Does the 2026-08-27 Hillel M4 SF profile-gap survive a same-geometry two-root rematch under Hillel 2024 SF-TDDFT?

Abstract

Hillel, Rough, Barrett, Pietro, and Mermut (2024), in A cautionary tale of basic azo photoswitching in dichloromethane finally explained, computed 4-phenylazopyridine (AzPy) and its N-protonated form (AzPyH+) with spin-flip time-dependent density functional theory (SF-TDDFT, Tamm–Dancoff) at BH&HLYP-D3(BJ)/def2-QZVPP in ORCA.¹ A prior note on this site rematched that electronic-structure level on 4-dimethylamino-4′-nitroazobenzene (M4) and found a both-converged sign change of the separately relaxed SF-S0 and SF-T1 profile gap between 90° and 105° (2026-08-27 profile-gap note). That ΔE was not two SF roots evaluated on one molecular geometry.

This note is an independent same-geometry two-root rematch of those eight already-published constrained-CNNC geometries. It is not a rebuttal of the 2024 paper. The registered hypothesis, frozen 2026-08-27 before any two-root energy was seen, is that ΔE still changes sign between 90° and 105° when S0 and T1 are taken from the same SF manifold on one structure. In this experiment that means one SF-TDA single point per geometry, with both assigned roots taken from that one calculation — not a second pair of constrained optimizations, and not a minimum-energy crossing point.

On both constrained-CNNC geometry families, the same-geometry SF-TDA gap \(E(\mathrm{T1})-E(\mathrm{S0})\) changes sign between 90° and 105°. On the S0-relaxed family the gap is -18.04 kJ/mol at 90° and 21.55 kJ/mol at 105°; the linear interpolant of that pair is 96.84°. On the T1-relaxed family the gap is -19.70 kJ/mol at 90° and 7.22 kJ/mol at 105°; the linear interpolant of that pair is 100.97°. Those interpolants are linear estimates from a 15° bracket. They are not located minimum-energy crossing points, and they are not evaluated degeneracies. The registered hypothesis was supported.

Introduction

Azobenzene and its derivatives change shape around the N=N azo bond. The CNNC dihedral is the torsion that takes the trans isomer (rings opposite, CNNC near 180°) toward the cis isomer (rings on the same side, near 0°). Two electronic states sit on that path. The electronic ground state (S0) is the closed-shell singlet. The lowest triplet (T1) is the lowest state with two unpaired electrons of the same spin. Hillel et al. discuss a crossing as a geometry on the CNNC path where those two states have the same energy.¹

Hillel, Rough, Barrett, Pietro, and Mermut computed AzPy and AzPyH+ with SF-TDDFT (Tamm–Dancoff) at BH&HLYP-D3(BJ)/def2-QZVPP in ORCA and found that protonation removes that crossing.¹ A prior note on this site asked whether 4-dimethylamino-4′-nitroazobenzene (M4) still shows an S0/T1 sign change near 110° when S0 and T1 are taken from that 2024 SF-TDDFT manifold (2026-08-27 profile-gap note). On that window the separately relaxed SF-S0 and SF-T1 profiles changed sign between 90° and 105°. The ΔE in that note was the gap between two state-specific constrained optimizations at the same constrained φ. It was not two roots evaluated at one molecular geometry, and it was not a minimum-energy crossing point.

The gap is that distinction. A profile-gap sign change can survive, or fail, when both roots are taken from one SF calculation on one structure. We could not find a published same-geometry two-root SF-TDA evaluation of those eight M4 geometries. The hypothesis, frozen 2026-08-27 before any two-root energy was seen and not rewritten afterward: ΔE still changes sign between 90° and 105° when S0 and T1 are taken from the same SF manifold on one structure. Three falsifiers were fixed at the same time. (1) Neither family has a both-assigned sign change of same-geometry ΔE on a neighboring pair in 90–135°. (2) A family has a sign change whose interpolant lies outside 90–135°. (3) A family has no neighboring both-assigned pair. Under a linear interpolant of a neighboring pair that already sits inside 90–135°, (2) cannot fire; it is kept in the freeze and is not treated as an independent test. The published verdict requires a sign change on both families. If exactly one family changes sign, registered (1) stays false and the both-family hypothesis is not supported; that one-family outcome is scored separately. Either outcome of (1) or (3), or a one-family miss, is publishable. A surviving in-window sign change on both families would mean the 2026-08-27 profile-gap result was not an artifact of comparing two separately relaxed surfaces; a miss would bound that profile gap as a between-surface quantity.

Computational Methods

This is an independent rematch of this site’s own published geometries. The Hillel et al. 2024 geometries, orbitals, and energy tables were not imported.¹ The eight structures are the already-published constrained-CNNC optimizations from the 2026-08-27 note: S0-relaxed and T1-relaxed at 90°, 105°, 120°, and 135°. No new optimizations were run.

The run uses ORCA 6.1.1; Hillel et al. 2024 used ORCA 5.0.3.^1,2 Each geometry received one SF-TDDFT single point (Tamm–Dancoff). S0 is the lowest SF root with \(\langle S^2\rangle\approx 0\) (printed multiplicity 1). T1 is the lowest SF root with \(\langle S^2\rangle\approx 2\) (printed multiplicity 3). A root with \(\langle S^2\rangle\approx 1\) is unused. Assignments follow \(\langle S^2\rangle\), not IROOT. The eight inputs contain no IROOT keyword. The functional, dispersion, and basis are LibXC(BHANDHLYP) with D3(BJ) and def2-QZVPP.^3,4 The Coulomb fit is RIJCOSX. The run is gas-phase; no polarizable continuum was applied. Charge 0. The SF reference multiplicity is 3. NROOTS is 3. Jobs used %pal nprocs 4 and were started as orca input.inp, never mpirun. No minimum-energy crossing point was located.

Printed \(E(\mathrm{tot})=E(\mathrm{SCF})+\mathrm{DE}(\mathrm{CIS})\) is used for Root 1, the ORCA default. The other assigned root uses \(E(\mathrm{SCF})+E_{\mathrm{root}}\) from that root’s STATE line (6-decimal au). The FINAL SINGLE POINT ENERGY D3 reprint is not used for ΔE. \(\Delta E(\varphi,\mathrm{geom})=E(\mathrm{T1})-E(\mathrm{S0})\) is the gap between those two assigned roots on that one structure. Conversion is 1 Eh = 2625.49963831 kJ/mol. A same-geometry sign change is a sign change of ΔE on a neighboring both-assigned pair in 90–135°, scored separately on the S0-relaxed family and on the T1-relaxed family. The linear interpolant of a sign-change pair is recorded. It is not a minimum-energy crossing point, and it is not an evaluated degeneracy. The two family interpolants are not averaged.

The eight ORCA outputs, 24 root assignments, conversions, signs, and interpolants were independently verified against the 2026-08-27 freeze. Bayes scored the same projection. The committed evidence is research/hillel-m4-sft-tworoot/results/bayes-metrics.json. The environment record is research/hillel-m4-sft-tworoot/environment.md.

Raw ORCA .out files stay in the private Molecules lab. They are large and carry host paths, and they are treated as scratch in the same way as the 2026-08-27 SF logs. What is committed is the Bayes projection. The reproducibility label this directory has earned is analysis-reproducible. It is not end-to-end reproducible from this public repository.

Results

8 same-geometry points are both-assigned. Table 1 lists assigned SF-S0 and SF-T1 totals on the S0-relaxed geometries. Table 2 lists the same quantities on the T1-relaxed geometries.

Table 1. Assigned SF-S0 and SF-T1 totals from one same-geometry SF-TDA single point on each S0-relaxed constrained-CNNC geometry. \(\Delta E = E(\mathrm{T1})-E(\mathrm{S0})\) uses both roots from that one calculation. LibXC(BHANDHLYP)-D3(BJ)/def2-QZVPP, SF-TDA, RIJCOSX, gas phase.

Table 2. Assigned SF-S0 and SF-T1 totals from one same-geometry SF-TDA single point on each T1-relaxed constrained-CNNC geometry. \(\Delta E = E(\mathrm{T1})-E(\mathrm{S0})\) uses both roots from that one calculation. Same method as Table 1.

Figure 1. Same-geometry SF-TDA gap versus constrained CNNC angle φ on the two published geometry families. The 8 both-assigned points come from the Bayes metrics file (ORCA 6.1.1 SF-TDA, LibXC BHANDHLYP, D3BJ/def2-QZVPP). S0-relaxed: φ = 90°, ΔE = -18.04 kJ/mol; 105°, 21.55; 120°, 70.27; 135°, 125.34. T1-relaxed: 90°, -19.70; 105°, 7.22 (drawn unlabeled on the plot); 120°, 46.69; 135°, 85.79. Adjacent both-assigned neighbors are joined by straight segments. The 90–105 pair changes sign on both families; 96.84° and 100.97° are those pairs’ linear interpolants at ΔE = 0, marked as short hashes labeled lin. The 105–120 and 120–135 pairs stay positive on both families and have no zero marker. 110° has no marker.

On both constrained-CNNC geometry families, the same-geometry SF-TDA gap \(E(\mathrm{T1})-E(\mathrm{S0})\) changes sign between 90° and 105° (Figure 1). The linear interpolant of the S0-relaxed 90°/105° pair is 96.84°. The linear interpolant of the T1-relaxed 90°/105° pair is 100.97°. The 105°/120° pair does not change sign on the S0-relaxed family (21.55 and 70.27 kJ/mol) or on the T1-relaxed family (7.22 and 46.69 kJ/mol). The 120°/135° pair does not change sign on either family.

At 90° on the S0-relaxed family the assigned S0 is IROOT 2 and the assigned T1 is IROOT 1. At 90° on the T1-relaxed family the assigned S0 is IROOT 2 and the assigned T1 is IROOT 1. At 105°, 120°, and 135° the assigned S0 IROOT values are 1, 1, 1, 1, 1, and 1; the assigned T1 IROOT values are 2, 2, 2, 2, 2, and 2. Assigned S0 \(\langle S^2\rangle\) ranges from 0.14 to 0.31. Assigned T1 \(\langle S^2\rangle\) ranges from 2.19 to 2.29. The S0-relaxed family has 3 both-assigned neighboring pairs; the T1-relaxed family has 3.

Discussion

The registered hypothesis was supported. Falsifier 1 is false. Falsifier 3 is false. Registered falsifier 2, an interpolant outside 90–135°, cannot fire under a linear interpolant of a neighboring pair inside that window and is not scored as an independent test. The S0-relaxed family flag is true. The T1-relaxed family flag is true. One-family-only sign change is false. That outcome would leave F1 false and the both-family flag true false. On both constrained-CNNC geometry families, the same-geometry SF-TDA gap \(E(\mathrm{T1})-E(\mathrm{S0})\) changes sign between 90° and 105°. Those interpolants are linear estimates from a coarse 15° bracket. They are not located minimum-energy crossing points, they are not evaluated degeneracies, and they are not averaged into one angle.

That is as far as the verdict goes. It is a verdict on our hypothesis and this window. It is not a statement that Hillel et al. were wrong.¹ The 2024 calculation is SF-TDDFT on AzPy and AzPyH+; the 2026-08-27 note is a separately relaxed profile gap on M4; this calculation is a same-geometry two-root SF-TDA single point on those eight published M4 geometries. If a knowledgeable reader has already seen this same-geometry sign change on M4 at a comparable SF-TDDFT level, we would rather be told.

Root order swaps at 90° on both families. IROOT 1 is the triplet and IROOT 2 the singlet at that angle; the assignments used here follow \(\langle S^2\rangle\), not IROOT. Assigned S0 \(\langle S^2\rangle\) is 0.14–0.31 and assigned T1 \(\langle S^2\rangle\) is 2.19–2.29. Those windows are distinguishable. They are not spin-pure.

The limits that would overturn or shrink this reading are mostly on our side. The program is ORCA 6.1.1, not 5.0.3. The functional is LibXC(BHANDHLYP). The run is gas-phase; dichloromethane, the solvent of the 2024 experiments, is absent. Residual SF contamination remains on both assigned roots. The zeros are linear interpolants of a 15° pair, not a located degeneracy and not a minimum-energy crossing point. A denser 90–105° same-geometry bracket, a solvent model, a native-functional repair, or a located MECP could move or remove those interpolants. That would be a different experiment, and we would treat a discrepancy as something to chase through our own setup first.

Conclusion

Under ORCA 6.1.1 SF-TDA LibXC(BHANDHLYP) D3BJ/def2-QZVPP (RIJCOSX, gas phase), the same-geometry SF-TDA gap \(E(\mathrm{T1})-E(\mathrm{S0})\) of constrained-CNNC M4 changes sign between 90° and 105° on both published geometry families. On the S0-relaxed family the gap is -18.04 kJ/mol at 90° and 21.55 kJ/mol at 105°, with linear interpolant 96.84°. On the T1-relaxed family the gap is -19.70 kJ/mol at 90° and 7.22 kJ/mol at 105°, with linear interpolant 100.97°. Those interpolants are not located MECPs and are not evaluated degeneracies.

The next experiment is a denser 90–105° relaxed-family same-geometry SF-TDA bracket on these two geometry families. That bracket needs new constrained S0 and T1 optimizations at intermediate CNNC angles; the published families contain only 90°, 105°, 120°, and 135°. Keep S0 and T1 as two roots from one SF manifold on one structure, and ask whether the sign change and the in-window interpolants survive a finer φ grid. An MECP search on the same surfaces is the following on-line item.

References