Principal Moments

For a rigid array of n + 1 particles, the radius of gyration tensor, S, can be expressed as a symmetric 3x3 matrix 

1 Masses, Sizes, and Shapes of Macromolecules from Multifunctional Monomers 

When the n + 1 particles are all of the same mass, the latter representation is generated from the vectors s, (each from the center of mass to atom i, s, = rQl-g) as 

Preparatory to the shape analysis, the coordinate system is rotated by a similarity transform so that the 3x3 representation of S for each individual conformation is rendered in diagonal form. 

The shape of a conformation can be characterized by various manipulations of the principal moments. The asymmetry of any one of the conformations is characterized by the dimensionless ratios 1 L lL L /L2 OJ34,35 Spherical symmetry requires L lL2 = Lh)l = 1. Averaging of the corresponding principal moments over all conformations permits discussion of the asymmetry of the population of conformations in terms of (L2)I(Li) and (L2)/(L2). Examples of these dimensionless ratios are presented in Table 1.2 for four types of macromolecules, unperturbed by long-range interactions. In none of the four cases do the conformations have spherical symmetry. 

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Equation XVI-21 provides for the general case of a molecule having n independent ways of rotation and a moment of inertia 7 that, for an asymmetric molecule, is the (geometric) mean of the principal moments. The quantity a is the symmetry number, or the number of indistinguishable positions into which the molecule can be turned by rotations. The rotational energy and entropy are [66,67]... 

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... 

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... 

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