Hyperspherical coordinates, for chemical

V. Aquilanti, G. Capecchi, and S. Cavalli, Hyperspherical coordinates for chemical reaction dynamics. Adv. Quant. Chem., 36 341-361, 1999. 

Hyperspherical Coordinates for Chemical Reaction Dynamics only one representation visualized by the tree in fig. 2. 

A historical account of the development of orthogonal coordinates for elementary chemical reactions has been given by one of the protagonists [12], The early hyper-spherical treatment for the helium atom as a tree-body quantum-mechanical problem [13,14], reviewed in Morse and Feshbach s treatise [15], was taken up in two basic papers by Fock [16]. They essentially used the parametrization referred to as asymmetrical in the following. Further important work in atomic physics [17,18] used the same hyperspherical parametrization, as did the investigations by Delves [19] on the breakdown of systems of many particles. 

Clapp and Delves rediscovered them for the field of nucJear physics (1949) [4] and within the last two decades the set of hyperspherical coordinates was introduced to the theory of molecular dynamics (i.e. molecular collisions (1960) [5], collision induced dissociation (1971) [6], photolysis of molecules (1974) [7], chemical reactions (1973)... 

We have shown how to obtain an accurate solution for three atom reaction. After introducing the democratic hyperspherical coordinates it is possible to solve the Schrodinger equation (within the Ritz approach). We obtain the rate constant for the state-to-state elementary chemical reaction. 

A useful method that can be used in both time-independent and time-dependent calculations on chemical reactions is to add a negative imaginary potential (NIP) to the overall potential in carefully selected regions. This enables reactive flux into particular channels to be absorbed by the NIP and it is then not necessary to integrate the SchrOdinger equation further in that channel as the total reactive flux into it has been obtained. This is useful when state-to-state information is not needed but an overall reactive cross-section is required. The NIP also obviates the need for complicated coordinate systems in chemical reactions (such as hyperspherical coordinates) but the most appropriate parameters for the NIP have to be carefully tested. 

Curvilinear internal bond coordinates versus rectilinear normal coordinates [9,10] is, of course, not the only choice to be made. There is a larger selection of coordinates to choose from Radau, Jacobi, hyperspherical, and so on, coordinates see, for example, Refs. 11-14, and the review by Bacic and Light [15] (and references therein). In addition to the rovibrational states of semirigid molecules, these can be used for different types of problems for example, for systems where molecular bonds are broken and formed, chemical reactions occur, and so on. It is clear that both kinetic energy operators and... 

---