Electronic and Vibrational Close-Coupling Method for Resonant Electron-Molecule Scattering

We report the development of a vibrational-electronic convergent close-coupling method for electron-molecule scattering with an ab-initio account of the coupling between the electronic and vibrational motions. The technique has been applied to scattering on molecular hydrogen, including coupling between vibrational levels in the first 11 electronic states. Distinct resonances associated with the temporary formation of the ${\mathrm{H}}_2^{-}$ ion are present between 10 and 14 eV for numerous transitions, including vibrational excitation of the $X{{}^1\mathrm{\Sigma }}_g^{+}$ state, dissociation via the $b{{}^3\mathrm{\Sigma }}_u^{+}$ state, and excitation of the $B{{}^1\mathrm{\Sigma }}_u^{+}$ state. With both resonant and nonresonant scattering treated in a single calculation, this method is capable of providing self-consistent sets of cross sections for electron-molecule scattering in regions where the adiabatic-nuclei approximation breaks down.

Figure 1

Electron-impact cross sections for excitation of the $b{{}^3\mathrm{\Sigma }}_u^{+}$ and $B{{}^1\mathrm{\Sigma }}_u^{+}$ states of ${\mathrm{H}}_2$ from the ground state, considering only the ${{}^2\mathrm{\Sigma }}_g$ scattering symmetry. The vertical dashed lines indicate the $C{{}^2\mathrm{\Sigma }}_g^{+}$ vibrational energy levels of the ${\mathrm{H}}_2^{-}$ ion determined experimentally by Comer and Read [23].

Figure 2

Illustration of the resonant and nonresonant processes contributing to the $X{{}^1\mathrm{\Sigma }}_g^{+}\to b{{}^3\mathrm{\Sigma }}_u^{+}$ transition. The ${\mathrm{H}}_2$ potential-energy curves are taken from Refs. [26, 27, 28], and the ${\mathrm{H}}_2^{-}$ curve from Ref. [19]. The vibrational energies of the $C{{}^2\mathrm{\Sigma }}_g^{+}$ state are taken from Ref. [23].

Figure 3

Electron-impact cross sections for excitation of the $b{{}^3\mathrm{\Sigma }}_u^{+}$, $B{{}^1\mathrm{\Sigma }}_u^{+}$, and $X{{}^1\mathrm{\Sigma }}_g^{+}$ states of ${\mathrm{H}}_2$ from the $X{{}^1\mathrm{\Sigma }}_g^{+}(v_i=0)$ state. The results are displayed separately for each of the four dominant scattering symmetries: ${{}^2\mathrm{\Sigma }}_u$, ${{}^2\mathrm{\Sigma }}_g$, ${{}^2\mathrm{\Pi }}_u$, and ${{}^2\mathrm{\Pi }}_g$. The $B{{}^1\mathrm{\Sigma }}_u^{+}$ cross section is summed over the final vibrational levels, while the $X{{}^1\mathrm{\Sigma }}_g^{+}$ cross section is summed over levels $v_f>0$.

Figure 4

Electron-impact cross sections in the ${{}^2\mathrm{\Sigma }}_g$ scattering symmetry for excitation of the $v_f=0\to 14$ vibrational levels from the $v_i=0$ level in the $X{{}^1\mathrm{\Sigma }}_g^{+}$ state of ${\mathrm{H}}_2$. Note the different vertical scale for the $v=0\to 0$ cross section.

Figure 5

Electron-impact cross sections in the ${{}^2\mathrm{\Sigma }}_g$ scattering symmetry for excitation of the $b{{}^3\mathrm{\Sigma }}_u^{+}$ state from the ground state of ${\mathrm{H}}_2$. Comparison is between calculations performed with the vibrational-electronic and fixed-nuclei MCCC methods (VE MCCC and FN MCCC, respectively).