Mean-square Amplitude Quantities Research Articles

Molecular vibrations of sulphur trioxide: Application of the Keating bendings

A new type of internal coordinates referred to as Keating's bending is defined. A generalized bending is also introduced and includes the ordinary (Decius') bending, Keating's bending and the nonbond interatomic distance deviation as special cases. The theory is applied to the planar symmetrical XY 3 molecular model, for which the mean-square amplitude quantities are treated in particular. The Keating Force Field (KFF) is defined in analogy with the well-known VFF (Valence Force Field) and CFF (Central Force Field). Numerical computations are performed for SO 3. From the calculated Coriolis constants it is deduced that the KFF is much superior to both of the other approximations. An “exact” force field is produced by means of an experimental Coriolis constant as additional data. The force field is presented in terms of the valence, central and Keating coordinates. The analysis confirms that the Keating coordinates (which involve Keating's bendings) in this case of SO 3 are best suited as the basis of a force field approximation. Calculated mean amplitudes of vibration and the Bastiansen—Morino shrinkage effect are reported.

Spectroscopic studies of root-mean-square amplitudes for an X2Y4 molecular model withC 2h symmetry: Application to diphosphorus tetrachloride and diphosphorus tetraiodide

A brief description of the vibrational and structural analyses of the diphosphorus tetrachloride and diphosphorus tetraiodide molecules possessing atrans configuration with aC 2h symmetry has been given. On the basis of group theoretical considerations, symmetry coordinates have been constructed and analytical expressions for the various mean-square amplitude quantities in terms of the symmetrized mean-square amplitude matrices have been obtained for an X2Y4 molecular model of atrans configuration with aC 2h symmetry. This method has been applied to diphosphorus tetrachloride and diphosphorus tetraiodide, and the root-mean-square amplitudes at the room temperature have been computed for both the bonded and nonbonded atom pairs by making use of the recent vibrational and structural data. The results have been discussed on the nature of the characteristic bonds of the two molecules and compared with those of the other related molecules having similar chemical bonds.

Universal Parameter for the Extrapolation of Force Constants, Mean-Square Quantities, and Bond Energies of Halides

A general formula of the type Y = kω + C, which is simultaneously applicable to force constants, the reciprocals of the mean-square amplitude quantities σr, bond dissociation as well as thermochemical bond energies of halides, has been presented. ω, moreover, is given by the expression ω = 16 | χm − χx | + 3.5(χm − χx)2 + θf, where χm and χx are the Pauling electronegativities of the atoms forming the M–X bond, and where θf is a constant for each of the aformentioned molecular parameters and may have a nonzero value only in the fluorides. The θf values for the three parameters studied were found to be 0, − 11, and − 15 or − 30, respectively. The choice of the latter depends on whether χm > 1 (− 15), or χm ≤ 1(− 30), except for the NaX series for which θf = − 15.

Mean Amplitudes of Vibration in Molecules with Internal Rotation. IV. Theory with Application to Disulfur Dichloride

A normal-coordinate analysis has been performed for disulfur dichloride, and used to calculate the mean amplitudes of vibration. A generalized definition of framework mean amplitudes is proposed, and is applicable to cases where the torsional mode (for internal rotation) is kinematically coupled with other normal modes of vibration. According to this definition the total mean-square amplitude is composed of a sum of the framework mean-square amplitude and the torsional mean-square amplitude; the latter quantity contains all contributions where the torsional coordinate is involved. Numerical results of framework and torsional mean amplitudes are reported for S2Cl2 at various temperatures as functions of the angle of rotation θ. The approximate formula for framework mean-square amplitudes, 〈Δρ2〉frm = (α + β cosθ + γ cos2θ) / ρ2, has been subjected to detailed studies. The α, β, and γ coefficients were evaluated in terms of certain mean-square amplitude quantities, and the following statement was deduced from the properties of their classical limits at high temperatures: α(θ, T), β(θ, T), and γ(θ, T) approach θ-independent values when T → ∞. Furthermore, it is explained why the above formula with constant coefficients may give a very good approximation even at low temperatures when the conditions of classical limits are not fulfilled. This feature was actually found for S2Cl2 even at absolute zero.

Mean amplitudes of vibration for an XY4 molecular model with D2d symmetry: application to phosphate ion

A method has been developed on the basis of symmetry considerations for the determination of mean amplitudes of vibration for an XY4 molecular model of slightly distorted tetrahedron possessing aD 2d symmetry and applied to the phosphate ion. The available four fundamental frequencies ofB 2 andE species have been considered and the mean-square amplitude quantities have been computed at room temperature. The results are discussed briefly.

Mittlere Schwingungsamplituden des Trithiocarbonat- und Triselenocarbonations

Mean-square amplitude quantities and mean amplitudes of vibration for CS3 2⊖ and CSe3 2⊖ at 298 °K and 500 °K are computed. The method of Cyvin utilizing symmetry coordinates has been employed. A brief discussion of the results is given.

Spectroscopic Studies of Molecular Constants in some Polyatomic Molecules

Force constants have been evaluated by the WILSON group theoretical method for various ij3 molecules possessing a planar trigonal symmetry. Mean-square amplitude quantities and mean amplitudes of vibration for the bonded as well as nonbonded atom pairs have been computed at the temperatures T=0 and T=298 °K for C3 , Si3 , and XeF2 by the CYVIN method. The molar thermodynamic functions from spectroscopic data have been calculated for Si3 and XeF2 on the basis of a rigid rotator, harmonic oscillator model. BASTIANSEN-MORINO shrinkage effects have been calculated for various linear asymmetrical ijk, linear symmetrical ij2 and tetrahedral ij4 molecules at the temperatures T=0 and T=298 °K.

Mean amplitudes of vibration in some tetrahedral tetrahalogeno complexes

The Cyvin method, utilizing symmetry coordinates, has been employed here to compute the mean-square amplitude quantities, generalized mean-square amplitudes (mean-square parallel amplitudes, mean-square perpendicular amplitudes and mean cross products) and mean amplitudes of vibration for the bonded as well as non-bonded atom pairs at the temperaturesT=298°K andT=500°K for the tetrahedral tetrahalogeno complexes of zinc, cadmium, gallium, indium, thallium and arsenic. The non-linear behaviour of the mean amplitudes of vibration in terms of the electronegativities of the central atoms has been discussed, followed by a discussion on the replacement of peripheral atoms.

Mean Amplitudes of Vibration, Thermodynamic Functions, Molecular Polarizability and Absolute Raman Intensities of Σ+ g Modes in Carbon Subnitride

Vibrational and structural informations for carbon subnitride, a molecule possessing a linear symmetrical structure with the point group D∞h, have been used to determine the mean-square amplitude quantities and mean amplitudes of vibration for the bonded and nonbonded atom pairs at the temperatures T=0 and T=298°K by the Cyvin method. The molar thermodynamic functions have been computed for the temperature range 100-6000°K on the basis of a rigid rotator, harmonic oscillator model. Bond polarizabilities, molecular polarizability and polarizability deriva­tives corresponding to absolute Raman intensities of Σg + modes in the ground electronic state have been calculated by the Lippincott-Stutman method employing the delta-function potentials

Mean amplitudes of vibration: Molecules of the linear and planar AXYZ type

Applying Cyvin’s secular equation method the mean-square amplitude quantities are calculated for molecules of the linear and planar AXYZ type belonging to the symmetry Cs. The mean amplitude values for HN3, DN3, HNCS, HNCO, HCOF and DCOF molecules are evaluated at 300° K.

Mean amplitudes of vibration for the planar symmetrical XY4 molecular model: Application to a few lons from Raman data

The theory for the determination of the mean-square amplitudes of vibration for the planar symmetrical XY 4 molecular model possessing the D 4 h symmetry using the symmetry coordinates has been given. Both in-plane and out-of-plane vibrations have been considered. Analytical expressions for the various mean-square amplitude quantities in terms of the symmetrized mean-square amplitude matrix elements (Σ) have been obtained. The method has been applied to determine the mean amplitudes of vibration from the observed Raman frequencies for the anions of tetrachloroperiodic acid, tetrachloro- and tetrabromoauric acids and tetrachloroplatinous acid at the temperatures T = 0 and T = 298°K.

Mean Amplitudes of Vibration and Thermodynamic Functions of the Dioxides of Chlorine, Sulphur and Selenium

AbstractAn elaborate discussion of the various methods regarding the molecular structure and vibrational spectral studies of the dioxides of chlorine, sulphur and selenium has been reported. The computed numerical values of (1) meansquare amplitude quantities, (2) generalized mean‐square amplitudes (meansquare 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 given for these molecules. Using the fundamental frequencies as well as normal frequencies the molar thermodynamic functions have been calculated on the basis of a rigid rotator, harmonic oscillator approximation for the temperature range from 100° to 1700°K.

Mean Amplitudes of Vibration of Some Octahedral XY6 Type Molecules

AbstractA brief survey on the vibrational analyses of the octahedral XY6 type molecules belonging to the symmetry point group Oh has been presented. For this type of molecules, the general theory of mean‐square amplitude matrices has been applied. The analytical expressions for the sixteen meansquare amplitude quantities as linear combinations of the symmetrized meansquare amplitude matrix elements have been obtained. The secular equations for the calculation of normal frequencies involving the symmetrized meansquare amplitude matrix elements have been given. For the twelve molecules namely, SF6, SeF6, TeF6, MoF6, WF6, UF6, NpF6, PuF6, ReF6, OsF6, IrF6 and PtF6, the computed values of (1) mean‐square amplitude quantities and (2) mean amplitudes of vibration at the temperatures T=0 and T=298°K.

Mean Amplitudes of Vibration of Some Tetrahedral XY4 Type Molecules‐Part I

AbstractThe general theory of mean‐square amplitude matrices has been briefly applied to the tetrahedral XY4 type molecule. The expressions for the twelve mean‐square amplitude quantities as linear combinations of the symmetrized mean‐square amplitude matrix elements have been given. The secular equations for the normal frequencies involving symmetry force constants and symmetrized mean‐square amplitude matrix elements have been obtained. The expressions for the generalized mean‐square amplitudes (mean‐square parallel amplitudes, mean‐square perpendicular amplitudes, and mean cross products) in terms of the Σ matrix elements have been given. For the molecules SiF4, OsO4, RuO4, CH4, CD4, CD4 and Cl4, the computed values of (1) mean‐square amplitude quantities, (2) generalized mean‐square amplitudes and (3) mean amplitudes of vibration at the temperatures T=O and T=298°K have been reported.

Vibrational mean-square amplitude matrices: Part VII. Treatment of tetrahedral XY4 molecules

The theory of mean-square amplitude matrices previously evaluated, is applied to the tetrahedral XY 4 molecular model. The elements of the symmetrized mean-square amplitude matrix, as well as a set of harmonic force constants, are related to the normal frequencies. The expressions are given for twelve mean-square amplitude quantities (nine of them being interaction terms) as linear combinations of the symmetrized mean-square amplitude matrix elements.

Vibrational mean-square amplitude matrices: Part IX. Germanium tetrachloride

The theory of mean-square amplitude matrices (Σ) is applied to the germanium tetrachloride molecule. The following quantities are computed from the spectroscopic vibrational frequencies, along with the electron-diffraction value for the ClCl mean-square amplitude of vibration: (a) Σ matrix elements at 298°K, (b) additional mean-square amplitude quantities at 298°K, including the GeCl mean-square amplitude of vibration. This value is quite compatible with the reported result from electron-diffraction. (c) L-matrix elements, (d) force constants, (e) perpendicular mean-square amplitudes at 298°K for the GeCl and ClCl atom pairs.

Vibrational mean-square amplitude matrices—II

The previously reported theory for treating mean-square amplitudes of vibration of polyatomic molecules is applied to the parallel vibrations of linear triatomic molecules. The mean-square amplitude quantities, as well as the harmonic force constants are related to the normal frequencies. Furthermore, approximate expressions are given for a set of mean-square amplitude quantities. It is pointed out that in the general case, three mean-square amplitudes of vibration might be detected from electron-diffraction, and represent sufficient informations for calculating all the vibrational constants of the parallel vibrations. A special treatment of the linear symmetrical XY2 molecular model is included.

Vibrational mean-square amplitude matrices—III Application to some linear triatomic molecules

The mean-square amplitude quantities of the following linear triatomic molecules have been studied: 12CO2, 13CO2, 14CO2, OCS, OCSe, CS2, SCSe, CSe2 and SCTe. The meansquare amplitudes of vibration for the bonded interatomic distances are found to possess characteristic values, namely: (298°K) 0·0012 2 for CO, about 0·0015 2 for CS, and 0·0016 2 for CSe. The computed value for CTe is about 0·0018 A2. Also the mean amplitudes of vibration (A units) for the bonded and non-bonded distances in all the present molecules are given.