Free-space electric-dipole spontaneous emission rate ($A_{21}={\omega }_{21}^3|{\mathbf{d}}_{12}{|}^2/(3\pi {ϵ}_0\hslash c^3)$)

For a specified electric-dipole transition, summing free radiation modes gives $A_{21}={\omega }_{21}^3|{\mathbf{d}}_{12}{|}^2/(3\pi {ϵ}_0\hslash c^3)$. Unresolved degenerate levels require sums over final substates and appropriate initial-state averages. Modified radiation environments alter the available modes.