Car-Parrinello (CPMD) and path integral molecular dynamics (PIMD) simulations were carried out for 1-(phenylazo)-2-naphthol (I) and 1-(4-F-phenylazo)-2-naphthol (II) (Sudan I) in vacuo and in the solid state at 298 K. The fast proton transfer (FPT) and tautomerism in the ketohydrazone-azoenol systems have been analyzed on the basis of CPMD and PIMD methods level. The two-dimensional free-energy landscape of reaction coordinate δ-parameter and RN···O distances shows the NH tautomer to be more favorable in the gas phase as well as in the solid state according to the CP and PI results, respectively. The hydrogen between the nitrogen and the oxygen atoms adopts a starkly asymmetrical position in the double potential well. The molecular geometry and energy barrier for the intramolecular proton transference were calculated, and the value found suggested a strong hydrogen bond with low barrier for FPT mechanism. These studies and the two-dimensional average index of π-delocalization ⟨λ⟩ landscape of time evolutions of RN1···O1 and RC1═O1 distances for both the crystals indicate that the hydrogen bonds in the crystals of 1-(phenylazo)-2-naphthol (I) and 1-(4-F-phenylazo)-2-naphthol (II) have characteristic properties for the type of bonding model: resonance-assisted hydrogen bonds and low-barrier hydrogen bonds, without the existence of equilibrium in the two tautomers. The infrared spectrum has been calculated, and a comparative vibrational analysis has been performed. The CPMD vibrational results appear to qualitatively agree with the experimental ones.