Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...

Articles Figures Tables About

Principal modes of vibration

Systems with two or more degrees of freedom vibrate in a complex manner where frequency and amplitude have no definite relationship. Among the multitudes of unorderly motion, there are some very special types of orderly motion called principal modes of vibration. [Pg.179]

During these principal modes of vibration, each point in the system follows a definite pattern of common frequency. A typical system with two or more degrees of vibration is shown in Figure 5-2. This system can be a... [Pg.179]

The principal modes of vibration of the anthracene molecule, active in absorption and in fluorescence, have been given in ref. 58. This subsection is intended to provide a consistent analysis of the whole structure of the crystal absorption (reflectivity) and to discuss its shape in terms of vibron-two-particle states and collective couplings and of the spectral and dynamical (relaxation) effects to which they give rise. These active vibration modes are here summarized in decreased order of strength ... [Pg.64]

Figure 4.3 Principal modes of vibration between carbon and hydrogen in an alkane (a) symmetrical stretching, (b) asymmetrical stretching and the bending vibrations, (c) scissoring,... Figure 4.3 Principal modes of vibration between carbon and hydrogen in an alkane (a) symmetrical stretching, (b) asymmetrical stretching and the bending vibrations, (c) scissoring,...
The CO2 laser is a near-infrared gas laser capable of very high power and with an efficiency of about 20 per cent. CO2 has three normal modes of vibration Vj, the symmetric stretch, V2, the bending vibration, and V3, the antisymmetric stretch, with symmetry species (t+, ti , and (7+, and fundamental vibration wavenumbers of 1354, 673, and 2396 cm, respectively. Figure 9.16 shows some of the vibrational levels, the numbering of which is explained in footnote 4 of Chapter 4 (page 93), which are involved in the laser action. This occurs principally in the 3q22 transition, at about 10.6 pm, but may also be induced in the 3oli transition, at about 9.6 pm. [Pg.358]

As a case study in the calculation of the relation between co and q (also known as phonon dispersion relations) for three-dimensional crystals, we consider the analysis of normal modes of vibration in fee Al. As will be evident repeatedly as coming chapters unfold, one of our principal themes will be to examine particular problems from the perspective of several different total energy schemes simultaneously. In the present context, our plan is to consider the dispersion relations in Al as computed using both empirical pair functional calculations as well as first-principles calculations. [Pg.226]

This Av corresponds to some vibrational frequency of the molecule as a whole. Now for any given molecular model it is possible to a first approximation to analyze all the various modes of vibration in terms of a set of principal modes each involving its own set of... [Pg.151]

More recently, adaptive dampers have been implemented on many occasions. Their principal advantages compared to passive dampers lie in the easy tuning (by adjusting the current or voltage) both before and after the installation and for different modes of vibration. [Pg.545]

The initial conditions for the vibrational modes of the (gly-FI)+ were chosen via the quasiclassical normal-mode method, with the energy for each normal mode of vibration selected from the mode s 300 K harmonic oscillator Boltzmann distribution. A 300 K rotational energy of RT/2 was added to each principal axis of rotation of the projectile. Initial conditions for the diamond surface were chosen by first equiHbrating the surface to a 300 K Boltzmann distribution with 2 ps of molecular dynamics and scaHng the atomic velocities. The structure and atomic velocities obtained from this equilibration process are then used as the initial conditions for an equilibration run at the beginning of each trajectory. [Pg.130]

FIGURE 2 la The three normal vibrational modes of 11,0. Two of these modes are principally stretching motions of the bonds, but mode v2 is primarily bending, (b) The four normal vibrational modes of C02. The first two are symmetrical and antisymmetrical stretching motions, and the last two are perpendicular bending motions. [Pg.217]

Vibrations of the symmetry class Ai are totally symmetrical, that means all symmetry elements are conserved during the vibrational motion of the atoms. Vibrations of type B are anti-symmetrical with respect to the principal axis. The species of symmetry E are symmetrical with respect to the two in-plane molecular C2 axes and, therefore, two-fold degenerate. In consequence, the free molecule should have 11 observable vibrations. From the character table of the point group 04a the activity of the vibrations is as follows modes of Ai, E2, and 3 symmetry are Raman active, modes of B2 and El are infrared active, and Bi modes are inactive in the free molecule therefore, the number of observable vibrations is reduced to 10. [Pg.44]

The principal weakness in the Hinshelwood model is the assumption that the molecule contains s harmonic vibrational modes of identical frequency. In addition the number of modes s in the theory is not strictly associated with the number of vibrational degrees of freedom of the molecule. In fact s is usually taken as a free parameter in the model that is adjusted to obtain the best agreement with experiment. Typically the optimal value of s is on the order of half of the actual number of vibrational frequencies in the molecule. Another deficiency of the Hinshelwood model is that it cannot account for the downward curvature in the plot of 1 / kuni versus 1 / [M] that was mentioned at the end of the previous section. [Pg.424]

Spectral data are highly redundant (many vibrational modes of the same molecules) and sparse (large spectral segments with no informative features). Hence, before a full-scale chemometric treatment of the data is undertaken, it is very instructive to understand the structure and variance in recorded spectra. Hence, eigenvector-based analyses of spectra are common and a primary technique is principal components analysis (PC A). PC A is a linear transformation of the data into a new coordinate system (axes) such that the largest variance lies on the first axis and decreases thereafter for each successive axis. PCA can also be considered to be a view of the data set with an aim to explain all deviations from an average spectral property. Data are typically mean centered prior to the transformation and the mean spectrum is used a base comparator. The transformation to a new coordinate set is performed via matrix multiplication as... [Pg.187]

Raman and infrared (IR) spectra of pentazoles are difficult to measure. The main problem is the separation of the pentazole from impurities of the azide decomposition products that are almost impossible to remove quantitatively in the synthesis and can be formed even during the measurement at low temperatures. In Table 7, an overview is given over the principal vibration modes of the pentazole anion at the CCSD level of theory. [Pg.750]

The extension of the trajectory calculations to a system with any number of atoms is straightforward except for the quantization of the vibrational and rotational states of the molecules. For a molecule with three different principal moments of inertia, there does not exist a simple analytical expression for the quantized rotational energy. This is only the case for molecules with some symmetry like a spherical top molecule, where all moments of inertia are identical, and a symmetric top, where two moments of inertia are identical and different from the third. For the vibrational modes, we may use a normal coordinate analysis to determine the normal modes (see Appendix E) and quantize those as for a one-dimensional oscillator. [Pg.87]

The nuclear wave functions h/v)- corresponding to different electronic terms, i.e. for different /u are not orthogonal with each other, (Xnjln ) 0 at ji v for any ns and. It is this non-orthogonality that is the principal reason for the population change in many modes of the vibration system (multi-phonon transition), as it was first noticed by Frenkel [3]. [Pg.14]

The calculation of vibration spectra in terms of force constants is similar to the calculation of energy bands in terms of interatomic matrix elements. Force constants based upon elasticity lead to optical modes, as well as acoustical modes, in reasonable accord with experiment, the principal error being in transverse acoustical modes. The depression of these frequencies can be understood in terms of long-range electronic forces, which were omitted in calculations tising the valence force field. The calculation of specific heat in terms of the vibration spectrum can be greatly simplified by making a natural Einstein approximation. [Pg.203]


See other pages where Principal modes of vibration is mentioned: [Pg.343]    [Pg.343]    [Pg.231]    [Pg.147]    [Pg.1209]    [Pg.48]    [Pg.206]    [Pg.553]    [Pg.197]    [Pg.295]    [Pg.111]    [Pg.1003]    [Pg.787]    [Pg.94]    [Pg.136]    [Pg.128]    [Pg.166]    [Pg.194]    [Pg.97]    [Pg.55]    [Pg.215]    [Pg.366]    [Pg.695]    [Pg.32]    [Pg.121]    [Pg.268]    [Pg.54]    [Pg.12]    [Pg.268]    [Pg.248]    [Pg.107]   
See also in sourсe #XX -- [ Pg.302 ]




SEARCH



Vibrational modes

© 2024 chempedia.info