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## 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 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. [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 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

 (1)

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

 (2)

with the D-matrices in the passive zyz convention for rotations.[79,80] is defined analogously to .

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

 = (3) = (4)

is equal to 0 or 1, depending on the spectroscopic parity[81,82] (-1)J+p. The same holds for . The maximum values of j and l in the bases are and , respectively. Because of the permutation inversion symmetry of the O2 molecule, only odd rotational j states are allowed in Eq. (3).[83] Physically, there is no such restriction on l in Eq. (4). (Note that we do not take the conical intersection in this reactive system into account. In that case the symmetry argument is not as straight forward).[84]

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 R-embedding to the coefficients of the parity adapted wave function in the r-embedding and for the inverse transformation:

 = (6) = (7)

where the symbols denote 3-jmsymbols.[85,86] From Eqs. (6) and (7) it is clear that given a certain value for , the triangular relations for the 3-jmsymbols lead to being equal to . Note also the symmetry between the transformation and the inverse transformation. The term in Eqs. (6) and (7) shows that there is indeed a selection rule for the lquantum numbers. For odd p one only gets even l's and for even p one only gets odd l's, given that j can only be odd for H + O2( ).[83] We wish to point out here as well that parity adaptation is necessary to get to these selection rules'', as is clear from the non-parity adapted transformation given in Ref. [74], Eq. (37). We also point out here that starting the calculation in a pure state in the R-embedding is equivalent to starting in a (coherent) superposition of (l,K) states in the r-embedding.

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

Next: Hamiltonian, Propagation, and Analysis Up: Theory and Computational details Previous: Theory and Computational details
Anthony J. H. M. Meijer
1998-10-14