Superposition of waves or densities: Which is the nature of chemical resonance?

Resonance is a fundamental and widely used concept in chemistry, but there exist two distinct theories of chemical resonance, based on quite different and incompatible premises: the wave-function-based resonance theory (WFRT), assuming the superposition of wave functions, versus the density-matrix-based resonance theory (DMRT), which interprets the resonance phenomenon as the superposition of density matrices. The latter theory, best known to the chemistry community as the natural resonance theory (NRT), has received much more popularity than the WFRT. In this contribution, the DMRT is shown to be inherently inadequate: (i) the exact density matrix expansion is mathematically impossible unless unphysical negative weights are introduced; (ii) any approximate density matrix representing the resonance hybrid lacks the idempotent property. Therefore, the validity of the NRT ansatz should be seriously questioned. The WFRT seems the only reasonable explanation of resonance so far, and has been shown to provide valuable insights into diverse chemical problems.