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When treating the exchange interaction of Wannier excitons, usually only theleading terms of the analytic and the nonanalytic exchange interaction areconsidered. However, higher order terms can lead to a splitting of excitonstates, for which reason a splitting of the $1S$~exciton in cuprous oxide$\left(\mathrm{Cu_{2}O}\right)$ depending on its total momentum $\hbar K$ hasbeen attributed to a $K$ dependent analytic exchange interaction by Dasbach etal [Phys. Rev. Lett. 91, 107401 (2003)]. Going beyond the common treatment ofthe exchange interaction, we derive the correct expressions for these $K$dependent higher order terms using $\boldsymbol{k}\cdot\boldsymbol{p}$perturbation theory. We prove that the appearance of a $K$ dependent exchangeinteraction is inseparably connected with a $K$ independent exchangeinteraction of $P$ and $D$ excitons. We estimate the magnitude of these termsfor $\mathrm{Cu_{2}O}$ from microscopic calculations and show that they are fartoo small to explain the observed $K$ dependent splitting. Instead, thissplitting has to be treated in terms of the dispersion of the excitons.Furthermore, we prove the occurence of a coupling between longitudinal andtransverse excitons in $\mathrm{Cu_{2}O}$ due to the $K$ dependent nonanalyticexchange interaction.

$K$ dependent exchange interaction of the $1S$ ortho exciton in Cu$_2$O