Analytical nuclear second derivatives for frozen-density embedding employing self-consistent field methods

We report the derivation and implementation of analytical nuclear ground-state second derivatives for uncoupled frozen-density embedding (FDEu) within Hartree-Fock and Kohn-Sham density functional theory. Exchange integrals are evaluated using the chain-of-spheres exchange approach, for which exact nuclear second derivatives are derived and implemented, including all grid derivative contributions, to assess the accuracy of new approximate derivatives. The accuracy of the FDEu Hessian is assessed for selected sample systems by comparison with corresponding supermolecule calculations, encompassing both weakly and strongly coupled subsystems to investigate strengths and limitations of the approach. Finally, the vibrational frequencies of a naphthalene dimer embedded in a crystalline naphthalene environment-comprising up to 44 molecules and a total of 792 atoms-are presented, demonstrating the method's applicability to extended molecular systems and its potential for the conceptual analysis of intermolecular dimer vibrations relevant to charge transport.