Mean-square Amplitude Matrix Research Articles

Planar Symmetrical XY3and Boron Trihalides: Mean-Square Amplitudes and Keating Coordinates

Abstract The E' block of the mean-square amplitude matrix (Σ) for the planar symmetrical XY3 molecular model is treated theoretically in terms of the Keating coordinates. A numerical example for 11BF3 is given. Also the mean amplitudes of vibration (ι) and Bastiansen-Morino shrinkage effect (Δ) are considered. New relations which connect the mean-square amplitudes and force constants are given. Previous works on mean amplitudes and related quantities for boron trihalides from vibrational spectroscopy and gas electron diffraction are reviewed, and some new calculations are reported.

General methods of analysing molecular vibrations

An acute need may arise to develop for the complete analysis of molecular vibrations practically convenient general methods based on coordinates other than “chemical coordinates”. One reason is the proven proposition: Among independent internal coordinates corresponding to a molecule, there cannot be one which describes a small displacement of a chemical group as a whole relative to a certain molecular plane, provided this group contains more than two linearly or three non-linearly arranged atoms. Two methods are presented in some detail. The first is based on the use of X 0 δ coordinates which are components of “bond vectors” in the “sown” (for each “bond”) Cartesian coordinate system. The second method utilizes X 0 coordinates, i.e. the components of atomic displacements in the “down” (for each atom) Cartesian coordinate system. Computation of the torsional vibration of transdichloroethane is given as an example illustrating the first method. The Mayants treatment of the symmetry of a molecule, proceeding from elementary considerations which do not use the group theory explicitly and are valid for any coordinates, is expounded in a somewhat improved version. The peculiarities arising when considering the mean-square amplitude matrix, Σ, in X 0 δ and X 0 coordinates are also discussed.

On some relations involving the mean-square amplitude matrix

On some relations involving the mean-square amplitude matrix

Secondary Isotope Effects on Amplitudes of Molecular Vibrations

Secondary isotope effects on amplitudes of bond oscillations in harmonically vibrating polyatomic molecules have been studied using normal coordinate theory. Morino's scheme for circumventing the L matrix by expanding the normal coordinate mean-square amplitude matrix in integral powers of the Λ matrix is extended and found to lead to some simple and useful inequalities. Upper and lower limits of bond amplitudes are established which are rigorously independent of masses of atoms other than the atoms in the bond. It is shown that the upper limit of secondary mass dependency vanishes with increasing temperature much more rapidly than does the primary mass dependency. Even at absolute zero, secondary isotope effects on bond amplitudes are found to be relatively very much smaller than secondary isotope effects on bond frequencies. Numerical results of ``exact'' and various approximate approaches for calculating root-mean-square amplitudes are presented for the C–I bonds in CH3I and CD3I, and for the C=C bonds in C2H4 and C2D4. The smallness of secondary isotope effects on amplitudes indicates that their coupling with Morse bond asymmetry is not a major source of some recently reported secondary isotope effects on bond lengths.

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 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.

Vibrational mean-square amplitude matrices: Part XIX. Coriolis coupling coefficients in planar symmetrical XY3 molecules

The ζ-values (Coriolis coefficients) for the planar symmetrical XY 3 molecular model have been expressed by means of (a) force constants and (b) mean-square amplitude matrix elements. In the latter case (b) a recently developed relation between the Σ and the C natrix (in Meal and Polo's notation) has been applied for the first time. Numerical ζ-values are reported for the four trihalides of boron.

Vibrational mean-square amplitude matrices: Part XX. Coriolis coupling coefficients in bent symmetrical XY2 molecules

The ζ-values (Coriolis coefficients) for the bent symmetrical XY 2 molecular model are studied. These quantities have been expressed in terms of (a) force constants, (b) mean-square amplitude matrix elements, and (c) inverse force constant matrix elements. Numerical values are reported for two isotopic species of nitrogen dioxide.

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 XVIII. The Bastiansen-Morino shrinkage effect of symmetrical XY3 and XY4 molecules

The nonlinear Bastiansen-Morino shrinkage effects are studied for (a) planar symmetrical XY 3, and (b) tetrahedral XY 4 molecular models. In both of these cases the so-called anharmonicity terms vanish. The shrinkage effects are expressed algebraically in terms of the mean-square amplitude matrix elements. Numerical values are given for the trihalides of boron, and germanium tetrachloride. In the case of GeCl 4 the result is compared with that from electron-diffraction data. Finally, the electron-diffraction r g values for this molecule are refined to fit accurately the presently calculated shrinkage effect.

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—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.