Changes in polarization dictate necessary approximations for modeling electronic deexcitation intensity: Application to x-ray emission

Accurate simulation of electronic excitations and deexcitations are critical for complementing complex spectroscopic experiments and can provide validation to theoretical approaches. Using a generalized framework, we contrast the accuracy and validity of orbital-constrained and linear-response approaches that build upon Kohn-Sham density functional theory (DFT) to simulate emission spectra of electronic origin and propose an efficient approximation, named many-body x-ray emission spectroscopy or MBXES, for simulating such processes. We show analytically as well as with computed examples that for electronic (de)excitation leading to an appreciable change in polarization (i.e., density rearrangement), the adiabatic approximation in a response-based formalism will be inadequate for the calculation of oscillator strength. Thus, such a change (e.g., in the net electrostatic dipole moment of a finite system) can be used as a metric for evaluating the applicability of the adiabatic response-based approach and can be particularly valuable in x-ray emission spectroscopy. On the other hand, MBXES, the flexible method introduced in this paper, can compute oscillator strengths accurately at a much lower computational expense on the basis of two DFT-based self-consistent field calculations. Using illustrative examples of emission spectra, the efficacy of the MBXES method is demonstrated by comparison with its parent theory, orbital-optimized DFT, and with experiments.

Figure 1

X-ray emission spectra for (top) Cl $K\beta$ from chloromethane (${\mathrm{CH}}_3\mathrm{Cl}$) and (bottom) O $K\alpha$ from phenol (${\mathrm{C}}_6{\mathrm{H}}_5\mathrm{OH}$) simulated using adiabatic LR-TDDFT (brown), ooDFT (blue), MBXES (green), and compared with experiment (black), from Refs [32] and [33], respectively. The associated change in dipole moment ($\mathrm{\Delta }D$) is indicated in each case. As detailed in the text, $\mathrm{\Delta }D$ is a diagnostic for errors in LR-TDDFT arising from the adiabatic approximation, which is not required in ooDFT or MBXES (an approximation to ooDFT). A rigid shift is added to each simulated spectrum to align the highest-energy peak with the experiment

Figure 2

Table showing experimental [32, 33, 45, 46, 47, 48] (black) adiabatic LR-TDDFT (brown) and MBXES (green) spectra of different molecules. The second column shows the change in the electrostatic dipole moment, in Debye units, of the remaining electrons following the relevant core ionization.

Figure 3

Emission spectra calculated using the MBXES (green line) and the GS (orange line) approach. Panel (a) shows Cl $K\beta$ emission from ${\mathrm{CH}}_3\mathrm{Cl}$, which is associated with a 1.43 Debye change in valence dipole moment, while panel (b) shows O $K\alpha$ emission from ${\mathrm{C}}_6{\mathrm{H}}_5\mathrm{OH}$, for which the valence dipole moment changes by 3.01 D. Yellow, cyan, red, and green spheres denote C, H, O, and Cl atoms, respectively. Each GS spectrum has been multiplied by a constant factor so that the intensity corresponding to the highest-energy peak matches the MBXES counterpart.

Figure 4

Panels (a), (c), and (e) display the auxiliary orbital, evaluated using, respectively, LR-TDDFT, ooDFT, and MBXES associated with the 11th deexcitation in the Cl $K$ β emission of ${\mathrm{CH}}_3\mathrm{Cl}$. Panels (b), (d), and (f) plot the same for the 23rd deexcitation in the O $K$ α emission of ${\mathrm{C}}_6{\mathrm{H}}_5\mathrm{OH}$. These plots demonstrate that MBXES effectively approximates ooDFT while, for phenol, LR-TDDFT exhibits errors associated with the adiabatic approximation.

Figure 5

Simulated emission spectra, computed with MBXES, ooDFT, and adiabatic LR-TDDFT, corresponding to electronic deexcitation of ${\mathrm{C}}_6{\mathrm{H}}_5\mathrm{OH}$ from an initial state with a hole in the 11th KS valence orbital. The associated change in valence polarization is 0.17 D.