  In the Cellular Dynamical Mean Field Theory (CDMFT), a strongly correlated
system is represented by a small cluster of correlated sites, coupled to an
adjustable bath of uncorrelated sites simulating the cluster's environment; the
parameters governing the bath are set by a self-consistency condition involving
the local Green function and the lattice electron dispersion. Solving the
cluster problem with an exact diagonalization method is only practical for
small bath sizes (8 sites). In that case the self-consistency condition cannot
be exactly satisfied and is replaced by a minimization procedure. There is some
freedom in the definition of the `merit function' to optimize. We use
Potthoff's Self-Energy Functional Approach on the one- and two-dimensional
Hubbard models to gain insight into the best choice for this merit function. We
argue that several merit functions should be used and preference given to the
one that leads to the smallest gradient of the Potthoff self-energy functional.
We propose a new merit function weighted with the self-energy that seems to fit
the Mott transition in two dimensions better than other merit functions.
