Principal axes of inertia

The coordinates p,Tx are called the principal axes of inertia symmetrized hyperspherical coordinates. The nuclear kinetic energy operator in these coordinates is given by... 

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of inertia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes (a, b, c). In order to determine the parity of the molecule through inversions in SF, we first rotate all the displacement vectors... 

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). 

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.

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of inertia, namely, by taking the coordinates (x, y, z) in Figure 1 coincided with the principal axes (a,b,c). In order to determine the parity of the molecule through inversions in SF, we first rotate all the electrons and nuclei by 180° about the c axis (which is perpendicular to the molecular plane) and then reflect all the electrons in the molecular ab plane. The net effect is the inversion of all particles in SF. The first step has no effect on both the electronic and nuclear molecule-fixed coordinates, and has no effect on the electronic wave functions. The second step is a reflection of electronic spatial coordinates in the molecular plane. Note that such a plane is a symmetry plane and the eigenvalues of the corresponding operator av then determine the parity of the electronic wave function. 

Sole (22) has calculated the moments of the distributions of the three principal orthogonal components of obtained by decomposing along the three principal axes of inertia of the chain, for certain star and comb molecules in addition to linear ones and rings he finds that branching or ring closure decreases the high asymmetry found for linear chains. 

The statistical analysis determines the principal axes of inertia of the clouds of data points. In a physical analogy, the axes of inertia of the solid defined by data points determine where the strongest relationships are, that is, which mutagenicity values of a month are more strongly related to a treatment. 

We will use the 7D CRS Hamiltonian which has been determined and analyzed in Ref. [10] (DFT/B3LYP, 6-31+G(d,p)). In short, the large-amplitude motion of the H/D atom is restricted to the (x,y) plane of the molecule (cf. Fig. 1). The origin of the molecule-fixed coordinate system is at the center of mass, with the axes pointing along the principal axes of inertia for the enol configuration. The H/D motion couples strongly to 5 in-plane skeleton modes, Q = (Q4, Q, Qu, Q26, Q3o)> which are described in harmonic approximation... 

V/a. The result will be a set of points forming a surface in three-dimensional space. It can be shown that this surface is an ellipsoid with center at the center of mass. The ellipsoid is called the ellipsoid of inertia, or the momental ellipsoid. The ellipsoid of inertia has three mutually perpendicular principal axes, which we designate a, b, and c. These axes are the principal axes of inertia of the molecule the corresponding moments of inertia about these three axes, Ia, Ib, and Ie, are the principal moments of inertia. The axes are labeled so that... 

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 multiplication table for the group G3v is given in Table 9.1. The three symmetry planes oa, ob, oc, are defined in Fig. 9.1 as making angles of 30°, 150°, and 270°, respectively, with the positive x axis. (The subscripts have no reference to the principal axes of inertia.) Consider the entry in the... 

Each selected configuration is translated and rotated in such a way that all of the solvent coordinates can be referred to a reference system centred on the centre of mass of the solute with the coordinate axes parallel to the principal axes of inertia of the solute. 

The r vectors are the principal axes of inertia determined by diagonalization of the... 

Principal Axes of Inertia Symmetrized Hyperspherical Coordinates... 

Let, in addition, GX Y Zik be another body-fixed frame IX whose axes are the principal axes of inertia of the three nuclei, and whose Euler angles with respect to GArsfYsfZsf are aK, bx, cx. The senses of those axes are chosen so that the IX system has the same handedness as the sf one and in a manner that results in a one-to-one correspondence between p, 4>x = ( x bK, cK, 0, space-fixed cartesian coordinates of Rx, and RXv Furthermore, the IX axes are labeled so that the corresponding principal moments of inertia lie in the order... 

The center of gravity of the molecule is placed at the origin of a Cartesian coordinate system. The molecule is oriented in this coordinate system so that the principal axes of inertia are along the coordinate axes. The vector radius R, of each mass point of the chain molecule (see Figure 4-13) can be resolved into the three orthogonal components (Ri)i, (Ri)2, and (Ri)3, where... 

Likewise, the radius of gyration can also be resolved into three components, namely, Rg,, Rg,2, and Rg,3. Since these three components stand in special relationship to the three principal axes of inertia of the molecule, they are often called the main components of the radius of gyration of the chain. 

Here the coordinates are again relative to the centre of mass. By choosing a suitable coordinate transformation, this matrix may be diagonalized (Section 16.2), with the eigenvalues being the moments of inertia and the eigenvectors called principal axes of inertia. 

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