Density Functional Theory Study of Reaction Equilibria in Signal Amplification by Reversible Exchange

Abstract

An in-depth theoretical analysis of key chemical equilibria in Signal Amplification by Reversible Exchange (SABRE) is provided, employing density functional theory calculations to characterize the likely reaction network. For all reactions in the network, the potential energy surface is probed to identify minimum energy pathways. Energy barriers and transition states are calculated and harmonic transition state theory is applied to calculate exchange rates that approximate experimental values. The reaction network energy surface can be modulated by chemical potentials that account for the dependence on concentration, temperature, and partial pressure of molecular constituents (hydrogen, methanol, pyridine) supplied to the experiment under equilibrium conditions. We show that, under typical experimental conditions, the Gibbs free energies of the two key states involved in pyridine-hydrogen exchange at the common Ir-IMes catalyst system in methanol are essentially the same, i.e., nearly optimal for SABRE. We also show that a methanol-containing intermediate is plausible as a transient species in the process.

Graphical Abstract

Density functional theory calculations are used to analyze the detailed mechanisms and reaction energies of Signal Amplification by Reversible Exchange (SABRE). Theoretical calculations rationalize typical reaction conditions and indicate that the pyridine-H₂ exchange process on the Ir-IMes catalyst follows a dissociative interchange mechanism. The presented framework can aid developments of new SABRE systems in the future.

Introduction

Magnetic resonance (MR) is a powerful tool for studying the structures of molecules in chemical, bio-chemical, and biomedical research. Conventional MR in the form of nuclear magnetic resonance (NMR) or magnetic resonance imaging (MRI) is restricted by low thermal spin polarization, generating low sensitivity and requiring large superconducting magnets.^[1–3] Hyperpolarization overcomes the low inherent sensitivity limitations of traditional MR by perturbing the polarization of nuclear spins far beyond thermal equilibrium conditions, enhancing the signals by four to nine orders of magnitude.^[4–7] Common techniques to achieve hyperpolarization include dynamic nuclear polarization (DNP)^[8,9], para-hydrogen induced polarization (PHIP)^[8–10], and spin exchange optical pumping (SEOP)^[8,11–13]. Among these techniques, PHIP is a robust, inexpensive method that utilizes singlet spin order of the para spin isomer in hydrogen gas (parahydrogen, p-H₂) to generate hyperpolarization on a wide range of substrates.^[14–16] Traditionally, hydrogenative addition of p-H₂ to a target substrate is utilized in PHIP.^[9,17] Signal Amplification by Reversible Exchange (SABRE) is an extension of PHIP where parahydrogen spin order is transferred to target substrates during reversible exchange processes on an organometallic catalyst.^[4,18,19] Critically, SABRE does not chemically modify the target substrate and therefore can provide p-H₂ derived hyperpolarization continuously in a room temperature solution.^[4,20]

In the first demonstrations of SABRE, pyridine and its common derivatives were studied as robust substrates to identify and characterize hyperpolarization effects due to the simplicity of the pyridine system.^[17] However, since the inception of this mechanism in 2009, the breadth of SABRE has greatly expanded with corresponding chemical and physical advances. These include the ability to hyperpolarize I=1/2 heteronuclei (e.g. ¹³C, ¹⁵N) directly with SABRE-SHEATH (SABRE in Shield Enables Alignment Transfer to Heteronuclei)^[21–23] and the ability to hyperpolarize substrates indirectly through exchangeable protons using SABRE-RELAY^[24–27]. In the meantime, a wide range of target compounds have also been investigated as SABRE substrates, with a substantial focus on compounds of biological interest for applications in hyperpolarized molecular imaging.^{[5,9,19,20,23,28,29]} Catalyst optimization has also been demonstrated for the several substrates.^{[5,17,19,20,23]} Nevertheless, the Ir-IMes catalyst (IMes = 1,3-bis(2,4,6-trimethylphenyl)-imidazole-2-ylidene)) used in many early experiments persists as one of the most robust catalysts for a wide range of substrates.^[17,20,23] Figure 1 shows a schematic illustration of the basic processes involved in SABRE, for the example of the Ir-IMes catalyst, pyridine as a substrate, and methanol as additionally present solvent molecules. This set of constituents is also the primary subject of the computational study presented in this work.

Figure 1.

The dynamic Signal Amplification by Reversible Exchange (SABRE) process, showing possible exchange processes of different ligands (pyridine, methanol, and hydrogen) at the Ir-IMes catalyst in solution. Color scheme: H atoms - light grey spheres, with or without red halo depending on depicted polarization state; C atoms - darker grey spheres; N atoms - blue spheres; O atoms - red spheres.

Understanding the chemical exchange and polarization transfer mechanisms behind the SABRE catalyst system is a crucial topic to facilitate novel substrate developments and catalyst optimization.^[20,30,31] Several previous reports have studied intermediate species in the substrate/hydrogen exchange processes that are relevant to the Ir-IMes catalyst.^[32–35] Specifically, the transient Ir-hydride species, a hallmark of the main catalytic complex [Ir(H)₂(IMes)(py)₃]+ (A1), schematically drawn in Figure 2 together with several other conceivable complexes, has been clearly identified.^[36] Chemical exchange saturation transfer (CEST) experiments confirmed the existence of low-concentration intermediates [Ir(Cl)(IMes)(py)₂(H)₂] and [Ir(CD₃OD)(IMes)(py)₂(H)₂] in room temperature solutions in the pyridine/Ir-IMes system.^[32] Solvent molecule coordination with the catalyst has also been supported by experimental evidence that observed polarization of aqueous protons in the solvent.^[37,38] In addition, multiple studies have experimentally investigated the chemical kinetics in the SABRE process to clarify the dynamics of the SABRE chemical system. One common theory assumes that the substrate exchange reactions are dissociative (S_N1-like), and that p-H₂ exchange is associative (S_N2-like), which are supported by experimental data.^[20,39] A “SABRE formula” that describes the dependence of substrate polarization on kinetic parameters has also been derived to integrate both chemical kinetics and spin evolution into a compact model for simulation.^[20,39,40] Notably, Cowley et al. built upon experimental kinetic results and used density functional theory (DFT) calculations to decipher atomic details of the intermediate species and reaction mechanism.^[17] Based on the common Ir-IMes and pyridine system, their study proposed a reaction mechanism centered around the [Ir(H)₂(η2-H₂)-(IMes)(py)₂]+ (A2) complex and involving an intermediate [Ir(H)₂(IMes)(py)₂]+ (C2). Zero-point and thermal free energies of ground states and transition states were calculated to reveal the energetics of the studied reactions.^[17]

Figure 2.

The Ir-IMes based complexes considered in this study and the rationales for considerations. Cross marks (unstable) and tick marks (stable) are used to indicate the absence or existence of locally stable minimum-energy geometries of the complexes at the DFT-PBE+TS theory level. Stable complexes (gray shaded text boxes) are further explored in the reaction network and minimum energy path (MEP) calculations in Figures 3, 4, and 5 below.

In the present paper, we conduct a theoretical study that adds to past research by considering a complete reaction network of pyridine, methanol, and hydrogen exchange reactions surrounding the main complex [Ir(H)₂(IMes)(py)₃]+ (A1) through electronic structure theory. A1 and several possible, related complexes are schematically visualized in Figure 2. Importantly, our computational analysis connects to experimentally controllable conditions via chemical potentials, accounting for temperature, partial pressures and concentration of different substrates. Detailed calculations of pathway energetics (as shown in Figure 3 and Section 3 of the SI) and video visualizations of the reactions (links provided in the caption of Figure 3) provide an atomic-scale understanding of the chemical exchange mechanisms in SABRE. The key step that emerges is a concerted py→H₂ exchange reaction step (A1→A2). For this step, rate constants were calculated from the energy barrier via harmonic transition state theory,^[41–43] showing excellent agreement with experimental results. Within the (A2) complex, H₂ exchange is rapid through a H₂-hydride conformational change. Analysis in terms of the reaction network and associated energetics shows that the presence of the C2 complex is not energetically favored in the dissociative interchange (I_D) mechanism of substrate exchange. We unravel its competition with methanol as a ligand through the lower-energy complex [Ir(H)₂(IMes)(py)₂(CH₃OH)]+ (A3). The Gibbs free energy-based reaction network including substrate chemical potentials shows that C2 and A3 remain competitive with one another as minority species, but they should have a much lower equilibrium concentration than the more favorable A1 and A2 species at common SABRE experimental conditions. Compared to A3, C2 is additionally disfavored since a higher kinetic barrier must be overcome to reach C2 from the main catalytic complex, A1. In short, our results indicate that concerted dissociative interchange mechanisms (py with H₂, py with methanol, methanol with H₂) drive the SABRE process, with an overall preference for the direct py↔︎H₂ exchange reaction (A1↔A2).

Figure 3.

Reactions and pathway energetics from minimum energy path (MEP) calculations. (LEFT COLUMN) Chemical equations for the exchange reactions a through i (see labels in the central column). The forward and reverse energies of activation are labelled. (CENTRAL COLUMN) The energy diagrams from MEP calculations. The x-axis is the image number in the MEP, and the y-axis is the relative total energy, where Eref is a sum of E(A1) and isolated gas-phase energies of species H2, py and MeOH needed to balance the number of atoms in the reactions shown (see Eq. (1) and Sec. 4.1 of the SI). Each image is marked with a cross, with the climbing image labelled as solid square. The end points are labelled with orange diamonds, where intermolecular interactions between separate molecules are not considered. Note that the reaction barriers are defined from the nearest minimum of the MEP on either side of the energy maximum. For example, the barrier to tear off a methanol molecule from state A3 in pathway g is given as 0.31 eV, with reference to the detached but still combined state (C2 + MeOH) where the methanol still has some van der Waals bonding to the Ir-IMes complex. The fully (infinitely) separated system, e.g., C2 in vacuum and MeOH in vacuum is shown as the orange symbol on the left and has a higher energy than the bound state A3. (RIGHT COLUMN) The geometries of the transition states. The MEP calculations are archived in the Novel Materials Discovery (NOMAD) repository (see Data Availability section). The MEPs are visualized as movies and deposited on Figshare. The DOIs are:

a) https://dx.doi.org/10.6084/m9.figshare.13185641

b) https://dx.doi.org/10.6084/m9.figshare.13219691

c) https://dx.doi.org/10.6084/m9.figshare.13219697

d) https://dx.doi.org/10.6084/m9.figshare.13219745

e) https://dx.doi.org/10.6084/m9.figshare.13219757

f) https://dx.doi.org/10.6084/m9.figshare.13219763

g) https://dx.doi.org/10.6084/m9.figshare.13219931

h) https://dx.doi.org/10.6084/m9.figshare.13219949

i) https://dx.doi.org/10.6084/m9.figshare.13219970

As detailed in the methods section, the reaction network energies in our study were computed on the Born-Oppenheimer potential energy surface (“total energy” for short) using the FHI-aims code package^[44,45], a high-precision implementation^[46,47] of all-electron, full-potential DFT, at the level of the Perdew-Burke-Ernzerhof (PBE) functional^[48] and additionally including the Tkatchenko-Scheffler (TS) term^[49] to capture van der Waals effects. This combination of methods has been shown to capture subtle conformational equilibria of complex molecular systems with high precision in the past^[50–54] and has also been applied to previous SABRE problems.^[4,44,55,56] The string method and the climbing image technique^[57–59] were used to probe and determine the minimum energy path for all the exchange reactions in the network. The reaction network was also adjusted via chemical potential terms to account for concentration, temperature, and pressure dependences of the substrates at typical experimental conditions. The understanding of the chemical exchange reaction mechanisms that is thus achieved will provide crucial insights into improving the hyperpolarization performance in SABRE and is also expected to aid future developments of alternative catalysts optimized for other substrates.

One point of note is that our analysis assesses equilibria with neutral ligands, but not with anion ligands that can also play a role, such as Cl⁻.^[32] The reason to omit Cl⁻ is purely technical. First, it is well known that anions are not well represented by many widely used approximate density functionals, including the PBE functional used here, due to their inherent delocalization error, as documented particularly for Cl⁻ in a solvated environment.^[60] Second, our calculations do not explicitly consider solvation effects (beyond chemical potentials that reflect solution-gas phase equilibria that can be experimentally controlled). Accounting for free-energy effects in solution by explicit simulations would require dynamics and sampling to a degree that, for DFT, significantly exceeds the range of computational feasibility for this study. The use of implicit solvation^[61], on the other hand, may introduce uncertainties due to the complex solvent-accessible surface of the Ir-IMes catalyst. Thus, our study relies instead on some degree of error cancellation between the effects of solvation on different species. This assumption is plausible for neutral species and for the magnitude of energy differences (several tenths of an eV) that are relevant in our work. For the strong electrostatic change introduced by a Cl⁻ anion, on the other hand, the same error cancellation may no longer be justified. Finding an effective way to faithfully include exchange reactions involving anions remains an important topic for future computational methodology developments.

Results and Discussion

An initial list of candidate intermediate structures that could conceivably arise during ligand association and dissociation to and from the main catalyst species [Ir(H)₂(IMes)(py)₃]+ (A1) was considered in this work. Assuming the standard Ir-IMes and pyridine system in methanol solvent, we examined possible coordination between the H₂, pyridine, and methanol molecules with the Ir-IMes catalyst. Figure 2 shows these structures and whether they are at least locally stable under computational geometry optimization using DFT-PBE+TS for the underlying potential energy surface (see section 2 of the SI). Among the structures considered in Figure 2, C1, A4, A5, and A6 contain non-classical hydrides, which are rare and do not form locally stable geometries, i.e., structures in which all ligands remain attached to the Ir center. The oxidation numbers of iridium in these complexes (I) are different from that in A1 (III). B4 is a 20-electron complex and will undergo ligand dissociation to minimize energy. For the remaining, stable complexes, energies at the local minimum on the Born-Oppenheimer potential energy surface were obtained.

For all chemical exchange reactions connecting the catalytic structures, energy profiles were investigated by minimum energy path (MEP) and transition state calculations, with the results shown in Figure 3. The MEPs were obtained via the String Method, and the transition states were further approached using the Climbing Image Technique (see methods for details).^[57–59] The results reveal the energy landscape of SABRE catalytic exchange reactions, as well as the atomic-scale mechanism of how the reactions proceed. Note that the reaction barriers defined in Figure 3 assume the barrier to be the difference between the nearest local energy minimum on either side of the reaction and the maximum energy barrier in between. The barrier is thus only related to changing the local state of the complex, and does not include the hypothetical energy needed to remove one of the components to infinite distance in vacuum. In the complex’s actual, solvated state, we expect the local reaction barriers to remain meaningful, whereas this might not be the case for a hypothetical, computationally assumed barrier between vacuum species. The relative energies of the hypothetical, infinitely separate species in vacuum are also given in Figure 3, as orange symbols in each plot. These calculated vacuum energies remain meaningful in the determination of equilibrium energies of each species (as opposed to the kinetic barriers), since the energies of equilibrium species can later be thermodynamically corrected by additional free-energy terms as shown below.

Based on the reaction pathways and energy barrier data, Figure 4 introduces a reaction network that sums up the pyridine/Ir-IMes SABRE system in the presence of H₂ and methanol at the DFT-PBE+TS Born-Oppenheimer potential energy surface (i.e., this network does not yet include temperature- or chemical potential-dependent energy contributions). The relative energies in Figure 4 are calculated with respect to the common reference energy of the lowest energy catalyst, A1. The energies of all other states are adjusted by adding or subtracting the corresponding substrate molecules (H₂, pyridine, CH₃OH in their gas-phase relaxed geometries) to match the number of atoms in each catalyst complex. For example, the energies of complexes A1 and A2 are adjusted according to Equations (1) to enable physical energy comparison between the two molecules.

Figure 4.

Reaction network for the Ir-IMes and pyridine SABRE catalyst system. Total energies relative to gas phase species for each complex are labeled on the y-axis. The calculated energy barriers for the exchange reactions are labeled above or below the corresponding reaction arrows. The reactions are labelled a-i, using the same scheme as in Figure 3.

$$E_{A1,adjusted}=E_{A1,BO}+E_{H_2,BO}+E_{C{H}_{3}OH,BO}$$

$$E_{A2,adjusted}=E_{A2,BO}+E_{py,BO}+E_{C{H}_{3}OH,BO}$$

$$E_{A2,adjusted}-E_{A1,adjusted}=E_{A2,BO}+E_{py,BO}-E_{A1,BO}-E_{H_2,BO}$$ (1)

The symbols $E_{X,BO}$ denote the calculated total energy of the isolated species X at the Born-Oppenheimer surface and $E_{X,adjusted}$ indicate sums of total energies to partially balance the presence or absence of additional species. As presented in Figure 4, the corresponding energies (relative to gas phase species) of the catalytic intermediates as well as reaction pathway activation energies already give some critical insights into the competing reaction pathways of SABRE chemical exchange.

Key takeaways from the MEP results and from the reaction network

The minimum energy pathways (Figure 3) show that substrate exchanges in the pyridine/Ir-IMes SABRE system appear to follow the dissociative interchange (I_D) mechanism. Interchange mechanisms (I) are between the pure associative (A) and dissociative (D) mechanisms, and can be characterized by an intermediate that involves manifestly partially bound versions of both species at once. In dissociative interchange (I_D) mechanisms, the ligand dissociates first to give room for the association of the nucleophile. To illustrate this point and to allow for similar considerations for all pathways in the reaction network, we have deposited interactive animations of the calculated pathways on Figshare. The links and DOIs are provided in the Data Availability section as well as in Section 3 of the supplementary information.

As shown in Figure 3, reaction pathways a, b, and c all account for substrate (i.e., pyridine) dissociation from the Ir-IMes catalyst, in line with the experimental and kinetics data in the literature for the same SABRE catalyst system.^[20,39,40] As already indicated in previous literature^[17,39], pathways a and b confirm that a 7-coordination intermediate does not occur, i.e., a fully associative mechanism can be ruled out. The present work suggests that a fully dissociative mechanism is also kinetically unfavorable. Figure 3 shows that the energy barriers associated with a and b are lower than that of pathway c, i.e., formation of the fully dissociated, 5-coordinated intermediate C2 is unfavorable. Instead, pathways a and b suggest a dissociative interchange (I_D) mechanism, in which the dissociation of an equatorial pyridine occurs in the presence of MeOH or H₂ in a concerted process. At the total-energy level only (Figure 4), the intermediate C2 (pyridine dissociation with no replacement by another species) seems thermodynamically out of reach; however, we note that including free-energy and chemical potential terms in Figure 5 can change this picture, as discussed further below. Nevertheless, the sheer kinetic barrier to simply tearing off a py molecule in pathway c is larger than the barriers in pathways a and b.

Figure 5.

The SABRE reaction network for different thermodynamically relevant energy expressions: (a) total energy at the Born-Oppenheimer approximation (BO), (b) free-energy expression including zero-point and vibrational energies at T=300 K (BO+ZPE+vib), and (c) Gibbs free energy, i.e., energies adjusted to include the approximate chemical potentials of H₂, py, and methanol at 300 K and a set of experimentally relevant conditions (BO+ZPE+vib+μ): 7 atm p-H₂ pressure, 100 mM pyridine concentration, and liquid-phase methanol as the solvent at its vapor pressure (see Section 4 in the SI for details). The reference energy in each subgraph was taken as the corresponding thermodynamic energy of the A1 complex (adjusted energy as defined in Eq. (1)). The small energy difference (~0.04 eV) shown between species A2 and A2’ is a consequence of minor conformational differences within the full complexes at the end points of reaction d (see Figure S4). The difference is an indication of the overall accuracy of the calculation in the face of a complex conformational energy landscape. Physically, this difference would vanish in an appropriate but computationally expensive thermodynamic average over the attainable configurational spaces of A2 and A2’, e.g., in a molecular dynamics simulation.

The reaction network (Figure 4) also outlines several hydrogen exchange reaction routes, through which polarization can be transferred from p-H₂ to the substrates via chemical exchange. Pathways a, d, e and h reveal the mechanisms and energetics for these hydrogen exchange reactions. Among them, reactions a and h show how the Ir-IMes catalyst incorporates the p-H₂ in the solution either through direct H₂ coordination or through pyridine-H₂ exchange. These pathways put p-H₂ and substrates in close proximity and enable the polarization transfer to the substrates. Reaction pathway d depicts an important H₂-hydride exchange pathway, through which polarization in the catalyst system can be refreshed with a very low energy barrier (0.209 eV). This pathway represents an important step for “p-H₂ coordination, H₂-hydride exchange, and o-H₂ departure” that can provide continuous polarization to the catalyst. The low energy barrier for this H₂-hydride exchange indicates that this step in unlikely to be the rate limiting step for parahydrogen repolarization on the catalyst, and ultimately hyperpolarization build-up. Parahydrogen repolarization on the catalyst is more likely to be limited by H₂ diffusion in the solution or by the pyridine exchange, as we detail below. Reaction e shows the H₂-H₂ exchange at A2, where a p-H₂ molecule in the solution exchanges with a ligating H₂ on the catalyst. It also provides a possible route for polarization refreshment through p-/o-H₂ association. The combination of these reactions demonstrates an equilibrium among free p-/o-H₂ in solution, ligating H₂, and hydrides in the Ir-IMes catalysts.

The reaction network also shows the effect of methanol coordination in lowering the energy barrier of pyridine-H₂ exchange reactions, as shown in Figures 3 and 4. Both equatorial (A3) and axial (A3*) coordination of methanol as a substrate in the octahedral conformation of the Ir-IMes catalyst were explored. The equatorial binding position (A3) for methanol is energetically favored because it has a lower total energy and a lower activation energy in the reaction leading to its formation (i.e. C2→A3 has a lower activation energy than C2→A3*). More importantly, the A3-assisted pyridine-H₂ exchange process (pathway b+f) could be competitive with the direct process, A1→A2. As shown in the reaction network, the solvent-assisted pathway b+f has a lower combined energy barrier (0.93 eV) than the dissociative pathway c+h (1.35 eV) where the stable intermediate C2 is formed. Thus, methanol involvement can play a role in SABRE while the formation of the high-energy C2 intermediate is kinetically disfavored. However, as we show below, adjusting the energies to the thermodynamically relevant Gibbs free energy differences at given supply (i.e., partial pressure) of each reactant can change this picture and bring down the Gibbs free energy of C2 compared to A1, A2, and A3, due to the relative scarcity of py in the system compared to methanol and H₂.

Adjustment of the reaction network energies

The energies discussed in Figures 3 and 4 are the total energies under the Born-Oppenheimer approximation at gas phase. However, these energies do not account for thermodynamics in the actual experiment, i.e., temperature, pressure, and concentration dependence of the energy of the various species in the reaction network. To approach the Gibbs free energies of reactions, we calculate the following terms and adjust the reaction network accordingly: zero-point energy and vibrational free energy contribution (both in the harmonic approximation) and chemical potential terms as given by concentrations, partial pressure and/or tabulated vapor pressures of the H₂, py, and methanol species under plausible experimental conditions. As shown in Figure 5, the original reaction network depicting total energies (“BO”; Figure 4, repeated in Figure 5a) was first adjusted by including the zero-point energy (ZPE) to approximate the system at temperature T=0 K and by adding the remaining vibrational energy terms (harmonic approximation) at T=300K (“BO+ZPE+vib”, Figure 5b). The chemical potentials of substrate molecules (hydrogen, pyridine, and methanol) were calculated using statistical thermodynamics (partition functions) to account for the concentration and pressure dependence of the reaction Gibbs energy, as shown in Equation (2) below (for nonlinear polyatomic molecule approximation) and further detailed in Section 4.5 of the SI.

$$\mu \left(T,P\right)-E=-k_{B}T·\left\{\frac{3}{2}\ln \left(\frac{2\pi m{k}_{B}T}{h^2}\right)+\ln \left(\frac{k_{B}T}{P}\right)+\ln \left(\frac{1}{\sigma }\sqrt{\frac{\pi T^3}{{\Theta }_{rot,1}{\Theta }_{rot,2}{\Theta }_{rot,3}}}\right)-{\displaystyle \sum }_i\ln \left(1-e^{-\frac{{\Theta }_{vib,i}}{T}}\right)\right\}$$ (2)

By varying the SABRE experimental conditions such as temperature, pressure of p-H₂, and concentration of pyridine, we can adjust the chemical potential terms μ_X of the substrates X. In turn, these variations will be reflected in the Gibbs free energies of the reactions, as well as the shape of the reaction network. In Figure 5c (“BO+ZPE+vib+μ”), we ground these values by taking common SABRE experimental conditions of 300K, 7atm pressure for p-H₂ gas, vapor pressure corresponding to liquid methanol at T=300K, and pyridine concentration at 100 mM, converted to the corresponding py vapor pressure using Henry’s Law (see section 4.5 of the SI).

In the chemical potential consideration of the reaction network, the SABRE exchange mechanism (i.e. the intermediate species that the system goes through) cannot be directly affected by the pyridine concentration, since all potential intermediates A2, A3, C2 contain the same number of associated py species (two). Varying pyridine chemical potential only changes the relative energy of A1 with respect to all other species, and thus the A1-A2 reaction is always the rate-limiting step. In other words, there must be an optimal pyridine concentration, such that the chemical potentials of A1 and A2 are equal, that maximizes the rate of the rate-limiting step. As shown in Figure 5c, where common SABRE experimental conditions were applied, the chemical potentials of A1 and A2 are already qualitatively identical within the level of numerical precision of the computational procedures used in this work. It can be expected that these conditions are close to optimal for the pyridine/Ir-IMes SABRE system, since the facile exchange between H₂ and pyridine is at the heart of the design principle that led to the invention of SABRE almost a decade ago.

A final comment concerns the relative Gibbs free energies of the two competing species C2 and A3, both of which remain considerably less favorable than A1 and A2. Compared to the purely vibrational free energies (“BO+ZPE+vib”, Figure 5b), the inclusion of chemical potential terms for py and methanol drastically lowers the Gibbs free energies of both C2 and A3, compared to A1 (which has three associated py substrates). However, even in its presence as a liquid solvent, methanol does not have a strong enough association tendency to render species A3 more favorable than C2 (which has no associated methanol ligands and thus an “open” site). Thus, under practical conditions, both species A3 and C2 remain competitive with one another as minority species in the solution, but neither can match (or prevent the formation of) the much more favorable, H₂-adsorbed species A2 at the typical experimental H₂ partial pressure of 7 atm.

Applying Harmonic Transition State Theory to Connect to Kinetic Data

In addition to chemical equilibria, kinetic barriers and rates determine the actual function of the SABRE system. Therefore, harmonic transition state theory (HTST)^[41–43] was applied to determine Gibbs free energies for the transition states of several of the reactions shown in Figure 3. We focus first on the rate-determining step of the reaction network (A1→A2) to obtain kinetic understandings of the substrate exchange process. The states A1 and A2 are of key importance to SABRE. A1 promotes the actual polarization transfer and A2 controls the hydrogen exchange. Importantly, the calculated kinetic parameters for the reaction (A1→A2) can be compared to existing SABRE experimental data in the literature, warranting a more precise computational approach for validation.^[17,62,63]

By applying Eyring’s equation, we obtained the Gibbs free energy of the exchange reactions and the rate constant for the rate-determining step of the reaction network, reaction A1→A2, as shown in Equation (3) and further detailed in Section 5 of the SI, where $k_r$ is the rate constant, $\mathrm{\Delta }{G}^{‡}$ is the Gibbs free energy barrier, and $q_{vib/rot,X}$ is the vibrational or rotational molecular partition functions for structure X. The activation energies ($\mathrm{\Delta }{E}_0$) are taken from the MEP results. Due to the solvation shell of the chemical system, translational degrees of freedom are restricted and thus not considered in the calculation of partition functions for the stable intermediates and the transition state ($‡$). Vibrational modes are calculated using the finite difference approach. For the transition states, the imaginary vibrational mode with negative frequency is omitted in the vibrational partition function according to the harmonic transition state approximation.^[41–43]

$$k_r=\frac{k_{B}T}{h·c^{\circ }}·e^{-\frac{\Delta G^{‡}}{k_{B}T}}=\frac{k_{B}T}{h·c^{\circ }}·\frac{\overline{q_{vib}^{‡}{q}_{rot}^{‡}}}{q_{vib,A1}{q}_{rot,A1}\times q_{vib,H_2}{q}_{rot,H_2}}·e^{-\frac{\Delta E_0}{k_{B}T}}$$ (3)

For the rate-determining step of the reaction network (reaction A1→A2), Eyring’s equation based on the DFT-PBE+TS computed data yields enthalpic, entropic, and Gibbs free energy values of $\mathrm{\Delta }{H}^{‡}=0.88eV,\mathrm{\Delta }{S}^{‡}=0.68meV{K}^{-1},\mathrm{\Delta }{G}^{‡}\left(300K\right)=0.68eV$, with a reaction rate of $25.7s^{-1}$. As shown in Table 1, these theoretical values are quantitatively close to the SABRE experimental values in the literature.

Table 1.

The comparison of theoretically calculated (this work) and experimental (literature) kinetic parameters of the SABRE substrate (py) exchange process in the Ir-IMes/pyridine catalyst system, i.e., the key reaction step A1→A2.

Pyridine Exchange at 300K	Rate constant ${\mathit{k}}_{\mathit{r}}$ (s−1)	$\mathrm{\Delta }{\mathit{H}}^{‡}$ (eV)	$\mathrm{\Delta }{\mathit{S}}^{‡}$ (meV K−1)	$\mathrm{\Delta }{\mathit{G}}^{‡}$ (eV)
Theoretical (this work)	25.7	0.88	0.68	0.68
Experimental	22.2 ± 0.5 (at 22 °C)[62,63]	0.964 ± 0.031[17]	1.01 ± 0.093[17]	0.663 ± 0.021[17]

For several other reactions considered in Figure 3, the Gibbs free energy reaction barriers ($\mathrm{\Delta }{G}^{‡}$) were calculated using the same workflow, as detailed in Section 5.5 of the SI. The comparison between total energy barriers and corrected Gibbs free energy barriers is shown in Table 2. Reactions c, g, h, and i appear to be simple “tear-off” processes, according to their MEP energies and transition state geometries. Therefore, harmonic transition state theory was not applied to these reaction transition states.

Table 2.

The comparison between total energy barriers ($\mathrm{\Delta }{E}_0$) and corrected free energy barriers ($\mathrm{\Delta }{G}^{‡}$) for several reactions in the SABRE reaction network, labeled according to their identification in Figure 3. For reactions with different forward and reverse energy barriers, both are listed. The free energy reaction barriers were corrected using the harmonic transition state theory and vibrational mode data calculated via FHI-aims, as detailed in Section 5 of the SI.

Reactions		Total energy barrier ($\mathrm{\Delta }{\mathit{E}}_0$) [eV]	Corrected free energy barrier ($\mathrm{\Delta }{\mathit{G}}^{‡}$) [eV]
a	A1 + H2 → A2 + py	0.88	0.68
	A2 + py → A1 + H2	0.67	0.64
b	A1 + MeOH → A3 + py	0.80	0.72
	A3 + py → A1 + MeOH	0.61	0.42
d	A2 → A2’	0.21	0.15
e	A2’ + H2 → A2’ + H2	0.72	0.52
f	A2 + MeOH → A3 + H2	0.71	0.50
	A3 + H2 → A2 + MeOH	0.33	0.24

The level of agreement between the theoretically computed kinetic values and experimental data for the rate-determining step (A1→A2) lends confidence in the qualitative validity of other local reaction barriers calculated in Figure 3 and shown in the total-energy based network of Figure 4, for which no direct comparison to kinetic data exists in the experimental literature. For example, the free energy barrier associated with the reaction from A1 to A3, i.e., the exchange of pyridine with methanol, is similar to that of the A1→A2 reaction. The low Gibbs free energy value of A3 in Figure 5c (compared to the direct A1→A2 barrier) and the low kinetic barrier between A3 and A2 thus imply that pyridine exchange with a methanol-associated intermediate species may well also play a role in the overall SABRE process. In the experimental literature data given in Table 1, the effect of this intermediate state A3 cannot be separated out on a microscopic level and would still be accounted for in the overall measured rate. Chemical exchange saturation transfer (CEST) experiments reported in Reference [32] show A3 as a short-lived intermediate. This is fully consistent with the Gibbs free energy balance (Figure 5c) and free energy barriers reported here, based on which A3 is reachable but should have a much lower equilibrium concentration than either A1 or A2. In contrast, the calculated barrier between A1 and C2 (mere py detachment without simultaneous replacement by another species) is much higher. Although C2 appears as a thermodynamically competitive species with A3 in Figure 5c, it is expected to be kinetically disfavored as an intermediate in the overall exchange between A1 and A2.

Conclusion

In this work, density functional theory calculations were used to probe the potential energy surface surrounding the SABRE Ir-IMes catalyst and a series of exchange reactions involving H₂, pyridine, and methanol as substrates. A reaction network was constructed from the minimum energy path results, and further adjusted by including different thermodynamically relevant energy terms (total energies, zero-point energies, vibrational energies, and chemical potentials). The relative energies of complexes in the reaction network reveals the selectivity among chemical species and how they relate to the SABRE technique.

The mechanistic details of the MEP calculations indicate that pyridine exchange reactions in this Ir-IMes/pyridine SABRE system follow the dissociative interchange (I_D) mechanism. The A1-A2 pyridine-H₂ exchange reaction is identified as the most favorable pathway to refresh py and as the rate-limiting step. The pure dissociative intermediate (species C2) is shown to be thermodynamically unfavorable at the total energy level, yet still achievable when the Gibbs free-energy terms are considered. However, the large calculated energy barrier between A1 and C2 makes C2 kinetically unfavored as an intermediate, and thus only exists as a minority species compared to the much more favorable intermediate A2.

Among the hydrogen exchange reactions explored in this work, the H₂-hydride exchange (reaction d) within the intermediate A2 outlines an important pathway for polarization refreshment in the catalyst system through p-H₂ association and o-H₂ dissociation. Its low energy barrier shows that this step is fast and unlikely to be the rate limiting step for parahydrogen repolarization on the catalyst, and ultimately hyperpolarization build-up.

Furthermore, methanol coordination to the Ir-IMes catalyst at the equatorial position (A3) is found to be more favored than the axial position (A3*). The MEP results also illustrate a role of A3 as a possible transient intermediate in a methanol-assisted pyridine-H₂ exchange process (pathway b+f). Compared to C2 as an intermediate, A3 lowers the energy barrier and becomes qualitatively competitive with the direct process A1→A2. Based on the low Gibbs free energy and energy barriers of A3, it is a reachable species in the overall SABRE process, but should have a much lower equilibrium concentration than either A1 or A2. This theoretical conclusion agrees with the experimental observations from CEST experiments that A3 is a short-lived intermediate.

The chemical potential-level adjustment of the reaction network further shows that, under common SABRE experimental conditions, the Gibbs free energies of A1 and A2 are qualitatively equal, leading to similar abundances of both key species in the H₂-py exchange process and making the overall SABRE process nearly optimal. As a final validation, harmonic transition state theory is applied the rate-limiting step (reaction a) to derive the kinetic parameters. The calculated rate constants, enthalpic, entropic, and Gibbs energy terms for substrate exchange yield values that match closely with experimental data in the literature.

To sum up, this work provides a theoretical understanding of the SABRE exchange reactions at a mechanistic and energetic level. This theoretical guidance will facilitate experimental optimizations of the SABRE process, and benefit future developments of alternative SABRE catalysts.

Computational Methods

(1). Ground State Energy Calculations

Born-Oppenheimer potential energies were calculated using the FHI-aims all-electron electronic structure code with the code’s preconstructed “tight” numerical settings for basis sets, integration grids and expansion order of the electrostatic potential.^[44] The PBE semilocal density functional was used,^[48] corrected by the Tkatchenko-Scheffler (TS) term to incorporate van der Waals interactions (PBE+TS)^[49]. The atomic zero-order regular approximation (ZORA) was employed as a scalar relativistic treatment^[44,64–66] using the specific form of Eqs. (55) and (56) of Ref. [44] and proven to match results from other scalar relativistic implementations in past benchmark work.^[46] The threshold magnitude of energy gradients below which structure optimizations were considered to have converged to a local potential-energy surface minimum was set to $5\times {10}^{-3}$ eV/Å. This set of methods, including the numerical settings, has been shown to address molecular conformational problems with high accuracy in past studies.^{[4,44,50–56]} The overall charge state for the Ir-IMes catalyst with different neutral ligands considered in the calculations is always +1.

(2). The String Method and Climbing Image Technique

The minimum energy paths (MEP) and transition state energies reported in this study were found using the string method and the climbing image technique implemented via the “aimsChain” code package in FHI-aims.^[44,57–59] The same density functional with van der Waals correction (PBE+TS), together with the atomic ZORA relativistic treatment and the FHI-aims code’s preconstructed “tight” numerical settings were used.^{[44,48,49,64–66]} In the context of MEP discussions in this paper and its supplementary information, the term “image” was used to denote the individual atomic structures along the minimum energy pathways.

The MEP computations started with linear interpolation of the pathways between minima on the BO potential energy surface, i.e., with both ends (the initial and final images) relaxed to their nearest local energy minimum. Each image was relaxed according to the projection of the negative gradient of the potential energy surface perpendicular to the estimated MEP. Cubic spline interpolations were then performed to re-interpolate the images to equal distance along the MEP. The iteration continues until the maximum residual projected potential force in the system is less than 0.01 eV/Å. This convergence threshold has been shown to provide very accurate representation of the MEP.^[4,44,55,56] Upon convergence of the MEP, the climbing image technique was applied to the highest energy state to accurately approach the true transition state. The convergence threshold was also set to 0.01 eV/Å of residual potential force in the transition state.

(3). Energy Jump Explanations

There exist sudden energy jumps or drops between adjacent images in some reaction pathways from the calculated MEP energetics. This phenomenon can be explained by the bonding energies and does not affect the accuracy and physicality of the reaction network. For example, the MEP of reaction c exhibits a steep jump between images 2 and 3, as shown in Figure 3. The detailed geometries of the two images (as shown in Supplementary Information) reveal that the breaking of the Ir-N bond accounts for the steep energy increase. This issue stems from the method of interpolation of used with the string method, as the images were interpolated to equal distance on the minimum energy path. Therefore, the reaction coordinate in the energy diagrams is not a physical axis, and gentle energy changes are not guaranteed with this method. All other steep energy changes in the reaction network can be explained in a similar manner.

(4). Vibrational Free Energy Calculations

The ground state energies were corrected by their vibrational energies (harmonic approximation) via the “vibrations” code package in FHI-aims^[44,53] The finite differences approach was taken to calculate the vibrational modes and corresponding energies. The small finite displacement of each atom was chosen to be 0.0025 Å. Rotational and ideal-gas translational energies were included for calculated equilibria with a gas phase species or known or assumed partial pressure (H₂, methanol, pyridine) but were not considered for Ir-IMes based species, which are inherently considered to be in solution in this work.

(5). Data Availability

All datasets and raw files generated via the FHI-aims-based calculations and described above in this work (including geometry optimizations, minimum energy paths, and vibrational free energy calculations) are hosted in the Novel Materials Discovery (NOMAD) repository at the following DOI link: https://dx.doi.org/10.17172/NOMAD/2021.07.08-2.

Optimized ground state geometries for the stable species in Figure 4 are visualized and deposited on Figshare, with the following DOIs. The DOIs are also provided in Section 2 of the SI.

• A1) https://doi.org/10.6084/m9.figshare.13322516

• A2) https://doi.org/10.6084/m9.figshare.13322519

• A2’) https://doi.org/10.6084/m9.figshare.13322522

• A3) https://doi.org/10.6084/m9.figshare.13322528

• A3*) https://doi.org/10.6084/m9.figshare.13322552

• C2) https://doi.org/10.6084/m9.figshare.13322558

The minimum energy path calculated for reactions a through i are further visualized as movies and deposited on Figshare, with the following DOIs. The DOIs are also provided in Section 3 of the SI.

1. https://dx.doi.org/10.6084/m9.figshare.13185641

2. https://dx.doi.org/10.6084/m9.figshare.13219691

3. https://dx.doi.org/10.6084/m9.figshare.13219697

4. https://dx.doi.org/10.6084/m9.figshare.13219745

5. https://dx.doi.org/10.6084/m9.figshare.13219757

6. https://dx.doi.org/10.6084/m9.figshare.13219763

7. https://dx.doi.org/10.6084/m9.figshare.13219931

8. https://dx.doi.org/10.6084/m9.figshare.13219949

9. https://dx.doi.org/10.6084/m9.figshare.13219970