Principal axis system coordinates

Rotational constants G = A, B or C are inversely proportional to principal moments of inertia Ia through the expressions G = h/Sn2Ia, where a refers to one of the three principal inertia axis directions a, b or c. The Ia are related to the coordinates of the atoms i in the principal axis system via the... 

In the coordinate system that diagonalizes g, the related D-tensor is also diagonal. Expanding the fine structure term in the principal axis system, we have ... 

Fig. 3. The principal axis system of the order tensor corresponds to the specific positioning of a set of coordinate axes relative to the molecule, as indicated by the Sxx, Syy and Szz labelled axes. The rotation which diagonalizes the order tensor also carries the initial molecular coordinate frame, specified by the x, and z labelled axes, into the PAS of the order tensor. This figure was prepared using the program Module.193...

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. 

Let us summarize the results. Using the principal-axis system, we defined the nuclear mass-weighted Cartesian coordinates qt by (6.1) and (6.3) ... 

The Zeeman splitting parameter matrices are called g both for electrons and nuclei. However, for electrons, it is not customary to depict the deviations from ge by expression g—yc.(U->, -cr). Matrices g can always be taken as real symmetric, to describe measured data, and it follows that then there exists a special spatial Cartesian coordinate system, the principal axis system (determined by the local symmetry, if any). Here g is diagonal, exhibiting its principal values (e.g. in EPR, see Ref. 7, Chapter 4). However, from theory, the true matrix g can turn out in some situations to be non-symmetric.10... 

The partitioned grand resistance matrix in Eq. (7.13) is a function only of the instantaneous geometrical configuration of the particulate phase. This consists of the fixed particle shapes together with the variable relative particle positions and orientations. As such, geometrical symmetry arguments (where such symmetry exists) may be used to reduce the number of independent, nonzero scalar components of the coefficient tensors in Eq. (7.13) for particular choices of coordinate axes (e.g., principal axis systems). 

The molecular coordinate system of a rigid molecule is defined as the principal axis system of the equilibrium configuration which is taken as as reference. Thus we write... 

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.

In the principal axis system (PAS) of a general polyatomic molecule with the origin of the molecule-fixed axes at the center of mass (COM), the principal moments of inertia arc related to the coordinates of the atoms of mass mj by... 

In the usual procedure a classical Hamiltonian function for the model is formulated. Attention should be given to the choice of the molecular coordinate system. As the frame and top are rigid, a convenient choice is the principal axis system of the whole molecule.8 As a consequence of the symmetry of the top, the orientation of the coordinate system within the molecule is independent of the torsion angle. Another choice, which is called the internal axis system, is defined in such a way that the angular momenta produced by internal rotation of the top and frame compensate each other.10 The Hamiltonian functions in both coordinate systems are related by a contact transformation, which guarantees the invariance of Poisson brackets11 and, subsequently, of the commutation relations. 

When using Eq. (11.11), care must be taken to rotate the constituent atomic susceptibility tensors, with their natural principal axis system oriented at the bond axes, into the principal inertia axis system of the molecule. With the components of the susceptibility tensor transforming like the corresponding coordinate products [compare Eq. (1.4)], the appropriate transformation is given by ... 

Since the dipole-dipole interaction is an internal interaction, the coordinate axes must correspond to the symmetry of the system being investigated. In a planar molecule, e.g. in naphthalene, the coordinate system x, y,z must therefore be identical to the principal-axis system of the molecule. In the Ti exciton, the coordinate system must reflect the symmetry of the crystal. The identification of the three axes X, y, and z, i.e. also of the three states T ), Ty), and Tf) relative to the molecular or crystal axes however requires a measurement in an applied magnetic field. 

Chemical shift tensor In NMR, the chemical shift anisotropy is described by a second-rank tensor (a 3 x 3 matrix). Can generally be expressed in a coordinate frame where all off-diagonal elements vanish. In this principal axis system, the chemical shift tensor is fully described by the three diagonal elements—the principal components, 5n, 822, and 533—and the three eigenvectors or Euler angles describing the orientation of the principal axes with respect to an arbitrary frame. 

Fig. 2 Spatial dependence of the dipolar shift anisotropy (a) and of the pseudocontact shift in the magnetic susceptibility principal axis system (b). The position of the nucleus is defined by its spherical coordinates (r, 6, tp) in the PAS of the x tensor. Violet surfaces represent isosurfaces of Aa, and blue and red surfaces represent respectively positive and negative isosurfaces of...

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