Projected tensors system parameters

Should one however, not transform the parameter the components of a projective scalar behave under coordinate transformation exactly like the components of an affine scalar. Not transforming the parameter means, so to speak, that we keep our space fixed in a specific state. If some scalar is given, each such state corresponds to a certain component of the scalar. As we shall soon see this applies to all our projective tensors as well. Each state is specifically associated with a certain coordinate system in each tangent space. 

The values of the rhombicity parameters are conventionally limited to the range 0 < EjD < 1/3 without loss of generality. This corresponds to the choice of a proper coordinate system, for which /)zz (in absolute values) is the largest component of the D tensor, and /) is smaller than Dyy. Any value of rhombicity outside the proper interval, obtained from a simulation for instance, can be projected back to 0 < EID < 1/3 by appropriate 90°-rotations of the reference frame, that is, by permutations of the diagonal elements of D. To this end, the set of nonconventional parameters D and EID has to be converted to the components of a traceless 3x3 tensor D using the relationships... 

The coefficients Blk are related to the components of the transition dipole moment in the molecule-fixed system and the 0 are the angular expansion functions defined in (11.5). The dipole moment transforms like a tensor of rank 1 which explains why it is expanded in terms of the angular functions for an angular momentum J = 1. Since its projection on the space-fixed z-axis is independent of the azimuthal angle pn, only functions with M = 0 are allowed. Furthermore, the dipole moment has the parity —1 so that the parameter p is restricted to +1 [remember that the parity is given by (—lj p]. 

We can call the terms shown explicitly in Equation (A3) quadratic terms other (higher-order) terms can exist [Ref. 7, Section 6.7]. The above spin-Hamiltonian is used to obtain the energies (and hence also transition energies) of the spin system considered. Another, similar spin-Hamiltonian (with the excitation magnetic-field amplitude vector B, replacing B in Equation (A3)), yields the transition relative intensities. The line positions and intensities obtained are expressed in terms of the scalar ( tensors of zeroth rank) parameters g, gn, D, P,A,..., derived as projections from the matrices g, gn, D, P and A.12... 

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