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## Coordinates and Basis set

We use the standard body-fixed (BF) Jacobi coordinates R, r, and , which are the length of the distance vector between H and the center-of-mass of O2, the length of the O2 internuclear distance vector , and the angle between and , respectively. is also the body-fixed z-axis. The overall orientation of the HO2 complex with respect to a space-fixed (SF) coordinate system is given by 3 Euler angles, collectively denoted by .

Good quantum numbers for the wave function are J, the total angular momentum quantum number, M, the projection of onto the SF z-axis and the parity p of the wave function under inversion of the SF nuclear coordinates. We use a Discrete Variable Representation (DVR), based on sinc-functions[55,56,57] for the radial coordinates and a Finite Basis Representation (FBR) for the angular coordinates. For a discussion of (sinc)-DVRs the reader is referred to e.g., Refs. [56,57,58,59,60]. The DVR-functions form an orthonormal set, characterized by the fact that a basis function is effectively non-zero only at the gridpoint , defined as with the spacing between gridpoints in the R-dimension. An equivalent relation holds for the functions in the r-dimension. Throughout this paper we will find it convenient to use instead of the proper Jacobi coordinate wave function to which it is related as [61]. This ensures that we can use the Hamiltonian in its form of Eq. (5) instead of the form given in e.g. Refs. [62] and [63]. We expand the wave function for the conserved quantum numbers J, M, and p in wrapped'' sinc-DVR functions[57] and for the radial coordinates and parity adapted angular basis functions for the angular coordinates:

 (4)

In Eq. (2) j is the rotational angular momentum of the O2 molecule. Because of the permutation-inversion symmetry of the O2 molecule only odd rotational states are allowed[64]. The quantum number is the projection of onto the BF z-axis and the projection of onto the BF z-axis. Therefore, the eigenvalues J and j must always be larger than or equal to . is 0 or 1, depending on the spectroscopic[65,66] parity (-1)J+p [see Eq. (3)]. The constants Nr and NR are the total number of DVR-functions in the vibrational and scattering coordinates, respectively. The functions are parity adapted angular functions, given by[67,68,69]

 (5)

where is an associated Legendre function in the phase convention of Condon and Shortley[70] and where is a normalized Wigner D-function, given by

 (6)

with the Wigner D-matrices defined in the passive zyz-convention for rotations[71,72]. The functions are eigenstates of , , , parity, and with eigenvalues J(J+1), M, , (-1)p, and j(j+1), respectively. Since all calculations are done for a specific J,M,p combination, we will now drop the J,M,p superscript.

Next: Hamiltonian and Propagation Up: Theory Previous: Theory
Anthony J. H. M. Meijer
1998-02-20