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.

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.

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.

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.

Trans-stilbene in n-hexane is excited with excess vibrational energy in the range 0-7000 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.

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.

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.

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.

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.

Exact time-dependent 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]

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.

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 "energy-loss deflection function" for scattering of particles from a periodic surface whose extrema lead to a new form-the "energy-loss rainbow" which appears as multiple maxima in the final energy distribution of the scattered particle. Energy-loss rainbows are caused by frictional phonon effects which induce structure in the energy-loss distribution instead of "washing it out." We provide evidence that they have been observed in Ne scattering on self-assembled monolayers.

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.

The time correlation functions for a Gaussian wave-packet 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.

We present a theoretical study of the S0 → S1 and S0 ← S1 vibronic spectra for trans-stilbene. Franck-Condon spectra in the harmonic approximation are generated for the complete system with 72 degrees of freedom by means of an analytic time-dependent approach accounting for Dushinsky rotations and thermal effects. The force fields are computed by means of density functional theory (DFT) and time-dependent DFT, on the one hand, and ab initio complete active space self-consistent field theory, on the other hand. The B3LYP functional shows that almost planar potential energy surface minima are found for the S0 and S1 state. Imposing C2h symmetry constraints, we obtain low-temperature high-resolution Franck-Condon spectra for both absorption and emission which are in reasonably good agreement with the experimental spectra measured by Syage [J. Chem. Phys. 81, 4685 (1984)] in supersonic jets. Due to thermal population of low-energy modes, the room temperature absorption spectrum is very broad. An almost structureless band which extends over several thousand cm-1 is obtained from the present theory and agrees with the experimental absorption band shape. Finally, within the harmonic model, we study the effect of photoexcitation on the energy distribution in the excited S1 state. We find noticeable cooling of ≈20 K within a frequency interval spanning from -400 to 200 cm-1 around the 0-0 transition. This indicates that photoinduced cooling must be taken into account when considering the dynamics of the photoinduced isomerization of trans-stilbene. Moreover, this is not the final word, as anharmonicity of the low frequency modes must be taken into account to obtain a full picture which would explain both the energy dependence of the isomerization rates as well as the dependence on the pressure of an external buffer gas.

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 classical-like 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.

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.

A second-order 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 second-order 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.

A general expression for thawed semiclassical initial-value representation propagators has been derived in the multidimensional form. The thawed Gaussian propagator of Heller and the coherent-state-averaged 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 higher-order terms of the averaged potential.

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 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.

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.

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 nature of the nascent vibrational distribution in the excited donor state in photoinduced electron transfer is shown to have a profound effect on the electron-transfer 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 electron-transfer 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 electron-transfer rate by variation of the excitation frequency.

Mixed quantum classical rate theory (MQCLT) was applied to the collinear hydrogen exchange reaction on the PK II potential energy surfaces and Liu-Siegbahn-Truhlar-Horowitz (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.

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.

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 generalization of Doetsch's formula [Math. Z. 42, 263 (1937)] is derived to develop a stable numerical inversion of the one-sided 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. [S0021-9606(99)00421-3].

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.

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.

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.

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.

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.

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 kx²y²/2.

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.

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 transition-state theory. A saddle-point 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.

The migration of adsorbed atoms on crystal surfaces is considered. To describe the adatom motion one often uses a generalized Langevin equation (GLE). The time-dependent friction, which enters the GLE, is caused by the interaction with the crystal excitations. However, the explicit form of the time-dependent 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 time-dependent site distribution and compare it with an experiment where correlated hops have been observed. We conclude that even when the motion is one-dimensional, correlated hops are to be expected only for sufficiently high temperatures or for physisorbed atoms.

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.

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.

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.

The variational-transition-state theory (VTST) approach to condensed-phase activated-rate 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.

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 all-points 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.

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.

It is demonstrated that a recent finite-barrier expansion for jump rates accounts quantitatively for the observed discrepancy between numerically determined exact rates and the Kramers estimates of these rates.

The one-dimensional 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 Fokker-Planck equation emerges naturally from this formulation.

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.

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 Kramers-Grote-Hynes reactive frequency. In this way, the Kramers-Grote-Hynes 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 Kramers-Grote-Hynes 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 non-Kramers 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.

A generalization of the Kramers-Grote-Hynes 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 Kramers-Grote-Hynes 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.

Keywords: Multidisciplinary Sciences

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.

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.

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.

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 memory-friction 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.

Classical mechanics is used to compute the low-resolution spectrum of Na3 in the classically chaotic regime. Excellent agreement between the numerical simulation and experiment enables an assignment of the experimental coarse-grained 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.

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 quantummechanical diffusion of a free particle.

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.

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.

It is shown that chaotic trajectories embedded in the continuum of the H⁺
₃ molecule may be described as a loosely bound H⁺ + H₂ vibrating rotating complex.

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 recently developed theory for tunneling in dissipative systems is rederived using quantal transition-state 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.

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.

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 H₃, HMuH, and IHI systems.

A recently proposed method, based on periodic orbits, for finding vibrationally adiabatic barriers and wells in collinear collisions is generalized to the full three-dimensional case. The main idea is a consistent use of the adiabatic approximation-one 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 bend-vibration adiabatic barrier levels for the H₂(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.

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.

A classical trajectory and adiabatic analysis of the IHI system predicts the existence of a single bound (J = 0) state of IHI on a minimum-free potential energy surface.

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

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 reactivity-selectivity 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.

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 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.

A quasiclassical model with no adjustable parameters is proposed for analysis of resonance widths of collinear atom-diatom 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.

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.

Under mild conditions on the long-range behavior of the potential, we show that classical transition state theory is exact at energy E for a collinear atom-diatom reaction if there is only one candidate for transition state at energy E-that is, only one periodic vibration of energy E across the interaction region.

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.