Letter pubs.acs.org/JPCL
Antisymmetric Couplings Enable Direct Observation of Chirality in Nuclear Magnetic Resonance Spectroscopy Jonathan P. King,*,†,‡ Tobias F. Sjolander,†,‡ and John W. Blanchard†,‡,§ †
Department of Chemistry, University of California, Berkeley, California 94720, United States Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States § Helmholtz-Institut Mainz, 55099 Mainz, Germany ‡
S Supporting Information *
ABSTRACT: Here we demonstrate that a term in the nuclear spin Hamiltonian, the antisymmetric J-coupling, is fundamentally connected to molecular chirality. We propose and simulate a nuclear magnetic resonance (NMR) experiment to observe this interaction and differentiate between enantiomers without adding any additional chiral agent to the sample. The antisymmetric J-coupling may be observed in the presence of molecular orientation by an external electric field. The opposite parity of the antisymmetric coupling tensor and the molecular electric dipole moment yields a sign change of the observed coupling between enantiomers. We show how this sign change influences the phase of the NMR spectrum and may be used to discriminate between enantiomers.
M
zero-field nuclear magnetic resonance and predict a signal amplitude within reach of current detectors. Although not yet observed, terms in the Hamiltonian arising from the antisymmetric J-coupling are predicted in a wide array of molecules from symmetry considerations,21,22 perturbational calculations,23−25 and quantum-chemical calculations.26,27 In addition to the connection to chirality made here, measurement of the antisymmetric J-coupling has been suggested as a means to observe molecular parity violation,28 which is predicted to cause a first-order energy shift to the antisymmetric J-coupling. The J-coupling interaction Hamiltonian (in frequency units) for two spins may be written
olecular chirality is an extremely important yet often difficult to measure property. Chiral molecules form many of the basic building blocks of living organisms, such as Lamino acids and D-sugars. Because of this biochirality, the effect of chiral pharmaceuticals can differ greatly between enantiomers,1 leading to interest in enantioselective synthesis and methods to analyze chiral products.2 Molecular chirality is also of interest in the study of fundamental symmetries where parity violation effects in molecules are predicted but have not yet been observed.3−5 The search for spectroscopic techniques to probe chirality is a field of great interest where many proposals and experiments may be found in the recent literature.6−9 Meanwhile, liquid-state NMR is the preeminent technique for chemical analysis owing to its generality and chemical specificity. The introduction of direct chiral detection to NMR, without the need to add derivatizing agents to the sample, would be a major step forward in the field of chiral analysis. Several proposals exist for direct observation of chirality via an electric-field-induced pseudoscalar term in the NMR Hamiltonian. These involve either the detection of a chiral nuclear magnetic shielding via induced oscillating molecular electric dipole moment10−16 or a magnetic dipole signal-induced via application of an electric field.16−19 Chiral effects may also manifest in the electric polarizability of the spin−spin J-coupling.20 We establish the connection between the permanent antisymmetric nuclear spin J-coupling tensor and molecular chirality. We show how the corresponding term in the nuclear spin Hamiltonian can generate an observable signal. When observed in oriented molecules these signals report on chirality. We then propose and simulate a prototype experiment using © 2016 American Chemical Society
H = I ·1·S
(1)
I and S are vector spin operators, and 1 is a tensor that contains the spatial coordinates. The total nuclear spin−spin coupling tensor may include terms from magnetic dipole− dipole coupling as well as the J-coupling, but the antisymmetric terms we are interested in can only arise from the J-coupling. This is because the J-coupling involves indirect interactions mediated by electron spins, which need not have local inversion symmetry, while the magnetic dipole−dipole coupling is symmetric under inversions and is described by a rank-2 irreducible spherical tensor. The 1 tensor describes the coupling between two angular momenta and may be decomposed into irreducible spherical tensors up to rank-2 Received: November 13, 2016 Accepted: December 28, 2016 Published: December 28, 2016 710
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The Journal of Physical Chemistry Letters 1 = 1 (0) + 1 (1) + 1 (2)
(2)
where the 1 (1) tensor is antisymmetric with respect to the spatial coordinates and therefore transforms under rotations and inversions as a pseudovector.29 Because 1 (1) is traceless, its components average to zero for unoriented molecules and thus cannot be observed in isotropic liquids. Alignment techniques can revive rank-2 terms, but to observe rank-1 interactions, molecular orientation is required (see the Supporting Information). Solid-state studies with magic-angle spinning can, in principle, detect antisymmetric couplings but in practice have insufficient resolution.30 Here we propose molecular orientation of liquid-state molecules by an applied electric field.31,32 1 (1) has three independent components, J(1) αβ , where α, β = x, y, z ⎛ 0 Jxy Jxz ⎞ ⎟ ⎜ 1 (1) = ⎜ −Jxy 0 Jyz ⎟ ⎟ ⎜ ⎟ ⎜− J − J 0 yz ⎠ ⎝ xz
Figure 1. (a) Schematic chiral molecule and 1 (1) tensor components for the 1H−13C J-coupling. μ is the molecular electric dipole that is oriented along the z axis by an applied electric field. (b) After reflection through the yz plane (perpendicular to the plane of the paper), the electric dipole is unchanged, but the xy component of 1 (1) has now gained a negative sign. This behavior is a result of pseudovectors and polar vectors transforming in the opposite sense under reflections. In both situations the molecule is rapidly rotating around the z axis, so only the Jxy (red) component of 1 (1) is observable.
(3)
It has been shown that the number of independent components of the 1 tensor depends on the local symmetry of the spin pair and that three independent components of 1 (1) exist only if the two nuclear sites have local C1 symmetry,21 which includes all chiral molecules and meso compounds. We now address the question: “Given 1 (1) in a moleculefixed coordinate system for a chiral molecule, how will it be different for the complementary enantiomer?” The transformation between enantiomers may be visualized as a reflection through a plane containing the molecular electric dipole vector (which we will call the z direction). The electric dipole will remain unchanged, as required by the orienting field, while the xy component of 1 (1) will acquire a negative sign (Jxy → J(−x)y = −Jxy, Figure 1). This sign change is a manifestation of the pseudovector nature of 1 (1). Because we are considering oriented liquids rapidly rotating around the z axis, the timeaveraged coupling tensor has only the single xy component, which is scaled by the degree of orientation (see the Supporting Information) ⎛ 0 ⎜ 1 (1) = ⎜− J ⎜ xy ⎜ ⎝ 0
Jxy 0 ⎞ ⎟ 0 0⎟ ⎟ ⎟ 0 0⎠
remainder of this work outlines methods to observe the sign of Jxy . Neglecting contributions from rank-2 terms (see Supporting Information), the spin Hamiltonian is H = JI ·S + i Jxy (I +S− − I −S+)
(5)
where J is the isotropic (rank-0) J-coupling constant. The term proportional to Jxy does not commute with the Zeeman Hamiltonian unless the spins have the same Larmor frequency. In typical high-field NMR, an identical Larmor frequency would imply that the spins be chemically equivalent, which is incompatible with local C1 symmetry. In practice, chiral effects could be seen if the chemical shift difference is not large relative to the antisymmetric J-coupling, but at zero magnetic field this condition is generally fulfilled. We therefore propose an experiment based on zero-field NMR33 of two coupled unlike spin- 1 nuclei in a chiral molecule oriented by an electric field. 2 Zero-field NMR spectroscopy is the detection of NMR signals generated by spin−spin couplings in the absence of an applied magnetic field, thus giving an identical Larmor frequency of zero for all spins. The J-coupling signals are generally in the sub-kHz regime and are detected by means such as atomic vapor cell magnetometers.34−37 In isotropic media, the signal gives the rank-0 (isotropic) coupling constant J. We show how, in oriented molecules, an additional signal whose sign and magnitude depend on Jxy may be observed along an axis orthogonal to the usual signal. The general principle is that the three axes defined by the initial spin state, detector axis, and electric field form a coordinate system with a definite handedness that yields chiral information (Figure 2). In our NMR experiment, the observable quantity is the magnetization along the detector axis, chosen here to be the x axis, and represented by the operator Mx. Note that for this experiment there will also be a large achiral signal along the y axis. The observable operator is
(4)
where the magnitude of Jxy is proportional to the degree of orientation and the sign depends on the handedness of the molecule. Therefore, the result of our analysis is that a measurement of the sign of Jxy yields the chiral signal. We note that 1 (1) is the projection of 1 (1) along the director axis, scaled by the degree of orientation. Because the orientation is defined by the electric field, the measured quantity is proportional to 1 (1)·E , which gives a pseudoscalar term in the Hamiltonian. We note that E may be replaced with another polar vector describing molecular orientation while still retaining the pseudoscalar Hamiltonian term, and therefore other methods of molecular orientation could be used. The 711
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Jxy . We choose an experimentally realizable initial condition containing coherences involving |η⟩ and |ζ⟩ (arrows in Figure 2b). Our chosen initial density operator is ρ(0) =
Bpℏ ⎡ γI + γS γ − γS ⎤ 1 (− Iy + Sy) + I (− Iy − Sy)⎥ + ⎢⎣ ⎦ 4 4kT 2 2 (10)
which corresponds to prepolarization of spins in a field Bp along the y axis at temperature T, followed by inversion of spin I. k is the Boltzmann constant. Similar initial conditions could also be prepared via other means, such as transition selective pulses.38 This initial condition gives the following predicted signal in the zero-field NMR experiment (see the Supporting Information for a full calculation of matrix elements and coherence amplitudes) ⟨Mx(t )⟩ =
Bpℏγ1γ2 Jxy kTN 2J
[cos(ωηt ) − cos(ωζ t )]
E −E
(6)
where γI and γS are the gyromagnetic ratios of the nuclei. The observable signal is given by ⟨Mx(t )⟩ = Tr{Mx†ρ(t )}
(7)
In the absence of Jxy , the energy eigenstates are three triplet states |T+1⟩, |T0⟩, |T−1⟩ and a singlet |S⟩ (Figure 2a). We consider Jxy as a perturbation, which is valid when |Jxy | ≪ |J | as is the case for weakly oriented molecules. We note that weak orientation is not strictly a necessary condition for chiral discrimination, but it is preferred to prevent residual dipolar couplings from suppressing the Jxy term. While the |T+1⟩ and
⟨My(t )⟩ = −
|T−1⟩ states are not affected by the presence of Jxy , the |T0⟩ and |S⟩ states are mixed so that the first-order eigenstates are ⎛ Jxy ⎞ 1 |η⟩ = ⎜⎜|T0⟩ − i |S⟩⎟⎟ 2J ⎠ N ⎝
⎛ ⎛ (γ12 − γ22)⎜1 + ⎜ ⎝ ⎝
Jxy ⎞ J
N2
2⎞
⎟ ⎠⎠ ⎟
[cos(ωηt ) + cos(ωζ t )] (12)
so that the ratio of the maximum amplitudes (Ax and Ay) is Ax = Ay
(8)
and ⎛ ⎞ Jxy 1 |ζ ⟩ = ⎜⎜|S⟩ − i |T0⟩⎟⎟ 2J ⎝ ⎠N
E −E
where ωη = η ℏ 1 and ωζ = ζ ℏ 1 . E1 is the energy of the triplet ±1 states. In our simulations, we consider the case of a chiral molecule with two spin- 1 nuclei (13C and 1H) with J = 100 Hz, typical of 2 a one-bond 1H−13C J-coupling. Because antisymmetric Jcouplings are similar in magnitude to the isotropic rank-0 term,26,27 we assume a residual Jxy of 1 Hz, corresponding to a orientational order parameter of ∼10−2, typical of electric-field orientation experiments.32 For a one-bond 13C−1H coupling with this degree of orientation, the alignment induced by the electric field results in a residual dipolar coupling of ∼0.7 Hz (see the Supporting Information). Oscillating magnetization emerges along the x axis with the sign of the signal determined by the sign of Jxy (Figure 3a,b), thereby distinguishing right and left enantiomers. The two curves in Figure 3a sum to zero at all points, meaning that no signal will be observed in a racemic mixture. The signal predicted by eq 11 is proportional to Jxy /J , which scales the signal by the degree of orientation. It is useful to compare the amplitude of this chiral signal to the achiral signal that evolves along the y axis. This achiral signal is the “usual” zero-field NMR signal33 and is given by
Figure 2. Schematic representation of a zero-field NMR experiment for chiral discrimination. (a) Without electric-field orientation, the energy eigenstates are pure singlet and triplets, and no signal emerges along the x axis. (There is an oscillating magnetization along the y axis.) (b) With electric-field orientation, the |S⟩ and |T0⟩ states mix and the coherences generate an oscillating transverse magnetization along the x axis as well as the y axis. The choice of initial magnetization, electric-field vector, and transverse detector vector define a handed coordinate system that yields chiral information in the sign of the detected signal. Small energy shifts are also present from both Jxy and residual dipolar couplings, but these are not necessary for chiral detection.
Mx = γIIx + γSSx
(11)
4γIγS
Jxy J
⎛ ⎛ Jxy ⎞2 ⎞ (γI2 − γS2)⎜1 + ⎜ J ⎟ ⎟ ⎝ ⎠⎠ ⎝
(13)
For our chosen parameters this ratio evaluates to 0.0108, essentially in agreement with the exact value of 0.0106 (see the Supporting Information). Therefore, we expect a signal reduced by a factor of 10−2 compared with standard zero-field NMR. A typical 100 μL sample contains ∼1021 1H−13C pairs, and, assuming a spherical geometry, the standard zero-field signal corresponds to a field of ∼50 fT at a distance of 0.5 cm from
(9)
where N is a normalization factor. The presence of a term linear in Jxy enables the creation of observable signals also linear in 712
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to search for molecular parity violation. For inequivalent spins these terms do not commute with the Zeeman Hamiltonian, and we propose a zero-field NMR experiment where the untruncated spin−spin coupling can be observed. This proposal combines previously established methods and, in principle, requires no new techniques. However, design constraints of our current zero-field NMR spectrometers do not allow the use of electric-field orientation cells. We therefore present this work as a guide to the design of future experiments.
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.6b02653. Derivation of order parameters for molecular orientation, derivation of the matrix elements used in the analytical calculation, and discussion of the numerical simulations. (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Figure 3. (a)Predicted x-magnetization for positive (black) and negative (red) Jxy with a magnitude of 1 Hz, J = 100 Hz, and a residual dipolar coupling of 0.7 Hz. High-frequency oscillations result from coherences between the |T±1⟩ and |ζ⟩ states, while low-frequency oscillations result from coherences between |T±1⟩ and |η⟩. (b) Fourier transform of (a) shows the phase relationship between peaks in the frequency spectrum for each enantiomer.
Jonathan P. King: 0000-0002-0875-8794 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank Professors Dmitry Budker, Mikhail Kozlov, Robert Harris, and Alexander Pines for helpful discussion. This work was supported by the National Science Foundation under award CHE-1308381.
the center of the sphere when prepolarized in 2 T field at 300 K. Given a demonstrated magnetometer sensitivity of 15 fT/ Hz 36 and accounting for the 10−2 factor, we expect an acquisition time of ∼5 min to achieve a signal-to-noise ratio greater than one. We note that systematic errors may occur when the three axes (magnetizing field, electric field, and detector) are not orthogonal or when the pulse field is not collinear with the detector. In this case, some component of the large achiral signal will emerge along the detector axis. Such systematic errors could potentially overwhelm the desired chiral signal. However, we note that the desired signal is proportional to the product 1 (1)·E , whose sign can be changed by reversal of the electric field. The spurious achiral signal has no such dependence and thus may be canceled by subtracting spectra acquired with reversed electric fields. In conclusion, we demonstrate that molecular chirality is a directly observable property in NMR spectroscopy through antisymmetric terms in the spin−spin coupling Hamiltonian. Electric-field orientation in liquids provides a method to observe these terms, owing to the interplay between the pseudovector nature of the antisymmetric coupling and the polar vector nature of a molecular electric dipole. The apparent sign change of the spin-coupling directly determines the sign of observed NMR signals, enabling chiral discrimination without adding additional chiral agents to the sample. Because the signal depends on pairwise spin−spin couplings, the spectrum gives local information and shows to what extent the local electronic structure is enantiomer-dependent. Furthermore, the observation of antisymmetric J-couplings will provide a new technique
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NOTE ADDED IN PROOF While this paper was in review, a paper by Garbacz and Buckingham was published proposing a high-field NMR experiment where a chirality-dependent oscillating electric dipole moment is generated in part by the antisymmetric Jcoupling.39
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