Volume 40, Issue 3, 01 February 1964

Off—Diagonal Hypervirial Theorem and Its Applications
View Description Hide DescriptionA hypervirial operator, by definition, is a time‐independent linear operator with an arbitrary functional structure expressed in terms of the dynamical variables of the system under consideration. In the energy representation, the diagonal matrix elements of the hypervirial operator are constant in time; this is known as the hypervirial theorem. The off‐diagonal matrix elements of the hypervirial operator in the energy representation oscillate in time with frequencies related to the energy differences between their corresponding stationary states of the system. This is known as the off‐diagonal hypervirial theorem.
The latter theorem is as powerful as the former for dealing with many quantum‐mechanical problems. Several aspects of the possible applications of the theorem to the problem of construction and improvement of wavefunctions are considered. As a special example, the restricted fourth alternative expression for dipole transition matrix elements is derived for pair functions not belonging to states with zero orbital angular momentum. A scheme is also outlined for improving wavefunctions of the excited states. The statistical implications of the sum rule, which is expressed in terms of an appropriate hypervirial operator, are discussed in connection with uncertainty principles.

Many—Body Aspects of Intermolecular Forces
View Description Hide DescriptionThe many‐body aspects of intermolecular forces are examined for the model based on a fluctuating reaction field. This is accomplished by expanding the reaction field in a series involving optical and translational parameters, and an expression is obtained for the cohesive free energy in terms of frequency‐dependent polarizabilities and dipole‐coupling tensors. When applied to a system of fixed translational parameters a series expansion is obtained for the configurational total potential energy, the leading term of which arises from pair interactions, the next higher term from triple interactions, etc. It is shown that the pair and triple potentials reduce to forms identical with the ones obtained by ordinary quantum‐mechanical second‐ and third‐order perturbation theory. The significance of the results is discussed.

Theory of the Diffuse Double Layer in Liquids
View Description Hide DescriptionThe theory of the diffuse double layer in liquids, as developed by Jaffe for plane, parallel, blocking electrodes, is extended to include the ionic charges as boundary conditions and the effects of a surplus of one ion. Dissociation and recombination effects are included. The results are expressed in terms of elliptic integrals and Jacobian elliptic functions. It is found that the electrically neutral condition can be divided into three regions designated field limited dielectrics, charge limited dielectrics, and electrolytes. Approximations of the exact expressions are developed for each region.

On the Relaxation of the Hard—Sphere Rayleigh and Lorentz Gas
View Description Hide DescriptionAs part of a study of the relaxation of nonequilibrium systems, the (translational) relaxation of a hardsphere Rayleigh and Lorentz gas is investigated. From a detailed analysis of the collision dynamics an exact expression is derived for the kernel A (x  x′) of the collision integral which gives the probability per unit time for a change of the reduced kinetic energy from x′ to x during a binary collision between a subsystem and a heat bath particle. A master equation, i.e., a linearized Boltzmann equation, incorporating this kernel is then formulated to represent the time variation of the distribution function of the subsystem particles. Making use of the special property of this kernel that it is a strongly peaked function around x—x′=0 for both the Rayleigh and Lorentz gas, a technique is developed for transforming this integral master equation into differential Fokker—Planck equations consistent in the order of the expansion parameter λ, the ratio of the mass of the heat bath particles to the subsystem particles. The Fokker—Planck equation for the Rayleigh gas is solved analytically and explicit solutions are presented for the relaxation of initial Maxwell and initial δ‐function distributions of the energy. It is shown that an initial Maxwell distribution of the energy (or speed) of the subsystem particles relaxes to the final equilibrium Maxwell distribution via a continuous sequence of Maxwell distributions. The mean energy of the subsystem particles is shown to relax exponentially to its equilibrium value independent of the form of the initial distribution. For the hard‐sphere Lorentz gas the Fokker—Planck equation is not susceptible of an analytical solution. Machine solutions are presented for various initial distributions which show that the Maxwell distribution is not preserved in the relaxation of the hard sphere Lorentz gas. Finally, a brief discussion is given of the relation between the hard‐sphere model (r ^{—∞}) considered here and the more general model of the Rayleigh and Lorentz gas with a r ^{—8} repulsive central force law.

Mass Spectrometric Measurement of Diffusion Coefficients
View Description Hide DescriptionBinary diffusion coefficients of CH_{4}–CO, CH_{4}–N_{2}, CH_{4}–CF_{4}, CH_{2}CH_{2}–N_{2}, CH_{3}CH_{3}–CH_{2}CH_{2}, CH_{3}CH_{3}–CHCH, CH_{2}CH_{2}–CH_{3}CH_{3} have been measured at 298.2°, 353.6°, and 382.6°K. By means of isotopic substitution, the self‐diffusion coefficients of CH_{4}, CH_{3}CH_{3}, CH_{2}CH_{2}, and CHCH have been determined at the same temperatures. In general, the depth of the potential minimum, ε, was less, when calculated from diffusionmeasurements, than when calculated from virial coefficients or viscosity measurements.

Quantum Calculation of the Sensitivity of Diffusion, Viscosity, and Scattering Experiments to the Intermolecular Potential
View Description Hide DescriptionCorrelations between the form of the intermolecular potential as a function of distance and the quantum mechanical phase shifts as functions of the angular momentum quantum number have served as the basis for estimates of the sensitivity of viscosity,diffusion, and low‐energy scattering experiments to the attractive and repulsive portions of the intermolecular potential. The sensitivities of the experimental techniques are approximately as follows: repulsive portion of the potential, diffusion>viscosity>scattering; to the attractive region of the potential, scattering>viscosity=diffusion.
Quantum diffusion and viscosity cross sections for the Lennard‐Jones potential have been calculated as functions of the reduced kinetic energy, K=μv ^{2}/2ε, and for decreasing values of the quantum parameter λ=h/σ(mε)^{½}. The cross sections for systems with λ corresponding to T_{2}, Ne, and CH_{4} are equal to the classical cross sections for K>1.2. At lower K's the deviations from classical behavior become larger as λ increases.

Model for the Lattice Dynamics of Metals. II. Application to Face‐Centered Cubic Metal Copper
View Description Hide DescriptionA model for the lattice dynamics of metals propounded by us earlier is applied to copper, a representative of face‐centered cubic metals. The force constants appearing in the secular equation for the lattice frequencies are estimated from the knowledge of the elastic constants. The phonon dispersion curves for the three symmetry directions [100], [110], and [111] are presented and are found to be in fair agreement with the neutronspectroscopic measurements. The frequency distribution function has been calculated. The specific heat computed from the frequency distribution function is found to be in good agreement with the experimental measurements.

Time‐Dependent Perturbation‐Variation Method. II
View Description Hide DescriptionIt is shown that the time‐dependent perturbation‐variation method recently developed by the author and that previously developed by Karplus and Kolker are equivalent.

First‐Order Dispersion Forces
View Description Hide DescriptionThe interaction energy between two atoms, one of which is excited, is investigated. The form of the interaction depends on the relation between the distance R separating the atoms and the wavelength k _{1} ^{—1} of the transition from the ground state to the excited state. If k _{1} R<1 the photon may be trapped between the two atoms leading to the well‐known dipole—dipole interaction. If k _{1} R>1 the interaction is small, the state of the interacting atoms is not a stationary state and the photon is emitted after a time determined by the spontaneous emission probability.

Optical Activity of Coordination Compounds. II. A Molecular Orbital Model and an Analysis of Experimental Data for Complexes of Trigonal Symmetry
View Description Hide DescriptionA molecular orbital model for the optical activity of trigonal coordination compounds is developed. It is shown that the combination of intensity data from solution and crystal spectra with optical activity data from solution allows one to calculate empirical values of the electric moment matrix elements for the one‐electron transition t _{2}→e. This empirical electric moment matrix is in good agreement with that predicted by the molecular‐orbital model, and it is in nearly as good agreement with that of the ionic model as far as the ratios of the matrix elements are concerned. Empirical values of the magnetic moment of the electronic transitions are interpreted qualitatively in terms of metal—ligand covalency. The problem of determining the absolute configuration from the sign of the rotational strength is discussed.

Surface‐Limited Vapor Solvent Growth of Crystals
View Description Hide DescriptionA general analysis is presented of surface reaction limitations in a vapor solvent system. A previous treatment of a multicomponent multireaction vapor diffusion limited system is extended to include finite supersaturations in the neighborhoods of the growing deposit and the etching source. The appropriate natural definitions of supersaturations are found to be in agreement with the usual conventions of irreversible thermodynamics.
A more detailed connection between the supersaturations above a growing crystal and the growth rate of the crystal is developed by means of an extension of the crystal growth theory of Burton, Cabrera, and Frank to apply to a multiple species multireaction vapor solvent system. It is found, at least in the limit of low supersaturations, that the flux of any particular species to the surface may be treated in terms of the formalism developed by Burton, Cabrera, and Frank for a monatomic crystal—monatomic vapor system, except that some of the parameters which characterize the process, e.g., diffusion length on the surface, lifetime on the surface, etc., have more phenomenological character. Relationships between the fluxes of different species are developed in the same manner as in the previously treated vapor diffusion limited system.
Quantitative calculation of the important parameters is apparently impossible in our present state of knowledge of surface processes. Qualitatively, we expect surfacediffusion effects to be far more important than in simple crystal—vapor systems. Other effects, however, are also expected to be important in vapor solvent systems, e.g., ``kink‐reaction'' limitations, impurity effects, etc. Disentanglement of these several processes is an unsolved problem and it is concluded that detailed quantitative understanding of surface‐limited crystal growth in a vapor solvent system is presently out of reach.

Solvation Approach^{*} to Ion Solvent Interaction
View Description Hide DescriptionFrom the hard‐sphere model proposed, i.e., that the ion and its solvation layer are incompressible, and from density and compressibility data, the variation of solvation number with concentration, the intrinsic volume of ions, and the mean distance of approach are obtained. These properties of the ions, and the molar refractions, the dipole moment of the ion pair, and the viscosity, are correlated via the solvated molar volume φ _{8}.

Absorption Spectrum of the ``Pink'' Afterglow of Nitrogen in the Vacuum Ultraviolet
View Description Hide DescriptionThe absorptionspectrum of the short‐lived ``pink'' afterglow in nitrogen was photographed in the vacuum ultraviolet from 1600 to 1150 Å. A large number of relatively intense absorption bands were observed. It is suggested that these may result from transitions between the ground state (X ^{1}Σ_{ g } ^{+}) and several excited electronic states (b ^{1}II_{ u }, b′ ^{1}Σ_{ u } ^{+}, and others) in which ground‐state vibrational levels of the ground electronic state are populated to very high levels (up to about v″=20). It appears that more than 10% of the nitrogen molecules are excited to vibrational levels greater than v″=8.

Spectra Excited in a Helium Afterglow
View Description Hide DescriptionThe selective excitation of spectra by collisions of the second kind with active helium species was examined in a flowing helium afterglow. Techniques for titrating for the active species in the afterglow are described. Conditions under which the metastable atom He(2^{3} S), the atomic ion He^{+}, or the molecular ion He_{2} ^{+} predominate in the burner are found and the spectra excited by each are listed.

Use of Field‐Sweep Double Resonance in the Unambiguous Analysis of Three‐Spin Spectra
View Description Hide DescriptionA systematic procedure for uniquely determining NMR parameters from a spectrum is applied to the case of three coupled spins, and several examples are treated. The analysis involves three steps: (1) use of frequency sum rules to determine consistent assignments; (2) removal of ambiguities in assignment by field‐sweep double resonance; and (3) iterative refinement of arbitrary initial parameters.

Metastable State of the Doubly Charged Carbon Dioxide Ion
View Description Hide DescriptionA metastable state in CO_{2} ^{++} dissociating into CO^{+}+O^{+} with a half‐life of 2.3±0.2 μsec has been observed. The corresponding metastable peaks are observed at (M _{1}/q _{1})^{*}=35.6 and (M _{2}/q _{2})^{*}=11.6, respectively. The widening of these peaks with accelerating voltage shows the repulsion energy to be 2.7 eV and the corresponding charge separation at fragmentation to be 5.3 Å. The appearance potential of CO_{2} ^{++} was found to be 38.0±0.2 eV, and this value is discussed in terms of the various repulsion energies and energy balances. The data is consistent with the hypothesis of an immediate fragmentation into CO^{+}(^{2}Σ) and O^{+} and the metastable fragmentation into CO^{+}(^{2}II_{ i }) and O^{+}. The fragmentation of CO_{2} ^{++} enriched in O^{18} has also been reinvestigated. It is shown that the unsymmetrical doubly charged molecule ion (CO^{16}O^{18++}) has a lower yield than the corresponding symmetrical molecule ions containing O^{16} and O^{18}, respectively.

Electron Spin Resonance of the Acetophenone and Benzaldehyde Anion Radicals
View Description Hide DescriptionThe electron spin resonance spectra of radicals obtained from acetophenone, benzaldehyde, and 4‐fluoroacetophenone were found in a previous investigation to be inconsistent with those of other carbonyl‐anion radicals. These compounds have been reinvestigated using new techniques for the electrolytic generation of the radicals. For the first two substances, unstable radicals were detected with spectra that were different from those previously obtained. The new spectra are completely consistent with the results for the other related radicals, and also show that the carbonyl groups are locked into the plane of the aromatic ring. The nature of the radicals investigated previously has not been ascertained. A new spectrum was not obtained for the 4‐fluoroacetophenone radical under the new conditions employed for the other two radicals presumably because this species is too unstable. A preliminary interpretation has been given to linewidth variations found in the spectra.

Nuclear Magnetic Resonance and Molecular Field Approximation in KCoF_{3}
View Description Hide DescriptionA molecular field calculation and a nuclear magnetic resonance study of F^{19} in paramagnetic KCoF_{3} are reported. The isotropic hyperfine interaction indicates presence of 0.57% unpaired 2s spins in F^{—} orbitals from each Co^{++} neighbor; this result is consistent with the data of Hall et al. and the data of Shulman et al.. The calculations are shown to be in good agreement with the F^{19} resonance data and also the susceptibility data of Hirakawa and co‐workers.

Oxygen‐17 Nuclear Magnetic Resonance in Paramagnetic MnO and CoO
View Description Hide DescriptionA nuclear magnetic resonance study of O^{17}resonance in paramagnetic MnO and CoO is reported. The internal fields at the oxygen nuclei were measured. Interpretation of the measured fields showed that the spin densities in the oxygen 2s orbitals were 0.76% for MnO and 0.66% for CoO. Nuclear relaxation times have also been measured and compared with Moriya's theory.

Effect of Light Molecules on Vibrational Relaxation in Oxygen
View Description Hide DescriptionSound absorption and velocity measurements have been made in O_{2} with added amounts of H_{2}, D_{2}, and He. The number of collisions Z _{10}(AB) necessary to de‐excite the first vibrational state of O_{2} has been calculated from the measuredrelaxation times. At 100°C: Z _{10}(O_{2}–H_{2})=2.0×10^{4}, Z _{10}(O_{2}—D_{2})=2.5×10^{5}. At 200°C: Z _{10}(O_{2}–H_{2})=1.1×10^{4}, Z _{10}(O_{2}—D_{2})=1.5×10^{5}, Z _{10}(O_{2}–He)=6.8×10^{4}. At 300°C: Z _{10}(O_{2}–He)=3.6×10^{4}. These values are in general agreement with values reported by Parker; by Holmes, Smith, and Tempest; and by White and Millikan. When logZ _{10}(AB) is plotted vs the cube root of the reduced mass times the square of the vibrational frequency divided by the absolute temperature, a linear relationship is found between the O_{2}–H_{2} and the O_{2}–He data. However Z _{10}(O_{2}—D_{2}) is unexpectedly larger than Z _{10}(O_{2}–He). When the collision numbers Z _{10}, for pure O_{2} and N_{2}, are plotted in this way they fall along a single curve.