Coherent evolution of para hydrogen induced polarisation using laser pump, NMR probe spectroscopy: theoretical framework and experimental observation

We recently reported a pump-probe method that uses a single laser pulse to introduce para hydrogen ( p -H 2 ) into a metal dihydride complex and then follows the time-evolution of the p -H 2 -derived nuclear spin states by NMR. We present here a theoretical framework to describe the oscillatory behaviour of the resultant hyperpolarised NMR signals using a product operator formalism. We consider the cases where the p -H 2 -derived protons form part of an AX, AXY, AXYZ or AA’XX’ spin system in the product molecule. We use this framework to predict the patterns for 2D pump-probe NMR spectra, where the indirect dimension represents the evolution during the pump-probe delay and the positions of the cross-peaks depend on the difference in chemical shift of the p -H 2 -derived protons and the difference in their couplings to other nuclei. The evolution of the NMR signals of the p -H 2 -derived protons, as well as the transfer of hyperpolarisation to other NMR-active nuclei in the product, is described. The theoretical framework is tested experimentally for a set of ruthenium dihydride complexes representing the different spin systems. Theoretical predictions and experimental results agree to within experimental error for all features of the hyperpolarised 1 H and 31 P pump-probe NMR spectra. Thus we establish the laser pump, NMR probe approach as a robust way to directly observe and quantitatively analyse the coherent evolution of p -H 2 -derived spin order over micro-to-millisecond timescales.


I. Introduction
Hyperpolarisation is an increasingly important area of development in magnetic resonance, particularly in fields such as biomedical NMR and MRI, where applications are often sensitivity limited. [1] Hyperpolarisation methods boost the sensitivity of magnetic resonance by increasing the population imbalance between nuclear energy levels from a few ppm/T at thermal equilibrium up to as much as several tens of percent. Dozens of hyperpolarisation schemes have been proposed. The most widely used of these are based on dynamic nuclear polarisation (DNP), [2][3][4][5] spin-exchange optical pumping (SEOP), [6] and parahydrogen-induced polarisation (PHIP). [7][8][9][10] These hyperpolarisation techniques differ in both the source of their polarisation and the method of transfer to the target nuclei.
In this work we focus on PHIP, which is a general term that refers to all methods where the source of polarisation is parahydrogen (p-H2). Parahydrogen is the nuclear spin isomer of molecular hydrogen in which the pair of spin-1/2 1 H nuclei form a singlet state.
The use of p-H2 as a route to hyperpolarisation was first predicted by Bowers and Weitekamp in 1986 [8] and exemplified shortly after. [7,9] This method takes advantage of the relatively easy route to prepare H2 in this thermodynamically preferred form.
However, p-H2 has no net angular momentum and so is NMR silent. If the symmetry of the chemical and/or magnetic environments of the 1 H nuclei in p-H2 is broken, for example by means of transporting them through oxidative addition at a metal centre into inequivalent environments, the singlet state will evolve into NMR observable triplet states in the product molecule. The NMR signals of the former p-H2-nuclei in the product can be enhanced by several orders of magnitude. [7,11,12] In some cases, NMR signal enhancements can also be observed for other NMR-active nuclei in the product molecule due to spontaneous or radio-frequency (rf) driven polarisation transfer from the protons originally on p-H2. [10,[13][14][15][16][17][18] This polarisation transfer phenomenon is harnessed in the signal amplification by reversible exchange (SABRE) approach, which uses a reversible exchange reaction to catalytically transfer polarisation from p-H2-derived protons to nuclei in a target substrate molecule without chemical alteration of the substrate. [10,19] Parahydrogen is a particularly attractive source of hyperpolarisation because it is relatively cheap and easy to produce by cooling H2 gas in the presence of a paramagnetic species (e.g. activated charcoal or iron oxide). Once generated at low temperature (typically between 20 -77 K depending on the desired level of p-H2 enrichment) p-H2 can be stored at room temperature for weeks or even months. [12] Since the initial experimental demonstrations by Bowers and Weitekamp, [7] and Eisenberg and co-workers, [9] PHIP has become a valuable tool for the study of reactivity in inorganic and organic chemistry, particularly for elucidating reaction mechanisms and for the identification of intermediates in hydrogenation chemistry. [20][21][22][23] More recently, PHIP and SABRE have been used for the development of hyperpolarised contrast agents for clinical MRI. [24][25][26] In most PHIP experiments, thermally-activated reactions are used to build-up the population of hyperpolarised molecules over a period of seconds. The asynchronous nature of this approach means that the p-H2 hyperpolarisation that is detected corresponds to a time-averaged response. Thus information about the coherent evolution of the p-H2 singlet state into NMR-observable triplet states is lost. [11] It should be noted that the coherent evolution of the p-H2-derived magnetic states in a product molecule could theoretically be observed following a thermally activated reaction if a suitable radio frequency (rf) irradiation scheme is applied during the reaction period to suppress evolution under chemical shift and/or spin-spin coupling. This approach was used by Goldman and Johannesson to maximise the efficiency of transfer of p-H2-derived 1 H spin order to a neighbouring heteronucleus (e.g. 13 C) in the product molecule. [27] Another approach is to use a photochemical reaction to unlock the hyperpolarisation potential of p-H2. In this method, a reaction with p-H2 is initiated photochemically within the NMR spectrometer. [28] Duckett, Perutz and co-workers have shown that in situ photolysis can be used to help elucidate chemical mechanisms and to observe intermediates in photochemically-activated reactions involving p-H2. [28][29][30][31] In addition, this approach has also been used to assess the purity of the p-H2-derived singlet state upon oxidative addition of p-H2 to a transition-metal complex. [32,33] Building on this previous work, we recently demonstrated a new method that harnesses p-H2 hyperpolarisation in a synchronised way. [34] This approach was inspired by pumpprobe methods that combine photochemical excitation with optical detection techniques such as UV/vis, [35,36] IR, [37] and Raman spectroscopy, [38][39][40] to perform timeresolved spectroscopy of reactivity on timescales from milliseconds down to femtoseconds. In the analogous time-resolved NMR experiment, the sample is pumped photochemically using a single in situ laser pulse that is synchronised with the NMR spectrometer. After a well-defined pump-probe delay a radio-frequency (rf) probe pulse is applied and the NMR response is recorded. The excess (unreacted) p-H2 in solution is NMR-silent; therefore, in the absence of any thermal reactivity of the p-H2, the hyperpolarised NMR response excited by the probe pulse corresponds exclusively to the products of the reaction with p-H2 that was initiated photochemically by the laser pump pulse. The time delay between the laser pump and the NMR probe is readily and precisely controlled by the 200 ns clock on the spectrometer, allowing for short pump-probe delays on the order of the duration of the NMR pulse i e s Typically a single scan pump-probe NMR spectrum provides sufficient signal-to-noise for analysis; however, signal averaging is possible in cases where the hyperpolarised species of interest is present at an extremely low concentration. The time-resolved NMR experiment with p-H2 hyperpolarisation presents the opportunity to (a) study chemical reactivity on microto-millisecond timescales, [41] and (b) observe the coherent evolution of the p-H2-derived spin states directly by conversion via rf excitation into observable NMR magnetisation.
Thus this photochemical pump, NMR probe method provides a unique test-bed for exploring the evolution of the p-H2-derived spin states in the product molecule. Of particular interest is the potential to directly probe the transfer of polarisation from the former-p-H2 1 H's to other nuclei, either through spontaneous transfer in the strong coupling regime or via rf-driven transfer.
Parahydrogen is of interest in NMR spectroscopy not only because of the potential for hyperpolarisation but also as an example of a nuclear singlet state. Singlet states are important in NMR because they can be exploited to prepare molecules with so-called long-lived states (LLS) [42] and long-lived coherences (LLC). [43] LLS and LLC have a wide range of potential applications, such as the preservation of NMR hyperpolarisation and line-narrowing in solution-state NMR. The theory behind the preparation and evolution of LLS and LLC has been discussed extensively [42,[44][45][46][47][48][49][50][51][52][53][54] and shares many features with the evolution of p-H2 hyperpolarisation. The fact that p-H2 is only one of a large number of molecules that can contain nuclei in long-lived nuclear spin states suggests that the applications of the photochemical pump, NMR probe methodology are not limited to reactions involving H2. The reactivity of any molecule prepared in a longlived state could be monitored using this time-resolved NMR approach as long as the reaction can (a) be initiated photochemically, and (b) the symmetry of the nuclei that make up the long-lived state is broken in an appropriate way in the reaction product. This feature of the method is particularly interesting since it has been shown that various hyperpolarisation methods, including dissolution DNP [55,56] and SABRE [57][58][59] can be used to prepare nuclei in a range of molecules in a hyperpolarised long-lived state.
In this paper we use the photochemical pump, NMR probe method to directly observe and analyse the effects of the coherent magnetic evolution of the p-H2-derived spin states on the observed hyperpolarised NMR signals. Starting from the standard description of of NMR relaxation on the observed NMR signals is discussed and the relaxation rates for our example systems are reported.

A. Laser pump, NMR probe spectroscopy with parahydrogen hyperpolarisation
The laser pump, NMR probe experiment, as applied to the oxidative addition of p-H2 to a transition metal complex, is illustrated schematically in Figure 1a. [ After a well defined pump probe delay a suitable NMR detection pulse sequence is applied ( Figure 1) and a single high-resolution NMR spectrum is recorded. The evolution of the system following the laser pulse is probed by acquiring a series of pump-probe NMR spectra for different delays where each D NMR spectrum is associated with a single laser pulse. This series of 1D NMR spectra can be Fourier Transformed to produce a 2D pump-probe NMR spectrum, where the indirect dimension provides information about the evolution of the system during the pump probe delay In contrast to other time-resolved spectroscopies, pump-probe NMR spectra can contain information about both the chemical evolution of the system (e.g. the kinetics of the oxidative addition of p-H2 to the metal complex) and the coherent evolution of the initial p-H2-derived state into observable magnetisation. The details of the coherent magnetic evolution will depend on the spin topology of the metal dihydride complex. In this work we consider complexes where the p-H2 addition rates are orders of magnitude faster than any magnetic evolution. Hence, we can assume that any changes in the NMR response as a function of the pump probe delay are due exclusively to magnetic evolution under coupling interactions, chemical shift or relaxation.

B. Magnetic evolution during pump-probe delay
The theory of parahydrogen hyperpolarisation [8] was reviewed by Natterer and Bargon in 1997 [11], by Bowers in 2007 [61], and by Green et al. in 2012. [12] Here we describe the basic principles of p-H2 hyperpolarisation to make the manuscript self-contained and to establish a consistent nomenclature. We note that the theoretical framework used here is consistent with that originally developed by Bowers and Weitekamp [7,8,62] and subsequently expanded to explore the use of field cycling and rf irradiation for the efficient transfer of p-H2-derived hyperpolarisation to other nuclei. [11] In a standard PHIP experiment, the reaction with p-H2 is initiated thermally and the p-H2labelled product is generated over a period of seconds. In this case, the observed NMR response is averaged over the reaction period between NMR excitation steps. In our approach, the photochemical pump step generates an ensemble of p-H2-labelled product molecules on a microsecond timescale. This allows us to validate the theoretical framework through direct observation of the coherent evolution of the p-H2-derived spin states in the resulting product over a micro-to-millisecond time window. In particular, we validate the predicted evolution of the zero-quantum coherences, something that has not been achieved using the standard time-averaged PHIP approach. We note that this laser pump, NMR probe methodology is analogous to the laser-initiated chemically induced dynamic nuclear polarisation experiment (photo-CIDNP), in which a spincorrelated radical pair is used to generate NMR signal enhancements on the order of 10 2 .
Coherent magnetic oscillations have been observed with CIDNP [63][64][65][66] and modelled using numerical simulations of the coherent evolution of the system driven by differences in chemical shift and/or J coupling interactions. [66] Photo-CIDNP has been used to observe short-lived radicals and to monitor reactivity on millisecond to second timescales. [67][68][69][70] As stated earlier, p-H2 is the spin isomer of H2 that exists in a nuclear singlet state. In the product operator basis, this state can be described by the density matrix, : where = , , and = , , are the total spin angular momentum operators (with associated Cartesian components) for the two 1 H nuclei in p-H2 and E is the identity operator. Without any loss of generality, we neglect the identity operator, which does not give rise to observable NMR signal, and consider only the second term in Eq. 1. For convenience, we use standard notation [11] to divide this term into two parts: the longitudinal term: (2 ) and the transverse term, zero-quantum-x (ZQx, Eq. 2).

= 2 + 2 (2)
If the 1 H nuclei are magnetically equivalent, as is the case in molecular hydrogen, this state is invariant to rotation by an rf pulse and is NMR silent. It is only by breaking the symmetry of the 1 H environments that we can exploit p-H2 to obtain orders-of-magnitude increases in NMR signal amplitudes. In p-H2-enhanced NMR, the required break in symmetry is typically achieved through the pairwise addition of p-H2 to a metal complex or through spin-conserved hydrogenation of an unsaturated substrate. [11,12] In the examples in this paper we use the former method but our results are not limited to this case, nor are they limited to the use of p-H2 as the source of the hyperpolarised singlet state. For example, a hyperpolarised 15 N-15 N singlet state could be used. [59] It has been shown previously that the p-H2 singlet state can be conserved upon oxidative addition of p-H2 at a metal centre. [33] If we assume this is the case and that the former p- where the density matrix at t = 0 is equal to the parahydrogen singlet state, (0) = .
Note here we are using the common convention in NMR where H is written in units of angular frequency.
The evolution of the system beyond this point will depend on the form of the internal spin Hamiltonian, H.

Chemical inequivalence (AX, AXY, AXYZ)
In our first example we consider a situation where the p-H2-derived protons are 'instantaneously' (i.e. on a timescale much faster than any magnetic evolution of the system) placed into two chemically distinct environments (Scheme 2a and 2b). The weakly-coupled AX case has been solved previously [8] and is provided in the supporting information.

Scheme 2. Coupling constants in (a) AX, (b) AXY and (c) AA'XX' spin systems.
Here we briefly summarise the result for an AXY spin system, where the p-H2-derived protons are placed into chemically inequivalent environments ( = 0) in the product and experience additional but different couplings to a third nucleus, Y ( = 0). This situation is encountered when a single NMR-active heteronucleus (e.g. 31 P) is coordinated to the metal centre trans to one of the hydride ligands (Scheme 2b) and reflects a common situation in inorganic chemistry. In this case, the relevant coupling is the difference between the couplings of each of the hydrides and the third nucleus: = .
The AXY class of spin system is of particular interest when considering the transfer of p-H2-derived 1 H spin order to other nuclei in the product molecule. This transfer of polarisation can occur spontaneously or under the influence of rf irradiation depending on the spin topology of the product molecule and the experimental conditions employed.
Many different approaches have been developed to optimise the transfer of spin order, including field cycling, [71] rf irradiation, [13,[15][16][17]72] exchange reactions, [10] and combinations thereof. In the following we focus specifically on the high-field case where the p-H2-derived protons are weakly coupled in the product and the difference in coupling to the third nucleus is significant (tens of Hz) but nevertheless is weaker than the difference in chemical shift between the p-H2-derived protons (hundreds of Hz The time-dependent density matrix under the influence of this Hamiltonian, where the initial density matrix is given by Eq. 1, can be obtained using Eq. 3 to obtain the expression in Eq. 5. Here we have used the Bloch approach to include the effects of NMR relaxation through the introduction of effective relaxation rates, , , and R2,ZQ, which describe the average NMR relaxation behaviour of the longitudinal two-spin-order term and the zeroquantum coherences, respectively. Note here we use the shorthand for the zeroquantum-y coherence, which is defined as = 2 . The longitudinal two-spin-order term, (2 ), commutes with the Hamiltonian (Eq. 4) at all points in time and so only evolves due to relaxation (Eq. 5b). The amplitudes of the ZQ coherences oscillate during the pump probe delay at angular frequencies of ± .
In the absence of a difference in coupling to a third nucleus (i.e. for an isolated AX spin system), this oscillation frequency is simply the difference in chemical shift between the p-H2-derived protons. The presence of couplings to additional nuclei will modulate this oscillation frequency in a manner that is directly analogous to that of the classic doublet observed in a 1D NMR spectrum, except here the doublet is centred at a frequency equal to the difference in chemical shift between the hydrides and the splitting is the difference in J coupling between each of the hydrides and the third nucleus J. If more than one additional nucleus couples differently to the hydrides, we would expect to observe additional frequency components in the oscillation due to the additional J coupling differences. The values and relative amplitudes of these frequency components will follow the well-known rules of multiplicity from standard 1D NMR, and the splittings will correspond to the difference in J coupling to the two hydrides.

Magnetic inequivalence (AA'XX')
In to include the effects of p-H2 hyperpolarisation for the case where the difference in coupling between the p-H2-derived A nuclei and the X nuclei is smaller than the homonuclear AA' and XX' couplings ( + ). [74,75] In this case, the transfer of spin-order from the p-H2 derived protons (AA') to the heteronuclei (XX') is driven by either the application of a low-power rf spin-lock field in resonance with the homonuclear couplings ( + ) in the high-field case [74] or by level anti-crossings in the ultralow-field case. [75][76][77] Similarly, high-field polarisation transfer from p-H2-derived A spins to X spins in an AA'XX' system has been demonstrated experimentally using highpower rf irradiation and analysed in the doubly rotating frame in terms of rf-induced level anti-crossings by Pravdivtsev et al. [78] In the situation considered in this paper + and we explore the coherent magnetic evolution that proceeds in the high-field case in the absence of rf irradiation.
The time-independent Hamiltonian for an AA'XX' spin system is given in Eq. 6 ( + ) ( + ) If we assume that the homonuclear X-X coupling is very small, and , the density matrix as a function of time can be written as in Eq. 7.
If the homonuclear coupling between the X spins is not negligible, additional magnetic states will be created through evolution under the influence of the fourth term of Eq. 6.
However, when the evolution of the system due to the strong coupling between the X nuclei can be neglected (i.e. if and ), the system can be solved analytically.
As in the chemically inequivalent case, the longitudinal two-spin-order term does not experience any coherent magnetic evolution in the high-field case and so the timedependent amplitude, ( ), can be written as: , where , is the effective longitudinal two-spin-order NMR relaxation rate.
As all of the terms in the density matrix (Eq. 7) do not commute, the evolution of ( ), ( ), and ( ) must be considered simultaneously, rather than sequentially. Using Eq.
6 and the Liouville-von-Neumann equation (Eq. 3) we can write down a series of coupled differential equations to describe the coherent magnetic evolution of the AA'XX' system.
Here we have included a single effective zero-quantum NMR relaxation rate, RZQ, to describe the NMR relaxation behaviour of the three exchanging states. This effective relaxation rate will depend on the individual relaxation rates for the three states, as well as the rate of exchange between the zero-quantum coherences, and 2 ( ), and the longitudinal term, 2 ( )( ).
This system of coupled equations is directly related to the case of two strongly-coupled, chemically-inequivalent p-H2-derived 1 H nuclei in low magnetic field (AB spin system) that has been solved previously by Natterer and Bargon for the initial conditions (0) = 1 and (0) = (0) = 0. [11] Here the difference in chemical shift, , is replaced by the difference in J coupling between the hydrides and the heteronuclei: . The solution to this system of equations is given in Eq. 9, where and = 2 . It should be noted that the amplitude of the 2 ( ) term, ( ), has the opposite sign in our result compared to the amplitude of in the result of Natterer and Bargon. [11] The signs in the differential equations in Eq. 8 and the solutions in Eq. 9 are in agreement with the sign conventions used in the equations throughout this paper.
The amplitudes b2, b3 and b4 oscillate during the pump probe delay at angular frequency = 2 . The contributions of the three terms in Eq. 9 to the overall density matrix is controlled by the ratio , which quantifies the relative strength of the hydride-hydride coupling. In a very strongly coupled system, in which is small but nonzero, the final term in Eq. 7, with amplitude b4, will dominate the density matrix.

C. NMR signal detection
To probe the evolution of the density matrix during the pump-probe delay, we can apply any appropriately adapted NMR pulse sequence at a fixed time, , following a single laser pulse (Figure 1a). The simplest detection pulse sequence consists of a single broadband (non-selective) 1 H rf pulse of tip angle, , applied at a time after the laser shot ( Figure   1b). The pulse rotates the ZQ magnetic states that make up the density matrix during the pump-probe delay into a combination of NMR observable (single-quantum, SQ) and NMR silent (double-quantum, DQ, and ZQ) states. The evolution of the SQ terms can be recorded as a free induction decay (FID) and Fourier Transformed to generate a highresolution 1 H NMR spectrum. The effect of a single 1 H pulse of tip angle, , on each of the states generated in the chemically inequivalent case (AX, AXY, AXYZ, Eq. 5) and the magnetically inequivalent case (AA'XX', Eq. 7) is summarized in Eq. S1 in the supporting information.
Using the results in Eq. S1, we can derive expressions for the single-quantum (SQ) density matrix (i.e. the NMR observable part) following the application of the rf pulse, where is used to denote the time immediately following the rf pulse. Here we assume that the rotations associated with the rf pulse are ideal and instantaneous. This approximation is only valid if the rf pulse duration is much shorter than the period of the oscillations of the zero-quantum coherences during the pump-probe delay. A single rf pulse, , can also be applied to the X channel, either instead of, or in addition to, the 1 H rf pulse (Figure 1c). This rf pulse will rotate the Tz and Rz terms by an angle in the usual way.
More complex pulse sequences, such as the OPSY (Only Parahydrogen SpectroscopY) family of sequences, which use the principles of multiple-quantum filtration with gradient pulses to selectively observe ZQ, DQ or higher-order coherences, [79,80] or selective excitation of only one of the hydride resonances, [33] can also be used to observe the different contributions to the p-H2-enhanced NMR signals.

Chemical inequivalence (AX, AXY, AXYZ)
The SQ density matrix for an AXY spin system following the application of simultaneous, non-selective rf pulses of tip angle to the 1 H nuclei (I and S) and to T is presented in Eq. 10. If we define the observation operator as the sum: we can predict the form of the NMR spectrum that will arise from each term in Eq. 10 as shown in Figure 2.  (Figure 2c). In other words, each contribution to the density matrix will produce two doublets centred on the chemical shift of the A and X spins, respectively, but the relative phases of these peaks will be different in each case. In addition, the amplitudes of the hyperpolarised contributions (A1 and A2) depend on the rf pulse angle 1 in different ways. A1 is proportional to sin 2 , while A2 is proportional to sin .
Therefore there are several methods that can be used to differentiate between the various contributions to the NMR signal.
If we apply an rf pulse with tip angle: , where n is any odd integer (i.e. odd multiples of 90°), the ZQx and longitudinal two-spin-order terms will produce a combination of NMRsilent ZQ and DQ terms, while the ZQy term will be rotated into NMR-observable SQ terms.
A FID recorded following this pulse will therefore contain information about the amplitude of the ZQy coherence exclusively, and have the form of Figure 2b. In contrast, a DQ or ZQ quantum filtered OPSY sequence [80] can be used to selectively probe the difference in amplitude between the ZQx and longitudinal two-spin-order term and will produce an NMR spectrum of the form shown in Figure 2a.
If we apply an rf pulse that is not odd integer multiples of 90° ( = 45°, for example) the SQ density matrix will contain a mixture of both terms in Eq. 10. However, we can exploit the differences in the relative phases of the hydride signals to isolate the contribution of each term to the density matrix through selective integration. To illustrate this behaviour, we first define the integral, ( ), of a peak centred at , over an integration range given by The individual contributions of the two terms in Eq. 10 to the observed NMR spectrum, This integration method has the advantage of minimizing errors in the spectral baseline and cancels any hydride signals originating from thermal magnetization if the anti-phase peaks are well resolved, i.e. for < 2 .
If a series of 1D 1 H pump probe NMR spectra are acquired with different delays they can be Fourier Transformed to produce a 2D spectrum, where the indirect dimension corresponds to evolution during the pump-probe delay (Figure 3a). In this 2D NMR spectrum, we would expect the first term in Eq. 10a to give rise to peaks at = ± /2 and = 0 Hz because it is modulated by [1 cos( )], while the second term in Eq.
10a will only give rise to peaks at = ± /2 because it is modulated by sin( ). As described previously, the relative contributions of these two terms to the 2D spectrum can be controlled by the choice of flip angle, 1. In practice, a flip angle of 1 = 90° is preferred because it generates observable NMR signals from the second term in Eq. 10a, exclusively, and so significantly simplifies the interpretation of the 2D spectrum. In the absence of Y decoupling, the evolution of the amplitudes of the various terms in Eq.
10 will be modulated by any difference in coupling to the third spin, Y, and will depend on the state of this spin. The amplitude of the peaks that correspond to the spin-up state of Y, ( = + ) will evolve with a frequency of: /2 /2, while the amplitude of the peaks associated with the spin-down state ( ) will evolvewith an oscillation frequency of: /2 /2. In a 2D pump-probe NMR spectrum this will correspond to a pair of doublets in the indirect dimension at ± /2 with a splitting equal to , as illustrated in Figure 3b. If there are other heteronuclear or homonuclear spins that couple to the hydrides, additional splittings will be observed in the indirect dimension of the 2D 1 H pump-probe NMR spectrum, which will give rise to the same peak multiplet patterns as seen in standard NMR spectroscopy. However, the peak separation will be equal to the difference in J coupling between each hydride and the additional nucleus.
The coupling of the hydrides to an additional spin provides an opportunity to observe p-H2-derived hyperpolarisation on this spin. In other words, we can probe the transfer of spin order from the p-H2-derived hydrides to other nuclei. Only the final term in Eq. 10 will give rise to observable signal for Y. This signal will be at a maximum for tip angles: In the indirect dimension of the resultant 2D pump-probe NMR spectrum we will observe peaks at = ±( /2 /2) due to the cosine modulation during the pump-probe delay. In the direct dimension, we will observe a pair of peaks separated by the sum of the scalar couplings ( = + ) and a pair of peaks of the opposite sense separated by the difference between the couplings ( = ). This is illustrated by the coupling pattern on the right in Figure 3b and is most readily apparent when Y is a heteronucleus.

Magnetic inequivalence (AA'XX')
The SQ density matrix for an AA'XX' spin system following the application of a nonselective 1 H rf pulse of tip angle is presented in Eq. 14. The first two terms are analogous to the AX case, where the difference in chemical shift is now replaced by the coupling term = 2 + . The final product, which involves two spin order terms across all four nuclei, will only be significant in the case of a strongly coupled spin system. In the indirect dimension of a 2D pump-probe spectrum of the A spins (I and S), this term will give rise to peaks at = ± /2 = ± + and at = 0. The amplitude of these peaks, relative to the dominant peaks arising from the first two terms, is determined by the ratio: J/JAA'.
If we instead apply an rf pulse, , to the X spins (T and R) then the SQ density matrix of the spin system immediately following the pulse will be as follows.
Due to the strong coupling between the hydrides, the two terms in Eq. 15 will interconvert with a frequency equal to the coupling between A spins, JAA' and observable signals will be seen in the X nucleus 2D pump-probe spectrum. The first term will result in a spectrum in the direct dimension that contains an anti-phase doublet of anti-phase doublets, with splittings of J and JAA', respectively. These peaks will appear at frequencies of = ± /2 in the indirect dimension. The second term will give rise to an anti-phase doublet of doublets with the same splittings: J (anti-phase) and JAA' (in-phase). These peaks will appear at = ± /2 and = 0 in the indirect evolution dimension The relative amplitudes of these two contributions to the 2D NMR spectrum will depend on . The expected peak patterns for a 2D pump-probe spectrum of this type are illustrated in Figure 3c.

III. Experimental Methods
All NMR spectra were recorded on a Bruker Avance widebore 600 MHz spectrometer fitted with a BBO probe. Laser photolysis was carried out with a pulsed Nd:YAG laser Hydrogen enriched in the para spin state to > 99% was prepared by cooling H2 to 27 K over activated charcoal using a system previously described in the literature. [33] Standard NMR pulse sequences were modified for use with p-H2 by including a synchronized laser initiation sequence prior to NMR excitation and detection (Figure 1a).
A purpose-written program was used to control the laser firing from the NMR console with the laser set on external triggering. The fire signals are sent to the laser via a BNC cable. The NMR pulse is initiated at a set delay time ( ) following the fire signal. The delay between sending the fire signal from the spectrometer and the actual firing of the laser pulse is controlled by a digital delay pulse generator (Stanford Research Systems Inc.) and was verified by a photodiode and an oscilloscope to be s This signal delay was incorporated into the pulse sequence such that synchronized measurements with a time delay, , were achieved by setting the spectrometer delay to: s The precision of this delay between the laser and rf pulses is controlled by the 200 ns clock of the spectrometer. The 90° rf pulse durations were 13.5 µs for the 1 H channel and 8.4 µs for 31 P. All laser-pump, NMR probe spectra were acquired in a single scan, i.e. without the use of signal averaging.

IV. Results and Discussion
The validity of our model for the evolution of p-H2-derived spin order following oxidative addition of p-H2 at a metal centre can be tested experimentally using our pump-probe approach with systems where the rate of oxidative addition of p-H2 to the transition metal complex is much faster than the magnetic evolution of the system. We have chosen to Scheme 3. The ruthenium dihydride complexes used in this study.

A. AX spin system
In the transition metal complex Ru(H)2(CO)2(dpae) (dpae = Ph2AsCH2CH2AsPh2) (1 in Scheme 3), the hydrides are chemically inequivalent, with a difference in chemical shift ranging from 0.2 to 0.4 ppm in three common NMR solvents (Figure 4a-c). The 1D pumpprobe 1 H NMR spectra ( 1 = 90°) of 1 in Figure 4 Figure 4; fit parameters in Table 1) reveals, as expected, that the oscillation frequency is equal to the differences in chemical shift given in Figure 4a-c. The high quality of the fits confirms the validity of our model and also highlights the fact that the pump-probe measurements are reproducible and quantitative. The solventdependent chemical shift differences allow for three independent validation measurements with the same complex. We note that on the timescale of these experiments, there was little observable relaxation (see Section D for relaxation time measurements). Table 1. Parameters for fits of Eqs. 10 and 13 to the data in Figure 4 (red lines). A series of 1D pump-probe NMR spectra can be Fourier transformed to yield a 2D spectrum as in Figure j 1 and k 1 For 1 = 90° (Figure 4j), we only detect NMR signals arising from the ZQy term and so we observe anti-phase peaks at = ± 2 = ±241 Hz see Figure a However for 1 = 45° (Figure 4k) we detect NMR signals arising from both terms in Eq. 10 and so peaks are also observed at f1 = 0 Hz, with destructive interference leading to partial-signal cancelation at = ± 2 = ±241 Hz . A comparison between Figures 4j and 4k demonstrates the advantage of using 1 = 90°, which gives rise to a 2D spectrum that is more easily interpreted.  Table 1. (j,k) 2D 1 H pump-probe NMR spectra of 1 in C6D6 acquired with (j) 1 = 90° and (k) 1 = 45°.
Positive intensity is plotted in black and negative intensity in red.

B. AXY and AXYZ spin systems
As a model of an AXY spin system, we studied the complex Ru(H)2(CO)(PPh3)(dppp) (dppp = Ph2PCH2CH2CH2PPh2) (2 in Scheme 3) in C6D6, which has a major mer isomer and a minor fac isomer that is only observed in the photochemical pump, NMR probe experiment. Coupling constants and chemical shifts for the two isomers of 2 are available in Table S2 of the supporting information. The mer isomer has a single 31 P nucleus coordinated to the metal in the same plane as the two hydrides, and two mutually trans 31 P nuclei. The couplings between the mutually trans 31 P and each of the hydrides are the same within a few Hz; therefore these couplings play only a very minor role in the evolution during the pump-probe delay on the timescale of our experiment. Thus, this isomer is a good model of an AXY spin system (Scheme 1b). The fac isomer has three 31 P nuclei coordinated to the metal, two trans to a hydride proton and a third trans to a CO ligand. This third 31 P couples similarly (to within a few Hz) to both hydride protons and so does not play a significant role in the evolution during the pump-probe delay.
Therefore, we can consider this isomer to be an AXYZ spin system. The NMR signals corresponding to the fac isomer are absent in a thermally polarised 1 H NMR spectrum (top in Figure 5a), but hyperpolarised signals for both isomers are visible (mer and -7.00, red; fac and blue in the D 1 H pump-probe NMR spectrum (bottom in Figure 5a, = 0.5 ms, 1 = 90°). Hz in good agreement with the predicted frequencies of J/2 (see Table 2 for tabulated values.) The coordination of a second 31 P nucleus to the metal in the same plane as the hydrides effectively renders the fac isomer an AXYZ spin system (see above). In direct analogy with the well-known multiplicity rules of NMR spectroscopy, the scalar coupling between the hydrides and the two 31 P nuclei gives rise to doublet of doublets, which resembles a triplet due to the similarity of the hydride-31 P coupling differences. In the 2D spectrum, the peaks occur at f1 = ±516 Hz, ±600 Hz, and ±684 Hz, in good agreement with predicted values (see Table 2). As predicted, 31 P hyperpolarisation can be observed if rf pulses are applied to the 1 H and 31 P channels simultaneously ( Figure 6). The 2D 31 P pump-probe NMR spectrum in Figure   6 was acquired under the influence of selective decoupling of the aromatic and CH2 protons in order to simplify the spectra and limit signal cancellation due to the overlap of positive and negative peaks. We note that under the conditions employed here (2 kHz waltz-64 1 H decoupling applied at 4 ppm) the hydride-phosphorus couplings appear to have been reduced by 8% relative to the values obtained from a fully coupled 1 H spectrum (see supporting information for numerical simulations). In the pump-probe 31 P spectra in Figure 6 we observe hyperpolarisation on the in-plane 31  Interestingly, we also observe a small amount of hyperpolarisation on the 31 P at (dppp), which has a difference in coupling to the hydrides of 4.5 Hz. The 31 P at (PPh3) has a difference in coupling of only 1.2 Hz and so no hyperpolarisation is observed for this nucleus on the timescale of the experiment ms Hyperpolarisation of the 31 P signal was also observed in pump-probe NMR spectra of 2 in C6D6 following the application of a rf pulse of 2 = 90° to the 31 P channel when continuous broadband rf irradiation was applied to the 1 H spins during the acquisition of the 31 P signal ( Figure 7). As in Figure 5, a doublet is observed in the indirect dimension at f1 = J/2 (f1 = ±246±5 Hz and ±336±5 Hz). The observation of 31 P hyperpolarisation in this case can be understood as follows. Immediately following the application of the single pulse to the heteronucleus (applied along y with a tip angle, ) the single-quantum density matrix will be: Under the influence of continuous wave (cw) irradiation applied to the hydride resonances, with an amplitude of , 2 will evolve according to Eq. 17. This residual coupling is essential for the observation of hyperpolarisation in this case.
The residual coupling is much smaller than the linewidth of the 31 P{ 1 H} peaks in a thermal spectrum (see Figure 7a). This confirms that a significant amount of signal cancellation has occurred, limiting the amount of observable 31 P hyperpolarisation. For this AA'XX' spin system we predict that there will be two contributions to a 2D pumpprobe NMR spectrum following a 90° rf pulse applied to either the 1 H spins (AA') or the 31 P spins (XX'). The major component, which corresponds to the first term in Eq. 15, will consist of peaks at = ± /2 = ± + = 83.9 Hz in the indirect dimension due to the modulation by sin during the pump-probe delay. The minor component, which corresponds to the second term in Eq. 15, will give rise to peaks at = ± /2 = ± + = 83.9 Hz and f1 = 0 Hz in the indirect dimension due to the ( cos ) modulation during the pump-probe delay. In previous work, we published a 2D pumpprobe 1 H NMR spectrum for 3 that displayed major peaks at = ±84 Hz in the indirect (pump-probe) dimension, [34] as predicted by the theoretical treatment in section II.B.2.
In addition, we showed that 31 P hyperpolarisation was observed when a 90° pulse was applied exclusively to the 31 P channel prior to signal acquisition. In contrast to the 1 H spectrum, the 2D 31 P pump-probe NMR spectrum displayed major peaks at = ±84 and at = 0 Hz. Furthermore, the lines were too broad to interpret the spectral features in the direct dimension. The theoretical development presented above suggests that, as in the 1 H spectrum, the 31 P pump-probe spectrum for an AA'XX' spin system should display a major component at = ± /2 = ± + due to the sinusoidal modulation of the second term in Eq. 15 during the pump probe delay. A much smaller contribution to the 31 P spectrum is expected at = ± /2 = ± + and at = 0 Hz due to the third term in Eq. 15, which is modulated by ( cos ). The relative contribution of this term is determined by the parameter, (~6% in this case). Figure 8a presents a 2D pump-probe 31

D. NMR Relaxation
The relaxation times for the ZQ coherences, T2,ZQ, and the longitudinal two-spin-order term, , , were determined for the three ruthenium di-hydrides 1, 2, and 3 in C6D6 (3 atm p-H2, 295 K) using the pulse sequence presented in Figure 1d. In this pulse sequence, the fixed delay immediately following the laser pulse maximises the evolution of the initial p-H2-derived singlet state into observable triplet states. In the case of an AX spin system this delay is set to = 1/ (4 ) such that the density matrix at the end of this delay contains contributions from only the longitudinal two-spin order term and ZQy. In the case of the AA'XX' system, the delay is set to = 1/ ). The fixed delay is followed by an echo, where a broadband 180° pulse is used to refocus any evolution of the ZQy coherence due to a difference in chemical shift between the hydrides and/or a difference in J-coupling between the hydride and other heteronuclei in the complex. This echo is not strictly necessary in order to measure the relaxation rates; however, it is convenient because it allows for the use of arbitrary pump-probe delays without the need to fully sample the ZQ coherence evolution frequencies. For the measurements of 2, broadband 31 P decoupling was applied during signal acquisition to simplify the hydride region of the 1 H NMR spectra and improve the signal-to-noise.
The relaxation rate of the various spin states can be determined in a single pseudo-2D experiment by repeating the pulse sequence in Figure 1d for a range of echo times, , with = 45°. If the initial delay is well calibrated, the density matrix at the end of the echo period will only contain contributions from ZQy and 2 . Spectra acquired following a 45° pulse can therefore be divided into contributions from the zero quantum coherence and the longitudinal two-spin order term through the selective integration of the hydride peaks as outlined in section II.C.1 (see Eq. 13). A summary of the relaxation times for the three complexes together with thermal (single-quantum) T1 values, measured using a standard inversion recovery sequence, are reported in Table 3. For the three complexes examined in this work, we observed minimal 1 H relaxation on the timescale of the magnetic oscillations during the pump-probe delay. This was because, as illustrated in Table 3, all three complexes have relaxation times on the order of 100 ms or longer, while the period of oscillation of the NMR signals ranged from 3 -12 ms. However, such comparatively long relaxation times are not a requirement of the method. In order to observe hyperpolarisation using the pump-probe approach, it is only necessary that the zero-quantum relaxation times are equal to or longer than the period of the magnetic oscillations that give rise to observable NMR signals (i.e , , , 2 ( ) for chemically inequivalent hydrides or , , , ( ) for magnetically inequivalent hydrides). Therefore, in contrast to the time averaged PHIP approach, in the synchronised pump-probe method it is possible to observe p-H2 hyperpolarisation for systems with zero-quantum relaxation times down to a few milliseconds. This was illustrated in the case of the complex Ru(H)2(CO)(PPh3)3 in Ref. [34], where hyperpolarisation was observed despite zero-quantum relaxation times on the order of 3 ms.

V. Conclusions
In this work, we have explored the use of laser pump, NMR probe spectroscopy with parahydrogen hyperpolarisation to directly observe the evolution of p-H2-derived spin order for a range of coupling networks. The laser initiation step generates an ensemble of hyperpolarised product molecules in a coherent magnetic state, allowing us to follow the evolution of this state on a fast timescale (tens to hundreds of µs) without the need for a time-consuming rf preparation step. We have presented a theoretical framework to describe the coherent magnetic evolution of the initial p-H2-derived singlet state when the p-H2 protons form part of an AX, AXY, AXYZ, or AA'XX' spin system in the product molecule. This model was validated through the study of three ruthenium dihydride complexes.
The complex Ru(H)2(CO)2(dpae) 1, a model for an AX spin system, was studied in three different NMR solvents for which the difference in chemical shift between the hydrides varies from 0.2 to 0.4 ppm (120 Hz to 240 Hz at 14 T). The magnetic oscillations observed during the pump-probe delay fit the predicted behaviour for all terms in the density matrix with a high degree of accuracy, demonstrating both the validity of our model and the reproducibility of the pump-probe technique.
2D 1 H and 31 P pump-probe NMR spectra of the complex Ru(H)2(CO)(PPh3)(dppp) 2 were presented where the second dimension corresponds to evolution during the probe delay.
These spectra were found to be in excellent agreement with our theoretical predictions for an AXY spin system. Both the major mer and minor fac isomers of 2 were observed in the hyperpolarised 1 H NMR spectra. The coupling patterns observed for the two isomers demonstrated the difference between the behaviour of an AXY and an AXYZ spin system and further validated our theoretical model. In accordance with the predictions, 31 P hyperpolarisation was observed for the mer isomer following simultaneous 90°rf excitation pulses applied to both the 31 P and 1 H nuclei. 31 P hyperpolarisation was also observed following the application of a single 90° pulse to the 31 P channel when broadband rf irradiation was applied to the 1 H channel during 31 P signal acquisition.
The complex cis-Ru(H)2(dppe)2 3 was studied as a model of an AA'XX' system, where the former p-H2 1 H's are chemically equivalent but magnetically inequivalent via the difference in J coupling to a pair of chemically-equivalent 31 P nuclei. A 2D 31 P pump-probe NMR spectrum of 3 was presented, where selective 1 H decoupling of the non-hydride resonances was used to narrow the 31 P lines such that anti-phase signal cancellation was reduced. The resultant high-resolution 2D 31 P spectrum displayed the same features as the 2D 1 H pump-probe spectrum (published previously) [34] in agreement with predictions from our theoretical analysis.
In all of these examples, we considered the evolution of the p-H2-derived nuclear spin states in the product molecule in the high-field regime and in the absence of rf irradiation during the pump-probe delay. However, this method could be extended to probe the transfer of p-H2-derived polarisation directly in a range of different scenarios. Of particular interest would be the transfer of polarisation via rf irradiation in the high field [74,78] and the spontaneous transfer of polarisation in the low-field regime, which has direct implications for optimising hyperpolarisation using the signal amplification by reversible exchange (SABRE) methodology. [10] In this work we have focused exclusively on the use of p-H2 as the source of the hyperpolarised long-lived nuclear spin state. However, p-H2 is just one of a growing number of examples of molecules that contain nuclei in a LLS. [42,55,84] Therefore this method is not fundamentally limited to the study of reactions of H2 but could also be used to monitor the reactivity of any molecule that contains nuclei prepared in a LLS, as long as the reaction can be initiated photochemically and the symmetry of the nuclei in the LLS is broken in a suitable way in the product molecule.