Tensor of inertia

With/ running over x, y and z all inertial products are constrained to zero, and consequently the molecular axes are principal axes of the instantaneous tensor of inertia. 

It is also worth mentioning that J is intermediate between 1° and the instantaneous tensor of inertia I. Approximately it holds that... 

Notice, that / ) is not related in any simple way to second derivatives of the instantaneous tensor of inertia, I, ... 

Fig. 6.9. Examples of four classes (d" t< (rotators). Hie numbers I. , lyy, Izz represent the eigenvalues of the tensor of inertia computed inaBFCS. There are fourpossibilities (a)alinearrotator(/v.v = lyy =0,/— 0) e.g., a diatomic or CD2 molecule (b) a spherical rotator (I y.y = lyy = /—) e.g., a sphere, a eube. a regular tetrahedron, or a methane molecule (c) a symmetric rotator (/y.v = lyy /—) e.g., a cylinder, a rectangular parallelepiped with square base, ammonia or benzene molecule (d) an asymmetric rotator (/t.t /w 7 f ) e.g., a general reetangular parallelepiped, a hammer, or water molecule.

Let us suppose that we have calculated the Descartes coordinates (xi, yi, Zi) of the atoms in a non-spherical structure of n atoms. Thus X, Y and Z are n-dimensional vectors containing the x, y and z coordinates of the atoms in order. Let us suppose further that the centre of mass of the molecule is in the origin of the coordinate system and the eigenvectors of its tensor of inertia are showing to the direction of the X, y and z axis. 

Equations 6.43 come from the conditions that the off-diagonal matrix elements of the tensor of inertia are zero because of the special position of the molecule. Let us suppose that the total energy... 

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. 

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... 

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. 

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