Time-Resolved EPR Spectra of Spin-Correlated Radical Pairs

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J. Phys. Chem. A 2010, 114, 162–171

Time-Resolved EPR Spectra of Spin-Correlated Radical Pairs: Spectral and Kinetic Modulation Resulting from Electron-Nuclear Hyperfine Interactions Qixi Mi, Mark A. Ratner,* and Michael R. Wasielewski* Department of Chemistry and Argonne-Northwestern Solar Energy Research (ANSER) Center, Northwestern UniVersity, EVanston, Illinois 60208-3113 ReceiVed: August 3, 2009; ReVised Manuscript ReceiVed: NoVember 13, 2009

This paper expands the established four-state model of spin-correlated radical pairs (SCRPs) to include local nuclear spins which are ubiquitous in real-world systems and essential for the radical pair intersystem crossing (RP-ISC) mechanism. These nuclei are coupled to the unpaired electron spins by hyperfine interaction and split their electron paramagnetic resonance (EPR) lines. Rather than enumerating all possible nuclear states, an algorithm is devised to sort out the net hyperfine offset 2Q, which, along with the electron spin-spin coupling 2J, characterizes the behavior of SCRPs. Using this algorithm, the EPR spectra of SCRPs coupled to arbitrary nuclear spins can be efficiently simulated with only 2J and the EPR spectra of individual radicals as the inputs. Particularly illustrative is the case of a SCRP resulting from photoinduced electron transfer comprised of a spectrally narrow anion radical signal having small hyperfine splittings and a broad cation radical signal having many large hyperfine splittings and a Gaussian width σ, where the EPR peak of the anion radical exhibits an effective splitting of 21/2J2/σ. For SCRPs having singlet and triplet pathways for charge recombination, their kinetic behavior is obtained concisely by considering the decay rate constants kS and kT as imaginary energies, while adhering to the existing derivation of the four-state model. These models are employed to interpret the diverse array of spectral and kinetic modulation patterns observed in the experimental EPR spectra of photogenerated SCRPs and to extract the 2J value, which reflects the donor-acceptor electronic coupling. During the first several hundred nanoseconds following photoexcitation, the spectral and time domain characteristics of the measured time-resolved EPR spectra manifest the consequences of the Uncertainty Principle, and the modulation patterns in either domain result from hyperfine splittings between the unpaired electron and the nuclear spins. Introduction Understanding the nature of electron-transfer reactions as a general phenomenon is critical to the development of molecular design criteria for preparing systems directed toward artificial photosynthesis as well as organic electronics and spintronics. An important aspect of photodriven, single-electron, multistep charge separation both in natural photosynthesis and in covalently linked donor-acceptor (D-A) systems is the fact that charge separation produces a radical ion pair (RP) whose initial spin state preserves the spin multiplicity of its excited state precursor. Following rapid charge separation, the initially formed singlet RP, 1(D+•-A-•), undergoes radical pair intersystem crossing (RP-ISC)1,2 induced by electron-nuclear hyperfine coupling within the radicals to produce the triplet RP, 3 (D+•-A-•). The spin-spin exchange (2J) and dipolar (D) interactions between the radicals also significantly influence the spin dynamics of 1,3(D+•-A-•), although the dipolar interaction is averaged to zero for most systems in fluid solution. The situation in which both 2J and D are small relative to the sum of the hyperfine interactions of the radicals is particularly important because this circumstance arises in photosynthetic charge separation, as well as in weakly coupled RPs in covalentlinked D-A systems of interest to organic electronics. These so-called spin-correlated radical pairs (SCRPs) are of particular relevance to understanding the molecular design requirements * To whom correspondence [email protected].

should

be

addressed.

E-mail:

for long-lived charge separation. The EPR transitions of SCRPs have been described previously by a four-state model;1–9 however, their experimental time-resolved EPR (TREPR) spectra often exhibit significant complexity as a result of numerous electron-nuclear hyperfine splittings within each radical. Explicit inclusion of a large number of nuclear spin states into this model proves to be computationally intensive. Our work described here explores an approach to this problem that offers insights into the physical nature of the numerous spin-spin interactions present in complex RPs, as well as providing a computationally efficient solution to this problem. The Four-State Model. Two S ) 1/2 spins, electron or nuclear, establish a manifold of four spin sublevels. The spin exchange interaction, Hexch ) -2JS1 · S2, makes the two spins indistinguishable such that only their total spin is a good quantum number, S ) 0 (singlet) or 1 (triplet). On the other hand, these spins are situated most often in dissimilar local environments, making them magnetically inequivalent with a frequency difference of ∆ν on a basis set of individual spin ups and downs. Following the standard SCRP formulation,1,3,4,6,7 we consider two electron spins with an exchange coupling of 2J and different g factors

H ) HeZ + Hexch ) βB(g1S1z + g2S2z) - 2JS1 · S2

(1)

where β is the Bohr magneton, B is the magnetic field, gi is the electron g factor, and Si is the electron spin operator, i ) 1, 2. In this paper, 2J is assumed to be constant, and the singlet-triplet basis set is adopted. Then, the g factor difference, ∆g ) g1 -

10.1021/jp907476q  2010 American Chemical Society Published on Web 12/08/2009

Spin-Correlated Radical Pairs Coupled to Nuclear Spins

J. Phys. Chem. A, Vol. 114, No. 1, 2010 163 the four-state model assume that J , Q, so that tan θ and therefore θ are large, resulting in large transition probabilities for 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 in most cases, only the new mixed eigenstates |S′〉 and |T0′〉 will be populated

SCHEME 1: Four-State Model of a SCRPa

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

F(|S′〉) ) cos2

θ 2

F(|T0′〉) ) sin2

θ 2

(5a,b)

Along with the transition probabilities TABLE 1: Four Equally Intense Transitions of a SCRP from a Singlet Precursor transition

energy

S′ f T-1 T0′ f T1 T0′ f T-1 S′ f T1

+ + -

B0 B0 B0 B0

Ω Ω Ω Ω

+ + -

probability J J J J

P(S′ f T(1) ) sin2

polarization

2

sin (θ/2) cos2(θ/2) cos2(θ/2) sin2(θ/2)

emissive absorptive emissive absorptive

g2, becomes the mixing term between the two nearly isoenergetic S and T0 states

(g1 - g2)|vV〉 - (g2 - g1)|Vv〉 2√2

∆g ) |T 〉 2 0

(

(2)

The full matrix form of eq 1 reads

-g1 - g2 0 0 0 2J ∆g βB 0 ∆g 0 2 0 0 0 g1

(

or in magnetic units

-B0 0 0 0 0 2J Q 0 0 Q 0 0 0 0 0 B0

)

0 0 0 + g2

|T-1〉 |S〉 |T0〉 |T1〉

)

|T-1〉 |S〉 |T0〉 |T1〉

(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

(3b)

〈S′|H|S′〉 ) J + Ω

(4a,b) θ θ |T0′〉 ) cos |T0〉 - sin |S〉 2 2

θ θ 1 Q2 cos2 ) 2 2 4 Q2 + J2

(3a)

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

θ θ |S′〉 ) cos |S〉 + sin |T0〉 2 2

θ 2 (6a,b)

P(T0′ f T(1) ) cos2

the concise result is obtained that all four transitions have equal intensities as a function of Q and J

|I| ) sin2

(g1S1z + g2S2z)|S〉 )

θ 2

〈T0′ |H|T0′〉 ) J - Ω

(4c,d) where Ω2 ) Q2 + J2 and tan θ ) Q/J, Scheme 1a,b. The transitions 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 Ω g |J| g 0, these four transitions can be grouped into two doublets, one centered at B0 + Ω and the other at B0 - Ω, with a common splitting of 2J. Previous discussions of

F(|S′〉) )

1 2θ sin 3 2

F(|T0′〉) )

θ 1 cos2 3 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| )

Q2 θ 1 2θ 1 sin cos2 ) 2 3 2 2 12 Q + J2

(10)

Nonetheless, assuming again that 2J > 0, there is a qualitative change 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 center field B0, its triplet precursor can have taken on an orientation

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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 by13,14

1 3 1 D E 2 ) (3AZ - 1) + (AX - AY) 3 15 B0 B0

F0 ) F(1

[

]

θ 2 θ I(S′ f T(1) ) (sin4 2

F(|T0′〉) ) cos2

θ 2

ISCRP )

(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 spin polarization 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 invert the 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 T0 of a triplet is populated, resulting from a reversed RP-ISC mechanism.15 Following the same scheme

F(|S′〉) ) sin2

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 2Q is limited by the EPR spectral widths of the individual radicals. For each 2Q within the limit, a subspectrum can be calculated and summed to an ensemble-averaged spectrum

θ 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 θ, 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 earlier for 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 than ∆g. Hence, the SCRP line shape as a function of Q 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

(14)

2Q

Here, the SCRP line shape Γ 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 B0 and 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, the statistical 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

(12a,b)

I(T0′ f T(1) ) (cos4

∑ P(B0, Q)XΓ(Q, J)

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 ) 2

∫ I1(B0 + Q)I2(B0 - Q) X Γ(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 hfcc’s 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 hfcc’s, 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 are considered for possible combinations of the nuclear states. Since radical 1 has no hfcc’s, it does not contribute to Σ(mI), while the nuclear spin states of radical 2 result in Σ(mI) ) +2 (red spectrum), +1 (blue spectrum), and 0 (magenta spectrum). Only the nuclear spin states for which Σ(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 Σ(mI) for each nuclear spin state configuration. Thus, referring to Scheme 2b, when Σ(mI) ) 0 (magenta), J . Q, so that the EPR transitions are forbidden and the line intensities vanish; when Σ(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 Σ(mI) ) +2 nuclear state has a low probability,


J. Phys. Chem. A, Vol. 114, No. 1, 2010 165

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

Figure 1. Simulated EPR spectrum of a SCRP consisting of a broad cation radical with a Gaussian width of σ ) 1 mT and a narrow anion radical, according to eq 20. The spin-spin exchange coupling 2J is equal to 1 mT.

and the line intensities of the four-line pattern are relatively weak. Lastly, when Σ(mI) ) +1 (blue), J ∼ Q, so that a good balance between transition probability and nuclear state population is achieved, and the line intensities are stronger. All fourline patterns including those for Σ(mI) < 0 are shown in Scheme 2c. It is obvious that the field position of the antiphase doublet from radical 1 (no hfcc’s) 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 Σ(mI) ) -1 nuclear state (orange) does not fully overlap with that of the Σ(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 B2 exp - 2 2σ σ√2π

(19)

with a standard deviation σ. The narrow anion can be idealized into a Dirac δ function, I2(B + B0) ) δ(B). Then, an analytical solution including an absorption and an emission term can be obtained for eq 18

I(B + B0) ) A(B - J) - A(B + J) (J2 /B2 - 1)2 1 A(B) ) · 2 2 4(J /B + 1) σ√2π

[

(J2 /B2 - 1)2 exp 2σ2 /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 function A(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 (σ > 2J), the expression reduces to 21/2J2/σ, Figure 1. Rather than the δ function, a binomial distribution of very narrow lines separated by a small hyperfine splitting a is a better model for actual anion radicals, Figure 2a. The EPR spectrum for 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 /σ > 3a

or

|2J| > √6√2σa ≈ 2.9√σa

(21)

In Figure 2b, two EPR spectra are simulated for the same SCRP with two 2J values. It is intriguing to observe that by tuning 2J around the transition point 2.9(σa)1/2, the SCRP spectrum makes a switch from displaying the anion hyperfine structure to mainly the four peaks similar to those in Figure 1. This useful property puts a limit on the value of 2J once σ and a are known, without the need for spectral simulation. Lifetime Broadening. A SCRP has two somewhat paradoxical 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 i |ψ〉 ) - H|ψ〉 dt p

F ) 〈ψ|ψ〉

(22a,b)

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

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Mi et al.

Figure 2. A model SCRP consisting of a broad cation radical with a Gaussian width of σ ) 1 mT and a narrow anion radical coupled to several protons 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 1.0 mT.

1 H|ψ〉 ) - ipk|ψ〉 2

1 〈ψ|H† ) ipk〈ψ| 2

θ θ |T0′〉 ) cos |T0〉 - sin |S〉 2 2 1 〈T0′ |H|T0′〉 ) J - ik+ - Ω 2

(23a,b) such that

d i i 〈ψ|ψ〉 ) - 〈ψ|(H|ψ〉) + (〈ψ|H†)|ψ〉 ) -k〈ψ|ψ〉 dt p p (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

(

)

1 2J - ikS Q 2J Q |S〉 1 kS 0 2 H) ) 1 Q 0 |T0〉 2 0 kT - ikT Q 2 (25)

(

)

( )

A preferred method to quantum mechanically treat population decay is to use density matrices and superoperators17,18

(

d F ) -i(HF - FH†) ) -iLF dt

)

(26)

-ikS -Q Q 0 |S〉〈S| 0 Q |S〉〈T0 | -Q 2J - ik+ L) -2J - ik+ -Q |T0〉〈S| Q 0 -ikT |T0〉〈T0 | 0 Q -Q kS ( kT (27a,b) k( ) 2 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

θ θ |S′〉 ) cos |S〉 + sin |T0〉 2 2 1 〈S′|H|S′〉 ) J - i/k+ + Ω 2

(28a,b)

(29a,b)

with the generalized parameters Ω2 ) (J - 1/2ik-)2 + Q2 and tan θ ) Q/(J - 1/2ik-), which are complex when k- * 0 or kS * kT. Since the angle θ and all coefficients in eqs 28b and 29b can 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 ω0 and rate k is conveniently noted by a Green’s function19

I(ω) )

i(ω - ω0) + k i i ) ) ω-L ω - ω0 + ik (ω - ω0)2 + k2 (30)

whose real part is a Lorentzian function centered at ω0, and the imaginary 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 × 106 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 from the Till-Hore model.22 Alternatively, a projection onto the new eigenstates |S′〉 and |T′〉 eliminates the modulations and leaves two exponential decays at rates kS′ and kT′ . It can be shown that S-T0 mixing always brings the two decay rates closer to each other

kS′ ) k+ - Im 2Ω > kS kT′ ) k+ + Im 2Ω < 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 each spectral line turns into a Lorentzian peak due to lifetime



Figure 3. (a) Simulated kinetics and (b) steady-state spectrum of a transient SCRP, whose triplet component decays faster and has a broader EPR line shape. In (a), the evolution of sublevel populations appears either as damped oscillations (s) or as exponential decays (- - -), depending on the representation. Inset: S-T0 mixing always brings the two decay rates closer to each other. Parameters: 2J ) 2Q ) 1 mT, kS ) 1 × 106 s-1, kT ) 5 × 106 s-1.

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

1 Γ(S′ f T(1) ) (kS′ + kT) 2 1 Γ(T0′ f T(1) ) (kT′ + kT) 2

I(ω, t) )

4πiω 3cp

∫

(32a,b)

(33)

where c is the speed of light, p is the reduced Planck constant, and µ(τ) is the system’s electric or magnetic moment. In the linear regime of continuous-wave EPR, I is proportional to the complex magnetic susceptibility χ ) χ′ + iχ′′, and µ is replaced by the x magnetization Sx. Ideally, only a spectrum or kinetic trace is necessary to fully characterize the time propagation of Sx(τ); in practice, the frequency and time domains complement each 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 automatically corresponds to damped or modulated kinetics and vice versa. The difference between eq 33 and its steady-state (SS) version

I(ω, SS) )

4πiω 3cp

∫0∞ eiωτ〈µ*(τ)µ(0)〉dτ

χ(ω, SS) ∝ F-1[〈Sχ*(τ)Sχ(0)〉] 〈Sχ*(τ)Sχ(0)〉 ∝ F[χ(ω, SS)]

or

(35a,b)

This can be formally substituted back in eq 33 to yield

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 rise and 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 〈µ*(τ)µ(0)〉 and an excitation electromagnetic wave at frequency ω24 t iωτ e 〈µ*(τ)µ(0)〉dτ 0

exactly half of the inverse FT of the autocorrelation function 〈Sx*(τ)Sx(0)〉

(34)

is simply that a steady-state spectrum is considered to be measured at a time long enough to establish thermal equilibrium, I(ω, SS) ) limtf∞ I(ω, t). This makes the integral in eq 33

χ(ω,t) ∝

∫0t eiωτF[χ(ω,SS)]dτ

(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 steadystate counterpart. Adhering to the FT formalism, a rectangular function is introduced to account for the time dependence

χ(ω, t) ∝ F-1[〈Sχ*(τ)Sχ(0)〉rect(τ/t)] rect(χ) )

{

1 |x| e 1 0 |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

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

t sinc(ωt) π

(38)

Here, the convolution with sinc(x) ) sin(x)/x effectively averages out all of the fine structure in χ(ω, SS) that has a spectral resolution of ∆ω < 2π/t. Rearrangement gives the famous Uncertainty Principle

∆E · t g 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 Planck’s constant. For example, only those spectra obtained after ∼360 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 above theoretical arguments, it has been taken for granted that the

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

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.25–27 Photoexcitation 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 χ′′(B, t) of D+•-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 µs, the spectrum is broad and featureless with an e/a polarization pattern. Later, when 0.2 < t < 0.4 µs, a 0.19 mT hyperfine splitting appears at the spectral center. In the last stage, when t > 0.4 µs, each of the hyperfine lines further splits into two with a much greater modulation depth, followed by an exponential 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 ∆E can 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 lowresolution 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/(∆υ)-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 of D+•-C-A-• is indeed a first-order and relaxationfree process within at least the first 0.4 and 0.2 µs, 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 of D+•-C is frozen, and only a broad Gaussian-like profile remains. By contrast, the acceptor anion



Figure 5. (a) Integrated and normalized cw EPR spectra of the individual moieties D+•-C and A-•. The experimental spectrum of A-• is depicted as 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). The transient cw EPR spectrum of D+•-C-A-• in toluene is taken at 210 K and 0.43 µs.

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 simulation parameters are tabulated in (d). The stars denote “shrugs” due to a faster rate for triplet charge recombination.

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 sometimes lead 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 µs reveals that its 14 central lines can be reconstructed from the 13 lines of A-• plus a 2J coupling 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 2J to be refined by numerical calculations. Besides 2J, there are more parameters involved in the simulation 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 proportionality 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, the time t defines the extent of uncertainty broadening, which is implemented in the simulations by nullifying all spectral components having an energy resolution of ∆E < h/t. Figure 6 shows the transient cw EPR spectra of D+•-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 µs adjusts 2J to 0.15 mT 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 pronounced than the simple model in Figure 1. Such a “shrug” effect can be understood by recognizing that a faster kT trims down the two inner, triplet-derived lines of the four-line spectral

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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 Fourier transform of the derivatives in (b) with their field positions indicated on the top, compared with a field-swept spectrum at 0.43 µs.

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 µs on top of an exponential decay. Some of them are so steep as to resemble a staircase function. Next, the traces are differentiated 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 quasiFID. In Figure 7C, a comparison between a field-swept spectrum and 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 of ∼28 MHz or a time resolution of ∼36 ns, which is typical for X-band transient cw EPR. Conclusions A spin-correlated radical pair resulting from an electrontransfer 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 about an 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, 21/ 2 2J /σ. This quantity competes with the hyperfine coupling in the anion radical, and it suffices as a rule of thumb that when |2J| > 2.9(σa)1/2, the SCRP spectrum will not reveal the fine features but take on the overall shape of four broadened peaks. Experimental SCRP EPR spectra for D+•-C-A-•, an intramolecular 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 2J value 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 Fouriertransform 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 timeresolved. Rather than rapidly rotating the z magnetization into the xy plane by application of a π/2 microwave pulse, the weak cw microwave field updates the EPR signal χ(t) incrementally by rotating a small fraction of the z magnetization as time

Spin-Correlated Radical Pairs Coupled to Nuclear Spins 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-pumpthaw 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 (λmax ) 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). Timeresolved 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 magnetization 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 and imaginary channels (quadrature detection). Sweeping the magnetic 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 (χ′′) and a dispersive part (χ′). 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 assistance in the EPR experiments and for helpful discussions. References and Notes (1) Closs, G. L.; Forbes, M. D. E.; Norris, J. R. J. Phys. Chem. 1987, 91, 3592–3599.

J. Phys. Chem. A, Vol. 114, No. 1, 2010 171 (2) Buckley, C. D.; Hunter, D. A.; Hore, P. J.; McLauchlan, K. A. Chem. Phys. Lett. 1987, 135, 307–12. (3) Hore, P. J.; Hunter, D. A.; McKie, C. D.; Hoff, A. J. Chem. Phys. Lett. 1987, 137, 495–500. (4) Hore, P. J. Analysis of Polarized Electron Paramagnetic Resonance Spectra. In AdVanced EPR in Biology and Biochemistry; Hoff, A. J., Ed.; Elsevier: Amsterdam, The Netherlands, 1989; pp 405-440. (5) Stehlik, D.; Bock, C. H.; Petersen, J. J. Phys. Chem. 1989, 93, 1612–1619. (6) Norris, J. R.; Morris, A. L.; Thurnauer, M. C.; Tang, J. J. Chem. Phys. 1990, 92, 4239–4249. (7) McLauchlan, K. A. Continuous-Wave Transient Electron Spin Resonance. In Modern pulsed and continuous-waVe electron spin resonance; Kevan, L., Bowman, M. K., Eds.; Wiley: New York, 1990; pp 285-363. (8) Tarasov, V. F.; Forbes, M. D. E. Spectrochim. Acta, Part A 2000, 56, 245–263. (9) Pople, J. A. High-resolution nuclear magnetic resonance; McGrawHill: New York, 1959. (10) Steiner, U. E.; Ulrich, T. Chem. ReV. 1989, 89, 51–147. (11) Hayashi, H. Introduction to Dynamic Spin Chemistry: Magnetic Field Effects upon Chemical and Biochemical Reactions; World Scientific: Singapore, 2004; p 250. (12) Prisner, T.; Dobbert, O.; Dinse, K. P.; Van Willigen, H. J. Am. Chem. Soc. 1988, 110, 1622–1623. (13) Wong, S. K.; Hutchinson, D. A.; Wan, J. K. S. J. Chem. Phys. 1973, 58, 985–989. (14) Atkins, P. W.; Evans, G. T. Mol. Phys. 1974, 27, 1633–1644. (15) Wiederrecht, G. P.; Svec, W. A.; Wasielewski, M. R.; Galili, T.; Levanon, H. J. Am. Chem. Soc. 2000, 122, 9715–9722. (16) Schulten, K.; Wolynes, P. G. J. Chem. Phys. 1978, 68, 3292–3297. (17) Schneider, D. J.; Freed, J. H. AdV. Chem. Phys. 1989, 73, 387– 527. (18) Kothe, G.; Weber, S.; Ohmes, E.; Thurnauer, M. C.; Norris, J. R. J. Am. Chem. Soc. 1994, 116, 7729–7734. (19) Mattuck, R. D. A guide to Feynman diagrams in the many-body problem, 2nd ed.; Dover Publications: New York, 1992. (20) Marcus, R. A. J. Chem. Phys. 1956, 24, 966–978. (21) Marcus, R. A. J. Chem. Phys. 1965, 43, 679–701. (22) Till, U.; Hore, P. J. Mol. Phys. 1997, 90, 289–296. (23) Reimers, J. R.; Wilson, K. R.; Heller, E. J. J. Chem. Phys. 1983, 79, 4749–4757. (24) Schatz, G. C.; Ratner, M. A. Quantum Mechanics in Chemistry, 2nd ed.; Dover Publications: Mineola, NY, 2002. (25) Deisenhofer, J.; Norris, J. R. The photosynthetic reaction center; Academic Press: San Diego, CA, 1993. (26) Blankenship, R. E. Molecular Mechanisms of Photosynthesis; Blackwell Science: Oxford, U.K., 2002. (27) Krauss, N. Curr. Opin. Chem. Biol. 2003, 7, 540–550. (28) Greenfield, S. R.; Svec, W. A.; Gosztola, D.; Wasielewski, M. R. J. Am. Chem. Soc. 1996, 118, 6767–6777. (29) Weiss, E. A.; Ratner, M. A.; Wasielewski, M. R. J. Phys. Chem. A 2003, 107, 3639–3647. (30) Hasharoni, K.; Levanon, H.; Greenfield, S. R.; Gosztola, D. J.; Svec, W. A.; Wasielewski, M. R. J. Am. Chem. Soc. 1996, 118, 10228–10235. (31) Mi, Q.; Weiss, E. A.; Ratner, M. A.; Wasielewski, M. R. Appl. Magn. Reson. 2007, 31, 253–270. (32) Weiss, E. A.; Tauber, M. J.; Ratner, M. A.; Wasielewski, M. R. J. Am. Chem. Soc. 2005, 127, 6052–6061. (33) Dance, Z. E. X.; Ahrens, M. J.; Vega, A. M.; Ricks, A. B.; McCamant, D. W.; Ratner, M. A.; Wasielewski, M. R. J. Am. Chem. Soc. 2008, 130, 830–832. (34) Polimeno, A.; Zerbetto, M.; Franco, L.; Maggini, M.; Corvaja, C. J. Am. Chem. Soc. 2006, 128, 4734–4741. (35) Susumu, K.; Frail, P. R.; Angiolillo, P. J.; Therien, M. J. J. Am. Chem. Soc. 2006, 128, 8380–8381. (36) Shine, H. J.; Padilla, A. G.; Wu, S. M. J. Org. Chem. 1979, 44, 4069–4075. (37) Kass, H.; Bittersmann-Weidlich, E.; Andreasson, L. E.; Bonigk, B.; Lubitz, W. Chem. Phys. 1995, 194, 419–432. (38) Smith, P. J.; Mann, C. K. J. Org. Chem. 1969, 34, 1821–1826.

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Time-Resolved EPR Spectra of Spin-Correlated Radical Pairs

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