Direct enantiomeric discrimination through antisymmetric hyperfine coupling

Abstract

Chiral open-shell molecules possessing permanent electric dipole moments have an EPR signal at the difference frequency of the electron and nuclear resonances, allowing direct enantiomeric discrimination by signal phase. The effect depends on the vector antisymmetry of the hyperfine coupling. Quantum chemistry suggests chiral bisfluorene methyl radical derivatives as promising for experiments.

---

The hyperfine coupling (HFC) between the spins of an unpaired electron and a nucleus provides important information about the molecular structure and interactions.^1,2 However, the techniques used in electron paramagnetic resonance (EPR) spectroscopy can so far only indirectly be applied to distinguish the enantiomers of chiral molecules, e.g., by transforming the molecule to a diastereoisomeric entity by chiral environment such as a gel or a chiral solvent, or by conducting a reaction with a chiral derivatizing agent.^3,4 In this letter we propose a new effect that allows direct chiral discrimination from the vector of antisymmetric HFC, A*. Typically the sensitivity of measurements of the electron spin transitions surpasses that of nuclear transitions by several orders of magnitude.⁵ Thus, the effect caused by A* puts us in a good position to achieve direct sensitivity to chirality in EPR. Ref. 6–11 discuss other chirality-sensitive nuclear spin effects. An alternative possibility provided by the chiral-induced spin selectivity effect (CISS) is presented in ref. 12 and 13.

The spin Hamiltonian for an unpaired electron S interacting with a nucleus I,¹⁴

includes the HFC of the corresponding spin operators Ŝ and Î,and the Zeeman interactions with the magnetic field B₀:Here, A (in MHz) is the HFC tensor, and g_S and g_I are the g-tensors of S and I. Overlooking the very small parity non-conservation effects,¹⁵eqn (1) holds for both enantiomers.

For a liquid sample of molecules with permanent electric dipole μ_e, its isotropic average due to at temperature T is

with Ω denoting the molecular orientation and assuming that the eigenvalues of are small compared to k_BT, and hence,

〈μ^c_e〉 = 〈μ^c,A_e〉 + 〈μ^c,B_e〉, (6)

〈μ^c,A_e〉 = −hA^cŜ × Î, (7)

〈μ^c,B_e〉 = (μ_Bg^c_SŜ + μ_Ng^c_IÎ) × B₀. (8)

with the pseudoscalarsThe two-index tensors A, g_S, and g_I can be decomposed as the sums of three irreducible tensors: the rank-0 isotropic (e.g., A^iso), the rank-1 antisymmetric (A^anti) and the rank-2 (traceless) symmetric (A^sym) parts.¹⁶ We have in eqn (9)–(11) expressed A^anti as the vector A* whose components are .

While μ_e is a polar vector, A* is axial, since the part of the Hamiltonian relevant to A* transforms under inversion as A*·(Ŝ × Î) →(−A*)·{−[(−Ŝ) × (−Î)]}.¹⁷ Therefore, the product μ_e·A* changes its sign under mirror reflection of the molecule and, consequently, the pseudoscalar A^c has opposite signs for enantiomers. It follows that, in an achiral dipolar molecule, μ_e·A* = 0, hence μ_e and A* are perpendicular to each other and A^c = 0. The same applies to the pseudoscalars g^c_S and g^c_I drawn from the g-tensors g_S and g_I (Fig. 1A).

Fig. 1 (A) Permanent electric dipole moment (μ_e) and the vector antisymmetries of the hyperfine coupling tensor (A*) and g-tensors of the electron and the nucleus ( and ) for a chiral molecule and its mirror image. (B) Expected signals of the enantiomers due to the precession of the magnetization (M, red color) and the oscillation of the chirality-sensitive electric polarization (P^c_A, blue, and P^c_B, green) for two different initial states of the spin system. The total signal shown in black.

When a sample is excited by electromagnetic field, the observed signal arises from the electron and nuclear magnetization

and the chirality-sensitive electric polarizationHere, [scr N, script letter N] is the number density of the molecules and [small rho, Greek, circumflex](t) is the density matrix of the spin system. The expected signal is proportional to the time derivative of the magnetization and the polarization asB₁ and E₁ are the amplitude vectors of the magnetic and electric field, respectively, which can be generated by the detector at frequency ν.¹⁸ The integration is over the volume of the sample.

Assume B₀ along the laboratory z axis. Due to the vector products in eqn (7) and (8), the observation of P^c is only possible for states whose spin vectors have a non-vanishing component perpendicular to B₀ (e.g., Ŝ_y) or possess mutually perpendicular components (e.g., Ŝ_yÎ_x).

The time-dependence of the density operator [small rho, Greek, circumflex](t) is found by solving the Liouville–von Neumann equation

with the appropriate initial condition [small rho, Greek, circumflex](0). At high field, the Hamiltonian [eqn (1)] can be approximated aswhere ħω_S = −μ_Bg^iso_SB₀ and ħω_I = −μ_Ng^iso_IB₀.¹⁹ Let us next consider two experiments beginning from the thermal equilibrium

[small rho, Greek, circumflex]_eq = 1/4 + 〈Ŝ_z〉_eqŜ_z + 〈Î_z〉_eqÎ_z + 〈Ŝ_zÎ_z〉_eqŜ_zÎ_z, (17)

where 〈Ŝ_z〉_eq = −ħω_S/(4k_BT), 〈Î_z〉_eq = −ħω_I/(4k_BT), and 〈Ŝ_zÎ_z〉_eq = −hA^iso/(4k_BT).

Case I. A pulse of the B₁ field on [small rho, Greek, circumflex]_eq generates the state

[small rho, Greek, circumflex]₁(0) = −〈Ŝ_z〉_eqŜ_y + 〈Î_z〉_eqÎ_z − 〈Ŝ_zÎ_z〉_eqŜ_yÎ_z. (18)

Solving eqn (15) one finds

[small rho, Greek, circumflex]₁(t) = [cos(πA^isot) sin(ω_St)Ŝ_x + sin(πA^isot) cos(ω_St)Ŝ_xÎ_z] 〈Ŝ_z〉_eq +… (19)

with the detection defined by B₁ = B₁e_x and E₁ = E₁e_y, M_x and P^c_y in eqn (12) and (13) may contribute to the signal. In eqn (19), both terms that (i) do not contribute to [scr S, script letter S](ν) and/or (ii) are much smaller than 〈Ŝ_z〉_eq, i.e., proportional to 〈Ŝ_zÎ_z〉_eq, are omitted. Then, the expected signal isThe signals of P^c appear at the same frequencies as M, forcing discrimination between the two require the measurement of the amplitude difference of the components of the EPR doublet (Fig. 1B, the first spectrum).

Case II. For a different initial spin state

[small rho, Greek, circumflex]₂(0) = −〈Ŝ_z〉_eqŜ_yÎ_x + 〈Î_z〉_eqÎ_x − 〈Ŝ_zÎ_z〉_eqŜ_y, (21)

which can be obtained by applying the pulse sequence on [small rho, Greek, circumflex]_eq, one finds the density matrixIf the electric field detection is changed into E₁ = E₁e_z, the signal becomes[small rho, Greek, circumflex]₂(t) only generates signal of the chirality-sensitive P^c_A at the difference frequency ω_S − ω_I. This allows to distinguish the postulated chirality-sensitive effect from other, nonchiral effects. P^c_A has the same direction as B₀ (Fig. 1B, the second spectrum).

The phases and amplitudes of B₁ and E₁ depend on the particular experimental implementation (see ref. 20 for examples). If the electronic relaxation time is long in comparison with the pulse, one can presume that the case II, i.e., [small rho, Greek, circumflex]₂(0) ∝ Ŝ_yÎ_x, represents a promising initial state. Spin relaxation toward the thermodynamic equilibrium causes Ŝ_z and Î_z to appear. However, being parallel to B₀, they do not generate any observable signals. One can hypothesize the inverse effect to that described above, i.e., that excitation by oscillating electric field would cause changes of the spin state that are dependent on the molecular chirality. Observing the inverse effect would be, however, complicated by dielectric heating since, due to the smallness of chirality-sensitive effects, an electric field of high amplitude is required.

We study neutral organic radicals depicted in Fig. 2: nitroxyls, i.e., 2,2,6,6-tetramethyl-1-piperidinyloxy (1a; TEMPO), the TEMPO derivative of (R)-alanine [(R)-1b], 1,3,5-trimethyl-6-oxoverdazyl (2a), and the verdazyl derivative of (R)-phenylalanine [(R)-2b], as well as carbon-centered radicals: triphenylmethyl (3a), bis(fluoren-9-yl)methyl (3b), and bisfluorene methyl derivative of (R)-alanine [(R)-3c]. 3a and 3b, whose rigid structures are non-superposable on their mirror images, do not exhibit chiral properties due to fast interconversion of their enantiomers. Chiral samples are 1b, 2b, and 3c.

Fig. 2 Antisymmetric parts of the hyperfine coupling tensors A* and permanent electric dipole moments μ_e of the studied radicals.

The molecular structures were computationally optimized as described in ESI.‡ The A tensors were calculated using unrestricted density-functional theory (DFT) at the fully relativistic matrix-Dirac–Kohn–Sham (mDKS) level^21,22 on the ReSpect code,^23,24 using mainly the PBE0 hybrid functional²⁵ and the pcH-2 large-component basis sets of ref. 26. See ESI‡ for further details. The results of computed eigenvalues of A are in a reasonable agreement which literature (Table S9 in the ESI‡). The experiments have been at various temperatures and physical states, whereas the computations were performed for single molecules at the equilibrium geometry in vacuo. However, such differences can easily be masked by the dependence of A on the chosen functional.^27,28

In contrast to the antisymmetry g_S* of the electronic g-tensor, which is a few times larger than the nuclear g_I* (Tables S10 and S11 in ESI‡ and ref. 29, 30), the antisymmetry A* of the HFC tensor is larger by about three orders of magnitude than the antisymmetric part of the indirect spin–spin coupling tensor in diamagnetic molecules.³¹ The direction of A* and its magnitude are mainly determined by the local electronic structure of the molecule and vary weakly when a chiral substituent is placed in a distal position with respect to the maximum of the unpaired electron density (Table 1; all components of HFC tensors and dipole moments are given in Tables S12–S20 of the ESI‡). E.g., A_N* of the nitroxyl nitrogen and A_C* of the adjacent carbons differ by less than 1% in 1a and (R)-1b. Similarly, the antisymmetries A_N* of radicals 2a and (R)-2b are almost unaffected by the chiral amino acid substituent.

Table 1 Permanent electric dipole moment μ_e (D), the isotropic part A_iso (MHz) and the length of antisymmetry vector, A* (kHz), of the hyperfine coupling tensor, and the pseudoscalar A^c (µHz m V⁻¹). A^c is only listed for chiral species whose enantiomers do not interconvert in solution

Radical	μ e	Nucleus/position	A iso	A ^*	A ^c
1a	3.03	^15N	−51.3	161.8
		^13C̲NO˙	−9.4	203.8
(R)-syn-1b	4.85	^15N	−51.4	166.9	−30.5
		^13C̲NO˙	−9.6	199.9	27.3
(R)-anti-1b	4.80	^15N	−51.6	166.9	49.4
		^13C̲NO˙	−9.5	200.2	−31.5
2a	1.12	^15N	−22.6	46.4
(R)-2b	2.23	^15N	−22.6	49.5	2.3
		^13C	−32.5	21.9	−8.3
3a	0.000193	^13C	−11.2	9.8
3b	0.0375	^13C̲C˙H	40.7	12.2
		CC˙^1H	42.8	23.3
(R)-3c	2.69	^13C̲C˙C	43.3	13.5	−6.4
		C^13C̲˙C	−40.7	11.3	4.4
		CC^13˙C̲	43.7	15.9	6.9

---

Chiral substituent may noticeably change the permanent electric dipole μ_e. For an achiral molecule, μ_e is perpendicular to A^* [eqn (9)]. The sum of the scalar products μ_e·A^* of two equivalent nuclei may vanish also in a chiral system. E.g., in 1a (Fig. 2), μ_e is in the σ_v plane of the molecule, whereas of the nitroxyl nitrogen is perpendicular to that plane. While the and of the adjacent carbons are not perpendicular to the σ_v plane, their projections onto this plane cancel, and the overall chirality-sensitive effect vanishes. However, if the molecule were isotopically ¹³C-labeled at the C position with ¹²C at the C′ position, 1a would be chiral and the predicted effect might be observed.

To maximize A^c, one has to orient the chiral substituent such that μ_e is maximally parallel to A*. For (R)-1b, the (R)-alanine substituent is placed at the opposite site to the nitroxyl group and its presence causes a tilt of μ_e. However, (R)-1b has two dominating conformers: syn with the carbonyl oxygen atom nearby the CH₃ group and anti, where the CO and CH₃ groups are distant (Fig. S1 in ESI‡). The dipole moments μ_e of the two conformers are oppositely tilted, thus A^c_N,syn ≈ −A^c_N,anti and the chirality-sensitive effect is greatly suppressed. While A^c_C and cancel out entirely for the isotopically unsubstituted molecule (vide supra), the effect of the substitution on A^iso_C is of the order of a fraction of MHz and, therefore, the chiral signal would presumably average to almost zero. A similar result is obtained for the nitrogen atoms of (R)-2b.

Comparison of 3a and 3b with (R)-3c shows that the carbon-centered radicals require a chiral substituent that ensures a sufficiently large μ_e. of 3b is only weakly affected by the (R)-alanine group but, due to more favorable orientation of μ_e with respect to , A^c_C reaches a much higher amplitude of 10–15 kHz. In this case, the amplitude of the expected chiral effect is at last 5 nHz m V⁻¹, meaning that its detection by the equipment currently used in electron-nucleus double resonance experiments is feasible. When X-band or W-band () EPR is applied, the signal of P^c_A will not overlap with any residual standard EPR signal, which could be present due to the finite purity of the desired initial state, i.e., −S_yI_x.

A further advantage of the 3c over 1b or 2b is the more efficient generation of the initial spin state, since the expected relaxation time of 3c is approx. 10 µs based on T₂ for an unsubstituted 3c.³² This is comparable with the length of the π/2 pulse, whereas the typical TEMPO relaxation time does not exceed several dozen ns. Assuming that the ratio between the electric and magnetic fields of the detector is cB₁/E₁= 10³ and selectively ¹³C-enriched 3c, one finds that the amplitude of the expected chirality-sensitive signal of P^c_A is of the order of 10⁻⁴ of the standard EPR signal.

To conclude, it is predicted that with a suitably chosen initial spin state, the electron-nucleus system generates an electric polarization P^c whose phase directly identifies the handedness of the molecule. P^c oscillates at the difference frequency of the electron and nuclear resonances in the direction of the main, static magnetic field. Largest magnitude of the effect is obtained if the permanent electric dipole and the antisymmetric HFC tensor, represented as a vector, are parallel to each other. This condition is fulfilled by a chiral derivative of bisfluorene 3c, which is a good candidate for forthcoming experiments. We believe that the predicted effect has potential for EPR investigations of chiral systems.

PG acknowledges the National Science Centre, Poland, for the financial support through OPUS 16 Grant No. 2018/31/B/ST4/02570 and Bartosz Kreft (U. Warsaw) for his help in computations. JV has received funding from the Academy of Finland (grant 331008) and U. Oulu (Kvantum Institute). Computations were carried at CSC-the Finnish IT Centre for Science and the Finnish Grid and Cloud Infrastructure project (persistent identifier urn:nbn:fi:research-infras-2016072533).

Note added after first publication

This article replaces the version published on 29th July 2021, which contained errors in eqn (2), (3), (4), (7), (8) and (9).

Conflicts of interest

There are no conflicts to declare.

Notes and references

1. K. Möbius, W. Lubitz, N. Cox and A. Savitsky, Magnetochemistry, 2018, 4, 50.
2. M. M. Roessler and E. Salvadori, Chem. Soc. Rev., 2018, 47, 2534–2553.
3. H. B. Stegmann, H. Wendel, H. Dao-Ba, P. Schuler and K. Scheffer, Angew. Chem., Int. Ed. Engl., 1986, 25, 1007–1008.
4. C. D. Stevenson, A. L. Wilham and E. C. Brown, J. Phys. Chem. A, 1998, 102, 2999–3001.
5. G. A. Rinard, R. W. Quine, S. S. Eaton and G. R. Eaton, in Frequency Dependence of EPR Sensitivity, ed. L. J. Berliner and C. J. Bender, Springer US, Boston, MA, 2004, p. 118.
6. P. Garbacz and A. D. Buckingham, J. Chem. Phys., 2016, 145, 204201.
7. A. D. Buckingham, J. Chem. Phys., 2014, 140, 011103.
8. P. Garbacz, J. Chem. Phys., 2016, 145, 224202.
9. J. P. King, T. F. Sjolander and J. W. Blanchard, J. Phys. Chem. Lett., 2017, 8, 710–714.
10. A. Soncini and S. Calvello, Phys. Rev. Lett., 2016, 116, 163001.
11. S. Calvello and A. Soncini, Phys. Chem. Chem. Phys., 2020, 22, 8427–8441.
12. R. Naaman, Y. Paltiel and D. Waldeck, Nat. Rev. Chem., 2019, 3, 250–260.
13. R. Naaman and D. Waldeck, Annu. Rev. Phys. Chem., 2015, 66, 263–281.
14. A. Schweiger and G. Jeschke, Principles of pulse electron paramagnetic resonance, Oxford University Press, Oxford, 2001, p. 29.
15. A. L. Barra and J. B. Robert, Mol. Phys., 1996, 88, 875–886.
16. D. A. Varshalovich, A. N. Moskalev and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1989, p. 63.
17. We found by quantum-chemical computations that the relation A*′ = det(R)RA* of the pseudovector transformation holds (R is a rotation matrix of radical coordinates). Thus, the coupling of the electron spin with molecular rotation may be neglected.
18. The signal depends on the sensitivity of the detector to the excitation by Ṁ and Ṗ^c. According to the reciprocity theorem, the sensitivity is proportional to the magnitude of the field that the detector would generate if voltage of the same amplitude as observed would be used for exciting the detector.
19. For an electron, g_S < 0, thus the angular frequency ω_S > 0 and the spin S precesses anticlockwise according to the right-hand rule. The reverse is true for a nucleus with g_I > 0 (such as ¹H and ¹³C); in this case the angular frequency ω_I < 0, i.e., the spin I precesses clockwise.
20. E. Reijerse and A. Savitsky, EPR Spectroscopy: Fundamentals and Methods, Wiley, 2018, pp. 242–251.
21. E. Malkin, M. Repiský, S. Komorovský, P. Mach, O. Malkina and V. G. Malkin, J. Chem. Phys., 2011, 134, 044111.
22. S. Gohr, P. Hrobárik, M. Repiský, S. Komorovský, K. Ruud and M. Kaupp, J. Phys. Chem. A, 2015, 119, 12892–12905.
23. M. Repiský, S. Komorovský, M. Kadek, U. Konecny, L. Ekström, E. Malkin, M. Kaupp, K. Ruud, O. L. Malkina and V. G. Malkin, J. Chem. Phys., 2020, 152, 184101.
24. M. Repisky, S. Komorovsky, V. G. Malkin, O. L. Malkina, M. Kaupp and K. Ruud, ReSpect, version 5.1.0 ( 2019); Relativistic Spectroscopy DFT program, with contributions from R. Bast, U. Ekström, M. Kadek, S. Knecht, L. Konecny, E. Malkin and I. Malkin-Ondik. See http://www.respectprogram.org.
25. C. Adamo and V. Barone, J. Chem. Phys., 1999, 110, 6158–6170.
26. P. Jakobsen and F. Jensen, J. Chem. Phys., 2019, 151, 174107.
27. M. Bühl, Chem. Phys. Lett., 1997, 267, 251–257.
28. J. Vaara, Phys. Chem. Chem. Phys., 2007, 9, 5399–5418.
29. P. Garbacz, J. Cukras and M. Jaszuński, Phys. Chem. Chem. Phys., 2015, 17, 22642–22651.
30. A. D. Buckingham, P. Lazzeretti and S. Pelloni, Mol. Phys., 2015, 113, 1780–1785.
31. P. Garbacz, Mol. Phys., 2018, 116, 1397–1408.
32. V. Meyer, S. S. Eaton and G. R. Eaton, Appl. Magn. Reson., 2014, 45, 993–1007.

---

Footnotes

† This article is dedicated to the memory of Prof. A. D. Buckingham (1930–2021), whose theoretical research laid the foundation for the magnetic resonance spectroscopy of chiral molecules.

‡ Electronic supplementary information (ESI) available. See DOI: 10.1039/d1cc02579a

---