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. 

When a data set is well-related, the first two or three eigenvalues—the axes of inertia—account for 95 or more of the total inertia. In this study, 80-95 of the inertia is contained in the first three axes. This condition indicates that mutagenic activity can be related to treatment efficiency. 

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... 

After a number of three-dimensional reconstructions are performed they may be averaged using the program SUPCOMB (Kozin and Svergun, 2001), which performs an initial alignment of structures based on their axes of inertia followed by measurement of their overlap by minimization of the normalized spatial discrepancy (NSD). For two sets of points Sj = 1,. .., and S2 = 1,. .N2 the NSD is defined as... 

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. 

Electric dipole moments components can be calculated from Stark effect measurements [Eqn. (3)], and the results for dimers are collected in Table 4. As shown by Eqn. (3), these experiments give the dipole moment components projected along the principle axes of inertia, but do not directly give the total moment of the molecule. Many of the molecules in Table 4 have A rotational constants which are very large, and thus the Stark effects are dominated by the a-component of the dipole moment in Eqn. (3). For example, p for H2S HF is accurately determined to be 2.6239(17)D, but only the combination px = (p + p2)1/2 = 0.97(20)D can be found for the remaining components. 341 The total molecular moment is then roughly 2.80(7)D. 

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

The librations (rotatory modes) of water molecules in solid hydrates (R) - hindered rotations around the three axes of inertia (see Fig. 3) - are normally observed in the relatively broad spectral region from 350 to 900 cm l The upper and lower limits found are 1123 (CsOH H20) and 335 cm (Bal2 H20) The H2O rotatory modes can be distinguished from other bands appearing in the spectral range under discussion, for example, internal modes of polyatomic ions, by deuteration experiments. 

Every body has three axes the use of which permits the kinetic energy to be expressed in a particularly simple form. These are called the princijxd axes of inertia. The moment of inertia about a principal axis is defined by the expression fprHr, in which p is the density of matter in a given volume clement dr, r is the perpendicular distance of this element from the axis in question, and the integration is over the entire volume of the solid. For a discussion of this question see J. C. Slater and N. II. Frank, Introduction to Theoretical Physics, p. 94, McGraw-Hill Book Company, Inc., New York, 1933. 

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