Matrix of inertia

The r vectors are the principal axes of inertia determined by diagonalization of the matrix of inertia (eq. (12.14)). By forming the matrix product P FP, the translation and rotational directions are removed from the force constant matrix, and consequently the six (five) trivial vibrations become exactly zero (within the numerical accuracy of the machine). 

A number of molecular shape descriptors can be used to characterize anisometry. The standard solution is to replace a OD descriptor by a two-dimensional one, specifically a matrix. The matrix of inertia I is the simplest choice.37 Similarly, one can construct the so-called matrix of the radius of gyration The radius Rg defined in terms of the eigenvalues of this... 

In order to determine the matrix thresholds, we present an expression of the coefficients dispersion that is related to the flattening of the cloud of the points around the central axis of inertia. The aim is to measure the distance to the G barycentre in block 3. So, we define this measure Square of Mean Distance to the center of Gravity as follow ... 

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... 

The eigenvectors extracted from the cross-product matrices or the singular vectors derived from the data matrix play an important role in multivariate data analysis. They account for a maximum of the variance in the data and they can be likened to the principal axes (of inertia) through the patterns of points that represent the rows and columns of the data matrix [10]. These have been called latent variables [9], i.e. variables that are hidden in the data and whose linear combinations account for the manifest variables that have been observed in order to construct the data matrix. The meaning of latent variables is explained in detail in Chapters 31 and 32 on the analysis of measurement tables and contingency tables. 

Fig. 31.2. Geometrical example of the duality of data space and the concept of a common factor space, (a) Representation of n rows (circles) of a data table X in a space Sf spanned by p columns. The pattern P" is shown in the form of an equiprobabi lity ellipse. The latent vectors V define the orientations of the principal axes of inertia of the row-pattern, (b) Representation of p columns (squares) of a data table X in a space y spanned by n rows. The pattern / is shown in the form of an equiprobability ellipse. The latent vectors U define the orientations of the principal axes of inertia of the column-pattern, (c) Result of rotation of the original column-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the score matrix S and the geometric representation is called a score plot, (d) Result of rotation of the original row-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the loading table L and the geometric representation is referred to as a loading plot, (e) Superposition of the score and loading plot into a biplot.

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). 

Write a computer program that finds the principal moments and principal axes of inertia for a molecule. Do not use matrix diagonalization instead, solve the secular equation by using the formula for the roots of a cubic equation. The input to the program is the set of atomic masses and coordinates in an arbitrary system with axes not necessarily at the center of mass. 

The moments of inertia A, B, C may be calculated from the optimized structures and the vibrational energy levels v,-, and these are available from the second-derivative matrix if a full-matrix Newton-Raphson refinement is used124,511. However, the approximations involved in the calculation of the entropies, that so far have been used for the computation of the conformational equilibria of coordination compounds, have led to considerable uncertainties163,86-881. 

Structural assignments for the dimer of aluminum tribromide are based on the electron diffraction data reported by Shen (5). These values are in good agreement with earlier diffraction studies by Palmer and Elliott (6) and Akishin et al. (7 ) The principal moments of inertia are I = 282.4212 x 10" , Ig - 502.7859 x 10" and I, - 624.3585 x 10" g cm. Beattie, Gilson and Ozin (8) measured the condensed phase IR and Raman bands for aluminum tribromlde while Beattie and Border (9) measured the gas phase Raman spectra. Perov et al (1 ) investigated the IR spectra of monomeric and dimeric aluminum tribromide in a xenon matrix at 20-30 K. 

Principal moments of inertia are the moments of inertia corresponding to that particular and unique orientation of the axes for which one of the three moments has a maximum value, another a minimum value, and the third is either equal to one or the other or is intermediate in value between the other two. The corresponding axes are called principal axes of a molecule (or principal inertia axes). Moreover, the products of inertia all reduce to zero and the corresponding inertia matrix is diagonal. Conventionally, principal moments of inertia are labeled as ... 

One possibility, used, e.g., by Rey and Gallego,49 is to use the eigenvalues of the matrix for the moments of inertia, Eq. (55). These quantities will first of all be able to catch transitions from a solid-like to a liquid-like behaviour, where the atoms change from being more or less confined at somewhat fixed positions... 

Figure 20 Different properties of NiN clusters related to the eigenvalues lua of the matrix with the moments of inertia. In the upper panel we show the average value together with points indicating whether clusters with overall spherical shape (lowest set of rows), overall cigar shape (middle set of rows), or overall lens shape (upper set of rows) are found for a certain size. Moreover, in each set of rows, the lowest row corresponds to the energetically lowest isomer, the second one to the energetically second-lowest isomer, etc. In the lower panel we show the maximum difference of the eigenvalues for the four different isomers...

---