Quantitative reconstruction of atomic orbital densities of neon from partial cross sections

The approach of photoemission orbital tomography, i.e., the orbital density reconstruction from photoemission of planar molecular layers by using a formalism equivalent to a Fourier transformation, is transferred to free atoms. Absolute radial orbital densities of neon $1s$, $2s$, and $2p$ orbitals are reconstructed with a central-field one-electron model, using well-known atomic photoionization data. The model parameters are optimized by a Markov chain Monte Carlo method with Bayesian inference from which uncertainties for the reconstructed orbital densities are derived. The presented model opens the path for photoemission orbital tomography as a powerful tool, as well as for a quantitative analysis.

Figure 1

(a) Coulomb wave functions $C_{l=1}(k=1,Z_C)$ as a function of the nuclear distance $r$ given in atomic units for $Z_C=\mathrm{const}=3$ and 1, as well as for $Z_f=3$ and $r_0=2a_0$, according to Eq. (11). For higher charge values, the wave function is drawn towards the core $(r=0)$ values. For $r_0=2a_0$ the transition from $Z_C=3$ to $Z_C=1$ is visible. (b) Associated charge $Z_C$.

Figure 2

Heatmap of the fit results for the parameters $Z_f$ and $r_0$ of the neon $1s$ orbital with color encoded normalization and size encoded likelihood. The best result is achieved at $(Z_f=7.9,r_0=5a_0)$ and marked by a white cross.

Figure 3

Corner plots for neon (a) $1s$, (b) $2s$, and (c) $2p$ with fit results and uncertainties for the parameters (a) $A_0=40.9\pm 0.6$ and $Z_i=7.4\pm 0.1$, (b) $A_0=4.1\pm 0.5$, $A_1=-23.2\pm 0.5$, and $Z_i=5.94\pm 0.01$, and (c) $A_0=18.3\pm 0.1$ and $Z_i=6.08\pm 0.01$.

Figure 4

Neon $1s$ result for $r_0=5a_0$ and $Z_f=7.9$ with experimental data from Saito and Suzuki [12]. (a) Reconstructed radial orbital density ${\left(rR\right)}^2$. Additionally, the Hartree-Fock solution is shown as a comparison, as calculated by Worsley [57]. (b) Fit result for the PCS, which is in good agreement with the experimental data ($•$). (c) Uncertainty range of the radial orbital density, generated from the distributions in Fig. 3. (d) Uncertainty range from the fit result of the PCS, marked by the dashed line and the deviation from the experimental values ($•$).

Figure 5

Neon $2p$ results for $Z_f=4.7$ and $r_0=2.6a_0$ with experimental data merged from Marr and West [59] and Wuilleumier and Krause [14, 15]. (a) Reconstructed radial orbital density ${\left(rR\right)}^2$. Additionally, the Hartree-Fock solution is shown as a comparison, as calculated by Worsley [57]. (b) Fit result for the PCS, which is in good agreement with the experimental data ($•$). (c) Uncertainty range of the radial orbital density, generated from the distributions in Fig. 3. (d) Uncertainty range from the fit result of the PCS, marked by the dashed line and the deviation from the experimental values ($•$).

Figure 6

Neon $2s$ results for $Z_f=4.3$ and $r_0=4.2a_0$ with experimental data merged from Wuilleumier and Krause [14, 15]. (a) Reconstructed radial orbital density ${\left(rR\right)}^2$. Additionally, the Hartree-Fock solution is shown as a comparison, as calculated by Worsley [57]. (b) Fit result for the PCS, which is in good agreement with the experimental data ($•$). (c) Uncertainty range of the radial orbital density, generated from the distributions in Fig. 3. (d) Uncertainty range from the fit result of the PCS, marked by the dashed line and the deviation from the experimental values ($•$).