Non-Markovian quantum evolution of the electronic subsystem in a laser-driven molecule is characterized through the appearance of negative decoherence rates in the canonical form of the electronic master equation. For a driven molecular system described in a bipartite Hilbert space H=Hel x Hvib of dimension 2 x Nv, we derive the canonical form of the electronic master equation, deducing the canonical measures of non-Markovianity and the Bloch volume of accessible states. We find that one of the decoherence rates is always negative, accounting for the inherent non-Markovian character of the electronic evolution in the vibrational environment. Enhanced non-Markovian behavior, characterized by two negative decoherence rates, appears if there is a coupling between the electronic states g, e, such that the evolution of the electronic populations obeys d(PgPe)/dt > 0. Non-Markovianity of the electronic evolution is analyzed in relation to temporal behaviors of the electronic-vibrational entanglement and electronic coherence, showing that enhanced non-Markovian behavior accompanies entanglement increase. Taking as an example the coupling of two electronic states by a laser pulse in the Cs2 molecule, we analyze non-Markovian dynamics under laser pulses of various strengths, finding that the weaker pulse stimulates the bigger amount of non-Markovianity. We show that increase of the electronic-vibrational entanglement over a time interval is correlated to the growth of the total amount of non-Markovianity calculated over the same interval using canonical measures and connected with the increase of the Bloch volume. After the pulse, non-Markovian behavior is correlated to electronic coherence, such that vibrational motion in the electronic potentials which diminishes the nuclear overlap, implicitly increasing the linear entropy of entanglement, brings a memory character to dynamics.
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