Normal-mode coordinate

Classical trajectories may also be calculated using instantaneous normalmode coordinates.The potential in a small region, defined by the trust radius around a point qg on the potential energy surface, can be expressed as 

This method has also been used for obtaining semiclassical equations of motion for atom-surface scattering by Jackson [108]. If only the equations for R(t) and P (r) are used we obtain classical dynamical equations of motions for the translational motion of the atom moving in an Ehrenfest average potential, i.e., a potential which has been averaged over the quantum coordinates of the system (here the solid). Thus the equations of motion are solved in a mean field potential of the solid. This mean field potential will be considered below. 

The transformation between displacement and normal mode coordinates is given as (see also section 2.3) 

We now quantize the vibrational modes r and write the normal mode coordinates in terms of the phonon creation/annihilation operators 

FIGURE 8.1 The phonon distribution g cD) = N(co)/N for a finite crystal as a function of in units of where t — 10 sec. The sum runs over phorK n frequencies in a small Ao range. 

The probability for being in quantum state rik for oscillator k is assumed to be given by a Boltzmann distribution, i.e.. 

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Figure 1, Coordinates used for describing the dynamics of a) H -I- H2 (6) NOCl, (c) butatriene, (a), (b) Are Jacobi coordinates, where and are the dissociative and vibrational coordinates, respectively, (c) Shows the two most important normal mode coordinates, Qs and Q a, which are the torsional and central C—C bond stretch, respectively.

Figure 5. Definition of the normal mode coordinates for a Dyi X3 molecule.

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

For a vibrating molecule to absorb radiation from an incident IR beam at the frequency of a particular normal mode it must be situated at a position of finite intensity and with an orientation such that there is a finite component of the dipole derivative du /dQ in the direction of the electric vector of the radiation field, where duj is the change of dipole for the change of normal mode coordinate dQ. At a... 

As before, we make the fundamental assumption of TST that the reaction is determined by the dynamics in a small neighborhood of the saddle, and we accordingly expand the Hamiltonian around the saddle point to lowest order. For the system Hamiltonian, we obtain the second-order Hamiltonian of Eq. (2), which takes the form of Eq. (7) in the complexified normal-mode coordinates, Eq. (6). In the external Hamiltonian, we can disregard terms that are independent of p and q because they have no influence on the dynamics. The leading time-dependent terms will then be of the first order. Using complexified coordinates, we obtain the approximate Hamiltonian... 

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 ... 

For each classical configuration xo 3Ar, the mass-scaled Hessian matrix is diagonalized to obtain a set of normal mode coordinates X° 3JV. 

The SQ method extracts resonance states for the J = 25 dynamics by using the centrifugally-shifted Hamiltonian. In Fig. 20, the SQ wavefunc-tion for a trapped state at Ec = 1.2 eV is shown. The wavefunction has been sliced perpendicular to the minimum energy path and is plotted in the symmetric stretch and bend normal mode coordinates. As anticipated, the wavefunction shows a combination of one quanta of symmetric stretch excitation and two quanta of bend excitation. The extracted state is barrier state (or quantum bottleneck state) and not a Feshbach resonance. 

Figure 1. Potential energy vs. normal mode coordinate for symmetrical electron transfer, AE = 0. As shown, the diagram is in the classical (high temperature) limit where h

In Equation 3.43 F now is the matrix of the force constants in valence coordinates. Again one finds normal mode coordinates, Qi, corresponding to the normal mode... 

Dividing surfaces are usually assumed to be perpendicular to the MEP (i.e., that is motion on the dividing surface involves no motion along the MEP) because this assumption saves much computational time. For the dividing surface of TST this result is automatically obtained because the normal mode coordinate of the frequency... 

The main difference between the two approaches is that PGH consider the dynamics in the normal modes coordinate system. At any value of the damping, if the particle reaches the parabolic barrier with positive momentum i n the unstable mode p, it will immediately cross it. The same is not true when considering the dynamics in the system coordinate for which the motion is not separable even in the barrier region, as done by Mel nikov and Meshkov. In PGH theory the... 

Normal mode coordinates are linear combinations of the atomic displacements (x, yt z,, which are the components of a set of vectors Q in a 3/V-dimcnsional vector space called... 

So far we have only discussed harmonic frequencies. The effect of anhar-monicity can be treated using either a Taylor expansion of the PES in terms of normal mode coordinates or by explicitly spanning the PES on a numerical grid. The discussion of anharmonic force constants is postponed to the following section. Here, we will focus on an explicit PES generated by means of the following correlation expansion, here written up to three-mode correlations, [53]... 

In this section we explore the second possibility to generate multidimensional PES, i.e. a Taylor expansion in terms of normal mode coordinate with respect to the geometry of the stable structure. Including terms up to fourth order we have (using dimensionless coordinates)... 

Here, the xy are normal mode coordinates. Note that off-diagonal electron-phonon coupling terms appear both in the representation Eq. (7) and in the excitonic site representation of Eq. (5). 

This degree of freedom is the reaction coordinate (note that this definition coincides with the definition in Chapter 3). In Appendix E, we show that a multidimensional system close to a stationary point can be described as a set of uncoupled harmonic oscillators, expressed in terms of so-called normal-mode coordinates. The oscillator associated with the reaction coordinate has an imaginary frequency, which implies that the motion in the reaction coordinate is unbound. 

Since harmonic oscillators are considered in this theory, the bonds will never break, so it is necessary to introduce an ad hoc criterion for when a reaction occurs. Reaction is normally defined to occur when a particular bond length attains a critical value. The bond length cannot be extracted directly from a particular normal-mode coordinate, since these coordinates, typically, involve the motion of several atoms in the molecule. The bond length can, however, be calculated quite readily, by noting that the displacement of a coordinate associated with an atom of mass mr is given by Eq. (E.5) ... 

Such a transformation will typically be a transformation to normal-mode coordinates around the TS for the activated complex leaving the solvent coordinates unchanged (see Sections 10.2.1 and 10.2.2 for details). This implies that... 

The standard procedure for introducing normal-mode coordinates (see Appendix E) is followed and we define a set of mass-weighted displacement coordinates r/j ... 

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