Volume 41, Issue 3, 01 August 1964

Variational Methods for Solving the Boltzmann Equation
View Description Hide DescriptionKnown variational methods are reviewed and their applicability to the Boltzmann equation when detailed balance does not hold is discussed. The relation between these variational methods and time‐reversal and space‐inversion symmetry is indicated.

Statistical Thermodynamics of Incompletely Specified Systems
View Description Hide DescriptionA formal theory of incompletely specified systems based on reasonable physical assumptions is presented. It is shown that the free energy for such systems involves the average of the logarithm of the partition function, rather than the average of the partition function itself. It is further shown that one cannot always replace the logarithm of the partition function by its average in all of the equations of statistical thermodynamics. The method differs from that of Mazo. We do not obtain a term corresponding to an entropy of mixing due to random elements. Finally, fluctuations as calculated here differ from those implied by Mazo's method. Due to the formal nature of the problem, a strict right—wrong distinction between the two methods cannot be made.

Lack of Charge Transfer in Aromatic Charge—Transfer Complexes
View Description Hide DescriptionHalogen pure electrical quadrupole resonance frequencies have been measured in the three 1:1 complexes, CBr_{4}·p‐xylene, CCl_{4}·p‐xylene, and Br_{2}·benzene. From an examination of the measured frequencies of the complexes and the halogen‐containing compounds (CBr, CCl_{4}, and Br_{2}) together with information from the x‐ray structure determinations for two of the complexes, it has been concluded that little if any charge transfer exists in the ground state of these complexes.

Intermolecular Quenching of Higher Excited States
View Description Hide DescriptionThe influence of β carotene on the fluorescence yield of chlorophyll a in 3‐methoxy‐ 1‐butanol is investigated. β carotene does not affect the fluorescence yield when chlorophyll a is excited in the red absorption band at 660 nm. However, a quenching of the red fluorescence of chlorophyll a is observed when excited at 415 nm. This is suggested to be due to a quenching of the second excited state of chlorophyll a by β carotene.

Separation of Sets of Variables in Quantum Mechanics
View Description Hide DescriptionSeparation of the Schrödinger equation for molecular dynamics into sets of variables can sometimes be performed when separation into individual variables is neither possible nor for certain purposes necesary. Sufficient conditions for such a separation are derived. They are the same as those found by Stäckel for the corresponding Hamilton—Jacobi problem, with an additional one which is the analog of the Robertson condition for one‐dimensional sets.
Expressions are also derived for operators whose eigenvalues are the separation constants. They provide a variational property for these constants. For use in aperiodic problems an expression is obtained for the probability current in curvilinear coordinates in an invariant form. Application of these results to reaction rate theory is made elsewhere.

Local Approximation of Potential‐Energy Surfaces by Surfaces Permitting Separation of Variables
View Description Hide DescriptionIn the immediate vicinity of a potential‐energy minimum or of a saddle point, it is shown that major topographical features of a ``nonseparable'' potential‐energy surface can be imitated by those of a surface permitting separation of variables. For each extremal path of descent or ascent to the cited critical point of the surface, there is an exact match of the tangent, the first curvature vector in configuration space, and the force constant along that path provided that the known curvature vector satisfies an equation containing the metric tensor of the selected coordinate system and known force constants. Because of the wide choice of coordinate systems available for selection, it is anticipated that this relation may be fulfilled for each extremal path, partly by choice of the coordinate system and partly by subsequent choice of the curvilinear coordinates of the critical point.
There are several possible applications of this local approximation, including those to problems involving anharmonic coupling of normal modes and those involving n‐dimensional tunneling and other calculations in reaction‐rate theory. Use will be made of the formalism to extend the activated complex theory in chemical kinetics. As a preliminary test of the local‐approximation concept, literature data on n‐ and one‐dimensional tunneling rates are compared. They are found to be fairly similar when proper cognizance is taken of zero‐point energies.

Crystal Field for Yb^{3+} in Garnets
View Description Hide DescriptionAn attempt has been made to determine the nine parameters required to define the crystal field acting on an Yb^{3+} ion in yttriumgalliumgarnet. Even for this most favorable case, there are at present insufficient data to determine the parameters from experiment alone, and we have therefore carried out an analysis combining seven independent and reliable experimental measurements with theoretical estimates based on an accurate point‐charge calculation. The results of the point‐charge calculations, made also for several other garnet lattices, indicate that distortions from a cubic potential are generally large, and that the second‐degree terms are poorly determined by the model. In fitting to the experimental data, the parameters V _{2} ^{0}, V _{2} ^{2}, V _{4} ^{0}, and V _{6} ^{0} (V_{n} ^{m} = A_{n} ^{m}〈r^{n}〉) were treated as adjustable, and the remaining parameters given the values expected from the point‐charge ratios. The set of parameters thus obtained fit the six experimental g values to within 0.2 and the two fitted energy splittings to within 2 cm^{—1}, and they also predict further energy levels in agreement with less definite experimental evidence. The remaining discrepancies reflect the inadequacies of the point‐charge model. The apparent success of previous analyses of data for Yb garnets in terms of a near‐cubic approximation is discussed, and it is shown how an insufficient number of parameters may incorrectly account for a limited set of experimental data.

Vibrational Interactions in Pariser—Parr Theory. I. Bandwidths and Transition Intensities
View Description Hide DescriptionIt is shown that bandwidths of some transitions can be calculated with good accuracy in the Pariser—Parr theory. For these calculations the critical point is a knowledge of the variation of the bond resonance integral β with respect to distance, which is given by . Calculations are carried out for the bandwidths of the N–V transition of ethylene and the ^{1} A _{1g }→^{1} E _{1u }, ^{1} E _{2g } transitions of benzene and the intensities of the ^{1} A _{1g }→^{1} B _{2u }, ^{1} B _{1u } transitions of benzene. Satisfactory agreement with experiment is obtained; the neglect of electronic repulsion terms is demonstrated to have a relatively small effect. Selection rules are established for the strength of vibrational coupling in alternant hydrocarbons.

Rotation—Vibration Interaction and Other Corrections to the Spin Doublet Separation in ^{2}Π Diatomic Molecules
View Description Hide DescriptionThe effect of rotation—vibration interaction on the spin doublet separation in ^{2}Π states of diatomic molecules is treated. Almy and Horsfall's refinement of Hill and Van Vleck's treatment is questioned and an alternative approach making use of Dunham's treatment of ^{1}Σ diatomic molecules is used. The J dependence of the spin‐orbit coupling constants is included in the discussion. Vibrational wavefunctions (including rotation—vibration interaction) and matrix elements are discussed. Additional corrections arising from the Λ doubling and from the spin—rotation interaction ΓN·S are briefly considered. A comparison of the results of this work with experimental spin doublet separation leads to better agreement between theory and experiment for the examples treated.

On the Continuous Absorption, Photoionization, and Fluorescence of H_{2}O, NH_{3}, CH_{4}, C_{2}H_{2}, C_{2}H_{4}, and C_{2}H_{6} in the 600‐to‐1000‐Å Region
View Description Hide DescriptionThe hydrogen‐containing molecules H_{2}O, NH_{3}, CH_{4}, C_{2}H_{2}, C_{2}H_{4}, and C_{2}H_{6} have strong continuous absorption spectra in the 12 to 21 eV (600‐to‐1000‐Å) region. Photoionization and absorption coefficients and cross sections have been computed from absorption and ionizationspectra using the Hopfield He continuum as a continuous background‐radiation source. A parallel plate combination absorption and ionization chamber was used behind the exit slit of a Seya scanning vacuum monochromator arranged for 0.5 Å bandwidth. Data were obtained with a windowless system using rare‐gas calibration to measure the absolute photon flux from which photoionization efficiencies were determined. In general, the cross sections increased with increasing number of hydrogen atoms, and for the process corresponding to electron loss by photoionization, the cross section was found to be proportional to the number of bonding electrons, varying from 17.5 Mb for H_{2}O to over 76 Mb for C_{2}H_{6} (1 Mb=10^{—18} cm^{2}). Inflections in the cross‐section curves indicate processes other than simple electron loss by photoionization. Preliminary results of the fluorescence from these gases indicate that NH_{3}, CH_{4}, and C_{2}H_{2} show measurable fluorescence. Maxima in the relative fluorescence yield curves are shown to be correlated with minima in the ionization efficiency curves, suggesting that the long‐wavelength photon emitted may be associated with ion removal, either by a molecular recombination process or by reaction leading to molecular products.

Triplet—Triplet Energy Transfer in Rigid Glasses: Lack of a Solvent Effect
View Description Hide DescriptionExperiments were conducted to study the possible effects of the nature of the solvent in triplet—triplet energy transfer between aromatic molecules dissolved in rigid glasses. Electron paramagnetic and emission spectroscopy were used to monitor the energy‐transfer processes. Results of the study indicate that the probability of triplet energy transfer is essentially independent of the solvent used.

Influence of Resonance Transfer on Luminescence Decay
View Description Hide DescriptionThe effects of dipole—dipole resonance transfer on the population and decay of excited donor molecules are discussed. The limitation of the previous treatments of this problem to flash excitation is demonstrated. The present treatment is applicable to systems which have achieved a steady state or have been flashed. Expressions are given for the donor quantum yields, the decay of excited donor molecules for flashed and steady‐state systems, and the steady‐state population of excited donor molecules.

Single‐Configuration Calculations on Excited States of Helium
View Description Hide DescriptionThe best one‐configuration wavefunctions for several excited states of helium have been found. The error in the energy obtained varies from 0.5% to 1.0% of the binding energy of the outer electron. The method used is shown to give a rigorous upper bound to the excited‐state energy.

Primary Photochemical Processes in Glyoxal at 4358 Å
View Description Hide DescriptionQuantum yields of CO and of H_{2} have been measured in the gas‐phase photolysis with 4358‐Å radiation at temperatures between 37° and 207°C. φ_{CO} is generally about 0.6; φ_{H2 } is about 0.01. The data suggest that about 70% of the excited molecules cross to the triplet state by a first‐order process, and the remaining excited singlet molecules return to the ground state with fluorescence. The principal reaction of triplet molecules is a dissociation into CO and HCHO, while a small fraction of triplet molecules are destroyed by collision with ground‐state glyoxal molecules. About 3% of the triplet molecules undergo a first‐order intersystem crossing to the ground state, forming molecules with about 55 kcal of vibrational energy. These either dissociate into H_{2} and CO or relax to vibrational equilibrium by collision with heat‐bath molecules. At pressures near 100 Torr, where higher concentrations of triplet molecules are formed because of increased absorption, reactions of free radicals produced in collisions between triplet molecules also contribute significantly to φ_{H2 }. The data suggest that temperature variation has little effect on the reactions destroying the triplet state, but that the rate of vibrational relaxation in the ground state increases markedly at higher temperatures.

Hard‐Sphere Model for the Dissociation of Diatomic Molecules
View Description Hide DescriptionA simple classical model for the activation of diatomic molecules interacting with a heat bath of monatomic particles is proposed. The diatomic molecules are considered as hard spheres connected by springs and the monatomic particles as hard spheres. The model permits strongly nonadiabaticinteraction, and because of its simplicity it makes possible a detailed calculation, which takes into account the energy transfer to translational, rotational, and vibrational degrees of freedom. The only parameters in the theory are the gas‐kinetic hard‐sphere radii of the atoms and the experimentally known activation energy. Comparison with experiments on the dissociation of molecules as I_{2}, Br_{2}, O_{2}, and H_{2} in the inert gases shows an excellent numerical agreement, as well as a satisfactory dependence of the rate constant on the mass and temperature.

Hole Photoconductivity in Ferrocene, Triphenylamine, and Trans‐Stilbene
View Description Hide DescriptionThe drift mobility of holes in single crystals of ferrocene, triphenylamine, and trans‐stilbene has been measured by the method of transient photoconductivity. At 20°C the mobilities are: ferrocene, 2.2×10^{—2} cm^{2}/V·sec; triphenylamine, 2×10^{—2} cm^{2}/V·sec; and trans‐stilbene, 2.4 cm^{2}/V·sec. In trans‐stilbene, the hole mobility decreases slightly with increasing temperature in the range of 100° to —50°C. In ferrocene and triphenylamine, the mobility increases with increasing temperature, but not in a manner that suggests hole motion by a thermally activated, hopping‐type motion. The photoconductive excitation spectra in these solids follow the optical absorption of the exciton states built from the π and π^{*} states of the molecules.

Structure of Water and Hydrophobic Bonding in Proteins. IV. The Thermodynamic Properties of Liquid Deuterium Oxide
View Description Hide DescriptionA model, proposed earlier for the theoretical derivation of the thermodynamic parameters of liquid water, is shown to be applicable to liquid deuterium oxide as well. The model, based on the ``flickering cluster'' concept proposed by Frank and Wen, and the partition function were described in a previous publication. In the application to D_{2}O, the intermolecular vibrational frequencies are changed in accord with the theoretical ratios for isotopic substitution. The energy of the hydrogen bond is the only adjustable parameter; its value is 0.24 kcal/mole higher than for H_{2}O. The calculated structural parameters indicate that more structural order exists in D_{2}O than in H_{2}O at a given temperature. The calculated values of the free energy,enthalpy, and entropy of the liquid have an over‐all average error of 6%, as compared with experimental data, over the range 0° to 65°C. The deviation of c_{v} is of the same magnitude as for H_{2}O. The temperature dependence of the molar volume is reproduced with an error of less than 0.4%, and the temperature of maximum density is obtained within one degree of the experimental value. Excellent agreement is also obtained between the calculated and observed compressibility. The calculation of the PVT data involved no new assumptions. Calculated first peak areas for the radial distribution functions of H_{2}O and D_{2}O are shown to be almost identical.

Ionization‐Efficiency Curves for Molecular and Fragment Ions from Methane and the Methyl Halides
View Description Hide DescriptionThe retarding potential difference method has been used to measure ionization efficiency curves of CH_{3}X^{+} (X=H, F, Cl, Br, I), CH_{3} ^{+}, and X^{+} from methane and the methyl halides in a time‐of‐flight mass spectrometer.Discontinuities between linear segments of the ionization efficiency curves are interpreted as appearance potentials (A.P.) of electronic states or processes. For all CH_{3} ^{+}, A.P.'s occur at 1.1, 1.7, 2.7, and about 3.4 eV above onset in the ion pair region. Empirical knowledge of this structure makes it possible to identify the A.P. for the ion—neutral pair, CH_{3} ^{+}—X, in their ground states. Combining these data and the A.P.'s for X^{+}/CH_{3}X gives the ionization potential for CH_{3} and the bond dissociation energies,D(CH_{3}—X).

Signs of the Isotropic Hyperfine Splittings in the Electron Spin Resonance Spectrum of the Dihydropyrazine Cation Radical
View Description Hide DescriptionLinewidth studies of the electron spin resonance spectrum of the dihydropyrazine cation radical have been employed to determine the signs of the isotropic hyperfine splittings. It was found that the nitrogen splitting a ^{N} is positive and the proton splitting a _{NH} ^{H} of the protons bonded to the nitrogen atoms is negative. The sigma—pi parameter Q _{NH} ^{H} in McConnell's relation for NH bonds (a _{NH} ^{H}=Q _{NH} ^{H}ρN^{π}) was found to be negative.

Alternating Linewidths and Related Phenomena in the Electron Spin Resonance Spectra of Nitro‐Substituted Benzene Anions
View Description Hide DescriptionStudies of linewidths in the electron spin resonance spectra of a number of nitro‐substituted benzene anions are reported. The radicals were generated electrolytically in N,N‐dimethylformamide solutions, and spectra were obtained over a range from above to below room temperature. Pronounced alternations in the linewidths of the nitrogen lines in many of the spectra were observed. If M _{N} is the total z component of the nuclear spinangular momentum in compounds containing two equivalent nitrogen nuclei, the alternating linewidth phenomenon causes a broadening of the lines for which M _{N}=±1 and a reduction in the amplitude of the central line. Large alternating linewidths were observed in the room‐temperature spectra of the anions of dinitrodurene, dinitromesitylene, and 2,6‐dinitro‐3,5‐dimethyl‐4‐acetyl‐t‐butylbenzene; and also in a spectrum which shows splittings from two equivalent nitrogen nuclei that was obtained from trinitromesitylene. Slight alternating linewidth effects at room temperature that became more marked at low temperatures were found in the spectra of the 2,6‐dinitrotoluene and m‐dinitrobenzene anions. No linewidth alternations were detectable even at low temperatures in the spectra of the o‐dinitrobenzene, p‐dinitrobenzene, and 2,6‐dinitrophenolate anions. The 2,6‐dinitroaniline anion, which was investigated only at room temperature, showed no evidence of an alternating linewidth effect. None of the spectra of the mononitrobenzene anions examined exhibited any anomalous linewidth phenomena. The experimental observations are in general agreement with the recently developed theory of linewidths. The theory attributes the alternation in widths in these radicals to an out‐of‐phase correlation of a modulation of the isotropic hyperfine splittings of the two equivalent nitrogen nuclei. This out‐of‐phase modulation probably arises from either fluctuating solvent complexes with the nitro groups, or from internal rotations of these groups relative to the plane of the benzene ring, and both types of motion may occur simultaneously. Steric hindrance of the nitro groups was found to enhance the magnitude of the alternations in width, as did lowering of the temperature.