Showing papers in "Journal of Chemical Physics in 1966"

On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods

Abstract: A method is suggested for the calculation of the matrix elements of the logarithm of an operator which gives the exact wavefunction when operating on the wavefunction in the one‐electron approximation. The method is based on the use of the creation and annihilation operators, hole—particle formalism, Wick's theorem, and the technique of Feynman‐like diagrams. The connection of this method with the configuration‐interaction method as well as with the perturbation theory in the quantum‐field theoretical form is discussed. The method is applied to the simple models of nitrogen and benzene molecules. The results are compared with those obtained with the configuration‐interaction method considering all possible configurations within the chosen basis of one‐electron functions.

Approximate Self‐Consistent Molecular Orbital Theory. III. CNDO Results for AB2 and AB3 Systems

Abstract: The approximate self‐consistent molecular orbital theory with complete neglect of differential overlap (CNDO) presented in earlier papers has been modified in two ways. (a) Atomic matrix elements are chosen empirically using data on both atomic ionization potentials and electron affinities. (b) Certain penetration‐type terms, which led to excess bonding between formally nonbonded atoms in the previous treatment, have been omitted. The new method (denoted by CNDO/2) has been applied to symmetrical triatomic (AB2) and tetratomic (AB3) molecules, for a range of bond angles. The theory leads to calculated equilibrium angles, dipole moments, and bending force constants which are in reasonable agreement with experimental values in most cases.

Infrared Study of Adsorbed Molecules on Metal Surfaces by Reflection Techniques

Abstract: The problem of obtaining the infrared spectrum of a molecular monolayer adsorbed on a bulk metal is discussed. The intensity of an infrared absorption band in radiation reflected from the surface is calculated for (a) various optical constants of the adsorbed layer and the metal, (b) various thicknesses of the adsorbed layer, (c) various angles of incidence, and (d) both states of polarization of the incident radiation. The absorption factor for infrared radiation polarized parallel to the plane of incidence typically has a peak at an incident angle of about 88°, where the absorption is 5000 times greater than at normal incidence. The absorption of a thin layer by the reflection technique, at optimum conditions, is calculated to be about 25 times greater than by transmission through the unsupported film at normal incidence.

Atomic View of Surface Self‐Diffusion: Tungsten on Tungsten

Abstract: Surface diffusion of tungsten adatoms on several smooth, low‐index planes of the tungsten lattice has for the first time been followed by direct observation of individual atoms in the field‐ion microscope. Contrary to expectation, the mobility at room temperature is found to increase in the order (211) > (321) ∼ (110) > (310) ∼ (111). Migrating atoms are reflected at the boundaries of the (110), (211), and (321) planes; on the latter two, motion along atomic rows is favored over diffusion across lattice steps. From quantitative determinations of the rate of change of the mean‐square displacement, diffusion coefficients are obtained as follows: (110), D=3×10−2exp(−22 000/RT)cm2/sec; (321), 1×10−3exp(−20 000/RT); (211), 2×10−7exp(−13 000/RT). Differences in diffusion on the (211) and (321), planes of very similar structure, suggest a weakening of interatomic forces at lattice edges.

Proton Resonance Shift of Water in the Gas and Liquid States

Abstract: Temperature‐dependence measurements have been made on the chemical shift of the proton of a water molecule in the liquid state and in the gas state at varying pressure. The problem of relating these experimental data to the intermolecular forces leading to cohesion and to hydrogen‐bond formation between water molecules is considered in detail. It is shown that a consistent treatment of the chemical shift, thermal, and dielectric data for water can be given based on a two‐state model involving an equilibrium between a hydrogen‐bonded ``icelike'' fraction and a ``monomer'' fraction whose interaction with the lattice arises entirely from London dispersion forces.Using semiempirically derived values of the chemical shift and energy associated with the condensation of water vapor to ``monomer,'' the magnitude of the shift associated with the transformation to ice is calculated. It is then shown that, on the assumption that the hydrogen bond is electrostatic in character, the ``polar'' contribution to this shif...

Thermodynamic and Conformational Properties of Polystyrene. I. Light‐Scattering Studies on Dilute Solutions of Linear Polystyrenes

Abstract: The results of an extensive study of some dilute solution properties of a series of linear, homogeneous (anionically prepared) polystyrenes (48<10−4M<440) over a wide temperature interval (10°

On the Rotational Diffusion of Molecules

Abstract: The Debye model of rotational diffusion by small angular steps is generalized to allow molecular reorientation through angular steps of arbitrarily large size. The generalized diffusion models are found to give a rather accurate representation of molecular reorientation in liquids and gases, as observed in the infrared and Raman spectra of simple molecules. One interesting feature of both the theoretical and experimental correlation functions is that the approach to rotational equilibrium often takes the form of a damped oscillation, rather than the monotonic decay which is usually assumed.

Random Sequential Addition of Hard Spheres to a Volume

Abstract: The configurations produced by placing spheres one at a time in a volume, subject to no overlap of the spheres but otherwise placing each at a point chosen at random, differ in a fundamental way from the configurations characteristic of a hard‐sphere system of the same density which is in thermodynamic equilibrium. The difference is illustrated in four examples. The density at which the system becomes jammed is found in each case. The distribution of gaps produced by the random sequential placement of spheres on an infinite line is found for all densities up to that of jamming.

Monte Carlo Study of a One‐Component Plasma. I

Abstract: A Monte Carlo study has been made of a plasma of heavy ions immersed in a uniform neutralizing background. Systems containing from 32 to 500 particles, with periodic boundary conditions, were used. The results of the study are presented in terms of a dimensionless parameter Γ= (4πn/3) ⅓[(Ze)2/kT], where n is the ion density (particles per cubic centimeter), T is the temperature (degrees Kelvin), k is the Boltzmann constant, e is the electronic charge, and Z is the atomic number. Thermodynamic properties and pair distribution functions were obtained for values of Γ ranging from 0.05 to 100.0 from the canonical ensemble by the Monte Carlo (MC) method.Two different methods were used to determine the potential energy of a configuration. The first is the ``minimum‐image convention'' employed in many previous MC calculations. Each particle is allowed to interact only with each other particle in the basic cell, or with the nearest periodic image of each other particle if the image is closer. In the second method...

On the Exponential Form of Time‐Displacement Operators in Quantum Mechanics

Abstract: We derive and discuss a formula, due to Magnus, for the exponential representation of the operator solution to Schrodinger's equation when the Hamiltonian is time dependent. The formula gives a unitary time‐displacement operator in every order of approximation. We study the usefulness of the first‐ and second‐order approximations for the kind of problem posed by the semiclassical theory of inelastic collisions, basing our discussion on two exactly soluble two‐state problems. The algebraic structure of the Magnus formula is in itself useful; to illustrate this, we solve exactly the problems of the linearly forced harmonic oscillator and the harmonic oscillator with time‐dependent force constant.

Exact Quantum-Mechanical Calculation of a Collinear Collision of a Particle with a Harmonic Oscillator

Abstract: Exact quantum‐mechanical calculations of the transition probabilities for the collinear collision of an atom with a diatomic molecule are performed. The diatomic molecule is treated as a harmonic oscillator. A range of interaction potentials from very hard to very soft are considered. It is found that for ``realistic'' interaction potentials the approximate calculations of Jackson and Mott are consistently high, even when the transition probabilities are low and good approximate results are expected. In some cases double and even triple quantum jumps are more important than single quantum jumps. Comparisons are made with exact classical calculations. A semiempirical formula is given for computing quantum‐mechanical transition probabilities from classical calculations.

Rate of Interconversion of Electronic and Vibrational Energy

Abstract: Radiationless transitions are treated from the viewpoint of the nonstationary character of the Born—Oppenheimer approximation. It is shown that the theory explains the fast rates of internal conversion and highly variable rates of intersystem crossing, accounts for the temperature dependence and selection rules of radiationless transitions, and explains the deuterium effect and external heavy‐atom effect on the radiationless transitions.

On the Analytical Mechanics of Chemical Reactions. Classical Mechanics of Linear Collisions

Abstract: The classical mechanics of chemically reactive linear collisions is investigated for vibrationally near‐adiabatic reactions. A coordinate system which passes smoothly from one suited to the reactants to one suited to the products is used. The Hamilton—Jacobi equation is then solved in the adiabatic approximation by introduction of an ``adiabatic‐separable'' method. Nonadiabatic corrections, which describe the probability of vibrational transitions, are also calculated. They involve the Fourier component of local internal centrifugal and vibration frequency‐change terms. The reaction coordinate for the adiabatic system is shown to be that curve on which local vibrational and internal centrifugal forces balance pointwise. Applications can be made to the role of translational—vibrational energy interchange in reactions, reaction‐cross‐section theory, bobsled effect, and other topics. The results may be compared with electronic computer calculations as they become available.

Cusp Conditions for Molecular Wavefunctions

Abstract: The conditions on the behavior of a wavefunction for a system of charged particles near the coalescence of any two of them are derived The new features of the derivation are: (a) the fixed‐nucleus approximation is not required; (b) the wavefunction is not spherically averaged; (c) the wavefunction can have a node at the singularity; and (d) the consequences of the two particles being identical are examined In addition to the general treatment, the electron—nucleus cusp conditions for diatomic molecules are discussed in detail

Gaussian Lobe Function Expansions of Hartree—Fock Solutions for the First‐Row Atoms and Ethylene

Abstract: Gaussian expansions of ground‐state Hartree—Fock solutions for the first‐row atoms are determined by a self‐consistent‐field minimization of atomic energies. Wavefunctions are constructed from a basis set designed primarily for use in molecular calculations which consists only of functions of the form exp(—ar2). Angular dependence is achieved by defining origins for the basis functions at points in space determined in part by the symmetry of the orbital to be expanded.A use of atomic information in molecular problems is discussed in a SCF treatment of the ethylene molecule.

Electronic Structure of Diatomic Molecules. III. A. Hartree—Fock Wavefunctions and Energy Quantities for N2(X1Σg+) and N2+(X2Σg+, A2Πu, B2Σu+) Molecular Ions

Abstract: The problem of the convergence of a sequence of Hartree—Fock—Roothaan wavefunctions and energy values to the true Hartree—Fock results is examined for N2(X1Σg+). This critical study is based on a hierarchy of Hartree—Fock—Roothaan wavefunctions which differ in the size and composition of the expansion basis set in terms of STO symmetry orbitals. The concluding basis set gives a total Hartree—Fock energy of −108.9956 hartree and Re(HF)=2.0132 bohr for N2(X1Σg+).Results are also presented from direct calculations for three states of the N2+ molecular ion (X2Σg+, A2Πu, B2Σu+) which are also thought to be very close approximations to the true Hartree—Fock values. The results give EHF=−108.4079, −108.4320, and −108.2702 hartree and Re(HF)=2.0385, 2.134, and 1.934 bohr for the X2Σg+, A2Πu, and B2Σu+ states of N2+, respectively. Extensive calculations for various R values establish that the X2Σg+ and A2Πu states are reversed in order relative to experiment, a short‐coming ascribed to the Hartree—Fock approximation.

Statistical Theory of Chemical Kinetics : Application to Neutral-Atom—Molecule Reactions

Abstract: Several processes involving uncharged species are discussed in terms of a statistical theory of strong‐coupling collisions. Specifically, we study the Mies—Shuler—Zwanzig model for vibrational excitation of a diatomic by impulsive collisions; the reactions of K with HBr, Cl with KH, Cl with Na2, and the subsequent reaction of vibrationally excited NaCl with Na; and the breakup of electronically excited H2O to H atom and OH (2Σ+) radical (as an example of a bad failure of the statistical model). Except in the last case, rotational and vibrational distributions in the product diatomics predicted by the statistical theory agree well with experiment (where the experiment has been done); the total reactive cross sections, although a little high, compare reasonably well with the measured cross sections.

Conformational Analysis of Macromolecules. III. Helical Structures of Polyglycine and Poly‐L‐Alanine

Abstract: A theoretical study of the regular conformations of isolated helices (i.e., with no intermolecular interactions) of polyglycine and poly‐L‐alanine has been carried out. The energy of each helical conformation was calculated by using semiempirical potential functions for the barriers to internal rotation about single bonds, nonbonded interactions, dipole—dipole interactions between the amide groups, and hydrogen bonding between backbone NH and CO groups. All bondlengths and bond angles were held fixed, and the amide group was fixed in the planar trans conformation. Within the accuracy of the calculations, the right and left‐handed α‐helical conformations were found to be those of lowest energy for polyglycine. For poly‐L‐alanine, the right‐handed α helix was found to be the most stable conformation. This conformation was stabilized not only by hydrogen bonding and dipole—dipole interactions, but also by the nonbonded interactions; in fact, it is the nonbonded energy which makes the right‐handed α helix more stable than the left‐handed one in poly‐L‐alanine, since dipole—dipole interactions and hydrogen bonding stabilize both structures about equally well. The effect of varying the parameters in the semiempirical potential functions was studied, and the results and conclusions of these calculations were compared with calculations made by other investigators and with experimentally determined polypeptide conformations.

Double‐Quantum Light Scattering by Molecules

Abstract: Double‐quantum light scattering by a system of molecules is discussed in this paper. Expressions have been obtained for the scattered light intensity considering both the coherent and incoherent contributions. In that coherent contributions are also considered in this treatment, it goes beyond the scope of previous studies. It is shown that, for molecules of low symmetry, elliptically polarized light must be used in order to determine five independent quadratic forms in the 18 symmetric components (βijk+βikj). According to the present results, the apparent discrepancy between the observed value of ⅓ for the depolarization ratio for CCl4 and the value to be expected from theory may be due to the fact that the coherent contribution had been neglected in previous theoretical considerations. In general, orientational correlation is essential if there is to be appreciable contribution from coherent scattering. For macromolecules, this constitutes a major difference between single‐ and double‐quantum scattering...

Extended Hartree—Fock Wavefunctions: Optimized Valence Configurations for H2 and Li2, Optimized Double Configurations for F2

Abstract: As a step beyond the Hartree—Fock technique in the search for better energies, wavefunctions, and the general description of molecular formation and dissociation, a configuration‐mixing method of the following nature is developed and illustrated for H2, Li2, and F2. The wavefunction Ψ= ∑ kAkΦk consists of several optimized configurations obtained by replacing one of the σ‐orbitals of the primary configuration (in our case the Hartree—Fock) by orbitals of different kinds. All the orbitals involved are orthonormalized, insuring orthonormalization of the configurations themselves. The present method determines energetically the optimum combination of both the mixing coefficients Ak and the linear orbital parameters by the solution of SCF‐type equations and insures dissociation of the molecule to two Hartree—Fock atoms. A program based upon this formalism has been constructed by the authors for the IBM 7094 computer and is capable of handling homonuclear diatomic molecules using as many as ten configurations ...

Anomalously Long Radiative Lifetimes of Molecular Excited States

Abstract: Certain excited states of molecules have been found to have anomalously long radiative lifetimes compared with the values calculated from integrated absorption coefficients. Using the properties of the stationary states, four mechanisms which may account for such anomalous lifetimes in collision‐free molecules are presented. Although the four mechanisms bear a one‐to‐one relationship to those which have been discussed using semiclassical concepts, certain consequences of the more exact treatment do not appear in the semiclassical approach. In particular, it is shown that the two mechanisms which are most important for polyatomic molecules lead to highly perturbed energy levels and correspondingly complex spectra. The spectra of NO2, SO2, and CS2 are discussed as examples of the proposed mechanisms.

Energy Transfer Between Rare‐Earth Ions

Abstract: Fluorescence quenching effects caused by multipolar energy‐transfer interactions between rare‐earth ions are examined. Transfer between remote ions by way of transitions that are matched in energy appears to be essentially dependent upon concentration alone (e.g., Sm3+ self‐quenching) while transfer by non‐resonant mechanisms can show a strong dependence upon crystal structure (e.g., Eu3+ self‐quenching). Examples of thermally dependent multipolar transfer interactions are also given.

Nuclear Magnetic Double Resonance with an Incoherent Radio-Frequency Field

Abstract: The decoupling of heteronuclear spin systems by means of a strong irradiation of rf magnetic random noise is explored theoretically and experimentally. It is shown that it is feasible to eliminate simultaneously all splittings due to one or more nuclear species. Double resonance with random noise is of particular advantage if the resonance frequencies of the nuclei causing the splittings cover a wide spectral range and if, at the same time, the couplings are relatively weak. In the case of a narrow spectral range and strong couplings, double resonance with a coherently frequency‐modulated rf field can give comparable decoupling effects. Double resonance with random noise has a strong similarity to the effects of chemical exchange and relaxation.

``Multiple-Scattering'' Model for Polyatomic Molecules

Abstract: The ``multiple‐scattering'' technique of determining energy bands in solids is modified for the problem of calculating bound one‐electron eigenstates of polyatomic molecules. The computational simplicity and applicability of this method to molecules of arbitrary complexity are dependent on the adoption of a model Hartree—Fock Hamiltonian based on the Slater approximation to the exchange potential and on the assumption of truncated spherically averaged potentials. The composite wavefunctions are expanded in rapidly convergent partial‐wave series around each atomic site. For molecules of moderate size, the calculations can easily be carried to self‐consistency within the spherical approximation, and first‐order perturbation theory can be applied to correct for the nonspherical components of the potential. Emphasis is placed on the advantages of this technique as a first step in calculating the theoretical electronic spectra of complex polyatomic systems.

Effect of High Pressure on the Lattice Parameters of Diamond, Graphite, and Hexagonal Boron Nitride

Abstract: The effect of pressure to several hundred kilobars has been measured on the lattice parameters of diamond, graphite, and hexagonal boron nitride. The linear compressibility of diamond is independent of pressure. The a axis of graphite (characteristic distance in the plane) is considerably more compressible than diamond at low pressure but above about 160 kbar is much less compressible than diamond. The c axis of graphite is much more compressible than the a axis. Assuming van der Waals forces between the graphite planes and using the lattice sums of Girifalco and Lad, it is possible to describe the compressibility of graphite to very high pressure from low‐pressure data. Hexagonal BN is much like graphite, with both a and c axes somewhat more compressible than the corresponding parameters of graphite.

Simple Classical Model for the Scattering of Gas Atoms from a Solid Surface

Abstract: A simple classical model for the scattering of gas atoms from a solid surface is proposed and its characteristics are determined. The principal assumptions are (1) the interaction of a gas atom with a surface atom is represented by an impulsive force of repulsion, (2) the gas—surface intermolecular potential is uniform in the plane of the surface (hence the interaction does not change the tangential velocity of the gas particle), (3) the surface atoms are represented by independent particles confined by square‐well potentials, (4) the surface atoms have a Maxwellian velocity distribution. The model incorporates many of the same features as that proposed by Goodman. As a result of these assumptions, the model is simplified to the extent that it contains no adjustable constants.Results are obtained for the angular distribution of the scattered particles. Significant quantitative results are not expected from such a simple model, but the qualitative behavior of the model does agree surprisingly well with the...

Nonbonded Potential Parameters Derived from Crystalline Aromatic Hydrocarbons

Abstract: Nonbonded (exp−6) potential parameters for C···C, C···H, and H···H interactions were derived from the crystal structures and properties of aromatic hydrocarbons. The exponent of the C···C repulsion term was taken from a calculation based on the interplanar spacing and compressibility of graphite. The exponent of the H···H repulsion was taken from a quantum‐mechanical calculation of the repulsion between two hydrogen molecules. The coefficients of the attractive and repulsive terms were fitted by weighted least squares to 77 observational equations involving the geometrical crystal structures, elastic constants, and sublimation energies of nine aromatic hydrocarbons. It was found necessary to externally estimate the H···H repulsion coefficient to obtain an appropriate partition between the C···C, C···H, and H···H energy terms. The geometrical mean combining law was applied for the attractive and repulsive coefficients.The resulting nonbonded potential parameters, when used to minimize the lattice energy by a steepest descent procedure, were found to reproduce the crystal structures of benzene, napthalene, and anthracene to within about 0.05 A in the carbon‐atom positions and within about 1% in the lattice constants. No assumption of anisotropic forces based on π‐electron polarizabilities was necessary to reproduce herringbone‐type packing. The position of the minimum in the best C···H nonbonded potential was less than expected, while the minimum in the H···H potential was larger than expected. The minima positions were found to be sensitive to the H···H repulsion coefficient. The correlation coefficients between the potential parameters found by this method were found to be large.

Energy Distribution Among Products of Exothermic Reactions. II. Repulsive, Mixed, and Attractive Energy Release

Abstract: A series of two‐dimensional classical kinematic computer calculations have been made on the hypothetical exothermic exchange reaction A+BC→AB+C, −ΔH=48.5 kcal mole−1. Product energy distribution (vibration, rotation, and translation; Evib, Erot, Etrans) was obtained as a function of initial position, impact parameter, and kinetic energy (αi, b, and Ekini). All eight different combinations of light (L=1 amu) and heavy (H=80 amu) masses were examined. Eight different potential‐energy hypersurfaces were explored. All were obtained from an empirical extension of the London—Eyring—Polanyi—Sato (L.E.P.S.) method. The hypersurfaces were categorized in terms of the percentage attraction, A⊥, and percentage repulsion R⊥, read off the collinear three‐dimensional surface. This categorization was shown to be helpful in arriving at a qualitative understanding of the product energy distribution to be expected from all the mass combinations reacting on these extended L.E.P.S. hypersurfaces. However, it was also shown th...

On the Eu3+ Fluorescence of Mixed Metal Oxides. IV. The Photoluminescent Efficiency of Eu3+‐Activated Oxides

Abstract: Four rules are proposed to explain the photoluminescent efficiency of Eu3+‐activated oxides: (a) uv absorbing groups in the host lattice should be nearest neighbors in the crystal structure to have sufficient wavefunction overlap; (b) the emission spectrum of the absorbing group should overlap the absorption spectrum of the group; (c) the configuration consisting of the center of the absorbing group, the intermediary O2− ion, and the Eu3+ ion should be as collinear as possible; (d) O2− ions surrounding the Eu3+ ion must feel a high potential fieldIf all these rules are satisfied, highly efficient Eu3+ phosphors can be expected; if one or more of the rules are violated, phosphors with low efficiency must be expectedThese ideas are applied to a large number of Eu3+‐activated oxides There is a very good agreement between experimental results and predictions from the rulesAs far as possible the physical background of the rules is discussed

ESR Linewidths in Solution. I. Experiments on Anisotropic and Spin—Rotational Effects

Abstract: The linewidths of the hyperfine components of the ESR spectra of vanadyl acetylacetonate in liquid toluene have been measured as a function of concentration, temperature, and applied field. The g‐tensor and hyperfine‐tensor components have been determined and used, along with the spectral data, to determine the Debye rotational correlation time τR. This analysis agrees very well with the experimental observations. A residual linewidth, not dependent upon the above mechanism, and independent of nuclear quantum number and applied field, has been studied and found to vary as T/η, where η is the viscosity. This residual linewidth is ascribed to a spin—rotational interaction.