Spherical tensor form of the Hamiltonian operator

The coordinates are illustrated in figure 4.1. ep is the charge of the p h proton in the nucleus, having position vector Rp, and ,- is the charge of the Ith electron (—e) with position vector R, in the remainder of the atom or molecule. 6ip is the angle between the vectors Rt and Rp. 

We now expand the Legendre polynomial (cos 0ip) using the spherical harmonic 

The electrostatic quadrupole term in this sum is that for which ( = 2, so that the quadrupole Hamiltonian is given by 

This may be written as the scalar product of two second-rank irreducible tensors, 

Some authors use equation (4.30) to represent the quadrupole interaction, as we will in this book, and others prefer to use the form 

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