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Coordinates and Basis set

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 O$_2$, the length of the O$_2$ internuclear distance ${\bf r}$, and the angle between ${\bf R}$ and ${\bf r}$, 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 $\bbox{\alpha}$.

Good quantum numbers 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[52] with ``wrapped'' basis functions[53] $\phi^{-}_\lambda(R)$ and $\phi^{-}_\nu(r)$ for the $R$ and $r$ coordinates, respectively.

For the angular coordinate, as in previous work, we employ a basis of parity adapted angular basis functions, $G^{J,M,p}_{j\Omega}(\bbox{\alpha},\vartheta)$. Here, $j$ is the rotational angular momentum of O$_2$ and $\Omega$ is the projection of both ${\bf j}$ and ${\bf J}$ on ${\bf R}$. The basis functions $G^{J,M,p}_{j\Omega}(\bbox{\alpha},\vartheta)$ are defined as

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

$\Theta^\Omega_j (\vartheta)$ are associated Legendre functions in the phase convention of Condon and Shortley.[54] The $F^{J}_{\Omega
M}(\bbox{\alpha})$ are normalized Wigner D-matrices and are defined as
\begin{displaymath}
F^{J}_{\Omega M}(\bbox{\alpha})=\sqrt{\frac{2J+1}{8\pi^2}}D^J_{\Omega M} (\bbox{\alpha})
\end{displaymath} (2)

with the D-matrices in the passive $zyz$ convention for rotations.[55,56]

This results in the following expression for the wave functions $\Psi^{J,M,p}$:


$\displaystyle \Psi^{J,M,p} (R,r,\bbox{\alpha},\vartheta;t)$ $\textstyle =$ $\displaystyle \sum^J_{\Omega=\Omega_{\rm min}}
\sum^{j_{max}}_{j\geq\Omega} \su...
...}_{\lambda}(R)
\phi^{-}_{\nu}(r) G^{J,M,p}_{j \Omega}(\bbox{\alpha},\vartheta).$ (3)

$\Omega_{\rm min}$ is equal to 0 or 1, depending on the spectroscopic parity[57,58] $(-1)^{J+p}$. Because of the permutation inversion symmetry of the O$_2$ molecule, only odd rotational $j$ states are allowed in Eq. (3).[59]

Since all calculations are independent of $M$, we drop the $M$ 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 2000-10-05