Molecules with Axial Symmetry

The molecules either have a generic infinite rotation axis (cones, rods, rotational ellipsoids, spherocylinders or discs) or acquire this average uniaxial form due to free rotation around the longitudinal molecular axis Then /(fl) becomes 4 -independent [13] / = /(i )/4ti with/(i ) =f(n — d), see Fig. 3.15. This figure shows that angle 0 and ti are equally and the most populated by molecules and these two angles correspond to the condition n = —n. The angles close to ti/2 are the less populated. Now our task is to find the form offi d) and relate it to experimentally measured parameters. 

As any axially symmetric function, f(d) can be expanded in series of the Legendre polynomials f i(cosi ) 

Recall that the Legendre polynomials of general formula 

The Legendre polynomials P ix) are tabulated for x = 0-1. In our case, the polynomials depend on angle d with x = cost and the integration should be made from K to 0. For even or odd m the polynomials are even or odd functions of cost), respectively  

Each function has a particular symmetry (like electron shells in atoms have flieir own symmetry p, d, etc.). The angular dependencies of the first two polynomials are plotted in Fig. 3.16. 

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This outline of the response theory has for simplicity been limited to molecules with axial symmetry of y and Aa and to the field on, field off cases, but can be extended in both respects without basic difficulties. Detailed comparisons with experiment have not yet been made, but it already is clear that Kerr effect relaxation data can now provide more valuable and better defined information about orientational dynamics of biopolymers and other molecules than was previously possible. With the increasing accuracy and time resolution of digital methods, it should be possible to study not only slow overall rotations of large molecules (microseconds or longer) but small conformational effects and small molecule reorientations on nano and picosecond time scales. Moreover, one can anticipate the possibilities, for simple problems at least, of extending response theory to other quadratic and higher order effects of strong electric fields on observable responses. 

Axial symmetry. In the case of molecules with axial symmetry along the k-axis, one has ... 

For a solution of rigid particles (or molecules) with axial symmetry of the optical properties (characteristic of a twisted chain molecule) and with a dipole-orientational type of EBF, the Kerr constant [19] is determined by the following equation ... 

Fortunately, in the case of a rotational diffusion tensor with axial symmetry (such molecules are denoted "symmetric top"), some simplification occurs. Let us introduce new notations D// = Dz and D = Dx = Dy. Furthermore, we shall define effective correlation times ... 

The dielectric anisotropy of long-chain fatty acid monolayers was analyzed. These fatty acids were considered as being oriented in a cylinder cavity with length (L) diameter (D). Considering each bond in these molecules as a polarization ellipsoid with axial symmetry about the -C-C- bonds, the mean polarizibtlity of the bonds was calculated. 

For molecules of axial symmetry such as diatomic molecules, if the molecular axis is aligned with the z-axis, we have Q20 = qi, the quadrupole strength all other ( 2m vanish. 

It has been shown that the main medianism responsible for EB in solutions of rigid-chain polymers is the rotation of their polar molecules as a whole whereas the anisotropy of the dielectric polarizability of the macromolecules only provides a small contribution to the Kerr effect. Hence, the general theory of the Kerr effect for rigid dipole particles with axial symmetry of the optical polarizability can be... 

The irreducible tensor method was originally developed by G. Racah in order to make possible a systematic interpretation of the spectra of atoms. In the present paper this method has been extended to irreducible sets of real functions that have the same transformation properties as the usual real spherical harmonics. Such an extension is particularly useful in the discussion of the spectra of molecules which belong to the finite point groups or to the continuous groups with axial symmetry. There are several reasons for this. 

The amount of different g values (actually the main components) observable in solid-state EPR spectra thus depends on the symmetry of the molecule. Molecules with low symmetry are likely to reveal three different g values, associated with the main axes of the molecule (rhombic g values gx gy = Sd-Axially symmetric molecules are likely to reveal two different g values (axial g-values gx gy gz) and highly symmetric species (e.g., tetrahedral complexes) give rise to isotropic g values gx gy g - Examples of such spectra are shown in Fig. 3. For EPR spectra recorded in solution, rapid tumbling of the molecule also results in isotropic g values, which are actually the weight average of the anisotropic g values of the species observed in frozen solution (Fig. 3). For transition metal complexes of concern to this chapter, containing only one unpaired electron (5 = 5 systems), species with exact isotropic g values gx = gy = gz) should not exist due to expected Jahn-Teller distortions. Therefore, any isotropic g values observed in frozen solutions are likely the result of poor resolution nonresolved g anisotropy due to broad lines. 

Let us consider what happens during an atomic collision between atoms i and j. At large internuclear distances (ry = r — rj —>oo) the atoms do not interact and the electrons of each atom should be spherically distributed about their respective nuclei. As the atoms approach each other, their electron clouds should begin to interact and distort each other. The electrons will then be distributed with axial symmetry about the internuclear vector (r y = r — tj). This axially symmetric charge distribution should give rise to an axially symmetric polarizability tensor, a(ry) and thereby to a depolarized component in the scattered light. The collisional pair can therefore be regarded as a quasilinear molecule, at least for the duration tc of the collision. 

There is a close connection between symmetry and the constants of the motion (these are properties whose operators commute with the Hamiltonian H). For a system whose Hamiltonian is invariant (that is, doesn t change) under any translation of spatial coordinates, the linear-momentum operator p will commute with H, and p can be assigned a definite value in a stationary state. An example is the free particle. For a system with H invariant under any rotation of coordinates, the operators for the angular-momentum components commute with H, and the toted angular momentum and one of its components are specifiable. An example is an atom. A linear molecule has axial symmetry, rather than the spherical synunetry of an atom here only the axial component of angular momentum can be specified (Chapter 13). 

For definite results it is useful to work with molecules having axial symmetry because they have only two main polarization directions— parallel and vertical to the axis. These two may, however, be calculated satisfactorily on the one hand from the average polarizability a determined from the refractive index (see p. 25) and from the degree of depolarization A so that two equations are available for the two unknowns. 

Nematic Liquid crystal phase in which the constituent molecules show partial orientational ordering, with axial symmetry, but no translational order. 

Woodward and Hoffmann [4, p. 147] characterize the reaction as an Homed [ 4a H- 2a]-cycloaddition, in which 5 /n-TC0D is formed rather than its anti isomer as a result of secondary orbital interactions. The reaction was analysed by OCAMS [5, p. 598] for a nearly-coplanar axial approach of the two CBD molecules, leading to an impossibly strained dimer with axial symmetry and a puckered central ring. The axial dimer , however, is not intended to depict a real molecule, but is a purely formal model produced by anasymmetrization to D2 of either the syn (Cs) or the anti (C2) isomer, and can thus represent them both. The correspondence diagram [5, Fig. 7] then shows that relaxation to the syn isomer involves a lower investment in distortional energy, in agreement with experiment. 

Ni aK-clinoptilolite showed O and P species due to the Nf-(NO) complex having axial symmetry - Table 8. The species Q, also with axial symmetry, was assigned to NO molecules adsorbed on the lattice. The 2-D ESEM study confirmed the Nf coordinations as above attributed. 

LS. In the LS phase the molecules are oriented normal to the surface in a hexagonal unit cell. It is identified with the hexatic smectic BH phase. Chains can rotate and have axial symmetry due to their lack of tilt. Cai and Rice developed a density functional model for the tilting transition between the L2 and LS phases [202]. Calculations with this model show that amphiphile-surface interactions play an important role in determining the tilt their conclusions support the lack of tilt found in fluorinated amphiphiles [203]. 

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