  We present a detailed derivation of a multi-site mean-field theory (MSMFT)
used to describe the Mott-insulator to superfluid transition of bosonic atoms
in optical lattices. The approach is based on partitioning the lattice into
small clusters which are decoupled by means of a mean field approximation. This
approximation invokes local superfluid order parameters defined for each of the
boundary sites of the cluster. The resulting MSMFT grand potential has a
non-trivial topology as a function of the various order parameters. An
understanding of this topology provides two different criteria for the
determination of the Mott insulator superfluid phase boundaries. We apply this
formalism to $d$-dimensional hypercubic lattices in one, two and three
dimensions, and demonstrate the improvement in the estimation of the phase
boundaries when MSMFT is utilized for increasingly larger clusters, with the
best quantitative agreement found for $d=3$. The MSMFT is then used to examine
a linear dimer chain in which the on-site energies within the dimer have an
energy separation of $\Delta$. This system has a complicated phase diagram
within the parameter space of the model, with many distinct Mott phases
separated by superfluid regions.
