arising from T. Georgiou et al. Nature Communications https://doi.org/10.1038/s41467-024-49966-8 (2024)

Distinguishing the enantiomers of small organic molecules is an industrially relevant problem with important implications for the health of the population1. In a recent publication, Bouchard and co-workers have suggested that large differences in indirect spin-spin (J) coupling constants between enantiomers are possible2. A close inspection of their work revealed significant flaws in their density functional theory (DFT) calculations and that the reported effects disappear with appropriate care. We thus conclude that enantiospecificity in spin-spin coupling constants has not be demonstrated either experimentally or theoretically.

The most common approach used to measure the population of enantiomers is chromatography using a chiral medium. Nuclear magnetic resonance (NMR) spectroscopy has long been known to be insensitive to enantiomers, because enantiomers yield identical NMR spectra. The resonance frequencies in these spectra are determined by the chemical shift and J coupling interactions of the various nuclear spins in the sample, which are forcibly identical for the two enantiomers. To separate the NMR signals from two enantiomers in a racemic mixture it is necessary to render them diastereomeric either using a chiral medium3, via the reaction with a second enantiopure molecule4, or by the application of an oscillating electric field5.

To the best of our knowledge, it has never been observed that the solution NMR spectrum of a racemic mixture yields two sets of resonances for the two enantiomers within an ordinary NMR apparatus. This observation is consistent with Ramsey’s theory for magnetic shielding6, and its extension to J coupling7, which depends on factors which are invariant with respect to mirror symmetry operations.

We were thus extremely surprised when the aforementioned communication reported DFT calculated predicted differences in 1H-1H, 1H-13C, and 13C-13C J coupling constants between enantiomers that differed by more than 10 Hz2. In solution-phase NMR spectra J couplings are typically reported to a precision of 0.1 Hz, thus such differences in J coupling constants for enantiomers should be experimentally observable in racemic mixtures of enantiomeric compounds. We believe that there are simpler explanations as to why their DFT calculations predicted different coupling constants for enantiomers.

The first issue is that the molecular geometries used for DFT calculations on the D and L molecules (and provided in their Supplementary data) had significant structural differences. The coordinates used for calculations did not have mirror symmetry, thus, the calculations were not performed on true enantiomers. Shown in Fig. 1 are overlays of the geometries used for the L and D isomers for the twelve compounds studied. The coordinates of the D enantiomer were subjected to a reflection operation and the structures were overlaid using the Chem3D program. Figure 1 clearly shows that there are significant differences in the conformation of the backbone carbons, the orientation of the NH2 and COOH moieties, and in one case (alanine) an extra fluoride ion was added to the D enantiomer. Only three structures could be said to approximate mirror symmetry (glutamic acid, phenylalanine, and tyrosine).

Fig. 1: Overlays of the structures for twelve of the D (inverted coordinates) and L molecules that were used for DFT calculations reported by Bouchard and co-workers (ref. 2).
figure 1

None of the structures are related by mirror symmetry and thus they are not truly enantiomers. Nine of the twelve structures further have quite significant structural differences, as outlined in the red boxes. Specifically, models differ in a the addition of extra atoms (alanine), b backbone conformations (arginine, glutamine, glyceraldehyde, threonine), c orientation of the NH2 (alanine, cysteine, methionine, serine), or the orientation of the carboxyl (aspartic acid).

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However, even for glutamic acid, phenylalanine, and tyrosine, significant differences in computed J coupling values were reported for stereoisomers. Investigation of the values reported in the supplementary material, revealed there are likely significant issues with the calculations themselves. Namely, one-bond 13C-13C J coupling constants, which typically are of approximately 40 Hz, are reported to be around 170 Hz. We thus repeated the DFT calculations of the J coupling constants in D and L-alanine (the compound discussed in the main text) at the PBE0/TZ2P level of theory8 using the Amsterdam Density Functional (ADF) software9 and its highest density integration grid10 (known colloquially as ‘excellent’) (see Table 1). Spin-orbit coupling was also added using the zeroth-order regular approximation (ZORA)11,12, given that it was implied in the theory. Our calculations yielded J coupling constants that are closer in line with experiment. Most importantly, we also did not observe any difference between the computed values for the two enantiomers (Fig. 2).

Fig. 2: Relative differences between calculated J coupling constants in D- and L-alanine.
figure 2

a Data taken from ref. 2. Calculated values in (b) and (e) were obtained from the independent optimizations of the two enantiomers and highlight minor geometrical differences that arise due to the integration grid and finite convergence criteria while those in (c) and (f) are exactly mirror symmetric. In (c) and (f) some J coupling values for the enantiomers deviate by up to 30 μHz, likely due to the finite integration grid. In (d) are plotted the correlations between the calculated J coupling constants from this work and ref. 2. This plot highlights non-physical deviations for certain calculated J coupling constants.

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Table 1 Comparison of the DFT-calculated 13C-1H and 13C-13C J coupling constants for D- and L-alanine from this work and the prior publication
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Shown in Fig. 2 are bar graphs of the normalized difference in J coupling constants between the D and L enantiomers of alanine from the prior publication and this work. As can be seen, we can obtain small differences in J coupling constants of a few percent when performing independent geometry optimizations of the two enantiomers. These mHz differences in predicted J coupling constants likely arise due to the finiteness of the integration grid and the convergence criteria, but none of the differences are as large as those that were reported. If both structures are made to be true enantiomers of one another by applying a mirror symmetry operation to the molecular coordinates, then the difference in calculated J coupling constants vanishes.

As such, when DFT calculations are performed with larger basis sets, finer integration grids, and using mirror-symmetric isomers, the conclusions from the Bouchard and co-workers report are not reproduced. Software programs, such as ORCA13 and ADF9 do not predict differing J coupling constants for enantiomers with fully converged calculations. Note that such errors are unlikely to occur in achiral molecules owing to the use of identical internal coordinates by the software.

From a theoretical perspective, the internuclear spin-spin elements (\({D}_{{ij}}^{{kl}}\)) computed above between nuclear dipoles k and l, which contains both through-space (dipolar) and through-bond (J) contributions, arise from the following energy derivative:

$${D}_{{ij}}^{{kl}}=\frac{1}{2}\frac{{\partial }^{2}E}{\partial {\mu }_{i}^{k}\partial {\mu }_{j}^{l}}$$
(1)

In Eq. 1, E is the total energy of the system while \({\mu }_{i}^{k}\) denotes component I of the dipole moment on atom k.

The Hohenberg-Kohn theorems show that the total energy E may be written exactly as

$$E\left[\rho \right]=T\left[\rho \right]+{V}_{{ee}}\left[\rho \right]+{V}_{{ext}}\left[\rho \right]$$
(2)

where each term is a functional of the electron density. The terms are the kinetic energy T, the electron-electron energy Vee, and the external potential Vext, which here represents the interactions of the electrons with the atomic nuclei. Notice that this energy expression is exact, despite the fact that it is not known how to compute it exactly. In any case, the only term in it that depends on the atomic positions is the external potential, which can be represented in atomic units as follows, assuming N atoms and M electrons

$${V}_{{ext}}=\sum\limits_{k=1}^{N}\sum\limits_{m=1}^{M}\frac{-{Z}_{k}}{\left|{\vec{r}}_{m}-{\vec{R}}_{k}\right|}$$
(3)

In this form, electron m is at position \({\vec{r}}_{m}\), and nucleus k carries charge Zk and is located at \({\vec{R}}_{k}\). Notice that the nuclear dipole energies are not included here; that is because they are so small that they do not renormalize the electron interactions (in other words, their effect is so weak compared to the electrostatic interactions that they may be added perturbatively).

For any isolated molecule, the Hamiltonian, or here, the external potential, is invariant under the operations of the point group of the molecule. For a pair of enantiomers, mirror symmetry is not one of these operations. Instead, each enantiomer under reflection is transformed into the partner molecule, which is distinguishable from it, but crucially, all internuclear distances (and therefore also angles) in the external potential are unchanged. Therefore, the electron density of the transformed molecule will be identical to that of the original, except mirrored, and all bond distances and angles will be identical. Therefore, the total energy will be unchanged.

Furthermore, the terms \({D}_{{ij}}^{{kl}}\) will also be unchanged (albeit transposed14,15, so that the eigenvalues of the \({D}_{{ij}}^{{kl}}\) matrix are invariant), because they depend precisely on the locations the nuclear dipole moments, which are located at the nuclear sites. Because all bond distances and angles are unchanged, the couplings are as well. Put more mathematically, the couplings as derivatives depend only on local data at each nuclear site, which is identical between the two partners. This conclusion is identical for both “through bond” and “through space” coupling mechanisms. Additionally, the fact that nuclear dipoles are pseudovectors and thus carry information on orientation, unlike regular vectors, is not a concern here because the couplings are bilinear in the dipole moments. Hence, the couplings always depend on a pair of moments and possible additional sign changes under reflections are always cancelled.

To attempt to develop a mechanism that could potentially show a difference between enantiomers, we can argue along the lines of recent work on the natural optical activity tensor, which is essentially the spatial dispersion of the dielectric function16,17. In that case, a third order response must be computed, of the following form:

$${E}^{\left(3\right)}=\frac{{\partial }^{3}E}{\partial {q}_{m}\partial {E}_{i}\partial {E}_{j}}$$
(4)

in which the wavevector \(\vec{q}\) describes the spatial dependence, and the Ei, Ej are components of the electric field of the light that generates the dielectric response function. In the present case, we would like to find the spatial dispersion of the nuclear dipole coupling D, and so one is led to consider

$${E}^{\left(3\right)}=\frac{{\partial }^{3}E}{\partial {q}_{m}\partial {\mu }_{i}^{k}\partial {\mu }_{j}^{l}}$$
(4)

The necessary density functional perturbation theory for such expressions has been fully developed18, and implemented for a variety of responses, including optical activity16. Due to the “2n + 1” theorem18, in all these cases the necessary third-order energy may be obtained from knowledge of the zeroth and first order wavefunctions only. In the optical activity case, the first order wavefunctions arise from perturbing the zeroth order wavefunctions and charge density by the electric field, which is a strong perturbation. In the nuclear dipole case, however, the first order perturbation is due to the nuclear dipole interaction. First order perturbation theory provides the simple estimate that the wavefunction perturbation has size ||H1 | | /ΔE, where ΔE is the energy gap to the excited states. For the nuclear dipole interaction, ||H1 | | ∼J∼102 Hz, while the gap is typically at least 1 eV (2.418 × 1014 Hz), if not more. This estimate then gives a ratio of 10−13 or less, so such a third order mechanism, at least of this type, is surely too weak to show any difference between enantiomers.

We lastly think that it is necessary to comment on the calculations of 1H-1H J couplings in a DNA molecule2. These calculations make the following assertions that disagree with experiment and theory on many fronts:

  1. 1.

    In Fig. 2a of ref.2, they calculate 1H-1H spin-dipole J coupling constants of the order of 100,000 Hz (as explicitly stated in the caption to Fig. 2 in the original publication) for spin pairs separated by a very large number of bonds. 3JHH couplings are known to be roughly in the range from -5 to +15 Hz.

  2. 2.

    They suggest that J coupling constants should differ in spin pairs related by rotational symmetry. Specifically, that J coupling values at coordinates (φ1, φ2) and (φ1 + θ, φ2 + θ) should be different. This arises from the use of wavefunction that describes DNA as a conductor.

  3. 3.

    They do not calculate the isotropic average of the spin-dipole contribution, which is anisotropic, unlike Fermi contact, and instead only calculate a single tensor element (the ‘zz’ component), which would of course depend on the orientation of the spin pair.

  4. 4.

    They suggest that J coupling constants are invariant of the internuclear distance, as seen by the calculation of large J coupling constants when Δφ is large (Fig. 2a–d in the original publication).

  5. 5.

    They calculate that the Fermi contact term, which depends on s-orbital overlap, is largest when Δφ is largest, namely, when the two nuclei are separated by the greatest distance (Fig. 2c, d in the original publication).

  6. 6.

    They predict that the spin-dipole mechanism should dominate for 1H and 13C, which goes against well-known periodic trends in J coupling mechanisms19. For 1H and 13C, Fermi contact contributions to J couplings are most significant.

Most importantly, however, they did not demonstrate that the J coupling constant would change for the opposite helicity and so these calculations do not show enantiospecificity.

To conclude, we believe there are several significant issues with the recent work of Bouchard and co-workers. Their DFT calculations were performed on pairs of chiral molecules that were not truly enantiomers because they had different conformations. In one case, calculations were even performed on molecules with different formulas. Here we demonstrated that performing calculations with high-quality integration grids on true enantiomers of alanine results in identical J coupling constants, consistent with prior literature precedent. Consideration of the theory underlying the DFT calculation of such J couplings indicate that they are not in fact enantiospecific and have no significant dispersive response.

Methods

All density functional theory (DFT) calculations were performed using the Amsterdam Density Functional (ADF) software (ver. 2021.104)9 using the highest density integration grid10 (known colloquially as ‘excellent’). Calculations used the hybrid functional PBE08 with the TZ2P basis set. Spin-orbit relativistic effects were included using the zeroth-order regular approximation (ZORA)11,12. The structures for D- and L-alanine were optimized independently, their coordinates are as Supplementary Data. A single-point J coupling calculation was further done on a D-alanine model created by the inversion of the x coordinates form the optimized L-alanine geometry to have perfect mirror symmetry. Differences between the two structures originate from the convergence criteria and the finite integration grid. The calculated J coupling constants are given in Table 1 whereas the full calculation input and output files are given as Supplementary Data.