Hyperspherical coordinate

There are many different ways of defining hyperspherical coordinates, which in the case of a collinear atom-diatom reaction are polar coordinates. The 

Unfortunately, however, the surface eigenfunctions and eigenvalues are themselves often quite difficult to obtain. The fundamental reason for this, which should be clear from Fig. 1.2c, is that the reactant and product arrangement channels are confined to smaller and smaller regions of the available 

While the collinear Delves hyperspherical coordinates are essentially unique, several different possibilities arise when one moves to 3D for triatomics. A highly detailed account of these possibilities is given in the introductory section of the work of Pack and Parker [45]. 

A recent overview of the hyperspherical view about dynamics on reactive PESs is given by Aquilanti et al [100]. Hyperspherical coordinates in four-atom systems is discussed by Clary and Echave [101] and Kuppermann [102]. 

For reactive scattering Johnson [27], Linderberg [34], Pack and Parker [45] and Hinze and Wolniewicz [103] have chosen so-called democratic hyperspherical coordinates. These coordinates differ from other coordinates (e.g. Fock, Launay and Lepetit [104]) in that all channels can be treated equally. The coordinates that Kuppermann [105] has chosen, are not so optimal for treating the Pauli-principle for identical nuclei. The hyperspherical coordinates are chosen such that the Euler angles a, and 7 define the orientation of the body-fixed frame, with the 2 -axis pointing in the direction of the vector product 

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One possibility is to use hyperspherical coordinates, as these enable the use of basis fiinctions which describe reagent and product internal states in the same expansion. Hyperspherical coordinates have been extensively discussed in the literature [M, 35 and 36] and in the present application they reduce to polar coordinates (p, p) defined as follows ... 

Hyperspherical coordinates have the properties that q motion is always bound since q = 0 and q = P correspond to cases where two of the three atoms are on top of one another, yielding a very repulsive potential. Also, p —> 0 is a repulsive part of the potential, while large p takes us to the reagent and product valleys. 

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. 

Kuppermann A 1997 Reactive scattering with row-orthonormal hyperspherical coordinates. 2. 

Aquilanti V and Cavalli S 1997 The quantum-mechanical Hamiltonian for tetraatomic systems in symmetric hyperspherical coordinates J. Chem. See. Faraday Trans. 93 801... 

Pogrebnya S K, Echave J and Clary D C 1997 Quantum theory of four-atom reactions using arrangement channel hyperspherical coordinates Formulation and application to OH + Hg < HgO + H J. Chem. Phys. 

Kuppermann A 1996 Reactive scattering with row-orthonormal hyperspherical coordinates. I. Transformation properties and Hamiltonian for triatomic systems J. Phys. Chem. 100 2621... 

In the presence of a phase factor, the momentum operator (P), which is expressed in hyperspherical coordinates, should be replaced [53,54] by (P — h. /r ) where VB creates the vector potential in order to define the effective Hamiltonian (see Appendix C). It is important to note that the angle entering the vector potential is shictly only identical to the hyperangle <]> for an A3 system. 

It is now convenient to introduce hyperspherical coordinates (p, 0, and <])), which specify the size and shape of the ABC molecular triangle and the Euler... 

When we wish to replace the quantum mechanical operators with the corresponding classical variables, the well-known expression for the kinetic energy in hyperspherical coordinates [73] is... 

The explicit expressions of the other terms in Eq, (27) can be evaluated in terms of hyperspherical coordinates using the results of Appendix C,... 

In hyperspherical coordinates, the wave function changes sign when <]) is increased by 2k. Thus, the cotTect phase beatment of the (]) coordinate can be obtained using a special technique [44 8] when the kinetic energy operators are evaluated The wave function/((])) is multiplied with exp(—i(j)/2), and after the forward EFT [69] the coefficients are multiplied with slightly different frequencies. Finally, after the backward FFT, the wave function is multiplied with exp(r[Pg.60]

As demonstrated in [53] it is convenient to incorporate the geometrical phase effect by adding the vector potential in hyperspherical coordinates. Thus we found that the vector potential gave three terms, the first of which was zero, the second is just a potential term... 

The vector potential is derived in hyperspherical coordinates following the procedure in [54], where the connections between Jacobi and the hyperspherical coordinates have been considered as below (see [67])... 

The coordinates p,Tx are called the principal axes of inertia symmetrized hyperspherical coordinates. The nuclear kinetic energy operator in these coordinates is given by... 

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.

The location of the crossing seam (or seam) for an X3 system is established from the requirement that /-ab = rec = r c, where j-ab, rec, and fAc are the interatomic distances. Since the goal are the the geometric properties produced by this seam, hyperspherical coordinates (p,0,atomic masses are equal, say iiiB = me, the seam is defined [5] by... 

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...

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