Hyperspherical coordinates potential energy surfaces

Figure 3. Relaxed triangular plot [68] of the U3 ground-state potential energy surface using hyperspherical coordinates. Contours, are given by the expression (eV) — —0.56 -t- 0.045(n — 1) with n = 1,2,..,, where the dashed line indicates the level —0.565 eV. The dissociation limit indicated by the dense contouring implies Li2 X Sg ) -t- Li.

Figure 11. Perspective view [60] of a relaxed triangular plot [68] for the two DMBE adiabatic potential energy surfaces of H3 using hyperspherical coordinates.

Figure 7, Schematic representation of the 1-TS (solid) and 2-TS (dashed) (where TS = transition state) reaction paths in the reaction Ha + HbHc Ha He + Hb- The H3 potential energy surface is represented using the hyperspherical coordinate system of Kuppermann [54], in which the equilateral-triangle geometry of the Cl is in the center (x), and the linear transition states ( ) are on the perimeter of the circle the hyperradius p = 3.9 a.u. The angle is the internal angular coordinate that describes motion around the CL...

Figure 14. Classical trajectories for the H + H2(v = l,j = 0) reaction representing a 1-TS (a-d) and a 2-TS reaction path (e-h). Both trajectories lead to H2(v = 2,/ = 5,k = 0) products and the same scattering angle, 0 = 50°. (a-c) 1-TS trajectory in Cartesian coordinates. The positions of the atoms (Ha, solid circles Hb, open circles He, dotted circles) are plotted at constant time intervals of 4.1 fs on top of snapshots of the potential energy surface in a space-fixed frame centered at the reactant HbHc molecule. The location of the conical intersection is indicated by crosses (x). (d) 1-TS trajectory in hyperspherical coordinates (cf. Fig. 1) showing the different H - - H2 arrangements (open diamonds) at the same time intervals as panels (a-c) the potential energy contours are for a fixed hyperradius of p = 4.0 a.u. (e-h) As above for the 2-TS trajectory. Note that the 1-TS trajectory is deflected to the nearside (deflection angle 0 = +50°), whereas the 2-TS trajectory proceeds via an insertion mechanism and is deflected to the farside (0 = —50°).

V. Aquilanti and S. Cavalli, Hyperspherical coordinates for molecular dynamics by the methods of trees and the mapping of potential energy surfaces for triatomic systems. J. Chem. Phys. 85, 1362-1375 (1986). 

In the laboratory frame the motion of the three particles depends on nine variables, three of which define the position of the center-of-mass. Other three coordinates are needed to describe the rotation of the system in the space and therefore the internal motion is described by the three remaining coordinates. For example, in molecular dynamics the potential energy surface in general is calculated and presented using geometrical coordinates, such the interparticle distances, or two bond distances and an angle. But it is convenient and necessary to use different coordinate systems to describe and understand the dynamics of the particles, because of the rotational terms which appear in the full Hamiltonian. In this context, we will present the transformation equations from the interparticle distances to coordinate sets of the hyperspherical and related types, successful in the treatment of the dynamics. 

The hyperspherical and related coordinates which have been considered in this work have served for the visualization of critical features of potential energy surfaces [91,92], crucial for the understanding of reactivity (role of the ridge [93] and the kinetic paths [94]). In [95], the PES for the O + H2 reaction was studied. A discrete hyperspherical harmonics representation is presented in [96] for proton transfer in malonaldehyde. 

V. Aquilanti, S. Cavalli, G. Grossi, and R.W. Anderson, Representation in hyperspherical and related coordinates of the potential-energy surface for triatomic reactions. J. Chem. Soc. Faraday Trans, 86(s) 1681-1687, 1990. 

J.D. Kress, Z. Bacic Z, G.A Parker, and R.T Pack, Quantum reactive scattering in 3 dimensions using hyperspherical (aph) coordinates. 5. comparison between 2 accurate potential energy surfaces for H + H2 and D + H2. J. Phys. Chem., 94 8055-8058, 1990. 

V. Aquilanti, S. Cavalli, G. Grossi, V. Pellizzari, M. Rosi, A. Sgamellotti, andF. Tarantelli, Potential energy surfaces in hyperspherical coordinate abinitio kinetic paths for the 0(3P) + H2 reaction. Chem. Phys. Lett., 162 179-184, 1989. 

The recent application of hyperspherical and related coordinates to treat the dynamics on a reactive potential energy surface offers, in fact, the possibility of exploring also those regions where reaction paths present sharp curvatures or bifurcations, taking into account of dynamical quantum effects like tunneling and resonances. Several reviews available [4-10] provide a useful introduction to various aspects of the hyperspherical approach. 

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