REVIEW 3 major objections 5 minor 16 references
Localised and Delocalised Charge Distribution in a Diamine Cation and Rydberg Excited State: A Challenging Test for Density Functionals
T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The paper establishes that the weight of Fock exchange—or equivalently the scaling of Perdew–Zunger self-interaction correction—controls whether the N,N'-dimethylpiperazine cation and its 3s Rydberg state localize charge, and that a…
desk verdict A thorough Jacob's-ladder benchmark of the DMP cation/Rydberg localization problem with a plausible central claim, but the Rydberg basis-set check is missing and needs to be addressed before publication. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central machinery is the two-dimensional energy surface of the six-membered ring, scanned over the two dihedral angles $d_1$ and $d_2$ that distinguish the delocalized (D) structure from the two equivalent localized (L) structures. The dial that carries the argument is the weight $\alpha$ of Fock exchange in hybrid functionals and the scaling factor of the Perdew–Zunger self-interaction correction; the paper's key identity is the rough correspondence $\alpha \approx 0.50 \leftrightarrow$ full SIC and $\alpha \approx 0.32 \leftrightarrow$ SIC scaled by 0.50. For the Rydberg state, the machinery is a variational orbital-optimization method that converges on saddle points of the electronic energy to obtain the excited state.
What would settle it
A direct test would be to compute the DMP$^+$ cation energy surface with a high-level, fully relaxed wave-function method that is unbiased toward symmetry breaking—for example, CCSDT(Q) with a larger basis—and check whether a localized minimum persists; if it does, the paper's conclusion that the cation has no localized charge state would be overturned. A second test is a systematic basis-set convergence study of the 3s Rydberg state of DMP (e.g., aug-cc-pVTZ and larger diffuse sets) to see whether the PBE0(.32) localized-hole minima survive.
Extended reading notes
Core claim
The paper's central claim is that the delicate balance between localized and delocalized charge in DMP$^+$ and its 3s Rydberg state is governed by how much self-interaction error is removed, and that this can be dialed by the Fock-exchange weight in hybrids or by the scaling of the Perdew–Zunger correction. Concretely, only delocalized charge is found for the cation with PBE0(.25) and PBE0(.32); the latter surface closely matches DM21. Localized minima appear only with a larger weight (0.50, as in BHLYP and PBE0(.50)) or with full SIC on PBE, and these surfaces are similar in shape and barrier height (0.02–0.05 eV). In the 3s Rydberg state, however, PBE0(.32) already produces localized-hole minima, of two distinct structural types, while PBE does not. The authors therefore conclude that a significant reduction of self-interaction error is needed to describe the Rydberg state, and that the Rydberg surface is not simply a copy of the cation surface.
Load-bearing premise
The load-bearing premise is that DM21, evaluated at BHLYP-optimized geometries because it lacks analytical forces, is the correct reference for the localization balance, and that the aug-cc-pVDZ basis set is diffuse enough to describe the 3s Rydberg orbital without artificial confinement.
Editorial extensions
If this is right
- If the DM21 surface is the accuracy benchmark, PBE0 with a Fock-exchange weight near 0.32 offers a practical global-hybrid substitute for cation surfaces in similar systems, without needing a machine-learned functional.
- Since PBE0(.50), BHLYP, and full PBE-SIC all produce localized minima with barriers of only 0.02–0.05 eV, any of these methods would predict a metastable, symmetry-broken cationic state that is not found with DM21 or PBE0(.32).
- The Rydberg state's PBE0(.32) surface shows that a highly diffuse electron can stabilize a localized hole even when the cation alone does not support a localized minimum, so Rydberg surfaces should not be assumed congruent with cation surfaces.
- The correspondence between FE weight and SIC scaling suggests a rationale for choosing hybrid weights: whatever value reproduces the fractional-charge-trained DM21 surface also matches half-scaled PZ-SIC.
- The double-hybrid functionals with high FE weight (B2GPPLYP, DSD-BLYP, DSD-PBEP86) predict localized minima, so users of double hybrids should check for artificial symmetry breaking.
Reading between the lines
- The rough correspondence between $\alpha=0.32$ and SIC scaled by 0.50 might be transferable: one could test whether other molecules with mixed-valence or diamine-like cores also show PBE0(.32) surfaces matching DM21, which would turn a single-molecule observation into a general calibration rule.
- Because the Rydberg electron stabilizes localization only when the cation surface is flat in the localized region, the paper implies a testable prediction: molecules whose cation surfaces are flatter (smaller curvature between D and L) should show stronger Rydberg-induced localization.
- A practical recipe for functional developers is to compare hybrid weights against DM21 on a small set of charge-localization test molecules rather than only against wave-function benchmarks, since DM21's fractional-charge training makes it sensitive to the delocalization error that drives these surfaces.
- The absence of a localized minimum in PBE0(.32) on the cation surface, combined with its presence in the Rydberg state, suggests that experimental probes of the cation (e.g., photoelectron spectroscopy) should see no symmetry-broken structure, while Rydberg spectroscopies should see it—an experimentally testable distinction.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper reports density functional calculations of the N,N'-dimethylpiperazine (DMP) cation and its 3s Rydberg excited state, comparing functionals across Jacob's ladder: LDA, GGA, meta-GGA, global/range-separated hybrids, double hybrids, the machine-learned DM21 family, and PBE with full or half-scaled Perdew-Zunger self-interaction correction. For the cation, only functionals with a large Fock-exchange weight (about 0.5, e.g., BHLYP, PBE0(.50)) and some double hybrids produce metastable localized-charge minima, while PBE0(.32) and DM21 show only a delocalized minimum with energy valleys toward localized structures. For the 3s Rydberg state, PBE0(.32) yields two localized-hole minima in addition to the delocalized minimum, whereas PBE does not. The paper concludes that a significant reduction of self-interaction error is needed to describe the Rydberg state and proposes a rough correspondence between a Fock-exchange weight of 0.50 and full SIC, and between about 0.32 and half-scaled SIC.
Significance. If the conclusions hold, this is a valuable benchmark for self-interaction error in density functionals, with implications for mixed-valence systems and Rydberg excited states. The manuscript has notable strengths: a broad and systematic functional survey, cross-checks between ORCA and GPAW for parts of the cation surfaces, and improved convergence (0.01 eV/A) for the PBE-SIC minimum-energy paths. The use of DM21 as a reference is also informative. However, the central Rydberg conclusion rests on a single basis set without demonstrated convergence for the diffuse 3s orbital, and the DM21-based anchoring of PBE0(.32) uses fixed BHLYP coordinates. These issues make the main qualitative claims plausible but not yet fully supported.
major comments (3)
- [Methods and Fig. 4] The central claim that PBE0(.32) yields localized-hole minima on the 3s Rydberg surface is computed with the aug-cc-pVDZ basis set. The Methods paragraph only states that "some of the calculations are done in both ways" and that only "insignificant difference" was found, without specifying whether this check included the PBE0(.32) Rydberg surface of Fig. 4(b). Given the manuscript's own warning, near Fig. 1 and citing ref. 42, that Rydberg orbitals can be artificially confined when diffuse basis functions are insufficient, a systematic convergence test with larger or more diffuse basis sets, or with a real-space grid representation, is required before this load-bearing conclusion can be accepted.
- [Fig. 1(b,c) and Methods] DM21 energies are evaluated only at BHLYP-optimized atomic coordinates because analytical forces are unavailable for DM21, and the PBE0(.32) surface is tuned to match DM21 on that same fixed coordinate grid. The comparison therefore does not independently validate PBE0(.32), and the recommended Fock-exchange weight of 0.32 is effectively fitted to DM21. A quantitative measure of the match (for example, RMS energy differences over the grid) and a sensitivity test with respect to the assumed geometry would make the choice of 0.32, and the claimed FE-weight/SIC correspondence, robust.
- [Table 1 and Fig. 3] The reported barriers between localized and delocalized minima are extremely small, 0.002-0.046 eV, yet no error estimates or convergence checks with respect to basis set or number of NEB images are provided. Since the paper uses these values to argue that functionals with a Fock-exchange weight of 0.50 and full SIC give "similar" energy surfaces and barriers, the numerical significance of these near-thermal-energy differences should be established.
minor comments (5)
- [Results, page 9] In the text discussing Fig. 2, "PBE(.32)" appears to be a typo for "PBE0(.32)"; please correct it.
- [Table 2] The notation B3LYP(.50) and PBE0(.50) should be defined explicitly, since these are modified functionals and readers need to know which coefficients are changed.
- [Fig. 4(b)] The red '+' marking the ground-state minimum is not visible in the reproduction; please enlarge it or annotate the figure so the marker can be identified.
- [Results, Rydberg section] The text states that PBE gives no localized minimum on the Rydberg surface, but later notes that Kaupp and coworkers found a very shallow PBE minimum in TDDFT; please clarify the difference in methods or conditions to avoid an apparent contradiction.
- [References, Table 2] Reference 64 is cited twice in Table 2 for two different functionals; please check whether the intended references are distinct and correct.
Circularity Check
No significant circularity: the 0.32 Fock-exchange weight is calibrated to the DM21 cation surface, but the headline Rydberg-state result is not part of that fit and retains independent content.
full rationale
The paper makes no claim to derive a functional from first principles; it compares fixed standard functionals plus deliberately modified variants (PBE0(.32), PBE0(.50), PBE-SIC, PBE-SIC/2). The only parameter adjusted to a target surface is the 0.32 Fock-exchange weight, explicitly chosen to reproduce the DM21 DMP+ energy surface (Fig. 1c). The headline Rydberg-state result, namely localized-hole minima on the PBE0(.32) 3s surface, is computed after this calibration and is not itself part of the fitting target, so it is an extrapolation rather than a rearrangement of the input. DM21 is an external machine-learned functional (DeepMind), not a self-citation, and the authors' own PZ-SIC and variational-excited-state references are methodological infrastructure with independent benchmarks. The MRCI+Q comparison is flagged as controversial and is accompanied by the independent Kaupp calculation. The basis-set-convergence and BHLYP-geometry concerns raised in the review are numerical-accuracy risks, not circular reductions: no equation in the paper defines the target result in terms of the fitted parameter. Accordingly no circular step is exhibited.
Assumptions & free parameters
free parameters (1)
- PBE0 Fock-exchange weight =
0.32
assumptions (5)
- domain assumption Kohn-Sham DFT with the tested exchange-correlation functionals gives qualitatively correct adiabatic energy surfaces for DMP+ and the 3s Rydberg state.
- domain assumption Spin purification Es = 2E(up-down) - E(up-up) recovers the singlet Rydberg energy from triplet and mixed-singlet variational DFT calculations.
- domain assumption DM21, trained on fractional charge and spin data, is an accurate reference for the localisation and delocalisation balance.
- ad hoc to paper BHLYP-optimized atomic coordinates are representative for the DM21 and PBE0(.32) surfaces in Figure 1.
- domain assumption The aug-cc-pVDZ basis set is sufficiently diffuse and complete for the 3s Rydberg orbital.
Cite this review
Pith. "Pith review of Localised and Delocalised Charge Distribution in a Diamine Cation and Rydberg Excited State: A Challenging Test for Density Functionals." pith.science (2026). https://pith.science/paper/PKPBOQVC
@misc{pith2026250605077,
author = {Pith},
title = {Pith review of: Localised and Delocalised Charge Distribution in a Diamine Cation and Rydberg Excited State: A Challenging Test for Density Functionals},
year = {2026},
howpublished = {\url{https://pith.science/paper/PKPBOQVC}},
note = {Machine review of arXiv:2506.05077}
}
abstract
The balance between localised and delocalised electron distribution in the N,N'-dimethylpiperazine (DMP) molecule in the 3s Rydberg excited state and in the fully ionised DMP$^+$ provides a valuable test of density functionals, in particular the weight of Fock exchange (FE) in hybrid functionals and the scaling of explicit orbital-based self-interaction correction (SIC) applied to less elaborate functionals. We present results of calculations using density functionals of all rungs of Jacob's ladder, ranging from LDA to the DM21 machine learned local hybrid as well as double hybrids. For DMP$^+$, the commonly used hybrid functionals, such as PBE0(.25) with FE weight of 0.25, as well as PBE0(.32) with a weight of 0.32 only produce the more delocalised charge. The latter mimics the DM21 energy surface well. However, hybrid functionals with stronger FE, such as BHLYP and PBE0(.50) as well as some double hybrid functionals, also produce local energy minima corresponding to localised charge. When full SIC is applied to the PBE functional, an energy surface analogous to hybrid functionals with FE weight of 0.50 is obtained, while the scaling of SIC by 0.50, which has previously been shown to give improved atomisation energy and band gap of solids, only produces the more delocalised charge. For the 3s Rydberg excited state of DMP, two types of configurations with a localised hole are obtained in calculations using the PBE0(.32) functional, in addition to the delocalised hole, but only the latter is found with the PBE functional, showing that a significant reduction of the self-interaction error is needed in order to obtain agreement between density functional calculations and experimental measurements of the Rydberg state
Figures
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Reference graph
Works this paper leans on
-
[1]
(1) Demadis, K. D.; Hartshorn, C. M.; Meyer, T. J. The Localized-to-Delocalized Transition in Mixed-Valence Chemistry. Chem. Rev. 2001, 101, 2655–2686. (2) Hankache, J.; Wenger, O. S. Organic Mixed Valence.Chem. Rev. 2011, 111, 5138–5178. (3) Heckmann, A.; Lambert, C. Organic Mixed-Valence Compounds: A Playground for Electrons and Holes. Angew. Chem. Int....
work page 2001
-
[9]
(80) Henkelman, G.; Uberuaga, B. P.; J´ onsson, H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys. 2000, 113, 9901–9904. 31 (81) Henkelman, G.; J´ onsson, H. Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points. J. Chem. Phys. 2000, 113...
work page 2000
-
[293]
(42) Sigurdarson, A. E.; Schmerwitz, Y. L. A.; Tveiten, D. K. V.; Levi, G.; J´ onsson, H. Orbital-optimized Density Functional Calculations of Molecular Rydberg Excited States with Real Space Grid Representation and Self-Interaction Correction. J. Chem. Phys. 2023, 159, 214109. (43) Perdew, J. P.; Ernzerhof, M.; Burke, K. Rationale for mixing exact exchan...
work page 2023
-
[448]
(76) Schmerwitz, Y. L. A.; Levi, G.; J´ onsson, H. Calculations of Excited Electronic States by Converging on Saddle Points Using Generalized Mode Following. J. Chem. Theory Comput. 2023, 19,
work page 2023
-
[944]
(21) Gudmundsd´ ottir, H.; Zhang, Y.; Weber, P. M.; J´ onsson, H. Self-interaction corrected density functional calculations of molecular Rydberg states. J. Chem. Phys. 2013, 139, 194102. (22) Zhang, Y.; Weber, P.; J´ onsson, H. Self-Interaction Corrected Functional Calculations of a Dipole-Bound Molecular Anion. J. Phys. Chem. Letters 2016, 7,
work page 2013
-
[1372]
(10) Kaupp, M.; Karton, A.; Bischoff, F. A. [Al2O4] −, a Benchmark Gas-Phase Class II Mixed-Valence Radical Anion for the Evaluation of Quantum-Chemical Methods. J. Chem. Theory Comput. 2016, 12, 3796–3806. (11) Klawohn, S.; Kaupp, M.; Karton, A. MVO-10: A Gas-Phase Oxide Benchmark for Lo- calization/Delocalization in Mixed-Valence Systems. J. Chem. Theor...
work page 2016
-
[2068]
(23) Lehtola, S.; J´ onsson, E. ¨O.; J´ onsson, H. Effect of complex-valued optimal orbitals on atomization energies with the Perdew–Zunger self-interaction correction to density functional theory. J. Chem. Theory Comput. 2016, 12,
work page 2016
-
[3634]
(77) Ziegler, T.; Rauk, A.; Baerends, E. J. On the calculation of multiplet energies by the Hartree-Fock-Slater method. Theor. Chim. Acta 1977, 43, 261–271. (78) Schmerwitz, Y. L. A.; Ivanov, A. V.; J´ onsson, E. O.; J´ onsson, H.; Levi, G. Variational Density Functional Calculations of Excited States: Conical Intersection and Avoided Crossing in Ethylene...
work page 1977
Show all 16 references
-
[3990]
V.; Schmerwitz, Y
(79) Ivanov, A. V.; Schmerwitz, Y. L. A.; Levi, G.; J´ onsson, H. Electronic Excitations of the Charged Nitrogen-Vacancy Center in Diamond Obtained Using Time-Independent Variational Density Functional Calculations. SciPost Physics 2023, 15,
2023
-
[4240]
(68) Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007–1023. (69) Kendall, R. A.; Dunning, T. H.; Harrison, R. J. Electron affinities of the first-row atoms revisited. Sy...
1989
-
[4296]
V.; J´ onsson, E
(24) Ivanov, A. V.; J´ onsson, E. ¨O.; Vegge, T.; J´ onsson, H. Direct Energy Minimization Based on Exponential Transformation in Density Functional Calculations of Finite and Extended System. Comp. Phys. Commun. 2021, 267, 108047. 25 (25) Ivanov, A. V.; Levi, G.; J´ onsson, E...
2021
-
[4733]
The diamine cation is not a chemical example where density functional theory fails
(40) Cheng, X.; Zhang, Y.; J´ onsson, E. ¨O.; J´ onsson, H.; Weber, P. M. Reply to: “The diamine cation is not a chemical example where density functional theory fails”. Nat. Comm. 2018, 9,
2018
-
[5034]
J.; et al
(26) Mortensen, J. J.; et al. , GPA W: Open Python Package for Electronic-structure Calcu- lations. J. Chem. Phys. 2024, 160, 092503. (27) Gudmundsd´ ottir, H.; J´ onsson, E.¨O.; J´ onsson, H. Calculations of Al dopant inα-quartz using a variational implementation of the Perde...
1996
-
[5348]
Rydberg electron stabilizes the charge localized state of the diamine cation
(41) Reimann, M.; Kirsch, C.; Sebastiani, D.; Kaupp, M. Rydberg electron stabilizes the charge localized state of the diamine cation. Nat. Comm. 2024, 15,
2024
-
[6968]
V.; J´ onsson, H
(75) Levi, G.; Ivanov, A. V.; J´ onsson, H. Variational Calculations of Excited States via Direct Optimization of the Orbitals in DFT. Faraday Discuss. 2020, 224,
2020
-
[9982]
27 (44) Kozuch, S.; Martin, J. M. L. DSD-PBEP86: In Search of the Best Double-Hybrid DFT with Spin-Component Scaled MP2 and Dispersion Correction. Phys. Chem. Chem. Phys. 2010, 12, 2940–2953. (45) Kozuch, S.; Martin, J. M. L. Spin-component-scaled double hybrids: An extensive ...
2010
Reviewed August 7, 2026 · model on record in the stance chip above.
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