Molecular frame

FigureBl.5.10 Euler angles and reference frames for the discussion of molecular orientation laboratory frame (v, y, z) and molecular frame (x y, z).

Most of these theoretical aspects are discussed in the following sections, pointing out the particuliar suitability of the very dissymmetrical molecular frame of thiazole for quantitative study, on the same species, of a large variety of fundamental organic reactions. 

Here u is a unit vector oriented along the rotational symmetry axis, while in a spherical molecule it is an arbitrary vector rigidly connected to the molecular frame. The scalar product u(t) (0) is cos 0(t) in classical theory, where 6(t) is the angle of u reorientation with respect to its initial position. It can be easily seen that both orientational correlation functions are the average values of the corresponding Legendre polynomials ... 

Here g and go are a set of angular variables, which define a molecular orientation at instants of time 0 and t, respectively and ft is the orientation at instant t which was g0 at t = 0. By difference of arguments we mean the difference of turns. In the molecular frame (MS), where the axes are oriented along the main axes of the inertia tensor, [Pg.86]

AMX, this is possible, because there are three nonselective relaxation-rate values for three unknown py values (pam. Pax, Pmx)- For a system in which y > 3 proton spins, this analysis cannot be unambiguously applied, because there are j(j — )/2 values of py to be determined from j measured R (ns) values. However, under favorable circumstances (see Section IV), depending on the relative disposition of the proton spins in the molecular frame, some Py values may be disregarded. This affords a good estimate of the appropriate Py values, and, hence, information about molecular geometry and conformation. 

In Eq. (12), l,m are the photoelectron partial wave angular momentum and its projection in the molecular frame and v is the projection of the photon angular momentum on the molecular frame. The presence of an alternative primed set l, m, v signifies interference terms between the primed and unprimed partial waves. The parameter ct is the Coulomb phase shift (see Appendix A). The fi are dipole transition amplitudes to the final-state partial wave I, m and contain dynamical information on the photoionization process. In contrast, the Clebsch-Gordan coefficients (CGC) provide geometric constraints that are consequent upon angular momentum considerations. 

When experimental results are later introduced, it will be seen that the significance of the final-state scattering in PECO measurements is confirmed by the observation that for C li core ionizations, which must therefore proceed from an initial orbital that is achiral by virtue of its localized spherical symmetry, there is no suggestion that the dichroism is attenuated. The sense of the chirality of the molecular frame in these cases can only come from final-state continuum electron scattering off the chiral potential. Generally then, the induced continuum phase shifts are expected to be of paramount importance in quantifying the observed dichroism. 

The electron and photon angular momentum projections, m, v, and the recoil direction, k, appearing in Eq. (A.3) are defined in the molecular frame, but our... 

Sixfold barriers to internal rotation occur in molecules such as toluene andp-fluoro-toluene whose molecular frame has C2v symmetry about the rotor axis. The simplest spectroscopic model of internal methyl rotation assumes a rigid, threefold symmetric methyl rotor attached to a rigid molecular frame with the C2 axis coincident with the rotor top axis.25 The effective one-dimensional sixfold torsional potential takes the traditional form,... 

Chemical substitution in a pattern that breaks the 2v symmetry of the molecular frame introduces a threefold component to the one-dimensional model torsional potential ... 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

Figure 1. Some examples of luminescent probes with intramolecular charge transfer (ICT) electronic excited states. The numbers in parenthesis indicate the typical wavelengths of the excitation/emission maximums for each of them in polar media however, introduction of chemical groups in the unsubstituted molecular frame or attachment to a solid support may shift those values.

Under MAS, the quadrupole splitting is modulated due to the reorientation of the molecular frame [see (14)] ... 

Only for a special class of compound with appropriate planar symmetry is it possible to distinguish between (a) electrons, associated with atomic cores and (7r) electrons delocalized over the molecular surface. The Hiickel approximation is allowed for this limited class only. Since a — 7r separation is nowhere perfect and always somewhat artificial, there is the temptation to extend the Hiickel method also to situations where more pronounced a — ix interaction is expected. It is immediately obvious that a different partitioning would be required for such an extension. The standard HMO partitioning that operates on symmetry grounds, treats only the 7r-electrons quantum mechanically and all a-electrons as part of the classical molecular frame. The alternative is an arbitrary distinction between valence electrons and atomic cores. Schemes have been devised [98, 99] to handle situations where the molecular valence shell consists of either a + n or only a electrons. In either case, the partitioning introduces extra complications. The mathematics of the situation [100] dictates that any abstraction produce disjoint sectors, of which no more than one may be non-classical. In view if the BO approximation already invoked, only the valence sector could be quantum mechanical9. In this case the classical remainder is a set of atomic cores in some unspecified excited state, called the valence state. One complication that arises is that wave functions of the valence electrons depend parametrically on the valence state. 

The first target of Inorganic Electrochemistry is therefore to study the effects of such electron addition/removal processes on the molecular frames. 

The lowest energy unoccupied orbital (LUMO - 27e) is anti-bonding with respect to the Rh-Rh bonds. As a consequence, the addition of electrons to [Rh4(CO)i2] would cause destruction of the molecular frame (see Chapter 8, Section 2.2). 

As we will discuss in Chapter 2, such an unsymmetrical pattern foreshadows fragmentation or severe structural reorganization of the original molecular frame. 

As we will discuss, such a symmetric profile is typical of an electron removal which does not lead to important structural changes. In fact, the 17-electron ferrocenium ion, [Fe(C5H5)2] +, generated upon oxidation, is a stable species which substantially maintains the original molecular frame (but for the fact that, because of the electron removal, the iron-carbon bonds are slightly weakened and hence elongated by about 0.04 A with respect to the neutral parent see Chapter 4, Section 1.1). 

The Chemical Meaning of an Electrochemically Irreversible Process. As a chemical consideration, the occurrence of an electrochemically irreversible process implies so large an activation barrier to the electron transfer that it is likely that (as discussed in the introductory section, Figure. 1.2) it causes breakage of the original molecular frame with formation of new species (see Chapter 7, Section 5). 

The e (antibonding), a (nonbonding) and e 2 (bonding) are assumed to be the frontier orbitals for metallocenes. The non-bonding nature of the a orbitals, the HOMO, is in agreement with the observation that for ferrocene (18 valence electrons terminal electronic configuration e2a f) the removal of one electron to form the ferrocenium ion (17 valence electrons) does not substantially destabilize the molecular frame. 

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