Published: July 18, 2011

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

ARTICLE

pubs.acs.org/JPCA

Theory of Solid-State Photo-CIDNP in the Earth’s Magnetic FieldGunnar 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

’ INTRODUCTION

Photochemically induced dynamic nuclear polarization(photo-CIDNP) arises from transfer of the electron spin orderassociated with an initial singlet or triplet state of a radical pairto polarization of nuclei that are hyperfine coupled to at leastone of the two electron spins (for review, see refs 1 and 2). As aresult, the NMR spectrum of the recombination products of theradical pair exhibits enhanced positive (absorptive) or negative(emissive) signals. These signals allow for selective, highly sen-sitive observation of the products of a photochemical reaction.Their signs and relative intensities provide information on theradical pair state.

Since the first observations of CIDNP effects in Bargon(1967),3,4 different mechanisms were revealed for generatingsuch hyperpolarization of nuclear spin transitions.2 By far, mostobservations pertain to liquid-state NMR at frequencies wherethe high-field approximation is valid for the electron spins in theradical pair state and for the nuclear spins in the product groundstates. In this regime, nuclear polarization is most often generatedby the radical pair mechanism (RPM).5,6 In many cases, thisnuclear polarization relies on spin sorting between differentreaction products from singlet and triplet radical pairs.

In cyclic reactions in solution, the RPM can create net nuclearpolarization7 if longitudinal nuclear relaxation differs betweenthe singlet and triplet branch. This allows observation of CIDNPenhancements at the surface of proteins8 and has been developedto a technique1 that can provide substantial insight into proteinfolding.9 This mechanism will also be operative in cyclic photo-chemical reactions in the solid state,10 has been termed thedifferential relaxation mechanism in this context,11 and explainedthe differences between photo-CIDNP spectra of wild-type and

carotenoid-less photosynthetic reaction centers (RC) of Rhodo-bacter sphaeroides bacteria.12

The first experimental observations of solid-state photo-CIDNP effects were made on photosynthetic RCs.13,14 Theywere later traced back to genuine solid-state photo-CIDNPmechanisms, which arise from anisotropy of the hyperfinecoupling. Due to the pseudosecular contribution of an aniso-tropic hyperfine coupling, electron polarization can be directlytransferred to nuclear polarization in a coherent process.15 In thedifferential decay (DD) mechanism,16 the electron polarization isgenerated from the initial state of the radical pair due to differentrecombination rates of singlet and triplet radical pairs, that is, as aresult of incoherent reaction dynamics. In the three-spin mixing(TSM) mechanism,17 it is generated by coherent transfer that isdriven by the pseudosecular contribution to the electron�electroncoupling. The electron�electron coupling can be of the exchange ordipole�dipole type. TheTSMmechanism is based on two coherentmagnetization transfers and can thus be understood and computedby exclusive use of spin quantum mechanics, without taking intoaccount any interaction of the spin system with its environment.

The DD and TSM mechanisms explain the majority of solid-state photo-CIDNP observations on native photosynthetic RCs,18

where radical pair recombination is very fast and donor tripletstates are also quenched fast, so that differential relaxation isnegligible. The DD and TSM mechanisms as formulated to daterequire matching of the hyperfine and nuclear Zeeman interac-tions15 and are thus inoperative at the earth field.

Received: May 26, 2011Revised: July 11, 2011

ABSTRACT: To date, solid-state photo-CIDNP experimentshave been performed only using magic angle spinning NMR in ahigh-field regime, which is not associated with physiologicallyrelevant spin dynamics. Here, we predict that nuclear spinpolarization up to 10%, almost 9 orders of magnitude larger thanthermal equilibrium polarization, can arise in cyclic photoreac-tions at the earth field due to a coherent three-spin mixingmechanism in the S�T� or S�T+ manifold. The effect ismaximal at a distance of about 30 Å between the two radicals,which nearly coincides with the separation between the donorand 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 likelyto become polarized. The theory predicts that only modest hyperfine couplings of a few hundred kilohertz are required to generatepolarization of more than 1% for radical�radical distances between 20 and 50 Å, that is, for a large number of radical pairs inelectron-transfer proteins.

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The Journal of Physical Chemistry A ARTICLE

Solid-state photo-CIDNP at high fields critically depends on amatching of three interactions in the case of the TSMmechanismand on amatching of two interactions and the radical pair lifetimein the case of the DD mechanism.11 While the solid-state photo-CIDNP effect has been found in various natural photsyntheticRCs,19�22 to date, no artificial donor�acceptor system has beenfound that exhibits this effect. In fact, only very recently was solid-state photo-CIDNP observed on any system that is not aphotosynthetic RC.23 Yet, solid-state photo-CIDNP is commonto different types of photosynthetic RCs of several bacteria andgreen plants in case electron transfer beyond the primary radicalpair is blocked by depletion or reduction of the secondaryacceptor quinone. This led to the speculation that occurrenceof high-field solid-state CIDNP is correlated to efficient electrontransfer in photosynthetic RCs.24 Such correlation may arisefrom the requirement of both processes for significant but nottoo strong overlap of molecular orbitals along the electron-transfer pathway.

These findings and considerations raise the question whethersolid-state photo-CIDNP can also arise under physiologicalconditions, that is, for unblocked photosynthetic RCs at theearth's magnetic field of approximately 50 μT and at tempera-tures where plants or photosynthetic bacteria grow. Recent workhas already shown that high-field solid-state photo-CIDNP isobservable on bacterial photosynthetic RCs at physiologicallyrelevant temperatures.25 Here, we address the question of apossible ultralow-field solid-state photo-CIDNP regime the-oretically. In particular, we consider a purely coherent TSMmechanism that does not rely on different reaction rates ofsinglet and triplet pairs nor on different nucler spin relaxationin the two branches.

The paper is organized as follows. In the Theory section, wefirst identify the characteristics of the low-field solid-state photo-CIDNP regime and construct a minimal spin Hamiltonian fordiscussing this effect. This minimal Hamiltonian is analyticallydiagonalized, and expressions for nuclear polarization are derivedby the product operator formalism.26 The Results andDiscussionsection starts with a discussion of sign rules and of the conditionsunder which the effect is maximized. An approximate doublematching condition is derived. We then use numerical densityoperator computations to relax the assumptions that weremade in constructing the minimal, analytically tractable Ha-miltonian. From the results of these numerical computations,dependence on the donor�acceptor distance and isotropichyperfine coupling is discussed, and consequences of hyper-fine anisotropy are considered. By numerical computations,we also test whether significant nuclear polarization is ex-pected for secondary radical pairs of photosynthetic RCs atthe earth field. Finally, we discuss the question whether theearth field solid-state CIDNP is expected to be common inelectron-transfer proteins.

’THEORY

Characteristics of the Ultralow-Field Solid-State Photo-CIDNP Regime. To find a possible earth field solid-state photo-CIDNP regime, we first consider the high-field solid-state photo-CIDNP effect. In most cases, this effect is based on mixingbetween the singlet state and the triplet substate with magneticquantum number MS = 0 (S�T0 mixing).5,6 Early liquid-statephoto-CIDNP experiments also established that for large ex-change coupling J, such as in biradicals, one of the other triplet

substates T� or T+ can be mixed with the singlet state.27 Whichstate is mixed with the singlet state depends on the sign of J,with S�T� mixing being the more common case. Transi-tions between the S and T� states can be induced by theisotropic hyperfine coupling. Unlike effects from S�T0 mix-ing, those from S�T� or S�T+ mixing are candidates for anearth field regime because they occur at a matching of theelectron Zeeman interaction and the exchange coupling. Forsufficiently small couplings, this matching may occur at theearth field.In early liquid-state photo-CIDNP work27�29 and in later

theoretical descriptions,30 effects due to S�T� mixing werediscussed in the context of diffusion and re-encounter of theradicals. The exchange coupling J was considered to be stochas-tically time-dependent. Later experimental work proved that theeffect can also be observed for cyclic reactions of rigid do-nor�acceptor pairs that transiently form a radical ion pairstate.31 Our initial numerical computations revealed thatS�T�mixing or S�T+mixing by a purely coherent TSMmecha-nism creates nuclear polarization in cyclic reactions.However, at the earth field, S�T�mixing due to the exchange

coupling is unlikely because this would require a couplingstrength of about 2.8 MHz, corresponding to a radical�radicaldistance of more than 15 Å. At such distances, dipole�dipolecoupling is much larger than exchange coupling32 and would thuscancel the matching. Hence, in the solid state, S�T� mixing isexpected to occur at even longer distances where the dipole�di-pole coupling matches the transition frequency between the Sand T� (or S and T+) levels and the exchange coupling isnegligibly small. Closer inspection reveals that the secular part ofthe dipole�dipole interaction is sufficient for the polarizationtransfer to occur. Although the dipole�dipole interaction ispurely secular only along the principal axes of the couplingtensor, this special case is of interest because it can be treatedanalytically.Diagonalization of the Spin Hamiltonian for a Special

Orientation. For such an analytical treatment, we consider asystem consisting of two electron spins S1 = 1/2 and S2 = 1/2 anda nuclear spin I = 1/2 coupled to S1 in a regime where the twoelectron Zeeman interactions, the coupling between the twoelectron spins, and the hyperfine coupling are all of the sameorder of magnitude. In this regime, the nuclear Zeeman interac-tion is negligibly small. If the two paramagnetic species areorganic radicals, such as, for instance, radical ions in a photo-synthetic RC, the difference between the two electron Zeemaninteractions 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 inter-action is ωS/2π ≈ 1.4 MHz. The dipole�dipole couplingbetween the two electron spins has this magnitude at adistance of about 30 Å. At such a distance, the exchangecoupling between two electron spins is usually negligible.32 Inthe following, we assume a point-dipole approximation forthe dipole�dipole coupling, so that the coupling tensor hasaxial symmetry with the unique axis being parallel to thespin�spin vector. Deviations from such axial symmetry due tofailure of the point-dipole approximation do not affect thegeneral conclusions drawn in this work or the principle of theanalytical 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 Iz. Forthe following analytical computation, we consider a special

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The Journal of Physical Chemistry A ARTICLE

orientation where the electron�electron spin�spin vector iseither parallel or perpendicular to the magnetic field. Thehyperfine interaction is either purely isotropic or one of theprincipal axes of the hyperfine tensor is parallel to the magneticfield. The static spin Hamiltonian in angular frequency units canthen be written as

H0 ¼ ωSS1z þ ωSS2z þ dS1zS2z þ AS1zIz

þ aisoðS1xIx þ S1yIyÞ ð1Þ

with the electron Zeeman frequency ωS, the electron�electrondipole�dipole coupling d, the isotropic hyperfine coupling aiso,and the secular hyperfine coupling A = aiso + T, where T is thedipole�dipole contribution to the hyperfine coupling. ThisHamiltonian can be considered as a minimal Hamiltonian forstudying the effect under consideration.Because the off-diagonal hyperfine terms do not connect the

mS2 = +1/2(S2R) andmS2 =�1/2(S2

β) subspaces, the Hamiltoniancan be diagonalized in the separated subspaces. The two sub-space Hamiltionians are

HR, β ¼ (ωS

2þ AS1zIz þ ωR, βS1z þ aisoðS1xIx þ S1yIyÞ

ð2Þwith

ωR, β ¼ ωS ( d=2 ð3ÞThe subspace Hamiltonians are diagonalized by the unitarytransformations

UR, β ¼ expf�iηR,βð2S1yIx � 2S1xIyÞg ð4Þwhere

ηR, β ¼ arctan � aiso

ωR, β þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

R, β þ a2isoq

0B@

1CA ð5Þ

In their eigenbasis, they take the form

HðdiaÞR, β ¼ (

ωS

2þ AS1zIz þ 1

2ðωR, β þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

R, β þ a2isoq

ÞS1z

þ 12ðωR, β �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

R, β þ a2isoq

ÞIz ð6Þ

For describing the spin evolution starting from a pure singletor triplet state of the radical pair, the Hamiltonian for the wholestate space needs to be constructed. This is necessary because inthe Cartesian basis, the singlet and triplet states correspond tozero-quantum coherence of the two electron spins. For this, thetwo commuting unitary transformations

U0R ¼ expf�iηRð2S1ySR2 Ix � 2S1xS

R2 IyÞg ð7Þ

U0β ¼ expf�iηβð2S1ySβ2 Ix � 2S1xS

β2 IyÞg ð8Þ

need to be applied to the Hamiltonian given by eq 1. The resultcan be inferred most easily from a slightly rewritten form of eq 6

HðdiaÞR, β ¼ (

ωS

2þ AS1zIz

þ ωR, βS1z � 12ðωR, β �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

R,β þ a2isoq

ÞS1z

þ 12ðωR, β �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

R, β þ a2isoq

ÞIz ð9Þ

Comparison with eqs 1 and 2 reveals that

HðdiaÞ0 ¼ ωSS1z þ ωSS2z þ dS1zS2z þ AS1zIz �ω

0RS1zS

R2

þω0RS

R2 Iz �ω

0βS1zS

β2 þ ω

0βS

β2 Iz ð10Þ

where

ω0R, β ¼ ωR, β -

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

R, β þ a2isoq

ð11Þ

By replacing the polarization operators S2R and S2

β by Cartesianoperators and then reordering terms, we obtain

HðdiaÞ0 ¼ ωS �

ω0R þ ω

0β

2

!S1z þ ωSS2z

þ ðd�ω0R þ ω

0βÞS1zS2z þ AS1zIz

þ ω0R þ ω

0β

2Iz þ ðω0

R �ω0βÞS2zIz ð12Þ

This expression can be simplified using the substitutions

ω0R þ ω

0β ¼ 2ωS �ωR

00 �ωβ00 ð13Þ

ω0R �ω

0β ¼ d�ωR

00 þ ωβ00 ð14Þ

with

ωR00 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðωS þ d=2Þ2 þ a2iso

qð15Þ

ωβ00 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðωS � d=2Þ2 þ a2iso

qð16Þ

Finally, we find for the diagonalized Hamiltonian

HðdiaÞ0 ¼ ωR

00 þ ωβ00

2S1z þ ωSS2z þ ðωR

00 �ωβ00ÞS1zS2z

þ AS1zIz þ ωS �ωR

00 þ ωβ00

2

!Iz

þ ½d� ðωR00 �ωβ

00Þ�S2zIz ð17ÞFor aiso f 0 (no state mixing), eqs 17 and 1 revert to the sameform, that is, the pseudonuclear Zeeman term (ωI

0 =ωS� (ωR00 +

ωβ00)/2) and the apparent hyperfine coupling to spin S2 (A0

2 =d � (ωR

00 � ωβ00)) vanish. For brevity, we make the additional

substitutions ωS0 = (ωR

00 + ωβ00)/2 and d0 = ωR

00 � ωβ00, so that the

full diagonalized Hamiltonian reads

HðdiaÞ0 ¼ ω

0SS1z þ ωSS2z þ d0S1zS2z þ AS1zIz

þ ω0I Iz þ A

02S2zIz ð18Þ

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The Journal of Physical Chemistry A ARTICLE

The unitary transformation U = UR0 + Uβ

0 that diagonalizes H0 isgiven by U = exp{�iA} with

A ¼ ηR þ ηβ2

ð2S1yIx � 2S1xIyÞ

þ ηR � ηβ2

ð4S1yS2zIx � 4S1xS2zIyÞ ð19Þ

For the following computations, it is most convenient to expressthis operator A as

A ¼ ηð2S1yIx � 2S1xIyÞ þ ξð3S1yS2zIx � 3S1xS2zIyÞ ð20Þwith

η ¼ ηR þ ηβ2

ð21Þ

ξ ¼ 2ðηR � ηβÞ3

ð22Þ

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

state E/4� ^S!

1 3^S!

2 or in a triplet state 3E/4 +^S!

1 3^S!

2. Theunit operator E is invariant under all transformations and doesnot correspond to observable magnetization; it can thus bedropped. In the Cartesian basis, the scalar product of the spinvector operators of the two electron spins

^S!

1 3^S!

2 ¼ S1xS2x þ S1yS2y þ S1zS2z ð23Þcan be interpreted as a sum of two-spin order

σTSO ¼ S1zS2z ð24Þand zero-quantum coherence

σZQC ¼ S1xS2x þ S1yS2y ¼ 1=2 SþS� þ S�Sþ� �

ð25Þ

Evolution of the zero-quantum coherence under the Hamil-tonian given by eq 1 does not generate nuclear polarization. Aftertransformation by the unitary operator U to the eigenbasis, two-spin order gives a linear combination of diagonal terms withoperators S1zS2z, S2zIz, S1z, and Iz and off-diagonal terms withoperators S1xIx, S1yIy, S1xS2zIx, and S1yS2zIy. The diagonal termsare constants of motion of the spin evolution of the radical pair.Together, they correspond to nuclear polarization

ÆIz, constæ ¼ 3 sinð4ηÞ½14 sinð2ξÞ þ 25 sinð4ξÞ�1024

ð26Þ

At time t = 0, this nuclear polarization is exactly canceled by thenuclear polarization that arises from the off-diagonal terms.Nuclear polarization is created by evolution of these off-

diagonal terms under the Hamiltonian H0(dia). The evolution

relevant for generation of nuclear polarization is a rotation in aphase space spanned by operators S1xIx + S1yIy, S1yIx � S1xIy,S1xS2zIx + S1yS2zIy, and S1yS2zIx � S1xS2zIy in the eigenbasis ofthe radical pair Hamiltonian.At time t = 0, the nonvanishing coefficients of these basis

operators are given by

ÆS1xIx þ S1yIyæð0Þ ¼ 38cos2 η sin2 ξ ð27Þ

and

Æ2S1xS2zIx þ 2S1yS2zIyæð0Þ ¼ 164

ð7 þ 25 cos 2ξÞ sin 2η

ð28ÞThe nuclear polarization at time t after generation of the

radical pair, starting from a singlet pair (negative two-spin order�S1zS2z), can now be computed as

ÆIzæðtÞ ¼ �TrfIzU�1 expð�iHðdiaÞ0 tÞUS1zS2zU�1 expðiHðdiaÞ

0 tÞUgð29Þ

We find

ÆIzæðtÞ ¼ C0 þ Cþ cosðωR00tÞ þ C� cosðωβ

00tÞ ð30Þwith

C0 ¼ �3 sinð4ηÞ½14 sinð2ξÞ þ 25 sinð4ξÞ�1024

ð31Þ

Cþ ¼ ð14 sinð2ηÞ þ sin½2ðη� ξÞ� þ 49 sin½2ðη þ ξÞ�Þ232768

ð32Þ

C� ¼ �ð14 sinð2ηÞ þ 49 sin½2ðη� ξÞ� þ sin½2ðη þ ξÞ�Þ232768

ð33ÞAs no nuclear polarization exists at t = 0, we expect C0 + C+ +C� = 0, which is indeed the case. Equations 30�33 were alsochecked by comparison with numerical computations.We now assume that the radical pair has a lifetime τ and

eventually recombines to a diamagnetic ground state. Moreprecisely, we assume that the hyperfine coupling of the nucleusis annihilated, so that no further polarization transfer betweenelectron spin and nuclear spin is possible. Such annihiliation canhappen in other ways than by recombination of the radical pair,for instance, by onward transfer of the electron to anotheracceptor for acceptor nuclei or by hydrogen abstraction froman amino acid residue such as tyrosin for donor nuclei. Lifetimesof singlet and triplet pairs are considered to be equal. In otherwords, the hyperfine coupling is annihilated in a non-spin-selective way. The nuclear polarization in the ground-statemolecule after complete recombination (t. τ) is then given by

ÆIzæ∞ ¼

Z ∞

0e�t=τ ÆIzæðtÞ dtZ ∞

0e�t=τ dt

¼ C0 þ Cþ1 þ ωR

002τ2þ C�

1 þ ωβ002τ2

ð34Þ

For ωR00τ,ωβ

00τ , 1, the polarization vanishes because C0 + C+ +C� = 0. ForωR

00τ,ωβ00τ. 1, the terms with coefficients C+ and C�

vanish. As these terms have a sign opposite to the one of C0, suchlong lifetimes correspond to maximum nuclear polarization,which assumes the value C0; see eq 31.The derivation of eqs 30�34 reveals the mechanism of earth

field solid-state photo-CIDNP. Radical pairs are born in a puresinglet or triplet state that involves negative or positive two-spin

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The Journal of Physical Chemistry A ARTICLE

order-S1zS2z, respectively. This two-spin order is transferred tonuclear polarization with opposite sign on the transitions |RS1RS2βIæ T |βS1RS2RIæ and |RS1βS2βIæ T |βS1βS2RIæ. The twotransitions are distinguished by a different state of the electronspin S2 that is not coupled to the nuclear spin. If the radical pairrecombines instantly or with a lifetime much shorter than thetypical frequencies of spin evolution, nuclear polarization willcancel.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 strengthin the RS2 and βS2 manifolds arises from the different levelsplittings ωS ( d/2. The mixing creates coherence on the twotransitions, which evolves with frequenciesωR

00 andωβ00 in the RS2

and βS2 manifold, respectively. This evolution changes theprojection of the coherence onto the eigenstates of the nuclearspin in the absence of hyperfine coupling. The eigenstates in theabsence of hyperfine coupling are the nuclear spin eigenstates inthe ground state (recombination product). Hence, at differenttimes after their creation, radical pairs recombine to productswith different, nonvanishing nuclear polarization.If ωR

00τ and ωβ00τ are both much larger than unity, the

contributions from evolution of coherence on the two transitionsvanish in the time average. In contrast, the fraction of nuclearpolarization of each transition that is not converted to coherenceby mixing is always transferred to the ground state (recombinationproduct). Due to the different extent of state mixing, this fractionof nuclear polarization is different in the RS2 and βS2 manifoldsand thus does not cancel upon radical pair recombination.Therefore, nuclear polarization of the recombination productafter a single photocycle approachesC0 at sufficiently long radicalpair lifetimes.

’RESULTS AND DISCUSSION

Sign Rules. The maximum nuclear polarization C0 is an oddfunction of angle η as well as an odd function of angle ξ. Thesetwo angles in turn depend on the electron Zeeman frequencyωS,which is generally positive, and on the electron�electron cou-pling d and the isotropic hyperfine coupling aiso via eqs 5, 21, and22. A sign change of aiso leads to a sign change in both ηR and ηβand thus to a sign change in both η and ξ, so that the sign of C0 isunchanged. Hence, the sign of nuclear polarization does notdepend on the sign of the isotropic hyperfine coupling.In contrast, a sign change of the electron�electron coupling d

causes an interchange of the values of ηR and ηβ, which leaves ηunchanged and leads to a sign change in ξ. This in turn changesthe sign of C0. Hence, the sign of nuclear polarization doesdepend on the sign of the electron�electron coupling.We also note that the secular hyperfine coupling A is of no

consequence for the sign and magnitude of the nuclear polariza-tion. This is because within each of the two pairs of mixed states,the secular hyperfine energy contribution is the same for bothstates. A difference in the Zeeman frequencies of the two electronspins is of no consequence for the extent of state mixing or for theevolution frequencies. Thus, it does not change the magnitudeor sign of the nuclear polarization. In radical ion pairs, such asthe ones observed in photosynthetic RCs, the sign of nuclearpolarization is therefore the same for donor and acceptorchromophores.

To this point, the discussion referred to the simple case of theminimal Hamiltonian, for which we have analytical expressions.The situation is more complicated for the orientation-averagedeffect of an ensemble of donor�acceptor systems with arbitraryorientation with respect to the earth’s field. In this case, differentorientations correspond to dipole�dipole couplings that differ inamplitude and may differ in sign. Hence, orientation averagingleads to some loss of polarization by cancellation of contributionsfrom orientations with a different sign of the dipole�dipolecoupling. Such cancellation is not complete, however, becausethe polarization is not proportional to the dipole�dipole coupling.If, in addition, hyperfine anisotropy is considered, no simple

sign rules can be formulated. This is because the amplitude ratiobetween polarization from orientations with positive and nega-tive polarization depends on the relative orientation of thehyperfine coupling tensor and the dipole�dipole coupling tensor,as well as on the ratio between hyperfine anisotropy and isotropichyperfine coupling. These issues are addressed below by numer-ical computations.Maximum Effect. The derivative of C0 with respect to η is

proportional to �cos 4η, and the second derivative is propor-tional to sin 4η. Consequently, the first maximum of C0 isattained for

ηmax ¼ �π=8 ¼ �22:5� ð35ÞThe derivative of C0 with respect to ξ is proportional to

7 cos 2ξ + 25 cos 4ξ, and the second derivative is proportional to7 sin 2ξ + 50 sin 4ξ. Both derivatives are also proportional tosin(4η). As the optimum value of η is negative, the secondderivative with respect to ξ is negative for small positive ξ. Thesmallest value of ξ corresponding to a maximum of C0 is

ξmax ¼ arccos110

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32ð31 þ ffiffiffiffiffiffiffi

561p Þ

rð36Þ

corresponding to an angle of 25.0831�. Maximum emissivenuclear polarization is obtained at (�ηmax,ξmax). Closer exam-ination shows that these parameters correspond to globalextrema because the extrema at larger absolute values of theangle η and ξ cannot be attained due to the definitions of ηR andηβ. The maximum absolute nuclear polarization after a singlephotocycle is found by substituting ηmax and ξmax into eq 31.We find

C0, max ¼9ð7 ffiffiffi

3p þ 3

ffiffiffiffiffiffiffi187

p Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ð817 þ 7

ffiffiffiffiffiffiffi561

p Þr

102400≈0:103551

ð37ÞThis maximum absolute polarization of about 10% can be

compared to numerical computations for the nuclear polariza-tion generated in a single photocycle at high field (4.7 T). In thiscase, we find a maximum polarization of 0.37% when we considerall 13C nuclei in the primary radical pair of photosynthetic RCs.Under such conditions high-quality solid-state photo-CIDNPmagic-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 electron�electron cou-pling d corresponds to nearly a matching situation |d| = 2ωS. Atthe earth field, the optimum electron�electron coupling corre-sponds to the dipole�dipole coupling between two electronspins almost exactly at a distance of 30 Å and to an isotropic

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The Journal of Physical Chemistry A ARTICLE

hyperfine coupling of 353 kHz. The relatively small value of aisofor optimum mixing is caused by the necessity to maximize thedifference in state mixing between the S2

R and S2β subspaces. A

large aiso leads to strong (and nearly equal) state mixing in bothsubspaces. However, the decrease of nuclear polarization towardlarger isotropic hyperfine couplings is relatively benign(Figure 1), even if the electron�electron coupling is kept fixedat its global optimum instead of considering the optimum valuefor each particular hyperfine coupling.The findings can be summarized by an approximate double

matching condition

jωSj≈jd=2j≈4jaisoj ð38ÞNumerical ComputationswithOrientationAverage. In the

following, we consider the same system consisting of twoelectron spins S1 = 1/2 and S2 = 1/2 and a nuclear spin I = 1/2coupled to S1 still in a regime where the two electron Zeemaninteractions, the dipole�dipole coupling between the two elec-tron spins and the purely isotropic hyperfine coupling, are all ofthe same order of magnitude. Again, the exchange interactionbetween the two electron spins is neglected, and the earth'smagnetic field is taken along z. The geometry of the system isnow characterized by a single angle θ between the magnetic fielddirection z and the electron�electron spin�spin vector. TheHamiltonian in angular frequency units is given by

H0, aniso ¼ ωSS1z þ ωSS2z þ aisoðS1xIx þ S1yIy þ S1zIzÞþ ð1� 3 cos2 θÞdS1zS2z þ ð3 cos2 θ� 2ÞdS1xS2xþ dS1yS2y � 3 sin θ cos θdðS1zS2x þ S1xS2zÞ

ð39Þwhere we have chosen the x axis of the electron spin frames in anway that situates the electron�electron spin�spin vector intothe xz plane. Due to the additional non-negligible off-diagonalelements, analytical expressions for the nuclear polarization canno 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 totalevolution time of 10 μs. After this time, 99.3% of all radical pairshad recombined to the diamagnetic ground state. Orientationswere averaged by integrating over sin θ dθ = �d cos θ. Thedipole�dipole coupling d and the isotropic hyperfine couplingaiso were varied.

The dependence of the absolute nuclear polarization ÆIzæ∞ ondipole�dipole coupling d at fixed isotropic hyperfine couplingaiso/2π = 2 MHz and the dependence on isotropic hyperfinecoupling aiso at fixed dipole�dipole coupling d/2π = 2 MHz areshown in Figure 2a and b, respectively, for ωS/2π = 1.4 MHz.Again, we see that significant nuclear polarization is generatedalready for isotropic hyperfine couplings of less than 1 MHz.Couplings of this magnitude are rather frequently observed inbiologically relevant radical pairs. Significant nuclear polarizationis generated over a broad range of hyperfine couplings. Thelargest hyperfine couplings to be expected in radicals in livingcells 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 dipole�dipole

coupling d is increased beyond the optimum value (Figure 2b).Nevertheless, the range of significant effects is rather broad alsowith respect to d, in particular, when considering that d isproportional to the inverse cube of the distance. As seen inFigure 3, significant nuclear polarization is created for typicalisotropic hyperfine couplings in a distance range from 18 and50 Å, which is rather typical for radical pairs in electron-transferproteins. The earth field nuclear polarization should thus be arather common phenomenon in biological electron-transferreactions that create radical pairs with a lifetime on the orderof 1 μs or longer.Influence of Anisotropic Hyperfine Coupling. Anisotropic

hyperfine coupling further modifies nuclear polarization as itleads to additional off-diagonal elements in the spin Hamiltonianand thus to modified level mixing. If we assume an axiallysymmetric hyperfine tensor with principal values (aiso + T,aiso + T, aiso � 2T), this influence depends on angle the θd,Tbetween the electron�electron spin�spin vector and the uniqueaxis of the hyperfine tensor. Furthermore, the influence dependson the magnitude T of the anisotropic component of thehyperfine coupling.We studied this influence by numerical simulations for a fixed

electron�electron distance of r = 28 Å, roughly corresponding tothe distance between the donor radical cation and the secondaryacceptor radical anion in photosynthetic RCs. The isotropichyperfine coupling was fixed at 2 or 5MHz, while θd,Twas variedbetween 0 and 90� in steps of 2.5� and T was varied between 0and 20 MHz in steps of 0.5 MHz.The maximummagnitude of the nuclear polarization found in

the presence of anisotropic hyperfine coupling is similar to themaximum magnitude found with only isotropic hyperfine cou-pling (Figure 4). However, in the presence of large anisotropichyperfine coupling, the nuclear polarization is positive, corre-sponding to enhanced absorptive NMR signals. This effect can betraced back to the mixing of additional level pairs by hyperfineanisotropy. If the NMR measurements are also performed at theearth field, where it may be difficult to separate signals fromdifferent nuclei, such sign inversion can lead to partial cancella-tion of the photo-CIDNP effect.Limitations of Current Theory. In realistic systems, several

magnetic nuclei are hyperfine coupled to the electron spins of thedonor and acceptor chromophores. It is evident that polarizationtransfer from the electron cannot be independent for theindividual nuclei because polarizations of several percent pernucleus would add up to more than the initial nonequilibriummagnetization of the radical pair. Hence, the multispin problemneeds to be treated explicitly. Unlike for the multispin problemsthat arise in electron spin�echo envelope modulation,33 such a

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

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treatment is not possible simply by factorization into two-spincontributions, that is, by deriving a product rule. Althoughnuclear�nuclear couplings are negligible for the shortest radicalpair lifetimes where the earth field solid-state CIDNP is expected,the factorization approach breaks down. This breakdown is dueto the influence of each individual nucleus on the quantizationaxis of the electron spin, which is in turn due to the similarmagnitude of the electron Zeeman and isotropic hyperfineinteraction. Therefore, nuclear spins will communicate with eachother via the electron spin in a nontrivial way, even if their directcouplings are negligible. Consequently, multispin effects need tobe addressed by a series of numerical computations that arebeyond the scope of the present paper. Work along these lines isin progress. We note that the difficulties that are commonlyencountered by such multispin computations pose a problem ininferring information on the radical pair state from the relativeCIDNP line intensities.Model Computations for the Secondary Radical Pair of

Photosynthetic RCs. To test for relevance of these effects inbiological systems, we numerically computed proton polariza-tions for the donor and acceptor in the secondary radical pair ofbacterial photosynthetic RCs of Rhodobacter sphaeroides wildtype. The two histidine side groups coordinated to the magne-sium ions of the special pair donor molecules, the residueshydrogen-bonded to the ubiquinone, and the backbone atoms

of two further residues close to the ubiquinone were included inthe computation (see also Computational Procedures). Alto-gether, the two chromophores contain 127 protons, of which 70exhibit an absolute polarization of more than 0.5% and 103 anabsolute polarization of more than 0.1% after a single photocycleat the earth field. Several nuclei attain absolute polarizationbetween 5 and 9%. Note that the largest absolute polarizationscomputed for high-field (4.7 T) solid-state 13CCIDNP are about1%. These polarizations correspond to experimental steady-stateenhancements with continuous illumination of about a factor of10 000. At the earth field, we compute enhancement factors for asingle photocycle of up to 7.44 � 108 with respect to the (verysmall) thermal equilibrium polarization. If the cycle rate exceedsthe longitudinal relaxation rate at the earth field, buildup of evenlarger steady-state polarization would be expected from the sameconsiderations that apply at high field.34

The large polarization created at the earth field or theoptimum ultralow field for a particular system of interest couldbe transferred to high field by previously established techniques.35

We would then expect a gain by a factor of about 9 for thestrongest signals and significant polarization for a much largernumber of nuclei than with polarization at high field. Many ofthese additionally polarized nuclei are more remote from thecenter of spin density, which increases the radius around theactive center fromwhich information byNMR can be obtained athigh sensitivity. The advantage arises from the fact that in theultralow-field regime, much smaller hyperfine couplings sufficefor 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 theproton from the center of the chromophore. For the ubiquinoneacceptor (blue squares), some of the closest protons exhibitthe largest emissive polarization. However, enhanced absorp-tive polarization larger than the largest polarization computed for13C nuclei under high-field conditions is found as far as 7.5 Åaway from the center of the quinone ring. For the special pairdonor (red diamonds), such large polarization is found as far as13.5 Å from the chromophore center. This is due to the broaderspatial distribution of the electron spin in the special paircompared to that of the ubiquinone.Although the absolute polarizations are expected to be mod-

ified by multispin effects, it is safe to assume that a substantialfraction of the initial nonequilibriummagnetization in the radicalis transferred to nuclear spins at distances up to 10 Å from the

Figure 2. Dependence of nuclear polarization ÆIzæ∞ (a) on isotropic hyperfine coupling aiso at fixed dipole�dipole coupling d/2π = 2MHz and (b) ondipole�dipole coupling d at fixed isotropic hyperfine coupling aiso/2π = 2MHz. The electron Zeeman frequencyωS/2π = 1.4 MHz corresponds to theearth’s field.

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

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centers of the electron spin densities of the two radicals. In thediamagnetic ground state, these nuclear spins, in particular, theprotons, are part of a dense network of dipole�dipole coupledspins. We thus expect that repeated photocycles in a continuousillumination experiment together with spin diffusion between thephotocycles will polarize an even larger spatial range around thechromophores. The size of the polarized range should depend onthe ratio between the spin diffusion rate and the longitudinalrelaxation rate of the protons. Relayed transfer throughout thewhole sample may require cooling below 100 K,36 that is, underphysiological conditions, the polarization is expected to beconfined 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 ofthe proton spin bath by spin diffusion, about 60% of this emissivepolarization is canceled by enhanced absorptive polarization ofother nuclei. This finding still suggests that continuous illumina-tion at the earth field should generate large polarization ofmagnetic nuclei in the chromophores of photosynthetic RCs aswell as substantial polarization of the protein matrix in thevicinity of the chromophores.These qualitative findings can probably be generalized to

many donor�acceptor pairs that take part in one-electron-transfer reactions in biological systems. In such reactions, whichare not necessarily initiated by light, radical pairs are generated in

an initial singlet state with a radical�radical distance in the rangewhere large effects are expected according to Figure 3. Biologicalelectron-transfer systems are usually optimized to reach suchlarge separations of the radicals to prevent fast back transfer ofthe electron. The radical species produced are of moderatereactivity to prevent unwanted side reactions. Hence, the radicalpair lifetime should usually also be sufficiently long to generatenuclear polarization by the earth field TSM mechanism. Therequired modest hyperfine couplings of a few hundred kilohertzare invariably found for magnetic nuclei in or around organicradicals or paramagnetic metal ions.These considerations suggest that the earth field CIDNP is an

ubiquitous phenomenon for electron-transfer proteins. Quanti-tative predictions of the polarization will rely on accounting forrelaxation and multiple-spin effects in the TSM polarizationbuild-up step, for spin diffusion effects, and for rotationaldiffusion at physiological temperatures. Among these complica-tions, rotational diffusion can be suppressed and spin diffusionenhanced if the electron-transfer chain can be operated atcryogenic temperatures, as is the case for photosynthetic RCs.Note also that for membrane proteins in living cells or recon-stituted into lipid bilayers, rotational correlation times are on theorder of 10 μs at ambient temperature.37,38 With a typical timescale of 1 μs for generation of TSM effects at the earth field,CIDNPmay well be common for electron-transfer proteins evenin living cells.This raises the question whether such nuclear polarization

could influence rates of biologically important reactions, forinstance, of charge transfer across the membrane in photosynth-esis. Mechanistically, such an influence cannot be excludedbecause substantial nuclear polarization will change the averagehyperfine field in the radical pair state, which in turn will changethe singlet�triplet conversion rate and, due to the differentrecombination rates of singlet and triplet primary radical pairs inphotosynthetic reaction centers, the fraction of photons that islost via unproductive triplet recombination. However, this frac-tion is small in any case. Furthermore, such nuclear polarizationeffects on reaction rates would strongly depend on the magneticfield, whereas previous attempts to find magnetic field effects onplant growth were rather inconclusive.39 It is also unclear whichadvantages would be conveyed to plants by magnetoreception,39

although the idea has been advanced that some biologicalrhythms in plants may be related to similar rhythms in thegeomagnetic field.40 If the accumulation of substantial nuclear

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

Figure 4. Contour plots of the dependence of nuclear polarization ÆIzæ∞ on the magnitude T of the anisotropic hyperfine coupling and the angle θd,Tbetween the electron�electron spin�spin 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.

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polarization in organisms under physiological conditions couldbe proved and its field dependence established, more preciseexperiments 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-transferproteins in general and photosynthetic reaction centers inparticular by NMR techniques. In conjunction with field cyclingtechniques,35 NMR spectra of the vicinity of active centers inelectron-transfer proteins could be obtained with high sensitivity.Field variation within the ultralow-field regime could, in princi-ple, provide information on the electron�electron dipole�dipole coupling and thus on the distance of radical pairs. Finally,large polarization obtained directly at the earth field would openup the way to zero-field41 or ultralow-field42,43 detection of NMRJ-spectra without prepolarization at high fields and with highersensitivity. Note also that at zero-field anisotropic broadening inNMR spectra vanishes because during free-spin evolution,only the magnetic field breaks the spherical symmetry ofspace. To benefit from this high-resolution solid-state NMRregime, the earth’s field would need to be compensated duringthe measurement.

’COMPUTATIONAL PROCEDURES

Analytical derivations were performed in Mathematica, usingthe Spin Operator Mathematica Environment (SOME) by SergeBoentges. Numerical computations were based on an extensionof the algorithm for high-field solid-state photo-CIDNPsimulations.18 The program is based on aMatlab implementationof the density operator formalism, using some routines from theEasySpin package.44 The new low-field version works with 8� 8Hilbert space matrices for the system consisting of two electronspins, S1 = 1/2 and S2 = 1/2, and one nuclear spin, I = 1/2. Thedensity operator is evolved in the laboratory frame for timeincrements Δt no longer than 1/50 of the radical pair lifetime orfor very long lifetimes, no longer than 20 ns. After coherentevolution for one time increment, the singlet and triplet con-tributions are projected out, and fractions Δt/τi of theseprojections are subtracted from the radical pair densityoperator and added to the ground-state density operator,where τi stands for the lifetime of singlet or triplet radicalpairs. In the current work, an equal lifetime of singlet andtriplet pairs was assumed as we could not find experimentaldata that would indicate multiplicity dependence of the life-time for the secondary pair of photosynthetic RCs.

The full expansion of the spin Hamiltonian into productoperators was taken into account, that is, nine terms each forthe hyperfine coupling and the electron�electron dipole�dipolecoupling and three terms each for the two electron Zeemaninteractions and for the electron�electron exchange couplingwere used.

The new program can be applied to computations for anyexternal magnetic field. It was tested by comparison with the oldprogram, which could be applied only to fields where the high-field approximation is valid for the two electron spins. Resultsagree within numerical precision at fields larger than 200 mT.The high-field approximation can be considered as acceptabledown to 25 mT for primary and secondary radical pairs ofphotosynthetic RCs. Below 25 mT, the old program misses thesignificant effects of low-field three-spin mixing.

The g tensor of the ubiquinone acceptor radical inRhodobactersphaeroides bacterial photosynthetic RCs was taken from the

high-field single-crystal measurements of Isaacson et al.45 Hy-perfine coupling tensors were computed using ORCA46 with theB3LYP density functional and the EPR-II basis set,47 based onPDB structure 1AIJ.48 Due to the requirement of accounting forhydrogen bonding effects on the semiquinone radical,49 weincluded the side group of His-219 (chain M) as a methylimida-zole ligand and Ala-260 as well as the backbone atoms of Asn-259and Thr-261 (all chainM) as structural context. Hydrogen atomswere added and their positions relaxed in an geometry optimiza-tion run with the SVP basis set and B3LYP functional. Heavyatom positions were fixed in this optimization. In the special pairdonor, the phytyl groups in the chlorophyll molecules werereplaced bymethyl groups, and the side groups ofHis-173 (chain L)and His-202 (chain M) were included as methylimidazole ligands,as in previous work.18 An electron�electron distance of 28.2 Åtogether with a point-dipole approximation was assumed to com-pute the dipole�dipole coupling tensor.

’CONCLUSION

Dipole�dipole coupling between two electron spins at dis-tances of around 30 Å matches the electron Zeeman interactionat the earth field. Under these conditions, isotropic hyperfinecouplings of about 350 kHz lead to very efficient coherent three-spin mixing in the S�T� or S�T+ manifold, which is distinctfrom the three-spin mixing observed at high fields in the S�T0

manifold. For radical pairs that are born in a pure singlet (ortriplet) state and decay by reaction dynamics on time scales of amicrosecond or longer, the three-spin mixing in turn causestransfer of a substantial part of the initial nonequilibriummagnetization of the two electron spins to hyperfine couplednuclear spins. The matching ranges for this transfer are suffi-ciently broad to include many radical pairs that are generated inelectron-transfer proteins, in particular, the secondary radical pairin photosynthetic RCs. This suggests that CIDNP effects at theearth field may be common in biological systems. Efforts forexperimental verification of these predictions by both fieldcycling and direct earth field observation techniques are inprogress. If the predictions are proved, the effect could providenew information on electron-transfer proteins by allowing high-sensitivity NMR measurements in a larger vicinity around theactive centers than with high-field CIDNP. Such measurementscould be performed with field cycling techniques and, possibly, atzero field by compensation of the earth’s field.

’APPENDIX

From the definitions of ηR and ηβ and trigonometric relations,we obtain

tan 2η ¼ aisoðωR þ ωβÞa2iso �ωRωβ

ð40Þ

while eq 35 corresponds to tan 2ηmax = �1. Likewise

tan 3ξ=2 ¼ aisoðωR �ωβÞa2iso þ ωRωβ

ð41Þ

With trigonometric relations, we find from eq 36

tan 3ξmax=2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi125� 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi681� 24

ffiffiffiffiffiffiffi561

pp125 þ 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi681� 24

ffiffiffiffiffiffiffi561

ppvuut 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

jaisojωS

� �max

¼ 0:252 ð43Þ

jdjωS

� �max

¼ 1:935 ð44Þ

’AUTHOR INFORMATION

Corresponding Author*E-mail: gjeschke@ethz.ch (G.J.); j.matysik@chem.leidenuniv.nl(J.M.).

’ACKNOWLEDGMENT

We thank P. J. Hore and an anonymous reviewer for helpfulcomments. B.E.B. is currently supported by a Feodor�Lynenfellowship by the Alexander von Humboldt Foundation financedby the German Federal Ministry of Education and Research. Thisresearch was supported by a Marie Curie Intra European Fellow-ship within the seventh European Community FrameworkProgramme. We thank for financial support by the ALW Grant818.02.019.

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