Jacobi coordinate

Which coordinate system for which quantum reactive procedure  

In reactive scattering in principle one needs coordinates which are convenient in describing reactant and product arrangements simultaneously and which have the correct asymptotic behaviour that does not lead to a coupling by the kinetic energy for the separated fragments (e.g. atom and diatom). But there is no such unique coordinate system and so one has to make a compromise between practicability, numerical efficiency, and physical insight and interpretation of the reaction dynamics. 

In the review of Manolopolous and Clary [14] the different quantum reactive procedures are classified by the coordinate systems used. In the following subsections we give a short introduction to the most popular coordinate systems. 

If one changes the coordinates in case of a collinear configuration in such a form that the kinetic energy operator T becomes simpler (i.e. no mixed derivatives), one arrives at Jacobi-coordinates or mass-scaled Jacobi-coordinates (see Fig. 4.1), 

The representation of the potential energy surface (PES) changes with the skewing angle (pa depending on the masses of the atoms, 

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Another reason why mass-scaled coordinates are useful is that they simplify the transfomiation to the Jacobi coordinates that are associated with the products AB + C. If we define. S as the distance from C to the centre of mass of AB, and s as the AB distance, mass scaling is accomplished via... 

We have expressed P in tenns of Jacobi coordinates as this is the coordmate system in which the vibrations and translations are separable. The separation does not occur in hyperspherical coordinates except at p = oq, so it is necessary to interrelate coordinate systems to complete the calculations. There are several approaches for doing this. One way is to project the hyperspherical solution onto Jacobi s before perfonning the asymptotic analysis, i.e. 

It would be convenient for obtaining the expressions of the gradient of the hyperangle physical region of the conical intersection in the following manner ... 

The gradient of v l with respect to Jacobi coordinates (the vector potential) considering the physical region of the conical intersection, is obtained by using Eqs. (C.6-C.8) and after some simplification ( ) we get,... 

Consider a triatomic system with the three nuclei labeled A, Ap, and Ay. Let the arrangement channel -1- A A be called the X arrangement channel, where Xvk is a cyclic permutation of apy. Let Rx,r be the Jacobi vectors associated with this arrangement channel, where r is the vector from A to and the vector from the center of mass of AyA to A . Let R i, rx be the corresponding mass-scaled Jacobi coordinates defined by... 

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.

Using Jacobi coordinates and reduced masses, the Hydrogen-Chlorine interaction is modeled quantum mechanically whereas the Ar-HCl interaction classically. The potentials used, initial data and additional computational parameters are listed in detail in [16]. 

Fig. 2. Collinear ArHCl-systera with the Jacobi-coordinates used.

We represent the four-atom problem in terms of diatom-diatom Jacobi coordinates R, the vector between the AB and CD centers of mass, and rj and r2, the AB and CD bond vectors. In a body-fixed coordinate system [19,20] with the z-axis chosen to R, only six coordinate variables need be considered, which we choose to be / , ri, and ra, the magnitudes of the Jacobi vectors, and the angles 01, 02, and (j). Here 0, denotes the usual polar angle of r, relative to the z-axis, and 4> is the difference between the azimuthal angles for ri and r2 (i.e., a torsion angle). 

To implement the vector potential in the Jacobi coordinate system (R,r,j), one proceeds as follows. The Jacobi kinetic energy operator splits into three parts [61] ... 

Figure 53. Two-dimensional Jacobi coordinates employed. Taken from Ref. [48].

Fig. 7. The probability density of the reactive resonance at Ec = 0.52 kcal/mol. In (a) the F-H-D collinear subspace is shown using the Jacobi coordinates (R,r). In (b), the probability density is sliced r = 2 Bohr and is shown in the (R, 7) coordinates. The plot clearly shows a state with 3 nodes along the asymmetric stretch and 0 nodes in the...

Fig. 20. A resonance state for the H + HD system at Ec = 1.2 eV at a total angular momentum of J = 25. In (a) the wavefunction is shown in the H + HD Jacobi coordinates for the collinear subspace. In (b) the wavefunction has been sliced perpendicular to the minimum energy path and is plotted in symmetric stretch and bend normal mode coordinates.

Fig. 1. Jacobi coordinates for a diatom-diatom reaction AB + CD. The angle is the out-of-plane torsion angle between the (r 1, R) plane and the (ro, R) plane.

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... 

Let us now illustrate the discretization process using the vibration of a triatomic molecule (ABC) as an example. The nuclear Hamiltonian with zero total angular momentum (J = 0) can be conveniently written in the Jacobi coordinates (h = 1 thereafter) ... 

When it comes to polyatomic molecules, there are two problems that complicate the issue, as already discussed in Note 1 of Chapter 3. One is the separation of the overall rotation of the molecule (Jellinek and Li, 1989). The other is that, depending on the choice of internal coordinates, certain coupling terms can be assigned to be kinetic or potential terms. A simple and familiar case is a linear triatomic, when one uses bond coordinates versus Jacobi coordinates. The case for Fermi coupling for a bending motion is discussed in Sibert, Hynes, and Reinhardt (1983). 

Jacobi Coordinates The Skew Angle in Three Center Collinear Reactions... 

By the introduction of the (x, y) coordinate system, one has reduced the problem to the motion of a particle of mass (i in a two-dimensional rectilinear space (x, y). Thus, the problem of the collision between an atom and a diatomic molecule in a collinear geometry has been converted into a problem of a single particle on the potential energy surface expressed in terms of the coordinates x and y rather than the coordinates rAB and rBc The coordinates x and y which transform the kinetic energy to diagonal form in such way that the kinetic energy contains only one (effective) mass are referred to as mass scaled Jacobi coordinates. 

Although diagrams like Fig. 6.1 are especially convenient to illustrate the qualitative features of TST and VTST, the solution of the equations of motion in (rAB,rBc) coordinates is complicated due to cross terms coupling the motions of the different species. It is for that reason we introduced mass scaled Jacobi coordinates in order to simplify the equations of motion. So, one now asks what does the potential function for reaction between A and BC look like in these new mass scaled Jacobi coordinates. To illustrate we construct a graph with axes designated rAB and rBc within the (x,y) coordinate system. In the x,y space lines of constant y are parallel to the x axis while lines of constant x are parallel to the y axis. The rAB and rBc axes are constructed in similar fashion. Lines of constant rBc are parallel to the rAB axis while lines of constant rAB are parallel are parallel to the rBc axis. From the above transformation, Equations 6.10 to 6.13... 

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