FE-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. 

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. 

In Table 4.8 we present the experimental results [4] for U Fe, as compared with the Hessian eigenvalues, based on extensive relativistic calculations [5]. The eigenfunctions of the Hessian matrix are the corresponding normal modes. The Hessian matrix will be block-diagonal over the irreps of the group and, within each irrep, over the individual components of the irrep. Moreover, the blocks are independent of the components. All this illustrates the power of symmetry, and the reasons for it will be explained in detail in the next chapter. As an immediate consequence, symmetry coordinates, which belong to irreps that occur only once, are exact normal modes of the Hessian. Five irreps fulfil this criterion the T g mode, which corresponds to the overall rotations, and the vibrational modes, A g + Eg + l2g + 72 . Only the T u irrep gives rise to a triple multiplicity. In this case, the actual normal modes will depend on the matrix elements in the Hessian. Let us study this in detail... 

Analytical Force and Hessian on Free Energy Surface (FES)... 

The force (vector) on the FES, i.e., the potential surface of mean force, can he calculated by time-averaging the instantaneous forces acting on each atom of a solute molecule over the equilibrium distribution for all solvent molecules. Furthermore, we can also evaluate the Hessian matrix on the FES, i.e., the second derivative matrix with respect to the solute coordinates on the FES. 

QM/MM-MD sampling 2. Calculate FE- and I NM-Hessian Fig. 8.1 Schematic flow chart of the dual approach... 

Table 8.1 Calculated vibrational frequencies (Normal vibrational frequencies (in cm ) scaled by 0.9418 [43]. It should be noted that these by the FE- and the CPCM-Hessian are calculated at the optimized structures by the FEG and the CPCM method, respectively) w (cm ) and their shifts from in gas phase to in aqueous solution Aco (cm ) (Reprinted table with permission from Kitamura et al. [31]. Copyright 2014 American chemical society)...

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