Instantaneous normal mode simulations

In the case of nonlinear polarizability coupling, the ratio of the cascaded to direct fifth-order response can be expressed in terms of the ratio of the first and second derivative of the polarizability [Equation (32)]. Using instantaneous normal mode simulations, Murry et al. (40) have calculated the relative ratios of a 11 and at2> for 5000 intermolecular modes in CS2. Although their results show this ratio to be somewhat randomly... 

In this contribution the concept of instantaneous normal modes is applied to three molecular liquid systems, carbon monoxide at 80 K and carbon disulphide at ambient temperature and two different densities. The systems were chosen in this way because pairs of them show similarities either in structural or in dynamical properties. The systems and their simulation are described in the following section. Subsequently two different types of molecular coordinates are used cis input to normal mode calculations, external, i.e. translational and rotational coordinates, and internal, i.e. vibrational coordinates of strongly infrared active modes, respectively. The normal mode spectra are related quantitatively to molecular properties and to those of liquid structure and dynamics. Finally a synthesis of both calculations is attempted on qualitative grounds aiming at the treatment of vibrational dephcising effects. 

For the calculation of the normal mode spectra external and internal coordinates were assumed to be dynamically decoupled. Translational and rotational coordinates were extracted from the trajectories while all vibrational coordinates were set to zero. Dynamical matrices were set up for 50 configurations generated by molecular dynamics simulation. Long-range Coulombic interactions were treated using the Ewald summation technique. In Figure 2 the instantaneous normal mode spectra are depicted while in Table 3 some of their integral properties are compiled. 

One of the important issues addressed in our simulations is the character of clusters under study. Are these clusters solid or liquid rmder experimental conditions If they arc liquid, then the distribution wc observe in the pick-up and consequently in the photodissociation simulations corresponds to a statistical distribution at a. given temperature. If, however, the cluster is solid then both in the simulations and in the cxj)eriment we observe a quasi-stationary state with a very long lifetime rather than an equilibrium thermodynamical state. This question can be resolved by means of the instantaneous normal modes (INM) density of states (DOS) spectrum. To calculate INM DOS wc construct the Hessian matrix in a mass-weighted atomic Cartesian coordinate basis of N atoms with /r=. r, y, z. The 3N eigenvectors in the form Ci -.Cjj,Cj-,C2, C2/,C2-,.c.vj.,ca/j,c.v de-... 

Figure 8. Instantaneous normal mode spectrum of liquid water. Solid, dashed, and dashed-dotted lines are calculated from the velocity correlation function, INM, and QNM, respectively. The system size in numerical simulations is 216.

Figure 9. Instantaneous normal mode spectrum of (a) liquid water H2O and (b) liquid deuterium D2O. The system size in numerical simulations was 64 and density of state was obtained over 79 sample averages. [Reprinted with permission from J. Chem. Phys. 87, 6070-6077 (1987). Copyright 1987 by American Institute of Physics.]...

Figure 6.2 shows an example where these ideas were applied to water as a solvent. The process under investigation is solvation dynamics (see Chapter 15), in this particular case—solvation of electron in water. Figure 6.2(a) shows the instantaneous normal mode density for water at 300 K obtained from numerical simulations. By... 

Fig. 6.2 (a) Instantaneous normal modes in room temperature water as obtained from molecular dynamics simulations. The negative frequency axis is used to show the density of imaginary frequencies. (b) The solvation response function (see Chapter 15) for electron solvation in water, calculated from direct classical MD simulations (full line), from the instantaneous normal mode representation of water (dash-dotted line), and from a similar instantaneous normal mode representation in which the imaginary frequency modes were excluded (dashed line). The inset in Fig. 6.2 shows the short time behavior of the same data. (From C.-Y. Yang, K. F. Wong, M. S. Skaf, and P. J. Rossky, J. Chem. Phys. 114, 3598 (2001).)... 

In the second step, we employ the above introduced instantaneous normal modes to construct appropriate initial conditions for the nonequilibrium MD simulations. To this end, we represent the solute normal modes in terms of classical action-angle variables %, < ) t ... 

For the lattice dynamical evaluation of external contributions to crystal heat capacities, see Filippini, G. Gramaccioli, C. M. Simonetta, M. Suffritti, G. B. Thermodynamic functions for crystals of rigid hydrocarbon molecules a derivation via the Born-von Karman procedure, Chem. Phys. 1975, 8, 136-146. Harmonic dynamics works for crystals thanks to reduced molecular mobility. By contrast, liquids exhibit so-called instantaneous modes (Stratt, R. M. The instantaneous normal modes of liquids, Acc. Chem. Res. 1995, 28, 201-207) the eigenvalues of an instantaneous hessian for a liquid has a spectrum of imaginary frequencies, since any instantaneous frame of liquid stmcture is far from mechanical equilibrium because of collisions. Therefore, it is impossible to estimate heat capacities of liquids in this way, and dynamic simulation is necessary. 

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. 

As discussed above in the chapter introduction, either Newton s or Hamiltonian s equations may be numerically integrated for direct dynamics simulations. There is also a choice of coordinate representation, such as Cartesian, internal, " or instantaneous normal modes. Though potential... 

Ma, A., Stratt, R.M. (2000). Fifth-order Raman spectrum of an atomic liquid Simulation and instantaneous-normal-mode calculation. Phys. Rev. Lett. 85 1004-1007. 

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