Volume 43, Issue 4, 15 August 1965

Variational Treatment of the HeH^{+} Ion and the β‐Decay in HT
View Description Hide DescriptionThe nonrelativistic energies of the HeH^{+} ion have been computed by the variational method for the ground state and for the first excited ^{1}Σ^{+} state. The wavefunction employed was a 64‐term generalized James—Coolidge expansion. The computed binding energy of the ground state amounts to D_{e} =16448.3 cm^{−1} at the equilibrium separation R_{e} =1.46324 a.u. The excited state is found to be repulsive.
The HeH^{+}wavefunctions are used to compare quantitatively the theory of β‐decay in molecules with experiment. It is shown that for the HT molecule, the theory predicts the dissociation in at least 17.8% of the events while the corresponding experimental number is 6.8%±1.9%.

Guillemin—Zener Energy of H_{2} ^{+}
View Description Hide DescriptionThe energy of the H_{2} ^{+}ground state 1sσ calculated with the Guillemin—Zener approximate wavefunction, lies within 0.0005 e ^{2}/a _{0} of the exact for the full range of internuclear separations. In the limit of zero separation, . The energy still lies within 0.0010 e ^{2}/a _{0} when the additional requirement is imposed that Hψ_{ s } remains finite at the nuclei. The Guillemin—Zener energy is less accurate for the first excited 2pσ state.

Perturbation Treatment of the Ground State of H_{2} ^{+}
View Description Hide DescriptionRayleigh—Schrödinger perturbation calculations are carried out for the ground state of the H_{2} ^{+} molecular ion. The zeroth‐order wavefunction for this problem is chosen to be the Guillemin and Zener variational functionThe energies through third order and the expectation value E(1) of the Hamiltonian calculated with the wavefunction accurate through first order are computed for various choices of the parameters. For the optimum choices of both α and β, we obtain for all separations values of E(1) which agree almost perfectly with the accurate series calculations of Bates, Ledsham, and Stewart. Also the values of the wavefunction accurate through the first order agree almost perfectly at most points in configuration space with Bates et al.

Perturbation Treatment of the Ground State of HeH^{+ +}
View Description Hide DescriptionRayleigh—Schrödinger perturbation calculations are carried out for the ground state of the HeH^{+ +} molecular ion. The zeroth‐order wavefunction isThe expectation value of the Hamiltonian calculated with the wavefunction accurate through first order, E(1), is compared with the accurate series calculations of Bates and Carson and found to agree up to five places. Also the values of the normalized wavefunction accurate through the first order agree quite well with Bates and Carson, particularly for small values of the internuclear separation.

Multiple Elastic Intramolecular Scattering in Gas Electron Diffraction
View Description Hide DescriptionThe effects of multiple elastic intramolecular scattering on gas‐electron‐diffraction patterns are calculated in the second Born approximation. Expressions for the first‐order double‐ and triple‐scattering contributions are obtained. An approximate expression convenient for computation, for the first‐order triple‐scattering term is also given. The relationships of the magnitude of the multiple scattering to atomic number, atomic size, and molecular structure are investigated. It is shown that corrections to the reduced molecular‐intensity function of the order of several percent can occur in the case of triple scattering from three or more heavy atoms in close proximity.

On the Evaluation of the Thermal‐Diffusion Coefficient of Heavy Particles Using a Theory of Brownian Motion in a Nonuniform Medium
View Description Hide DescriptionAn extension of the theory of Brownian motion of heavy particles in a condensed medium of light particles is presented, which accounts for deviations of the state of the medium from equilibrium. The case of media under a thermal gradient is especially considered and the coefficients of diffusion and thermal diffusion of the heavy component are identified by solving the generalized kinetic equation for the Brownian particle distribution. The first terms of a systematic expansion of the thermal‐diffusion coefficient in powers of the mass ratio are explicitly given, and their sign is found by an approximate further evaluation. It is suggested that the thermal‐diffusion coefficient follows an asymptotic saturation law as far as the isotopic effect is concerned.

Exact Solutions for Many‐Level Multiple‐Resonance Problems
View Description Hide DescriptionThe matrix elements of the time‐development operator are obtained exactly for the influence of many resonant or near‐resonant fields on a system with a finite number of levels for arbitrary initial conditions in a wide class of problems. Perturbative techniques are presented which enable the exact treatment of all perturbations to a desired order. As examples, a three‐level double‐resonance and two eight‐level triple‐resonance systems are given. The applications of these exact solutions to the treatment of perturbations, including relaxation phenomena, are presented using both standard‐perturbation‐theory and density‐matrix techniques.

Theory of the Linear Heisenberg Antiferromagnet. Application to Paramagnetic Excitations in Organic Crystals
View Description Hide DescriptionWe consider a linear array of tightly bound electrons that are exchange coupled to their neighbors. The Hamiltonian, in the Heisenberg model, iswhere δ is the alternation parameter and the exchange integral J is positive, corresponding to antiferromagnetic coupling. We investigate the solutions for the one‐dimensional antiferromagnet, for any δ≤1, by transforming the Hamiltonian first to Fermi creation and annihilation operators and then to pseudospin operators similar to the ones used by Anderson in connection with superconductivity. We thus take into account the coherence between antiparallel electrons in a self‐consistent manner. We exploit the similarity between the linear antiferromagnet and superconductivity theory to obtain the ground‐state energy and the excitation spectrum for arbitrary δ and temperature. We thus obtain a complete thermodynamic description. In particular, we calculate the paramagnetic susceptibility and compare it with experiment.
At 0°K, the solutions we present agree with or improve upon previous calculations, for either the regular (δ=0) or the alternating (δ≠0) antiferromagnet, of the ground‐state energy, the excitation spectrum, and the short‐range order. Using the finite temperature extension of the theory, we find good agreement between the calculated and the experimental paramagnetic susceptibilities of organic crystals whose structures, so far as spin properties are concerned, may be approximated by either the regular or the alternating antiferromagnet. The temperature dependence of the excitation energies, a many‐body effect describing the decrease of coherence between antiparallel electrons with increasing temperature, agrees with the observed behavior of the singlet—triplet energy gap in paramagneticexciton systems. Thus, we obtain the extension of excitontheory to arbitrary temperature and alternation. The electron pairs forming triplet excitations are, for any alternation, increasingly separated at high temperatures, where, as expected, the electrons behave like a paramagnetic gas.

Internal Energy of Reaction Products by Velocity Analysis. II. Scattered KBr^{*} from the Crossed Molecular Beam Reaction K+Br_{2}
View Description Hide DescriptionThe velocity distribution of excited KBr^{*} formed in reactive collision between crossed beams of velocity‐selected K and thermal Br_{2} has been measured at several laboratory scattering angles and initial relative kinetic energies of 1.26, 2.14, and 3.27 kcal/mole. From these observations qualitative information may be inferred regarding the internal energydistribution function and the angular distribution of reactive scattering in the center‐of‐mass (c.m.) system. It is found that (1) the KBr^{*} internal energy distribution is broad, extending over a range of at least 30 kcal/mole, with most of the 46 kcal/mole exothermicity of the reaction appearing in the form of internal excitation of KBr^{*}, and (2) the angular distribution of reactive scattering in the ``forward'' direction (KBr preferring the direction of the incident K in the c.m. system), is characterized by a cone in the c.m. with an angle of generation of at least 30°.

Paramagnetic Resonance of Trivalent ^{57}Fe in Double Nitrate Crystals
View Description Hide DescriptionThe paramagnetic resonance absorption spectra of trivalent iron in double nitrate crystals have been identified and fitted by two spin Hamiltonians which correspond to the two sites into which the iron enters these crystals. The two sites appear to be the same as those occupied by trivalent chromium ions. The spin‐Hamiltonian constants are reported for lanthanumzinc and ceriumzinc double nitrates at temperatures of 4.2°, 77°, and 193°K. The hyperfine interaction constant for ^{57}Fe in either site is (10.9±0.3)×10^{−4} cm^{−1} which is very close to the value anticipated for a hydrated complex.

Crystal Structure of Tellurium at High Pressures
View Description Hide DescriptionPowder x‐ray‐diffraction studies on tellurium up to 150 kbar show only two structural transitions at 40 and 70 kbar in agreement with Bridgman's work. The structure of the 40–70‐kbar phase is unsolved. Above 70 kbar, it is proposed that tellurium is isomorphous with β‐polonium and is rhombohedral with a=3.002±15 Å, α=103.3°±3° (P=115 kbar), space group with an atom in 1a (0, 0, 0).

Luminescence of Rare‐Earth‐Activated Zinc Sulfide
View Description Hide DescriptionThe line structure of the photoluminescence of ZnS:Tb, ZnS:Nd, ZnS:Tm, and ZnS:Dy has been studied under various preparative conditions. There is a strong tendency for rare‐earth pairing with other lattice defects in the presence of excess S. This pairing is manifested by the appearance of new emission lines and excitation bands. Of the many distinct sites for the rare‐earth ion, one seems to stand out as most important in electroluminescence, although it gives only a minor contribution to the photoluminescence. Using a variety of luminescent phenomena, we are able to count at least four distinct sites for trivalent rare‐earth ions in ZnS, and there is strong evidence that more exist.

Shear‐Relaxation Processes in Liquids
View Description Hide DescriptionBoth cooperative behavior, involving breakup and re‐formation of molecular aggregates, and environmental dissimilarities, attributed to incompleteness of short‐range order, have been proposed earlier by two of the authors as possible mechanisms giving rise to the distribution of shear‐relaxation times exhibited by most liquids. Experimental evidence is presented here giving qualitative indication that environmental‐dissimilarity effects indeed can be operative in a liquid together with cooperative phenomena. Shear‐relaxation characteristics are studied at a number of temperatures for a host liquid, hexachlorobiphenyl, having methanol and toluene in various concentrations as separate impurity liquids. Small amounts of the additive liquids are found to lessen the decay rate of cooperative behavior with increasing temperature, a result attributed to both degraded cooperative behavior and an environmental‐dissimilarity mechanism. Although cooperative behavior predominates throughout the temperature and concentration range investigated, in the low‐temperature region the impurity additive primarily decomposes aggregates of cooperative molecules, whereas at higher temperatures the impurity markedly enhances environmental dissimilarities and the distribution function widens with increasing (small) concentration. Toluene impurity additive displays greater degradation effect on cooperative behavior as well as greater enhancement of environmental dissimilarities than does methanol. Critical solution temperatures are noted as a matter of interest, and a method is presented for determining analytically the distribution of relaxation times associated with our experimentally determined components of shear modulus.

Photoionization‐Efficiency Curves. II. False and Genuine Structure
View Description Hide DescriptionSpecific examples are given of artifacts in photoionization‐efficiency curves that result from the use of a discrete‐line light source. A comparative technique is used to distinguish this false structure from genuine structure marking energy levels above the ionization potential (I.P.). More accurate I.P.'s than previously available are given for HCl, CH_{3}Cl, CH_{3}NO_{2}, CH_{3}CN, C_{2}H_{4}, C_{3}H_{6}, C_{2}H_{2} and new values for CH_{4}, CD_{4}, C_{2}H_{6}, C_{3}H_{8}.

Normal Coordinate Analyses of Hydrogen‐Bonded Compounds. IV. The Acid Carbonate Ion
View Description Hide DescriptionThe infrared spectra of potassiumacidcarbonate and its deutero analog have been obtained from 4000 to 160 cm^{−1}. The Raman spectrum of the latter has also been obtained in the crystalline state. A normal coordinate analysis has been carried out to estimate the force constants as well as to make theoretical band assignments. The infrared bands of the nondeuterated compound at ∼2620, 1405, and 248 cm^{−1} have been assigned to the O–H stretching, O–H···O in‐plane bending, and O···H stretching coupled with C=O bending modes, respectively. The corresponding force constants are: O–H stretching, 3.20, O–H···O bending, 0.22, and O···H stretching, 0.76 mdyn/Å. Plots of these force constants versus O–H···O distances for three compounds thus far investigated yield a linear relationship for each force constant.

Paramagnetic Resonance in L‐Cysteine Hydrochloride Irradiated at 77°K
View Description Hide DescriptionAn electron paramagnetic resonance study has been made on a single crystal of L‐cysteine hydrochloride irradiated with a 1.5‐MeV electron beam at 77°K. The hyperfine structure of the main resonance pattern was found to be an almost isotropic double doublet, whereas the g factor was found to be highly anisotropic but axially symmetrical, the principal values being 2.29, 1.99, and 1.99. The main resonance has been attributed to the free radicalwith the unpaired electron occupying one of the two nearly degenerate orbitals of the sulfur atom. A remarkable broadening of the resonance spectrum was observed at elevated temperatures, and evidence is given that this broadening is due to a short spin—lattice relaxation time due to a large spin—orbit coupling in this free radical.

Physical‐Adsorption Isotherm Based on a Triangular‐Site Model
View Description Hide DescriptionA multilayer physical‐adsorption‐isotherm theory is developed; it assumes that a gas atom cannot be physically adsorbed in an upper layer unless there is a triangular array of three adsorbed atoms in the layer below forming a site for adsorption. Neighbor interactions via Lennard‐Jones intermolecular potentials are included in the calculation of the adsorption energy. The resultant isotherm is linear for low coverages, indicates a horizontal interaction correction to the Langmuir adsorption isotherm when there is only first‐layer coverage, and may yield a series of steps for sufficiently low temperature. Numerical adsorption isotherms for He on glass at 4.28° and 77°K are presented. The former isotherm exhibits distinctive steps when each layer is condensed, whereas the latter isotherm is quite smooth.

Microwave Spectrum, Dipole Moment, and Ring‐Puckering Vibration of β‐Propiolactone
View Description Hide DescriptionThe microwave spectrum of β‐propiolactone has been observed in the vibrational ground state and in the first four excited states of the ring‐puckering vibrational mode. The variation of the rotational constants with vibrational quantum number and the relative intensities of the observed vibrational satellites indicate that the vibration is essentially harmonic with ν_{0}=160 cm^{−1} and with no appreciable barrier at the planar configuration. The dipole moment is 4.17±0.03 D, with μ_{ a }=3.67±0.03 D, μ_{ b }=1.99±0.02 D, and μ_{ c }=0 in both the ground and first excited states.

Effect of Temperature on Fluorescence Quantum Yields in Solution
View Description Hide DescriptionFluorescence quantum yields and lifetimes have been measured as a function of temperature for anthracene and 9,10‐dichloroanthracene in hexane and ethanol. It was found that the radiative transition probabilities are very slightly temperature dependent. However, for the nonradiative processes, 9,10‐dichloroanthracene shows a strong temperature dependence, whereas anthracene shows none.

Monte Carlo Equation of State for Hard Spheres in an Attractive Square Well
View Description Hide DescriptionThe equation of state for hard spheres in a short‐range attractive square well has been calculated for several values of the well strength by the Monte Carlo method. For shallow wells, it has the same general form but lies everywhere below the equation of state for the simple hard‐core potential. As the well strength increases, two van der Waals loops appear, one at a density higher than the density of the order—disorder transition and one at a lower density.