Hyperspherical coordinate symmetrized

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

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... 

Manz and Romelt (1981). Rm and 7 hi are the two I-H bond distances. The heavy point marks the saddle point and the shaded area indicates schematically the Franck-Condon region in the photodetachment experiment. The arrow along the symmetric stretch coordinate (f Hi = -Rih) illustrates the early motion of the wavepacket and the two heavy arrows manifest dissociation into the two identical product channels, (b) The same PES as in (a) but represented in terms of hyperspherical coordinates (p, i9) defined in (7.33). The horizontal and the vertical arrows illustrate symmetric and anti-symmetric stretch motions, respectively, as indicated by the two insets. 

The hyperspherical coordinates are simply the polar coordinates in the (R, r)-plane. The Jacobi coordinates (R,r) defined in Figure 2.1 are not appropriate for symmetric molecules because they cannot simultaneously describe both product channels, A + BC and AB + C. The set of Jacobi coordinates appropriate for one dissociation channel is inappropriate for the other one and vice versa. The hyperspherical coordinates, on the other hand, describe both channels equally well and they can be used at short distances as well as in the asymptotic regions. The two-dimensional Hamiltonian for the linear triatomic molecule is given in hyperspherical coordinates by... 

The model potential displayed in Figure 8.2 had originally been used by Kulander and Light (1980) to study, within the time-independent R-matrix formalism, the photodissociation of linear symmetric molecules like C02. It will become apparent below that in this and similar cases the time-dependent approach, which we shall pursue in this chapter, has some advantages over the time-independent picture. The motion of the ABA molecule can be treated either in terms of the hyperspherical coordinates defined in (7.33) or directly in terms of the bond distances Ri and i 2 The Hamiltonian for the linear molecule expressed in bond distances... 

The symmetric parametrization can be achieved by taking as internal reference system the one that diagonalizes the inertia tensor, placing the principal axis in correspondence with that of maximal inertia [6,64], The symmetric hyperspherical coordinates can be calculated from the asymmetric hyperspherical coordinates ... 

When cos (2 ) vanishes O is undefined. The asymmetric hyperspherical coordinates from the symmetric hyperspherical coordinates can be calculated by ... 

It can be useful to know the relationship between these new coordinate sets. So here we present some inverse formulae. For example, to pass from the f, r], and f set to symmetric hyperspherical coordinates one can use ... 

The interparticle distances are related with symmetric hyperspherical coordinates by ... 

The symmetric hyperspherical coordinates from the interparticle distances, where p is given by the Eq. (84), can be written as ... 

V. Aquilanti and S. Cavalli, The quantum-mechanical hamiltonian for tetra-atomic systems in symmetric hyperspherical coordinates. J. Chem. Soc. Faraday Trans., 93 801-809, 1997. 

Consider then an adiabatic well in the hyperspherical coordinate system. Classically, the motion of the periodic orbit at the well would be an oscillation from a point on the inner equipotential curve in the reactant channel to a point on the same equipotential curve in the product channel. This is qualitatively the motion of what are termed "resonant periodic orbits" (RPO s). For example the RPO s of the IHI system are given in Fig. 5. Thus, finding adiabatic wells in the radial coordinate system corresponds to finding RPO s and quantizing their action. Note that in Fig. 5 we have also plotted all the periodic orbit dividing surfaces (PODS) of the system, except for the symmetric stretch. By definition, a PODS is a periodic orbit that starts and ends on different equi-potentials. Thus the symmetric stretch PODS would be an adiabatic well for an adiabatic surface in reaction path coordinates. However, the PODS in the entrance and exit channels shown in Fig. 5 may be considered as adiabatic barrieres in either the radial or reaction path coordinate systems. Here, the barrier in radial coordinates, has quantally a tunneling path between the entrance and exit channels. 

We can calculate the right-hand side of (22) using the definition of Hna (q) together with (17). In the vicinity of the intersection of interest it is appropriate to retain in (22) only the lowest-order terms in each of the Qk (k = 1, 2,.. ., 3N — 6). In addition, in order to explore the point group symmetry properties of ij/ (rel q0) and v /j(rel q0) it is desirable to choose for the Qk coordinates which display simple transformation properties under the operations of that group, such as normal mode coordinates [13,15] or, in some circumstances, symmetrized hyperspherical coordinates [12,16,17]. Such choices lead to simple point group symmetries for the / (rel q0) and (rel q0) and permit the identification of which. (q0) and q0) vanish due to symmetry [18]. 

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