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Notes: We compare the sensitivities to initial conditions for both direct (regular) and long-lived (chaotic) trajectories in classical scattering calculations with the corresponding properties of trajectories of position and momentum expectation values for quantum wave packets. The collinear H+H2 reaction is used as an example. The results show that the high sensitivity seen in chaotic trajectories is not reflected in the quantum dynamics. We conclude that it is possible for a classical ensemble consisting of only regular trajectories to respond trajectory by trajectory to perturbations in much the same way as a quantum wave packet. (There will of course be cases that are exceptions to this rule.) The response of an ensemble consisting of chaotic trajectories may on the average be similar to that of a wave packet, but not at the level of individual trajectories. In addition, the sensitivities of these trajectories to variations in the potential are analyzed. We conclude that the large contributions to the sensitivities from particular long-lived trajectories must approximately cancel when an exact ensemble average is taken. An algorithm is presented to smoothly account for the contributions to the sensitivities from these trajectories.

Notes: This paper presents a method for inverting temporal experimental data from chemical models to obtain estimates of unknown parameters. Most of the models under consideration are deterministic and we assume that the measurements obtained from experimental observations are represented as the solution of a differential equation containing the variables of the model. To incorporate any extraneous laboratory effects that are not included in the model, we assume that these equations are perturbed by a white noise process so that the measurements become time-dependent stochastic variables. A particular measurement is then equivalent to a realization of these variables and applying stochastic estimation theory this realization can be used to obtain estimates of the unknown parameters in the model. As an example of this estimation method, we consider chemical kinetics models with various observational equations and construct an estimator for the unknown reaction rate constants. We also show the estimators for the structural constants in a laser model depending on the representation of the experimental data. In some cases the observations are simulated numerically and we present the parameter estimates as a function of time. The efficiency of the estimation process is calculated as the ratio of the a posteriori variance of the parameter estimator and the Rao–Cramer lower bound. Some issues in the numerical implementation of the filtering equations are discussed and a comparison is made between the minimum least-square estimation method and the filtering method.

Notes: We used a randomly corrugated hard wall model and the sudden approximation to analyze two experiments on atom scattering from disordered surfaces. In one, the structural surface disorder was caused by ion bombardment. In the other, the disorder was due to an incomplete overlayer of adsorbed atoms. We also present a study of the scattering of a rigid rotor by a randomly corrugated hard wall using the sudden approximation.

Notes: Functional sensitivity analysis is used to study the effect of potential structure upon the elastic scattering of He atoms from a one-dimensional surface. The calculations are implemented by computing the total scattering wave functions from a wave packet calculation by a Møller wave operator method. The functional sensitivities of the various diffraction probabilities for several angles of incidence and surface corrugation are studied. The method is extended to examine the role of potential structure for a surface with adsorbed impurities. It was observed that the various diffraction processes draw from local regions of the potential in very different ways. At high angles of incidence for back scattering and particularly for strong surface corrugation, the large protruding portions of the surface cast a "shadow'' of lower dynamical sensitivity. Results of this type should ultimately be insightful for the inversion of experimental data to obtain the interaction potential.

Notes: The corresponding effects of features in the potential on classical, semiclassical, and quantum mechanics are probed using the technique of functional sensitivity analysis. It is shown that the classical and quantum functional sensitivities are equivalent in the classical (small (h-dash-bar)) and harmonic limits. Classical and quantum mechanics are known to react in qualitatively similar ways provided that features on the potential are smooth on the length scale of oscillations in the quantum wave function. By using functional sensitivity analysis, we are able to show in detail how the classical and quantum dynamics differ in the way that they sense the potential. Two examples are given, the first of which is the harmonic oscillator. This problem is well understood by other means but is useful to examine because it illustrates the detailed information about the interaction of the potential and the dynamics which can be provided by functional sensitivity analysis, simplifying the analysis of more complex systems. The second example is the collinear H+H2 reaction. In that case there are a number of detailed and striking differences between the ways that classical and quantum mechanics react to features on the potential. For features which are broad compared to oscillations in the wave function, the two react in qualitatively the same way. The sensitivities are oscillatory, however, and there are phasing differences between the classical and quantum sensitivity functions. This means that using classical mechanics plus experimental data in an inversion scheme intended to find the "true'' potential will necessarily introduce sizeable errors.

Notes: The method of classical functional sensitivity analysis (CFSA) is applied to the collinear F+H2 reaction and its isotopic analogues. The effects of features in the Muckerman 5 (M5) potential upon product vibrational energy are examined using constant energy ensembles (1 and 5 kcal/mol) with H2(v=0) and H2(v=1). The same is done using thermal averaged ensembles at 300, 400, 1000, and 1500 K. Two significant results are reported. First, at low energies and at even relatively high temperatures (1000 K), the region of the potential to which the reactive product vibrational energy is most sensitive is the vicinity of the saddle point. In addition, the sensitivity function is smooth and nonoscillatory there. Second, at high temperatures and energies, the sensitivity function becomes oscillatory in magnitude and sign over the entire interaction region. The implications which these results have for methods which fit a potential to experimental data are discussed.

Notes: A recently developed global, nonlinear map-facilitated quantum inversion procedure is used to obtain the interaction potential for Ar–HCl(v=0) based on the rotationally resolved state-to-state inelastic cross sections of Lorenz, Westley, and Chandler [Phys. Chem. Chem. Phys. 2, 481 (2000)] as well as rovibrational spectral data. The algorithm adopted here makes use of nonlinear potential→observable maps to reveal the complete family of surfaces that reproduce the observed scattering and spectral data to within its experimental error. A nonlinear analysis is performed on the error propagation from the measured data to the recovered family of potentials. The family of potentials extracted from the inversion data is compared to the Hutson H6(4,3,0) surface [Phys. Chem. 96, 4237 (1992)], which was unable to fully account for the inelastic scattering data [Phys. Chem. Chem. Phys. 2, 481 (2000)]. There is excellent agreement with H6(4,3,0) in the attractive well, where Hutson's surface is considered most reliable. There is also good long-range agreement. However, it is shown that H6(4,3,0) predicts too soft a wall for the linear Ar–HCl configuration and significantly too steep a wall for linear Ar–ClH. These differences account for the systematically backscattered inelastic cross sections computed using the H6(4,3,0) surface. The new, nonlinear inversion results provide a global Ar–HCl interaction potential with reliable error bars that are consistent with all of the experimental data. © 2001 American Institute of Physics.

Notes: Three-body interactions in a homonuclear van der Waals bound trimer (the 1 4A2′ state of Na3) are studied spectroscopically for the first time using laser induced emission spectroscopy on a liquid helium nanodroplet coupled with ab initio calculations. The van der Waals bound, spin polarized sodium trimers are prepared via pickup by, and selective survival in, a beam of helium clusters. Laser excitation from the 1 4A2′ to the 2 4E′ state, followed by dispersion of the fluorescence emission, allows for the resolution of the structure due to the vibrational levels of the lower state and for the gathering of precise information on the three-body interatomic potential. From previous experiments on Na2 we know that the presence of the liquid helium perturbs the spectra by a very small amount [see J. Higgins et al., J. Phys. Chem. 102, 4952 (1998)]. Ab initio potential energy calculations are carried out at 42 geometries of the lowest quartet state using the coupled cluster method at the single, double, and noniterative triple excitations level [CCSD(T)]. The full potential energy surface is obtained from the ab initio points using an interpolation procedure based on a Reproducing Kernel Hilbert Space (RKHS) methodology. This surface is compared to a second, constructed using an analytical model function for both the two-body interaction and the nonadditivity correction. The latter is calculated as the difference between the CCSD(T) points and the sum of the two-body interactions. The bound vibrational states are calculated using the two potential energy surfaces and are compared to the experimentally determined levels. The calculated bound levels are combined with an intensity calculation of the ν2″ mode of E′ symmetry derived from a Jahn–Teller analysis of the excited electronic state. The calculated frequencies of ν1″ and ν2″ are found to be 37.1 cm−1 and 44.7 cm−1, respectively, using the RKHS potential surface while values of 37.1 cm−1 and 40.8 cm−1 are obtained from the analytical potential. These values are found to be in good to fair agreement with those obtained from the emission spectrum and to be significantly different from any values calculated from additive potential energy surfaces. The 1 4A2′ Na3 potential energy surface is characterized by a D3h symmetry minimum of −850 cm−1 (relative to the three 3 2S Na atom dissociation limit) with a bond distance of 4.406 Å. This bond distance differs by about 0.8 Å from the value of 5.2 Å found for the sodium triplet dimer. This means that approximately 80% of the binding energy at the potential minimum is due to three-body effects. This strong nonadditivity is overwhelmingly due to the deformability of the valence electron density of the Na atoms which leads to a significant decrease of the exchange overlap energy in the trimer. © 2000 American Institute of Physics.

Notes: The collinear dynamics of a model diatom–diatom system is investigated. The collision partners are harmonic oscillators for which the masses and force constants are chosen to correspond to those of the nitrogen and oxygen molecules. The interaction between the molecules arises from a Lennard-Jones 6-12 potential acting between the inside atoms in the collinear system. Quantum mechanical close coupled calculations are performed for several collision energies ranging from 1.0 to 2.25 eV. The state-to-state transition probabilities which are extracted from these calculations are then used as a benchmark for comparison. Semiclassical calculations are performed within the framework of a classical path approximation. A simple scheme to modify the classical path to reflect energy exchange between the collision coordinate and the internal degrees of freedom is found to improve the results. On the whole, the agreement between the semiclassical and the quantum mechanical results is surprisingly good. The classical trajectory calculations correctly display many of the qualitative features of the collisions but the numerical agreement is not as close. Unexpectedly, the classical results do not appear to be improving as the collision energy is increased.

Notes: This work describes an extension of the Reproducing Kernel Hilbert Space (RKHS) method, in conjunction with the Tikhonov regularization, for constructing potential energy surfaces, with correct asymptotic forms, from high quality experimental measurements. The method is applied to the construction of new, global potential energy curves of the two lowest states X 1Σg+ and a 3Σu+ of the sodium dimer using rovibrational spectral measurements. The exchange interaction of Na2 at intermediate and long ranges is accordingly derived and adopted for determining the ionization energy of the corresponding valence electron. It is found that the resulting ground-state X 1Σg+ dissociation energy 6022.025 (±0.049) cm−1 of Na2 agrees within the experimental errors with the most recent experimental value [6022.0286 (±0.0053) cm−1, Jones et al., Phys. Rev. A 54, R1006 (1996)]. The well depth of the a 3Σu+ state is determined to be 174.96 (±1.18) cm−1, compared to the Rydberg–Klein–Rees (RKR) value of 174.45 (±0.36) cm−1 [Li et al., J. Chem. Phys. 82, 1178 (1985)]. Moreover, the equilibrium positions of both RKHS potential curves, 3.0796 (±0.0010) Å for the X 1Σg+ state and 5.089 (±0.062) Å for the a 3Σu+ state, are in excellent agreement with previously determined RKR results of 3.079 53 Å [Babaky and Hussein, Can. J. Phys. 67, 912 (1989)] and 5.0911 Å (Li et al.), respectively. The experimentally determined values of the equilibrium position and well depth for the a 3Σu+ state differ from recent theoretical values of 5.192 Å and 177.7 cm−1 obtained by highly accurate ab initio calculations [Gutowski, J. Chem. Phys. 110, 4695 (1999)]. Finally, both RKHS potential curves at large distances reproduce very recent theoretical dispersion coefficients within 1.0×10−5 percentage errors. © 2000 American Institute of Physics.