  Azulene is a prototypical molecule with an anomalous fluorescence from the
second excited electronic state, thus violating Kasha's rule, and with an
emission spectrum that cannot be understood within the Condon approximation. To
better understand photophysics and spectroscopy of azulene and other
non-conventional molecules, we develop a systematic, general, and efficient
computational approach combining semiclassical dynamics of nuclei with ab
initio electronic structure. First, to analyze the nonadiabatic effects, we
complement the standard population dynamics by a rigorous measure of
adiabaticity, estimated with the multiple-surface dephasing representation.
Second, we propose a new semiclassical method for simulating non-Condon
spectra, which combines the extended thawed Gaussian approximation with the
efficient single-Hessian approach. S$_{1} \leftarrow$ S$_0$ and S$_{2}
\leftarrow$ S$_0$ absorption and S$_{2} \rightarrow$ S$_0$ emission spectra of
azulene, recorded in a new set of experiments, agree very well with our
calculations. We find that accuracy of the evaluated spectra requires the
treatment of anharmonicity, Herzberg--Teller, and mode-mixing effects.
