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Analytic first-order non-adiabatic coupling matrix elements of spin-adapted open-shell time-dependent density functional theory

T0 review · 1 major / 1 minor · reviewed 2026-07-01 · grok-4.3

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Pith's one-line read X-TDDFT reduces NACME errors of U-TDDFT by one-third to two-thirds for open-shell molecules

desk verdict X-TDDFT now has analytic NACMEs that reduce U-TDDFT errors by 1/3-2/3 versus multireference values and shift IC rates and pathways in open-shell cases, but the reference convergence is not demonstrated. read the letter →

arxiv 2605.26594 v2 pith:JAGGSJLI submitted 2026-05-26 physics.chem-ph

Xiaoli Wang , Xingwen Wang , Zikuan Wang , Wenjian Liu This is my paper · ORCID
classification physics.chem-ph
keywords non-adiabaticcouplingmatrixelementsspin-adaptedTDDFTX-TDDFTopen-shellmoleculesinternalconversionratesdensityfunctionaltheoryphotochemistrytransitionmetalcomplexes
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 derives the analytic expressions and implements the first non-adiabatic coupling matrix elements for the spin-adapted open-shell TDDFT method known as X-TDDFT. These NACMEs apply to both ground-to-excited and excited-to-excited transitions while incorporating the effects of the restricted open-shell reference and doubly excited determinants. When benchmarked against multireference results, X-TDDFT lowers the errors seen in U-TDDFT NACMEs by one-third to two-thirds. This accuracy gain produces corrections to internal conversion rates of up to two orders of magnitude and changes the assessed importance of different relaxation pathways in copper(II) porphyrin.

What carries the argument

The analytic NACME formulas for X-TDDFT that incorporate spin adaptation through the restricted open-shell Kohn-Sham reference and the contribution of doubly excited determinants.

What would settle it

An experimental measurement of internal conversion rates and pathway preferences in copper(II) porphyrin that aligns with the X-TDDFT predictions rather than the U-TDDFT ones.

Watch

Extended reading notes

Core claim

X-TDDFT can supply analytic NACMEs for spin-conserving excitations in open-shell molecules. The implementation reuses U-TDDFT code with adjustments for the ROKS reference and implicit double excitations. Benchmarks show that the resulting NACMEs cut the error relative to multireference values by a factor of one-third to two-thirds, which in turn revises internal conversion rates by as much as two orders of magnitude and alters the ranking of excited-state relaxation pathways for copper(II) porphyrin.

Load-bearing premise

The multireference NACME values used as reference are accurate and complete enough to benchmark the two TDDFT variants across the molecules studied.

Editorial extensions

If this is right

  • Internal conversion rates receive corrections of up to two orders of magnitude.
  • The relative importance of excited state relaxation pathways changes qualitatively for copper(II) porphyrin.
  • Substituent effects on the internal conversion rates are revised.
  • X-TDDFT NACMEs are useful for photophysics and photochemistry studies of open-shell systems such as radicals and transition metal complexes.

Reading between the lines

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

  • Unrestricted TDDFT errors in NACMEs are large and unsystematic, pointing to the need for spin adaptation in coupling calculations.
  • The method opens the door to more reliable non-adiabatic dynamics simulations for open-shell transition metal complexes.
  • Similar accuracy gains may apply when computing other derivative properties in spin-adapted TDDFT.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 1 minor

Summary. The manuscript derives and implements analytic first-order NACMEs for ground-to-excited and excited-to-excited transitions within the spin-adapted open-shell TDDFT framework (X-TDDFT). The implementation builds on an existing U-TDDFT NACME code by incorporating the ROKS reference and implicit doubly excited determinants. Benchmark results are presented showing that X-TDDFT NACMEs reduce errors relative to U-TDDFT by factors of 1/3–2/3 when compared against multireference reference values; these improvements are reported to produce corrections to internal conversion rates of up to two orders of magnitude and qualitative changes in pathway ordering and substituent effects for copper(II) porphyrin.

Significance. If the numerical claims hold, the work would be significant as the first reported analytic NACME implementation for any spin-adapted TDDFT method. The modest computational overhead and reuse of U-TDDFT infrastructure are practical strengths. The reported impact on IC rates for open-shell systems would be relevant to photophysics studies of radicals and transition-metal complexes.

major comments (1)
  1. [Benchmark calculations] Benchmark calculations: The central numerical claim—that X-TDDFT reduces U-TDDFT NACME error by 1/3–2/3 versus multireference values, producing order-of-magnitude rate changes and qualitative pathway revisions for Cu(II) porphyrin—rests on the assumption that the multireference NACME references are converged and more accurate than the DFT results. No explicit convergence tests (active-space enlargement, basis-set extrapolation, state-averaging sensitivity, or cross-validation against a second multireference method) are described for the reference values. This is load-bearing for the error-reduction percentages and downstream rate revisions.
minor comments (1)
  1. [Abstract] The abstract states the numerical improvements but does not reference the explicit NACME working equations or the benchmark data tables; readers must reach the main text to locate these.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment on the benchmark section. We address the point below.

read point-by-point responses

  1. Referee: [Benchmark calculations] Benchmark calculations: The central numerical claim—that X-TDDFT reduces U-TDDFT NACME error by 1/3–2/3 versus multireference values, producing order-of-magnitude rate changes and qualitative pathway revisions for Cu(II) porphyrin—rests on the assumption that the multireference NACME references are converged and more accurate than the DFT results. No explicit convergence tests (active-space enlargement, basis-set extrapolation, state-averaging sensitivity, or cross-validation against a second multireference method) are described for the reference values. This is load-bearing for the error-reduction percentages and downstream rate revisions.

    Authors: We agree that explicit convergence tests for the multireference NACME reference values are not described in the manuscript and that this information is important for supporting the reported error reductions. In the revised manuscript we will add a dedicated subsection presenting convergence checks with respect to active-space size, basis-set quality, state averaging, and cross-validation against a second multireference method for the key benchmark systems, including copper(II) porphyrin. These additional data will be used to confirm that the reference values are sufficiently converged for the comparisons presented. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity in NACME derivation chain

full rationale

The paper presents an analytic derivation of ground-to-excited and excited-to-excited NACMEs for X-TDDFT, implemented as an extension of an existing U-TDDFT code that incorporates the ROKS reference and implicit doubly excited determinants. No load-bearing step reduces the NACME expressions to fitted parameters, self-citations, or prior ansatzes by construction; the benchmark comparisons to external multireference values serve as independent validation rather than definitional inputs. The derivation chain is therefore self-contained.

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

Abstract supplies no information on free parameters, background axioms, or new postulated entities.

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Cite this review

Pith. "Pith review of Analytic first-order non-adiabatic coupling matrix elements of spin-adapted open-shell time-dependent density functional theory." pith.science (2026). https://pith.science/paper/JAGGSJLI

@misc{pith2026260526594,
  author       = {Pith},
  title        = {Pith review of: Analytic first-order non-adiabatic coupling matrix elements of spin-adapted open-shell time-dependent density functional theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JAGGSJLI}},
  note         = {Machine review of arXiv:2605.26594}
}
read the original abstract

While spin-adapted time-dependent density functional theory (TDDFT) approaches significantly improve the excitation energies and gradients of open-shell molecules, the effect of spin-adaptation on non-adiabatic coupling matrix elements (NACMEs) remains unknown for spin-conserving excitations. In this article, we report the derivation, implementation and benchmark studies of the ground state-excited state and excited state-excited state NACMEs of our spin-adapted TDDFT method, X-TDDFT; to our best knowledge, this represents the first implementation of the analytic NACMEs of a spin-adapted TDDFT method. Similar to the X-TDDFT analytic gradients, X-TDDFT NACMEs can be easily implemented on top of an existing U-TDDFT NACME implementation taking into account the restricted open-shell Kohn-Sham (ROKS) reference and the implicit involvement of doubly excited determinants, with acceptable computational overhead. Benchmark calculations reveal that X-TDDFT reduces the error of U-TDDFT NACMEs by 1/3-2/3 (referenced against high-level multireference NACMEs), which leads to large corrections of internal conversion rates (up to two orders of magnitude). In particular, for copper(II) porphyrin, X-TDDFT leads to qualitative revisions of the relative importance of the excited state relaxation pathways, as well as the substituent effects of the internal conversion (IC) rates, suggesting that the error of U-TDDFT NACMEs is not only large but also unsystematic. It is therefore expected that X-TDDFT NACMEs will prove useful in the photophysics/photochemistry studies of open-shell systems such as radicals and transition metal complexes.

Figures

Figures reproduced from arXiv: 2605.26594 by the authors.

Figure 1
Figure 1. Heatmaps of the norm errors of the U-TDDFT and X-TDDFT NACMEs of CH [PITH_FULL_IMAGE:figures/full_fig_p036_1.png] view at source ↗
Figure 2
Figure 2. Heatmaps of the angles between the U-TDDFT/X-TDDFT NACMEs computed [PITH_FULL_IMAGE:figures/full_fig_p037_2.png] view at source ↗
Figure 3
Figure 3. Molecular structures of CuP, CuOEP, and CuTPP. [PITH_FULL_IMAGE:figures/full_fig_p041_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: NACME vectors obtained from U-TDDFT and X-TDDFT for the molecules CuP, [PITH_FULL_IMAGE:figures/full_fig_p043_4.png]
Figure 5
Figure 5. Figure 5: 2T1 state configuration coefficients of selected excitations of CuP, CuOEP, and CuTPP at the 2T1 geometries, and the plots of the involved canonical orbitals (PBE0/x2c￾TZVPall, isovalue: 0.03). The dominant excitation (which is always a CV(1) excitation), the accompany…
Figure 6
Figure 6. Figure 6: ROKS frontier molecular orbital energy levels and isosurface plots of CuOEP, CuP [PITH_FULL_IMAGE:figures/full_fig_p046_6.png]
Figure 7
Figure 7. Figure 7: Relative concentrations of different states as a function of time, starting from the [PITH_FULL_IMAGE:figures/full_fig_p050_7.png]

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Forward citations

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