Harmonic Oscillator Approximation Research Articles

Vibrational Spectra of Molecules and Atoms on Surfaces

The vibrational spectrum of a monolayer film chemisorbed on a single-crystal metal substrate is discussed in the harmonic-oscillator approximation. Optical modes must be totally symmetric with respect to translation, and be of an appropriate symmetry species under the unit cell (or factor) group. The spectral implications of the presence of anharmonicity, quadrupolar-induced transitions, and non k=0 processes are discussed in considerable detail. This theory is applied to the case of CO and H2 or D2 adsorbed on nickel (100), (110), and (111) crystallographic surfaces.

Application of the S.C.F. Method of Lattice Dynamics to the Coulomb Lattice

AbstractAnharmonic effects of the Coulomb‐lattice vibrations are investigated by the time‐deendent s.c.f.‐method. In contrast to the traditional treatment using the lattice theory of Born there are higher branches in the dispersion relation of the lattice vibrations which represent anharmonic elementary excitations. In the harmonic oscillator approximation for the low s.c.f.‐states, which is justified for sufficient lylow density, Kohn's sum rule is generalized to include the anharmonic branches. Taking into account anharmonic branches, the instability of the lattice state above a certain density is indicated by imaginary eigenfrequencies in the transverse vibration mode. The contribution of the anharmonic branches to the zero‐point energy is evaluated.

Temperature Dependence of the Total Intensity of Difference-Band Systems: The 10-μ Band System of CO2

The temperature dependence of the total intensity of difference-band systems is discussed and a general expression is given in the harmonic-oscillator approximation for difference transitions between two nondegenerate vibrations. The results are compared with experimental measurements of the sum of the total intensities of the (ν3—ν1) and (ν3−2ν2) difference bands of CO2 (the 10-μ band system). New data are presented at temperatures higher than previously investigated (up to 2300°K). In the temperature range studied, the main contribution to the temperature dependence comes from the Boltzmann term.

Reduction Parameters for an Unspecified Intermolecular Potential. A Harmonic-Oscillator Approximation

The reduction parameters for an unspecified pair interaction potential have been calculated for argon and krypton. The method used is a sphericalized equivalent of the method first proposed by Corner. It is similarly limited, but only in the present harmonic version, to systems for which the zero-point energy is very small. The parameters obtained from solid-state data allow an extremely accurate corresponding states comparison of second virial coefficient data. Third virial and also viscosity data similarly appear to be reduced to single classical curves. This suggests that any nonadditivity of potential present in the multiparticle-interaction situation of the solid phase must be very small.

Molecular constants of SiH3C ≡ CH and SiD3C ≡ CH

A normal co-ordinate analysis of silylacetylene and silylacetylene-d3 has been carried out following Wilson'sF-G matrix method. The potential energy constants obtained therefrom have been used to evaluate rotational distortion constants and mean square amplitudes of vibration for these molecules. Thermodynamic functions, such as heat content, free energy, entropy heat capacity for the ideal gaseous state at one atmosphere pressure and with the usual rigid rotor harmonic oscillator approximation, have also been calculated for 12 temperatures from 100°K to 1000°K.

Spectral Absorption of the 46-μ, Bands of N_2O

The spectral absorption due to parallel bands of nitrous oxide is computed in the region of the ν3 fundamental for temperatures of 200°K and 300°K. The harmonic-oscillator approximation was used to compute the relative intensities of vibrational transitions for which no published data are available. The total integrated intensity of all of the transitions considered in the calculation was normalized to a value of 1850 cm−2 atm−1 at 300°K Values of the spectral absorption computed using the quasi-statistical band model and a half-width of 0.05 cm−1 at 300°K are compared with published experimental data for various amounts of absorbing gas and total pressures; the results are seen to be in good agreement. A comparison of the spectral absorption obtained using the quasi-statistical band model and the strong-line and weak-line approximations is also given.

Approximate band absorption and total emissivity calculations for H 2O

A highly simplified procedure, utilizing symmetrical band contours and intensity estimates derived from harmonic oscillator approximations, may be used to calculate band emissivities for H 2O at sufficiently high pressures where the rotational fine structure has been removed. Agreement with experimental data is excellent. The empirically determined dependence of total emissivities on the partial pressure of water vapor, at constant total pressure, may be accounted for quantitatively by using a model of randomly distributed dispersion lines falling in the square root region of the curves of growth.

Temperature Dependence of the Total Integrated Intensity of Vibrational—Rotational Band Systems

The temperature dependencies of the total integrated intensities of vibrational—rotational band systems are discussed and general expressions are given for the temperature dependencies for the harmonic oscillator approximation. The derived dependencies indicate that while the total integrated intensity of a fundamental is independent of temperature, the integrated intensity of an overtone or combination band system increases with increasing temperature. The theoretically derived temperature dependencies are compared with available experimental data and good agreement is found where self absorption within the test gas does not influence the experimental results.

Franck—Condon Factors for Polyatomic Molecules

General expressions are derived for calculating Franck—Condon factors for most transitions of polyatomic molecules (excluding transitions between linear and bent configurations) in the harmonic oscillator approximation. The derivation employs the method of generating functions and also linear transformation of normal mode coordinates between initial and final states. Input data required in the general case are geometries, frequencies, and vibrational force fields for the initial and final states. Explicit algebraic expressions are presented for the transition between linear, symmetric, triatomic initial and final states. The results are used to show that fragmentation cannot result from direct vertical ionization to the ground electronic states of CS2+ and CO2+ or to the excited 2IIu state of CO2+. An attempt was made to fit the experimental photoionization data for C2H2 and C2D2. This resulted in an estimated increase of 0.05 Å in the equilibrium C–C bond distance and no change in the C–H bond distance in C2H2+ as compared to C2H2.

Integrated Intensity Measurements of the 5.3-μ Fundamental and 2.7-μ Overtone Bands of NO between 1400° and 2400°K

Measurements have been made of the total integrated band intensities of the 5.3-μ fundamental and the 2.7-μ first overtone bands of NO in the temperature range 1400° to 2400°K. The NO gas sample was studied at high temperatures using a reflected shock-wave technique. Within the temperature range used in the present experiments, the integrated band intensity of the NO fundamental does not vary. However, the integrated intensity of the first overtone band of NO shows a small temperature dependence. The temperature dependencies of the two bands are in agreement with theoretical temperature dependencies determined assuming a harmonic oscillator approximation to the NO molecule. Using the theoretical temperature dependencies the experimental data have been extrapolated to give values of the integrated intensities α° at STP. The values of α° obtained for the fundamental and first overtone bands of NO are 76±7 cm—2·atm—1 and 2.8±0.5 cm—2·atm—1, respectively.

Approximate band absorption and total emissivity calculations for CO 2

A highly simplified procedure, using symmetrical band contours and intensity estimates derived from harmonic oscillator approximations, has been used to calculate band and total emissivities for CO 2. Agreement with experimental data and with more accurate calculations is excellent.

Integrated Intensity Measurements of the 1.9-μ Bands of CO2 in the Temperature Range 1400° to 2500°K

Measurements have been made of the total integrated band intensity of the 1.9-μ bands of CO2 in the temperature range 1400° to 2500°K. The gas was heated to the high temperature by a shock wave reflected from the rigid end plate of a shock tube. The experiment determines the total integrated band emission as a function of optical path length. The total emission is related to the integrated band intensity in a simple way. The intensity in the 1.9-μ region of the CO2 spectrum arises from three combination bands, namely the (v3+4v2), (v3+2v2+v1), and (v3+2v1) bands. These band systems are in strong Fermi resonance. The bands have not been resolved; the total integrated intensity of the three bands was measured as a function of temperature. The temperature dependence of the absolute intensity is discussed in terms of a simple model using the harmonic oscillator approximation to the CO2 molecule. The results indicate that the intensity in the 1.9-μ resonant triplet of CO2 originates in the (v3+2v1) band. An extrapolation of the data using the derived temperature dependence gives an integrated band intensity of 2.07 (cm—2 atm—1) at STP for the total 1.9-μ CO2 band.

Substituted methanes: Part XXXIV. Raman and infrared spectral data and calculated thermodynamic properties for CH 2Cl 2, CHDCl 2, and CD 2Cl 2

Raman displacements, semiquantitative relative intensities, and approximate depolarization ratios were obtained for liquid CH 2Cl 2, CHDCl 2, and CD 2Cl 2; and wave numbers and estimated relative intensities for the infrared bands from 400 to 4000 cycles/cm were obtained for both the liquid and the gas. Vibrational assignments for the deuteriated compounds were made by comparison with the results for CH 2Cl 2, and by application of the complete isotopic substitution rules of Brodersen and Langseth. The results confirm the infrared work of Shimanouchi and Suzuki for these compounds, except for the σ 2 fundamental of CD 2Cl 2 which was assigned to the 1052 cycles/cm Raman line. Thermodynamic properties (heat content function, free energy function, entropy, and heat capacity) were calculated for twelve temperatures from 100 to 1000°K to a rigid-rotor, harmonic-oscillator approximation.

Potential Energy Constants, Rotation Distortion Constants and Thermodynamical Properties of GeH3CN and GeD3CN

AbstractThe force constants, obtained from a normal coordinate analysis of the molecules GeH3CN and GeD3CN using Wilson's F.G. matrix method, have been made use in the evaluation of rotational distortion constants for these molecules. Molar thermodynamic properties of these molecules for the ideal gaseous state at one atmospheric pressure, with rigid rotor harmonic oscillator approximation have also been obtained.

Molecular vibrations of HCOF and DCOF

The harmonic wave numbers for HCOF and DCOF have been calculated using the observed fundamentals and the force constants have been evaluated on the basis of the general quadratic and Urey-Bradley force fields. The out-of-plane fundamental in HCOF, which has not been reported so far, is calculated. The thermodynamic properties of HCOF and DCOF have been calculated for fourteen temperatures ranging from 200° K. to 1500° K. for the ideal gaseous state at one atmosphere pressure and for a rigid rotor, harmonic oscillator approximation.

Potential Constants and Thermodynamic Functions of Iodine Pentafluoride

AbstractAn orthonormalized set of symmetry coordinates satisfying the transformation properties has been constructed for the iodine pentafluoride molecule having the tetragonal pyramidal structure with the symmetry point group C4v. Utilizing these symmetry coordinates and a potential function consisting of the ordinary valance force terms and central force terms, the elements of potential and kinetic energy matrices have been obtained. From the observed Raman and infrared fundamental frequencies the potential constants and thermodynamic functions such as heat content, free energy, entropy and heat capacity have been calculated for a rigid rotator, harmonic oscillator approximation.

Molecular constants of SiH 3NCS and SiD 3NCS

Potential energy constants and mean amplitudes of vibration have been calculated for silyl isothiocyanate and silyl isothiocyanate- d 3 from vibrational spectral data. Values of heat content, free energy, entropy, and heat capacity also are given for the rigid rotor, harmonic oscillator approximation, at 14 temperatures between 100°K and 1200°K.

Potential constants and thermodynamic functions of the tetrachlorides of titanium and vanadium

AbstractUsing the observed Raman and infrared absorption fundamental frequencies of gaseous and liquid tetrachlorides of titanium and vanadium, the potential constants on the basis of Wilson's group theoretical method and the molar thermodynamic functions on the basis of a rigid rotator, harmonic oscillator approximation have been calculated.

Mean Amplitudes of Vibration and Thermodynamic Functions of the Dichlorides of Oxygen and Sulphur

AbstractThe various stages of improvement in the vibrational as well as molecular structure studies for the dichlorides of oxygen and sulphur have been reported. The computed numerical values of (1) mean‐square amplitude quantities, (2) generalized mean‐square amplitudes (mean‐square parallel amplitudes, mean‐square perpendicular amplitudes, and mean cross products) and (3) mean amplitudes of vibration for the temperatures at T=0 and T=298°K have been presented for these two molecules. Further, the molar thermodynamic functions have been calculated by using the fundamental vibrational frequencies with the assumption of a rigid rotator, harmonic oscillator approximation for the temperature range from 50° to 1600°K. Besides these, the computed numerical values of the generalized mean‐square amplitudes have also been reported for the difluorides of oxygen and silicon.

Raman spectra, normal coordinate treatment, and calculated thermodynamic properties for hydrazine and tetradeuterohydrazine

Raman displacements, relative intensities, and depolarization factors have been obtained for liquid hydrazine at room temperature. All available data concerning the structure, Raman and infrared spectra, frequency assignments, and dipole moment measurements were collected and examined for the most reliable values. Results of normal coordinate treatment based on the C 2 model are given and frequency assignments are reviewed. Also, calculated thermodynamic properties for 273.16°K to 1000°K on the basis of the rigid rotator harmonic oscillator approximation are listed. The C 8 model (suggested by others) also was considered.