Publications

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 ℏ² 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.

The uniform semiclassical expression for the energy-dependent 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 ℏ²ⁿ expansion of the thermal transmission coefficient. The central result of this paper is that adding an ℏ² 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 \u201cstandard\u201d 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 one-dimensional theories are also generalized to many-dimensional systems. Their effect on the thermal instanton theory is discussed.

The quantum version of Kramers turnover theory is generalized beyond the parabolic barrier approximation. The result is a uniform instanton-based 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.

Instanton-based rate theory is a powerful tool that is used to explore tunneling in many-dimensional systems. Yet, it diverges at the so-called \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 many-dimensional 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 well-known 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

The central claim of Gavassino and Disconzi [Gavassino and Disconzi, Phys. Rev. A 107, 032209 (2023)2469-992610.1103/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.

Nine decades after Wigner's formulation of quantum rate theory, his celebrated result was recently generalized to the asymmetric barrier using an exact first-order expansion of the transmission probability in terms of ℏ2. This paper extends the first-order quantum correction to second-order 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 VPT-4 (vibrational perturbation theory - fourth order) and will potentially need a VPT-6 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.

Single-molecule 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.

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 time-domain into quantum effects in nonadiabatic scattering.

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 non-negative 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 forwardbackward 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 by-product, this example shows that one may obtain \u201cgood\u201d tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.

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 well-known 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 Temples lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temples results. In this paper, we further improve the SCLBT and compare its quality with Lehmanns theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmanns 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 Lehmanns lower bound theory. The novel iSCLBT exhibits a significant improvement over the previous version. Both Lehmanns theory and the SCLBT variants provide significantly better lower bounds than those obtained from Weinsteins and Temples 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 low-lying excited states as well. Compared to Lehmanns 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.

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 ButtikerLandauer 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 well-defined only for incident states which are monochromatic in energy, others are well-defined 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 ill-posed) question: how much time does it take to tunnel through a barrier?

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 Temples 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 CauchySchwartz 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 self-consistent lower bound theory (J. Chem. Phys.2020,115, 244110) are identical. Examples include implementation to the hydrogen and helium atoms.

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 self-consistent 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 double-well potential. The tight bounds are due to the very high accuracy of the lower bounds obtained for the energy levels, using the self-consistent 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.

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 self-consistent, 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 self-consistently 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.

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.

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 approximations-can they account for temporal interference? Published by AIP Publishing.

The vibronic absorption spectrum of the electric dipole forbidden and vibronically allowed S₁(¹ A₂) ← S₀(¹ A₁) 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 Herman-Kluk semiclassical initial value representation (HK-SCIVR) are compared to assess the accuracy and convergence of these methods, benchmarked against numerically exact time-dependent 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 HK-SCIVR approach with ∼10⁵ 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 10-100 cm^-1 occur, underscoring that both electronic structure calculations and dynamical approaches remain challenging in the calculation of typical small-molecule excited-state spectra by trajectory-based methods.

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 time-energy uncertainty principle and time-energy 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 cost-it 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.

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 one-dimensional time-independent 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 one-dimensional tunneling and reinforces the conclusion that the tunneling flight time vanishes.

The transition-path-time 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 δ-function potential, a square-barrier potential, and symmetric-double-well dynamics at very low temperature. These studies exemplify extreme nonlocality for motion in δ-function potentials, vanishing tunneling times for the square-barrier potential, and varying transit times in the symmetric-double-well 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.

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.

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.

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 (diffractive-black line) and classical Wigner (red circles) angular distributions associated with the scattering along the [100] surface direction.

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 one-dimensional 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.

A second-order 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 in-plane scattering of Ar atoms from a LiF(100) surface. The resulting diffraction intensities are compared with second-order perturbation theory classical distributions and close-coupling results for two incident energies of 300 and 700 meV. The previous first-order perturbation theory predicts a symmetric diffraction pattern about the elastic peak, while the second-order semiclassical perturbation theory accounts correctly for the asymmetry in the diffraction pattern.

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.

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.

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 in-plane 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.

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 lipoxygenase-1. 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.

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 first-order 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.

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.

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 cos²(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.

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 single-particle ansatz. We apply the method to the argon trimer and investigate the dissociation process of the cluster. The results clearly show a classical-like 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 low-temperature 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.

Measurements of the atomic-scale 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 transition-state theory description of the quantum contribution to the motion.

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 leading-order 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 double-well 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 path-integral evaluation demonstrates that the Gaussian based perturbation series representation converges much faster.

In-plane 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 higher-order terms can also increase the number of rainbow scattering angles. Three rainbows are found for a second-order sine term in the corrugation, four symmetrically spaced rainbow angles are found when adding in a second-order 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).

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 phase-space integrals appearing in the semiclassical expressions and therefore leads to a large reduction in the computational effort. In addition, other energy-dependent 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, Herman-Kluk, 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 Herman-Kluk 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.

The anharmonic S₀ → S₁ vibronic absorption spectrum of the formaldehyde molecule is computed on the fly using semiclassical dynamics. This first example of an on-the-fly 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.

We present a theoretical study of the S(0)-> S(1) and S(0)

We present a computation of the classical momentum and velocity correlation functions of Br₂ 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 mass-weighted 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.

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 forward-backward semiclassical initial value representation expression for thermal correlation functions. (c) 2007 American Institute of Physics.

The Rayleigh-Ritz functional is used in conjunction with an approximate time evolution to improve ab initio estimates of ground-state 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 coherent-state 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 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.

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.

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'nikov-Pollak-Grabert-Hänggi [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.

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 Herman-Kluk 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.

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.

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 Herman-Kulk 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.

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.

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 Herman-Kluk propagator.

The dynamics and dissipative tunneling in a symmetric quartic double well potential was studied with the aim of numerically investigating the Wigner-Fokker-Planck (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.

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.

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 Hansen-Andersen approach on the other. It is shown that MQCLT gives the exact first couple of nonzero initial time derivatives of the flux-flux autocorrelation function, hence, it falls under the category of approximations considered by Hansel and Andersen.

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.

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.

The room temperature photoinduced fluorescence decay of isolated trans-stilbene and trans-stilbene 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 (S-0) to the ground vibrational state of the S-1 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 trans-stilbene upon excitation at the omega(00) frequency. The experimental results are in good agreement with theoretical analysis. (C) 2000 American Institute of Physics. [S0021-9606(00)01809-2].

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_BT) 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 10⁸-10⁹V m^-1.

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 ground-state frequencies and if position shifts are not too large, then there exist excitation frequencies for which the excited-state 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.

Kramers' theory is used to derive simple expressions for the hopping distribution in multidimensional activated surface diffusion. The expressions are tested against one- and two-dimensional 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_BT or more. The ratio of double hops to single hops is shown to obey an Arrhenius-like 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.

Many unimolecular reactions are initiated by photoexcitation of a polyatomic molecule at room temperature from its S₀ ground state to an electronically excited S₁ 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 trans-stilbene. Application to the decay dynamics of the trans-stilbene molecule shows that an initial temperature of 230 K for trans-stilbene in the excited S₁ state suffices for good agreement between the theoretical and experimental survival probability measured at a gas temperature of 300 K.

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 trans-stilbene 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.

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_BT≥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.

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 L² 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.

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 two-dimensional system studied previously by Straub and Berne [J. Chem. Phys. 85, 2999 (1986)]. The theory, which is the multidimensional generalization of the one-dimensional 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.

A theoretical investigation of the experimental measurements of the isomerization rate of trans-stilbene 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.

A numerical study of the effect of dissipation on the radiationless transition rate in the adiabatic and solvent-controlled 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.

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.

Application of the Newton method locating stable periodic orbits is extended to include nonotating and rotating triatomic molecules in 3D. A Monte Carlo-Newton method search for stable periodic orbits of the H₃⁺ molecular ion at the dissociation energy into H⁺ + H₂ 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 H₃⁺ is the most important stable orbit. An additional new stable orbit is also discovered. Implications for quantum computations and experimental results are discussed.

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 one-dimensional potential energy surface. The classical reactive flux method has been modified to account for semiclassical tunneling and above-barrier 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.

A new approach is suggested for evaluation of the radiationless transition rate for the curve-crossing problem in the presence of dissipation. The rate is evaluated by using the conventional Landau-Zener theory but for a collective system - bath coordinate, which is characterized by a maximal mean-free 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.

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.

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 Fokker-Planck equation which is generalized to include time dependent friction; (b) the Kramers' stationary flux distribution function; (c) the stochastic separatrix.

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.

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.

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.

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.

A dynamically corrected variational transition-state 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 Hamiltonian-equivalent formulation of the generalized Langevin equation. The dynamical corrections are obtained by utilizing the reactive-flux method in which the choice of dividing surface is guided by minimization of the transition-state flux. Analytic correction formulas, valid for memory friction, are obtained for the Kramers-Grote-Hynes estimate of the rate in the range from moderate friction to the large-friction 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 mean-first-passage-time formulation. The dynamically corrected variational transition-state-theory expansion is shown to be identical to the mean-first-passage-time result.

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.

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 two-degree-of-freedom 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.

New developments in the application of variational transition-state 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 temperature-dependent 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 transition-state 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.

Keywords: Multidisciplinary Sciences

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 two-degrees-of-freedom Hamiltonian involving the unstable normal mode and a newly defined collective bath mode. Explicit treatment of strong nonlinearities or low-barrier systems present no special problem.

Quantum-mechanical 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.

A study of the semiclassical analogue of hyperspherical quantum resonances found by Manz and co-workers 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 collision-induced dissociation.

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.

The word \u201cmetastable\u201d has traditionally been used to denote states that decay with a much slower rate than what is \u201cnormally\u201d 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.

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.

Using a sudden transition state theory we find that photodissociation of long-lived resonances of D₂H⁺ 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 zero-point energies between D₂ and HD explains an additional aspect of the Carrington-Kennedy experiments.

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 well-known 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.

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 MuD₂, 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.

Recently measured isomerization rates of trans-stilbene 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.

The quantum theory of activated events in condensed phases, developed by Wolynes using path integral techniques, is derived via harmonic quantum state theory.

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 Born-Oppenheimer potential energy surface.

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 FH₂ system are compatible with thermal rate and molecular beam experiments.

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+H₂(n=1), the BCRLM rates are a factor of 6-10 smaller than the most recent experimental values.

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.

Classical and quantal total reaction probabilities are compared with adiabatic and sudden predictions for a model of the H + H₂ (ν = 1) reaction. Near the adiabatic threshold, a distinct transition is found from dominately adiabatic dynamics to dominately sudden dynamics.

A vibrationally adiabatic infinite order sudden approximation (IOSA) is formulated in terms of angle dependent adiabatic barriers. These barriers are determined by half-integer 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.

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 species-an adiabatic molecule.

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.

Quantal wavefunction, density and flux maps have been computed for the hydrogen exchange reaction. We find that the threshold for appearance of quantal.

A three dimensional and collinear quasiclassical study of the Li + HF → LiF + H reaction is performed on the Zeiri-Shapiro semi-empirical 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 forward-backward 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 periodic-orbit-dividing surfaces which predict correctly the onset of the various dynamic barriers.

The PODSs of the H₃ 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.

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.

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 so-called "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.

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.

Keywords: Chemistry, Physical; Physics, Atomic, Molecular & Chemical

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 + H₂ and F + H₂ reactions.

Experimental and trajectory-generated products' translational energy distributions are shown to be well represented by expressing ω(E_T) = P(E_T/P⁰(E_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_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 'Franck-Condon'-like momentum transfer constraint is discussed.