Inertia tensor

For molecules that are non-linear and whose rotational wavefunctions are given in terms of the spherical or symmetric top functions D l,m,K, the dipole moment Pave can have components along any or all three of the molecule s internal coordinates (e.g., the three molecule-fixed coordinates that describe the orientation of the principal axes of the moment of inertia tensor). For a spherical top molecule, Pavel vanishes, so El transitions do not occur. 

Here g and go are a set of angular variables, which define a molecular orientation at instants of time 0 and t, respectively and ft is the orientation at instant t which was g0 at t = 0. By difference of arguments we mean the difference of turns. In the molecular frame (MS), where the axes are oriented along the main axes of the inertia tensor, [Pg.86]

Azurmendi, H. F., Bush, C. A. Tracking alignment from the moment of inertia tensor (TRAMITE) of biomolecules in neutral dilute liquid crystal solutions. J. Am. Chem. Soc. 2002, 124, 2426-2427. 

A better estimate of the shape of the polymer molecules, since they are highly anisotropic, is a representation of each molecule in terms of an equivalent spheroid with the same moment of inertia [45,46]. This is achieved by diagonalizing the moment of inertia tensor to obtain the eigenvectors a, b, and c and the principal moments 7, I/,/, and Icc. The moment of inertia tensor of molecule j is given by... 

Denoting the vector from the reference point to the center of mass of the rigid body k by Yi<, its mass by and its inertia tensor by h (Fig. 2.1b), the kinetic energy can be computed in a linear loop over all rigid bodies... 

Here I denotes the moment of inertia tensor defined with nuclear masses i cM is the position vector of the center of nuclear mass. The electrons, with position vectors have to he treated quantum mechanically which implies that their contribution is obtained as expectation value of the corresponding electronic operator over the ground state wavefunction... 

In practical applications of the expressions for the rotational g tensor, equations (2), (4), (5), or (6), the nuclear masses in the nmment of inertia tensor I are generally approximated with atomic masses and is approximated with the center of atomic masses. This introduces a correction term to the moment of inertia tensor which is actually closely related to the nuclear contribution to the rotational g tensor [3,11,38]. Going to second order of perturbation theory for the electronic contributions one would obtain a further correction term to the moment of inertia tensor which is similar to the electronic contribution to the rotational g tensor [3,4]. 

All together one would obtain an effeetive moment of inertia tensor which includes the rotational g tensor again. This correction is normally ignored for polyatomic molecules, but allows to estimate the rotational g factor of diatomic molecules from field-free rotation-vibration spectra [5,10,11]. 

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). 

A kinetic argument shows that ay - oy always. Any imbalance between these two would lead to an angular acceleration of a volume element. If this volume element were shrunk, then the torque would reduce in proportion to the linear dimension cubed, but the moment of inertia would reduce in proportion to the fifth power of the linear dimension, so that the angular acceleration would increase as the reciprocal of the square of the size of the volume element, becoming infinite in the limit. Thus reductio ad absur-dum, ay = cry. Hence there are only six independent components of the stress tensor. 

Averaged principal moments, a , of the Inertia tensor are obtained for polypeptides undergoing a helix-coil transition. Polypeptide chains containing 101, 201,401,... 

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). 

Identity element, 387-388 Identity operation, 54, 395 Improper axis of symmetry, 53 Improper rotation, 396 Index of refraction, 132 INDO method, 71, 75-76 and ESR coupling constants, 380 and force constants, 245 and ionization potentials, 318 and NMR coupling constants, 360 Induced dipole moment, 187 Inertial defect, 224-225 Inertia tensor, 201... 

---