A new density functional theory (DFT) for calculations of van der Waals (vdW) complexes is presented. In this scheme, a long-range-corrected exchange-correlation functional is combined with a new damping method of the Andersson-Langreth-Lundqvist (ALL) vdW correlation functional for dealing with molecules containing various types of atoms. The long-range correction (LC) scheme for the exchange functional was used in this method. This LC-DFT þ ALL method was applied to calculations of rare-gas dimers and vdW complexes, consisting of diatomic molecules and He atoms. The present method reproduces very accurate potential energy surfaces of these systems. In particular, for CO-He complexes, LC-DFT þ ALL gives an equilibrium bending structure in agreement with experiment without using a particular basis function, while the collinear structure is given by conventional DFT and high-level ab initio molecular orbital methods without using a bond function. By comparing LC-DFT and ALL component energies, it was found that LC plays a major role in the determination of vdW complex structures. This LC-DFT þ ALL method has proved to be a powerful tool for quantitative calculations of large weakly bound systems because it requires much less computational time than high-level ab initio methods.