  By quantizing the semiclassical motion of excitons, we show that the Berry
curvature can cause an energy splitting between exciton states with opposite
angular momentum. This splitting is determined by the Berry curvature flux
through the $\bm k$-space area spanned by the relative motion of the
electron-hole pair in the exciton wave function. Using the gapped
two-dimensional Dirac equation as a model, we show that this splitting can be
understood as an effective spin-orbit coupling effect. In addition, there is
also an energy shift caused by other "relativistic" terms. Our result reveals
the limitation of the venerable hydrogenic model of excitons, and highlights
the importance of the Berry curvature in the effective mass approximation.
