Jacobi coordinate system

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. 

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] ... 

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... 

The rovibrational Hamiltonian in the Jacobi coordinate system has the form... 

This coupling potential is smooth everywhere, which allows numerical calculations with high precision. There is no nonadiabatic coupling since the basis functions [0< )( 2C) are independent of p in each sector. The solution I Wf/o, 2C) is connected smoothly, in principle, from sector to sector by a unitary frame transformation from the /th set of channels to the (/ + l)st set [97-99]. The coordinate system is transformed from the hyperspherical to the Jacobi coordinates at some large p, beyond which the conventional close-coupling equations are employed for determining the asymptotic form of the wavefunction appropriate for the scattering boundary condition [100]. 

The results of HSCC calculations have proved much more rapid convergence with the number of coupled channels than the conventional close-coupling equations in terms of the independent-particle coordinates or the Jacobi coordinates based on them. This is considered to be because of the particle-particle correlations considerably taken into account already in the choice of the hyperspherical coordinate system. The results suggest an approximate adiabaticity with respect to the hyperradius p, even when the mass ratios might appear to violate the conditions for the adiabaticity, for example, for Ps- with three equal masses. Then, it makes sense to study an adiabatic approximation with p adopted as the adiabatic parameter. 

For energies below the dissociation threshold we can use various coordinate systems to solve the nuclear Schrodinger equation (2.32). If the displacement from equilibrium is small, normal coordinates are most appropriate (Wilson, Decius, and, Cross 1955 ch.2 Weissbluth 1978 ch.27 Daudel et al. 1983 ch.7 Atkins 1983 ch.ll). However, if the vibrational amplitudes increase so-called local coordinates become more advantageous (Child and Halonen 1984 Child 1985 Halonen 1989). Eventually, the molecular vibration becomes unbound and the molecule dissociates. Under such circumstances, Jacobi or so-called scattering coordinates are the most suitable coordinates they facilitate the definition of the boundary conditions of the continuum wavefunctions at infinite distances which we need to determine scattering or dissociation cross sections (Child 1991 ch.l0). Normal coordinates become less and less appropriate if the vibrational amplitudes increase they are completely impractical for the description of unbound motion in the continuum. 

In a collision process, it is the relative position of the atoms that matters, not the absolute positions, when external fields are excluded, and the potential energy E will depend on the distances between atoms rather than on the absolute positions. It will therefore be natural to change from absolute Cartesian position coordinates to a set that describes the overall motion of the system (e.g., the center-of-mass motion for the entire system) and the relative motions of the atoms in a laboratory fixed coordinate system. This can be done in many ways as described in Appendix D, but often the so-called Jacobi coordinates are chosen in reactive scattering calculations because they are convenient to use. The details about their definition are described in Appendix D. The salient feature of these coordinates is that the kinetic energy remains diagonal in the momenta conjugated to the Jacobi coordinates, as it is when absolute position coordinates are used. 

If we consider the potential energy as a function of the Jacobi coordinates X and X2 and draw the energy contours in the X1-X2 plane, then the entrance and exit valleys will asymptotically be at an angle to one another and in the mass-weighted skewed angle coordinate system parallel to its axes. So the idea with this coordinate system is that it allows us to directly determine the atomic distances as they develop in time and that it shows us the asymptotic directions of the entrance and exit channels. 

For three-body collinear systems, various types of coordinate systems allow for this type of kinetic energy (Jacobi type of coordinates). 

Figure 12. Structure control of a Ne-HBr cluster by a DC electric filed, x-axis cos0, y-axis wavcfunction(0 is the angular part of the Jacobi coordinate system).

Exploiting a four-dimensional rotation group analysis, the transformation between harmonic expansions in the two coordinates systems was given explicitly [32], as well as the most general representation in terms of Jacobi functions [2], In practice, however, the two representations are in one form or another those being used in all applications and specifically in recent treatments of the elementary chemical reactions as a three-body problem [11,33-36]. For example, Eqs. (29)-(31) and Eqs. (47)-(49) permitted to establish [37] the explicit connection between coordinates for entrance and exit channels to be used in sudden approximation treatments of chemical reactions [38],... 

Figure 1. Main systems of coordinates heliocentric Jacobi s (right).

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 ... 

Figure 5. Plot of the partition function Q as a function of the hyperradius R of the great sphere enclosing the molecules H 0 and D20. A mass-weighted Jacobi coordinate system is used in each case. One Fourier coefficient is used per degree of freedom, and 10s Monte Carlo samples were used for each calculation. The Monte Carlo samples were drawn uniformly on the interior of the great sphere in coordinate space and from P(a) in Fourier coefficient space.

We will introduce the Jacobi coordinates (Fig. 7.2 cf. p. 897) three components of vector R pointing to C from the origin of the coordinate system (the length R and angles and ,... 

Now let us write down the Hamiltonian for the motion of the nuclei in the Jacobi coordinate system (with the stiff AB molecule with AB equilibrium distance equal to req f. ... 

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

Each of the coordinate systems (let us label them k = 1,2, 3) highlights two atoms (i, j) close to each other and a third (k) that is distant . Fig. 14.4a. Now, let us choose a pair of vectors rk, Rk for each of the choices of the Jacobi coordinates by the following procedure... 

Fig. 14.4. (a) The three equivalent Jacobi coordinate systems (b) the Euler angles show the mutual orientation of the two Cartesian... 

The three Jacobi coordinate systems are related by the foUowing formnlas (cf., Fig. 14.4) ... 

When a chemical reaction proceeds, the role of the atoms changes and nsing the same Jacobi coordinate system aU the time leads to technical problems. In order not to favor any of the three atoms despite possible differences in their masses, we introduce hypersphericcd democratic... 

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... 

The actual choice of the reference vibrational potential depends on the particular application. In the RXNID program/ a quadratic reference potential is chosen in the NCC and Jacobi coordinate systems, and the functions F and G form a harmonic oscillator basis. In hyperspherical coordinates, we use the entire potential and determine the basis H by a finite difference approach. 

In the RXNID program, both the NCC and. Jacobi coordinate systems are used. We begin at the collinear matching surface (M in Fig. 1) with a sector R matrix as the first "global" R matrix, and propagate all four blocks of the R matrix outwards toward the ot-channel asymptotic region, and then switch to a Jacobi coordinates when R ==RV For asymmetric sys... 

Using the Jacobi coordinate system as an example, we seek the overlap 9C of the scattering wavefunction M in Eq. (24) with a bounded... 

Jacobi coordinate system (p. 279) angular momenta addition (p. 281) rovibrational spectrum (p. 283) dipole moment (p. 283) sum of states (p. 283) force field (p. 284)... 

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