We propose to express corrections to the standard approximations of the exchange-correlation functionals in the form of multiatom-centered functionals that can be related to the atoms-in-molecules theory of Bader and a multicenter density expansion. Unlike conventional functionals, the variables of these multicenter functionals are the positions and identities of the nuclei, which can unambiguously be determined from the topology of the electronic density. The proposed multicenter correction is used to provide a theoretical basis for the dispersion-corrected atom-centered potential (DCACP) approach used to improve van der Waals interactions in conventional density functional theory. Convergence to the correct long-range asymptotic behavior of the multicenter expansion for DCACPs is demonstrated in the test case of the H-2 van der Waals dimer.