Spin angular momentum operators

[ehrenfest]

I posted this is in the QM section but maybe here would have been better. I don't think it is a hard question for anyone who knows QM:

https://www.physicsforums.com/showthread.php?t=181220

 

[ehrenfest]

Okay, I want to know the answer enough that I will repost the question here:

The dipolar coupling Hamiltonian expressed in the lab frame (units of Hz) is

$$H ^D_{ij} = - constants/r_{ij}^3 * I_{iz} * I_{jz} * P_2(\cos(\theta))$$

where r_ij is the internuclear distance between spins, ci and cj are the gyromagnetic ratios of spins i and j, and I_kz are spin angular momentum operators. The angular portion of the
dipolar Hamiltonian is described using the second rank Legendre function, P_2(cos h(t)),
which is a function of the angle h subtending the magnetic field and the ijth internuclear
vector.

Somehow they go from that equation to the following one:

$$D^{resultant}_{ij} = constants * < P_2(cos(\theta(t))/r_{ij}^3)>$$