Molecule-fixed frame

As mentioned above, the electron spin system is strongly coupled to the classical degrees of freedom, in the first place the orientation of a molecule-fixed frame with respect to the lab frame, through anisotropic interactions. We concentrate here on the case of S > 1, where the main anisotropic interaction is denoted as the zero-field splitting, ZFS. In the language of spin Hamiltonians (8,80), the ZFS interaction is written ... 

P(Qol, t) is the conditional probability of the orientation being at time t, provided it was Qq a t time zero. The symbol — F is the rotational diffusion operator. In the simplest possible case, F then takes the form of the Laplace operator, acting on the Euler angles ( ml) specifying the orientation of the molecule-fixed frame with respect to the laboratory frame, multiplied with a rotational diffusion coefficient. Dr. Equation (44) then becomes identical to the isotropic rotational diffusion equation. The rotational diffusion coefficient is simply related to the rotational correlation time introduced earlier, by tr = 1I6Dr. 

This leads to Eq. (38) taking on a correspondingly more complicated form 93-96). In essence, the spectral density becomes dependent on the angles 0 and ( ) specifying the relative orientations of the two relevant molecule-fixed frames the principal frame of the static ZFS and that of the dipole-dipole interaction. Disregarding the possibility of internal motions, the 0 and ( > angles are time-independent. 

In contrast, the n ==> Jt transition has a ground-excited state direct product of B2 x Bj = A2 symmetry. The C2V s point group character table clearly shows that the electric dipole operator (i.e., its x, y, and z components in the molecule-fixed frame) has no component of A2 symmetry thus, light of no electric field orientation can induce this n ==> Jt transition. We thus say that the n ==> 7t transition is El forbidden (although it is Ml allowed). 

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.

We now turn to the electronic part of the quadrupole interaction in equation (9.11). Since this part of the interaction is most sensibly described in a molecule-fixed frame,... 

Here we have partitioned the sums over all atoms a and /3 in the molecules P and P in the following manner. First, we sum over equivalent atoms within the same class a E a and (3 E b, which have the same chemical nature X = Xa and Xp = X and the same distance da = da and dp = db to the respective molecular center of mass. Next, we sum over classes a E P and b E P. The orientations da and dp of the position vectors of the atoms d and dp, relative to the molecular centers of mass, are still given with respect to the global coordinate frame. If we denote the polar angles of da and dp in the molecule fixed frames by d°a and dp and remember that the molecular frames are related to the global frame by rotations through the Euler angles o)P and to/., respectively, we find that... 

The Wigner rotations describe the coordinate transformations from the principal axis frame (P ) in which the tensor describing the interaction X is diagonal, via a molecule-fixed frame (C) and the rotor-fixed frame (R) to the laboratory frame (L) as illustrated in an ORTEP representation in Fig. 1. 

When very accurate dipole moments are deduced, it is proper to query the significance of a breakdown of the Bom-Oppenheimer approximation. This approximation justifies the assignment of molecular property tensors, such as dipole moments and polarizabilities, to specific directions in a molecule-fixed frame and supports the use of a property function or surface representing the variation of the property with nuclear position. The dipole moment of HD (5.85 X 1(T4 D)26 arises solely from the breakdown of the approximation and may have the sense H D-.27-29 In HC1 and DC1, there is an isotope effect on the dipole moment that has been attributed to a violation of the Born-Oppen-heimer approximation30 there is an apparent difference of 0.0010 0.0002 D between the dipole functions of HC1 and DC1, with HC1 having the bigger moment. This result is in accord with a recent theoretical analysis by Bunker.31... 

As stated above, this theory assumes that translational diffusion is isotropic that is, in a molecule-fixed frame, the diffusion constant parallel to the long molecular axis is the same as that perpendicular to it. For highly anisotropic large molecules this is probably not a good assumption. Maeda and Saito (1969) have calculated the spectrum taking into account the anisotropy of the translational diffusion constant. Their resulting expressions are rather complex and will not be given here. Their results are expressed as a power series in the translational diffusion coefficient anisotropy,... 

When, as explained in Sect. 5.2, the molecule-fixed frame is attached to two vectors, the ro-vibrational basis functions for a molecule with 5 atoms are... 

It is still true that choosing molecules with reduced anisotropy in the molecule-fixed frame is beneficial. After all, if there is no anisotropy in the molecule-fixed frame, the helium interaction in the lab frame is spherical for all rotational states. It is therefore desirable to seek molecules with short bond lengths, which should reduce the anisotropy of the electron charge cloud in the molecule frame for a fixed helium approach distance. There is potential for the Zeeman relaxation of superficially anisotropic molecules to be suppressed by spatially large, roughly spherical electron wavefunctions, much like was found with the submerged-shell rare earth elements. 

The multipole moment operators in Eq. (7) are still referred to the global coordinate frame. We now transform them to the local or molecule-fixed frame ... 

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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