Perturbation Theory Treatment of Hyperfine Term

If we expand the hyperfine term of the spin Hamiltonian and write the operators in terms of raising and lowering operators  

Thus the zero-, first-, and second-order energies are  

Thus we must express iq and F2 in parameters of the original spin Hamiltonian. 

Starting with the two ways of expressing the hyperfine Hamiltonian term, we equate the coefficients of Sx, Sy and Sz  

Z2 IX + Z22Iy + Z22Iy = Dylx + Eyly, + FyIzn Z2 IX + Z22Iy + Z22Iy = K 

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