of both methods improves as the potential is smoothed. VTST and the CW expression are also found to give a reasonable description of asymmetric reactions, provided that the asymmetry is not too large.
1995

(1995) Journal of Chemical Physics. 103, 19, p. 85018512 Abstract
Variational transition state theory (VTST) is applied for the first time to a chemical reaction in a liquid. The theory provides accurate estimates of reaction rates and leads to well defined microscopic friction functions. The structure of the optimized planar dividing surface provides insight into the range of solutesolvent interactions for which there is an appreciable effect on the reaction dynamics. The VTST method also allows for separation of the frictional effects of solvent translation, rotation, and stretch modes. The numerical cost is less than an analogous molecular dynamics reactive flux computation and the insight gained is greater. (C) 1995 American Institute of Physics.

(1995) Chemical Physics Letters. 242, 12, p. 5461 Abstract
Application of the Newton method locating stable periodic orbits is extended to include nonotating and rotating triatomic molecules in 3D. A Monte CarloNewton method search for stable periodic orbits of the H_{3}^{+} molecular ion at the dissociation energy into H^{+} + H_{2} is presented. Using the convergence volume of the Newton method as an importance criterion, we find that the 'horsehoe' orbit used previously to assign the experimental coarse grained photodissociation spectra of H_{3}^{+} is the most important stable orbit. An additional new stable orbit is also discovered. Implications for quantum computations and experimental results are discussed.

(1995) The Journal of chemical physics. 103, 3, p. 973980 Abstract
A quantum theory of activated rate processes applicable to nonlinear potentials of interaction is developed. The central premise is that the rate is determined by the point of maximal quantum free energy separating reactants and products. The quantum free energy is defined in terms of a quantum centroid potential. The resulting rate expressions reduce to known limits for generalized Langevin equations and their Hamiltonian representation. They also reduce in the classical limit to previous results derived using an optimal planar dividing surface classical variational transitionstate theory. A saddlepoint estimate of the quantum rate leads to a generalization of Wolynes' high temperature rate expression valid for nonlinear system bath interactions and anharmonic baths. Maximizing the free energy leads to a quantum friction function. Application to realistic systems demands the computation only of centroid densities.

(1995) The Journal of chemical physics. 102, 10, p. 40374055 Abstract
We have implemented a semiclassical dynamics simulation method to investigate the effects of finite barrier heights and nonlinear potentials on the rate of diffusion of a particle which is coupled to a frictional bath and is traveling on a onedimensional potential energy surface. The classical reactive flux method has been modified to account for semiclassical tunneling and abovebarrier reflection. A novel perturbation theory treatment of the semiclassical dynamics is developed to simulate the motion of the particle when the coupling to the frictional bath is small and the particle's motion is nearly conservative. Our simulation results support the theoretical prediction that the diffusion constant increases as friction decreases. We also find supporting evidence for an inverse isotope effect, as the diffusion constant for a classical particle can be larger than that of a corresponding quantum mechanical particle. The escape rate and the average energy of escaping particles are also found to be in good agreement with theoretical predictions.

(1995) The Journal of chemical physics. 102, 17, p. 69096918 Abstract
The migration of adsorbed atoms on crystal surfaces is considered. To describe the adatom motion one often uses a generalized Langevin equation (GLE). The timedependent friction, which enters the GLE, is caused by the interaction with the crystal excitations. However, the explicit form of the timedependent friction is not well known. We show that if the damping is associated with acoustic phonons and the coupling of the adatom with the surface is not too strong, then the friction is Ohmic. An explicit expression for the friction coefficient is given in terms of the basic physical parameters of the crystal and in terms of the activation energy of the adatom on the surface. We find that usually the diffusion occurs in the intermediate damping regime. In this regime nearest neighbor hops are most probable and transition state theory gives an excellent estimate for the rate of escape and for the diffusion coefficient. Using the recently developed turnover theory for surface diffusion [Phys. Rev. E 49, 5098 (1994)] we derive explicit expressions for the evolution of the timedependent site distribution and compare it with an experiment where correlated hops have been observed. We conclude that even when the motion is onedimensional, correlated hops are to be expected only for sufficiently high temperatures or for physisorbed atoms.

(1995) The Journal of chemical physics. 103, 20, p. 89108920 Abstract
Escape of a particle from a metastable potential, whose motion is governed by the generalized Langevin equation, is a common model of many chemical and physical activated processes in condensed phase. In the intermediatetostrong damping regime the rate of escape is controlled by the particle dynamics near the barrier top. Since Kramers, the parabolic barrier approximation is commonly used to get the expression for the rate in this regime. We consider the influence of anharmonic corrections to the potential barrier on the quantum rate and get leading order corrections in terms of the inverse barrier height. New terms appearing in the quantum expression for the rate are associated with tunneling through the barrier and become important at low temperatures. The analytic theory is compared with recent numerically exact quantum simulations [M. Topaler and N. Makri, J. Chem. Phys. 101, 7500 (1994)].

(1995) The Journal of chemical physics. 103, 18, p. 79127926 Abstract
A new approach is suggested for evaluation of the radiationless transition rate for the curvecrossing problem in the presence of dissipation. The rate is evaluated by using the conventional LandauZener theory but for a collective system  bath coordinate, which is characterized by a maximal meanfree path in the vicinity of the crossing point. Variational transition state theory (TST) is employed for determination of this quasiballistic mode. The resulting uniform rate expression bridges between the known nonadiabatic, solvent controlled and TST limits. The main effect of dissipation is the reduction of the slope difference of the potential of mean force along the quasiballistic mode compared to that along the original reaction coordinate. This results in an increase of the reaction adiabaticity. Application of the theory is illustrated for the symmetric normal crossing of two parabolic diabatic terms with Ohmic dissipation. Explicit results for the rate in the relevant physical limits are derived. The theory is also used to analyze resonant electron transfer reactions in Debye solvents.

(1995) Physical Review E. 51, 3, p. 18681878 Abstract
In the framework of the transition state theory (TST), the rate of thermally activated escapes from a locally stable state in phase space is determined by the unidirectional flux through a conveniently chosen dividing surface. It is known that the occurrence of trajectories that recross this surface renders the true rate smaller than the TST rate by the socalled transmission factor. By means of a statistical theory we show how the mean number of recrossings can be related to the transmission factor. Formulas are derived for the average number of recrossings at the top of a parabolic barrier and through an energy surface in phase space. The former case is relevant for the spatial diffusion regime and the latter for the energy diffusion regime. The resulting transmission factors are in good agreement with the exact ones.
1994

(1994) Physics Letters B. 327, 12, p. 6769 Abstract
It is shown that earlier claims, identifying the so called fundamental subsystem of YangMills classical mechanics as a Kolmogorov Ksystem, are not true.

(1994) Chemical Physics. 180, 23, p. 191197 Abstract
The turnover theory for activated rate processes is generalized to include multidimensional strongly coupled systems. A uniform expression for the rate, valid for all damping values is derived and applied to numerical simulations results of Straub and Berne. The theory is applicable to both space and time dependent friction.

(1994) The Journal of chemical physics. 100, 1, p. 334339 Abstract
Kramers' treatment of activated rate processes is based on the Langevin equation of motion for the escaping particle. The stochastic dynamics may be cast equivalently as the dynamics of a particle interacting bilinearly with a bath of harmonic oscillators. This paper explores the connection between the solutions of Kramers' problem (and its generalization to include memory friction) obtained in the framework of these two approaches. We demonstrate their equivalence for the specific case of a parabolic barrier potential. The Hamiltonian representation is used to construct (a) a nontrivial eigenfunction of the FokkerPlanck equation which is generalized to include time dependent friction; (b) the Kramers' stationary flux distribution function; (c) the stochastic separatrix.

(1994) The Journal of chemical physics. 101, 8, p. 71747176 Abstract
Variational transition state theory (VTST) is applied for the first time to a realistic simulation of a reaction in a liquid. Rate information is obtained from Monte Carlo thermal classical centroid averages of second derivatives of the full potential of interaction. A well defined friction function is computed and found to be in excellent agreement with previous approximate prescriptions leading to identical rate constants.

(1994) Physical Review E. 49, 2, p. 12161224 Abstract
The variationaltransitionstate theory (VTST) approach to condensedphase activatedrate processes is extended to include bent planar dividing surfaces. This allows removal of formal divergences which arise when applying VTST, based on simple planar dividing surfaces, to unrestricted potentials. Practical applications are demonstrated for the cubic and strongly asymmetric quartic potentials.

(1994) The Journal of chemical physics. 101, 9, p. 78117822 Abstract
A general theory is presented for the thermally activated rate constant in systems influenced by spatially dependent and time correlated friction. The theory is valid at all damping strengths and goes uniformly from the energy diffusion limit to the spatial diffusion limit. Results of the theory for a model system with an exponentially time correlated and spatially dependent friction kernel are compared with results from a numerically exact solution of the equivalent generalized Langevin equation. Predictions of the theory are found to be in excellent agreement with the numerical simulation results. The phenomenon of memory suppression of the rate is observed for long time scale frictions and its modification due to the spatial dependence of the friction is discussed. The effects of spatially dependent friction can be understood through a quantity called the "average spatial modification" of the coupling between the reaction coordinate and the environment.

(1994) The Journal of chemical physics. 100, 8, p. 58945904 Abstract
Monte Carlo methods are combined with a Newton method to construct an efficient numerical procedure for locating stable periodic orbits embedded in a largely chaotic system. We find that the Newton method effectively enlarges the basin of attraction of the stable orbit by orders of magnitude relative to the stable region surrounding the orbit. Three variants of the Newton method are tested. We conclude that an allpoints finite difference version is the optimal choice. Use of a Monte Carlo search with importance sampling and combined with the Newton method proves to be the most efficient search procedure. Application to the two and three dimensional quartic oscillator leads to previously unknown stable orbits.

(1994) Physical Review E. 49, 6, p. 50985102 Abstract
A semiclassical theory for the diffusion of a particle moving on a periodic potential, coupled to a dissipative heat bath, is presented. The resulting expressions for the diffusion coefficient, mean squared path length, and hopping length distribution are valid for memory friction and provide a theory which goes uniformly from the underdamped to the strongly damped limit. In the underdamped limit, quantum tunneling and reflection cause the quantum diffusion coefficient to be lower than the classical, leading to an inverse isotope effect; the diffusion of D atoms should be faster than the diffusion of H atoms.

(1994) The Journal of chemical physics. 101, 6, p. 47784789 Abstract
Variational transition state theory (VTST) is applied to the study of the activated escape of a particle trapped in a multidimensional potential well and coupled to a heat bath. Special attention is given to the dependence of the rate constant on the friction coefficients in the case of anisotropic friction. It is demonstrated explicitly that both the traditional as well as the nontraditional scenarios for the particle escape are recovered uniformly within the framework of VTST. Effects such as saddle point avoidance and friction dependence of the activation energy are derived from VTST using optimized planar dividing surfaces.

(1994) Physical Review E. 50, 4, p. 26462653 Abstract
Activated rate processes are often described in terms of a generalized Langevin equation. The concept of an optimized planar dividing surface in conjuction with variational transition state theory has been demonstrated to be useful in understanding the effects of nonlinearities on reaction rates. A different approach is based on the Rayleigh quotient method, in which one varies the trial functions. We prove a restricted identity of the two methods. The restrictions are that the variational transition state theory method is limited to planar dividing surfaces. The Rayleigh quotient method is restricted to the class of Kramers functions. These functions are constructed by replacing the true potential with a parabolic barrier and using the known eigenfunction for the parabolic barrier. The parameters of the parabolic barrier are used as variational parameters in the Rayleigh quotient for the true nonlinear potential.

(1994) p. 311329 Abstract
This paper summarises recent work on rate theory of activated proceses in condensed phases. Various studies have shown that nonlinearities in the potential of the reacting particle or its interaction with the bath may lead to substantial reductions of the rate constant relative to predictions of the standard theories. It is shown that the optimized planar dividing surface variational transition state theory can account correctly for all these observed suppressions.
1993

(1993) Chemical Physics Letters. 207, 46, p. 309316 Abstract
An analytic theory is presented for the thermally activated rate constant in systems which exhibit spatially dependent and timecorrelated friction along the reaction coordinate motion. The theory is valid over the entire range of damping strengths, including in the region of the Kramers turnover. It is compared to the results of computer simulations on a model nonlinear system and excellent agreement is obtained. The present work significantly generalizes existing theories for the activated rate constant which are based on the generalized Langevin equation for the reaction coordinate motion and spatially independent friction.

(1993) Chemical Physics. 170, 3, p. 265273 Abstract
A quantum version of classical variational transition state theory suggested by McLafferty and Pechukas is refined. In this new quantum version, the variational property of the theory leads to the identification of an optimal smeared dividing surface. This optimal function is shown to be the eigenfunction associated with the lowest eigenvalue of a positive quantum transition state theory operator. The lowest eigenvalue is the optimal bound on the quantum rate. Application of the theory to the parabolic barrier provides better bounds but does not give an essential improvement when compared to previous quantum transition state theories.

(1993) The Journal of chemical physics. 98, 12, p. 95329543 Abstract
A detailed study of memory and temperature induced suppression of activated rate processes is presented. Numerical computations demonstrate that long memory in the presence of moderate barriers can induce noticeable deviation of the reaction rate constant from the predictions of the KramersGroteHynes theory. A canonical variational transition state theory, based on finding the optimized planar dividing surface, is shown to account quantitatively for the observed suppression of the rate. The suppression is associated with an almost perpendicular rotation of the optimal dividing surface away from the usual one. A further generalization of the PollakGrabertHänggi theory for the Kramers turnover is presented and shown to account for the computed rate constants for the whole range of damping at a fixed bath memory time.

(1993) Physical Review E. 47, 2, p. 922933 Abstract
A dynamically corrected variational transitionstate theory is formulated for the thermally activated escape of a particle trapped in a potential well separated from a different well or continuum by a barrier and coupled to a heat bath. The theory is based on the Hamiltonianequivalent formulation of the generalized Langevin equation. The dynamical corrections are obtained by utilizing the reactiveflux method in which the choice of dividing surface is guided by minimization of the transitionstate flux. Analytic correction formulas, valid for memory friction, are obtained for the KramersGroteHynes estimate of the rate in the range from moderate friction to the largefriction limit. The analytic expansion is in terms of the inverse barrier height (1/V). For the special case of an extended Smoluchowski equation containing finite damping corrections, the exact expansion is also obtained using the meanfirstpassagetime formulation. The dynamically corrected variational transitionstatetheory expansion is shown to be identical to the meanfirstpassagetime result.

(1993) Physical Review E. 47, 1, p. R21R23 Abstract
It is demonstrated that a recent finitebarrier expansion for jump rates accounts quantitatively for the observed discrepancy between numerically determined exact rates and the Kramers estimates of these rates.

(1993) Physical review letters. 70, 21, p. 32993302 Abstract
Scanning tunneling microscopy observations of long hops in the diffusion of Pb atoms on Ge surfaces are explained by the model of a Brownian particle in a periodic potential. The classical turnover theory for barrier crossing predicts a large correlated hopping probability in the underdamped limit, consistent with experiment and in agreement with simulations. The corresponding quantum theory predicts that in the underdamped limit the rate is dominated by tunneling. This causes the quantum correlated hopping probability to vanish in this limit and may be thought of as a new form of quantum localization.

(1993) The Journal of chemical physics. 99, 2, p. 13441346 Abstract
The onedimensional stochastic equation of motion for a particle in the presence of space and time dependent friction involves multiplicative fluctuations and a nonlinear friction kernel. We show how this rather complicated equation may be significantly simplified. Introduction of an auxiliary mode leads to a set of two nonlinearly coupled equations with space and time independent damping. An exact FokkerPlanck equation emerges naturally from this formulation.
1992

(1992) PHYSICA D. 56, 4, p. 368380 Abstract
A numerical method aimed at locating stable periodic orbits in strongly chaotic systems is presented. The method is based on the selection of trajectory segments which are characterized by a relatively low positive local Lyapunov exponent. Once such a selection is made, convergence to stable (or weakly unstable) periodic orbits is obtained by a Newton method. The algorithm is rather general and can be used for systems with more than two degrees of freedom. The proposed approach is tested on the quartic oscillator model and on the potential of the hydrogen atom in a strong magnetic field. In the latter case new stable periodic orbits are found in the region of strong chaotic motion. The possible quantum localization on these orbits is discussed briefly.

(1992) Journal of Statistical Physics. 66, 34, p. 975990 Abstract
Upper bounds for the classical escape rate of a particle trapped in a metastable well and interacting with a dissipative medium are derived based on the periodic orbits of a reduced twodegreeoffreedom Hamiltonian involving the unstable normal mode and a collective bath mode. It is shown that even in what is usually thought of as the spatial diffusion limit the reactive flux can involve an energy diffusion term due to energy transfer from the dissipative media, in addition to the standard spatial diffusion term.

(1992) The Journal of chemical physics. 96, 12, p. 88778888 Abstract
A numerical solution for the canonical variational dividing surface of two degree of freedom conservative systems is presented. The method is applied to reaction rates in dissipative systems whose dynamics is described by a generalized Langevin equation. Applications include a cubic and a quartic well using Ohmic and memory friction. For Ohmic friction, we find that in almost all cases, curvature of the optimal dividing surface may be neglected and the Kramers spatial diffusion limit for the rate is in practice an upper bound. For a Gaussian memory friction and a cubic oscillator, we compare the present theory with numerical simulations and other approximate theories presented by Tucker et al. [J. Chem. Phys. 95, 5809 (1991)]. For the quartic oscillator and exponential friction, we discover a strong suppression of the transmission coefficient and the reaction rate whenever the reduced static friction is of the same order of the reduced memory time. We also show that in this case, there is a strong suppression of the energy diffusion process in the reactants' well.

(1992) The Journal of chemical physics. 97, 4, p. 24222437 Abstract
The variational transition state theory approach for dissipative systems is extended in a new direction. An explicit solution is provided for the optimal planar dividing surface for multidimensional dissipative systems whose equations of motion are given in terms of coupled generalized Langevin equations. In addition to the usual dependence on friction, the optimal planar dividing surface is temperature dependent. This temperature dependence leads to a temperature dependent barrier frequency whose zero temperature limit in the one dimensional case is just the usual KramersGroteHynes reactive frequency. In this way, the KramersGroteHynes equation for the barrier frequency is generalized to include the effect of nonlinearities in the system potential. Consideration of the optimal planar dividing surface leads to a unified treatment of a variety of problems. These are (a) extension of the KramersGroteHynes theory for the transmission coefficient to include finite barrier heights, (b) generalization of Langer's theory for multidimensional systems to include both memory friction and finite barrier height corrections, (c) Langer's equation for the reactive frequency in the multidimensional case is generalized to include the dependence on friction and the nonlinearity of the multidimensional potential, (d) derivation of the nonKramers limit for the transmission coefficient in the case of anisotropic friction, (e) the generalized theory allows for the possibility of a shift of the optimal planar dividing surface away from the saddle point, this shift is friction and temperature dependent, (f) a perturbative solution of the generalized equations is presented for the one and two dimensional cases and applied to cubic and quartic potentials.

1991

(1991) Journal of Chemical Physics. 95, 8, p. 58095826 Abstract
Rate constants evaluated from (1) the energyloss turnover theory of Pollak, Grabert, and Hänggi (PGH), (2) the GroteHynes extension of Kramers theory (GH), and (3) the microcanonical variational transition state theory for dissipative systems of Tucker and Pollak (μVTST) are compared with rate constants determined from direct computer simulations of generalized Langevin dynamics. The comparisons are made for a cubic oscillator under the influence of a slow bath characterized by a Gaussian friction kernel. In the μVTST calculations, which are based on an effective two degree of freedom, Hamiltonian, barrier crossing due to energy transfer from the bath to the effective Hamiltonian is neglected. This neglect is significant only at very strong coupling, where it causes the μVTST results to drop below the simulation results. Both GH and μVTST theories fail (as expected) in the energy diffusion regime, while PGH theory is only moderately successful. The μVTST results agree extremely well with the simulation results in the spatial diffusion regime, providing a significant improvement over the GH results at intermediate coupling strengths and over the PGH results at strong coupling strengths. This improvement is a result of nonlinear effects which are included in the μVTST approach but neglected in the PGH and GH theories.

(1991) Journal of Physical Chemistry. 95, 25, p. 1023510240 Abstract
New developments in the application of variational transitionstate theory to activated rate processes in dissipative media are reported. A variational solution for the optimal dividing surface in configuration space is found. The canonical flux is proportional to the classical action along a classical trajectory evolving under the dynamics of a temperaturedependent 2 degrees of freedom Hamiltonian. This result is of general validity for 2 degrees of freedom systems and so of interest also for thermal reaction rates in conservative systems. An application of variational transitionstate theory to a cubic oscillator in the presence of ohmic dissipation is presented. Here, the dividing surface is curved; however, we find that the Kramers estimate for the rate is valid for almost all parameter regimes.

(1991) The Journal of chemical physics. 95, 1, p. 533539 Abstract
A generalization of the KramersGroteHynes theory for reaction rates in the spatial diffusion limit is derived for a general class of Hamiltonians. Previous restrictions to harmonic baths and bilinear system bath couplings are removed. The key ingredient is the systematic use of variational transition state theory (VTST) to identify the optimal dividing surface. A pair of collective modes are defined as a linear combination of all system and bath modes. A free energy surface is defined in the two degree of freedom collective mode phase space. The VTST estimate for the rate of reaction on this surface is shown to be an upper bound to the exact rate. The optimal definition of the collective modes is obtained by minimizing the rate. The resulting rate expression is formally identical to the KramersGroteHynes theory. However, the minimization procedure leads to a new definition of the time dependent friction. In consistence with transition state theory, this time dependent friction is constructed from equilibrium properties of the composite system and does not call for any dynamical computations. The friction parameters are determined from equilibrium centroid averages of partial derivatives of the full potential at the barrier of the potential of mean force. This removes previous ambiguities as to the definition of time dependent friction in condensed matter systems. A procedure is presented for finding collective modes along which the friction exerted by the bath is minimized. This result may be of substantial interest in the study of complex dynamical systems in biology, chemistry, and physics.
1990

(1990) Chemical Physics Letters. 174, 34, p. 325332 Abstract
Autocorrelation functions for a chaotic coupled quartic oscillator system are found to have typical recurrence times. Phase space decomposition of the associated spectrum at the peak frequency shows that the peak comes from well defined localized regions bounded by the stable and unstable manifolds of a class of periodic orbits. This localization may be thought as a classical analog of quantum scars.

(1990) PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY OF LONDON SERIES AMATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES. 332, 1625, p. 343359 Abstract
Keywords: Multidisciplinary Sciences

(1990) The Journal of chemical physics. 92, 5, p. 30053017 Abstract
Two degrees of freedom quantum mechanical calculations on the bound states of H^{+}_{3} are presented. Two different potential energy surfaces are employed. The effect of rotational excitation is analyzed. For J=0, the high energy region is composed largely of states that cannot be assigned. However, two regularly spaced series are observed, corresponding to ‘‘horseshoe’’ states predicted previously by classical calculations. In addition we find a new assignable series of inverted hyperspherical states. Conversely, for high orbiting angular momentum (l=20) in which the proton rotates about the diatom, assignable normal mode states persist up to dissociation. Semiclassical periodic orbit quantization is found to give excellent agreement for the regular quantum states. The significance of these results for the interpretation of the H^{+}_{3} photodissociation spectrum is discussed.

(1990) PHYSICS REVIEW LETTERS. 65, 12, p. 13991402 Abstract
Variational upper bounds are derived for the escape rate of a particle trapped in a metastable well and interacting wth a dissipative medium. The theory leads to a reduced twodegreesoffreedom Hamiltonian involving the unstable normal mode and a newly defined collective bath mode. Explicit treatment of strong nonlinearities or lowbarrier systems present no special problem.

(1990) The Journal of chemical physics. 93, 2, p. 11161124 Abstract
A variational transition state theory is formulated for the decay rate of a particle trapped in a metastable potential well and coupled to a heat bath. Rigorous upper bounds are derived for the transmission coefficient and the rate constant. The variational theory leads to minimization of the flux of an effective two degree of freedom Hamiltonian whose parameters depend on the system potential and the time dependent friction kernel. An explicit solution for the canonical variational dividing surface in the presence of nonlinearities in the system potential is provided. The Kramers expression for the rate in the spatial diffusion limit and its generalization to memory friction, is shown to give upper bounds provided that the nonlinearity in the system potential is positive definite. However, the variational result can still lead to substantially lower bounds for the rate. An application of VTST to a symmetric cusped double well potential provides a new expression for the rate, valid for arbitrary friction kernels and damping strength.

(1990) The Journal of chemical physics. 92, 6, p. 33773386 Abstract
A study of the bound states of the H+3 molecular ion at zero total angular momentum is presented. Wave functions are shown for the accurate ab initio Meyer–Botschwina–Burton potential energy surface and the more approximate diatomics in molecules (DIM) surface. The qualitative behavior is similar for the two potentials. The analytic form of the DIM surface enables a study that reaches energies as high as the dissociation threshold. Quantum states are found to localize regularly around the horseshoe periodic orbits found in previous classical studies. There is good agreement between a semiclassical periodic orbit quantization formula and the exact quantum energies. The antisymmetric stretch frequency with respect to the orbit is estimated classically and quantum mechanically and found to be in agreement with a previous estimate. A three‐dimensional stability analysis of the horseshoe orbit is presented and used as a basis for the semiclassical theory. The implications on the assignment of the coarse grained photodissociation spectrum measured by Carrington and Kennedy are discussed.

(1990) Physical Review A. 41, 10, p. 53665382 Abstract
The quantummechanical version of the Kramers turnover problem is considered. The multidimensional character of the problem is taken into account via transformation to normal modes. This eliminates the coupling to the bath near the barrier top allowing the use of a simple harmonic transmission coefficient for the barrier dynamics. The well dynamics is described by a continuum form of a master equation for the energy in the unstable normal mode. Within firstorder perturbation theory, the equations of motion for the stable normal modes have the form of a forced oscillator. The transition probability kernel is found using the known solution for the quantum forced oscillator problem. An expression for the quantum escape rate is derived. It encompasses all previously known limiting results in the thermally activated tunneling regime. The depopulation factor, which accounts for the nonequilibrium energy distribution is evaluated. The quantum transition probability kernel is broader than the classical and is skewed towards lower energies. Interplay between these two effects, together with a positive tunneling contribution, determines the relative magnitude of the quantum rate compared to the classical one. The theory is valid for arbitrary dissipation. Its use is illustrated for the case of a cubic potential with Ohmic (Markovian) dissipation.

(1990) Physical Review B. 41, 4, p. 22102220 Abstract
Quantummechanical resonance energies and their corresponding decay rates (inverse lifetimes) for the metastable system of a cubic potential coupled to a harmonic oscillator are computed numerically via the complex scaling method. This system, which mimics tunneling in dissipative media, is investigated for different barrier heights and a variety of coupling strengths. The large number of computed resonances allows one to calculate thermally averaged decay rates for temperatures up to the crossover temperature. The numerical results are compared to the sudden theory of dissipative tunneling, and rather good agreement is found. The suppression of the rate with increasing dissipation and the thermal enhancement of the rate, as predicted by the instanton method for dissipative tunneling, are also confirmed. When the time scale of the bath oscillator exceeds the time scale of the system, an interesting, counterintuitive observation is that the temperature for the crossover between tunneling and thermally activated escape increases with increasing coupling strength. This is in contrast to the usual behavior for ohmiclike dissipative systems. The numerical results in this work can be used as a benchmark to test other theories of dissipative tunneling.
1989

(1989) Chemical Physics. 139, 23, p. 471487 Abstract
The effect of dissipation on quantum states whose energy is in a range where the classical dynamics is largely chaotic is studied in detail. The model system used in the HénonHeiles Hamiltonian where the coupling to a bath is modeled in terms of a generalized Langevin equation (GLE). The bath employed has a lowfrequency cutoff relative to the harmonic frequencies of the HénonHeiles Hamiltonian. The GLE is recast in terms of a Hamiltonian. A frozen bath approximation leads to a Hamiltonian which is a function of stochastic Gaussian variables whose variance determines the coupling strength. We find that (a) the level spacing distribution of averaged eigenvalues nears a delta function at the strongest coupling studied, (b) dissipation tends to make the quantum densities more classical, (c) different states have varying stability with respect to coupling to the bath, (d) the most stable states are those which can be associated with the class of stable precessing periodic orbits. These results are in qualitative agreement with recent studies on coarse grained spectra of highly excited polyatomic molecules.

(1989) Chemical Physics. 134, 1, p. 3745 Abstract
A study of the semiclassical analogue of hyperspherical quantum resonances found by Manz and coworkers in A + BA scattering on a coupled Morse oscillator potential energy surface is presented. Semiclassical (EBK) quantization based on Liapunov stability analysis of hyperspherical periodic orbits provides an excellent estimate for the quantum resonance energies. However, the regular region of phase space surrounding the orbits is too small to support a single quantum state. The periodic orbit analysis indicates the possible existence of quantum resonances in collisioninduced dissociation.

(1989) Physical Review A. 39, 8, p. 37763793 Abstract
We present a detailed study of the classical and quantum mechanics of a strongly chaotic quartic oscillator. The topology of the motion is such that there is a channel in which one has good separation of time scales. Many quantum states are found to scar along these channels. An adiabatic breakup for the action of the periodic orbits based on adiabatic stability of orbits is used to derive an approximate, integrable Hamiltonian. Semiclassical quantization of this Hamiltonian yields accurate energies for all states scarred along the channels.

(1989) The Journal of chemical physics. 91, 7, p. 40734087 Abstract
An analytical theory is formulated for the thermal (classical mechanical) rate of escape from a metastable state coupled to a dissipative thermal environment. The working expressions are given solely in terms of the quantities entering the generalized Langevin equation for the particle dynamics. The theory covers the whole range of damping strength and is applicable to an arbitrary memory friction. This solves what is commonly known as the Kramers turnover problem. The basic idea underlying the approach is the observation that the escape dynamics is governed by the unstable normal mode coordinate  and not the particle system coordinate. An application to the case of a particle moving in a piecewise harmonic potential with an exponentially decaying memoryfriction is presented. The comparison with the numerical simulation data of Straub, Borkovec, and Berne [J. Chem. Phys. 84, 1788 (1986)] exhibits good agreement between theory and simulation.

(1989) Physical review letters. 63, 12, p. 12261229 Abstract
The distribution of local Lyapunov exponents is used to analyze power spectra of conservative dynamical systems. It is shown that sharp and broad peaks in the spectra can be related to well defined regions in phase space, associated with algebraic and exponential stretching of distances, respectively.

Classical mechanical analysis of the experimental highenergy spectrum of the sodium trimer molecule(1989) Physical review letters. 62, 18, p. 20962099 Abstract
Classical mechanics is used to compute the lowresolution spectrum of Na3 in the classically chaotic regime. Excellent agreement between the numerical simulation and experiment enables an assignment of the experimental coarsegrained spectrum, which is based not on periodic orbits but on weakly unstable, reduced dimension, quasiperiodic motion. In contrast to previous work on similar systems, there is no need to use a semiclassical approximation based on the Gutzwiller summation formula.

(1989) Israel Journal of Chemistry. 29, 4, p. 355359 Abstract
The normal mode transformation of the Hamiltonian equivalent of the Generalized Langevin Equation for the free particle is used to provide explicit solutions for the classical and quantum‐mechanical diffusion of a free particle.

(1989) Accounts of Chemical Research. 22, 6, p. 223229 Abstract
The word “metastable” has traditionally been used to denote states that decay with a much slower rate than what is “normally” expected. For example, in atomic spectroscopy, states that emit photons of visible light at a rate much slower than the usual 108 s'1 are called metastables. Metastability in atomic spectroscopy is generally due to selection rules.Mass spectra of molecular ions consist of fragmentation patterns of a mother ion, generally produced by electron impact or by laser photoionization. Mother ions that undergo slow decay (i.e., in microseconds instead of picoseconds) are called metastable ions,1,2 and they are observable as fractional masses in the mass spectrometer. Metastable unimolecular decay of excited neutral molecules is also expected, but measurement is more difficult than in ion fragmentation. However, in principle there is no difference between neutrals and ions in the dynamics described below.

(1989) Physical Review B. 40, 4, p. 21382146 Abstract
The effect of a lowfrequency dissipative medium on the tunneling rate of a particle trapped in a metastable potential is investigated with the aid of a frozenbath sudden approximation. The sudden theory is formulated for arbitrary timedependent friction and for all temperatures at which the escape process is dominated by tunneling. The validity criterion of the theory is only that the bath frequency spectrum be substantially lower than the system frequency. It is applicable both in the weak and strongdamping limits. The sudden theory is applied to tunneling in a cubic potential and a piecewise harmonic potential where the barrier frequency differs from the well frequency. In contrast to the ImF method, the sudden theory, when valid, can provide estimates of tunneling rates from welldefined excited resonance states and is not limited to estimates of only thermal rates. We find that, if the barrier is thin relative to the well, dissipation can serve to enhance tunneling rates from excited states.

(1989) Journal of Physical Chemistry. 93, 6, p. 23192328 Abstract
The unimolecular decay of HD_{2}^{+} and H_{2}D^{+} after IR excitation, as observed by Carrington and Kennedy, is discussed. Since the incorporation of zeropoint energies in the classical trajectory tunneling (CTT) method employed in our earlier work on H_{3}^{+} is difficult, we develop a sudden (with respect to bending angle) transitionstate theory (TST). Good agreement is found between a classical version of sudden TST (i.e., without zeropoint energies) and CTT. We then apply the quantum version of sudden TST to HD_{2}^{+} and H_{2}D^{+}, and compute the decay rate and the average translational energy release as a function of total energy E, total angular momentum J, and channel (H^{+} or D^{+}). We find that, if one fixes the lifetime range to the experimental window, the contributions of both isotopic channels depend critically on zeropoint energy, come from almost nonoverlapping J ranges, and produce different translational energy distributions. These results are in agreement with the experimental findings (different spectra for H^{+} and D^{+}, predominance of H^{+} over D^{+} in the dissociation of HD_{2}^{+}).
1988

(1988) Chemical Physics Letters. 151, 6, p. 557564 Abstract
The tunneling splitting in the localmode spectrum of an isolated symmetric molecule is destroyed in the condensed phase. This change should lead to an experimental identification of local modes using temperaturedependent overtone spectroscopy in liquids.

(1988) Chemical Physics Letters. 146, 5, p. 353357 Abstract
Using a sudden transition state theory we find that photodissociation of longlived resonances of D_{2}H^{+} leads to the H^{+} fragment for total angular momentum J≤27 while D^{+} is formed for J≥30. This quantum result which is critically dependent on the difference of zeropoint energies between D_{2} and HD explains an additional aspect of the CarringtonKennedy experiments.

(1988) Chemical Physics. 120, 1, p. 3749 Abstract
The semiclassical model of Waite and Miller for tunneling rates in unimolecular decomposition is reformulated and further developed. A practical numerical algorithm is obtained, enabling application of the model to threedimensional (and larger) systems. The model is then applied to the unimolecular decomposition of H_{3}^{+}. Tunneling rates obtained are in agreement with the experimental photodissociation rate measured by Carrington and Kennedy. The method is also used to obtain product state distributions.


(1988) The Journal of chemical physics. 88, 3, p. 19591966 Abstract
A technique is developed for solving the generalized Langevin equation (GLE) describing enharmonic oscillators in the weak coupling limit. The GLE is rewritten as a Hamiltonian with a nonlinear system coupled to an infinite bath of harmonic oscillators. A normal mode transformation followed by a perturbation technique is used to obtain the fluctuating system frequency. When the method is applied to a single oscillator with cubic anharmonicity, both the classical and quantal dephasing rates are shown to be equal to the wellknown result of Oxtoby. The technique is also applied to a system with more than one vibrational degree of freedom (linear triatomic molecules) to obtain the dephasing rates for the symmetric and antisymmetric normal modes. The effects of system anharmonicity on frequency shifts are investigated.

(1988) The Journal of chemical physics. 88, 9, p. 56435656 Abstract
Recently, Kennedy and Carrington found new quasibound states of H _{3}^{+}, which lie up to 1 eV above the dissociation limit with lifetimes as long as 1 μs. In an effort to understand the structure of these states, we investigate classically bound states embedded in the dissociative continuum of this molecule. In the first part, we assume J = 0, and specialize to one of the two symmetries, C _{∞ V} or C _{2V}. Poincaré surfaces of section are used to demonstrate the existence of a small region of bound phase space in these 2D problems, but stability analysis of the periodic orbits show that most of them are unstable in 3D. We conclude that J = 0 or, more generally, low J states cannot explain the experiments. In the second part we treat the case J > 0. A total angular momentum centrifugal barrier provides a classically rigorous boundary, which separates the phase space into two parts: a dissociative and a bound region. Wells and double wells exist. Trajectories in these wells show quasiperiodic or chaotic character, depending on the total angular momentum, and on the energy relative to the bottom of the well. Quantally, these states can dissociate by tunneling. One finds long lifetimes in qualitative agreement with the experiments. The volume of the bound part of the phase space is determined by Monte Carlo integration. Typically, several thousand resonance states are found for any J between 20 and 50. This suffices (in principle) to explain the very large number of experimentally observed lines.
1987

ORDER OUT OF CHAOS IN THE H3+ MOLECULE(1987) Chemical Physics Letters. 138, 3Feb, p. 125130 Abstract

(1987) Chemical Physics Letters. 138, 23, p. 125130 Abstract
It is shown that chaotic trajectories embedded in the continuum of the H^{+}
_{3} molecule may be described as a loosely bound H^{+} + H_{2} vibrating rotating complex. 
(1987) Chemical Physics Letters. 137, 2, p. 171174 Abstract
The periodic orbit reduction method is used to analyse timedelay peaks in exact 3D quantal computations. Of the six observed peaks in the H + H_{2} reaction three are related to resonances, the other three to adiabatic barriers.

(1987) The Journal of chemical physics. 87, 3, p. 15961603 Abstract
It was recently suggested that vibrational excitation of van der Waals molecules could lead to prereaction instead of predissociation. A mechanism for vibrational prereaction based on tunneling of a light atom is proposed and tested. The effect of van der Waals wells on the reactivity of collinear MuD_{2}, ClHBr, and ClHCl systems is studied. We find that vibrational prereaction is sensitive to the ratio of tunneling and vibrationally nonadiabatic interaction. If tunneling dominates, prereaction will take place. This is the case for the ClHBr and ClHCl systems. The interplay between reaction probability and photodissociation cross sections is studied. We conclude that vibrational prereaction will lead to an increase in reaction probability and an increase in the formation of products in photodissociation. This study suggests that vibrational prereaction could be observed at least in principle in light atom transfer systems.


(1987) The Journal of chemical physics. 86, 7, p. 39443949 Abstract
Recently measured isomerization rates of transstilbene and diphenylbutadiene over a very large pressure range indicate that a medium can seemingly increase the rate beyond the gas phase high pressure limit. A model based on a generalized Langevin equation is proposed and solved using transition state theory. Rates obtained are in good agreement with experiment. The model incorporates a solvent shift which lowers the barrier to reaction. The model is based on the unification of two different approaches to the description of a dissipative harmonic bath.

(1987) The Journal of chemical physics. 87, 2, p. 10791088 Abstract
This paper shows how to quantize Hamiltonians of symmetric ABA molecules using energies and stability frequencies of simple normal and local mode periodic orbits. It is shown that the quantization can be based either on the idea of adiabatic dynamical potentials or on the hindered rotor representation. In the former case, the stable periodic orbits correspond to the wells and the unstable ones to the barriers of the adiabatic potentials. In the latter case the normal mode periodic orbits correspond to the equilibria of the hindered rotor Hamiltonian, and the local mode periodic orbits correspond to the rotor's orbits with the maximal allowed "angular momentum." Results of extensive numerical testing of both approaches are presented for the H _{2}O model Hamiltonian used by Sibert et al., and for the DavisHeller potential.
1986

(1986) International Journal of Chemical Kinetics. 18, 9, p. 10871100 Abstract
Recent dynamical computations on the LSTH and PK(II) potential energy surfaces are analyzed in terms of different properties of the surfaces. Differences in the bend level structure of resonances are found tobe due to the weaker vibrational force constant at the saddle point of the LSTH surface. The importance of including van der Waals wells in the potential energy surface is demonstrated by analysis of quantal resonances in thecollinear Mu + D_{2} reaction.

(1986) Chemical Physics Letters. 127, 2, p. 178182 Abstract
The quantum theory of activated events in condensed phases, developed by Wolynes using path integral techniques, is derived via harmonic quantum state theory.

(1986) The Journal of chemical physics. 85, 2, p. 865867 Abstract
The generalized Langevin equation of motion for a particle trapped in a onedimensional well with a barrier height V_{0} and coupled to a dissipative medium is modeled by a harmonic bath. Using the properties of the bath and a normal mode analysis we prove that the reactive frequency defined by Grote and Hynes for averaged motion across the barrier is actually a renormalized effective barrier frequency. We then show that the KramersGroteHynes expression for the rate of escape over the barrier is just the continuum limit of the usual gas phase harmonic transition state theory expression.

(1986) Journal of Physical Chemistry. 90, 16, p. 36193624 Abstract
The structure of resonances in the hydrogen exchange reaction is studied by stability analysis of resonant periodic orbits. An adiabatic approximation using bond length bond angle coordinates is shown to explain existing differences between 3D resonances on two potential energy surfaces. Comparison with other approximate methods is provided. We conclude that resonance spectroscopy provides in principle an important experimental tool for obtaining information on the BornOppenheimer potential energy surface.

(1986) Physical Review A. 33, 6, p. 42444252 Abstract
The recently developed theory for tunneling in dissipative systems is rederived using quantal transitionstate theory. As predicted by Caldeira and Leggett, we find an exponential damping of the tunneling rate at 0 K. The exponential rate enhancement at low temperatures as well as the crossover temperature are also obtained with this approach. Moreover, the rate enhancement is given explicitly in terms of energy transfer from the bath to the dissociative mode. The present derivation includes memory effects.

(1986) Chemical Physics Letters. 123, 4, p. 352354 Abstract
An analysis of exact quantal collinear reaction probabilities of the Mu + D_{2} system suggests the feasibility of laserinduced resonant enhancement of bimolecular reactions as well as a new method for spectroscopy of van der Waals complexes.
1985

(1985) Chemical Physics. 99, 1, p. 1533 Abstract
A collinear model for symmetric light atom transfer reactions is constructed and generalized analytically to 3D scattering via a sudden and an adiabatic approach. The model predicts oscillations in the energy dependence of the full 3D integral cross section in the threshold region. The conditions for oscillatory behaviour are a light atom transfer reaction which occurs on a collinearly dominated, lowbarrier potential energy surface. At higher energies there is a switch from a loose to a tight transition state which prevents further oscillations in the cross section. For reactions with large activation energy, such as the Cl + HCl exchange reaction, already at threshold the transition state is tight so that one does not find an oscillatory integral cross section. For the Cl+HCl reaction on the LastBaer potential energy surface, the model rate constants are in good agreement with experiment. Implications of our study for future 3D quantal and classical computations are discussed in detail.

(1985) Chemical Physics Letters. 119, 1, p. 98104 Abstract
A sudden transition state theory is used to show that the recent ab initio computations of Frisch. Binkley and Schaefer of the saddle point properties of the FH_{2} system are compatible with thermal rate and molecular beam experiments.

(1985) Chemical Physics Letters. 113, 6, p. 585588 Abstract
By reinterpreting the IOS cross sections of the D + HH (n = 1) reaction as rotationally averaged cross sections, we find thermal rate constants that are in good agreement with all available experimental results.

(1985) The Journal of chemical physics. 83, 6, p. 28512856 Abstract
The large differences between sudden and adiabatic approximate reactive cross sections are removed by rotationally averaging the bending corrected rotating linear model (BCRLM) cross sections and by shifting the sudden cross sections by the zero point bend energy at the transition state. For D+H _{2}(n=0) we find that the BCRLM rotationally averaged rate constants are in excellent agreement with experiment. For D+H_{2}(n=1), the BCRLM rates are a factor of 610 smaller than the most recent experimental values.

(1985) J. Chem. Phys.. 82, 1, p. 106112 Abstract
Harmonic tunneling corrections are incorporated within semiclassical adiabatic and sudden transition state theory. Good agreement is obtained with CS and sudden quantal computations. Analysis of the transition state theories at the level of rotationally averaged cross sections leads to a new interpretation of the quantal sudden computation and to convergence of the adiabatic and sudden approximations in reactive scattering.

(1985) The Journal of chemical physics. 82, 10, p. 45004508 Abstract
The EisenbudWigner time delay matrix is used to study the dynamics of reaction close to vibrationally adiabatic barrier energies. Maxima in the time delay are predicted and are found to be in excellent agreement with vibrationally adiabatic barrier energies determined by quantized pods. The actual time spent in the vicinity of the barriers is estimated by separating out the free particle time. This "real time" is then used to analyze the validity of the adiabatic and sudden approaches to reactive scattering in the 3D H+H_{2} and D+H_{2} reactions.

(1985) Surface Science. 149, 1, p. 146156 Abstract
A model is formulated and applied to the recently measured vibrational and translational accommodation of NO scattered from a Pt(111) crystal surface. The model assumes that the initial adsorption of NO occurs via a precursor state. Experimentally observed memory of the incident energy by the scattered molecules is a result of competition between chemisorption and desorption from the precursor state.

(1985) The Journal of chemical physics. 83, 3, p. 11111120 Abstract
The concept of time in quantal tunneling processes is reexamined. We find that the WignerEisenbud definition of real time and the definition of imaginary time may be understood in terms of stationary phase analysis in complex time of the microcanonical fluxflux correlation function. This analysis explains why the real time should not be used to justify adiabatic approximations for perpendicular degrees of freedom at tunneling energies. A semiclassical analysis shows that, as suggested by Buettiker and Landauer, the imaginary time should be used to determine the validity of the adiabatic approximation for tunneling processes. Numerical examples for the hydrogen exchange reaction are provided. The implications on adiabatic and sudden approximations in reactive scattering are discussed. A theory unifying the two approaches is outlined.
1984

(1984) Chemical Physics Letters. 111, 45, p. 473480 Abstract
The equivalence of classical microcanonical vibrational sudden and adiabatic transition state theory is established. An optimal coordinate system for the sudden theory is defined by periodic orbit dividing surfaces.

(1984) Chemical Physics Letters. 110, 4, p. 340345 Abstract
Classical and quantal total reaction probabilities are compared with adiabatic and sudden predictions for a model of the H + H_{2} (ν = 1) reaction. Near the adiabatic threshold, a distinct transition is found from dominately adiabatic dynamics to dominately sudden dynamics.

(1984) PHYSICS REVIEW LETTERS. 53, 2, p. 115118 Abstract
We show that the collision time may be interpreted as the time average of a fluxflux correlation function. This interpretation leads quantum mechanically to a complex time whose real part is identical to the usual definition as provided by F. T. Smith. The imaginary part is identical, in the semiclassical limit, to the imaginary time associated with tunneling.

(1984) The Journal of chemical physics. 80, 8, p. 36133622 Abstract
The orthogonal curvilinear coordinate system defined by periodic orbits is used to analyze the quantum mechanical approximation of vibrational adiabaticity in collinear reactive scattering. We show that with this system, the quantal close coupled equations decouple at an adiabatic barrier or well. The classical adiabatic frequency of a periodic orbit is shown to be the classical analog of the quantal adiabatic frequency at a barrier or well. Finally, we find that the success of the recently formulated DIVAH theory, in quantitative prediction of quantal resonance energies, is due to the fact that the diagonal correction terms appearing in this theory serve as curvature corrections. Adding the diagonal terms in the adiabatic approximation in radial coordinates compensates to a large extent for the difference in curvature of these coordinates and the system defined by periodic orbits. Numerical examples include the H_{3}, HMuH, and IHI systems.


(1984) The Journal of chemical physics. 81, 4, p. 18011812 Abstract
Spectroscopic properties of resonances in the 3D H+H_{2} and F+H_{2} reactions are predicted through application of a semiclassical adiabatic theory. The theory is based upon an assumed time scale separation between translationvibration, bending, and overall rotational motions. In the first step, bending and rotational coordinates are frozen, and translationvibration periodic orbits are semiclassically quantized. In the second step, the quantized translationvibration energy (parametrized by the bending angle) serves as an effective potential for the slower bending motion, which is also semiclassically quantized. Finally, average rotational constants are derived from the bending and translationvibration periodic orbits and the total rotational energy is quantized. A novel result is the prediction of bend level structure for excited resonance states.
1983

(1983) Chemical Physics Letters. 102, 5, p. 416420 Abstract
A classical and semiclassical study of a model lightheavylight collinear system predicts the possibility of vibrational bonding on a surface with a saddle point. A comparison is made with vibrational bonding in heavylightheavy systems.

(1983) Chemical Physics Letters. 94, 1, p. 8589 Abstract
A classical trajectory and adiabatic analysis of the IHI system predicts the existence of a single bound (J = 0) state of IHI on a minimumfree potential energy surface.


(1983) The Journal of chemical physics. 78, 6, p. 30143020 Abstract
A vibrationally adiabatic infinite order sudden approximation (IOSA) is formulated in terms of angle dependent adiabatic barriers. These barriers are determined by halfinteger action periodic orbits. We show that in the F+HH reaction, the adiabatic theory describes quantitatively the exact IOSA total reactive cross section over a relatively large range of translational energies. A detailed analysis points out why in many cases transition state theory becomes more accurate as one goes from the collinear to the 3D world. Inversion of the quasiclassical IOSA provides a good estimate of the angular dependence of the adiabatic barrier parameters.

(1983) The Journal of chemical physics. 78, 7, p. 44644476 Abstract
A recently proposed method, based on periodic orbits, for finding vibrationally adiabatic barriers and wells in collinear collisions is generalized to the full threedimensional case. The main idea is a consistent use of the adiabatic approximationone first solves for the fast vibrational motion to obtain an effective Hamiltonian for the slower bend motion which in turn is solved to obtain an effective Hamiltonian for the overall rotation. The method is applied to the hydrogen exchange reaction. We find the bendvibration adiabatic barrier levels for the H_{2}(v=1) state. The zero point motion in the bend degree of freedom is found to be substantial (0.1 eV) and is a source for nonnegligible discrepancies between approximate theories such as the infinite order sudden and quasiclassical trajectory approach and exact quantal scattering computations. Having found the barrier levels we are able to evaluate the collision cross section. Our analysis points out that differences between experimental cross sections and theoretical predictions may be due to inaccuracy in the potential energy surfaces. The available surfaces probably overestimate the adiabatic barrier height.

(1983) The Journal of chemical physics. 78, 3, p. 12281236 Abstract
A classical direct reaction theory is formulated and shown to account for the recently observed quantal oscillations in the reaction probability of light atom transfer reactions. Quasiperiodic orbits and irregular orbits are found for the systems although the potential energy surfaces used have only saddle points in the interaction region. These orbits imply the existence of a new type of bound speciesan adiabatic molecule.

(1983) The Journal of chemical physics. 79, 6, p. 28142821 Abstract
Two algorithms are presented for a direct determination of the boundary between reactive and unreactive portions of phase space in collinear collisions. These algorithms provide a fast and highly accurate determination of classical reactant and product distributions. Since boundary trajectories originate at variational transition states, this method provides new insight to an old problem: the relationship between product and reactant state distributions and the transition state of a chemical reaction.

(1983) The Journal of chemical physics. 78, 7, p. 44144422 Abstract
A detailed forward and reverse quasiclassical trajectory computation for the FHH reaction is presented. An adiabatic analysis of the results shows that to a large extent the differences between HF(v=3) and HF(v=2) product distributions are due to the existence of an exit channel adiabatic barrier for the v=3 state. A sideways peak in the angular distribution for HF(v=2,j) is found in the reverse quasiclassical computation. Total cross sections computed from reverse quasiclassical trajectories are in good agreement with the quantal 1_{in} reactive infinite order sudden approximation. We conclude that many of the discrepancies between forward quasiclassical results and quantal computations are not due to quantal resonances but rather to the large boxing of vibrational states.
1982

(1982) Chemical Physics Letters. 93, 2, p. 184187 Abstract
The IHI system has four vibrationally bonded collinear bound states. They are located in the saddle point region of a minimumfree potential energy surface.

(1982) Chemical Physics. 70, 3, p. 207221 Abstract
Quantal wavefunction, density and flux maps have been computed for the hydrogen exchange reaction. We find that the threshold for appearance of quantal.

(1982) Chemical Physics Letters. 86, 1, p. 2632 Abstract
Isotopic substitution of hydrogen by muonium in the collinear H + H2 reaction causes dramatic changes in the resonance patterns of the quantum reaction probabilities. These changes are explained by a classical model. Using the systems' resonant orbits, we show that resonances appear close to energetic thresholds of new vibrational channels.

TRANSITIONSTATE THEORY AND BEYOND  A CONSTRAINED PHASESPACE APPROACH(1982) Berichte Der BunsenGesellschaftPhysical Chemistry Chemical Physics. 86, 5, p. 458464 Abstract



(1982) Journal of Physical Chemistry. 86, 25, p. 49314937 Abstract
A unified approach to statistical theories of reaction probabilities and to the specificity and selectivity of molecular collisions is presented with applications to collinear reactive collisions in classical mechanics. Statistical theories have previously been derived by using the distribution of trajectories in terms of their number of crossings of a critical surface. Selectivity and specificity refer to the distribution of trajectories in terms of the action variables of phase space (which have quantal analogues). Here we consider the joint distribution, in both number of crossings and the action variables. The result is an expression for the reaction probability which does take the role of selectivity and specificity into account. It reduces to known previous results (e.g., the unified statistical theory, transition state theory, phase space theory, the branching ratio in terms of the entropy deficiency) in suitable limits. Computational examples are provided. An appendix provides an explicit proof of the reactivityselectivity theorem in the framework of statistical theories: For a given flux of all trajectories through a critical surface, the flux of reactive trajectories increases as the selectivity/specificity of the reactive trajectories diminishes.
1981

(1981) Chemical Physics. 61, 3, p. 305316 Abstract
It has recently been shown that periodic orbits define uniquely the vibrationally adiabatic barriers and wells of reactive systems. This analysis is extended here to show how one may define adiabatic frequencies for motion perpendicular to the periodic orbits. In terms of the repulsive and attractive characterization of periodic orbits we find that attractive orbits correspond to adiabatic wells and so may be termed adiabatically stable, repulsive orbits correspond to barriers and are adiabatically unstable. It is shown that the adiabatic frequencies are actually the first order Magnus approximation to the characteristic exponents of a periodic orbit. The adiabatic frequencies of the hydrogen exchange reaction are evaluated numerically. Good agreement is found between the present purely classical computation and adiabatic quantum mechanical results. This is an indication of the general use of the present method in analysing classical and quantal correspondence. The adiabatic frequencies should provide important curvature corrections to the existing vibrationally adiabatic transition state theories.

(1981) Chemical Physics. 60, 2, p. 239247 Abstract
A three dimensional and collinear quasiclassical study of the Li + HF → LiF + H reaction is performed on the ZeiriShapiro semiempirical potential energy surface. The results are compared to a recent experimental study. In agreement with experiment ≈43% of available energy is channeled to the internal (vibrational and rotational) mode of the LiF product, most of which, we show, ends up as rotation (≈33%) and only the remaining 10% as vibration. The differential cross section is shown to peak in the forwardbackward directions as also observed experimentally. The calculated cross section is more backward peaked than the experimental one. The accuracy of the quasiclassical method for this system is assessed by comparison of collinear results with available quantal calculations. It is shown that the average product vibrational distributions are adequately given by the quasiclassical method. The results are analyzed using periodicorbitdividing surfaces which predict correctly the onset of the various dynamic barriers.

(1981) Chemical Physics. 60, 1, p. 2332 Abstract
Quantal collinear reactive scattering computations have shown that in the vicinity of thresholds of reactant or product vibrational states, one finds resonances in the state to state reaction probability. We find that these resonances can be explained classically in terms of energy transfer between adiabatic reactant and product channels. This transfer is attributable to resonant periodic orbits, resonating between reactants and products. The classical condition for a quantal resonance is that the action of the orbit over one period be an integer (in units of h) and that the energy at which this occurs be lower than the adiabatic barrier heights of the resonating states. These conditions suffice for a prediction of the location of the quantal resonance within a 1% accuracy!

(1981) Chemical Physics Letters. 80, 1, p. 4550 Abstract
The PODSs of the H_{3} system and its isotopic derivatives are used for computing exact adiabatic barriers. We find good agreement with experimentally measured ground and vibrationally excited activation energies. The theoretical results should stimulate further measurements of activation energies of vibrationally excited species.

(1981) The Journal of chemical physics. 74, 10, p. 55865594 Abstract
A necessary and sufficient condition for the existence of a classical vibrationally adiabatic barrier or well in collinear systems is the existence of periodic orbit dividing surfaces. Knowledge of all pods immediately provides all adiabatic barriers and wells. Furthermore, the classical equation connecting the barriers and wells to the masses and potential energy surface of the system is shown, under mild conditions, to be identical in form to the corresponding quantal equation. The only difference is in the determination of the vibrational state which is obtained by WKB quantization classically. The classical barriers and wells can therefore be used to analyze quantal computations. Such analysis is provided for the hydrogen exchange reaction and the F + HH system. A novel result is the existence of vibrationally adiabatic barriers even where no saddle point exists on the static potential energy surface. These barriers are an outcome of competition between the increase of potential energy and decrease of vibrational force constant along the reaction coordinate. Their existence is therefore of general nature  not limited to the specific structure of a given potential energy surface. The experimental significance of these barriers is discussed. The implications on the use of forward or reverse quasiclassical computations is analyzed. A definite conclusion is that one should not average over initial vibrational action in such calculations.

(1981) The Journal of chemical physics. 74, 12, p. 67636764 Abstract
A proof of a classical spectral theorem recently conjectured by P. Brumer, D. E. Fitz, and D. Wardlaw [J. Chem. Phys. 72, 386 (198O)] is provided. The theorem states that the amount of time spent by a set of classical trajectories in phase space is proportional to the volume in phase space swept out during that time. Implications of the theorem are briefly discussed.

(1981) The Journal of chemical physics. 74, 12, p. 67656770 Abstract
A quantal transition state bound on the rate of chemical reactions in any dimension is derived. The magnitude of the bound is dependent on the location of a dividing surface and thus is variational. Analysis of the quality of the bound in one and two dimensions shows that the quality of the bound improves with increasing dimensionality of the problem. The new theory requires only the computation of diagonal density matrix elements and so should prove to be computationally useful.

(1981) The Journal of chemical physics. 76, 12, p. 58435848 Abstract
A quasiclassical model with no adjustable parameters is proposed for analysis of resonance widths of collinear atomdiatom reactions. We find two important contributions to the widths. One comes from tunneling through adiabatic exit channel barriers. The other involves the stability frequency of resonant periodic orbits. This frequency, if it is imaginary, is a measure of the nonadiabatic coupling in the system. We find that the resonances of the H+HH exchange reaction are determined by this nonadiabatic coupling. The higher lying resonances of the H+MuH system are controlled by the tunneling mechanism. We find that the resonant periodic orbit of the HMuH reaction is stable over a large energy range. The implications of this stability on analysis of quantal computations are discussed in detail.

(1981) The Journal of chemical physics. 75, 9, p. 44354440 Abstract
A classical prediction on the existence of adiabatic barriers even where no saddle point exists on the potential energy surface is verified using a purely quantal calculation. The adiabatic surfaces are then used for a vibrationally adiabatic transition state theory computation of reaction probabilities. Comparison with exact quantal results shows that the barriers suffice for explaining the socalled "dynamic barriers" to reaction. Since the barriers are in a region where the adiabatic assumption is valid, the adiabatic transition state theory provides an approximate upper bound to the exact reaction probabilities. Finally, it is shown that adiabatic transition state theory coupled with a purely classical transmission factor suffices for explaining most of the oscillatory nature of the exact quantal probability.
1980

(1980) The Journal of chemical physics. 73, 9, p. 43654372 Abstract
An analytical theory for the origin and dynamical implications of multiple trapped periodic trajectories on reactive surfaces is developed, and compared with numerical calculations. The dynamical motion is visualized in an orthogonal curvilinear coordinate system determined by the forms of the trapped trajectories, a device which leads naturally to the introduction of a generating function to determine the number and positions of possible trapped trajectories at any given energy. The connection between this function and the potential surface is examined in detail. This shows that the pattern of trapped trajectories may be deduced from knowledge of the combined variation of the potential energy and the transverse vibrational frequency along the reaction coordinate. This generating function is used to show that the lines of the trapped trajectories correspond to turning points of dynamical flux with respect to position along the reaction coordinate. It also provides a static explanation for the recently observed alternate repulsive and attractive character of successive trapped trajectories.

(1980) The Journal of chemical physics. 73, 9, p. 43734380 Abstract
Instead of finding regions of reactivity in the asymptotic reactants (products) phase space, involving a two dimensional search, one may directly evaluate the boundary of reactivity bands. Here we provide a practical method, for the regime in which transition state theory is not exact, for directly evaluating such boundaries. The method is iterative, convergent and at each iteration step provides improved upper and lower bounds to the reaction probability. A numerical application to the hydrogen exchange reaction giving product distributions and reaction probabilities over a wide range of energies is provided. We find that the existence of two bounded trajectories that are not periodic is crucial to understanding the dynamics of the system.

(1980) The Journal of chemical physics. 72, 3, p. 16691678 Abstract
We derive a rigorous lower bound to the microcanonical reaction probability in classical collinear atomdiatom collisions. This lower bound complements the upper bound provided by transition state theory, and the information needed to calculate the bound is acquired automatically in the search for the periodic orbit dividing surfaces that are possible transition states for the reaction. Numerical calculations for F+H_{2} and H+Cl_{2} over a wide energy range show that the lower bound provides the best available estimate of the reaction probability, short of a full dynamical calculation.

(1980) The Journal of chemical physics. 72, 4, p. 24842494 Abstract
The decline of the reaction probability due to selectivity of the reaction with respect to the initial, reagents, states or with respect to the final, products, states is demonstrated analytically and by computational results for the H + H_{2}, F + H_{2}, and H + Cl_{2} collinear reactive collisions. Upper and lower bounds on the reaction probability in terms of the entropy deficiency of the observed or computed products state distribution (or of the relative rates for state selected reagents) are derived and applied to the results of the collinear computations. For very selective processes, the maximal suprisal provides a tight bound for the (logarithm of the) reaction probability.
1979

(1979) The Journal of chemical physics. 70, 1, p. 325333 Abstract
Miller's unified statistical theory for bimolecular chemical reactions is tested on the collinear H+H_{2} exchange reaction, treated classically. The reaction probability calculated from unified statistical theory is more accurate than that calculated from ordinary transition state theory or from variational transition state theory; in particular, unified statistical theory predicts the highenergy falloff of the reaction probability, which transition state theory does not. A derivation of unified statistical theory is presented that emphasizes the dynamical and statistical assumptions that are the foundation of the theory. We show how these assumptions unambiguously define the "collision complex" in unified statistical theory, and we test these assumptions in detail on the H+H_{2} reaction. Finally, a lower bound on the reaction probability is derived; this bound complements the upper bound provided by transition state theory and is significantly more accurate, for the H+H_{2} reaction, than either transition state theory or unified statistical theory.

(1979) The Journal of chemical physics. 71, 5, p. 20622068 Abstract
Under mild conditions on the longrange behavior of the potential, we show that classical transition state theory is exact at energy E for a collinear atomdiatom reaction if there is only one candidate for transition state at energy Ethat is, only one periodic vibration of energy E across the interaction region.

(1979) The Journal of chemical physics. 70, 8, p. 39953996 Abstract
Keywords: Chemistry, Physical; Physics, Atomic, Molecular & Chemical

(1979) The Journal of chemical physics. 72, 5, p. 29902997 Abstract
Statistical theories are particularly appropriate when one can define a strong interaction regime. We consider the distribution of classical trajectories which enter or exit from this regime. That distribution of trajectories which is of maximal entropy subject only to total conservation of flux is shown to lead to the familiar "phasespace" expression for the reaction probability. By including more refined conservation conditions as constraints one obtains improved statistical theories. As an example the "unified" statistical theory of Miller and the HirschfelderWigner expression for the reaction probability are derived by imposing one more conservation constraint. Transition state theory is derived as a special case corresponding to a particular, extreme, numerical value of the constraint. Phasespace theory is obtained when the value of the constraint is at the other extreme (in which case the constraint is not informative). Essentially, exact results for the reaction probability in the collinear H+H_{2} reactive collision are obtained using two conservation conditions (beside the conservation of total flux). In general, it is shown that the procedure is variational, i.e., that including additional constraints can only improve the results.
1978

(1978) Journal of the American Chemical Society. 100, 10, p. 29842991 Abstract
For many years there has been some question whether one should correct for the effects of molecular symmetry, in the rate expressions of transition state theory, by simply using symmetry numbers, as one does in the equilibrium expressions of statistical thermodynamics; several authors have asserted that the correct rate expressions should instead contain “statistical factors”, which are dynamically defined numbers characteristic of the reaction mechanism. We show that the use of symmetry numbers is always correct, and that statistical factor rate expressions—when they differ from their symmetry number counterparts—are wrong. Special attention is given to reactions involving optically active species, and to symmetric reactions, where it is easy to make mistakes in writing down transition state theory rate expressions. The implications for the Boensted relationsof acidbase catalysis are discussed.

(1978) The Journal of chemical physics. 68, 2, p. 547554 Abstract
The statistical theories of reaction rates play a major role in the formulation and use of the information theoretic approach in reaction dynamics. The statistical theory is used as the prior theory to which the actual rates should be compared. Since one can formulate many different statistical theories, one is faced with the problem of which to use in the information theoretic analysis. Four different theories are reviewed and analyzed for a collinear reaction. It is shown that of these the theory based on the assumption of equal rates for equal product flux in phase space leads to difficulties. Furthermore, a classical collinear calculation of a reaction on a potential surface with a deep well shows that the product state distribution of a reaction involving a long lived complex is well characterized by the statement of equal probability for equal product density in phase space. The implications of these findings on the information theoretic approach are discussed.

(1978) The Journal of chemical physics. 69, 3, p. 12181226 Abstract
We show that the best choice of transition state, for the atom exchange reaction in a classical collinear collision of an atom with a diatomic, is a classical bound state embedded in the continuum: a periodic vibration of the triatomic system across the interaction region of the potential surface. These unstable bound states also serve as limit sets of the trapped trajectories that form the boundary of reactivity bands in molecular collisions, and we comment on the implications of this result for calculation of product state distributions. Numerical calculations of transition states are presented for the collinear H + H_{2} and F + H_{2} reactions.
1977

(1977) Chemical Physics Letters. 47, 3, p. 513516 Abstract
The linear surprisal plots for rotational cross sections of the H + H_{2} exchange reaction are shown to be a manifestation of the transfer of momentum constraint. As a result, it is shown that the complete reactive cross section matrix at a given total energy is a function of the variable E_{j} + E_{j}′. The surprisal of the matrix elements as a function of this variable is linear as predicted from the constraint.

(1977) Chemical Physics. 21, 1, p. 6180 Abstract
Experimental and trajectorygenerated products' translational energy distributions are shown to be well represented by expressing ω(E\u2032_{T}) = P(E\u2032_{T}/P^{0}(E\u2032_{T}) as a gaussian in the final momentum. (A quadratic surprisal plot.) It is also verified that the entropy deficiency of the actual distributions is quite close to the minimal value of the entropy deficiency (subject to the constraints). The dependence of the product E\u2032_{T} distribution on the translational energy of the reagents is found to be simple. Microscopic reversibility is employed to study this point and to relate the surprisal parameters of the forward and reversed reactions. The physical interpretation of the results for direct reactions in terms of a 'FranckCondon'like momentum transfer constraint is discussed.

(1977) The Journal of chemical physics. 67, 12, p. 59765977 Abstract
Keywords: Chemistry, Physical; Physics, Atomic, Molecular & Chemical

(1977) Chemical Physics. 22, 1, p. 151166 Abstract
Keywords: Chemistry, Physical; Physics, Atomic, Molecular & Chemical
1976

(1976) Chemical Physics Letters. 39, 2, p. 199204 Abstract
The surprisal analysis of trajectory generated rate constants of three endoergic reactions is employed to compare the actual effect of reagent vibrational excitation to that expected on purely energetic grounds. It is shown that the enhancement of the reaction rate is not a function of the potential energy surface alone. Rather, reagent vibrational energy has an opposite effect (as compared to the statistical limit) in the endothermic and exothermic regimes of a given endoergic reaction.
1975

(1975) Chemical Physics Letters. 33, 2, p. 201206 Abstract
A FranckCondon type argument, which requires the least transfer of momenta to the nuclei during a collision is outlined and applied to the analysis of translational energy disposal and its dependence on the initial translational energy. Using the maximal entropy procedure of information theory we are able to proceed directly from the assumed (model) constraint to the product state distribution.
1974

(1974) PHYSICS REVIEW A. 9, 6, p. 23982408 Abstract
Upper and lower bounds for transition probabilities in multichannel collision theory are derived, with numerical applications. The approach provides a general scheme for bounding a unitary matrix which depends on a parameter. Particular choices of the parameter include the radial approach coordinate of the colliding systems, with application to closecoupling expansions (both quantum mechanical and semiclassical); the strength parameter of the potential, with applications to the exponential approximation; the time, with applications to transitions induced by a timedependent potential (a case previously treated by Spruch); and, in general, any parameter in the interaction potential. The method is based on two key steps: (i) the derivation of an integral equation for the dependence of the scattering matrix on the parameter, which can then be bounded, and (ii) the use of the matrix H (all whose elements are unity) as the initial upper bound for the (absolute value of the) scattering matrix. This initial bound is then improved by iterating the integral equation. Computational examples for rotational excitation in an atomrotor collision are provided.