Quantum chemistry methods have been enacted to characterize the second-order nonlinear optical (NLO) properties of a multi-state molecular switch combining dihydroazulene and spiropyran units. They have demonstrated that the first hyperpolarizabilities constitute a novel output signal from the viewpoint of molecular logic, owing to the contrasts of NLO responses between the different forms of the dyad. On the one hand, the DFT calculations demonstrate that switching either unit has only a negligible impact on the geometrical parameters of the other one, as well as on the thermodynamics of their transformations. Similarly, as determined by time-dependent DFT calculations, the UV/vis absorption spectra are in good approximation a superposition of those from the parent units. On the other hand, this additivity trend is not consistently observed for the NLO responses. However, an interpretation of these NLO responses has been proposed by using the unit sphere representation of the first hyperpolarizability tensors to address their orientational aspects as well as by resorting to the few-state approximation and the missing state analysis to highlight the localized or charge-transfer nature of the dominant excited states.