Showing papers in "Journal of Chemical Physics in 1975"

Time‐dependent approach to semiclassical dynamics

Abstract: In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.

A random‐walk simulation of the Schrödinger equation: H+3

Abstract: A simple random‐walk method for obtaining ab initio solutions of the Schrodinger equation is examined in its application to the case of the molecular ion H+3 in the equilateral triangle configuration with side length R=1.66 bohr. The method, which is based on the similarity of the Schrodinger equation and the diffusion equation, involves the random movement of imaginary particles (psips) in electron configuration space subject to a variable chance of multiplication or disappearance. The computation requirements for high accuracy in determining energies of H+3 are greater than those of existing LCAO–MO–SCF–CI methods. For more complex molecular systems the method may be competitive.

Metric geometry of equilibrium thermodynamics

Abstract: It is shown that the principal empirical laws of equilibrium thermodynamics can be brought into correspondence with the mathematical axioms of an abstract metric space. This formal correspondence permits one to associate with the thermodynamic formalism a geometrical aspect, with intrinsic metric structure, which is distinct from that arising from graphical representations of equilibrium surfaces in phase space.

Graph theory and molecular orbitals. XII. Acyclic polyenes

Abstract: A graph‐theoretical study of acyclic polyenes is carried out with an emphasis on the influence of branching on several molecular properties. A definition of branching is given and several branching indices are analyzed. The case of polyenes without a Kekule structure is discussed briefly. The main conclusions are: (a) thermodynamic stability of conjugated polyenes decreases with branching, but (b) reactivity, in general, increases with branching.

The effect of intramolecular quantum modes on free energy relationships for electron transfer reactions

Abstract: A general quantum mechanical description of exothermic electron transfer reactions is formulated by treating such reactions as the nonradiative decay of a ’’supermolecule’’ consisting of the electron donor, the electron acceptor, and the polar solvent. In particular, the role of the high‐frequency intramolecular degrees of feedom on the free energy relationship for series of closely related reactions was investigated for various model systems involving displacement of potential energy surfaces, frequency shift, and anharmonicity effects. The free energy plots are generally found to pass through a maximum and to be asymmetric with a slower decrease in the transition probability with increasing energy of reaction. For high‐frequency intramolecular modes this provides a rationalization of the experimental observation of ’’activationless’’ regions. Isotope effects are discussed as also are the oscillatory free energy relationships, predicted for low temperatures and high frequencies, and which are analogous t...

Semiclassical limit of quantum mechanical transition state theory for nonseparable systems

Abstract: The semiclassical limit of quantum mechanical transition state theory is derived by invoking the classical path approximation for the Boltzmann density operator and making use of the stationary phase approximation; separability of motion along a reaction coordinate is not assumed. The resulting expression for the rate constant bears an interesting similarity to that of conventional transition state theory, although all quantities in it refer to the full classical dynamics on the potential energy surface. In place of the vibrational frequencies of the ’’activated complex’’ which appear in the conventional theory, for example, the semiclassical expression contains characteristic frequencies related to the stability properties of a periodic classical trajectory. Conservation of total angular momentum is easily accounted for in a rigorous manner so that the semiclassical model can be applied to three−dimensional dynamical systems.

Second and third order hyperpolarizabilities of organic molecules

Abstract: We have used the technique of static field induced second harmonic generation in liquids to determine the second order hyperpolarizability β, and the third order hyperpolarizability γ, for a wide range of conjugated and nonconjugated organic molecules. The adequacy of the Onsager local field formulation was tested as was the accuracy of the bond additivity approximation. The large π‐electron nonlinearities which are possible in substituted benzene molecules were investigated by measuring a large number of benzene derivatives and subtracting out the σ‐electron contributions.

Dynamic effects of pair correlation functions on spin relaxation by translational diffusion in liquids

Abstract: It is shown how the equilibrium pair correlation function between spin‐bearing molecules in liquids may be incorporated as an effective force in the relative diffusion expressions, and how one may solve for the resulting time correlation functions and spectral densities needed for studies of spin relaxation by translational diffusion. The use of finite difference methods permits the solution no matter how complex the form of the pair correlation function (pcf) utilized. In particular, a Percus–Yevick pcf as well as one corrected from computer dynamics, both for hard spheres, are utilized. Good agreement with the experiments of Harmon and Muller on dipolar relaxation in liquid ethane is obtained from this analysis. Effects of ionic interactions in electrolyte solutions upon dipolar relaxation are also obtained in terms of Debye–Huckel theory for the pcf. Analytic solutions are given which are appropriate for the proper boundary‐value problem for the relative diffusion of molecules (i.e., a distance of mini...

Theory of inhomogeneous polymers: Fundamentals of the Gaussian random‐walk model

Abstract: In earlier work a theory of inhomogeneous polymers was developed from a mean field theory point of view. Applications were made to polymer−polymer interfaces and to the microdomain structure of block copolymers. Here a connection is made between the intuitive mean field arguments for these problems and more fundamental statistical mechanics. Functional integral techniques are employed extensively.

Raman spectroscopic investigation of the structure of silicate glasses. I. The binary alkali silicates

Abstract: The Raman spectra of some binary alkali silicate glasses and crystals have been measured and the results interpreted. The glasses are of composition M2O⋅xSiO2, 1⩽x⩽4 and M = Li, Na, K. There is a strong resemblance between glassy and crystalline spectra which is consistent with a considerable amount of structural disorder of the glass. The reason is that the disorder acts as a perturbation on the spectrum of an elementary vibrating unit from which the glass is largely constructed. As a result, the general nature of the glass structure may be determined. Furthermore, glasses of the same x but with different alkali have different amounts of short‐range order. This is demonstrated by showing that the widths of certain Raman peaks increase with increasing disorder of the Si–O network of the glass. A strong argument is made against the microcrystallite theory of glass structure. For any composition x, glasses increase in disorder in the direction K → Na → Li. The crystallization rates increase in the same dire...

Energy gap law in the solvent isotope effect on radiationless transitions of rare earth ions

Abstract: Data on the fluorescent yields of tervalent ions of Pr, Sm, Eu, Gd, Tb, Dy, and Tm in H2O and D2O, containing perchlorate ions, and isotope effects on the fluorescence are reported. The results are interpreted in terms of the mechanisms responsible for radiationless processes in solutions of rare earth ions [Y. Haas and G. Stein, J. Phys. Chem. 75, 3668 (1971); 76, 1093 (1972)]. It is shown that along the series of the closely related rare earth ions, all of which exhibit well shielded f−f electronic transitions, the results can be correlated in terms of a single variable: the energy gap between the lowest fluorescent and highest nonfluorescent level. For several rare earth ions, the matching of this gap by a single high energy (OH or OD) vibration of one solvent molecule is the decisive factor accounting for observed yields and isotope ratios.

PNO–CI and CEPA studies of electron correlation effects. III. Spectroscopic constants and dipole moment functions for the ground states of the first‐row and second‐row diatomic hydrides

Abstract: A systematic investigation of the ground state potential curves and dipole moment functions has been performed for the diatomic hydrides LiH to HCl on the basis of variational configuration interaction wavefunctions PNO–CI and the coupled electron pair approximation CEPA. The basis sets of Gaussian‐type orbitals are derived by simple rules from optimized atomic sets available in the literature. Between 95% (LiH) and 85% (HCl) of the valence shell correlation energies are accounted for in the CEPA calculations. Core–valence intershell correlation has been included for several members of each row, and its effect on the potential curves is analyzed. Comparison of the spectroscopic constants derived from the CEPA potential curves with experiment shows a high reliability of the theoretical values. The standard deviations over both rows are as follows: re:0.003 A, ωe:14 cm−1, αe:0.005 cm−1, and ωexe:1.5 cm−1. The errors of the calculated dissociation energies go up to 0.3 eV but behave very regularly. They are believed to allow for predictions which are correct to about ±0.05 eV. The following new D0 values are recommended (empirical values in parenthesis): NH:3.40 (3.2±0.16); NaH:1.88 (2.05±0.2); MgH:1.23 (2.0±0.5); PH:3.02 (3.5±0.3). Various vibrational matrix elements have been calculated from the dipole moment curves. The μ0 values show errors of 0.02–0.04 D. The differences between diagonal elements as well as the transition elements for the first few levels deviate by only about 10% from the available experimental data.

Theory of unsymmetric polymer–polymer interfaces

Abstract: Solutions have been obtained to equations which described the interface between two immiscible polymers and are more general than the equations first introduced by Helfand and Tagami. Gaussian random−walk statistics are assumed for the macromolecules. As a consequence of the present work, limitations of the earlier theory are removed, particularly the assumption that the properties of the two polymers when pure are identical. Calculations are performed for a variety of polymers and comparison with experiment is made.

Approximations for the exchange potential in electron scattering

Abstract: Four new exchange potentials (the semiclassical exchange approximation, the asymptotically adjusted free‐electron–gas exchange approximation, the second‐order free‐electron–gas exchange approximation, and the high‐energy exchange approximation) are derived. Calculations are performed for elastic electron scattering from helium and argon. The results are compared to one another and to calculations using Hara’s free‐electron–gas approximation and the exact nonlocal exchange potential. Three of the approximations to exchange are in good agreement with the exact exchange —except at very low energy— but are much easier to use. Thus they should be very useful in electron–atom and electron–molecule scattering calculations.

A new computational approach to Slater’s SCF–Xα equation

Abstract: A new computational scheme is presented for the performance of LCAO−MO calculations in the SCF−Xα model. The scheme is intended to be applicable for large systems and to be more accurate than the scattered−wave SCF−Xα method. The Xα potential is fitted by least−squares to a linear combination of Gaussians, and the approximated SCF−Xα equation is solved by the conventional Rayleigh−Ritz variational method. The muffin−tin approximation is avoided, and matrix elements are calculated analytically in contrast to the discrete variational scheme. Some illustrative results are given for the ionization energies and equilibrium geometries of small molecules. It is found that over−all performance of the method is satisfactory for both ionization energies and equilibrium geometries.

A computer simulation of charged particles in solution. I. Technique and equilibrium properties

Abstract: A computer technique is presented for simulating the translational motion of ions in a liquid solution. In the model the diffusive motion of each ion is perturbed by the electrostatic force of the surrounding ions. Several polyelectrolyte systems of spherical polyions (10–50 A in radius) and small ions (∼1 A in radius) have been studied. For each system the polyion electrostatic shielding length and the average potential energy of each ion species was calculated. When the shielding length was sufficiently short, the computer results and the predictions of the zero polyion concentration Debye–Huckel theory were in good agreement.

Calculation of ionization potentials from density matrices and natural functions, and the long-range behavior of natural orbitals and electron density

Abstract: Given an exact eigenfunction ψ for some system with N electrons, a procedure is developed for determining ionization potentials of the system to various states of the corresponding system having N−p electrons. The long−range behavior of the electron density and of the natural spin orbitals is shown to involve a set of eigenvalues which are obtained by this procedure. Following is the procedure. First determine the pth−order natural functions for ψ, X m (p), and their occupation numbers n m (p), by diagonalization of the pth−order density matrix Γ(p). Calculate the quantities where ? ≡ (x 1 x 2⋅⋅⋅x p ) and ?ξ ≡ (ξ1ξ2⋅⋅⋅ξ p ). Then diagonalize the Hermitian matrix μ(p) k l = λ(p) k l / (n (p) k n (p) l )1/2. The resultant eigenvalues μα (p) are approximations to the negative of the p−electron ionization potentials of the system, with successive μα (p) (from the highest to the lowest) being lower bounds to the negative successive p−electron ionization potentials (from the lowest to the highest). For an approximate eigenfunction ?, the same procedure is recommended, except that the Hermitian part of the approximate matrix ?(p) k l is diagonalized. For ? a Hartree−Fock approximation to Ψ, the procedure for the case p = 1 is the classic method of Koopmans. It is shown that in general all natural spin orbitals χ m (x) in a system have long−range expontential fall−off governed by μ(1) max, i.e., χ m (x) ∼ exp[−(−2μmax (1))1/2 r], and that the electron density behaves similarly, π (x) ∼ exp[−2(−2μmax (1))1/2 r]. The behavior naively expected, namely π (x) ∼ exp[−2(2I min)1/2 r], results if μmax (1) is equal to −I min.

Theory of angular momentum decoupling approximations for rotational transitions in scattering

Abstract: A systematic method is discussed for decoupling the internal angular momentum of molecules involved in a collision from their relative angular momentum. This leads to a large class of rotational approximations of varying degrees of complexity and accuracy. These approximations may be used directly for computing rotational transitions or they may be used for reducing the rotational complexity involved in accurate vibrational calculations. It is shown how this approach may be used to study the infinite−order sudden approximation and how that approximation may be extended to more complex potentials. It is shown also how one may use results of the jz−conserving approximation to obtain more complete information on the scattering matrix. The present approach may be used to deduce new angular momentum decoupling approximations and analyze such approximations arrived at through other considerations.

Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. III. Analysis of the approximations

Abstract: In Parts I and II of this series [J. Chem. Phys. 59, 3873, 3879 (1973)] we developed a new method of sensitivity analysis for large sets of coupled nonlinear equations with many parameters. In developing this theory and in carrying out the computer calculations involved in this analysis we made a number of approximations. We present here a quantitative analysis of these approximations and, where applicable, develop rigorous error bounds. Our analysis shows that we can specify the approximations which enter into our theory so as to obtain sensitivity measures of known accuracy. On this basis we feel that the techniques developed in this series of papers provide a useful and efficient method of sensitivity analysis of large systems with many parameters.

Measurement of the dispersion in polarizability anisotropies

Abstract: We have measured the dispersion of the gas phase depolarization ratio of Rayleigh scattered light for 12 linear and symmetric top molecules Combining these data with known refractive index data we obtain the frequency dependence of the polarizability anisotropy For all molecules studied we find that the polarizability anisotropy increases more rapidly with increasing frequency than the bulk polarizability We have correlated this behavior with the oscillator strength and direction of the first electronic transition We have also compared our zero frequency extrapolated anisotropies with the anisotropies determined from static fields with the Kerr effect and the Stark effect in microwave spectroscopy

Molecular dynamics study of the structure and thermodynamic properties of argon microclusters

Abstract: Microclusters of 2–100 argon atoms were studied using molecular dynamics. The microclusters were ordered solids at low temperatures and energies and disordered liquids at high temperatures and energies. The melting transition occurred considerably below the bulk melting temperature. Radial density functions, interference functions, diffusion coefficients, and surface energies were calculated for both the solid and liquid phases. The surface energies of the microclusters could be expressed as a function of the form Ai2/3+Bi1/3 where i is the number of atoms in the cluster. For 0 °K solid clusters A and B have the values 0.26×10−12 and −0.12×10−12 erg, respectively; for liquid clusters at 40 °K, A and B have the values 0.17×10−12 and −0.045×10−12 erg, respectively. Clusters containing 7 or more atoms melted with the occurence of a first‐order‐like transition. This transition was studied further through Monte Carlo calculations. A possible model for the first‐order‐like transition is proposed.

Rough hard sphere theory of the self‐diffusion constant for molecular liquids

Abstract: A simple theory based on a hard sphere model is used to calculate the self−diffusion constant and the shear viscosity of liquid carbon tetrachloride. The theory accounts for the coupling between translational and rotational motions of molecules. This feature distinguishes the hard sphere theory presented herein with that given recently by other workers. It is shown that the coupling has a large (factor of 2) effect on the diffusion constant, and thus the coupling should not be neglected even for molecules as spherical as carbon tetrachloride. The approximations that are needed to arrive at a hard sphere theory for transport coefficients are discussed in detail. This analysis reveals the strengths and limitations of such a theory.

The fluorescence excitation spectrum of rotationally cooled NO2

Abstract: The fluorescence excitation spectrum of NO2 was measured in the region 5708–6708 A using a tuneable dye laser as an excitation source. The NO2 was cooled to a rotational temperature of ∼3 K by expansion with argon as a carrier gas through a supersonic nozzle. This cooling drastically reduced the rotational structure and thereby permitted a clear separation and analysis of 140 vibronic bands found in this 1000 A region of the spectrum. The results indicate that most of the fluorescence in this region is due to a 2B2 electronic state, and that this state is so heavily perturbed by high lying levels of the ground electronic state that a substantial fraction of all ground state levels of appropriate vibronic symmetry in this region have appreciable fluorescence intensity.

Rotational excitation in H2–H2 collisions: Close‐coupling calculations

Abstract: Rotational excitation in molecule−molecule collisions has been treated for the first time by accurate quantum close−coupling scattering calculations, employing an expansion basis set of two to three rotational levels for each molecule and correctly accounting for exchange of identical particles. Elastic and inelastic cross sections have been computed for collisions of para−para, ortho−ortho, and para−ortho hydrogen molecules assuming an intermolecular potential suggested previously. The accuracy of recent ’’effective potential’’ calculations is demonstrated by comparison with the exact quantum results.

Temporary negative ions and electron affinities of benzene and N‐heterocyclic molecules: pyridine, pyridazine, pyrimidine, pyrazine, and s‐triazine

Abstract: Electron transmission spectroscopy is used to study shape resonances (temporary negative ions) in benzene and some isolectronic N−heterocyclic molecules (pyridine, diazines, and s−triazine), in the energy range 0−6 eV. The lowest shape resonance in each of these molecules exhibits vibrational structure which is interpreted in all cases as the totally symmetric C−C stretch mode. The ground vibrational level of this lowest shape resonance is accessible by electron impact only in benzene and pyridine. Thus, their electron affinities can be determined from the present experiment (−1.15 eV for C6D6 and −0.62 eV for C5H5N). Only excited vibrational levels are accessible in the diazines and s−triazine, indicating that the electron affinities for these molecules have positive values. For benzene, pyridine, and some other aromatic hydrocarbons, we compare the electron affinities established in the gas phase with the polarographic potentials established in the liquid phase and we find a linear relationship. Using this correlation in conjunction with the measured values of the polarographic potentials, we estimate the electron affinities for pyridazine (0.25 eV), pyrimidine (0 eV), pyrazine (0.40 eV) and s−triazine (0.45 eV).

PNO–CI (pair natural orbital configuration interaction) and CEPA–PNO (coupled electron pair approximation with pair natural orbitals) calculations of molecular systems. I. Outline of the method for closed‐shell states

Abstract: The methods of configuration interaction with double substitutions to pair natural orbitals (PNO−CI) and of the coupled electron pair approximation (CEPA) proposed by W. Meyer are improved by combination with a new scheme of the calculation of the pair natural orbitals (PNO) and an efficient iterative scheme for the diagonalization of the CI matrix. The relevant matrix elements for the closed shell case are tabulated, the quantities that are pertinent for an analysis of the correlation energy are defined, and the organization of the computer programs is described.

Macromolecular dimensions obtained by an efficient Monte Carlo method without sample attrition

Abstract: The statistical dimensions of macromolecular chains of fixed contour length can be rapidly calculated by Monte Carlo methods applied to a model consisting of dynamic self‐avoiding random chains on a lattice. This ’’slithering snake’’ model involves moving the head of a chain one space in a lattice with all other elements of the chain moving forward along the old contour. Possible moves of the head are selected at random, but if such a move is precluded by double occupancy, the old configuration is retained, with head and tail interchanged, and then counted as if a move were made. This technique gives unbiased statistical results except for the effect of double cul‐de‐sacs. The method can also be applied to interacting chains, either free or confined to a box. Calculations have been made for 10‐link chains on a square planar lattice for two different concentrations in infinite space and for two concentrations in a small box.

Collisional ionization of Na, K, and Cs by CO2, COS, and CS2: Molecular electron affinities

Abstract: The negative ion products resulting from collisions between orthogonal beams of alkali metal atoms (Na, K, Cs) and the linear triatomic molecules CO2, COS, and CS2 have been studied from threshold to ∼400 eV (lab). Ions with masses corresponding to the parent molecules CO2, COS, and CS2 are detected for all collision permutations except for Na colliding with CO2. The following electron affinities are deduced from measurements of the threshold for the ion pair production reactions: CO2(−0.60±0.2 eV), COS(+0.46±0.2 eV), and CS2(1.0±0.2 eV). The CO−2 ion was found to be metastable with respect to autodetachment. This result is compatible with the negative electron affinity for CO2 and in agreement with our earlier observations of CO−2* and with recent theoretical calculations. The lifetime of CO−2* (9±2×10−5 sec) was measured to be independent of collision energy over the region of energy studied (threshold to ∼20 eV c.m.). The fragment ions O−/CO2, O−/COS, S−/COS, and S−/CS2 were detected at a threshold ene...