Multilevel DFT Response Theory
We present a general computational protocol for the evaluation of extensive molecular response properties in complex environments within a polarizable quantum embedding framework. The approach extends multilevel density functional theory (MLDFT) to response theory by formulating the coupled-perturbed Kohn-Sham (CPKS) equations for the MLDFT Hamiltonian. The method is further coupled to an additional polarizable molecular mechanics layer based on the fluctuating-charge (FQ) force field, which allows an accurate yet computationally efficient description of long-range interactions. We apply this new protocol to compute static and frequency-dependent linear polarizabilities and first hyperpolarizabilities of para-nitroaniline (PNA) in 1,4-dioxane and 3-hydroxybenzoic acid (HBA) in aqueous solution. The framework enables physicochemical insight into solute-solvent interactions by disentangling the competing roles of electrostatics, mutual polarization, and quantum confinement (Pauli repulsion). The results match available experiments, demonstrating the reliability and robustness of the proposed approach and providing a viable route for response properties within quantum embedding methods.