Jacobi coordinates dynamics

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.

The vibrational dynamics of this system can be adequately studied by a two degrees of freedom model, with the C-N distance kept frozen at its equilibrium value of re = 2.186 a.u. The vibrational (total angular momentum J = 0) Hamiltonian in scattering or Jacobi coordinates is given by... 

The coupling between the angle y and the dissociation coordinate R is always large if Jacobi coordinates are used. At low energies deep inside the well, this coupling is linear and normal coordinates are usually better suited for interpretation and assignment than are Jacobi coordinates. However, if the molecular dynamics above the dissociation threshold is studied, the normal-mode picture breaks down and scattering coordinates have to be employed. 

In order to simplify the evaluation of overlap integrals between bound and continuum wavefunctions, it is advisable (although not necessary) to describe both wavefunctions by the same set of coordinates. Usually, the calculation of continuum, i.e., scattering, states causes far more problems than the calculation of bound states and therefore it is beneficial to use Jacobi coordinates for both nuclear wavefunctions. If bound and continuum wavefunctions are described by different coordinate sets, the evaluation of multi-dimensional overlap integrals requires complicated coordinate transformations (Freed and Band 1977) which unnecessarily obscure the underlying dynamics. 

One of the main assets of the time-dependent theory is the possibility of treating some degrees of freedom quantum mechanically and others classically. Such composite methods necessarily lead to time-dependent Hamiltonians which obviously exclude time-independent approaches. We briefly outline three approximations that are frequently used in molecular dynamics studies. To be consistent with the previous sections we consider the collinear triatomic molecule ABC with Jacobi coordinates R and r. 

Fig. 10.7. The bond coordinates used to describe the photodissociation of H2O2.

The N x N matrix AmT1 AT will in general not be diagonal, so there will be cross terms of the kind PiPj in the expression for the kinetic energy. This may sometimes be inconvenient, and we shall see in the following how one may choose the matrix A in such a way that the kinetic energy is still diagonal in the new momenta. This leads to the so-called Jacobi coordinates that are often used in reaction dynamics calculations. 

It is important to note that the dynamics is done in the Jacobi coordinates, because they make the equations of motion (see Eqs (D.19) and (D.20)) particularly simple. The idea of choosing the as as in Eq. (D.17) is that all masses Mj. equal one, so the equations of motion are analogous to those for a particle of mass one with coordinates X and X2-... 

The wavepacket dynamics were carried out using the Lanczos method with real wavepackets [121] employing Jacobi coordinates to describe the relative positions of the three nuclei in the body fixed plane. As mentioned the novel results... 

Ultimately, the exact dynamics study for dissociative adsorption of a diatomic molecule on a corrugated, static surface should include the lateral motion of the center of mass of the diatom, i.e., include six degrees of freedom. The 6D Hamiltonian could be written in terms of the gas-phase Jacobi coordinates (molecular coordinates) as... 

In Fig. lb we show an alternative coordinate system for the same four atoms. In the absence of an external force, the center of mass remains stationary. There is therefore no need to include the coordinate in the motion. The dynamics reduces to 18 equations of motion in three coordinates, r12, r34, and R. These are called the Jacobi coordinates. Instead of the masses of the individual atoms appearing in the Hamiltonian, we now have the reduced masses, x12, jx34, and x ... 

Jacobi coordinate system (p. 341) kinetic minimum (p. 353) Langevin dynamics (p. 371)... 

Coupled-channel equations arise in scattering dynamics when all but one of the degrees of freedom of the system are expanded in a square integral basis (of "channels"). The coupled channel equations are then solved numerically and describe motion in the unbound, or scattering coordinate. The principal difficulty of any reactive scattering calculation is that the coordinate system which best describes the asymptotic motions of reactants differs from the coordinate system best suited for products. Consequently, computational methods commonly use different coordinate systems in different parts of configuration space. Boundary conditions are expressed in terms of Jacobi coordinates (sometimes referred to as "cartesian coordinates"), where in the A -BC arrangement... 

For AB + CD systems the most straightforward choice of coordinates to describe the dynamical system is the Jacobi coordinates corresponding to the diatom-diatom arrangement. As shown in Fig. 1, the three vectors (R,ri,r2) denote, respectively, the vector R from the center of mass (CM) of diatom AB to that of CD, the AB diatomic vector F, and the CD diatomic vector F2. The full Hamiltonian expressed in this set of coordinates is written as... 

After this and for the convenience of extracting state-to-state dynamical quantities, the initial nuclear wave packet is immediately transferred from reactant Jacobi coordinates to product Jacobi coordinates by ... 

After the extraction of the reactive resonance wave function in certain convenient coordinates used in the propagation, one may need to transform it into another optimal coordinates to facilitate the observation of its resonance quantization structure, such as normal mode near the transition state region or product/reactant Jacobi coordinates or hyperspherical coordinates. In this way, the dynamics origin of the reactive resonance wave functions may be clarified to us. 

Fig. 3.1. Potential energy contours for an atom-diatom reaction. The Jacobi coordinates of arrangement a are sketched. (Reprinted, by permission, from Launay, J.M. in Dynamical Processes in Molecular Physics Delgado-Barrio, G. (Ed.) (Institute of Physics Publishing, Bristol and Philadelphia, 1991), p.97. Copyright 1991 Institute of Physics Publishing.)...

Since time here is an ignorable variable, it can be eliminated from the dynamics by subtracting ptt from the modified Lagrangian and by solving H = E for t as a function of the spatial coordinates and momenta. This produces Jacobi s version of the principle of least action as a dynamical theory of trajectories, from which time dependence has been removed. The modified Lagrangian is... 

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