Mean Square Amplitudes Of Vibration Research Articles

Generalized mean-square amplitudes of vibration in some XY 4 type ions

The generalized mean-square amplitudes of vibration for ten ions ZnCl 2- 4, CdBr 2- 4, HgI 2- 4, PO 3- 4, MoO 2- 4, WO 2- 4, SiO 4- 4, FeO 2- 4 and TcO - 4 have been calculated using Cyvin's method at three temp eratures T=0°K, T=298°K, and T=500°K.

Hexachlorodisilane: vibrational spectrum, structure and internal rotation

The infrared and Raman spectra of liquid Si2Cl6 combined with accurate polarization measurements eliminate the high barrier staggered D3d structural model and favor the D3h or D3h' models for this molecule. The eleven active fundamentals for the D3h or the D3h' point groups were located and assigned. These results agree with the low internal barrier model found from electron diffraction studies and provide information for the evaluation of the mean square amplitudes of vibration. These can be used with the diffraction results to more accurately assess the barrier to internal rotation.

Verfeinerung der Kristallstruktur von Benitoit BaTi[Si3O9

The crystal structure of benitoite has been refined on the basis of powder and single-crystal diffractometer data. Zachariasen's original structure has been fully confirmed. For the least-squares refinement, three-dimensional intensity data with l = 2n + 1 were used. Thus, the Si3O9 part of the structure could be refined with high accuracy neglecting the presence of heavy cations. The standard deviations of atomic distances were in the range of .002 to .004 Å, those of bond angles were .1 to .5°. Anisotropic mean-square vibrational amplitudes were determined with σ between 3 and 6 percent. With these results, a rough relative estimation of bindung forces could be obtained. The effect of thermal vibration on bond lengths, computed on a purely geometrical basis, has been demonstrated.

The structure of 4-chloronortricyclene

The structure of 4-chloronortricyclene in the gas phase was determined by electron diffraction. Interatomic distances and mean square amplitudes of vibration were evaluated. The cyclopropyl ring bonds are 1.510 Å, typical of C 3 rings. A possible correlation between increased cyclopentyl bond lengths and increased flap angle is noted, as is the apparent decrease in these bond lengths with substitution of electronegative groups. A summary of CCl bond lengths in a variety of molecular environments completes the discussion.

Molecular constants of dicyanoacetylene

The potential energy constants of dicyanoacetylene have been evaluated using a general valence force field. Applying the theory of Mean-square amplitudes of vibration, the parallel and perpendicular mean square amplitudes have been calculated. The shrinkage effects are obtained for the different atomic distances. Coriolis constants are determined for the various possible couplings. The thermodynamic functions are estimated assuming a rigid rotor, harmonic oscillator approximation for the temperature of 100 to 1000° K.

Molecular constants of non-bridged X2Y6 systems :

The generalised mean-square amplitudes of vibration of seven molecules Ge2 H6 , Ge2 D6 , Si2 H6, Si2 D6 , Si2 Cl6 , C2 F6 and C2 Clc have been evaluated by the Σ matrix method. The Coriolis coupling coefficients have also been computed using the method of Meal and Polo. The above quantities have been compared with the available experimental data.

Intermolecular bonding and characteristic lattice temperatures in organotin halides

A set of Debye temperatures and characteristic temperatures of organotin halides is given based on Mössbauer spectroscopic measurements and Thirring's expansion of frequency moments, confirming that the second negative moment is directly related to the mean-square vibrational amplitude in the Debye–Waller factor and that the extent and types of intermolecular forces in strongly co-ordinated polymers are quite different from those in monomers.

Force Constants, Generalised Mean Square Amplitudes and Coriolis Coupling Coefficients of P2I4

Normal co-ordinate analysis has been carried out for diphosphorous tetra iodide (P2I4) on the basis of general valence force field and the force constants have been evaluated. The values of the generalised mean square amplitudes of vibration calculated at 300 °K indicate that the vibrational amplitudes are not large even though the frequencies of vibrations are rather low. The Coriolis coupling coefficients also have been evaluated.

Test of a Melting Criterion for Cubic Metals

The electron gas model for the lattice dynamics of cubic metals propounded by one of us earlier has been used to test Lindemann's melting criterion by calculating the ratio γm of the mean-square amplitude of thermal vibrations to the square of the interatomic spacing at the melting point for a number of cubic metals belonging to different groups of the periodic table. The calculated γm has roughly the same value for elements belonging to one particular group, but the spread in γm values is quite large from one group to another. It is concluded that Lindemann's hypothesis of constant γm for all solids is inadequate.

Force field, Coriolis coupling coefficients, generalized mean square amplitudes of vibration and shrinkage constants of XeF4 and XeOF4

Normal coordinate analyses of XeF4 and XeOF4 have been carried out using the General Valence and Urey —Bradley Force Fields, respectively. The Coriolis coupling coefficients, generalized mean square amplitudes and shrinkage constants have also been evaluated. The values obtained in both cases have been discussed in relation to each other.

Mean amplitudes of vibration and coriolis coupling constants of some XY3Z type silicon compounds

A theoretical treatment of the generalised mean-square amplitudes of vibration of some silicon compounds of XY3Z type molecules is presented. The numerical values evaluated for the parallel and perpendicular mean-square amplitudes for all the bonded and non-bonded atom pairs are tabulated. The L matrix elements and Coriolis coefficients have also been evaluated for these molecules.

Generalised Mean-square Amplitudes of Vibration and Coriolis Coupling Coefficients for C3H6 and C3D6

Generalised vibrational mean-square amplitudes for cyclopropane and cyclopropane-d6 are calculated using spectroscopic data, and the values at 300 °K are reported. A complete set of the Coriolis coupling coefficients for these molecules has been evaluated.

Calculation of mean-square vibrational amplitudes

Calculation of mean-square vibrational amplitudes

Effective debye temperature for surface vibrations of a crystal

Recent studies show that the mean square amplitude of vibrations of atoms on the surface of a crystal is almost twice that in the bulk. We show that this result follows from simple physical arguments.

Effect of Molecular Vibrations on Gas Electron Diffraction. II. Probability Distribution Function and Phase Parameter for Polyatomic Molecules

Abstract A method has been presented for calculating the first, second, and third moments of the displacement of an internuclear distance in a polyatomic molecule, 〈Δr〉, 〈Δr2〉, and 〈Δr3〉. The moments, which depend on the frequencies of normal vibrations, the cubic potential constants, and the temperature of the system, have been related to the extent of distortion of the probability distribution function of the internuclear distance from a Gaussian function caused by the anharmonicity in the potential function. An approximate expression has been obtained for the phase parameter κ in the molecular intensity of gas electron diffraction in terms of the moments. Numerical results are given for CO2, CS2, SO2, H2O, D2O, CH4, and CD4 by the use of experimental or estimated cubic potential constants. The phase parameters for bonded distances agree with simple estimates based on “diatomic approximation” to the accuracy of current experimental studies, while the phase parameters for nonbonded distances are similar in magnitude to those for bonded distances in spite of their larger mean-square amplitudes of vibration.

Radial distribution functions in solid cobalt

Radial distribution functions have been calculated from X-ray diffraction data for Co-α(h.c.p.) and Co-β(f.c.c.) near the allotropie transformation temperature. These functions provide values of lattice parameters, mean square amplitudes of vibration, vibrational coupling factors, and the mean force potential function. The mean square amplitude of vibration, u ̄ s 2 , for α and β at 417° C are 0.0155 Å 2 and 0.0168 Å 2, which correspond to Debye temperatures of 332°K and 320°K, respectively. The Debye temperature and coupling factors obtained for β at 417° C were consistent with a second experimental distribution function determined at 678° C, indicating that those parameters are essentially invariant over that range of temperature. From the mean force potential functions the elastic modulii for nearest neighbor directions have been determined. In both α- and β-phase the measured value is approximately 2.0 × 10 12 dyn/ cm 2. The α-phase modulus may be estimated from other data and is found to be in excellent agreement. The difference between the nearest neighbor potential minima in the α- and β-phases at 417° C is 100 cal/g-atom. Since this energy difference is almost exactly the heat of transformation, it appears that pertinent thermodynamic values may be derived from the X-ray results.

Fundamental assignments and molecular properties of CF 2ClBr

The fundamental assignments suggested by Plyler and Acquista for difluorochlorobromomethane were re-examined, and new assignments satisfying Rayleigh's rule were recommended. On the basis of the new assignments, the sum of the fundamentals fits in very well with Bernstein's substitution rule, and also the medium intensity band at 977 cm −1 is accounted for very satisfactorily. A normal coordinate calculation was carried out on the basis of the new assignments, which favored our assignment of the 440-cm −1 band to the symmetric ν 4 fundamental rather than to the antisymmetric CF 2 rocking ν 8, as previously suggested by Plyler and Acquista. The calculated values of all the potential constants were compared with those of the corresponding ones in similar substituted halomethanes. They compared very closely, both in magnitude and sign, for all constants but one. This lent further support to the principle of transferability of potential constants. The calculated PED showed considerable coupling among the CCl stretching, the CBr stretching, and the low-lying bending A′ vibrations. The mean square amplitudes of vibration for the bonded and nonbonded atoms, the Coriolis coupling coefficients, and the thermodynamic properties for the ideal gas were calculated on the basis of the new assignments.

Kraftkonstanten tetraedrischer Oxoanionen des Mangans (MnO4-, MnO42-, MnO43-) und Rutheniums (RuO4, RuO4-, RuO42-)

Force constants of a modified valence force field (M.V.F.F.), a general valence force field (G.V.F.F.) and the Urey-Bradley force field (U.B.F.F.) are calculated for MnO4 -, MnO4 2-, MnO4 3-, RuO4, RuO4 -, and RuO4 2-. In the series of the manganates as well as the ruthenates the metal-oxygen stretching force constants show a nearly linear dependence on the formal oxydation number of the central metal atom. This is explained on the basis of molecular orbital theory. The L matrix method of Morino is used to calculate the mean square amplitudes of vibration for the same ions.

Generalized Mean-Square Amplitudes of Vibration, Mean Amplitudes of Vibration, and Shrinkage Effects of Some Trigonal Bipyramidal XY5 and XY3Z2 Molecules.

Generalized Mean-Square Amplitudes of Vibration, Mean Amplitudes of Vibration, and Shrinkage Effects of Some Trigonal Bipyramidal XY5 and XY3Z2 Molecules.

Environmental Effects on the Infrared Frequencies of Polyatomic Molecules. Evaluation of Perturbation-Theory Parameters for HCN and DCN

Buckingham's equations describing solvent-induced vibrational frequency shifts for polyatomic solute molecules are applied to experimental solvent-shift data for HCN and DCN. The equations were transformed to internal coordinates such that the expressions for the frequency shifts involve the first (Vi′) and second (Vii″) derivatives of the solute—solvent interaction energy with respect to the internal vibrational coordinates. Numerical values for these energy derivative terms were determined from the experimentally observed shifts of the six fundamentals of HCN—DCN in a series of solvents in a manner such that the problem was overdetermined. (Note that this method fails when applied to diatomic molecules.) It is found that second-derivative terms make significant and often dominant contributions to the observed shifts and that solvent perturbation of a given internal vibration in a molecule materially affects the shifts of other vibrational bands in the molecule. The equations of Huong et al. relating the relative frequency shifts of the stretching vibrations of HCN and DCN are shown to be a consequence of the equations of Buckingham. Expressions are given for the solvent perturbation contribution to 〈Qi〉 and 〈Qi2〉. Quantitative estimates of average bond extensions and mean-square vibrational amplitudes are given for certain cases.