  We introduce an approach for calculating perturbative corrections to the
ring-polymer instanton approximation to tunneling splittings (RPI+PC), by
computing higher-order terms in the asymptotic expansion in $\hbar$. The
resulting method goes beyond standard instanton theory by using information on
the third and fourth derivatives of the potential along the tunneling path to
include additional anharmonic effects. This leads to significant improvements
both in systems with low barriers and in systems with anharmonic modes. We
demonstrate the applicability of RPI+PC to molecular systems by computing the
tunneling splitting in full-dimensional malonaldehyde and a deuterated
derivative. Comparing to both experiment and recent quantum-mechanical
benchmark results, we find that our perturbative correction reduces the error
from -11% to 2% for hydrogen transfer and performs even better for the
deuterated case. This makes our approach more accurate than previous
calculations using diffusion Monte Carlo and path-integral molecular dynamics,
while being more computationally efficient.
