Initial-state and final-state vibrational effects in the treatment of molecular photoionization dynamics

The influence upon molecular photoionization dynamics of vibrational motion of the nuclei in both the initial (neutral) state and the final (molecular ion) state is examined using a consistent theoretical treatment applied to the wide range of existing experimental data for the benchmark CO C $1s^{-1}$ $K$-shell ionization. This allows comparisons to be made against cross sections, lab-frame $\beta$-parameter measurements, and molecule-frame photoelectron angular distributions that have all been recorded both with, and without, ion vibrational-state resolution. A relatively simple multiple-scattering treatment works well in all these applications, its performance comparing very favorably with alternative relaxed core Hartree-Fock methods. The calculations are extended to examine possible effects of vibrational excitation in the neutral, and show marked effects that extend to energies lying away from the obvious center of the CO σ${}^{*}$ shape resonance.

Figure 1

C $1s^{-1}$ photoionization cross-section (top) and $\beta$-parameter (bottom) curves calculated for fixed internuclear separations, $r_{\mathrm{CO}}$, that span the $v$${}^{″}=0$ motion of CO. The curves at the equilibrium separation, $r_{\mathrm{e}}$$=$1.128 Å, are distinguished by shading.

Figure 2

Cross-section (top) and $\beta$-parameter (bottom) curves for C $1s^{-1}$ ionization to vibrationally unresolved CO${}^{+}$. The CMS-X$\alpha$ $r_{\mathrm{CO}}$-averaged results (solid lines) are contrasted with the fixed-nuclei results for $r_{\mathrm{e}}$$=$1.128 Å (broken lines). Experimental data taken from Ref [44] (open triangles); Ref. [20] (filled circles); Ref. [45] (filled inverted triangles).

Figure 3

Fixed-molecule PADs for CO C $1s^{-1}$ photoionization to nonresolved ion vibrational states. The CO molecule is oriented as shown with the oxygen atom towards the right (at 0°). Left-circularly polarized light, at the photon energies indicated, propagates into the plane of the page. Theoretical results: CMS-X$\alpha$ calculations (solid blue line), this work; fixed-nuclei ($r_{\mathrm{e}}$) RCHF(TS) calculations, Ref. [47] (red dotted line); RCHF calculations, Ref. [17] (orange dot-dashed line). Experimental data: from Ref. [46] (circles); from Ref. [17] (diamonds). For the shape resonance at 306.1 eV the PAD also includes a single, fixed-nuclei CMS-X$\alpha$ calculation at $r_{\mathrm{e}}$ (green dashed line). Each distribution is scaled for plotting to minimize its χ${}^2$ residual with the experimental data.

Figure 4

CDAD asymmetry factors, ${\mathrm{A}}_{\mathrm{CD}}$. The angle θ is measured from the C→O direction. Theoretical results: vibrationally averaged CMS-X$\alpha$ (blue solid line); fixed-nuclei ($r_{\mathrm{e}}$) CMS-X$\alpha$ (green dashed line, lower left panel); fixed-nuclei ($r_{\mathrm{e}}$) RCHF(TS) calculations, Ref. [47] (red dotted line); RCHF calculations, Ref. [17] (purple dot-dashed line). Experimental data: from Ref. [46] (circles); from Ref. [17] (diamonds).

Figure 5

CMS-X$\alpha$ calculated cross sections for C $1s^{-1}$ ionization to vibrationally resolved CO${}^{+}$. A $v$${}^{\prime }=0$ anharmonic vibrational wave function (Ref. [43]) was used for the neutral state in all these calculations. For the CO${}^{+}$ ion, anharmonic vibrational wave functions, χ${}^{+}$, were evaluated using parameters ($r_e,{\omega }_e,{\omega }_e{x}_e)$ from either Ref. [49] (curves labeled Anh1) or from Ref. [51] (curves labeled Anh2); harmonic oscillator ion wave functions ($r_e,{\omega }_e$from Ref. [48]) were employed for the curves labeled Hrmn. Experimental data (circles) taken from Ref. [20]. Note the change in vertical scale between top and bottom row.

Figure 6

CMS-X$\alpha$ calculated $\beta$-parameter curves for C $1s^{-1}$ ionization to vibrationally resolved CO${}^{+}$. Details as for Fig. 5.

Figure 7

Molecule-frame PADs for resolved ion vibrational states. The linear polarization and the CO molecule are both aligned along the horizontal axis (with the O atom to the right, pointing at 0°): CMS-X$\alpha$ calculations, this work (blue solid line); RCHF calculations, Ref. [16] (red dotted line). Experimental data also from Ref. [16]. Theoretical distributions have been scaled to minimize their χ${}^2$ residual with the corresponding experimental data before plotting.

Figure 8

C $1s^{-1}$ MF-PADs for left-circularly polarized radiation (at indicated photon energies, below the shape resonance) propagating into the plane of the page. The CO molecule is aligned horizontally, O atom to the right. Each row shows data for a resolved ion vibrational state. CMS-X$\alpha$ calculations (blue solid line); experimental data (circles), and RCHF calculations (red dotted line), both from Ref. [17]. Theoretical distributions have been scaled to minimize their χ${}^2$ residual with the corresponding experimental data before plotting.

Figure 9

CDAD asymmetry, ${\mathrm{A}}_{\mathrm{CD}}$, at $h$ν$=$298.3 eV for resolved $v_{0,1,2}^{+}$ vibrational levels of CO${}^{+}$. CMS-X$\alpha$ calculations before (gray dashed line) and after (blue solid line) allowance for finite angular acceptance in the experiment. Experimental data (diamonds) and RCHF calculations (orange dot-dashed line) are from Ref. [17].

Figure 10

CMS-X$\alpha$ calculated MF-PADs for CO C $1s^{-1}$ ionization by linearly polarized light aligned parallel to the molecular axis (shown) and from neutral states having ${\nu }^{″}=2$ (blue solid line); ${\nu }^{″}=1$ (green dashed line); ${\nu }^{″}=0$ (red dotted line). Calculations are at the indicated electron kinetic energies and for ionization to the ${\nu }^{+}=0$,1,2 states in the top, middle, and bottom rows, respectively. For reference the experimental data for ${\nu }^{″}=0$ ionization (Ref [16]) are also included (circles). All PADs are normalized to the peak in the ${\nu }^{″}=2$ calculation to emphasize comparison of their shapes.

Figure 11

CMS-X$\alpha$ MF-PADs for ionization by left-circularly polarized light propagating into the plane of the page and from neutral CO having ${\nu }^{″}=2$ (blue solid line); ${\nu }^{″}=1$ (green dashed line); ${\nu }^{″}=0$ (red dotted line). Experimental data for the ${\nu }^{″}=0$ ionization (Ref. [17]) are included as a reference. The top row shows data for nonresolved ion vibrational levels, the rows beneath are for the indicated ion vibrational level, and are at the indicated electron kinetic energies. All distributions are normalized to the peak in the ${\nu }^{″}=2$ calculation.