Principal inertial axes

There are other important properties tliat can be measured from microwave and radiofrequency spectra of complexes. In particular, tire dipole moments and nuclear quadmpole coupling constants of complexes may contain useful infonnation on tire stmcture or potential energy surface. This is most easily seen in tire case of tire dipole moment. The dipole moment of tire complex is a vector, which may have components along all tire principal inertial axes. 

Figure 5.1 Principal inertial axes of (a) hydrogen cyanide, (b) methyl iodide, (c) benzene, (d) methane, (e) sulphur hexafluoride, (f) formaldehyde, (g) s-lraws-acrolein and (h) pyrazine...

An improvement on the rg structure is the substitution structure, or structure. This is obtained using the so-called Kraitchman equations, which give the coordinates of an atom, which has been isotopically substituted, in relation to the principal inertial axes of the molecule before substitution. The substitution structure is also approximate but is nearer to the equilibrium structure than is the zero-point structure. 

If we start out with the principal inertial axes coincident with (.X, Y, Z), that is,... 

The assertion that the PAS is convenient for separating rotations and vibrations can be rejected, therefore. We shall see below (Sect. 4) that the small amplitude vibrations are always treated most simply using Eckart conditions, whereas large amplitude motions must be specially taken care of. Principal inertial axes may only be relevant in relation to the reference structure of the Eckart conditions. 

Examination of the high resolution microwave spectrum of COF j enabled the spin-rotation constants along the principal inertial axes = -19 kHz = -13 kHz ... 

More recently, improved resolution (between 5 and 20 times better) was obtained by examining several of the rotational transitions in a molecular beam maser spectrometer [2145]. This gave more precise values of the F spin-rotation constants along the principal inertial axes (A/gj = -19.77 kHz = -13.46 kHz = -7.80 kHz) [2145], and the F... 

In this equation Av, Bv, and Cv are the rotational constants and La, Lb, and Lc are the projections of the rotational angular momentum on the principal inertial axes of the molecule. The Hamiltonian in Eq. (1) is often referred to as a rigid-rotor Hamiltonian, even though significant vibrational effects appear in the rotational constants. To good approximation... 

The Q, a, and 31 tensors are all defined in the principal inertial axes systems. Qzz is the scalar quadrupole moment of the nucleus [defined by the convention in Eq. (11)] and Q is the field-gradient tensor at the nucleus described again in the principal inertial axes systems. All other terms have been defined previously. 

With W/ = trij, the mass of atom i, ej, 62, 3 are the principal inertial axes of the molecule. The matrix M mj) is closely related to the inertial matrix I, namely,... 

Molecular structures may be described and compared in terms of external or internal coordinates. The question of which is to be preferred depends on the type of problem that is to be solved. For example, one problem that is much easier to solve in a Cartesian system is that of finding the principal inertial axes of a molecule indeed, if only internal coordinates are given then, in general, the first step is to convert them to Cartesian ones and then proceed as described in Section 1.2.4. Similarly, the optimal superposition of two or more similar molecules or molecular fragments, i.e. with the condition of least-squared sums of distances between all pairs of corresponding atoms, is best done in a Cartesian system. On the other hand, systematic trends in a collection of molecular structures and correlations among their structural parameters are more readily detectable in internal coordinates. 

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... 

The rotational dependence of q j, can be calculated by transformation to the principal inertial axes system (g = a,b, c) yielding... 

Pg) principal inertial axes system taken over the unperturbed rotational state Jj Pg is in units of... 

When the principal inertial axes system (PAS) is used as the coordinate system, the inertial tensor of the whole molecule is diagonal, and thus... 

Basic structures of (a) the anti- and (b) the -isomer in their principal inertial axes systems. There is a switch of inertial axes on isomerisation. 

Microwave spectroscopy of rotational transitions in PHgD gave an upper limit of 20 kHz to the components epaaQ + eqbbQ QaaQ + eciccQ b, c=principal inertial axes) [10]. 

The PH2D microwave spectrum for two rotational transitions showed a hyperfine structure due to 31P spin-rotation interaction with (Caa + Cbb)/2 = (Caa+Ccc)/2 = -98 3 kHz (a, b, c=principal inertial axes) [13]. 

Components of the Q tensor along the principal inertial axes a, b, c (b bisects the bond angle, c 1 molecular plane) from the first- and second-order Zeeman effect of several pure rotational transitions [26, 27,32,37] areQaa = -1.6 1.4, Qbb= + 2.1 1.1, and Qcc = -0.5 1.9. 

The components along the principal inertial axes were derived from the molecular g values, the magnetic susceptibility anisotropies, and the rotational constants [1] ... 

Elements of the g tensor for the rotational magnetic moment in units of the nuclear magneton and referred to the principal inertial axes from the linear and quadratic rotational Zeeman effect [1, 11] ... 

Molecules in the tetrahedral, octahedral, and icosahedral point groups behave like spherical electron distributions in this respect the induced dipole moment is independent of the molecule s orientation in the electric field. Most molecules, however, are more easily polarized along one axis than another. The polarizability in this case is actually represented by a matrix called the polarizability tensor, with elements that describe the polarizability along the molecule s principal inertial axes. 

These axes are not directly related to the principal rotation axis introduced in Chapter 6, which is the (often unique) rotation axis for the C proper rotation with the greatest value of n. However, the principal rotation axis often corresponds to one of the principal inertial axes, as weTl see shortly. 

For symmetric molecules, there may be more than one way to orient the principal inertial axes, as in the following example. 

The principal inertial axes a, b, and c are the rotational axes that we choose for defining the rotational motion of a molecule. 

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