Translational Coordinates Research Articles

A novel wave function factorization simplifying the matrix representation of the Schrödinger equation

In quantum scattering theory, coordinate systems with nontrivial Jacobians may arise and cause difficulty in the reduction of close coupled equations to the desired, simple, obviously Hermitian form {−I(d2/dx2)+W(x)}f=0 with W=W°. We consider x to be the translational coordinate, orthogonal to the surface coordinates, and the Schrödinger equation is represented in a basis of surface functions. We introduce a novel wave function factorization which permits reduction to the above form if the basis is (locally) independent of x for arbitrary Jacobian and general weight function for the surface functions. This factorization is compared with the more common factorization where all coordinates are treated equally. Applications to wave function matching and the calculation of surface integrals are mentioned. Several three-dimensional, orthogonal coordinate systems provide examples simplified by the novel factorization.

Quantum theory of energy exchange in direct encounters of polyatomic molecules with nonrigid surfaces

Recently developed effective mass theory of vibrational, rotational and translational energy exchange in molecular collisions is extended naturally to polyatomic molecules colliding with nonrigid surfaces. Considering the processes where the short-range forces are of prevailing importance, we transform the Hamiltonian into the form which enables us to eliminate the coordinates which are cyclic in the limit of zero range forces and can thus be expected to be approximately ignorable when forces are of short, but finite, range. Instead of three rotational and three translational coordinates we are thus left with one only relevant generalized coordinate. The mass corresponding to this coordinate can easily be calculated; it depends on the mass and the moments of inertia of the molecule, and on the segment of the potential surface on which the system evolves. The motion of the only relevant generalized coordinate is then quantized. The inelastic scattering problem is treated within first-order in the distorted wave expansion. In the cases where this may be expected to be inadequate, more advanced approximation schemes developed in scattering theory could be applied as well. The excitations of bulk phonons and also of vibrational modes localized on the surface are included in the theory. As an application, energy transfer in collisions of CO 2 molecules with Pt and Ag surfaces is studied (since this is the experimentally most studied case involving molecules of more than two atoms), and the results are compared with the experiments and with the recent semiclassical stochastic trajectory calculations in which the transfer of vibrational energy to rotations is neglected and the solid's vibrations are treated by a quasiclassical approximation. Excellent agreement was found between the present theory and the trajectory calculations mentioned for all three gas temperatures considered (290, 580 and 1160 K) in the case of small change in vibrational energy and high surface temperature, i.e., under the conditions where the agreement between the two methods could possibly be expected because of the approximations involved.

Quantum theory of energy exchange in direct encounters of polyatomic molecules with nonrigid surfaces

Abstract Recently developed effective mass theory of vibrational, rotational and translational energy exchange in molecular collisions is extended naturally to polyatomic molecules colliding with nonrigid surfaces. Considering the processes where the short-range forces are of prevailing importance, we transform the Hamiltonian into the form which enables us to eliminate the coordinates which are cyclic in the limit of zero range forces and can thus be expected to be approximately ignorable when forces are of short, but finite, range. Instead of three rotational and three translational coordinates we are thus left with one only relevant generalized coordinate. The mass corresponding to this coordinate can easily be calculated; it depends on the mass and the moments of inertia of the molecule, and on the segment of the potential surface on which the system evolves. The motion of the only relevant generalized coordinate is then quantized. The inelastic scattering problem is treated within first-order in the distorted wave expansion. In the cases where this may be expected to be inadequate, more advanced approximation schemes developed in scattering theory could be applied as well. The excitations of bulk phonons and also of vibrational modes localized on the surface are included in the theory. As an application, energy transfer in collisions of CO2 molecules with Pt and Ag surfaces is studied (since this is the experimentally most studied case involving molecules of more than two atoms), and the results are compared with the experiments and with the recent semiclassical stochastic trajectory calculations in which the transfer of vibrational energy to rotations is neglected and the solid's vibrations are treated by a quasiclassical approximation. Excellent agreement was found between the present theory and the trajectory calculations mentioned for all three gas temperatures considered (290, 580 and 1160 K) in the case of small change in vibrational energy and high surface temperature, i.e., under the conditions where the agreement between the two methods could possibly be expected because of the approximations involved.

Quantum dynamical calculations of linear exchange reactions in combined coordinates

A new method of quantum dynamical calculations for light-atom transfer reactions in linear triatomic systems is proposed. The method is based on the use of a special translational—vibrational coordinate system where the potential surface cuts at a fixed translational coordinate represent a double-well potential. A double set of harmonic oscillator functions localized in the reagent and product valleys are used as an analytical vibrational basis. The two-dimensional Schrödinger equation is reduced to a one-dimensional matrix differential equation by averaging over the vibrational coordinate. The equation is solved by the propagation Wigner R -matrix technique ensuring an accurate conservation of the probability flux at any size of the vibrational basis. The dynamics of the hydrogen and deuterium atoms transfer in the reactions of the methyl radical with methane and toluene was calculated. The total probabilities of these reactions in the tunnelling energy region were shown to depend only slightly on the resonance misfit. The Arrhenius activation energy is ≈ 3 kcal mol −1 lower than the quantum barrier height.

Computed structure of small benzene clusters

The structures of small benzene clusters (C 6H 6) n , n = 2–7, have been calculated employing potential-energy minimization with respect to molecular translational and rotational coordinates, using exp-6-1 non-bonded atom-atom potential functions. The influence of the adopted point-charge model is discussed for the dimer structure.

Quantum theory of vibrational, rotational, and translational energy exchange in collisions of polyatomic molecules. Application to methyl halides

We derive a quantum theory of inter- and intramolecular transfer of vibrational, rotational, and translational energy in collisions of polyatomic molecules, in the case that short range forces between the collision partners are of prevailing importance. For this purpose, we first transform the Hamiltonian into the form which enables us to eliminate the coordinates which are cyclic in the limit of zero range forces and can thus be expected to be approximately ignorable when forces are of short, but finite, range. Instead of six rotational and three translational coordinates, we are then left with one single relevant generalized coordinate. The corresponding mass can easily be calculated; it depends on the usual reduced mass of the colliding molecules, on their moments of inertia, and on the geometry of the potential surface. The motion of the only relevant generalized coordinate is then quantized. To calculate the probabilities for the collision induced vibrational-rotational transitions, one can make use of any of a large number of the approximation schemes which have been developed in scattering theory. In the present work, we use the distorted wave expansion, for its simplicity and also because it leads, in the corresponding limit, directly to the results obtained recently by the semiclassical theory. As an application, we study vibrational deactivation probabilities of methyl halides colliding with inert gases: He, Ne, Ar, Kr, and Xe. Calculations are performed for two intermolecular potentials: exponential repulsive and Morse potential. The results obtained by the two potentials are very similar and agree well with the measured vibrational transition probabilities.

Charge distribution from least-energy structure in Ca - Mg orthosilicates

Calcium-olivine, γ-Ca2SiO4, larnite, β-Ca2SiO4, merwinite, Ca3Mg(SiO4)2, and monticellite, CaMgSiO4, are considered. According to a rigid oxyanion scheme, eulerian orientation angles of the SiO4 tetrahedra and translation coordinates of Ca and Si atoms are specified as structural variables τk. All derivatives of the static energy (Born model) contain atomic charges and repulsive parameters as unknowns; the minimum energy conditions ∂E L/∂τk=0 yield 34 equations which are solved by a least-squares method. The set of energy parameters fitting structural properties of all four phases together is: z Ca=1.50, z o=−1.10 e, r Ca=1.05, ρ=0.25 A; the Mg charge was fixed at 1.38 e, from a previous study on forsterite. An average shift of 0.04 A is observed between experimental and least-energy calculated atomic positions. Results are compared with those of Mg2SiO4, where the fit was based both on thermoelastic and on structural properties. If no charge values were fixed “a priori”, just ratios between charges could be determined by fitting them to structural data only.

Dynamics of translations and rotations in molecular crystals: Macroscopic and microscopic approaches

Starting from a Hamiltonian for an orientationally disordered crystal with bilinear coupling between translational and orientational coordinates, a nonequilibrium macroscopic free energy is calculated. It is used to obtain coupled dynamic equations for translations and rotations by methods of irreversible thermodynamics. The corresponding dynamic response functions are in full agreement with those derived using a microscopic Mori-type projection-operator method. The coefficients of the macroscopic free energy are related to the static susceptibilities obtained from the Hamiltonian in the framework of molecular-field theory. Dynamic sum rules for translational-rotational motion are given.

ChemInform Abstract: VIBRATIONS AND FORCE CONSTANTS OF THE HEXAGONAL LITHIUM IODATE CRYSTAL

Force constants of the α‐LiIO3 crystal are calculated based on the thirteen k=0 infrared and Raman active vibrations. Setting up a generalized valence force field for the IO3 group and both valence and central forces for the external interactions has allowed us to obtain good agreement between experimental and calculated frequencies. The introduction of translational and librational coordinates related to the normal coordinate analysis leads to a precise assignment of the frequencies and shows that the degenerate bending vibrations of the IO3 groups are strongly coupled with the translational vibrations of the lithium atoms.

Theory for rotation and translation of islands on a substrate

A theory is developed for rotation and translation of islands deposited on a substrate by assuming the harmonic potentials for both an island and the substrate themselves and a Gaussian type potential between them. The probability density that the island takes a set of values for the rotational and translational coordinates on the substrate surface is obtained by calculating the partition function in which integration is performed over all the other variables. The integration over the vibrational coordinates results in a set of effective parameters for the interaction potential, which differ from the original ones by few percent in typical cases. The integration over the rotational and translational coordinates other than those on the surface results in a pre-exponential factor, which affects the relative probability within one order of magnitude in typical cases.

Vibrations and force constants of the hexagonal lithium iodate crystal

Force constants of the α-LiIO3 crystal are calculated based on the thirteen k=0 infrared and Raman active vibrations. Setting up a generalized valence force field for the IO3 group and both valence and central forces for the external interactions has allowed us to obtain good agreement between experimental and calculated frequencies. The introduction of translational and librational coordinates related to the normal coordinate analysis leads to a precise assignment of the frequencies and shows that the degenerate bending vibrations of the IO3 groups are strongly coupled with the translational vibrations of the lithium atoms.

Applications of the RISM equation to diatomic fluids: the liquids nitrogen, oxygen and bromine

The reference interaction site model (RISM) integral equation is used to study the equilibrium pair correlation for one-component liquids composed of homonuclear diatomic molecules. The integral equation is first tested by comparing the results obtained from it with those of computer simulation calculations. An internal consistency test is developed which seems to provide an a priori measure of the accuracy of the RISM equation. Then the theory is used to interpret the neutron scattering structure factor data taken on the liquids nitrogen and oxygen. For both of these liquids, a satisfactory explanation of the data is obtained by assuming that the short ranged repulsive forces between molecules are mimicked by a two-site hard core model. However, this simple model does not provide a satisfactory explanation of the data taken from liquid bromine. But, it is shown that a slightly more sophisticated model for the short-ranged repulsion between Br 2 molecules does provide an adequate explanation. With the molecular models determined by fitting the neutron scattering data, the RISM equation provides a method for determining the atom, atom to center-of-mass, and center-of-mass to center-of-mass intermolecular distribution functions in the diatomic liquids. From these functions, the local structures in the three liquids are analyzed. While orientational pair correlations are nearly negligible in both the liquids nitrogen and oxygen, these correlations are fairly substantial in liquid bromine. Furthermore, even when the orientation of a molecule does not greatly influence the orientation of its neighbors, it is found that the orientation of a molecule does have an important effect on the location of its neighbors. Thus, the coupling of translational and orientational coordinates is significant, even in the liquids nitrogen and oxygen.

New type of cluster theory for molecular fluids: Interaction site cluster expansion

A new cluster series, called the interaction site cluster expansion, is derived for classical molecular fluids. For a general class of molecular models, the series provides an exact formula for the equilibrium pair correlations between interaction sites on different molecules. In the models considered, each molecule contains m of these interaction sites. The total intermolecular potential between two molecules is the sum of m2 site–site potentials. A site–site potential depends on the separation between the sites only. However, because the sites are not necessarily located at the centers of molecules, the total molecular pair interaction can depend strongly on molecular orientations. Even though the interaction site cluster expansion is exact, the cluster integrals in the series involve the translational coordinates of the interaction sites only. The molecular orientational coordinates are removed by a transformation which introduces intramolecular correlation functions into the series. Thus, the new cluster series expresses the intermolecular site–site correlation functions in terms of diagrams involving site–site cluster bonds and intramolecular correlation bonds. The interaction site cluster expansions are used to derive several important results. It is shown that when the site–site interactions are hard core potentials, the pair correlation functions rigorously contain not only discontinuities but also cusps. The locations for these singularities are derived. A method is presented which shows how these singularities are smoothed when the hard core potentials are softened. The singularities as well as special limiting results derived herein are used to provide a justification for the reference interaction site model (RISM) equation.

Normal coordinates and Cartesian displacement coordinates

The problem of constructing analytically a unique set of normal coordinates in Cartesian basis is discussed. It is shown, with examples, that the translational and rotational normal coordinates are easy to construct and hence the remaining normal coordinates. The method avoids the problem of redundant coordinates.

On the Theory of an Impurity Molecule. I. Kinetic Energy Operator

AbstractFor an impurity molecule the translational, rotational, and vibrational coordinates are introduced by means of the Eckart conditions. The kinetic energy operator is transformed from Cartesian to new coordinates mentioned with the help of the Frobenius theorem and the Gaussian algorithm of matrix algebra.

The physical interaction of gases with crystalline solids: II. Two-dimensional second and third virial coefficients

The virial adsorption isotherm for a gas on a perfect crystal lattice surface is considered. The formal expressions for the two-dimensional second and third virial coefficients of a classical monatomic adsorbate are put into a form suitable for numerical evaluation. This is accomplished by expanding the Boltzmann factors in the general expressions as Fourier series in the translational symmetry coordinates of the surface lattice. Numerical calculations of the requisite Fourier transforms are presented for two fcc lattice planes interacting with gas atoms of different sizes. In addition, Fourier transforms of the Mayer function are presented for mutual interactions of adsorbed atoms given by a Lennard-Jones function. These results were used to calculate second and third virial coefficients for atoms adsorbed on several surfaces, including cases where the atoms are completely localized and where they are perfectly mobile. These results are compared with previous calculations, and some errors are corrected. In addition, the second virial coefficient for argon adsorbed on the (111) face of a solid xenon lattice is calculated and compared with experiment.

Rigid-Body Rotor Dynamics: Dynamic Unbalance and Lubricant Temperature Changes

The case of a symmetric rotor supported on two identical, rigidly mounted, self-aligning, finite-length (L/D = 1) fluid-film journal bearings is considered. Rotor position is described by two translation coordinates in a plane perpendicular to the bearing line of centers, and by three Euler angles. Introduction of various amounts of dynamic unbalance via the inertia tensor off-diagonal terms (products of inertia) allows determination of angular velocity and static eccentricity ratio combinations leading to bearing “failure” defined for arbitrary maximum allowable eccentricity ratios. Instability hysteresis, defined here as the persistence, during rotor deceleration, of instability to speeds below which it first appeared, is considered by means of the above model. Equations and methods developed for the unbalance investigation are adapted to a variable-speed analysis. With both constant and variable mean bearing temperatures, variable-speed simulations terminating at constant speed are observed to be stable when the terminating point is below the instability threshold curve on the angular velocity-static eccentricity ratio parameter plane and unstable when above. The slope of the threshold curve and the shape of the equilibrium-condition path on the parameter plane (single-line path for constant temperature, closed curve for variable temperature) apparently combine to produce hysteresis in the variable temperature case and none at constant temperature.

General Internal Motion of Molecules, Classical and Quantum-Mechanical Hamiltonian

The dynamics and the Hamiltonian of a general asymmetric-top molecule undergoing almost arbitrary deformations are discussed. s vector notations are used for translational, rotational, and internal-velocity coordinates. The kinetic energy is formulated by generalizing the G-matrix technique known from the theory of molecular vibrations. A geometrical definition of the rotational coordinates referring to the instantaneous principal axis system is compared with a dynamical definition involving the over-all angular momentum. States of general internal motion are associated by definition with zero linear momentum and zero over-all angular momentum.

Consistent mass Matrix for Distributed Mass Systems

A technique is developed for constructing a consistent mass matrix for vibration analysis of general structural configurations. The approach is consistent with standard procedures for stiffness matrix formulation of structural problems, but accounts for the actual mass distribution within the structure in a manner similar to the Rayleigh-Ritz formulation. The basic equation for computing the mass matrix coefficients is given a physical interpretation. It is shown that the natural frequencies obtained by using the consistent mass matrix are upper bounds to the exact solution. The procedure is applicable to general dynamic response analysis and is demonstrably superior to the usual procedure of physical mass lumping by application to frequency analysis of free-free and simply-supported prismatic beams with uniformly distributed mass. It is observed that if lateral translation coordinates alone are used to represent the distortion of a uniform beam, the number of individual beam segments must exceed by one the number of natural mode frequencies it is desired to closely approximate.