  We apply parquet-diagram summation methods for the calculation of the
superfluid gap in $S$-wave pairing in neutron matter for realistic
nucleon-nucleon interactions such as the Argonne $v_6$ and the Reid $v_6$
potentials. It is shown that diagrammatic contributions that are outside the
parquet class play an important role. These are, in variational theories,
identified as so-called "commutator contributions". Moreover, using a
particle-hole propagator appropriate for a superfluid system results in the
suppression of the spin-channel contribution to the induced interaction.
Applying these corrections to the pairing interaction, our results agree quite
well with Quantum Monte Carlo data.
