Inertial tensor

Geometric Integrators for Rigid Body Simulation 355 where I = diag(/i,/2,/a) is the (diagonalized) inertial tensor,... 

In the molecular approximation used in (14) only the L = 3W — 6 (W is the number of atoms) discrete intramolecular vibrations of the molecular complex in vacuo are considered. In general these vibrations correspond to the L highest optical branches of the phonon spectrum. The intermolecular vibrations, which correspond to the three acoustical branches and to the three lowest optical branches are disregarded, i.e., the center of mass and - in case of small amplitudes - the inertial tensor of the complex are assumed to be fixed in space... 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

It can be noticed that at least two independent relaxation parameters in the symmetric top case, and three in the case of fully anisotropic diffusion rotation are necessary for deriving the rotation-diffusion coefficients, provided that the relevant structural parameters are known and that the orientation of the rotational diffusion tensor has been deduced from symmetry considerations or from the inertial tensor. 

Kuz min et al. (15) pointed out a standard result of classical mechanics If a configuration of particles has a plane of symmetry, then this plane is perpendicular to a principal axis (19). A principal axis is defined to be an eigenvector of the inertial tensor. Furthermore, if the configuration of particles possesses any axis of symmetry, then this axis is also a principal axis, and the plane perpendicular to this axis is a principal plane corresponding to a degenerate principal moment of inertia (19). 

Let us first examine a few special cases that cover most common point groups. A linear molecule, such as HCN (point group Coov) or acetylene (Dxl), will lie along one principal axis, say the z axis, so that the first eigenvalue of the inertial tensor vanishes and the other is doubly degenerate alternatively, by the second case in Eq. 3 x, = v, = 0 for all i, and thus % = 0. 

The q matrix is the negative of the electric-field gradient. Like the inertial tensor and the polarizability tensor, q is symmetric (since the order of partial differentiation is immaterial), and we can make an orthogonal transformation to a new set of axes a, ft, y such that q is diagonal, with diagonal elements qaa, q, q. Note, however, that the origin for q is at the nucleus in question and the axes for which q is diagonal need bear no relation to the principal axes of inertia (unless the nucleus happens to lie on a symmetry element). 

For each of the following, state whether the principal axes of the inertial tensor have the same orientation in the molecule as the principal axes of the polarizability tensor, (a) H20 (b) HDO (c) D20. 

As pointed out by Gwinu and Gaylord19), the solutions of the eigenvalue problems associated with a vibration-rotation problem do not and must not depend on the choice of the rotating axis system as long as an adequate Hamiltonian is used. What do depend on the axis system used are the numerical values of elements of the inertial tensor or vibration-rotation interaction constants determined from analysing the data. 

The symbol I is used for the latter product sum because this quantity may be an inertial tensor element. This turns out to be the case in both examples discussed below. 

This factorization of the ju-tensor has also been observed in the standard theory of small amplitude motion57, S8, where I = 1°, the inertial tensor of the equilibrium configuration. Below it is shown how this particular result is following from the Eckart conditions. 

The 1° appearing here is the generalized inertial tensor of the semirigid model [Eq. (4.5)]. The fact that Eq. (3.50) could be used in general was pointed out by Newton and Thomas68 as early as 1948. 

In the PAS centered at the COM, the off-diagonal components of the inertial tensor I vanish ... 

Inertial tensor of the reference molecule If x,y, z are orthogonal coordinates in the reference system with origin at the center of mass ... 

Under the assumptions that the intermolecular interactions in molecular crystals can be represented as the sum of the atom-atom interactions (Table 2.2 and Fig. 2.6), and that the molecules are rigid, i.e. their masses and their inertial tensors are constant, Pawley [14] carried out the first analytical formulation of the lattice dynamics. In the following, we wiU sketch the physical fundamentals of his model and describe its results in the harmonic approximation in comparison with the experimental phonon dispersion relations (Fig. 5.8). As we shall see, the model of an oriented gas describes the experimental results in detail, and furthermore, it yields both the similarities of the lattice dynamics of different molecular crystals and their differences qualitatively, even when the same parameters for the intermolecular atom-atom potentials are used for aU of the molecular crystals (Table 2.2 and Fig. 2.6). 

When the principal inertial axes system (PAS) is used as the coordinate system, the inertial tensor of the whole molecule is diagonal, and thus... 

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