  The derivation of linear response theory within polarizable embedding is
carried out from a rigorous quantum-mechanical treatment of a composite system.
Two different subsystem decompositions (symmetric and nonsymmetric) of the
linear response function are presented, and the pole structures as well as
residues of the individual terms are analyzed and discussed. This theoretical
analysis clarifies which form of the response function to use in polarizable
embedding, and we highlight complications in separating out subsystem
contributions to molecular properties. For example, based on the nonsymmetric
decomposition of the complex linear response function, we derive conservation
laws for integrated absorption cross sections, providing a solid basis for
proper calculations of the intersubsystem intensity borrowing inherent to
coupled subsystems and how that can lead to negative subsystem intensities. We
finally identify steps and approximations required to achieve the transition
from a quantum-mechanical description of the composite system to polarizable
embedding with a classical treatment of the environment, thus providing a
thorough justification for the descriptions used in polarizable embedding
models.
