We use the standard body-fixed (BF) Jacobi coordinates , , and , which are the length of the distance vector between H and the center-of-mass of O, the length of the O internuclear distance , and the angle between and , respectively. The overall rotation of the complex with respect to a space fixed (SF) coordinate system is given by 3 Euler angles, collectively denoted by .
Good quantum numbers are the total angular momentum quantum number, , the projection of onto the SF -axis, , and the parity of the wave function under inversion of the SF nuclear coordinates, . We expand the wave function as a function of these quantum numbers. We use a sinc-DVR[52] with ``wrapped'' basis functions[53] and for the and coordinates, respectively.
For the angular coordinate, as in previous work, we employ a
basis of parity adapted angular basis functions,
. Here, is the rotational angular
momentum of O and is the projection of both and
on . The basis functions
are defined as
This results in the following expression for the wave functions :
Since all calculations are independent of , we drop the superscript from now on.