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Efficient implementation of core-excitation Bethe Salpeter equation calculations
K. Gilmore, John Vinson, E. L. Shirley, D. Prendergast, C. D. Pemmaraju, J. J. Kas, F. D. Vila, J. J. Rehr
TL;DR
Core-level spectra require quantitative treatment of localized core holes in extended systems, but prior BSE implementations were computationally limited. This work improves the ocean GW-BSE implementation with efficient basis functions, parallelization, and memory reductions, enabling calculations on much larger systems. The implementation reaches a 500-atom, 3200-valence-electron SrTiO3 supercell in 12.5 hours on 128 cores and supports spectra relevant to dilute and non-periodic systems.
Problem
Quantitative core-level spectroscopy must model localized core holes in extended systems, while previous BSE calculations were limited to relatively small systems.
Method
The ocean implementation combines DFT, GW corrections, PAW transition matrix elements, and a two-particle BSE solver with optimal basis functions, parallelization, and memory reductions.
Results
A 500-atom, 3200-valence-electron SrTiO3 supercell required 12.5 hours on 128 cores, while optimal-basis interpolation provided a 2-2.5 speed-up for large systems.
Takeaways & Limitations
The enhanced implementation makes core-level spectral calculations feasible for dilute, amorphous, and other non-periodic systems, and supports XAS, XES, and inelastic x-ray scattering spectra.
Takeaways & Limitations
BSE remains a two-particle approximation that can describe many-body physics less completely than multiplet and cluster approaches.
Abstract
from arXiv · showhide
We present an efficient implementation of the Bethe-Salpeter equation (BSE) method for obtaining core-level spectra including x-ray absorption (XAS), x-ray emission (XES), and both resonant and non-resonant inelastic x-ray scattering spectra (N/RIXS). Calculations are based on density functional theory (DFT) electronic structures generated either by abinit or Quantumespresso, both plane-wave basis, pseudopotential codes. This electronic structure is improved through the inclusion of a GW self energy. The projector augmented wave technique is used to evaluate transition matrix elements between core-level and band states. Final two-particle scattering states are obtained with the NIST core-level BSE solver (NBSE). We have previously reported this implementation, which we refer to as ocean (Obtaining Core Excitations from Ab initio electronic structure and NBSE) [Phys. Rev. B 83, 115106 (2011)]. Here, we present additional efficiencies that enable us to evaluate spectra for systems ten times larger than previously possible; containing up to a few thousand electrons. These improvements include the implementation of optimal basis functions that reduce the cost of the initial DFT calculations, more complete parallelization of the screening calculation and of the action of the BSE Hamiltonian, and various memory reductions. Scaling is demonstrated on supercells of SrTiO_3 and example spectra for the organic light emitting molecule Tris-(8-hydroxyquinoline)aluminum (Alq_3 ) are presented. The ability to perform large-scale spectral calculations is particularly advantageous for investigating dilute or non-periodic systems such as doped materials, amorphous systems, or complex nano-structures.
1. Introduction
Core-level spectra offer element- and orbital-specific information, but quantitative interpretation requires predictive calculations beyond qualitative reference-spectrum comparisons. The paper motivates BSE while addressing its computational cost and limited system sizes.
- Motivation: Core-level XAS and XES probe unoccupied and occupied densities of states, while XANES reveals oxidation, spin, crystal-field, and bonding information.The extended absorption region can reconstruct local coordination shells.
- Motivation: Reliable spectral interpretation is needed because reference-spectrum comparisons are at best qualitative.The paper argues that quantitative, predictive calculations are preferable.
- Motivation: Accurate core-level excitation modeling must combine a localized core hole with the extended condensed system.This creates a scale-versus-accuracy challenge for computational spectroscopy.
- Prior approaches: Single-particle approaches can treat large systems but may lose accuracy near edges or when angular-momentum and many-electron effects matter.The cited limitations include non-spherical potentials, nonzero core-hole angular momentum, and multi-electron excitations.
- BSE challenge: BSE introduces particle-hole interactions and self-energy corrections, but its computational cost historically limited calculations to a few tens of atoms.The paper identifies ground-state electronic structure, GW corrections, screening, and BSE spectral evaluation as major costs.
- Paper approach: The implementation targets larger calculations through more efficient electronic-structure preparation, parallel screening, and parallel BSE-Hamiltonian evaluation.The reported demonstrations include SrTiO3 supercells and spectra for Alq3.
2. Formalism
The formalism expresses core-level spectroscopy through a many-body loss function and a two-particle BSE Green’s function. The implementation combines GW-BSE ingredients with approximations and targeted improvements to make the calculation more efficient.
- Formalism: The loss function is written as σ(q, ω) = −Im ǫ^-1(q, ω), with the dielectric response depending on photon energy and momentum transfer.This provides the formal spectroscopic observable before the BSE approximation is introduced.
- Formalism: The photon-interaction operator depends on the process, using eiq·r for NRIXS and a polarization-dependent expansion for XAS.The polarization vector is denoted by ˆe.
- Formalism: The excited-state Green’s function is approximated in a two-particle form using the Bethe-Salpeter Hamiltonian.The BSE description includes core-hole and excited-electron degrees of freedom.
- Hamiltonian: The BSE Hamiltonian combines core-hole energies and lifetime effects, a GW-corrected excited-electron Hamiltonian, screened direct attraction, and bare exchange.The core term includes ǫc, spin-orbit interaction χ, and lifetime Γ.
- Implementation: The ocean implementation uses pseudopotentials, a many-pole GW self-energy approximation, and hybrid local-plus-model screening to reduce computational effort.The local response is evaluated near the absorbing site, while long-range screening uses a model dielectric function.
- Computational improvements: Earlier bottlenecks were ground-state DFT and core-hole screening, with BSE-Hamiltonian evaluation also limiting calculations involving many atomic sites.The improvements therefore focus on DFT efficiency and parallelizing screening and Hamiltonian evaluation.
3. Implementation
The implementation reduces the cost of core-level BSE spectra by combining basis reduction, localized representations, iterative Hamiltonian application, and parallelization across spatial points and k-points.
- The code supports DFT wavefunctions from Quantumespresso as an alternative to abinit for plane-wave electronic-structure calculations.
- Optimal basis functions reduce Bloch-function generation through k-space interpolation and basis reduction, while retaining a self-consistent density calculation.
- The BSE Hamiltonian acts on core-hole–conduction-electron vectors, with spectra obtained iteratively without explicitly constructing or storing the full Hamiltonian.
- 3.2.1. Long-range: The long-range direct interaction is reduced to an electron-coordinate-only function by integrating out core-hole dependence and exploiting its angular-momentum structure.
- 3.2.2. Short-range: Short-range interactions use localized angular-momentum-resolved basis functions, isolated-atom radial solutions, precomputed projections, and selection rules to limit multipole terms.
- 3.3. Screening: Screening uses RPA in real space near the core hole, static screening, and a model dielectric function for long-range behavior beyond approximately r = 8 a.u.
4. Results
The ocean improvements preserve spectral fidelity while extending BSE calculations to larger systems through optimal basis functions, parallelization, and memory reductions. Tests on SrTiO3 and Alq3 demonstrate improved scaling, large-supercell spectra, and generally favorable agreement with experiment.
- 4. Results: SrTiO3 benchmarks span 5 to 320 atoms and 32 to 2048 valence electrons across increasingly large supercells.The study uses cubic supercells with 1, 2, 3, and 4 repetitions of the conventional cell in each direction.
- 4.1. Single Processor Calculations: The BSE stage dominates screening at larger sizes, while both stages show quadratic scaling with system size in the reported benchmarks.Including the growing number of sites leads to overall volume-cubed scaling for both stages.
- 4.1. Single Processor Calculations: Projection of plane-wave DFT states onto the radial grid contributes to screening’s quadratic behavior, motivating improved localized-basis projection methods.Both the number of plane waves and bands grow with system size, increasing projection cost.
- 4.2. Reduced Basis: Optimal basis functions reduce large-cell Bloch-function generation to less than half the runtime of the estimated non-OBF approach.The reported savings are strongest for the 3^3 and 4^3 cells and depend strongly on required k-point sampling.
- 4.3. BSE Scaling: Parallel BSE execution reduces a benchmark from slightly over 8.5 hours on one thread to approximately 3.5 minutes on 192 cores, a 114x speedup.Hybrid MPI/OpenMP threading improves speed over pure MPI, although further work is needed to remove bottlenecks preventing linear scaling.
- 4.5. Example Spectra: The improved code reproduces SrTiO3 spectral fidelity for a 100-formula-unit supercell and gives generally favorable Alq3 XAS agreement with experiment.The Alq3 calculations average spectra over 10 molecular-dynamics configurations and atomic sites; remaining differences may reflect condensed-phase effects and neglected excited-state vibronic coupling.
5. Conclusion
The authors’ improvements extend ocean’s core-level BSE spectra calculations from systems of a few tens of atoms to a 500-atom SrTiO3 supercell, while achieving substantial computational speedups. The method supports large-scale spectra for dilute and amorphous systems, but the two-particle BSE approximation remains limited relative to more complete many-particle approaches.
- Scalability: 500 atoms and 3200 valence electrons were treated in a 5x5x4 SrTiO3 supercell using 12.5 hours on 128 cores.The calculation would permit direct simulation of doping at the 1 % level.
- Efficiency improvements: Optimal-basis interpolation accelerated large-system NSCF DFT calculations by 2-2.5x.Parallelized screening and BSE-Hamiltonian evaluation provided additional savings.
- Parallel scaling: Hybrid MPI/OpenMP execution achieved a 114x speedup on 192 processors, while screening scaled only to a few dozen processors.The screening limitation was considered minor because that stage remained relatively inexpensive compared with initial DFT.
- Applications: The enhanced capability makes spectral calculations on amorphous and dilute systems feasible.The authors also identify applications including in-operando studies of fuel cells, photocatalysts, sensors, energy-storage materials, and liquids.
- Scope and limitations: BSE remains a predictive first-principles approach whose two-particle formulation can be a crude approximation to the full many-body problem.The authors note that multiplet and cluster methods capture many-body physics more completely, while future BSE improvements should incorporate additional many-particle effects.