Volume 57, Issue 8, 15 October 1972

Unified Treatment of Two‐Center, Two‐Electron Integrals. I. Electron‐Repulsion, Spin‐Spin, and Spin‐Orbit Integrals Over Slater Functions
View Description Hide DescriptionUtilizing the Neumann expansions of the electron‐repulsion operator, all two‐electron two‐center spin‐spin, spin‐orbit, and electron‐repulsion, hybrid, Coulomb, and exchange integrals arising in computations of expectation values of the respective operators over wavefunctions written as linear combinations of Slater functions are expressed in terms of a simple formula requiring, effectively, a one‐dimensional numerical integration. A general formula for one‐center integrals is also derived by utilizing the Laplace expansion of the electron‐repulsion operator.

Excited‐State Dipole Moments, Polarizabilities, and Local Fields in Molecular Crystals; Polarizabilities of Benzophenone in its Lowest Singlet and Triplet nπ Excited States
View Description Hide DescriptionA refined calculation that accounts for molecular polarizabilities, crystal reaction fields and resulting field induced moments in polar molecular crystals allows excited‐state dipole moments and polarizabilities to be derived from optically determined Stark splitting versus applied electric field results. The determination of these excited‐state properties in a dipolar lattice is particularly dependent on the evaluation of the local field at a molecular site. For this case, the local field is evaluated by introduction of an ellipsoidal cavity approximation and by using the Ewald‐Kornfeld lattice summation method for dipolar arrays. The dipolar benzophenone crystal was chosen to demonstrate this method. The dipole moments of the lowest ^{1} nπ* and ^{3} nπ* states of benzophenone have been determined and are 1.23 D and 1.72 D, respectively; these are the zero field values and refer to the benzophenone molecules in the absence of the crystal environment. The polarizabilities of benzophenone in its ground, ^{1} nπ* and ^{3} nπ* states are, in units of 10^{−23} cm^{3}, 0.532, 0.898, and 0.56, respectively. The excited‐state moments are nearly parallel to the ground state moment in the crystal environment and the benzophenone dipole does not reverse direction in going from the ground to the lowest singlet and triplet states.

Some Theoretical Results for the Photochemical Decomposition of Large Molecules
View Description Hide DescriptionThe expressions describing photochemicaldissociation derived by Rice, McLaughlin, and Jortner are evaluated analytically for molecules satisfying and for times , where is the width of the initial state due to interaction with an intermediate manifold, ε is the level spacing of the manifold, and r is essentially the ratio of the manifold level widths to ε. Excitation is found to decay exponentially from the initial state with rate . In contrast to the behavior originally perdicted, the decay is found to be nonsequential, and a constant ratio, equal to r, is maintained between the populations of the continuum and the manifold. The results should be applicable to large molecules having a single decomposition mode.

Coupling of Optical Phonons to the Transition in p‐Dibromobenzene Crystals
View Description Hide DescriptionHigh resolution spectra of the transition in p‐dibromobenzene crystals have revealed that the coupling of optical phonons to the transition is much weaker than believed previously. By accounting for phonon coupling, an upper limit of about 15 cm^{−1} has been established for the total electronic band width of the T _{1} state. The zero phonon lines are very sharp and are more intense than the phonon addition bands. In fact, the zero phonon line of the (0, 0) transition is so sharp and intense that it can be observed in a crystal thick. The major difference between this work and previous work is the spectroscopic resolution. At lower resolution where the spectral linewidth is narrower than the slit width, crystals of several millimeters thickness are required to observe the transition, and the spectrum is distorted in that the broad phonon lines are relatively enhanced. Also, the previously reported temperature dependence of the zero phonon lines can be ascribed to line broadening and not to intensity enhancement with increasing temperature. The order of magnitude increase in sensitivity obtained by going to high resolution may be a quite general phenomenon for triplet states of aromatic molecules making them much more accessible to spectroscopic study than believed previously.

Lattice Model for Biaxial Liquid Crystals
View Description Hide DescriptionThe possibility that the repulsive interaction between platelike molecules might give rise to biaxial liquid crystal order is examined by considering a molecular‐field lattice model in which molecules are represented by rectangular plates. We find that plates which are neither very square nor very rodlike in shape may exist in a phase characterized by biaxial long‐range order in the orientation of the plates. This biaxial phase occurs at high pressure. At some lower pressure the biaxial phase undergoes a second order phase transition to a uniaxial phase. At a still lower pressure the uniaxial phase undergoes a transition to an isotropic phase. This last phase transition is usually first order in that there are discontinuities in the density and the orientational order in going through the transition. However the size of the discontinuities depend strongly on the shape of the plates. For example for plates of length 6 and width approximately 2.22 and thickness 1, the discontinuities are zero, and the transition is accidentally second order. The discontinuities at this uniaxial‐isotropic transition are always very small for plates which have the potential to give rise to a biaxial phase. This last result is shown to have a direct physical basis which should transcend the limitations imposed by the approximate nature of our calculation. Thus we conclude that for real substances likely to have biaxial phases of the type we have studied, the value of the nematic order parameter at the nematic‐isotropic transition should be much lower than for other nematic liquid crystals.

Vibrational Spectrum and Force Field of Xenon Tetroxide
View Description Hide DescriptionInfrared and Raman spectra are reported for Xe^{16}O_{4} and Xe^{18}O_{4} vapor. The infrared band contours are analyzed to give values for the Coriolis constants and the xenon isotopic frequency shift, and anharmonicity corrections are estimated for the fundamental frequencies. General quadratic force constants are calculated, using as constraints the and frequency shifts and the Coriolis constants. The force field is significantly different from those of RuO_{4} and OsO_{4}. Valence stretching force constants are and the negative value of the interaction constant f_{rr} indicates that the bonding differs appreciably from that in the transition metal tetroxides. Standard‐state thermodynamic functions for XeO_{4} vapor are calculated for the temperature range 250–400°K.

Theoretical Interpretation of the Optical and Electron Scattering Spectra of Polyatomic Molecules. III. N_{2}O and the Discovery of Resonant Phenomena in the B Region at 6.8 eV
View Description Hide DescriptionEnergies and parts of potential surfaces are calculated for the first eleven excited electronic states of nitrous oxide; the states within ten electron volts of the ground state. The assignment of the energies and symmetries of these states which is carried out with the aid of semiempirical INDO calculations, is shown to be consistent with all available optical absorptionspectra,electron scattering data, and with photolysis and photosensitization experiments. The weak, diffuse vibrational bands in the 6.8 eV B region have been interpreted as resonant interaction between the continuum levels of the 1 ^{1}II state and bound vibrational levels of the underlying ^{1}Σ^{−} state. The mixing is made electronically allowed through bending in the excited states. Supporting experimental and theoretical evidence for this assignment are given, and experiments are suggested to confirm several of the other assignments made.

Measurement of the Quadrupole Coupling Constant and Asymmetry Parameter of the Ring Deuterons in Several Deuterated Benzene Derivatives
View Description Hide DescriptionMeasurements are reported of the magnitude and asymmetry of the ring deuteron quadrupole coupling tensor in several simple deuterated benzene derivatives, i.e., benzene, toluene, ethylbenzene, as well as naphthalene, pyridine, and cyclohexane, using polycrystalline specimens at 77°K and in a magnetic field strength of 24 kOe. The e ^{2} qQ/h and η values obtained are: benzene, ; toluene, ; ethylbenzene, ; pyridine, ; cyclohexane, ; naphthalene, .

Formation of Calcium Ions by Charge Transfer
View Description Hide DescriptionCharge transfer cross sections have been measured in the energy range from 1 to 500 eV for nine common atmospheric ions in collision with neutral calcium atoms. The ions studied are O^{+}, N^{+}, , NO^{+}, , H_{2}O^{+}, H_{3}O^{+}, N_{2}O^{+}, and Ca^{+}. In all cases the cross sections are found to be large indicating near resonant processes. Mechanisms consistent with this observation are given. Extrapolation of the measured cross sections to thermal energies has also been carried out.

Formation of Iron Ions by Charge Transfer
View Description Hide DescriptionReactions of iron atoms with common atmospheric ions have been studied in the energy range from 2 to 500 eV. The ions considered are H^{+}, O^{+}, N^{+}, , NO^{+}, , H_{2}O^{+}, and H_{3}O^{+}. With the exception of one proton transferreaction, all of the processes studied involved charge transfer. All but one of the charge transferreactions were found to have large cross sections over the total energy range and probably proceed by near resonant energy paths. Where possible the measured cross sections have been extrapolated to thermal energies.

Electron‐Spin Resonance Studies of Crystal Field Parameters in Mn^{2+}: LiNbO_{3}
View Description Hide DescriptionESR studies were performed on a single crystal of ferroelectric LiNbO_{3}doped with manganese. Mn^{2+} in an axially symmetric site is described by the spin Hamiltonian. Values for reported here, have not been reported previously. By examining the temperature dependence of and a value for the fractional change in the optical parameter has been calculated. A discussion of the temperature dependence of is also given.

Crossed Beam Collision Mechanics: Reactions of Ca, Sr, and Ba with HI and Limits on for CaI, SrI, and BaI
View Description Hide DescriptionAngular distributions of MI (M=Ca, Sr, or Ba) products scattered from crossed beams of HI and M are reported and compared with derived expressions for the angular distributions of the velocities of the center‐of‐mass. These comparisons indicate that the reaction threshold relative kinetic energies, E*, are 5, 4, and 2.5 kcal/mole for the Ca, Sr, and reactions, respectively. Energy conservation and these measured E* values establish rigorous lower bounds for of 64 (CaI), 65 (SrI), and 66 (BaI) kcal/mole.

Comparison of Calculated Chain‐Length Dependence of the Anisotropy and Depolarization Ratio in n‐Alkane Gases with Measurements of the Laser Light Scattering by Photon Counting
View Description Hide DescriptionTheoretical optical anisotropy and depolarization ratio calculations of n‐alkanes are critically reviewed. New depolarization ratio measurements on gaseous n‐alkanes are reported from a new method employing a continuously rotating analyzer, an argon ion laser, and photon countingelectronics. The experimental and theoretical depolarization ratios are compared. The experiments show an increase of depolarization ratio with increasing number of bonds in contrast to the theoretical results which show a general decrease of depolarization ratio. A new formula for depolarization ratio calculation of ethane which is a nonlinear symmetric top molecule is suggested. This formula reduces ethane's theoretical depolarization ratio by 25%. Our measurements suggest nonzero depolarization ratios for the spherical molecules methane and neopentane in contrast to the theoretical predictions. Possible sources of errors are discussed.

Analysis of the Internal Rotation of the Radical as Observed by ESR in Irradiated Trifluoroacetamide Single Crystals
View Description Hide DescriptionThe temperature‐dependent ESR spectrum of the (found in x‐irradiated single crystals of trifluoroacetamide) has been measured at both 9 and 35 GHz over the temperature range from 77 to 290°K. A model is proposed which explains the observed temperature dependence in terms of a torsional oscillatory motion about the bond with a barrier height, V _{0}=3100 cal. Nonadiabatic spin exchange is used to account for the doublet splitting of the central line at 77°K changing to a single sharp line at all angles at 290°K. The exchange rate is described by the empirical equation .

Photoionization of N_{2}, O_{2}, NO, CO, and CO_{2} by Soft X Rays
View Description Hide DescriptionThe total and relative abundances of thermal energy ions produced by absorption of 0.28, 1.25, and 1.5 keV x rays in N_{2}, O_{2}, NO, CO, and CO_{2} and the dissociativeionization cross sections at these energies were measured using a mass‐spectrometric technique. The results demonstrate that there is a significant change in the character of the ionization at the K‐absorption edges of the atomic constituents. At these energiesdissociativeionization predominates for all the molecules accounting for over 95% of the ions formed at energies above the K edge. Quasistable doubly charged molecular ions make a significant contribution to the thermal energy ions at energies above the K edge. Our results for the total abundances of thermal singly and doubly charged molecular ions are compared with recent measurements by Carlson and Krause.

Son et Lumière or the Optoacoustic Effect in Multilevel Systems
View Description Hide DescriptionThe theory of the periodically driven as well as of the pulse spectrophone is given for a gas with an arbitrary number of collisional excitation and de‐excitation reactions. The basic kinetic quantities are the rate constants which provide the most simple means of comparison with other experimental methods. No use is made of the methods of irreversible thermodynamics, but the approach is closely related to that theoretical concept. A matrix notation is used throughout since it provides the least complicated expressions and so eliminates possible sources of algebraic errors as well as programming errors in the case of numerical evaluations. Compared to the two‐level theory no additional assumptions are made. A link to the wealth of theoreticalinformation available for the acoustic relaxation experiment is installed, especially for the case of mixtures of harmonic oscillators. The results are compared to those obtained by other means and examples are given.

Reorientation and Vibrational Relaxation as Line Broadening Factors in Vibrational Spectroscopy
View Description Hide DescriptionThe theory of Gordon which relates the shape of the depolarized components of certain Raman bands to a molecular reorientation correlation function is extended to a number of types of molecular vibrations not previously covered. In addition, the contribution to the line shape for vibrational relaxation and reorientation is obtained for the three types of vibrational spectroscopy, infrared, Raman, and hyper‐Raman, and it is shown that the form of the vibrational relaxation contribution is the same for each spectroscopy. Conditions and assumptions are given under which the line broadening contributions of reorientation and vibrational relaxation can be separated and experimentally determined.

On Moment Conserving Decoupling Techniques for Electron Propagators
View Description Hide DescriptionMoment‐conserving decoupling procedures for the one‐electron Green's function (electron propagator) are investigated. By analysis of one particular system (Hubbard model) it becomes clear that the moment‐conserving techniques so far proposed for interacting systems are all special cases of [1, 0], [2, 1], or [3, 2] Padé approximants differing only in which terms they omit, how they define self‐consistency, and which operator algebra they utilize. The number of formally conserved moments appears not to be a particularly useful criterion of the accuracy of a decoupling procedure, since a two‐moment scheme (Hartree‐Fock) may be numerically closer to the exact answer than some three‐moment procedures. Use of the complete Padé approximant, as suggested by Lukman and Goscinski for the particle‐hole propagator, appears to be the most consistent method.

Vibrational Relaxation of Hydrogen
View Description Hide DescriptionThe rates of vibrational relaxation of D_{2} and HD have been studied by laser‐induced vibrational fluorescence technique. The relaxation times of D_{2}/HCl, HD/HCl, D_{2}/H_{2}/HCl, and D_{2}/HD/HCl mixtures at 296°K show cross sections of , R energy transfer asCross section for energy transfer from HCl to HD asComparison of vibrational relaxation rates among hydrogen isotopes and direct correlation between high temperature shock tube results and room temperature data reveal the importance of both rotational and translational motion in the process of vibrational energy transfer of hydrogen.

Trnasition Dipole Moments in Extreme Renner Effect Molecules, with Application to the Visible Bands in CH_{2}, NH_{2}, and BH_{2}
View Description Hide DescriptionThe theoretical structure of transitiondipole moments in extreme Renner effect molecules is considered. Pople and Longuet‐Higgins vibronic wavefunctions, which explicitly correlate nuclear and electronic motion, are employed. Constructive and destructive interference of wavefunctions causes the transition moment to vary, as a function of K and Λ, between the molecular b̂ and ĉ axes. A few unassigned lines in the spectrum are tentatively identified as type btransitions. It is concluded that in the dihydride radicals CH_{2}, NH_{2}, and BH_{2}, l is decoupled from Λ at the large bending angle where transitions occur.