Photon absorption recoil correction ($h\nu =\Delta E+\Delta E^2/(2M{c}^2)$)

An initially stationary isolated chemical atom of rest mass $M$ absorbing a single photon must recoil. If the internal excitation changes its rest mass by $\Delta M=\Delta E/c^2$, conservation of energy and conservation of momentum give $M{c}^2+h\nu =\sqrt{(M{c}^2+\Delta E{)}^2+(h\nu {)}^2}$ and hence the exact formula in the title. The additional positive energy supplies recoil. An external body sharing the momentum, or an initially moving chemical atom, changes this laboratory resonance condition.