INM-Hessian

To investigate vibrational properties of solute molecules in solution, we have proposed a new theoretical method as a direct extension of the FEG one, i.e., the dual approach to the vibrational frequency analysis (VFA) [31]. By employing the dual VFA approach, we can simultaneously obtain the effective vibrational normal modes and the vibrational spectra in solution, which uses complementarily two kinds of Hessian matrices obtained by the equilibrium QM/MM-MD trajectories, that is, a unique Hessian on the FES (i.e., the FE-Hessian) and a sequence of instantaneous ones (i.e., the instantaneous normal mode Hessians INM-Hessians). Figure 8.1 shows a schematic chart of the dual VFA approach. First, we execute the QM/MM-MD simulation and collect many solvent conformations around the solute molecule being fixed at q, sequentially numbered. Second, we obtain an FE-Hessian as the average of instantaneous Hessian matrices at those conformations and then, by diagonalizing the FE-Hessian (cf. Eq. (8.11 a)), we can obtain a set of FE normal coordinates Qi and FE vibrational frequencies coi of the solute molecule in solution. 

On the other hand, one can calculate the approximate vibrational spectra via the instantaneous normal mode (INM) analysis [32-35]. The l-th INMs are defined by diagonalizing an instantaneous Hessian ( INM-Hessian ) matrix at the Z-th solvent conformation = Rm in the sequence of conformations. The corresponding l-th eigenvalues (Z = 1, , 3N) are obtained by... 

For an application to the vibrational spectroscopy analysis, we took an H2O molecule in liquid water [42]. Initially, the structure of the H2O molecule in water was optimized by the standard FEG method for the H2O geometry to satisfy the zero-FEG condition (cf. Eq. (8.19)) using the FE-Hessian matrix (cf. Eq. 8.10). Then, to estimate INM-Hessian matrices for the vibrational frequency analysis (VFA) at the optimized stmcture q on FES, we executed ab initio QM/MM-MD simulation to apply the dual VFA (cf. Sect. 8.2.2.3) approach to the present H2O system. 

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