starting from Eq. ( ). For small molecules, the characteristic frequencies in Eq. ( ) are typically further approximated by ionization energies (I a and I b ), as indicated, which affords the textbook result. The single-frequency model in Eq. ( ) is tantamount to the Unsöld approximation, in which energy denominators are replaced by a single effective excitation energy so that a sumover-states expression for the polarizability reduces to a single term. This approximation enters London's second-order treatment of dispersion 32 and is found to work well, hence there is a natural consistency to its use in the present context. It is worth noting that a single-frequency or "plasmon-pole" approximation for the inverse dielectric function is used to construct both the vdW-DF and VV10 nonlocal correlation functionals. The vdW-TS scheme uses C 6,ab in Eq. ( ) within the DFT+D(0) model of Eq. ( ), although it has also been implemented as a self-consistent functional based on a C 6,ab [ρ(r)] functional. Like XDM, the vdW-TS approach affords pairwise C 6 coefficients that depend on the molecular environment, meaning that type-A nonadditivity is included in the model. We will not consider the vdW-TS scheme in the present work because it is largely superseded by the MBD model developed subsequently by Tkatchenko and co-workers. Starting from the vdW-TS polarizability model in Eq. ( ), the MBD approach further accounts for type-B nonadditivity based on selfconsistent screening of the atomic polarizabilities. These many-atom dispersion effects may be especially important for large, conjugated molecules and in materials having numerous polarizable centers. In these nanoscale materials, many-body effects lead to deviations from R -6 behavior in the long-range interactions. Development of the MBD method starts from the atomic polarizability model in Eq. ( ), arranged as a vector ᾱTS of dimension N atoms . (In polarizable force fields, even anisotropic molecular polarizabilities are found to be well described by isotropic atom-centered distributed polarizabilities, as the short-range damping that is necessary to avoid divergence also has the effect of reducing the polarizability along the direction of covalent bonds. ) Allowing for mutual dipoledipole interactions between the atomic centers, which is consistent with a model that includes only dipole polarizabilities, leads to a self-consistent screening (SCS) ansatz, 61