Towards detection of molecular parity violation via chiral co-sensing: the $^1$H/$^{31}$P model system

Andr\'es Emilio Wedenig; Danila A. Barskiy; Denis Kargin; Dmitry Budker; Erik Van Dyke; James Eills; John Blanchard; Kirill Sheberstov; Konstantin Gaul; Manfred Wagner

REVIEW 2 major objections 6 minor 11 references

Towards detection of molecular parity violation via chiral co-sensing: the $^1$H/$^{31}$P model system

T0 review · 2 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Co-sensing $^{31}$P and $^1$H NMR in one chiral probe resolves tens-of-millihertz splittings, shows the racemic-point residual is systematic rather than parity violation, and clears the way for high-$Z$ tests.

desk verdict A careful, honest systematic-error study of chiral co-sensing for PV NMR; the main conclusion holds, but the residual intercept is less reproducible than the paper implies. read the letter →

arxiv 2412.20997 v1 pith:RU2RYOER submitted 2024-12-30 physics.chem-ph

Erik Van Dyke , James Eills , Kirill Sheberstov , John Blanchard , Manfred Wagner , Robert Graf , Andrés Emilio Wedenig , Konstantin Gaul

show 5 more authors

Robert Berger Rudolf Pietschnig Denis Kargin Danila A. Barskiy Dmitry Budker

This is my paper · ORCID

classification physics.chem-ph PACS 33.25.+k11.30.Er

keywords molecularparityviolationchiralco-sensingNMRspectroscopydiastereomericsplittingcomagnetometry31Psolvatingagentparity-violatingchemicalshift

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

This paper is a systematic-error study for a proposed route to the first observation of parity violation in a molecule. The route is chiral co-sensing: a chiral probe molecule carries two NMR-visible nuclei, a heavier one ($^{31}$P) that should feel the parity-violating weak interaction as a tiny handedness-dependent chemical shift, and a lighter one ($^1$H) that acts as an internal magnetometer tracking the chiral environment. By titrating the handedness of the chiral solvating agent and extrapolating the $^{31}$P diastereomeric splitting to the racemic point (equal left- and right-handed forms), a residual splitting is obtained that would reveal parity violation if systematic errors were absent. The paper measures residuals of $-190 \pm 80$ mHz (first titration) and $-56 \pm 61$ mHz (second titration) against a computed parity-violating benchmark of $-0.7 \mu$Hz, attributes the offset to concentration-dependent systematic effects, and concludes that no fundamental obstacle blocks experiments with high-$Z$ nuclei where the effect could reach millihertz. A sympathetic reader would care because this is the most detailed error budget yet for a strategy that could make molecular parity violation measurable.

What carries the argument

The load-bearing object is the comagnetometry plot: measured $^{31}$P diastereomeric splitting plotted against methyl-$^1$H diastereomeric splitting across a titration of the S:R enantiomeric ratio of the chiral solvating agent. In the rapid chemical-exchange regime the probe samples both chiral environments many times per $T_2^*$, so the splittings vary smoothly with the enantiomeric ratio, and the racemic point is the location where the $^1$H splitting vanishes; a two-axis linear regression of the $^{31}$P splitting on the $^1$H splitting then returns a y-intercept that would contain the parity-violating shift plus any uncompensated systematic offset. The assumption that both splittings respond linearly to the enantiomeric ratio, together with error propagation through a quadratic total-error term, carries the extrapolation. The BIRD pulse sequence, which edits the $^1$H spectrum through the 16.8 Hz $^{31}$P–$^1$H coupling, supplies the precision gain in the second titration, and two-component ZORA density-functional calculations supply the predicted $-0.7\,\mu$Hz benchmark that the intercept must beat.

What would settle it

Repeat the enantiodiscriminatory titration at several CSA-to-probe ratios that lie inside the linear binding regime (the paper's own data place the chosen 1:1 ratio outside it) and compare the extrapolated racemic-point intercepts: an intercept that shifts or vanishes as the ratio changes identifies the residual as concentration-driven systematic error, while an intercept that holds fixed at the millihertz level across ratios would point to a real chirality-dependent effect rather than sample preparation.

Watch

Extended reading notes

Core claim

The central claim is that the chiral co-sensing protocol — extrapolating the $^{31}$P diastereomeric splitting against the $^1$H splitting to the racemic point of the chiral solvating agent — functions for an intermediate-$Z$ nucleus and reaches sub-linewidth precision, with the meaning of the residual resting on systematic-error analysis. Two titrations give racemic-point intercepts of $-190 \pm 80$ mHz (weighted average over two proton resonances) and $-56 \pm 61$ mHz (with BIRD editing of overlapping resonances), while the conformer-averaged quasi-relativistic calculation predicts a parity-violating splitting of only $-0.7\,\mu$Hz at 20 T. Because the proton–proton intercept ($3 \pm 36$ mHz) is consistent with zero whereas the $^{31}$P intercept is not, the residual is located in the $^{31}$P channel and is attributed to systematic error, chiefly the nonlinear concentration dependence of the diastereomeric splitting at the chosen 1:1 ratio and sample-preparation variability. The paper's stated conclusion is that there are no show-stoppers for high-$Z$ experiments, although concentration dependence must be carefully controlled.

Load-bearing premise

The method assumes the 31P and the 1H NMR splittings both change linearly with the ratio of left- to right-handed dissolving agent, so that plotting one against the other erases common systematic errors; the paper's own data show the chosen 1:1 concentration sits outside this linear range.

Editorial extensions

If this is right

Diastereomeric splittings are extracted at tens-of-millihertz precision in 20 T spectra, far below the natural NMR linewidth, so sensitivity is not the limiting factor for heavier nuclei.
The best racemic-point intercept, $-56 \pm 61$ mHz, quantifies the current systematic floor: roughly three orders of magnitude above the computed $-0.7\,\mu$Hz parity-violating splitting for $^{31}$P.
Dropping inverse-gated proton decoupling from the $^{31}$P acquisition removes a candidate systematic arising from decoupling-field leakage into $B_0$, while BIRD editing shows spectral cleanup sharply reduces frequency-estimation error.
Future experiments should seek systems or conditions where both splitting values are linear in the concentration and handedness ratio of the chiral solvating agent, and should move to high-$Z$ nuclei such as $^{203,205}$Tl or $^{207}$Pb, where predicted parity-violating splittings reach the millihertz range.
Multi-dimensional co-sensing, correlating several diastereomeric-split proton multiplets with the heavy nucleus, is demonstrated as a route to diagnosing which nucleus carries the systematic error.

Reading between the lines

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

An implication the authors leave implicit is that the residual intercept may scale with the strength of binding-induced chemical-shift nonlinearity rather than with nuclear charge; if so, a heavier nucleus in the same 1:1 stoichiometry would inherit a similar systematic floor, and reaching mHz parity precision may require a redesigned binding regime, not merely a heavier probe.
The contrast between a zero proton–proton intercept and a nonzero $^{31}$P intercept suggests the residual is nucleus-specific rather than chiral contamination of the solvent, which would have shifted both channels; a co-sensor with two heavy nuclei could cancel this class of systematics directly.
The paper's reasoning implies a control experiment: an achiral structural analogue of the probe should reproduce the same concentration-driven intercept if the effect is purely systematic, and should show none if the residual carries genuine chirality-dependent physics.
Because the racemic-point residual is obtained by extrapolation in the fast-exchange regime, a slow-exchange measurement at lower temperature that resolves the four diastereomer lines separately would provide an independent, non-extrapolated check of the intercept.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read the Van Dyke et al. paper on 1H/31P chiral co-sensing for molecular parity violation. Worth your time if you follow the PV-NMR roadmap.

What is actually new: first application of the Eills et al. co-sensing idea to a 31P probe, and the first detailed error budget for the approach. The experimental work is genuinely careful: two full titrations with different pulse schemes, BIRD suppression of overlapping cyclohexyl resonances, explicit sample-preparation reproducibility tests, and an honest catalog of failure modes (decoupling field leakage, temperature drift, sample degradation, tautomerization). The PV predictions come from independent quasi-relativistic DFT calculations with standard constants and lambda_PV = -1 fixed, not fitted to the intercept. The separation between computed PV and measured intercept is clean.

The main message is well supported: what looks like a residual splitting at the racemic point is dominated by systematic effects, and nothing here rules out trying high-Z nuclei. The predicted PV shift is -0.7 microHz, four orders of magnitude below their resolution; nobody claims a detection.

The soft spot is the quantitative intercept. The two titrations give -190 +/- 80 mHz and -56 +/- 61 mHz. The difference is about 1.3 sigma combined, so not formally inconsistent, but it is the same size as the residuals themselves. The extrapolation to the 1H-axis origin assumes the 31P-versus-1H diastereomeric splitting is linear all the way to the racemic point. The paper's own SI (Fig. 7) shows the 10 mM/10 mM system is outside the linear concentration regime, and the residual plot in Fig. 4 has structure. The first titration omitted the near-racemic points (56.25 to 43.75% S) entirely, so the fit is anchored by points farther from the origin; a small curvature in the unmeasured interval could shift the intercept by the observed amount. The post hoc error rescaling (scaling down by sqrt(chi2/dof)=0.58 because chi2=4.2 at dof=12) is defensible, but it does not address the between-titration scatter.

So the paper does not quite demonstrate that the residual is a stable, understood systematic error; it demonstrates it is currently uncontrolled at the ~100 mHz level. That is still a useful result, and the authors are appropriately cautious in the conclusions.

Who should read it: anyone planning a PV-NMR experiment with heavier nuclei, and people working with chiral co-sensing/comagnetometry in NMR. It deserves a serious referee. I would send it out. In review I would ask for a more direct treatment of the linearity assumption — show the near-racemic data, or fit a mild nonlinear model and report how much the intercept moves, and frame the intercept as a range across titrations rather than a single number.

Recommendation: accept with minor to moderate revision.

Referee Report

2 major / 6 minor

Summary. The paper reports a systematic-error study for a proposed NMR route to molecular parity violation (PV), using an intramolecular co-magnetometry scheme in which the diastereomeric splittings of a heavy nucleus (31P) and a light nucleus (1H) on the same sensor molecule are measured simultaneously as a function of the enantiomeric ratio of a chiral solvating agent. The residual 31P splitting at the racemic point, obtained by linear extrapolation of the 31P-vs-1H comagnetometry curve, is reported as -190±80 mHz from a first titration and -56±61 mHz from a second, improved titration, while quasi-relativistic DFT calculations predict an actual PV splitting of only -0.7 µHz. The authors conclude that the observed nonzero intercept is dominated by uncontrolled systematic errors rather than PV, and that the approach shows no fundamental obstacles for future high-Z experiments, provided concentration nonlinearities are addressed.

Significance. If the systematic-error analysis is sound, this is a valuable contribution to the ongoing search for molecular parity violation, a long-standing open problem. The paper is unusually transparent: it reports detailed experimental procedures, sample-degradation observations, explicit error budgets, and a clear statement of the critical assumption that co-sensing cancels sample-preparation errors. The PV predictions are obtained from independent quasi-relativistic DFT calculations with standard constants and a fixed coupling parameter (lambda_PV = -1), not fitted to the measured intercept, which avoids circularity. The work also delivers practical guidance for choosing future high-Z systems. However, the central empirical claim—that the extrapolated intercept is a reproducible, understood systematic offset—is not yet fully supported by the data presented.

major comments (2)

[§3.3.1, Fig. 4, SI Fig. 7, end of §2.3] The intercept of the comagnetometry plot is obtained by linear extrapolation across the unmeasured near-racemic region, but the paper itself provides evidence that this extrapolation may be biased. Section 2.3 states that the compositions 56.25%–43.75% FBTrp-S were omitted from the first titration, and the second titration (Fig. 18 of the SI) also has no points in that interval. SI Fig. 7 shows that at the chosen 10 mM/10 mM concentrations the system is outside the linear concentration regime, and Section 3.3.1 concedes that nonlinear changes in Δd are not compensated by nuclear co-sensing. Figure 4's residual plot shows systematic structure. A small curvature in the unmeasured 25%-wide enantiomeric-ratio interval could shift the extrapolated intercept by the observed amount; the present data do not bound this curvature.
[§3.2 and §4 (intercept values), §3.3.2] The two titrations yield intercepts of -190±80 mHz and -56±61 mHz, a difference of about 130 mHz that is comparable to the individual uncertainties and to the residual itself. Since the first titration used inverse-gated 1H decoupling during 31P acquisition—a potential source of uncompensated systematic error identified in Section 3.3.2—and the second did not, the two measurements cannot be considered a demonstration of reproducibility. The conclusion that there are 'no show-stoppers' relies on the residual being a stable, understood systematic error; the present data show instead that the intercept is not reproducible under the assumed linear model. This point is load-bearing for the paper's central claim and needs to be addressed, for example by including near-racemic points, modeling the nonlinearity, or reporting an explicit systematic-uncertainty budget for the extrapolation.

minor comments (6)

[Introduction, §1.4] Typo: 'as apposed to a chiral solvent' should be 'as opposed to a chiral solvent'.
[§2.3] Typo: 'seperate' should be 'separate'.
[Eq. (1)] The notation 'Tr[AAA]' in Eq. (1) appears corrupted; the trace should be of the matrix product shown, and the surrounding text has some garbled symbols (e.g., matrix definitions). Please correct the typesetting.
[§2.4] The 'popt experiment in TopSpin' is not defined; please provide a brief explanation or reference so that the pulse-sequence calibration is reproducible.
[References, Ref. 41] Reference 41 lists 'S. Purity, J. A. Dale and H. S. Mosher' with no journal name; this appears to be a corrupted citation. Please verify and correct it, since the reference is used to support the fitting procedure.
[§2.6] The error-scaling procedure is described in the text (Chapter 2.6) but the caption of Fig. 4 states that a reduced chi-squared value is used to calibrate errors 'such that χ2red = 1'; consider clarifying in the caption that the errors were scaled by sqrt(χ2/dof) = 0.58, which is less than 1, indicating overestimated uncertainties or correlations.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PV prediction is an independent DFT calculation with fixed constants, and the comagnetometry intercept is an empirical fit that the paper itself attributes to systematics.

full rationale

The paper's central quantitative comparison is between (i) a quasi-relativistic DFT prediction of the parity-violating 31P splitting and (ii) an experimentally determined comagnetometry intercept. These are not linked by construction. The DFT value ΔPV(31P) = −0.7 μHz is computed with a fixed coupling parameter λPV = −1 and standard constants (GF, sin^2 θ_W); the paper explicitly states these are effective parameters to be scaled by nuclear-structure terms, and the computation is benchmarked against conventional chemical shifts and J-couplings that deviate by 9–28% from experiment, so the PV number is not tuned to the measured intercept. The comagnetometry intercept is obtained by linear regression of measured 31P diastereomeric splittings on measured 1H splittings; it is a fitted observable, not a quantity defined to equal the fitted parameter. The paper does not claim the intercept is the PV effect; it states 'there clearly arises a shift away from zero, which is unexpected' and attributes it to systematic errors, which is a consistency argument rather than a derivation. The assumption that co-sensing removes sample-preparation errors is explicitly labeled as an assumption and is tested with replicate samples; the paper's own admission that the system is outside the linear concentration regime (SI Fig. 7) is a limitation or correctness risk, not a circular step. The self-citation of Ref. 35 (Eills et al., with overlapping authorship) introduces the co-sensing concept, but the present paper independently validates it, e.g., via the 1H-1H intercept consistent with zero and reproducibility measurements. No load-bearing step reduces to its own input by definition or by self-citation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central measurement rests on (1) the rapid-exchange/motional-narrowing assumption, (2) the linearity and common-mode rejection assumptions of the comagnetometry regression, (3) the DFT methodology for PV shift estimates, and (4) the Z-scaling assumption for the proton control. The fitted quantities are the regression slope/intercept and the binding constant; lambda_PV is a scaling choice consistent with prior work.

free parameters (4)

Comagnetometry slope a = not explicitly stated in text; fitted slope of 31P vs 1H delta_d regression
Fitted slope used to extrapolate the 31P splitting to zero 1H splitting; auxiliary result.
Comagnetometry y-intercept b = -190±80 mHz (titration 1), -56±61 mHz (titration 2)
The residual 31P splitting at the racemic point; the central measured quantity that is compared to PV predictions.
Dissociation constant Kd = 9.0 ± 1.5 mM
From chemical shift fits in the SI; used to characterize binding but not central to the main claim.
lambda_PV = -1 for all nuclei
Effective coupling strength parameter chosen for consistency with previous PV-NMR studies; actual nuclear-structure scaling is left to future work.

assumptions (5)

domain assumption PV effects scale as Z^a with 2 < a < 5 for NMR shielding (see Ref. 21).
Used to argue the 1H PV contribution is negligible relative to 31P; verified numerically for this system (Section 3.1).
domain assumption The system is in rapid chemical exchange (motional narrowing), so diastereomeric complexes are characterized by the enantiomeric ratio of the CSA.
Needed for the absence of line splitting at the racemic point and for the comagnetometry model (Section 1.3).
domain assumption Systematic errors affecting 31P delta_d are largely compensated by 1H delta_d (common-mode rejection).
The critical assumption of the co-sensing approach, explicitly stated in Section 3.3.1; shown to be only partially valid due to nonlinear concentration dependence.
domain assumption The comagnetometry relationship between 31P delta_d and 1H delta_d is linear over the measured range.
Needed for the linear regression and intercept extraction; the residual plot shows deviation from linearity (Fig. 4).
domain assumption DFT methodology (2c-ZORA-BHandH) accurately captures relative PV NMR shieldings.
Used for predicted PV splittings; authors note errors in chemical shifts (25% for 31P), but argue PV shifts are less gauge dependent and sufficiently accurate for order-of-magnitude estimates (Section 3.1).

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Cite this review

Pith. "Pith review of Towards detection of molecular parity violation via chiral co-sensing: the $^1$H/$^{31}$P model system." pith.science (2026). https://pith.science/paper/RU2RYOER

@misc{pith2026241220997,
  author       = {Pith},
  title        = {Pith review of: Towards detection of molecular parity violation via chiral co-sensing: the $^1$H/$^31$P model system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RU2RYOER}},
  note         = {Machine review of arXiv:2412.20997}
}

abstract

Fundamental weak interactions have been shown to violate parity in both nuclear and atomic systems. However, observation of parity violation in a molecular system has proven an elusive target. Nuclear spin dependent contributions of the weak interaction are expected to result in energetic differences between enantiomers manifesting in nuclear magnetic resonance (NMR) spectra as chemical shift differences on the order of $10^{-6}$ Hz to $10^{-3}$ Hz for high-$Z$ nuclei. By employing simultaneous measurements of the diastereomeric splittings for a light and a heavy nucleus in solution-state NMR, residual chemical shift differences persisting in non-chiral environment between enantiomers of chiral compounds smaller than the typical linewidth of high-field NMR may be resolved. Sources of error must be identified and minimized to verify that the observed effect is, in fact, due to parity violation and not systematic effects. This paper presents a detailed analysis of a system incorporating \textsuperscript{31}P and \textsuperscript{1}H NMR to elucidate the systematic effects and to guide experiments with higher-$Z$ nuclei where molecular parity violation may be resolved.

Figures

Figures reproduced from arXiv: 2412.20997 by the authors.

**Figure 4.** figure 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

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Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [10]

b Helmholtz Institute Mainz, 55128 Mainz, Germany; GSI Helmholtz Center for Heavy Ion Research, Darmstadt, Germany

From the early days of atomic PV, it has been recognized that parity violation should also manifest in molecules 11,12; in partic- a Institute for Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany. b Helmholtz Institute Mainz, 55128 Mainz, Germany; GSI Helmholtz Center for Heavy Ion Research, Darmstadt, Germany. c Institute of Biological ...

[2] [31]

Nuclear magnetization distributions were assumed to be point-like in all calculations

The Coulomb potential of the nuclei was modeled in all calculations assuming a finite spherical Gaussian-shaped nuclear charge density distribution eZAρA (⃗r) = eZA ζ 3/2 A π3/2 e−ζA|⃗r−⃗rA|2 with ζA = 3 2r2nuc,A and the root-mean-square radius rnuc,A was cho- sen as suggested by Visscher and Dyall 67, where nuclear mass numbers where chosen as nearest in...

work page arXiv

[3] [76]

show-stoppers

4 Conclusions The goal of this study was to asses sources of error in determin- ing frequency estimates from NMR experiments using diastere- omeric complexes containing an intermediate-Z atom (31P). Min- imizing sources of error is crucial to observing parity violating contributions of the weak interaction to the chemical shift tensor of molecules, which ...

work page 1956

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2 C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes and R. P. Hudson, Physical Review, 1957, 105, 1413–1415. 3 Y. B. Zel’dovitch, Sov. Phys. JETP, 1959,

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25 A. Soncini, F. Faglioni and P. Lazzeretti, Phys. Rev. A , 2003, 68, 033402. 26 V. Weijo, P. Manninen and J. Vaara,J. Chem. Phys., 2005, 123, 054501. 27 G. Laubender and R. Berger, Phys. Rev. A, 2006, 74, 032105. 28 R. Bast, P. Schwerdtfeger and T. Saue, J. Chem. Phys. , 2006, 125, 064504. 29 V. Weijo, R. Bast, P. Manninen, T. Saue and J. Vaara, J. Chem...

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12 S. F. Mason, Nature, 1984, 311, 19–23. 13 I. B. Khriplovich, Parity nonconservation in atomic phenomena, Philadelphia: Gordon and Breach Science Publishers,

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Hasert, H

4 F. Hasert, H. Faissner, W. Krenz, J. Von Krogh, D. Lanske, J. Morfin, K. Schultze, H. Weerts, G. Bertrand-Coremans, J. Lemonne, J. Sacton, W. Van Doninck, P. Vilain, C. Bal- tay , D. Cundy , D. Haidt, M. Jaffre, P. Musset, A. Pullia, S. Natali, J. Pattison, D. Perkins, A. Rousset, W. Venus, H. Wachsmuth, V. Brisson, B. Degrange, M. Haguenauer, L. Kluber...

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23 J. B. Robert and A. L. Barra, Chirality, 2001, 13, 699–702. 24 G. Laubender and R. Berger, ChemPhysChem, 2003, 4, 395–

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