Qixi Mi, Mark A. Ratner and Michael R. Wasielewski- Time-Resolved EPR Spectra of Spin-Correlated Radical Pairs: Spectral and Kinetic Modulation Resulting from Electron-Nuclear Hyperfine

8/3/2019 Qixi Mi, Mark A. Ratner and Michael R. Wasielewski- Time-Resolved EPR Spectra of Spin-Correlated Radical Pairs: S

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g2, becomes the mixing term between the two nearly isoenergetic

S and T0 states

(g1S1z + g2S2z)|S )(g1 - g2)|vV - (g2 - g1)|Vv

22

)g

2|T0 (2)

The full matrix form of eq 1 reads

B2

(-g1 - g2 0 0 0

0 2J

g 00 g 0 0

0 0 0 g1 + g2 )|T-1

|S|T0

|T1

(3a)

or in magnetic units

(-B0 0 0 0

0 2J Q 0

0 Q 0 0

0 0 0 B0)

|T-1

|S|T0

|T1

(3b)

in which B0 is the center field and 2Q is the field difference

between two resonant peaks. Note that symbols 2J and 2Q

signify their physical meanings, while J and Q are used for

mathematical convenience. Diagonalization of the S-T0

block

gives the new eigenstates and their respective eigenvalues

|S ) cos

2|S + sin

2|T0 S|H|S ) J +

(4a,b)

|T0 ) cos

2|T0 - sin

2|S T0 |H|T0 ) J -

(4c,d)

where 2 ) Q2 + J2 and tan ) Q/J, Scheme 1a,b. Thetransitions from either of these new eigenstates to the intact

T(1 states, a total of four listed in Table 1, are now partly

allowed. Since g |J| g 0, these four transitions can be grouped

into two doublets, one centered at B0 + and the other at B0- , with a common splitting of 2J. Previous discussions of

the four-state model assume that J , Q, so that tan andtherefore are large, resulting in large transition probabilitiesfor each line. However, the four-state model is more general

and gives reasonable results even if J Q, albeit with greatly

decreased transition probabilities.

Transient species are usually not populated according to the

Boltzmann distribution. Instead, their sublevel occupancies are

dictated by the precursor and the populating mechanism. A

SCRP resulting from ultrafast electron transfer inherits the

overall spin state of its precursor. If this is a singlet state, as inmost cases, only the new mixed eigenstates |S and |T0 willbe populated

F(|S) ) cos2

2F(|T0) ) sin

2

2(5a,b)

Along with the transition probabilities

P(S f T(1) ) sin2

2P(T0 f T(1) ) cos

2

2(6a,b)

the concise result is obtained that all four transitions have equal

intensities as a function of Q and J

|I| ) sin2

2cos

2

2)

1

4

Q2

Q2

+ J2

(7)

In terms of the signs, two of the transitions appear in absorption

and the other two in emission. Given a positive 2J, the transitions

are sorted in Table 1 from high to low energies, and they will

appear in a field-swept EPR spectrum from low to high fields.

This polarization pattern is denoted by e/a/e/a, or even shorter

by e/a for each doublet, Scheme 1.

SCRP from a Triplet Precursor. A triplet precursor can be

spin-polarized prior to charge separating to give a SCRP. In

the simplest case, the precursor is under thermal equilibrium

with almost equal populations in each sublevel

F(T-1) F(T0) F(T1) 1

3(8)

After charge separation, the T0 population is redistributed

between the new eigenstates

F(|S) )1

3sin

2

2F(|T0) )

1

3cos

2

2(9a,b)

Substituting these into the derivation in the last section, we once

more get four lines with the same but weaker intensity

|I| )1

3sin

2

2cos

2

2)

1

12

Q2

Q2

+ J2

(10)

Nonetheless, assuming again that 2J > 0, there is a qualitativechange in the spectrum in that the polarization pattern inverts

to a/e.

A main source of triplet states is spin-orbit intersystem

crossing (SO-ISC) from excited singlet states.10,11 The spin-orbit

interaction and thus the population distribution within a SO

triplet state are purely anisotropic. The EPR spectra of randomly

ordered SO triplets in solid solution exhibit a variety of polarized

powder patterns since at each field value, only the triplets with

a specific orientation are in resonance. However, after the triplet

precursor undergoes charge separation, the zero-field splitting

(zfs) interaction vanishes and, the very wide powder pattern

suddenly collapses into the narrow SCRP line shape. 12 Hence,

although the latter is detected only at the vicinity of the centerfield B0, its triplet precursor can have taken on an orientation

SCHEME 1: Four-State Model of a SCRPa

a (a) Populations and transition probabilities are indicated by the linethickness. (b) The trigonometric relation between Q, J, , and . (c)A schematic four-line spectrum of the SCRP.

TABLE 1: Four Equally Intense Transitions of a SCRPfrom a Singlet Precursor

transition energy probability polarization

S f T-1 B0 + + J sin2( /2) emissive

T0 f T1 B0 + - J cos2( /2) absorptive

T0 f T-1 B0 - + J cos2( /2) emissive

Sf

T1 B0 -

- J sin2

( /2) absorptive

Spin-Correlated Radical Pairs Coupled to Nuclear Spins J. Phys. Chem. A, Vol. 114, No. 1, 2010 163


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that corresponds to anywhere within the broad triplet spectrum.

In other words, the resonant fields of an individual triplet

molecule before and after charge separation are completely

unrelated to each other; therefore, an ensemble-averaged

population of the SO triplet state can be used to evaluate the

spin polarization of its SCRP successor.

It has been shown for the triplet mechanism (TM) of

chemically induced dynamic electron polarization (CIDEP) that

the sublevel populations of a SO triplet are given by 13,14

F0 )1

3

F(1 )1

3-

2

15[(3AZ - 1)D

B0+ (AX - AY)

E

B0] (11a,b)where the center field B0 is around 0.34 T for X-band EPR, D

and E are the zfs parameters, and AX,Y,Z are the anisotropic

population distributions for the SO-ISC mechanism. For a typical

organic triplet state, E , D < B0 /4, and the limit of spinpolarization in a SO triplet is estimated to be Fi )(1/3) - (i/

15) (i ) -1, 0, 1), or a 20% excess in the T-1 sublevel. Such

a ratio is much less than the spin polarization due to radical

pair intersystem crossing (RP-ISC). It renders two of the four

SCRP lines more intense than the other two but does not invertthe sign of the spin polarization. In a word, the polarization

pattern developed earlier in this section for triplets in thermal

equilibrium also holds for SO triplets in an isotropic medium.

There is a third possibility that only the center sublevel T0of a triplet is populated, resulting from a reversed RP-ISC

mechanism.15 Following the same scheme

F(|S) ) sin2

2F(|T0) ) cos

2

2(12a,b)

I(S f T(1) ) (sin4

2I(T0 f T(1) ) (cos

4

2(13a,b)

Thus, the two transitions within the triplet manifold are

substantially stronger, forming a polarization pattern e/A/E/a,

where the capital letters A and E denote enhanced line

intensities. With a small Q/J ratio and thus a small , the e/A/E/a pattern becomes essentially A/E, possibly giving the

incorrect impression that an inversion in the sign of the

polarization has occurred. However, various factors contributing

to line broadening make a convoluted spectrum difficult to judge

by simple inspection, so that spectral simulations are needed to

verify the ISC mechanism.

Simulation Methods

Spectral Simulations. The four-state model developed earlierfor a SCRP is succinct and elegant, yet the X-band EPR spectra

of SCRPs are frequently difficult to interpret without full

simulation because the hyperfine splittings of the radicals are

usually larger thang. Hence, the SCRP line shape as a function

ofQ and J, eqs 4 and 7, should be applied as a spectral kernel

in more complicated situations where all of the factors causing

a field shift can be combined into an effective 2Q term.16

A nuclear spin induces a weak, local magnetic field that offsets

the resonant field of an electron spin by 2Q ) an, where a is the

hyperfine coupling constant (hfcc) and n is the nuclear magnetic

quantum number. For an individual SCRP with a specific nuclear

state (nk), the hyperfine offsets can be summed up and directly

substituted into the four-state model. However, in an ensemble of

molecules, the state of each nuclear spin is unrelated to any other.Enumerating each combination and then averaging over the

ensemble is highly inefficient and scales exponentially as the

number of nuclei increases. There are also situations involving

g-factor and/or hyperfine anisotropies, or even spin dynamics, in

which 2Q becomes intrinsically a continuum.

To simplify these complexities in real-world systems, it

should be emphasized that only the difference in resonance

fields, 2Q, matters in eqs 4 and 7. Despite the huge number of

nuclear states, their overall contribution to the range of 2 Q is

limited by the EPR spectral widths of the individual radicals.

For each 2Q within the limit, a subspectrum can be calculatedand summed to an ensemble-averaged spectrum

ISCRP ) 2Q

P(B0, Q)X(Q,J) (14)

Here, the SCRP line shape is depicted in Scheme 1; the line

position B0 is the center of the four lines, the probability P is

the statistical weight of the hyperfine states that give rise to B0and Q, and X denotes convolution. Therefore, under such a

nuclear configuration, the resonant fields of the radicals on their

own are

B1 ) B0 + Q B2 ) B0 - Q (15a,b)

Because nuclear spins are independent of each other, thestatistical weight of the combined nuclear configuration can be

divided into two parts

P(B0, Q) ) P(B1,B2) ) P1(B1)P2(B2) (16)

It is important to realize that the individual probability Pi(Bi),

i ) 1 or 2, is just a synonym for the EPR spectrum Ii(B) of the

radical. Consequently, eqs 15 and 16 can be rewritten into

P(B0, Q) ) I1(B0 + Q)I2(B0 - Q) (17)

In a computer simulation program, this is carried out by shifting

the two EPR spectra I1,2(B) relative to each other and then taking

a pointwise multiplication. When the increment 2Q is sufficiently

small, eq 14 becomes an integral

ISCRP ) 2I1(B0 + Q)I2(B0 - Q) X (Q,J)dQ (18)

with 2J and the two EPR spectra I1,2(B) as the input. There is

no need for hyperfine information such as the hfccs or nuclear

spin; eq 18 serves better as a functional that blends two known

spectra into a third, convoluted one.

Scheme 2 illustrates the basis of this procedure using a simple

example. Scheme 2a shows the stick plots of two organic

radicals with the same g factor; radical 1 has no hfccs, and

radical 2 is split by four identical protons with aH ) 1.5 mT.

We assume that radicals 1 and 2 constitute a SCRP with 2J )

1.0 mT. In Scheme 2b, the resulting four-line patterns areconsidered for possible combinations of the nuclear states. Since

radical 1 has no hfccs, it does not contribute to (mI), while

the nuclear spin states of radical 2 result in (mI) ) +2 (red

spectrum), +1 (blue spectrum), and 0 (magenta spectrum). Only

the nuclear spin states for which (mI) > 0 are shown for clarity.It is important to note that since the intensities of each line

depend on Q2/(Q2 + J2) (eq 7), when two radicals have the

same g factor, the intensity of each spectral line depends only

on the contribution to Q from the value of(mI) for each nuclear

spin state configuration. Thus, referring to Scheme 2b, when

(mI) ) 0 (magenta), J . Q, so that the EPR transitions are

forbidden and the line intensities vanish; when (mI) ) +2 (red),

J, Q, so that the four-line pattern is composed of two antiphase

doublets separated by approximately 2Q (3 mT in this case).However, the (mI) ) +2 nuclear state has a low probability,

164 J. Phys. Chem. A, Vol. 114, No. 1, 2010 Mi et al.


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and the line intensities of the four-line pattern are relatively

weak. Lastly, when (mI) ) +1 (blue), J Q, so that a good

balance between transition probability and nuclear state popula-

tion is achieved, and the line intensities are stronger. All four-

line patterns including those for (mI) < 0 are shown in Scheme2c. It is obvious that the field position of the antiphase doublet

from radical 1 (no hfccs) stays relatively fixed, whereas the

other antiphase doublets from the various nuclear spin states of

radical 2 are shifted to other fields based on aH with their line

intensities governed by the binomial nuclear spin state statistics

and eq 7. As a result, the hyperfine split antiphase doublets from

radical 1 generally sum constructively, while those of radical 2

tend to cancel each other. On the other hand, if Q is finite, none

of these antiphase doublets are exactly centrosymmetric about

the origin, so that, for example, the doublet due to the (mI) )

-1 nuclear state (orange) does not fully overlap with that of

the (mI) ) +1 nuclear state (blue), and similarly for the red

and green states.

Inhomogeneous Line Broadening. Building on the concepts

illustrated in Scheme 2, which focuses on a simple binomial

distribution of nuclear spins states, and taking advantage of eq

18, we now consider a simple combination of a broad,

featureless cation radical spectrum and a narrow anion radical

spectrum. The cation line shape, assumed to result from

inhomogeneous broadening due to a large number of hyperfine

splittings, is described by a Gaussian function

I1(B + B0) )1

2exp(- B

2

22) (19)

with a standard deviation . The narrow anion can be idealizedinto a Dirac function, I2(B + B0) ) (B). Then, an analyticalsolution including an absorption and an emission term can be

obtained for eq 18

I(B + B0) ) A(B - J) - A(B + J)

A(B) )(J

2/B

2- 1)

2

4(J2/B

2+ 1)

1

2

exp

[-

(J2/B

2- 1)

2

22

/B2

](20a,b)

An example plotted in Figure 1 shows that all of the basic

elements of the four-line spectral kernel are still present; the anion

is split into four sharp lines, and the wide wings result from the

cation. However, the enormous contrast between the line intensities

of the cation and anion makes the latter dominate the spectrum,

and the apparent polarization pattern becomes e/e/a/a.

A quantitative analysis can also be performed on the functionA(B) in eq 20. First of all, there are a few blind spots in the

SCRP spectrum, despite the broad, unresolved nature of the

cation radical spectrum. These spots are located at B ) 0, (J,

and (2J, such that A(B ( J) ) 0, and they significantly help

retain the resolution of the sharp center lines. Furthermore, the

gap between the pair of e/e or a/a peaks can be derived by

solving (d/dB)A(B) ) 0. The result is, in general, very

complicated, but under the condition that the cation radical signal

is broad enough ( > 2J), the expression reduces to 21/2J2/,Figure 1.

Rather than the function, a binomial distribution of verynarrow lines separated by a small hyperfine splitting a is a better

model for actual anion radicals, Figure 2a. The EPR spectrumfor this improved model absorbs the binomial pattern by an

approximate convolution with eq 20, which essentially serves

as a line-broadening process or a low-pass bandwidth filter. It

can be shown that when the doublet gap in Figure 1 exceeds 3

times the hyperfine splitting in Figure 2a, the latter will be

overwhelmed as a result of the convolution. That is to say

2J2/> 3a or |2J| > 62a 2.9a (21)In Figure 2b, two EPR spectra are simulated for the same SCRP

with two 2Jvalues. It is intriguing to observe that by tuning 2J

around the transition point 2.9(a)1/2, the SCRP spectrum makesa switch from displaying the anion hyperfine structure to mainly

the four peaks similar to those in Figure 1. This useful propertyputs a limit on the value of 2Jonce and a are known, withoutthe need for spectral simulation.

Lifetime Broadening. A SCRP has two somewhat paradoxi-

cal qualities; its spin dynamics is described by quantum

mechanics, while the electron-transfer rates fall into the classical

chemical kinetics regime. Nonetheless, the Correspondence

Principle requires that quantum mechanical results of a large

object approximate its classical properties. For instance, the

decay of a population F at a first-order rate k is expressed by

(d/dt)F ) -kF. Alternatively, in quantum mechanical language

d

dt| ) -

i

pH| F ) | (22a,b)

In order to reconcile these definitions, one simply needs to assignan imaginary energy term -ipk /2 to the Hamiltonian

SCHEME 2: Contribution of Nuclear Spin States toSCRP Line Intensities As Described in the Text

Figure 1. Simulated EPR spectrum of a SCRP consisting of a broadcation radical with a Gaussian width of) 1 mT and a narrow anionradical, according to eq 20. The spin-spin exchange coupling 2J isequal to 1 mT.



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H| ) -1

2ipk| |H )

1

2ipk|

(23a,b)

such that

ddt| ) - ip|(H|) + ip(|H)| ) -k|

(24)

Henceforth, p is dropped for brevity, and it is noteworthy that

hermitian properties no longer apply to H.

On the basis of the simple Hamiltonian in eq 1, we consider

spin-selective charge recombination rates kS and kT that diminish

the SCRP populations in the S and T0,(1 sublevels, respectively.

Then, the new hybrid Hamiltonian reads

H ) (2J QQ 0

)|S|T0

-

1

2(

kS 0

0k

T

))

(2J -

1

2ikS Q

Q -1

2ikT

)(25)A preferred method to quantum mechanically treat population

decay is to use density matrices and superoperators 17,18

d

dtF ) -i(HF - FH

) ) -iLF (26)

L ) (-ikS -Q Q 0

-Q 2J - ik+ 0 Q

Q 0 -2J - ik+ -Q

0 Q -Q -ikT)

|SS||ST0|

|T0S|

|T0T0|

k( )kS ( kT

2(27a,b)

Here, the superoperator L is identical to the one derived by

Hore4 directly from basic density matrix definitions, which

validates our treatment of kinetic rates as imaginary frequencies

in the Hamiltonian, eq 23.

The hybrid Hamiltonian in eq 25 can be diagonalized like a

hermitian one in the usual way to yield the new eigenstates

and eigenvalues

|S ) cos

2|S + sin

2|T0

S|H|S ) J - 12

i/k+ + (28a,b)

|T0 ) cos

2|T0 - sin

2|S

T0 |H|T0 ) J -1

2ik+ - (29a,b)

with the generalized parameters 2 ) (J - 1/2ik-)2 + Q2 and

tan ) Q/(J - 1/2ik-), which are complex when k- * 0 or kS* kT. Since the angle and all coefficients in eqs 28b and 29bcan also be complex, the physical significance of a complex

eigenenergy warrants interpretation; its real part corresponds

to the usual energy level, whereas the imaginary part equals

half of the decay rate as introduced earlier in eq 23. While this

interpretation makes conceptual sense, it proves completely

unnecessary in calculations. All mathematical operations in eqs

28 and 29 are self-consistent on treating a complex number as

an integral entity, with both the energetic and kinetic information

contained naturally in a single term. For example, the EPR line

shape associated with a relaxation process at frequency 0 andrate k is conveniently noted by a Greens function19

I() ) i - L

)i

- 0 + ik)

i( - 0) + k

( - 0)2

+ k2

(30)

whose real part is a Lorentzian function centered at 0, and theimaginary part is a dispersive line shape. Equations 28 and 29

are exemplified by a numerical simulation with the parameters

2J ) 2Q ) 1 mT, kS ) 1 106 s-1, and kT ) 5 10

6 s-1.

Here, the singlet charge recombination rate kS is assumed to be

much slower because the process is usually deeply in the Marcus

inverted region.20,21 As presented in Figure 3a, the spin

population shows a damped oscillation between the pure singlet

and triplet configurations, driven by the mixing term 2Q. Once

again, these results prove to be the same as those obtained fromthe Till-Hore model.22 Alternatively, a projection onto the new

eigenstates |S and |T eliminates the modulations and leavestwo exponential decays at rates kS and kT . It can be shown thatS-T0 mixing always brings the two decay rates closer to each

other

kS ) k+ - Im 2 > kSkT ) k+ + Im 2 < kT (31a,b)

as opposed to their wider energy gap.

Figure 3b depicts the corresponding EPR spectrum of the

SCRP. Compared with the simplest four-line spectrum in

Scheme 1c, the e/a/e/a polarization pattern characteristic of a

singlet precursor and a positive 2J is retained, whereas eachspectral line turns into a Lorentzian peak due to lifetime

Figure 2. A model SCRP consisting of a broad cation radical with a Gaussian width of ) 1 mT and a narrow anion radical coupled to severalprotons with a hyperfine splitting of a ) 0.1 mT. (a) Schematic EPR spectra of its components. (b) Simulated EPR spectra with 2J ) 0.8 or 1.0mT.



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broadening, and the shorter-lived triplet component results in a

broader and weaker pair of peaks at the spectral center.

Specifically, the Lorentzian width of a transition is equal to the

average of the decay rates of its initial and final states

(S f T(1) ) 12(kS + kT)

(T0 f T(1) )1

2(kT + kT) (32a,b)

Consequently, even though the two spin-selective decay rates

kS and kT differ by a factor of 5, they do not produce distinct

EPR line widths in Figure 3b.

Transient Continuous-Wave (CW) EPR Spectroscopy. The

non-Boltzmann spin distribution within a SCRP results from

ultrafast charge separation, which is orders of magnitude faster

than spin relaxation. The significant spin polarization generates

enhanced absorption and emission lines, so that formation of a

relatively small yield of SCRPs in a dilute sample solution can

yield very intense EPR signals. Also, an analysis of the riseand decay kinetics of these EPR lines provides additional

information.

According to the time-dependent theory of spectroscopy,23 a

spectrum in both the frequency and time domains reflects the

relationship between the autocorrelation function *()(0)and an excitation electromagnetic wave at frequency 24

I(, t) )4i

3cp

0

te

i*()(0)d (33)

where c is the speed of light, p is the reduced Planck constant,

and () is the systems electric or magnetic moment. In thelinear regime of continuous-wave EPR, I is proportional to the

complex magnetic susceptibility ) + i, and is replacedby the x magnetization Sx. Ideally, only a spectrum or kinetictrace is necessary to fully characterize the time propagation of

Sx(); in practice, the frequency and time domains complementeach other to reach higher resolutions and signal-to-noise ratios.

Equation 33 also establishes a quasi-Fourier-transform (FT)

relationship between the two domains; a spectral peak automati-

cally corresponds to damped or modulated kinetics and vice

versa.

The difference between eq 33 and its steady-state (SS) version

I(, SS) )4i

3cp

0

e

i*()(0)d (34)

is simply that a steady-state spectrum is considered to be

measured at a time long enough to establish thermal equilibrium,I(, SS) ) limtf I(, t). This makes the integral in eq 33

exactly half of the inverse FT of the autocorrelation function

Sx*()Sx(0)

(,SS) F-1[S*()S(0)] or

S*()S(0)

F[(, SS)] (35a,b)

This can be formally substituted back in eq 33 to yield

(,t) 0

te

iF[(,SS)]d (36)

Keeping in mind that the integrand here is simply the free

induction decay (FID), eq 36 reveals that to first-order, a

transient cw kinetic trace is essentially the integrated version

of a FID measured at the same microwave frequency. Given

infinite time resolution and sensitivity, a FID fully characterizes

the single-quantum transitions of a system, and so does a

transient cw kinetic trace.

A transient cw spectrum can also be related to its steady-

state counterpart. Adhering to the FT formalism, a rectangular

function is introduced to account for the time dependence

(, t) F-1[S*()S(0)rect(/t)]

rect() ) {1 |x| e 10 |x| > 1 (37a,b)The convolution theorem states that the FT of a product equals

the convolution of individual FTs. Then, the above two

equations combine to afford

(, t) ) (, SS) X F-1[rect(/t)] )

(, SS) Xt

sinc(t) (38)

Here, the convolution with sinc(x) ) sin(x)/x effectivelyaverages out all of the fine structure in (, SS) that has aspectral resolution of < 2/t. Rearrangement gives thefamous Uncertainty Principle

E tg h (39)

in which t equals the time interval between charge separation

and spectrum acquisition, E stands for the highest spectral

resolution in energy units, and h is Plancks constant. For

example, only those spectra obtained after360 ns can resolve

a hyperfine splitting of 0.1 mT or 2.8 MHz.

Results and Discussion

2D Transient CW EPR Spectra. In all of the abovetheoretical arguments, it has been taken for granted that the

Figure 3. (a) Simulated kinetics and (b) steady-state spectrum of a transient SCRP, whose triplet component decays faster and has a broader EPRline shape. In (a), the evolution of sublevel populations appears either as damped oscillations (s) or as exponential decays (- - -), depending on therepresentation. Inset: S-T0 mixing always brings the two decay rates closer to each other. Parameters: 2J ) 2Q ) 1 mT, kS ) 1 10

6 s-1, kT )5 106 s-1.



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electron spin-spin exchange interaction 2J is a constant. To

meet this requirement experimentally, the electron donor and

acceptor must be kept within a fixed distance, similar to the

alignment of the bacteriochlorophyll special pair and the

ubiquinones in photosynthetic reaction centers.2527 Photoexci-

tation of the covalent donor-chromophore-acceptor (D-C-A)

triad (shown below) has been shown to produce a SCRP,28,29

which mimics the spin dynamics characteristic of SCRP

formation in photosynthetic reaction centers.30 Figure 4 shows

the 2D density plots of transient cw EPR spectra (B, t) ofD+-C-A- in a toluene solution at two temperatures, 210 and

295 K. In the following, this SCRP is employed as a benchmark

for validating the theoretical models discussed above.

First of all, the uncertainty relationship between the energy

(i.e., magnetic field) and the time domains is investigated. At

both temperatures, it is evident that more and more fine spectral

features are resolved as time elapses. In Figure 4a, this

progression falls into three discrete stages. For 0 < t < 0.2 s,

the spectrum is broad and featureless with an e/a polarizationpattern. Later, when 0.2 < t < 0.4 s, a 0.19 mT hyperfinesplitting appears at the spectral center. In the last stage, when

t > 0.4 s, each of the hyperfine lines further splits into twowith a much greater modulation depth, followed by an expo-

nential decay of the signal. A similar process is also present in

Figure 4c, although only two stages of evolution can be

recognized. The modulations in the spectrum are rather shallow

but cover almost the whole spectral range.

A better method to quantitatively determine E, the highest

spectral resolution, is by Fourier transformation along the field

axis. Then Ecan be directly read out as the highest component

in units of inverse field, which converts to time according to

the identity (1 mT)-1 ) 35.7 ns. Plotted in Figure 4b,d are the

corresponding field-wise FTs of the transient cw EPR spectra

in Figure 4a,c. A side-by-side comparison of the corresponding

FT pair sheds light on the trend that each new stage in the

evolution of hyperfine patterns is simply due to the introduction

of a higher-resolution spectral component. Moreover, all of these

components plotted as the dark areas appear first from the low-

resolution side and then to the high-resolution side, and they

altogether form a linear envelope that runs diagonally from the

origin of the FT plots. Rearranging eq 39 gives t/()-1 g 1,in which the ratio on the left-hand side is directly represented

by the slope of the envelope. In Figure 4b,d, the slope is

determined to be 1.02 at 210 K and 1.06 at 295 K, just slightly

above the theoretical limit, which confirms that the spin

dynamics ofD+-C-A- is indeed a first-order and relaxation-

free process within at least the first 0.4 and 0.2 s, respectively,for the two temperatures.

Spectral Analyses. According to the theoretical discussion

given above, the EPR spectrum of a SCRP incorporates the

hyperfine structures of both radical constituents as well as the

four-line spectral kernel, eq 18. In the case of the triad D-C-A,

two reference molecules, D-C and A, were chemically

converted into their corresponding radicals, D+-C and A-,

and their individual cw EPR spectra are presented in Figure

5a. As studied earlier,31 the donor cation exhibits a resolved

array of hyperfine peaks at 295 K thanks to motional averaging

of the two possible methoxy group orientations. At 210 K, the

conformational dynamics ofD+-C is frozen, and only a broadGaussian-like profile remains. By contrast, the acceptor anion

Figure 4. Experimental transient cw EPR spectra (B, t) of photoinduced SCRP D+-C-A- in toluene at (a,b) 210 and (c,d) 295 K. TheUncertainty Principle limits are indicated by the red lines.



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radical is a rigid, planar aromatic system, which keeps its EPR

spectrum free of dynamic effects and always resolved in liquid

solution.

The combination of a broad, poorly defined cation radical

signal and a narrow, resolved anion radical signal can sometimeslead to a simple situation, in that their spectral features cover

separate energy ranges and remain uncontaminated by each other

even in a convoluted spectrum. Specifically, A- contributes only

to signals near the spectral center where D+ behaves like a

background signal. Then, each hyperfine line of A- is split by

electron spin-spin coupling into an e/a doublet with a constant

spacing of 2J. In Figure 5, a careful inspection of the

D+-C-A- spectrum at 210 K and 0.43 s reveals that its 14central lines can be reconstructed from the 13 lines of A- plus

a 2Jcoupling equal to 3/2 times the hyperfine splitting, roughly

0.14 mT. Other half-integer ratios such as 1, 2, or 5/2 will

produce completely different patterns. This estimation gives a

good initial value of 2Jto be refined by numerical calculations.

Besides 2J, there are more parameters involved in thesimulation of SCRP spectra, charge recombination rates kS,T and

time t. Since all of the EPR results are in arbitrary units, the

average decay rate k+ ) (kS + kT)/2 appears in the proportional-

ity factor and scales the whole spectrum uniformly. Conversely,

the rate difference kT - kS preferentially depletes the triplet

character of the SCRP, as in eqs 28 and 29. In addition, thetime t defines the extent of uncertainty broadening, which is

implemented in the simulations by nullifying all spectral

components having an energy resolution ofE < h/t.Figure 6 shows the transient cw EPR spectra ofD+-C-A-

under several typical conditions and the simulation results.

Besides an e/a polarization pattern, they all exhibit hyperfine

splittings, which serve as alignment marks to help fine-tune the

simulation parameters. At 210 K, optimization of the fits for

the experimental spectra at 0.25 and 0.43 s adjusts 2J to 0.15mT from the early estimate. At 295 K, this value more than

triples to 0.48 mT. For all of the cases in Figure 6, the two

shoulder peaks labeled by stars are considerably more pro-

nounced than the simple model in Figure 1. Such a shrug

effect can be understood by recognizing that a faster kT trimsdown the two inner, triplet-derived lines of the four-line spectral

Figure 5. (a) Integrated and normalized cw EPR spectra of the individual moieties D+-C and A-. The experimental spectrum of A- is depictedas a stick plot to illustrate the number of lines that comprise it. (b) A close-up of the spectral centers of D+-C-A- (top) and A- (bottom). Thetransient cw EPR spectrum of D+-C-A- in toluene is taken at 210 K and 0.43 s.

Figure 6. Experimental (black) and simulated (red) transient cw EPR spectra of D+-C-A- in toluene at (a,b) 210 and (c) 295 K. The simulationparameters are tabulated in (d). The stars denote shrugs due to a faster rate for triplet charge recombination.



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kernel and thus diminishes the central peaks in the simulated

spectra. The rate difference kT - kS is found to be around 1

106 s-1 and quite insensitive to temperature changes.Temporal Analyses. In the linear regime, a transient cw EPR

trace is related to the FID of pulse EPR spectroscopy, in the

sense that the spectral information in a kinetic trace is encoded

as the time integral of the FID, eq 36. To test this relationship,

several transient cw EPR traces of D+-C-A- were obtained

at a series of field positions. As Figure 7 demonstrates, all of

the traces feature abrupt turning points at around 0.2 and 0.4

s on top of an exponential decay. Some of them are so steepas to resemble a staircase function. Next, the traces are differ-

entiated against time to yield quasi-FIDs, which bear the familiar

oscillatory and rhythmic appearances. Finally, a spectrum is

reconstructed by the inverse Fourier transform of each quasi-

FID. In Figure 7C, a comparison between a field-swept spectrumand the FT spectra reveals that their fine features match on a

one-to-one basis, including the polarizations, line widths and

positions. This again illustrates that the frequency and time

domains are simply two reciprocal representations of the same

spectra, even under continuous-wave excitation. Nonetheless,

the line intensities on the FT spectra are severely distorted; only

the signal within 1 mT of the field position can be properly

reproduced. This is equivalent to a bandwidth of28 MHz or

a time resolution of36 ns, which is typical for X-band transient

cw EPR.

Conclusions

A spin-correlated radical pair resulting from an electron-transfer reaction is characterized by its three components, the

cation radical spectrum, the anion radical spectrum, and, most

importantly, the spin-spin exchange interaction 2J. In the

simplest scenario, the two unpaired spins split each other into

weighted or polarized doublets. When each radical is spin

coupled to neighboring nuclei, the SCRP is decorated with a

myriad of possible arrangements of the nuclear states, and its

spectrum becomes a convolution of the two radical spectra and

the four-line SCRP pattern.

In order to extract the 2J parameter and leave out the

contributions from the nuclei, invariant properties of the SCRP

spectrum need to be identified. One such property is that only

the resonant field gap between the two radicals, 2Q, makes a

difference to the four-line pattern. This realization brings aboutan efficient simulation algorithm to handle all of the hyperfine

states statistically. In addition, for the combination of a broad

cation radical signal and a very narrow anion radical signal,

the effective line splitting is found to be nearly a constant, 2 1/2J2/. This quantity competes with the hyperfine coupling inthe anion radical, and it suffices as a rule of thumb that when

|2J| > 2.9(a)1/2, the SCRP spectrum will not reveal the finefeatures but take on the overall shape of four broadened peaks.

Experimental SCRP EPR spectra for D+-C-A-, an in-

tramolecular SCRP with a well-defined 2J coupling, were

analyzed. The cation radical D+ can be described roughly by a

Gaussian width of 1.2 mT, and the anion radical A- features a

primary hyperfine splitting of 0.095 mT. These two numbers

set a maximum 2J for the system of 1.0 mT, if hyperfine

structure appears in its EPR spectrum. Indeed, at 210 K,

D+-C-A- has a small 2J value of 0.15 mT and exhibits

marked hyperfine splittings near the spectral center. At 295 K,conformational gating of the donor-acceptor coupling becomes

thermally activated, and 2J rises to 0.48 mT.32 As a result, the

EPR spectrum is only slightly modulated by motion of the

methoxyprotonsofD+.Furthermore,ananalogousdonor-acceptor

triad33 with a closer donor-acceptor distance is reported to have

a 2Jvalue of 4.7 ( 0.3 mT, which is strong enough to erase all

fine structure in its EPR spectrum.

Generally speaking, time and energy/frequency are two sides

of the same coin. To fully characterize a dynamic system, only

the information from one of the two sides is required, whichever

is easier to implement experimentally. Therefore, in Fourier-

transform spectroscopy, time domain signals are obtained as a

FID or an interferogram, even though a spectrum is eventually

presented versus the frequency axis. On the other hand, the rate

of a kinetic process including relaxation34 and exchange35 can

be conveniently derived from the line shape of a steady-state

spectrum. In this study, such time-energy dualism is further

extended so that a classical first-order decay rate is treated as

the imaginary part of energy. This concept may not be as

universal when compared to the density matrix formalism, but

it greatly helps to simplify the mathematical complexities, while

presenting the results in a physically meaningful way.

Transient continuous-wave EPR is normally considered to

be a type of 2D spectroscopy that is both field- and time-

resolved. Rather than rapidly rotating the z magnetization into

the xy plane by application of a /2 microwave pulse, the weak

cw microwave field updates the EPR signal (t) incrementallyby rotating a small fraction of the z magnetization as time

Figure 7. (a) Quadrature-detected transient cw EPR traces of D+-C-A- in toluene at 210 K and (b) their time derivatives. (c) Inverse Fouriertransform of the derivatives in (b) with their field positions indicated on the top, compared with a field-swept spectrum at 0.43 s.



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elapses. In the linear regime, this is equivalent to a time integral

of the FID. As a result, the transient spectra experience

uncertainty broadening, and the kinetic traces are stamped with

staircase kinks. In terms of technical difficulties, current EPR

instrumentation faces limitations mainly in bandwidth and time

resolution; the magnetic field is still the foremost variable to

be tuned over a large dynamic range. From this point of view,

the utility of transient cw EPR spectroscopy is still apparent,

even when a variety of pulse experiments are taking EPR to a

new level.

Experimental Section

The molecular triad D-C-A was synthesized described

earlier28 and purified by preparative TLC (1:4 EtOAc/DCM,

silica gel). Its saturated toluene solution (0.2 mM) was loaded

in 2 mm ID quartz tubes and subjected to several freeze -pump-

thaw degassing cycles on a vacuum line (10 -4 mBar). The

samples were then fused with a hydrogen torch and kept in the

dark when not being used. The cation radical D+-C was

prepared by titrating a sub-mM dichloromethane solution of

D-C with an acetonitrile solution of 1:2 AgClO4 and I236,37

under an oxygen-free atmosphere until the mixture turned deep

brown (max ) 487 nm). The anion radical A- was photoreduced

by triethylamine38 in DMF under 355 nm illumination.

To generate the SCRP state, a sample was excited by 416

nm, 1 mJ, 7 ns laser pulses from the Raman shifted output of

a Q-switched Nd:YAG laser (Quanta Ray DCR-2). Time-

resolved EPR experiments were carried out using a Bruker

Elexsys E580 X-band EPR spectrometer with a variable-Q split

ring resonator (Bruker ER 4118X-MS5), fitted with a dynamic

continuous flow cryostat (Oxford Instruments CF935) and

cooled with liquid nitrogen. Kinetic traces of transient magne-

tization were accumulated following photoexcitation under 6.3

mW cw microwave irradiation. Field modulation was disabled

for a high time resolution, and microwave signals in emission

(e) and absorption (a) were registered in both the real andimaginary channels (quadrature detection). Sweeping the mag-

netic field gave 2D complex spectra versus time and magnetic

field. For each kinetic trace, the signal acquired prior to the

laser pulse was set to zero. EPR signals recorded at off-resonant

fields were considered background noise, whose average was

subtracted from all kinetic traces. The spectra were finally

phased into a Lorentzian part () and a dispersive part ().

Acknowledgment. This work was supported by the National

Science Foundation, under Grant No. CHE-0718928 (M.R.W.).

M.A.R. thanks the NSF for partial support under the CHE and

MRSEC divisions as well as ONR-Chemistry. We thank Dr.

Zachary E. X. Dance and Michael T. Colvin for their assistancein the EPR experiments and for helpful discussions.

References and Notes

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JP907476Q


Documents

Qixi Mi, Mark A. Ratner and Michael R. Wasielewski- Time-Resolved EPR Spectra of Spin-Correlated Radical Pairs: Spectral and Kinetic Modulation Resulting from Electron-Nuclear Hyperfine