Instantaneous normal-mode coordinates

The two most widely implemented numerical integration techniques within MD are the Verlet algorithm and the use of instantaneous normal mode coordinates. The Verlet algorithm begins by writing the Taylor expansion for a coordinate at time t+ Af and f- Af ... 

We first focus on the properties of the bound molecule and consider the effects of non-separability on the molecular partition function and density of states. The potential non-separability is reflected by the variation of the normal mode frequencies as a function of t. Projecting out the torsional motion from the Hessian matrix, we obtain the quadratic form for the potential in terms of the instantaneous normal mode coordinates, ( = 1-5) ... 

To render the KP theory feasible for many-body systems with N particles, we make the approximation of independent instantaneous normal mode (INM) coordinates [qx° 3N for a given configuration xo 3W [12, 13], Hence the multidimensional V effectively reduces to 3N one-dimensional potentials along each normal mode coordinate. Note that INM are naturally decoupled through the 2nd order Taylor expansion. The INM approximation has also been used elsewhere. This approximation is particularly suited for the KP theory because of the exponential decaying property of the Gaussian convolution integrals in Eq. (4-26). The total effective centroid potential for N nuclei can be simplified as ... 

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 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-... 

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. 

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... 

It is noteworthy that these two modes actually correspond to the same molecular vibration. This occurs, though the material is isotropic, just because it has been put in the form of a thin layer. The origin of the effect can be intuitively explained as follows (Eig. 6.13a) in a homogeneous isotropic material, the synchronous vibration of the polar species produces a dielectric polarization P= Ne Sz in the medium (where Sz represents the instantaneous value of the normal coordinate associated with the vibration mode under consideration). If the sample is shaped in the form of a thin layer and the direction of the vibration is perpendicular to the layer, a depolarizing field -PIb Bq will appear, acting as a restoring force -e PlBa ,Bo on the vibrators (Eig. 6.13a). This amounts to increasing the usual... 

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