Vibrational displacement

The appearance of the (normally small) linear term in Vis a consequence of the use of reference, instead of equilibrium configuration]. Because the stretching vibrational displacements are of small amplitude, the series in Eqs. (40) should converge quickly. The zeroth-order Hamiltonian is obtained by neglecting all but the leading terms in these expansions, pjjjf and Vo(p) + 1 /2X) rl2r and has the... 

The molecular structures were rendered with good-quality shading on a blue background. Isosurfaces produced from cube files or checkpoint files also looked nice. Molecular vibrations can be animated on screen and vibrational displacement vectors displayed. The vibrational line spectrum may be displayed too, but the user has no control over the axes. There is no way to set the background color. The display can be saved using several image file formats. 

As for the change of dipole moment, the change of polarizability with vibrational displacement x can be expressed as a Taylor series... 

Noncontacting eddy-current vibration displacement probe adjacent to ... 

Sliop test facilities should include instrumentation with the capability of continuously monitoring and plotting rpm, peak-to-peak displacement, and phase angle (X-Y-Y ). Presentation of vibration displacement and phase marker by use of an oscilloscope makes visualization easier. 

In a review of the subject, Ubbelohde [3] points out that there is only a relatively small amount of data available concerning the properties of solids and also of the (product) liquids in the immediate vicinity of the melting point. In an early theory of melting, Lindemann [4] considered that when the amplitude of the vibrational displacements of the atoms of a particular solid increased with temperature to the point of attainment of a particular fraction (possibly 10%) of the lattice spacing, their mutual influences resulted in a loss of stability. The Lennard-Jones—Devonshire [5] theory considers the energy requirement for interchange of lattice constituents between occupation of site and interstitial positions. Subsequent developments of both these models, and, indeed, the numerous contributions in the field, are discussed in Ubbelohde s book [3]. 

During the test, the rotor s speed, vibration displacement, and corresponding phase angle, filtered to rotor speed (IX), shall be measured and recorded. 

Figure 7. Steicoprojeclions of the 514, 490, 398, and 388-cm normal modes of showing the equilibrium (white circles and black bonds) and vibrationally displaced molecules (shaded circles and white bonds). Beneath each mode is a schematic representation of the predominant pattern of motion and the linear combination of achiral cyclohexanone motions that generate the vibration. (Reproduced with permission from ref. 77. Copyright 1984 American Chemical Society.)...

The vibrational displacements corresponding to the anharmonic terms in the potential are most pronounced in the directions away from the stronger bonding interactions, in which restoring forces are weaker. Thus, for the tetrahedral site symmetry of the diamond structure, the anharmonicity causes a larger mean-square displacement in directions opposite to the covalent bonds. At lower... 

That the vibrational displacements of the valence shell electrons may be smaller than those of the core electrons can be qualitatively understood by considering the vibrations of two identical, strongly bonded atoms. When the atoms vibrate in phase, they behave as a rigid body, so all shells will vibrate equally. But when they vibrate out of phase, the density near the center of the bond will be stationary, assuming the average static overlap density to be independent of the vibrations. This apparently invariant component of the valence density would contribute to a lowering of the outer-shell temperature... 

Since any general displacement is a superposition of translational, rotational and vibrational displacements it is possible to redescribe the motion of the molecule in terms of overall translations and rotations, and the normal modes of vibration. Using the character tables, it is possible to decompose the symmetry of the general displacement into the symmetries of the different types of motion. 

Deformation density, as originally conceptualized represents the difference between crystallographically observed electron density and calculated densities of the spherical atoms, which consitute the so-called promolecule. The effects of vibrational displacement, represented by ellipsoids in Figure 5.19, and ignored when defining a promolecule by spherical atoms, are most likely... 

The ratio of yHF to yDF was assumed to be 21/4 in deriving Eqn. (10). For yHF — 20-30°,Eqn.(10)givesadipolemomentdifferenceof0.03-0.08 D, as observed. Thus, the dipole moment data and the hyperfine interactions support a nearly linear equilibrium hydrogen bond, with large amplitude vibrational displacements from equilibrium. It must be emphasized that none of the experimental evidence discussed so far, including this, precludes the hydrogen bond from having a small departure, on the order of 5°, from linearity at equilibrium. 

This is the basic equation used to derive normal coordinate analysis [110] as well as to define the vibrational quantities to be calculated using molecular orbital theory [79,94], The coefficients, g , are the forces acting on the nuclei, which are zero at equilibrium geometiy. This leaves the quadratic terms Vs the first term in the change of potential energy with instantaneous vibrational displacement. The quadratic terms Fy, are conveniently ordered as a matrix which is known as the force field or force matrix. These terms correspond to the derivatives of the potential energy V ... 

Interatomic (internuclear) distances and vibrational displacements are often expressed in the non-SI unit Angstrom, where A = 10"lom = 0.1 nm = 100pm. 

If it is assumed that the interaction is an exponential repulsion [equation (1)], where x is the distance between the impinging atom and the molecule, and that the amplitude of the vibrational displacement is much less than /, then one may show that [13]... 

Figure 3.9 Moore s vibrator-rotator collision model, used to adapt semi-classical V-T theory to the explanation of V-R processes. AR is the vibrational displacement. The velocity of impact negiects transiational contributions.

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