  We calculate Heisenberg-type magnetic exchange interactions for SrMnO$_3$
under isotropic volume expansion using an approach that is based on total
energy variations due to infinitesimal spin rotations around a given reference
state. Our total energy calculations using density functional theory (DFT)
indicate a transition from antiferromagnetic to ferromagnetic coupling for
increasing interatomic distances, corresponding to a sign change of the nearest
neighbor exchange interaction. This sign change cannot easily be understood
from a standard superexchange mechanism. Furthermore, the exchange interaction
strongly depends on the corresponding reference state. This "non-Heisenberg"
behavior increases with increasing volume and is also confirmed through
non-collinear DFT calculations. An orbital- and energy-resolved decomposition
of the exchange coupling suggests that an increased partial occupancy of $e_g$
orbitals near the Fermi level is crucial both for the sign change and the
non-Heisenberg behavior of the nearest neighbor interaction. Furthermore, even
though both $e_g$ and $t_{2g}$ contributions to the exchange interactions decay
exponentially for large inter-atomic distances, the $e_g$ contribution remains
surprisingly strong over relatively large distances along the crystal axes.
