  Analytic second derivatives of electrostatic embedding (EE) quantum
mechanics/molecular mechanics (QM/MM) energy are important for performing
vibrational analysis and simulating vibrational spectra of quantum systems
interacting with an environment represented as a classical electrostatic
potential. The main bottleneck of EE-QM/MM second derivatives is the solution
of coupled perturbed equations for each MM atom perturbation. Here, we exploit
the Q-vector method {[J. Chem. Phys., 151, 041102 (2019)]} to workaround this
bottleneck. We derive the full analytic second derivative of the EE-QM/MM
energy, which allows to compute QM, MM and QM-MM Hessian blocks in an efficient
and easy to implement manner. To show the capabilities of our method, we
compute the normal modes for the full \textit{arabidopsis thaliana} plant
cryptochrome. We show that the flavin adenine dinucleotide vibrations (QM
subsystem) strongly mix with protein modes. We compute approximate vibronic
couplings for the lowest bright transition, from which we extract spectral
densities and the homogeneous broadening of FAD absorption spectrum in protein
using vibrationally resolved electronic spectrum simulations.
