Mass-scaled Jacobi coordinates

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... 

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... 

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 ... 

The hyi)ersi)herical coordinates used in the RBU model are defined bj the mass-scaled Jacobi coordinates R .r .R-. r- ) as... 

At pa the XJMlU (p, Hx) are matched to asymptotic atom-diatom wave functions expressed in the usual mass-scaled Jacobi coordinates Rk, yk, iK, R 2, 0, asymptotic analysis one obtains the reactance matrix R7 1 and from it the scattering matrix S7111. This is done for all T and both parities II = 0 and 1 and a sufficient number of partial waves (i.e., values of J) for the resulting cross sections of interest have converged to the desired degree of accuracy. 

We let the coordinates x be a set of convenient Cartesian-type coordinates specifying the geometry of a single molecule (e.g., in previous work we have variously used mass-scaled Jacobi coordinates [25,100-105] and mass-scaled Cartesian coordinates [46]) with the origin at the molecular center of mass. The domain D thus confines the atoms to stay within some maximum distance from the center of mass. Truncating to finite , we have... 

Figure 2 Kinetic paths (broken curves show the two valley bottoms and dots the ridge) and the minimum energy path (solid curve) as a function of the mass scaled Jacobi coordinates, obtained as described in Sec.4. The saddle of the surface is marked by a cross.

The six mass-scaled Jacobi coordinates (S xjS xj tj t) are the space-fixed (SF) Jacobi coordinates whereas the coordinates (5x,5t,0t)Ot)/ t,7t) are the body-fixed (BF) Jacobi coordinates. The angles Sr and Sr are the two sets of polar and azimuthal angles associated with the vectors Sx and Sx respectively, and the angle between these vectors is... 

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),... 

Definitions by Pack, Parker and coworkers. [45] Six mass scaled Jacobi coordinate components Sr Sr-,Sr-,Sr) are the space-fixed Jacobi coordinates, whereas the coordinates (5r, Sr, r, It) are the boby-fixed Jacobi... 

It should be clear from Fig. 2b (Part I) that either set of mass-scaled Jacobi coordinates alone provides a complete description of the available collinear coordinate space. However, it should be equally clear that while Ra and Va are better suited to describing translational and vibrational motions in the reactant channel, Rc and Tc are more appropriate for a corresponding description of the products. It therefore seems natural to retain both sets of coordinates at once, using each set for convenience as required. Moreover, formulations of quantum reactive scattering based on this idea are quite easy to construct. Indeed a comprehensive account of such a formulation, for the... 

Instead of working directly with these coordinates, it is more convenient to work with the mass-scaled Jacobi coordinates Ra and of the reactants ... 

The reason for this is that the collinear reactive scattering Hamiltonian can be written equivalently in terms of the mass-scaled Jacobi coordinates of either arrangement a = a or c as... 

Indeed the (collinear) mass-scaled Jacobi coordinates of the two arrangements are related by the orthogonal transformation... 

A -)- BC AB + C reaction as a contour plot in mass-scaled Jacobi coordinates. Notice in particular that the presence of a single reduced mass fj. in equation (8) means that there is no preferred direction in these coordinates and that Figure 2 therefore provides a kinematically democratic picture of the reaction. 

The Kohn variational method described above for potential scattering extends in a straightforward way to the collinear reactive scattering problem described in Section 2 without the need to introduce any special coordinates (i.e., one can continue to work with the optimum mass-scaled Jacobi coordinates of the reactant and product arrangements). Moreover, the extensions that are required are comparatively minor, and the overall structure of the method remains the same. 

Xc using a standard discrete variable representation in the reactant mass-scaled Jacobi coordinates and thereby eliminating the need for any special coordinates that swing smoothly from reactants to products, and one can perform the time-evolution in equation (59) using any one of a variety of... 

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