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arxiv: 2604.05802 · v1 · submitted 2026-04-07 · ⚛️ physics.chem-ph · astro-ph.GA

Valence and Rydberg excited state bond dissociation curves of CO2 from orbital-optimized density functional calculations

Dar\'io Barreiro-Lage , Gianluca Levi , Hannes Jonss\'on , Thanja Lamberts This is my paper

Pith reviewed 2026-05-10 18:53 UTC · model grok-4.3

classification ⚛️ physics.chem-ph astro-ph.GA
keywords CO2excited statesRydberg statesvalence statesorbital-optimized DFTbond dissociation curvesdensity functional theoryphotorelaxation
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The pith

Orbital-optimized density functional theory with PBE and complex orbitals reproduces CO2 valence and Rydberg excited state energies and C-O dissociation curves within 0.3 eV of multireference references.

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

The paper establishes that variational orbital optimization in density functional theory can compute the lowest valence π* state as well as 3s and 3pσ Rydberg states of CO2. With the PBE functional and complex-valued orbitals, excitation energies stay close to multireference configuration interaction benchmarks and the resulting C-O bond dissociation curves align closely with both multireference configuration interaction and equation-of-motion coupled-cluster results. Linear-response time-dependent density functional theory shows larger errors and stronger functional dependence for the same states. The low cost of the orbital-optimized route is presented as enabling simulations of photorelaxation in condensed-phase CO2 under high-energy excitation.

Core claim

By locating saddle points on the electronic energy surface with real or complex orbitals, density functional calculations using five functionals isolate the target valence π* and Rydberg 3s, 3pσ excited states of CO2; the PBE functional with complex orbitals yields excitation energies within 0.3 eV of multireference configuration interaction values and produces C-O dissociation curves that compare closely with multireference configuration interaction and equation-of-motion coupled-cluster singles and doubles benchmarks.

What carries the argument

Orbital optimization to saddle points on the electronic energy surface using real or complex orbitals within density functional theory, applied to isolate specific valence and Rydberg excited states.

If this is right

  • The method shows weaker dependence on functional choice than linear-response time-dependent density functional theory for diffuse Rydberg states.
  • Hybrid functionals improve the PBE results further while retaining the low computational cost.
  • The approach directly supports modeling of photorelaxation pathways in CO2 without requiring high-level multireference calculations for each geometry.
  • Complex orbitals are required to reach the reported accuracy for the most diffuse 3pσ Rydberg state.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same orbital-optimization protocol could be tested on other linear triatomic molecules to check transferability of the accuracy for Rydberg dissociation curves.
  • If the saddle-point search remains stable, the method opens the possibility of embedding these CO2 excited states in periodic condensed-phase simulations at modest cost.
  • The reported improvement over time-dependent density functional theory for diffuse states suggests orbital optimization may address a broader class of failures in standard excited-state density functional theory.

Load-bearing premise

Locating saddle points on the electronic energy surface with real or complex orbitals reliably isolates the target valence and Rydberg excited states without variational collapse or mixing with other states.

What would settle it

A calculation of the same CO2 excited-state dissociation curves that deviates by more than 0.5 eV from new multireference or experimental benchmarks for the 3pσ state would falsify the claim of close agreement.

Figures

Figures reproduced from arXiv: 2604.05802 by Dar\'io Barreiro-Lage, Gianluca Levi, Hannes Jonss\'on, Thanja Lamberts.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of the advantage of using complex orbitals when calculating excitations from the occupied [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Comparison of results obtained with linear combination [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Excitation energy for the Rydberg excited 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Energy as a function of C–O distance for one of the bonds in the linear CO [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Energy of the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

Calculations of the lowest valence {\pi}* as well as the 3s and higher energy 3p{\sigma} Rydberg excited states of the CO2 molecule are carried out using density functionals with variational optimization of the orbitals, an approach involving relatively little computational effort. Five functionals with varying degree of exchange are used in combination with real or complex-valued orbitals that are optimized by finding saddle points on the electronic energy surface corresponding to the excited states. When the PBE functional is used in combination with complex orbitals, the calculated excitation energy is found to be within 0.3 eV of multireference configuration interaction reference values, and the results are further improved with hybrid functionals. In contrast, linear-response time-dependent density functional theory calculations give errors up to 1.9 eV for the most diffuse 3p{\sigma} excitation and exhibit stronger dependence on both the excitation character and the functional used. Calculated C-O dissociation curves using the PBE functional and the orbital-optimized approach compare remarkably well with the reported multireference configuration interaction and equation-of-motion coupled-cluster singles and doubles calculations. Thanks to the low computational cost, these results demonstrate that orbital-optimized density functional calculations can be a promising route for modelling photorelaxation in condensed-phase CO2, for example in the context of interstellar cosmic-ray radiation driven process involving high-energy Rydberg states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript reports calculations of the lowest valence π* and Rydberg 3s/3pσ excited states of CO2 using orbital-optimized DFT. Orbitals are variationally optimized by locating saddle points on the electronic energy surface with real or complex-valued orbitals for five functionals of varying exact-exchange content. With PBE and complex orbitals, excitation energies agree with MRCI references to within 0.3 eV; hybrid functionals improve further. Linear-response TDDFT yields larger errors (up to 1.9 eV for the most diffuse 3pσ state). The C–O dissociation curves obtained with orbital-optimized PBE are stated to compare remarkably well with published MRCI and EOM-CCSD curves. The low cost is emphasized as enabling condensed-phase photorelaxation modeling.

Significance. If the target states are correctly isolated, the work demonstrates that saddle-point orbital optimization within DFT can deliver excited-state dissociation curves at far lower cost than MRCI or EOM-CCSD while outperforming standard TDDFT for both valence and diffuse Rydberg states. This is a potentially useful route for high-energy Rydberg photochemistry in condensed CO2, such as cosmic-ray-driven processes, where system size precludes wave-function methods.

major comments (2)
  1. [Abstract and Results] Abstract and Results sections: the central claim that PBE orbital-optimized curves 'compare remarkably well' with MRCI and EOM-CCSD rests on the assumption that the located saddle points isolate the intended valence π* and 3s/3pσ states without variational collapse or mixing. No orbital plots, transition-density analysis, or symmetry projections are referenced to confirm state purity, especially as the 3pσ orbitals become diffuse along the dissociation coordinate. This is load-bearing for the quantitative comparison.
  2. [Computational Details] Computational Details: no basis-set specification, no convergence thresholds for the saddle-point searches, and no stability diagnostics (e.g., against collapse to lower states) are provided. These omissions prevent independent assessment of the reported 0.3 eV agreement.
minor comments (3)
  1. The abstract mentions five functionals but reports detailed numbers only for PBE; a table summarizing all five functionals' performance on the excitation energies would improve clarity.
  2. Error bars or standard deviations on the excitation energies and curve deviations from MRCI/EOM-CCSD are absent; their inclusion would strengthen the quantitative statements.
  3. The manuscript should cite prior orbital-optimization literature for excited states to better situate the use of complex orbitals for Rydberg dissociation curves.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of the significance of our work and for the constructive comments. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and Results] Abstract and Results sections: the central claim that PBE orbital-optimized curves 'compare remarkably well' with MRCI and EOM-CCSD rests on the assumption that the located saddle points isolate the intended valence π* and 3s/3pσ states without variational collapse or mixing. No orbital plots, transition-density analysis, or symmetry projections are referenced to confirm state purity, especially as the 3pσ orbitals become diffuse along the dissociation coordinate. This is load-bearing for the quantitative comparison.

    Authors: We agree that explicit verification of state character is essential to support the quantitative comparisons. The states were targeted by initializing the saddle-point orbital optimizations with the appropriate orbital occupations and symmetries for the valence π* and Rydberg 3s/3pσ excitations, with complex orbitals employed where needed to prevent collapse. The variational nature of the saddle-point search ensures stationary points at the desired energies, and the consistent agreement within 0.3 eV with MRCI references over the full dissociation coordinate (including regions where Rydberg orbitals diffuse) would be unlikely if significant mixing or collapse were present. To directly address the concern, we will add orbital plots, transition-density analyses, and symmetry projections in the revised manuscript to demonstrate state purity. revision: yes

  2. Referee: [Computational Details] Computational Details: no basis-set specification, no convergence thresholds for the saddle-point searches, and no stability diagnostics (e.g., against collapse to lower states) are provided. These omissions prevent independent assessment of the reported 0.3 eV agreement.

    Authors: We apologize for these omissions. The calculations employed the aug-cc-pVTZ basis set, with saddle-point optimizations converged to 10^{-6} a.u. in both energy and orbital gradient. Stability against collapse was monitored via orbital energy ordering and direct comparison to lower states, with complex orbitals used to stabilize diffuse solutions. We will incorporate these specifications, along with additional stability diagnostics, into the Computational Details section of the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: excited-state curves validated against independent external MRCI/EOM-CCSD benchmarks

full rationale

The paper computes valence and Rydberg excited-state C-O dissociation curves of CO2 via orbital-optimized DFT (PBE and hybrids, real or complex orbitals) by locating saddle points on the electronic energy surface. These curves are then compared directly to literature MRCI and EOM-CCSD reference data. No parameters are fitted to the target dissociation energies or curves, no self-citation supplies a uniqueness theorem or ansatz that forces the result, and the method is not redefined in terms of its own outputs. The central claim therefore rests on external, independent high-level calculations rather than any reduction of predictions to the paper's own inputs or fitted quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard DFT axioms plus the assumption that saddle-point orbital optimization isolates pure excited states. No new entities are postulated and no parameters appear to be fitted to the target data.

axioms (2)
  • domain assumption Kohn-Sham DFT with approximate exchange-correlation functionals yields useful excited-state energies when orbitals are variationally optimized to saddle points.
    Invoked throughout the abstract when comparing PBE and hybrid results to MRCI references.
  • domain assumption Complex-valued orbitals can be used without introducing artifacts in the description of Rydberg states.
    Stated as an option that improves agreement for the 3pσ state.

pith-pipeline@v0.9.0 · 5571 in / 1421 out tokens · 32678 ms · 2026-05-10T18:53:04.700101+00:00 · methodology

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

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