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Book Chapter

Chapter 8: Chirality-sensitive Effects Induced by Antisymmetric Spin–Spin Coupling^{^†}

P. Garbacz

aFaculty of Chemistry, University of Warsaw, Pasteura 1, 02-093, Warsaw, Poland [email protected]

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;

J. Vaara

bNMR Research Unit, University of Oulu, P.O. Box 3000, FI-90014 Oulu, Finland [email protected]

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Doi:

https://doi.org/10.1039/BK9781839169588-00204

Published:

20 Sep 2024
Special Collection: 2024 eBook Collection

Series: New Developments in NMR

Product Type: Open Access

Page range:

204 - 242

This chapter is subject to a Creative Commons CC-BY-NC-ND 4.0 International license.

The chapter introduces the concepts of direct and indirect spin–spin coupling as well as hyperfine spin–spin coupling. It is shown that a chiral molecule with a spin pair coupled by an antisymmetric interaction carries an induced electric dipole moment whose direction depends on the handedness of the molecule. Finding this induced dipole, in turn, serves as the starting point for the derivation of the nuclear magnetoelectric resonance Hamiltonian and analysis of spin dynamics under the influence of an externally applied electric field. Four effects that enable direct chirality sensing in nuclear magnetic resonance spectroscopy are also described with emphasis on the relationships between chirality, antisymmetry of spin–spin coupling, electric dipole moment, and molecular structure. Several capacitive devices capable of generating an electric field with properties suitable for nuclear magnetoelectric resonance experiments, in which the central role is played by spin–spin couplings, are described.

Funding Group:

Award Group:
- Funder(s):
  National Science Centre
- Award Id(s):
  2018/31/B/ST4/02570
Award Group:
- Funder(s):
  Research Council of Finland
- Award Id(s):
  331008
Award Group:
- Funder(s):
  University of Oulu
Funding Statement(s):
Financial support from the National Science Centre, Poland (through OPUS 16 grant No. 2018/31/B/ST4/02570), Academy of Finland (project 331008) and University of Oulu is acknowledged

8.1 Introduction

The mathematical framework, theoretical depiction, and relevant electronic properties required for the description of the interactions between spins of a chiral molecule, are closely related to those mediated by nuclear shielding, as described in Chapter 7. Therefore, in this chapter, we will refer quite often to the results obtained in the previous one, and consequently reading the previous chapter before the current one is recommended.

Many fundamental particles and nuclei have magnetic moments that can interact with each other, so depending on the context, one may name these interactions differently. For instance, in the domain of nuclear magnetic resonance (NMR) spectroscopy, the interactions between two nuclear magnetic dipole moments are called spin–spin coupling. In contrast, electron paramagnetic resonance (EPR) spectroscopists prefer to use the term hyperfine coupling for the interaction between the magnetic dipole moments of an electron and a nucleus or interaction between the magnetic dipole moments of two electrons.^1,2 These two terms are frequently used, although both cases involve interactions between the nuclear and electron magnetic moments resulting from their spins. These are hyperfine interactions, i.e., they take place between the magnetic moments and magnetic fields in the molecule. Hereafter, the term “dipole” will be left out in examples where its omission will not introduce ambiguity.

Similarly, we will follow the common convention and use the terms spin and magnetic moment interchangeably, although the magnetic moment is perceived as a concept of classical physics and the spin is considered as a purely quantum-mechanical entity; their correspondence is expressed by the equation $μ_{I} = g_{I} \hat{I}$ ⁠, where μ_I is the magnetic moment expressed in Bohr magnetons for electrons, μ_B = 9. 274 009 994 × 10⁻²⁴ J T⁻¹ or nuclear magnetons for nuclei μ_N = 5.0 507 837 461 × 10⁻²⁷ J T⁻¹, g_I is its g-factor (dimensionless), and $\hat{I}$ is the spin operator (a dimensionless vector of Pauli matrices, in the case of a one-half spin). Alternatively, the proportionality constant between the magnetic moment and the spin operator may be expressed using the gyromagnetic ratio γ (in rad T⁻¹ s⁻¹); $μ_{I} = γ_{I} ℏ \hat{I}$ ⁠, where ℏ = 1.054571817 × 10⁻³⁴ J s rad⁻¹ is the reduced Planck constant, and the magnetic moment μ_I is expressed in J T⁻¹.

Two spins may interact in at least two ways. One is direct coupling, which, from the classical physics viewpoint is an interaction between two magnetic dipoles through their magnetic field.^3,4 The other is an indirect coupling resulting from the influence of the magnetic field of the magnetic dipole on the molecule’s electrons. The distinction between direct and indirect spin–spin couplings lies in the mediating contribution of the electrons of the molecule, rather than in the presence or absence of a chemical bond. For example, through-space contributions to indirect spin–spin coupling between ¹⁹F nuclei due to two lone-pair orbitals on intramolecularly crowded molecules are relatively common.⁵ In EPR spectroscopy, unlike the convention used in NMR, both interactions are considered together as one interaction.

Spin–spin couplings may be direction-dependent similarly to magnetic shielding, and it is convenient to describe their anisotropic nature with tensors: D for direct spin–spin coupling, J for indirect spin–spin coupling, and A for hyperfine coupling. Usually, such tensors are expressed in Hz. Frequently, from the point of view of quantum-chemical computations, the interactions between two magnetic moments are considered as a second (mixed) derivative of the molecular energy

E

with respect to two magnetic moments (μ₁ and μ₂) taking part in the interaction,^6–8

D_{12} + J_{12} = μ_{1} μ_{2} (\begin{matrix} \frac{\partial^{2} E}{\partial μ_{1 x} \partial μ_{2 x}} & \frac{\partial^{2} E}{\partial μ_{1 x} \partial μ_{2 y}} & \frac{\partial^{2} E}{\partial μ_{1 x} \partial μ_{2 z}} \\ \frac{\partial^{2} E}{\partial μ_{1 y} \partial μ_{2 x}} & \frac{\partial^{2} E}{\partial μ_{1 y} \partial μ_{2 y}} & \frac{\partial^{2} E}{\partial μ_{1 y} \partial μ_{2 z}} \\ \frac{\partial^{2} E}{\partial μ_{1 z} \partial μ_{2 x}} & \frac{\partial^{2} E}{\partial μ_{1 z} \partial μ_{2 y}} & \frac{\partial^{2} E}{\partial μ_{1 z} \partial μ_{2 z}} \end{matrix}) / h .

(8.1)

Therefore, this kind of interaction is described by a tensor which is represented by a 3 × 3 matrix, and which gives the energy of the interaction between the magnetic moments. For instance, the energy of the magnetic moments μ₁ and μ₂ interacting through indirect spin–spin coupling is

{\hat{H}}_{J} / h = \begin{matrix} (\begin{matrix} {\hat{I}}_{1 x} & {\hat{I}}_{1 y} & {\hat{I}}_{1 z} \end{matrix}) \end{matrix} (\begin{matrix} J_{x x} & J_{x y} & J_{x z} \\ J_{y x} & J_{y y} & J_{y z} \\ J_{z x} & J_{z y} & J_{z z} \end{matrix}) (\begin{matrix} {\hat{I}}_{2 x} \\ {\hat{I}}_{2 y} \\ {\hat{I}}_{2 z} \end{matrix}) .

(8.2)

The form of the Hamiltonian

{\hat{H}}_{J}

follows a slightly different convention than that adopted for the interaction of nuclear magnetic moments with a magnetic field, wherein the dependence on the gyromagnetic ratio is often explicitly included in the Hamiltonian. Analogous equations to apply to the direct and hyperfine tensors.

J-coupling is comparable to D-coupling in rare and extreme cases, e.g., thallium–platinum cyanide complexes.⁹ Usually, the strength of direct spin–spin coupling exceeds that of indirect coupling by at least an order of magnitude. The direct spin–spin coupling, whose spatial dependence is entirely determined by the unit vector e₁₂ from the first interacting nucleus to the second,

{\hat{H}}_{D} / h = b_{12} [3 ({\hat{I}}_{1} \cdot e_{12}) ({\hat{I}}_{2} \cdot e_{12}) - {\hat{I}}_{1} \cdot {\hat{I}}_{2}],

(8.3)

has the dipole–dipole coupling constant

b_{12} = - \frac{μ_{0}}{4 π} \frac{γ_{1} γ_{2}}{{r_{12}}^{3}} h

(8.4)

on the order of several kHz, while the indirect coupling rarely exceeds several hundreds of Hz. In , μ₀ = 1.25663706212 × 10⁻⁶ T m A⁻¹ is the vacuum permeability and r₁₂ is the distance between two spins (typically a few Å). For example, the dipole–dipole coupling constant of molecular hydrogen b_HH is about 300 kHz, while the isotropic indirect spin–spin coupling is approximately 280 Hz.¹⁰ Since the direct coupling is traceless, it does not manifest itself in the NMR spectrum taken in an isotropic liquid sample, unless its contribution to nuclear relaxation is considered.

As was shown in Chapter 7, the antisymmetric component of a tensor [see eqn (7.3b) and ] is essential for linking molecular chirality with the magnetic properties of a spin. The direct and indirect couplings differ notably concerning the antisymmetric component: the tensor J possesses an antisymmetric component (J*) if the symmetry of the molecule does not imply that it must be zero, while the tensor D is symmetric by definition and, consequently, cannot have an antisymmetric part. Therefore, the tensors J and D of an n-spin system can be written using spherical tensor decomposition as follows:

{\hat{H}}_{D} / h = \sum_{i = 1}^{n} \sum_{\begin{array}{l} j = 1 \\ j > i \end{array}}^{n} {\hat{I}}_{i} \cdot D_{i j}^{sym} \cdot {\hat{I}}_{j},

(8.5)

{\hat{H}}_{J} / h = \sum_{i = 1}^{n} \sum_{\begin{array}{l} j = 1 \\ j > i \end{array}}^{n} (J_{i j}^{iso} {\hat{I}}_{i} \cdot {\hat{I}}_{j} + J_{i j}^{*} \cdot ({\hat{I}}_{i} \times {\hat{I}}_{j}) + {\hat{I}}_{i} \cdot J_{i j}^{sym} \cdot {\hat{I}}_{j}) .

(8.6)

One can observe that the cross-product term, i.e.,

{\hat{I}}_{i} \times {\hat{I}}_{j}

in , bears resemblance to the antisymmetric exchange interaction, known as the Dzyaloshinskii–Moriya interaction known from solid-state studies of magnetic materials.^11–14 However, this chapter focuses on liquids rather than solids, and spins are not necessarily alike, so the correspondence is not strict.

8.2 Chirality-sensitive Spin–Spin Coupling

Let us consider the effect of the spin–spin coupling energy on the average orientation of a molecule. We will follow the derivation given by Garbacz and Buckingham.¹⁵ If the molecule possesses a permanent electric dipole moment μ^e, then the sample polarization, which is the sum of electric moments per sample volume, will not vanish because some orientations of the molecule are more probable than others. According to statistical thermodynamics, the induced electric dipole moment by the presence of the spin–spin coupling is¹⁶

〈 μ^{c} 〉 = \frac{\int μ^{e} \exp (- \frac{{\hat{H}}_{J}}{k_{B} T}) d Ω}{\int \exp (- \frac{{\hat{H}}_{J}}{k_{B} T}) d Ω},

(8.7)

where k_B = 1.380649 × 10⁻²³ J K⁻¹ is the Boltzmann constant, and T is temperature. In , the solid angle Ω parametrizes the orientations of the molecules with respect to the laboratory frame of reference, and the integral is taken over all orientations of the molecule. The energy associated with the Hamiltonian

{\hat{H}}_{J}

is much smaller than the thermal energy k_BT. E.g., the internal energy of one mole of molecular hydrogen is about 6.25 kJ at temperature T = 300 K, while the contribution of the indirect coupling between protons of H₂ slightly exceeds 0.1 µJ. Therefore, the fraction

- \frac{{\hat{H}}_{J}}{k_{B} T}

is small compared to unity, and one can expand the exponential function into a power series keeping only the lowest-order term, which gives

〈 μ_{α}^{c} 〉 = \frac{h}{k_{B} T} 〈 - μ_{α}^{e} J_{12, β γ} {\hat{I}}_{1 β} {\hat{I}}_{2 γ} 〉 .

(8.8)

The 3-index term

μ_{α}^{e} J_{12, β γ}

contains quantities that vary with molecular orientation, while the 2-index term

{\hat{I}}_{1 β} {\hat{I}}_{2 γ}

is essentially independent of the orientation of the molecule. Let us make the not-so-limiting assumption that the spin state is independent of molecular tumbling. In this case, the isotropic average of the induced electric dipole moment 〈μ^c〉 has to be proportional to the Levi-Civita tensor defined in eqn (7.8), because the Levi-Civita tensor is the only one third rank-tensor that is isotropic, and then becomes

〈 μ_{α}^{c} 〉 = \frac{h}{k_{B} T} (- \frac{1}{3} ε_{α' β' γ'} μ_{α'}^{e} J_{12, β' γ'}) (ε_{α β γ} {\hat{I}}_{1 β} {\hat{I}}_{2 γ}) .

(8.9)

can be written in a compact form using the cross products and the antisymmetry J* of the indirect spin–spin tensor as

〈 μ^{c} 〉 = - h J_{12}^{c} {\hat{I}}_{1} \times {\hat{I}}_{2},

(8.10)

where the pseudoscalar

J_{12}^{c}

J_{12}^{c} = μ^{e} \cdot J_{12}^{*} / (3 k_{B} T) .

(8.11)

The pseudoscalar J^c (we omit indices if it does not introduce ambiguity) has the following properties: it exhibits reversed signs for enantiomers, it is equal to zero for achiral molecules, and

J_{12}^{c} = - J_{21}^{c}

⁠. The last property follows directly from because the exchange of rows and columns reverses the signs of the antisymmetric elements of the spin–spin coupling matrix.

Eqn (8.9) can be directly related to Section 1.2, which discusses the relationship between the chiral tetracube handedness and the reference frame in which tetracube’s parameters are described. In eqn (8.9), the indirect spin–spin coupling tensor J corresponds to the parameter specifying the handedness of the tetracube, and information about the handedness of the Cartesian coordinate system is provided by the Levi-Civita tensor. The components of the Levi-Civita tensor for the left-handed Cartesian coordinate system have the same magnitude but opposite signs compared to those given in eqn (7.8) for the right-handed system. Consequently, the sign of the pseudoscalar $J_{12}^{c}$ for an (R)-enantiomer in a right-handed coordinate system is the same as that of an (S)-enantiomer considered in a left-handed coordinate system. For left-handed and right-handed molecules placed in either the left or to the right-handed reference frame, the signs of the pseudoscalar $J_{12}^{c}$ are opposite, analogous to the situation shown in Figure 1.2 for the chiral tetracube.

Further insight into the chirality-sensitive contribution resulting from the antisymmetric part of the indirect spin–spin coupling tensor may be obtained by observing that (i) the energy associated with the interaction of an induced electric dipole 〈μ^c〉 with an (internal) electric field is −〈μ^c〉 · E, meaning that

{\hat{H}}_{J, c} = h J_{12}^{c} [({\hat{I}}_{1} \times {\hat{I}}_{2}) \cdot E],

(8.12)

and (ii) the vector triple product

({\hat{I}}_{1} \times {\hat{I}}_{2}) \cdot E

remains invariant under cyclic permutation of its components. Therefore, one may view the energy −〈μ^c〉 · E equivalently as the energy of a magnetic dipole

- J_{12}^{c} h ({\hat{I}}_{2} \times E)

induced by an (external) electric field interacting with the magnetic moment of spin I₁. The third view on the chirality-sensitive contribution to the Hamiltonian of the spin system is that it is the interaction of a magnetic dipole

- J_{12}^{c} h (E \times {\hat{I}}_{1})

induced by an (external) electric field interacting with the magnetic moment of the spin I₂. It is worth noting that the order of the components in the induced magnetic dipoles, i.e.,

{\hat{I}}_{2} \times E

and

E \times {\hat{I}}_{1}

⁠, is reversed, since

J_{12}^{c} h (E \times {\hat{I}}_{1}) = J_{21}^{c} h ({\hat{I}}_{1} \times E)

⁠.

In terms of the dielectric properties of liquids, the contribution ${\hat{H}}_{J, c}$ appears because of the partial orientation of molecules bearing a permanent electric dipole moment under the influence of an externally applied electric field. The partial orientation prevents the complete averaging out of the anisotropic parts of the indirect spin–spin coupling tensor J between spins I₁ and I₂ observed in an isotropic phase.

8.3 Relationship Between Chirality and Antisymmetric Spin–Spin Coupling

While the antisymmetric part of the indirect spin–spin coupling tensor J* is a pseudovector, the molecular permanent dipole moment μ^e is a polar vector. The transformation properties of their scalar product, a pseudoscalar J^c, may be analyzed in two equivalent ways: by selecting a particular frame of reference determining the plane of reflection or by considering a space inversion quantified by the parity operator

\hat{P}

⁠. Let us consider first the effect of a mirror reflection of a molecule in a plane that is defined by vectors J* and μ^e, which are not parallel to each other. In such a case, the pseudoscalar

J_{12}^{c}

reverses its sign since the reflection changes the direction of the pseudovector J* to its opposite. Alternatively, the application of space inversion, i.e., the parity operator

\hat{P}

⁠, yields

(μ^{e} \cdot J_{12}^{*}) \cdot [({\hat{I}}_{i} \times {\hat{I}}_{j}) \cdot E] \to ({- μ^{e}} \cdot J_{12}^{*}) \cdot [({\hat{I}}_{i} \times {\hat{I}}_{j}) \cdot {- E}] .

(8.13)

Therefore, although energy quantified by the Hamiltonian remains the same, the sign of the pseudoscalar

J_{12}^{c}

directly distinguishes enantiomers.

Following a similar approach, one can deduce the effect of mirror reflection on the component of the Hamiltonian associated with the antisymmetric part of the indirect spin–spin coupling tensor, J*. If the plane of the reflection is defined by the pseudovector J* and the pseudovector

{\hat{I}}_{i} \times {\hat{I}}_{j}

⁠, then the effect of the reflection in that plane on their scalar product is

J_{12}^{*} \cdot ({\hat{I}}_{i} \times {\hat{I}}_{j}) \to {- J_{12}^{*}} \cdot {- ({- {\hat{I}}_{i}} \times {- {\hat{I}}_{j}})} .

(8.14)

This is the result stated in ref. 17.

8.4 Time-evenness of the Pseudoscalar J^c

In , the condition that the spectroscopic quantity suitable for studying chiral molecules has to be a pseudoscalar that reverses its sign upon space inversion and remains invariant over time was imposed. The spin operators

{\hat{I}}_{i}

and

{\hat{I}}_{j}

are P-even and T-odd quantities. Thus, assuming that the antisymmetric indirect spin–spin coupling Hamiltonian

{\hat{H}}_{J^{*}} / h = J_{12}^{*} \cdot ({\hat{I}}_{1} \times {\hat{I}}_{2})

[see ] preserves parity, one can deduce that the antisymmetry

J_{12}^{*}

is a time-even pseudovector, since

\begin{matrix} {\hat{H}}_{J^{*}} & \propto & J_{12}^{*} & \cdot & ({\hat{I}}_{1} & \times & {\hat{I}}_{2}) \\ ↑ & ↑ & ↑ & ↑ \\ P & + 1 & + 1 & + 1 & + 1 \\ T & + 1 & + 1 & - 1 & - 1 \end{matrix}

(8.15)

Similarly, analysis of the whole chirality-sensitive contribution,

\begin{matrix} {\hat{H}}_{J, c} & \propto & (μ^{e} & \cdot & J_{12}^{*}) & [({\hat{I}}_{1} & \times & {\hat{I}}_{2}) & \cdot & E] \\ ↑ & ↑ & ↑ & ↑ & ↑ & ↑ \\ P & + 1 & - 1 & + 1 & + 1 & + 1 & - 1 \\ T & + 1 & + 1 & + 1 & - 1 & - 1 & + 1 \end{matrix}

(8.16)

reveals that the pseudoscalar

J^{c} \propto μ^{e} \cdot J_{12}^{*}

is time-even, fulfilling the requirement for a spectroscopic quantity indicating the chirality sense of the molecule, because the vectors μ^e and E are P-odd and T-even.

8.5 Relationship Between Chirality and the Permanent Electric Dipole Moment

As clearly seen from considerations given in this chapter up to now, as well as in Chapter 7, the molecular permanent electric dipole moment, μ^e, plays a crucial role in the occurrence of nuclear magnetoelectric resonance (NMER) chirality-sensitive effects. Based on the symmetry arguments given in Section 8.3 and the quantum-mechanics view on chirality given in Section 1.4, we are now in a good position to address the discrepancy in views on μ^e within the communities of chemists and physicists.

The only vector degree of freedom for a fundamental particle (such as an electron) is its spin $\hat{S}$ and, consequently, the possible electric dipole moment d of the particle may only be oriented either parallel or antiparallel to the spin $\hat{S}$ ⁠.^18–20 Consequently, the dipole moment has to be proportional to the spin, $d \propto d \hat{S}$ ⁠, and the Hamiltonian of its interaction with an electric field E takes the form $H \propto - d \hat{S} \cdot E$ ⁠. However, under space inversion and time reversal operations, such a Hamiltonian reverses its sign, i.e., $H$ equals, $- d \hat{S} \cdot (- E)$ and $- d (- \hat{S}) \cdot E$ ⁠, respectively.²¹ Therefore, an elementary particle possessing both the spin and electric dipole moments violates P-even and T-even symmetries expected for the Hamiltonian $H$ ⁠. Measurement for electric dipole moments of electrons,²² neutrons,²³ and molecules²⁴ indicate that the electric dipole moment is either zero or immeasurably small; see also ref. 25 for an extensive review of the performed experiments. The same argument can be used for molecules since they are composed of elementary particles.

This observation could raise reasonable doubts regarding the arguments put forward that NMER phenomena (and many others) would allow chirality sensing. On the other hand, the use of molecular electric dipole moments in the chemistry community is very common,^26,27 and their existence is frequently considered as an indisputable experimental fact.

So, can molecules have permanent electric dipole moments? The discrepancy in nomenclature results largely from how we define a “molecule”: an object resembling a rigid body embedded in a cloud of diffuse electric charge (the chemical perspective) or an object described by a wave function, consisting of elementary particles that constitute it (the physical perspective). To answer this question, stated and discussed by Klemperer et al. in ref. 28, we have to look deeper into the quantum-mechanical description of a chiral molecule.

If the nuclear spin Hamiltonian is assumed to be parity-conserving, then the energies of both enantiomers are the same. This does not mean, however, that the parity of any initial state of the molecule is preserved over a arbitrarily long period of time. Eigenstates of the Hamiltonian $| ψ_{+} 〉 = \frac{1}{\sqrt{2}} (| ψ_{R} 〉 + | ψ_{S} 〉)$ and $| ψ_{-} 〉 = \frac{1}{\sqrt{2}} (| ψ_{R} 〉 - | ψ_{S} 〉)$ have, indeed, parity +1 and −1, respectively (see Section 1.4). Depending on the rate of interconversion between the enantiomers, the wavefunction |ψ_R〉 of a chiral molecule successively evolves in time into |ψ₊〉, |ψ_S〉, −|ψ₋〉, … , and returns to the initial wavefunction |ψ_R〉 (Figure 8.1). The period of the time evolution of the quantum state of the chiral molecule can vary significantly depending on the molecule under consideration, ranging from fractions of a second for hydrogen peroxide^29,30 to practically unmeasurably long durations, such as for solid alanine.

Figure 8.1

The time evolution of the quantum state of a chiral molecule, | ψ 〉 = 1 2 [ cos ( ω t ) | ψ + 〉 + sin ( ω t ) | ψ − 〉 ] , in terms of its parity and handedness. On the red axes, the probabilities of two “racemic” states are shown: one of the positive parity, |〈ψ|ψ+〉|2 (the horizontal axis) and the other of the negative parity, |〈ψ|ψ−〉|2 (the vertical axis). The axes rotated by 45° represent probabilities of finding a molecule in a state attributed arbitrarily to the (R)-enantiomer, | ψ R 〉 = 1 2 ( | ψ + 〉 + | ψ − 〉 ) , and the (S)-enantiomer, | ψ S 〉 = 1 2 ( | ψ + 〉 − | ψ − 〉 ) . The higher the energetic barrier, the slower the state of the molecule rotates in the plane, resulting in a smaller uncertainty of enantiomeric excess δee in a given time interval δt. The clockwise sense of rotation, assumed in the figure, is also arbitrarily chosen, since it depends on the handedness of the chosen coordinate system.

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The time evolution of the quantum state of a chiral molecule, $| ψ 〉 = \frac{1}{\sqrt{2}} [\cos (ω t) | ψ_{+} 〉 + \sin (ω t) | ψ_{-} 〉]$ ⁠, in terms of its parity and handedness. On the red axes, the probabilities of two “racemic” states are shown: one of the positive parity, |〈ψ|ψ₊〉|² (the horizontal axis) and the other of the negative parity, |〈ψ|ψ₋〉|² (the vertical axis). The axes rotated by 45° represent probabilities of finding a molecule in a state attributed arbitrarily to the (R)-enantiomer, $| ψ_{R} 〉 = \frac{1}{\sqrt{2}} (| ψ_{+} 〉 + | ψ_{-} 〉)$ ⁠, and the (S)-enantiomer, $| ψ_{S} 〉 = \frac{1}{\sqrt{2}} (| ψ_{+} 〉 - | ψ_{-} 〉)$ ⁠. The higher the energetic barrier, the slower the state of the molecule rotates in the plane, resulting in a smaller uncertainty of enantiomeric excess δee in a given time interval δt. The clockwise sense of rotation, assumed in the figure, is also arbitrarily chosen, since it depends on the handedness of the chosen coordinate system.

In the case of eigenstates |ψ₊〉 and |ψ₋〉, the (permanent) electric dipole moment of the molecule vanishes. It is precisely these (pure) states that the conclusions drawn on the basis of symmetry arguments, rooted in physical considerations, are concerned with. From a chemist’s point of view, however, it is more useful to consider the mixed states |ψ_R〉 and |ψ_S〉, which are interpreted as enantiomers. In these cases, the molecule actually has an electric dipole moment. We note that the electric field, unlike the magnetic field, mixes molecular states of opposite parities; hence, a linear shift in levels (Zeeman effect) in the presence of a magnetic field is experimentally observed, whereas there is no evidence for the first-order Stark effect in a molecule in its eigenstate. This does not preclude the observation of a first-order Stark effect for a mixed state induced by an external electric field, reported routinely in rotational spectroscopy.^31–34 In such a case, from the perspective of molecular eigenstates, the effect of the electric field on the molecule can be more precisely called a second-order Stark effect.³⁵

In the diagram shown in Figure 8.1, the points corresponding to the states |ψ₊〉 (parity +1) and |ψ₋〉 (parity −1) do not have any electric dipole moment μ^e, although the antisymmetry J* is non-zero. Consequently, the pseudoscalar J^c vanishes. For convenience, let us assume that the antisymmetry pseudovector points upwards. Since the antisymmetry is P-even, its direction is the same for all states shown in the diagram. The states of the molecule, except for points on the horizontal and vertical axes, do have a dipole moment. If the states in the half-circle above the horizontal axis have the electric dipole moment pointing upwards, resulting in a positive pseudoscalar J^c, then the states in the half-circle below the horizontal axis have the electric dipole moment pointing downwards, yielding a negative pseudoscalar J^c. It is worth noting that, from the assumption of the parity conservation of the Hamiltonian, it follows that neither left-handed nor right-handed coordinate systems are preferred, so the direction of the circulation in Figure 8.1 is arbitrary and depends on the choice of the handedness of the coordinate system. If this were not true, e.g., we could define on the basis of fundamental laws that the enantiomer that first changes the state to that of positive parity |ψ₊〉 is designated as the (R)-enantiomer.

8.6 Relationship Between the Molecular Structure and Pseudoscalar J^c

Similar to the antisymmetric part of the shielding tensor σ*, the direction and amplitude of the J* antisymmetry are related to molecular symmetry. The relationships between nuclear site symmetry and the components of the antisymmetric part of the indirect spin–spin coupling tensor have been discussed by Buckingham and his co-workers in ref. 36 and 37. Bryce and Wasylishen analyzed the symmetry properties of indirect nuclear spin–spin coupling tensors of simple fluorine-containing molecules, CFCl₃ and F₂O.³⁸ An extensive survey of small and medium-sized molecules, with emphasis on the relationship between the molecular structure and directions of the permanent electric dipole moment μ^e and the antisymmetry J*, can be found in ref. 39. Despite the generally accepted existence of the antisymmetric part of the J tensor resulting from strong theoretical foundations,⁴⁰ this quantity has not yet been measured experimentally, although experiments aimed at measuring it have been reported.^41,42

Water is one of the simplest examples of a molecule with an antisymmetric indirect spin–spin coupling. The Fermi contact, spin–dipole, paramagnetic and diamagnetic spin–orbit contributions to the total J-coupling tensor between the protons and oxygen are listed in Table 8.1.

Table 8.1

Indirect spin–spin coupling of water (the molecule is in the yz-plane) computed in the CFOUR program using the coupled cluster singles and doubles (CCSD) method in the aug-pcJ-2 basis set; the initial molecular geometries were taken from Computational Chemistry Comparison and Benchmark Data Base and further optimized at the CCSD/aug-pCVTZ level. Tensor elements that contain antisymmetric parts are boldfaced.

	¹J(¹⁷O, ¹H_a)	¹J(¹⁷O, ¹H_b)	²J(¹H_a, ¹H_b)
Fermi-contact and correlated FC-SD	$(\begin{matrix} - 68.96 & 0.00 & 0.00 \\ 0.00 & - 68.96 & 0.00 \\ 0.00 & 0.00 & - 54.00 \end{matrix})$	$(\begin{matrix} - 68.96 & 0.00 & 0.00 \\ 0.00 & - 68.96 & 0.00 \\ 0.00 & 0.00 & - 54.00 \end{matrix})$	$(\begin{matrix} - 9.83 & 0.00 & 0.00 \\ 0.00 & - 15.81 & - 4.52 \\ 0.00 & - 4.52 & - 8.59 \end{matrix})$
Spin–dipole	$(\begin{matrix} - 1.64 & 0.00 & 0.00 \\ 0.00 & 0.64 & - 0.17 \\ 0.00 & 1.89 & - 0.57 \end{matrix})$	$(\begin{matrix} - 1.64 & 0.00 & 0.00 \\ 0.00 & 0.64 & 0.17 \\ 0.00 & - 1.89 & - 0.75 \end{matrix})$	$(\begin{matrix} - 1.54 & 0.00 & 0.00 \\ 0.00 & 1.70 & 1.49 \\ 0.00 & - 1.49 & 2.62 \end{matrix})$
Paramagnetic spin–orbit	$(\begin{matrix} - 18.47 & 0.00 & 0.00 \\ 0.00 & - 6.79 & 22.74 \\ 0.00 & 19.92 & - 10.27 \end{matrix})$	$(\begin{matrix} - 18.47 & 0.00 & 0.00 \\ 0.00 & - 6.79 & - 22.74 \\ 0.00 & - 19.92 & - 10.27 \end{matrix})$	$(\begin{matrix} 11.61 & 0.00 & 0.00 \\ 0.00 & - 6.14 & - 11.03 \\ 0.00 & 11.03 & 22.62 \end{matrix})$
Diamagnetic spin–orbit	$(\begin{matrix} 7.76 & 0.00 & 0.00 \\ 0.00 & - 6.67 & - 11.51 \\ 0.00 & - 11.08 & - 1.16 \end{matrix})$	$(\begin{matrix} 7.76 & 0.00 & 0.00 \\ 0.00 & - 6.67 & 11.51 \\ 0.00 & 11.08 & - 1.16 \end{matrix})$	$(\begin{matrix} - 14.39 & 0.00 & 0.00 \\ 0.00 & 12.79 & 11.38 \\ 0.00 & - 11.38 & - 20.06 \end{matrix})$
Total indirect spin–spin coupling	$(\begin{matrix} - 81.31 & 0.00 & 0.00 \\ 0.00 & - 81.79 & 11.05 \\ 0.00 & 10.73 & - 66.01 \end{matrix})$	$(\begin{matrix} - 81.31 & 0.00 & 0.00 \\ 0.00 & - 81.79 & - 11.05 \\ 0.00 & - 10.73 & - 66.01 \end{matrix})$	$(\begin{matrix} - 14.16 & 0.00 & 0.00 \\ 0.00 & - 7.47 & - 2.68 \\ 0.00 & - 6.37 & - 3.41 \end{matrix})$

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In contrast to the other contributions, the Fermi-contact contribution is isotropic, so it does not contribute to the antisymmetry J*. We observe that different contributions may cancel out. For instance, the paramagnetic and diamagnetic terms of ¹J(¹⁷O, ¹H_a) and ¹J(¹⁷O, ¹H_b) couplings are almost equal in magnitude, but their signs are opposite. The amplitudes of total antisymmetries derived at the coupled-cluster level of the theory are ¹J*(¹⁷O, ¹H) = 0.16 Hz and ²J*(¹H, ¹H) = 1.85 Hz.

The orientation of the antisymmetry with respect to the molecular frame of reference can be deduced from the fact that the molecule is planar. Thus, the only non-vanishing component of the antisymmetry J* must be perpendicular to the plane in which the water molecule lies. As expected, for a non-chiral molecule such as water, the antisymmetry J* is perpendicular to the dipole moment μ^e and the pseudoscalar J^c is zero. From the C_2v symmetry of the water molecule, it follows that the directions of antisymmetric components of the couplings ¹J(¹⁷O, ¹H_a) and ¹J(¹⁷O, ¹H_b) are opposite each other. These structural relationships are also observed in larger molecules of symmetry close to C_2v.

Let us take as an example of a molecule of norborn-5-en-2,3-diol shown in Figure 8.2. The two pairs of protons, one methylene (marked by the red color) and the other vicinal (marked by the blue color) are located in quite distinct regions of the molecule. The direction of the antisymmetry ²J*(¹H, ¹H) for the coupling between the methylene protons is perpendicular to the CH₂ plane, aligning with the expected direction based on that for the protons of water. Moreover, the amplitude is comparable to the case of protons of water: ²J*(¹H, ¹H) = 1.08 Hz. On the other hand, a simple reasoning based only on the analyzed fragment may not give the correct direction of antisymmetry. An example is the antisymmetry ³J*(¹H, ¹H) for the coupling between the vicinal protons, which approximately lies in the HC═CH plane rather than perpendicular to it, as observed in ethene.

Figure 8.2

From left to right: molecular structures of norborn-5-en-2,3-diol (top and side views), water, and ethene-d2. The electric dipole moments μe (green arrows), as well as the antisymmetric parts of the proton–proton indirect spin–spin coupling tensors J* (blue and red arrows) are shown in the bottom row. The lengths of these vectors are not to scale. The directions of the vectors are based on the results of density-functional theory computations. Optimal geometries were obtained using the Turbomole computer program43 using the PBE0+D3(BJ) functional44–46 with the def2-QZVPP basis, and spin–spin couplings were computed in the Dalton computer program47 using the PBE0 functional with the aug-pcJ-2 basis.48

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From left to right: molecular structures of norborn-5-en-2,3-diol (top and side views), water, and ethene-d₂. The electric dipole moments μ^e (green arrows), as well as the antisymmetric parts of the proton–proton indirect spin–spin coupling tensors J* (blue and red arrows) are shown in the bottom row. The lengths of these vectors are not to scale. The directions of the vectors are based on the results of density-functional theory computations. Optimal geometries were obtained using the Turbomole computer program⁴³ using the PBE0+D3(BJ) functional^44–46 with the def2-QZVPP basis, and spin–spin couplings were computed in the Dalton computer program⁴⁷ using the PBE0 functional with the aug-pcJ-2 basis.⁴⁸

A similar structural relationship between the direction of the antisymmetry of the spin–spin coupling is observed in a much bigger molecule, a chiral derivative of 1,3-bisdiphenylene-2-phenylallyl radical, the so-called the Koelsch radical,⁴⁹ frequently abbreviated as BDPA, shown in Figure 8.3. Since the molecule is a radical, the dominating antisymmetric coupling between spins originates from the coupling with the unpaired electron. The highest probability of finding an unpaired electron is in the middle of the molecule, near the carbon atoms joining three groups of rings (two of the tricyclic aromatic hydrocarbon – fluorene, and one of the phenyl substituent). Therefore, the strongest coupling with the electron is exhibited by, among others, the two adjacent carbon nuclei of the fluorenes. The directions of the antisymmetric components of their hyperfine coupling tensors (blue and red arrows in Figure 8.3) are, to a good approximation, not influenced by the labile bond that allows the alanine substituent bridged by the square acid unit to rotate. In contrast, the direction of the permanent electric dipole moment μ^e of the molecule depends strongly on the conformation that the molecule adopts. If the molecule adopts conformation A shown on the left side of Figure 8.3, the angle between the vectors μ^e and A* is 47° for the coupling with one of the carbon-13 nuclei, while 127° is obtained for the other. However, if the molecule adopts conformation B shown on the right side of Figure 8.3, the angles between the vectors μ^e and A* become close to the right angle (94° and 102°, respectively), and the magnitude of the pseudoscalar A^c is, therefore, greatly reduced. It is worth noting that, similar to antisymmetries ¹J*(¹⁷O, ¹H) of water, the antisymmetries of the hyperfine coupling of these two ¹³C nuclei exhibit opposite directions in the case of the chiral derivative of BDPA shown in Figure 8.3, resulting in opposite signs of the pseudoscalars A^c of the two carbons [A^c is defined in an analogous way to J^c – see eqn (8.11)].

Figure 8.3

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The structures of two dominating conformers of 1,3-bisdiphenylene-2-phenylallyl radical functionalized with a square acid and alanine. The permanent electric dipole moments μ^e (green arrows) and the antisymmetries of hyperfine coupling tensors A* between an unpaired electron and carbon-13 atoms (blue and red arrows that are attached to these atoms) are shown. The arrow near the square acid unit indicates the bond about which the substituent exhibits a hindered rotation. The optimal geometry was found using the Turbomole computer program using the PBE0+D3(BJ) functional with the def2-QZVPP basis set. The hyperfine couplings were computed in the ReSpect computer program^50,51 using the PBE0 functional with the decontracted Jensen’s pcH-2 basis set.⁵² Atoms are colored according to the Corey–Pauling–Koltun convention: hydrogen – white, carbon – black, oxygen – red, and nitrogen – blue.

The last example is trifluoropropan-2-ol shown in Figure 8.4. This molecule is not rigid since the hydroxyl group may rotate in the solution. Quantum chemistry computations indicate that two of the three fluorine nuclei in the CF₃ group coupled with the CH̲(OH) proton have antisymmetries of ca. 2 Hz. Their amplitudes vary less than 10% when the group OH rotates, which is much less than the corresponding changes in the magnitude of the permanent electric dipole moment of the molecule. On average, one of the antisymmetries ³J*(¹⁹F, ¹H) forms an angle of 69° with the moment μ^e while the other is at the angle of 133°. Hence, effectively a fraction of the maximum possible amplitude of the pseudoscalar J^c, corresponding to antisymmetry ³J*(¹⁹F, ¹H) oriented parallel to the electric dipole moment μ^e, could be observed. This situation is also observed in the antisymmetric hyperfine coupling of the chiral derivative of BDPA, as well as in water ¹J*(¹⁷O, ¹H) antisymmetries.

Figure 8.4

The variation of the permanent electric dipole moment μe (green arrows) and antisymmetries of proton–fluorine indirect spin–spin coupling, 3J*(19F, 1H) (the CH̲(OH) proton; red and blue arrows), with rotation of the hydroxyl group (the black arrow) for the (R)-trifluoropropano-2-ol molecule. The third proton–fluorine coupling is an order of magnitude smaller than the two depicted ones and, consequently, is not shown in the figure. The same molecule is shown from two points of view for better visibility. On the far right: diagrams illustrating the relationship between local symmetry and directions of antisymmetries for 3J*(19F, 1H) of trifluoropropan-2-ol (the upper part) and 1J*(17O, 1H) of water (the lower part). Atoms are colored according to the Corey–Pauling–Koltun convention: hydrogen – white, carbon – black, oxygen – red, and fluorine – green.

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The variation of the permanent electric dipole moment μ^e (green arrows) and antisymmetries of proton–fluorine indirect spin–spin coupling, ³J*(¹⁹F, ¹H) (the CH̲(OH) proton; red and blue arrows), with rotation of the hydroxyl group (the black arrow) for the (R)-trifluoropropano-2-ol molecule. The third proton–fluorine coupling is an order of magnitude smaller than the two depicted ones and, consequently, is not shown in the figure. The same molecule is shown from two points of view for better visibility. On the far right: diagrams illustrating the relationship between local symmetry and directions of antisymmetries for ³J*(¹⁹F, ¹H) of trifluoropropan-2-ol (the upper part) and ¹J*(¹⁷O, ¹H) of water (the lower part). Atoms are colored according to the Corey–Pauling–Koltun convention: hydrogen – white, carbon – black, oxygen – red, and fluorine – green.

In all three cases, the relative orientation of the vectors μ^e, J* (or A*) originates from the presence of a nucleus (or an unpaired electron) on the local two-fold axis/plane of symmetry and a nucleus located off the axis/plane. These symmetry features (C₂ – the two-fold axis, a curved arrow, and σ-the mirror plane, a continuous line) are shown diagrammatically for the chiral BDPA derivative, trifluoroisopropan-2-ol, and water on the right sides of Figures 8.3 and 8.4. The influence of other types of local symmetry on the value of the pseudoscalar J^c can be analyzed in a similar way. It seems, however, that the occurrence of local symmetry close to the two-fold axis or plane is one of the most common occurrences in molecules and clearly illustrates the role of local symmetry in the case of the pseudovector J* and the global distribution of electric charge, manifested by the vector μ^e. See several further examples of achiral and chiral radicals in Figure 8.5.

Figure 8.5

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Neutral radicals with their dipole moments (the green arrows), antisymmetries of some of their hyperfine coupling tensors. The red, blue, and yellow arrows represent the coupling of the unpaired electron with ¹⁵N, ¹³C, and ¹H nuclei, respectively. Vectors and pseudovectors are not to scale. Nitroxyls: 1a – TEMPO, i.e., 2,2,6,6-tetramethyl-1-piperidinyloxy, (R)-1b – the TEMPO derivative of (R)-alanine, 2a – 1,3,5-trimethyl-6-oxoverdazyl, (R)-2b – the verdazyl derivative of (R)-phenylalanine; carbon-centered radicals: 3a – triphenylmethyl, 3b – bis(fluoren-9-yl)methyl, and (R)-3c – bisfluorene methyl derivative of (R)-alanine. The radicals 1b, 2b, and 3c are considered chiral from the point of view of EPR and NMR spectroscopy measurement time scales. Reproduced from ref. 17 with permission from the Royal Society of Chemistry.

8.7 Nuclear Magnetoelectric Effective Hamiltonian

8.7.1 Rotating Frame Transformation

The spin precession frequency in a high magnetic field (several to hundreds of MHz) is many orders of magnitude greater than the effects of nuclear magnetic shielding and the interaction between spins (tens of kHz). Therefore, it is convenient to describe spin dynamics in a rotating reference frame rather than in the laboratory one. In particular, let us transform the Hamiltonian

{\hat{H}}_{E}

that includes chirality-sensitive terms to the frame that rotates with the precession frequency of each spin. In terms of quantum-mechanical concepts, this essentially is the transformation from the Schrödinger picture (in which the operators are constant and the states evolve in time) to the interaction picture (in which both the operators and states change in time). The transformation rule that allows us to express the laboratory frame Hamiltonian

{\hat{H}}_{E}^{lab}

in the frame that rotates at frequency ω_r, is given by the following formula:^53,54

{\hat{H}}_{E}^{rot} = {\hat{U}}^{†} {\hat{H}}_{E}^{lab} \hat{U} + \frac{d}{d t} {\hat{U}}^{†} \hat{U},

(8.17)

where the transformation operator is

\hat{U} = \exp (- ı ω_{r} t \sum_{i} {\hat{I}}_{i z})

⁠;

{\hat{U}}^{†}

is an adjoint of

\hat{U}

⁠, i.e., a conjugate transpose matrix,

{U_{i j}}^{†} = {\bar{U}}_{j i}

(overbar here means a complex conjugate).

For a two-spin system, the chirality-sensitive nuclear spin Hamiltonian in a chiral molecule possessing a permanent electric dipole moment μ^e, placed in an electric field E, is

{\hat{H}}_{E}^{lab} / ℏ = σ_{1}^{c} [E \cdot (γ_{1} B \times {\hat{I}}_{1})] + σ_{2}^{c} [E \cdot (γ_{2} B \times {\hat{I}}_{2})] + 2 π J_{12}^{c} [E \cdot ({\hat{I}}_{1} \times {\hat{I}}_{2})] + \dots,

(8.18)

i.e., including the terms of shielding and coupling antisymmetries,

{\hat{H}}_{CS, c} + {\hat{H}}_{J, c}

[see eqn (7.17) and ]. The Hamiltonian is time-dependent, i.e., E = E(t) and B = B(t), so finding an effective time-independent Hamiltonian that retains the essential features would offer a noticeable simplification in the calculation of the time evolution of the nuclear spin state. In the following rotating-frame discussion it is sufficient to limit oneself to only this part of the Hamiltonian, which does not vary with time, and to approximate the Hamiltonian

{\hat{H}}_{E}^{rot}

using its time-averaging integral

{\hat{H}}_{E}^{eff} = \frac{1}{τ} \int_{0}^{τ} {\hat{H}}_{E}^{rot} (t) d t,

(8.19)

where the period of the rotating frame is

τ = \frac{2 π}{ω_{r}}

⁠. In high magnetic fields, this assumption is not restrictive and, for most systems, gives results that are indistinguishable from the exact solution. Let us consider several special cases that illustrate the similarities and distinctive properties of the standard spin Hamiltonian used in NMR and EPR communities and the chirality-sensitive contributions to that Hamiltonian.

8.7.2 Antisymmetric Nuclear Magnetic Shielding Effective Hamiltonian

In the first case, let us assume that the chirality-sensitive interaction is dominated by the contribution of the pseudoscalar σ^c, in the

{\hat{H}}_{CS, c}

term of the Hamiltonian. The electric field is oriented along the y-axis and oscillates with the precession frequency of the first spin,

E = E \cos (ω_{1} t + \frac{π}{2}) e_{y}

⁠. The static magnetic field B₀ is along the z-axis. If the precession frequency of the first spin differs much from that of second one, then applying the transformation given by with ω_r = ω₁, one obtains the following effective chirality-sensitive Hamiltonian:

{\hat{H}}_{CS, c}^{eff} / h = \frac{ω_{1}^{E}}{2 π} {\hat{I}}_{1 y},

(8.20)

where

ω_{1}^{E} = - \frac{1}{2} σ_{1}^{c} E ω_{1}

⁠.

For instance, for trifluoropropan-2-ol placed in a magnetic field B₀ of strength 11.75 T and an electric field of amplitude 10 V mm⁻¹, the nutation frequency $\frac{ω_{E}}{2 π}$ for ¹⁹F nuclei slightly exceeds 1 mHz. The effective Hamiltonian in eqn (8.20) has the same form as that resulting from the cosinusoidal magnetic field oscillating with frequency ω₁ along the x-axis that corresponds to a radiofrequency pulse rotating nuclear magnetization in a standard NMR experiment. However, the effect of the magnetic field on spin dynamics is noticeably stronger in comparison with the influence of the electric field because, in typical NMR experiments, the nutation frequency due to the magnetic field, $ω_{B}^{1} = - \frac{1}{2} γ_{1} B_{1}$ ⁠, amounts to several kHz.

Similar to the magnetic field, one can summarize the influence of the resonant electric field on the spin state by introducing a rotation superoperator (an operator acting on operators, e.g., spin operators), which is especially convenient in the analysis of complex spin systems,

{\hat{R}}_{φ} (θ) = \exp (- ı θ \hat{I} \cdot e_{φ}) .

(8.21)

In , the tilt angle is θ = ω_Eτ_p, where the time τ_p is the time for which the electric field was applied, and the unit vector e_φ is the axis about which the nuclear magnetization is tilted; this vector is in the plane perpendicular to the magnetic field B₀. For instance, the superoperator

{\hat{R}}_{y} (θ)

acts on magnetization that is initially along the z-axis in such a way that it tilts the magnetization towards the direction of the x-axis by an angle θ.

Let us make a brief comment regarding the relationship between the unwanted magnetic field B₁ and the electric field E₁ generating it, which is applied in experimental protocols aiming to observe the $E 1$ effect described in Section 7.9 of the previous chapter. Maxwell’s equations (Faraday’s law, in particular) dictate that the magnetic field B₁ and the electric field E₁ have to be perpendicular to each other and shifted in time by a quarter of a period. For instance, if the laboratory frame of reference is suitably chosen, then these fields may be written as B₁ = B₁ cos(ω₁t)e_x and $E_{1} = E_{1} \cos (ω_{1} t + \frac{π}{2}) e_{y}$ ⁠. Consequently, the electric and magnetic fields tilt the magnetization into perpendicular directions: –y-axis and the x-axis, respectively.

Let us assume that the reference frame is such that the magnetic field is B₁ = B₁ cos(ω₁t)e_x and examine step-by-step how the cross product $B \times {\hat{I}}_{1}$ ⁠, the phase shift and spatial orthogonality between the B₁ and E₁ fields affect the effective rotation operator of the electric field derived from Maxwell’s equations, $E_{1} = E_{1} \cos (ω_{1} t + \frac{π}{2}) e_{y}$ ⁠. This net effect follows from three independent factors among which two cancel out each other. First, if we ignore the phase shift and spatial orthogonality between B₁ and E₁, then the cross product $B \times {\hat{I}}_{1}$ in the Hamiltonian ${\hat{H}}_{CS, c}$ (compare with the scalar product $B \cdot {\hat{I}}_{1}$ in a purely magnetic spin Hamiltonian) results in the rotation operator ${\hat{R}}_{y} (θ)$ ⁠, obtained from eqn (8.17) applied to the electric field E₁ = E₁ cos(ω₁t)e_x. Next, let us take into account the second factor, the phase shift between the electric field E₁ and magnetic field B₁, i.e., consider the electric field $E_{1} = E_{1} \cos (ω_{1} t + \frac{π}{2}) e_{x}$ ⁠. Such a field corresponds to the rotation operator ${\hat{R}}_{x} (θ)$ ⁠. Finally, by introducing the third factor, the orthogonality of the fields E₁ and B₁ in space, i.e., the electric field $E_{1} = E_{1} \cos (ω_{1} t + \frac{π}{2}) e_{y}$ ⁠, one obtains the rotation operator ${\hat{R}}_{y} (θ)$ ⁠; see eqn (8.20) and (8.21).

8.7.3 Antisymmetric Spin–Spin Coupling Effective Hamiltonian

In the second case, i.e., when the primary goal is to observe the effects induced by the presence of the antisymmetric part of the indirect spin–spin coupling tensor J*, one can consider several factors that influence the design of the experiment. The most basic of these are (i) the choice of the spin system, i.e., homonuclear vs. heteronuclear two-spin system, (ii) the relative orientation of the electric E and magnetic field B₀, i.e., one can orient the electric field either parallel or perpendicular to the static magnetic field B₀, and (iii) the relative amplitude of the coupling between spins, quantified by the isotropic indirect spin–spin coupling J, to the amplitude of the interaction of the nuclear magnetic moment with the magnetic field, quantified by the spin precession frequency ω. Let us illustrate the various experimental possibilities offered by the Hamiltonian given in eqn (8.18) by three particular examples.

8.7.3.1 Case I

In the case of a very strongly coupled homonuclear pair of spins placed in a high magnetic field, the Hamiltonian that governs their dynamics is

{\hat{H}}_{E, J}^{rot} / h = J {\hat{I}}_{1} \cdot {\hat{I}}_{2} + J^{c} [E \cdot ({\hat{I}}_{1} \times {\hat{I}}_{2})] .

(8.22)

The same Hamiltonian may be written for a general pair of spins (including the heteronuclear pair) under a nearly-zero magnetic field. In , the frame of reference rotates with a frequency equal to the average of the spin precession frequencies of the spins I₁ and I₂. The terms that correspond to nuclear magnetic shielding are presumed to be sufficiently small to be neglected.

Regardless of whether the terms related to the interaction between the magnetic field and nuclear magnetic moments are present or not, the initial state ${\hat{ρ}}_{0} = {\hat{I}}_{1 z} - {\hat{I}}_{2 z}$ (vide infra) is not affected by the n-fold application of a spin echo, i.e., the pulse sequence τ–180°–τ, where the delay duration is τ = 1/(4J). Let us assume that the electric field E is static and oriented along the magnetic field B₀. Then, the initial state ${\hat{ρ}}_{0}$ transforms into a singlet–triplet population difference, ∣S₀〉〈S₀∣ − ∣T₀〉〈T₀∣, with a period of (2J^cE)⁻¹, where $| S_{0} 〉 = \frac{1}{\sqrt{2}} (| ↑ ↓ 〉 - | ↓ ↑ 〉)$ and $| T_{0} 〉 = \frac{1}{\sqrt{2}} (| ↑ ↓ 〉 + | ↓ ↑ 〉)$ ⁠. The advantage of using the state ∣S₀〉〈S₀∣ − ∣T₀〉〈T₀∣ is that it is resistant to dipolar relaxation, enabling a longer period of application of the exciting electric field E in comparison with other states of the system, e.g., ${\hat{I}}_{1 x} + {\hat{I}}_{2 x}$ ⁠. Therefore, by using a number of echoes equal to the natural number nearest to J(4J^cE)⁻¹, and assuming a sufficiently long relaxation time, one can potentially transform the state ${\hat{ρ}}_{0}$ completely into the state ∣S₀〉〈S₀∣ − ∣T₀〉〈T₀∣. The stronger the system of coupled spins, the lesser the influence of the term related to the interaction of the spin with the magnetic field on the dynamics of the spin system. Hence, the most strongly coupled systems are beneficial for obtaining maximal electric field-induced transfer from the ${\hat{I}}_{1 z} - {\hat{I}}_{2 z}$ state to the ∣S₀〉〈S₀∣ − ∣T₀〉〈T₀∣ state.

8.7.3.2 Case II

Let us compare the above case with the case of a weakly coupled heteronuclear spin system in a high magnetic field, and apply a radiofrequency electric field E = E cos((ω₁ − ω₂)t)e_z. As in case I, the direction of the electric field aligns with that of the magnetic field B₀ (both taken here being along the z-axis). In this case, the transformation to the rotating frame with the difference frequency ω₁ − ω₂ yields

{\hat{H}}_{E, J}^{rot} / h = J {\hat{I}}_{1 z} {\hat{I}}_{2 z} + J^{c} E ({\hat{I}}_{1 x} {\hat{I}}_{2 y} - {\hat{I}}_{1 y} {\hat{I}}_{2 x}) + \dots

(8.23)

In , the three dots denote all terms that oscillate with high frequencies, on the order of several MHz, and consequently average out due to the low bandwidth of the detector (several kHz typically). The use of a static electric field in the case of a pair of weakly coupled spins would not be advantageous because the chirality-dependent term is averaged out after transformation from the laboratory to the rotating frame of reference. Similar to the case of the strongly coupled spin system, the chirality-sensitive contribution to the Hamiltonian given by affects the initial state

{\hat{ρ}}_{0} = {\hat{I}}_{1 z} - {\hat{I}}_{2 z}

and transforms it into the state

{\hat{I}}_{1 x} {\hat{I}}_{2 x} + {\hat{I}}_{1 y} {\hat{I}}_{2 y}

⁠. If the initial state is

{\hat{ρ}}_{0} = {〈 {\hat{I}}_{1 z} 〉}_{eq} {\hat{I}}_{1 z} \pm {〈 {\hat{I}}_{2 z} 〉}_{eq} {\hat{I}}_{2 z}

⁠, where

{〈 {\hat{I}}_{1 z} 〉}_{eq}

and

{〈 {\hat{I}}_{2 z} 〉}_{eq}

are proportional to the magnetization of spins I₁ and I₂ in the thermodynamic equilibrium state of the system, then the amplitude of the

{\hat{I}}_{1 x} {\hat{I}}_{2 y} - {\hat{I}}_{1 y} {\hat{I}}_{2 x}

state is proportional to that of

{〈 {\hat{I}}_{1 z} 〉}_{eq} \mp {〈 {\hat{I}}_{2 z} 〉}_{eq}

⁠. Therefore, the smaller the difference frequency, the more advantageous it is to use

{〈 {\hat{I}}_{1 z} 〉}_{eq} {\hat{I}}_{1 z} - {〈 {\hat{I}}_{2 z} 〉}_{eq} {\hat{I}}_{2 z}

as the initial state instead of the thermodynamic equilibrium state

{〈 {\hat{I}}_{1 z} 〉}_{eq} {\hat{I}}_{1 z} + {〈 {\hat{I}}_{2 z} 〉}_{eq} {\hat{I}}_{2 z}

⁠. Again, assuming relaxation to be negligible, the transformation from the state

{\hat{I}}_{1 z} - {\hat{I}}_{2 z}

to the state

{\hat{I}}_{1 x} {\hat{I}}_{2 x} + {\hat{I}}_{1 y} {\hat{I}}_{2 y}

takes place within a period of (2J^cE)⁻¹. A close correspondence between cases I and II becomes clearly visible when we compare the chirality-sensitive terms in and . In fact, the operators that are responsible for the chirality-sensitivity of the spin system are the same:

\frac{1}{2} ({\hat{I}}_{1 x} {\hat{I}}_{2 y} - {\hat{I}}_{1 y} {\hat{I}}_{2 x}) = - ı (| S_{0} 〉 〈 T_{0} | - | T_{0} 〉 〈 S_{0} |) .

(8.24)

Depending on the context, the notation in highlights the spin system properties relevant to the weak spin–spin coupling limit, ω₁ − ω₂ ≫ J, on the left side of , and strong coupling limit, ω₁ − ω₂ ≪ J, on the right side of .

8.7.3.3 Case III

The last case is analogous to case II, except for the orientation of the electric field and its frequency dependence. Let us consider an electric field oscillating at the frequency of one of the spins in a direction perpendicular to the magnetic field, e.g., E = E cos(ω₁t)e_y. Applying the transformation to the rotating frame of reference with the frequency of ω₁, one obtains

{\hat{H}}_{E, J}^{rot} / h = J {\hat{I}}_{1 z} {\hat{I}}_{2 z} - J^{c} E {\hat{I}}_{1 x} {\hat{I}}_{2 z} + \dots .

(8.25)

For the initial state

{\hat{ρ}}_{0} = {〈 {\hat{I}}_{1 z} 〉}_{eq} {\hat{I}}_{1 z} + {〈 {\hat{I}}_{2 z} 〉}_{eq} {\hat{I}}_{2 z}

⁠, the chirality-sensitive term

{\hat{I}}_{1 x} {\hat{I}}_{2 z}

induces a doublet antiphase signal (one of its components points up, whereas the other is down). Unlike in cases I and II, the coincidence of the electric field and spin precession frequencies necessitates the suppression of the unwanted magnetic field component perpendicular to the main magnetic field B₀. Consequently, the perpendicular arrangement of the E and B₀ fields offers little (or even a complete lack of) advantage for observing the

E 1

effect induced by the antisymmetric nuclear magnetic shielding, because of the typically smaller amplitude of J^cE in comparison to γσ^cE. Nonetheless, one can expect a relatively small difference in the amplitudes of multiplet components in the NMER spectrum due to the overlapping of the antiphase multiplet induced by the antisymmetry J* and the in-phase multiplet induced by the antisymmetry σ*.

8.8 A General Description of the Chirality-sensitive Electric Polarization

The chirality-sensitive electric polarization that is, on the one hand, generated by the precession of nuclear magnetic moments (see ) and, on the other hand, by the antisymmetry of the indirect spin–spin coupling tensor can be described in the same framework. This is done by introducing an operator χ that maps the density matrix

ρ (t)

of the spin system into a polarization vector P(t) according to the formula

P (t) = Tr [ρ (t) χ] .

(8.26)

The operator χ is

χ = N (\sum_{i} ℏ σ_{i}^{c} ({\hat{I}}_{i} \times γ_{i} B) + \sum_{i > j} h J_{i j}^{c} ({\hat{I}}_{i} \times {\hat{I}}_{j})),

(8.27)

where

N

is the number of chiral molecules per unit volume.

In the next sections, a few effects denoted by $E 4$ ⁠, $E 5$ ⁠, $E 6$ ⁠, and $E 7$ that could facilitate chiral discrimination by NMR, are described and briefly summarized in Table 8.2. Each of them is provided with a brief description on how it could be generated, its experimental and hardware requirements, why the effect is important compared to other effects discussed here, and the feasibility of its experimental observation (Table 8.2).

Table 8.2

Predicted NMER effects.

Effect	Number of spins	Nuclear properties	Experimental conditions		Favorable sample	Expected signal magnitude^a
Effect	Number of spins	Nuclear properties	Angle between E and B₀	Frequency of P/E oscillations	Favorable sample	Expected signal magnitude^a
$E 4$	2	J*	0°	ω_H − ω_F	1,1,1-Trifluoro-propan-2-ol	10⁻³
$E 5$	2	A*	0°	ω_e − ω_C	¹³C-BDPA-alanine	10⁻⁴
$E 6$	2	J*	N/A	ω_H − ω_F	1,1,1-trifluoro-propan-2-ol	10⁻⁵
$E 7$	3	D	0°	ω_H − ω_C	¹³C-alanine	10⁻⁶

Effect	Number of spins	Nuclear properties	Experimental conditions		Favorable sample	Expected signal magnitude^a
Effect	Number of spins	Nuclear properties	Angle between E and B₀	Frequency of P/E oscillations	Favorable sample	Expected signal magnitude^a
$E 4$	2	J*	0°	ω_H − ω_F	1,1,1-Trifluoro-propan-2-ol	10⁻³
$E 5$	2	A*	0°	ω_e − ω_C	¹³C-BDPA-alanine	10⁻⁴
$E 6$	2	J*	N/A	ω_H − ω_F	1,1,1-trifluoro-propan-2-ol	10⁻⁵
$E 7$	3	D	0°	ω_H − ω_C	¹³C-alanine	10⁻⁶

The magnitude with respect to that obtained after the excitation of the sample with a 90° pulse; E = 1 kV mm⁻¹, B₀ = 10 T.

View Large

8.9 Chirality-sensitive Coherences Induced by the Antisymmetric Part of the J-coupling Tensor

8.9.1 Description of the $E 4$ Effect

The spin dynamics on which the $E 4$ effect is based arises from the quantum-mechanical treatment described in Section 8.7.3 for case II. In particular, the application of a radiofrequency electric field on the two-spin system placed in a chiral molecule induces zero-quantum coherences, whose phase differs for the two enantiomers of that molecule.

The $E 4$ effect relies on the partial alignment of the polar chiral molecules. Therefore, for its occurrence, it is necessary for the molecules to be capable of reorienting under the influence of the E field on a time scale comparable to the time variation of the electric field. Such conditions, for low enough frequency, are met in chiral substances dissolved in the liquid phase and in some molecular solids allowing reorientation in a lattice site, e.g., the chiral derivatives of fullerenes. Typically, the higher the field frequency, the more hindered the reorientation of the sample becomes due to internal friction in a liquid, which manifests itself as the dissipation of energy (sample heating). The frequency dependence strongly varies with the solvent – numerical values of the imaginary part of the dielectric permittivity, on which the losses mainly depend, are listed for several common organic solvents in ref. 55. Moreover, excitation at the differential frequency, as compared to the spin precession frequency, allows for longer molecular reorientation times; the high value of these times is a limiting factor in the case of medium-viscosity solvents.

Even more importantly, dielectric losses increase with the square of the electric field amplitude, which may significantly limit experiments, as shown by the mathematical models in Figure 8.6.

Figure 8.6

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The simulated dielectric heating effect due to the application of a radiofrequency electric field on an NMER probe assembly. The probe head generating the electric field through two parallel plates of the capacitor (marked by the orange color), together with the sample placed inside the coil surrounded by the capacitor (on the left side), the temperature distribution in a vertical cross-section of the assembly (in the middle), the temperature profile along the symmetry axis of the sample depending on the flow and temperature of the cooling gas flowing from the source located at the bottom of the assembly (on the right side).

8.9.2 Experimental Protocol for the $E 4$ Effect

The experimental protocol for effect $E 4$ is shown in Figure 8.7. The chiral sample is polar and has a heteronuclear pair of spins (I₁ and I₂). First, the electric field oscillating at the difference frequency, E₁ = E₁ cos((ω₁ − ω₂)t)e_z, excites the sample for a duration of τ = T₂, where T₂ is the spin–spin relaxation time. Then, the generated chirality-dependent coherences are converted into an observable magnetization of the spin I₂ (Figure 8.7A). In the next step, the chirality-sensitive magnetization induced by the E field of the capacitor is detected by the coil (not shown), generating a signal with opposite phases for the enantiomers (Figure 8.7B).

Figure 8.7

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Chirality-sensitive coherences induced by the antisymmetric part of the J-coupling tensor (the $E 4$ effect): (A) the pulse sequence with the states of the spin system marked after (i) application of the electric field oscillating at the difference frequency (the gray rectangle) and (ii) radiofrequency pulses at spin-precession frequencies ω₁ (spin I₁) and ω₂ (spin I₂) flanked by pulses of magnetic-field gradient along the z-axis (the white rectangles); (B) the arrangement of the main magnetic field B₀ and electric field E delivered by a capacitor surrounding the sample from above and below and the expected difference in the signals of the chiral molecule enantiomers. In the pulse sequence, the delay τ′ equals 1/(4J). The positive phase of the red signal is attributed to the (R)-enantiomer, and the negative one of the blue signal is attributed to the (S)-enantiomer only for illustrative purposes.

8.9.3 Significance and Estimation of the Magnitude of the $E 4$ Effect

The promising samples for experimental observation of the $E 4$ effect are light fluoroalcohols considering, e.g., the two-bond fluorine-proton coupling of (1,1,1)-trifluoropropan-2-ol and other compounds that contain fluorine, e.g., the three-bond fluorine-fluorine coupling of 1,1,1,2-tetrafluoro-2-chloroethane. In these molecules, the pseudoscalar $J_{HF}^{c}$ for the proton–fluorine spin system is on the order of 1 nHz m V⁻¹. Let us assume that the spin system is the pair of the proton and fluorine nuclei, the permanent electric dipole moment of the molecule is μ^e = 1 D, and the sample is placed in the static magnetic field B₀ = 10 T, the exciting electric field has a strength of E = 100 V mm⁻¹, and the longitudinal relaxation time T₂ is 1 s for both nuclei. Then, the estimated magnitude of the chirality-sensitive nuclear magnetization is 10⁻⁴ of the nuclear magnetization signal obtained after applying a 90° pulse to the sample at thermodynamic equilibrium.

Applying the oscillating electric field at the difference frequency minimizes the perturbation of the system by the time-dependent magnetic field generated by the E field. Considering that the magnitude of the effect is proportional to the strength of the electric field E, and the dielectric heating is proportional to the square of the strength of the electric field, one finds that a relatively low-frequency excitation is optimal for the experiment (e.g., for the ¹H–¹⁹F pair the difference frequency is approx. 30 MHz in a magnetic field of 10 T).

8.10 Electric Polarization Induced by the Antisymmetric Part of Indirect Spin–Spin Coupling

8.10.1 Description of the $E 5$ Effect

A liquid sample containing chiral molecules that have non-vanishing permanent electric dipole moment μ^e and a two-spin system (spins I₁ and I₂) interacting by the indirect spin–spin coupling, exhibits the following chirality-sensitive electric polarization

P = - N h J_{12}^{c} {\hat{I}}_{1} \times {\hat{I}}_{2},

(8.28)

where

N

is the number density of spins; compare with from which follows. The pseudoscalar

J_{12}^{c}

exhibits opposite signs for the two enantiomers and vanishes if the molecule is not chiral. Therefore, the phase of the radiofrequency signal indicated by the electric polarization is an indicator of the handedness of the molecule.

8.10.2 Experimental Protocol for the $E 5$ Effect

The experimental protocol for the $E 5$ effect is shown in Figure 8.8. In the first step, the desired quantum state of the two-spin system is obtained using the modified insensitive nuclei enhanced by the polarization transfer (INEPT) pulse sequence (Figure 8.8A). Then, the obtained state, which is described by the density matrix $\hat{ϱ}$ ⁠, generates an oscillating electric polarization P at the sum frequency ω₁ + ω₂. Depending on the enantiomers, the induced electric polarization P has a parallel or an antiparallel direction with respect to the field B₀ (Figure 8.8B).

Figure 8.8

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Electric polarization induced by the antisymmetric part of indirect spin–spin coupling (the $E 5$ effect). On the left side, the pulse sequence is shown. The narrow rectangles are 90° pulses, whereas the wide ones are 180° pulses. The red arrow indicates the final state of the spin system generating chirality-sensitive electric polarization of the sample. The radiofrequency pulses are applied on-resonance for both spins I₁ and I₂. In the pulse sequence, the delay τ′ equals 1/(4J). On the right side, one of the spin state coherences after application of the pulse sequence is depicted. This coherence is the same for both enantiomers. The difference in the directions of electric polarisation P (upwards vs. downwards) arises from the different transformation properties of the vector μ^e and the pseudovector J* under mirror reflection.

8.10.3 Significance and Estimation of the Magnitude of the $E 5$ Effect

The favourable samples for the observation of the $E 5$ effect are the same as those for the $E 4$ effect described in the previous section. The pseudoscalar $J_{HF}^{c}$ for the proton–fluorine spin system is on the order of 1 nHz m V⁻¹, resulting in a chirality-sensitive electric polarization P ≈ 0.1 aC m⁻² in a magnetic field of strength B₀ = 10 T. In this case, the voltage generated by this electric polarization P in a resonance circuit containing the capacitor is approximately five orders of magnitude smaller than that of the nuclear magnetization corresponding to the equilibrium state of the sample.

The $E 5$ effect can be observed without the need to apply an electric field. Consequently, dielectric heating, which may become an obstacle in observing the effects induced by the electric field E, does not affect the sample. In contrast to the $E 4$ effect, the induced electric polarization P oscillates at the sum frequency and, therefore, does not overlap with the precession frequencies of the spins I₁ and I₂, simplifying the experimental setup. The detector is a parallel-plate capacitor in the radiofrequency resonant circuit.

8.11 Chirality-sensitive Effects Induced by Antisymmetry of the A-coupling Tensor

The antisymmetric interactions between spins are not limited to the interactions between nuclear spins. In fact, for each of the proposed effects that may occur for nuclear spins, such as those described in Sections 8.9 (the $E 4$ effect) and 8.10 (the $E 5$ effect), an analogous effect can be proposed by replacing one or both nuclear spins with, for instance, electron spins. In this section, we will focus on such interactions that are in the domain of EPR spectroscopy or at the borderline between NMR and EPR. It is tempting to presume that such a generalization is evident due to the similarity of the mathematical framework used to describe the antisymmetries J* and A*; however, the very different technical challenges posed by EPR compared to NMR, lead to the opposite conclusion. Moreover, there is a noticeable gap between the energy scales of the spin–spin interactions between, on the one hand, nuclei and electrons and, on the other hand, among nuclei only. Typically, the sensitivity of the measurements of the electron spin transitions surpasses that of nuclear transitions by several orders of magnitude. Thus, the effects caused by A* place us in a favorable position to achieve direct sensitivity to chirality by EPR.

8.11.1 Description of the $E 6$ Effect

In general terms, the mathematical description of the effect mediated by A* follows those described in Sections 8.9 and 8.10 (vide supra). Similar to the two effects described in those sections, the antisymmetry of the hyperfine coupling yields a chirality-sensitive electric polarization if the initial quantum state of the system is suitably chosen and the electric field, oriented along the main magnetic field, oscillates at the difference frequency and induces zero-quantum coherences; the phase of these coherences depends on the handedness of the studied molecule.

8.11.2 Experimental Protocol for the $E 6$ Effect

The experimental protocols for the effects due to the antisymmetry A* are shown in Figure 8.9.

Figure 8.9

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Two chirality-sensitive effects induced by antisymmetric hyperfine coupling, which lifts the intrinsic insensitivity of electron paramagnetic resonance spectroscopy to detection of molecular chirality (called collectively in the text as the $E 6$ effect). On the top: (R)-BDPA-Ala (in the red color) and (S)-BDPA-Ala (in the blue color) radicals and the chirality-sensitive EPR electric polarization signals that are at a frequency equal to the difference between the resonance frequencies of the ¹³C nucleus and the electron. An absorption spectrum is shown, in which the frequency is given relative to the electronic transition, and the isotropic hyperfine electron-¹³C coupling is $A_{C}^{iso}$ ⁠. On the bottom: the EPR/NMR pulse sequence generating the spin state ${\hat{I}}_{1 x} {\hat{I}}_{1 y} - {\hat{I}}_{1 y} {\hat{I}}_{1 x}$ (⁠ ${\hat{I}}_{1}$ is the electron, while, ${\hat{I}}_{2}$ the ¹³C nuclear spin operator), whose phase indicates which enantiomer is present in the sample. The spectra on the far right side illustrate the expected signal lineshape in the dispersion mode, i.e., according to the convention used in EPR spectroscopy.

The spin precession frequencies of any nucleus and an electron differ a lot, so the use of initial states 〈I_1z〉_eqI_1z ± 〈I_2z〉_eqI_2z becomes irrelevant. The difference between the angular frequency of nuclear spin precession (exemplified here by carbon-13), ω_C, and that of the electron, ω_e, is comparable to the isotropic hyperfine coupling, $A_{C}^{iso}$ ⁠. Therefore, there is a risk of a poor spectral separation of the difference frequency from the spin transition of the system. Although the separation is in the range of MHz at fields corresponding to the X-band (e.g., B₀ = 0.33 T) or the W-band (ca. 3 T), the duration of the microwave pulses results in a relatively broad excitation. A sufficiently selective excitation provides a major advantage in the absence of a sample response, if there are no chirality-sensitive effects. This happens either because of a suitably chosen initial state resulting in electric polarization, or excitation by the external electric field. Therefore, there is no need to eliminate the interfering achiral background signal.

8.11.3 Significance and Estimation of the Magnitude of the $E 6$ Effect

The success of an experiment heavily depends on the degree of fulfillment of required conditions imposed by the experiment protocol, in its hardware implementation, which may be technically challenging. From the point of view of the optimal sample, the lengthening of the electron relaxation time seems to be crucial. Comparing the nuclear relaxation times (that are on the order of seconds) with those typical in EPR spectroscopy (a few to dozens of ns), we can see that creating a short radio/microwave pulse is a harder task for EPR than for NMR. Therefore, relatively slow relaxation of the chiral radical is very convenient. For, e.g., in BDPA derivatives, one cannot expect significantly longer electron relaxation time than a few microseconds. The second issue is isotope labeling. In several cases, nuclei related by an approximate C₂ symmetry axis would give canceling signals, so non-selective labeling does not offer much advantage over samples that are not enriched at all. Taking into account such considerations, one can find that, if the relaxation time of the chiral radical is ca. 10 µs, the ratio between the electric and magnetic fields at the detector is $\frac{c B}{E} = 10^{3}$ ⁠, and when the radical is selectively ¹³C-enriched, the amplitude of the expected chirality-sensitive electric polarization is on the order of 10⁻⁴ of the standard EPR signal.

8.12 Interference Between Dipolar Relaxation Mechanisms in an Electric Field

In contrast to previously postulated chirality-sensitive NMR effects, i.e., the magnetization induced by the electric field (the $E 1, E 3$ ⁠, and $E 4$ effects) and the electric polarization induced by precessing nuclear magnetic moments (the $E 2$ ⁠, $E 5$ ⁠, and $E 6$ effects), the $E 7$ effect described in this section does not require computation of the components of the nuclear magnetic shielding tensor and the indirect spin–spin coupling tensor for determination of the sense of molecular chirality. A sufficient condition for determining the absolute configuration of a molecule by observing the dipolar coupling effect is having knowledge about the relative orientation of the observed three nuclei with respect to the permanent electric dipole moment of the molecule.

8.12.1 Description of the $E 7$ Effect

The following discussion will be limited to the interference involving the electric field-induced terms of the direct coupling for only one pair of spins, I₁ and I₂. Without loss of generality, the other contributions are neglected since their influence on the spin dynamics may be suppressed by suitably chosen experimental conditions.

In order to achieve a chirality-sensitive effect, it is necessary to ensure that the interaction is linear in the electric field, which may be achieved for interference between an E-field induced relaxation mechanism and one that is not perturbed by the electric field. For isotropic tumbling of the molecule, only relaxation mechanisms of the same tensorial rank contribute to cross-correlation effects; thus, only the second-rank part of the contribution $(μ^{e} \cdot E) ({\hat{I}}_{1} \cdot D_{12} \cdot {\hat{I}}_{2}) / (k_{B} T)$ ⁠, couples with dipolar relaxation, parametrised by a second-rank tensor.

Let us assume that the electric field E = Ee_z is along the z-axis of the laboratory frame, and the vector r₁₂ from the spin I₁ to the spin I₂ is along the z-axis of the molecular frame. Then, the contribution

(μ^{e} \cdot E) ({\hat{I}}_{1} \cdot D_{12} \cdot {\hat{I}}_{2}) / (k_{B} T)

has a component

{\hat{I}}_{1} \cdot U_{12} \cdot {\hat{I}}_{2}

⁠, where U₁₂ is the second-rank tensor

U_{12} = - \frac{b_{12}}{2} \frac{E}{k_{B} T} (\begin{matrix} 0 & 0 & μ_{x}^{e} \\ 0 & 0 & μ_{y}^{e} \\ μ_{x}^{e} & μ_{y}^{e} & 0 \end{matrix}),

(8.29)

if μ^e and r₁₂ are not parallel to each other, and zero otherwise.

Let us consider an interference between the contribution induced by the electric field

{\hat{I}}_{1} \cdot U_{12} \cdot {\hat{I}}_{2}

and the nonperturbed dipolar contribution

{\hat{I}}_{1} \cdot D_{13} \cdot {\hat{I}}_{3}

⁠. If the vector r₁₃ is from spin I₁ to spin I₃ in the yz-plane in the molecular frame, and it is at an angle θ with respect to the z-axis, then the tensor of the dipolar interaction between spins I₁ and I₃ is

D_{13} = b_{13} (\begin{matrix} - 1 & 0 & 0 \\ 0 & \frac{1}{2} (1 - 3 \cos 2 θ) & \frac{3}{2} \sin 2 θ \\ 0 & \frac{3}{2} \sin 2 θ & \frac{1}{2} (1 + 3 \cos 2 θ) \end{matrix}) .

(8.30)

The spatial orientation of the tensors U₁₂ and D₁₃ is shown in Figure 8.10.

Figure 8.10

Interference between dipolar relaxation mechanisms in an electric field (the E 7 effect). The tensors U12 (in the blue color) and D13 (in the red colour) represented as ellipsoids oriented along their eigenvectors and of semiaxes equal to the absolute values of their eigenvalues, i.e., the ratios of the semiaxes are 0 : 1 : 1 and 1 : 1 : 2, respectively. The orientation of the tensors, which is shown in the figure ( φ = 0 , θ = π 4 , and ψ = π 2 ) corresponds to the maximum value of the amplitude of the interference. The permanent electric dipole moment μe is shown as a green arrow.

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Interference between dipolar relaxation mechanisms in an electric field (the $E 7$ effect). The tensors U₁₂ (in the blue color) and D₁₃ (in the red colour) represented as ellipsoids oriented along their eigenvectors and of semiaxes equal to the absolute values of their eigenvalues, i.e., the ratios of the semiaxes are 0 : 1 : 1 and 1 : 1 : 2, respectively. The orientation of the tensors, which is shown in the figure $(φ = 0, θ = \frac{π}{4}, and ψ = \frac{π}{2})$ corresponds to the maximum value of the amplitude of the interference. The permanent electric dipole moment μ^e is shown as a green arrow.

The amplitude of the interference between relaxation mechanisms described by any two traceless second-rank tensors is proportional to the trace of the product of their matrix representations; therefore, the contribution of the tensors U₁₂ and D₁₃ to the relaxation matrix has an amplitude (expressed in Hz) given by

A_{DxED}^{c} = \frac{μ^{e} E}{k_{B} T} b_{12} b_{13} \sin 2 θ \cos φ \sin ψ τ_{2} .

(8.31)

The amplitude

A_{DxED}^{c}

reaches its maximum for

θ = \frac{π}{4}

⁠, φ = 0, φ = 0, and

ψ = \frac{π}{2}

⁠, i.e., when the vectors r₁₂ and r₁₃ are at an angle of 45° and the permanent electric dipole moment of the molecule μ^e is perpendicular to the plane spanned by these two vectors.

8.12.2 Experimental Protocol for the $E 7$ Effect

In order to achieve a chirality-sensitive response from the sample, and one that has sufficient amplitude for experimental observation, it is beneficial to use an oscillating electric field. If the molecule tumbles isotropically and much faster than the speed at which the nuclear spins precess in the magnetic field B₀, i.e., at the extreme narrowing limit, the relaxation matrix becomes block diagonal. The excitation of the double and triple coherences requires the field to oscillate at a higher frequency than the spin-precession frequency. At such high frequencies, dielectric heating may severely limit the available amplitude of the electric field. The unwanted magnetic field generated by the electric field oscillating at the spin-precession frequency excites single quantum coherences that are much larger than the postulated effect. Therefore, the most convenient choice is electric field oscillating at the frequency equal to the difference between the precession frequencies of the two spins, which perturb the blocks of the zero quantum coherences.

In order to avoid unnecessary complexity of the solution, let us limit to the case where only the direct dipole–dipole relaxation mechanism contributes to the relaxation matrix. This mechanism is the most pronounced one under typical experimental conditions that prevail in liquid-state physicochemical NMR studies. Moreover, we assume that the dipole–dipole relaxation time for each pair of nuclei is T₁. In this case, one finds that the maximal amplitude of the chirality-sensitive signal is reached by the application of the electric field E on the spin state $({\hat{I}}_{1 x} {\hat{I}}_{2 y} - {\hat{I}}_{1 y} {\hat{I}}_{2 x}) {\hat{I}}_{3 z}$ ⁠, over the period τ = T₁ and then by observation of the phase of the chirality-dependent magnetization of the spin I₃. The amplitude is numerically roughly two orders smaller than $A_{DxED}^{c} T_{1}$ ⁠.

8.12.3 Significance and Estimation of the Magnitude of the $E 7$ Effect

Suppose the electric field E is oscillating at a frequency equal to the difference between the spin precession frequencies of two of the three spins, and that the field E is aligned along the magnetic field B₀. In this case, the magnitude of the predicted effect for B₀ = 10 T and E = 5 kV mm⁻¹ is 1% of the signal generated by the interference between dipole–dipole relaxation pairs measured without using the electric field E. Although the $E 7$ effect is small in amplitude, its significance lies in the fact that it enables the determination of the absolute configuration of a molecule provided that the relative orientation of these three nuclei with respect to the permanent electric dipole moment of the molecule μ^e, is known.

8.13 Sources of Radiofrequency Electric Field for J-NMER Experiments

Basically, the magnetic field needed to observe the $E 4 - E 7$ effects should be parallel to the static magnetic field of the spectrometer, so it requires a different experimental arrangement than that described in Section 7.14 for the $E 1 - E 3$ effects. Note that, unlike much more complicated systems that suppress the undesirable transverse component of the magnetic field generated by the electric field, in the design of the capacitor relevant to J-NMER experiments, it is sufficient to consider the highest possible homogeneity and amplitude of the generated electric field as primary design objectives. One of the radiofrequency electric field sources is a capacitor with parallel plates, between which the sample is placed, aligning its axis of symmetry along the axis of the capacitor. The advantageous aspect of such an electric field source is not only that the sample can be placed inside the capacitor, but also the surrounding transceiver coil. This type of arrangement is shown in Figure 8.11.

Figure 8.11

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The circular parallel-plate capacitor (in the color of metallic copper) together with the transmitting and receiving saddle coil located inside. The sample (not visible) is placed along the vertical axis of symmetry of the assembly, inside the coil. The remaining elements serve as a frame positioning the individual elements of the system.

One can optimize the dimensions of the capacitor using the electric field distribution found by integrating the charge density on the capacitor’s plates; see the corresponding classical electrostatics formula in ref. 56. The capacitor shown in Figure 8.11 has holes that make such a configuration similar to a Helmholtz coil in terms of optimizing its shape. As in the case of a Helmholtz coil, where the highest homogeneity of the magnetic field is achieved when the distance between the coils equals their radii, one can find optimal values of the distance between the capacitor plates L, the radius the plates R and the radius of the holes r in them.

Let us assume that the symmetry axis of the capacitor is the z-axis. The degree of homogeneity of the electric field at the center of the capacitor can be determined by analysing the second derivative of the electric field in a given direction. Let us consider as the criterion for electric-field homogeneity the vanishing of the second derivatives of the electric field (i) along the axis of the symmetry of the capacitor, i.e.,

\frac{\partial^{2} E_{z}}{\partial z^{2}}

⁠, and (ii) in the radial direction that is perpendicular to the z-axis, i.e.,

\frac{\partial^{2} E_{z}}{\partial r^{2}}

⁠. Then, by differentiation of the electric field distribution between the plates of the circular capacitor given in ref. 57–59 one finds that, the highest homogeneity of the electric field along the z-axis is obtained if

\frac{L}{R} = 2 \sqrt{\frac{{(\frac{r}{R})}^{\frac{4}{5}} + {(\frac{r}{R})}^{\frac{6}{5}} + {(\frac{r}{R})}^{\frac{8}{5}}}{1 + {(\frac{r}{R})}^{\frac{2}{5}}}} .

(8.32)

This solution holds for

0 \leq L / R \leq \sqrt{6}

and 0 ≤ r/R ≤ 1. On the other hand, the highest homogeneity of the electric field in the radial direction is satisfied in the parameter range of

0 \leq L / R \leq \sqrt{10}

for

\frac{L}{R} = 2 \sqrt{\frac{{(\frac{r}{R})}^{2} - {(\frac{r}{R})}^{\frac{4}{7}}}{{(\frac{r}{R})}^{\frac{4}{7}} - 1}} .

(8.33)

Clearly, one cannot satisfy and simultaneously – see Figure 8.12. Therefore, a compromise between these two conditions is indispensable in practical applications. For instance, one can adopt an as easy-to-memorize configuration in which the distance between the plates is twofold the radius of the plate, L = 2R, and the radius of the hole is one half of the radius of the plate, r = R/2. The parameters of such a configuration are shown as a black dot in Figure 8.12.

Figure 8.12

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The optimum relationships between the height L and the radius r of the holes both normalized to the radius R of the plates of the capacitor in the capacitor-coil assembly of Figure 8.11. The optimal dimensions in the direction along and perpendicular to the symmetry axis of the capacitor are shown as continuous and dashed lines, respectively. Gray lines mark the asymptotic cases reached for a ring made of a charged wire, which are $\frac{L}{R} = \sqrt{6}$ for the direction parallel to the symmetric axis of the capacitor and $\frac{L}{R} = \sqrt{10}$ for the direction perpendicular to the symmetric axis of the capacitor.

For a capacitor to generate an electric field that oscillates at radio frequencies, its plates must be connected to an appropriate excitation unit. For the capacitor shown in Figures 8.11 and 8.12, this could be a transmission line with a standing wave whose anti-node, i.e., a point where the amplitude of the wave reaches maximum, is at the capacitor plate. Another solution, geometrically compatible with direct connection to the coaxial cable, is shown in Figure 8.13. There is a capacitor with circular and parallel plates, the lower plate of which is connected to the core and the upper plate is connected to the shield of the coaxial cable. Such a solution could be used together with a standard NMR probe. In this case, the capacitor is immersed in a chiral liquid that is placed in a standard NMR tube. The disadvantage incurred in this case is the reduced homogeneity of the electric field and, above all, the static magnetic field due to the mismatch between the magnetic susceptibility of a chiral liquid and the metallic elements constituting the plates, the core, and the shield of the transmission line.

Figure 8.13

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A capacitor directly connected to the coaxial cable (on the left side) together with the cross-section of electric field distribution inside it, taken along the plane containing the symmetry axis of the system (on the right side). The red color marks a high electric-field amplitude, while the field of a low amplitude is shown in the blue color. Arrows mark the direction of the electric field.

8.14 Summary and Conclusions

Although from an experimental point of view it would be difficult to consider each of the NMER effects described in this and previous chapter individually, there is an inevitable difference in their amplitudes. If we limit ourselves to the chirality-sensitive effects that are induced by the changes in the spin state over time, the NMER spectrum is a superposition of the $E 2$ effect due to σ* and $E 5$ resulting from J*. The dominant contribution stems from the effect of antisymmetric magnetic nuclear shielding, rather than from antisymmetry of the indirect spin–spin coupling. Therefore, in the domain of experiments aiming for the detection of chirality-sensitive polarization that is small compared to the achiral nuclear magnetization, antisymmetric spin–spin coupling does not offer any features that make the experiment more feasible (Figure 8.14, the upper trace). However, the situation changes if we consider the frequencies that lie far beyond the tiny fragment of the NMR spectrum that is excited on-resonance. Especially the region near the difference frequency seems to be well-suited for achieving the experimental aim of observing chirality-sensitive NMER effects. In this case, the most difficult requirement to fulfill, i.e., suppression of the unwanted magnetic field, is lifted, which permits us to drastically simplify the construction of the device generating the radiofrequency electric field (Figure 8.14, the lower trace). This applies to the effects $E 3$ ⁠, $E 4$ ⁠, and $E 5$ ⁠. Among them, the coherences induced by the antisymmetric J* seem to be the most promising because of their relatively high amplitude.

Figure 8.14

View large Download slide

The various contributions to the chirality-sensitive polarization in a system consisting of a nucleus and an electron interacting through hyperfine coupling. The antisymmetries due to the interaction of the magnetic field with nuclear magnetic moment (the blue $g_{I}^{*}$ pseudovector) and the electron magnetic moment (the red $g_{S}^{*}$ pseudovector), hyperfine coupling between them (the black A* pseudovector) are shown for a chiral molecule and its mirror image (A). The green arrow shows the permanent electric dipole moment μ^e of the molecule. Expected signals of the enantiomers due to the precession of the magnetization M (the red color) and the oscillation of the chirality-sensitive electric polarizations: $P_{A}^{c}$ (a sum of effects induced by $g_{I}^{*}$ and $g_{S}^{*}$ ⁠) and $P_{B}^{c}$ originating from A*. The spin states are: $ρ_{1} (t) = \cos (π A^{iso} t) \sin (ω_{1} t) {\hat{I}}_{1 x} + \sin (π A^{iso} t) \cos (ω_{1} t) {\hat{I}}_{1 x} {\hat{I}}_{2 z}$ and $ρ_{2} (t) \propto \cos ((ω_{1} - ω_{1}) t) ({\hat{I}}_{1 x} {\hat{I}}_{1 y} - {\hat{I}}_{1 y} {\hat{I}}_{1 x})$ ⁠. The total signal is shown in black. Reproduced from ref. 17 with permission from the Royal Society of Chemistry.

Moreover, in some special cases, such as in a very strongly coupled spin system, a static electric field could be applied that can be higher in amplitude than the radiofrequency one. A certain obstacle in this case is the difficulty posed by the molecular structure because, in addition to the condition that the spins have to be nearly equivalent, the condition that the permanent electric dipole vector μ^e and the pseudovector J* cannot be perpendicular to each other, must also be met.

Looking cross-sectionally at the effects described in this section, i.e., $E 3 - E 7$ ⁠, one can notice that certain quantum spin states appear repeatedly regardless of the specific experimental protocol. An example of such a state is ${\hat{I}}_{1 x} {\hat{I}}_{1 y} - {\hat{I}}_{1 y} {\hat{I}}_{1 x}$ ⁠, which is antisymmetric when the spins are interchanged. The frequent occurrence of such a quantum state of the system results from its connection with the antisymmetric tensorial properties of the molecule. This leads to the conclusion that the analysis of potential effects depending on the molecular chirality could also be carried out in the order opposite to that presented in this book, i.e., by searching for a quantum state in which chirality can manifest itself directly and then finding its connection with molecular properties.

It could also be observed that one of the effects induced by the spin–spin coupling, namely, the perturbation of dipolar relaxation in a three-spin system by the external electric field (effect $E 7$ ⁠), offers a unique feature: the possibility of determination of the absolute configuration of the molecule, without the need for any quantum-mechanical computation of pseudoscalars. This predicted effect would offer a very rare opportunity for a direct insight into the molecular structure at a comparable level of detail, as mentioned in Chapter 1, such as Coulomb explosion imaging and atomic force microscopy of chiral molecules.

Besides the reasons mentioned above, which motivate experiments aiming for the observation of chirality-sensitive effects, one can perceive studies of such effects not only as a possible practical tool for studying molecular chirality, but also as a means of determining their structure beyond the fact that chiral molecules are distinguishable from their mirror images. Each chirality-sensitive NMER effect has a different dependence on the molecular structure, e.g., $E 4$ vs. $E 7$ or $E 1$ vs. $E 3$ ⁠. This allows us to see the molecular structure and dynamics from different perspectives, possibly revealing its different physicochemical properties.

Acknowledgements

PG acknowledges the National Science Centre, Poland, for the financial support through OPUS 16 grant No. 2018/31/B/ST4/02570, the Max Planck Society for providing access to computational resources (in particular, the COMSOL program), and Dr Grzegorz Łach (Faculty of Physics, University of Warsaw) for helpful discussions about the permanent electric dipole moment of a chiral molecule. JV thanks the Academy of Finland (project 331008) and University of Oulu (Kvantum Institute) for the financial support, as well as CSC-the Finnish IT Centre of Science and the Finnish Grid and Cloud Infrastructure project (persistent identifier urn:nbn:fi:researchinfras-2016072533) for computational resources.

†

This chapter is subject to a Creative Commons CC-BY-NC-ND 4.0 International license. Financial support from the National Science Centre, Poland (through OPUS 16 grant No. 2018/31/B/ST4/02570), Academy of Finland (project 331008) and University of Oulu is acknowledged.

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Log in

Chapter 8: Chirality-sensitive Effects Induced by Antisymmetric Spin–Spin Coupling†

8.1 Introduction

8.2 Chirality-sensitive Spin–Spin Coupling

8.3 Relationship Between Chirality and Antisymmetric Spin–Spin Coupling

8.4 Time-evenness of the Pseudoscalar Jc

8.5 Relationship Between Chirality and the Permanent Electric Dipole Moment

8.6 Relationship Between the Molecular Structure and Pseudoscalar Jc

8.7 Nuclear Magnetoelectric Effective Hamiltonian

8.7.1 Rotating Frame Transformation

8.7.2 Antisymmetric Nuclear Magnetic Shielding Effective Hamiltonian

8.7.3 Antisymmetric Spin–Spin Coupling Effective Hamiltonian

8.7.3.1 Case I

8.7.3.2 Case II

8.7.3.3 Case III

8.8 A General Description of the Chirality-sensitive Electric Polarization

8.9 Chirality-sensitive Coherences Induced by the Antisymmetric Part of the J-coupling Tensor

8.9.1 Description of the E 4 Effect

8.9.2 Experimental Protocol for the E 4 Effect

8.9.3 Significance and Estimation of the Magnitude of the E 4 Effect

8.10 Electric Polarization Induced by the Antisymmetric Part of Indirect Spin–Spin Coupling

8.10.1 Description of the E 5 Effect

8.10.2 Experimental Protocol for the E 5 Effect

8.10.3 Significance and Estimation of the Magnitude of the E 5 Effect

8.11 Chirality-sensitive Effects Induced by Antisymmetry of the A-coupling Tensor

8.11.1 Description of the E 6 Effect

8.11.2 Experimental Protocol for the E 6 Effect

8.11.3 Significance and Estimation of the Magnitude of the E 6 Effect

8.12 Interference Between Dipolar Relaxation Mechanisms in an Electric Field

8.12.1 Description of the E 7 Effect

8.12.2 Experimental Protocol for the E 7 Effect

8.12.3 Significance and Estimation of the Magnitude of the E 7 Effect

8.13 Sources of Radiofrequency Electric Field for J-NMER Experiments

8.14 Summary and Conclusions

Acknowledgements

References

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Chapter 8: Chirality-sensitive Effects Induced by Antisymmetric Spin–Spin Coupling^{^†}

8.4 Time-evenness of the Pseudoscalar J^c

8.6 Relationship Between the Molecular Structure and Pseudoscalar J^c

8.9.1 Description of the $E 4$ Effect

8.9.2 Experimental Protocol for the $E 4$ Effect

8.9.3 Significance and Estimation of the Magnitude of the $E 4$ Effect

8.10.1 Description of the $E 5$ Effect

8.10.2 Experimental Protocol for the $E 5$ Effect

8.10.3 Significance and Estimation of the Magnitude of the $E 5$ Effect

8.11.1 Description of the $E 6$ Effect

8.11.2 Experimental Protocol for the $E 6$ Effect

8.11.3 Significance and Estimation of the Magnitude of the $E 6$ Effect

8.12.1 Description of the $E 7$ Effect

8.12.2 Experimental Protocol for the $E 7$ Effect

8.12.3 Significance and Estimation of the Magnitude of the $E 7$ Effect