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REVIEW 3 major objections 4 minor 101 references

Accurate and Efficient Prediction of Double Excitation Energies Using the Particle-Particle Random Phase Approximation

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper shows that ppRPA with about 10–20% exact exchange predicts double excitation energies with roughly 0.4 eV mean absolute error, comparable to CCSDT and CASPT2, at much lower cost.

desk verdict A useful, honest benchmark of ppRPA for double excitations, with a real but fixable apples-to-oranges problem in the headline MAEs. read the letter →

arxiv 2411.16599 v1 pith:XCIXTWWK submitted 2024-11-25 physics.chem-ph

Jincheng Yu , Jiachen Li , Tianyu Zhu , Weitao Yang This is my paper · ORCID
classification physics.chem-ph
keywords particle-particlerandomphaseapproximationdoubleexcitationsexcitationenergiesdensityfunctionaltheoryexactexchangepointdefectsdelta-SCFbenchmark
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 argues that double excitation energies, which standard time-dependent DFT misses and high-level wave function methods compute only at steep cost, can be obtained cheaply from the particle-particle random phase approximation (ppRPA). Across a 21-molecule double-excitation benchmark, ppRPA built on Kohn-Sham orbitals from functionals with about 10–20% exact exchange reaches mean absolute errors near 0.35–0.40 eV, an accuracy the paper places alongside CCSDT and CASPT2. It also shows ppRPA describing double-excitation-character states in periodic point defects, and introduces a $\Delta$SCF-based starting point that lets the $(N-2)$-electron reference be an excited determinant when orbital ordering shifts, as in acrolein. If these results hold, ppRPA becomes a practical alternative for studying dark, doubly excited states in molecules and solids.

What carries the argument

The machinery is the pairing-matrix fluctuation propagator $K_{pqrs}(\omega)$, the dynamic fluctuation of $\langle\Psi_0^N|\hat a_p \hat a_q|\Psi_0^N\rangle$, which obeys a Dyson-like equation $K = K^0 + K^0 V K$ and becomes a generalized eigenvalue problem with matrix blocks $A_{ab,cd}=\delta_{ac}\delta_{bd}(\epsilon_a+\epsilon_b)+\langle ab||cd\rangle$ and analogous $B$ and $C$ blocks. The eigenvalues are two-electron addition or removal energies of the $(N\pm2)$-electron system, and the difference between two such eigenvalues is the $N$-electron double excitation energy. This object carries the argument because it treats the two added or removed electrons in a subspace-configuration-interaction manner while the remaining electrons are described by DFT; active-space truncation and density fitting keep the computational cost at $O(N_{\rm act}^4)$.

What would settle it

Recompute the 21-molecule benchmark against full configuration interaction or EOM-CC4 reference energies for a subset of the molecules; if ppRPA's mean absolute error with TPSSh or B3LYP rises above about 0.5 eV while the reported CCSDT and CASPT2 errors stay at their benchmark values, the claim of comparable accuracy would be refuted.

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Extended reading notes

Core claim

The central discovery is that the ppRPA eigenvalue problem, solved on top of an $(N-2)$-electron Kohn-Sham reference, yields vertical double excitation energies of the $N$-electron system whose average error can be as low as 0.350 eV (TPSSh) and around 0.39–0.43 eV for B3LYP and HSE03, close to the reported errors of CCSDT and CASPT2 and better than SA-CASSCF and CC3 on the same benchmark. The accuracy holds for both genuine and partial double excitations, and for the two periodic defects tested, NV$^-$ and VC in diamond, ppRPA reproduces the measured $1A_1$ and $1E$ energies with errors much smaller than TD-DFT. The paper also demonstrates that when removing two electrons changes the orbital order, the $(N-2)$-electron calculation can be seeded from a $\Delta$SCF excited determinant, extending ppRPA to references that are not ground states.

Load-bearing premise

The accuracy claims for molecules rest on the assumption that the reference double-excitation energies from the benchmark of Ref. 13 are the true values; if those references are systematically wrong, the reported mean absolute errors, and the comparison to CCSDT and CASPT2, lose their foundation.

Editorial extensions

If this is right

  • ppRPA with hybrid functionals in the 10–20% exact-exchange range gives double-excitation mean absolute errors of about 0.35–0.40 eV on the 21-molecule benchmark, close to CCSDT and CASPT2 rather than to SA-CASSCF and CC3.
  • The method also produces double-excitation energies for bulk point defects, including the $1A_1$ state of NV$^-$ and the $1E$ state of VC in diamond, with errors around 0.1–0.5 eV, smaller than TD-DFT's roughly 1 eV error.
  • Starting the $(N-2)$-electron SCF from a non-ground-state determinant, justified by $\Delta$SCF, lets ppRPA handle cases where simple removal from the HOMO would misalign orbitals, as in the acrolein $1A'$ state when using B3LYP.
  • Active-space truncation and density fitting keep the cost at $O(N_{\rm act}^4)$ with iterative diagonalization, making the approach feasible for systems where CCSDT and CASPT2 are prohibitively expensive.
  • ppRPA treats genuine and partial double excitations with comparable accuracy, reflecting its two-electron-in-CI treatment of the particle-particle channel.

Reading between the lines

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

  • An implication the paper leaves implicit is that the 10–20% exact-exchange window might transfer to other density-based two-particle methods, so the optimal fraction could be tuned without re-benchmarking 21 molecules; testing ppRPA on a second benchmark set would reveal whether the window is universal.
  • A testable extension is to start from $\Delta$SCF $(N-2)$-electron states beyond the acrolein example, for instance orbitals from a different occupied manifold, which would probe whether the method's accuracy survives when the reference determinant is not the ground state.
  • A caution implied by the benchmark design is that the 21-molecule and two-defect sample is small, so the reported mean absolute error is a small-sample estimate; a broader test set including larger conjugated systems would show whether the 0.35–0.40 eV floor persists.
  • If the accuracy claim holds, ppRPA becomes a practical screening tool for double-excitation spectroscopy in extended $\pi$-systems and defect qubits where CASPT2-scale costs are prohibitive; this is a forward-looking consequence the paper points to but does not itself demonstrate.
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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

3 major / 4 minor

Summary. The manuscript benchmarks the particle-particle random phase approximation (ppRPA) for computing vertical double excitation energies in 21 molecular systems and two point defects in diamond, using eight density functional approximations as starting points. The authors report mean absolute errors (MAEs) of about 0.35–0.40 eV for functionals with 10–20% exact exchange, claim accuracy comparable to high-level wave function methods such as CCSDT and CASPT2, demonstrate a ΔSCF-based ppRPA calculation for the acrolein double excitation, and apply ppRPA to the NV− and VC defects in bulk periodic systems. The central claim is that ppRPA is an accurate and efficient alternative for double excitations in both molecular and periodic systems.

Significance. If the accuracy claims hold, the paper would provide a practical low-scaling method for double excitations that is competitive with much more expensive wave function methods. The work also extends ppRPA to a new starting-point construction (ΔSCF) and to periodic defect systems, which are of genuine interest in materials chemistry. A strength is that the benchmark uses the well-established theoretical best estimates (TBEs) from Ref. 13 and does not fit any parameters to target energies. However, the central accuracy comparison is weakened by missing data and by an incomplete documentation of the wave function reference values, so the paper's quantitative conclusions are not fully supported as written.

major comments (3)
  1. [Table I and Section IV A] The MAEs and MSEs in Table I are computed over different subsets of the 26 molecular states because many (N−2)-electron SCF calculations failed to converge (N/A entries). For example, TPSSh, which gives the lowest MAE (0.350 eV), is missing at least acrolein, benzoquinone, cyclopentadienethione, both cyclopentadienone states, diazete, and both pyrazine states. If these missing states have large errors analogous to those seen for other functionals (e.g., cyclopentadienethione errors near −1 eV), the reported MAE could increase substantially. The authors should report the number of states included for each functional, recompute MAEs on the common subset of states for all functionals, and state explicitly how the comparison with reference wave function methods is affected by the differing subsets.
  2. [Section IV A, comparison with WFT methods] The text states that ppRPA 'provides similar accuracy as CCSDT and CASPT2' and that ppRPA is more accurate than SA-CASSCF and CC3, but the only numerical MAE values given for comparison are 0.48 eV (SA-CASSCF) and 0.56 eV (CC3), quoted globally from Ref. 13. No MAE values for CCSDT or CASPT2 are provided, and the global values from Ref. 13 are not restricted to the states for which ppRPA results are available. The claim of parity with CCSDT/CASPT2 therefore cannot be verified. The authors should tabulate the MAEs of CCSDT, CASPT2, and the other WFT methods on the same subset of states used for the ppRPA MAEs, and, if possible, provide per-state errors to enable a rigorous comparison.
  3. [Section IV B and Table III] The defect excitation energies are obtained using a two-point supercell-size linear extrapolation E(1/N) = E∞ + a/N, but the manuscript reports only the extrapolated values without error bars or the individual supercell data. Since the experimental ranges are narrow (e.g., 1.76–1.85 eV for NV−), the spread of 1.67–2.09 eV across functionals is material to the claim of small starting-point dependence. The authors should provide the raw supercell results, the fitted values, and an estimate of the extrapolation uncertainty, or at least include the two-point data in the Supporting Information.
minor comments (4)
  1. [Section IV A, paragraph on functional trends] The statement that ppRPA with 10–20% exact exchange 'can provide more accurate double excitation energies compared to GGAs and meta-GGAs' is not strongly supported by Table I: the MAE for PBE (0.378 eV) is close to that of B3LYP (0.393 eV) and TPSSh (0.361 eV), and the differences may be within statistical noise given the differing subsets. The authors should soften this statement or support it with a statistical analysis.
  2. [Section IV A and Figure 1] The discussion of the acrolein ΔSCF calculation could be clearer. Figure 1 shows qualitative energy levels, but the text does not explain how the maximum overlap method (MOM) was used to converge the (N−2)-electron state, nor how the resulting ppRPA calculation differs from the standard ground-state-based calculation. A brief description of the MOM procedure and the convergence criteria would help readers reproduce the result.
  3. [Throughout] There are minor typographical errors, including 'results form ppRPA' (should be 'from') in Section IV A, and 'nitrosomethan' in Table I (should be 'nitrosomethane'). The abbreviation 'DFA' is used without definition; it should be defined at first use (e.g., density functional approximation).
  4. [Table III] In Table III, the column header 'dominant configuration in 1A1' applies to the ground state for VC and to the excited state for NV−; this is confusing because for NV− the 1A1 is the excited state of interest, while for VC the 1A1 is the ground state. The authors should clarify the row/state labels to avoid ambiguity.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-citation for the Delta-SCF justification; the central double-excitation benchmark is independent and not circular.

  1. self citation load bearing [Section IV.A (acrolein Delta-SCF discussion); abstract's 'for the first time' claim; supported solely by Ref. 87 (arXiv:2403.04604).]
    "This is as well justified as ppRPA calculations starting from on a DFT ground state, given the recent theoretical work establishing the foundation of the ΔSCF method 87."

    The novel demonstration that ppRPA can start from an excited (N-2)-electron state computed by Delta-SCF is justified by invoking Ref. 87, an arXiv preprint authored by the corresponding author of the present paper. No independent proof, machine-checked verification, or external validation of that foundation is supplied here, so the methodological novelty claim rests on an unverified self-citation. This does not affect the main molecular benchmark, whose MAEs are computed against the independent theoretical best estimates of Ref. 13 and experimental defect values.

full rationale

The core benchmark is not circular: Eq. (3) defines ppRPA excitation energies as eigenvalue differences of a matrix built from DFT orbital energies and two-electron integrals, and no parameter is fitted to the target excitation energies. The reference values are the independent theoretical best estimates from Ref. 13 (Kossoski et al.) and experiment for point defects, so the MAE claims are externally grounded. The self-citations to the ppRPA formalism (Refs. 60-62, 65, 81) supply working equations and active-space methods but do not encode the benchmark answers. The one notable self-referential dependency is the Delta-SCF foundation: the paper justifies the new excited-(N-2)-state starting point solely by Ref. 87, a preprint by the corresponding author and P. W. Ayers, with no proof reproduced and no external verification cited. That is load-bearing only for the 'first time' demonstration, not for the main accuracy comparison. The incomplete Table I subsets due to N/A convergence entries affect the comparability of functional MAEs and the WFT comparison, but this is a data-completeness and statistical-comparison concern, not circularity; no predicted energy is defined in terms of the target energy by construction.

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

The paper introduces no new physical entities. Its free parameters are computational choices (supercell extrapolation slope, active-space sizes) that are not reported in full. The key background assumptions are the validity of ppRPA as a two-electron addition/removal formalism, the accuracy of the external benchmark references, the soundness of Delta-SCF for excited determinants, and the two-point supercell extrapolation model.

free parameters (2)
  • Supercell extrapolation slope a = not reported
    In E(1/Natom) = E_inf + a/Natom, the slope a is fitted to two supercell sizes for each defect state; the paper does not report the supercell sizes or the fitted slope values.
  • Active-space sizes for defect ppRPA = not reported
    The active-space ppRPA for point defects constrains occupied and virtual indices to N_occ,act and N_vir,act, but these values are not specified in Section IV B.
assumptions (4)
  • domain assumption The ppRPA eigenvalue equation (Eq. 3) correctly yields two-electron addition/removal energies for the (N-2)/(N+2)-electron reference.
    Used throughout; derived in prior works (Refs. 60, 61), invoked in Section II. Without this, the excitation energies from the pp/hh channels are not meaningful.
  • domain assumption The theoretical best estimates (TBEs) from Ref. 13 are accurate reference values for double excitation energies.
    The MAE/MSE figures in Table I and the comparison to CCSDT/CASPT2 depend on the correctness of these external references, as discussed in Section IV A.
  • domain assumption The Delta-SCF method, cited to Ref. 87, provides a valid excited (N-2)-electron determinant for use as a ppRPA starting point.
    Used for the acrolein 1A' state in Section IV A; the paper relies on the recently established foundation, co-authored by one of the present authors.
  • domain assumption The linear supercell extrapolation model E = E_inf + a/Natom is accurate for the defect excitation energies.
    Used for NV- and VC in Section IV B; a two-point fit assumes the leading finite-size error is linear in 1/Natom.

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Pith. "Pith review of Accurate and Efficient Prediction of Double Excitation Energies Using the Particle-Particle Random Phase Approximation." pith.science (2026). https://pith.science/paper/XCIXTWWK

@misc{pith2026241116599,
  author       = {Pith},
  title        = {Pith review of: Accurate and Efficient Prediction of Double Excitation Energies Using the Particle-Particle Random Phase Approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XCIXTWWK}},
  note         = {Machine review of arXiv:2411.16599}
}
abstract

Double excitations are crucial to understanding numerous chemical, physical, and biological processes, but accurately predicting them remains a challenge. In this work, we explore the particle-particle random phase approximation (ppRPA) as an efficient and accurate approach for computing double excitation energies. We benchmark ppRPA using various exchange-correlation functionals for 21 molecular systems and two point defect systems. Our results show that ppRPA with functionals containing appropriate amounts of exact exchange provides accuracy comparable to high-level wave function methods such as CCSDT and CASPT2, with significantly reduced computational cost. Furthermore, we demonstrate the use of ppRPA starting from an excited ($N-2$)-electron state calculated by $\Delta$SCF for the first time, as well as its application to double excitations in bulk periodic systems. These findings suggest that ppRPA is a promising tool for the efficient calculation of double and partial double excitation energies in both molecular and bulk systems.

Figures

Figures reproduced from arXiv: 2411.16599 by the authors.

Figure 1
Figure 1. FIG. 1. Valence energy levels in [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Natural transition orbitals of double excitations for the [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Illustration of defect energy levels and ground-state electron configurations of NV [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

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