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Density functional theory for core-level X-ray absorption

Seokkyu An, Taisuke Ozaki

TL;DR

This work develops a formally rigorous DFT framework for core-level X-ray absorption spectroscopy by anchoring core-excited states to Gunnarsson–Lundqvist constrained-search principles and implementing an explicit-core ΔSCF with a penalty projector to achieve shift-free absolute edge alignment. It introduces a selection-rule–aware basis (SRB) that reformulates the dipole matrix element, reducing the computational scaling from $O(N^4)$ to $O(N^3)$ while preserving accuracy and enabling efficient, site-resolved XAS in large systems. The combination of GL-based variational theory, the core-occupancy penalty, and the SRB-derived dipole formulation yields robust predictions of line shapes, polarization anisotropies, and absolute onsets for C, B, O, and Li K-edges in both molecules and solids, without reliance on empirical energy shifts. The approach significantly improves scalability and reliability of first-principles XAS simulations in DFT, with potential extensions to L/M edges, non-collinear magnetism, and multiple core excitations, thereby broadening the applicability of XAS modeling to complex materials and large supercells.

Abstract

We establish a rigorous density functional theory (DFT) framework for core-level X-ray absorption spectroscopy (XAS) by formulating a constrained search for core-excited states based on the Gunnarsson-Lundqvist theorem. Within this framework, the explicit-core Delta SCF scheme enables shift-free absolute edge alignment and a consistent treatment of L/M edges with spin-orbit-resolved projectors. In addition, by exploiting dipole selection rules, we recast the evaluation of the dipole matrix elements, which otherwise requires many independent Slater determinant calculations, into a compact single determinant form. This reduces the computational scaling from $\mathcal{O}(N^4)$ to $\mathcal{O}(N^3)$, where $N$ is the number of electrons, without introducing additional approximations. Across representative C, B, O, and Li K-edge benchmarks in molecules and solids, the method reproduces line shapes, polarization anisotropies, and absolute onsets without empirical shifts, providing a robust and scalable route to quantitatively reliable XAS simulations within DFT.

Density functional theory for core-level X-ray absorption

TL;DR

This work develops a formally rigorous DFT framework for core-level X-ray absorption spectroscopy by anchoring core-excited states to Gunnarsson–Lundqvist constrained-search principles and implementing an explicit-core ΔSCF with a penalty projector to achieve shift-free absolute edge alignment. It introduces a selection-rule–aware basis (SRB) that reformulates the dipole matrix element, reducing the computational scaling from to while preserving accuracy and enabling efficient, site-resolved XAS in large systems. The combination of GL-based variational theory, the core-occupancy penalty, and the SRB-derived dipole formulation yields robust predictions of line shapes, polarization anisotropies, and absolute onsets for C, B, O, and Li K-edges in both molecules and solids, without reliance on empirical energy shifts. The approach significantly improves scalability and reliability of first-principles XAS simulations in DFT, with potential extensions to L/M edges, non-collinear magnetism, and multiple core excitations, thereby broadening the applicability of XAS modeling to complex materials and large supercells.

Abstract

We establish a rigorous density functional theory (DFT) framework for core-level X-ray absorption spectroscopy (XAS) by formulating a constrained search for core-excited states based on the Gunnarsson-Lundqvist theorem. Within this framework, the explicit-core Delta SCF scheme enables shift-free absolute edge alignment and a consistent treatment of L/M edges with spin-orbit-resolved projectors. In addition, by exploiting dipole selection rules, we recast the evaluation of the dipole matrix elements, which otherwise requires many independent Slater determinant calculations, into a compact single determinant form. This reduces the computational scaling from to , where is the number of electrons, without introducing additional approximations. Across representative C, B, O, and Li K-edge benchmarks in molecules and solids, the method reproduces line shapes, polarization anisotropies, and absolute onsets without empirical shifts, providing a robust and scalable route to quantitatively reliable XAS simulations within DFT.

Paper Structure

This paper contains 14 sections, 3 theorems, 50 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. $\Delta$SCF with an explicit core projector
  4. Reformulation of the matrix element
  5. Multiple excited state
  6. Implementation
  7. General
  8. KS equation with LCPAO
  9. Dipole matrix element
  10. Results
  11. Method verification
  12. Case study
  13. Conclusion
  14. ACKNOWLEDGMENTS

Key Result

Theorem 1

Let $\hat{H}$ be the many-body Hamiltonian and $\hat{{A}}$ an operator such that $[\hat{H},\hat{{A}}]=0$. For a fixed eigenvalue $\lambda$ of $\hat{{A}}$, let $\mathbb{H}_\lambda:=\{\,|\Psi\rangle:\hat{{A}}|\Psi\rangle=\lambda|\Psi\rangle\,\}$ denote the corresponding subspace and let $E^{(\lambda)}

Figures (7)

  • Figure 1: Geometrical interpretation of the matrix element between Slater determinants ($N=2$). The horizontal plane (yellow) stands for the initial state, spanned by $\psi_\mathrm{i0}$ and $\psi_\mathrm{i1}$, while the perpendicular plane (blue) for the final state spanned by $\psi_\mathrm{f0}$ and $\psi_\mathrm{f1}$. (a) shows $\braket{\Phi_\mathrm{f}|\Phi_\mathrm{i}}=\braket{\Phi_\mathrm{f}|\hat{n}_\mathbf{c}|\Phi_\mathrm{i}}=0$, and (b) shows $\braket{\Phi_\mathrm{f},\phi_\mathbf{c}|\Phi_\mathrm{i},\hat{O}\phi_\mathbf{c}}\neq0$ in general.
  • Figure 2: Computational time to evaluate a single dipole matrix element using CHB (Eq. \ref{['eq:CHB']}) and SRB (Eq. \ref{['eq:newformulation']}) formulations (Left: seconds (zoomed). Right: minutes (extended)). Measurements were performed on a single CPU core using the LAPACK LU factorization routine (zgetrf). SRB (green) values are measured wall-clock times from runs on a priori randomly generated $\mathbb{C}^{N\times N}$ matrices, whereas CHB (purple) values are estimated by multiplying single-term cost with $N$ to account for the summation. Black dots indicate exact values of representative materials with SRB (averaged over 10 runs), whereas gray dots indicate estimation with CHB for comparison.
  • Figure 3: (a) Linear acetylene (C2H2) and (b) ethane (C2H6) with the incident polarization indicated (C: brown, H: bright brown). (c) and (d) show the corresponding carbon K-edge XAS; polarization-resolved and averaged spectra are plotted with peak labels P1–-P10. Experimental reference spectra (dashed line) were digitized from published figures (Ref. BesleyNoble2007JPCC) and replotted for comparison. The lower panels display isosurfaces of the final Kohn--Sham states corresponding to peaks P1–P10, with the core hole located on the right C atom; P1--P2, P6--P7, and P9--P10 are degenerate. Yellow and cyan denote opposite phases of the eigenstate.
  • Figure 4: Projected density of states (PDOS) of the B atom which retains the 1$s$ core hole. Panels (a,b) show BN and (c,d) show MgB2; (a,c) are ground state (initial), while (b,d) include a B–1$s$ core hole (final). Each dashed line denotes the Fermi level.
  • Figure 5: (a,b) Crystal structures of (a) c-BN and (b) MgB2 (B: green, N: gray, Mg: orange). (c,d) Isosurface of the density difference between final (core-excited) and initial (ground) state calculations of (c) BN and (d) MgB_2. Isolevel = $+$0.003 (cyan) and $-$0.003 (yellow). (e,f) Spherically averaged density difference from the core hole center. (g,h) B K-edge XAS of (g) c-BN and (h) MgB2. The inset in (g) shows the result from a smaller unit cell ($N=320$), while the main trace corresponds to a large supercell ($N=2560$). Experimental reference spectra (dashed line) were digitized from published figures (Refs. Jayawardane2001PRBZhu2002PRL) and replotted for comparison.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Theorem 1: Gunnarsson--Lundqvist (GL) GL
  • Theorem 2: Penalty functional
  • proof : Proof of Theorem \ref{['theorem2']}
  • Theorem 3: Density functional (Levy's constrained search Levy1982_PRA_ConstrainedSearch)
  • proof : Proof of Theorem 3