  We construct the effective lowest-band Bose-Hubbard model incorporating
interaction-induced on-site correlations. The model is based on ladder
operators for local correlated states, which deviate from the usual Wannier
creation and annihilation, allowing for a systematic construction of the most
appropriate single-band low-energy description in the form of the extended
Bose-Hubbard model. A formulation of this model in terms of ladder operators
not only naturally contains the previously found effective multibody
interactions, but also contains multibody-induced single-particle tunneling,
pair tunneling, and nearest-neighbor interaction processes of higher orders. An
alternative description of the same model can be formulated in terms of
occupation-dependent Bose-Hubbard parameters. These multiparticle effects can
be enhanced using Feshbach resonances, leading to corrections which are well
within experimental reach and of significance to the phase diagram of ultracold
bosonic atoms in an optical lattice. We analyze the energy-reduction mechanism
of interacting atoms on a local lattice site and show that this cannot be
explained only by a spatial broadening of Wannier orbitals on a single-particle
level, which neglects correlations.
