Simulation of UV-vis absorption spectra of cryptochromes and flavoproteins requires an explicit account of vibrations of the flavin chromophore embedded in protein.

In electrostatic embedding mixed quantum and molecular mechanics (QM/MM) approaches, the QM charge distribution is polarized by the electrostatic interaction with the MM environment. Analytic derivatives of expectation values of operators are required to extract properties such as vibrational spectra. These derivatives usually require solving a set of coupled perturbed equations for each nucleus/atom in the system, thus becoming prohibitive when the MM subsystem contains thousands of atoms. In the context of Electrostatic Potential Fitting (ESPF) QM/MM, we can easily overcome this bottleneck by defining a set of auxiliary coupled perturbed equations called the Q-vector equations. The Q-vector method scales only with the size of the QM subsystem, producing an effective charge tensor that leads to the atomic charge derivative after contraction with the MM electrostatic potential gradient. As an example, we use the charge derivatives as an analysis tool to identify the most important chromophore-polarizing amino-acids in plant cryptochrome. This finding opens up the route of defining polarizable force fields and simulating vibrational spectroscopy using ESPF QM/MM electrostatic embedding at an affordable computational cost.

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 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 vibrationally resolved absorption spectrum of FAD in protein.