Principal moment of inertia

When the three principal moment of inertia values are identical, the molecule is termed a spherical top. In this case, the total rotational energy can be expressed in terms of the total angular momentum operator J2... 

Molecules for which two of the three principal moments of inertia are equal are called symmetric tops. Those for which the unique moment of inertia is smaller than the other two are termed prolate symmetric tops if the unique moment of inertia is larger than the others, the molecule is an oblate symmetric top. 

Again, the rotational kinetic energy, which is the full rotational Hamiltonian, can be written in terms of the total rotational angular momentum operator J2 and the component of angular momentum along the axis with the unique principal moment of inertia ... 

The rotational eigenfunctions and energy levels of a molecule for which all three principal moments of inertia are distinct (a so-called asymmetric top) can not easily be expressed in terms of the angular momentum eigenstates and the J, M, and K quantum numbers. However, given the three principal moments of inertia la, Ib, and Ic, a matrix representation of each of the three contributions to the rotational Hamiltonian... 

For the purposes of studying the rotational spectra of molecules it is essential to classify them according to their principal moments of inertia. 

For a symmetric rotor, or symmetric top as it is sometimes called, two of the principal moments of inertia are equal and the third is non-zero. If... 

A spherical rotor has all three principal moments of inertia equal ... 

An asymmetric rotor has all principal moments of inertia unequal ... 

As in diatomic molecules the structure of greatest importance is the equilibrium structure, but one rotational constant can give, at most, only one structural parameter. In a non-linear but planar molecule the out-of-plane principal moment of inertia 4 is related to the other two by... 

AB and ABC are the products of the principal moments of inertia. Moments of inertia are calculated from bond angles and bond lengths. Many values are given by Landolt-Bornsteiu. is Avogadro s number, and M is the molecular weight of the molecule. Stuper et al. give a computerized method for prediction of the radius of gyration. 

A considerable variety of experimental methods has been applied to the problem of determining numerical values for barriers hindering internal rotation. One of the oldest and most successful has been the comparison of calculated and observed thermodynamic quantities such as heat capacity and entropy.27 Statistical mechanics provides the theoretical framework for the calculation of thermodynamic quantities of gaseous molecules when the mass, principal moments of inertia, and vibration frequencies are known, at least for molecules showing no internal rotation. The theory has been extended to many cases in which hindered internal rotation is... 

For a nonlinear molecule the rotational energy levels are a function of three principal moments of inertia /A, /B and /c- These are moments of inertia around three mutually orthogonal axes that have their origin (or intersection) at the center of mass of the molecule. They are oriented so that the products of inertia are zero. The relationship between the three moments of inertia, and hence the energy levels, depends upon the geometry of the molecules. 

Finally, an asymmetric top is one in which all three principal moments of inertia are different. The energy levels are given by... 

In terms of principal moments of inertia A, B and G, and the molecular mass M, the entropy term is then given by equation (14) (cf. also Leffek and Matheson, 1971). 

Rotational constants G = A, B or C are inversely proportional to principal moments of inertia Ia through the expressions G = h/Sn2Ia, where a refers to one of the three principal inertia axis directions a, b or c. The Ia are related to the coordinates of the atoms i in the principal axis system via the... 

I = principal moment of inertia) respectively. It then follows for the ratio of the products of the A s of two isotopomers... 

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). 

Microwave spectroscopy provides oidy three data points (the principal moments of inertia)... 

Molecules for which all three principal moments of inertia (the Ij s) are equal are called spherical tops. For these species, the rotational Hamiltonian can be expressed in terms of the square of the total rotational angular momentum J2 ... 

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