ALBERT

Notes: A new method for the quantization of classical Hamiltonian systems is presented. This method is based upon the correspondence between the Liouville formulation of classical mechanics and the Liouville–von Neumann formulation of quantum mechanics. It does not distinguish between integrable and nonintegrable systems, and consequently, is equally applicable to both types of systems. Further, it treats the indistinguishability of identical particles correctly, and thus, the semiclassical eigenstates have the correct symmetry properties. Application of the method is illustrated by a series of examples. The results are in excellent agreement with quantum mechanics and represent an improvement over results obtained using the uniform semiclassical approximation.

Notes: The implications of approximate dynamical constants of motion for statistical analysis of highly excited vibrational spectra are investigated. The existence of approximate dynamical constants is related to localized chaos and partial assignability of a "chaotic spectrum.'' Approximate dynamical constants are discussed in a dynamical symmetry breaking formulation of the transition from periodic to quasiperiodic motion, and from quasiperiodic to chaotic motion. Level repulsion, leading to a Wigner distribution in the case of a strongly chaotic system, is shown to originate in dynamical symmetry breaking via the noncrossing rule that states of the same symmetry do not cross. It is argued that quantum numbers for dynamical constants must be correctly assigned to detect localized chaos in statistical spectroscopy. Two possible kinds of approximate constants, for a "total polyad number'' and a bend normal mode, are discussed in relation to two coupling schemes that could govern the transition to chaos in H2O.

Notes: A time-independent quantum mechanical adiabatic switching algorithm is presented. This algorithm, which is based upon Born's quantum mechanical adiabatic theorem, is used to calculate the energy levels and eigenstates of the Henon–Heiles Hamiltonian system. The relationship between this algorithm and similar classical algorithms are discussed with particular attention focused upon the correspondence between avoided crossings and classical nonlinear resonances.

Notes: A comprehensive first principles theoretical investigation of the gas phase reaction Ca+HF→CaF+H is reported. The overall study involves three distinct elements: (a) generation of an accurate ab initio potential energy surface for the ground electronic state of the Ca–F–H system, (b) careful fitting of the computed surface to an analytical form suitable for three-dimensional reactive scattering calculations, and (c) execution of classical trajectory calculations for Ca+HF collisions using the fitted potential surface. Ab initio potential energy calculations were performed for 175 Ca–F–H geometries using an MCSCF-CI method with a large Gaussian orbital basis set. The error in the computed endothermicity for the reaction of Ca and HF is less than 1 kcal/mol and the errors in the computed saddle point energies are believed to be less than 3 kcal/mol. The potential energy surface is dominated by a deep well corresponding to a stable linear H–Ca–F intermediate with an extremely small bending force constant. The calculations clearly demonstrate that the preferred geometry for Ca attack on HF is markedly noncollinear. The saddle point for both fluorine exchange reaction and insertion into the H–Ca–F well occurs for a Ca–F–H angle of 75° and has an energy of 16.1 kcal/mol relative to Ca+HF. The energy barrier for collinear reaction, 30.0 kcal/mol, is nearly twice as high. The analytical representation of the ab initio potential energy surface isbased on a polynomial expansion in the three diatomic bond lengths that reproduces the values of the computed energies to within a root mean square deviation of 1.2 kcal/mol and reduces to the appropriate diatomic potentials in the asymptotic limits. Classical trajectory calculations for Ca+HF(v=1) utilizing the fitted surface establish the fact that the H–Ca–F potential well dominates the collision dynamics thus qualifying Ca+HF as a bona fide example of a chemical insertion reaction. Because of the extensive sampling of the H–Ca–F well, many trajectories formed rather long-lived intermediate complexes before reaching diatomic end products. A significant number of these trajectories were not converged with respect to changes in the integration time step. Despite uncertainties associated with the ultimate fates of the nonconverged trajectories, the results obtained support a number of generalizations relating to microscopic features of Ca+HF collisions. Among these are: (1) at fixed total collision energy, excitation of HF to v=1 is much more effective in promoting reaction than is placing the corresponding amount of energy in Ca,HF translation, (2) at fixed initial translational energy, reaction cross sections increase with increasing HF rotational quantum number J, (3) for trajectories which enter the H–Ca–F well, escape to form products is favored by increasing initial HF rotation and escape back to reactants is favored by increasing the initial relative translational energy, and (4) the CaF fractional product energy disposals are remarkably independent of initial collision conditions. These conclusions are compatible with the observation that significant intermode vibrational energy transfer does not occur in the H–Ca–F intermediate on the collision time scale (1–2 ps).

Notes: We present a modification of the time-independent adiabatic switching algorithm which is applicable to nonintegrable Hamilton systems. This algorithm transforms the Hamiltonian into a generalization of the Birkoff–Gustavson normal form. We discuss in detail the difficulties which occur when singular motions are encountered and explore the dynamical nature of the adiabatically switched tori using traditional methods of classical mechanics. A quantization scheme based upon this algorithm is proposed and numerical results, i.e., the semiclassical energies, are reported for a simple model of water.

Notes: The F+H2→HF+H potential energy hypersurface has been studied in the saddle-point and entrance channel regions. Using a large [5s 5p 3d 2f 1g/4s 3p 2d] atomic natural orbital basis set, we obtain a classical barrier height of 1.86 kcal/mol at the CASSCF/multireference CI level (MRCI) after correcting for basis set superposition error and including a Davidson correction (+Q) for higher excitations. Based upon an analysis of the computed results, the true classical barrier is estimated to be about 1.4 kcal/mol. We also compute the location of the bottleneck on the lowest vibrationally adiabatic potential curve, and determine the translational energy threshold from a one-dimensional tunneling calculation. Using the difference between the calculated and experimental threshold to adjust the classical barrier height on the computed surface yields a classical barrier in the range of 1.0–1.5 kcal/mol. Combining the results of our direct estimates of the classical barrier height with the empirical values obtained from our approximate calculations of the dynamical threshold, we predict that the true classical barrier height is 1.4±0.4 kcal/mol. Arguments are presented in favor of including the relatively large ((approximate)1 kcal/mol)+Q correction obtained when nine electrons are correlated at the CASSCF/MRCI level.

Notes: A time independent algorithm based upon Ehrenfest's adiabatic hypothesis for the construction of the transformation from the physical variables to a set of action-angle variables is presented. At the heart of this algorithm is the recognition that the switching parameter need not be treated as a function of time. The calculations of semiclassical energy levels and dipole matrix elements for the simple quartic oscillator are presented to illustrate this approach.

Notes: The classical-quantum correspondence is developed in its most natural setting, i.e., that of phase space distributions. Specifically, the Wigner–Weyl representation of the two fundamental quantum densities ||n(approximately-greater-than)〈n|| and ||n(approximately-greater-than)〈m||, which are eigenstates of the quantum Liouville operator, are shown, for integrable systems, to yield stationary and nonstationary classical Liouville eigenstates in the (h-dash-bar)→0 limit. The former are uniform distributions on semiclassically quantized tori, whereas the latter are nonuniform distributions on similar tori at intermediate actions. Applications to semiclassical mechanics in action-angle variable, the Heisenberg correspondence for matrix elements and the computation of vibrational intensities are provided.

Notes: The adsorption geometry, binding energy and electronic structure of alkali metal overlayers on the MgO (001) surface have been studied by means of density functional theory, using Gaussian-type orbitals to expand the wave functions and electronic charge density. A two-dimensionally periodic slab of MgO with alkali metal adsorbed at one surface was used to model the semi-infinite system. Li, Na, and K were considered at both half- and quarter-monolayer coverage. Results were compared for the local density approximation and for two different forms of the generalized gradient approximation. In all cases Li was found to interact with the surface approximately twice as strongly as Na and three times as strongly as K. The epitaxial binding energies were, however, always less than or close to the bulk cohesive energies of the respective alkali metals, suggesting an instability of the adsorbed film toward the formation of two- or three-dimensional islands, in agreement with experiment. Spin polarized and unpolarized calculations were compared to detect metal–insulator transitions in the alkali overlayer. Only Li adsorbed at 1:4 coverage was found to have lower energy in a spin polarized (hence nonmetallic) state. © 2000 American Institute of Physics.