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Zero Quantum Coherence in a Series of Covalent Spin-Correlated Radical Pairs Jordan N. Nelson, Matthew D. Krzyaniak, Noah E. Horwitz, Brandon K. Rugg, Brian T Phelan, and Michael R. Wasielewski J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b00587 • Publication Date (Web): 03 Mar 2017 Downloaded from http://pubs.acs.org on March 5, 2017

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The Journal of Physical Chemistry

Zero Quantum Coherence in a Series of Covalent Spin-Correlated Radical Pairs Jordan N. Nelson, Matthew D. Krzyaniak, Noah E. Horwitz, Brandon K. Rugg, Brian T. Phelan, and Michael R. Wasielewski* Department of Chemistry and Argonne-Northwestern Solar Energy Research (ANSER) Center, Northwestern University, Evanston, Illinois 60208-3113 *email: [email protected] ABSTRACT Photo-initiated sub-nanosecond electron transfer within covalently linked electron donoracceptor molecules can result in the formation of a spin-correlated radical pair (SCRP) with a well-defined initial singlet spin configuration. Subsequent coherent mixing between the SCRP singlet and triplet ms = 0 spin states, the so-called zero quantum coherence (ZQC), is of potential interest in quantum information processing applications because the ZQC can be probed using pulse electron paramagnetic resonance (pulse-EPR) techniques. Here, pulse-EPR spectroscopy is utilized to examine the ZQC oscillation frequencies and ZQC dephasing in three structurally well-defined D-A systems. While transitions between the singlet and triplet ms = 0 spin states are formally forbidden (∆ms = 0), they can be addressed using specific microwave pulse turning angles to map information from the ZQC onto observable single quantum coherences. In addition, by using structural variations to tune the singlet-triplet energy gap, the ZQC frequencies determined for this series of molecules indicates a stronger dependence on the electronic g-factor than on electron-nuclear hyperfine interactions.

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INTRODUCTION Coherent spin states of electrons and nuclei are being actively investigated for quantum information processing (QIP) applications using pulse magnetic resonance techniques.1-4 An important challenge in this field is the development of spin systems in which the initial spin states are well defined. Photoexcitation of covalently linked donor (D) - chromophore (C) acceptor (A) triads (D-C-A) has been shown to result in sub-nanosecond non-adiabatic electron transfer leading to D+•-C-A-• spin-correlated radical pairs (SCRPs) with well-defined initial singlet spin states (|).5-21 The initially formed singlet SCRP, 1(D+•-C-A-•), undergoes hyperfine coupling-induced intersystem crossing in a few nanoseconds to produce the triplet SCRP, 3(D+•C-A-•), provided that the electron spin-spin exchange (J) and dipolar (D) interactions between the two radicals are comparable to, or smaller than, the electron-nuclear hyperfine interactions in the radicals. Application of a magnetic field, B0, results in Zeeman splitting of the SCRP triplet

sublevels, which at the high fields typical of EPR spectroscopy are the | 〉, | 〉, and | 〉

eigenstates that are quantized along B0, while the | state energy remains field invariant (Figure 1A).22-39 A

T+1

T+1

B a

S

a

ωZQC

T0

ΦB

ΦA e e

T−1

T−1

Figure 1. (A) Zeeman splitting of RP energy levels (J > 0); (B) Relative energies of the SCRP states and transitions in the high magnetic field limit following mixing of |〉 and | 〉 to yield ΦA and ΦB, which are initially populated. The arrows indicate the allowed microwave-induced transitions responsible for the observed spin-polarized EPR spectrum.


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The magnitude of J decreases exponentially with SCRP distance r, while that of D diminishes by 1/r3. Thus, in the high magnetic field limit at long inter-radical distances, typically ≥15-20Å, population of the SCRP triplet state occurs exclusively by coherent mixing between

the | and | states for which ms = 0, the so-called zero quantum coherence (ZQC), which is

of potential interest to QIP because the spin states can be probed using pulse electron paramagnetic resonance (pulse-EPR) techniques. The ability to probe these states derives from

the fact that the mixed states |Φ 〉 and |Φ 〉 both have triplet character (Figure 1B), so that a

finite transition probability from these mixed states to the initially unpopulated | 〉 and | 〉

states exists.40-44 The resulting experimentally observed EPR spectrum of the SCRP is a superposition of the four-line spectra for all possible nuclear spin states of the SCRP, as well as

all orientations with respect to B0.41, 43-44 The ZQC mixing frequency ( ) indicated in Figure 1B is a function of the g-factor difference between the two radicals, the difference in the sums of their hyperfine interactions, J, and D (see Theory section below).41, 43-44

Since microwave-induced transitions between the | and | states are formally forbidden

(∆ms = 0), probing in SCRPs is challenging, yet remains a necessary step towards

characterizing spin coherence in SCRPs.45-46 Accurate measurements of complement the

electronic and magnetic information furnished from long-lived single quantum coherences (SQC) in photogenerated SCRPs traditionally measured by transient EPR spectroscopy using continuous microwave irradiation (TCW-EPR).47 In this report, we will refer to the observable EPR signal to which ZQC is transferred as either SQC or transverse magnetization. In the SCRP model, each radical experiences different internal magnetic environments,

which allows for coherent interconversion between the | and | states at a frequency

until a stationary mixed state is achieved. Thus, this forbidden transition can be detected as a 3 ACS Paragon Plus Environment


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time-dependent modulation of the signal obtained from the allowed SQC in the EPR spectra observed following SCRP generation using a single laser pulse. For example, in photosynthetic reaction center proteins, quantum beats attributed to ZQC were first observed using TCW-EPR48 and later using pulse-EPR spectroscopy.49 This phenomenon can be more generally thought of as a signature of photogenerated SCRPs because it has also been observed in the time-resolved EPR spectra of donor-acceptor molecules used to model photosynthesis.50-53 An early perturbation treatment predicted ZQC frequencies from 3-20 MHz due to

instrument and spin relaxation considerations.54 Numerical calculations of with an oscillation period of about 60 ns based on spectral parameters of the secondary SCRP in green

plant Photosystem I are within this frequency range.48 If the oscillation period is similar to the ZQC dephasing time, a fairly common occurrence, the determination of is difficult. ZQC

dephasing times in SCRPs are generally only available as upper or lower limits in the experimental literature. Additionally, when quantum beats of SCRPs are observed by TCW-EPR spectroscopy to obtain , special care must be taken to eliminate other contributions to the signal using experimental or data analysis techniques.51, 55 At the early times most relevant to

short-lived ZQCs, the exposure of a transient species to continuous microwave radiation induces background transient nutations (Torrey oscillations).56-58 Additionally, electron-nuclear coherences in TCW-EPR59 or in selective (long) pulse-EPR60 spectra, although promising for applications to hybrid magnetic resonance QIP, otherwise complicate or prevent identification of electron-electron ZQC. As a consequence, comparisons of between different experimental

studies are difficult. These effects can be avoided using broadband pulse-EPR, which is advantageous compared to TCW-EPR spectroscopy for quantifying and the ZQC dephasing time, as well as providing the potential for selecting specific two dimensional (2D) 4 ACS Paragon Plus Environment

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correlations and fixed pulse turning angles. More importantly, with respect to QIP, pulse techniques are already established in EPR spectroscopy as benchmarks for quantum control and error correction.61 In this work, we demonstrate the use of calibrated microwave pulse turning angles to observe ZQC in a series of structurally related SCRPs. THEORY Hamiltonian. We adopt here the spin Hamiltonian of Tang and Norris,62 which considers the electron-electron exchange (), the spin-spin dipolar coupling () and the electron-nuclear hyperfine interaction. In the rotating frame with the z-axis defined along the static magnetic field , and ℏ ≡ 1

ℋ = ( − )! + (# − )#! + ! ∑& %& '&! + #! ∑& %#& '&! + (# − 2 ∙

2# + + # ((3 -./ # 0 − 1)[!# − 2 ( + 1)]

(1)

where and # are the g-values for each radical, is the Bohr magneton,

is the static

microwave frequency, % is the hyperfine coupling constant (e.g. between nucleus 4 and electron

1), 0 is the angle between the principal axis of dipolar interaction and external magnetic field,

and and ' are the usual spin operators.

The first four terms in ℋ represent differences in the internal magnetic field experienced by

the two radicals, which allow for ZQC. In the rotating frame of the external magnetic field, ,

the hyperfine effects and the electron Zeeman interactions for each radical in the | and |

spin configurations are subtractive, such that ∆ or ∆% give the effective contributions. These

first two terms are additive for | and | , such that they are energetically inaccessible to mixing at high field (ca. 350 mT for X-band EPR). The last two terms describe an energy difference that exists for any two radicals interacting at a given distance. Accordingly, when 5 ACS Paragon Plus Environment


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and are on the same order of magnitude or less compared to ∆ and hyperfine interactions,

mixing of | and | occurs.

Eigenbasis. Neglecting the hyperfine interaction, there are four spin states for a radical pair and the Hamiltonian in this basis is represented by a 4x4 matrix. For an SCRP in an applied external magnetic field in the rotating frame the states |, | are not eigenstates of ℋ––that is, there

are nonzero off-diagonal elements 6 |ℋ| and 6|ℋ| , and so | and | are not

stationary. To form the appropriate eigenbasis, two stationary, mixed states |Φ7 and |Φ8

replace | and | . These mixed states can be determined by diagonalizing ℋ with a unitary matrix whose columns are normalized eigenvectors of ℋ:63 1 0 0 0 -./ > /?@ > : = -./ > ;0 0 0

where > = 2 CD@−1 (6 1

such that

260 |E|0

0 |E|0 − 60 |E|0

0 0B 0 1A

0 G1 −2 HI +∑J(%1J −%2J )KJ

+ = 2 CD@−1 F 1

1 2+ (3-./2 L −1) 3

(2)

M=

N O

|Φ7 = cos > | + sin > |

|Φ8 = cos > | − sin > |

(3)

(4) (5)

This state-mixing explains the presence of four antiphase transitions (two for each radical) commonly observed in TCW-EPR spectroscopy of SCRPs. As > represents the normalized

phase coherence between | and | , it is again clear that and in the denominator (O ) must be on the same order of magnitude or less than ∆ and ∆% (N) for maximal mixing.

The difference in the eigenvalues for |Φ7 and |Φ8 gives the frequency of ZQC and the

energy gap UVW − UVX between the two mixed states in angular frequency units. 6 ACS Paragon Plus Environment

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∆UVW VX = 2Y = ZN2 + O2

(6)

Thus, the frequency of ZQC and the phase coherence > are intimately related, but a small > does not necessarily indicate a small .

Two-dimensional (2D) time domain measurements. Because the change in net magnetization from zero-quantum transitions cannot be detected directly with the standard EPR resonator geometry (∆K[\ = 0), it becomes necessary to monitor the time evolution of the ZQC indirectly by first transferring it to a SQC using an appropriate microwave pulse. This becomes possible by monitoring the evolution of the free induction decay (FID) in time τ following a single microwave pulse as a function of the time T between the laser pulse that generates the SCRP and the microwave pulse. The density matrix formalism describing the time evolution of the spin systems for the one, two, and three pulse experiments is described elsewhere.64-65 In these theoretical treatments, the transverse magnetization for the quadrature detection of the FID is determined analytically or

numerically from the trace of the ] or ^ operator acting on the fully-evolved density matrix. The analytical expression for the FID following a single microwave pulse is known, considering a single dipolar orientation 0 and uniform hyperfine field.66 However, in our experiments,

multiple nuclear spins result in a range of values. A more general insight is that ZQCs are generated only by the initial laser flash, while SQCs and double quantum coherences (DQCs) are generated by the first microwave pulse. Therefore, oscillations in the SQC as a function of the delay time () after the laser flash can only be attributed to ZQC. Assessing ZQC using pulse turning angle dependencies. To understand observed SQCs following a microwave pulse it is useful to examine the density matrix. For the compounds



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studied here, the initial spin density matrix _(0) begins as a pure singlet state. The time evolution of the density matrix can be described in the singlet/triplet basis after the laser flash as: _() = : ‡ exp −?ℋ : _(0) : ‡ exp ?ℋ :

(7)

where the diagonal form of ℋ is utilized for convenience in determining analytical expressions

for the spin density matrix after the laser flash _(). This expression gives four nonzero matrix elements with populations _dd (), _ee () on the diagonal and ZQC on the off-diagonal.

_de () = _ed ()‡ = sin# (2>) [cos 2 >f1 − cosG∆UVW VX H + ? sinG∆UVW VX Hg] #

_ee () = # sin#(2>) [1 − cos(UVW VX )]

(9)

_dd () = 1 − _ee ()

] :

(8)

(10)

Following the effect of a microwave pulse, hi , with an arbitrary turning angle j and phase 0 hi = exp ?j ] = exp ?j k0 1 0

0 0 0 0

1 0 0 1

the θ dependence of the components of the spin density matrix is clear: vv qrs[yt] y√y qrs[t] qv o _ (, j) √y n ruvq (w) 1 − _ee () -./[j]_de () n −qrs[t] √y ‡ _(j) = hi _()hi = n ruvv (w) # n−qrs[yt] y√y -./[j]_ed () _ee () cos[j] n ru (w) ruvv (w) z{ qrs[t] vq qrs[yt] y√y m √y

ru (w)

ru (w)

0 0l 1 0

(11)

z{

~ −qrs[t] √y ~ ruvv (w)~ −qrs[yt] y√y ~ ~ _|| (, j) } ruqv (w)

(12)

where the double quantum coherence (DQC) and some of the populations are abbreviated, but can be determined with straightforward matrix multiplication. Only the SQCs, noted in bold in eq 12, result in an observable signal following time

evolution in τ when ] or ^ is ultimately applied to the spin density matrix. There are two

turning angle dependencies for these terms: the contribution from ZQC and that from the 8 ACS Paragon Plus Environment

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populations which have sin j and sin 2j dependence, respectively. As such, a perfect Y/2 pulse maximizes transference of ZQC to the observable SQCs. The utilization of a distinct turning angle pulse is therefore of great advantage to measuring and identifying ZQC of SCRPs with pulse-EPR spectroscopy. EXPERIMENTAL SECTION The synthesis of 1 and 2 are described in the Supporting Information, and the synthesis of 3 has been previously described.67 Optical Spectroscopy. A Shimadzu UV-1800 spectrometer was used to measure the steady state optical absorption spectra of solution samples. Femtosecond transient absorption experiments were performed employing a regeneratively amplified Ti:sapphire laser system operating at 828 nm and a 1 kHz repetition rate as previously described.68-69 The output of the amplifier was frequency-doubled to 414 nm using a BBO crystal and that light was used to pump a laboratorybuilt collinear optical parametric (OPA) amplifier70 for visible-light excitation. Approximately 13 mW of the fundamental was focused onto a sapphire disk to generate the visible white-light probe spanning 430-850 nm, or into a 5 mm quartz cuvette containing a 1:1 mixture of H2O:D2O to generate a UV/visible white light probe spanning 385-750 nm, or onto a proprietary medium (Ultrafast Systems, LLC) to generate the NIR white-light probe spanning 850-1620 nm. The total instrument response function was 300 fs. Experiments were performed at a randomized pump polarization to suppress contributions from orientational dynamics. Spectral and kinetic data were collected with a CMOS or InGaAs array detector for visible and NIR detection, respectively, and a 8 ns pump-probe delay track (customized Helios, Ultrafast Systems, LLC). Transient spectra were averaged for at least 3 seconds. Solution samples had an absorbance of 0.2-0.7 at the excitation wavelength and were



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irradiated in 2 mm quartz cuvettes with 0.4-0.8 µJ/pulse focused to ca. 0.2 mm diameter spot. Samples were stirred to avoid effects of local heating or sample degradation. The samples were prepared in the glovebox and degassed by multiple freeze-pump-thaw cycles prior to analysis. For nanosecond transient absorption (nsTA) experiments at 295 K, samples were prepared in toluene with an optical density of 0.4-0.7 in a 2 mm cuvette that was free-pump-thaw degassed three times to 10-4 torr. Samples for low temperature (200K) nsTA experiments were prepared in toluene at 0.4-0.7 optical density at 414 nm and sealed under nitrogen atmosphere in a glovebox between two quartz windows, separated with a 2 mm Teflon spacer. This cell was secured in a Janis VNF-100 cryostat and the temperature was monitored with a Cryo-Con 32B temperature controller. The frequency-tripled output of a Continuum Precision II 8000 Nd-YAG laser was used to pump a Continuum Panther OPO to generate a pump excitation beam with a 7 ns, 416 nm, 1.5 mJ pulse focused to a 5 mm diameter at the sample. A xenon flashlamp (EG&G ElectroOptics FX-249) generated the probe pulse, which was then overlapped on the sample with the pump spot size focused to slightly larger than the probe. Kinetic traces were recorded with an oscilloscope (LeCroy Wavesurfer 42Xs) interfaced with a customized LabVIEW program (LabVIEW v.8.6.1), and acquired using a monochromator and Hamamatsu R928 photomultiplier tube, acquired in 5 nm intervals from 430–800 nm. Spectra were constructed by merging the kinetic traces (averaged over 10-20 ns, 150 shots per kinetic trace). Factoring of the two-dimensional (signal vs time & frequency) data set by Singular Value Decomposition (SVD) was performed as implemented in the MATLAB software package.71 This factoring produces an orthonormal set of basis spectra that describe the wavelength dependence of the species and a corresponding set of orthogonal vectors that describe the time-dependent amplitudes of the basis spectra.72 These kinetic vectors were fit using a first-order kinetic model


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with a rate matrix that is solved numerically with MATLAB. The solutions were convoluted with a Gaussian instrument response function before least-squares fitting using a LevenbergMarquardt method was employed to match the kinetic data. EPR Spectroscopy. Compounds were dissolved in toluene to achieve an optical density of 0.5 in 2 mm cuvette at 416 nm (ca. 10–4 M). To dry quartz tubes (4.0 mm o.d. × 3.0 mm i.d.), 0.5 mL of this solution was loaded, degassed with three freeze-pump-thaw cycles, and flame-sealed under vacuum (

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