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Core binding energies of solids with periodic EOM-CCSD

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Periodic EOM-CCSD predicts core binding energies of solids to about 2 eV, the same accuracy it reaches for molecules.

desk verdict First periodic EOM-CCSD core binding energies for solids: honest, useful, but the 2 eV claim rests on a finite-size extrapolation the authors themselves flag as non-asymptotic. read the letter →

arxiv 2508.00168 v1 pith:CPD7MVOA submitted 2025-07-31 physics.chem-ph cond-mat.mtrl-sci

Ethan A. Vo , Timothy C. Berkelbach This is my paper
classification physics.chem-phcond-mat.mtrl-sci
keywords corebindingenergyEOM-CCSDperiodiccoupledclusterX-rayphotoelectronspectroscopycore-valenceseparationthermodynamiclimitextrapolationMP2naturalorbitalscompositecorrections
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether periodic equation-of-motion coupled-cluster theory with single and double excitations can compute core-electron binding energies in solids with the same fidelity it has in molecules. It reports all-electron calculations for six K- and L-edge transitions in silicon, silicon carbide, aluminum phosphide, diamond, and cubic boron nitride, combining a triple-zeta basis with core correlation, k-point sampling up to $4\times4\times4$, composite corrections, and extrapolation to the thermodynamic limit. The final EOM-CCSD numbers are systematically too large by about 2 eV relative to experiment, matching the typical molecular CCSD error, while the cheaper P-EOM-MP2 approximation is too large by about 3 eV. A reader should care because this establishes a wave-function reference for core-level X-ray photoelectron spectroscopy in periodic systems, outside the GW and density-functional approximations usually applied to solids.

What carries the argument

The argument is carried by periodic equation-of-motion coupled-cluster theory with single and double excitations applied to ionization potentials, with the core binding energy formed as the difference between a core IP and the valence-band-maximum IP. Two practical devices make the calculation feasible: the core-valence separation approximation, which restricts the EOM-CCSD diagonalization to core-hole states and introduces errors of 0.1 eV or less, and a truncated MP2 natural-orbital basis, which compresses the virtual space so that 20-40 active orbitals recover the triple-zeta result to within 0.1-1.2 eV. A composite correction combines a full-basis calculation at a coarse k-point mesh with active-space calculations at denser meshes, and the final numbers are extrapolated to the thermodynamic limit assuming finite-size errors decaying as $N_k^{-1/3}$. The central identity is simply the energy-difference definition of a binding energy; the machinery exists to make that difference computable with periodic boundary conditions.

What would settle it

Recompute the six core binding energies at denser k-point meshes, such as $N_k=125$ and $N_k=216$, with the same composite correction, and compare the $N_k^{-1/3}$ extrapolated values with the $N_k=64$ values; if the extrapolated diamond C 1s (286.46 eV) or Si 2p (100.94 eV) shifts by more than roughly 0.5 eV, the extrapolation model is the weak link. A complementary test is to run composite-corrected EOM-CCSDT for one transition and check whether the error drops toward the 0.2 eV level that molecular transferability predicts.

Watch

Extended reading notes

Core claim

The central claim is that periodic EOM-CCSD, with the core binding energy defined as $\mathrm{CBE} = \mathrm{IP}_{\mathrm{core}} - \mathrm{IP}_{\mathrm{VBM}}$ and evaluated as a difference of two ionization potentials, reproduces experimental K- and L-edge core binding energies of simple semiconductors and insulators with a mean absolute error of 2.03 eV and a consistent positive bias. On the six transitions in Table I, Hartree-Fock overestimates by 12.17 eV, $G_0W_0$ on a PBE reference underestimates by 5.95 eV, and the composite-corrected, thermodynamic-limit-extrapolated EOM-CCSD values sit 2.03 eV above experiment. This is the same level of accuracy previously observed for molecular core ionization potentials, and the paper notes that molecular CCSDT reduces those errors to about 0.2 eV, suggesting a similar path for solids. The authors view the result as establishing EOM-CCSD as a viable all-electron method for core binding energies in periodic systems, with accuracy comparable to its molecular counterpart rather than to the best available $G_0W_0$ or ADC variants.

Load-bearing premise

The load-bearing premise is that finite-size errors in the correlated core binding energies follow the $N_k^{-1/3}$ law on the k-point meshes used here, even though the paper notes that periodic CCSD excitation energies on comparable meshes have not settled into that asymptotic regime, so that extrapolated values and the quoted accuracy would shift if subleading finite-size terms dominate.

Editorial extensions

If this is right

  • Periodic EOM-CCSD can act as a benchmark reference for core-level binding energies in simple semiconductors and insulators, with errors comparable to molecular CCSD.
  • Because molecular CCSDT shrinks core-IP errors to roughly 0.2 eV, the same transferability would make composite-correction EOM-CCSDT calculations approach experimental precision for these solid-state core binding energies.
  • The low-scaling P-EOM-MP2 approximation, at about 3 eV error, provides a cheaper route to core binding energies when full EOM-CCSD is too expensive, though with a larger systematic overestimate.
  • On this six-transition set, EOM-CCSD is more accurate than G0W0@PBE and less accurate than G0W0@PBE45 or ADC(2)-X, so method choice for solid-state core-level XPS remains a trade-off between cost, systematic bias, and accuracy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the roughly constant positive bias across all six transitions suggests that a system-specific empirical shift could turn EOM-CCSD into a practical predictor of relative chemical shifts, even before triple excitations are added.
  • Editorial inference: the composite-correction strategy should transfer to core-level spectral intensities and satellite features, where EOM-CCSD's known overestimate of double-excitation character could be tested against experimental XPS line shapes.
  • Editorial inference: the validity of the $N_k^{-1/3}$ extrapolation for the CBE difference is directly checkable by repeating one material at a much denser mesh, for example $N_k=125$, and comparing the extrapolated value with the raw $N_k=64$ result.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper reports periodic equation-of-motion coupled-cluster with singles and doubles (EOM-CCSD) calculations of core binding energies (CBEs) for six K- and L-edge transitions in Si, SiC, AlP, diamond, and cubic BN. Using all-electron cc-pCVTZ calculations with k-point sampling up to 4x4x4, natural orbital truncation, and a composite correction for basis-set incompleteness, the authors extrapolate to the thermodynamic limit assuming Nk^{-1/3} finite-size scaling. The final EOM-CCSD values have a mean absolute error of 2.03 eV relative to experiment, systematically overestimating the CBEs; a low-scaling P-EOM-MP2 approximation gives a 3.02 eV MAE. The paper compares these results to G0W0 and ADC(2) calculations from the literature.

Significance. If the extrapolated values are reliable, this is the first demonstration that periodic EOM-CCSD can predict core binding energies of solids with an accuracy similar to that of molecular CCSD (~2 eV), establishing a new reference for core-level XPS simulations. The work is also useful for benchmarking cheaper methods, and the explicit comparison with GW and ADC puts the EOM-CCSD results in context. A notable strength is that the active-space sizes and the extrapolation exponent are not fitted to the experimental target data, so the benchmark is not circular. However, the significance is conditional on the validity of the finite-size extrapolation, which is the main weakness.

major comments (2)
  1. [Section II, Fig. 4 and the paragraph following Fig. 3] The thermodynamic limit values are obtained by fitting a pure Nk^{-1/3} law to only three k-point meshes (Nk = 8, 27, 64). The authors themselves cite Refs. 20 and 21, which show that CCSD valence IPs at accessible meshes are not in the asymptotic regime and that subleading corrections are expected. No evidence is provided that core IPs or the CBE difference converge faster than the valence IPs studied there. The stated 0.5 eV finite-size error bar is the difference between the Nk = 64 value and the intercept of the same assumed Nk^{-1/3} line, so it cannot detect curvature or an incorrect functional form. If the true scaling differs, the extrapolated CBEs and hence the 2.03 eV MAE could shift by more than 0.5 eV. I recommend testing the sensitivity of the extrapolation to the assumed functional form (e.g., Nk^{-1} or Nk^{-1/3} with a subleading term) or adding at least one denser mesh (Nk = 125) for one or two representative materials.
  2. [Section II, Eq. (2)] The composite correction transfers the basis-set correction from the coarse mesh (Nk = 8) to the finer meshes, assuming the virtual-orbital truncation error is independent of k-point sampling. The paper states that the convergence behavior with the number of virtual orbitals is "similar" for denser meshes, but no quantitative comparison is shown. If the basis-set error varies with Nk, the composite-corrected data fed into the extrapolation are systematically biased. The authors should demonstrate the Nk-independence of the basis-set correction explicitly (e.g., by a full-TZ calculation at Nk = 27 for at least one system) or assign an uncertainty to this approximation.
minor comments (5)
  1. [Section I and II] There are several typographical errors: "dependendent" should be "dependent" in the Introduction; "Speci cially" should be "Specifically", "as can been" should be "as can be seen", and "modifed" should be "modified" in Section II; and the Fig. 1 caption says "four four of the six transitions" instead of "four of the six transitions."
  2. [Section II, notation] The notation "Nk = 23", "33", and "43" is ambiguous in the text; these should be written as 2^3, 3^3, 4^3 (or 8, 27, 64) to avoid confusion with decimal numbers.
  3. [Section II, Eq. (2)] The subscripts 1 and 2 and the labels L and S in Eq. (2) are not explicitly defined; please clarify that 1 and 2 denote different k-point meshes and L and S denote large and small virtual orbital spaces.
  4. [Section II, CVS approximation] The statement that the CVS approximation introduces errors of 0.1 eV or less would benefit from a supporting figure or table, or a sentence describing how this was tested.
  5. [Section II, Table I] Table I reports point estimates without uncertainties; adding the estimated finite-size and basis-set uncertainties (e.g., the stated 0.5 eV) would make the "about 2 eV" claim more quantitative.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: EOM-CCSD CBEs are benchmarked against independent experimental values with no fitted parameters.

full rationale

The paper's derivation chain is: compute core and VBM ionization potentials with periodic EOM-CCSD, form CBE = IP_core - IP_VBM, apply the composite basis correction of Eq. (2), extrapolate to the thermodynamic limit assuming an Nk^{-1/3} finite-size decay, and compare the resulting values to experimental CBEs in Table I. No parameter is fitted to the experimental CBEs: the active-space sizes (20, 40, full TZ) are determined by convergence tests against the full triple-zeta basis (Fig. 2), and the extrapolation exponent is taken from independent finite-size studies (Refs. 20-21), stated explicitly as an assumption with a conservative error bar. The experimental values are external, and the paper's central claim is an error benchmark, not a derivation of the experiments. Self-citations (Refs. 17, 19, 32) provide method provenance and prior valence-excitation benchmarks, but they are not load-bearing for the 2 eV claim, which is assessed against independent experimental data and against independent GW and ADC calculations. The acknowledged limitations (finite-size extrapolation uncertainty, CVS error, relativistic corrections, possible subleading finite-size terms) are correctness risks rather than circularity, because they do not make the prediction equivalent to its inputs by construction. Therefore no specific circular reduction can be exhibited, and the paper is self-contained against external benchmarks.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No entities are invented; the central claim rests on the formalisms from prior work and on three methodological assumptions (composite additivity, finite-size scaling, VBM at Gamma), plus hand-chosen active space sizes. No parameter is fitted to the experimental CBE values, so circularity burden is low.

free parameters (2)
  • Active space sizes for composite corrections = 20 and 40 active orbitals (occupied + NO virtuals)
    Chosen by hand from convergence tests (Fig. 2) to balance cost and accuracy; the final composite-corrected values depend on the additivity of corrections across these truncations, though the truncations are not fitted to experimental CBEs.
  • Finite-size extrapolation exponent = -1/3
    Assumed for EOM-CCSD and P-EOM-MP2 CBE extrapolations based on literature on valence excitation energies; changing this exponent would shift the thermodynamic-limit estimates by amounts comparable to or larger than the stated 0.5 eV uncertainty.
assumptions (4)
  • domain assumption Periodic EOM-CCSD with the core-valence separation (CVS) approximation yields core IPs as interior eigenvalues of the similarity-transformed Hamiltonian with errors at the 0.1 eV level in these solids.
    The paper relies on the periodic EOM-CCSD formalism from Refs. 17 and 19 and asserts CVS errors of 0.1 eV or less based on unspecified testing.
  • ad hoc to paper The composite correction E(Nk,2,L) ≈ E(Nk,2,S) + [E(Nk,1,L) - E(Nk,1,S)] (Eq. 2) is additive, i.e., basis-set and k-point-sampling errors separate cleanly.
    The paper justifies this by similar convergence behavior across k-point meshes, but it is an unproved separability assumption that supports the composite-corrected final numbers.
  • domain assumption Finite-size errors in EOM-CCSD CBEs decay as Nk^{-1/3}, permitting linear extrapolation of the data to the thermodynamic limit.
    The paper explicitly uses this scaling for final predictions and notes that accessible meshes are not in the asymptotic regime (Refs. 20,21), making the extrapolation the main source of uncertainty.
  • domain assumption The valence band maximum occurs at the Gamma point in all five materials, so evaluating IP_VBM at Gamma is exact.
    Stated in Section II without derivation; standard for these materials but load-bearing for the CBE definition.

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Cite this review

Pith. "Pith review of Core binding energies of solids with periodic EOM-CCSD." pith.science (2026). https://pith.science/paper/CPD7MVOA

@misc{pith2026250800168,
  author       = {Pith},
  title        = {Pith review of: Core binding energies of solids with periodic EOM-CCSD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CPD7MVOA}},
  note         = {Machine review of arXiv:2508.00168}
}
read the original abstract

We report the core binding energies of K-edge and L-edge transitions in simple semiconducting and insulating solids using periodic equation-of-motion coupled-cluster theory with single and double excitations (EOM-CCSD). In our all-electron calculations, we use triple zeta basis sets with core correlation, and we sample the Brillouin zone using up to 4x4x4 k-points. Our final numbers, which are obtained through composite corrections and extrapolation to the thermodynamic limit, exhibit errors of about 2 eV when compared to experimental values. This level of accuracy from CCSD is about the same as it is for molecules. A low-scaling approximation to EOM-CCSD performs marginally worse at lower cost, with errors of about 3 eV.

Figures

Figures reproduced from arXiv: 2508.00168 by the authors.

Figure 1
Figure 1. FIG. 1. Convergence of the Hartree-Fock core binding energy [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Convergence of the core IP, valence band maximum (VMB) IP, and core binding energy (CBE) with the number of correlated virtual [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. The core binding energy (CBE) as function of [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

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Forward citations

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Reference graph

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