  We show how an embedded many-body expansion (EMBE) can be used to calculate
accurate \emph{ab initio} energies of water clusters and ice structures using
wavefunction-based methods. We use the EMBE described recently by Bygrave
\emph{et al.} (J. Chem. Phys. \textbf{137}, 164102 (2012)), in which the terms
in the expansion are obtained from calculations on monomers, dimers, etc. acted
on by an approximate representation of the embedding field due to all other
molecules in the system, this field being a sum of Coulomb and
exchange-repulsion fields. Our strategy is to separate the total energy of the
system into Hartree-Fock and correlation parts, using the EMBE only for the
correlation energy, with the Hartree-Fock energy calculated using standard
molecular quantum chemistry for clusters and plane-wave methods for crystals.
Our tests on a range of different water clusters up to the 16-mer show that for
the second-order M\o{}ller-Plesset (MP2) method the EMBE truncated at 2-body
level reproduces to better than 0.1 m$E_{\rm h}$/monomer the correlation energy
from standard methods. The use of EMBE for computing coupled-cluster energies
of clusters is also discussed. For the ice structures Ih, II and VIII, we find
that MP2 energies near the complete basis-set limit reproduce very well the
experimental values of the absolute and relative binding energies, but that the
use of coupled-cluster methods for many-body correlation (non-additive
dispersion) is essential for a full description. Possible future applications
of the EMBE approach are suggested.
