Normal mode representation

Substituting Eq. (25) into Eq. (23) we find the expression for the propagator of free diffusion in the normal mode representation ... 

A normal-mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix, which in turn generates a reduced-dimension potential-energy surface in terms of the mass-weighted coordinates of the reaction path [64] ... 

While the electronic structure calculations addressed in the preceding Section could in principle be used to construct the potential surfaces that are a prerequisite for dynamical calculations, such a procedure is in practice out of reach for large, extended systems like polymer junctions. At most, semiempirical calculations can be carried out as a function of selected relevant coordinates, see, e.g., the recent analysis of Ref. [44]. To proceed, we therefore resort to a different strategy, by constructing a suitably parametrized electron-phonon Hamiltonian model. This electron-phonon Hamiltonian underlies the two- and three-state diabatic models that are employed below (Secs. 4 and 5). The key ingredients are a lattice model formulated in the basis of localized Wannier functions and localized phonon modes (Sec. 3.1) and the construction of an associated diabatic Hamiltonian in a normal-mode representation (Sec. 3.2) [61]. 

A normal mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix Ffmai, which... 

The major limitation of the approaches to multiscale modeling discussed thus far is the timescale. In each of these examples, there are atomic vibrations (on the order of 10 seconds) that need to be followed. This pins down the total simulation time to 0(10 seconds for reasonable calculations. There are many clever multiple time step methods for improving efficiency (e.g., Nakano 1999) by using a quatemion/normal mode representation for atoms that are simply vibrating or rotating, but this buys only a factor of 0(10). 

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).)... 

The overbar specifies actions and angles in the normal mode representation, which differs from the local mode representation only by a rotation of tt/2 about the y axis). Thus,... 

The normal mode representation of the phase space trajectories contains the same information as the local mode representation. However, the resonance region on the normal mode phase space map contains the local mode trajectories (la, lb 2a, 2b) and the stable fixed points Ca and C t,. The trajectories contained within the resonance zone are not free to explore the entire 0 < tp < n range whereas the trajectories outside the resonance zone do explore the 0 < ip < n range and are therefore classified as normal mode trajectories. The fixed point B (Iz = I = +2) is unstable, because it lies on a separatrix, and is located at the north pole of the normal mode polyad phase sphere. The stable fixed point A (7Z = — I = —2) is located at the south pole. 

Since this transformation to normal coordinates is invertible, one can readily determine the functional dependencies of the terms in Eq. (1) using either the normal or internal coordinates. Interestingly, in our study of vibrational states of the well-known local mode molecule H20 and its deuterated analogs we found only minor differences between the results of CVPT in the internal and normal mode representations (46). The normal mode calculations, however, required significantly less computer time to run, since many terms in the Hamiltonian are constrained to zero by symmetry. For this reason we chose to use the normal mode coordinates for all subsequent studies. 

Although the normal mode representation is very useful, the physically observable coordinate is usually the system coordinate q. In the Langevin representation of the dynamics, the trajectory q(t) is characterized by a random force which affects its motion. In the equivalent Fokker-Planck representation of Langevin dynamics, all dynamical information lies in the joint probability distribution function that the particle at time t has position and velocity q, v, given that at time r = 0 its position and velocity were q, v. ... 

The same approach has been used in Ref. 68 to derive an eigenfunction of the Fokker-Planck operator as well as the stochastic separatrix in phase space (71-73). The dynamics in the normal mode representation was also used in conjunction with the re-... 

Vo = T[r(0]-j JicOkal, we finally obtain the instantaneous normal-mode representation of the vibrational energy... 

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