Delves coordinate

G. Hauke, J. Manz, J. Romelt, CoUinear triatomic reactions described by polar Delves coordinates, J. Chem. Phys. 73 (1980) 5040-5044. 

Considering the literature it has to be realized that the hyperspherical coordinates (sometimes called Delves coordinates or mass weighted polar coordinates, too) have guite a long standing tradition in describing three body problems in a variety of physical fields. Originally they seem to have appeared in studies of the helium atom (1932) [1] and since then a continuous stream of publications indicate their application to the treatment of two electron atoms [2] and the molecule [3]. 

Recently though, Launay and LeDorneuf and Rbmelt have shown that if one uses the radial Delves coordinate system and solves the diagonal part of the Schrbdinger equation including all Miagonal non-adiabatic terms, then one finds quantitative agreement with exact quantal computations. ... 

As stressed earlier the classical signature of an adiabatic barrier or well is a periodic orbit. Adiabatic wells, in the Delves coordinate... 

Here p is the hyperradius. No subscript is put on p because p is independent of arrangement channel. However, the Delves hyperangle 0d does depend on the arrangement channel r. In addition to p and the four SF or four BF angles of the r arrangement complete the Delves coordinate set. 

For some rearrangement scattering processes, it may be convenient to use the three sets of Delves coordinates everywhere in the strong interaction region and then project onto functions of the three Jacobi sets at large p. 

At this point we note that Kuppermann[la,19 21] and co-workers use what we shall call Doubled Delves hyperspherical (DDH) coordinates. They are the same as the Delves coordinates just discussed except that the angle Bdt is replaced by Kr =... 

Here the three possible arrangement channels r/ are distinctly included and the angular functions y are exactly the same as those used in the Jacobi wavefunctions of Eqs. (7) and (8). The T( d p) are vibrational wavefunctions which depend parametrically upon p. We use sector-adiabatic basis functions of the Delves coordinates. That is, the basis functions change from sector to sector but not within a sector. On the sector we choose the vibrational functions T to satisfy... 

Currently we are using Delves coordinates only in the moderate interaction region where there is strong rovibrational coupling but negligible coupling between different arrangement channels. 

For systems with long range potentials it is computationally faster to project the Delves solutions onto Jacobi solutions, and propagate those solutions on out to where the boundary conditions can be applied. However, it is conceptually simpler and often practicable to propagate the Delves solutions out to the asymptotic region and apply the boundary conditions directly in Delves coordinates. To allow that, we transform the boundary conditions into Delves coordinates. We note that the T are obtained at a finite set of p values, which are the centers of the propagation sectors. 

The APH coordinates[4,22] are used in the strong reactive region. As with Delves coordinates they are simply obtained from the scaled Jacobi coordinates... 

In addition to p and Odt, the four space-fixed or four boby-fixed angles of the arrangement complete the Delves coordinate set. 

In 1986 Kuppermann and Hipes reported the successful application of a method based on doubled Delves coordinates to the hydrogen exchange reaction, H+H2( f—0, for total angular momentum J=0... 

In their early calculations Hipes and Kuppermann used the finite element method to solve the surface eigenvalue problem at each required value of the hyperradius p. However, like Parker and co-workers, they have since found that other techniques may make their calculations easier to do. Specifically, Cuccaro, Hipes, and Kuppermann have published two papers in which they describe a variational approach to the surface eigenvalue problem [131] and apply it to the calculation of J=0 and J=1 reaction probabilities and collision lifetime matrices for the PK2 and LSTH potential energy surface representations of H-hH2 this work provides the first practical demonstration of the doubled Delves coordinate method for J>0. 

Fig. 25. Contour diagram of the Porter-Karplus potential energy surface for the collinear H 4- H2 system in Delves coordinates The solid curves are equipotentials whose energies (with respect to the bottom of the isolated H2 well) are indicated at the lower right side of the figure. The dashed line is the minimum energy path and the cross along it is the saddle point. The polar coordinates p,a of a general point P in this configuration space are also indicated. The circular arcs centered at the origin are lines of constant p.

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