Coordinate systems Jacobi coordinates

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. 

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... 

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] ... 

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.

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... 

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. 

Fig. 2.1. Definition of Jacobi coordinates R and r for the linear triatomic molecule ABC. S denotes the center-of-mass of the total system and s marks the center-of-mass of the diatom BC.

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. 

If we fix the intramolecular distance of the diatom, the system is described by the two Jacobi coordinates R and 7, where R is the distance from A to the center-of-mass of BC and 7 is the orientation angle of the diatom with respect to the scattering vector R. The appropriate Hamiltonian is given by (see Section 11.1 for a more detailed discussion)... 

In the following we will illustrate this qualitative picture by a two-dimensional model for the photodissociation of FNO(5i) which is characteristic for other systems as well (Ogai et al. 1992). The coordinates which we include in the model are the Jacobi coordinates R, the distance between F and the center-of-mass of NO, and 7, the orientation angle of R with respect to the N-0 bond the latter is fixed. The two-dimensional PES (see the top of Figure 10.13), which is constructed from three-dimensional ab initio calculations, has the general shape illustrated in Figure 10.12(b), i.e., a shallow trough at short distances, where the excited complex is trapped for several internal periods, and a repulsive... 

Fig. 11.1. Center-of-mass or Jacobi coordinates R and r used to describe the fragmentation of a triatomic molecule ABC into A and BC. S and s are the centers-of-mass of ABC and BC, respectively, v is the relative velocity of the recoiling fragments in the center-of-mass system. The space-fixed z-axis is parallel to the vector Eo of the electric field, while the body-fixed z -axis is parallel to the scattering vector R at all times. The azimuthal angle ip, which is not indicated in the figure, describes rotation in the plane perpendicular to R.

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. 

The Jacobi coordinates used for the propagation of the system are not convenient for such an analysis, since they were based on reactants A and BC and therefore include the BC distance and the A-BC distance rather than the AB and AB C distances. It is therefore convenient to replace the coordinates in Eq. (4.66),... 

We have already seen that a convenient set of coordinates for the relative motion of the atoms is given by the so-called Jacobi coordinates in which the kinetic energy is diagonal with no cross terms between different momenta. A systematic way of deriving these coordinates is given in Appendix C, and applied to this system we get... 

Here we recognize the mass associated with coordinate R as the reduced mass Mi for a two-particle system (as in Eq. (2.25)). With these generalized Jacobi coordinates, the internal kinetic energy has the simple form without cross terms according to Eq. (D.15) ... 

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 6. Wavefunctions of states 315 and 206 of the HCN <- CNH system, with respective energies = 18,069 cm and = 15,750 above the quantum mechanical ground state. The figures show one particular contour I (S,r, y) = const, where (R,r, y) are the Jacobi coordinates. (Left) The adiabatically delocalized state 315, which is assigned as (1,40, l) j. (Right) The non-adiabatically delocalized state 206, which can be assigned as (1,16, 1)jj[ n l tt displays the nodal structure of (0,24,0)ctm CNH side.

Mass-scaled Jacobi coordinates associated to a generic arrangement X — a for A -I- BC, /I for B + CA and ) for C + AB) cU c denoted by r (diatom vector) and R (atom-molecule vector). They are used in the definition of hyperspherical coordinates which parametrize the nuclear motion of the system, namely the principal axis body frame hyperspherical coordinates [3, 4, 5]. These coordinates are ... 

Figure 3. Jacobi coordinates for the X + YCZ3 system, as exemplified by H + CH4.

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],... 

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