Lattice approach for $\alpha +{{\mathrm{H}}_2}^{+}$ collisions

We solve the time-dependent Schrödinger equation for an ion-molecule collision using a lattice approach. This method carries over naturally from similar lattice treatments of ion-atom collisions, and is free from the multicenter problems that make extensions of other ion-atom methods difficult.

Figure 1

The coordinate system.

Figure 2

The $1s\sigma$ Born-Oppenheimer potentials for ${{\mathrm{H}}_2}^{+}$: the heavy line is calculated with the present lattice approach and the light line is an accurate solution. The inset compares the shape of the two potentials by shifting the present result to give the correct asymptotic threshold.

Figure 3

The total electron capture probability as a function of impact parameter $b$ for $p+\mathrm{H}$ at an impact energy of 5 keV $(v_p=0.447\mathrm{a}.\mathrm{u}.)$. The circles mark the results from the present lattice approach; and the squares, accurate calculations from Ref. 19.

Figure 4

Convergence tests for the electron capture probability for $\alpha +{{\mathrm{H}}_2}^{+}$. In (a), $\delta =0.06\mathrm{a}.\mathrm{u}.$ and $\varphi =0°$; the circles mark $\Delta x=0.2\mathrm{a}.\mathrm{u}.$ while the triangles mark $\Delta x=0.1\mathrm{a}.\mathrm{u}.$ In (b), $\Delta x=0.2\mathrm{a}.\mathrm{u}.$ and $b=2.0\mathrm{a}.\mathrm{u}.$; the circles mark $\delta =0.06\mathrm{a}.\mathrm{u}.$ while the triangles mark $\delta =0.03\mathrm{a}.\mathrm{u}.$ The other parameters are $v_{\alpha }=0.41\mathrm{a}.\mathrm{u}.$, $R=2.2\mathrm{a}.\mathrm{u}.$, and $\theta =90°$.

Figure 5

The differential capture cross section $\sigma (R,\theta ,v_{\alpha })$ Eq. (3) as a function of $\theta$ at $R=2.2\mathrm{a}.\mathrm{u}.$, the equilibrium of ${\mathrm{H}}_2^{+}$ in our model (see Fig. 2).

Figure 6

The differential capture cross section $\sigma (R,\theta ,v_{\alpha })$ Eq. (3) as a function of $\theta$ at $R=1.4\mathrm{a}.\mathrm{u}.$, the equilibrium value of ${\mathrm{H}}_2$.

Figure 7

The differential capture cross section $\sigma (R,\theta ,v_{\alpha })$ Eq. (3) as a function of $\theta$ at several values of $R:1.4\mathrm{a}.\mathrm{u}.$ (squares and solid line), 1.6 a.u. (diamonds and dashed line), 1.8 a.u. (circles and dashed line), 2.0 a.u. (circles and solid line), and 2.2 a.u. (filled circles and solid line).

Figure 8

The total electron capture probability as a function of impact parameter $b$ for $\alpha +\mathrm{H}$ at $v_{\alpha }=0.477\mathrm{a}.\mathrm{u}.$ calculated with the the present lattice approach.