Compounded or product spin wavefunctions

The first example concerns a system with an electron spin and a nuclear spin, and for simplicity we take S = 1/2 and / = 1/2. Actual examples would be localized radicals 13C or 15N , and mononuclear low-spin 57Fein or 183WV. The spin Hamiltonian is 

The spin wavefunctions are compounded one partrefers to the electron spinandanother part to the nuclear spin, Ims m) (an alternative name is product wavefunctions)  

Just like the electron spin-raising and spin-lowering operators defined in Equations 7.15 and 7.16, we have the analogous operators for the nuclear spin I  

When letting all the spin operators in the Hamiltonian of Equation 7.57 work on the compounded spin functions in Equation 7.58, note that. S, only work on the first part of the spin function, leaving the second part unchanged, and, equivalently, /, works only on the second part leaving the first part unchanged, e.g., 

With all operations worked out, and using the definitions G, = fHUjgJl and a lgfAJg, we obtain the full energy matrix 

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