Volume 41, Issue 6, 15 September 1964

Theory of Normal Vibrations of Chain Molecules with Finite Length
View Description Hide DescriptionThis paper is concerned in the end effects on the normal vibrations of a chain molecule with finite length. In Part I we study the mathematical problem of how to solve the eigenvalue problem of the GF matrix of the chain molecule with finite length. Making use of the fact that the force range can be regarded finite we reduce the problem to solving a determinantal equation usually of much lower order than that of the GF matrix through the introduction of auxiliary variables and transfer matrices. We obtain the simple formula for giving an admissible phase difference in which the end effect is taken into account. In Part II, using this formula we analyze the infrared data of Snyder on crystalline normal paraffins C_{3}H_{8} through C_{30}H_{62}. Our analyses indicate that the formula is generally useful for making extrapolation, interpolation, assignment, and determination of frequency‐phase relations from the experimental data of homologous series of chain molecules.

Hyperfine Structure in the Microwave Spectrum of NH_{2}D
View Description Hide DescriptionHyperfine structure in the 3_{13}→3_{03} rotation—inversion transition of deuterated ammonia NH_{2}D at 18 807.7 Mc/sec has been studied with a beam masermicrowave spectrometer giving a linewidth of 7 kc/sec. All details of the observed hyperfine structure have been accounted for in terms of the various quadrupole, spin—rotation, and spin—spin interactions of the four coupling nuclei. The hyperfine coupling constants which have been determined with the aid of a digital electronic computer are, for the 3_{13} state: (eq_{J}Q)_{D}=—62.5±1.0 kc/sec, C _{D}=—2.7±0.3 kc/sec, C _{H}=—13.6±0.5 kc/sec; and for the 3_{03} state: (eq_{J}Q)_{D}=—73.9±1.0 kc/sec, C _{D}=—2.7±0.3 kc/sec, C _{H}=—12.3±0.5 kc/sec. On the assumption that the electric field at the deuteron is cylindrically symmetric about the N—D bond direction ξ, we find that (eV_{ξξ}Q)_{D}=282±12 kc/sec, or V _{ξξ}=1.38±0.06×10^{15} statV/cm^{2}, where V _{ξξ} is the second derivative of the electrostatic potential along ξ.

Far‐Ir Spectrum of Dimethylacetylene: Internal Rotation and Evidence for a D _{6h } Effective Symmetry
View Description Hide DescriptionThe spectrum of dimethylacetylene vapor, liquid, and solid phases has been measured in the region 50 to 450 cm^{—1}. Therefrom, together with previous work, a unique and ``complete'' vibrational assignment is derived. Rotational fine structure seems to confirm ``free internal rotation,'' i.e., a barrier below 100 cal/mole.
Dimethylacetylene is an ethane‐type molecule with practically free internal rotation. The new data strongly confirm the previous conclusion that neither D _{3}, D _{3h }, or D _{3d } are appropriate symmetry groups. The selection rules are consistent with a D _{6h } effective symmetry.
An appendix deals with the relative intensities in degenerate vibrational transitions.

Normal Coordinate Analyses of Hydrogen‐Bonded Compounds. I. Monomeric Formic Acid and Acetic Acid
View Description Hide DescriptionIn order to obtain the Urey—Bradley force constants, normal coordinate analyses have been carried out for monomeric formic acid and acetic acid including their deutero analogs. Previous empirical band assignments have been confirmed or corrected on the basis of the calculation of the potential‐energy distribution in each normal vibration.

Normal Coordinate Analyses of Hydrogen‐Bonded Compounds. II. Dimeric Formic Acid and Acetic Acid
View Description Hide DescriptionNormal coordinate analyses have been made both for dimeric formic acid and acetic acid, including their deutero analogs. The Urey—Bradley force constants of the dimers have been compared with those of the corresponding monomers to study the effect of hydrogen bonding on individual bonds. Theoretical band assignments have been made for all the in‐plane vibrations based on the potential energy distribution. The results reveal the nature of vibrational couplings which have hitherto been assumed only on an empirical basis.

Thermodynamic Functions of Morse Oscillators
View Description Hide DescriptionPartition functions and thermodynamic functions are calculated for one‐dimensional classical and quantal Morse and harmonic oscillators to investigate anharmonic and quantal effects. Results are shown graphically for oscillators corresponding to H_{2}, D_{2}, and N_{2}; results on Br_{2} are described briefly.

Computation of Character Tables for Nonrigid Molecules
View Description Hide DescriptionA simple method is described, by means of which it is possible to calculate character tables for the symmetry groups of molecules consisting of a number of methyl groups attached to a rigid framework. Character tables are given for the symmetry groups of neopentane C(CH_{3})_{4}, and octahedral metal hexammine ions, M(NH_{3})_{6}. The method can be generalized to apply to other nonrigid molecules.

Collision‐Induced Microwave Absorption in Compressed Gases. III. CO_{2}—Foreign‐Gas Mixtures
View Description Hide DescriptionMeasurements of the increase in dielectric loss accompanying the addition of various foreign gases (He, H_{2}, Ar, N_{2}, CH_{4}, C_{2}H_{6}, and SF_{6}) to CO_{2} have been made at a frequency near 24 GHz and a temperature of 25°C. This loss is attributed to the dipoles induced in the foreign gas by the molecular quadrupole field of CO_{2}. Excluding mixtures with He and H_{2}, for which the effect is very small, the value of the quadrupole moment of CO_{2} derived from these losses is about 4.4×10^{—26} esu in all cases. This is in good agreement with the directly measured value of 4.1×10^{—26} esu reported by Buckingham and Disch.

Some Excited States of the Hydrogen Molecule. II. ^{1}Π_{ g }(1s2pπ), ^{3}Π_{ g }(1s2pπ), ^{1}Δ_{ g }(1s3dδ), ^{3}Δ_{ g }(1s3dδ), ^{1}Δ_{ u }(1s3dδ), ^{3}Δ_{ u }(1s3dδ)
View Description Hide DescriptionComplete theoretical potential curves for the bound ^{1}Δ_{ g }(1s3dδ) and ^{3}Δ_{ g }(1s3dδ) states of H_{2} and for the long‐range portions of the repulsive ^{1}π_{ g }(1s2pπ), ^{3}π_{ g }(1s2pπ), ^{1}Δ_{ u }(1s3dδ), and ^{3}Δ_{ u }(1s3dδ) states of H_{2} are reported. Comparison of these results plus results from Paper I of this series to the conventional perturbation theory computations for the long‐range interactions of those states shows that the conventional perturbation results may be misleading if ``valence'' and ``overlap'' forces are not considered. The computed total energies at R_{e} for the ^{1}Δ_{ g }(1s3dδ) and ^{3}Δ_{ g }(1s3dδ) states are —0.6570143 (—0.65778), and —0.6571785 (—0.65872 a.u.), respectively. The values in parentheses are experimental values. Spectroscopic constants and expectation values for some one‐electron operators for the ^{1}Δ_{ g }(1s3dδ) and ^{3}Δ_{ g }(1s3dδ) states are also presented. Two‐center and one‐center wavefunctions yielding comparable energies for the ^{1}Δ_{ g }(1s3dδ) state are compared through expectation values for a series of one‐electron operators.

Magnetic and Structural Characteristics of Lanthanide—Nickel Compounds
View Description Hide DescriptionThe magnetic and structural characteristics of 11 intermetallic compounds represented by the formula LnNi are reported. The five compounds with Ln=Dy, Ho, Er, Tm, and Y are isomorphous and occur in the FeB structure. The others, with Ln=Ce, Pr, Nd, Sm, Gd, and Tb, form in the CrB structure, YNi and CeNi exhibit only Pauli paramagnetism down to 4.2°K. The others order magnetically with Curie temperatures ranging from 8°K for TmNi to 71°K for GdNi. Nickel appears to be nonmagnetic in the series and CeNi is nonmagnetic and hence Ce is quadripositive in CeNi. Curie—Weiss behavior is obeyed in the paramagnetic region for the compounds containing a magnetic Ln atom, except for TmNi, which exhibits a tendency toward Van Vleck‐type temperature‐independentparamagnetism at low temperatures. This and the fact that the observed saturation moments are well below the free tripositive lanthanide ion moments suggests that appreciable crystal field quenching of the orbital contribution occurs in these compounds. Except for TmNi, effective moments in the paramagnetic region are as expected for the free tripositive ions.

Pressure Dependence of the Resistance of VO_{2}
View Description Hide DescriptionThe resistance of VO_{2} through the semiconductor‐to‐metal transition has been measured as a function of hydrostatic pressure. The transition has been studied in pure and partially hydrated single crystals of VO_{2} and in powdered VO_{2}. The temperature of the transition is shifted by less than 0.5 C° by pressures up to 6×10^{3} bars. The conductivity in both the semiconducting and metallic states is decreased by about 10^{—3}%/bar. The effect of pressure on both the resistance and transition temperatures of VO_{2} is much smaller than that reported in previous measurements on V_{2}O_{3}. The relation between the pressure dependence of the energy gap in the semiconducting state and the pressure dependence of the transition temperature is examined and is shown to be consistent with Mott's mechanism for the transition to the metallic state.

Energy‐Distribution Function for Hot Atoms Produced by Nuclear Transformations
View Description Hide DescriptionThe energy‐distribution function for hot atoms produced by nuclear transformations is examined in terms of the fraction of the initial hot‐atom energy. Energy dependent and independent asymmetric scattering is specifically considered. For gaseous tritium generated by the ^{3}He(n, p) process, the asymptotic solution probably can serve as a reasonable approximation of the distribution function in the energy range 2–20 eV. For hot atoms produced with an initial distribution of energies, such as ^{79}Br(n, γ)‐produced ^{80}Br, the asymptotic solution will be less valid than for tritium, but may still be a valid approximation.

Comparison of Experimental and Theoretical Vibrational Relaxation Times for Diatomic Gases
View Description Hide DescriptionA comparison of theoretical and experimental vibrational relaxation times has been carried out for the halogens, nitrogen, oxygen and carbon monoxide. Reasonably good agreement is obtained for all gases except oxygen. For oxygen the predicted variation of the vibrational collision number with temperature appears to be correct; however, experimental values at high temperatures appear to be low by a factor of 0.55. Values obtained at room temperature on the other hand agree well with theory. It would seem therefore that an anomalous decrease in the vibrational collision number takes place in the temperature range from 300° to 500°K.
Since data for carbon monoxide could be fitted very well by theory, it is apparent that the effect of radiation is negligible, at least in the temperature range above 1000°K. This is consistent with the value of 0.03 sec for the radiative lifetime of carbon monoxide in the first excited vibrational state.
An empirical relationship is established between the interaction constant α and the molecular diameter σ, which reproduces the experimentally derived values quite closely.

Infrared Spectra in the Characterization of Some Molecular Complexes of the Dative Type
View Description Hide DescriptionThe infrared spectra of six diamine—quinone complexes and eight azine—quinone complexes were examined and compared with those of the component quinones and the semiquinone salts. The absence of the quinone but the presence of the semiquinone ion was proved in these complexes; therefore, the complexes may be characterized as essentially of the dative type. It is pointed out that most, if not all, of the known complexes having relatively low electrical resistivity are of this type.

Near‐Infrared Emission Spectrum of NO
View Description Hide DescriptionThe emission spectrum of the vibration—rotation bands of the electronic ground state of nitric oxide has been observed in the 1.83–1.95‐ and 2.63–2.81‐μ regions under moderately high resolution. The source of the spectrum was a high‐tension discharge maintained through streaming nitric oxide gas at pressures of less than 1 atm. A detailed analysis of the 2–0, 3–;1, and 4–;2 bands, occurring at 2.69, 2.73, and 2.77 μ, respectively, has been carried out. Accurate values obtained for many of the rotational and vibrational constants associated with the ^{2}II_{½} and substates of the electronic ground state are presented, including a value for the spin—orbit coupling constant A. Also, the effective rotational temperature of the emitting gas has been determined.

Microwave Rotation Spectra of Hydrogen‐Bonded Molecules
View Description Hide DescriptionThe pure rotation spectra have been observed for the hydrogen‐bonded bimolecules CF_{3}COOH–HCOOH, CF_{3}COOH–CH_{3}COOH, and CF_{3}COOH–CH_{2}FCOOH. For CF_{3}COOH–HCOOH, the O···O distance R _{H} was found to be 2.69±0.02 Å. The O···O distance becomes 0.011 Å longer when D is substituted for H in the hydrogen bond, i.e., R _{D}—R _{H}=0.011 Å. The heat of formation was found to be —ΔH=15.8 kcal/mole. Similar values, R _{H}=2.67 Å and R _{D}—R _{H}=0.012 Å, were obtained for CF_{3}COOH–CH_{3}COOH, but in this molecule the spectrum of the singly deuterated species shows that the D atom tunnels between the two partners in the bimolecule at a rate faster than the rotational frequency. For a symmetric double minimum potential function the barrier to tunneling is found to be less than 5000 cm^{—1}. The rapid tunneling also implies that only a sixfold barrier to the internal rotation of the CF_{3} and CH_{3} groups can exist, and therefore the nearly free rotation of these groups in the bimolecules and in dimers can be expected. This free rotation would explain the failure to detect the low K lines in the pure rotation spectra. The rapid tunneling of the D (and H) atoms in the hydrogen bond also implies an inversion doubling which would contribute to the width of the v(OH) bands in the carboxylic acid dimer infrared spectra. The difference R _{D}—R _{H} predicts a shortening of the O···O distance of 0.073 Å when the O–H vibration is excited, which provides a mechanism to couple the low hydrogen‐bond stretching frequency and the v(OH) frequency to produce the broad dimer bands.

Variational Solutions to the Brillouin—Wigner Perturbation Differential Equations
View Description Hide DescriptionVariational techniques for the Brillouin—Wigner (BW) perturbation theory, analogous to the Hylleraas and Sinanoğlu principles in Rayleigh—Schrödinger (RS) theory, are derived. A practical method of applying this approach to BW theory, which does not require the knowledge of the exact BW wavefunctions, is discussed. Using this method one obtains an upper bound to the exact total energy for systems in the lowest energy state of a given symmetry. Finally, a convenient matrix method of applying the variational principles is suggested and degenerate BW theory is discussed briefly.

Degree of the Critical Isotherm
View Description Hide DescriptionAn argument is presented which indicates that if g is the degree of the critical isotherm, d the degree of the coexistence curve, and f the power of ∥ T _{ c }—T ∥^{—1} with which the isothermal compressibility becomes infinite as the critical point is approached, then g=1+fd. The relation of this result to other theoretical and experimental facts is discussed.

Reformulation of the Virial Series for Classical Fluids
View Description Hide DescriptionThe usual graphical representation of the virial coefficients is reformulated in terms of graphs containing not only Mayer f functions, but also f̃ functions .
This reformulation has three main advantages:
(1) The number of integrals of topological graphs contributing to the virial coefficients is reduced; this simplifies numerical calculations.
(2) In Mayer's formulation none of the star integrals contributing to the virial coefficients (for hard potentials, at least) could be ignored; each made a nonnegligible contribution. In the new formulation (again, for hard potentials) many integrals make negligible (or even zero) contributions; the extensive cancellation of positive and negative terms found in Mayer's formulation is reduced.
(3) Several new ways of summing the virial series by successive approximation are suggested by the new formulation. One such way is worked out, in the first three approximations, for gases of hard parallel squares and cubes; the third approximation reproduces the first five virial coefficients exactly.
The reformulation is not restricted to the virial series alone. We also generalize our treatment to the radial distribution function. It can be applied to any series whose coefficients are integrals of graphs.

One‐Center Wavefunction for the Ground State of the HeH^{+} Molecular Ion
View Description Hide DescriptionThe HeH^{+} molecular ion is a member of that class of diatomic hydride molecular ions which are one‐center systems at R=0 and at R=∞. The total electronic energy of the ground state of HeH^{+} has been calculated over a wide range of R with a 30‐term orbital product wavefunction centered on the helium nucleus. The results are compared with a James and Coolidge‐type calculation and with a two‐center orbital product calculation. The results in the order Evett, Anex, and present calculation are R _{ e }=1.432, 1.446, 1.464 a.u.; ω_{ e }=3600, 3378, 3184 cm^{—1}; and E(R=1.4 a.u.)=—2.9730, —2.9742, —2.9691 a.u. The energy of the present calculation is the lowest for R greater than about 2 a.u. and is lower than the classical polarizationenergy for 2<R<4 a.u. For R>4 a.u. the one‐center interaction energies [E(R)—E(∞)] lie quite close to the classical polarizationenergies.