Excitation and charge transfer in low-energy hydrogen-atom collisions with neutral atoms: Theory, comparisons, and application to Ca

A theoretical method is presented for the estimation of cross sections and rates for excitation and charge-transfer processes in low-energy hydrogen-atom collisions with neutral atoms, based on an asymptotic two-electron model of ionic-covalent interactions in the neutral atom-hydrogen-atom system. The calculation of potentials and nonadiabatic radial couplings using the method is demonstrated. The potentials are used together with the multichannel Landau–Zener model to calculate cross sections and rate coefficients. The main feature of the method is that it employs asymptotically exact atomic wave functions, which can be determined from known atomic parameters. The method is applied to $\text{Li}+\text{H}, \text{Na}+\text{H}$, and $\text{Mg}+\text{H}$ collisions, and the results compare well with existing detailed full-quantum calculations. The method is applied to the astrophysically important problem of $\text{Ca}+\text{H}$ collisions, and rate coefficients are calculated for temperatures in the range 1000–20 000 K.

Figure 1

Coordinate system for the $X+\text{H}$ hydride system in the covalent configuration. The hydrogen atom is centered on A and the target atom $X$ is centered on B. The atom $X$ is considered as a charged core with a single active electron. The two electrons considered explicitly are labeled 1 and 2. The dashed circles delineate the two atoms, and their relevant quantum numbers.

Figure 2

Coordinate system for the $X+\text{H}$ hydride system in the ionic configuration, $X^{+}+{\text{H}}^{-}$. The dashed circle encompasses the ${\mathrm{H}}^{-}$ ion, and the full circle encompasses the bare core of atom $X$; in both cases the relevant quantum numbers are specified.

Figure 3

Comparisons of calculations in the LCAO asymptotic model for $\text{Li}+\text{H}, \text{Na}+\text{H}$, and $\text{Mg}+\text{H}$ with results from full-quantum-scattering calculations and the quantum chemistry data they are based on. The first, second, and third columns are for the $\text{Li}+\text{H}, \text{Na}+\text{H}$, and $\text{Mg}+\text{H}$ systems, respectively. The top row shows the adiabatic potentials for the LiH (${}^1{\mathrm{\Sigma }}^{+}$), NaH (${}^1{\mathrm{\Sigma }}^{+}$), and MgH (${}^2{\mathrm{\Sigma }}^{+}$) systems; the quantum chemistry potentials are from Refs. [6, 41] for LiH, from Ref. [44] for NaH, and from Ref. [8] for MgH. The middle row shows the coupling $H_{12}$; in the quantum chemistry calculations these are derived from the adiabatic potentials via the two-state Landau–Zener model (except for the long-range crossing data for $\text{Li}+\text{H}$, which are extrapolations of the data from shorter range, see Ref. [6]). The bottom row shows the ratio of rate coefficients at 6000 K, as a function of the full-quantum data; the full-quantum-scattering results are from Refs. [6, 12] for $\text{Li}+\text{H}$, from Refs. [7, 39] for $\text{Na}+\text{H}$, and from Refs. [9, 40] for $\text{Mg}+\text{H}$. The error bars show the “fluctuations,” analogous to the uncertainty, calculated from the variation of alternate calculations as discussed in the text.

Figure 4

Comparison of radial couplings for the LiH(${}^1{\mathrm{\Sigma }}^{+}$) quasimolecule. The nonadiabatic radial coupling matrix elements between the states X, A, C, D ${}^1{\mathrm{\Sigma }}^{+}$ (labelled 1, 2, 3, 4 here) are compared with data from Ref. [6] based on quantum chemistry calculations of Ref. [41], which are improvements on earlier work [42, 43].

Figure 5

The potentials energies for $\text{Ca}+\text{H}$. The top row is ${}^2{\mathrm{\Sigma }}^{+}$, with cores ${\mathrm{Ca}}^{+}\left(4s\right), {\mathrm{Ca}}^{+}\left(3d\right)$, and ${\mathrm{Ca}}^{+}\left(4p\right)$. The bottom row are ${}^2\mathrm{\Pi }$, with cores ${\mathrm{Ca}}^{+}\left(3d\right), {\mathrm{Ca}}^{+}\left(4p\right)$, and ${}^2\mathrm{\Delta }$, with core ${\mathrm{Ca}}^{+}\left(3d\right)$.

Figure 6

The LZ parameters for $\text{Ca}+\text{H}$. The top row is ${}^2{\mathrm{\Sigma }}^{+}$, with cores ${\mathrm{Ca}}^{+}\left(4s\right), {\mathrm{Ca}}^{+}\left(3d\right)$, and ${\mathrm{Ca}}^{+}\left(4p\right)$. The bottom row are ${}^2\mathrm{\Pi }$, with cores ${\mathrm{Ca}}^{+}\left(3d\right), {\mathrm{Ca}}^{+}\left(4p\right)$, and ${}^2\mathrm{\Delta }$, with core ${\mathrm{Ca}}^{+}\left(3d\right)$.

Figure 7

Graphical representation of the rate coefficient matrix $\langle \sigma v\rangle$ (in ${\mathrm{cm}}^3$/s) for inelastic $\text{Ca}+\text{H}$ and ${\text{Ca}}^{+}+{\text{H}}^{-}$ collisions at temperature $T=6000$ K. Results are from the LCAO asymptotic model. The logarithms in the legend are to base 10.

Figure 8

The rate coefficients $\langle \sigma v\rangle$ for $\text{Ca}+\text{H}$ collision processes at 6000 K, plotted against the asymptotic energy difference $\mathrm{\Delta }E$ between initial and final molecular states. The data are shown for endothermic processes, i.e., excitation and ion-pair production. The legend labels the initial state of the transition, and processes leading to a final ionic state (ion-pair production) are circled. The points show the results of the LCAO asymptotic model, with the bars showing the fluctuations.