Comparative study of inelastic squared form factors of the vibronic states of B1Σu+, C1Πu, and EF1Σg+ for molecular hydrogen: Inelastic x-ray and electron scattering

A joint experimental and theoretical investigation of the valence-shell excitations of hydrogen has been performed by the high-resolution inelastic x-ray scattering and electron scattering as well as the multireference single- and double-excitation configuration-interaction method. Momentum-transfer-dependent inelastic squared form factors for the vibronic series belonging to the $B{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Sigma}}}_{u}^{+},\phantom{\rule{0.28em}{0ex}}C{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Pi}}}_{u}$, and $EF{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Sigma}}}_{g}^{+}$ electronic states of molecular hydrogen have been derived from the inelastic x-ray scattering method at an impact photon energy around 10 keV, and the electron energy-loss spectra measured at an incident electron energy of 1500 eV. It is found that both the present and the previous calculations cannot satisfactorily reproduce the inelastic squared form-factor profiles for the higher vibronic transitions of the $B{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Sigma}}}_{u}^{+}$ state of molecular hydrogen, which may be due to the electronic-vibrational coupling for the higher vibronic states. For the $C{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Pi}}}_{u}$ state and some vibronic excitations of $EF{\phantom{\rule{0.16em}{0ex}}}^{1}{\mathrm{\ensuremath{\Sigma}}}_{g}^{+}$ state, the present experimental results are in good agreement with the present and previous calculations, while the slight differences between the inelastic x-ray scattering and electron energy-loss spectroscopy results in the larger squared momentum-transfer region may be attributed to the increasing role of the higher-order Born terms in the electron-scattering process.