  Recent experiments revealed the importance of higher-band effects for the
Mott insulator (MI) -- superfluid transition (SF) of ultracold bosonic atoms or
mixtures of bosons and fermions in deep optical lattices [Best \emph{et al.},
PRL \textbf{102}, 030408 (2009); Will \emph{et al.}, Nature \textbf{465}, 197
(2010)]. In the present work, we derive an effective lowest-band Hamiltonian in
3D that generalizes the standard Bose-Fermi Hubbard model taking these effects
as well as nonlinear corrections of the tunneling amplitudes mediated by
interspecies interactions into account. It is shown that a correct description
of the lattice states in terms of the bare-lattice Wannier functions rather
than approximations such as harmonic oscillator states is essential. In
contrast to self-consistent approaches based on effective Wannier functions our
approach provides a quantitative explanation of the observed reduction of the
superfluid phase for repulsive interspecies interactions.
