  We propose a new, controlled approximation scheme that explicitly includes
the effects of non-local correlations on the $D=\infty$ solution. In contrast
to usual $D=\infty$, the selfenergy is selfconsistently coupled to two-particle
correlation functions. The formalism is general, and is applied to the
two-dimensional Falicov-Kimball model.
  Our approach possesses all the strengths of the large-D solution, and allows
one to undertake a systematic study of the effects of inclusion of k-dependent
effects on the $D=\infty$ picture. Results for the density of states
$\rho(\omega)$, and the single particle spectral density for the 2D
Falicov-Kimball model always yield positive definite $\rho(\omega)$, and the
spectral function shows striking new features inaccessible in $D=\infty$. Our
results are in good agreement with the exact results known on the 2D
Falikov-Kimball model.
