Translational mode

In the crystal, the total number of vibrations is determined by the number of atoms per molecule, N, and the nmnber of molecules per primitive cell, Z, multiplied by the degrees of freedom of each atom 3ZN. In the case of a-Sg (Z =4, N =8) this gives a total of 96 vibrations ( ) which can be separated in (3N-6)—Z = 72 intramolecular or "internal" vibrations and 6Z = 24 intermo-lecular vibrations or lattice phonons ("external" vibrations). The total of the external vibrations consists of 3Z = 12 librational modes due to the molecular rotations, 3Z-3 = 9 translational modes, and 3 acoustic phonons, respectively. 

In the case of translational modes simply the mass of the rings has to be considered, while in the case of librational modes the moment of inertia has to be taken into account... 

The reaction coordinate that describes the adsorption process is the vibration between the atom and the surface. Strictly speaking, the adsorbed atom has three vibrational modes, one perpendicular to the surface, corresponding to the reaction coordinate, and two parallel to the surface. Usually the latter two vibrations - also called frustrated translational modes - are very soft, meaning that k T hv. Associative (nondissociative) adsorption furthermore usually occurs without an energy barrier, and we will therefore assume that A = 0. Hence we can now write the transition state expression for the rate of direct adsorption of an atom via this transition state, applying the same method as used above for the indirect adsorption. 

In general a nonlinear molecule with N atoms has three translational, three rotational, and 3N-6 vibrational degrees of freedom in the gas phase, which reduce to three frustrated vibrational modes, three frustrated rotational modes, and 3N-6 vibrational modes, minus the mode which is the reaction coordinate. For a linear molecule with N atoms there are three translational, two rotational, and 3N-5 vibrational degrees of freedom in the gas phase, and three frustrated vibrational modes, two frustrated rotational modes, and 3N-5 vibrational modes, minus the reaction coordinate, on the surface. Thus, the transition state for direct adsorption of a CO molecule consists of two frustrated translational modes, two frustrated rotational modes, and one vibrational mode. In this case the third frustrated translational mode vanishes since it is the reaction coordinate. More complex molecules may also have internal rotational levels, which further complicate the picture. It is beyond the scope of this book to treat such systems. 

Most liquids do have a defined vapor pressure which means that molecules can and do escape from the surface of the liquid to form a gas. This is another reason that the properties of a liquid vary from those of the gaseous state. Hence, we still have the vibrational and rotational degrees of freedom left in the liquid, but not those of the translational mode. A representation of water molecules in the liquid state is presented in the following diagram, shown as 1.2.4. on the next page. 

The low-frequency shift and the broadening of the CO spectra at 0 ps suggest that the low-frequency modes of adsorbed CO, that is, stretching, frustrated rotation, and frustrated translation modes of Pt-CO, were thermally excited by pump pulses, as reported by Bonn et al. [82] Thus, it is concluded that the transient site migration of adsorbed CO on the Pt electrode surface was caused by a transient rise in the surface temperature of Pt induced by pump pulses. 

If the center of mass of the molecular probe coincides with the Mossbauer nucleus, then the low-energy part of the spectrum monitors exclusively translational modes of the probe molecule thus providing a selective probe for fast translational processes on the lengthscale of several molecular diameters and larger. If, however, the center of mass does not coincide with the Mossbauer nucleus, then hindered rotations, i.e., librations, will contribute to the low-energy DOS. If... 

Show that, for the bimolecular reaction A + B - P, where A and B are hard spheres, kTsr is given by the same result as jfcSCT, equation 6.4-17. A and B contain no internal modes, and the transition state is the configuration in which A and B are touching (at distance dAR between centers). The partition functions for the reactants contain only translational modes (one factor in Qr for each reactant), while the transition state has one translation mode and two rotational modes. The moment of inertia (/ in Table 6.2) of the transition state (the two spheres touching) is where p, is reduced mass (equation 6.4-6). 

We must warn on the fact that in Table IV, we have reported, for the liquid experimental and MD results, the frequencies corresponding to the maximum positions of the bending and stretching bands, which, because of thermal disorder, are quite broaden in addition the two sets of data present also librational and translational modes, not discussed here. 

Here (7, is the surface tension, and Sq is the permittivity of free space. The mode n = 0 corresponds to the equilibrium sphere, and n = 1 is a purely translational mode. The first unstable mode is n = 2. The critical charge for this mode is given by setting Eq. (37) to zero, which yields the Rayleigh limit of charge,... 

The motion of atoms in the lattice can be depicted as a wave propagation (phonon). By dispersion we mean the variation in the wave frequency as reciprocal space is traversed. The propagation of sound waves is similar to the translation of all atoms of the unit cell in the same direction hence the set of translational modes is commonly defined as an acoustic branch. The remaining vibrational modes are defined as optical branches, because they are capable of interaction with light (see McMillan, 1985, and Tossell and Vaughan, 1992, for more exhaustive explanations). 

The efiect is thus not related to geometrical constraints induced on complexes anchored in mesoporous charmels (sometimes also called as confinement efiect, even if this definition is not properly correct), neither to shape-selectivity effects as possible in zeolites, since the size of mesoporous charmels is much larger than those of micro-porous materials. Instead, an effective modification on the characteristics of the fluids is observed due to the electrostatic field generated by the charmel walls. This is an enthalpic effect versus an entropic effect as observed when the modification is instead related to limitations in the translation modes of molecules. Recently, it was also demonstrated that wall curvature influence the molecular orientation of the... 

Measurements have been made by helium atom scattering of the vibration of benzene against the Cu(110) surface near 150 K (139) and of the frus-trated-translation mode of benzene parallel to the surface on Rh(lll) at 160 K (327). 

According to literature on time scales of about 1 ps overdamped, collective translational modes of the bulk liquid occur as well as heavily damped restricted translational modes of the liquid with hydrogen bond bending and breaking character [9,10]. We interpret our longer decay time (-750 fs) to trace the dynamics of these processes. The significant reduction of... 

We determined earlier (page 63) that the irreducible components of this representation are three A, one A2. two and three B3 species. To obtain from this total set the representations for vibration only, it is necessary to subtract (he representations for the other two forms of motion rotation and translation. We can identify them by referring to the C2t character table. The three translational modes will belong to the same representations as the x, y, and z basis functions, and the rotational modes will transform as Ry, and Rs. Subtraction gives... 

The actual values of AS6 may be considered in relation to each of three clear-cut situations. In the first the intracrystalline molecules are assumed to have two translational modes (2 T) and one simple harmonic vibrational mode (IV) with respect to the local environment, for which AS6 = ASi] the second assumes IT and 2V for which ASe = ASn and the third assumes 3V with AS6 = ASm6. Then (23)... 

The infra-red spectrum of Li-6 hydroxide was discussed by Decius and Lilley 69> who report that the major feature at 3681 400 cm"1 in the LiOH spectrum (which was previously assumed to involve a librational mode) is partly due to the absorption at 3681 290 cm 1 which involves a translator mode in which there is a significant amount of Li7-participation the rest may be a two-phonon process (see Oehler and Gunthard S6>). 

We present in Section 2 the formalism giving the equations for the reduced density operator and for competing instantaneous and delayed dissipation. Section 3 presents matrix equations in a form suitable for numerical work, and the details of the numerical procedure used to solve the integrodiffer-ential equations with the two types of dissipative processes. In Section 4 on applications to adsorbates, results are shown for quantum state populations versus time for the dissipative dynamics of CO/Cu(001). The fast electronic relaxation to the ground electronic state is shown first without the slow relaxation of the frustrated translation mode of CO vibrations, for comparison with previous work, and this is followed by results with both fast and slow relaxation. In Section 5 we comment on the general conclusions that can be reached in problems involving both vibrational and electronic relaxation at surfaces. 

---