Irreducible spherical tensorial representations

With this general common form, the behavior of all internal Hamiltonians under rotations can be nicely described using irreducible spherical tensorial representations for the tensor and vector products given in Equation (2.7.24) [4,17]. This formalism allows the complete understanding of the behavior and evolution of all these Hamiltonians during complex experimental manipulations carried out in high-resolution NMR, all of which involves rotations performed either spatially or in the spin space. 

In our description of spin reorientational relaxation processes, tensorial quantities are used for which it is necessary to know the transformation properties concerning rotation. A clear and compact formulation is obtained by replacing the cartesian components with a representation in terms of irreducible spherical components. It is known that any representation of the group of rotations can be developed into a sum of irreducible rqpre-sentations D of dimension 2/ +1. If for the description of general rotation R(U) we use the Euler angles Q = (a, p, y), this rotation will be defined by... 

However, for many purposes it is desirable to extend the theory of irreducible tensorial sets to the real irreducible representations that are carried by sets of real spherical harmonics. In the present paper this has been put into effect by using the properties of the invariant irreducible products considered by Fano and Racah. In particular, the phase conventions proposed in that work have been preserved so that invariants (reduced matrices, W- and -coefficients) are identical in the two representations. 

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