Research ArticleCHEMICAL PHYSICS

Infinite-order perturbative treatment for quantum evolution with exchange

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Science Advances  07 Aug 2020:
Vol. 6, no. 32, eabb6874
DOI: 10.1126/sciadv.abb6874
  • Fig. 1 SABRE provides an ideal system to challenge the limits of an exchange model, given the complexity of the underlying dynamics.

    (A) SABRE transfers the singlet order of parahydrogen to a target ligand via reversible interactions with an iridium catalyst and exhibits nonlinear dynamics that are highly dependent on the relative concentrations of each species. (B) The coherent hyperpolarization dynamics can be probed by interleaving pulses at or near the SABRE resonance condition (red) with periods far off resonance (B = − 22.5 μT) to allow for exchange. (C) The (15N,13C)-acetonitrile SABRE system demonstrates rich dynamical information that varies with the resonant field as a result of a complex coupling network between 15N, 13C, and 1H. Lines are meant as visual guides only.

  • Fig. 2 Comparison of the traditional master equation solution (Eq. 7, black) and our DMEx models for G2 and G3 pseudorotations.

    We use s-trioxane ring inversion to model G2 pseudorotation (A) and tert-butyl rotation in t-BuPCl2 to model G3 pseudorotation (B) The graphs at the bottom compare the root mean square deviation (RMSD) of the generated magnetization as a function of the time evolution such as in Fig. 1C using Δt = 10 μs (which is taken as the ground truth). The DMEx models retain good fidelity with no additional computational overhead, even with step sizes commonly being 10 times larger than were possible with traditional solutions.

  • Fig. 3 Evaluation of model errors in 15N-SABRE simulations using our SABRE-specific DMExFR2 model, which adds free ligand effects, rebinding, and relaxation (FR2) to the DMEx.

    An example of SABRE hyperpolarization dynamics is shown for reference, calculated on a six-spin 15N SABRE-SHEATH system (A). Comparing the DMEx and a QMC treatment, which is viewed as the “gold standard” but is computationally inefficient, indicates that there is a genuine but small difference of 0.142 ± 0.018%, on average, between the two solutions [red data, (B)]. The convergence error of the QMC is indicated by the black line. This error in the DMEx is attributed to the loss of the time orderings between the quantum and exchange degrees of freedom that are retained in the QMC. Even with nonlinear effects incorporated in the simulation, the DMExFR2 exhibits a larger radius of convergence over the traditional implementation by approximately a factor of 4 (C) and an improved self-consistency (parameterized by σk, the error in the predicted exchange rate) (D), which uses Δt = 10 μs solutions as ground truth.

  • Fig. 4 Importance of modeling SABRE systems using complete models.

    (A) To reduce computational overhead, virtually all calculations to date remove ancillary spins from the system, such as artificially reducing the number of protons on the pyridines in the bis-(15N) SABRE complex from 10 to 2 (A). Doing so alters the hyperpolarization dynamics (blue) as compared to the explicit 14-spin calculation (black), which is stable using the DMExFR2 models for step sizes even up to 5 ms (red). (B) Fitting the 14-spin calculated time evolution with this smaller model produces incorrect values of the exchange rates. (C) Including all relevant exchange pathways when modeling SABRE systems is also crucial for predicting accurate exchange parameters. Here, we fit the experimental (15N,13C)-acetonitrile hyperpolarization dynamics to DMExFR2 models with (solid) and without (dashed) coligand exchange effects. When neglecting these exchange pathways, the predicted exchange rates differ from the correct values by 44 to 92%.

Supplementary Materials

  • Supplementary Materials

    Infinite-order perturbative treatment for quantum evolution with exchange

    Jacob R. Lindale, Shannon L. Eriksson, Christian P. N. Tanner, Warren S. Warren

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