Nonresonant SABRE provides an additional versatile hyperpolarization approach in magnetic resonance

Abstract

Hyperpolarization approaches in magnetic resonance overcome sensitivity limitations imposed by thermal magnetization and are integral to many modern applications. We begin with an overview of hyperpolarization. We then focus on one method, SABRE, which uses parahydrogen to form hydride transition metal complexes, followed by reversible exchange to polarize target molecules in solution. SABRE has produced large signal enhancements (up to 10⁴) on many different molecules, cheaply and rapidly. Here, we demonstrate an unconventional SABRE strategy, applicable to a wide range of targets, that produces field-independent spin order in target molecules which efficiently converts to magnetization. The observed signal is even independent of field direction, and hyperpolarization can be achieved on a lab bench with no field control. We show that this signal arises from creation of two-spin order in targets and discuss ways this strategy should expand SABRE generality and efficiency. We also show that, often, the standard assumption that low-field SABRE requires a singlet-only starting state leads to incorrect results.

INTRODUCTION

Over eight decades after the first observations of bulk spin signals (1, 2), nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) have evolved into powerful and versatile tools with broad applications in chemistry, physics, biology, materials science, and medicine. However, these applications are almost always limited by sensitivity. Sometimes, that limitation is subtle: In MRI, for example, achievable resolution is many orders of magnitude finer than the exciting wavelength but, in practice, limited by the magnetization in each individual volume element. Sometimes, the limitation is more direct: NMR spectra characterize organic molecules much more cleanly than mass spectra do, but metabolomics is dominated by mass spectroscopy because the required concentrations are much smaller. The low sensitivity at thermal equilibrium arises because the energy difference between spin states is almost always very small compared to the Boltzmann factor kT for accessible magnetic fields and room temperatures, which implies that the detected signal (proportional to the bulk magnetization) arises from only a small fraction of the spins.

This sensitivity challenge has, for decades, driven the development of methods to enhance nuclear magnetization/polarization (the terms are used interchangeably) far past its equilibrium value. This is commonly called “hyperpolarization,” and in many cases, the produced magnetization is 100 to 10,000 times the room temperature thermal value. This paper begins with a broad overview of multiple methods, including current and projected strengths and limitations. We then focus on one of the newer hyperpolarization methods, variants of SABRE (signal amplification by reversible exchange), and identify a substantial omission in the general understanding of this method. Correcting this omission should lead to signal enhancements and new experimental strategies.

The starting point for discussing hyperpolarization is the fractional magnetization, which is $\text{tanh}\left(\frac{\hslash \mathrm{\gamma }{B}_{\mathrm{o}}}{2\mathit{kT}}\right)$ at thermal equilibrium for a spin-¹/₂ system. For ¹H at 30 T and T = 300 K, this is ≈10⁻⁴; for lower fields or isotopes with lower γ (¹³C and ¹⁵N), it can be much worse (see Fig. 1). The most obvious way to enhance the magnetization is drastic cooling (millikelvin to kelvin) in a strong magnetic field [“brute-force” hyperpolarization (3)], but this is slow; achieving equilibrium at these temperatures (as measured by the spin-lattice relaxation time T₁) can take months (4). Instead, most hyperpolarization technologies share a common theme: transfer in nuclear spin order from something else, usually unpaired electron spin order. One way to get electron spin order is to introduce a paramagnetic species, sometimes by ionizing (5) or ultraviolet radiation (6), more often by doping the target of interest with a radical. Electrons have vastly larger fractional magnetization than nuclei (γ_e/γ_H ≈ 658), and magnetization is transferred by irradiating at or near electron resonance frequencies. These methods, first demonstrated in 1953 (7–9), are generally called dynamic nuclear polarization (DNP) (7, 10–15). Even if the total number of unpaired electrons is relatively small (millimolar concentration), electron T₁ values are commonly far shorter than their nuclear counterparts, so a single electron can be reused to polarize many nuclear spins.

Fig. 1. Sources of spin order for nuclear hyperpolarization.

This figure shows the usable spin order as a function of temperature at equilibrium in a 10-T magnet. For spins, this is the population difference between spin-up and spin-down. For parahydrogen (p-H₂), as discussed in the text, this is the difference between singlet population (the even J rotational states) and one-third of the triplet population (the odd J rotational states). The difference between the three spin cases arises solely because of the different gyromagnetic ratios; the p-H₂ differences arise because the rotational spacing effectively controls the spin populations.

Room temperature DNP, limited by γ_e/γ_H to a gain of several hundred, has recently become commercially available and is becoming popular in solid-state and protein NMR. It is worth noting that the many-decade delay between discovery and wide applications is partly because the original postulated mechanisms scale very unfavorably to modern high fields. Only with a better understanding of different mechanisms (which led, for example, to biradicals with electron line splittings comparable to nuclear spin frequencies) did high-field experiments become practical. Today, DNP can be done directly in the same magnet used for acquiring an NMR spectrum, although modern proton resonance frequencies above 1 GHz (hence electron resonance frequencies approaching 1 THz) present severe technical challenges, so often lower fields are used. A major frontier will be the transition from continuous-wave DNP to pulse methods, which can take advantage of the astonishing versatility of pulse sequences inherited from NMR and likely improve performance (16).

T₁ for electrons typically remains <<1 s even at low temperatures, and near-unity electron spin magnetization can be achieved in a modest field. This means DNP can create very large magnetization enhancement in cryogenic samples without the problems facing brute-force magnetization discussed earlier. In a typical application, a glassy matrix of the target molecule, solvent, and a free radical polarizing agent is cooled to about 1 K and irradiated for many minutes. The cold sample can then be rapidly dissolved into a body temperature (310 K) solution without depolarization (this is called dissolution DNP or d-DNP) (15). Hundreds of different molecules have been polarized (17–25), and commercial instruments have been available for over two decades. The first clinical DNP application (26), metabolism of ¹³C-labeled pyruvate to alanine or lactate, has been validated as a prostate cancer marker and is currently in clinical trials, and other molecules such as HCO₃⁻ also show promise (27–29). In common applications, samples with roughly 10 mmol of polarized pyruvate are produced in a few tens of minutes (30, 31). The most important limitation is that polarization is costly per sample because of the required drastic cooling, rapid heating, and long microwave irradiation, and further scalability is limited because millimeter-wave fields penetrate bulk samples poorly.

Nuclear spin polarization can also be produced by optical pumping, mediated by electron spin states. For example, in spin-exchange optical pumping (SEOP) (32–37), a circularly polarized laser polarizes electron spins in Rb vapor, which then collide to polarize nuclear spins in ³He, ⁸³Kr, or ¹²⁹Xe (38–43). The origins of this field are even older than DNP (32, 33), but again what was needed was decades of careful experimental and theoretical work, for example, to develop gas mixtures (including N₂) that increased the efficiency and high-power solid-state lasers at the right frequency. The outcome was that ¹²⁹Xe lung imaging has actually received Food and Drug Administration (FDA) approval (44), but no molecular targets are possible. It is also possible to pump nitrogen vacancy (NV) centers in diamond (45) or triplet states in organic molecules (photochemically induced DNP or photo-CIDNP) (46–51) and take advantage of intrinsic differences in relaxation rates to create electron (and ultimately nuclear) polarization. Although these methods are subjects of active research, the range of targets is somewhat restricted, and they are generally limited in producing large magnetization by the einstein (E) (the traditional name for a mole of photons). For visible light (λ = 600 nm), 1 E is 200,000 joules, and almost all of which is lost in the polarization process as the energy difference between nuclear spin states is trivial. For SEOP, where the sample is a mixture of gases, this problem is surmounted by massive powers (often >100 W) and long exposure times, leading to typical delivered volumes of 100 mmol of ¹²⁹Xe at >50% polarization in an hour (52). The other optical methods cannot use comparable powers without sample destruction and are restricted to much smaller numbers of excited spins.

The focus of this paper is on yet another source of spin order, parahydrogen [p-H₂; the singlet-state $S\equiv (\mid \mathrm{\alpha \beta }⟩-\mid \mathrm{\beta \alpha }⟩)/\sqrt{2}$ of gaseous H₂]. ¹H nuclei are fermions, thus antisymmetric with respect to exchange. Even rotational states are symmetric, so the ground rotational state J = 0 is purely singlet spin state. Odd J states are antisymmetric, so they pair with the “ortho” nuclear states $T_{+}\equiv \mid \mathrm{\alpha \alpha }⟩,T_0\equiv (\mid \mathrm{\alpha \beta }⟩+\mid \mathrm{\beta \alpha }⟩)/\sqrt{2},T_{-}\equiv \mid \mathrm{\beta \beta }⟩$ (o-H₂). The energy separation between J = 0 and J = 1 is E/k = 175 K, 10⁴ times the ¹H spin splitting in a large magnetic field, so p-H₂ dominates at equilibrium at low temperatures. Figure 1 shows the temperature dependence of the spin order (the total singlet population minus one-third of the total triplet population as the triplet states interconvert readily), which is far higher than even electron magnetization at moderately low temperatures and is still larger than ¹H nuclear polarization at room temperature. With a suitable catalyst to break the symmetry and allow singlet-triplet interconversion (53–56) (even rust works well), a system of 40 K can create liters per hour of 95% p-H₂. Industrial-scale preparation of p-H₂ has been extensively demonstrated as conversion to the para state is essential for stability of liquid H₂ rocket fuel.

However, the feature which makes p-H₂ truly interesting is that, at room temperature without catalyst or oxygen, excess p-H₂ can be stable for months (57). Thus, p-H₂ is an extraordinary “quantum reagent” in a metastable, pure spin state, which can be made in large quantities at high spin purity and low cost (58, 59). Rotational Raman shows that the ortho-para separation is retained for H₂ in solution (60), which is fundamental for the NMR methods below to work.

Until recently, p-H₂ was mostly used in NMR by addition across inequivalent carbon double or triple bonds, and then pulse sequences converted singlet order into nuclear polarization (61–66). This so-called parahydrogen–induced polarization (PHIP) initially lagged DNP in generality because each target molecule requires different optimization (often including a different catalyst), but again this is a rapidly evolving field. For example, pyruvate can now be hyperpolarized by synthesizing a “side arm” derivative with a double bond, adding p-H₂ across that bond, relaying magnetization, and cleaving the side arm (67). Target molecules are limited to those which have a double or triple bond, and optimization of hyperpolarization parameters is time and material intensive.

This paper focuses on a different method, SABRE (Fig. 2) introduced by Duckett and co-workers in 2009 (68–72), sometimes called nonhydrogenative PHIP. SABRE works by reversibly binding p-H₂ and target ligands (such as pyridine in Fig. 2) to an octahedral transition metal complex, usually iridium-based. Under the right circumstances (see Fig. 2), as discussed later, para order in the hydrides is converted to negative ligand (and positive hydride) magnetization. After both p-H₂ and ligand exchange (typically ligand exchange is much faster), the Ir atom can catalyze another transfer. SABRE directly polarizes ¹H on the ligand target, typically at a field around 6.5 mT. Methods for direct heteronuclear polarization such as SABRE-SHEATH (73) go by the general name X-SABRE and typically use fields around 0.6 μT, about 1/100 of Earth’s field. There are now at least a dozen such methods with a wide range of applications. This paper will show that those field constraints can be readily overcome.

Fig. 2. Comparison of different SABRE variants.

In all cases, p-H₂ and a target ligand (here pyridine) bind reversibly in the equatorial positions of an Ir complex. Typically, one axial position has an N-heterocyclic carbene, as illustrated here. In SABRE, four-bond couplings between the hydride and target ¹H atoms create nuclear spin polarization, commonly 1000 to 3000 times larger than what is observed at thermal equilibrium in even a large magnet (only one proton is highlighted for simplicity). In X-SABRE methods, the much larger two-bond couplings between the hydride and ¹⁵N create polarization. Here, we demonstrate a third mechanism, nonresonant SABRE, which directly creates heteronuclear two-spin order in the ligands. This has vastly different properties from the first two approaches.

An “ideal” hyperpolarization method would be simple, rapid, readily scalable, and completely general. For real methods, Table 1 summarizes the currently demonstrated (and projected) advantages and disadvantages. At the moment, DNP leads the other molecular methods in terms of quantity produced, but PHIP and SABRE are often within an order of magnitude of DNP and should in principle be more scalable. The cost difference between the p-H₂ systems and DNP systems is substantial (today, 100 to 1000 cost ratio). This vastly limits routine DNP applications, and it is hard to see that changing anytime soon due to continued reliance on high frequency microwave irradiation and temperature cycling. DNP is a very general polarization strategy with relatively few constraints as to target. A double/triple bond and appropriate catalyst (PHIP) or lone pairs (SABRE) are needed in the p-H₂ methods, but those appear widely. In addition, side-arm hydrogenation for PHIP (67) and relayed magnetization through exchangeable protons for SABRE (74) further expand the range of interesting molecular targets. NV-based approaches might eventually end up becoming quite general if the challenges with getting polarization out of the diamond sample can be overcome. PHIP and SABRE often produce nearly maximum polarization in seconds, whereas the other methods commonly use minutes to hours.

Table 1. Comparison of common methods for hyperpolarization.

This table summarizes the relative strengths and weaknesses of current hyperpolarization methods as described in the main text, with + indicating greater strength and − indicating greater weakness. Likely potential capabilities are in (parentheses). There is no single best method as the strengths are complementary, and technological developments are proceeding rapidly in all techniques.

HP method	DNP	SEOP	PHIP	SABRE	Photoinduced (NV)	CIDNP/photo-CIDNP
Scalability	++	++	+/(+++)	+/(+++)	−	−
Simplicity	− − −	−	++	++	+	+
Generality	+++	− − −	+	++	−/(++)	−
Speed	− −	−	+	+	−	−

Thus, there is no single best hyperpolarization method, but SABRE and X-SABRE (where we focus from now on) avoid all cooling, can hyperpolarize in seconds, are not as limited in sample volume as DNP, and have been applied to a wide variety of molecules. Today, many groups work on different aspects including methods that use shaped three-dimensional (3D) magnetic fields or work in a high-field magnet using radio frequency (rf) pulses (75–92). However, the “workhorse” experiment since the beginning has used a constant magnetic field and no rf irradiation. There are two basic regimes: the traditional SABRE case, where this constant field is matched to make the hydride scalar coupling comparable to the frequency difference between hydride protons and target protons (≈6.5 mT), and the SABRE-SHEATH case, where the field is matched to the frequency difference between hydride protons and target heteronuclei such as ¹⁵N and ¹³C [≈0.5 μT, achieved in a mu-metal shield (73)]. In both cases, the matching condition is generally justified as an example of level anticrossings, although this does not quite work in the limit of large couplings as has been detailed elsewhere (93). Still, as Fig. 3 shows, well-validated simulations (93, 94) reproduce the experimental field dependence well. SHEATH experiments are constrained by inconvenient field strengths that require a mu-metal shield to achieve, and the high-field experiments are very sensitive to experimental complications such as resonance frequency difference between the two hydride spins.

Fig. 3. ¹⁵N Signal enhancement (ϵ) for free ¹⁵N-pyridine in an Ir(H)₂(IMes)(py)₃ sample with continuous application of a static polarizing field.

The light red line represents a simulated Ir(H)₂(IMes)(py)₃ system (k_L = 16 s⁻¹, k_H = 2 s⁻¹, [catalyst]:[ligand] = 1:20, and 5-s evolution). Red data points show experimental magnetization enhancement relative to thermal magnetization measured at 1 T with error bars reflecting a 95% confidence interval (N = 3 for each field). Magnetization is measured as the amplitude of the FID immediately after a 90° pulse.

Here, we show (to be honest, to our surprise) that this picture is an oversimplification. For a wide range of target ligands that have both protons and heteronuclei (including the most heavily studied molecules in the field), there is a highly efficient path to two-spin order in the target, which can be quantitatively converted to observable magnetization competitive with SABRE and X-SABRE. Over the field regime between SABRE and X-SABRE, this produces magnetization independent of field strength and direction—in other words, inverting the field has no effect on the magnetization. For this reason, we call this effect nonresonant SABRE. Magnetization can even be produced by bubbling directly on a tabletop with no external magnet or shield, further simplifying the SABRE method. Although antiphase line shapes resulting from hyperpolarized heteronuclear two-spin order have been observed in ¹³C, ¹⁵N, and ¹⁹F in studies as early as 2009 (69, 73, 83, 95), previous conceptualizations of this signal treat the two-spin order as a field-dependent by-product of the hyperpolarization process, rather than a potential target. To our knowledge, the mechanisms which lead to this two-spin order have never been considered in detail, and strategically generating two-spin hyperpolarization may lead to many options for SABRE improvement in other regimes.

RESULTS

Experimental demonstration of nonresonant SABRE

We start by presenting experimental evidence for nonresonant SABRE. Figure 4 shows hyperpolarized free induction decays (FIDs) and spectra for 320 mM ¹⁵N-pyridine at two specific fields: one which matches the well-known SABRE-SHEATH condition and one that is about a factor of 100 away. As the field strength increases, the spectrum transitions smoothly from in-phase to antiphase and then remains constant over a wide variety of field strengths. Far from resonance, the FID is zero immediately after a 90 pulse (implying there is no one-spin z magnetization), but the FID grows in at later times. The first maximum occurs at time 1/(4J_NH) where J_NH is the coupling between the pyridine ¹⁵N and the ortho protons (≈−10 Hz).

Fig. 4. Experimental one-scan ¹⁵N signal from 320 mM ¹⁵N-pyridine after excitation at the peak of the SABRE-SHEATH resonance (0.55 μT) and a factor of 100 away from that resonance (50.0 μT).

The off-resonance signal is nearly zero immediately after the pulse, implying that no z magnetization was directly produced. However, substantial signal grows in at later times. The amplitude and phase of the FID of the off-resonance signal are nearly independent of field over the range of 5 to 500 μT, including inverting the field. ¹H or ¹⁵N π/2 pulses applied after the polarization time produce nearly equal ¹H or ¹⁵N fractional magnetization, respectively (see the Supplementary Materials).

Figure 5 shows this field dependence over about three orders of magnitude. As shown in the Supplementary Materials, detection of ¹H magnetization instead of ¹⁵N magnetization at fields far from the SHEATH condition reveals signal comparable to the ¹⁵N magnetization, with similar antiphase behavior.

Fig. 5. Experimental ¹⁵N-pyridine data showing ¹⁵N off-resonance signal and total effective magnetization as a function of polarizing field.

In (A), the signal at a time $t=1/\left(4J_{\mathit{NH}}\right)$ is plotted; this time corresponds to the maximum signal in the off-resonance case. In (B), it shows the “effective magnetization,” corresponding to the maximum value of the FID (which would also be the largest z magnetization available after an additional π/2 pulse). This is mathematically equivalent to acquiring the spectrum, optimizing the zero-order and first-order phase, and taking its integral. The gray bars indicate the 95% confidence interval for the mean out-of-phase signal at high (>25 μT) fields. Error bars indicate the 95% confidence interval for a given field condition (N = 3). Enhancement ϵ was calculated on the basis of the maximum intensity of a thermally polarized reference sample (see Materials and Methods).

We note several experimental observations which contribute to the analysis below:

First, we stopped the bubbling for a variable period before pulsing the sample after moving it from the polarizing field to the 1-T measurement magnet and saw much faster attenuation of the off-resonance signal (T₁ = 12.46 ± 0.26 s after polarization buildup at 50 μT) than was observed for the conventional SABRE-SHEATH signal (T₁ = 96.91 ± 5.86 s after polarization at 0.5 μT). When left in the polarizing field for variable time before transferring to the 1 T, the two signals showed fast and roughly comparable relaxation, with the 50-μT signal relaxing with a time constant of 7.82 ± 0.50 s and the 0.5-μT signal with 4.95 ± 0.10 s.

Second, we performed SABRE at three different points in the lab, without a shield, all with roughly 80-μT field aligned substantially away from the z axis of the magnet used to detect the magnetization, and all gave comparable antiphase signal.

Last, as seen in Fig. 5, inverting the direction of the field has no effect on the off-resonance FID.

Theoretical basis for nonresonant SABRE

In this section, we will focus on the simplest spin topology that reveals the nonresonant SABRE effect. This shows that the experiments above provide compelling evidence of a hitherto unappreciated SABRE mechanism which directly and efficiently produces two-spin z order in the target ligand.

We start by summarizing the theoretical approach of ref. (93) for SABRE-SHEATH as the modifications needed to understand nonresonant SABRE follow directly. Figure 6 illustrates a model SABRE system. The initial density matrix is assumed to have singlet order between the two hydride protons, derived from binding parahydrogen from solution, but otherwise is random

$${\hat{\mathrm{\rho }}}_0=(\frac{1}{4}\hat{1}-{\stackrel{\rightharpoonup }{I}}_1·{\stackrel{\rightharpoonup }{I}}_2)\otimes \hat{1}$$ (1)

Fig. 6. Simplified schematic to explain both resonant and nonresonant SABRE.

In the simplest case, p-H₂ and one NMR-active ligand are bound in the equatorial plane of the octahedral Ir complex. The Ir is commonly bound to a lone pair (e.g., ¹⁵N or a carbonyl group) on the ligand. In both SABRE and SABRE-SHEATH, this model includes one ligand spin L that is scalar coupled to hydrogen I_1. Because of this geometry, only the linear H-Ir-ligand geometry supports a scalar coupling. For nonresonant SABRE, we assume that L is ¹H and S is a heteronucleus on the ligand, ¹⁵N or ¹³C. In (A), it illustrates the topology for pyridine, one of the molecules studied extensively in this paper; however, all that matters in our analysis is the coupling network, shown more explicitly in (B).

The simplest SABRE case is a three-spin system, where the spin operator of the target nucleus is labeled as L (which is ¹H for SABRE and a heteronucleus, typically ¹⁵N or ¹³C, for SABRE-SHEATH) and the two hydride spins are ${\stackrel{\rightharpoonup }{I}}_1$ and ${\stackrel{\rightharpoonup }{I}}_2$. As shown in ref. (93), the three-spin system (which is equivalent to assuming only one of the equatorial positions is occupied by an NMR-active species) gives similar results at this level to the four-spin system (which assumes both positions are occupied), so for simplicity, we will stay with this model here.

The nuclear spin Hamiltonian has the form

$$\begin{array}{cc}\hfill \hat{\mathcal{H}}=& -\left(B_0\right)[{\mathrm{\gamma }}_H({\hat{I}}_{1z}+{\hat{I}}_{2z})+{\mathrm{\gamma }}_L{\hat{L}}_z]\hfill \\ & +2\mathrm{\pi }{J}_{\mathit{HH}}({\stackrel{\rightharpoonup }{I}}_1·{\stackrel{\rightharpoonup }{I}}_2)+2\mathrm{\pi }{J}_{\mathit{HL}}({\stackrel{\rightharpoonup }{I}}_1·\stackrel{\rightharpoonup }{L})\hfill \end{array}$$ (2)

here in natural units (ℏ = 1). For simplicity, we couple the ligand spin to only one of the hydride spins (spin I₁) as this is consistent with experimental observations that only the hydride directly opposite the ligand has an observable coupling. We can rearrange this Hamiltonian as

$$\begin{array}{cc}\hfill \hat{\mathcal{H}}=& -\left(B_0\right){\mathrm{\gamma }}_H({\hat{I}}_{1z}+{\hat{I}}_{2z}+{\hat{L}}_z)+(∆{\mathrm{\omega }}_{\mathit{HL}}){\hat{L}}_z\hfill \\ & +2\mathrm{\pi }{J}_{\mathit{HH}}({\stackrel{\rightharpoonup }{I}}_1·{\stackrel{\rightharpoonup }{I}}_2)+2\mathrm{\pi }{J}_{\mathit{HL}}({\stackrel{\rightharpoonup }{I}}_1·\stackrel{\rightharpoonup }{L})\hfill \end{array}$$ (3)

in which $∆{\mathrm{\omega }}_{\mathit{HL}}=B_0({\mathrm{\gamma }}_H-{\mathrm{\gamma }}_L)$. The first term commutes with the rest of the Hamiltonian (and with the operators we are going to prepare), so we drop it in what follows for simplicity; this is equivalent to viewing the system in the ¹H hydride rotating frame but without explicit assumptions on the field strength. We will thus use the reduced Hamiltonian

$${\hat{\mathcal{H}}}^{\prime }=(∆{\mathrm{\omega }}_{\mathit{HL}}){\hat{L}}_z+2\mathrm{\pi }{J}_{\mathit{HH}}({\stackrel{\rightharpoonup }{I}}_1·{\stackrel{\rightharpoonup }{I}}_2)+2\mathrm{\pi }{J}_{\mathit{HL}}({\stackrel{\rightharpoonup }{I}}_1·\stackrel{\rightharpoonup }{L})$$ (4)

The coherence transfer pathway leading from singlet order on the hydrides to magnetization on the target nucleus goes through four commutators starting with the initial density matrix in Eq. 1. Thus, a standard Taylor series expansion for the time evolution of the density matrix at $t=0$ $(d\hat{\mathrm{\rho }}/\mathit{dt}=i[{\hat{\mathrm{\rho }}}_{\mathrm{o}},\hat{\mathcal{H}}];d{\hat{\mathrm{\rho }}}^n/{\mathit{dt}}^n=i[d{\hat{\mathrm{\rho }}}^{n-1}/{\mathit{dt}}^{n-1},\hat{\mathcal{H}}])$ shows that the first term in the density matrix which has ligand magnetization ${\hat{L}}_z$ appears in the fourth derivative. Pathways that produce this term originate from ${\hat{I}}_{1z}{\hat{I}}_{2z}$ (Eqs. 5 to 8) or from ${\hat{I}}_{1x}{\hat{I}}_{2x}$ and ${\hat{I}}_{1y}{\hat{I}}_{2y}$ (Eqs. 9 to 12)

$$\begin{array}{cc}\hfill {\hat{I}}_{1z}{\hat{I}}_{2z}\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\to }& -i{\hat{I}}_{1x}{\hat{I}}_{2z}{\hat{L}}_y\stackrel{J_{\mathit{HH}}\left({\hat{I}}_{1z}{\hat{I}}_{2z}\right)}{\to }\hfill \\ & -{\hat{I}}_{1y}{\hat{L}}_y\stackrel{\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HL}}{\hat{L}}_z}{\to }-i{\hat{I}}_{1y}{\hat{L}}_x\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\to }{\hat{L}}_z\hfill \end{array}$$ (5)

$$\begin{array}{c}{\hat{I}}_{1z}{\hat{I}}_{2z}\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\to }-i{\hat{I}}_{1x}{\hat{I}}_{2z}{\hat{L}}_y\stackrel{\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HL}}{\hat{L}}_z}{\to }{\hat{I}}_{1x}{\hat{I}}_{2z}{\hat{L}}_x\stackrel{J_{\mathit{HH}}\left({\hat{I}}_{1z}{\hat{I}}_{2z}\right)}{\to }\\ -i{\hat{I}}_{1y}{\hat{L}}_x\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\to }{\hat{L}}_z\end{array}$$ (6)

$$\begin{array}{c}{\hat{I}}_{1z}{\hat{I}}_{2z}\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\to }i{\hat{I}}_{1y}{\hat{I}}_{2z}{\hat{L}}_x\stackrel{J_{\mathit{HH}}\left({\hat{I}}_{1z}{\hat{I}}_{2z}\right)}{\to }\\ -{\hat{I}}_{1x}{\hat{L}}_x\stackrel{\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HL}}{\hat{L}}_z}{\to }i{\hat{I}}_{1x}{\hat{L}}_y\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\to }{\hat{L}}_z\end{array}$$ (7)

$$\begin{array}{c}{\hat{I}}_{1z}{\hat{I}}_{2z}\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\to }i{\hat{I}}_{1y}{\hat{I}}_{2z}{\hat{L}}_x\stackrel{\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HL}}{\hat{L}}_z}{\to }{\hat{I}}_{1y}{\hat{I}}_{2z}{\hat{L}}_y\stackrel{J_{\mathit{HH}}\left({\hat{I}}_{1z}{\hat{I}}_{2z}\right)}{\to }\\ i{\hat{I}}_{1x}{\hat{L}}_y\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\to }{\hat{L}}_z\end{array}$$ (8)

$$\begin{array}{c}{\hat{I}}_{1x}{\hat{I}}_{2x}\underrightarrow{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}i{\hat{I}}_{1z}{\hat{I}}_{2x}{\hat{L}}_y\underrightarrow{J_{\mathit{HH}}\left({\hat{I}}_{1x}{\hat{I}}_{2x}\right)}\\ -{\hat{I}}_{1y}{\hat{L}}_y\underrightarrow{\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HL}}{\hat{L}}_z}-i{\hat{I}}_{1y}{\hat{L}}_x\underrightarrow{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\hat{L}}_z\end{array}$$ (9)

$$\begin{array}{c}{\hat{I}}_{1x}{\hat{I}}_{2x}\underrightarrow{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}i{\hat{I}}_{1z}{\hat{I}}_{2x}{\hat{L}}_y\underrightarrow{\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HL}}{\hat{L}}_z}\hfill \\ -{\hat{I}}_{1z}{\hat{I}}_{2x}{\hat{L}}_x\underrightarrow{J_{\mathit{HH}}\left({\hat{I}}_{1x}{\hat{I}}_{2x}\right)}-i{\hat{I}}_{1y}{\hat{L}}_x\underrightarrow{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\hat{L}}_z\hfill \end{array}$$ (10)

$$\begin{array}{c}{\hat{I}}_{1y}{\hat{I}}_{2y}\underrightarrow{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}-i{\hat{I}}_{1z}{\hat{I}}_{2y}{\hat{L}}_x\underrightarrow{J_{\mathit{HH}}\left({\hat{I}}_{1y}{\hat{I}}_{2y}\right)}\hfill \\ -{\hat{I}}_{1x}{\hat{L}}_x\underrightarrow{\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HL}}{\hat{L}}_z}i{\hat{I}}_{1x}{\hat{L}}_y\underrightarrow{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\hat{L}}_z\hfill \end{array}$$ (11)

$$\begin{array}{c}{\hat{I}}_{1y}{\hat{I}}_{2y}\underrightarrow{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}-i{\hat{I}}_{1z}{\hat{I}}_{2\mathrm{y}}{\hat{L}}_x\underrightarrow{\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HL}}{\hat{L}}_z}\hfill \\ -{\hat{I}}_{1z}{\hat{I}}_{2y}{\hat{L}}_y\underrightarrow{J_{\mathit{HH}}\left({\hat{I}}_{1y}{\hat{I}}_{2y}\right)}i{\hat{I}}_{1x}{\hat{L}}_y\underrightarrow{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\hat{L}}_z\hfill \end{array}$$ (12)

These pathways result in direct magnetization on the target nucleus ${\hat{L}}_z$ with a t⁴ dependence (93)

$$\stackrel{....}{\mathrm{\rho }}\frac{t^4}{4!}=8{\mathrm{\pi }}^3{J}_{\mathit{HL}}^2\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HL}}{J}_{\mathit{HH}}{\frac{t}{4!}}^4\left(2{\hat{L}}_z\right)+\dots$$ (13)

This approach shows why a nonzero external magnetic field is necessary. It comes into this derivation in the Δω_HL term by generating a phase shift that is needed to create the final magnetization. Mathematically, this happens because it creates a resonance frequency difference between the hydride and the ligand. Note that in every case in Eqs. 5 to 12, the “flip-flop” component (xx+yy) of the coupling J_HL appears as the first and fourth commutator, with the Δω_HL and J_HH terms appearing in between (in either order, as they commute with each other).

The expansion above shows how it is possible to create ligand magnetization from hydride singlet order and predicts the short-time dependence. It does not show the optimal field, which was originally found by invoking level anticrossing arguments. However, as ref. (93) shows, this prediction generally works when the J_HH coupling dominates (as in SABRE) but can be off by as much as an order of magnitude in the SABRE-SHEATH case. It also does not show the ultimate limit of the magnetization transfer under realistic experimental conditions, where polarization is typically allowed to build up for many seconds because, in that case, a typical catalyst complex has undergone 100 or more ligand exchanges. However, well-validated numerical calculations starting from the equilibrium density matrix agree with Eq. 13 and agree with experimental data (Fig. 3). Figure 7 shows the results of such a calculation and validates the t⁴ prediction for the most important one-spin and two-spin operators. Complementary simulations that show the evolution of all major one-spin and two-spin operators on the bound species in the absence of exchange are shown in fig. S6.

Fig. 7. Expectation values of ${\hat{\mathit{S}}}_{\mathit{z}}$, ${\hat{\mathit{L}}}_{\mathit{z}}$, and ${\hat{\mathit{L}}}_{\mathit{z}}{\hat{\mathit{S}}}_{\mathit{z}}$ on both bound and free ¹⁵N-pyridine during SABRE buildup.

In (A), buildup is simulated at 0.5 μT,while (B and C) show buildup at 50 μT. The vertical gray line indicates the average lifetime of a bound complex in this system, roughly 62.5 ms. At 0.5 μT, ${\hat{S}}_z$ buildup outcompetes ${\hat{L}}_z{\hat{S}}_z$ by roughly two orders of magnitude at long times. At 50 μT, $\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HS}}$ is about 100 times larger than the largest coupling in the system, so ${\hat{S}}_z$ buildup is not observed at long times. This allows ${\hat{L}}_z{\hat{S}}_z$ to dominate. In plot (B), the nonsecular terms of the J couplings to ¹⁵N are omitted to clarify the t⁴ time dependence of ${\hat{L}}_z{\hat{S}}_z$ buildup for the bound operators; the free spin operators in solution rise initially as t⁵ because an additional exchange is needed to create them. In (C), it includes all couplings to ¹⁵N.

We now turn to the nonresonant SABRE case. The nucleus L is now assumed to be a proton in the ligand, and we now add one more nucleus S, which is assumed to be a heteronucleus on the ligand. This adds the terms in the square bracket below

$$\begin{array}{c}{\hat{\mathcal{H}}}^{\prime }=2\mathrm{\pi }{J}_{\mathit{HH}}{\stackrel{\rightharpoonup }{I}}_1·{\stackrel{\rightharpoonup }{I}}_2-\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HL}}{\hat{L}}_z+2\mathrm{\pi }{J}_{\mathit{HL}}({\stackrel{\rightharpoonup }{I}}_1·\stackrel{\rightharpoonup }{L})\hfill \\ +\left[2\mathrm{\pi }{J}_{HS}({\hat{I}}_{1x}{\hat{S}}_x+{\hat{I}}_{1y}{\hat{S}}_y)+2\mathrm{\pi }{J}_{\mathit{HS}}{\hat{I}}_{1z}{\hat{S}}_z+2\mathrm{\pi }{J}_{\mathit{LS}}({\hat{S}}_x{\hat{L}}_x+{\hat{S}}_y{\hat{L}}_y)\right.\hfill \\ \hfill \left.+2\mathrm{\pi }{J}_{\mathit{LS}}{\hat{S}}_z{\hat{L}}_z-\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HS}}{\hat{S}}_z\right]\end{array}$$ (14)

For simplicity, we will begin by considering a field regime where the resonance frequency difference between the hydride and ligand protons Δω_HL can be ignored, and the resonance frequency differences between the protons and heteronucleus are large enough to truncate the nonsecular terms in those couplings, $2\mathrm{\pi }{J}_{\mathit{HS}}({\hat{I}}_{1x}{\hat{S}}_x+{\hat{I}}_{1y}{\hat{S}}_y)$ and $2\mathrm{\pi }{J}_{\mathit{LS}}({\hat{S}}_x{\hat{L}}_x+{\hat{S}}_y{\hat{L}}_y)$. This would imply fields on the order of 10 μT to 1 mT. In other field regimes, there will be additional competing terms. This leaves us with

$$\begin{array}{cc}\hfill {\hat{\mathcal{H}}}^{\prime }=& 2\mathrm{\pi }{J}_{\mathit{HH}}{\stackrel{\rightharpoonup }{I}}_1·{\stackrel{\rightharpoonup }{I}}_2+2\mathrm{\pi }{J}_{HL}({\hat{I}}_{1x}{\hat{L}}_x+{\hat{I}}_{1y}{\hat{L}}_y)\hfill \\ & +2\mathrm{\pi }{J}_{\mathit{HL}}{\hat{I}}_{1z}{\hat{L}}_z-\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HS}}{\hat{S}}_z+2\mathrm{\pi }{J}_{\mathit{HS}}{\hat{I}}_{1z}{\hat{S}}_z+2\mathrm{\pi }{J}_{\mathit{LS}}{\hat{L}}_z{\hat{S}}_z\hfill \end{array}$$ (15)

We now use Eq. 15 with the initial density matrix in Eq. 1 and do a Taylor expansion as before for the evolution. Equations 16 to 23 illustrate eight of the pathways that lead, in the fourth order, to two-spin order ${\hat{L}}_z{\hat{S}}_z$. It is important to note that this order lies in the ligand itself and thus survives dissociation of the ligand from the transition metal complex

$$\begin{array}{c}{\hat{I}}_{1z}{\hat{I}}_{2z}\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\to }i{\hat{I}}_{1y}{\hat{I}}_{2z}{\hat{L}}_x\stackrel{J_{\mathit{HH}}\left({\hat{I}}_{1z}{\hat{I}}_{2z}\right)}{\to }\\ -{\hat{I}}_{1x}{\hat{L}}_x\stackrel{J_{\mathit{LS}}\left({\hat{L}}_z{\hat{S}}_z\right)}{\to }i{\hat{I}}_{1x}{\hat{L}}_y{\hat{S}}_z\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\to }{\hat{L}}_z{\hat{S}}_z\end{array}$$ (16)

$$\begin{array}{c}{\hat{I}}_{1z}{\hat{I}}_{2z}\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\to }i{\hat{I}}_{1y}{\hat{I}}_{2z}{\hat{L}}_x\stackrel{J_{\mathit{HH}}\left({\hat{I}}_{1z}{\hat{I}}_{2z}\right)}{\to }\\ -{\hat{I}}_{1x}{\hat{L}}_x\stackrel{J_{\mathit{HS}}\left({\hat{I}}_z{\hat{S}}_z\right)}{\to }i{\hat{I}}_{1y}{\hat{L}}_x{\hat{S}}_z\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\to }-{\hat{L}}_z{\hat{S}}_z\end{array}$$ (17)

$$\begin{array}{c}{\hat{I}}_{1z}{\hat{I}}_{2z}\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\to }i{\hat{I}}_{1y}{\hat{I}}_{2z}{\hat{L}}_x\stackrel{J_{\mathit{LS}}\left({\hat{L}}_z{\hat{S}}_z\right)}{\to }{\hat{I}}_{1y}{\hat{I}}_{2z}{\hat{L}}_y{\hat{S}}_z\\ \stackrel{J_{\mathit{HH}}\left({\hat{I}}_{1z}{\hat{I}}_{2z}\right)}{\to }i{\hat{I}}_{1x}{\hat{L}}_y{\hat{S}}_z\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\to }{\hat{L}}_z{\hat{S}}_z\end{array}$$ (18)

$$\begin{array}{c}{\hat{I}}_{1z}{\hat{I}}_{2z}\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\to }i{\hat{I}}_{1y}{\hat{I}}_{2z}{\hat{L}}_x\stackrel{J_{\mathit{HS}}\left({\hat{I}}_{1z}{\hat{S}}_z\right)}{\to }\\ -{\hat{I}}_{1x}{\hat{I}}_{2z}{\hat{L}}_x{\hat{S}}_z\stackrel{J_{\mathit{HH}}\left({\hat{I}}_{1z}{\hat{I}}_{2z}\right)}{\to }i{\hat{I}}_{1y}{\hat{L}}_x{\hat{S}}_z\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\to }-{\hat{L}}_z{\hat{S}}_z\end{array}$$ (19)

$$\begin{array}{cc}\hfill {\hat{I}}_{1z}{\hat{I}}_{2z}\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\to }& -i{\hat{I}}_{1x}{\hat{I}}_{2z}{\hat{L}}_y\stackrel{J_{\mathit{HH}}\left({\hat{I}}_{1z}{\hat{I}}_{2z}\right)}{\to }\hfill \\ & -{\hat{I}}_{1y}{\hat{L}}_y\stackrel{J_{\mathrm{L}S}\left({\hat{L}}_z{\hat{S}}_z\right)}{\to }-i{\hat{I}}_{1y}{\hat{L}}_x{\hat{S}}_z\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\to }{\hat{L}}_z{\hat{S}}_z\hfill \end{array}$$ (20)

$$\begin{array}{cc}\hfill {\hat{I}}_{1z}{\hat{I}}_{2z}\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\to }& -i{\hat{I}}_{1x}{\hat{I}}_{2z}{\hat{L}}_y\stackrel{J_{\mathit{HH}}\left({\hat{I}}_{1z}{\hat{I}}_{2z}\right)}{\to }\hfill \\ & -{\hat{I}}_{1y}{\hat{L}}_y\stackrel{J_{\mathit{HS}}\left({\hat{I}}_z{\hat{S}}_z\right)}{\to }-i{\hat{I}}_{1x}{\hat{L}}_y{\hat{S}}_z\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\to }-{\hat{L}}_z{\hat{S}}_z\hfill \end{array}$$ (21)

$$\begin{array}{cc}\hfill {\hat{I}}_{1z}{\hat{I}}_{2z}\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\to }& -i{\hat{I}}_{1x}{\hat{I}}_{2z}{\hat{L}}_y\stackrel{J_{\mathit{LS}}\left({\hat{L}}_z{\hat{S}}_z\right)}{\to }{\hat{I}}_{1x}{\hat{I}}_{2z}{\hat{L}}_x{\hat{S}}_z\stackrel{J_{\mathit{HH}}\left({\hat{I}}_{1z}{\hat{I}}_{2z}\right)}{\to }\hfill \\ & -i{\hat{I}}_{1y}{\hat{L}}_x{\hat{S}}_z\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\to }{\hat{L}}_z{\hat{S}}_z\hfill \end{array}$$ (22)

$$\begin{array}{cc}\hfill {\hat{I}}_{1z}{\hat{I}}_{2z}\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1y}{\hat{L}}_y\right)}{\to }& -i{\hat{I}}_{1x}{\hat{I}}_{2z}{\hat{L}}_y\stackrel{J_{\mathit{HS}}\left({\hat{I}}_{1z}{\hat{S}}_z\right)}{\to }\hfill \\ & -{\hat{I}}_{1y}{\hat{I}}_{2z}{\hat{L}}_y{\hat{S}}_z\stackrel{J_{\mathit{HH}}\left({\hat{I}}_{1z}{\hat{I}}_{2z}\right)}{\to }-i{\hat{I}}_{1x}{\hat{L}}_y{\hat{S}}_z\stackrel{J_{\mathit{HL}}\left({\hat{I}}_{1x}{\hat{L}}_x\right)}{\to }-{\hat{L}}_z{\hat{S}}_z\hfill \end{array}$$ (23)

As shown explicitly in the Supplementary Materials, there are a total of 20 pathways which proceed from ${\stackrel{\rightharpoonup }{I}}_1·{\stackrel{\rightharpoonup }{I}}_2$ (the eight above which proceed from ${\hat{I}}_{1z}{\hat{I}}_{2z}$, plus six each from ${\hat{I}}_{1x}{\hat{I}}_{2x}$ and ${\hat{I}}_{1y}{\hat{I}}_{2y}$). In all of the pathways, the first step is creation of three-spin order with both hydride protons and one of the ligand nuclei. The net effect of the next two steps is to transform this into three-spin order with only the coupled hydride proton and two of the ligand nuclei, with either a two-spin or a four-spin state as the intermediate. The last step strips the hydride term. All of the many terms that can be written out are proportional to $J_{\mathit{HH}}{J}_{\mathit{HL}}^2$, but there are pathways proportional to either J_HS or J_LS.

All of these pathways would have been dismissed in past work because they do not result in observable ${\hat{L}}_z$ or ${\hat{S}}_z$. However, the critical point we make here is that the two-spin ${\hat{L}}_z{\hat{S}}_z$ order in the ligand itself can efficiently create magnetization. After a standard 90 pulse, it becomes antiphase magnetization (${\hat{L}}_z{\hat{S}}_y$ or ${\hat{L}}_y{\hat{S}}_z$), which converts completely into in-phase transverse magnetization ${\hat{S}}_x$ or ${\hat{L}}_x$, respectively, after a time 1/(2J_LS)

$$\begin{array}{c}{\hat{L}}_z{\hat{S}}_z\stackrel{90_S,\text{phase} x}{\to }{\hat{L}}_z{\hat{S}}_y\stackrel{ \text{free evolution} }{\to }{\hat{L}}_z{\hat{S}}_y\text{cos}\left(\mathrm{\pi }{J}_{\mathit{LS}}t\right)-\frac{1}{2}{\hat{S}}_x\text{sin}\left(\mathrm{\pi }{J}_{\mathit{LS}}t\right)\\ \stackrel{90_L,\text{phase} x}{\to }{\hat{L}}_y{\hat{S}}_z\stackrel{ \text{free evolution} }{\to }{\hat{L}}_y{\hat{S}}_z\text{cos}\left(\mathrm{\pi }{J}_{\mathit{LS}}t\right)-\frac{1}{2}{\hat{L}}_x\text{sin}\left(\mathrm{\pi }{J}_{\mathit{LS}}t\right)\end{array}$$ (24)

Thus, it does not matter at all that it takes a fifth coupling to convert to observable magnetization as long as J_LS is not vanishingly small. Note that we are still working with a four-spin system, which is equivalent to having a ligand with AX geometry. Evolution of an AX₂ model, which is a better match for pyridine, has a different time dependence and is described later in the text.

A qualitative way to understand nonresonant SABRE is that the newly added couplings $2\mathrm{\pi }{J}_{\mathit{HS}}{\hat{I}}_{1z}{\hat{S}}_z+2\mathrm{\pi }{J}_{\mathit{LS}}{\hat{L}}_z{\hat{S}}_z$ perturb the local environment of I₁ and L depending on the spin state of S. When spin S is up, a positive J coupling will result in a larger local field experienced by L (or I). So long as J_HS ≠ J_LS, this creates a resonance frequency difference between spin I₁ and spin L, just as the term Δω_HL did in the original SABRE derivation. Thus, the coupling plays the same role as the magnetic field did in the SABRE case. If J_HS-J_LS > 0 and spin S is up, the hydride spin is at the higher frequency; if spin S is down, the hydride spin is at the lower frequency and the produced L magnetization is inverted. This means that the order is two-spin rather than one-spin.

Antiphase magnetization is not directly observed in magnetic resonance. In normal NMR experiments, only transverse magnetization ${\hat{L}}_x$, ${\hat{L}}_y$, ${\hat{S}}_x$, or ${\hat{S}}_y$ induces a voltage in the detection coil. Such a signal is created immediately after a π/2 pulse from ${\hat{L}}_z$ or ${\hat{S}}_z$ and gives an absorptive line shape (nonzero area after Fourier transformation of the FID). In contrast, Eq. 24 shows that these two-spin operators only create signal after a time comparable to 1/(2J_LS) and the area of the peak is zero (see Fig. 4). This can be fixed by delaying the start of acquisition to the maximum signal time, but this is not an issue as long as that time is short compared to the homogeneous relaxation time T₂^*. This is commonly the case for most SABRE imaging targets (¹³C-pyruvate and its derivatives, for example, show relaxation on the order of 500 ms or longer). Inhomogeneous broadening can be eliminated by simply inserting simultaneous π pulses on both L and S at time 1/(4J_LS), similar to a refocused INEPT experiment.

Figure 7 and fig. S6 show simulation data on a four-spin ¹⁵N-pyridine system where the expectation values of ${\hat{S}}_z$, ${\hat{L}}_z$, or ${\hat{L}}_z{\hat{S}}_z$ are taken for both the bound and free species during SABRE buildup at either 0.5 or 50 μT. The t⁴ time dependence of ${\hat{S}}_z$ and ${\hat{L}}_z{\hat{S}}_z$ observed at short times (before exchange occurs) provides evidence for the proposed dominant pathways. The sharp oscillations at 50 μT arise from the roughly 2.3-kHz resonance frequency difference between ¹H and ¹⁵N. Complementary simulations that show the evolution of all major one-spin and two-spin operators on the bound species in the absence of exchange are shown in fig. S6. It is important to note that, although many multispin operators will be produced in a fourth-order commutator, the only multispin operators that are productive for SABRE are those that are contained on the ligand and thus are preserved when chemical exchange occurs. In addition, only the operators that do not evolve under the applied magnetic field will coherently build up. Therefore, we focus on the three operators that fit those criteria: ${\hat{S}}_z$, ${\hat{L}}_z$, and ${\hat{L}}_z{\hat{S}}_z$.

This derivation reproduces the experimental features in Figs. 4 and 5. It is independent of the applied field magnitude and direction (subject to the constraint above, which we are only considering the secular part of the J_LS coupling). Inverting the field, or pointing it in an arbitrary direction, makes no difference. This mechanism is equally effective at producing L or S magnetization. In the regime where neither SABRE nor SABRE-SHEATH is efficient (e.g., 10 to 1000 μT), this pathway will dominate; in other regimes, it competes. Last, because it arises from two-spin order, the nonresonant pathway does not share one important advantage SABRE-SHEATH has over SABRE: the ability to store magnetization on a longer-lived spin. In the 10- to 1000-μT regime, ¹H relaxation is much faster than ¹³C or ¹⁵N relaxation, so in practice, the two-spin order decays away with essentially the ¹H relaxation time. At lower fields, couplings connect different spins, and heteronuclear magnetization leaks into neighboring protons, so the buildup and decay dynamics are often different.

On the basis of these results, we can also do a more detailed interpretation of the experimental data. For our purposes, it is a good approximation to model ¹⁵N pyridine as an AX₂ spin system at high fields, here retaining only the two ortho protons and their −10-Hz coupling to the nitrogen. Adding in labels L₁ and L₂ for the two ortho protons, we expect the resonant pathway to create ${\hat{S}}_z$ and the nonresonant pathway to create ${\hat{L}}_{1z}{\hat{S}}_z+{\hat{L}}_{2z}{\hat{S}}_z$. Giving a 90y pulse to the nitrogen and then calculating evolution under the operators $J_{\mathit{LS}}({\hat{L}}_{1z}{\hat{S}}_z+{\hat{L}}_{2z}{\hat{S}}_z)$ gives the time dependence of these two pathways as

$$\begin{array}{c}{\hat{S}}_z\stackrel{90_{S,y}+\text{free evolution} }{\to }{\hat{S}}_x{\text{cos}}^2\left(\mathrm{\pi }{J}_{\mathit{LS}}t\right)\text{+}2{\hat{S}}_y({\hat{L}}_{1z}+{\hat{L}}_{2z})\hfill \\ \hfill \text{cos}\left(\mathrm{\pi }{J}_{\mathit{LS}}t\right)\text{sin}\left(\mathrm{\pi }{J}_{\mathit{LS}}t\right)-4S_x\left({\hat{L}}_{1z}{\hat{L}}_{2z}\right){\text{sin}}^2\left(\mathrm{\pi }{J}_{\mathit{LS}}t\right)\end{array}$$ (25)

$$\begin{array}{c}{\hat{S}}_z({\hat{L}}_{1z}+{\hat{L}}_{2z})\stackrel{90_{S,y}+\text{free evolution} }{\to }{\hat{S}}_x({\hat{L}}_{1z}+{\hat{L}}_{2z})\text{cos}\left(2\mathrm{\pi }{J}_{\mathit{LS}}t\right)\\ +2{\hat{S}}_y{\hat{L}}_{1z}{\hat{L}}_{2z}\text{sin}\left(2\mathrm{\pi }{J}_{\mathit{LS}}t\right)+\frac{1}{2}{\hat{S}}_y\text{sin}\left(2\mathrm{\pi }{J}_{\mathit{LS}}t\right)\end{array}$$ (26)

Thus, the maximum observable signal for the nonresonant case is around 25 ms ($t=1/\left(4J_{\mathit{LS}}\right)$) as seen experimentally. At that time, the resonant SABRE-SHEATH signal is half of its initial value. This lets us combine the $t=0$ and $t=1/\left(4J_{\mathit{NH}}\right)$ signals to measure the relative size of one-spin and two-spin z order, as visualized in Fig. 8. We may extrapolate the magnitude of ${\hat{L}}_z{\hat{S}}_z$ through the following steps: First, we take the value of the FID at $t=1/\left(4J_{\mathit{NH}}\right)$ and subtract half of the value at $t=0$ from it to isolate signal coming from the nonresonant pathway from the residual signal from the resonant pathway. From Eq. 26, we observe that there is twice the magnitude of ${\hat{S}}_x({\hat{L}}_{1z}+{\hat{L}}_{2z})$ at $t=0$ than ${\hat{S}}_y$ at $t=1/\left(4J_{\mathit{LS}}\right)$. Thus, the total amount of ${\hat{L}}_z{\hat{S}}_z$ produced during SABRE buildup is four times the experimental nonresonant signal. Figure 8 suggests that the production of two-spin order is about as efficient as one-spin order at these experimental conditions, but the resulting observable magnetization is about four times smaller in the nonresonant case compared to the resonant case (Fig. 4).

Fig. 8. Inferred one-spin and two-spin order from the Ir(H)₂(IMes)(py)₃ experimental data.

The one-spin order can be read as the first point in the FID (as in Fig. 3). The two-spin order is found by subtracting half of the $t=0$ signal from the $t=1/\left(4J_{\mathit{NH}}\right)$ signal and then multiplying by four to account for the two L spins and the factor of ½ in Eq. 26 as described in the text.

There are a few nuances to how this magnetization may be used experimentally. Note from Eq. 24 that, at time $t=0$, there will be no observable signal in the FID. This indicates that the integral of the resulting Fourier-transformed NMR spectrum will be zero, so either the signal must be converted to in-phase or an alternative metric must be used to quantify it. There are several methods that have been applied historically to convert antiphase magnetization to net magnetization, including the use of a refocused INEPT sequence in many common 2D NMR techniques, so we do not repeat the work here.

The 20 pathways (in the Supplementary Materials) from ${\stackrel{\rightharpoonup }{I}}_1·{\stackrel{\rightharpoonup }{I}}_2$ to ${\hat{L}}_z{\hat{S}}_z$ include 8 terms of the form $J_{\mathit{HL}}^2{J}_{\mathit{HH}}{J}_{\mathit{LS}}$ (all of which lead to positive ${\hat{L}}_z{\hat{S}}_z$) and 12 terms of the form $J_{\mathit{HL}}^2{J}_{\mathit{HH}}{J}_{\mathit{HS}}$ (six leading to positive ${\hat{L}}_z{\hat{S}}_z$ and six to negative ${\hat{L}}_z{\hat{S}}_z$), so the expected functional dependence is $J_{\mathit{HL}}^2{J}_{\mathit{HH}}{J}_{\mathit{LS}}$. However, our long-time simulations, which are relevant for generating experimental signal, show the functional dependence is $J_{\mathit{HL}}^2{J}_{\mathit{HH}}(J_{\mathit{LS}}-J_{\mathit{HS}})$. Figure 9 shows an example of this. Simulations show a broad resonance creating usable magnetization with two gaps: one around $J_{\mathit{LS}}=0$ (where the antiphase order cannot convert back into magnetization) and one where $J_{\mathit{HS}}=J_{\mathit{LS}}$. This aligns with the qualitative description mentioned earlier where, if J_HS and J_LS are equal, the local fields of I₁ and L are perturbed equally, and there is no resonance frequency difference created between the two.

Fig. 9. Predicted efficiency of generating usable ¹⁵N and ¹H magnetization at 50 μT as a function of ${\mathit{J}}_{\text{LS}}$ for a four-spin pyridine system.

(A) shows simulated ¹⁵N magnetization while (B) shows ortho-¹H magnetization. In this system, $J_{\mathit{HH}}=-7$ Hz, $J_{\mathit{HS}}=19.6$ Hz, and $J_{\mathit{HL}}=1.2$ Hz, and usable magnetization is defined as the maximum value of the FID after 5 s of polarization buildup. Polarization efficiency varies smoothly except around $J_{\mathit{LS}}=0$ and around $J_{\mathit{LS}}=J_{\mathit{HS}}$. This agrees with the predicted $J_{\mathit{HL}}^2{J}_{\mathit{HH}}(J_{\mathit{LS}}-J_{\mathit{HS}})$ dependence starting from ${\hat{I}}_{1z}{\hat{I}}_{2z}$. The orange curves assume the catalyst starts with singlet hydrides and is always replenished with singlet. The blue represents an extreme case where the hydrides and hydrogen in solution have only the ${\hat{I}}_{1z}{\hat{I}}_{2z}$ order. The signals are nearly identical, with a slight shift of the zero, reflecting some action by a mechanism proportional to $J_{\mathit{HL}}^2{J}_{\mathit{HH}}{J}_{\mathit{LS}}$ from the ${\hat{I}}_{1y}{\hat{I}}_{2y}$ and ${\hat{I}}_{1z}{\hat{I}}_{2x}$ order. Simulations at the SABRE-SHEATH peak (0.5 μT) also show relatively little effects from the ${\hat{I}}_{1y}{\hat{I}}_{2y}$ and ${\hat{I}}_{1x}{\hat{I}}_{2x}$ order (see the Supplementary Materials). This insensitivity comes because, in the usual limit where hydrides exchange far slower than ligands, the ${\hat{I}}_{1y}{\hat{I}}_{2y}$ and ${\hat{I}}_{1x}{\hat{I}}_{2x}$ order is quenched almost all the time (as discussed in the text).

This functional dependence reveals another interesting feature of the nonresonant mechanism. The eight pathways from ${\hat{I}}_{1z}{\hat{I}}_{2z}$ to ${\hat{L}}_z{\hat{S}}_z$ in Eqs. 16 to 23 include four terms of the form $J_{\mathit{HL}}^2{J}_{\mathit{HH}}{J}_{\mathit{LS}}$ (all of which lead to positive ${\hat{L}}_z{\hat{S}}_z$) and four terms of the form $J_{\mathit{HL}}^2{J}_{\mathit{HH}}{J}_{\mathit{HS}}$ (all of which lead to negative ${\hat{L}}_z{\hat{S}}_z$), which is this exact functional dependence. This strongly suggests that the only term that matters in the initial density matrix for the long-time behavior is the ${\hat{I}}_{1z}{\hat{I}}_{2z}$ term, which is equivalent to a 50:50 mixture of singlet (αβ-βα) and triplet (αβ+βα). This was verified numerically by replacing the initial condition in Eq. 1 with

$${\hat{\mathrm{\rho }}}_0=(\frac{1}{4}\hat{1}-{\hat{I}}_{1z}{\hat{I}}_{2z})\otimes \hat{1}$$ (27)

and replenishing magnetization with the same 50:50 mixture, which, as Fig. 9 shows, has virtually no impact on the predicted signal. This is also true to a lesser extent in normal SABRE-SHEATH (data shown in the Supplementary Materials). In effect, what the simulation is showing is that, when ligand exchange is much more rapid than hydride exchange, the effect of the $J_{\mathit{HL}}{\hat{I}}_{1z}{\hat{L}}_z$ and $J_{\mathit{HS}}{\hat{I}}_{1z}{\hat{L}}_z$ couplings (which commute with ${\hat{I}}_{1z}{\hat{I}}_{2z}$ but not ${\hat{I}}_{1x}{\hat{I}}_{2x}$ or ${\hat{I}}_{1y}{\hat{I}}_{2y}$) is to convert the hydride ${\hat{I}}_{1x}{\hat{I}}_{2x}$ and ${\hat{I}}_{1y}{\hat{I}}_{2y}$ order into three-spin terms (e.g., ${\hat{I}}_{1y}{\hat{I}}_{2x}{\hat{L}}_z$), which do not survive ligand dissociation. Thus, in the usual limit where the hydride dissociation is at least an order of magnitude slower than the ligand dissociation, only the ${\hat{I}}_{1z}{\hat{I}}_{2z}$ term is an effective hyperpolarization source most of the time. This is also a known issue at high fields due to para-ortho interconversion (96, 97).

DISCUSSION

Inclusion of the additional pathways that create two-spin magnetization has several practical consequences that have not yet been fully explored but are likely meaningful for SABRE applications:

The pathways are independent of magnetic field over a broad range, but additional terms come in at frequencies lower than the SABRE-SHEATH or higher than the SABRE condition. This will lead to additional optimal frequencies, particularly in the SABRE case. For example, the C─H coupling in a methyl group ¹³CH₃ is typically about 130 Hz. This implies that, if the carbon is spin-down, the methyl protons have an intrinsic 130-Hz difference between spin up and spin down, which is the same as the frequency difference those protons would have from the hydride protons at 120 mT [assuming a chemical shift difference of 25 parts per million (ppm)]. In the normal SABRE condition, the frequency difference of 25 ppm between the hydride and methyl protons gives about a 7-Hz frequency difference at 6.5 mT. However, this suggests an additional resonance would exist offset by this amount from 120 mT, where this field cancels out the J_CH coupling. In this case, spin-down ¹³C nuclei allow polarization of ¹H (because the ligand-hydride resonance frequency difference is matched to the traditional SABRE condition) but spin-up ¹³C nuclei produce a large frequency difference so they are inactive. Mathematically, this implies normal ¹H and antiphase ¹H-¹³C magnetization are produced in equal quantities. This resonance will move with the chemical shift difference (so it provides some selectivity) and is near the range of commercially available low-field MRI scanners (89). This effect is also likely responsible for anomalous observations of SABRE resonances near 100 mT in ¹³C-labeled pyridines (83).

Although we have only explicitly considered the case of ¹H plus a heteronucleus, there are likely extensions to two heteronuclei. One example is the ¹⁵N-⁷⁷Se case, which has recently been explored experimentally (98) and to direct pumping of homonuclear singlets such as in diazirine (99, 100). In the latter case, direct fourth-order pathways from hydride singlet order to ligand singlet order (e.g., ${\stackrel{\rightharpoonup }{I}}_1·{\stackrel{\rightharpoonup }{I}}_2\stackrel{J_{\mathit{HL}}}{\to }{\hat{I}}_{1z}{\hat{I}}_{2y}{\hat{L}}_x\left(\text{cyc}\right)\stackrel{J_{\mathit{HH}}}{\to }{\stackrel{\rightharpoonup }{I}}_1·\stackrel{\rightharpoonup }{L}\stackrel{J_{\mathit{LS}}}{\to }{\hat{I}}_{1x}{\hat{L}}_y{\hat{S}}_z\left(\text{cyc}\right)\stackrel{J_{\mathit{HL}}}{\to }\stackrel{\rightharpoonup }{S}·\stackrel{\rightharpoonup }{L}$) are also field independent.

Most of the terms in the Hamiltonian in Eq. 14 are readily controlled, inverted, or cancelled by rf pulses in the standard ways known in NMR for over half a century. This provides a different strategy for doing high-field SABRE than has been commonly explored. Bubbling a SABRE sample at high fields, say 1 T, will not result in observable nonresonant signal because the resonance frequency difference between the hydride protons and the ligand proton (over 1 kHz at 1 T) dominates the system and truncates the transverse components of the $\stackrel{\rightharpoonup }{I}·\stackrel{\rightharpoonup }{L}$ coupling. This may easily be circumvented by using multiple echo sequences on both the proton and the heteronucleus during SABRE buildup to remove chemical shift evolution while preserving the full scalar coupling. Note that these 180° pulses must be applied to I, L, and S to preserve the ${\hat{L}}_z{\hat{S}}_z$ being built up while averaging out the $\mathrm{\Delta }{\mathrm{\omega }}_{\mathit{HL}}$ term. This brings us back into the regime where the rest of the nonresonant SABRE mechanism may proceed as usual.

There are likely substantial advantages to this strategy for high-field SABRE experiments because most existing methods are highly sensitive to the population difference between hydride singlet (αβ-βα) and the triplet state (αβ+βα). For example, the first such experiment (LIGHT-SABRE) (101) showed explicitly that polarization is created by weak irradiation at the nitrogen frequency, interchanging states with those two different hydride characters. As noted earlier, the fragility of the ${\hat{I}}_{1x}{\hat{I}}_{2x}$ and ${\hat{I}}_{1y}{\hat{I}}_{2y}$ order implies a strong tendency to equalize those populations (${\hat{I}}_{1z}{\hat{I}}_{2z}$ has an equal population of αβ-βα and αβ+βα), which cancels the LIGHT-SABRE effect. Thus, nonresonant SABRE works fine with the ${\hat{I}}_{1z}{\hat{I}}_{2z}$ order, whereas most existing high-field SABRE approaches do not. This has been addressed in some recent papers (97, 102, 103) with relatively complex pulse sequences, but using the nonresonant mechanism will likely improve pulse sequence strategies.

Last, extensive optimization of SABRE-SHEATH approaches by computer has led to robust, 3D fields that are accessible in the laboratory (104). To our knowledge, none of these calculations have ever tried to optimize anything other than magnetization ($⟨{\hat{S}}_z⟩$). Feeding two-spin as well as one-spin order into optimization approaches is likely to improve results as the pathways are largely independent of one another.

We have demonstrated a hitherto unappreciated parahydrogen–based hyperpolarization mechanism, which involves direct creation of two-spin zz order in the target ligand. Although here we have presented data mostly for pyridine, the mechanism should be accessible for essentially every molecule that has been hyperpolarized by SABRE and has a heteroatom near the hydrogen atoms, and we have found experimental signals in other common molecules such as acetonitrile (see the Supplementary Materials). One reason this may have gone undetected until now is relatively simple. In normal hyperpolarization experiments, magnetization is detected by the signal directly after a 90 pulse, so inhomogeneous broadening in the spectrometer is not important. Thus, SABRE testing is often done without removing the bubbling apparatus (which causes susceptibility effects). This is a critical problem for detecting the antiphase magnetization in nonlinear SABRE, particularly if (unlike the pyridine case) the coupling J_LS is small as the signal disappears before it can refocus. This is of course easily solved in the laboratory; it was just that it was not realized that it might be important.

The nonresonant mechanism has multiple desirable features. These include near independence of external field strength over multiple orders of magnitude (it can be done quite well in an uncontrolled lab setting in Earth’s field) and low sensitivity to the coupling-induced interconversion between hydride singlet and triplet order. It also suffers from a few drawbacks, including fast relaxation (compared to traditional X-SABRE experiments) and the need for an echo pulse to counteract inhomogeneous broadening in imaging applications. Overall, this mechanism likely provides multiple new strategies for SABRE optimization at a wide variety of fields. This amplifies the advantages of SABRE (a cheap and simple source of spin order as well as simple and low-cost apparatus) as the current best candidate for a truly general and versatile hyperpolarization strategy.

MATERIALS AND METHODS

Sample preparation

¹⁵N-Pyridine SABRE samples were prepared with 160 mM ¹⁵N-pyridine (Cambridge Isotope Laboratories, NLM 305-0.5, CAS 34322-45-7) and 10 mM Ir-IMes catalyst [IMes = 1,3-bis(2,4,6-trimethylphenyl)imidazol-2-ylidene; synthesized by a collaborator] in methanol-d₄ (Cambridge Isotope Laboratories, DLM-24-50, CAS 811-98-3) solvent. For the representative spectra shown in Fig. 4, samples additionally contained 10% (v/v) methanol (Sigma-Aldrich, 34860-4L-R, CAS 67-56-1) so that the measurement magnet could be shimmed on the sample to help reduce inhomogeneity resulting from the presence of the H₂ gas bubbler in the NMR tube. The sample tube was a Wilmad 5 mm sapphire NMR tube (WG-507-7) custom fitted to adapt to the H₂ gas line which was pressurized to 120 psi. The Ir-IMes catalyst was activated by bubbling H₂ gas through the sample tube at a rate of 16.15 cm³/min for 1 hour at 40°C. Roughly half of the sample volume was lost to evaporation during activation, causing the final pyridine concentration to be ∼320 mM.

Procedure for nonresonant SABRE experiments

For each SABRE experiment, the sample was bubbled with 80 to 90% p-H₂ gas (produced with a Bruker BPHG 90 generator) at a rate of 43.46 cm³/min for 5 s in the presence of the polarizing field. After this polarization buildup period, bubbling was shut off, and the electromagnet was triggered to output a field with a magnitude of ≈50 μT to control the transfer into the lab field. The sample was then transferred into a Magritek Spinsolve benchtop 1-T spectrometer and ¹⁵N and ¹H spectra were acquired back-to-back. A repetition time of 200 s was allowed between SABRE experiments to allow for hyperpolarized ¹⁵N magnetization to fully relax.

A thermal reference sample containing 2.72 M ¹⁵N-pyridine in methanol-d₄ was created, and a 250-scan-averaged reference spectrum was recorded using the same NMR tube and measurement magnet as the presented experiments. The signal intensity from the thermal reference was scaled down by a factor of 8.5 to control for concentration differences. Enhancement values were obtained by dividing the SABRE sample signal by this concentration-scaled thermal value.

Simulation methodology

The simulation framework used here to model SABRE dynamics has been reported in ref. (94). The ¹⁵N-pyridine SABRE complex was modeled as either a four-spin (Fig. 7 and figs. S6 to S8) or a five-spin (Figs. 3 and 9, and figs. S2 and S4) system, which includes two ¹H (H) from bound p-H₂ as well as one ¹⁵N (S) and one or two ortho ¹H (L) from a bound pyridine. The SABRE complex was treated as a Y system, although it is known that pyridine creates an X geometry in which it binds both available equatorial positions of the SABRE catalyst. Simulations that accounted for the X geometry showed analogous dynamics to the Y geometry (just with twice the efficiency because pyridine is bound to both positions), so Y geometry was preferred for computational efficiency. Unless otherwise specified, this system used the following J-coupling values: $J_{\mathit{HH}}=-7$ Hz, $J_{\mathit{HS}}=19.6$ Hz, $J_{\mathit{HL}}=1.2$ Hz, $J_{\mathit{LS}}=-10.06$ Hz, and $J_{\mathit{LL}}=-0.16$ Hz. Simulations included a chemical shift of 30.75 ppm between H and L. The longitudinal relaxation times were assumed to be: $T_{1,H}=1$ s, $T_{1,L}=4$ s, and $T_{1,S}=40$ s. As stated in the main text, the simulations accounted for a ligand exchange rate of 16 s⁻¹ a hydride exchange rate of 2 s⁻¹, a 1:20 catalyst to ligand ratio, and assumed that p-H₂ flowed continuously ([Ir]/[H₂] is negligible). Unless otherwise specified, SABRE performance was evaluated for a given test condition by allowing the system to evolve in the polarizing field for 5 s before a 90y pulse was applied to either ¹⁵N or ¹H and a simulated FID was acquired by allowing a further 5 s of evolution to proceed at $B_0=1$ T.

Supplementary Materials

This PDF file includes:

Supplementary Text

Figs. S1 to S8