Long-lived nuclear spin states far from magnetic equivalence

Abstract

Clusters of coupled nuclear spins may form long-lived nuclear spin states, which interact weakly with the environment, compared to ordinary nuclear magnetization. All experimental demonstrations of long-lived states have so far involved spin systems which are close to the condition of magnetic equivalence, in which the network of spin–spin couplings is conserved under all pair exchanges of symmetry-related nuclei. We show that the four-spin system of trans-[2,3-¹³C₂]-but-2-enedioate exhibits a long-lived nuclear spin state, even though this spin system is very far from magnetic equivalence. The 4-spin long-lived state is accessed by slightly asymmetric chemical substitutions of the centrosymmetric molecular core. The long-lived state is a consequence of the locally centrosymmetric molecular geometry for the trans isomer, and is absent for the cis isomer. A general group theoretical description of long-lived states is presented. It is shown that the symmetries of coherent and incoherent interactions are both important for the existence of long-lived states.

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1 Introduction

Clusters of coupled nuclear spins may form long-lived states (LLS) with decay time constants T_LLS much longer than the conventional spin–lattice relaxation time T₁.^1–22 Such long-lived states have been used to enhance the study of slow processes such as diffusion and flow by NMR, and to enhance the contrast in ligand binding investigations.^21,22 Long-lived states hold particular promise for extending the range of nuclear hyperpolarization experiments,^23–28 in which the small (∼10⁻⁵) thermal polarization of nuclear spins is temporarily enhanced by many orders of magnitude. In the case of spin systems containing only two nuclear spins-1/2, the long-lived state is known as singlet order.¹⁰ This represents the difference in population between the singlet state of the spin-1/2 pair, which is antisymmetric with respect to spin exchange, and the mean population of the triplet states, which are symmetric with respect to exchange. Singlet order is immune to intra-pair dipolar relaxation, which is often the dominant mechanism for the T₁ process. Singlet order lifetimes T_LLS exceeding ∼60T₁ have been reported.¹³ Long-lived singlet order may also be observed for spin-1/2 isotopes of different type (heteronuclear singlet order).²⁹

In systems of more than two coupled spins-1/2, the existence of long-lived singlet order (and LLS in general) depends strongly on the geometrical arrangement of the nuclei, and the relative magnitudes of spin–spin couplings and chemical shift differences. This problem was examined both theoretically¹⁴ and numerically.³⁰ Multiple-spin states that are protected against intramolecular relaxation mechanisms were predicted to exist, in the case that the rigid geometrical arrangement of nuclei displays local inversion symmetry.³⁰ However, no experimental demonstrations of such geometrically imposed long-lived states were provided.

The absence of local geometrical centrosymmetry does not preclude the existence of long-lived spin orders when either the geometrical remoteness of the central spin pair from the other members of the spin system is provided,^{16–19,25–27} or in the presence of fast intramolecular dynamics.³¹

In the following discussion, we provide a clear experimental demonstration of a geometrically-imposed long-lived nuclear spin state in an asymmetrically substituted derivative of ¹³C₂-fumarate (trans-[2,3-¹³C₂]-but-2-enedioate). As discussed below, this system exhibits local centrosymmetry but is far from the regime of magnetic equivalence. The confirmed existence of a long-lived state in this system verifies the analyses in ref. 14 and 30, and shows that near-magnetic-equivalence is not a necessary condition for generating and observing long-lived nuclear spin states.

A theoretical framework accounting for the symmetry properties of the coherent and fluctuating terms in the hamiltonian is introduced for predicting the existence of long-lived spin order. This theory emphasizes the interlocking symmetries of both the coherent and fluctuating parts of the nuclear spin Hamiltonian, as opposed to recent work which only takes the coherent spin interactions into account.^16–19

2 Near magnetic equivalence

Consider a pair of nuclei denoted I_j and I_k, with chemical shifts δ_j and δ_k, so that their chemical shift frequencies in a magnetic field B⁰ are given by ω⁰_j = −γ_jB⁰(1 + δ_j) where γ_j is the magnetogyric ratio of I_j, and similarly for I_k. Their mutual J-coupling is denoted J_jk. The nuclei are considered to be chemically equivalent if the symmetry of the molecule ensures equal chemical shifts δ_j = δ_k. If there are no other nuclei in the system, chemical equivalence also implies magnetic equivalence. However, if there are other nuclei {I_l…} in the system, with finite couplings to one or both of I_j and I_k, then a chemically-equivalent pair of nuclei is only magnetically equivalent if all out-of-pair couplings are equal, i.e. J_jl = J_kl ∀ l ∉ {j,k}.³²

We define a pair of nuclei I_j and I_k to exhibit near magnetic equivalence if the following condition holds:

(ω⁰_j − ω⁰_k)² + π²(J_jl − J_kl)² ≤ π²J_jk² ∀ l ∉ {j,k} (1)

Under conditions of near magnetic equivalence, the mixing of singlet and triplet states by the symmetry-breaking interactions is strongly suppressed by the intra-pair J-coupling – a phenomenon called J-stabilization.⁵ In this regime, efficient decay of singlet order between spins I_j and I_k is induced neither by correlated fluctuations of local magnetic fields at the sites of the two spins, nor by fluctuations of the mutual dipole–dipole coupling of I_j and I_k. The suppression of these important relaxation mechanisms often leads to a long-lived state with a relaxation time much longer than the conventional magnetization relaxation time, T₁.⁵

The condition (1) is depicted graphically in Fig. 1. The horizontal axis indicates the difference in chemically shifted resonance frequencies of the two spins in the pair, while the vertical axis represents the difference of out-of-pair J-couplings to other spins in the same molecule. The dashed circle indicates the intra-pair J-coupling. Spin pairs may be characterized as being chemically equivalent (corresponding to points on the vertical axis) or magnetically equivalent (point at the origin). Spin pairs with parameters within the dashed circle present near magnetic equivalence.

Fig. 1 Pictorial representation of the concepts of chemical and magnetic equivalence for a pair of spins labelled j and k. The label l identifies homonuclear spin species coupled with the pair. The dashed circle has a radius |πJ_jk|. The filled circle indicates the point of exact magnetic equivalence. Points on the vertical axis (gray) correspond to systems whose nuclei are chemically equivalent. The filled hexagon represents a spin system whose nuclei are chemically equivalent but slightly magnetically inequivalent.^16–19 The filled square indicates a spin system formed by two nuclei which are slightly chemically inequivalent.^9,11 The open square, falling well outside the dashed circle, represents spin systems in the regime of strong magnetic inequivalence, as studied in this work.

There have been extensive investigations of isolated spin-1/2 pairs with slightly different chemical shifts, represented by the filled square in Fig. 1. In some cases, the chemical shift difference is sufficiently small that the near-magnetic-equivalence condition (1) is satisfied without intervention in high magnetic fields.^9,12 In other cases is satisfied by transporting the sample into a region of low applied magnetic field^6,8 or by applying resonant radiofrequency fields.³³ In the regime of near magnetic equivalence, long-lived order may be accessed by carefully timed pulse trains^9,34 or by continuous radiofrequency fields of suitable amplitude.^20,35 Appropriate modifications of the terms ω⁰_j and ω⁰_k in eqn (1) should be used in these cases.

It is also possible to attain near-magnetic-equivalence by a deviation in the vertical direction, corresponding to small differences in J-couplings to spins outside the pair, providing that eqn (1) is still satisfied. This has been demonstrated for chemically equivalent systems (points along the vertical axis).^16–19

In this publication we demonstrate the existence of long-lived order in a four-spin-1/2 system which does not conform to condition (1). This spin system displays strong magnetic inequivalence with parameters represented by an open square falling well outside the dashed circle in Fig. 1. By comparing two strongly magnetic inequivalent isomers, one exhibiting local centrosymmetry of a four-spin system, and one without centrosymmetry, we show that only the centrosymmetric system displays a long-lived state. In this special case, the long-lived state is geometrically imposed by the local centrosymmetry of the rigid molecular structure, as suggested in ref. 14 and 30. These cases highlight the important role of symmetry in both the coherent and fluctuating spin Hamiltonian, in determining the existence of LLS.

3 Experimental

3.1 Samples

The molecular structures of the samples used in this study are shown in Fig. 2. Both materials are di-esters of [2,3-¹³C₂]-but-2-enedioate which contains a central spin system comprising two ¹³C nuclei and two ¹H nuclei; the fumarate derivative (panels (a) and (c)) has a trans double bond, while the maleate derivative (panes (b) and (d)) has a cis double bond. In both cases, the two ester groups are different, causing the two ¹³C sites (and their attached protons) to have slightly different chemical shifts. As discussed below, this slight chemical asymmetry is necessary for accessing the long-lived state. All ester groups are deuterated in order to reduce their relaxation contributions.

Fig. 2 Molecular structure of (a) ¹³C₂-AFD and (b) of ¹³C₂-AMD. Filled circles denote ¹³C nuclei. For both molecules R¹ = CD₂CD₃ and R² = CD₂CD₂CD₃. (c) The local spin system of ¹³C₂-AFD, with geometrical inversion centre marked by an asterisk. (d) The local spin system of ¹³C₂-AMD.

In the following discussion, the [2,3-¹³C₂]-fumarate derivative in Fig. 2a [1-(ethyl-d5) 4-(propyl-d7)(E)-but-2-enedioate-2,3-¹³C₂] is referred to as ¹³C₂-AFD (asymmetric fumarate diester); the [2,3-¹³C₂]-maleate derivative in Fig. 2b [1-(ethyl-d5) 4-(propyl-d7)(Z)-but-2-enedioate-¹³C₂] is referred to as ¹³C₂-AMD (asymmetric maleate diester).

¹³C₂-AFD contains a 4-spin-1/2 system displaying a local centre of inversion, midway between the two ¹³C nuclei (Fig. 2c); the local 4-spin-1/2 system in ¹³C₂-AMD, on the other hand, displays a reflection plane but is not locally centrosymmetric (Fig. 2d).

The samples were synthesized as described in the ESI.† In both cases 30 mg of the diester were dissolved in 0.5 ml of CD₃OD, added to a Wilmad low pressure/vacuum 5 mm NMR tube, degassed thoroughly using the freeze and pump technique (5 cycles), and hermetically sealed.

Fig. 3 NMR 1D-spectra of ¹³C₂-AFD: (a) simulated ¹³C spectrum; (b) experimental ¹³C spectrum. (c) Simulated ¹H spectrum; (d) experimental ¹H spectrum.

3.2 NMR experiments

All NMR experiments were performed at a magnetic field of 11.75 T on a Bruker 500 MHz Avance III spectrometer using a Bruker 5 mm triple-resonance liquid-state probe.

Fig. 4 NMR 1D-spectra of ¹³C₂-AMD: (a) simulated ¹³C spectrum; (b) experimental ¹³C spectrum. (c) simulated ¹H spectrum; (d) experimental ¹H spectrum.

The ¹³C and ¹H 1D-NMR spectra were fitted using Spinach simulation software (Fig. 3 and Fig. 4).^36,37 The obtained chemical shift and coupling parameters are given in Table 1.

Table 1 Spin system and pulse sequences parameters for ¹³C₂-AFD and ¹³C₂-AMD in CD₃OD. J couplings and chemical shift differences were obtained by fitting the 1D-NMR spectra with Spinach.^36,37 The pulse sequence parameters Δ(¹³C), Δ(¹H), n₁(¹³C), n₁(¹H), ω^nut_ev(¹³C)/2π, ω^nut_ev(¹H)/2π were optimized experimentally for both molecules. Relaxation time constants T₁ were measured by standard inversion-recovery at a magnetic field of 11.75 T

		^13C2-AFD	^13C2-AMD
Simulation	J 23	71.0 ± 0.6 Hz	71.6 ± 0.8 Hz
	J 13 = J24	−2.8 ± 0.03 Hz	−1.16 ± 0.02 Hz
	J 12 = J34	166.7 ± 0.2 Hz	166.5 ± 0.2 Hz
	J 14	15.7 ± 0.2 Hz	11.95 ± 0.4 Hz
	Δδ23	62 ± 5 ppb	92 ± 4 ppb
	Δδ14	7 ± 0.9 ppb	1.8 ± 0.2 ppb
Experiment	Δ(^13C)	3.5 ms	3.7 ms
	Δ(^1H)	16 ms	21 ms
	n 1(^13C)	14	14
	n 2(^13C)	7	7
	n 1(^1H)	6	18
	n 2(^1H)	3	9
	ω ^nutev(^1H)/2π	520 Hz	520 Hz
	ω ^nutev(^13C)/2π	700 Hz	700 Hz
	T 1(^13C)	6.0 ± 0.1 s	6.0 ± 0.1 s
	T 1(^1H)	6.0 ± 0.1 s	4.5 ± 0.1 s

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The spin systems in both ¹³C₂-AMD and ¹³C₂-AFD are far from magnetic equivalence. The coupling difference |J₁₂ − J₁₃| is more than twice |J₂₃|, well outside the near-magnetic-equivalence condition expressed in eqn (1).

Experimental investigation of the long-lived states in the four-spin-1/2 systems of ¹³C₂-AFD and ¹³C₂-AMD was accomplished by the double-resonance pulse sequences shown in Fig. 5. These pulse sequences are based on the magnetization-to-singlet (M2S) and singlet-to-magnetization (S2M) pulse sequences used to study LLS in homonuclear 2-spin-1/2 systems^8,9 and in near-equivalent heteronuclear systems.^16–19 In the current case, each of the M2S and S2M sequences may be performed on either the ¹³C or the ¹H channel, giving rise to the four pulse sequence combinations reported in Fig. 5.

Fig. 5 Pulse sequences used to access the Δ_gu state in ¹³C₂-AFD and ¹³C₂-AMD. Excitation and detection of Δ_gu is done using M2S and S2M pulse sequences, respectively. M2S and S2M are run on the same RF channel in (a) and (b) or on different channels in (c) and (d). The exact scheme of an M2S sequence is shown in (e). The S2M pulse sequence is obtained by running M2S in reverse time order. The echo duration is given by τ_e ∼ 1/(2J₁₄) when the sequence is applied to the ¹H channel, and τ_e ∼ 1/(2J₂₃) when using the ¹³C channel. The number of loops is always n₁ = 2n₂. During the variable evolution time τ_ev a weak continuous wave irradiation of 800 μW is applied on both RF channels (ω^nut_ev(¹³C)/2π = 700 Hz, ω^nut_ev(¹H)/2π = 520 Hz) to impose chemical equivalence. A WALTZ-16 decoupling sequence³⁸ is applied to the ¹H channel during detection of the ¹³C signal; WALTZ-16 is applied to the ¹³C channel when the ¹H signal is detected.

All four sequence variants access the same collective state denoted Δ_gu, as reported by the simulation in Fig. 6 performed using SpinDynamica.³⁹ This operator represents the population difference between spin states that are symmetric (g) and antisymmetric (u) with respect to simultaneous exchanges of the ¹H and the ¹³C nuclei, as described below. In the case of ¹³C₂-AFD this four-spin state is long-lived. In all cases, the M2S sequence converts either ¹H or ¹³C magnetization into Δ_gu. This state evolves during a variable evolution interval τ_ev, during which an unmodulated irradiation field (∼800 μW) is applied on both RF channels (nutation frequencies reported in Table 1). This suppresses the small chemical shift difference induced by the different ester substituents.³³ An S2M sequence, applied on either the ¹H or ¹³C channel, converts the state Δ_gu into observable transverse magnetization, which is then detected.

Fig. 6 Simulated trajectories for the transfer of longitudinal magnetization into carbon transverse magnetization I_2x + I_3x in blue, proton transverse magnetization I_1x + I_4x in red and Δ_gu spin order in black under the pulse sequence in Fig. 5a for ¹³C₂-AFD. The spin system is assumed to evolve under the influence of with parameters reported in Table 1 and τ_ev = 0.5 s. No relaxation was included in the simulations.

A WALTZ-16 decoupling sequence³⁸ is applied on the ¹H channel during ¹³C observation, to collapse the spectrum into a single peak. Conversely, the ¹H NMR signal is detected in the presence of ¹³C WALTZ-16 decoupling.

The signal amplitudes for the two compounds, using the four different pulse sequence variants, are plotted against τ_ev in Fig. 7. The decay time constants reported in Table 2 are consistent among the four different experiments. The experimental points were fitted using a single exponential decay curve when exciting and detecting the spin system on different RF channels. A double exponential decay curve was instead applied when excitation and detection occured on the same RF channel. Further details and the fitting functions are reported in the ESI.† In the case of ¹³C₂-AFD, the state Δ_gu is found to have a decay time constant of ∼10T₁, marking the existence of a four-spin LLS protected against dipole–dipole relaxation and the other mechanisms contributing to the rapid decay of longitudinal magnetization. In the case of ¹³C₂-AMD, where a local centre of inversion is absent, the state Δ_gu decays faster than T₁ and hence cannot be designated as a long-lived state. The experimental results confirm the link between local molecular geometry and the existence of LLS in multi-spin systems.

Fig. 7 Experimental results are shown for pulse sequence in (a) Fig. 5a, (b) Fig. 5b, (c) Fig. 5c and (d) Fig. 5d. Black triangles refer to ¹³C₂-AFD, and gray circles to ¹³C₂-AMD. The dashed lines are best fits to bi-exponential decays in (a) and (b) and single exponential decays in (c) and (d). The time constants for ¹³C₂-AFD are: (a) 61.01 ± 0.1 s; (b) 63.2 ± 2.5 s; (c) 59.9 ± 3.2 s; (d) 61.6 ± 2.0 s. The time constants for ¹³C₂-AMD are: (a) 2.5 ± 0.2 s; (b) 2.1 ± 0.1 s; (c) 2.5 ± 0.2 s; (d) 2.7 ± 0.2 s.

Table 2 Experimental decay time constants for the spin operator term Δ_gu as determined by fitting the experimental results in Fig. 7. In the case of ¹³C₂-AFD, the operator Δ_gu is a LLS

Pulse sequence	^13C2-AFD/s	^13C2-AMD/s
M2S(^13C)–S2M(^13C)	61.0 ± 1.0	2.5 ± 0.2
M2S(^1H)–S2M(^1H)	63.2 ± 2.5	2.1 ± 0.1
M2S(^13C)–S2M(^1H)	59.9 ± 3.2	2.5 ± 0.2
M2S(^1H)–S2M(^13C)	61.6 ± 2.0	2.7 ± 0.2

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4 Theory

A nuclear spin system experiences a spin Hamiltonian containing coherent and fluctuating contributions: The coherent terms are time-independent and uniform over the ensemble, whereas the fluctuating terms are time-dependent and have different instantaneous values for different ensemble members. In general is associated with the frequencies and amplitudes of the NMR peaks, while leads to relaxation. In solution NMR, the symmetry of is associated with chemical and magnetic equivalence; the symmetry of is responsible for the existence of long-lived forms of spin order. The existence of LLS in near magnetic equivalent multi-spin systems has been recently analysed in terms of the symmetry of the coherent Hamiltonian alone.^17,19 As shown below, this can be misleading. The interplay of the various symmetries is now described. Although the discussion will be conducted for the specific cases described in this paper, the general principles may be used to analyze a wide variety of molecular geometries.

4.1 Symmetry of the coherent hamiltonian

We write the coherent Hamiltonian as follows:where respects the idealized symmetry of the molecular spin system and is a small perturbation due to the asymmetric remote substituents. In isotropic liquids, each of these is a superposition of Zeeman and J-coupling terms, i.e. and , where in the current cases Here ω⁰_C and ω⁰_H are the Larmor frequencies of ¹³C and ¹H, and δ_i represents the chemical shift of nucleus I_i. The terms and respect the local symmetry of the four-spin system, while the terms and contain the small symmetry-breaking terms induced by the remote asymmetric ester groups. In the experiments described here, the asymmetric term is exploited during the M2S and S2M sequences to access the state Δ_gu. However, during the evolution interval τ_ev, the effect of is suppressed by the applied RF field.

Spin permutation operators denoted (i₁i₂…i_n) may be defined, which lead to a cyclic permutation of n spin labels when applied to a spin state or a spin operator.⁴⁰ Different spin permutations may be multiplied. Examples are:

(12)I_1zI_3z(12)^† = I_2zI_3z (8)

(12)(34)I_1zI_3z(34)^†(12)^† = I_2zI_4z (9)

The symmetry group of the coherent Hamiltonian, denoted , is defined here as the group of spin permutation operations which commutes with .

In the current case, the group consists of just two operations, for both molecules:

where E is the identity. The presence of the permutation (14) indicates that spins I₁ and I₄ are chemically equivalent (as far as the idealized Hamiltonian is concerned). The fact that the permutation (14) does not appear in the group as an isolated element, but as a product with a different permutation (23), indicates that spins I₁ and I₄ are magnetically inequivalent.

The group has two irreducible representations denoted A_g and B_u, with opposite parity under the operation (14)(23).

In the absence of any symmetry breaking interactions, the NMR spectral peaks are generated by transitions within the A_g or B_u manifolds.

4.1.1 Singlet–triplet product basis. We follow Warren et al.¹⁶ by considering a basis set which is a direct product of the singlet–triplet bases formed by homonuclear spin pairs:

[Doublestruck B]_ST = {ST}₂₃ ⊗ {ST}₁₄ (11)

A permutation (ij) acting upon a ket belonging to {ST}_ij gives:

(ij)|ψ_ij〉 = p^(ij)|ψ_ij〉 (12)

|ψ_ij〉 is said to be symmetric or gerade (g) when p^(ij) = +1; antisymmetric or ungerade (u) when p^(ij) = −1.

As different spin permutations may be multiplied, a four-spin state can be either symmetric (g) or antisymmetric (u) under the homonuclear spin permutation (ij)(kl), that is p^(ij)(kl) = +1 (for g) or p^(ij)(kl) = −1 (for u).

States belonging to [Doublestruck B]_ST can be classified (see Table 2) according to the magnetic quantum number M and their parities under the homonuclear spin permutations:

(I_1z + I_2z + I_3z + I_4z)(|ψ₁₄〉 ⊗ |ψ₂₃〉) = M(|ψ₁₄ ⊗ |ψ₂₃〉) (14)(|ψ₁₄〉 ⊗ |ψ₂₃〉) = p⁽¹⁴⁾(|ψ₁₄ ⊗ |ψ₂₃〉) (23)(|ψ₁₄〉 ⊗ |ψ₂₃〉) = p⁽²³⁾(|ψ₁₄ ⊗ |ψ₂₃〉) (14)(23)(|ψ₁₄〉 ⊗ |ψ₂₃〉) = p⁽¹⁴⁾p⁽²³⁾(|ψ₁₄〉 ⊗ |ψ₂₃〉) = p^(14)(23)(|ψ₁₄〉 ⊗ |ψ₂₃〉) (13)

The 16 states reported in Table 3 form an orthonormal basis set and are classified according to their symmetry with respect to the permutation (14)(23) into irreducible representations A_g and B_u:

Γ_spin = 10A_g ⊕ 6B_u (14)

Table 3 The first column lists the components of the basis set [Doublestruck B]_ST in eqn (11) used for describing both ¹³C₂-AFD, ¹³C₂-AMD. Each ket is classified according to its parity under exchange of the two ¹³C nuclei, exchange of the two ¹H nuclei, and simultaneous exchange of both homonuclear pairs. The last column indicates the total magnetic quantum number

[Doublestruck B] ST	p ^(23)	p ^(14)	p ^(14)(23)	M
|T1^23T1^14〉	g	g	g	2
|T0^23T1^14〉	g	g	g	1
|T1^23T0^14〉	g	g	g	1
|T1^23T−1^14〉	g	g	g	0
|T0^23T0^14〉	g	g	g	0
|T−1^23T1^14〉	g	g	g	0
|T0^23T−1^14〉	g	g	g	−1
|T−1^23T0^14〉	g	g	g	−1
|T−1^23T−1^14〉	g	g	g	−2
|S0^23S0^14〉	u	u	g	0
|T1^23S0^14〉	g	u	u	1
|T0^23S0^14〉	g	u	u	0
|T−1^23S0^14〉	g	u	u	−1
|S0^23T1^14〉	u	g	u	1
|S0^23T0^14〉	u	g	u	0
|S0^23T−1^14〉	u	g	u	−1

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4.2 Symmetry of the fluctuating hamiltonian

Consider now the fluctuating spin interactions which cause relaxation. An example of such interactions is given by the dipole–dipole interaction between the nuclei, which may be written as follows: where the dipole–dipole coupling constant between a pair of nuclei is b_ij = −(μ₀/4π)γ_iγ_jℏr_ij⁻³, D² is a second-rank Wigner rotation matrix, Ω^ij_PL represents the set of three Euler angles defining the orientation of the internuclear vector between I_j and I_k in the laboratory frame, and T^ij₂ is a second-rank irreducible spherical tensor.

At an arbitrary time point t, molecular vibrations distort the molecular geometry away from its equilibrium configuration, breaking the geometrical symmetry. Hence, in general, the instantaneous value of (and other fluctuating terms) does not display symmetry. However, since rapid vibrations are usually too fast to cause significant NMR relaxation, the relevant spin Hamiltonian for relaxation purposes may be locally averaged over molecular vibrations (typically on the sub-picosecond timescale), denoted here . This vibrationally averaged spin Hamiltonian reflects the geometrical symmetry of the equilibrium molecular structure, and displays the corresponding spin permutation symmetry.

In general, may be calculated from eqn (15) by using nuclear coordinates from the equilibrium molecular geometry, but with small adjustments to the interaction strengths caused by vibrational averaging.

As previously done for we will consider in this section the nuclear spin permutations that are symmetry operations for in both molecular systems.

In the case of ¹³C₂-AFD, the permutation (14)(23) is a symmetry operation for the vibrationally-averaged dipole–dipole spin Hamiltonian because of the inversion geometry of the local molecular structure. The double permutation always exchanges pairs of nuclei with parallel internuclear vectors, so that the spatial interaction tensors are identical for all molecular orientations. Similar considerations apply for other interaction terms such as the chemical shift anisotropy.

The group of the vibrationally-averaged fluctuating Hamiltonian is therefore given by

where contains the group of spin permutation operators that commute with 〈〉_vib. In the case of the cis isomer, the equilibrium geometry is not sufficient to impose symmetry on the vibrationally averaged spin Hamiltonian. We therefore get a trivial symmetry group in this case,Fig. 8 illustrates the symmetry properties of the vibrationally averaged intramolecular dipole–dipole interaction, in the two molecular systems. The figure shows matrix representations of 〈H_DD〉_vib in the basis [Doublestruck B]_ST. In the case of ¹³C₂-AFD (Fig. 8b), the matrix representation of 〈H_DD〉_vib is block-diagonal in this basis, since the vibrationally averaged dipole–dipole Hamiltonian has the same symmetry as (eqn (10) and (16)). In the case of ¹³C₂-AMD (Fig. 8d), on the other hand, off-diagonal elements connecting different irreducible representations appear, since 〈H_DD〉_vib has a lower symmetry than (eqn (17)).

Fig. 8 Matrix plot representations of and hamiltonians for ¹³C₂-AFD in panels (a), (b) and for ¹³C₂-AMD in panels (c) and (d) respectively. hamiltonian is block diagonal for ¹³C₂-AFD (panel (b)). The basis is [Doublestruck B]_ST, and the ordering follows the convention adopted in Table 3.

The sole consideration of the symmetry properties of the coherent hamiltonian (Fig. 8a and c) to infer the existence of LLS could be deceptive. The symmetry group for is identical for both ¹³C₂-AFD and ¹³C₂-AMD (eqn (10)) but a long-lived order is expected only in one case.

In general, if the molecular structure is rigid, and the equilibrium geometry of the local spin system is centrosymmetric, the group has dimension 2. If the molecule is rigid but non-centrosymmetric, has dimension 1. The symmetry properties of the coherent and fluctuating hamiltonians can be combined to derive the correct symmetry properties for the Liouvillian superoperator.

4.3 Idealized Liouvillian group and long-lived states

We define the idealized Liouvillian group of the spin system as the intersection of the groups and : For the two molecules discussed here, we have: The coherent and fluctuating Hamiltonians cannot induce transitions between states which belong to different irreducible representations of the group . Hence the spin evolution must preserve a population difference between states belonging to different irreducible representations of . Such a population difference is a constant of motion and does not evolve, providing that the realistic spin interactions conform exactly the idealized symmetry assumed in the theoretical model. In practice, deviations from ideal symmetry invariable cause slow relaxation, but the lifetime of such a state may nevertheless be long compared to the relaxation of most other states.

Suppose that has irreducible representations. There are combinations of populations which do not evolve, in the ideal case. However one of these constants of motion is the total spin system population, which is trivially conserved in all circumstances, for a closed system.

Hence the number of non-trivial long-lived states (N_LLS) is given by − 1.

In the case of ¹³C₂-AFD, which has a two-dimensional idealized Liouvillian group (eqn (19)), there is a LLS given by the difference in mean population of states with A_g and B_u symmetry: This corresponds to the following combination of spherical tensor operators:where T^ij₀₀ = −3^−1/2I_i·I_j is the rank-0 spherical tensor operator between nuclei i and j. Eqn (23) highlights the invariance of the LLS under rotations of both spin ¹H and ¹³C spin states, and can also be written as: The operator Δ_gu is a LLS in the case of ¹³C₂-AFD, since the double spin permutation (14)(23) is an element of (see eqn (19)). In the case of ¹³C₂-AMD, the idealized Liouvillian group is one-dimensional even in a regime of perfect chemical equivalence (see eqn (20)) and there is no long-lived state, except for the trivial mean of all spin state populations that is always a conserved quantity for an isolated spin system.

4.4 Torsional angle dependence of the decay rate

The dependence of the LLS on the equilibrium molecular geometry is now investigated. The arrangements of the four nuclear spins in ¹³C₂-AMD and ¹³C₂-AFD may be obtained from each another by rotating one of the heteronuclear ¹³CH single bonds around the axis passing through the ¹³C₂ double bond.

The standard approach in LLS analysis consists in the diagonalization of the Liouvillian superoperator that can be set up to include all the relevant relaxation mechanisms. We will restrict our considerations to the relaxation induced by the intra-molecular dipole–dipole interaction (eqn (25)) for simplicity. When the dimension of the Hilbert space is N the Liouvillian superoperator has a set of N² eigenvalues–eigenoperators pairs {L_q,Q_q}. In general is not hermitian so the eigenvalues may be complex.

L_q = −λ_q + iω_q (26)

Eqn (25) is the Liouville–von Neumann equation. Eigenoperators Q_q with complex eigenvalues correspond to coherences oscillating at frequency ω_q and decaying to equilibrium with a rate constant λ_q. Eigenoperators Q_q with real eigenvalues (ω_q = 0) correspond to spin state populations decaying with a rate λ_q. The eigenvalue λ_q = 0 corresponds to the identity operator, corresponding to the conservation of the sum of all spin-state populations.

Numerical simulation performed with SpinDynamica³⁹ in Fig. 9b shows the dependence of the smallest non-zero rate constant λ_min on the torsional angle θ. It was assumed the J coupling parameters to be independent of the torsional angle and equal to those reported in Table 1 for ¹³C₂-AFD. This approximation appears to be reasonable given the small difference in parameters as obtained by fitting the 1D-NMR spectra and listed in Table 1. The relaxation superoperator takes into account all dipole–dipole couplings in the 4-spin system and assumes rigid molecular geometry and isotropic molecular tumbling with a correlation time τ_C = 15 ps. As expected, a long-lived state with infinite relaxation time is predicted only when θ = 180°, which is the geometric equilibrium configuration for ¹³C₂-AFD.

Fig. 9 (a) The geometric configurations obtainable by rotation of the single ¹³CH bond through the ¹³C₂ double bond axis. The front view and the through ¹³C₂ double bond view are given. θ = 0° and θ = 180° correspond to equilibrium spin systems configurations of ¹³C₂-AMD and ¹³C₂-AFD respectively. (b) The smallest non-zero decay rate constant λ_min plotted against torsional angle θ, obtained by numerical analysis of the Liouvillian superoperator. Only dipole–dipole relaxation is included, with a rotational correlation time of τ_C = 15 ps. A zero value, signing the existence of a infinitely LLS, is present only when θ = 180°: the geometric configuration of ¹³C₂-AFD.

4.5 Symmetry breaking

Since applied fields are symmetric for all spins, the LLS described by eqn (22) can only be accessed by breaking the idealized Liouvillian symmetry . In the current case, the symmetry-breaking occurs naturally through the term , which is associated with the asymmetric ester substituents. The true symmetry group of the coherent Hamiltonian is therefore given by , instead of the idealized group in eqn (10). The true Liouvillian group is given bywhich in the current case is simply for both molecular systems. In the case of ¹³C₂-AFD, the symmetry-breaking perturbation generates terms connecting the irreducible representations of , allowing experimental access to the LLS, through pulse sequences such as M2S, S2M, and relatives.^8,9,20 In the study described here, the symmetry-breaking perturbation induced by the asymmetric ester substituents is large enough to provide experimental access to the LLS, but sufficiently small that a theoretical description based on the idealized Liouvillian group provides a good approximation.

Following the notation used throughout the script where ¹H nuclei are labelled 1 and 4, and ¹³C nuclei are labelled 2 and 3, the theory developed could be also applied to the molecular AA‘XX’ 4-spin systems discussed for example in ref. 17, 18 and 25–27. The coherent spin Hamiltonian displays near-magnetic-equivalence, and the central spin pair is sufficiently remote from other participating spins that the symmetry group {E,(14),(23),(14)(23)} is a reasonable approximation for both the coherent and the fluctuating Hamiltonians. This group has four irreducible representations, leading to three non-trivial long-lived states. Two of these may be accessed without breaking the chemical equivalence.

5 Conclusions

We have demonstrated the existence of a long-lived nuclear spin state in a multiple spin system, far from the usual conditions of near magnetic equivalence. A state of this kind is only supported by molecules with centrosymmetric local molecular geometry, such as fumarate. A group theoretical description of the conditions leading to long-lived states in multiple spin systems has been given. This theoretical approach is likely to be useful for understanding a variety of related problems, such as long-lived states in chemically equivalent spin systems^{16–19,26,27} and long-lived states in rapidly rotating methyl groups.³¹

The existence of long-lived states in fumarate derivatives may also have practical relevance to hyperpolarized NMR studies of fumarate metabolism, in the context of in vivo cancer detection.^41,42 We are currently exploring the possibility of generating the long-lived population imbalance between the A_g and B_u manifolds directly through solid-state dynamic nuclear polarization (DNP), as has been demonstrated for singlet order in spin-pair systems.⁴³

Acknowledgements

This research was funded by EPSRC (UK) and the European Research Council. We are grateful to Salvatore Mamone, Jean-Nicolas Dumez, Michael C. D. Tayler and Kevin Brindle for discussions and suggestions, and to Ole G. Johannessen for instrumental support.

References

1. M. Carravetta, O. Johannessen and M. Levitt, Phys. Rev. Lett., 2004, 92, 153003.
2. M. Carravetta and M. H. Levitt, J. Am. Chem. Soc., 2004, 126, 6228–6229.
3. M. Carravetta and M. H. Levitt, J. Chem. Phys., 2005, 122, 214505.
4. R. Sarkar, P. R. Vasos and G. Bodenhausen, J. Am. Chem. Soc., 2007, 129, 328–334.
5. G. Pileio and M. H. Levitt, J. Magn. Reson., 2007, 187, 141–145.
6. G. Pileio, M. Carravetta, E. Hughes and M. H. Levitt, J. Am. Chem. Soc., 2008, 130, 12582–12583.
7. M. C. D. Tayler, S. Marie, A. Ganesan and M. H. Levitt, J. Am. Chem. Soc., 2010, 132, 8225–8227.
8. G. Pileio, M. Carravetta and M. H. Levitt, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 17135–17139.
9. M. C. D. Tayler and M. H. Levitt, Phys. Chem. Chem. Phys., 2011, 13, 5556–5560.
10. M. H. Levitt, Annu. Rev. Phys. Chem., 2012, 63, 89–105.
11. G. Pileio, J. T. Hill-Cousins, S. Mitchell, I. Kuprov, L. J. Brown, R. C. D. Brown and M. H. Levitt, J. Am. Chem. Soc., 2012, 134, 17494–17497.
12. G. Pileio, S. Bowen, C. Laustsen, M. C. D. Tayler, J. T. Hill-Cousins, L. J. Brown, R. C. D. Brown, J. H. Ardenkjaer-Larsen and M. H. Levitt, J. Am. Chem. Soc., 2013, 135, 5084–5088.
13. J.-N. Dumez, J. T. Hill-Cousins, R. C. D. Brown and G. Pileio, J. Magn. Reson., 2014, 246, 27–30.
14. A. K. Grant and E. Vinogradov, J. Magn. Reson., 2008, 193, 177–190.
15. W. S. Warren, E. Jenista, R. T. Branca and X. Chen, Science, 2009, 323, 1711–1714.
16. Y. Feng, R. M. Davis and W. S. Warren, Nat. Phys., 2012, 8, 831–837.
17. Y. Feng, T. Theis, T.-L. Wu, K. Claytor and W. S. Warren, J. Chem. Phys., 2014, 141, 134307.
18. K. Claytor, T. Theis, Y. Feng and W. Warren, J. Magn. Reson., 2014, 239, 81–86.
19. K. Claytor, T. Theis, Y. Feng, J. Yu, D. Gooden and W. S. Warren, J. Am. Chem. Soc., 2014, 136, 15118–15121.
20. S. J. DeVience, R. L. Walsworth and M. S. Rosen, Phys. Rev. Lett., 2013, 111, 173002.
21. N. Salvi, R. Buratto, A. Bornet, S. Ulzega, I. Rentero Rebollo, A. Angelini, C. Heinis and G. Bodenhausen, J. Am. Chem. Soc., 2012, 134, 11076–11079.
22. R. Buratto, D. Mammoli, E. Chiarparin, G. Williams and G. Bodenhausen, Angew. Chem., Int. Ed., 2014, 53, 11376–11380.
23. P. R. Vasos, A. Comment, R. Sarkar, P. Ahuja, S. Jannin, J.-P. Ansermet, J. a. Konter, P. Hautle, B. van den Brandt and G. Bodenhausen, Proc. Natl. Acad. Sci. U. S. A., 2009, 106, 18469–18473.
24. P. Ahuja, R. Sarkar, S. Jannin, P. R. Vasos and G. Bodenhausen, Chem. Commun., 2010, 46, 8192–8194.
25. L. Buljubasich, M. B. Franzoni, H. W. Spiess and K. Münnemann, J. Magn. Reson., 2012, 219, 33–40.
26. M. B. Franzoni, L. Buljubasich, H. W. Spiess and K. Münnemann, J. Am. Chem. Soc., 2012, 134, 10393–10396.
27. Y. Zhang, P. C. Soon, A. Jerschow and J. W. Canary, Angew. Chem., Int. Ed., 2014, 53, 3396–3399.
28. T. Theis, M. Truong, A. M. Coffey, E. Y. Chekmenev and W. S. Warren, J. Magn. Reson., 2014, 248C, 23–26.
29. M. Emondts, M. P. Ledbetter, S. Pustelny, T. Theis, B. Patton, J. W. Blanchard, M. C. Butler, D. Budker and A. Pines, Phys. Rev. Lett., 2014, 112, 077601.
30. H. J. Hogben, P. J. Hore and I. Kuprov, J. Magn. Reson., 2011, 211, 217–220.
31. B. Meier, J.-N. Dumez, G. Stevanato, J. T. Hill-Cousins, S. S. Roy, P. H kansson, S. Mamone, R. C. D. Brown, G. Pileio and M. H. Levitt, J. Am. Chem. Soc., 2013, 135, 18746–18749.
32. M. H. Levitt, Spin dynamics. Basics of Nuclear Magnetic Resonance, 2001.
33. G. Pileio and M. H. Levitt, J. Chem. Phys., 2009, 130, 214501.
34. G. Pileio, M. Carravetta and M. H. Levitt, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 17135–17139.
35. T. Theis, Y. Feng, T.-l. Wu and W. S. Warren, J. Chem. Phys., 2014, 140, 014201.
36. H. J. Hogben, M. Krzystyniak, G. T. P. Charnock, P. J. Hore and I. Kuprov, J. Magn. Reson., 2011, 208, 179–194.
37. I. Kuprov, J. Magn. Reson., 2011, 209, 31–38.
38. A. Shaka, J. Keeler and R. Freeman, J. Magn. Reson., 1983, 53, 313–340.
39. M. H. Levitt, Spin Dynamica, 2013.
40. P. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy, Academic Press, London, 2 edn, 2006, p. 721.
41. M. Karlsson, A. Gisselsson, S. K. Nelson, T. H. Witney, S. E. Bohndiek, G. Hansson, T. Peitersen, M. H. Lerche and K. M. Brindle, Proc. Natl. Acad. Sci. U. S. A., 2009, 106, 19801–19806.
42. K. M. Brindle, S. E. Bohndiek, F. a. Gallagher and M. I. Kettunen, Magn. Reson. Med., 2011, 66, 505–519.
43. M. C. D. Tayler, I. Marco-Rius, M. I. Kettunen, K. M. Brindle, M. H. Levitt and G. Pileio, J. Am. Chem. Soc., 2012, 134, 7668–7671.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c4cp05704j

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