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 . This ensures that we can use the Hamiltonian in its form of Eq. (5) instead of the form given in e.g. Refs.  and . We expand the wave function for the conserved quantum numbers J, M, and p in ``wrapped'' sinc-DVR functions and for the radial coordinates and parity adapted angular basis functions for the angular coordinates: