Multichannel active-space embedding of atomic multiplets in plane-wave DFT/PAW for core-level spectroscopies

TL;DR

This work presents a multichannel active-space embedding framework that coherently couples localized atomic multiplets with continuum plane-wave photoelectrons in a plane-wave DFT/PAW setting. The core idea is an entangled wavepacket formalism, where a sum over ionic configurations tensored with plane-wave channels is propagated to yield spectra via a Fourier-transformed correlation function, computed with Lanczos or Kernel Polynomial methods. It introduces Pauli projection and a redistribution operator to handle redundancy across equivalent open-shell representations, and defines specialized ionic and PW Hamiltonians to capture both multiplet structure and continuum effects. Demonstrated on Ce $N_{4,5}$ edges, the method reproduces high-$Q$ multiplet peaks and the low-$Q$ GDR continuum, including a characteristic low-energy shoulder, providing a robust, open-source route (xspectruplet) for ab initio core-level spectroscopies of open-shell absorbers.

Abstract

We introduce an active-space embedding framework for core-level spectroscopies that connects localized atomic multiplets to continuum resonances within a plane-wave DFT/PAW description. The approach is complementary to widely used core-level Bethe--Salpeter implementations based on a two-particle (core-exciton) picture with typically static screening: here a correlated multiplet manifold of the absorber (including the core hole and open-shell configurations) is coherently coupled to a plane-wave photoelectron, enabling a unified treatment of localized multiplet structure and continuum lineshapes. Spectra are computed in a general time-domain formulation equivalent to Fermi's golden rule: a transition operator tailored to the specific spectroscopy technique is applied to the correlated ground state to generate an excited wavepacket, and the corresponding wavepacket autocorrelation function is evaluated without explicit real-time propagation, using Lanczos tridiagonalization or the kernel polynomial method; the spectral intensity follows from its Fourier representation. We validate the method at the Ce \(N_{4,5}\) edges, reproducing in quantitative agreement with experiment both high-\(Q\) multiplet features and the low-\(Q\) giant dipole resonance continuum, including a characteristic low-energy shoulder relevant for robust Ce valence assignments. The implementation is available open-source within the Quantum ESPRESSO XSPECTRA package (xspectruplet mode), together with reproducible inputs and scripts.

Figures (2)

• Figure 1: Bird's-eye view of the leading exchange contribution to the dipolar resonance energy at the Ce $N_{4,5}$ edges. (a) Schematic excitation step: a photon of multipolarity $(l_{\mathrm{ph}},m_{\mathrm{ph}})$ promotes a $4d$ electron into the $4f$ manifold, creating a coherent superposition of core--photoelectron pairs. (b) The exchange vertex. (c) The corresponding first-order exchange shift can be written as the trace of the excited-state density matrix times an exchange vertex of rank $K$ (snake line), where solid arrows denote electronic angular-momentum channels. In spherical symmetry (and after averaging over $m_{\mathrm{ph}}$), orthogonality of Wigner $3j$ symbols selects $K=l_{\mathrm{ph}}$, strongly enhancing the dipole ($K{=}1$) contribution.
• Figure 2: XRS at the Ce $N_{4,5}$ edge for CeO$_2$ (left) and $\mathrm{Ce_2(SO_4)_3}$Das2024 (right): experiment (top), xspectruplet (middle), multiplet (bottom). Red: high-$Q$ ($9.5$ Å$^{-1}$); blue: low-$Q$ ($3.5$ Å$^{-1}$). The entangled propagation reproduces localized peaks and continuum GDR structure with correct energy and lineshape, including low-energy shoulders otherwise absent in pure multiplet.