Small amplitude motions

One may compare this result with that of Section 1.2. The vibrational part of (1.13) is again identical to Eq. (1.68). The rotational part is, however, missing in the one-dimensional problem. It is worth commenting on this special feature of the vibrational problem. It arises from the fact that molecular potentials usually have a deep minimum at r = re. For small amplitude motion (i.e., for low vibrational states) one can therefore make the approximation discussed in the sentence following Eq. (1.13) of replacing r by re in the centrifugal term. In this most extreme limit of molecular rigidity, the vibrational motion is the same in one, two and three dimensions. 

Equation (11-11) depends on neglect of inertial terms in the Navier-Stokes equation. Neglect of inertia terms is often less serious for unsteady motion than for steady flow since the convective acceleration term is small both for Re 0 (Chapters 3 and 4), and for small amplitude motion or initial motion from rest. The second case explains why the error in Eq. (11-11) can remain small up to high Re, and why an empirical extension to Eq. (11-11) (see below) describes some kinds of high Re motion. Note also that the limited diffusion of vorticity from the particle at high cd or small t implies that the effects of a containing wall are less critical for accelerated motion than for steady flow at low Re. 

Any small-amplitude motion is excluded for it could not lead to the large observed dipolar tensor reduction, as well as 90° flips about the 1,4 axis or... 

A number of methods have been developed for assessing nitroxide dynamics based on the cw-EPR spectrum (see review by Sowa and Qin, 2008). In the semiquantitative approach, parameters measured directly from the EPR spectrum, such as the central fine-width (AHpp, Fig. 15.9A), the splitting of the resolved hyperfine extrema (2AeS, Fig. 15.9A), and the second moment (H2, characterizing how broad the spectrum is), are used to characterize nitroxide dynamics (Columbus and Hubbell, 2002, 2004 Mchaourab et al., 1996). These parameters report on the nitroxide mobility, which describes a combined effect of the rate and the amplitude of motion. For example, a broad center fine gives a small (AHpp) 1 value and indicates low mobility, which can result from low frequency but large amplitude motions, or small amplitude motions with fast rates. The line-shape parameters can be easily measured on a properly processed EPR spectrum, and... 

Group Theory for non-rigid molecules considers only large amplitude mou-vements ignoring the small amplitude motions, such as vibrations. In the following, the Non-Rigid Molecule Group (NRG) will be strictly defined as the complete set of the molecular conversion operations, which commute... 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

This is a large amplitude momentum arising from the small amplitude motion in a way similar to that of the vibrational angular momenta [Eq. (3.38)]. The momentum vanishes in the reference configuration (all 2 = 0) in accordance with our aim of removing zeroth order coupling effects. 

Next we can consider the small amplitude motions which present a standard GF-eigenvalue problem, but with p as a free parameter. G°-elements corresponding to a basic set of internal valence coordinates, Rt where t= 1,2,... 3N-1, are derived from s°-vectors using Eq. (3.41), and thus they vary with p according to the variation of the first derivatives of Eq. (3.3). Also the force constants, Fn, may be functions of p and contribute to the general functional properties of L- and /-elements as well as of the eigenvalues, Xk (Sect. 4.7). 

Here nonrigidity will be considered under the perturbation treatment as well by excluding terms in the large amplitude coordinates and momenta from the zeroth order Hamiltonian. The resulting effective semirigid rotor Hamiltonians are therefore operators confined to the separate eigenspaces of a zeroth order Hamiltonian with terms of the small amplitude motion only [Eq. (4.36)]. 

The Eckart- and Sayvetz-conditions constitute a set of conventions for the reference structures which are particularly useful, since they allow us to use rectilinear coordinates for the small amplitude motions (Sect. 3.3). However, the introduction of reference structures, depending on the large amplitude coordinates only, leaves us with the question of how the molecular axes should be oriented within an arbitrary set of atomic reference positions. This question was only briefly commented on in Sect. 4.6, since it is special to the molecule under consideration. Some examples may illustrate types of solutions. 

For polyatomic molecules, the problem is more complicated. For small-amplitude motion, certainly one can decompose the motion into independent harmonic (normal mode) motion and treat each of these as was done for a single oscillator. If the system has significant anharmonicity, then the good action-angle variables must first be found. Such techniques are available in the literature (8,21-25). 

In terms of intramolecular flexibility, the poly(2,6-disub-stituted-1,4-phenylene oxides) are freely rotating chains [71] however, intermolecular steric effects may limit phe-nylene rotation in the solid state and periiaps account for the absence of detectable sub-Tg relaxational processes. For example, results of NMR measurements indicate that the phenylene rings of PPO can execute only small amplitude motions due to the relative stiffness and dense packing of the PPO chain and blockage from rings on adjacent chains. 

We can learn a great deal about the motions of the global atmosphere by examining the normal modes of the atmosphere. The atmosphere is a vibrating system and has natural modes of oscillations, like a musical instrument. Although the atmospheric equations are nonlinear, they can be linearized if we are interested in small-amplitude motions such as the perturbations around the atmosphere at rest with no external forcing and heating. Solutions of such a system with appropriate boundary conditions are referred to as normal modes. 

At 35 K, the atoms in the cluster make mainly small-amplitude motions [28, 29]. The cluster thus has a definite geometric structure and can be considered to be a large molecule, and the powerful methods of quantum chemistry can be used [30-32]. For n <6 and for n = 9, the agreement between experiment and theory is good [30]. For all other clusters... 

A detailed study of crystals of macromolecules 20,21) and their melting under equilibrium conditions revealed that the entropy of fusion, ASf, is often about 7-12 J/(K mol) per mobile unit or "bead" (22). This entropy is linked mainly to the conformational disorder (A and mobility that is introduced on fusion. Sufficiently below the melting temperature, disorder and thermal motion in crystals is exclusively vibrational. While vibrations are small-amplitude motions that occur about equilibrium positions, conformational, orientational, and translational motions are of large amplittide. These types of large-amplitude motion can be assessed by their contributions to heat capacity (23), entropy (22), and identified by relaxation times of the nuclear magnetization 24), Orientational and positional entropies of fusion ASQ ent trans importance to describe the fusion of small molecules. They can be deriv from the many data on fusion of the appropriate rigid, small molecules of nonspheiical and spherical shapes [nonspheiical molecules Walden s rule (1908), ASf = AS j ent AStrans 20-60 J/(K mol) and spherical molecules Richards rule 0 97), ASf = trans = 2-14 J/(K mol)].. The contributions of ASQ ent melting of... 

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