Coordinates, Basis set, and Coordinate transformations

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 O_{2}, the length of the O_{2} 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 HO_{2} 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. [74], 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[76] with ``wrapped'' basis
functions[77]
and
for
the *R* and *r* coordinates, respectively. For the angular coordinate we use
parity adapted angular basis functions[69]
for the *R*-embedding and
for the *r*-embedding. Here, *j* is the
rotational angular momentum of O_{2} 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 O_{2}and *K* is the projection of both
and
on the BF *z*-axis (in
this case ). The basis functions
are defined as

are associated Legendre functions in the phase convention of Condon and Shortley.[78] The are normalized Wigner D-matrices and are defined as

with the D-matrices in the passive

This results in the following expression for the wave functions
and
in the *R*-embedding and *r*-embedding,
respectively.

is equal to 0 or 1, depending on the spectroscopic parity[81,82] (-1)

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[74]

(5) |

Using this relation, the orthogonality of Wigner D-matrices and associated Legendre functions, and the relation for the integral over a product of 3 Wigner D-matrices [See Ref. [85], page 97, Eq. (8)], we get the following expressions for the transformation of the coefficients of the parity adapted wave function in the

where the symbols denote 3-

Since all the calculations are done for specific *J*,*M*,*p* combinations, we drop
the *J*,*M*,*p* superscript from now on.