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Analytic first and second derivatives of the energy are developed for thefragment molecular orbital method interfaced with molecular mechanics in theelectrostatic embedding scheme at the level of Hartree-Fock and densityfunctional theory. The importance of the orbital response terms isdemonstrated. The role of the electrostatic embedding upon molecular vibrationsis analyzed, comparing force field and quantum-mechanical treatments for anionic liquid and a solvated protein. The method is applied for 100 proteinconformations sampled in MD to take into account the complexity of a flexibleprotein structure in solution, and a good agreement to experimental data isobtained: frequencies from an experimental IR spectrum are reproduced within 17cm$^{-1}$.

Analytic First and Second Derivatives for the Fragment Molecular Orbital Method Combined with Molecular Mechanics