Molecular frame axis

Figure 27. Molecular frame axis convention for the C2V NO dimer. The y axis is along the N—N bond. Both pump and probe laser polarizations are parallel to the y axis.

The 6D fully correlated TRCIS data set may be cut, projected, or filtered to reveal both scalar and vector correlations as a function of time. We restrict our discussion here to angular correlations. The molecular frame axis convention for the NO dimer is shown in Fig. 27. Note that the pump and probe laser polarizations were parallel to each other in these experiments. 

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 ... 

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,... 

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... 

Consider the vector that lies along the z-axis of the molecular frame. It has orientation QL = (uP) in the laboratory frame. The conditional probability per unit solid angle, g(i2/ ft00), that this vector with orientation 20 at time... 

In order to extract some more information from the csa contribution to relaxation times, the next step is to switch to a molecular frame (x,y,z) where the shielding tensor is diagonal (x, y, z is called the Principal Axis System i.e., PAS). Owing to the properties reported in (44), the relevant calculations include the transformation of gzz into g x, yy, and g z involving, for the calculation of spectral densities, the correlation function of squares of trigonometric functions such as cos20(t)cos20(O) (see the previous section and more importantly Eq. (29) for the definition of the normalized spectral density J((d)). They yield for an isotropic reorientation (the molecule is supposed to behave as a sphere)... 

The antisymmetric tensor is generally not observable in NMR experiments and is therefore ignored. The symmetric tensor is now diagonalized by a suitable coordinate transformation to orient into the principal axis system (PAS). After diagonalization there are still six independent parameters, the three principal components of the tensor and three Euler angles that specify the PAS in the molecular frame. 

Figure 3 shows several examples of 13C chemical shift tensors of carbe-nium ions oriented in the molecular frame as indicated. If the molecule or ion has a rotational axis of C3 or higher order that passes through the nucleus in question, symmetry demands that the two components perpendicular to the C3 or higher order rotational axes be identical in such cases the identical components are designated SL. The unique component that lies along the... 

For linear molecules, we choose the molecular axis to be along the z-axis of the molecular frame. In this case, Q/ = 0 for all n f 0 because the Y(m depend on cp as exp imcp. As a consequence, Q( becomes e-ma Q/o, which means that only Q/o = qe is invariant under rotation. In other words there is only one independent multipole moment for every order L Laboratory-fixed components thus become simply... 

Figure 1. Schematic of the mesomorphous structure in a lamellar phase. The different coordinate systems used in the text are outlined laboratory frame (L), director frame (D), and molecular frame (M). 0LD and 0DM are angles between the z axis in laboratory-director systems and director-molecular systems, respectively (13).

Axial symmetry according to Bleaney s approach is maintained in polymetallic lanthanide complexes when the metals lie on the molecular threefold or fourfold axes. For n magnetically non-coupled lanthanide ions packed along the symmetry axis, contact and pseudo-contact contributions can be considered as additive and the original model-free equation (eq. (47)) is transformed into eq. (61) in which the sum runs over the n paramagnetic centres, each being located at the origin of its own reference frame associated with a specific set of axial coordinates O 1 and r" (the z axis corresponds to the molecular symmetry axis, fig. 54),... 

The single-particle wave function for the free photoelectron may be expressed as an expansion in angular momentum partial waves characterized by an orbital angular momentum quantum number l and and associated quantum number X for the projection of l on the molecular frame (MF) z axis [22, 23, 63-66],... 

Therefore, according to Eq. (12.57), the molecule feels a LIP AW that de] the orientation of the molecular Z axis relative to the field polarization r azimuthal rotation angle of the molecular frame about the Z axis. 

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