Vibrational dynamics of polyatomic molecules in solution: assignment, time evolution and mixing of instantaneous normal modes

Intramolecular vibrational dynamics of polyatomic molecules in solution can be addressed through normal mode analysis based on either equilibrium normal modes (ENMs) or instantaneous normal modes (INMs). While the former offers a straightforward way of examining experimental spectra, the latter provides a decoupled short-time description of the vibrational motions of the molecule. In order to reconcile both representations, a realistic assignment of the INMs in terms of the ENMs is needed. In this paper, we describe a novel method to assign the INMs using the ENMs as templates, which provides a unique relationship between the two sets of normal modes. The method is based specifically on the use of the so-called Min-Cost or Min-Sum algorithm, duly adapted to our problem, to maximize the overlaps between the two sets of modes. The identification of the INMs as the system evolves with time then allows us to quantify the vibrational energy stored in each INM and so monitor the flows of intramolecular vibrational energy within the solute molecule. We also discuss the degree of mixing of the INMs and characterize the way they change with time by means of the corresponding autocorrelation functions. The usefulness of the method is illustrated by carrying out equilibrium molecular dynamics (MD) simulations of the deuterated N-methylacetamide (NMAD) molecule in D2O solution.