Democratic coordinates

The hainiltonian of the ABC system in body-frame hyperspherical democratic coordinates is written as [4] ... 

The expression (10) is not suitable to determine collisional wavcfunctions because their asymptotic part is determined in Jacobi space-fixed coordinates specific for each arrangement A. The transformation from body-frame hypcrsphcrical democratic coordinates (p,6,(j),S) to space-fixed (Rx,rx) Jacobi coordinates is performed in several steps. It necessitates the introduction of Fock internal hvjier-spherical coordinates p,oJx., ii ) to analyze the rovibrational character of the surface functions in the fragmentation region at huge p. 

Eq. (19), which is called the kinematic or democratic rotation, generally brings about a change in molecular shape exchanging the positions of the constituent atoms in a democratic manner. It is proved [13] that the range of the hyper-angle (p is limited to 0 < (p < n for tp to preserve one-to-one correspondence between molecular shape and the internal coordinates. 

The Jacobi coordinates will now be nsed to define what is called the (more convenient) hyperspheiical democratic coordinates. 

When a chemical reaction proceeds, the role of the atoms changes and nsing the same Jacobi coordinate system aU the time leads to technical problems. In order not to favor any of the three atoms despite possible differences in their masses, we introduce hypersphericcd democratic... 

The axes are oriented along the main axes of the moments of inertia, with lyy > hx-Finally, we introduce hyperspherical democratic coordinates equivalent to the RMCS ... 

As chosen, the hyperspherical democratic coordinates which cover all possible atomic positions within the plane z = 0 have the following ranges 0 < p < oo. 0 < 6 < j.O < < 47T. 

The hyperspherical democratic coordinates represent a useful alternative for RMCS from Appendix 1 available at booksite.elsevier.com/978-0-444-59436-5 (they themselves form another RMCS), and therefore do not depend on the orientation with respect to the body-fixed coordinate system (BFCS). However, the molecule has somehow to be informed that it rotates (preserving the length and the direction of the total angular momentum), because a centrifugal force acts on its parts and the Hamiltonian expressed in BFCS (cf.. Appendix I available at booksite.elsevier.com/978-(M44-59436-5) bas to contain information about this rotation. 

The exact kinetic energy expression for a polyatomic molecule in a space-fixed coordinate system (SFCS cf. Appendix I available at booksite.elsevier.com/978-0-444-59436-5) bas been given in Chapter 6 (Eq. 6.39). After separation of the center-of-mass motion, the Hamiltonian is equal to H = T + V. where V represents the electronic energy playing the role of the potential energy for the motion of the nuclei (an analog of (/ ) from Eq. (6.8), and we assume the Bom-Oppenheimer approximation). In the hyperspherical democratic coordinates, we obtain ... 

We have shown how to obtain an accurate solution for three-atomic reactions. After introducing the hyperspher-ical democratic coordinates, it is possible to solve the Schiddinger equation (within the Ritz approach). We obtain the rate constant for the state-to-state elementary chemical reaction. 

Coriolis coupling (p. 906 and 912) critical points (p. 888) cross section (p. 901) curvature coupling (p. 906 and 914) cycloaddition reaction (p. 944) democratic coordinates (p. 898) diabatic and adiabatic states (p. 949) donating mode (p. 914) early and late reaction barriers (p. 895) electrophilic attack (p. 938) entrance and exit channels (p. 895) exo- and endothermic reactions (p. 909) femtosecond spectroscopy (p. 889) Franck-Condon factors (p. 962) intrinsic reaction coordinate (IRC) (p. 902) inverse Marcus region (p. 954) mass-weighted coordinates (p. 903)... 

Darwin solution, 135 decoherence, 40,45 deexcitations, 94, 636 democratic coordinates,... 

Finally, we introduee democratic hyperspherical coordinates equivalent to RMCS ... 

The hyperspherical democratic coordinates represent a useful alternative for RMCS from Appendix I (they themselves form another RMCS), and therefore... 

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