Atom motions normal mode contributions

Each set of values /ajg for normal mode p forms a N x N matrix, where N is the number of nuclei. A diagonal term Jaa p represents the contribution which atom a makes to Jp, and the sum Ja/3 p + the contribution due to the coupled motion of the pair of nuclei a and / . The graphical representation as full and empty circles, depending on the sign, in a matrix, in upper triangular form, maps the way nuclear motion creates Raman and ROA intensity in the vibrating molecule [42],... 

Kelvin (the zero point motion). This latter effect is explained by quantum mechanics, and it can in turn explain absorption features of impurities in crystalline matrices. The presentation of the fundamental vibrational modes of crystals is based on the harmonic approximation, where one only considers the interactions between an atom or an ion and its nearest neighbours. Within this approximation, an harmonic crystal made of N ions can be considered as a set of 3N independent oscillators, and their contribution to the total energy of a particular normal mode with pulsation ivs (q) is ... 

Phonons are normal modes of vibration of a low-temperature solid, where the atomic motions around the equilibrium lattice can be approximated by harmonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled normal modes (phonons) if a harmonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

Figure 1 The normal modes of motion for the three stretch modes of DCCH. (Adapted from T. A. Holme and R. D. Levine, Chem. Phys. Lett. 150 393 (1988).) The displacement of the atoms in each mode is shown by an arrow. Note that while all atoms contribute to all modes, the respective contributions do vary and the v, mode is almost a localized CH stretch. For recent studies of the overtone spectroscopy of HCCH and its isotopomers see J. Lievin, M. Abbouti Temsamani, P. Gaspard, and M. Herman, Chem. Phys. Lett. 190 419 (1995) M. J. Bramley, S. Carter, N. C. Handy, and 1. M. Mills, J. Mol. Spectrosc. 160 181 (1993) B. C. Smith and J. S. Winn, J. Chem. Phys. 89 4638 (1988) K. Yamanouchi, N. Ikeda, S. Tuschiya, D. M. Jonas, J. K. Lundberg, G. W. Abramson, and R. W. Field, J. Chem. Phys. 95 6330 (1991).)...

A single molecule consisting of n atoms exhibits 3n degrees of freedom associated with translational, rotational, and vibrational motion. A nonlinear molecule will exhibit 3n-6 normal vibrational modes, Qj, (3n-5 for a linear molecule), each of a particular frequency W (cm l) that is determined by the masses of the atoms contributing to that particular normal mode and the interaction force constants (k) between the atoms. For a diatomic molecule, the relationship in equation (1) applies, where p is the reduced mass of the oscillator, and c is the speed of light. 

Note from Fig. 5.1 that Za and zb are opposite in sign (the liquid model used here gives limfc o[ A(fc), ajB(fc)] = [—0.11, 0.99] when the norm of the eigenvector is normalized to unity). Thus the atoms A and B contribute to this mode with the out-of-phase fashion in terms of Eq. (5.146). Since the magnitude of Za gauges the efficiency of the atom a for participating in the orientational motion, the optical mode is evidently related to the rotational motion of the molecules. This fact is also obvious by noting that Eq. (5.148) depends on the moment of inertia of the molecule. 

Polyatomic molecules vibrate in a very complicated way, but, expressed in temis of their normal coordinates, atoms or groups of atoms vibrate sinusoidally in phase, with the same frequency. Each mode of motion functions as an independent hamionic oscillator and, provided certain selection rules are satisfied, contributes a band to the vibrational spectr um. There will be at least as many bands as there are degrees of freedom, but the frequencies of the normal coordinates will dominate the vibrational spectrum for simple molecules. An example is water, which has a pair of infrared absorption maxima centered at about 3780 cm and a single peak at about 1580 cm (nist webbook). 

D17.1 An approximation involved in the derivation of all of these expressions is the assumption that the contributions from the different modes of motion are separable. The expression = kT/hcB is the high temperature approximation to the rotational partition function for nonsymmetrical linear rotors. The expression q = kT/hcv is the high temperature form of the partition function for one vibrational mode of the molecule in the haimonic approximation. The expression (f- =g for the electronic partition function applies at normal temperatures to atoms and molecules with no low lying excited electronic energy levels. 

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