Radau coordinate

For the triatomic case with one dimensional (1-D) groups of coordinates (i.e. 7 ], 7-2 and the interior angle B in Jacobi or Radau coordinates), the explicit zero order Hamiltonians and the AU and AT terms can be written as ... 

Figure. 1 Potential energy profiles together with some of the reactant energy levels for the H + H2O —> OH + H2 (a) and the H + H F —> F + H2 reaction (b). On panel (a) the energies of the stationary points are derived from the WSLFH PES [11], the initial energy levels for water are calculated with a DVR method using Radau coordinates, and the final levels for the products are obtained from the Morse potentials corresponding to the PES. On panel (b) the energies of the stationary points are obtained from the 6-SEC PES [13] both the initial and final energy levels are calculated from the Morse parameters obtained by fitting a Morse curve to the potential for the separated FIF and H2 oscillators.

H. Wei and T. Carrington, Jr., An exact Eckart-embedded kinetic energy operator in Radau coordinates for triatomic molecules. Chem. Phys. Lett. 287, 289—300 (1998). 

For triatomic molecules it is possible to find orthogonal coordinates in terms of which the kinetic energy operator has no cross terms. Three possibilities are Jacobi coordinates, Radau coordinates, and hyperspherical coordinates, The form of the kinetic energy operator in Jacobi and Radau coordinates is the same. 

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... 

Besides this coordinate sets, other sets of orthogonal vectors have been considered in the literature. Kinematic Rotations by mass-dependent matrices allows to relate different particle couplings in the Jacobi scheme, and to build up alternative systems such as those based on the Radau-Smith vectors and hyperspherical coordinates [1,3]. The Radau-Smith vectors RSi, RS2 and the angle cos Urs (0 < )rs < n), showed in Figure 3 (the D point is defined by OD = OE x OA, where O is the center-of-mass of the BC couple, E is the center-of-mass of the three particles and A is the position of the A particle), can be calculated from the Jacobi vectors xa and X using ... 

To illustrate this method we consider in some detail the resonances of the 3D FHH system on the Muckerman l surface ( O) As noted in the previous section, collinearly the resonance may be identified with an RPO having Ah action. Defining the bend angle Y via the Natanson-Smith-Radau (62,63) coordinate system, we find the Ah action RPO at fixed y. This provides the curve E.(y) (shown in Fig. 7) where Ej (Y ) is the energy of the y dependent" RPO. 

The polyspherical approach provides, for a particular class of curvilinear coordinates, general expressions for the elements of the G, C and F matrices, whatever the number of atoms and the set of - 1 vectors (Jacobi, Radau, valence,...) used to parametrize the system. In addition, the use of a specific definition for the BF frame ensures that the KEO has the so-called product form (see Sect. 4.2.2), i.e. it is expressed as a sum of products of operators acting on a single coordinate. This definition of the BF frame is as follows the axis is parallel to one of the — 1 vectors used to parametrize the system and the (xz) BF half-plane, with x > 0 is parallel to another vector. A detailed presentation of the method can be found in Ref. [6]. 

Fig. 8.2 Radau polyspherical coordinates for the NHD2 molecule. CP denotes the canonical point (see Ref. [40] for a definition). Figure reproduced from Ref. [2]...

Consider first the matrix-vector product for a triatomic sequential diagonalization-truncation basis of the type discussed in Section 4. In this section we label with j, functions of q and q2 obtained by diagonalizing a two-dimensional Hamiltonian for each DVR point (<73 )y for coordinate q. If the two-dimensional Hamiltonian is diagonalized in a direct product q q2 basis and a and are DVR labels for the q and q2 DVR basis functions, then the matrix of eigenvectors is the transformation matrix whose elements are (Instead of diagonalizing the two-dimensional Hamiltonian in a direct product DVR basis one might use products of optimized ID functions for q and DVR functions for q2 ) In a basis of functions labeled by j and y the Hamiltonian (written in Radau, symmetrized Radau, or Jacobi coordinates) matrix elements are,... 

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