  When treating the exchange interaction of Wannier excitons, usually only the
leading terms of the analytic and the nonanalytic exchange interaction are
considered. However, higher order terms can lead to a splitting of exciton
states, for which reason a splitting of the $1S$~exciton in cuprous oxide
$\left(\mathrm{Cu_{2}O}\right)$ depending on its total momentum $\hbar K$ has
been attributed to a $K$ dependent analytic exchange interaction by Dasbach et
al [Phys. Rev. Lett. 91, 107401 (2003)]. Going beyond the common treatment of
the 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 exchange
interaction is inseparably connected with a $K$ independent exchange
interaction of $P$ and $D$ excitons. We estimate the magnitude of these terms
for $\mathrm{Cu_{2}O}$ from microscopic calculations and show that they are far
too small to explain the observed $K$ dependent splitting. Instead, this
splitting has to be treated in terms of the dispersion of the excitons.
Furthermore, we prove the occurence of a coupling between longitudinal and
transverse excitons in $\mathrm{Cu_{2}O}$ due to the $K$ dependent nonanalytic
exchange interaction.
