Index of content:
Volume 63, Issue 2, 15 July 1975

Direct correlation function: Hard sphere fluid
View Description Hide DescriptionThe direct correlation functionc (r) and in particular the ’’tail’’ of the direct correlation functiond (r) are calculated for hard spheres from reliable expressions for the radial distribution function of the hard sphere fluid. This permits a direct examination of the Percus–Yevick (PY) approximation d (r) =0. It is found that the PY approximation is very poor over‐all but fairly reasonable in the most important region r ? σ. Finally a simple but relable parametrization of d (r) for hard spheres is proposed.

A ^{14}N nuclear relaxation study of hydrogen bonding in diethylamine–alcohol solutions
View Description Hide DescriptionThe rotational correlation time τ_{ q } of ^{14}N in diethylamine has been measured in a number of solutions containing normal alcohols as solute. Using the nuclear absorption linewidth of ^{14}N to follow changes in τ_{ q }, contributions associated with hydrogen bonding and with alcohol chain hindrance were identified. Solutionviscosities are also reported and correlation times derived from them are compared with the corresponding τ_{ q } in order to contrast their responses to solute additions of this type.

Magnetic circular dichroism in hemoglobin
View Description Hide DescriptionMagnetic circular dichroism(MCD) and absorption have been measured in various derivatives of hemoglobin in an applied field of 16 kG at temperatures ranging from 77 to 294 °K. The visible and Soret bands have been studied. We develop a spin–orbit coupling model to help explain the sign, magnitude, and 1/T temperature dependence of the MCD of deoxyhemoglobin. The results for deoxyhemoglobin are compared with those for unligated separated alpha and beta chains. Differences in spectra occur which can only be attributed to the different temperature dependent heme environments for chains and for tetramers. Cyanomethemoglobin and oxyhemoglobin are also discussed.

New‐type correlated wavefunctions for three‐body systems with Coulomb interaction
View Description Hide DescriptionSimple correlated wavefunctions are reported for the ground states of three selected Coulombic systems, namely (1) the positronium negative ion e ^{−} e ^{+} e ^{−}, (2) the hydrogen negative ion H^{−}, and (3) the mesic molecular ion pμp. They give quite accurate energy results, and are analytically simple enough to be used widely in applications. Two characteristic features of the proposed wavefunctions are that (1) the correlation between the two particles with the same sign of the charge is explicitly included in the form G= x ^{ n } _{3} e ^{−γx } ^{ 3 }, and (2) the variational parameters to be optimized are mostly nonlinear (exponential).

Generalization of Slater’s transition state concept
View Description Hide DescriptionWe present a generalization of the transition state technique introduced by Slater for the calculation of many‐electron relaxation effects accompanying electronic excitations in molecules and molecular simulations of solids. By making use of ground state information (which is generally available but not used in the Slater formulation) and transition states which are computationally cheaper (due to being closer to the ground state), the generalization permits the evaluation of excitation energies to be improved in any of three ways: (1) comparable accuracy for less computation; (2) improved accuracy for comparable computation; and (3) full Δ‐SCF accuracy can be approximated with arbitrary precision with additional computation. In particular, we show that excitation energies of somewhat greater accuracy are obtained from self‐consistent calculations performed for transition states corresponding to 2/3 of the transition rather than 1/2 of the transition as in the original formulation by Slater.

A theory of the hard sphere solid
View Description Hide DescriptionA ’’self‐consistent field’’ theory based on the Bethe approximation and previously shown to reproduce the exact partition function for one‐dimensional hard spheres at all densities and to give good results for classical harmonic and nearly harmonic crystals is applied to the three‐dimensional hard‐sphere system. Solutions of the resulting equations which are approximately valid at high densities are obtained. There are two possible solutions, one of which gives values for the entropy and one‐particle distribution function in good agreement with results for the hard‐sphere solid derived from computer simulations. The other solution may be associated with the fluid phase.

Self‐consistent field calculation of the electronic structure of the uranyl ion (UO_{2} ^{++})
View Description Hide DescriptionUsing the multiple‐scattering Xα method the electronic ground state (^{1}Σ^{+} _{ g }) of the UO^{++} _{2} molecule has been calculated self‐consistently. The self‐consistent calculations were performed at three U–O bond distances [3.269, 4.269, and 5.269 (a.u.)]. The four highest occupied one‐electron levels (σ_{ g }, σ_{ u }, Π_{ g }, Π_{ u }) of the ground state, which have received so much attention in the past, change ordering with decreasing bond distance but are so close in energy that the ordering may be unimportant in determining the absorption spectra. We have also computed the electronic transition energies for the excitations ^{1}Σ^{+} _{ g } → ^{1,3}Φ_{ g }(σ^{1} _{ u }φ^{1} _{ u }), ^{1}Σ^{+} _{ g } → ^{1,3}Δ_{ g }(σ^{1} _{ u }φ^{1} _{ u }), ^{1}Σ^{+} _{ g } → ^{1,3}Π_{ u }, ^{1,3}Φ_{ u }(π^{3} _{ g }δ^{1} _{ u }), ^{1}Σ^{+} _{ g } → ^{1,3}Δ_{ g }, ^{1,3}Γ_{ g }(π^{3} _{ u }φ^{1} _{u}), and ^{1}Σ^{+} _{ g } → ^{1,3}Φ_{ g }, ^{1,3}Π_{ g }(π^{3} _{ u }φ^{1} _{ u }). All of these excitation energies fall in the region of the experimental absorptionspectrum.

Energy dependence and isotope effect for the total reaction rate of Cl+HI and Cl+HBr
View Description Hide DescriptionA laser initiated chemical reaction method has been used to determine the total reaction rates for Cl+HI and Cl+HBr and the dependence of the rates on collision energy and H–D isotopic substitution. The rate constants are k=1.64×10^{−10} cm^{3} molecule^{−1}⋅sec^{−1} (σ=33.5 Å^{2}) for Cl+HI and k=7.4×10^{−12} cm^{3} molecule^{−1}⋅sec^{−1} (σ=1.44 Å^{2}) for Cl+HBr. Measurements were all done at 295 °K in slowly flowing gases. Substituting H by D decreases the rate constant by a factor of 1.84 in the case of HI and by 1.5 in the case of HBr. The isotope effect may be a result of tunneling on corner cutting trajectories. The cross section decreases with increasing collision energy. This, together with the large reaction cross section, indicates the importance of an attractive potential in these systems.

Statistical mechanical theory of polymers. II. Thermodynamic functions of a single ring polymer
View Description Hide DescriptionStatistical thermodynamics of a single ring polymer with an intersegmental interaction is developed. Volume in this system is defined in terms of the mean‐square radius of gyration. The concept of compressed and expanded states of the polymer is developed by introducing a coefficient‐of‐expansion σ. In this way, the volume of the single polymer can be varied and change of free energy with this variation in volume can be studied. Tension is now defined as the infinitesimal change in free energy due to an infinitesimal change of volume, i.e., infinitesimal expansion of the system. The chemical potential is the change in free energy due to charging (or severing of) one of the segments of the polymer chain. Using these definitions and methods analogous to statistical mechanical theory of fluids, explicit expressions are derived for the equation of state as well as other thermodynamic properties in terms of the binary intersegmental correlation functiong ^{(2)}(R) and the intersegmental interaction u (R). The techniques developed here are applicable to the study of gas–liquid type phase transitions observed in polymer solutions and Monte Carlo simulation experiments.

Statistical mechanical theory of polymers. III. Equation of state for the hard sphere model of a single ring polymer
View Description Hide DescriptionThermodynamic functions for a single ringpolymer with hard‐sphere binary intersegmental interactions are studied. The integral equation of the binary intersegmental correlation functiong ^{(2)}(R) developed earlier is expressed in terms of a parameter λ in a manner analogous to Kirkwood, Maun, and Alder for the case of fluids. The parameter λ is related to the coefficient of expansion σ (and therefore density) by a thermodynamic identity. The binary intersegmental correlation functiong ^{(2)}(R) is now computed numerically by an iterative procedure. Numerical values of various thermodynamic functions are computed. The data are in qualitative agreement with Monte Carlo calculations of Mazur and McCrackin at high temperatures.

Weibull distribution and kinetics of heterogeneous processes
View Description Hide DescriptionThe rate equation Q _{ t }/Q _{∞} = 1 − e ^{−} b t ^{ n }, where Q _{ t }/Q _{∞} is the fractional yield of an interface reaction at time t, was shown to be formally analogous to the Weibull distribution functions. Further correlation between the Weibull and the normal distributions did allow for a direct statistical interpretation of surface processes. Rates are thus defined by the well known parameters μ and σ. Geometrically incompatible processes were standardized on the basis of the exponent n and the shape parameter β of the Weibull distribution. For 3<n<4, processes obeyed the normal distribution, and for n<0.7, they obeyed the log normal distribution. By using the linearized probability plots, ambiguities imposed by the multiple logarithming of the above rate equation are avoided.

Statistical foundations of heterogeneous kinetics
View Description Hide DescriptionA mathematical analysis of the correlation between Weibull and gamma distributions is presented. On this basis, processes obeying the relation Q _{ t }/Q _{∞} = 1 − e ^{−b t } ^{ n } (Q _{ t }/Q _{∞} being the fractional yield of an interface reaction at time t) can be explained better. Selected values of n, found experimentally, reflect the specific rate dependence of the distribution of the surface potential on active sites. By using the corresponding gamma distribution functions we show that most of these processes obey either directly or indirectly the normal distribution of the activity of reacting sites.

Exact quantum, quasiclassical, and semiclassical reaction probabilities for the collinear F+H_{2} → FH+H reaction
View Description Hide DescriptionExact quantum, quasiclassical, and semiclassical reaction probabilities and rate constants for the collinear reaction F+H_{2} → FH+H are presented and compared. The exact quantum results indicate a large degree of population inversion of the FH product with P ^{ R } _{02} and P ^{ R } _{03} being the dominant reaction probabilities. The energy dependence of these two probabilities at low translational energies are quite different. P ^{ R } _{02} shows an effective threshold of 0.005 eV which can largely be interpreted as resulting from tunneling through a vibrationally adiabatic barrier. P ^{ R } _{03} has a much larger effective threshold (0.045 eV) apparently resulting from dynamical effects. Quasiclassical probabilities for the collinear F+H_{2}reaction were calculated by both the forward (initial conditions chosen for reagent F+H_{2}) and reverse (initial conditions for product H+FH) trajectory methods. The results of both calculations correctly indicate that P ^{ R } _{03} and P ^{ R } _{02} should be the dominant reaction probabilities. However, the threshold behavior of the quasiclassical forward P ^{ R } _{03} disagrees strongly with the corresponding exact quantum threshold energy dependence. By contrast, there is good agreement between the reversed trajectory results and the exact quantum ones. The uniform semiclassical results also agree well with the corresponding exact quantum ones indicating that the quasiclassical reverse and the semiclassical methods are preferable to the quasiclassical forward method for this reaction. The important differences between the threshold behavior of the exact quantum and quasiclassical forward reaction probabilities are manifested in the corresponding rate constants primarily as large differences in their activation energies. Additional exact quantum results at higher total energies indicate that threshold effects are no longer important for reactions with vibrationally excited H_{2}. Resonances play an important role in certain reaction probabilities primarily at higher relative translational energies.

Exact quantum, quasiclassical, and semiclassical reaction probabilities for the collinear F+D_{2} → FD+D reaction
View Description Hide DescriptionExact quantum, quasiclassical, and semiclassical reaction probabilities and rate constants for the collinear reaction F+D_{2} → FD+D are presented. In all calculations, a high degree of population inversion is predicted with P ^{ R } _{03} and P ^{ R } _{04} being the dominant reaction probabilities. In analogy with the F+H_{2}reaction (preceding paper), the exact quantum 0→3 and 0→4 probabilities show markedly different energy dependence with P ^{ R } _{03} having a much smaller effective threshold energy (E _{ T }=0.014 eV) than P ^{ R } _{04} (0.055 eV). The corresponding quasiclassical forward probabilities P ^{ R } _{03} and P ^{ R } _{04} are in poor agreement with the exact quantum ones, while their quasiclassical reverse and semiclassical counterparts provide much better approximations to the exact results. Similar comparisons are also made in the analysis of the corresponding EQ, QCF, QCR, and USC rate constants. An information theoretic analysis of the EQ and QCF reaction probabilities indicates nonlinear surprisal behavior as well as a significant isotope dependence. Additional quantum results at higher energies are presented and discussed in terms of threshold behavior and resonances. Exact quantum reaction probabilities for the related F+HD → FH+D and F+DH → FD+H reactions are given and an attempt to explain the observed isotope effects is made.

The use of discontinuous trial functions in the computation of the electronic structure of molecules: Calculation of the ESCA spectrum of tetrathiofulvalene (TTF)
View Description Hide DescriptionIt is shown that discontinuous trial functions can be easily adopted in the computation of the electronic structure of molecules; in fact, when used in conjunction with a model of the molecule based on intersecting spheres, they permit very fast and accurate calculations to be performed. As an example, the computation of the ESCA spectrum of a large organic molecule (TTF) is presented.

Three‐dimensional natural coordinate asymmetric top theory of reactions: Application to H + H_{2}
View Description Hide DescriptionA particular partitioning of the Hamiltonian in natural collision coordinates is shown to lead to the use of hindered asymmetric top basis functions to represent all rotational motion (triangle tumbling plus internal bending) during reaction. These functions (along with perturbed Morse oscillator functions) are used as an adiabatic basis for expansion of the scattering wavefunction. The theory is discussed for both one and two reaction path potentials. The close coupled equations for the translational wavefunctions are then solved for the H + H_{2}reaction at total angular momentumJ = 0. Wavefunctionbifurcation and matching at the reactant–product boundary surface is considered in detail. Finally, numerical results (reaction probabilities, probability conservation, detailed balance, energy dependence of reactive S‐matrix elements, probability density, wavefunction real part, and flux) are presented and comparisons are made with other quantum mechanical, semiclassical, and statistical reaction studies.

Thermodynamic perturbation theory for potentials of multipolar symmetry. I
View Description Hide DescriptionEarlier work in simple polar fluids is extended to higher multipolar terms. Some general properties of multipolar expansions and of the three‐body terms that appear in them are considered. Illustrative application is made to a system of quadrupolar spheres. A method of incorporating the effects of nonspherical molecular cores into our formalism is outlined, and some preliminary estimates are given for the relative size of such effects compared to that of the quadrupolar terms.

The kinetics of ionic/metallic ion‐exchangers
View Description Hide DescriptionA model for the kinetics of ionic/metallic ion‐exchangers is developed from the Nernst–Planck equation for ionic diffusion and the chemical reaction rate equations assumed to be proportional to the electrochemical affinity of each species. The model is demonstrated for a number of hypothetical systems, and the uses and limitations of the model are discussed.

Correlation function formula for the intrinsic viscosity of dilute polymer solutions
View Description Hide DescriptionThe correlation function formalism for the intrinsic viscosity of polymers is studied. A controversy concerning the correct force to use in the momentum flux is resolved. It is shown that when the diffusion equation is used in the full configuration space of polymer segments the forces entering the momentum flux are purely mechanical and there is no entropic contribution. A comparison is made with Kirkwood’s theory of viscoelastic behavior. The correlation function expression we advocate is shown to yield the correct high frequency limiting behavior for the case of elastic dumbbells.

Temperature dependence of third order ion molecule reactions. The reaction H^{+} _{3}+2H_{2} = H^{+} _{5}+H_{2}
View Description Hide DescriptionThe rate constantsk _{1} for Reaction (1): H^{+} _{3}+2H_{2} = H^{+} _{5}+H_{2} were measured in the temperature range 100–300 °K. The temperature dependence of k _{1} has the form k _{1}∝T ^{−n }, where n=2.3. Pierce and Porter have reported a much stronger negative temperature dependence with n=4.6. The difference arises from a determination of k _{1} at 300 °K obtained by Arifov and used by Porter. The present k _{1} (300 °K) =9×10^{−30} (cm^{6} molecules^{−2}⋅sec^{−1}). This is more than an order of magnitude larger than the Arifov value. The temperature dependence of third body dependent association reactions like (1) is examined on the basis of the energy transfertheory and the recently proposed trimolecular complex transition state theory by Meot‐Ner, Solomon, Field, and Gershinowitz. The temperature dependence of the rate constant for the reverse reaction (−1) is obtained from k _{1} and the previously determined temperature dependence of the equilibria (1). k _{−1} gives a good straight line Arrhenius plot leading to k _{−1} =8.7×10^{−6} exp(−8.4/R T) cm^{3} molecules^{−1}⋅sec^{−1}. The activation energy is in kcal/mole. The preexponential factor is much larger than the rate constant for Langevin collisions. This is typical for pyrolysis of ions involving second order activation.