Chapter 7: Antisymmetric Nuclear Magnetic Shielding as an Indicator of Molecular Chirality^†

Nuclear magnetic resonance (NMR) spectroscopy investigates the interactions of nuclear dipole moments in the presence, at least for the typical studies on the structure and physicochemical properties of molecules, of a strong magnetic field with a magnitude of several teslas. This experimental approach has been successfully applied, over several years, in studies aiming to determine the atomic connectivity and three-dimensional structures of molecules in the liquid phase, which allow us, e.g., to deduce biochemical properties in an environment similar to in vivo conditions.¹  However, more subtle structural properties, such as molecular chirality, may be inferred from the results of NMR spectroscopy only in an indirect way, specific to a given group of chemical compounds.²  The range of information obtained from NMR studies can be significantly extended by allowing the presence of an external perturbation in the system under consideration, particularly that induced by an electric field. In this case, according to theoretical predictions discussed further in detail in this chapter, the limitation of NMR spectroscopy in studies of molecular chirality is overcome, and the postulated effects provide a novel method to determine the molecular structural parameters. After a brief introduction to the mathematical formalism used in this work, each of these effects is described in detail incorporating the proposed experimental protocol for the observation of the effect, estimation of the magnitude of the anticipated signal, and discussing the significance of the effect for the determination of molecular structure.

Tensor analysis is frequently referred to in the following discussions. Therefore, let us briefly summarize the relevant tensor properties from the viewpoint of this work. Further details of tensor properties in the context of nuclear magnetic shielding tensor and molecular chirality are given in Section 7.3. In the following text, variables (in italics) that are tensors of rank at least one (e.g., vectors) are boldfaced in contrast to those variables that are plain numbers.

A two-index tensor decomposes to a sum of irreducible components of, at most, the second rank under rotations in three-dimensional space.⁴  The ranks of the isotropic, antisymmetric, and traceless symmetric parts of tensor $T$ are zero, one, and two, respectively.^‡ The total number of tensor elements equals the sum of the numbers of tensor elements into which it has been decomposed under rotations in three-dimensional space, e.g., $T iso$ has one element, $T anti$ has three, and $T sym$ has five, giving a total of nine elements. Thus, the decomposition in isotropic, antisymmetric, and traceless symmetric parts becomes ambiguous for tensors of ranks higher than two. For instance, a reducible third-rank tensor has one zeroth rank component (proportional to the so-called Levi-Civita tensor), three first-rank components, two second-rank components, and one third-rank component.

Usually only the isotropic parts, mainly in liquid-state NMR, and the symmetric parts of the relevant tensors, mainly in solid-state NMR, are experimentally studied.⁷  However, as will be shown, the presence of the antisymmetric component induces interesting phenomena in chiral molecules.

The hat above the Hamiltonian symbol, $H ˆ$ ⁠, emphasize that the Hamiltonian is an operator, e.g., the way it acts on the quantum state of nuclear spins allows us to determine how the spin state changes in time. Thus, strictly speaking, one can view eqn (7.6) as the interaction of a magnetic dipole known from classical physics (the resultant of the multiplication is a number), but since the form of the equation is the same as in its quantum-mechanical counterpart, classical and quantum interpretations will be considered as equivalent in the further text – see Section 7.8 for details.

The energy of the interaction between a mole of ¹H magnetic moments $( μ H m =1.41× 10 − 26 J T − 1 )$ in a water molecule (shielding ∼3 × 10⁻⁵) and a magnetic field of strength B = 11.75 T, is approximately 3 mJ, which is nearly six orders of magnitude smaller than the thermal energy at T = 300 K.

From eqn (7.6), it follows that the components of the effective magnetic field, (1₃ − σ) · B, interacting with the magnetic moment μ^m may not only have a different amplitude compared to the magnetic field B, which occurs via the diagonal elements of the tensor σ, but also experience changes in direction due to off-diagonal elements. This observation may be described more rigorously with the aid of the irreducible tensor decomposition mentioned in Section 7.2. Mathematical details of the decomposition are given in ref. 4. Here, we limit ourselves to the application of the irreducible decomposition of a two-index tensor, an example of which is the nuclear magnetic shielding tensor σ.

To deepen our understanding of tensor antisymmetry, let us look at how its coordinates change under the influence of rotations in three-dimensional space. If R is the rotation matrix, then the components of the two-index tensor transform $T$ under a basis change according to the equation $R⋅[T]⋅ R T$ ⁠. The symbol R^T represents the transposition (an interchange of the rows and columns) of the matrix R.

The nuclear site symmetry determines the form of the nuclear magnetic shielding tensor σ, typically resulting in particular components of the tensor becoming zero. For example, highly symmetric nuclear sites such as T_d (tetrahedral) and O (octahedral) have only one independent component of σ; in this case, σ_iso = σ_xx = σ_yy = σ_zz holds true, so such shielding tensors do not have antisymmetric parts. However, less symmetric nuclear environments do not exclude the antisymmetry of the tensor σ. A few examples of the magnetic shielding tensors of nuclei in simple planar molecules are listed in Table 7.1.

The first two molecules shown in Table 7.1, water and formaldehyde, have a two-fold symmetry axis, resulting in the absence of an antisymmetric component of the oxygen and carbon shielding tensors, since these nuclei are placed along the C_2v axis. The antisymmetric parts do not vanish for the protons, but the planarity of water and formaldehyde molecules constrains the direction of their antisymmetry, σ*(¹H). The rotation about the C_2v axis, which coincides with the z-axis of the reference frame chosen for the shielding tensor elements given in the table, interchanges the protons. Hence, any element $( σ y * , σ z * )$ of the σ*(¹H) vector in the yz-plane, in which the atoms are, vanishes. Thus, from computations, it follows that (i) the shielding tensors of protons in a water molecule do have antisymmetry that is perpendicular to the plane containing the two protons and the oxygen atom of the molecule (along the x-axis), and in a water molecule, the antisymmetric part of the shielding tensor of the first proton is oriented antiparallel to the antisymmetric part of the shielding tensor of the second proton, σ*(¹H_a) = −σ*(¹H_b); see Figure 7.1.

Nuclei of less symmetric molecules, such as formic acid, may all have antisymmetric components in their shielding tensors. In a very crude approximation, the magnitude of the shielding antisymmetry is correlated with the variation of the shielding for a given element. For example, typically in an organic molecule, the nuclear shielding of proton (¹H), carbon (¹³C), and oxygen (¹⁷O) varies in the range of ten, a hundred, and several hundred ppm, respectively. This trend is exemplified in the case of formic acid nuclear shielding antisymmetries, which are less than 3 ppm for protons, about 10 ppm for carbon, and reach almost 40 ppm for oxygen. Several other examples of various nuclear shielding antisymmetries of benzene derivatives are provided in Figure 7.2.

Let us note that if a molecule is non-rigid, then the impact of the flexibility of the molecule may greatly vary with the position of the nucleus. For example, the antisymmetry σ*(¹H_a) is almost the same for the two conformers of formic acid, cis-HCOOH and trans-HCOOH, in contrast to the antisymmetry σ*(¹H_b) that is more than two times larger for trans-HCOOH in comparison to cis-HCOOH. An analogous observation applies to the pair of ¹⁷O_a and ¹⁷O_b atoms in the formic acid molecule. Antisymmetry is a much more local property than the permanent molecular electric dipole moment. This is shown by the example of chiral phosphaferrocene depicted on the right side of Figure 7.1. While the dipole moment of the molecule (μ^e) changes significantly depending on the opening angle between the acrylic substituents, the shielding antisymmetries of the phosphorus nuclei σ*(³¹P) remain approximately constant and the opening angle only affects their directions.

The next case of a flexible molecule is hydrogen fluoroperoxide, whose σ*(¹H) and σ*(¹⁹F) antisymmetries are shown in Figure 7.3. As in the case of formic acid, the antisymmetry of the nucleus with a large variability of magnetic shielding, fluorine-19, is roughly an order of magnitude larger than that of nuclei with small shielding variation, i.e., proton. Hydrogen fluoroperoxide is an unstable molecule whose enantiomers interconvert into each other (by the rotation of the oxygen–oxygen bond) on a fast time scale from the point of view of NMR. FOOH molecules with dihedral angles ∠(FOOH) ranging from −180° to 0° are (R)-enantiomers, and those with ∠(FOOH) ranging from 0° to 180° are (S)-enantiomers. Although it is true that the directions of the antisymmetries σ*(¹H) and σ*(¹⁹F) depend on the molecular conformation and, consequently, on the chirality of a molecule, one cannot use this difference to discriminate between the FOOH enantiomers. The reason for this is that, in a standard liquid-state NMR experiment, only the amplitude of antisymmetry can be measured, e.g., by measuring the rate of nuclear relaxation, which is the same for the positive and negative values of the dihedral angle ∠(FOOH). Moreover, the significant variation of the antisymmetries σ*(¹H) and σ*(¹⁹F) with the molecular shape indicates that conformational flexibility may be one of the strongest factors determining the direction and amplitude of the antisymmetry, which, in order to reach an agreement with experimental observations, has to be averaged over all the conformations of the molecule that are available at a given temperature.

As mentioned at the end of Section 7.3, even though the antisymmetry of the nuclear magnetic shielding tensor is a key component required for direct discrimination between enantiomers, its mere presence is not sufficient for designing an experiment, where the outcome will solely depend on the absolute configuration of a molecule. The significance of nuclear shielding antisymmetry lies in its transformation properties under space inversion.

Let us consider a mirror image of a solenoid through which a current flows. If the axis of the solenoid is vertically oriented, parallel to the surface of the mirror and the current flows counterclockwise, then the magnetic field generated by the solenoid is directed upwards. In the mirror image of a solenoid, the direction of current flow is reversed, i.e., current flows clockwise, and consequently, the magnetic field is directed downwards (Figure 7.4A). Comparing the magnetic-field behaviour with an electric field of a capacitor that has its plates perpendicular to the mirror plane, one finds that the electric field does not alter its direction upon mirror reflection (Figure 7.4B). So, although the magnetic and electric fields are vector quantities, they exhibit different transformation properties under mirror reflection. The manner in which particular elements of the vector change depends on the vector orientation with respect to the mirror plane. If the solenoid axis is perpendicular to the plane, then the direction of the magnetic field will remain the same. However, for a capacitor whose plates are parallel to the plane of the mirror, the direction of the electric field will be reversed. Therefore, it is convenient to use space inversion instead of mirror symmetry since, if the former acts on the magnetic and electric fields, we find that B → B and E → −E.

If we would like to introduce mirror reflections into the rule defining a vector A, i.e., its components are transformed according to R · A, where R is the rotation matrix, we need to differentiate proper and improper rotations, where the latter begins with a mirror reflection followed by a proper rotation. A proper rotation matrix has the determinant +1, while the determinant of an improper rotation matrix is −1. Thus, a polar vector, e.g., an electric field, transforms under any rotation (proper or improper) according to the formula R · A, while a pseudovector (alternatively called an axial vector), e.g., magnetic field, transforms as det(R)R · A. These transformation rules allow us to reproduce the result presented at the end of the previous paragraph because the improper rotation matrix corresponding to inversion through a point (S₂ = σ_hC₂) is −1₃ (minus unity 3 by 3 matrix).

The amplitude of the pseudoscalar σ^c may be maximized by making the favourable properties of the antisymmetric part of the shielding tensor and the molecular electric permanent dipole moment μ^e, i.e., minimization of the angle between μ^e and σ*, and by maximization of the amplitudes of μ^e and σ*. In order to illustrate the significance of both mentioned factors, let us take an example of permutations of the substituents in a rigid molecule of a triangular shape, a halogen derivative of cyclopropene HFClBrC₃. The magnitude of the permanent electric dipole moment of HFClBrC₃ is roughly the same (1.7–2.7 Debye) for its isomers obtained by permutations of hydrogen, fluorine, chlorine, and bromine: HFC–CBr═CCl (where Br and Cl are attached to carbons forming a double bond; the C–C bond closing the ring is implied), ClBrC–CH═CF, and FBrC–CH═CCl. Quantum-chemical computations of ¹⁹F nuclear magnetic shielding in HFClBrC₃ at the DFT (density-functional theory) level show the following.

• In HFC–CBr═CCl, μ^e and σ*(¹⁹F) are perpendicular to the cyclopropane ring plane, but the antisymmetry of fluorine shielding is relatively small (a few ppm); thus, even though the relative orientation of μ^e and σ*(¹⁹F) is favorable, $∡ ( μ e , σ * ) ≈12°$ ⁠, the pseudoscalar is small, ∣σ^c∣ < 2 pm V⁻¹.

• In ClBrC–CH═CF, μ^e is in the cyclopropane ring plane, while its σ*(¹⁹F) is almost perpendicular to the ring’s plane, $∡ ( μ e , σ * ) ≈91°$ ⁠, so althought σ*(¹⁹F) reaches nearly 70 ppm, the pseudoscalar is even smaller than in the case of HFC–CBr═CCl.

• In FBrC–CH═CCl σ*(¹⁹F) is as large as ClBrC–CH═CF and simultaneously the angle $∡ ( μ e , σ * )$ is slightly smaller than 30°, resulting in a 30 times larger value of |σ^c∣ in comparison with HFC–CBr═CCl and ClBrC–CH═CF (3500 pm V⁻¹).

One can view $F ̲ Br C ̲ – CH ̲ ═CCl$ and $ClBrC– CH ̲ ═ CF ̲$ as molecular scaffolds for a two-spin ¹H–¹⁹F system embedded in an electronic environment resembling an open-book topology of FOOH (also compare with H₂O₂ antisymmetries shown in Figure 7.1). This is because, in both cases, other atoms, such as heavy halogens, carbon, and oxygen are considered to be either NMR-inactive (¹²C, ¹⁶O) or self-decoupled isotopes (³⁵Cl, ⁷⁹Br). In the case of ClBrC–CH═CF, the FCCH part of the molecule is planar, and the pseudoscalar σ^c almost vanishes, similar to the achiral planar conformation of FOOH, for which σ*(¹⁹F) is perpendicular to the molecular plane and its permanent electric dipole moment μ^e is in the direction from the fluorine to the hydrogen atom. In contrast to the case of ClBrC–CH═CF, in FBrC–CH═CCl, the dihedral angle FCCH is about 90°, corresponding to a chiral conformation of FOOH.

Apart for the choice of substituents, the second factor influencing the magnitudes of μ^e and σ* and the angle $∡ ( μ e , σ * )$ is molecular flexibility. Frequently, the value of the pseudoscalar σ^c varies with torsional angles, and in an extreme case, the pseudoscalar σ^c may reverse its sign for some values of the torsional angle. An example illustrating the influence of molecular flexibility on the sign of the pseudoscalar σ^c is given in Figure 7.5.

The pseudoscalars $σ N c$ and $σ C c$ of nitrogen-15 and carbon-13 of (R)-alanine tripeptide (all three alanine groups are (R)-enantiomers) strongly vary with the conformation of the oligopeptide. The figure shows two regions of the Ramachandran plot: at the bottom, the range of the ϕ and ψ torsional angles corresponds to the structure of a β-sheet, while the top of the plot corresponds to the structure of an α-helix of a protein. Both regions have segments that exhibit positive and negative values of chirality-sensitive pseudoscalars; therefore, determining which enantiomer is present in the sample requires simultaneous knowledge about the conformation of the (R)-alanine tripeptide. On the other hand, the strong dependence of the pseudoscalar value on the structure of the molecule can be useful because it carries information about the detailed three-dimensional structure of a molecule within a certain structural motif. The usefulness lies in the fact, that by finding the values of chirality-sensitive pseudoscalars of nitrogen and carbon nuclei, one can determine the relative orientation of the chemical groups containing ¹⁵N and ¹³C nuclei, e.g., peptide bonds, relative to the permanent electric dipole moment of the molecule. Consequently, this strategy of spatial structure measurement would provide complementary data to those obtained from widely-used biomolecular structure determination techniques such as those utilizing the nuclear Overhauser effect, which gives distances between the atoms, or the Karplus relationship providing the torsional angles.

The relationship between the molecular structure and the amplitude and direction of the antisymmetric part of the nuclear magnetic shielding tensor in Section 7.3 and the current section allows us, to propose a general strategy for maximizing the amplitude of the pseudoscalar σ^c. This strategy utilizes the facts that the quantities μ^e and σ* are almost independent in many molecules, and that an achiral molecule can be transformed into a chiral one by attaching a chiral substituent. As mentioned earlier, the chirality-sensitive NMR effects reach maximal amplitude if the angle between the molecular dipole moment μ^e and the antisymmetric shielding σ* is optimal, and if the amplitudes of the vectors μ^e and σ* are large. Additionally, the molecule contains other advantageous structural features, e.g., it may be hyperpolarized, which amplifies the chirality-sensitive signals to a level at which they will be easily observed. While in the case of the pseudoscalar σ^c the optimal angle is 0°, for other effects described further in this chapter, the angle may take other optimal values, since the dependence of the amplitude of the chirality-sensitive effect may be related to the pseudoscalar in another more complicated way (vide infra).

In particular, the strategy is exemplified in Figure 7.6. The core of the chiral molecule (blue sphere) has two substituents, one of which has a large electric dipole moment (polar chemical groups; green cylinder) while the second one is hyperpolarisable and contains a nucleus with a large antisymmetry σ*, (indicated by a red disc). Examples suitable for a SABRE (signal amplification by reversible exchange) experiment are camphor substituted with pyridine (Figure 7.6A) and α-amino acid derivative with a triazole ring (Figure 7.6B). Other target molecules are light fluorinated alcohols and their derivatives with a substituent that has a triple carbon–carbon bond permitting the usage of ALTADENA (adiabatic longitudinal transport after dissociation engenders nuclear alignment) – Figure 7.6C (the latter is not shown in the figure). Note that, in practice, one has to make a trade-off between the molecular complexity and molecular weight since the more molecular features are taken into account, the heavier the molecule becomes, and the molar concentration decreases to which the expected chirality-signal is proportional. This point is clearly visible if we contrast the molecules shown in Figure 7.6A and C. Moreover, this approach is more suitable for a rigid molecule than a flexible one. In the case of flexible molecules, signals may partially cancel out as it happens for (R)-alanine tripeptide shown in Figure 7.5. This is the price that one has to pay for the possible insight into the dynamical behaviour of the molecule, based on the determination of the amplitude of the pseudoscalar σ^c.

Let us briefly summarize the results described in the chapter up to now, in view of the other approaches to directly see chirality by NMR. Parity non-conservation does not influence the NMR spectra measured using the currently used NMR spectrometers.^13,14  Consequently, one assumes that the Hamiltonian $H ˆ CS$ given in eqn (7.13) is even under the inversion through a point. Therefore, one cannot directly discriminate between the enantiomers through NMR spectroscopy unless the molecule is transformed into a diastereomer (e.g., via Mosher’s method)¹⁵  or a chiral environment is present. This limitation is lifted if the electric field operator, which is odd under inversion, is involved in the Hamiltonian.^16,17  In this case, some contributions to the Hamiltonian $H ˆ E$ defined in eqn (7.14), e.g., $H ˆ CS , c$ (see eqn (7.17)), change their sign under the inversion, which may be used for a direct discrimination between the enantiomers. Note that the effects that are linear in the electric field are chirality-sensitive, and they vanish for an achiral sample. In contrast, the quadratic effects are present both in the achiral and the chiral samples. We will focus only on the linear effects and show that all of these effects generate NMR signals of the opposite phases (i.e., phase-shifted by 180°) for the enantiomers of a given molecule. The phase dependence of the effects induced by the electric field on chirality differentiates the chirality-sensitive effects described here from the other effects proposed in the literature, which rely on the shift of the NMR lines due to molecular chirality.

In further discussion, it is convenient to introduce the concept of NMR phenomena in which the presence of the electric field is essential. Among others, the effects described in this chapter can be termed nuclear magnetoelectric resonance (NMER; [nmɪr], i.e., NM-ee-R). The adjective magnetoelectric (rather than “electromagnetic”) emphasizes that we are dealing with NMR phenomena either generating an electric field (excluding the electric field associated with the time-varying magnetic field resulting from the Larmor precession of magnetic moments) or induced by the application of an external electric field. This means that interactions such as quadrupolar coupling in which an internal molecular electric field is involved, are also ruled out. NMR is based on the magnetic response of the spin system, while NMER also utilizes the electric response originating from the partial orientation of the sample due to the presence of the antisymmetric part of the nuclear magnetic shielding tensor, σ*. Although the definition outlined above is not entirely strict, the bottom line is that at the essence of NMER lies the slight perturbations of the dynamics of the nuclear spin system associated with the presence of an electric field. Not all NMER signals are chirality-dependent, as may be exemplified in the case of the quadratic effects.

Although work on experimental NMER spectra is still in progress, let us use a ¹H NMER spectrum of sucrose calculated using quantum chemistry methods, to indicate the unique features of NMER spectroscopy (see Figure 7.7).

Let us have a look at the first region of the spectra marked by rectangle I. The signals corresponding to NMER signals have opposite phases for d-sucrose and l-sucrose, i.e., for one, the signals are up, while for the other, they are down. In particular, the NMER signals of enantiomers are phase-shifted by ±90° with respect to the NMR signals; the NMER spectra mentioned above are additionally shifted by −90° for clarity. In contrast to NMER, the ¹H NMR spectrum of sucrose is the same for both enantiomers. In the case of NMER, the relative intensities of lines and, in particular, integrals of the peaks do not correspond to the molar ratios of protons in the molecule as it is for NMR (see the region marked by rectangle II). Rather than providing information about the number of protons in the individual chemical groups, their intensities depend on the orientation of the given group of protons with respect to the permanent dipole moment of the molecule. Based on this dependence, one may infer the absolute configuration of the molecule through NMER. Finally, rectangle III shows the region with a strong solvent signal (water). This signal would not cover the spectrum of the studied compound in the case of NMR since water is, similar to many other commonly used solvents, an achiral substance. The common feature of NMR and NMER signals is that, in both cases, the frequency of the signal equals the spin precession frequency. The points differentiating NMER from NMR are summarized in Table 7.2.

It is convenient to use second-order perturbation theory to find tractable analytical solutions of eqn (7.18), averaged over the ensemble of tumbling molecules. This approach is used in the Bloch–Redfield–Wangsness theory.^20,21  For the considered case, i.e., isotropic rotational diffusion, the changes in the density matrix ϱ(t) over time may be evaluated using the irreducible spherical tensor decomposition without the need to explicitly average over molecular orientations. This approach was used in the standard NMR case, i.e., without the electric field, in ref. 22. If the coefficients of the obtained relaxation matrix are time-dependent, as in the case of a modulated electric field, then the equation is further approximated by transforming the reference frame to a frame that rotates around the z-axis with the spin precession frequency of each spin, expanding the obtained fundamental matrix of the linear ordinary differential equation system in the Floquet–Magnus series, and truncating the series after the few first terms.²³ 

Taking into account the incoherent spin system dynamics, the number of irreducible spherical components of the Hamiltonian $H ˆ E$ is larger than 30, even for a two-spin system, which gives at least a thousand of interactions described by the tensors. Therefore, a careful selection of the desired interactions is mandatory. This selection is based on the following three criteria: (i) the amplitude of the effect is sufficiently large for detection in the magnetic and electric fields of strengths accessible by the currently used experimental equipment, (ii) an appropriately designed experiment enables the separation of the effect from other simultaneously present effects, and (iii) experimental observation of the effect reveals interesting information about the molecular structure for chemists and physicists. Based on the general solution of eqn (7.18) and these three criteria, one can propose three chirality-sensitive effects described in subsequent sections: magnetization induced by antisymmetry of the shielding tensor (⁠ $E1$ ⁠, see Section 7.9), electric polarization induced by the antisymmetry of shielding tensor (⁠ $E2$ ⁠, see Section 7.10), and the interference between chemical shift anisotropy and dipolar relaxation in an electric field (⁠ $E3$ ⁠, see Section 7.11). In the following text, we assume that the sample is a liquid and the molecules tumble considerably faster than the electric field varies in time. The postulated effects are listed from those with the highest amplitude $( E 1 )$ to those with the lowest amplitude $( E 3 )$ ⁠.

The inductively coupled NMR signal detection requires precession of the nuclear magnetization, which takes place if the vector M has components perpendicular to the main static field B₀, used in the experiment. The external magnetic field B₀ is commonly assumed to be along the z-axis of the laboratory frame of reference, and we will follow this convention in further text. However, it is worth noting that in some cases it is convenient to orient the field in a different way due to technical reasons, e.g., along the x-axis – this is the case with low magnetic field experiments and, partially, in NMR imaging. In the case of B₀ = B₀e_z, the vector M rotates in the xy-plane with the angular frequency ω₀ = γ_NB₀ (in rad Hz).

In order to rotate the magnetization M from its initial orientation resulting from thermodynamic equilibrium, one can use an oscillating magnetic field B₁, which is perpendicular to the main field B₀ and changes with frequency ω_B (called a resonance frequency) over time. Both components of the term B + σ^cB × E in eqn (7.19) may be responsible for the rotation of the magnetization M if one applies external, oscillating magnetic B₁(t) and electric E₁(t) fields. However, even for a sample placed in an electric field of strength E₁ = 1 kV mm⁻¹ and for which the pseudoscalar σ_c is 1 fm V⁻¹, the term σ_cE₁ is considerably smaller than unity. Thus, the angle of rotation of the nuclear magnetization induced by the time-dependent electric field remains a few orders of magnitude smaller than that caused by the time-varying magnetic field.

The details of the experimental protocol are shown in Figure 7.8. The electric field E₁, which oscillates with the spin precession frequency ω₀ for the duration τ > 3T₁, rotates the nuclear magnetization (Figure 7.8A). A capacitor may generate an electric field E₁; however, the oscillating-in-time B₁ field saturates the magnetization and decreases the amplitude of the chirality-sensitive signals (more details are given in ref. 17). Therefore, a sufficiently effective suppression of the unwanted B₁ requires a dedicated resonator. The resonator depicted in Figure 7.8C consists of a capacitor modified by adding two loops (i.e., loop-gap resonator). The magnetic field generated by these loops cancels the transverse component of the magnetic field generated by the capacitor, and the resulting magnetic field is oriented in the same direction as the field B₀ of the NMR spectrometer. The volume of the sample placed at the central point between the plates, where unwanted excitation does not saturate the magnetization, is marked by the red colour (κ < 10⁻⁵) in Figure 7.8C, and is sufficiently large for generating a signal approximately 10% of that generated in the standard NMR experiment (Figure 7.8B). The signals obtained from the enantiomers exhibit different phases, contributing to the spectrum as both positive and negative peaks. The positive peak is arbitrarily attributed to the (R)-enantiomer (Figure 7.8C).

Favourable samples for this experiment are the derivatives of cyclopropane (e.g., 1,3-diphenyl-2-fluoro-3-trifluoromethylcyclopropene) and light alcohols (e.g., 1,1,1-trifluoropropan-2-ol), which contain the fluorine nucleus. The latter are more convenient as they are stable at room temperature. Taking into account the possible imperfections in the fabrication of the resonator, the estimated chirality-induced NMR signal that would reach 1–5% of the standard achiral NMR signal, is predicted for an electric field of the strength E₁ = 10 V mm⁻¹ and the magnetic field B₀ = 10 T.

The $E1$ effect allows one to discriminate between the enantiomers, and it has the largest amplitudes among the chirality-sensitive effects described in this and the next chapter. The main limitation in detecting it is the suppression of the unwanted magnetic field generated by the oscillating-in-time electric field, along with convection in the sample caused by dielectric heating from the E₁ field.

The experimental protocol for $E2$ effect is shown in Figure 7.9. Initially, the magnetization M is rotated from its equilibrium position along the direction of the magnetic field B₀ to the xy-plane, followed by precession at the frequency ω (Figure 7.9A). Its precession induces the chirality-sensitive electric polarization P, which, in the reference frame rotating with the frequency ω, aligns with the x-axis (Figure 7.9B). For a sample placed between the plates of the resonator, the signals $S(P)$ of polarization P exhibit opposite signs for the enantiomers (shown arbitrarily as a positive peak for the (R)-enantiomer and a negative peak for the (S)-enantiomer). If the symmetry of the resonator is preserved, then the unwanted signal $S(M)$ of the magnetization M integrated over the sample is zero (Figure 7.9C). See ref. 17 for more details about the resonator.

The favourable samples for the $E2$ effect are the same as those for the effect $E1$ ⁠. The estimated magnitude of the chirality-sensitive electric polarization for these samples is P = 100 aC m⁻² at a magnetic field of strength B₀ = 10 T. Similar to the $E1$ effect, the $E2$ effect allows one to discriminate between enantiomers. This is one of the strongest predicted effects. The main advantage of the observation of the $E2$ effect is that the sample is not perturbed by dielectric heating. If the sign of the product μ^e·σ* is known from quantum-chemical computations, then the experimental determination of the sign of the signal $S(P)$ allows one to not only distinguish between the enantiomers but also to determine the absolute configuration of the molecule.

The allowed transitions related to $E3$ occur in the populations/zero coherence block (M₀) and the single-quantum coherence block (M₁) of the relaxation matrix of the spin system. From the viewpoint of the experimenter, the transitions within the M₀ block are more convenient for detection as they require excitation by the electric field at the difference frequency. The maximal amplitude of the $E3$ effect is achieved if the electric field E and the field B₀ are parallel to each other. The experimental protocol for the $E3$ effect is shown in Figure 7.10. At the beginning of the experiment, the initial state of the spin system, I_yS_x + I_xS_y, is created using the modified INEPT pulse sequence (Figure 7.10A). Then, this state evolves under the influence of the electric field $E 1 = E 1 sin [ ( ω I − ω S ) t ] z ˆ$ ⁠. The final state induced by field E₁ is I_z of amplitude 〈I_z〉_c, which can be distinguished from the I_z component of amplitude 〈I_z〉_ac created by the dipolar and CSA relaxation mechanism by subtracting the signal acquired with electric field E₁ from that obtained without the E₁ field.

In contrast to $E1$ and $E2$ effects, that vanish if the antisymmetric part of the nuclear shielding tensor is zero, the $E3$ effect is non-zero even for σ* = 0 (as long as ψ₂ ≠ 0); thus, it may be observable in a broader range of chiral molecules. The modulation of the electric field at the difference frequency permits the use of a simpler resonator for excitation than that required in the experiment aiming to observe the $E3$ effect, i.e., a parallel-plate capacitor.

Let us briefly summarize several pieces of information that may be obtained in the studies of the above-described effects $E1$ ⁠, $E2$ ⁠, and $E3$ ⁠.

• The effects $E1–E3$ are linear in the electric field; consequently, they are only present in chiral samples.

• The measurements of the amplitudes of $E1–E3$ effects allow us to determine the three-dimensional structure of the molecule. In particular, they offer insight into the relative orientation of local nuclear magnetic properties, i.e., the nuclear magnetic shielding tensors and the permanent electric dipole moment of the molecule. Furthermore, each effect depends in a different manner on the structural parameters and tensor components; thus, a combination of these effects provides comprehensive information about the molecular structure.

• The $E1–E3$ effects allow one to distinguish directly between the enantiomers. Measurements of $E1–E3$ combined with the results of simple quantum-chemical computations permit the determination of the absolute configuration of the molecule. The quantitative measurement, i.e., determination of the sign (phase) of the signals of the $E1–E3$ effects, is sufficient for assigning the absolute configuration, which is less experimentally demanding.

The predicted effects for a single-spin and a two-spin system differ in terms of their optimal angle between the externally applied electric field E and the field B₀, and the frequency of the electric field E, that varies with time. The application of the oscillatory electric field amplifies the $E1–E3$ effects by at least six orders of magnitude as compared to the application of a static field. These factors, favourable samples for the first experiments (predicted from quantum-chemical computations), and the expected signals generated by the effects, are summarized in Table 7.3.

The available strength of the electric field limits the highest achievable signals of the predicted effects; therefore, a careful selection of the sample for the first set of experiments is required. The fluorine nucleus has favourable properties as it has a high gyromagnetic ratio, large natural abundance, and notable antisymmetry of the nuclear magnetic shielding tensor. The magnitude of the effects for the same electric-field strength field may vary depending on the particular sample. However, for a given group of chemical compounds, one can predict that their magnitudes decrease in the following series: $E1>E2>E3$ ⁠. The general requirements for the experimental setup, assuming a molecule with a permanent electric dipole moment of μ^e = 1 D, are a magnetic field B₀ on the order of 10 T and an electric field E = 0.1–1 kV mm⁻¹ which, for favourable samples mentioned in Table 7.3 (e.g., fluorine derivatives of light alcohols), generate expected signals from 5 × 10⁻¹ (the $E1$ effect) to 5 × 10⁻⁶ (the $E3$ effect) of the ¹H NMR signal, obtained after excitation of the sample by a 90° pulse at thermodynamic equilibrium. For effects with relatively large amplitudes, it is sufficient to use, at the beginning of the experiment, the nuclear magnetization at thermodynamic equilibrium, whereas in the case of the effects with the smallest amplitude, an application of hyperpolarization techniques may be required, to increase the initial nuclear magnetization (see Chapter 10 for details of hyperpolarization techniques applied to chiral molecules).

Besides the effects that are chirality-sensitive, the studies of the spin dynamics governed by the Hamiltonian $H ˆ E$ and eqn (7.18), have revealed several other effects induced by an external electric field E, but which are not related to molecular chirality. Thus, such phenomena are potentially observable for all NMR-active nuclei. The two effects that are quadratic in the electric field are (i) the transverse relaxation rate induced by the isotropic nuclear magnetic shielding and (ii) induction by the electric field of the transverse component of the magnetization. A detailed description of these effects lies beyond the scope of this book, so the interested reader is referred to the source articles in which these effects are described.^11,29 

Briefly speaking, if one takes into account the partial orientation of the sample under consideration due to the application of the electric field, then, unlike in the isotropic case, the isotropic part of the magnetic nuclear shielding tensor can lead to nuclear relaxation. Only the magnetization components that are perpendicular to the magnetic field B₀ are affected by the electric field. The optimal conditions for the observation of such an effect are as follows: the magic angle between the electric and the magnetic fields, i.e., $arccos(1/ 3 )≈ 54.7 °$ ⁠, application of a strong, static electric field, i.e., the amplitude of E at least in the order of kVs per mm, and as small a ratio between the isotropic part of the shielding tensor and its anisotropy as possible. As a promising sample for such an experiment, a Pt(CN)₆Tl complex containing a ²⁰⁵Tl nucleus was proposed.¹¹ 

The second effect arises when the nuclear magnetization returns from the direction that is opposite to the magnetic field.²⁹  Then, the electric field E induces the transverse component of magnetization, which is not present in the absence of the electric field. The emergence of the additional transverse component stems from the disruption of the independence between the longitudinal and transverse nuclear relaxation processes, which is maintained in the absence of the externally applied electric field E. To increase the magnitude of the effect, E can be made to oscillate at one half of the spin precession frequency ω₀. Optimally, the electric field is generated by the capacitor tilted by 45° with respect to the field B₀, i.e., E₁ = E₁ sin(ω₀t)(e_y + e_z), and applied for a duration that is equal to the longitudinal relaxation time of the sample measured in the absence of the electric field E₁, i.e., τ = T₁. For B₀ = 10 T, and E₁ = 5 kV mm⁻¹, the estimated amplitude of the effect for the ¹⁹⁵Pt nucleus in the cisplatin molecule is 10 mHz, which corresponds to a signal approximately four orders of magnitude smaller than that obtained from the same sample in thermodynamic equilibrium after the application of a 90° pulse without the electric field E.

One may find higher-order effects induced by an external electric field in NMR spectroscopy, but the main limiting factor is a fast decrease in the amplitude of the effects with their order, since the factor $μ e E k B T$ is much smaller then unity in the electric fields of strength that does not exceed the breakdown field in liquids (several kV mm⁻¹). A similar remark applies to the comparison of the magnitudes of the effects, that are due to the coherent dynamics of nuclear spins, and the effects caused by nuclear relaxation – typically, the former have a much larger amplitude than the latter.

As mentioned in Sections 7.9 and 7.10, the interconnection between time-dependent electric and magnetic fields resulting from Maxwell’s equations is the main challenge faced in the experimental detection of chirality-sensitive NMER effects. Although an undesirable magnetic field manifests itself differently in each of the chirality-sensitive phenomena, a coherent description of how to minimize its influence on the NMER signals can be obtained for the analysis of the κ parameter. In the $E1$ effect, the oscillating electric field generates an unwanted magnetic field, which results in saturation of the NMER signal, while in the $E2$ effect, the source of the time-dependent magnetic field is the precessing nuclear magnetization. In both cases, the lowest possible value of κ parameter is desirable.

Any radiofrequency system that generates an electric field will generate a magnetic field oscillating at the same frequency. Hence, one cannot discriminate between them based on their time dependence. However, one can take advantage of the electric and magnetic fields spatial dependence and, consequently, at a given place, the electric field may dominate over the magnetic field. From this, it follows that the primary aim is to optimise the spatial distribution of E₁ and B₁ fields by minimizing the E₁/B₁ ratio and the angle between B₁ and B₀. For instance, ideally, a capacitor-like source of the electric field with plates perpendicular to the y-axis would have the following fields: E₁(t)e_y and B₁(t)(x² + z²)(e_x − e_z) in the sample volume. In practice, the construction of such a device is technically not possible. The main reason is that even a uniform radiofrequency electric field generates a circulating magnetic field, which increases in amplitude with the distance from the centre of the space containing the electric field. Moreover, it is hard to shield the region where the sample is placed from the magnetic field generated by the other parts of the resonance circuit. In the subsequent discussion, let us explore the different possible approaches to achieve a system generating an electromagnetic field distribution with a low value of the κ parameter. The optimal value of the κ parameter is, at most, one hundred thousandths in the volume of at least several cubic millimetres. The maximal acceptable value of the κ parameter depends on how precisely the phase of the NMR signal can be determined in the experimental search of the $E1$ effect, alternatively how large is the single-to-noise if we aim for the detection of the $E2$ effect. A large signal-to-noise ratio, e.g., S/N ∼ 1000, or sensitive phase detection, e.g., with a resolution better than 0.1°, would increase the maximum acceptable value of the parameter κ by two orders of magnitude, giving κ ≤ 0.001.

The close relationship between the properties of the sample and electromagnetic field distribution quantified by eqn (7.21) and (7.23), is illustrated in Figure 7.11.