Robust transformation of singlet order into heteronuclear magnetisation over an extended coupling range

Several important NMR procedures involve the conversion of nuclear singlet order into het- eronuclear magnetisation, including some experiments involving long-lived spin states and para-hydrogen-induced hyperpolarization. However most existing sequences suﬀer from a limited range of validity or a lack of robustness against experimental imperfections. We present a new radio-frequency scheme for the transformation of the singlet order of a chemically-equivalent homonuclear spin pair into the magnetisation of a heteronuclear coupling partner. The proposed radio-frequency (RF) scheme is called gS2hM (generalized singlet-to-heteronuclear magnetiza- tion) and has good compensation for common experimental errors such as RF and static ﬁeld inhomogeneities. The sequence retains its robustness for homonuclear spin pairs in the interme-diatecouplingregime,characterisedbythein-paircouplingbeingofthesameorderofmagnitudeasthediﬀerencebetweentheout-of-paircouplings.Thisisasubstantialimprovementtotheva-lidityrangeofexistingsequences.Analyticalsolutionsforthepulsesequenceparametersareprovided.Experimentalresultsareshownfortwotestcases.


Introduction
Nuclear magnetic resonance (NMR) techniques are often limited by low polarisation levels and the restricted lifetime of nuclear spin order. These limitations put strong restrictions on many powerful NMR techniques such as correlation spectroscopy, diagnostic imaging, diffusion imaging and kinetic analysis [1][2][3][4][5][6]. Some of these issues may be circumvented by utilising nuclear long-lived spin states and hyperpolarization techniques. Nuclear long-lived spin states are spin ensemble configurations with exceptionally long relaxation lifetimes, often exceeding longitudinal relaxation time constants 1 by large factors [7][8][9]. Hyperpolarization techniques generate strong perturbations of the thermal equilibrium population distribution of the spin ensemble, leading, in favourable cases, to NMR signals enhanced by many orders of magnitude [10][11][12][13][14][15]. Long-lived states may be combined with hyperpolarization techniques in order to generate strongly enhanced spin order that persists for a relatively long time [16][17][18][19][20][21][22][23][24][25][26].
The seminal example of a nuclear long-lived spin state is the nuclear singlet order (SO) of coupled spin-1/2 pairs of the same isotopic type [9,[27][28][29]. Nuclear singlet order indicates a population imbalance between the nuclear singlet and triplet states. Although singlet order is not associated with observable magnetization, several techniques have been developed for converting observable magnetization into singlet order, and back again [30][31][32][33][34][35][36][37][38][39][40]. These techniques include the M2S (magnetization-to-singlet) and S2M (singlet-to-magnetization) pulse sequences [32,33]. As originally designed, the M2S/S2M sequences operate best for spin systems in the near-equivalence regime, meaning that the members of the spin pair display a much stronger mutual coupling than the difference between their chemically-shifted resonance frequencies [41,42]. Recent improvements have extended the operational range of the M2S/S2M sequences, leading to the generalized M2S (gM2S) and general-coupling M2S (gcM2S) methods [38][39][40]. The gcM2S method has been used for the spectral editing of complex biomolecular NMR spectra [39].
It is also possible to convert singlet order of a spin pair into magnetization of a third spin, which is of a different isotopic type [43]. Several different pulse sequences have been developed for this task, especially in the context of parahydrogen induced polarisation (PHIP) [35,36,[44][45][46][47][48][49][50][51]. Although most such methods involve "two-channel" irradiation, i.e. the application of radiofrequency pulses at the resonance frequencies of both species, the requisite transformation may be achieved in a robust fashion by single-channel irradiation at the resonance frequency of the target spin alone [36,50,[52][53][54][55][56]. The relevant pulse sequence is called S2hM (singlet-to-heteronuclear magnetization) [50,57] and is closely Generalised S2hM sequence Figure 1: Schematic diagram of a AA ′ X system. The homonuclear spins shown in blue are chemically equivalent but magnetically inequivalent. They have identical resonance frequencies, but differing coupling constants to the heteronuclear spin shown in grey. The coupling regime depends on the angle ST , defined by equation 1. related, but not identical, to the homonuclear S2M sequence [32,33]. Like the S2M sequence, from which it is derived, the S2hM sequence is best adapted to the near-equivalence regime.
To facilitate the discussion, onsider a AA ′ X system of the type shown in figure 1. The homonuclear spins are assumed to be of the same isotopic type , such as 1 H for example, whereas the out-of-pair partner belongs to a different isotopic type , such as 13 C. The -spins are assumed to have identical resonance frequencies ∕2 , which is satisfied for chemically equivalent spins, or for chemically inequivalent spin pairs in low magnetic field [58][59][60][61][62]. The members of the homonuclear spin pair have different couplings ( 13 ≠ 23 ) to the out-of-pair partner. The difference in the heteronuclear couplings induces singlet-triplet mixing of the -spin pair. The coupling regime of the homonuclear pair may be characterised by a singlet-triplet mixing angle ST , defined as follows: where 12 represents the homonuclear coupling, and 13 , 23 describe the out-of-pair couplings to the heteronucleus. A mixing angle of ST = 0 corresponds to exact magnetic equivalence. The near-equivalence regime corresponds to small values of ST . A mixing angle of ST ∼ ∕4 indicates the intermediate coupling regime, while a mixing angle of ST ≃ ∕2 represents the case of strong inequivalence, often known as "weak coupling" [41,42]. The amplitude for the transformation of homonuclear singlet order into heteronuclear magnetization is denoted here by the symbol . An explicit definition of is given below. It has a maximum achievable value of = 1. The operational regime of a pulse sequence is conveniently represented by plotting numerical evaluations of against the mixing angle ST .
Plots of against ST are shown for three different pulse sequences in figure 2. One of the first pulse sequences proposed for this task was designed by Goldman and Jóhannesson [46]. Its performance is shown by the black line in figure 2(a). As seen in the plot, the Goldman-Jóhannesson sequence performs well for the intermediate-coupling regime, but its performance declines sharply in both the near-equivalence and strong inequivalence regimes. The performance of the S2hM sequence [50,57] is shown by the blue curve. For small mixing angles ST (the nearequivalence regime), the S2hM sequence is able to efficiently transform singlet polarisation into heteronuclear Zeeman polarisation with an amplitude close to the theoretical maximum. However, the performance of S2hM rapidly declines at larger mixing angles.
The pulse sequence described in the current paper is called gS2hM (generalized S2hM). Its performance is shown by the orange line in figure 2(a). For small mixing angles ST , the S2hM and gS2hM sequences both transforms singlet order into heteronuclear Zeeman polarisation with efficiencies close to the theoretical maximum. However, the operational regime of the gS2hM sequence is considerably wider than that of S2hM, covering a range of mixing angles spanning 0 • < ST ≲ 50 • , albeit with prominent dips in performance at certain ST values. However, as discussed below, these dips may be flattened out by modifying the echo numbers.
When comparing pulse sequences, it is also important to take into account their robustness with respect to common experimental imperfections. Figure 2(b) compares the performance of the three pulse sequences again, but this time in the presence of a distribution of static magnetic fields. A normal distribution of resonance offsets is simulated, with standard deviation equal to twice the characteristic frequency = 2 { 2 12 + 1 4 ( 13 − 23 ) 2 } 1∕2 . This quantity Generalised S2hM sequence  represents the magnitude of the relevant part of the spin Hamiltonian, independent of the angle ST , which varies across the plot. As may be seen in figure 2(b), the performance of the S2hM and gS2hM sequences is almost unchanged, while the performance of the Goldman-Jóhannesson sequence is strongly degraded. The response of the different pulse sequences to experimental imperfections is discussed in more detail below. The rest of this paper presents the theory and construction principles of gS2hM. Experimental results are shown which illustrate the performance of gS2hM in realistic contexts. A more thorough comparison of gS2hM with other pulse sequences, especially those used in the context of para-hydrogen induced polarization, is given in the discussion section.

Theory
The following theory was developed with the assistance of SpinDynamica software [64].

Transformation Amplitude
Consider a three-spin-1/2 system with spins (1, 2) being of isotopic type and spin (3) being of isotopic type . The state of the spin ensemble is described by a density operator . Following reference [50], the degree of -spin singlet order in the spin ensemble is denoted , and is given by: where the symbol ⟨ ← ← → ⟩ is called the operator amplitude of in , and is defined as follows: The singlet polarisation level operator is given by [50] = − 1 Similarly, the degree of -spin Zeeman polarization is denoted , and is given by: where the Zeeman polarisation level operator [50] is defined as follows: The singlet order and Zeeman polarisation are bounded, as shown below: The transformation amplitude of operator into operator by a unitary transformation is defined by The transformation amplitude under is constrained by the eigenvalue spectra of the two operators and . The corresponding bounds are known as the unitary constraints for the transformation of into [65,66].
Our main interest is the transformation amplitude of -spin singlet order into spin Zeeman polarisation which is bounded by −1 ≤ ≤ 1 [66]. This indicates that any amount of singlet order may in principle be fully converted into -spin Zeeman polarisation by coherent evolution under a pulse sequence.
The AA ′ spins are of isotopic type and the X spin is of isotopic type . The resonance frequency of the twospins is assumed to be identical so that both -spins display identical resonance offset behaviour described by a single resonance offset frequency Ω 0 = 0 − ref given in rad s −1 . Similarly, Ω 0 = 0 − ref represents the resonance offset frequency of spin in rad s −1 . The scalar coupling constants in Hz are denoted by . The Hamiltonian 0 may be decomposed as follows Here, Δ = 2 12 denotes the homonuclear coupling, Δ = 2 ( 13 − 23 ) denotes the difference between the out-of-pair and Σ = 2 ( 13 + 23 ) the sum of the out-of-pair scalar couplings (all in rad s −1 ). The terms and commute which may be used to show that the Hamiltonian takes no active role in the manipulation of singlet order. We will therefore focus on the Hamiltonian . The Hamiltonian may be separated into two independent parts acting on the and subspace of the spin, respectively = + , The operators and are the corresponding projectors onto the and subspace of spin [41] = 1 2 A slight shift of notation reinforces the algebraic similarities between the Hamiltonian of an AA ′ X system and an AA ′ spin system where we have defined Ω Δ = 1 2 Δ .
The terms and are identical to the Hamiltonians of a coupled spin-1/2 pair with Ω Δ taking the role of the (rotating-frame) resonance offset frequency [40,42]. Motivated by this observation we introduce the singlet-triplet mixing angle ST and an effective angular frequency : The mixing angle is defined in such a way that magnetically equivalent -spins display a mixing angle ST = 0 and strongly inequivalent -spins display a mixing angle ST ≃ ∕2. The Hamiltonians and take the form

Singlet-triplet evolution
Further analysis is conveniently carried out by a convenient choice of basis. Consider first a basis formed by nuclear singlet-triplet states for spin and a basis formed by the conventional Zeeman states for spin It proves convenient to add a multiple of the unity operator to the Hamiltonian : This is possible since the unity operator commutes with all other operators and its presence in the Hamiltonian has no effect on the spin dynamics. This allows the terms and to be expressed as follows: with being a single-transition operator along the Cartesian axis and 1 the identity on { , } [67,68].
For both̃ and̃ the non-trivial part induces transitions between the | | 0 ⟩ and | | 0 ⟩ states of the spins, whereas the identity part causes pure phase evolution. Within the single-transition operator formalism [67,68], the free evolution propagator 0 ( ) is given by where the individual propagators are The operator ( , ) represents a sequence of rotations around the ( , , ) axes, while Φ ( ) generates pure phase evolution through the angle :

Heteronuclear spin echoes
A detailed overview of the S2hM and gS2hM sequence are given in figure 3. Both sequences make heavy use of spin echo blocks of the type { − − }, where the -pulse is solely applied to the -spins. The propagator for a single echo event may be simplified by expressing the -spin -pulse generator in terms of single transition operators generated by the basis states of equation 22 The operator for a rotation of the -spin through an angle around the -axis may be written as follows: This may be combined with the free evolution propagator of equation 26 to give the following form for the -spin echo propagator: The effective echo propagators SE may be shown to induce fictitious rotations within their respective subspaces [40] with angles and given by For simplicity we suppress the explicit time-dependence of ( ) and ( ).
A physical interpretation of the angles and is as follows: The angle represents a tilt angle defining the orientation of the effective rotation axis in the -plane, with = 0 corresponding to rotation around the -axis. The angle represents the net rotation angle of the echo propagator.

The S2hM sequence
Extensive descriptions of the S2hM sequence may be found in references 50 and 57. Here we briefly highlight some key aspects illustrating the limitations of the S2hM approach. As shown in figure 3a, the S2hM sequence converts homonuclear singlet order into heteronuclear magnetisation by two sets of heteronuclear spin echoes synchronised with the effective angular frequency of the system, bracketing an additional delay and a central ∕2 pulse [50,57]. The S2hM echo delay 1 , and the phase evolution delay 2 , are given by Generalised S2hM sequence The loop number of the S2hM sequence depends on the mixing angle ST as follows: As discussed in reference 57, the derivation of equations 34 and 35 relies on small mixing angles ST ≪ 1, corresponding to the near-equivalence regime. Under this assumption the tilt angle of the spin echo propagator is given approximately by The desired tilt angle of = ∕2 may be generated even outside the regime of small ST by setting the echo evolution delay 1 as follows: which reduces to 1 = ∕(2 ) as ST tends towards zero. However, this equation does not admit any physical solutions for mixing angles ST ≥ ∕4. The S2hM sequence is therefore fundamentally limited to spin systems with mixing angles ST < ∕4. Figure 2 indicates that the performance of S2hM rapidly decreases as a function of ST , well before the fundamental limit of ST = ∕4.
Although the S2hM sequence converts -spin singlet order into -spin magnetization, the resulting magnetization is not distributed evenly amongst the -spin transitions. Indeed, as shown in references 50 and 57, the -spin magnetization is concentrated on the inner transitions of the -spin multiplet. The density operator transformation achieved by the S2hM pulse sequence may therefore be written: (38) where "IT" stands for "inner transitions". The operator for the selective -polarization of the inner transitions of the -spin multiplet is given by: The selective transfer of -spin singlet order into the central peak of the -spin multiplet is prominently visible in, for example, figure 1d of reference 50. Consider the detailed breakdown of the gS2hM sequence in figure 4. The spin density operators at the indicated time points are denoted 0 , 1 … 5 . The operation of the pulse sequence may be explained by expressing the density operators at the different time points in terms of single-transition operators [68,69], using the basis states given in equation 22. Assume an initial state of pure -spin singlet order. The density operator 0 may be written as follows: Assume that the final state 5 is given approximately by the operator ,IT , as defined in equation 39. This implies that the penultimate state before the final -spin ∕2 pulse should be given by where the operator for selective -magnetization on the inner -spin transitions is given by This operator may be written in terms of single-transition operators as follows: By comparing equations 40 and 43, it may be seen that the desired transformation of -spin singlet order into innertransition -spin magnetization is accomplished by a pulse sequence whose propagator leads to the following transformations: 1. The operator 56 should be inverted in sign; 2. The operator 12 should be preserved without sign change; 3. The states {|3⟩ , |4⟩ , |7⟩ , |8⟩} may each accumulate an arbitrary phase shift, but should not mix with other states.
Equations 26 and 30 show that intervals of free evolution, and -spin pulses with phase 0, both induce selective rotations in the {|1⟩ , |2⟩} and {|5⟩ , |6⟩} subspaces, and hence fulfil the last of the three conditions above. An appropriate pulse sequence may therefore be constructed from a sequence of evolution intervals and -spin x-pulses which invert the sign of the -operator for the {|5⟩ , |6⟩} subspace, at the same time as preserving the sign of the -operator for the {|1⟩ , |2⟩} subspace. The problem of pulse sequence design reduces to finding the combination of -spin x-pulses and evolution delays which implement these two simultaneous operator transformations, for as wide a range of ST values as possible. The gS2hM sequence accomplishes these transformations in four steps, which may be summarized as follows: where the bracketed numbers refer to the time points in figure 4. The individual steps in equation 44 are implemented as follows.
• Step 1. This is implemented by applying a series of heteronuclear spin echoes of the form { 1 − − 1 } whose effective rotation axis has a tilt angle of = ∕4, instead of the tilt angle of = ∕4 used in the S2hM sequence. A similar approach was used in the design of the gM2S sequence [40].
The tilt angle of = ∕4 is imposed by choosing the echo delay such that 1 = * 1 , where * 1 is defined as follows: * The effective frequency is given by equation 17. For this particular choice of echo delay 1 , the tilt angle is given by = ∕4, and the effective rotation angle of the spin echo element is equal to * ∶= ( * 1 ) = 2 sec −1 According to equation 31, the overall propagator of a single echo block { * 1 − − * 1 } is given by Such echo blocks may be constructed for mixing angles 0 < ST ≤ 67.5 • . A series of consecutive spin echoes of the form * 1 − − * 1 has the following propagator: Now assume that the echo number satisfies the following condition: Since is an integer and * is real, this condition may only be satisfied exactly for certain values of ST . Hence, the following analysis is only approximate, in general. Assuming that equation 49 is satisfied exactly, the following argument shows that the rotation operators ( * , ∕4) produce ( ∕2) rotations within their respective subspaces (within a phase factor): The overall propagator of the echo block may therefore be written as follows: with 34 and 78 generating unimportant phase evolution within subspaces {|3⟩ , |4⟩} and {|7⟩ , |8⟩}: The echo block propagator in equation 51 generates the desired transformations: [ SE ( * 1 )] 56 This generates the crucial sign difference between the operators in the {|1⟩ , |2⟩} and {|5⟩ , |6⟩} subspaces.
• Step 3. This step converts the single-transition -operators into single-transition -operators, without a sign change. This is accomplished, to a reasonable approximation, by a single spin echo of the form 2 − − 2 , with a suitable choice of the interval 2 .
According to equation 32, the spin density components 12 and 56 undergo identical evolution under the spin echo: The spin density operator components 12 and 56 are approximately aligned with the -axis by choosing 2 so as to minimise the function The function ( 2 ) is minimized at 2 = * 2 , where * 2 is defined as follows: * The four steps may be chained together as in equation 44, giving the desired transformation property of the complete gS2hM sequence:  Table 2 Chemical structures of 1-13 C-fumaric acid (trans-[1-13 C]-but-2-endioic acid) and the 1-13 C-maleate anion (cis-[1-13 C]-but-2-endioate), which are the dominant species in the preparations used for the experiments. The relevant nuclei and their spin-spin couplings are indicated. The magnitude of the J-couplings were determined by fitting experimental data with numerical simulations (see sections 4.1 and 4.2). All couplings are assumed to be positive, as predicted by Gaussian calculations [70]. The M2S parameters for 1-13 C-fumaric acid and 1-13 C-maleate were optimised empirically. which is associated with the finite value of the function ( 2 ), at the echo delay 2 = * 2 (equation 59). Despite these defects, the gS2hM sequence provides a reasonably accurate transformation of -spin singlet order into heteronuclear magnetization over a much broader range of ST values than the S2hM sequence, including the important intermediate coupling regime. In addition, the gS2hM sequence is robust with respect to common instrumental imperfections, as discussed below.
An alternative method for choosing the echo numbers in the gS2hM sequence is given in Appendix 1. This approach stabilises the performance of the gS2hM sequence and smooths out the worst sawtooth dips. However this method involves a more complicated algorithm for choosing the echo numbers and leads in general to a longer pulse sequence duration. The "basic" parameter choices, as summarized in table 1, are assumed for the remainder of this article.

Samples
NMR experiments were performed on 13 C-labelled fumaric acid (trans-[1-13 C]-but-2-endionate) and 13 C-labelled maleic acid (cis-[1-13 C]-but-2-endionate), both purchased from Sigma-Aldrich. A 8.3 mM 1-13 C-fumaric acid solution was prepared by dissolving 0.5 mg of the substance in 0.5 ml of acetoned 6 , and transferring to a conventional 5 mm NMR tube. Under these conditions, the dissolved fumaric acid remains predominantly in the undissociated form, as shown by the structure in table 2 (left column). The -coupling parameters were estimated by fitting simulations of the S2hM transfer to experimental data presented in the following sections, and are in good agreement with previously reported values [50,57]. The signs of the coupling constants were determined by Gaussian calculations [70]. The singlet-triplet mixing angle of ST ≈ 4.5 • places the 1 H spin pair of 1-13 C-fumaric acid in the near-equivalence regime.
A 6.6 mM sample of 1-13 C maleic acid was prepared by dissolving 0.54 mg of the substance in 0.7 ml of D 2 O, followed by transfer to a conventional 5 mm NMR tube. In aqueous solution, the labile protons of maleic acid are transferred to water forming maleate anions and hydronium cations. The structure of the 1-13 C-maleate anion is shown in table 2 (right column), together with the scalar coupling parameters, which were estimated by fitting simulations of the S2hM transfer to experimental data presented in the following sections. The signs were determined by Gaussian Generalised S2hM sequence Figure 5: Pulse sequence protocol. Singlet order preparation on the spin channel consists of a M2S block and a singlet order filtration element (T 00 ). Singlet order is converted into heteronuclear magnetisation through application of an S2hM/gS2hM block on the spin channel. The signal detection is initiated by application of a read out sequence. The singlet filter is given by a sequence of radio-frequency pulses and field gradient pulses. The phase angle "ma" indicates the magic angle ≃ 54.7 • [71,72]. The read-out sequence consists of a field gradient pulse for suppressing potential antiphase signal components followed by excitation of transverse magnetisation by a 90 pulse.
calculations [70]. The singlet-triplet mixing angle of ST ≈ 24.28 • places the 1 H spin pair of 1-13 C-maleate in the intermediate coupling regime.

Instrumental
All experiments were carried out on a 400 MHz Bruker Avance Neo system. The recycling delays between transients were set to 20 s and 30 s for fumaric and maleic acid respectively. The pulse nutation frequencies were ∼18.9 kHz for 1 H and ∼22.3 kHz for 13 C. The data from 1-13 C-fumaric acid was averaged over 2 transients with 131 k sampling points and a spectral width of 160 ppm. The data from 1-13 C-maleic acid was averaged over 8 transients with 131 k sampling points and a spectral width of 160 ppm.

Pulse Sequences
An overview of the experimental protocol is given in figure 5. The pulse sequence steps operate as follows: 1. The initial M2S block excites homonuclear singlet order starting from a thermally polarised state. The M2S pulse sequence parameters for the two compounds are summarised in table 2. A formal description of the M2S sequence in AA ′ X systems may be found in reference 57. 2. The T 00 block acts as a filter suppressing all NMR signals not passing through singlet order [71]. The field gradient parameters are summarised in table 3. 3. The S2hM or gS2hM block on the heteronuclear channel converts homonuclear singlet order into longitudinal heteronuclear magnetisation. 4. The read-out block on the heteronuclear channel consists of a field gradient pulse which suppresses unwanted single-quantum coherences, followed by excitation of transverse magnetization by a single ∕2 pulse, and acquisition of the NMR signal. The suppression of transverse magnetization before the final ∕2 pulse reduces the phase distortions of the resulting spectrum. The field gradient parameters are summarised in table 3.
All individual pulses for the M2S, S2hM and gS2hM sequences were replaced by their composite pulse counterparts, as follows: 180 pulses were replaced by 90 90+ 180 90 90+ sequences; 90 pulses were replaced by 180 97.2+ 360 291.5+ 180 97.2+ 90 sequences [73,74]. The free evolution delays were adjusted to account for the longer pulse element durations. For example, an echo delay was shortened by half the duration of the composite refocusing pulse element. The echo blocks for the S2hM and gS2hM sequences were additionally supercycled using a repeated (0,180 • ) phase scheme [75].

1-13 C-Fumaric acid
The experimental procedure in figure 5 was applied to 1-13 C-fumaric acid in order to study the relative performance of S2hM and gS2hM in a nearly-equivalent AA ′ X system. Figures 6(a,b) show the dependence of the 13 C signal amplitudes for 1-13 C-fumaric acid on the echo delays 1 and 2 . Experimental results (orange) are complemented by SpinDynamica simulations (blue) [64]. Experimental data sets and simulations were both normalized by dividing the data by the maximum amplitude in the series. The 1 -dependence displays a rather complex behaviour without a straightforward interpretation. The oscillatory behaviour in Figure 6(b) represents the rotational motion of the spin density operator components 12 and 56 around their respective -axes during step 2 of the gS2hM sequence. The good agreement between theory and experiment is gratifying. Figures 6(a,b) indicate optimal gS2hM performance for to their optimal values. The results shown in figure 6(c) indicate a slowly oscillating pattern. This is because the small Generalised S2hM sequence value of ST for 1-13 C-fumaric acid only allows the necessary spin order transformations to be built up in a cumulative fashion [33,57]. Figure 7 shows 13 C spectra of 1-13 C-fumaric acid solution, obtained by the protocol in figure 5, using the M2S sequence to generate 1 H singlet order, and either the S2hM or gS2hM sequences for the transformation of 1 H singlet order into 13 C magnetization. As expected, the performances of S2hM and gS2hM are very similar for this nearequivalent system. The central peak appears to be slightly less intense for the gS2hM spectrum, as compared to S2hM, but the outer peaks of the 13 C multiplet are more prominent for gS2hM and more in phase with the central transition. The ratio of integrated spectral amplitudes is given by gS2hM ∕ S2hM = 0.92.
The procedure given in Appendix 2 was used to obtain an experimental estimate of the transformation amplitude for the two sequences. This procedure is similar to that used to evaluate the S2hM sequence [50]. The estimates of the experimental transformation amplitudes are = 0.81 ± 0.05 for gS2hM and = 0.88 ± 0.06 for S2hM, in the case of 1-13 C-fumaric acid solution. The small deviations from the theoretical values are attributed to relaxation losses and experimental imperfections.

1-13 C-Maleic acid
The relative performance of the S2hM and gS2hM sequences for an intermediate-regime AA ′ X system was studied by applying the experimental protocol in figure 5 to the solution of 1-13 C-maleic acid. The 13 C signal amplitudes for 1-13 C-maleic acid solution as a function of the echo delays 1 and 2 are displayed in figures 8(a,b). The experimental results (orange) are complemented by SpinDynamica simulations (blue) [64]. Experimental data sets and simulations

Table 5
Theoretical and experimental pulse sequence parameters for 1-13 C-maleic acid solution. The notation is as in were both normalized by dividing the data by the maximum amplitude in the series. The normalised experimental data and simulations are in good qualitative agreement. Maximal 13 C signal amplitudes were observed for echo delays exp 1 = 16.6 ms and exp 2 = 7.5 ms. Table 5 shows good agreement between empirically optimised echo delays and theoretical predictions. The experimentally optimised echo delay exp 2 = 7.5 ms is roughly 2 ms shorter than the analytic prediction * 2 = 10.1 ms. The experimental optimum does agree with the simulations shown in Figure 8, so we attribute the discrepancy to the approximations used in the analytical theory. Figure 8c displays the dependence of the 13 C signal amplitude on the echo number for the optimised intervals exp 1 =16.6 ms and exp 2 =7.5 ms. In contrast to the 1-13 C-fumaric acid solution, the echo number dependence for 1-13 C-maleic acid solution displays an irregular behaviour. This is a direct consequence of the larger mixing angle displayed by 1-13 C-maleate. For large mixing angles increasing or decreasing the echo number even by a single unit strongly influences the performance of the sequence, as can be seen for echo numbers = 10 and = 11, for example. Figure 9 shows 13 C spectra of 1-13 C-maleic acid solution obtained via the experimental protocol in figure 5 using either the S2hM or gS2hM sequence for the final spin order transfer step. The echo delays and echo numbers for the S2hM and gS2hM sequences are given by the experimentally optimised values in table 5. The gS2hM sequence clearly produces a more intense spectrum than the S2hM sequence for this intermediate-regime spin system. Additionally the outer peaks of the 13 C multiplet for the gS2hM display similar intensities, whereas the outer peaks for the S2hM spectrum are slightly asymmetric. The ratio of integrated spectral amplitudes is given by gS2hM ∕ S2hM = 1.20.
An estimate of the experimental transformation amplitudes for the two sequences was obtained by the procedure given in Appendix 2. For 1-13 C-maleic acid solution the experimental transformation amplitudes are = 0.83 ± 0.05

Discussion
Many experimental schemes have been proposed for the conversion of spin-pair singlet order into magnetization of a third nucleus, especially in the context of parahydrogen-enhanced NMR [44-48, 50, 51, 56, 57, 63, 76-78]. It is not our intention to thoroughly review all of these schemes here. A review and comparison of many pulse sequences, suitable for application in the high-field NMR context, was given by Bär et al. in 2012 [63]. A more recent review has been given by Stevanato [43]. A brief summary of the current landscape is as follows: • The proposed pulse sequences divide into those that require resonant irradiation on both the -spin (usually 1 H) and -spin (often 13 C) channels.
• Most of the single-channel pulse sequences are very sensitive to deviations in the static magnetic field strength, or equivalently, to resonance offset effects. This undesirable feature applies, for example, to the single-channel versions of the sequences by Goldman and Jóhannesson [46], and by Kadlecek [47]. The same criticism applies to SLIC (Spin-Lock-Induced Crossing) type methods [34] in the heteronuclear context [50,[52][53][54], and  to the heteronuclear ADAPT (Alternating Delays Achieve Polarization Transfer) sequence [56]. The S2hM sequence [57] and the current gS2hM sequence are rare examples of single-channel pulse sequences which are insensitive to resonance offset and static field deviations.
• Many single-channel pulse sequences may be rendered insensitive to resonance offset and static field deviations by inserting pulses on both the -spin and the -spin channels. This applies for example to the Goldman-Jóhannesson [46] and Kadlecek [47] sequences. However, the improvement in robustness is achieved at the expense of an increase in complexity and the introduction of additional error sources.
• Schemes which involve smooth variations in the radio-frequency field strength and/or frequency, instead of discrete strong pulses, hold much promise in this area. One group of methods involves near-adiabatic transformations which may be highly robust [35,36]. However, such methods often require individual numerical optimizations of the shape for each experimental context. Analytical solutions are not generally available.
The issue of robustness with respect to static field deviations or resonance offsets is explored in figure 10. This shows simulations of against -spin resonance offset Ω for the S2hM and gS2hM sequence, as well as the singlechannel version of the Kadlecek(2b) sequence [47]. On resonance, the Kadlecek(2b) sequence converts singlet order into heteronuclear magnetisation with an efficiency close to the theoretical maximum. However, the heteronuclear magnetisation oscillates in sign as a function of resonance offset, while the performance of the S2hM and gS2hM sequences is almost unchanged. As mentioned above, the Kadlecek sequence may be compensated for resonance offsets by including numerous two-channel pulses [63], but this comes at the expense of increased complexity and provision of an additional rf channel. Figure 2 illustrates the same point, but this time for the single-channel version of the sequence by Goldman and Jóhannesson [46], using pulse sequence parameters given by the analytic solutions in reference 63. The black curve in figure 2(a) shows that the Goldman-Jóhannesson sequence performs well in the intermediate coupling regime ST ≃ ∕4. However, figure 2(b) shows that its performance is greatly degraded when a normal distribution of resonance offsets is included in the simulation, while the performances of the S2hM and gS2hM sequences are essentially unchanged.
The simulations in figure 2(b) include a normal distribution of resonance offsets with standard deviation = 2 , where = 2 { 2 12 + 1 4 ( 13 − 23 ) 2 } 1∕2 , in order to provide a comparable spread in offsets as the angle ST changes Generalised S2hM sequence across the plot. Consider for example 1-13 C-maleic acid. Table 2 indicates that maleic acid displays a mixing angle of ST ≃ 24 • and an effective frequency of ∕(2 ) ≃ 13.5 Hz. The corresponding standard deviation is given by ∕(2 ) ≃ 27 Hz. Conventional imaging magnets and low-field NMR setups however typically only reach static magnetic field homogeneities on the order of parts-per-million. At magnetic fields strengths ranging from 0.5 T to 3 T, 13 C resonance offset distributions with standard deviations of several tens of Hertz are therefore not unusual and robustness against offset errors is vital. Within this regime the gS2hM sequence has significant advantages over the other strong pulse methods, while at the same time preserving a relatively low degree of complexity.

Conflicts of interest
There are no conflicts to declare.

Appendices Appendix 1. Optimal echo numbers
As discussed in section 2.4, the gS2hM sequence displays dips in performance for certain mixing angles ST . These dips occur whenever the effective rotation angle lies approximately halfway between two whole numbers, where is an integer. For such cases, the "round" instruction in table 1 does not represent a good compromise. Instead it is possible to make use of the periodicity of the spin dynamics to define a family of approximate echo numbers * = round((2 + 1) ∕ * ) with ∈ {0, 1, 2, … }, characterised by the integer . The integer provides some additional flexibility and may be used to increase the performance of the gS2hM sequence. Figure 11(a) illustrates the performance of the gS2hM sequence for different echo number families * as a function of the mixing angle ST . Simulations have been performed for values of ranging from 0 to 9. The evolution delays 1 and 2 of the gS2hM sequence have been determined according to the analytic solutions summarised in table 1. For the sake of clarity, figure 11(a) only highlights transformation amplitudes for ∈ {0, 4, 9} given by the transparent curves. As can be seen neither of the values on their own provide a flat transformation amplitude profile for the full range of mixing angles. Instead individual families follow a saw-tooth pattern with occasional dips. For example, a large dip may be seen at ST ≃ 57 • for the = 0 family shown in red. As pointed out above, dips of this type occur whenever the ratio ∕ * lies close to the halfway point between two integers.
The black line in figure 11(a) shows a compilation of the best transformation amplitudes achieved within the considered set ∈ {0, … , 9}. As can be seen, the additional flexibility provided by the optional integer stabilises the performance of the gS2hM sequence, and the transformation amplitudes remain above 85% for all mixing angles 0 < ST < 67.5 • . However, one drawback of this procedure is that a numerical search is required to find the optimal number for a given mixing angle ST .
For practical applications there is a clear payoff between a small echo number * , accurate theoretical performance, and a long pulse sequence duration leading to relaxation losses. The total echo numbers corresponding to the optimal theoretical amplitudes are indicated in figure 11(b). For the most part the echo numbers stay well below * = 15. A selected few regions require echo numbers above > 20. These may still be generated by making use of equation 65, albeit with larger values of .

Appendix 2. Experimental estimation of transformation amplitudes
To isolate the S2hM and gS2hM transformation amplitudes we follow the line of reasoning presented in reference 50. For simplicity we make the following abbreviations The amplitude of a single pulse acquire spectrum on the spin channel is directly proportional to the spin equilibrium magnetisation The transformation amplitudes for M2S and gS2M are related as shown below [50] exp and may be used to express M2S in a more concise way The transformation amplitude for a single M2S step is then given by Similarly the amplitude of a single pulse acquire spectrum on the spin channel is given by with being an instrumental constant for detection on the spin channel. The transformation amplitude for the M2S-gS2hM (M2S-gS2hM) block is decomposed as follows Generalised S2hM sequence The transformation amplitude ⟨ gS2hM ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → ⟩ for spin singlet order to spin Zeeman order step may then be extracted as shown below Experimentally obtained transformation amplitudes for the instrumental setup discussed in section 3 are summarised in table 6.