Mean-square Amplitude Matrices Research Articles

ChemInform Abstract: SHRINKAGES OF THE INTERNUCLEAR DISTANCES IN SOME HEXAFLUORIDES OF OCTAHEDRAL SYMMETRY

AbstractEine kurze Analyse der Verkürzung der interatomaren Abstände in molekularen Systemen durch thermische Bewegungen wird gegeben.

Shrinkages of the internuclear distances in some hexafluorides of octahedral symmetry

A brief analysis of the shrinkages or shortening of the internuclear distances in many molecular systems has been made. Analytical expressions for the linear shrinkage and nonlinear shrinkage have been derived in terms of the meansquare perpendicular amplitudes. Analytical expressions for the generalized mean-square amplitudes (mean-square parallel amplitudes, mean-square perpendicular amplitudes, and mean cross products) for an octahedralXY6 molecule have been given in terms of the mean-square amplitude matrices. The linear and nonlinear shrinkages for the hexafluorides of sulfur, selenium, tellurium, molybdenum, technetium, ruthenium, rhodium, tungsten, rhenium, osmium, iridium, platinum, uranium, neptunium, and plutonium have been computed at 298°K and 500°K, and the results have been briefly discussed.

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.

Potential Constants, Mean Amplitudes of Vibration, Shrinkage Constants and Coriolis Coupling Coefficients of Monohalo Acetylenes

The potential constants are obtained for the monohaloacetylenes using the general valence force field. The theory of the mean-square amplitude matrices is applied to these molecules and the parallel and perpendicular mean-square amplitudes for the interatomic distances are calculated. The perpendicular amplitudes have been used for calculating the shrinkage effect. The Coriolis coupling coefficients are also obtained for all these molecules.

Mean amplitudes of vibration: Planar X2Y4 molecules

The theory of mean-square amplitude matrices is applied to the ethylene type of molecules. The mean amplitudes of thermal vibrations of atomic distances in these molecules are evaluated at 300‡K from their vibrational frequencies.

Mean amplitudes of vibration: Bent XYZ molecules

The theory of mean-square amplitude matrices is applied to the bent XYZ molecules. The principal mean amplitudes of vibration for the molecules investigated here are reported for 300° 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: Part VIII. Mean-square parallel and perpendicular amplitudes of tetrahedral XY 4 molecules

The theory of computing mean-square parallel and perpendicular amplitudes from spectroscopic data, developed by Morino and Hirota, is summarized. A slightly modified method is proposed, connecting these quantities to the concept of mean-square amplitude matrices. As a particular case, the tetrahedral XY 4 molecular model is considered, and all the nonvanishing generalized mean-square amplitudes for the bonded and nonbonded distance types are given. The quantities are expressed as linear combinations of the symmetrized mean-square amplitude matrfix.

Vibrational mean-square amplitude matrices—XI Cyclopropane and cyclopropane- d2

The theory of mean-square amplitude matrices (Σ) is applied to the cyclopropane and cyclopropane- d 6 molecules. Numerical values at 298°K are given for (i) the symmetrized Σ matrix elements, and (ii) the six mean amplitudes of vibration for the various bonded and non-bonded atom pairs.

Vibrational mean-square amplitude matrices—I

Mean-square amplitudes of vibration in the theory of small harmonic vibrations of polyatomic molecules are studied. The quantities are treated as a kind of vibrational constants, similarly to the familiar treatment of harmonic force constants. After including certain interaction terms to form the mean-square amplitude matrix Σ, it was possible to develop two secular equations, the former one involving the matrices Σ and G−1, the latter one the matrices Σ and F. Here F and G are used for the energy matrices in Wilson's notation. Applications of the secular equations, including the ordinary one involving F and G, are discussed, and an isotope rule for the mean-square amplitude matrix elements is evaluated. Ultimately some approximation methods for computing mean-square amplitude matrices are reported, and are based on approximate expressions for the hyperbolic cotangent.

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.

Vibrational Mean-Square Amplitude Matrices. V. Treatment of Bent Symmetrical XY2 Molecules with Application to Nitrogen Dioxide.

Vibrational Mean-Square Amplitude Matrices. V. Treatment of Bent Symmetrical XY2 Molecules with Application to Nitrogen Dioxide.

Vibrational Mean-Square Amplitude Matrices. IV. Treatment of Tetrahedral X4 Molecules with Application to Phosphorus.

Vibrational Mean-Square Amplitude Matrices. IV. Treatment of Tetrahedral X4 Molecules with Application to Phosphorus.