  Quantum-mechanical (QM) and classical embedding models approximate a
supermolecular quantum chemical calculation. This is particularly useful when
the supermolecular calculation has a size that is out of reach for present QM
models. Although QM and classical embedding methods share the same goal, they
approach this goal from different starting points. In this study, we compare
the polarizable embedding (PE) and frozen-density embedding (FDE) models. The
former is a classical embedding model, whereas the latter is a density-based QM
embedding model. Our comparison focuses on solvent effects on optical spectra
of solutes. This is a typical scenario where super-system calculations
including the solvent environment become prohibitively large. We formulate a
common theoretical framework for PE and FDE models and systematically
investigate how PE and FDE approximate solvent effects. Generally, differences
are found to be small, except in cases where electron spill-out becomes
problematic in the classical frameworks. In these cases, however, atomic
pseudopotentials can reduce the electron-spill-out issue.
