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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 $\bbox{R}$ between H and the center-of-mass of O2, the length of the O2 internuclear distance vector $\bbox{r}$, and the angle between $\bbox{R}$ and $\bbox{r}$, respectively. $\bbox{R}$ is also the body-fixed z-axis. The overall orientation of the HO2 complex with respect to a space-fixed (SF) coordinate system is given by 3 Euler angles, collectively denoted by $\bbox{\alpha}$.

Good quantum numbers for the wave function are J, the total angular momentum quantum number, M, the projection of $\bbox{J}$ onto the SF z-axis and the parity p of the wave function under inversion of the SF nuclear coordinates. We use a Discrete Variable Representation (DVR), based on sinc-functions[55,56,57] for the radial coordinates and a Finite Basis Representation (FBR) for the angular coordinates. For a discussion of (sinc)-DVRs the reader is referred to e.g., Refs. [56,57,58,59,60]. The DVR-functions form an orthonormal set, characterized by the fact that a basis function $\phi^{-}_{\lambda}(R)$ is effectively non-zero only at the gridpoint $R_\lambda$, defined as $R_\lambda=\lambda*\Delta_R$ with $\Delta_R$ the spacing between gridpoints in the R-dimension. An equivalent relation holds for the functions $\phi^{-}_{\nu}(r)$ in the r-dimension. Throughout this paper we will find it convenient to use $\Psi^{J,M,p}(R,r,\vartheta;t)$ instead of the proper Jacobi coordinate wave function $\Phi^{J,M,p} (R,r,\vartheta;t)$ to which it is related as $\Phi^{J,M,p} (R,r,\vartheta;t)=\Psi^{J,M,p}
(R,r,\vartheta;t)/(Rr)$[61]. This ensures that we can use the Hamiltonian in its form of Eq. (5) instead of the form given in e.g. Refs. [62] and [63]. We expand the wave function for the conserved quantum numbers J, M, and p in ``wrapped'' sinc-DVR functions[57] $\phi^{-}_{\nu}(r)$ and $\phi^{-}_{\lambda}(R)$ for the radial coordinates and parity adapted angular basis functions for the angular coordinates:

...hi^{-}_{\nu}(r) G^{J,M,p}_{j \Omega}(\bbox{\alpha},\vartheta).
\end{displaymath} (4)

In Eq. (2) j is the rotational angular momentum of the O2 molecule. Because of the permutation-inversion symmetry of the O2 molecule only odd rotational states are allowed[64]. The quantum number $\Omega $ is the projection of $\bbox{J}$ onto the BF z-axis and the projection of $\bbox{j}$ onto the BF z-axis. Therefore, the eigenvalues J and j must always be larger than or equal to $\Omega $. $\Omega_{min}$ is 0 or 1, depending on the spectroscopic[65,66] parity (-1)J+p [see Eq. (3)]. The constants Nr and NR are the total number of DVR-functions in the vibrational and scattering coordinates, respectively. The functions $G^{J,M,p}_{j
\Omega}(\bbox{\alpha},\vartheta)$ are parity adapted angular functions, given by[67,68,69]

G^{J,M,p}_{j \Omega}(\bbox{\alpha},\vartheta) =
...) + (-1)^{J+\Omega+p}
F^{J}_{-\Omega M}(\bbox{\alpha})\right],
\end{displaymath} (5)

where $\Theta^\Omega_j (\vartheta)$ is an associated Legendre function in the phase convention of Condon and Shortley[70] and where $F^{J}_{\Omega M}(\bbox{\alpha})$ is a normalized Wigner D-function, given by

F^{J}_{\Omega M}(\bbox{\alpha}) =
\sqrt{\frac{2J+1}{8\pi^2}}D^J_{\Omega M} (\bbox{\alpha})
\end{displaymath} (6)

with the Wigner D-matrices $D^J_{\Omega M} (\bbox{\alpha})$ defined in the passive zyz-convention for rotations[71,72]. The functions $G^{J,M,p}_{j
\Omega}(\bbox{\alpha},\vartheta)$ are eigenstates of $\hat{\bbox{J}}^2$, $\hat{J}_z^{SF}$, $\hat{J}_z^{BF}$, parity, and $\hat{\bbox{j}}^2$ with eigenvalues J(J+1), M, $\Omega $, (-1)p, and j(j+1), respectively. Since all calculations are done for a specific J,M,p combination, we will now drop the J,M,p superscript.

next up previous
Next: Hamiltonian and Propagation Up: Theory Previous: Theory
Anthony J. H. M. Meijer