Instantaneous normal mode analysis

LaNave. E., Scala, A., Starr, E W., Sciortino, E, and Stanley, H. E., Instantaneous normal mode analysis of supercooled water. Phys Rev. Lett 84,4605 (2000). 

Wu, T, and Tsay, S., Instantaneous normal mode analysis of liquid Na. I. Chem. Phys 105,9281 (1996). 

To decompose the vibrational energy of a flexible molecule into single-mode contributions, it is useful to perform an instantaneous normal-mode analysis - of the vibrational dynamics. In this approach, we choose a structure at the instantaneous position r(t) and consider the normal mode vibrations around this reference structure. We expand the molecular potential energy up to second order... 

Ryu, S., Stratt, R.M. (2004). A case study in the molecular interpretation of the optical Kerr efifeet speetra Instantaneous-normal-mode analysis of the OKE speetrum of liquid benzene. J. Phys. Chem. B 108 6782-6795. 

Ryu, S. Stratt, R. M. (2004). A case study in the molecular interpretation of optical Kerr effect spectra Instantaneous-normal-mode analysis of the OKE spectrum of liquid benzene. Journal cf Physical Chemistry B, 108,6782-6795 Seki, S. Hayamizu, K. Tsuzuki, S. Fujii, K Umebayashi, Y. Mitsugi, T. Kobayashi, T. Ohno, Y. Kobayashi, Y. Mita, Y. Miyashiro, H. Ishiguro, S. (2009). Relationships between center atom species (N, P) and ionic conductivity, viscosity, density, selfdiffusion coefficient of quaternary cation room-temperature ionic liquids. Physical Chemistry Chemical Physics, 11,3509-3514... 

A more mechanistic approach, Instantaneous Normal Mode (INM) theory [122], can be used to characterize the collective modes of a liquid. Ribeiro and Madden [123] applied this theory to a series of fused salts, including both noncoordinating and coordinating species. They found that the INM analysis provided a good estimate of the diffusion constants for noncoordinating fused salts. For coordinating ions, however, the situation was complicated by the existence of transient, quasimolecular species. While a more detailed analysis is possible [124], the spectrum becomes sufficiently complicated that it would be difficult to characterize specific motions in the system. 

In the simplest case, we can use a normal mode analysis for the fast q vibrations. Since the x mode is potentially a high amplitude degree of freedom, we should employ instantaneous normal modes. In this treatment, the frequencies and normal mode coordinates are obtained by diagonalizing the projected force constant matrix ... 

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... 

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