Molecular symmetry axis frame

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 N CSA principal axis system (PAS) is a frame in which the N CSA is diagonal, with principal components o-ji < molecular symmetry axis (MSA) system is used as a reference frame, the Euler angles... 

Consider a degenerate ( ) vibrational mode in benzene the dipole or polarizability derivative then lies in the plane of the hexagon and the normal modes correspond to in-plane translational motion of the nuclei. Rotation around the molecular symmetry axis will produce a Coriolis force which is in the plane of the molecule (but perpendicular to the direction of translational motion). As a consequence, the dipole or the polarizability derivative (which would otherwise be fixed in the molecular frame) will rotate rapidly in the plane. Specifically, it will have a rotational velocity equal to where Q is the Coriolis coupling constant... 

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 ... 

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... 

Let us proceed to define appropriate coordinate systems (see Fig. 1 and Table 1). We use four different levels of coordinate frames in the averaging procedures of our model construction scheme to describe the molecular arrangement in the scattering volume. On the first level, System M describes the real structure of a segment, whereby the origin is fixed to a well-defined molecular unit System MC (second level) is defined in cylindrical coordinates with the symmetry axis given by the long axis of the rod-like molecular... 

Note that the direction of the z axis is now perpendicular to the molecular plane, in agreement with the notion that the symmetry of the thpy fragment is Cg. It should be added, however, that fits of similar good quality for the angular dependencies of the ODMR lines in a magnetic field was obtained choosing the direction of the principal axes as the x, y, z axis frame in Fig. 13. Previously, from optical experiments [72], the orientation of the transition moment of the lowest-energy excitation was found to lie approximately parallel to the x axis shown in Fig. 13. Anyhow, the ODMR results clearly show that the fine structure z axis is perpendicular to the molecular thpy plane and that the x andy axes are not lifted out of the thpy plane. 

With reference to the molecular axis frame of Fig. 16, it is recalled from the ODMR results that the ligand-localized triplet state excitation in Rh(TTB)+ is on the fragment (of Cj symmetry) marked with the single asterisk. We consider now, as in Sect. 6.2, that the transition moment for radiative decay of the triplet state is determined by (Hsqc) of Eq. (19), with a major contribution from the SOC... 

In the case of a uniaxially symmetric distribution of the molecular axes M, the intensities corresponding to the P and A directions lying along the fixed-frame axes (Z corresponds to the symmetry axis) can be conveniently expressed through the following quantities ... 

The g- and superhyperfine tensors, in general, are eharacterized on the basis of symmetry, and their relative orientation is fixed in the moleeular referenee frame (usually defined by molecular symmetry axes). The Hamiltonian that describes the electronic Zeeman interaction in a principal axis frame (no terms of the form gij, where i J) is... 

The order parameters S, D and the principal axes of the order tensor can be determined by numerous methods. S, D for the 1,1 -binaphthyls (Figure 3.5) are obtained from NMR and C NMR spectroscopy and confirmed by the polarized ultraviolet (UV) spectroscopy (Figure 3.7). One of the principal axes of the order tensor is parallel to the C2 symmetry axis. The two remaining principal axes are rotated around the Ci-axis by a few degrees [46]. The temperature dependence of the orientation of these axes with respect to the molecular frame between 25 °C and 70 °C is less than 2°. For R-B4, R-B5, and R-B6 the Vj-axis is approximately parallel to the naphthyl-naphthyl bond, whereas is parallel to the C2 symmetry axis. The orientation axis of R-B7 is switched [43] in comparison to R-B4, R-B5, and R-B6. Then is parallel to the C2 symmetry axis, whereas is approximately parallel to the naphthyl-naphthyl bond. For R-Bl and R-B3 the axes can be obtained by exchanging the axes of R-B4 (vj, v, V3) to (v, vj , v ) and (vj, Vj, v ), respectively. For R-B3 there is an unsolved... 

The lowest molecular singlet transition for 6T is from a state of Ag symmetry to a Bu-state. Due to the Cah symmetry of the crystal point group the synunetry representations are the same as in the molecular symmetry frame and for clarification we will use lower case letters with respect to the crystal framework. Since the site occupied by a 6T molecule has no special symmetry operations, the site synunetry is Ci. Within the unit cell (space group P2i/ ) there are 4 molecular sites which are related to each other by different symmetry operations 1 identity, 2 inversion, 3 glide plane, 4 two-fold axis. These symmetry operations correspond to the factor group whose irreducible representations are displayed in Table 2 [110]. The molecular wavefunctions have thus to be written as follows, to require the synunetry properties of the crystal (see characters in Table 2) [110]. 

A phenomenological description of the differential cross-section for emission of photoelectrons into solid angle O in the lab frame can be written, assuming random molecular orientation and an axis of cylindrical symmetry defined by the photon polarization, as... 

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,... 

Rotating single-crystal measurements also permitted the extraction of the orientation of the magnetic tensor in the molecular reference frame and the experimental easy axis was found to coincide with the idealized tetragonal axis of the coordination dodecahedron of Dy. Crystal field calculations assuming idealized tetragonal symmetry permitted the reproduction of magnetic susceptibility data for gz = 19.9 and gxy 0 [121]. More elaborated calculations such as ab initio post Hartree-Fock CASSCF confirmed this simple analysis [119]. 

N Is the number of molecules per unit volume (packing density factor), fv Is a Lorentz local field correction at frequency v(fv= [(nv)2 + 2]/3, v = u) or 2u). Although generally admitted, this type of local field correction Is an approximation vdilch certainly deserves further Investigation. IJK (resp Ijk) are axis denominations of the crystalline (resp. molecular) reference frames, n(g) Is the number of equivalent positions In the unit cell for the crystal point symmetry group g bjjj, crystalline nonlinearity per molecule, has been recently Introduced 0.4) to get general expressions, lndependant of the actual number of molecules within the unit cell (possibly a (sub) multiple of n(g)). 

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... 

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