Charge-transfer cross sections in collisions of ground-state Ca and ${\mathrm{H}}^{+}$

We have investigated collisions of $\mathrm{Ca}\left(4s^2\right)$ with ${\mathrm{H}}^{+}$ in the energy range of $200\mathrm{eV}∕\mathrm{u}-10\mathrm{keV}∕\mathrm{u}$ using the semiclassical molecular-orbital close-coupling (MOCC) method with 18 coupled molecular states ($11{}^1\Sigma ^{+}$ and seven ${}^1\Pi ^{+}$ states) to determine charge-transfer cross sections. Except for the incoming channel $6{}^1\Sigma ^{+}$, the molecular states all correspond to charge-transfer channels. Inclusion of ${\mathrm{Ca}}^{2+}\text{−}{\mathrm{H}}^{-}$ is crucial in the configuration-interaction calculation for generating the molecular wave functions and potentials. Because of the Coulomb attraction, the state separating to ${\mathrm{Ca}}^{2+}\text{−}{\mathrm{H}}^{-}$ creates many avoided crossings, even though at infinite separation it lies energetically above all other states that we included. Because of the avoided crossings between the incoming channel $6{}^1\Sigma ^{+}$ and the energetically close charge-transfer channel $7{}^1\Sigma ^{+}$ the charge-transfer interaction occurs at long range. This makes calculations of charge-transfer cross sections by the MOCC method very challenging. The total charge-transfer cross sections increase monotonically from $3.4\times {10}^{-15}{\mathrm{cm}}^2$ at $200\mathrm{eV}∕\mathrm{u}$ to $4.5\times {10}^{-15}{\mathrm{cm}}^2$ at $10\mathrm{keV}∕\mathrm{u}$. Charge transfer occurs mostly to the excited ${\mathrm{Ca}}^{+}\left(5p\right)$ state in the entire energy range, which is the sum of the charge transfer to $7{}^1\Sigma ^{+}$ and $4{}^1\Pi ^{+}$. It accounts for $\sim 47\%$ of the total charge transfer cross sections at $200\mathrm{eV}∕\mathrm{u}$. However, as the energy increases, transfer to ${\mathrm{Ca}}^{+}\left(4d\right)$ increases, and at $10\mathrm{keV}∕\mathrm{u}$ the charge-transfer cross sections for ${\mathrm{Ca}}^{+}\left(5p\right)$ and ${\mathrm{Ca}}^{+}\left(4d\right)$ become comparable, each giving $\sim 38\%$ of the total cross section.

Figure 1

Adiabatic potentials of the ${\mathrm{CaH}}^{+}$ molecule. (a) The $n{}^1\Sigma ^{+}$ states from $n=4$ at the bottom to $n=11$ at the top, and (b) the $n{}^1\Pi ^{+}$ states from $n=3$ at the bottom to $n=7$ at the top. The order of the states is as shown in Table .

Figure 2

(a) Magnified potential curves for $6{}^1\Sigma ^{+}$ (long-dashed curve), $7{}^1\Sigma ^{+}$ (dot-dashed curve), and diabatic potential curves for $6{}^1\Sigma _{dia}^{+}$ (dotted curve), $7{}^1\Sigma _{dia}^{+}$ (solid curve). The diabatic potentials cross each other. (b) Radial coupling matrix element between $7{}^1\Sigma ^{+}$ and $6{}^1\Sigma ^{+}$.

Figure 3

Examples of the coupling matrix elements. (a) Radial coupling matrix elements: solid line, $5{}^1\Sigma ^{+}$–$6{}^1\Sigma ^{+}$; dotted line, $8{}^1\Sigma ^{+}$–$6{}^1\Sigma ^{+}$. (b) Rotational coupling matrix elements: solid line, $3{}^1\Pi ^{+}$–$6{}^1\Sigma ^{+}$; dotted line, $4{}^1\Pi ^{+}$–$7{}^1\Sigma ^{+}$; dot-dashed line, $4{}^1\Pi ^{+}$–$6{}^1\Sigma ^{+}$.

Figure 4

Total and partial charge-transfer cross sections. Total charge-transfer cross section, line with circles, and the labeling of each curve indicates the final products of ${\mathrm{Ca}}^{+}\left(nl\right)$ ions. The partial cross sections of the degenerate states were added to give the cross sections for ${\mathrm{Ca}}^{+}\left(nl\right)$.

Figure 5

Product of probability and impact parameter versus impact parameter for the $7{}^1\Sigma ^{+}$ (full line), $5{}^1\Sigma ^{+}$ (dotted line), and $4{}^1\Pi ^{+}$ (dot-dashed line) states. The collision energy is $2.232\mathrm{keV}∕\mathrm{u}$ (the velocity $0.3\mathrm{a}.\mathrm{u}.$)

Figure 6

(a) Time evolution of the $7{}^1\Sigma ^{+}$ probability ($v=0.3\mathrm{a}.\mathrm{u}.$ and $b=0.5\mathrm{a}.\mathrm{u}.$) using the adiabatic basis set. The dotted line is for $4{}^1\Pi ^{+}$. (b) The same as in (a), using the diabatic basis set.

Figure 7

(a) Radial coupling between $7{}^1\Sigma ^{+}$ and $6{}^1\Sigma ^{+}$ used for obtaining $\omega \left(R\right)$. (b) $\omega \left(R\right)$ (solid line), $\sin \omega \left(R\right)$ (dotted line), and $\cos \omega \left(R\right)$ (dot-dashed line). (c) The diabatic potentials for $6{}^1\Sigma ^{+}$ (full line) and $7{}^1\Sigma ^{+}$ (dotted line). They are enveloped by the adiabatic $6{}^1\Sigma ^{+}$ potential (dashed line) and the adiabatic $7{}^7\Sigma ^{+}$ potential (chain line).