Exact subsystem time-dependent density-functional theory

In this communication, we show that coupled subsystem time-dependent density functional theory (subsystem TDDFT) [J. Neugebauer, J. Chem. Phys. 126, 134116 (2007)] in combination with projection-based embedding (PbE) is an exact subsystem theory in the sense that supermolecular TDDFT excitation energies can exactly be restored. A correct handling of the kernel contribution due to the enforced orthogonality is crucial in this context, which leads to different PbE kernel contributions in the A and B matrices of the general TDDFT eigenvalue problem. Although this formalism has been proposed before [D. V. Chulhai and L. Jensen, Phys. Chem. Chem. Phys. 18, 21032 (2016)], the symmetric eigenvalue problem used in that work implicitly introduces an approximation concerning this kernel contribution. We show that our treatment numerically exactly reproduces supermolecular results for the previously investigated helium dimer and for the fluoroethane molecule as a more challenging case with a partitioning of a covalent bond. We also demonstrate that the symmetric approximation can lead to significant deviations, including a wrong ordering of electronic transitions.