= Secular approximation for scalar-coupled spins
{title2=$H_{\mathrm{eff}}=(\omega_1ZI+\omega_2IZ+JZZ)/2$}
The leading-order averaging of rapidly oscillating exchange terms in the <interaction picture> of two nuclear <Zeeman energy> terms. For $\Delta=\omega_1-\omega_2$,
$$
V_{\mathrm{int}}(t)=J\left(e^{i\Delta t}|01\rangle\langle10|+e^{-i\Delta t}|10\rangle\langle01|\right).
$$
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.
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