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. 

Prior to solvation, the solute is oriented according to its inertial axes such that the box size needed to accommodate it is minimized (minimizing the number of water molecules). The principal inertial axis is oriented along the viewer s Z axis, for example. Then water molecules are eliminated if any of the three atoms are closer to a solute atom than the contact distance you specify. 

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...

Dipole moments of asymmetric rotors or, strictly, their components along the various inertial axes, may be determined using the Stark effect. 

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. 

In a molecule such as the asymmetric rotor formaldehyde, shown in Figure 5.1(f), the a, b and c inertial axes, of lowest, medium and highest moments of inertia, respectively, are defined by symmetry, the a axis being the C2 axis, the b axis being in the yz plane and the c axis being perpendicular to the yz plane. Vibrational transition moments are confined to the a, b or c axis and the rotational selection mles are characteristic. We call them... 

Also included in this work is a comparison of the direction of the dipole moments for each compound using two different measurement techniques. The first technique involves comparing the vector components against a standard Cartesian coordinate system, where the center of mass of the molecular model rests at the origin and the inertial axes of the molecule are oriented along X, Y, and Z axes, respectively. This represents the dipole in coordinate space, as shown in Figure 1. 

The rotational energies represent the spinning motions of a molecule, when the entire molecule rotates around one of its inertial axes. This should not be confused with internal rotation which is the rotational motion of one part of a molecule with respect to some other part of the same molecule. 

In low-symmetry molecules, diagonal and off-diagonal matrix elements of the electronic dipolar coupling tensor may contribute to ( h[)0 ) J ssl b ). Therefore, they are specified mostly in terms of their Cartesian components. If symmetry is C2V or higher, the off-diagonal matrix elements of the tensor operator in Eq. [163] vanish (i.e., the principal axes diagonalizing the SCC tensor coincide with the inertial axes). For triplet and higher multiplicity states, one then obtains... 

Figure 2.16. Raman-scattering spectra131 of an anthracene crystal at 5 K. One distinguishes the three frequency doublets, corresponding roughly to the three main molecular inertial axes and to the two components (A, and BJ of the unit cell. In this spectrum, only the frequency of the lower At mode (49.35cm-1) is determined with the required accuracy ( + 0.10cm-1)-...

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

The subscripts I, m and n represent the molecular inertial axes a, b and c) aim and iM represent the Raman and dipole vibrational transition elements, respectively... 

Spectra of molecules in the crystalline state, i.e., of molecular crystals, are obtained from molecules which are at fixed positions (sites) in the lattice (Fig. 2.6-1C). Normal (first-order) infrared and Raman spectra can be seen as spectra of hyper molecules , the unit cells (Schneider 1974, Schneider et al., 197.5). As a consequence, any molecular vibration is split into as many components as there are molecules present in the unit cell. Their infrared and Raman activity is determined by the symmetry of the unit cell. In addition, the translational and rotational degrees of freedom of molecules at their sites are frozen to give rise to lattice vibrations translational vibrations of the molecules at their sites and rotational vibrations about their main inertial axes, so-called librations. 

A molecule-fixed Cartesian coordinate system is oriented such that its origin coincides with the center of mass and the axes coincide with the main inertial axes of the molecule. A rotation axis can thus be identified by a Cartesian axis, for example C (z). The main axis is usually defined as the 2 axis. Planes are identified by the axes they contain a(xy) or tr.y. 

Recently, van den Heuvel et al. [378 a] studied the bioconcentration of [ H]avermectin Bj in an 28-d uptake flow-through test with bluegill sunfish Lepomis macrochirus). Avermectin Bj (see Fig. 16), the major component of abamectin, possesses a molecular mass of 872. The molecular dimensions are 17.0 X 18.7 X 18.4 A and were determined by Nachbar (cited in [378]) by finding the smallest parallelepiped whose faces were centered on the inertial axes of the molecule and would enclose the van der Waals surface of the molecule. A van der Waals radius of 1.2 A for hydrogen was used and the atomic coordin-... 

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... 

The rotational spectra of the four CWA simulants are shown in Figure 2. The upper two panels show the rotational spectmm of pinacolone (PC) and pinacolyl alcohol (PCA), respectively and are precursors to the nerve agent soman. The rotational constants, distortion terms, and dipole moment components (determined by ab initio calculations ) have been reported elsewhere. The rotational constants and projection of the permanent dipole moment onto the inertial axes are reported in Table 1. 

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. 

For cyanamide, the components of electric dipolar moment/10 C m parallel to inertial axes are = 14.14 and = 3.03 [33], slightly larger than our calculated values = 13.33 and p = 2.878 the total moment from experiment is 14.46 X 10 C m, correspondingly larger than our calculated value in Table 16-3. We compare our calculated rotational parameters with the effective rotational parameters/m for the vibrational ground state — A = 1041.19325, B = 33.78923475 and C = 32.90918052 [34], which are all a little larger than the calculated values. Our calculations indicate that both these isomers have a nonlinear spine, consistent with experiment for both structural isomers those calculated internuclear distances, in Table 16-3, agree satisfactorily with the values deduced from experiment for cyanamide. 

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. 

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