Principal inertial axis system

The(2/ + l)-fold degeneracy ofan energy level with rotational quantum number J can be partially lifted by placing the molecule in a static electric field. This so-called Stark effect can be utilized to measure the magiutudes of the dipole moment components, fit, Mo in die principal inertial axis system of the molecule. For this purpose, Stark electrodes can be mounted outside the microwave cavity to generate a... 

To calculate the rotational energy due to the Hamiltonian in Eq. (7), we write Q and X in the principal inertial axis system with the direction cosine transformation,, leading to... 

For convenience in application we have obtained both atom and bond values. In the atom approach we have assumed that an atom in a particular bonding situation (particular hybridization) will always contribute the same amount to the molecular susceptibility. This contribution consists of the three principal components as shown in Table 8 under atom susceptibilities. To evaluate the molecular susceptibility, the atom or bond values in Table 8, which are principal values, are rotated into the principal inertial axis system (a, b, and c) of the molecule. The atom and bond susceptibilities were determined by least squares fitting the experimental molecular susceptibility components of the 14 common nonstrained, nonaromatic molecules shown in Table 7. 

Table II.3. Diagonal values of molecular susceptibilities referred to the principal inertial axis system in units of 10 S erg/(G2 mole). In each case the first reference is for the susceptibilities, the second for the structure. The local values Xaa(loc), Xbb(loc), Jfcc(ioc) have been calculated according to Eq. (11.12), using the known structure and the atom susceptibilities listed in Table II.2...

Table 4 Atom coordinates in the principal inertial axis system and substitution (r ) structure for the diketo tautomer of uracil. Distances are given in A and angles in degrees... 

The coordinate system is now drawn such that the origin is at the center of mass of the molecule. In order to describe the rotational motion of a nonlinear molecule, three angular coordinates are needed resulting in three moments of inertia. The position of each atom is now expressed in a coordinate system whereby the center of mass of the molecule is at the origin and each atom is along the axes labeled by convention as a, b, and c. This coordinate system is called the principal inertial axis system. The three moments of inertia that result from the principal inertial axis system are called the principal moments of inertia. 

Pitfalls still remain in determining the geometry of molecules and crystals, even for relatively simple systems. Pairs of interatomic distances that are not well separated cause problems in diffraction techniques, and atoms near a principal inertial axis are difficult to locate by spectroscopic measurements. On the other hand, quantitative structure determinations on large molecules have made major strides in recent years, and further progress can be expected. It is hoped that NMR, Raman, and infrared spectroscopy will prove valuable in complementing the information derived from x-ray diffraction. 

I is the moment of inertia tensor if the x, y, z axes are chosen to be the principal inertial axes of the molecule a, b, c), I is then diagonal with principal components ha, hb, he For a linear molecule (including diatomics), ha = 0 and hb = he- In the inertial axis system equation (8.76) becomes simply... 

For asymmetric top molecules, the principal axis system of the inertia tensor and the field gradient tensor do not coincide in general. In the case of a completely non-symmetric position of the quadmpolar nucleus in the molecule, none of the components Xgg- of the field gradient tensor eqnals zero. If a nucleus lies on a plane which contains the principal inertial axes g and g and which is a symmetry plane of the molecule, then the off-diagonal elements Zgg and Xg g" vanish. 

In many cases the coupling tensor cannot be determined completely. In order to discuss bond properties, one frequently assumes that the bond direction coincides with one of the principal axes of the quadmpole coupling tensor (bond axis system), yielding information if the position of the principal inertial axes is known. 

The first step in the study of a rotational spectrum is to evaluate the moments of inertia, or rotational constants, from which the rigid rotor spectrum (discussed in the following section) can be predicted. The rotational problem is treated mathematically in terms of a molecule-fixed axis system with its origin at the center of mass of the molecule and its axes oriented along the principal inertial axes. With respect to these axes the moments of inertia are constant, and the intertia matrix is diagonal. The principal axes of inertia are designated by a, b, and c. The corresponding moments of inertia are denoted by la, h, and Ic, where, by convention, the inertial axes are labeled so that la = h = Join terms of the coordinates of the atoms in the principal axis system, the principal moments of inertia are defined by... 

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