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.
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