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Coordinates, Basis set, and Coordinate transformations

We use the standard body-fixed (BF) Jacobi coordinates R, r, and $\vartheta$, which are the length of the distance vector ${\bf R}$ between H and the center-of-mass of O2, the length of the O2 internuclear distance ${\bf r}$, and the angle between ${\bf R}$ and ${\bf r}$, respectively. We use two different choices for the z-axis (``embeddings'') for the coordinates. In the first embedding (``R-embedding'') ${\bf R}$ 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 ${\bf r}$ 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 $\vartheta$ around the y-axis. These two frames are defined identically to the BF$_{\tau_1}$ and BF$_{\tau_2}$ 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 $\bbox{\alpha}_R$ and $\bbox{\alpha}_r$ 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 ${\bf J}$ 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] $\phi^{-}_\lambda(R)$ and $\phi^{-}_\nu(r)$ for the R and r coordinates, respectively. For the angular coordinate we use parity adapted angular basis functions[69] $G^{J,M,p}_{j\Omega}(\bbox{\alpha}_R,\vartheta)$ for the R-embedding and $H^{J,M,p}_{lK}(\bbox{\alpha}_r,\vartheta)$ for the r-embedding. Here, j is the rotational angular momentum of O2 and $\Omega $ is the projection of both ${\bf j}$ and ${\bf J}$ on the BF z-axis (in this case ${\bf R}$). Similarly, l is the orbital angular momentum quantum number for the rotation of H around O2and K is the projection of both ${\bf l}$ and ${\bf J}$ on the BF z-axis (in this case ${\bf r}$). The basis functions $G^{J,M,p}_{j\Omega}(\bbox{\alpha}_R,\vartheta)$ are defined as


 \begin{displaymath}
G^{J,M,p}_{j\Omega}(\bbox{\alpha}_R,\vartheta)=
\left[2\left...
... +
(-1)^{J+\Omega+p} F^{J}_{-\Omega M}(\bbox{\alpha}_R)\right]
\end{displaymath} (1)

$\Theta^\Omega_j (\vartheta)$ are associated Legendre functions in the phase convention of Condon and Shortley.[78] The $F^{J}_{\Omega
M}(\bbox{\alpha}_R)$ are normalized Wigner D-matrices and are defined as


 \begin{displaymath}
F^{J}_{\Omega M}(\bbox{\alpha}_R)=\sqrt{\frac{2J+1}{8\pi^2}}D^J_{\Omega M} (\bbox{\alpha}_R)
\end{displaymath} (2)

with the D-matrices in the passive zyz convention for rotations.[79,80] $H^{J,M,p}_{lK}(\bbox{\alpha}_r,\vartheta)$is defined analogously to $G^{J,M,p}_{j\Omega}(\bbox{\alpha}_R,\vartheta)$.

This results in the following expression for the wave functions $\Psi^{J,M,p}_R$ and $\Psi^{J,M,p}_r$ in the R-embedding and r-embedding, respectively.


  
$\displaystyle \Psi^{J,M,p}_R (R,r,\bbox{\alpha}_R,\vartheta;t)$ = $\displaystyle \sum^J_{\Omega=\Omega_{\rm min}}
\sum^{j_{max}}_{j\geq\Omega} \su...
..._{\lambda}(R)
\phi^{-}_{\nu}(r) G^{J,M,p}_{j \Omega}(\bbox{\alpha}_R,\vartheta)$ (3)
$\displaystyle \Psi^{J,M,p}_r (R,r,\bbox{\alpha}_r,\vartheta;t)$ = $\displaystyle \sum^J_{K=K_{\rm min}}
\sum^{l_{max}}_{l\geq K} \sum_{\nu=1}^{N_r...
...^{-}_{\lambda}(R)
\phi^{-}_{\nu}(r) H^{J,M,p}_{l K}(\bbox{\alpha}_r,\vartheta).$ (4)

$\Omega_{\rm min}$ is equal to 0 or 1, depending on the spectroscopic parity[81,82] (-1)J+p. The same holds for $K_{\rm
min}$. The maximum values of j and l in the bases are $j_{\rm max}$ and $l_{\rm max}$, 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]


\begin{displaymath}F^{J}_{K'M}(\bbox{\alpha}_r)=\sum_{\Omega'=-J}^{\Omega'=J}
D^{J}_{K'\Omega'}(0,\vartheta,0) F^{J}_{\Omega' M}(\bbox{\alpha}_R)
\end{displaymath} (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:


  
$\displaystyle A^{J,M,p}_{\lambda\nu lK} (t)$ = $\displaystyle \left[2\left(1+\delta_{K0}\right)\right]^{-\frac{1}{2}}
\sum_{\Om...
...}}
\left(\begin{array}{ccc}l & J & j \\  0 & \Omega & -\Omega\end{array}\right)$  
    $\displaystyle \sqrt{(2l+1)(2j+1)}
\left(\begin{array}{ccc}l & J & j \\  -K & K & 0\end{array}\right)
\left[ 2 + 2(-1)^{p+l+j} \right]$ (6)
$\displaystyle C^{J,M,p}_{\lambda\nu j \Omega} (t)$ = $\displaystyle \left[2\left(1+\delta_{\Omega 0}\right)\right]^{-\frac{1}{2}}
\su...
...}}
\left(\begin{array}{ccc}l & J & j \\  0 & \Omega & -\Omega\end{array}\right)$  
    $\displaystyle \sqrt{(2l+1)(2j+1)}
\left(\begin{array}{ccc}l & J & j \\  -K & K & 0\end{array}\right) \left[ 2 + 2(-1)^{p+l+j} \right],$ (7)

where the symbols ${\left(\begin{array}{ccc} . & . &
. \\ . & . & . \end{array} \right)}$ denote 3-jmsymbols.[85,86] From Eqs. (6) and (7) it is clear that given a certain value for $j_{\rm max}$, the triangular relations for the 3-jmsymbols lead to $l_{\rm max}$ being equal to $j_{\rm max} + J$. Note also the symmetry between the transformation and the inverse transformation. The $\left[ 2 + 2(-1)^{p+l+j} \right]$ 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( $^3\Sigma^-_g$).[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 $(j,\Omega)$ 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 up previous
Next: Hamiltonian, Propagation, and Analysis Up: Theory and Computational details Previous: Theory and Computational details
Anthony J. H. M. Meijer
1998-10-14