Hyperspherical and related coordinates

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. 

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. 

M. Ragni, A.C.P. Bitencourt, V. Aquilanti, Hyperspherical and related types of coordinates for the dynamical treatment of three-body systems, Prog. Theor. Chem. Phys. 16 (2007) Part 1,123. 

HYPERSPHERICAL AND RELATED TYPES OF COORDINATES FOR THE DYNAMICAL TREATMENT OF THREE-BODY SYSTEMS... 

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 Fock transformation of variables consists in projecting the momentum vector p with coordinates Px, py, pz and modulus p in momentum space on a tetradimensional hypersphere of unit radius. The momentum pq = V-2E is directly related to the energy spectrum. A point on the hypersphere surface has coordinates ... 

Further remarks on hyperspherical coordinates, e.g. Fock [41] coordinates, Smith-Whitten [42] democratic coordinates and relations between Smith-Whitten and Fock coordinates are given in the work of Launay [2]. 

Let us now consider the overlap between the spherical and the Stark basis. For the latter, the momentum space eigenfimctions, which in configuration space correspond to variable separation in parabolic coordinates, are similarly related to alternative hyperspherical harmonics [2]. The connecting coefficient between spherical and 5 torA basis is formally identical to a usual vector coupling coefficient (from now on n is omitted from the notation) ... 

Besides this coordinate sets, other sets of orthogonal vectors have been considered in the literature. Kinematic Rotations by mass-dependent matrices allows to relate different particle couplings in the Jacobi scheme, and to build up alternative systems such as those based on the Radau-Smith vectors and hyperspherical coordinates [1,3]. The Radau-Smith vectors RSi, RS2 and the angle cos Urs (0 < )rs < n), showed in Figure 3 (the D point is defined by OD = OE x OA, where O is the center-of-mass of the BC couple, E is the center-of-mass of the three particles and A is the position of the A particle), can be calculated from the Jacobi vectors xa and X using ... 

Figure 17.8 Relaxed triangular plot in hyperspherical coordinates (dimensionless) of the CHIPRI PES for groxmd state H3. Solid contours are at intervals of 0.005 Eh, starting at the separated atom-diatom energy. Indicated are the three symmetry-related collinear saddle points (solid dots) and the conical intersection (open circles). Diametrically opposed to the former are the atom-united-atom of H2 limits and, nearby, the H- -H2 vdW minima.

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