Theory of Solid-State Photo-CIDNP in the Earth's Magnetic Field - The

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Theory of Solid-State Photo-CIDNP in the Earth’s Magnetic Field Gunnar Jeschke,*,† Ben C. Anger,‡ Bela E. Bode,‡ and J€org Matysik*,‡ † ‡

Laboratory of Physical Chemistry, ETH Z€urich, Z€urich, Switzerland Leiden Institute of Chemistry, Leiden, The Netherlands ABSTRACT: To date, solid-state photo-CIDNP experiments have been performed only using magic angle spinning NMR in a high-field regime, which is not associated with physiologically relevant spin dynamics. Here, we predict that nuclear spin polarization up to 10%, almost 9 orders of magnitude larger than thermal equilibrium polarization, can arise in cyclic photoreactions at the earth field due to a coherent three-spin mixing mechanism in the ST or ST+ manifold. The effect is maximal at a distance of about 30 Å between the two radicals, which nearly coincides with the separation between the donor and secondary acceptor in natural photosynthetic reaction centers. Analytical expressions are given for a simple limiting case. Numerical computations for photosynthetic reaction centers show that many nuclei in the chromophores and their vicinity are likely to become polarized. The theory predicts that only modest hyperfine couplings of a few hundred kilohertz are required to generate polarization of more than 1% for radicalradical distances between 20 and 50 Å, that is, for a large number of radical pairs in electron-transfer proteins.

’ INTRODUCTION Photochemically induced dynamic nuclear polarization (photo-CIDNP) arises from transfer of the electron spin order associated with an initial singlet or triplet state of a radical pair to polarization of nuclei that are hyperfine coupled to at least one of the two electron spins (for review, see refs 1 and 2). As a result, the NMR spectrum of the recombination products of the radical pair exhibits enhanced positive (absorptive) or negative (emissive) signals. These signals allow for selective, highly sensitive observation of the products of a photochemical reaction. Their signs and relative intensities provide information on the radical pair state. Since the first observations of CIDNP effects in Bargon (1967),3,4 different mechanisms were revealed for generating such hyperpolarization of nuclear spin transitions.2 By far, most observations pertain to liquid-state NMR at frequencies where the high-field approximation is valid for the electron spins in the radical pair state and for the nuclear spins in the product ground states. In this regime, nuclear polarization is most often generated by the radical pair mechanism (RPM).5,6 In many cases, this nuclear polarization relies on spin sorting between different reaction products from singlet and triplet radical pairs. In cyclic reactions in solution, the RPM can create net nuclear polarization7 if longitudinal nuclear relaxation differs between the singlet and triplet branch. This allows observation of CIDNP enhancements at the surface of proteins8 and has been developed to a technique1 that can provide substantial insight into protein folding.9 This mechanism will also be operative in cyclic photochemical reactions in the solid state,10 has been termed the differential relaxation mechanism in this context,11 and explained the differences between photo-CIDNP spectra of wild-type and r 2011 American Chemical Society

carotenoid-less photosynthetic reaction centers (RC) of Rhodobacter sphaeroides bacteria.12 The first experimental observations of solid-state photoCIDNP effects were made on photosynthetic RCs.13,14 They were later traced back to genuine solid-state photo-CIDNP mechanisms, which arise from anisotropy of the hyperfine coupling. Due to the pseudosecular contribution of an anisotropic hyperfine coupling, electron polarization can be directly transferred to nuclear polarization in a coherent process.15 In the differential decay (DD) mechanism,16 the electron polarization is generated from the initial state of the radical pair due to different recombination rates of singlet and triplet radical pairs, that is, as a result of incoherent reaction dynamics. In the three-spin mixing (TSM) mechanism,17 it is generated by coherent transfer that is driven by the pseudosecular contribution to the electronelectron coupling. The electronelectron coupling can be of the exchange or dipoledipole type. The TSM mechanism is based on two coherent magnetization transfers and can thus be understood and computed by exclusive use of spin quantum mechanics, without taking into account any interaction of the spin system with its environment. The DD and TSM mechanisms explain the majority of solidstate photo-CIDNP observations on native photosynthetic RCs,18 where radical pair recombination is very fast and donor triplet states are also quenched fast, so that differential relaxation is negligible. The DD and TSM mechanisms as formulated to date require matching of the hyperfine and nuclear Zeeman interactions15 and are thus inoperative at the earth field. Received: May 26, 2011 Revised: July 11, 2011 Published: July 18, 2011 9919

dx.doi.org/10.1021/jp204921q | J. Phys. Chem. A 2011, 115, 9919–9928

The Journal of Physical Chemistry A Solid-state photo-CIDNP at high fields critically depends on a matching of three interactions in the case of the TSM mechanism and on a matching of two interactions and the radical pair lifetime in the case of the DD mechanism.11 While the solid-state photoCIDNP effect has been found in various natural photsynthetic RCs,1922 to date, no artificial donoracceptor system has been found that exhibits this effect. In fact, only very recently was solidstate photo-CIDNP observed on any system that is not a photosynthetic RC.23 Yet, solid-state photo-CIDNP is common to different types of photosynthetic RCs of several bacteria and green plants in case electron transfer beyond the primary radical pair is blocked by depletion or reduction of the secondary acceptor quinone. This led to the speculation that occurrence of high-field solid-state CIDNP is correlated to efficient electron transfer in photosynthetic RCs.24 Such correlation may arise from the requirement of both processes for significant but not too strong overlap of molecular orbitals along the electrontransfer pathway. These findings and considerations raise the question whether solid-state photo-CIDNP can also arise under physiological conditions, that is, for unblocked photosynthetic RCs at the earth's magnetic field of approximately 50 μT and at temperatures where plants or photosynthetic bacteria grow. Recent work has already shown that high-field solid-state photo-CIDNP is observable on bacterial photosynthetic RCs at physiologically relevant temperatures.25 Here, we address the question of a possible ultralow-field solid-state photo-CIDNP regime theoretically. In particular, we consider a purely coherent TSM mechanism that does not rely on different reaction rates of singlet and triplet pairs nor on different nucler spin relaxation in the two branches. The paper is organized as follows. In the Theory section, we first identify the characteristics of the low-field solid-state photoCIDNP regime and construct a minimal spin Hamiltonian for discussing this effect. This minimal Hamiltonian is analytically diagonalized, and expressions for nuclear polarization are derived by the product operator formalism.26 The Results and Discussion section starts with a discussion of sign rules and of the conditions under which the effect is maximized. An approximate double matching condition is derived. We then use numerical density operator computations to relax the assumptions that were made in constructing the minimal, analytically tractable Hamiltonian. From the results of these numerical computations, dependence on the donoracceptor distance and isotropic hyperfine coupling is discussed, and consequences of hyperfine anisotropy are considered. By numerical computations, we also test whether significant nuclear polarization is expected for secondary radical pairs of photosynthetic RCs at the earth field. Finally, we discuss the question whether the earth field solid-state CIDNP is expected to be common in electron-transfer proteins.

’ THEORY Characteristics of the Ultralow-Field Solid-State PhotoCIDNP Regime. To find a possible earth field solid-state photo-

CIDNP regime, we first consider the high-field solid-state photoCIDNP effect. In most cases, this effect is based on mixing between the singlet state and the triplet substate with magnetic quantum number MS = 0 (ST0 mixing).5,6 Early liquid-state photo-CIDNP experiments also established that for large exchange coupling J, such as in biradicals, one of the other triplet

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substates T or T+ can be mixed with the singlet state.27 Which state is mixed with the singlet state depends on the sign of J, with ST mixing being the more common case. Transitions between the S and T states can be induced by the isotropic hyperfine coupling. Unlike effects from ST 0 mixing, those from ST or ST + mixing are candidates for an earth field regime because they occur at a matching of the electron Zeeman interaction and the exchange coupling. For sufficiently small couplings, this matching may occur at the earth field. In early liquid-state photo-CIDNP work2729 and in later theoretical descriptions,30 effects due to ST mixing were discussed in the context of diffusion and re-encounter of the radicals. The exchange coupling J was considered to be stochastically time-dependent. Later experimental work proved that the effect can also be observed for cyclic reactions of rigid donoracceptor pairs that transiently form a radical ion pair state.31 Our initial numerical computations revealed that ST mixing or ST+ mixing by a purely coherent TSM mechanism creates nuclear polarization in cyclic reactions. However, at the earth field, ST mixing due to the exchange coupling is unlikely because this would require a coupling strength of about 2.8 MHz, corresponding to a radicalradical distance of more than 15 Å. At such distances, dipoledipole coupling is much larger than exchange coupling32 and would thus cancel the matching. Hence, in the solid state, ST mixing is expected to occur at even longer distances where the dipoledipole coupling matches the transition frequency between the S and T (or S and T+) levels and the exchange coupling is negligibly small. Closer inspection reveals that the secular part of the dipoledipole interaction is sufficient for the polarization transfer to occur. Although the dipoledipole interaction is purely secular only along the principal axes of the coupling tensor, this special case is of interest because it can be treated analytically. Diagonalization of the Spin Hamiltonian for a Special Orientation. For such an analytical treatment, we consider a system consisting of two electron spins S1 = 1/2 and S2 = 1/2 and a nuclear spin I = 1/2 coupled to S1 in a regime where the two electron Zeeman interactions, the coupling between the two electron spins, and the hyperfine coupling are all of the same order of magnitude. In this regime, the nuclear Zeeman interaction is negligibly small. If the two paramagnetic species are organic radicals, such as, for instance, radical ions in a photosynthetic RC, the difference between the two electron Zeeman interactions will also be negligibly small. In particular, we are interested in a radical pair in the earth's magnetic field (∼50 μT), where the electron Zeeman interaction is ωS/2π ≈ 1.4 MHz. The dipoledipole coupling between the two electron spins has this magnitude at a distance of about 30 Å. At such a distance, the exchange coupling between two electron spins is usually negligible.32 In the following, we assume a point-dipole approximation for the dipoledipole coupling, so that the coupling tensor has axial symmetry with the unique axis being parallel to the spinspin vector. Deviations from such axial symmetry due to failure of the point-dipole approximation do not affect the general conclusions drawn in this work or the principle of the analytical derivations below. We take the earth field along the z axis and select a Cartesian basis built from the eigenstates of operators ^S1z, ^S2z, and Î z. For the following analytical computation, we consider a special 9920

dx.doi.org/10.1021/jp204921q |J. Phys. Chem. A 2011, 115, 9919–9928

The Journal of Physical Chemistry A

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orientation where the electronelectron spinspin vector is either parallel or perpendicular to the magnetic field. The hyperfine interaction is either purely isotropic or one of the principal axes of the hyperfine tensor is parallel to the magnetic field. The static spin Hamiltonian in angular frequency units can then be written as ^ 0 ¼ ωS ^S1z þ ωS^S2z þ d^S1z ^S2z þ A^S1zÎ z H þ aiso ð^S1xÎ x þ ^S1yÎ y Þ

ð1Þ

need to be applied to the Hamiltonian given by eq 1. The result can be inferred most easily from a slightly rewritten form of eq 6 ^ ðdiaÞ ¼ (ωS þ A^S1zÎ z H R, β 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ωR, β^S1z ðωR, β ω2R, β þ a2iso Þ^S1z 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 1 þ ðωR, β ω2R, β þ a2iso ÞÎ z ð9Þ 2 Comparison with eqs 1 and 2 reveals that

with the electron Zeeman frequency ωS, the electronelectron dipoledipole coupling d, the isotropic hyperfine coupling aiso, and the secular hyperfine coupling A = aiso + T, where T is the dipoledipole contribution to the hyperfine coupling. This Hamiltonian can be considered as a minimal Hamiltonian for studying the effect under consideration. Because the off-diagonal hyperfine terms do not connect the mS2 = +1/2(SR2 ) and mS2 = 1/2(Sβ2 ) subspaces, the Hamiltonian can be diagonalized in the separated subspaces. The two subspace Hamiltionians are ^ R, β ¼ (ωS þ A^S1zÎ z þ ωR, β^S1z þ aiso ð^S1xÎ x þ ^S1yÎ y Þ H 2 ð2Þ

0 ^ ðdiaÞ H ¼ ωS ^S1z þ ωS^S2z þ d^S1z ^S2z þ A^S1zÎ z ωR ^S1z^SR2 0

þ ωR ^SR2 Î z ωβ^S1z ^Sβ2 þ ωβ^Sβ2 Î z 0

0

where 0

ωR, β ¼ ωR, β -

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2R, β þ a2iso

þ

ð3Þ

UR, β ¼ expfiηR, β ð2^S1yÎ x 2^S1xÎ y Þg

ηR, β

B ¼ arctan@

ωR, β

aiso C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 þ ωR, β þ aiso

0

^ ðdiaÞ H R, β

0

ð5Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωS 1 þ A^S1zÎ z þ ðωR, β þ ω2R, β þ a2iso Þ^S1z ¼( 2 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 1 þ ðωR, β ω2R, β þ a2iso ÞÎ z ð6Þ 2

For describing the spin evolution starting from a pure singlet or triplet state of the radical pair, the Hamiltonian for the whole state space needs to be constructed. This is necessary because in the Cartesian basis, the singlet and triplet states correspond to zero-quantum coherence of the two electron spins. For this, the two commuting unitary transformations U R ¼ expfiηR ð2^S1y ^SR2 Î x 2^S1x ^SR2 Î y Þg 0

ð7Þ

Î z þ ðω0R ω0β Þ^S2zÎ z

0

0

ωR ωβ ¼ d ωR00 þ ωβ00 with

In their eigenbasis, they take the form

2

ωR þ ωβ ¼ 2ωS ωR00 ωβ00

ð4Þ 1

0

ωR þ ωβ

ð12Þ

This expression can be simplified using the substitutions

where 0

ð11Þ

0 0 þ ðd ωR þ ωβ Þ^S1z ^S2z þ A^S1zÎ z 0

The subspace Hamiltonians are diagonalized by the unitary transformations

ð10Þ

By replacing the polarization operators ^SR2 and ^Sβ2 by Cartesian operators and then reordering terms, we obtain ! 0 0 ωR þ ωβ ðdiaÞ ^S1z þ ωS^S2z ^ 0 ¼ ωS H 2

with ωR, β ¼ ωS ( d=2

0

ð13Þ ð14Þ

ωR00 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðωS þ d=2Þ2 þ a2iso

ð15Þ

ωβ00 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðωS d=2Þ2 þ a2iso

ð16Þ

Finally, we find for the diagonalized Hamiltonian ^ ðdiaÞ ¼ H 0

ωR00 þ ωβ00

^S1z þ ωS ^S2z þ ðωR00 ωβ00 Þ^S1z^S2z ! ωR00 þ ωβ00 Î z þ A^S1zÎ z þ ωS 2 2

þ ½d ðωR00 ωβ00 Þ^S2zÎ z

ð17Þ

For aiso f 0 (no state mixing), eqs 17 and 1 revert to the same form, that is, the pseudonuclear Zeeman term (ω0I = ωS (ω00R + ω00β)/2) and the apparent hyperfine coupling to spin S2 (A0 2 = d (ω00R ω00β)) vanish. For brevity, we make the additional substitutions ω0S = (ω00R + ω00β)/2 and d0 = ω00R ω00β, so that the full diagonalized Hamiltonian reads 0 ^ ðdiaÞ H ¼ ωS ^S1z þ ωS^S2z þ d0^S1z ^S2z þ A^S1zÎ z 0

U β ¼ expfiηβ ð2^S1y ^Sβ2 Î x 2^S1x ^Sβ2 Î y Þg 0

0 0 þ ωIÎ z þ A2 ^S2zÎ z

ð8Þ 9921

ð18Þ



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^ 0 is The unitary transformation U = U0R + U0β that diagonalizes H ^ } with given by U = exp{iA ^ ¼ A

ηR þ ηβ ð2^S1yÎ x 2^S1xÎ y Þ 2 ηR ηβ ð4^S1y ^S2zÎ x 4^S1x ^S2zÎ y Þ þ 2

ð19Þ

For the following computations, it is most convenient to express ^ as this operator A ^ ¼ ηð2^S1yÎ x 2^S1xÎ y Þ þ ξð3^S1y ^S2zÎ x 3^S1x^S2zÎ y Þ A

and 1 Æ2^S1x ^S2zÎ x þ 2^S1y ^S2zÎ y æð0Þ ¼ ð7 þ 25 cos 2ξÞ sin 2η 64 ð28Þ The nuclear polarization at time t after generation of the radical pair, starting from a singlet pair (negative two-spin order ^S1z^S2z), can now be computed as ^ ^ 1 expðiH ^ ðdiaÞ ^ ðdiaÞ ÆÎ z æðtÞ ¼ TrfÎ z U 1 expðiH 0 tÞU S1z S2z U 0 tÞUg

ð29Þ

ð20Þ We find

with η¼

ξ¼

ηR þ ηβ 2

ð21Þ

2ðηR ηβ Þ 3

ð22Þ

ÆÎ z æðtÞ ¼ C0 þ Cþ cosðωR00 tÞ þ C cosðωβ00 tÞ with C0 ¼ Cþ ¼

Analytical Expressions for Nuclear Polarization in a Special Orientation. The radical pair is created either in a singlet

! ^ ! ^ ^ ^ ^/4 + ! ^/4 ! S 1 3 S 2. The state E S 1 3 S 2 or in a triplet state 3E ^ is invariant under all transformations and does unit operator E not correspond to observable magnetization; it can thus be dropped. In the Cartesian basis, the scalar product of the spin vector operators of the two electron spins ! ^ ^ ! ð23Þ S 1 3 S 2 ¼ ^S1x^S2x þ ^S1y ^S2y þ ^S1z^S2z can be interpreted as a sum of two-spin order σTSO ¼ ^S1z^S2z

ð24Þ

and zero-quantum coherence

σZQC ¼ ^S1x ^S2x þ ^S1y ^S2y ¼ 1=2 ^Sþ ^S þ ^S ^Sþ

ð25Þ

Evolution of the zero-quantum coherence under the Hamiltonian given by eq 1 does not generate nuclear polarization. After transformation by the unitary operator U to the eigenbasis, twospin order gives a linear combination of diagonal terms with operators ^S1z^S2z, ^S2zÎ z, ^S1z, and Î z and off-diagonal terms with operators ^S1xÎ x, ^S1yÎ y, ^S1x^S2zÎ x, and ^S1y^S2zÎ y. The diagonal terms are constants of motion of the spin evolution of the radical pair. Together, they correspond to nuclear polarization ÆÎ z, const æ ¼

3 sinð4ηÞ½14 sinð2ξÞ þ 25 sinð4ξÞ 1024

ð26Þ

At time t = 0, this nuclear polarization is exactly canceled by the nuclear polarization that arises from the off-diagonal terms. Nuclear polarization is created by evolution of these off^ (dia) . The evolution diagonal terms under the Hamiltonian H 0 relevant for generation of nuclear polarization is a rotation in a phase space spanned by operators ^S1xÎ x + ^S1yÎ y, ^S1yÎ x ^S1xÎ y, ^S1x^S2zÎ x + ^S1y^S2zÎ y, and ^S1y^S2zÎ x ^S1x^S2zÎ y in the eigenbasis of the radical pair Hamiltonian. At time t = 0, the nonvanishing coefficients of these basis operators are given by 3 Æ^S1xÎ x þ ^S1yÎ y æð0Þ ¼ cos2 η sin2 ξ 8

ð30Þ

ð27Þ

3 sinð4ηÞ½14 sinð2ξÞ þ 25 sinð4ξÞ 1024

ð31Þ

ð14 sinð2ηÞ þ sin½2ðη ξÞ þ 49 sin½2ðη þ ξÞÞ2 32768 ð32Þ

C ¼

ð14 sinð2ηÞ þ 49 sin½2ðη ξÞ þ sin½2ðη þ ξÞÞ2 32768 ð33Þ

As no nuclear polarization exists at t = 0, we expect C0 + C+ + C = 0, which is indeed the case. Equations 3033 were also checked by comparison with numerical computations. We now assume that the radical pair has a lifetime τ and eventually recombines to a diamagnetic ground state. More precisely, we assume that the hyperfine coupling of the nucleus is annihilated, so that no further polarization transfer between electron spin and nuclear spin is possible. Such annihiliation can happen in other ways than by recombination of the radical pair, for instance, by onward transfer of the electron to another acceptor for acceptor nuclei or by hydrogen abstraction from an amino acid residue such as tyrosin for donor nuclei. Lifetimes of singlet and triplet pairs are considered to be equal. In other words, the hyperfine coupling is annihilated in a non-spinselective way. The nuclear polarization in the ground-state molecule after complete recombination (t . τ) is then given by Z ∞ et=τ ÆÎ z æðtÞ dt 0 ÆÎ z æ∞ ¼ Z ∞ et=τ dt 0

¼ C0 þ

Cþ C þ 1 þ ωR002 τ2 1 þ ωβ002 τ2

ð34Þ

For ω00Rτ,ω00βτ , 1, the polarization vanishes because C0 + C+ + C = 0. For ω00Rτ,ω00βτ . 1, the terms with coefficients C+ and C vanish. As these terms have a sign opposite to the one of C0, such long lifetimes correspond to maximum nuclear polarization, which assumes the value C0; see eq 31. The derivation of eqs 3034 reveals the mechanism of earth field solid-state photo-CIDNP. Radical pairs are born in a pure singlet or triplet state that involves negative or positive two-spin 9922


The Journal of Physical Chemistry A order -^S1z^S2z, respectively. This two-spin order is transferred to nuclear polarization with opposite sign on the transitions | RS1RS2βIæ T |βS1RS2RIæ and |RS1βS2βIæ T |βS1βS2RIæ. The two transitions are distinguished by a different state of the electron spin S2 that is not coupled to the nuclear spin. If the radical pair recombines instantly or with a lifetime much shorter than the typical frequencies of spin evolution, nuclear polarization will cancel. The isotropic hyperfine coupling aiso mixes state |RS1RS2βIæ with |βS1RS2RIæ on the one hand and state |RS1βS2βIæ with |βS1βS2RIæ on the other hand but does so to a different extent. This different extent of state mixing at the same coupling strength in the RS2 and βS2 manifolds arises from the different level splittings ωS ( d/2. The mixing creates coherence on the two transitions, which evolves with frequencies ω00R and ω00β in the RS2 and βS2 manifold, respectively. This evolution changes the projection of the coherence onto the eigenstates of the nuclear spin in the absence of hyperfine coupling. The eigenstates in the absence of hyperfine coupling are the nuclear spin eigenstates in the ground state (recombination product). Hence, at different times after their creation, radical pairs recombine to products with different, nonvanishing nuclear polarization. If ω00Rτ and ω00βτ are both much larger than unity, the contributions from evolution of coherence on the two transitions vanish in the time average. In contrast, the fraction of nuclear polarization of each transition that is not converted to coherence by mixing is always transferred to the ground state (recombination product). Due to the different extent of state mixing, this fraction of nuclear polarization is different in the RS2 and βS2 manifolds and thus does not cancel upon radical pair recombination. Therefore, nuclear polarization of the recombination product after a single photocycle approaches C0 at sufficiently long radical pair lifetimes.

’ RESULTS AND DISCUSSION Sign Rules. The maximum nuclear polarization C0 is an odd function of angle η as well as an odd function of angle ξ. These two angles in turn depend on the electron Zeeman frequency ωS, which is generally positive, and on the electronelectron coupling d and the isotropic hyperfine coupling aiso via eqs 5, 21, and 22. A sign change of aiso leads to a sign change in both ηR and ηβ and thus to a sign change in both η and ξ, so that the sign of C0 is unchanged. Hence, the sign of nuclear polarization does not depend on the sign of the isotropic hyperfine coupling. In contrast, a sign change of the electronelectron coupling d causes an interchange of the values of ηR and ηβ, which leaves η unchanged and leads to a sign change in ξ. This in turn changes the sign of C0. Hence, the sign of nuclear polarization does depend on the sign of the electronelectron coupling. We also note that the secular hyperfine coupling A is of no consequence for the sign and magnitude of the nuclear polarization. This is because within each of the two pairs of mixed states, the secular hyperfine energy contribution is the same for both states. A difference in the Zeeman frequencies of the two electron spins is of no consequence for the extent of state mixing or for the evolution frequencies. Thus, it does not change the magnitude or sign of the nuclear polarization. In radical ion pairs, such as the ones observed in photosynthetic RCs, the sign of nuclear polarization is therefore the same for donor and acceptor chromophores.

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To this point, the discussion referred to the simple case of the minimal Hamiltonian, for which we have analytical expressions. The situation is more complicated for the orientation-averaged effect of an ensemble of donoracceptor systems with arbitrary orientation with respect to the earth’s field. In this case, different orientations correspond to dipoledipole couplings that differ in amplitude and may differ in sign. Hence, orientation averaging leads to some loss of polarization by cancellation of contributions from orientations with a different sign of the dipoledipole coupling. Such cancellation is not complete, however, because the polarization is not proportional to the dipoledipole coupling. If, in addition, hyperfine anisotropy is considered, no simple sign rules can be formulated. This is because the amplitude ratio between polarization from orientations with positive and negative polarization depends on the relative orientation of the hyperfine coupling tensor and the dipoledipole coupling tensor, as well as on the ratio between hyperfine anisotropy and isotropic hyperfine coupling. These issues are addressed below by numerical computations. Maximum Effect. The derivative of C0 with respect to η is proportional to cos 4η, and the second derivative is proportional to sin 4η. Consequently, the first maximum of C0 is attained for ηmax ¼ π=8 ¼ 22:5

ð35Þ

The derivative of C0 with respect to ξ is proportional to 7 cos 2ξ + 25 cos 4ξ, and the second derivative is proportional to 7 sin 2ξ + 50 sin 4ξ. Both derivatives are also proportional to sin(4η). As the optimum value of η is negative, the second derivative with respect to ξ is negative for small positive ξ. The smallest value of ξ corresponding to a maximum of C0 is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 3 ð31 þ 561Þ ð36Þ ξmax ¼ arccos 10 2 corresponding to an angle of 25.0831. Maximum emissive nuclear polarization is obtained at (ηmax,ξmax). Closer examination shows that these parameters correspond to global extrema because the extrema at larger absolute values of the angle η and ξ cannot be attained due to the definitions of ηR and ηβ. The maximum absolute nuclear polarization after a single photocycle is found by substituting ηmax and ξmax into eq 31. We find rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 pffiffiffi 9ð7 3 þ 3 187Þ ð817 þ 7 561Þ 2 C0, max ¼ ≈0:103551 102400 ð37Þ This maximum absolute polarization of about 10% can be compared to numerical computations for the nuclear polarization generated in a single photocycle at high field (4.7 T). In this case, we find a maximum polarization of 0.37% when we consider all 13C nuclei in the primary radical pair of photosynthetic RCs. Under such conditions high-quality solid-state photo-CIDNP magic-angle spinning NMR spectra were measured. As shown in the Appendix, the maximum amplitude of the nuclear polarization is attained at |aiso| = 0.252ωS and |d| = 1.935ωS. Unsurprisingly, the optimum electronelectron coupling d corresponds to nearly a matching situation |d| = 2ωS. At the earth field, the optimum electronelectron coupling corresponds to the dipoledipole coupling between two electron spins almost exactly at a distance of 30 Å and to an isotropic 9923



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Figure 1. Dependence of the absolute nuclear polarization ÆÎ zæ on the relative hyperfine coupling |aiso|/ωS at optimum relative electron electron coupling |d|/ωS = 1.935.

hyperfine coupling of 353 kHz. The relatively small value of aiso for optimum mixing is caused by the necessity to maximize the difference in state mixing between the SR2 and Sβ2 subspaces. A large aiso leads to strong (and nearly equal) state mixing in both subspaces. However, the decrease of nuclear polarization toward larger isotropic hyperfine couplings is relatively benign (Figure 1), even if the electronelectron coupling is kept fixed at its global optimum instead of considering the optimum value for each particular hyperfine coupling. The findings can be summarized by an approximate double matching condition jωS j≈jd=2j≈4jaiso j

ð38Þ

Numerical Computations with Orientation Average. In the following, we consider the same system consisting of two electron spins S1 = 1/2 and S2 = 1/2 and a nuclear spin I = 1/2 coupled to S1 still in a regime where the two electron Zeeman interactions, the dipoledipole coupling between the two electron spins and the purely isotropic hyperfine coupling, are all of the same order of magnitude. Again, the exchange interaction between the two electron spins is neglected, and the earth's magnetic field is taken along z. The geometry of the system is now characterized by a single angle θ between the magnetic field direction z and the electronelectron spinspin vector. The Hamiltonian in angular frequency units is given by

^ 0, aniso ¼ ωS^S1z þ ωS ^S2z þ aiso ð^S1xÎ x þ ^S1yÎ y þ ^S1zÎ z Þ H þ ð1 3 cos2 θÞd^S1z ^S2z þ ð3 cos2 θ 2Þd^S1x^S2x þ d^S1y^S2y 3 sin θ cos θdð^S1z ^S2x þ ^S1x ^S2z Þ ð39Þ where we have chosen the x axis of the electron spin frames in an way that situates the electronelectron spinspin vector into the xz plane. Due to the additional non-negligible off-diagonal elements, analytical expressions for the nuclear polarization can no longer be derived. Numerical computations were performed for ωS/2π = 1.4 MHz, τ = 2 μs with a time increment of 10 ns for a total evolution time of 10 μs. After this time, 99.3% of all radical pairs had recombined to the diamagnetic ground state. Orientations were averaged by integrating over sin θ dθ = d cos θ. The dipoledipole coupling d and the isotropic hyperfine coupling aiso were varied.

The dependence of the absolute nuclear polarization ÆÎ zæ∞ on dipoledipole coupling d at fixed isotropic hyperfine coupling aiso/2π = 2 MHz and the dependence on isotropic hyperfine coupling aiso at fixed dipoledipole coupling d/2π = 2 MHz are shown in Figure 2a and b, respectively, for ωS/2π = 1.4 MHz. Again, we see that significant nuclear polarization is generated already for isotropic hyperfine couplings of less than 1 MHz. Couplings of this magnitude are rather frequently observed in biologically relevant radical pairs. Significant nuclear polarization is generated over a broad range of hyperfine couplings. The largest hyperfine couplings to be expected in radicals in living cells are on the order of 20 MHz. Even for such large couplings, an absolute polarization of about 2% is expected (Figure 2a). Polarization reduces more strongly when the dipoledipole coupling d is increased beyond the optimum value (Figure 2b). Nevertheless, the range of significant effects is rather broad also with respect to d, in particular, when considering that d is proportional to the inverse cube of the distance. As seen in Figure 3, significant nuclear polarization is created for typical isotropic hyperfine couplings in a distance range from 18 and 50 Å, which is rather typical for radical pairs in electron-transfer proteins. The earth field nuclear polarization should thus be a rather common phenomenon in biological electron-transfer reactions that create radical pairs with a lifetime on the order of 1 μs or longer. Influence of Anisotropic Hyperfine Coupling. Anisotropic hyperfine coupling further modifies nuclear polarization as it leads to additional off-diagonal elements in the spin Hamiltonian and thus to modified level mixing. If we assume an axially symmetric hyperfine tensor with principal values (aiso + T, aiso + T, aiso 2T), this influence depends on angle the θd,T between the electronelectron spinspin vector and the unique axis of the hyperfine tensor. Furthermore, the influence depends on the magnitude T of the anisotropic component of the hyperfine coupling. We studied this influence by numerical simulations for a fixed electronelectron distance of r = 28 Å, roughly corresponding to the distance between the donor radical cation and the secondary acceptor radical anion in photosynthetic RCs. The isotropic hyperfine coupling was fixed at 2 or 5 MHz, while θd,T was varied between 0 and 90 in steps of 2.5 and T was varied between 0 and 20 MHz in steps of 0.5 MHz. The maximum magnitude of the nuclear polarization found in the presence of anisotropic hyperfine coupling is similar to the maximum magnitude found with only isotropic hyperfine coupling (Figure 4). However, in the presence of large anisotropic hyperfine coupling, the nuclear polarization is positive, corresponding to enhanced absorptive NMR signals. This effect can be traced back to the mixing of additional level pairs by hyperfine anisotropy. If the NMR measurements are also performed at the earth field, where it may be difficult to separate signals from different nuclei, such sign inversion can lead to partial cancellation of the photo-CIDNP effect. Limitations of Current Theory. In realistic systems, several magnetic nuclei are hyperfine coupled to the electron spins of the donor and acceptor chromophores. It is evident that polarization transfer from the electron cannot be independent for the individual nuclei because polarizations of several percent per nucleus would add up to more than the initial nonequilibrium magnetization of the radical pair. Hence, the multispin problem needs to be treated explicitly. Unlike for the multispin problems that arise in electron spinecho envelope modulation,33 such a 9924



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Figure 2. Dependence of nuclear polarization ÆÎ zæ∞ (a) on isotropic hyperfine coupling aiso at fixed dipoledipole coupling d/2π = 2 MHz and (b) on dipoledipole coupling d at fixed isotropic hyperfine coupling aiso/2π = 2 MHz. The electron Zeeman frequency ωS/2π = 1.4 MHz corresponds to the earth’s field.

Figure 3. Contour plot of the dependence of nuclear polarization ÆÎ zæ∞ on isotropic hyperfine coupling aiso/2π = 2 MHz and distance r between the two electron spins. Shades of gray correspond to levels of nuclear polarization as indicated by the labels. The electron Zeeman frequency ωS/2π = 1.4 MHz corresponds to the earth’s field.

treatment is not possible simply by factorization into two-spin contributions, that is, by deriving a product rule. Although nuclearnuclear couplings are negligible for the shortest radical pair lifetimes where the earth field solid-state CIDNP is expected, the factorization approach breaks down. This breakdown is due to the influence of each individual nucleus on the quantization axis of the electron spin, which is in turn due to the similar magnitude of the electron Zeeman and isotropic hyperfine interaction. Therefore, nuclear spins will communicate with each other via the electron spin in a nontrivial way, even if their direct couplings are negligible. Consequently, multispin effects need to be addressed by a series of numerical computations that are beyond the scope of the present paper. Work along these lines is in progress. We note that the difficulties that are commonly encountered by such multispin computations pose a problem in inferring information on the radical pair state from the relative CIDNP line intensities. Model Computations for the Secondary Radical Pair of Photosynthetic RCs. To test for relevance of these effects in biological systems, we numerically computed proton polarizations for the donor and acceptor in the secondary radical pair of bacterial photosynthetic RCs of Rhodobacter sphaeroides wild type. The two histidine side groups coordinated to the magnesium ions of the special pair donor molecules, the residues hydrogen-bonded to the ubiquinone, and the backbone atoms

of two further residues close to the ubiquinone were included in the computation (see also Computational Procedures). Altogether, the two chromophores contain 127 protons, of which 70 exhibit an absolute polarization of more than 0.5% and 103 an absolute polarization of more than 0.1% after a single photocycle at the earth field. Several nuclei attain absolute polarization between 5 and 9%. Note that the largest absolute polarizations computed for high-field (4.7 T) solid-state 13C CIDNP are about 1%. These polarizations correspond to experimental steady-state enhancements with continuous illumination of about a factor of 10 000. At the earth field, we compute enhancement factors for a single photocycle of up to 7.44 108 with respect to the (very small) thermal equilibrium polarization. If the cycle rate exceeds the longitudinal relaxation rate at the earth field, buildup of even larger steady-state polarization would be expected from the same considerations that apply at high field.34 The large polarization created at the earth field or the optimum ultralow field for a particular system of interest could be transferred to high field by previously established techniques.35 We would then expect a gain by a factor of about 9 for the strongest signals and significant polarization for a much larger number of nuclei than with polarization at high field. Many of these additionally polarized nuclei are more remote from the center of spin density, which increases the radius around the active center from which information by NMR can be obtained at high sensitivity. The advantage arises from the fact that in the ultralow-field regime, much smaller hyperfine couplings suffice for efficient polarization transfer than at high fields. In Figure 5, simulated absolute earth field proton polarizations after a single photocycle are plotted versus the distance of the proton from the center of the chromophore. For the ubiquinone acceptor (blue squares), some of the closest protons exhibit the largest emissive polarization. However, enhanced absorptive polarization larger than the largest polarization computed for 13 C nuclei under high-field conditions is found as far as 7.5 Å away from the center of the quinone ring. For the special pair donor (red diamonds), such large polarization is found as far as 13.5 Å from the chromophore center. This is due to the broader spatial distribution of the electron spin in the special pair compared to that of the ubiquinone. Although the absolute polarizations are expected to be modified by multispin effects, it is safe to assume that a substantial fraction of the initial nonequilibrium magnetization in the radical is transferred to nuclear spins at distances up to 10 Å from the 9925



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Figure 4. Contour plots of the dependence of nuclear polarization ÆÎ zæ∞ on the magnitude T of the anisotropic hyperfine coupling and the angle θd,T between the electronelectron spinspin vector and the unique axis of the hyperfine tensor for two different isotropic hyperfine couplings, (a) aiso/2π = 2 MHz and (b) aiso/2π = 5 MHz. The electron Zeeman frequency ωS/2π = 1.4 MHz corresponds to the earth’s field.

Figure 5. The earth field proton polarization of the donor (red diamonds) and acceptor (blue squares) in the secondary radical pair of bacterial photosynthetic RCs of Rhodobacter sphaeroides wild type (simulation). The dependence on distance of the proton from the center of the corresponding chromophore is shown.

centers of the electron spin densities of the two radicals. In the diamagnetic ground state, these nuclear spins, in particular, the protons, are part of a dense network of dipoledipole coupled spins. We thus expect that repeated photocycles in a continuous illumination experiment together with spin diffusion between the photocycles will polarize an even larger spatial range around the chromophores. The size of the polarized range should depend on the ratio between the spin diffusion rate and the longitudinal relaxation rate of the protons. Relayed transfer throughout the whole sample may require cooling below 100 K,36 that is, under physiological conditions, the polarization is expected to be confined to the vicinity of the chromophores. By addition of the results for the individual spins, we find that emissive polarization dominates and that in an equilibration of the proton spin bath by spin diffusion, about 60% of this emissive polarization is canceled by enhanced absorptive polarization of other nuclei. This finding still suggests that continuous illumination at the earth field should generate large polarization of magnetic nuclei in the chromophores of photosynthetic RCs as well as substantial polarization of the protein matrix in the vicinity of the chromophores. These qualitative findings can probably be generalized to many donoracceptor pairs that take part in one-electrontransfer reactions in biological systems. In such reactions, which are not necessarily initiated by light, radical pairs are generated in

an initial singlet state with a radicalradical distance in the range where large effects are expected according to Figure 3. Biological electron-transfer systems are usually optimized to reach such large separations of the radicals to prevent fast back transfer of the electron. The radical species produced are of moderate reactivity to prevent unwanted side reactions. Hence, the radical pair lifetime should usually also be sufficiently long to generate nuclear polarization by the earth field TSM mechanism. The required modest hyperfine couplings of a few hundred kilohertz are invariably found for magnetic nuclei in or around organic radicals or paramagnetic metal ions. These considerations suggest that the earth field CIDNP is an ubiquitous phenomenon for electron-transfer proteins. Quantitative predictions of the polarization will rely on accounting for relaxation and multiple-spin effects in the TSM polarization build-up step, for spin diffusion effects, and for rotational diffusion at physiological temperatures. Among these complications, rotational diffusion can be suppressed and spin diffusion enhanced if the electron-transfer chain can be operated at cryogenic temperatures, as is the case for photosynthetic RCs. Note also that for membrane proteins in living cells or reconstituted into lipid bilayers, rotational correlation times are on the order of 10 μs at ambient temperature.37,38 With a typical time scale of 1 μs for generation of TSM effects at the earth field, CIDNP may well be common for electron-transfer proteins even in living cells. This raises the question whether such nuclear polarization could influence rates of biologically important reactions, for instance, of charge transfer across the membrane in photosynthesis. Mechanistically, such an influence cannot be excluded because substantial nuclear polarization will change the average hyperfine field in the radical pair state, which in turn will change the singlettriplet conversion rate and, due to the different recombination rates of singlet and triplet primary radical pairs in photosynthetic reaction centers, the fraction of photons that is lost via unproductive triplet recombination. However, this fraction is small in any case. Furthermore, such nuclear polarization effects on reaction rates would strongly depend on the magnetic field, whereas previous attempts to find magnetic field effects on plant growth were rather inconclusive.39 It is also unclear which advantages would be conveyed to plants by magnetoreception,39 although the idea has been advanced that some biological rhythms in plants may be related to similar rhythms in the geomagnetic field.40 If the accumulation of substantial nuclear 9926


The Journal of Physical Chemistry A polarization in organisms under physiological conditions could be proved and its field dependence established, more precise experiments could be devised to prove or reject such hypotheses. The ultralow-field solid-state CIDNP effect could also be useful for obtaining more information on electron-transfer proteins in general and photosynthetic reaction centers in particular by NMR techniques. In conjunction with field cycling techniques,35 NMR spectra of the vicinity of active centers in electron-transfer proteins could be obtained with high sensitivity. Field variation within the ultralow-field regime could, in principle, provide information on the electronelectron dipole dipole coupling and thus on the distance of radical pairs. Finally, large polarization obtained directly at the earth field would open up the way to zero-field41 or ultralow-field42,43 detection of NMR J-spectra without prepolarization at high fields and with higher sensitivity. Note also that at zero-field anisotropic broadening in NMR spectra vanishes because during free-spin evolution, only the magnetic field breaks the spherical symmetry of space. To benefit from this high-resolution solid-state NMR regime, the earth’s field would need to be compensated during the measurement.

’ COMPUTATIONAL PROCEDURES Analytical derivations were performed in Mathematica, using the Spin Operator Mathematica Environment (SOME) by Serge Boentges. Numerical computations were based on an extension of the algorithm for high-field solid-state photo-CIDNP simulations.18 The program is based on a Matlab implementation of the density operator formalism, using some routines from the EasySpin package.44 The new low-field version works with 8 8 Hilbert space matrices for the system consisting of two electron spins, S1 = 1/2 and S2 = 1/2, and one nuclear spin, I = 1/2. The density operator is evolved in the laboratory frame for time increments Δt no longer than 1/50 of the radical pair lifetime or for very long lifetimes, no longer than 20 ns. After coherent evolution for one time increment, the singlet and triplet contributions are projected out, and fractions Δt/τi of these projections are subtracted from the radical pair density operator and added to the ground-state density operator, where τi stands for the lifetime of singlet or triplet radical pairs. In the current work, an equal lifetime of singlet and triplet pairs was assumed as we could not find experimental data that would indicate multiplicity dependence of the lifetime for the secondary pair of photosynthetic RCs. The full expansion of the spin Hamiltonian into product operators was taken into account, that is, nine terms each for the hyperfine coupling and the electronelectron dipoledipole coupling and three terms each for the two electron Zeeman interactions and for the electronelectron exchange coupling were used. The new program can be applied to computations for any external magnetic field. It was tested by comparison with the old program, which could be applied only to fields where the highfield approximation is valid for the two electron spins. Results agree within numerical precision at fields larger than 200 mT. The high-field approximation can be considered as acceptable down to 25 mT for primary and secondary radical pairs of photosynthetic RCs. Below 25 mT, the old program misses the significant effects of low-field three-spin mixing. The g tensor of the ubiquinone acceptor radical in Rhodobacter sphaeroides bacterial photosynthetic RCs was taken from the

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high-field single-crystal measurements of Isaacson et al.45 Hyperfine coupling tensors were computed using ORCA46 with the B3LYP density functional and the EPR-II basis set,47 based on PDB structure 1AIJ.48 Due to the requirement of accounting for hydrogen bonding effects on the semiquinone radical,49 we included the side group of His-219 (chain M) as a methylimidazole ligand and Ala-260 as well as the backbone atoms of Asn-259 and Thr-261 (all chain M) as structural context. Hydrogen atoms were added and their positions relaxed in an geometry optimization run with the SVP basis set and B3LYP functional. Heavy atom positions were fixed in this optimization. In the special pair donor, the phytyl groups in the chlorophyll molecules were replaced by methyl groups, and the side groups of His-173 (chain L) and His-202 (chain M) were included as methylimidazole ligands, as in previous work.18 An electronelectron distance of 28.2 Å together with a point-dipole approximation was assumed to compute the dipoledipole coupling tensor.

’ CONCLUSION Dipoledipole coupling between two electron spins at distances of around 30 Å matches the electron Zeeman interaction at the earth field. Under these conditions, isotropic hyperfine couplings of about 350 kHz lead to very efficient coherent threespin mixing in the ST or ST+ manifold, which is distinct from the three-spin mixing observed at high fields in the ST0 manifold. For radical pairs that are born in a pure singlet (or triplet) state and decay by reaction dynamics on time scales of a microsecond or longer, the three-spin mixing in turn causes transfer of a substantial part of the initial nonequilibrium magnetization of the two electron spins to hyperfine coupled nuclear spins. The matching ranges for this transfer are sufficiently broad to include many radical pairs that are generated in electron-transfer proteins, in particular, the secondary radical pair in photosynthetic RCs. This suggests that CIDNP effects at the earth field may be common in biological systems. Efforts for experimental verification of these predictions by both field cycling and direct earth field observation techniques are in progress. If the predictions are proved, the effect could provide new information on electron-transfer proteins by allowing highsensitivity NMR measurements in a larger vicinity around the active centers than with high-field CIDNP. Such measurements could be performed with field cycling techniques and, possibly, at zero field by compensation of the earth’s field. ’ APPENDIX From the definitions of ηR and ηβ and trigonometric relations, we obtain tan 2η ¼

aiso ðωR þ ωβ Þ a2iso ωR ωβ

ð40Þ

while eq 35 corresponds to tan 2ηmax = 1. Likewise tan 3ξ=2 ¼

aiso ðωR ωβ Þ a2iso þ ωR ωβ

With trigonometric relations, we find from eq 36 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 125 3 681 24pffiffiffiffiffiffiffi 561 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan 3ξmax =2 ¼ t pffiffiffiffiffiffiffi 125 þ 3 681 24 561 9927

ð41Þ

ð42Þ



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From these equations, we can calculate the ratios aiso/ωS and d/ωS, where maximum nuclear polarization is attained. These ratios are jaiso j ¼ 0:252 ð43Þ ωS max

jdj ωS

¼ 1:935

ð44Þ

max

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (G.J.); [email protected] (J.M.).

’ ACKNOWLEDGMENT We thank P. J. Hore and an anonymous reviewer for helpful comments. B.E.B. is currently supported by a FeodorLynen fellowship by the Alexander von Humboldt Foundation financed by the German Federal Ministry of Education and Research. This research was supported by a Marie Curie Intra European Fellowship within the seventh European Community Framework Programme. We thank for financial support by the ALW Grant 818.02.019. ’ REFERENCES (1) Hore, P.; Broadhurst, R. Prog. Nucl. Magn. Reson. Spectrosc. 1993, 25, 345–402. (2) Goez, M. In Photo-CIDNP Spectroscopy; Webb, G. A., Ed.; Annual Reports on NMR Spectroscopy; Academic Press: New York, 2009; Vol. 66; Chapter 3, pp 77147. (3) Bargon, J.; Fischer, H.; Johnsen, U. Z. Naturforsch., A: Phys. Sci. 1967, 22, 1551–5. (4) Ward, H.; Lawler, R. J. Am. Chem. Soc. 1967, 89, 5518–5519. (5) Kaptein, R.; Oosterhoff, L. J. Chem. Phys. Lett. 1969, 4, 214–216. (6) Closs, G. L. J. Am. Chem. Soc. 1969, 91, 4552–4554. (7) Closs, G. L. Chem. Phys. Lett. 1975, 32, 277–278. (8) Kaptein, R.; Dijkstra, K.; Nicolay, K. Nature 1978, 274, 293–294. (9) Mok, K. H.; Kuhn, L. T.; Goez, M.; Day, I. J.; Lin, J. C.; Andersen, N. H.; Hore, P. J. Nature 2007, 447, 106–109. (10) Goldstein, R. A.; Boxer, S. G. Biophys. J. 1987, 51, 937–946. (11) Jeschke, G.; Matysik, J. Chem. Phys. 2003, 294, 239–255. (12) Prakash, S.; Alia; Gast, P.; de Groot, H. J. M.; Matysik, J.; Jeschke, G. J. Am. Chem. Soc. 2006, 128, 12794–12799. (13) Zysmilich, M.; McDermott, A. J. Am. Chem. Soc. 1994, 116, 8362–8363. (14) Zysmilich, M.; McDermott, A. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 6857–6860. (15) Jeschke, G. J. Chem. Phys. 1997, 106, 10072–10086. (16) Polenova, T.; McDermott, A. J. Phys. Chem. B 1999, 103, 535–548. (17) Jeschke, G. J. Am. Chem. Soc. 1998, 120, 4425–4429. (18) Prakash, S.; Alia; Gast, P.; de Groot, H. J. M.; Jeschke, G.; Matysik, J. J. Am. Chem. Soc. 2005, 127, 14290–14298. (19) Alia; Roy, E.; Gast, P.; van Gorkom, H. J.; de Groot, H. J. M.; Jeschke, G.; Matysik, J. J. Am. Chem. Soc. 2004, 126, 12819–12826. (20) Diller, A.; Roy, E.; Gast, P.; van Gorkom, H. J.; de Groot, H. J. M.; Glaubitz, C.; Jeschke, G.; Matysik, J.; Alia, A. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 12767–12771. (21) Roy, E.; Alia; Gast, P.; van Gorkom, H.; de Groot, H. J. M.; Jeschke, G.; Matysik, J. Biochim. Biophys. Acta 2007, 1767, 610–615. (22) Roy, E.; Rohmer, T.; Gast, P.; Jeschke, G.; Alia, A.; Matysik, J. Biochemistry 2008, 47, 4629–4635.

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Theory of Solid-State Photo-CIDNP in the Earth's Magnetic Field - The

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