Hyperspherical potentials

Figure 4.13 Adiabatic hyperspherical potentials, Eq. (101), in a.u. without the adiabatic correction term for He of symmetries 1,3SC and 1,3P° converging to the asymptotic limit e + He+(n = 2), plotted against the hyperradius R in a.u. Each potential supports an infinite number of Rydberg states of Feshbach resonance, of which the lowest level is indicated by a horizontal line. Figure from Ref. [90], Note the difference in notation.

According to Eq. (81) with /S = 6, the three a values for H in Figure 4.14 are —3.708, 2.000, and 9.708. Thus, the asymptotically lowest hyperspherical potential supports an infinite series of Feshbach resonances in the nonrelativistic approximation, although only three lowest members remain as resonances after corrections for the relativistic and radiative effects [80, 82], as was mentioned in Section 3.1.2. Only the lowest member is indicated in the figure by a horizontal line. This resonance is supported by the diabatic potential with A = — 1 connecting from the lowest curve for large p to the middle curve for small p. 

Figure 4.15 Hyperspherical potentials without the adiabatic correction term for Hef P0) converging to the asymptotic limits e + He+(n = 4,5,6). Each potential supports an infinite Rydberg series of Feshbach resonances. Some of them exhibit typical cases of inter-series overlapping resonances illustrated in Figures 4.9 and 4.10. Figure from Ref. [69].

Figure 4.16 Hyperspherical potentials without the adiabatic correction term (full curves) for systems of unnatural parity e+He+(Pc) (left) and e+He+(D°) (right) converging to the asymptotic limits e+ + He+(n = 5-8) and He2+ + Ps(n = 2,3). The diabatic broken curves are for the He2+ + Ps configurations only see text. The vertical positions of the symbols He+(n) and Ps(n) on the right roughly indicate the asymptotic threshold energies. The calculated resonance levels are shown by horizontal bars, some of which are unexpected from the adiabatic potentials. Adapted from Ref. [66].

All in all, adiabatic and diabatic hyperspherical potentials have proved to be quite useful in detailed interpretation of the resonance information obtained by the time-delay analysis of the scattering parameters calculated accurately from the HSCC equations. 

Hyperspherical coordinates provide another powerful tool for precise three-body calculations, on one hand, and for visual understanding of QBS dynamics in terms of hyperspherical potentials playing a role similar to molecular adiabatic potentials. Adiabatic hyperspherical potentials exhibit many avoided crossings, which often must be cast into diabatic potentials to extract essential physics. 

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. 

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. 

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])... 

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°).

Radial basis function networks (RBF) are a variant of three-layer feed forward networks (see Fig 44.18). They contain a pass-through input layer, a hidden layer and an output layer. A different approach for modelling the data is used. The transfer function in the hidden layer of RBF networks is called the kernel or basis function. For a detailed description the reader is referred to references [62,63]. Each node in the hidden unit contains thus such a kernel function. The main difference between the transfer function in MLF and the kernel function in RBF is that the latter (usually a Gaussian function) defines an ellipsoid in the input space. Whereas basically the MLF network divides the input space into regions via hyperplanes (see e.g. Figs. 44.12c and d), RBF networks divide the input space into hyperspheres by means of the kernel function with specified widths and centres. This can be compared with the density or potential methods in pattern recognition (see Section 33.2.5). 

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