Translational vibration

Hayden B E and Lament C L A 1989 Coupled translational-vibrational aetivation in dissoeiative hydrogen adsorption on Cu(110) Phys. Rev. Lett. 63 1823... 

Before considering other concepts and group-theoretical machinery, it should once again be stressed that these same tools can be used in symmetry analysis of the translational, vibrational and rotational motions of a molecule. The twelve motions of NH3 (three translations, three rotations, six vibrations) can be described in terms of combinations of displacements of each of the four atoms in each of three (x,y,z) directions. Hence, unit vectors placed on each atom directed in the x, y, and z directions form a basis for action by the operations S of the point group. In the case of NH3, the characters of the resultant 12x12 representation matrices form a reducible representation... 

Molecular enthalpies and entropies can be broken down into the contributions from translational, vibrational, and rotational motions as well as the electronic energies. These values are often printed out along with the results of vibrational frequency calculations. Once the vibrational frequencies are known, a relatively trivial amount of computer time is needed to compute these. The values that are printed out are usually based on ideal gas assumptions. 

This argument can be generalized to any number of subsystems and energy levels. For the case of a molecular system in a given electronic state, the factorization into translational, vibrational, and rotational contributions gives... 

Summarizing, in the crystal there are 36 Raman active internal modes (symmetry species Ug, hig, 2g> and 26 infrared active internal modes (biw b2w hsu) as well as 12 Raman active and 7 infrared active external vibrations (librations and translations). Vibrations of the type are inactive because there appears no dipole moment along the normal coordinates in these vibrations of the crystal. 

Continuum models are being increasingly used to study protein-ligand recognition [115]. Most studies have considered series of similar ligands or protein mutants and focussed on binding free energy differences. This leads to partial cancelation of some troublesome contributions, especially rotational/translation/vibrational entropy... 

The transition state theory of reaction rates [21] provides the link between macroscopic reaction rates and molecular properties of the reactants, such as translational, vibrational, and rotational degrees of freedom. For an extensive discussion of transition state theory applied to surface reactions we refer to books by Zhdanov [25] and by Van Santen and Niemantsverdriet [27]. The desorption of a molecule M proceeds as follows ... 

The partition functions Q contain separate terms for translation, vibration and rotation ... 

And third, energy is possessed by virtue of the potential energy, and the translational, vibrational, rotational energy states of the atoms and bonds within the substance, be it atomic, molecular or ionic. The energy within each of these states is quantized, and will be discussed in greater detail in Chapter 9 within the subject of spectroscopy. These energies are normally much smaller than the energies of chemical bonds. 

BC bond is fixed. It should be noted that PESs represent the energy of the collection of atoms at absolute zero, i.e., they provide no information on the translational, vibrational, or rotational energies that are normally present. The latter forms of energy clearly must be considered to determine thermochemistry under conditions of practical interest. 

With 4o/44Ca- and 92/ioojio- data the differentiation of the translational and librational modes was obtained for the molybdates and tungstates with scheelite structure 98, 100, 101). Tables 29 and 30 reproduce these results. In contrast to early findings, the lower frequency bands were found to be librations and not translations. Furthermore, the band associated with Vi for CaMo04 and CaW04 cannot be expressed as a pure deformational vibration of the MOl group, but this mode is coupled with the translational vibrations, as indicated in Table 29. The relationship between the translational vibration, E, and the square root of the mass of the cation for compounds of the type AMO 4 (A = Ca, Sr, Ba, Pb M = Mo, W) was determined to be linear (100). 

The interpretation of the lattice vibrations for scheeUte type molybdates or tungstates with relatively light cations, Ca or Sr, has indicated that the lowest translational vibrations are produced by Mo—Mo or W—W motions respectively, while those at higher frequency are from cation-cation motions 98). This has not been found, however, in the case of the barium or lead compounds. The librational frequencies have been found to decrease hnearly with the ionic radius of the cation for AMO4 type compounds, where A = Ca, Sr, Ba, or Pb and M =Mo or W 98). 

Solid state spectra can be measured by the Attenuated Total Reflectance or Multiple Internal Reflection methods as well as by simple transmission techniques (222). Brooker (223) has reported the discovery of the translational vibrations of NaN02, NaNOs, and CaCOs (calcite). The orientation of the bonds in (U02)(N0s)2. 6 H2O has been ascertained from the spectra obtained in this manner (223). 

Methylene is formed in the primary step in the photolysis of CH2N296 and CH2C0.95 There is evidence that the translational, vibrational, and electronic energies may depend on the precursor and on the wavelength of photolysis.18 h. 

The translational vibration frequencies of n-C H2 +2 (n 6,8,10) against Cu(110) surfaces at temperatures near 150 K have been measured by helium atom scattering (139). 

The Boltzmann distribution of the populations of a collection of molecules at some temperature T was discussed in Section 8.3.2. This distribution, given by Eq. 8.46 or 8.88, was expressed in terms of the quantum mechanical energy levels and the partition function for a particular type of motion, for instance, translational, vibrational, or rotational motion. It is useful to express such population distributions in other forms, particularly to obtain an expression for the distribution of velocities. The velocity distribution function basically determines the (translational) energy available for overcoming a reaction barrier. It also determines the frequency of collisions, which directly contributes to the rate constant k. 

In Section 5.1, we noted that to a good approximation the nuclear motion of a polyatomic molecule can be separated into translational, vibrational, and rotational motions. If the molecule has N nuclei, then the nuclear wave function is a function of 3/V coordinates. The translational wave function depends on the three coordinates of the molecular center of mass in a space-fixed coordinate system. For a nonlinear molecule, the rotational wave function depends on the three Eulerian angles 9, principal axes a, b, and c with respect to a nonrotating set of axes with origin at the center of mass. For a linear molecule, the rotational quantum number K must be zero, and the wave function (5.68) is a function of 6 and only only two angles are needed to specify the orientation of a linear molecule. Thus the vibrational wave function will depend on 3N — 5 or 3N — 6 coordinates, according to whether the molecule is linear or nonlinear we say there are 3N — 5 or 3N — 6 vibrational degrees of freedom. 

Understanding chemical reactions has been a major preoccupation since the historical origins of chemistry. A main difficulty is to reconcile the macroscopic description in which reactions are rate processes ruling the time evolution of populations of chemical species with the microscopic Hamiltonian dynamics governing the motion of the translational, vibrational, and rotational degrees of freedom of the reacting molecules. 

A correct understanding of the ice-water transition came when it was recognized that when ice melts not only does H increase by 6.008 kj mol-1, as the molecules acquire additional internal energy of translation, vibration, and rotation, but also the molecules become more disordered. Although historically entropy was introduced in a different context, it is now recognized to be a measure of "microscopic disorder." When ice melts, the entropy S increases because the structure becomes less ordered. 

Different kinds of molecules have different degrees of translational, vibrational, and rotational freedom and, hence, different average degrees of molecular disorder or randomness. Now, if for a chemical reaction the degree of molecular disorder is different for the products than for the reactants, there will be a change in entropy and AS0 A 0. 

The photofragmentation that occurs as a consequence of absorption of a photon is frequently viewed as a "half-collision" process (16)- The photon absorption prepares the molecule in assorted rovibrational states of an excited electronic pes and is followed by the half-collision event in which translational, vibrational, and rotational energy transfer may occur. It is the prediction of the corresponding product energy distributions and their correlation to features of the excited pes that is a major goal of theoretical efforts. In this section we summarize some of the quantum dynamical approaches that have been developed for polyatomic photodissociation. For ease of presentation we limit consideration to triatomic molecules and, further, follow in part the presentation of Heather and Light (17). 

These methods provide an accurate means of investigating translation-vibration and translation-rotation transfer. The passage of a sound wave through a gas involves rapidly alternating adiabatic compression and rarefaction. The adiabatic compressibility of a gas is a function of y, the ratio of the specific heats, and the classical expression for the velocity, V, of sound in a perfect gas is... 

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