  We extend our recently proposed Deep Learning-aided many-body dispersion
(DNN-MBD) model to quadrupole polarizability (Q) terms using a generalized
Random Phase Approximation (RPA) formalism, thus enabling the inclusion of van
der Waals contributions beyond dipole. The resulting DNN-MBDQ model only relies
on ab initio-derived quantities as the introduced quadrupole polarizabilities
are recursively retrieved from dipole ones, in turn modelled via the
Tkatchenko-Scheffler method. A transferable and efficient deep-neuronal network
(DNN) provides atom in molecule volumes, while a single range-separation
parameter is used to couple the model to Density Functional Theory (DFT). Since
it can be computed at a negligible cost, the DNN-MBDQ approach can be coupled
with DFT functionals such as PBE,PBE0 and B86bPBE (dispersionless). The
DNN-MBQ-corrected functionals reach chemical accuracy while exhibiting lower
errors compared to their dipole-only counterparts.
