  The diffusion Monte Carlo (DMC), auxiliary field quantum Monte Carlo (AFQMC),
and equation-of-motion coupled cluster (EOM-CC) methods are used to calculate
the electron binding energy (EBE) of the non-valence anion state of a model
(H$_2$O)$_4$ cluster. Two geometries are considered, one at which the anion is
unbound and the other at which it is bound in the Hartree-Fock (HF)
approximation. It is demonstrated that DMC calculations can recover from the
use of a HF trial wave function that has collapsed onto a discretized continuum
solution, although larger electron binding energies are obtained when using a
trial wave function for the anion that provides a more realistic description of
the charge distribution, and, hence, of the nodal surface. For the geometry at
which the cluster has a non-valence correlation-bound anion, both the inclusion
of triples in the EOM-CC method and the inclusion of supplemental diffuse d
functions in the basis set are important. DMC calculations with suitable trial
wave functions give EBE values in good agreement with our best estimate EOM-CC
result. AFQMC using a trial wave function for the anion with a realistic
electron density gives a value of the EBE nearly identical to the EOM-CC result
when using the same basis set. For the geometry at which the anion is bound in
the HF approximation, the inclusion of triple excitations in the EOM-CC
calculations is much less important. The best estimate EOM-CC EBE value is in
good agreement with the results of DMC calculations with appropriate trial wave
functions.
