Subscriber access provided by University of Washington | Libraries Physical Insights into Quantum Phenomena and Function Quantitative Structure-Based Prediction of Electron Spin Decoherence in Organic Radicals Elizabeth R. Canarie, Samuel M. Jahn, and Stefan Stoll J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.0c00768 • Publication Date (Web): 13 Apr 2020 Downloaded from pubs.acs.org on April 16, 2020

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1 2 3 4 5 6 7 Quantitative Structure-Based Prediction of Electron 8 9 10 11 12 Spin Decoherence in Organic Radicals 13 14 15 16 Elizabeth R. Canarie‡, Samuel M. Jahn‡, Stefan Stoll* 17 18 19 20 Department of Chemistry, University of Washington, Seattle, Washington, United States, 98195 21 22 23 Corresponding Author 24 25 *E-mail: [email protected] 26 27 28 29 ABSTRACT The decoherence, or dephasing, of electron spins in paramagnetic molecules limits 30 31 32 sensitivity and resolution in electron paramagnetic resonance (EPR) spectroscopy, and it repre- 33 34 sents a challenge for utilizing paramagnetic molecules as qubit units in quantum information de- 35 36 vices. For organic radicals in dilute frozen aqueous solution at cryogenic temperatures, electron 37 38 spin decoherence is driven by neighboring nuclear spins. Here, we show that this nuclear-spin- 39 40 41 driven decoherence can be quantitatively predicted from molecular structure and solvation geom- 42 43 etry of the radicals. We use a fully deterministic quantum model of the electron spin and up to 44 45 2000 neighboring protons with a static spin Hamiltonian that includes nucleus-nucleus couplings. 46 47 48 We present experiments and simulations on two nitroxide radicals and one trityl radical, which 49 50 have decoherence time scales of 4-5 μs below 60 K. We show that nuclei within 12 Å of the 51 52 electron spin contribute to decoherence, with the strongest impact from protons 4-8 Å away. 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment 1 The Journal of Physical Chemistry Letters Page 2 of 15

1 2 3 TOC GRAPHICS 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 KEYWORDS 23 24 25 26 nuclear spin diffusion, cluster correlation expansion, phase memory time, coherence, EPR, DNP 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment 2 Page 3 of 15 The Journal of Physical Chemistry Letters

1 2 3 The spins of unpaired electrons in organic radicals and metal ions are extensively used in 4 5 6 pulse electron paramagnetic resonance (EPR) spectroscopy to probe the structure and dynamics of 7 8 the nano-environment around the electrons.1–3 Molecule-based unpaired electrons are also inves- 9 10 tigated as potential building blocks for devices useful to quantum information science (QIS).4–6 In 11 12 13 this context, they are often referred to as “molecular spin qubits”. In both cases, a key limitation 14 15 is the fact that excited electron spins lose coherence over time. This process, called decoherence, 16 17 dephasing, or transverse relaxation, results in the loss of signal. In EPR, decoherence limits sen- 18 19 sitivity and spectral resolution. In QIS applications, it impacts the efficient transfer of information 20 21 22 between coupled qubits, and it limits the complexity of algorithms that can be executed. Extending 23 24 coherence times is therefore an important development goal in both fields. For this, a detailed 25 26 understanding of the physical origins of decoherence is crucial. 27 28 29 There are many processes that drive electron spin decoherence, including molecular mo- 30 31 tions and magnetic interactions7. It is possible to eliminate the effect of motions (librations, thermal 32 33 methyl rotations, etc.) on decoherence by operating at low temperatures. Magnetic interactions 34 35 36 among electron spins, and between electron spins and nearby nuclear spins, also contribute to 37 38 decoherence. Processes due to couplings between electron spins can be eliminated by dilution. At 39 40 sufficiently low temperatures and low electron spin concentrations, decoherence is driven by 41 42 nearby nuclear spins.8,9 This mechanism, often called nuclear spin diffusion, has been described 43 44 45 semi-classically as arising from stochastic energy-conserving flip-flops of pairs of nuclei, leading 46 47 to spectral diffusion of the electron spin resonance frequency and consequently to decoherence.10– 48 49 12 The problem with this stochastic model is that it is not predictive and does not provide insight 50 51 52 into the physical origin of the assumed flip-flop rate. 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment 3 The Journal of Physical Chemistry Letters Page 4 of 15

1 2 3 Here, we experimentally deter- 4 5 6 mine the decoherence behavior of three 7 8 prototypical organic radicals in frozen 9

10 aqueous solutions, at temperatures and Figure 1. Two-pulse echo pulse sequence. For a decoherence experi- 11 ment, the spin echo amplitude is recorded as a function of increasing τ. 12 13 concentrations low enough to eliminate 14 15 contributions of any motional or electron-electron processes to decoherence. We show that the 16 17 nuclear-spin-driven decoherence of the radicals can be quantitatively predicted directly from their 18 19 molecular geometry and solvation structure, using a combination of molecular dynamics (MD) 20 21 22 and quantum spin dynamics, without the need for any adjustable free parameters. 23 24 Electron spin decoherence can be measured using various pulse EPR techniques. The most 25 26 straightforward is the two-pulse echo decay (Figure 1). This technique uses the pulse sequence 27 28 29 2 echo and records the decay of the echo amplitude resulting from increasing 30 31 πthe⁄ inter− 𝜏𝜏-pulse delay − π −𝜏𝜏 . 32 33 We investigated𝜏𝜏 the decoherence characteristics of three different radicals: 2,2,6,6-tetra- 34 35 36 methylpiperidine-1-oxyl (TEMPO), its perdeuterated isotopologue (d18-TEMPO), and a perdeu- 37 13 38 terated trityl radical (p1TAM) (Figure 2). TEMPO and d18-TEMPO both are N/O-centered radi- 39 40 cals with four neighboring methyl groups and differ only in that all 1H atoms in TEMPO are re- 41 42 2 43 placed with H atoms in d18-TEMPO. The trityl 44 45 radical, p1TAM, is a C-centered radical and has 46 47 12 deuterated methyl groups. The concentra- 48 49 tions of the radicals were kept low enough to 50 51

52 minimize additional decoherence effects aris- 53 Figure 2. Radicals used in this study. From left to right, TEMPO, 54 ing from instantaneous diffusion mediated by d18-TEMPO, and p1TAM. 55 56 57 58 59 60 ACS Paragon Plus Environment 4 Page 5 of 15 The Journal of Physical Chemistry Letters

1 2 3 14,15 electron-electron couplings. Electron T1 val- 4 5 6 ues are on the order of 0.1–1 s and do also not 7 8 contribute to decoherence. All samples were pre- 9 10 pared in a solution of 1:1 (w:w) H2O:glycerol 11 12 13 and were snap frozen in liquid nitrogen. The ex- 14 15 periments were performed from 20-60 K at Q- 16 17 band frequencies (ca. 33 GHz). 18 19 Figure 3 shows the experimental two- 20 21 Figure 3. Experimental decoherence behavior of the molecules pulse echo decays at 20 K, presented in black. used in this study (black), stretched exponential (SE) fits (gray), 22 and the structure-based simulations (red). From top to bottom, ap- 23 proximately 10 μM p1TAM, 100 μM d18-TEMPO, and 200 μM 24 For all samples, the coherence decays on a simi- TEMPO. The experiments were performed in 1:1 (w:w) H2O 25 :glycerol at 20 K and ca. 33 GHz. The SE fits gave x values of 3.37, 2.81, and 2.77 and TM values of 5.37, 4.23, and 4.33 μs for 26 lar timescale and is completely lost within 10 μs. 27 p1TAM, d18-TEMPO, and TEMPO, respectively. 28 29 The decays are phenomenologically fitted well by stretched exponentials of the form (2 ) = 30 0 31 exp( (2 / ) ), shown in gray. Here, is the phase memory time, and is a stretching𝑉𝑉 𝜏𝜏 expo-𝑉𝑉 ⋅ 32 𝑥𝑥 33 nent.− This𝜏𝜏 stretched𝑇𝑇M -exponential form is 𝑇𝑇predictedM by the semiclassical model𝑥𝑥 that uses stochastic 34 35 10–12 36 nuclear flip-flops to describe the loss of electron spin coherence. 37 38 The experimental decay shape of p1TAM (Figure 3, top) is the closest of the three mole- 39 40 cules used in this study to a pure stretched exponential. p1TAM does not have hydrons (protons or 41 42 43 deuterons) neighboring the C-centered unpaired electron; the closest hydrons are about 4 Å away. 44 45 The d18-TEMPO decay (Figure 3, middle) shows oscillations at short times resulting from nuclear 46 47 electron spin echo envelope modulations (ESEEM). This effect arises from the pseudo-secular 48 49 parts of the hyperfine coupling to nearby nuclei, the deuterons on the radicals in this case.2,16 No 50 51 52 ESEEM oscillations are visible in the TEMPO sample (Figure 3, bottom). Instead, the TEMPO 53 54 decay shows a slight deviance from the shape of a stretched exponential at early times that may be 55 56 57 58 59 60 ACS Paragon Plus Environment 5 The Journal of Physical Chemistry Letters Page 6 of 15

1 2 3 17,18 the result of thermal or tunneling methyl rotations. The p1TAM decay is slightly slower than 4 5 6 those of TEMPO and d18-TEMPO, since the closest magnetic nuclei are farther from the electron 7 8 spin in p1TAM. 9 10 To investigate the temperature dependence of the decoherence behavior at cryogenic tem- 11 12 13 peratures, we performed experiments from 20-60 K; the decoherence behavior is unaffected by 14 15 temperature in this range (Figures S2 and S3). This is consistent with previous experimental re- 16 17 sults.8 18 19 To predictively model the experimental decoherence behavior, we performed explicit 20 21 22 structure-based spin quantum dynamics simulations of the two-pulse echo decays. Structures of 23 24 the solvated radicals were generated using molecular structures optimized by density functional 25 26 theory (DFT) and solvation geometries obtained by molecular dynamics (MD). The system of 27 28 29 spins used in the spin dynamics calculation comprises the unpaired electron, all magnetic nuclei 30 31 on the radicals, and up to 2000 solvent protons within a radius of up to 20 Å from the electron. 32 33 Full nucleus-nucleus coupling tensors as well as both secular and pseudo-secular parts of hyperfine 34 35 36 coupling tensors between nuclei and the electron spins were included in the static spin Hamiltonian 37 38 39 40 = + 𝑁𝑁 , + ( ) + + 𝑁𝑁−1 𝑁𝑁 , , . 41 T T T 42 𝐻𝐻�0 𝜇𝜇B𝑔𝑔e𝐵𝐵0𝑆𝑆̂𝑧𝑧 ��−𝜇𝜇N𝑔𝑔𝑛𝑛𝐵𝐵0𝐼𝐼̂𝑧𝑧 𝑛𝑛 𝑆𝑆̂𝑧𝑧 𝒛𝒛 𝑨𝑨𝑛𝑛 𝑰𝑰�𝑛𝑛 𝑰𝑰�𝑛𝑛𝑷𝑷𝑛𝑛𝑰𝑰�𝑛𝑛� � � 𝛿𝛿𝑔𝑔𝑛𝑛 𝑔𝑔𝑚𝑚𝑰𝑰�𝑚𝑚𝒃𝒃𝑚𝑚 𝑛𝑛𝑰𝑰�𝑛𝑛 43 The individual terms𝑛𝑛=1 in this Hamiltonian mostly follow standard notation𝑚𝑚=1 𝑛𝑛=𝑚𝑚+1 and are described in the 44 45 Supporting Information. Microwave pulses were modelled as ideal (i.e. infinitely short). 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment 6 Page 7 of 15 The Journal of Physical Chemistry Letters

1 2 3 To handle the enormous spin Hil- 4 5 6 bert space, we employed the ensemble clus- 7 8 ter correlation expansion (CCE) method19,20 9 10 that has been developed for predicting the 11 12 13 decoherence behavior of spin centers in 14 21–24 15 crystals. It is conceptually similar to the 16 17 truncated Liouville space methods em- 18

19 ployed in nuclear magnetic resonance 20 Figure 4. Illustration of the truncation parameters, which are the sys- 21 tem size (purple circle and arrow), the cluster sizes used (rust ar- 25–29 (NMR). The method simulates spin dy- eas), and the clustering based on neighbor cutoff (green arrows). 22 𝑆𝑆𝑆𝑆𝑆𝑆 23 𝑟𝑟 24 namics in Hilbert space using a truncated expansion approach that calculates the signal of the total 25 26 spin system as a combination of the signals from many small clusters of spins (Figure 4). Algo- 27 28 29 rithmic truncation parameters of the CCE method are the size of the total spin system, a neighbor 30 31 cutoff that eliminates clusters with negligible internal couplings, the maximum cluster size used, 32 33 and the number of orientations of the magnetic field to be averaged over. They are shown in Figure 34 35 36 4 and have been optimized to assure convergence of the result. Further details are given in the 37 38 Supporting Information. It is important to note that the simulation is exclusively structure-based 39 40 and does not contain any additional free physical parameters, neither static nor dynamic. 41 42 The simulated echo decays are shown in red in Figure 3. In all, the decoherence behaviors 43 44 45 of all three radicals are quantitatively predicted both in shape and timescale. All three simulations 46 47 result in a stretched exponential decay with oscillations from ESEEM at early times. Eliminating 48 49 nucleus-nucleus couplings from the simulation completely eliminates the decays, showing that 50 51 52 these couplings are key to the mechanism of nuclear-spin-driven electron spin decoherence. The 53 54 slight timescale discrepancies between simulated and experimental decays may arise from errors 55 56 57 58 59 60 ACS Paragon Plus Environment 7 The Journal of Physical Chemistry Letters Page 8 of 15

1 2 3 in the MD prediction of the solvation geometries and solvent proton densities and from potential 4 5 6 residual contributions of other decoherence mechanisms. The p1TAM simulation is in most agree- 7 8 ment with experiment with or without the inclusion of ESEEM-generating terms in the Hamilto- 9 10 nian. The d18-TEMPO simulation required the inclusion of ESEEM-generating terms in the Ham- 11 12 13 iltonian to reproduce the oscillations at early times in the experimental decays. Inclusion of 14 15 ESEEM in the TEMPO simulation did not reproduce the early time shape of the experimental 16 17 decay. The simulations show increased ESEEM oscillation amplitudes that are likely due to in- 18 19 complete averaging over solvation geometries. A measurement and simulation of d18-TEMPO at 20 21 22 X-band (Fig. S4) demonstrate that the model works at different magnetic fields. 23 24 To gain more insight into the structural aspects that determine decoherence timescales, we 25 26 devised a method to attribute decoherence effects to individual nuclei in the systems. To do so for 27 28 29 a particular nucleus i, we simulated the echo decay of the system without nucleus i (i.e. assuming 30 31 it as non-magnetic). This results in a slightly prolonged echo decay compared to the full system. 32 33 We capture this by the difference in phase memory times, , , between the full system and the 34 35 Δ𝑇𝑇M 𝑖𝑖 36 reduced system (Figure S7); , quantifies by how much the presence of nuclear spin i shortens 37 M 𝑖𝑖 38 the phase memory time. It is Δ𝑇𝑇important to note that individual values are not additive, as the 39 40 decoherence mechanism is a cooperative effect driven by nucleusΔ𝑇𝑇M-nucleus couplings. Still, they 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment 8 Page 9 of 15 The Journal of Physical Chemistry Letters

1 2 3 allow the assessment of the relative im- 4 5 6 portance of individual nuclei to electron 7 8 spin decoherence, within their context 9 10 of neighboring nuclear spins. 11 12 13 The results of this analysis for 14 15 TEMPO are depicted in Figure 5. Fig- 16 17 ure 5(a) shows a 3D cartoon of the solv- 18 19 ated radical, with each proton colored 20 21 22 according to , , where darker indi- 23 M 𝑖𝑖 24 cates larger values.Δ𝑇𝑇 No clear geometric 25 26 patterns are discernible, although it is 27 28 29 evident that some protons in the vicinity 30 31 of the electron contribute much more 32 Figure 5. (a) Map of decoherence effects ΔTM of individual nuclei for 33 than others. Figure 5(b) shows a scatter- TEMPO in H2O demonstrating which nuclei contribute most to the 34 dephasing of the electron spin. The color scale corresponds to that in 35 (b), with darker blue indicating larger values of ΔTM. (b) Individual 36 plot of for all nuclei as a function ΔTM values as a function of distance (blue, left vertical scale) and cal- 37 culated TM as a function of system size (red, right vertical scale). M 38 of their Δ𝑇𝑇distance from the electron spin. 39 40 The protons 4-8 Å from the electron have the largest effect upon electron spin decoherence. Their 41 42 43 presence each shortens the predicted phase memory time by up to 28 ns. Protons that are less than 44 45 about 4 Å from the electron, such as the methyl protons, contribute much less to decoherence. This 46 47 volume of reduced decoherence contributions corresponds to the notion of the diffusion barrier 48 49 50 observed in nuclear spin diffusion. The value obtained here (4 Å) is somewhat smaller than those 51 8,11 30 31 32,33 52 reported for nuclear diffusion barriers (7-10 Å, 4-6.6 Å , < 6 Å ). However, this direct 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment 9 The Journal of Physical Chemistry Letters Page 10 of 15

1 2 3 comparison is not entirely valid, since we model loss of electron spin coherence, whereas the dif- 4 5 6 fusion barrier relates to spatial transfer of polarization. Beyond 8 Å, the individual taper off 7 M 8 rapidly with increasing distance. Figure 5(b) also plots the calculated overall asΔ𝑇𝑇 a function of 9 M 10 system size and shows that in order to obtain a converged , all nuclei up to𝑇𝑇 at least 12 Å from 11 12 M 13 the electron spin need to be included. The conclusions are𝑇𝑇 similar for d18-TEMPO and p1TAM 14 15 (Figure S8). 16 17 In combination with the experimentally validated simulations, the ability to examine deco- 18 19 20 herence effects due to individual nuclei greatly enhances insight into the structural origins of nu- 21 22 clear-spin-driven electron spin decoherence. As mentioned, this decoherence mechanism has tra- 23 24 ditionally been described semi-classically by a stochastic process of nuclear flip-flops involving 25 26 pairs of surrounding nuclei, with an ad hoc flip rate constant. Within the quantum model presented 27 28 29 herein, however, electron spin decoherence is the result of a coherent, deterministic evolution of a 30 31 large closed system of nuclear spins coupled to each other and to the electron spin, without any 32 33 external stochastic influence. Even after electron spin coherence has decayed, the coherent evolu- 34 35 36 tion of the spin system continues, even though it is not observable. Eventually, T1 processes destroy 37 38 the coherence, and the total spin system is brought back to equilibrium. 39 40 Extending the methodology to fully deuterated solvents is not straightforward. Deuterons 41 42 43 lead to prolonged time scales of the echo decays, and the presence of nuclear quadrupole couplings 44 45 results in deeper echo modulations. The longer time scales pose experimental challenges of isolat- 46 47 ing nuclear-spin-driven dephasing from other processes such as spectral diffusion and instantane- 48 49 ous diffusion. Both the longer time scales as well as the increased modulation depths create con- 50 51 34 52 vergence challenges for the CCE method. 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment 10 Page 11 of 15 The Journal of Physical Chemistry Letters

1 2 3 In summary, we have shown that nuclear-spin-driven electron spin decoherence of radicals 4 5 6 in sufficiently dilute frozen aqueous solutions at cryogenic temperatures can be quantitatively pre- 7 8 dicted directly from the solvation structure of the radicals using explicit spin quantum dynamics 9 10 with a static spin Hamiltonian, without any free parameters. This methodology can be useful for 11 12 13 designing molecules and platforms that maximize electron spin coherence lifetimes, both to in- 14 15 crease sensitivity and resolution in EPR spectroscopy as well as to improve molecular spin qubits 16 17 for potential QIS applications. 18 19 20 21 22 ASSOCIATED CONTENT 23 24 25 26 Supporting Information 27 28 The Supporting Information is available free of charge. Detailed description of experimental pa- 29 rameters, MD simulations, summary of theory, temperature dependence experiments, X-band ex- 30 periment and simulation of d18-TEMPO, decoherence effect maps and plots of TEMPO, d18- 31 TEMPO, and p1TAM (PDF). 32 33 34 AUTHOR INFORMATION 35 36 37 Notes 38 ‡ 39 These authors contributed equally to this work. 40 The authors declare no competing financial interests. 41 42 ORCID 43 Elizabeth R. Canarie: 0000-0002-8393-3927 44 Samuel M. Jahn: 0000-0002-0197-681X 45 Stefan Stoll: 0000-0003-4255-9550 46 47 48 49 50 ACKNOWLEDGMENT 51 52 53 We thank Drs. V. V. Khramtsov and B. Driesschaert (In vivo Multifunctional Magnetic Resonance 54 55 Center, West Virginia University) for providing the p1TAM radical and Drs. Sandra S. Eaton and 56 57 58 59 60 ACS Paragon Plus Environment 11 The Journal of Physical Chemistry Letters Page 12 of 15

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