  A new and thorough derivation of linear-response subsystem time-dependent
density functional theory (TD-DFT) is presented and analyzed in detail. Two
equivalent derivations are presented and naturally yield self-consistent
subsystem TD-DFT equations. One reduces to the subsystem TD-DFT formalism of
Neugebauer [J. Chem. Phys. 126, 134116 (2007)10.1063/1.2713754]. The other
yields Dyson type equations involving three types of subsystem response
functions: coupled, uncoupled, and Kohn-Sham. The Dyson type equations for
subsystem TD-DFT are derived here for the first time. The response function
formalism reveals previously hidden qualities and complications of subsystem
TD-DFT compared with the regular TD-DFT of the supersystem. For example,
analysis of the pole structure of the subsystem response functions shows that
each function contains information about the electronic spectrum of the entire
supersystem. In addition, comparison of the subsystem and supersystem response
functions shows that, while the correlated response is subsystem additive, the
Kohn-Sham response is not. Comparison with the non-subjective partition DFT
theory shows that this non-additivity is largely an artifact introduced by the
subjective nature of the density partitioning in subsystem DFT.
