  Standard forms of density-functional theory (DFT) have good predictive power
for many materials, but are not yet fully satisfactory for solid, liquid and
cluster forms of water. We use a many-body separation of the total energy into
its 1-body, 2-body (2B) and beyond-2-body (B2B) components to analyze the
deficiencies of two popular DFT approximations. We show how machine-learning
methods make this analysis possible for ice structures as well as for water
clusters. We find that the crucial energy balance between compact and extended
geometries can be distorted by 2B and B2B errors, and that both types of
first-principles error are important.
