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Final analysis

After the calculation we analyze the wave packet to get the energy dependent reaction probability. To calculate this quantity we use the method taken from Ref. [33]. The energy dependent reaction probability is defined as


 \begin{displaymath}
P_r(E) = \frac{\hbar}{\mu_e} {\rm Im} \left\langle \Psi^{+}(...
...left\vert
\hat{F}(r_s) \right\vert \Psi^{+}(E) \right\rangle ,
\end{displaymath} (12)

where $\hat{F}(r_s)$ is the flux-operator for the flux through a surface at rs, defined in Ref. [33]. Its matrix elements for the wrapped sinc-DVR are easy to obtain, using the definition of the flux-operator, the chain-rule and l'Hospitals theorem,


 \begin{displaymath}
\left\langle\phi^{-}_m (r) \vert F (r_s) \vert \phi^{-}_n (r...
...forall & r_s = r_m & \bigwedge & m
\neq n, \end{array} \right.
\end{displaymath} (13)

$\mu_e$ in Eq. (10) is the scattering mass for the exit channel. The scattering wave function $\Psi^{+}(E)$ is defined as


 \begin{displaymath}
\Psi^{+}(E) = \frac{1}{a(E)} \int_{-\infty}^{\infty} \exp(iEt) \Psi(t) \,dt
\end{displaymath} (14)

a(E) is defined as the overlap between the initial wave packet and a free wave. It can be seen as the probability of finding a specific energy E in the wave packet. As such, it can be shown to be independent of the total angular momentum quantum number J. Therefore, we can use the form of a(E) derived for the case of J=0 also for J>0. Thus, a(E) is given as[80]


 \begin{displaymath}
a(E) = - \sqrt[4]{(8\pi\beta)} \left(\frac{\mu_R}{k_E}\right)^\frac{1}{2}
\exp\left[-\beta\left(k_E-k_0\right)^2\right]
\end{displaymath} (15)

with $k_E=\sqrt{2\mu_RE}$ and k0 the initial linear momentum of the wave packet [See Eq. (14)]. The parameter $\beta$ is a measure for the width of the initial wave packet [see also Eq. (14)].

Eq. (13) can also be derived rigorously rather than with the hand-waving argument presented here. However, the proof is elaborate, albeit straightforward. Therefore, we will only outline it here[81]. First, take a plane wave along the SF z-axis, expand it into spherical Bessel functions[82] and rotate it to the BF-frame. Then, project the plane wave onto a specific (J,M) state and normalize this state. Next, write the spherical Bessel functions in their asymptotic expression and take the inner product with the initial wave packet. As in Ref. [80] one gets two integrals, one of which is negligible. Neglegting this integral¸ one obtains Eq. (13), apart from a complex factor which disappears in calculating Pr(E).

By taking the surface not at rs, but at Rs we can use this method also to calculate Pnr(E), the non-reaction probability. This last quantity is interesting, because we can use it to assess the convergence of the calculation. If the results are converged, Pr(E) + Pnr(E) are equal to 1.


next up previous
Next: Computational aspects Up: Theory Previous: Hamiltonian and Propagation
Anthony J. H. M. Meijer
1998-02-20