Search

We discuss here a convenient numerical approach for the construction of double-ended classical trajectories. This problem occurs frequently in both semiclassical and numerical path integral applications. The present approach utilizes a combination of simulated annealing and path integral methods.

Spatial averaging Monte Carlo (SA-MC) is an efficient algorithm dedicated to the study of rare-event problems. At the heart of this method is the realization that from the equilibrium density a related, modified probability density can be constructed through a suitable transformation. This new density is more highly connected than the original density, which increases the probability for transitions between neighboring states, which in turn speeds up the sampling. The first successful investigations included the diffusion of small molecules in condensed phase environments and characterization of the metastable states of the migration of the CO ligand in myoglobin. In the present work, a general and robust implementation including rotational and torsional moves in the CHARMM molecular modeling software is introduced. Also, a procedure to estimate unbiased properties is proposed in order to compute thermodynamic observables. These procedures are suitable to study a range of topical systems including Lennard-Jones clusters of different sizes and the blocked alanine dipeptide (Ala)(2) in implicit and explicit solvent. In all cases, SA-MC is found to outperform standard Metropolis simulations in sampling configurational space at little extra computational expense. The results for (Ala)(2) in explicit solvent are in good agreement with previous umbrella sampling simulations.

A detector for the detection of laser photons backscattered off an incident high-energy electron beam for precision Compton polarimetry in the 3.5 GeV electron stretcher ring ELSA at Bonn University has been developed using individual photon counting. The photon counting detector is based on a silicon microstrip detector system using dedicated ASIC chips. The produced hits by the pair converted Compton photons are accumulated rather than individually read out. A transverse profile displacement can be measured with μm accuracy rendering a polarization measurement of the order of 1% on the time scale of 10– 15 min possible.

We study the 38-atom Lennard-Jones cluster with parallel tempering Monte Carlo methods in the microcanonical and molecular dynamics ensembles. A new Monte Carlo algorithm is presented that samples rigorously the molecular dynamics ensemble for a system at constant total energy, linear and angular momenta. By combining the parallel tempering technique with molecular dynamics methods, we develop a hybrid method to overcome quasi-ergodicity and to extract both equilibrium and dynamical properties from Monte Carlo and molecular dynamics simulations. Several thermodynamic, structural and dynamical properties are investigated for LJ$_{38}$, including the caloric curve, the diffusion constant and the largest Lyapunov exponent. The importance of insuring ergodicity in molecular dynamics simulations is illustrated by comparing the results of ergodic simulations with earlier molecular dynamics simulations., Comment: Journal of Chemical Physics, accepted

Vibrational line shapes for a hydrogen atom on an embedded atom model (EAM) of the Ni(111) surface are extracted from path integral Monte Carlo data. Maximum entropy methods are utilized to stabilize this inversion. Our results indicate that anharmonic effects are significant, particularly for vibrational motion parallel to the surface. Unlike their normal mode analogs, calculated quantum line shapes for the EAM potential predict the correct ordering of vibrational features corresponding to parallel and perpendicular adsorbate motion.

Described in this work is an approach to computing quantum dynamics which utilizes a conditional (Bayesian) transition probability. Using this approach the problem separates into an equilibrium and a dynamical component. The equilibrium portion of the problem is treated quantum mechanically using Fourier path integral methods, and the dynamical component is treated classically. This approach is shown to give exact quantum dynamics for polynomial potentials of orders one and zero. For more complicated problems the method is shown to be numerically exact for short-times. This Bayesian approach is then used to develop a theory for quantum instantaneous normal modes (INM's).

Two new approaches to numerical QFT are presented., Comment: Lattice2002(theoretical), 3 pages

An efficient experimental scheme for the generation of a monochromatic, slow and bright atomic beam is reported. It combines the good efficiency of a Zeeman-slower for medium slow final velocities with an optimized atomic funnel design. The two severe problems of a Zeeman-slower, as there is the high final divergence and the vanishing efficiency for very low final velocities, are overcome. The total efficiency in flux of the complete device has been measured to more than 10% of the thermal atomic beam. Using a beam of metastable neon atoms (Ne∗) a beam with a transverse velocity spread of σx < 0.34 m/s and a mean longitudinal velocity of 19 m/s with a spread of σv < 1.5 m/s is prepared. By this preparation technique a purely metastable atomic beam is obtained, all Ne atoms in the 3s[3/2]2-state.

Research on optimal control of quantum systems has been severely restricted by the lack of experimentally feasible control pulses. Here, to overcome this obstacle, optimal control is considered with the help of chirped pulses. Simulated annealing is used as the optimizing procedure. The examples treated are pulsed population inversion between electronic levels, and optimization of vibronic excitation in the presence of another electronic level. In the problem of population inversion effective potentials of displaced harmonic oscillators are used. For optimizing vibronic excitation the CsI model potential is used.

The J‐walking (or jump‐walking) method is extended to quantum systems by incorporating it into the Fourier path integral Monte Carlo methodology. J walking can greatly reduce systematic errors due to quasiergodicity, or the incomplete sampling of configuration space in Monte Carlo simulations. As in the classical case, quantum J walking uses a jumping scheme to overcome configurational barriers. It couples the usual Metropolis sampling to a distribution generated at a higher temperature where the sampling is sufficiently ergodic. The J‐walker distributions used in quantum J walking can be either quantum or classical, with classical distributions having the advantage of lower storage requirements, but the disadvantage of being slightly more computationally intensive and having a more limited useful temperature range. The basic techniques are illustrated first on a simple one‐dimensional double well potential based on a quartic polynomial. The suitability of J walking for typical multidimensional quantum Mo...

As summarized by Hamming's motto, "the purpose of computing is insight, not numbers." In the spirit of this dictum, we describe here recent algorithmic developments in the theory of quantum dynamics. Through the use of the somewhat unlikely combination of modern numerical simulations and a visualization device borrowed from 19th century optics, the present efforts suggest the existence of an important, underlying structure in the general problem. This structure, verified for a relatively simple class of model problems, provides broad guidelines for the Keywords: Monte Carlo, stationary phase, quantum dynamics.

Clusters of rare gas atoms provide an interesting setting for the study of the issue of quantum mechanical localization. The properties of these clusters of 2–7 atoms are calculated using variational Monte Carlo methods. To our knowledge, this is the first variational Monte Carlo study of localized clusters and new solidlike wave function forms, including shadow, multiple bond length, and Boltzmann‐like wave functions, are reported. Diffusion Monte Carlo methods provide an independent, exact value of the ground state energy, useful as a check of the variational results. The properties of the variational wave functions, when analyzed in terms of probability distribution functions, quench studies, and visual examination of the wave functions, indicate delocalized helium and localized argon and neon clusters.

A method is introduced that is easy to implement and greatly reduces the systematic error resulting from quasi‐ergodicity, or incomplete sampling of configuration space, in Monte Carlo simulations of systems containing large potential energy barriers. The method makes possible the jumping over these barriers by coupling the usual Metropolis sampling to the Boltzmann distribution generated by another random walker at a higher temperature. The basic techniques are illustrated on some simple classical systems, beginning for heuristic purposes with a simple one‐dimensional double well potential based on a quartic polynomial. The method’s suitability for typical multidimensional Monte Carlo systems is demonstrated by extending the double well potential to several dimensions, and then by applying the method to a multiparticle cluster system consisting of argon atoms bound by pairwise Lennard‐Jones potentials. Remarkable improvements are demonstrated in the convergence rate for the cluster configuration energy, ...

The numerical advantage of quantum Monte Carlo simulations of rigid bodies relative to the flexible simulations is investigated for some simple systems. The results show that if high frequency modes in molecular condensed matter are predominantly in the ground state, the convergence of path integral simulations becomes nonuniform. Rigid body quantum parallel tempering simulations are necessary to accurately capture thermodynamic phenomena in the temperature range where the dynamics are influenced by intermolecular degrees of freedom; the stereographic projection path integral adapted for quantum simulations of asymmetric tops is a significantly more efficient strategy compared with Cartesian coordinate simulations for molecular condensed matter under these conditions. The reweighted random series approach for stereographic path integral Monte Carlo is refined and implemented for the quantum simulation of water clusters treated as an assembly of rigid asymmetric tops.

We describe searches for $B$ meson decays to the charmless vector-vector final states $\ensuremath{\omega}{K}^{*}$, $\ensuremath{\omega}\ensuremath{\rho}$, $\ensuremath{\omega}\ensuremath{\omega}$, and $\ensuremath{\omega}\ensuremath{\phi}$ with $233\ifmmode\times\else\texttimes\fi{}{10}^{6}$ $B\overline{B}$ pairs produced in ${e}^{+}{e}^{\ensuremath{-}}$ $B\overline{B}$ annihilation at $\sqrt{s}=10.58\text{ }\text{ }\mathrm{GeV}$. We also search for the vector-scalar $B$ decay to $\ensuremath{\omega}{f}_{0}$. We measure the following branching fractions in units of ${10}^{\ensuremath{-}6}$: $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}\ensuremath{\omega}{K}^{*0})=2.4\ifmmode\pm\else\textpm\fi{}1.1\ifmmode\pm\else\textpm\fi{}0.7\text{ }\text{ }(l4.2)$, $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}\ensuremath{\omega}{K}^{*+})={0.6}_{\ensuremath{-}1.2\ensuremath{-}0.9}^{+1.4+1.1}\text{ }\text{ }(l3.4)$, $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\rho}}^{0})=\ensuremath{-}0.6\ifmmode\pm\else\textpm\fi{}{0.7}_{\ensuremath{-}0.3}^{+0.8}\text{ }\text{ }(l1.5)$, $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\rho}}^{+})=10.6\ifmmode\pm\else\textpm\fi{}{2.1}_{\ensuremath{-}1.0}^{+1.6}$, $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}\ensuremath{\omega}\ensuremath{\omega})={1.8}_{\ensuremath{-}0.9}^{+1.3}\ifmmode\pm\else\textpm\fi{}0.4\text{ }\text{ }(l4.0)$, $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}\ensuremath{\omega}\ensuremath{\phi})=0.1\ifmmode\pm\else\textpm\fi{}0.5\ifmmode\pm\else\textpm\fi{}0.1\text{ }\text{ }(l1.2)$, and $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}\ensuremath{\omega}{f}_{0})=0.9\ifmmode\pm\else\textpm\fi{}{0.4}_{\ensuremath{-}0.1}^{+0.2}\text{ }\text{ }(l1.5)$. In each case the first error quoted is statistical, the second systematic, and the upper limits are defined at the 90% confidence level. For ${B}^{+}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\rho}}^{+}$ decays we also measure the longitudinal spin component ${f}_{L}=0.82\ifmmode\pm\else\textpm\fi{}0.11\ifmmode\pm\else\textpm\fi{}0.02$ and the charge asymmetry ${\mathcal{A}}_{CP}=0.04\ifmmode\pm\else\textpm\fi{}0.18\ifmmode\pm\else\textpm\fi{}0.02$.

We present studies of the potential energy landscape of selected binary Lennard-Jones thirteen atom clusters. The effect of adding selected impurity atoms to a homogeneous cluster is explored. We analyze the energy landscapes of the studied systems using disconnectivity graphs. The required inherent structures and transition states for the construction of disconnectivity graphs are found by combination of conjugate gradient and eigenvector-following methods. We show that it is possible to controllably induce new structures as well as reorder and stabilize existing structures that are characteristic of higher-lying minima. Moreover, it is shown that the selected structures can have experimentally relevant lifetimes., 12 pages, 14 figures, submitted to J. Chem. Phys. Reasons for replacing a paper: figures 2, 3, 7 and 11 did not show up correctly

A UIMS (user interface management system) that is completely based on nonproprietary software has been developed. A central part of the UIMS are processes (mapper) that act as universal X-clients for each specified X-server. Mapper (graphic server) and applications (graphic clients) exchange requests by an event-driven interface. The communication protocol is free from any graphical information. The most powerful mapper client is a former interpreter that can be programmed to act as an equipment access server. The concept of a graphics server and a form interpreter allows the man-machine interface to be developed in a descriptive way. Appropriate to the level of abstraction in the UIMS, a specific UIDL (user interface definition language) had to be introduced. >

Collision-induced absorption spectra for mixed neon-argon clusters are investigated in the temperature range from 1 K to 250 K using classical molecular dynamics simulations. With increasing cluster size the spectra reproduce the results known from experiment. For smaller clusters $(n+m\ensuremath{\approx}10),$ the simulated spectra depend strongly on the composition of the clusters. Pairwise, Ne-Ne, Ne-Ar, and Ar-Ar interactions designed to accurately describe macroscopic and microscopic properties are used. The influence of two different parametrizations of the dipole moment function is investigated and discussed. For the largest clusters $({\mathrm{Ne}}_{10}{\mathrm{Ar}}_{90}),$ the maximum intensity of the collision-induced infrared spectrum as a function of the temperature closely follows the experimental observations.

Using a common technique for approximating distributions [generalized functions], we are able to use standard Monte Carlo methods to compute QFT quantities in Minkowski spacetime, under phase transitions, or when dealing with coalescing stationary points., Comment: 3 pages, 9 figures, Lattice2002(theoretical)

In this paper we present a method for locating transition states and higher-order saddles on potential energy surfaces using double-ended classical trajectories. We then apply this method to 7- and 8-atom Lennard-Jones clusters, finding one previously unreported transition state for the 7-atom cluster and two for the 8-atom cluster., Journal of Chemical Physics, 13 pages

In the present paper we describe relaxation methods for constructing double-ended classical trajectories. We illustrate our approach with an application to a model anharmonic system, the Henon-Heiles problem. Trajectories for this model exhibit a number of interesting energy-time relationships that appear to be of general use in characterizing the dynamics., Comment: (12 pages, submitted to Chemical Physics Letters. Figures are too large for convenient e-mail access. they are available via anonymous ftp on willie.chem.brown.edu and reside in the directory pub/chem-ph/9407 as the compressed tar file 9407001.tar.Z. If you have difficulty retrieving the figures, please contact J. Doll (doll@carlo.chem.brown.edu) for assistance)

Using a Hubbard-Stratonovich transformation coupled with Fourier path integral methods, expressions are derived for the numerical evaluation of the microcanonical density of states for quantum particles obeying Boltzmann statistics. A numerical algorithmis suggested to evaluate the quantum density of states and illustrated on a one-dimensional model system., Comment: Journal of Chemical Physics

Several stochastic simulations of the TIP4P [W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L. Klein, J. Chem. Phys. 79, 926 (1983)] water octamer are performed. Use is made of the stereographic projection path integral and the Green's function stereographic projection diffusion Monte Carlo techniques, recently developed in one of our groups. The importance sampling for the diffusion Monte Carlo algorithm is obtained by optimizing a simple wave function using variational Monte Carlo enhanced with parallel tempering to overcome quasiergodicity problems. The quantum heat capacity of the TIP4P octamer contains a pronounced melting peak at 160 K, about 50 K lower than the classical melting peak. The zero point energy of the TIP4P water octamer is 0.0348+/-0.0002 hartree. By characterizing several large samples of configurations visited by both guided and unguided diffusion walks, we determine that both the TIP4P and the SPC [H. J. C. Berendsen, J. P. Postma, W. F. von Gunsteren, and J. Hermans, (Intermolecular Forces, Reidel, 1981). p. 331] octamer have a ground state wave functions predominantly contained within the D(2d) basin of attraction. This result contrasts with the structure of the global minimum for the TIP4P potential, which is an S(4) cube. Comparisons of the thermodynamic and ground-state properties are made with the SPC octamer as well.

We examine the efficiency of the partial averaged Fourier path integral method for a model of the (H2)22 system. We show errors in thermodynamic average energies asymptotically approach zero as the inverse of the square of the number of path variables. Using common estimators, we also find the efficiency of the Fourier method for the (H2)22 application comparable to that reported recently [J. Chem. Phys. 109, 2123 (1998)] for effective-potential, discretized approaches.

A Monte Carlo path integral method to study the coupling between the rotation and bending degrees of freedom for water is developed. It is demonstrated that soft internal degrees of freedom that are not stretching in nature can be mapped with stereographic projection coordinates. For water, the bending coordinate is orthogonal to the stereographic projection coordinates used to map its orientation. Methods are developed to compute the classical and quantum Jacobian terms so that the proper infinitely stiff spring constant limit is recovered in the classical limit, and so that the nonconstant nature of the Riemann Cartan curvature scalar is properly accounted in the quantum simulations. The theory is used to investigate the effects of the geometric coupling between the bending and the rotating degrees of freedom for the water monomer in an external field in the 250 to 500 K range. We detect no evidence of geometric coupling between the bending degree of freedom and the orientations.

We present a numerically exact procedure for the calculation of an important class of finite temperature quantum mechanical time correlation functions. The present approach is based around the stationary phase Monte Carlo (SPMC) method, a general mathematical tool for the calculation of high dimensional averages of oscillatory integrands. In the present context the method makes possible the direct numerical path integral calculation of real‐time quantum dynamical quantities for times appreciably greater than the thermal time (βℏ). Illustrative applications involving finite temperature anharmonic motion are presented. Issues of importance with respect to future applications are identified and discussed.

We consider in the present paper an extension of numerical path integral methods for use in computing finite temperature time correlation functions. We demonstrate that coordinate rotation techniques extend appreciably the time domain over which Monte Carlo methods are of use in the construction of such correlation functions.

By expanding Feynman path integrals in a Fourier series a practical Monte Carlo method is developed to calculate the thermodynamic properties of interacting systems obeying quantum Boltzmann statistical mechanics. Working expressions are developed to calculate internal energies, heat capacities, and quantum corrections to free energies. The method is applied to the harmonic oscillator, a double‐well potential, and clusters of Lennard‐Jones atoms parametrized to mimic the behavior of argon. The expansion of the path integrals in a Fourier series is found to be rapidly convergent and the computational effort for quantum calculations is found to be within an order of magnitude of the corresponding classical calculations. Unlike other related methods no special techniques are required to handle systems with strong short‐range repulsive forces.

Path‐integral Monte Carlo calculations in quantum statistical mechanics have been performed using either discretized methods or Fourier methods. In each of these methods the internal energy has been calculated using either temperature differentiation or direct operation on the density matrix by the Hamiltonian. It is shown that the variance of the internal energy calculated by operation of the Hamiltonian on the density matrix in the Fourier method is independent of the number of Fourier components included in the expansion of the paths for a number of systems. The variance of the internal energy obtained from the other methods is shown to grow with the size of the expansion used for all systems.

We present here prototype studies of the effects of quantum mechanics on the melting of clusters. Using equilibrium path integral methods, we examine the melting transition for small rare gas clusters. Argon and neon clusters are considered. We find the quantum‐mechanical effects on the melting and coexistence properties of small neon clusters to be appreciable.

The effects of anharmonicity on the integrated state densities required by RRKM formulations are examined. A PitzerGwinn-like approximation is developed in which the desired state densities are related to those for a solvable reference system by an energy-dependent, classical correction factor. A Monte Carlo calculation procedure is outlined.

Steele's model of elastic gas/surface scattering is solved semiclassically. The model is reasonably successful in reproducing the diffractive data of Smith, O'Keefe, Saltsburg and Palmer.

As illustrated by the diverse range of problems considered in the present volume, Monte Carlo based path-integral methods have widespread use in condensed-phase, many-body studies. The present note considers their use for the study of simple quantum fluids. In particular, we consider a model of a Lennard-Jones fluid for the purpose of examining the numerical convergence characteristics of a number of different path-integral approaches. The system chosen is liquid helium, which has been studied in detail by Pollock and Ceperley./1) As a primitive model of the numerical characteristics of this fluid we consider a one-dimensional "cage" of two fixed helium atoms separated by a distance of twice the reciprocal of the cube root of the fluid density. A third and movable helium atom is located within this cage and interacts with the other two atoms via a two-term Gaussian fit ~2) to the helium Lennar~l-Jones potential (a = 2.556 ~, e = 10.22K). Using various path-integral methods described below, we then compute the free energy of this model system for various temperatures :and note the relative convergence rates of the different methods. Table I lists the free energy of our model system computed at two temperatures for a density of pa3=0.365. The free energies were computed using the "primitive" discretized method which iterates first-order estimates of the "short-time" density matrix. ~3) The Trotter index, P, corresponds to the number of subintervals into which the interval (0,/~h) is divided. The

The generalized Langevin equation approach for gas/solid‐surface scattering is applied to the study of gas translational to surface vibrational energy transfer. A local Brownian approximation to the exact dynamics is used for these studies. Finite temperature lattice studies are facilitated by directly solving the appropriate Fokker–Planck equation.

Numerical brownian dynamics techniques are extended to the case of general gaussian random noise sources. No special form is assumed for either the random force correlation function or the interaction potentials present.

The angular and velocity distributions for gas/solid‐surface collisions are examined. It is shown that the envelope of the scattered phase‐space distribution is quite sensitive to the gas/surface interaction potential.

Monte Carlo based variational electronic structure techniques are examined. The nature of the convergence of such methods, associated extrapolation methods and techniques for wavefunction optimization are discussed. These techniques are illustrated with an application to the H + 3 system. Compact and accurate correlated wavefunctions for this system are reported.

A new Monte Carlo simulation procedure is developed which is expected to produce more rapid convergence than the standard Metropolis method. The trial particle moves are chosen in accord with a Brownian dynamics algorithm rather than at random. For two model systems, a string of point masses joined by harmonic springs and a cluster of charged soft spheres, the new procedure is compared to the standard one and shown to manifest a more rapid convergence rate for some important energetic and structural properties.

Work is presented that examines the effect of substrate motion on the surface self‐diffusion of hydrogen and its isotopes on the Cu(100) surface. Lattice motion, represented as a sum of Lennard‐Jones interactions, is found to increase the diffusion constant of hydrogen and its isotopes at all temperatures examined. The increase varies from 1.3 to 4.0 over the temperature range from 1000 to 110 K. The results agree with the recent calculations of Lauderdale and Truhlar above 150 K. The quantum contribution to the isotope effect is enhanced relative to the values for the frozen substrate. These conclusions are based on approximate path integral calculations in which quantum‐mechanical effects are treated in a semiclassical manner using temperature‐dependent effective potentials. The differences between the present results and those of Lauderdale and Truhlar are attributed to a breakdown of these semiclassical approximations at low temperatures. In the temperature range considered, commonly accepted harmonic corrections to the classical results afford slightly poorer agreement with Lauderdale and Truhlar than the present results.

General numerical techniques suitable for the treatment of Gaussian generalized Langevin dynamics are developed. The specific problem considered is that of gas/solid–surface collisions. The basic result is that minor modifications of the methods commonly utilized in classical‐mechanical trajectory approaches allow their use in generalized Langevin studies. Sample numerical results are presented for the Ne/W and Ar/W systems.

Thermal desorption is examined using a combination of recently developed generalized Langevin methods and techniques related to Bennett’s induced trajectory approach. The combined methods appear well suited for the treatment of the dynamics of infrequent, many‐body events.

The time domain Langevin kernels necessary for the treatment of inelastic gas–solid surface collisions are examined for several model systems. General techniques are presented for construction of these time domain kernels. A preliminary examination of results for disordered lattices is made.

A general dynamical system of N degrees of freedom is considered as a zeroth order separable part plus a nonseparable perturbation, and explicit formulas are derived for the classical‐limit approximation to the eigenvalues through second order in the nonseparable interaction. As a numerical example, the first and second order perturbation energies of the helium atom are calculated semiclassically, and agreement with the exact quantum mechanical values is quite reasonable.

The elastic scattering of atoms from solid surfaces is examined within the semiclassical framework. Explicit expression for diffraction intensities are obtained which utilize classical trajectory information as the computational device. Effect of lattice disorder are examined.The results of Beeby and Weinberg are considered.An alternative method for extracting the surface atom displacements and the attractive well depth of the atom-surface interaction is discussed.

Complex‐valued classical trajectories have been computed by direct numerical integration of the equations of motion for three‐dimensional collisions of He and H2, and from such trajectories classical S‐matrix elements for transitions between specific rotational‐vibrational states of H2 have been constructed. At the collision energies employed (∼1–2 eV) all vibrationally inelastic transitions are classically forbidden, thus the need for analytically continued, complex‐valued trajectories. Comparison with the quantum mechanical calculations of Eastes and Secrest shows excellent agreement between the quantum and uniform semiclassical transition probabilities. Since, however, for three‐dimensional collision systems one is seldom concerned with individual S‐matrix elements, but rather sums over many of them, an important practical feature is developed which shows how one can combine the usual Monte Carlo classical treatment of some of the internal degrees of freedom with a semiclassical state‐by‐state description of others; i.e., one can ``quantize'' only those degrees of freedom that are highly quantumlike (e.g., vibration). This ``partial averaging'' approach also greatly simplifies the practical aspects of applying classical S‐matrix theory to systems with several internal degrees of freedom.

Closed form expressions are obtained for the vibrational transition probabilities in a semiclassical formulation for the inelastic hard sphere collision of an atom with a harmonic oscillator. The uniform semiclassical expression is found to be in excellent agreement with the numerically computed quantum mechanical transition probabilities. Expressions are also obtained for the average energy transfer in this collision and for the scattering of atoms from a hard cube model surface.

Energy transfer between a gas and some other system is a process of interest to workers in a number of areas. Depending on whether the second system is another gas molecule, a small aggregate of molecules, or a solid surface; gas-phase kinetics, aerosol microphysics, or solid-state studies are involved. Energy-transfer theories are best developed for the gas-phase case where currently available methods provide reasonably reliable treatment for both the interactions and the collision dynamics.