Predicting photoemission intensities and angular distributions with real-time density-functional theory

Photoemission spectroscopy is one of the most frequently used tools for characterizing the electronic structure of condensed matter systems. We discuss a scheme for simulating photoemission from finite systems based on time-dependent density-functional theory. It allows for the first-principles calculation of relative electron binding energies, ionization cross sections, and anisotropy parameters. We extract these photoemission spectroscopy observables from Kohn-Sham orbitals propagated in real time. We demonstrate that the approach is capable of estimating photoemission intensities, i.e., peak heights. It can also reliably predict the angular distribution of photoelectrons. For the example of benzene we contrast calculated angular distribution anisotropy parameters to experimental reference data. Self-interaction free Kohn-Sham theory yields meaningful outer valence single-particle states in the right energetic order. We discuss how to properly choose the complex absorbing potential that is used in the simulations.

Figure 1

Blue arrows indicate the seven different directions of the light polarization that were chosen to simulate the angularly averaged photoemission spectrum of the benzene molecule. In the chosen coordinate system shown in the figure, the polarization directions indicated by the blue arrows correspond to the vectors $(1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,1,1), (0,1,1)$, and $(1,0,1)$.

Figure 2

Photoemission spectrum of benzene. Black dashed line: Experimental spectrum from Ref. [52]. Orange full line: Photoemission spectrum as predicted by TD-SIC. Dotted blue line: Ground-state SIC eigenvalues folded by Gaussian functions with a width of 0.4 eV. Blue bars indicate the positions of the SIC eigenvalues. The peak position and height at the smallest binding energy were scaled to match the corresponding experimental peak for all theoretical spectra.

Figure 3

Photoemission spectrum of pyridine. Black dashed line: Experimental spectrum from Ref. [52]. Orange full line: Photoemission spectrum as predicted by TD-SIC. Dotted blue line: Ground-state SIC eigenvalues folded by Gaussian functions with a width of 0.4 eV. Blue bars indicate the positions of the SIC eigenvalues. The peak position and height at the smallest binding energy were scaled to match the corresponding experimental peak for all theoretical spectra.

Figure 4

Left: Angle-resolved photoemission intensity from the benzene HOMO as predicted by TD-SIC. The direction of the light polarization is indicated by the orange arrows. Photoemission parallel to the light polarization corresponds to $\theta =0^{\circ }$ and ${180}^{\circ }$; $\theta ={90}^{\circ }$ corresponds to perpendicular emission. The calculated intensity (black crosses) was fitted with the help of Eq. (4), resulting in $\beta =0.75$ (blue line). Emission occurs predominantly parallel to the light polarization. Right: Angle-resolved photoemission intensity from the benzene HOMO-7 (green-white orbital in Fig. 5). Here the emission is predominantly perpendicular to the light polarization, with $\beta =-0.20$.

Figure 5

Lower panel: Experimentally recorded photoemission spectrum for benzene from Ref. [52] (black) and photoemission spectrum from TD-SIC (orange). In difference to Fig. 2, we here stretched the theoretical spectrum by a factor of $s=1.11$ to ease comparison to experiment. Top panel: Anisotropy parameter $\beta$ as a function of energy. Triangles depict the calculated values obtained for the emission from individual orbitals, i.e., corresponding to the peaks shown right below in the lower panel. Different experimental $\beta$ values are designated by small symbols as designated in the plot, with the data taken from Liu et al. [52], Carlson et al. [53], Mattsson et al. [65], and Sell and Kuppermann [66].

Figure 6

PES spectra obtained using absorbing potentials with different effective length. We show the kinetic energy of photoelectrons from benzene for one relative orientation of light polarization and molecule.

Figure 7

Anisotropy parameter for three different choices of the detector radius: $R_{\mathrm{D}}=14.5a_0$ (red triangles), $16.0a_0$ (blue stars), and $17.5a_0$ (orange triangles). Positions of the TD-SIC $\beta$ values are aligned and scaled by $s=1.11$. The experimental data is taken from Liu et al. [52], Carlson et al. [53], Mattsson et al. [65], and Sell and Kuppermann [66].

Figure 8

Photoemission spectrum of benzene, predicted by TD-SIC, for three different choices of the detector radii: $R_{\mathrm{D}}=17.0a_0$ (orange), $16.0a_0$ (blue), and $14.5a_0$ (red). Experimental spectrum taken from Ref. [52].

Figure 9

Photoemission spectrum of pyridine, predicted by TD-SIC, for three different choices of the detector radii: $R_{\mathrm{D}}=17.0a_0$ (orange), $16.0a_0$ (blue), and $14.5a_0$ (red). Experimental spectrum taken from Ref. [52].

Figure 10

Orange: Photoelectron spectrum of benzene as predicted by TD-LDA. Black: Experimental photoemission spectrum from Ref. [52]. The position and height of the TD-LDA peak corresponding to the smallest binding energy was aligned to its experimental counterpart. The agreement between TD-LDA and experiment is not good.

Figure 11

Directions of the light polarization in the first octant with respect to benzene that is lying in the $x-y$ plane. The red pentagons correspond to the directions chosen for calculating the PES spectra of benzene. For calculating $\beta$ we additionally ran the TD-DFT simulations with the orientations corresponding to the blue dots.