Max Wolfsberg

Investigations of a nonrandom numerical method for multidimensional integration

A numerical integration technique based upon the use of nonrandom number sequences is examined with test integrations of a simple, analytical function. A comparison of the nonrandom technique with the familiar Monte Carlo method shows that the error of the new method decreases faster as more points are used in the calculation. Moreover, the new method needs fewer points to calculate an integral to an accuracy of 10% than does the Monte Carlo method.

Isotope Effects on Internal Frequencies in the Condensed Phase Resulting from Interactions with the Hindered Translations and Rotations. The Vapor Pressures of the Isotopic Ethylenes

Interaction of the hindered translations and rotations with the internal vibrations in the condensed phase leads to isotope‐dependent shifts of the internal frequencies. This isotope dependence is a necessary consequence of the fact that the coordinates representing the translations and rotations of the molecule as a whole are, in general, isotope dependent. This external—internal interaction is investigated and the XH2 system is employed as a simple example to demonstrate the nature of the interaction. It is shown that the experimentally determined vapor pressures of the isotopic ethylenes may be rationalized when the external—internal interaction is taken into account. A force field based on a simple cell model for liquid ethylene is obtained which yields good agreement with the experimentally determined vapor‐pressure isotope effects. The force field, while not unique, fits (and is in fact partially based on) other available data on liquid ethylene.

Further empirical testing of the suitability of a nonrandom integration method for classical trajectory calculations

A nonrandom method of approximating multidimensional integrals is compared to the traditional Monte Carlo method in the determination of average energy transfer values from classical trajectories. Two atom–diatomic molecule collision systems are studied. Estimates of error show that, for a given number of trajectories, the nonrandom method tends to be more accurate than the Monte Carlo method.

A quasiclassical trajectory study of the energy transfer in CO2–rare gas systems

Computational methods are presented for the study of collisions between a linear, symmetric triatomic molecule and an atom by three‐dimensional quasiclassical trajectory calculations. Application is made to the investigation of translational to rotational and translational to vibrational energy transfer in the systems CO2–Kr, CO2–Ar, and CO2–Ne. Potential‐energy surfaces based on spectroscopic and molecular beam scattering data are used. In most of the calculations, the CO2 molecule is initially in the quantum mechanical zero‐point vibrational state and in a rotational state picked from a Boltzmann distribution at 300°K. The energy transfer processes are investigated for translational energies ranging from 0.1 to 10 eV. Translational to rotational energy transfer is found to be the major process for CO2–rare gas collisions at these energies. Below 1 eV there is very little translational to vibrational energy transfer. The effects of changes in the internal energy of the molecule, in the masses of the collidants, and in the potential‐energy parameters are studied in an attempt to gain understanding of the energy transfer processes.

About the influence of lattice vibrations on the diffusion of methane in a cation-free LTA zeolite

Abstract The influence of lattice vibrations on the diffusion of methane in a cation-free zeolite of structure Type LTA is examined. It is shown that contrary to earlier published results the self-diffusion coefficients obtained with flexible and with rigid lattices are practically the same. This finding is true over a wide range of temperatures and for different interaction parameters. The reason why earlier papers did not state this independence of D on the lattice vibrations is explained.

Variational calculations of rotational–vibrational energy levels of water

Calculations are carried out of the rotational–vibrational energy levels of H2O and D2O for J≤10 by a variational method. The full Watson Hamiltonian is employed, with the potential function in valence displacement coordinates and with the integrations over normal coordinates carried out by Gauss–Hermite quadrature. The basis set consists of products of vibrational functions and symmetric top functions; the vibrational functions diagonalize the Hamiltonian for J=0. Comparison is made with experiment, and also results obtained with different force fields are compared. The mixing of different vibrational functions into a wave function for a given rotational–vibrational state is studied; mixing is found to be quite prevalent.

Corrections to the Born‐Oppenheimer approximation and electronic effects on isotopic exchange equilibria. II

Corrections to the Born‐Oppenheimer electronic energy are calculated for several first and second row diatomic molecules. These corrections are evaluated in the adiabatic approximation at the experimental internuclear distance and thereby constitute the leading terms in the non‐Born‐Oppenheimer molecular energy. The corrections to the electronic energy are functions of the nuclear masses and are found to produce significant electronic isotope effects on the equilibrium constants of certain exchange reactions. In particular, at 300 °K the electronic isotope effect on the equilibrium constant of the reaction HX + HD = DX + H2 where X is Li, B, N, or F is found to range from 1.029 for Li to 1.101 for B. By partitioning the correction to the Born‐Oppenheimer electronic energy into a sum of atomic and overlap terms, the electronic isotope effect is shown to be dependent upon certain characteristics of the molecules, the single most important of which is the X atom electronic angular momentum.

Variational calculation of lower vibrational energy levels of formaldehyde X̃ 1A1

The energies of lower lying vibrational states (J=0) of formaldehyde in its ground electronic state are calculated variationally for H2CO and D2CO. The functions of the basis set correspond to products of harmonic oscillator functions. The full Watson Hamiltonian is used and integrals are evaluated by Gauss–Hermite numerical quadrature. Two different potential functions in internal displacement coordinates, which have been given in the literature, are employed in the calculations. Calculations are carried out for A1, B2, and B1 symmetries with 196, 165, and 108 basis functions, respectively. The forms of some of the eigenvectors are reported.

Effect of Vibrational Anharmonicity on the Isotopic Self‐Exchange Equilibria H2X+D2X=2HDX

The anharmonicity correction to the zero‐point energy change in the triatomic dihydride isotopic self‐exchange equilibria, H2X+D2X=2HDX, is discussed. It is shown that the inclusion of the usually ignored term G0 in the expression for the anharmonic correction to the zero‐point energy (G0 + 14Σxij) causes the anharmonicity correction to the zero‐point energy change to be quite small for the cases considered (X is O, S, Se), but when this term is ignored the correction is found to be nonnegligible. Equilibrium constants are calculated, and it is found that the agreement between theory and experiment is considerably improved by including the G0 factor.

Molecular dynamics consideration of the mutual thermalization of guest molecules in zeolites

Abstract Molecular dynamics (MD) is applied to simulate methane self-diffusion in a model zeolite ZK4. It is found that the mutual interaction of the adsorbed molecules is sufficient to guarantee their thermalization, so that in first-order considerations, the energy exchange with the zeolite framework may be neglected.