Weak coupling condition for two NMR spins ($|{\omega }_1-{\omega }_2|\gg |J|$)

For the two-spin Hamiltonian operator $H=({\omega }_{1}ZI+{\omega }_{2}IZ+J(XX+YY+ZZ))/2$, juxtaposition of two Pauli matrices denotes their tensor product. Put $\Delta ={\omega }_1-{\omega }_2$. The exchange matrix element between $|01⟩$ and $|10⟩$ is $J$, whereas their diagonal separation is $\Delta$. Nondegenerate perturbation theory therefore gives admixtures $\pm J/\Delta$ and second-order frequency shifts $\pm J^2/\Delta$. Neglecting the exchange terms is justified to leading order when $|J/\Delta |\ll 1$. A large common Zeeman energy is insufficient. To approximate accumulated phases by this leading-order Hamiltonian operator, one also needs $J^{2}t/|\Delta |\ll 1$ over the observation time, or must include the second-order shifts.

For fixed nonidentical chemical shifts, increasing the static magnetic field increases $|\Delta |$ in angular-frequency units while the NMR scalar coupling changes little. If $\Delta =0$ remains exact, the approximation fails at every nonzero $J$.

The leading-order averaging of rapidly oscillating exchange terms in the interaction picture of two nuclear Zeeman energy terms. For $\Delta ={\omega }_1-{\omega }_2$,Their first-order time integrals are bounded by $2|J/\Delta |$, so they contribute little when $|\Delta |\gg |J|$, while the stationary Ising coupling $JZZ/2$ survives. Higher orders give frequency shifts of order $J^2/\Delta$. This approximation is controlled by the difference of resonance frequencies, not their common magnitude.