Electron Transfer between Biphenyl andBiphenyl Anion Radicals: ReorganizationEnergies and Electron TransferMatrix Elements

XIANG-YUAN LI, FU-CHENG HEDepartment of Applied Chemistry, Sichuan Union University, Chengdu 610065, China

Received 12 June 1998; accepted 23 November 1998

Ž .ABSTRACT: Intermolecular electron transfer ET between the parallel benzeneanion radical and neutral benzene is studied at the UHFr4-31G level. It is found

Ž y1 .that the diabatic activation energy remains almost invariant 14.2 kJ molwhen d, the distance between the two parallel benzene rings, is ) 0.45 nm. Theexponential fall-off of the ET matrix element, V , with d is examined. On ther pbasis of the calculated results of the ET matrix element for the system of twoparallel benzenes, it is concluded that direct calculation of the ET matrixelement, based on the two-state model, is more accurate than that based on theKoopmans theorem. Ab initio calculations are performed in the investigation ofthe ET reaction between biphenyl anion radical and neutral biphenyl. By using

Ž . Ž .the Dunning’s 9s, 5p r 3s, 2p basis set with polarization functions on all atomsŽ .DZP , the reorganization energy for the gas phase intermolecular ET is shownto be 109.2 kJ moly1. Using the UHFrSTO-3G method and direct calculation ofthe two-state model, V values of 2.055 kJ moly1 and 0.429 kJ moly1 arer pobtained for cyclohexylenyl- and decalenyl-mediated ET systems. When we usethe Koopmans theorem instead of the direct calculation, these V values arer pshown to be 1.55 kJ moly1 and 0.326 kJ moly1 for the two correspondingsystems, respectively. Q 1999 John Wiley & Sons, Inc. J Comput Chem 20:597]603, 1999

Keywords: electron transfer; reorganization energy; electron transfer matrixelement; electron-localized diabatic state

Correspondence to: X.-Y. Li; e-mail: xyli@scuu.edu.cnContractrgrant sponsors: National Natural Science Founda-

tion of China; State Key Laboratory of Theoretical and Compu-tational Chemistry in JiLin University

( )Journal of Computational Chemistry, Vol. 20, No. 6, 597]603 1999Q 1999 John Wiley & Sons, Inc. CCC 0192-8651 / 99 / 060597-07

LI AND HE

Introduction

etals, semiconductors, and insulators areM represented in terms of band structures.For metals, the band is about half-filled with elec-trons and hence no activation energy is necessaryfor the generation of charge carriers. In contrast,the band is completely filled with electrons forboth semiconductors and insulators, and thereforeactivation is required for the transition of electronsfrom the valence band to the conduction band togenerate charge carriers. For this reason, doping isneeded to generate charge carriers for the conduct-ing organic polymers. The theoretical research onconducting polymers has been concerned mainlywith radical and ionic sites, referred to as neutralradical and charged defects, respectively. To de-scribe the movements of defects, the concept ofsolitary wave has been used mathematically. Sucha concept is described as a soliton in the languageof field theory. The radical defect is referred to as aneutral soliton, and the anion defect or the cationdefect is referred to as a charged soliton.1, 2 Manyinvestigations of neutral and charged defects inconjugated polymers have been presented on thebasis of quantum chemical Huckel calculations,which are known to have quantitative limitations.More sophisticated treatments of neutral andcharged defects in various conjugated polymers,including ab initio calculations, have been carriedout.3, 4 Recently, the two-site model has been usedfor the investigation of interchain electron trans-

Ž .fer ET in polyacetylene, and ET matrix elementand other ET kinetic parameters have been in-bvestigated using quantum chemical ab initiocalculation.5 ] 7

Recently, there has been considerable interest inthe kinetic study of ET reactions. When biphenyl istaken as the donor or the acceptor in an ET reac-tion, the reorganization energy is partly con-tributed from the torsion motion of the molecule.ET rate constants between the biphenyl anion radi-cal and a series of acceptors have been measuredexperimentally.8 To examine the influence ofbiphenyl torsion motion upon ET reorganizationenergy, experiments have been performed usingboth biphenyl and fluorene as the electron donors,8a

because the former will undergo torsion of about458, whereas the latter remains planar in the ETprocess due to the existence of the tetrahedralcarbon. A reorganization value of 0.13 eV of en-

ergy contributed from the interring torsion hasbeen obtained from the difference in ET rate con-stants in these two cases.8a

The purpose of the present study is to investi-gate theoretically the reorganization energy andthe ET matrix element for the spacer-mediated ETfrom the biphenyl anion radical to the neutralbiphenyl. In describing the transition state of theself-exchange reaction, we employed the linear re-action coordinate and the electron-localizedinitial-guess-induced SCF technique. In comparingdifferent methods for calculating the ET matrixelement, the ET reaction between parallel benzeneanion radical and neutral benzene has been inves-tigated and the distance dependence of the ETmatrix element is discussed.

Electron Transfer between ParallelBenzene Anion Radical andNeutral Benzene

The mechanism of interchain conductivity ofŽ . Ž .poly p-phenylene PPP can be described by the

interchain hopping of bipolaron as shown in Fig-ure 1a. To realize the quantum chemical calcula-tion for such conducting polymers, the ET betweenparallel acelene and acelene anion radical,6 and theET between parallel allyl and allyl anion radicalwere investigated by Rodriguez-Monge et al.5 Theconductivity of a polymer is essentially an ETprocess. An excess electron or a hole appears afterthe virgin polymer has been doped. In this work,we take benzene, the monomer of PPP, into ac-count, and investigate the electron transfer be-tween parallel benzene anion radical and neutralbenzene. In our calculations for the gas phase ET

Žreaction Fig. 1b, the distance, d, between the donorand the acceptor is assumed to remain constant.

( )FIGURE 1. a Interchain hopping of bipolaron in PPP.( )b Scheme for the ET reaction between benzene anionradical and neutral benzene.

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Marcus’s theory9 has been used widely to in-vestigate ET processes. According to transitionstate theory, the non-adiabatic ET rate constant fora self-exchange reaction can be expressed as9c, 10a:

V 2 p 3r2r p Ž . Ž .k s exp yE rRT 1c1r2h Ž .k TEB c

where V is the ET matrix element, h the Planckr pconstant, k Boltzmann’s constant, T the tempera-Bture, and E the diabatic activation energy. The ETc

Ž .ybetween a benzene anion radical C H and a6 6neutral benzene C H is a self-exchange reaction6 6Ž .cf. Fig. 1b . We use F and F to denote the twor pelectron-localized diabatic states. These two statescan be expressed as two Slater determinants10b, c

and have the same energy at the crossing of dia-batic potential surfaces. Linear reaction coordi-nates are employed in determining the transitionstate geometry,10a ] d that is:

Ž r p . Ž .Q s Q q Q r2 2i i i

Žwhere Q refers to the ith internal coordinate bondi.length, bond angle, or dihedral angle , and r and

p refer to the reactant and product, respectively.² < : ² < < :By defining S s F F and H s F H Fr p r p i j i e j

Ž .i, j s r, p, and H is the electronic Hamiltonian ,ethe ET matrix element can be written as10 :

Ž .V s E y E r2r p 1 0

2Ž . Ž .s H y S H q H r2 r 1 y S 3Ž .r p r p r r p p r p

when the two-state model has been invoked. In eq.Ž .3 , E and E are the two adiabatic potential0 1energies of the ground state and the first excitedstate, respectively.

On the other hand, the Koopmans theorem canbe used to estimate the value of V by calculatingr p

the transition energy from LUMO and the nextŽ .lowest unoccupied molecular orbital LUMO1 for

Ž . 5, 6, 11the neutral system C H ; that is :6 6 2

Ž .D s 2V s « y « 4r p LUMO1 LUMO

where D is the energy splitting factor at the cross-ing point, and « is the eigenvalue of the canonicalSCF MO.

We use the symmetry constraint of D to per-6 hform the geometry optimization for neutral C H ,6 6and use D constraint and D constraint to per-6 h 2 hform the geometry optimization for anion radicalŽ .yC H at the 4-31G level. For the system6 6Ž .yH H , the D constraint gives one optimized6 6 6 hgeometry with a minimum energy of y230.15434

a.u. whereas the D constraint gives the other2 hoptimized geometry with an energy of y230.26627a.u. Therefore, the D benzene anion radical, be-2 hcause of its lower total energy, has been applied toform the donor]acceptor couple with the neutralbenzene in the ET reaction.

For electron localization, we increase the dis-tance between the donor and the acceptor with theoptimized geometries of the neutral molecule andthe anion, so that the SCF calculation gives a set ofMOs in which an excess electron is entirely local-ized in the anion. With this set of electron-local-ized MOs to induce the UHF SCF calculations atthe 4-31G level, we obtained the total energies, E ,rof the reactant and the E of the diabatic transi-trantion state, at different donor]acceptor distances dŽ .Fig. 2a . The transition state has been determined

Ž .with eq. 2 . The diabatic activation energy can becalculated using:

Ž .E s E y E 5c tran r

From variation of E with d, we did not find arminimum of E along d. When d G 0.45 nm, Er cremains almost invariant at approximately 14.2 kJmoly1. When d - 0.45 nm, E decreases withcthe decreasing d. The results of the ET reactionbetween two parallel ethene molecules by Ro-driguez-Monge et al.6 produced a similar conclu-sion that the ET activation energy is of approxi-mately constant value of 6 kJ moly1 when d, thedistance between the two parallel ethenemolecules, is greater than 0.5 nm. In the determi-

Ž .nation of an ET matrix element V , both eq. 3r pŽ .and eq. 4 are used for comparison. Values of Vr p

at different distances d are shown in Fig. 2b. It iswidely believed that the relationship between Vr pand d can be described by 9, 12 :

0 w Ž .x Ž .V s V exp yb d y d 6r p r p 0

where V 0 is the ET matrix element at d s d andr p 0b is a constant. From Figure 2b one can see thatthese results, obtained by solving the secular equa-

w Ž .xtion of two states eq. 3 , are more appropriateŽ . Ž .for fitting eq. 6 , whereas eq. 4 fails to describe

these results in the region of small d. To sum, theenergy splitting between two MOs used in theestimation of V is not much more reliable thanr pthe approach of direct calculation based on thetwo-state model. A least-square method has been

Ž .applied to fit the constant b by transferring eq. 6into the form of:

0 Ž . Ž .ln V s ln V y b d y d 7r p r p 0

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( )FIGURE 2. Total energy and ET matrix element at different donor ]acceptor distances d. a The total energy, Er( ) ( ) ( )filled circles of the reactant, and the total energy E open circles , of the diabatic transition state. b The ET matrix,tran

( )V . Open triangles designate those calculated using eq. 3 , and filled triangles refer to those calculated using therp[ ( )]splitting energy between LUMO and LUMO1 cf. eq. 4 .

A linear fit for V of direct calculation gives ar pvalue of 24.149 nmy1 for b ; that is:

5 Ž . w y1 xV s 4.5976 = 10 exp y24.149 drnm kJ molr p

Ž .8

The correlation coefficient turns out to be y0.9966.

ET between Spacer-Mediated BiphenylAnion Radical and Neutral Biphenyl

Biphenyl, the dimer of PPP, is a typical systemin the family of PPP oligomers. Its molecular struc-ture, electronic properties, and reactivity have beenstudied intensively through MO calculations.13 ] 17

For neutral biphenyl, we used both AM1 and abinitio calculations to investigate the torsion angleand the height of the rotational barrier. AM1semiempirical method optimization gives a valueof 44.08 for the torsion angle. On the other hand,UHFr4-31G optimization gives a torsion anglevalue of 45.18, and the internal rotation has abarrier height of 17.64 kJ moly1 when the torsionangle changes from 45.18 to 08. These results areconsistent with those of experimental estimationsand theoretical calculations by other researchers.For example, values of 44.748 for torsion angle and13.5 kJ moly1 for the internal rotation barrier heightat the level of 6-31G have been obtained byHafelinger et al.,16 whereas the 6-31GUU calcula-

tion performed by Tsuzuki et al.14 resulted in aŽ . Ularger torsion angle 46.268 , and MP2r6-31G rr

HFr6-31GU calculation performed by the sameinvestigators resulted in a value of 16.05 kJ moly1

for the barrier height.14 At the level of UHFr4-31G,we obtained C—C bond length between the twophenyl rings of 0.1486 nm, which is obviouslymore reliable than the 0.1462 nm obtained by AM1calculations. In geometry optimization, a C sym-2metry constraint has been applied.

At the level of UHFr4-31G, a planar structurefor the biphenyl anion radical has been optimized.Such coplanarity for the biphenyl anion radical orthe biphenyl cation radical has also been predictedby others.18, 19 The C—C bond length between thetwo phenyl rings is 0.1472 nm. This value of bondlength implies that the anion radical is more likelya p-system, because the conjugated p electronsforce the C—C bond close to a double bond.

To investigate ET from the biphenyl anion radi-cal to the neutral molecule, two ET systems havebeen designed, as shown in Figure 3. ET reactionsbetween the biphenyl anion and a series of accep-tors have been investigated experimentally in fluidsolution by Miller et al.8 In this work, we optimizethe geometries of biphenyl anion radical and neu-tral biphenyl, as well as the two spacers at theUHFr4-31G level, and then investigate the ETactivation energies and ET matrix elements for thetwo ET reactions demonstrated in Figure 3.

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FIGURE 3. Self-exchange ET between the biphenyl( y) ( )anion radial B and natural biphenyl B mediated by

( ) ( ) ( ) ( )a cyclohexylenyl S and b decalenyl S . d is the1 2distance between the mass centers of the donor and theacceptor.

Internal Reorganization Energy andTorsion Potential Barrier

The reorganization energy for a gas phase ETbetween the biphenyl anion radical and neutralbiphenyl is different from the torsion barrier as thetorsion angle varies from 45.18 to 08, becausechanges of bond lengths and bond angles takeplace when the system varies from the twistedneutral molecule with the torsion angle of 45.18 toa coplanar anion radical. To calculate the reorgani-zation energy, let us consider the intermolecularET between the biphenyl anion radical and the

Ž .neutral molecule Scheme 1 .

SCHEME 1.

We divide the internal reorganization energy intotwo parts: one from the torsion of the neutralbiphenyl and the anion radical, and the other fromthe changes of all internal coordinates except thetorsion motions. At the 4-31G level, we found that

Ž .the total reorganization energy for ET Scheme 1is 111.39 kJ moly1. For neutral biphenyl, the en-ergy barrier of the simple twist from 45.18 to 08 is17.64 kJ moly1, as illustrated. For the biphenylanion radical a simple torsion motion from 08 to45.18 gives a value of 36.11 kJ moly1 for the barrierheight, which is about two times higher than thatof the neutral molecule. We found that these twoparts contribute a total of 53.75 kJ moly1 to theinternal reorganization of the intermolecular reac-

Ž .tion Scheme 1 . Therefore, we may say that onehalf of the reorganization energy comes from thetorsion motion of both donor and acceptor. Such aresult means that neither the torsion motion northe bond length and bond angle changes can beneglected in the estimation of the internal reorga-nization of an ET reaction, which contains biphenylas the donor or the acceptor. One study 8 experi-mentally estimated the contribution for the reorga-nization energy from the interring torsion of the

Ž y1 .biphenyl fragment to be 0.13 eV 12.55 kJ mol ,which is lower than the directly calculated result

Žin the present work for both biphenyl 17.64 kJy1 . Ž y1 .mol and its anion radical 36.11 kJ mol .

For comparison we employ UHFr4-31G-opti-mized geometries to perform the internal reorgani-zation energy calculation with different basis setsŽ .UHFr4-31G, UHFrSTO-3G, and UHFrDZP .These calculated reorganization energies are listedin Table I. It can be seen that the internal reorgani-zation energy, l , calculated using a STO-3G, isi

higher than that calculated with 4-31G and DZPbasis sets by about 30 kJ moly1. Clearly, we shouldconsider the DZP result more reliable. When aharmonic oscillator is employed to describe thesymmetric double-well potential, the activation en-

Ž .ergy E is equal to l r4. These values are alsoc ishown in Table I.

TABLE I.( ) ( )Internal Reorganization Energy l and Activation Energy E for Gas Phase Intermolecular ET betweeni c

Biphenyl Anion Radical and Neutral Biphenyl.

Basis set UHF / STO-3G UHF / 4-31G UHF / DZP

y1( )l kJ mol 143.53 111.39 109.21iy 1( )E kJ mol 35.88 27.85 27.30c

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Spacer-Mediated ET between BiphenylAnion Radical and Neutral Biphenyl

Two spacers, cyclohexylenyl and decalenyl, asshown in Figure 3, are isolately optimized at theRHFr4-31G level. In linking the donor and accep-tor to the spacers, we use the originally optimized

Ž .geometries at the appropriate positions Fig. 3 andassume a value of 0.1533 nm for the C—C bond,

Ž .which connects the donor or acceptor and thespacer. Similar to the case of the ET reaction be-tween benzene anion radical and neutral benzene,the ET transition state is determined with the

w Ž .xlinear reaction coordinate eq. 2 . The spacers areassumed to be rigid. In determining the reorgani-zation energy and ET matrix element, we per-formed the initial-guess-induced UHFSCF calcula-tions at the STO-3G level for the two ET systems,as shown in Figure 3. The calculated results, in-cluding net charge, overlap integral between thetwo electron-localized Slater determinants F andrF , etc., are given in Table II.p

The cyclohexylenyl- or decalenyl-mediated ETŽ y.between the biphenyl anion radical B and the

Ž .naphthalene molecule N have been measuredexperimentally by Closs et al. Their results8b forV for these two ET systems, By]S ]N andr p 1By]S ]N, are given in Table II for comparison.2

y y ŽSome similarity between B ]S]B and B ]S]N S.represents S or S can be predicted because of1 2

Ž 0the negligible free energy change DG s y0.05.8c yeV for the B ]S]N system.From Table II it can be seen that, in these two

intramolecular ET systems, By]S ]B and By]S ]B,1 2the activation energies, E , are almost the samecand close to the UHFrSTO-3G results given in

Table I. Such a similarity of activation energies inboth the intermolecular and intramolecular ET re-actions is reasonable because we assumed thechanges of bond lengths and bond angles in thedonor and acceptor fragments to be the same as inthe intermolecular ET. This simplification is help-ful in the theoretical calculation of an intramolecu-lar ET in which a long chain intervenes betweenthe donor and the acceptor. In the By]S ]B and1By]S ]B ET systems, we can use the more reliable2DZP values of E instead of the UHFrSTO-3Gcresults in further kinetic calculations. In compari-son with experimentally fitted results of V in ther pET reaction between biphenyl anion radical andneutral naphthalene mediated with the same spac-ers,8b the values of V determined in the presentr pwork are acceptable if we neglect the contributionof solvents to V .r p

All ab initio calculations were performed byusing the HONDO95.6 package.20

Acknowledgment

The authors thank Dr. M. Dupuis for his help inthe use of HONDO computer program.

References

Ž .1. a Frommer, J. E.; Chance, R. R. In Kroschwitz, J. I., ed.Electrical and Electronic Properties of Polymers: A State-of-the-Art Compendium; John Wiley & Sons: New York; 1988,

Ž .p 84. b Shirota, Y. In Takemoto, K., et al., Eds. FunctionalMonomers and Polymers; Marcel Dekker: New York; 1987,

Ž .p 283. c Sato, N., Seli, K., Inokuchi, H. J Chem Soc FaradayŽ .Trans II 1981, 77, 1621. d Bredas, J. L., Elsenbaumer, R. L.,

Ž .Chance, R. R. J Chem Phys 1983, 78, 5656. e Bredas, J. L.,Chance, R. R., Silbey, R. J Chem Phys 1981, 75, 255.

TABLE II.( ) ( ) ( )Overlap Integral between F and F , ET Matrix Element V , Activation Energy E , Net Charges P and Pr p rp c D A

( )of Transition State, and Distance d between Two Mass Centers of Donor and Acceptor for ET Systemsy y a y 1( )B – S – B and B – S – B Energies in kJ mol .1 2

b c ( )Systems E S V P P d nmc rp rp D A

y d e fB ]S ]N 1.686 1.181y d e fB ]S ]N 0.407 1.402y y 3 g h i( )B ]S ]B 36.35 y5.399 = 10 2.055 1.555 y0.94 y0.06 1.2871y y 3 g h i( )B ]S ]B 36.60 1.132 = 10 0.429 0.326 y0.95 y0.05 1.6462

a UHF / STO-3G results. b Net charge on donor. c Net charge on acceptor. d Cyclohexylenyl- and decalkenyl-mediated ET reactionsbetween biphenyl anion radical and neutral naphthalene appear in ref. 8b. e Fitted from the experimentally measured rate

8b f 8b g ( ) hconstant. MM2 force field optimized results. ET matrix elements using eq. 3 . ET matrix elements using energy splitting[ ( )] ibetween LUMO and LUMO1 in a neutral system cf. eq. 4 . Distance between the two mass centers of the donor and the

acceptor.

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2. Chance, R. R.; Bredas, J. L.; Silbey, R. Phys Rev B 1984, 29,4991.

3. Fukomoto, H.; Sasai, M. Prog Theor Phys 1982, 67, 41; 1983,69, 1; 1983, 69, 373.

4. Bredas, J. L.; Themans, B.; Fripiat, J. G., et al. Phys Rev B1984, 29, 6761.

5. Rodriguez-Monge, L.; Larsson, S. Int J Quantum Chem1997, 61, 847.

6. Rodriguez-Monge, L.; Larsson, S. J Phys Chem 1996, 100,6298.

7. Rodriguez-Monge, L.; Larsson, S. J Chem Phys 1995, 102,7106.Ž .8. a Miller, J. R.; Paulson, B. P.; Bal, R., et al J Phys Chem

Ž .1995, 99, 6923. b Closs, G. L.; Calcaterra, L. T.; Green, N. J.,Ž .et al. J Phys Chem 1986, 90, 3676. c Miller, J. R.; Calcaterra,

Ž .L. T.; Closs, G. L. J Am Chem Soc 1984, 106, 3047. d Liang,N.; Miller, J. R.; Closs, G. L. J Am Chem Soc 1989, 111, 8740.Ž .e Liang, N.; Miller, J. R.; Closs, G. L. J Am Chem Soc 1990,112, 5353.Ž . Ž .9. a Marcus, R. A. J Chem Phys 1956, 24, 966. b Marcus,

Ž .R. A. Dis Faraday Soc 1960, 29, 21. c Siddarth, P.; Marcus,R. A. J Phys Chem 1993, 97, 6111.Ž .10. a Li, X.-Y.; Tian, A.-M.; He, F.-C., et al. Chem Phys Lett

Ž .1995, 233, 227. b Farazdel, A.; Dupuis, M.; Clementi, E.,

Ž .et al. J Am Chem Soc 1990, 112, 4206. c Li, X.-Y.; Tian,Ž .A.-M.; He, F.-C., et al. J Comput Chem 1997, 17, 1108. dŽ .Farazdel, A.; Dupuis, M. J Comput Chem 1991, 12, 276. e

Ohta, K.; Morokuma, K. J Phys Chem 1987, 91, 401.

11. Ladik, J. J. Quantum Theory of Polymers as Solids; Plenum:New York; 1988, p 199.Ž . Ž .12. a Liang, C.; Newton, M. D. J Phys Chem 1993, 97, 3199. bLiang, C.; Newton, M. D. J Phys Chem 1992, 96, 2855.

13. Farmer, B. L.; Chapman, B. R.; Pudis, D. S., et al. Polymer1993, 34, 1588.

14. Tsuzuki, S.; Tanabe, K. J Phys Chem 1991, 95, 139.

15. Perakyla, M.; Pakkanen, T. A. J Chem Soc Perkin Trans 2,1995, 1045.

16. Hafelinger, G.; Regelmann, C. J Comput Chem 1987, 8,1057.Ž .17. a Rubio, M.; Merchan, M.; Orti, E. Theor Chim Acta 1995,

Ž .91, 17. b Rubio, M.; Merchan, M.; Orti, E., et al. ChemPhys Lett 1995, 234, 373.

18. Takei, Y.; Yamaguchi, T.; Osamura, Y., et al. J Phys Chem1988, 92, 577.

19. Buntinx, G.; Poizat, O. J Phys Chem 1989, 91, 2153.

20. Dupuis, M.; Marquez, A.; Davidson, E. R. HONDO95.6; IBM:Kingston, NY; 1995.

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