Extending orbital-optimized density functional theory to L-edge XPS and beyond: Spin-orbit coupling via non-orthogonal quasi-degenerate perturbation theory

TL;DR

This paper extends orbital-optimized density functional theory to accurately predict L-edge X-ray photo-electron spectroscopy by incorporating spin-orbit coupling through non-orthogonal quasi-degenerate perturbation theory, achieving near-quantitative agreement with experiments.

Contribution

It introduces a novel method combining scalar-relativistic OO-DFT with NOCI to include spin-orbit coupling effects at L-edges, improving accuracy over previous approaches.

Findings

NO-QDPT/SCAN predicts L-edge CEBEs within 0.2 eV of experimental values for 3rd row atoms.

The method accurately models doublet splitting due to spin-orbit coupling.

Accuracy decreases for 4th-row transition metals, especially with increasing atomic number.

Abstract

Quantum mechanical calculations of core electron binding energies (CEBEs) leading to 2p hole states are relevant to interpreting L-edge x-ray photo-electron spectroscopy (XPS), as well as higher edges. Orbital-optimized density functional theory (OO-DFT) accurately predicts K-edge CEBEs but is challenged by the presence of significant spin-orbit coupling (SOC) at L- and higher edges. To extend OO-DFT to L-edges and higher, our method utilizes scalar-relativistic, spin-restricted OO-DFT to construct a minimal, quasi-degenerate basis of core-hole states corresponding to a chosen inner-shell (e.g. ionizing all six possible 2p spin orbitals). Non-orthogonal configuration interaction (NOCI) is then used to make the matrix elements of the full Hamiltonian including SOC in this quasi-degenerate model space of determinants. Using a screened 1-electron SOC operator parametrized with the Dirac-Coulomb-Breit (DCB) Hamiltonian results in doublet splitting (DS) values for 3rd row atoms that are nearly in quantitative agreement with experiment. The resulting NOCI eigenvalues are shifted by the average of the (scalar) OO-DFT CEBEs to yield CEBEs (split by SOC) corrected for dynamic correlation. Comparing calculations on gas phase molecules with experimental results establishes that NO-QDPT with the SCAN functional (NO-QDPT/SCAN), using the DCB screened 1-electron SOC operator is accurate to about 0.2 eV for L-edge CEBEs of molecules containing 3rd row atoms. However, this NO-QDPT approach becomes less accurate for 4th-row elements starting in the middle of the 3d transition metal series, especially as the atomic number increases.