NMR and chiralityNMR, as described by D. Buckingham7, is “blind” to chirality since none of its standard parameters appear to be sensitive to it. Enantiomers display identical NMR spectra in an achiral environment. Thus, differentiating enantiomers using standard NMR techniques in the absence of a chiral resolvent or probe is challenging. We are aware of three methods to indirectly detect chirality by NMR: (1) chiral derivatizing agents (CDAs)8,9: These compounds react with a chiral substrate to produce diastereomers, which have distinct NMR spectra. For example, when a chiral alcohol reacts with a CDA like Mosher’s acid, the resultant diastereomeric esters can be distinguished by their NMR chemical shifts, revealing the absolute configuration of the alcohol. (2) Chiral Solvents9,10: In these solvents, enantiomers present slight differences in their NMR spectra due to unique interactions with the chiral environment. These differences can help deduce enantiomeric excess and sometimes the absolute configuration. (3) Chiral Lanthanide Shift Reagents11,12,13,14: These metal complexes can cause shifts in the NMR spectra of chiral compounds. Lanthanide ions, especially Eu, Yb, and Dy, have been used to distinguish the NMR signals of enantiomers by forming diastereomeric complexes detectable due to their differing chemical shifts. However, each of these methods has limitations. Mainly, they are indirect molecular effects that rely on external agents. To determine chirality conclusively, complementary analytical methods are often necessary. Alternatively, Buckingham, Harris, Jameson, and colleagues have proposed using electric fields for chiral discrimination7,15,16,17,18,19, though this remains to be demonstrated in experiments.Indirect NMR methods to distinguish enantiomers are less accessible and often more cumbersome than non-NMR methods such as chiral chromatography, high-performance liquid chromatography, gas chromatography, capillary electrophoresis, circular dichroism spectroscopy, optical rotatory dispersion, X-ray crystallography or vibrational circular dichroism. The development of a method to directly probe the chirality of a molecule using NMR, without reliance on external agents or indirect techniques, would be an important development in the fields of stereochemistry and analytical chemistry. Direct enantiomeric detection via NMR would uniquely combine non-destructive, quantitative capabilities with reproducibility, while concurrently bypassing the need for reactive chiral derivatizing agents, chiral solvents, and chromatography columns.CP and enantiospecificityThe experiments performed in refs. 4,5 demonstrated the existence of an enantiospecific response in CP. This effect was also observed recently by Bryce and co-workers20, but the authors argued that experimental artifacts such as particle size could contribute. Rossini and colleagues6 suggested that impurities, crystallization, and particle size effects likely contribute to the observation. Although such factors may influence the measurements, the data presented in refs. 6,20 does not rule out the contribution from CISS. As to CP, it is the bread-and-butter of solid-state NMR thanks to its ability to dramatically increase the sensitivity of experiments involving nuclei in low concentrations. CP is a technique where magnetization is transferred from an abundant, high gamma nucleus (I1) to a low gamma, dilute nucleus (I2) that is coupled to the I1 spin bath during a certain “contact” period21,22,23. During the contact time, radiofrequency (r.f.) fields for both I1 and I2 are turned on. Usually, the dominant magnetic coupling between pairs of nuclei is due to the magnetic dipole interaction. In the simplest solid-state NMR experiment, the enhanced magnetization of the dilute isotope is then detected while the abundant protons, or any other reference nuclei, are decoupled. The maximum gain in sensitivity is equal to the ratio of gyromagnetic ratios between the two nuclei (e.g., for 1H and 13C this ratio is approximately 4:1; a factor of 4 implies 16-fold SNR gains).The method of using heteronuclear double resonance to transfer coherence between nuclei in a two-spin system was introduced by Hartmann and Hahn21,22,23,24 and has since become widely employed in solid-state NMR. It is possible to do highly selective recoupling among nuclei25,26. Spectroscopists can also modulate the amplitude of spin-locking pulses to enhance CP dynamics, perform Lee-Goldburg decoupling to reduce homonuclear proton couplings during spin-locking or apply multiple-quantum CP to half-integer quadrupole systems27,28,29. CP is a highly useful experiment that facilitates high-resolution NMR in the solid state encompassing key principles of dipolar coupling (decoupling/recoupling) and MAS30.The working principle of CP is illustrated in Fig. 1a. If two nuclear spins I1 and I2 with gyromagnetic ratios γI1 and γI2, respectively, are placed in an external magnetic field B0, they will be able to absorb r.f. photons at frequencies γI1B0 and γI2B0, respectively, according to the Zeeman effect. They will not be able to exchange energy spontaneously, since the two frequencies γI1B0 and γI2B0 are different. If instead a bimodal oscillating r.f. field is applied at these two frequencies, with amplitudes such that ωI1 = ωI2 (Hartmann-Hahn condition24), where ωI1 = γI1B1,I1 and ωI2 = γI2B1,I2. In the “doubly rotating frame” generated by these frequencies both nuclear spins I1 and I2 appear stationary. Photons can now be absorbed by either of these spins at the same frequency ωI1 = ωI2, the condition for resonant energy transfer.Fig. 1: Indirect nuclear spin-spin (J) coupling enables cross-polarization in NMR.The CISS effect gives rise to delocalized conduction bands. Delocalized electrons can in turn mediate indirect nuclear spin-spin couplings. a In the cross-polarization experiment of solid-state NMR energy transfers between heteronuclei are forbidden in the lab frame. Application of a bichromatic RF field oscillating at the resonance frequencies of both nuclei, enables energy transfer. At the Hartmann-Hahn condition γI1B1,I1 = γI2B1,I2, resonant energy transfer will lead to transfer of polarization from the cold to hot spin systems. b Indirect spin-spin coupling between two nuclei is mediated by conduction electrons. c Model for DNA helix, helicoidal coordinates (a, b, φ) and two nuclear spins I1, I2 and their corresponding positions φ1, φ2 along the helix. R is the distance whereas Δφ is the angle between consecutive nucleotides. a is the helix radius and b is its pitch.The CP experiment is often described using the concept of spin temperature31,32,33. The abundant spin system is prepared with an artificially low temperature. This is typically done by applying a π/2-pulse on the abundant nuclei, followed by a spin-locking field31. One then allows the dilute system to come into thermal contact with the cold system of abundant spins. Contact is typically established through the magnetic dipole-dipole interaction between nuclei. Heat flows from the sparse spin system to the cold abundant spins, which produces a drop in the temperature of the sparse spins. Physically, we observe resonance energy transfer if the natural frequencies of the two systems are close. This was Hahn’s ingenious concept24. This experiment requires the heat capacity of the abundant system to be larger than that of dilute spins. In the context of such experiments, to say that the spin temperature has dropped is nearly equivalent to the statement that population difference between the ground \(\left\vert g\right\rangle\) and excited \(\left\vert e\right\rangle\) states is increased, which leads to an increased sensitivity of the NMR experiment.In the ref.4,5 different efficiencies of CP were obtained depending on the choice of enantiomer. The existence of an enantiospecific bilinear coupling (see Fig. 1b) of the form \({{{\bf{I}}}}_{1} \cdot {{{\mathbf{\sf{F}}}}} \cdot {{{\bf{I}}}}_{2}\) was postulated in those papers, where the coupling tensor \({{{\mathbf{\sf{F}}}}}\) depends on Rashba SOC, an interaction which is itself enantiospecific. Herein we argue that bond polarization and SOC provides a possible mechanism to mediate the interaction between two nuclear spins through the creation of enantiospecific delocalized electron conduction bands which, in turn, enable these electrons to couple to both nuclear spins simultaneously via magnetic dipole interaction.A summary of all known CP results on enantiomers published to date (see refs. 4,5) is shown in Table 1. According to the traditional view, NMR parameters are not supposed to depend on the handedness of enantiomers; therefore, the ratio I(D)/I(L) should be 1. Instead, for all CP-MAS results a clear trend I(D)/I(L) > 1 is observed indicating that the D enantiomer consistently inherits more polarization compared to the L enantiomer. This goes against all known mechanisms describing nuclear spin interactions in a diamagnetic molecule.Table 1 Summary of solid-state NMR experimental results from refs. 4 and 5 on CP of enantiomers of several different molecular structuresSantos and co-workers4,5 postulated the existence of an effective nuclear spin-spin interaction mediated by SOC because the effective strength of the SOC interaction in molecules exhibiting CISS is enantiospecific (Fig. 1b), leading to different transmission probabilities for the two values of electronic spin. However, the precise mechanism remains elusive, as SOC itself does not couple directly to nuclear spins, as far as fundamental interactions are concerned. SOC only directly affects the electronic wavefunction. We must then turn our attention to the nature of effective interactions affecting these nuclear spins. The hyperfine interaction defines the manner in which nuclear spins couple to electron spins. From this emerges a possible mechanism. The electron-nuclear hyperfine interaction is made up of three contributions: Fermi contact, electron-nuclear dipole interaction, and nuclear spin-electron orbital angular momentum. The first two interactions provide a possible mechanism for spin-spin coupling, albeit indirectly. Indirect couplings in NMR, also known as J couplings, were discovered independently by Hahn and Maxwell34 as well as McCall, Slichter, and Gutowski35. While initially discovered in liquids, Slichter36 has presented a theory for the solid state. For 3D Bloch wavefunctions in a solid the case of the Fermi contact interaction is discussed in ref. 36 whereas the case of the dipole-dipole interaction is discussed in Bloembergen and Rowland37. These theories, however, do not incorporate in any way the effects of chirality. We propose instead to investigate the following two pathways:$${{{\rm{nuclear}}}}\, {{{\rm{spin}}}}\,1 \quad \mathop{\longleftrightarrow}\limits^{{{\rm{dipole-dipole}}}} \quad \mathop{{{{\rm{conduction}}}}\, {{{\rm{electron}}}}\, {{{\rm{spins}}}}}\limits^{{({{{\rm{wavefunction}}}}\, {{{\rm{is}}}}\, {{{\rm{enantiospecific}}}})}} \quad \mathop{\longleftrightarrow}\limits^{{{\rm{dipole-dipole}}}} \quad {{{\rm{nuclear}}}}\, {{{\rm{spin}}}}\,2$$ and$${{{\rm{nuclear}}}}\, {{{\rm{spin}}}}\,1 \quad \mathop{\longleftrightarrow}\limits^{{{{\rm{Fermi}}}}\, {{{\rm{contact}}}}} \quad \mathop{{{{\rm{conduction}}}}\, {{{\rm{electron}}}}\, {{{\rm{spins}}}}}\limits^{{({{{\rm{wavefunction}}}}\, {{{\rm{is}}}}\, {{{\rm{enantiospecific}}}})}} \quad \mathop{\longleftrightarrow}\limits^{{{{\rm{Fermi}}}}\, {{{\rm{contact}}}}} \quad {{{\rm{nuclear}}}}\, {{{\rm{spin}}}}\,2.$$A classic example of chiral molecule is the DNA helix. DNA is also amenable to simple modeling. The hypothetical case of indirect coupling of nuclear spins I1 and I2 in a DNA molecule is illustrated in Fig. 1c.The key observation in the present work is that the electronic wavefunction in CISS differs from normal 3D Bloch wavefunctions (e.g.,36) in that it is enantiospecific38,39. Enantiospecificity is related to the SOC interaction and helicity, which takes into account the direction of electron propagation. Another difference is the 1D nature of helical molecules, giving rise to 1D wavefunctions in a band structure model38,39. The physics of one-dimensional systems involves unique mathematical considerations. In Supplementary Text S1 we present a detailed theoretical treatment of the indirect coupling between pairs of nuclear spins in a helical molecule based on spin-dependent mechanisms (electron-nuclear dipole-dipole, Fermi contact). The main result is that both interactions are sufficiently strong to cause observable CP. The electron-nuclear dipolar contribution to the effective coupling tensor (derived in Supplementary Text S1) depends on chirality. Amplitude estimates are shown in Fig. 2a, where coupling strengths between pairs of nuclear spins (assumed to be protons for simplicity) can reach amplitudes that generate observing measurable effects by CP3 for specific positions of the nuclear spins. The coupling strength depends on the position (φ1, φ2) of the nuclear spins along the helix. We remark that this calculation should not be considered quantitative due to the one-dimensional nature of the problem, which leads to the emergence of divergences. This calculation should instead serve to establish the plausibility of the mechanism. As to the Fermi contact interaction, it is generally weaker than dipole-dipole (see Fig. 2b), yet sufficiently strong to produce measurable effects3. Weak Fermi contact interactions are generally due to low overlap of the electronic wavefunction at the site of the nuclei, possibly due to a stronger contribution from p-wave character of the wavefunction38,39 than s-wave36. However, as explained in SI for the case of high-field NMR the Fermi contact tensor is not enantioselective. The dipole-dipole term, on the other hand, is. This analysis applies to the DNA toy model only. The situation could be different for real chiral molecules and an independent analysis is warranted on a case-by-case basis.Fig. 2: Plots of indirect nuclear spin-spin coupling tensor component \({{{{{{\boldsymbol{{F}}}}}}}}_{zz}\) as a function of the position of nucleus 1 and 2 along the DNA chain (φ1, φ2 ∈ [0, 2π], with 0 indicating the start and 2π the end of the helix).Multiplication by 0.01 gives the coupling strength in Hz. a Contribution from the magnetic dipole interaction. Peak coupling strengths attain 100 kHz (white regions, right panel). b Contribution from the Fermi contact interaction (values should be multiplied by 0.01 to get units of Hz).We sketch the main steps of the derivation presented in SI. An effective Hamiltonian is derived using second-order perturbation theory:$$\begin{array}{rcl} {{{\mathcal{H}}}}_{eff} &=& \underbrace{\left(\frac{2 \mu_0}{3} \right)^{2} \gamma_I^{2} \gamma_{S}^{2} {\hbar}^{4} \sum_j {{{\mathbf{I}}}}_1 \cdot \frac{ \langle{ 0 | \sum_l {{{\mathbf{S}}}}_l \delta^{(3)}({{{\mathbf{r}}}}_l – {{{\mathbf{R}}}}_1) |\,j}\rangle \langle\;{j| \sum_l {{{\mathbf{S}}}}_l \delta^{(3)}({{{\mathbf{r}}}}_l – {{{\mathbf{R}}}}_2)|0}\rangle }{ E_0 – E_j } \cdot {{{\mathbf{I}}}}_2+c.c.}_{{{{\mathcal{H}}}}_{eff}^{FC}}. \\ &&+\underbrace{\left( \frac{\mu_0}{4\pi}\right)^{2} \gamma_{I}^{2} \gamma_{S}^{2} {\hbar}^{4} \, p.v. {\sum}_{\alpha,\alpha^{\prime}} {\sum}_{\beta,\beta^{\prime}} \sum_j I_{1}^{\alpha} \frac{ \left\langle{0 \left| \sum_l \frac{ \delta_{\alpha\beta} – 3 {\hat{\tilde{r}}}_{1l,\alpha} {\hat{\tilde{r}}}_{1l,\beta}}{ |{{{\mathbf{R}}}}_1-{{{\mathbf{r}}}}_l|^3} S_l^\beta \right| \,j }\right\rangle \left\langle{ j \left|\sum_l \frac{ \delta_{\alpha^{\prime}\beta^{\prime}} – 3 {\hat{\tilde{r}}}_{2l,\alpha^{\prime}} {\hat{\tilde{r}}}_{2l,\beta^{\prime}}}{ |{{{\mathbf{R}}}}_2-{{{\mathbf{r}}}}_l|^3} S_l^{\beta^{\prime}} \right|0}\right\rangle}{E_0-E_j} I_2^{\alpha^{\prime}}+c.c.}_{{{{\mathcal{H}}}}_{eff}^{DD}} \end{array}$$
(1)
The term on the first line describes the effects of the Fermi contact interaction (\({{{{{{\mathcal{H}}}}}}}_{eff}^{FC}\)), whereas term on the second line, the effects of the dipole-dipole interaction (\({{{{{{\mathcal{H}}}}}}}_{eff}^{DD}\)). The Varela spinors38,39, which were recently obtained by solving a minimal tight-binding model constructed from valence s and p orbitals of carbon atoms, describe the molecular orbitals of helical electrons in DNA molecules. These spinors can be used to compute the summations by considering them as the electronic states \(\left\vert j\right\rangle\):$${{{{{{\boldsymbol{\psi }}}}}}}_{n,s}^{\nu,\zeta }=\left[\begin{array}{c}{F}_{A}{e}^{-i\varphi /2}\\ \zeta {F}_{B}^{*}{e}^{i\varphi /2}\end{array}\right]{e}^{i\nu \tilde{n}\varphi },\quad {F}_{A}=\frac{\sqrt{s}}{2}(s{e}^{i\theta /2}+{e}^{-i\theta /2}),\quad {F}_{B}=\frac{\sqrt{s}}{2}(s{e}^{-i\theta /2}-{e}^{i\theta /2}),$$where \(\tilde{n}\) is analogous to a wavenumber, φ is the angular coordinate along the helix, θ depends on SOC and is the tilt of the spinor relative to the z axis, s = ± 1 is the electron spin orientation and ζ = ± 1 labels the enantiomer. By keeping track of ζ we can determine which contribution(s) depend on enantiomer. This leads to the result$${{{{{{\mathcal{H}}}}}}}_{eff}^{DD}={\left(\frac{{\mu }_{0}}{4\pi }\right)}^{2}{\gamma }_{I}^{2}{\gamma }_{S}^{2}\sum\limits_{n,{n}^{{\prime} }}\sum\limits_{\alpha,\beta }\sum\limits_{{\alpha }^{{\prime} },{\beta }^{{\prime} }}{I}_{1}^{\alpha }\frac{{M}_{1,\beta }^{\alpha \beta }({\tilde{n}}^{{\prime} },\tilde{n}){M}_{2,{\beta }^{{\prime} }}^{{\alpha }^{{\prime} }{\beta }^{{\prime} }}(\tilde{n},{\tilde{n}}^{{\prime} })}{| T| ({n}^{{\prime} }-n)}{I}_{2}^{{\alpha }^{{\prime} }}f(\tilde{n})[1-f({\tilde{n}}^{{\prime} })]+c.c.$$
(2)
where μ0, γI, γS, ∣T∣ are constants, \(f(\tilde{n})\) is a Fermi function, \({I}_{1}^{\alpha }\) are nuclear-spin operators (see SI) and explicit expressions for the matrices \({M}_{1,\beta }^{\alpha \beta }({\tilde{n}}^{{\prime} },\tilde{n})\)’s are given in Supplementary (SI) equations 2–4.This expression for the indirect coupling is enantiospecific. The effective Hamiltonian contains a product \({M}_{1,\beta }^{\alpha \beta }({\tilde{n}}^{{\prime} },\tilde{n}){M}_{2,{\beta }^{{\prime} }}^{{\alpha }^{{\prime} }{\beta }^{{\prime} }}(\tilde{n},{\tilde{n}}^{{\prime} })\). Explicitly, this term is:$$\mathop{\sum}\limits_{\beta,{\beta }^{{\prime} }}{M}_{1,\beta }^{\alpha \beta }({\tilde{n}}^{{\prime} },\tilde{n}){M}_{2,{\beta }^{{\prime} }}^{{\alpha }^{{\prime} }{\beta }^{{\prime} }}(\tilde{n},{\tilde{n}}^{{\prime} }) = \left[{M}_{1,x}^{\alpha,1}({\tilde{n}}^{{\prime} },\tilde{n})+{M}_{1,y}^{\alpha,2}({\tilde{n}}^{{\prime} },\tilde{n})+{M}_{1,z}^{\alpha,3}({\tilde{n}}^{{\prime} },\tilde{n})\right]\hfill\\ \times \left[{M}_{2,x}^{{\alpha }^{{\prime} },1}(\tilde{n},{\tilde{n}}^{{\prime} })+{M}_{2,y}^{{\alpha }^{{\prime} },2}(\tilde{n},{\tilde{n}}^{{\prime} })+{M}_{2,z}^{{\alpha }^{{\prime} },3}(\tilde{n},{\tilde{n}}^{{\prime} })\right].$$
(3)
While Mi,z is independent of ζ, both Mi,x and Mi,y depend linearly on ζ (see equations 2-4 in SI). The term \({M}_{1,z}^{\alpha,3}({\tilde{n}}^{{\prime} },\tilde{n}){M}_{2,z}^{{\alpha }^{{\prime} },3}(\tilde{n},{\tilde{n}}^{{\prime} })\) does not depend on ζ, since neither factor depends on ζ. Neither do \({M}_{1,x}^{\alpha,1}({\tilde{n}}^{{\prime} },\tilde{n}){M}_{2,x}^{{\alpha }^{{\prime} },1}(\tilde{n},{\tilde{n}}^{{\prime} })\) and \({M}_{1,y}^{\alpha,2}({\tilde{n}}^{{\prime} },\tilde{n}){M}_{2,y}^{{\alpha }^{{\prime} },2}(\tilde{n},{\tilde{n}}^{{\prime} })\) since ζ2 = 1. On the other hand, terms such as M1,zM2,x depend linearly on ζ. The effect of enantiomer handedness is to flip the sign of this term, leading to a change in the magnitude of the indirect coupling mediated by dipole-dipole interaction. As explained in SI (and as seen in Fig. 2a) the magnitude of this term depends on the exact relative positions of the two nuclei of interest along the helix.As mentioned earlier, the spin-dependent coupling mechanism could be different for real chiral molecules. For the amino acids in Table 1, analytical expressions for the spinors of electronic states, which are essential for the computation of J couplings, are not available to us. We can instead use DFT calculations. In Fig. 3 we present calculations of J couplings between 1H and 13C nuclei for the two (D, L) enantiomers of alanine. As seen in the bar plot of Fig. 3a, significant relative differences in the J couplings between enantiomers can be observed. In SI we include DFT results for the remaining amino acids: phenylalanine, arginine, aspartic acid, cysteine, glutamic acid, glutamine, glyceraldehyde (non-amino acid), methionine, serine, threonine, tyrosine, and valine. There, we find that J coupling values depend on the choice of enantiomer for all the molecules.Fig. 3: Differences in NMR J couplings between (\({\mathsf{D}},{\mathsf{L}}\)) enantiomers can be quantified using the J coupling stereochemical deviation, \([J({\mathsf{L}})-J({\mathsf{D}})]/[J({\mathsf{L}})+J({\mathsf{D}})]/2\).Nonzero values of this relative difference constitute evidence of chiral selectivity of the scalar coupling. J couplings between 1H and 13C nuclei were computed by DFT using ORCA for the amino acids: alanine, arginine, aspartic acid, cysteine, glutamic acid, glutamine, glyceraldehyde, methionine, phenylalanine, serine, threonine, tyrosine and valine (see SI for results). The case of alanine is shown here: a J coupling stereochemical deviation b labeling of atoms in alanine.
Enantiospecificity in NMR enabled by chirality-induced spin selectivity
