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 , and the angle between and , respectively. We use two different choices for the z-axis (``embeddings'') for the coordinates. In the first embedding (``R-embedding'') is the BF z-axis. The y-axis is perpendicular to the plane of the HO2 complex. The x-axis is chosen such that the coordinate system becomes right-handed. In the second embedding (``r-embedding'') the internuclear distance vector is chosen to be the BF z-axis. Again, the y-axis is perpendicular to the plane of the complex and the x-axis makes the coordinate system right-handed. The R-embedding transforms to the r-embedding by a rotation over around the y-axis. These two frames are defined identically to the BF and BF frames in Ref. , respectively. Other (equally valid) definitions can be found in the literature as well.[74,75] The overall rotation of the complex with respect to a space fixed (SF) coordinate system is given by 3 Euler angles, collectively denoted by and for the R-embedding and r-embedding, respectively.
Good quantum numbers for both embeddings are the total angular momentum quantum number, J, the projection of onto the SF z-axis, M, and the parity of the wave function under inversion of the SF nuclear coordinates, p. We expand the wave function as a function of these quantum numbers. We use a sinc-DVR with ``wrapped'' basis functions and for the R and r coordinates, respectively. For the angular coordinate we use parity adapted angular basis functions for the R-embedding and for the r-embedding. Here, j is the rotational angular momentum of O2 and is the projection of both and on the BF z-axis (in this case ). Similarly, l is the orbital angular momentum quantum number for the rotation of H around O2and K is the projection of both and on the BF z-axis (in this case ). The basis functions are defined as
This results in the following expression for the wave functions and in the R-embedding and r-embedding, respectively.
Since the number of states should be conserved in any embedding, we studied the transformation between the R-embedding and the r-embedding to find a ``selection rule'' for l. For the normalized Wigner D-matrices from Eq. (2) the transformation between the two embeddings is given as
Since all the calculations are done for specific J,M,p combinations, we drop the J,M,p superscript from now on.