Molecule-fixed coordinate system frame

For an isolated system, treatment of the intramolecular Jahn-Teller effect is relatively simple. As the system is isolated, we may ignore molecular rotation and consider a molecule-fixed coordinate system. Within this frame of reference, the electronic and vibrational states can be formulated in terms of the irreducible representations (irreps) of the reference configuration. Overall, the system Hamiltonian is generally written in the form... 

The dipole operator d is a vector defined in the body-fixed frame of the molecule. Consequently, the transition dipole moment /a defined in (2.35) is a vector field with three components each depending — like the potential — on R, r, and 7. For a parallel transition the transition dipole lies in the plane defined by the three atoms and for a perpendicular transition it is perpendicular to this plane. Following Balint-Kurti and Shapiro, the projection of /z, which is normally calculated in the body-fixed coordinate system, on the space-fixed z-axis, which is assumed to be parallel to the polarization of the electric field, can be written as... 

Fig. 1. ORTEP-type representation of a spatial second-rank anisotropic interaction tensor in its principal-axis system (P j, a molecule (or crystallite) fixed coordinate system (C), the rotor-fixed coordinate system (R), and the lahoratory-lixed coordinate system (L) along with the Euler angles fixr = ctxYy Pxyj yxY describing transformation between the various frames X and T. Reproduced from Ref. 36 with pemtission.

The orientational distribution fimction P (cos 0) enters the shape of the wideline spectrum 5(f2) in a slightly hidden way. The angular dependence of the resonance frequency is given by (3.1.23) via the orientation of the magnetic field in the principal axes system XYZ of the coupling tensor (cf. Fig. 3.1.2), while the orientational distribution function specifies the distribution of the preferential direction n in a molecule-fixed coordinate frame (Fig. 3.2.2(a)). Figure 3.2.3 shows the relationship between the different coordinate frames and the definition of the relative orientation angles. 

Fig. 3.2.3 Relationships between coordinate systems for the description of molecular order. The orientation of the laboratory coordinate system in the principal axes system of the coupling tensor determines the angular dependence of the resonance frequency. The orientations of the preferential sample direction in a molecule-fixed coordinate frame determines the orientational distribution function.

Now consider the case where 7)/ is a tensor in the laboratory-fixed coordinate frame. Then the spherical components in Eq. (7.C.13) are also in the laboratory frame, and we denote this by writing these elements as Tf (L). The elements Tf (L) that appear in Eq. (7.C.13) can be related through Eq. (7.C.8) to the spherical components in the molecule or body-fixed coordinate system... 

As the wavelength is much larger than the dimension of the molecule (e.g., 5000 A with respect to 10 A) the space-depending factors can be put equal to one, meaning that they are constant over the whole molecule. However, the mathematical treatment becomes more subtle when it comes to introducing the absorption coefficient k (that will later be related to the molar absorption coefficient e). A subscript has to be introduced, meaning that at the absorption frequency v y corresponding to the frequency of the transition from level a to level j or in a close vicinity, the expressions (e " / ) and (e / ) have a fixed value (see Piepho and Schatz, 1983, pp. 11 and 17). Within the frame of the fixed coordinate system of a molecule, the space-dependent factors may be expressed as a... 

Figure 8.11. Cartesian coordinate system for describing the position vectors of the particles (electrons and nuclei) in a molecule. 0(X, Y, Z) is the laboratory-fixed frame of arbitrary origin, and c.m. is the centre-of-mass in the molecule-fixed frame. For the purposes of illustration four particles are indicated, but for most molecular systems there will be many more than four.

Let us consider the electron-vibrational matrix element. As is usually done, we consider two coordinate systems, the origins of which are located at the center of mass of the molecule. The first coordinate system is fixed in space, while the second system (the rotational one) is fixed to the molecule. For describing the orientation of the rotational system with respect to the fixed frame we use the Eulerian angles 6 = a, / , y. In the Born-Oppenheimer approximation, the motion of nuclei may be decomposed into the vibrations of the nuclei about their equilibrium position and the rotation of the molecule as a whole. Accordingly, the wave function of the nuclei X (R) is presented as a product of the vibrational wave function A V(Q) and the rotational wave function... 

Schrodinger s Hamiltonian describing the molecule as a system of N charged particles in a coordinate frame fixed in the laboratory is... 

Reference Frames and Fluxes, in speaking of a diffusional flux, it is necessary to specify a reference frame from which the diffusion process is to be observed. Generally, a reference velocity such as the mass, molar, or volume average bulk velocity (v,v, or v, respectively) in the system is selected, and movement of the component of interest relative to this reference velocity is defined to be true diffusion. For practical reasons, a fixed reference frame is generally also considered to relate the mathematical treatment of this molecular scale transport process to an actual physical system with well-defined dimensions. In this case, the total flux relative to the fixed reference frame is partitioned into two parts bulk flow and true molecular diffusion. This partitioning is necessary, because even in the absence of an externally imposed bulk flow, interdiflfusion of molecules with respect to each other produces an effective bulk flow relative to fixed coordinates if the molecules have different masses (1,2). 

It is apparent that the 7 take the place in this formulation of the interaction tensors T of the conventional Cartesian formulation, but it should be emphasized once again that all the formulae given here refer to multipole moment components in the local, molecule-fixed frame of each molecule, whereas the corresponding Cartesian formulae deal in space-fixed components throughout and require a separate transformation between molecule-fixed and space-fixed frames. ( Space-fixed is perhaps a misleading term here, since the calculation is commonly carried out in a coordinate system with one of its axes along the intermolecular vector. However, the point is that in the Cartesian tensor notation there has to be a common set of axes for the system as a whole, and this can be the molecule-fixed frame for at most one of the molecules involved.)... 

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