Volume 51, Issue 8, 15 October 1969

Two‐Electron Homopolar Molecule: A Test for Spin‐Density Waves and Charge‐Density Waves
View Description Hide DescriptionThe eigenstates of the one‐band Hamiltonian for a two‐electron homopolar molecule can be exactly determined. The exact solution for the ground state and all its properties can thus be used as a standard to assess the accuracy and validity of several commonly used approximations: the normal molecular‐orbital ground state, the Heitler–London states, the spin‐density waves and the charge‐density waves. The most general unrestricted self‐consistent field approximations are studied in both the Hartree and Hartree–Fock approximations and are compared with one another as well as with the exact results. New and simple correlated states are also proposed and tested: a symmetrized spin‐density wave and a symmetrized charge‐density wave. These new states are easily obtained from the corresponding Hartree–Fock states, and whenever they exist, they are extremely good approximations to the exact ground state. General consequences and extensions to more complicated systems can be inferred from this study.

Reactions of H_{2}O_{3} in the Pulse‐Irradiated Fe(II)–O_{2} System
View Description Hide Descriptionis measured in pulse‐irradiated O_{2}‐saturated solutions of 20 to 160 μMFe(II), at the pH's 0.46, 1.51, and 2.74 H_{2}SO_{4} and HClO_{4} and with dose rates between 1 and 8 krad/1 μsec pulse. Based on homogeneous kinetics, the results are interpreted by a system of 18 reactions. The formation of the hydrogen sesquioxide H_{2}O_{3} as an intermediate in the reaction OH + HO_{2}→H_{2}O_{3}→H_{2}O + O_{2} is confirmed. In the absence of Fe(II), varies from 2.04 at pH 0.46 to 1.57 at pH 2.74. The rate constant at the pH's studied. The direct reaction, OH + HO_{2} = H_{2}O + O_{2}, does not take place. In H_{1}SO_{4}, at pH 0.46, 151, and 2.74 and in HClO_{4} at pH 0.46 and 1.51, the complexes of Fe(III) produced by the radiation reach equilibrium before they react with HO_{2}. However, equilibrium is not reached in HClO_{4} at pH 2.74, where the effective rate constant is 6.2 times its value when Fe(III) is in its equilibrium form. This ratio is constant over the dose rate range studied. With dose rates up to 64 krad/1.4 μsec pulse and were measured in the air saturated 1‐mM FeSO_{4} dosimeter and in the O_{2} saturated 10‐mM FeSO_{4} dosimeter, each in . decreases in both dosimeters less rapidly with increasing dose rate than was found by earlier workers. The principal reason for the decrease is the failure of O_{2} to scavenge the H atoms at high dose rates. Excellent agreement is obtained between experiment and theory except at high dose rates, where the calculated are slightly too high in both dosimeters.

Ab Initio Studies of Orbital Hybridization in Polyatomic Molecules
View Description Hide DescriptionSingle‐determinant molecular orbital wavefunctions are constructed for a variety of polyatomic molecules using well‐known techniques of hybridization and bond orbital formation. Such concepts generally are employed for qualitative purposes; however, in the present study the emphasis is on a quantitative comparison of energies and wavefunctions with results of ab initio SCF treatments. All electrons are included in the formulation, and all integrals over basis functions are evaluated accurately in order to permit rigorous use of the variational theorem. Minimization of the total energy of a molecule with respect to hybrid angles and bond polarity parameters is found to give energies and wavefunctions in rather good agreement with the minimal LC(Hartree–Fock) AO SCF limit for the systems investigated. Hybridization and bond polarity parameters are reported for a number of systems, and these results provide an assessment of the validity of certain widely used qualitative concepts. The applicability of the procedure in predicting useful molecular wavefunctions and SCF starting vectors is discussed.

On the Unrestricted Hartree–Fock Wavefunction
View Description Hide DescriptionThe unrestricted Hartree–Fock (UHF) wavefunction is analyzed and interpreted in configuration‐interaction (CI) language. The results of the present study are as follows. (i) The UHF wavefunction includes only one type of the singly excited configurations [Eq. (20)], and thus the correlation effects included are very limited ones, compared with the usual CI treatment. (ii) The weight of the lowest contaminating spin function, included in the UHF wavefunction, decreases with increasing spin multiplicity. (iii) The annihilation of the lowest contaminating spin function little affects the electron density distributions and other physical quantities, the operators of which commute with the annihilation operator. (iv) In the UHF method, the “spin‐appearing” (spin‐polarization and spin‐delocalization) mechanisms are clearly divided, and an approximate method to separate these contributions is generalized, and some discussions about spin annihilation are made.

Evaluation of the Lamb Shift for the Hydrogen Molecule–Ion
View Description Hide DescriptionThe Lamb shift of the hydrogen molecule–ions H_{2} ^{+} and HD^{+} have been calculated to the lowest order in the radiative corrections. It is found to depend sensitively on internuclear separation and therefore to shift the various vibrational levels by a different amount. Possible experimental tests are considered for the detection of these quantum electrodynamical effects.

NMR Spectra of Some CF_{3}‐Substituted Triphenylphosphines. Evidence for “Through‐Space” P–F Spin–Spin Coupling
View Description Hide DescriptionIn the ortho‐substituted compounds , and , is 55.00 ± 0.17 Hz, 52.82 ± 0.12 Hz, and 53.41 ± 0.20 Hz, respectively. In contrast, we could not observe any phosphorus–fluorine coupling in the compounds and . These results, together with the very short phosphorus–fluorine distance in the ortho‐substituted compounds, strongly suggest that the P–F coupling results from a “through‐space” mechanism in the ortho‐substituted compounds. This hypothesis is supported by the strong positive temperature dependence found for , which is similar to that found for in compounds in which the “through‐space” mechanism is believed to make a major contribution to the fluorine–fluorine coupling constant.

Conditionally Convergent Integrals Arising from Electromagnetic Interactions
View Description Hide DescriptionThe treatment of conditionally convergent electromagnetic‐interaction integrals is described for the coordinate systems usually employed for molecular integrals. The change in the value of an integral due to the exclusion of some infinitesimal volume other than a sphere about one of the particles is evaluated for several choices. These contributions to the integrals can be expressed at delta‐function terms necessary in expansions of the interaction operator.

Molecular Values, Magnetic Susceptibility Anisotropies, Molecular Quadrupole Moments, and the Second Moment of the Electronic Charge Distribution in OCF_{2}, CH_{2}CF_{2}, cis‐CHFCHF, and CH_{2}F_{2}
View Description Hide DescriptionThe rotational molecular Zeeman effect is observed under high resolution and high magnetic fields in OCF_{2}, CH_{2}CF_{2}, cis‐CHFCHF, and CH_{2}F_{2}. The absolute values of the molecular values and the signs and magnitudes of the magnetic susceptibilityanisotropies in the principal inertial axes are obtained and listed. Arguments are presented to show that all molecular values are negative leading to the molecular quadrupole moments. For ease of comparison, the values are given below in an arbitrary and axis system. The and axes are in the heavy‐atom molecular plane, and the axis bisects the FCF angle in all of the molecules. The units of are 10^{−6} erg/G^{2}·mole and the units of are 10^{−26} esu·cm^{2}.
The diagonal elements in the paramagneticsusceptibilitytensor and the anisotropies in second moment of the electronic charge distributions are computed from the above results and the known molecular structures. The above results are used along with additivity relations for the second moment of the electronic charge distribution to give estimates of the diamagnetic and total magnetic susceptibilitytensor elements.

Energies and Expectation Values for Be by the Transcorrelated Method
View Description Hide DescriptionFurther calculations are reported using the transcorrelated method for energies and wavefunctions recently proposed by Boys and the author. In this method, approximate solutions to the Schrödinger equation are found by obtaining approximate solutions to the transcorrelated wave equation, where and . The advantage of working with wavefunctions which explicitly include electron correlation seems to outweigh the difficulties of working with a nonvariational procedure. Two arbitrary parameters, similar to the Slater orbital exponents, occur in the expansion functions used here for . For nearly all values of these parameters, a nine‐term expansion for gives at least 82% of the correlation energy, and a 19‐term expansion gives 88%. The transcorrelated orbitals φ are within 6% of the self‐consistent field orbitals and arguments are given which suggest that the true transcorrelated orbitals are much closer than this. It is also suggested that the method used here and in the previously reported LiH calculations is not the best, but that the variant used in the Ne calculations is preferable. In the second part, it is demonstrated how expectation values can be determined for transcorrelated wavefunctions without having to evaluate 3N‐dimensional integrals, and without using cluster expansion‐type formula and approximations for them. The method used the Hellmann–Feynman formula, and this is considered reasonable because of the great accuracy of the transcorrelated wavefunction. The general agreement between the expectation values reported here for Be by this method, and those obtained by configuration‐interaction‐type procedures, upholds this view. The transcorrelated method can therefore be used for the ab initio determination of highly accurate energies, wavefunctions, and expectation values.

Electric Field Effects on the Optical Rotatory Power of a Compensated Cholesteric Liquid Crystal
View Description Hide DescriptionBy studying the optical rotatory power (OR) of a compensated cholesteric helical structure, it is possible to observe both thermally induced helical inversion and electric field perturbations of the helix. For a cholesteryl chloride–cholesteryl myristate 1.75:1 mixture, the OR changes sign at increasing to infinity prior to the inversion. Upon application of a dcelectric field parallel to the helix axis, the OR decreases continuously and reversibly by a factor of 4 in the field range , where denotes the critical field required to induce a nematic transition. This effect cannot be explained as a macroscopic reordering of the material, but is rather due to a decrease in pitch accompanying a conical helical perturbation. This perturbation arises from bending forces tending to align the dipole moments of cholesteryl chloride in the field direction. When a dc field is applied perpendicular to the helix axis, the pitch diverges logarithmically to infinity as approaches the critical value for inducing a transition.

Mössbauer Studies of Iron Organometallic Complexes. VII. The Iron (−II) Tetrahedral Compounds
View Description Hide DescriptionMagnetic perturbation was used to determine the electric‐field gradient parameters and for several tetrahedral Fe(− II) compounds. Because of the spherical symmetry of the configuration, only ligand dispositions and strength are measured. Distortions from tetrahedral symmetry are sensitively measured by the magnitude of .

Far‐Infrared Spectra of Ring Compounds. V. Ring‐Puckering Potential Functions of Some Oxygen‐Containing Molecules
View Description Hide DescriptionThe infrared absorption spectra of gaseous 2,5‐dihydrofuran, 3‐oxetanone, 2‐methyl‐4,5‐dihydrofuran, diketene, cyclopentanone, and cyclohexanone have been observed in the range 250–25 cm^{−1}. Under high resolution the spectrum of 2,5‐dihydrofuran shows a satellite series of branches in addition to those observed by Shimanouchi and Ueda. The satellites arise from transitions between ring‐puckering energy levels in an excited ring‐twisting state with altered spacing due to higher‐order cross terms in the potential energy. The spectrum of 3‐oxetanone is that of a planar molecule, and the energy levels of the ring‐puckering mode fit a potential function of the form . The absorption of 2‐methyl‐4,5‐dihydrofuran is complicated by several low‐frequency modes, but the spectrum can be fairly well interpreted with a double‐minimum potential function of the form with a barrier height of 98 cm^{−1}. Seven sharp branches were observed for cyclopentanone, which were fitted to a hindered pseudorotational potential function of the form . Although one might also expect diketene and cyclohexanone to have anharmonic out‐of‐plane ring modes, both compounds gave far‐infrared spectra which appear to be essentially harmonic.

Rate and Heat of Vaporization of Graphite above 3000°K
View Description Hide DescriptionThe rate of vaporization of graphite filaments has been measured from 2500 to 3400°K in vacuum and in He by pulse heating techniques. At 3400°K, the rate of vaporization was sufficiently high that a direct determination of the heat of vaporization could be obtained. The experimental results have been correlated with thermodynamic data obtained at lower temperatures on graphite and its vapor species; the predominance of C_{2} in vaporizing graphite is shown to satisfy the experimental results. The specific surface area of the filaments was measured and interpreted.

Band Structures of Metal‐Free Phthalocyanine in the β Phase
View Description Hide DescriptionThe excess electron and hole‐band structures of β‐metal‐free phthalocyanine are calculated with the tight‐binding approximation, using the Hückel molecular orbitals obtained previously by this author. The heteroatomic nature of the molecule is emphasized, and hence a new form of molecular potential, which reflects the nonuniform distribution of the pi electrons in the molecule, replaces the Goeppart–Mayer–Sklar potential used in previous band‐structure calculations of hydrocarbons. The interaction between the two electron bands formed from the two nearly degenerate lowest empty molecular orbitals has also been considered. The resulting band structures are qualitatively similar to those of anthracene, but with more pronounced anisotropy in the bandwidths. The velocity tensors and , where is the group velocity and are its components, associated with a particular band are computed from the band structures for different values of vibronic overlaps. The applicability of the simple band model of charge transport is discussed.

Limiting Laws and Counterion Condensation in Polyelectrolyte Solutions. III. An Analysis Based on the Mayer Ionic Solution Theory
View Description Hide DescriptionThe interaction with point ions of a line charge supplemented by a distance of closest approach is studied by means of the Mayer theory of ionic solutions. A number of points involved in a recent theory of polyelectrolytesolutions (Papers I and II of this series) are thereby clarified. It is demonstrated that the counterion condensation phenomenon occurs in the limit of zero concentration, that the Debye–Hückel approximation is applicable to the coions and uncondensed counterions, and that the relevant dielectric constant is that of the pure bulk solvent.

Energy Transfer in Tb^{3+}‐Activated Cerium(III) Compounds
View Description Hide DescriptionThe luminescence of Tb^{3+}‐activated CeF_{3}, CePO_{4}, CeBO_{3}, and CeAlO_{3} has been investigated. Energy transfer from the Ce^{3+} host lattice to the Tb^{3+} ion occurs by Ce^{3+}–Ce^{3+} and Ce^{3+}–Tb^{3+} transfer. From a study of the Ce^{3+}luminescence in the analogous La compounds it can be concluded that the probability of the former transfer is different for these host lattices. An analogy with Eu^{3+}‐activated host lattices is pointed out.

ESR Study of HĊO in Single Crystals of Formic Acid at 77°K
View Description Hide DescriptionFormyl radicals, HĊO, are formed when crystals of formic acid which have been gamma irradiated at 77°K are warmed to 195°K. An ESR study of these radicals in single crystals of formic acid has been made at 77°K. The tensor (principal values 2.0037, 2.0023, 1.9948) and the hyperfinetensors for the proton (principal values 337, 346, 378 MHz) and for ^{13}C (principal values 341, 438, 317 MHz) were determined. These tensors are discussed in relation to the structure of the formyl radical.

Collision Transfer of Vibrational Energy from Nitrogen and Methane to the Carbon Dioxide Molecule
View Description Hide DescriptionThe sound absorption of CO_{2} has been measured at 95°C from 10 to 5 × 10^{6} Hz/atm with 99%–92.5% N_{2} and at 80°C from 2 × 10^{4} to 4 × 10^{7} Hz/atm with 10%–90% CH_{4} admixed. N_{2} has been chosen because of its good resonance to the asymmetric valence‐bond vibration of CO_{2} (ν_{3}), while CH_{4}resonates approximately with the CO_{2} levels at about 1900°K. The results were analyzed by a modified “binary two‐level” theory of the quantum exchange mechanism between the two molecules. From the CO_{2}–N_{2}measurements, the reciprocal de‐excitation probabilities of were obtained to be in collisions with N_{2} and CO_{2}, respectively. The de‐excitation of the first bending vibration level in collisions with nitrogen needs collisions. The same analysis applied to the CO_{2}–CH_{4} mixtures yielded the following transition probabilities: de‐excitation of CO_{2} (01^{1}0) by CH_{4}, ; resonance production of two states (01^{1}0) from one at 1900°K in collision with ground‐state CO_{2} molecules, ; quantum exchange from excited CH_{4} to the CO_{2} levels at 1900°K, ; de‐excitation of CH_{4} in collisions with CO_{2}, . The fact that the relaxing molar heat capacity of the CO_{2}–CH_{4} system exceeded its total vibrational heat capacity is explained as an influence of rotational–vibrational transitions, in which preferentially the vibration is de‐excited and the rotation excited. Such processes also would explain the high efficiency of the (fast‐rotating) methane for the de‐excitation of the CO_{2} bending vibration.

Highly Anharmonic Crystal
View Description Hide DescriptionThe free energy of the constrained hard‐sphere crystal is evaluated by Hermite expansion techniques applied to the Boltzmann factors of the partition function. An independent‐oscillator comparison potential generates a Hermite polynomial basis set in which exp is expanded. A first approximation results from retention of the first term, the Gaussian average. A second approximation results from retention and exponentiation of the succeeding two terms, and corresponds to a coupled‐oscillator potential. For both approximations the pressure deviation from the molecular dynamics pressure is close to the statistical irregularities of the latter, for densities such that the unconstrained solid is thermodynamically stable. The free energy in second approximation, significantly improved over the first, is excessive by at high densities and by up to at low and intermediate densities, where the constraint is required to stabilize the lattice. The methods, readily applicable to smooth potentials (and more promising for them) are similar to those earlier applied to polymertheory. Related methods, based on expansions of the potential itself, have been devised by Koehler and others for weak or moderate crystal anharmonicity.

Charge‐Carrier Injection and the Pre‐exponential Factor in Semiconducting Organic Substances
View Description Hide DescriptionA charge‐carrier injection model is shown to lead to an approximately linear dependence of the preexponential factor on activation energy for small organic semiconductors, without assuming a linear temperature dependence for the activation energy.