Article pubs.acs.org/JPCA
Radical−Triplet Pair Mechanism of Electron Spin Polarization. Detailed Theoretical Treatment A. I. Shushin* Institute of Chemical Physics, Russian Academy of Sciences, GSP-1, Kosygin Street 4, 117977 Moscow, Russia ABSTRACT: Specific features of net chemically induced dynamic electron spin polarization (CIDEP) Pn, generated in liquid-phase triplet−radical (TR) quenching, are analyzed in detail within the general model, which allows for fairly simple analysis of CIDEP both numerically and analytically. This model enables one to accurately treat nonadiabatic transitions between the terms of TR-pair spin Hamiltonian, resulting in CIDEP generation. The proposed theory predicts fairly simple analytical dependence of Pn on parameters of the model. In particular, it is shown that within the wide region of parameters the Pn dependence on the coefficient of relative TR diffusion Dr is described by simple linear relation P−1 n (Dr) ≈ Q0 + qn̅ Dr (Q0 and qn̅ are independent of Dr). It is also demonstrated that obtained numerical and analytical results are very helpful for the analysis of experimental data, which is demonstrated by analyzing the experimental Dr-dependence of Pn.
I. INTRODUCTION The mechanism and specific features of net chemically induced dynamic electron spin polarization (CIDEP) Pn of radicals (R), generated in liquid-phase quenching of triplet (T) excited states of molecules by radicals, are studied in a large number of works both experimentally and theoretically (see the review in ref 1 and references therein). These studies have allowed for obtaining important information on this mechanism, called in what follows radical−triplet pair one (RTPM).2,3 For more than 20 years of experimental investigations a lot of TR-systems have been analyzed.1 The results of these investigations have made it possible to clarify the molecular energetics of a number of processes as well as important details of intermolecular mechanisms of quenching, resulting in CIDEP. Recent progress in experimental studies of CIDEP generation in different TR-quenching processes, some of which are of great interest for possible applications,1,4−6 has motivated further developments of the theory of these processes. So far theoretical considerations of the RTPM have mainly been restricted to numerical calculations based on exact numerical solution of the stochastic Liouville equation (SLE) as well as very simplified analytical semiquantitative estimations.6−14 In general, the SLE allows one to correctly describe the spin dynamics in recombining TR-pairs, controlling CIDEP generation. The key process of the TR spin evolution is nonadiabatic transitions between the terms of the TR spin Hamiltonian, induced by the zero field splitting (ZFS) interaction in the T-state of the molecule. Unfortunately, the ZFS is a strongly fluctuating interaction due to stochastic rotational motion of the molecule, which results in the essential complexity of the general SLE, describing the transitions. Because of this complexity, only simplified approximate variants © 2014 American Chemical Society
of the SLE have usually been applied in CIDEP calculations.10−13 Despite the great importance of approximate SLE-based numerical calculations for studying some characteristic properties of the RTPM, they, however, can hardly completely clarify these properties because of a large number of parameters of the model: the rates of nonadiabatic transitions at different distances, the relative diffusion coefficient Dr, and reactivity of TR-pairs, etc. In the presence of so many parameters the accurate and simple analytical expressions for CIDEP Pn would certainly be quite helpful. In our work we propose the new approximate variant of the SLE for fairly simple numerical calculation of the observables under study. We also derive some analytical formulas for the CIDEP Pn and TR-quenching rate, which allow us to accurately describe the dependence on previously mentioned parameters of the model. Special attention is paid to the dependence of Pn on Dr. It is found that in the wide region of Dr-values the dependence Pn(Dr) is described by the linear relation Pn−1(Dr) ≈ Q0 + qn̅ Dr with Q0 and qn̅ independent of Dr. This relation is shown to agree with the experimental dependence Pn(Dr). Very simple analytical expressions are also obtained for quantitative analysis of experimental data in the limit of large and intermediate values of Dr.
II. FORMULATION OF THE PROBLEM In general, TR-quenching process is represented by the kinetic scheme1−3 Received: September 11, 2014 Revised: November 9, 2014 Published: November 10, 2014 11355
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Article
kq̂
K diff
T + R ←→ [TR]Q , D → [S0R*]D → S0 + R*
(2.1)
in which Kdiff is the rate of diffusion-controlled TR encounters/ dissociation, [TR]Q,D denotes the intermediate state of the TRpair in the quartet (Q) or doublet (D) spin state, and kq̂ is the rate of (spin selective) TR quenching in D states (formulas for both rates are given below). The RTPM of CIDEP in the process (2.1) is based on the assumption that CIDEP generation results from nonadiabatic transitions between terms of the spin Hamiltonian H(r) of the TR-pair, where r is the vector of relative T−R coordinate. For simplicity, we will consider the case of spherically symmetric Tand R-particles, in which H(r) depends only on the TR distance r = |r)|: H(r) ≡ H(r). The spin Hamiltonian is assumed to be written as7−9 H(r ) = H0(r ) + VZFS
Figure 1. Schematic picture of terms UM(r) (M = Q±3/2, Q±1/2, or D±1/2) of the spin Hamiltonian H0(r) 2.3 of the TR-pair [as well as the terms of D-states 2.7 of the S0R*-pair]. In this picture r1, r2,3, and ra are the coordinates of c- and a-regions (see eqs 2.9 and 2.10).
(2.2)
where
Q −TR3/2D+TR1/2 (1);
⎛1 ⎞ 4 H0(r ) = ωBSz − J0 e−α(r − d)⎜ + ST SR ⎟ ⎝3 ⎠ 3
(2.8)
⎛ ⎞ 1 VZFS = D⎜STζ 2 − ST 2⎟ + E(STξ 2 − STη 2) ⎝ ⎠ 3
r1 = d + α −1 ln(|J0 | /ωB),
r ≈ ra = d + α −1[ln(2|J0 |/Dr α 2) + 1.15]
1 , 2
(2.4)
|γ±⟩ = |±1⟩ ±
1 2
(2.5)
while at r ∼ d they reduce to the eigenstates of Sz |1⟩ = |Q +TR3/2⟩ = |γ+⟩,
|2⟩ = |Q +TR1/2⟩ =
1 2 |3⟩ = |Q −TR1/2⟩ = |β ⟩ + |α−⟩, 3 − 3 2 1 |5⟩ = |D+TR1/2⟩ = |β ⟩ − |α+⟩, 3 + 3 2 1 |6⟩ = |D−TR1/2⟩ = |β ⟩ − |α−⟩ 3 − 3
1 |β ⟩ + 3 +
σ ̇ = Dr ∇r 2 σ − i(Hσ − σH ) − K̂ qσ
2 |α+⟩, 3
1 , 2
|4⟩ = |Q −TR3/2⟩ = |γ−⟩,
|D−SR1/2⟩ = |0S⟩ −
⎛1 ⎞ K̂ q(r )σ = ⎜ kq ⎟e−αq(r − d)(PDσ + σPD) ⎝2 ⎠
(2.6)
1 2
(2.10)
(2.11)
where ∇r ≡ ∂/∂r, Dr is the coefficient of relative diffusion of Tand R-particles, and K̂ q(r) is the superoperator (operator in the space of spin-density matrix elements) of the highly localized quenching rate, defined by (2.12)
−1
with αq ≫ d , in which PD = ∑v=±1/2|Dv⟩⟨Dv| is the operator of projection on D spin states of the TR-pair. The SLE 2.11 should be solved with the reflective boundary condition at the distance of closest approach d: ∇rσ − d−1σ|r=d = 0. As to the outer boundary condition at r → ∞ and the initial condition, they depend on processes considered, geminate or bulk (see below). The efficiency of reaction can be characterized by the radius of TR-quenching in D-states, which, for example, for fast quenching, kq/Drαq2 > 1, is written as7−9,11,12
There are also two final D-states of S0-molecule-radical pair (S0R*-pair): |D+SR1/2⟩ = |0S⟩ +
(2.9)
where Dr is the coefficient of relative diffusion. In what follows we will consider the limit of fairly strong exchange interaction: |J0| > ωB = gβB, in which all c-regions are located at distances r1, r2,3 > d (see Figure 1). We also suggest the realistic relation r1, r2,3 ≲ ra, which is satisfied for ωB ≳ Drα2. The c- and a-regions play the key role in processes under study. The fact is that nonadiabatic transitions in them (induced by the ZFS interaction) essentially determine the CIDEP generation. The kinetics of this nonadiabatic transition affected diffusioncontrolled quenching and CIDEP generation is, in general, described by the spin-density matrix ρ(r,t) of the TR-pair, which, for the sake of convenience, is represented as ρ(r,t) = r−1σ(r,t). The matrix σ(r,t) satisfies the SLE (ℏ = 1):
is the ZFS interaction in the T-molecule, represented in the molecular frame of reference (ξ,η,ζ). In what follows we will assume that D ≫ E and use the approximation VZFS ≈ D(STζ2 − (1/3)ST2). In the Hamiltonian (2.2) the contribution of the weak hyperfine interaction is neglected. At r → ∞ the eigenstates of the H0 coincide with those of noninteracting TR-pair: |β±⟩ = |±1⟩ ∓
r2,3 = r1 + α −1 ln 2
In addition, shown are two regions of approaching terms TR (denoted as a-regions) (|QTR +1/2⟩,|D+1/2⟩) ≡ (|2⟩,|5⟩) and (| TR QTR ⟩,|D ⟩) ≡ (|3⟩,|6⟩)] (see Figure 1), located at7−9 −1/2 −1/2
15
1 , 2
Q −TR3/2D−TR1/2 (3)
located at r1,r2 and r3, respectively, (r2 = r3 ≡ r2,3):
(2.3)
is the part of the spin Hamiltonian, diagonal in the total spin S = ST + SR, including the Zeeman interaction with ωB = gβB (the first term), in which Sz = STz + SRz is the z-component of the spin (along the magnetic field B), and the exchange interaction (the second term), whose value is determined by J0 at the distance of closest approach d (J0 < 0). The second term in eq 2.3
|α±⟩ = |0⟩ ±
Q −TR1/2D+TR1/2 (2);
(2.7)
In eq 2.7 |0S⟩ is the ground S0-state of the molecule. The terms of the spin Hamiltonian H0(r) are schematically displayed in Figure 1. This figure shows that there exist three regions of terms crossing (called c-regions) 11356
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Here λq̂ is the matrix of quenching radii in D-states of the TRpair (see eq 2.13). As to the outer boundary condition, it depends on the process considered:17,18 for geminate TR-quenching this condition is written as n(r → ∞) = 0, while for bulk process this condition is replaced by
(2.13)
In our further analysis we will discuss both limits of weak and strong reactivity kq, corresponding to rq < r1,r2,3 and rq ≈ r1,r2,3, respectively. Formulas of the type of eq 2.13 can be used only for rough estimations. In reality, in our analysis we will consider the radius rq as the adjustable parameter (see section III).
r −1n(r )|r →∞ = ne =
III. METHODS AND APPROXIMATIONS In general, the solution of the SLE 2.11 is a very complicated problem. The goal of our work is to propose some simple but rigorous and accurate approximations which allow for fairly simple numerical and (in some cases) analytical analysis of TRquenching and CIDEP generation. Some general approximations have already been developed and applied in a number of works.7−9,11,12,16 These approximations turn out, however, to be fairly cumbersome and not quite convenient for the analysis of experimental results. In this work we discuss simplified variants of approximations, which allow for deeper insight into the problem. The approximations are valid in the limit of not very low viscosity of the solution, in which the correlation time τV of fluctuations of VZFS(t) is so long, that ωBτV ≫ 1 and thus the relaxation transitions between states of the Zeeman part of the Hamiltonian H0 are negligibly weak. A. Balance Equation for Crossing c-Regions. In the considered case of not very low viscosity the effect of nonadiabatic transitions in c-regions on TR-quenching and CIDEP generation can be described by balance equations for populations of spin eigenstates |j⟩ (2.6) of the spin Hamiltonian H0(r)
+ − Dr ∇r 2 σa − i(Hâ + Hâ )σa = 0
Ha± = Ha±0 + Ja±
In eq 3.1 we use the notation ∥j⟩⟩ = |j⟩⟨j| for (diagonal) spin states in the Liouville space (the space of density matrix elements). In principle, these equations have been discussed in the literature.11,12 In our work we will analyze the simplified variant of equations, in which the difference between the distances (2.8) of the location of term crossing regions is neglected; i.e., we assume that
2 with K̂ r = (pt /rt)Tt̂ δ(r − rt)
(3.2)
(3.3)
Here the transition rate matrix T̂ t is given by formula ̂ , Tt̂ = 3T53̂ + 2T54̂ + T64
where Tiĵ = ||ij⟩⟩⟨⟨ij||
(3.4)
with ∥ij⟩⟩ = ∥i⟩⟩ − ∥j⟩⟩, ⟨⟨ij∥ = ⟨⟨i∥ − ⟨⟨j∥, and pt = Dt /Dr
with Dt =
2π 2 (D rt /αωB) 45
(3.5)
In eq 3.3 the quenching process is quite adequately described by proper inner boundary condition9,17,18 [(rq − λq̂ )∇r n − n]|r = rq = 0,
where λq̂ = rqPD̂
(3.9)
± with H±a0 = ωV[(1/3)(PDD − P±QQ) + ((2√2)/3)(P±QD + P±DQ)] −α(r−d) ± ± and Ja = J0e (PQQ − P±DD), where P±MN = |M±1/2⟩⟨N±1/2|, (M,N = Q,D). The characteristic interaction ωV(ϑ) = (1/2)D cos(2ϑ) depends on the orientation of T-molecules, i.e., on the angle ϑ between z and ζ axes, and, therefore, the effect of aregions should, in principle, be averaged over orientations (over ϑ). In our analytical analysis, however, [due to weak dependence of calculated observables on ωV11,12 (as ωV1/2)] we simplify the problem by solving eq 3.8 with ωV replaced by the average ω̅ V, defined as the square root of the mean square ⟨ωV2⟩ϑ = (1/2)∫ π0 dϑ sin(ϑ) ω2V(ϑ): ω̅ V = ⟨ω2V⟩ϑ1/2 ≈ (1/3)D. The SLE 3.8 is still fairly complicated for analytical solution. It can, nevertheless, quite accurately be solved approximately within the approach of sudden change of J±a (r),17,18 in which the solution predicts the modification of TR-quenching and CIDEP-generation radii17,18 [represented in a matrix form (see section III.C)]. C. Quenching and CIDEP-Generation Radii. In what follows we will mainly consider the bulk processes, the kinetics of which is determined by matrices L̂ e and L̂ q of radii of CIDEP-generation and quenching, respectively. These matrices can be obtained from the behavior of n(r) at r → ∞ and r → rq:
In this approximation the balance equations can be represented in fairly compact form. In particular, the steadystate variant of these equations, which will mainly be used in the analysis, is written as 2 (K̂ r − ∇r 2 )n = n 0
(3.8)
for spin-density matrix σa(r) [in the spin subspaces of two pairs of terms (Q±1/2,D±1/2)], in which the spin Hamiltonians Ĥ a± = [Ha± , ...] are given by
(3.1)
r2,3 ≈ r1 ≈ rt ≈ d + α −1 ln(|J0 | /ωB)
j=1
1 (1, 1, 1, 1, 1, 1)T 6 (3.7)
∑ nj||j⟩⟩ j=1
6
∑ ||j⟩⟩ ≡
In the case of geminate quenching n0 = ni(2πri)−1δ(r − ri), where ri is the initial TR-distance and ni is the initial population of spin states of the TR-pair, assumed to correspond to equipopulation, ni = ne. For bulk reactions one should put ni = 0. B. Effect of Approaching a-Regions. Quantum tranTR TR sitions between terms (|QTR +1/2⟩,|D+1/2⟩) ≡ (|2⟩,|5⟩) and (|Q−1/2⟩, TR |D−1/2⟩) ≡ (|3⟩,|6⟩) in a-regions (at r ≳ ra ≳ rt) can markedly manifest themselves, reducing the efficiency of CIDEP generation. Transitions are induced by fluctuating V ZFS interaction (2.4). Exact treatment of these transitions is, in general, a fairly complicated problem. Quite accurate estimation of the effect of transitions in (a‑regions can, however, be obtained in the steady-state approach, assuming VZFS(t) to be independent of time within a-regions.9 In this approach the effect of transitions is described by the steady-state SLE9
6
n=
1 6
(3.6) 11357
r −1n(r → ∞) = [1 − (Ln̂ /r )]ne
(3.10)
r −1n(r < rt) = [1 − (Lq̂ /r )]nq
(3.11)
dx.doi.org/10.1021/jp509199m | J. Phys. Chem. A 2014, 118, 11355−11363
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where ne and nq are some constant vectors. In the approximation 3.2 matrices of radii L̂ n and L̂ q can be derived in a simple matrix form. General formulas for L̂ n and L̂ q (taking into account all previously discussed effects) are conveniently represented in terms of radii ln̂ = rt[1 − Pĉ(1 + 7t̂ Pĉ)−1]
and
Note that in our work we will mainly discuss the absolute value of the CIDEP P̅μn , obtained with eq 3.22, which ensures a positive sign of P̅μn for the process under study.
IV. RESULTS A. General Remarks. In our analysis of CIDEP we will use the commonly accepted definition of the probability of CIDEP generation applied in recent experimental works1
lq̂ = rtPr̂(1 + 7t̂ Pĉ)−1 (3.12)
describing the effect of transitions in crossing regions only, in which 7t̂ = pt Tt̂ ,
Pr̂ = λq̂ /rt ,
and
Pĉ = 1 − Pr̂
Pn = K n/Kq = Pn̅ /Pq = Pne + pD Pnq
Pμn
are the matrices of efficiencies of transition in c-regions (7t̂ ), as well as probabilities of quenching (P̂ r) and escaping (P̂c) of TRpairs created at the distance rt: (3.14)
Lq̂ = lq̂ (1 + kL̂ n̂ ) = rtPr̂(1 + 7ŝ Pĉ)−1
(3.15)
In these expressions Λ̂a = ra − ln̂ , 7ŝ = 7t̂ + 7â
and
̂ + T36̂ ) k ̂ = κa(T25
(3.16)
where matrices T̂ ij are defined in eq 3.4,
Pne ≈ 2PnQ
7â = rtkĝ â
with gâ = [1 + (ra − rt)k]̂
(3.17)
κa(Dr ) =
2 2 3 (ω̅ V /Dr )1/2 = (D/Dr )1/2 9 27
(3.18)
−1
and
Pq = Kq /Kdiff
with Kdiff = 4πrtDr
(3.19)
Within the proposed mechanism the rate and probability of CIDEP generation can be written as sums Zn =
Zne
+
pD Znq
(Z = K , P ̅ )
(3.20)
of contributions Zen and pDZqn of radicals R (escaped from quenching) and R* (appeared after quenching), respectively. In these sums the second term is represented as a product of the CIDEP Zqn of radicals R, which quenched T-particles in D-states of TR-pairs, multiplied by the efficiency pD ≤ 1 of transfer of the CIDEP to that of R*-radicals in the process TR → S0R*. The probabilities P̅μn (μ = e, q) and Pq (as well as the rates Kμn and K q ) are straightforwardly expressed in terms of corresponding probability matrices e 7̂ n
= 1 − Ln̂ /rt
and
q 7̂ n
= 7q̂ = Lq̂ /rt
μ Pn̅ μ = 2⟨Szμ⟩ = ⟨⟨2Szμ 7̂ n ne⟩⟩,
Pn̅ q =
⎤ 1⎡1 ⎢⎣ ϕ(3pt ) + (p− /p+ )ϕ(p+ ) + ϕ(pa )⎥⎦ 6 3
(4.4)
Pq =
1 [2 + ϕ(3pt ) + ϕ(p+ ) + ϕ(pa )] 6 p± = 3pt ± pa
ϕ(x) = x /(1 + x)
(4.5)
(4.6)
(4.7)
Formulas 4.3−4.5 are helpful for analyzing specific features of dependences of P̅μn (μ = e, q) and Pq (and thus Pn) on the parameters of the model. Of special interest is the behavior of these observables at small and large values of pa (pa ≪ 1 and pa > 1). 1. Weak Transitions in a-Regions (pa ≪ 1). In the limit of weak transitions in a-regions, when pa ≪ 1, the expressions 4.3−4.6 predict
where μ = e, q; ⟨⟨ne∥ = ∑6j=1⟨⟨j∥ ≡ (1, 1, 1, 1, 1, 1), and
(3.24)
(4.3)
[with κa defined in eqs 3.16 and 3.18], and
(3.22)
⟨⟨2Sqz || = (0, 0, 0, 0, +1, −1)
⎤ 1⎡ 1 1 ⎢⎣ϕ(3pt ) + ϕ(p+ ) − ϕ(pa )⎥⎦ 6 3 3
pa = κara ,
Pq = ⟨⟨ne 7q̂ ne⟩⟩
(3.23)
Pn̅ e =
where
(3.21)
⎛ 1 1 1 1⎞ ⟨⟨2Sez || = ⎜ +1, + , − , −1, − , + ⎟ ⎝ 3 3 3 3⎠
(4.2)
In reality, even this more correct relation is valid only for low viscosities. At high viscosities there is no simple relation between Pen and PnQ (see below). B. Analytical Expressions. The analysis of manifestation of nonadiabatic transitions in c-regions can easily be made in the realistic case of not very large distances of term crossings rt and approaching of terms ra: ra ≈ rt ≈ rq, which is quite accurate in the limit α−1 ≪ ra,t,q and for which ĝa ≈ 1 (eq 3.17). In this case fast reactive depopulation of D-states at r < rq results in effective decoupling of terms ∥j⟩⟩ in the region r ≈ rt, thus reducing the solution of the system of eqs 3.6 to solving four decoupled differential equations. This solution results in
D. Evaluation of Observables. The observables under study are the rates of quenching Kq and CIDEP generation Kn, as well as corresponding probabilities (for geminate TR-pairs, created at ri = rt with ni = ne) Pn̅ = K n/Kdiff
(4.1)
P̅μn
where = (μ = e, q), with probabilities and Pq defined in eqs 3.19−3.22, and pD ≤ 1 is the efficiency of CIDEP transfer to R*-radicals (see eq 3.20). It is important to note that in some works1 the experimental CIDEP Pen (4.1) was associated with the CIDEP PnQ observed in geminate quenching of TR-pairs in the initial isotropic quartet state (corresponding to S = 3/2), i.e., for ni = niQ = (1/4)(1, 1, 1, 1, 0, 0)T,8 suggesting Pn = PnQ. This relation, however, is not quite correct, which can easily be demonstrated in the case of strong reactivity rq ≈ rt and weak transitions in (c- and aregions (i.e., in the limit of not very high viscosity). In that case P̅en ≈ (2/3)PnQ and Pq ≈ 1/3; therefore
(3.13)
Ln̂ = (1 + Λ̂ak)̂ −1 ln̂ = rtgâ [1 − Pĉ(1 + 7ŝ Pĉ)−1]
P̅μn /Pq
11358
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The Journal of Physical Chemistry A Pn̅ e ≈ Pn̅ q ≈
2 ϕ(3pt ) 9
and
Pq ≈
Article
1 [1 + ϕ(3pt )] 3
except formulas for Pqn(pt) in the limit pt ≫ 1 (see section IV.C]. Expressions 4.10−4.13 show that the weak reactivity (small rq < rt and pq = rq/rt < 1) manifests itself in the decrease of CIDEP generation and quenching probabilities. At small pq ≪ 1 the dependence Pqn(pq) ∼ pq is stronger than that Pen(pq) ∼ 2 + pq, predicting a negligibly small contribution of R*-radicals to CIDEP in the case of weak quenching. C. Interpolation Formulas. Analytical expressions, obtained earlier, demonstrate that in a number of different limits both Pen(pt) and Pqn(pt) and, therefore, the experimentally measured net CIDEP Pn(pt) can be represented in the universal form
(4.8)
According to formulas 4.8 the CIDEP P̅en(pt) and P̅qn(pt) increase with pt, reaching maximum values P̅enm = P̅qnm = 2/9 at pt ≫ 1 (for high viscosities, i.e., small Dr). These formulas also show that, at large Dr, corresponding to pt < 1, one gets Pen = 2pt = 2PnQ (in agreement with the previous discussion in section IV.A), while, in the opposite limit of small Dr (for pt > 1), this relation is not valid. The maximum values P̅e,q nm =2/9 and Pqm = 2/3 can easily be interpreted, taking into account that in the limit pt ≫ 1 the CIDEP (Pe,q nm ) and quenching probability (Pqm) are determined by escaping in two nonreactive states |1⟩ and |2⟩ and quenching in four strongly reactive states (|3⟩, |4⟩, |5⟩, |6⟩), respectively. 2. Strong Transitions in a-Regions (pa > 1). In the case pa ≫ 1 strong transitions in a-regions (of terms approaching) TR lead to equipopulation of pairs of terms (QTR ±1/2, D±1/2). For pa > 1 the expressions 4.3−4.5 predict simple limiting formulas: Pn̅ e
≈
3Pn̅ q
1 ≈ ϕ(3pt ) 6
and
1 Pq ≈ [4 + ϕ(3pt )] 6
Pn(pt ) ≈ Pn0ϕ(le pt )
where ϕ(x) is defined in eq 4.7, and Pn0 = Pn(pt ≫ 1)
(4.9)
Pn0 =
Pnq ≈ 2pq pt ,
Pq ≈
1 p 3 q
Pn0 =
1 , 3
Pnq = 0,
Pq ≈
1 bq p 3 q
Pnq ≈
1 bq p p , 8 q t
Pq ≈
1 , 5
Pnq = 0,
Pq ≈
1 cqp 3 q
1 (3 + pD ) 15
Q 0 = 1/Pn0
for pa ≪ 1
and
le =
15 , 4
for pa ≫ 1
with xr = Dr /Dt
and
qn = Q 0/le = 1(lePn0)
(4.20)
The representation 4.19 reduces the study of CIDEP to the analysis of specific features of two parameters Q0 and qn only. Unfortunately, analytical expressions for Q0(pa, pq) and qn(pa, pq) cannot be derived, in general. These dependences can, however, be described by fairly simple interpolation formulas. It is easily seen that Q0 and qn are expressed in terms of similar parameters for TR- and S0R*-contributions Qμ0 and qμn (μ = e, q),
(pt ≫ 1)
(pt ≪ 1)
(pt ≫ 1)
Q0 = (4.13)
where pq = rq/rt ≤ 1,
Q 0eQ 0q pD Q 0e + Q 0q
and
qn =
qneqnq pD qne + qnq
(4.21)
for which we get:
⎛ 1 ⎞ bq = ⎜1 − pq ⎟ , ⎝ 2 ⎠
TR-contribution terms
⎛ 3 ⎞ cq = ⎜1 − pq ⎟ ⎝ 5 ⎠
−1
−1
aq = 2 + pq ,
le = 6,
and Dt given by eq 3.5. In this formula Q0 and qn are the characteristic parameters, expressed as
(4.12)
Pne ≈
and
(4.19)
(pt ≪ 1)
1 bq p 3 q
1 (1 + pD ) 3
Q (xr ) = Pn−1(xr ) ≈ Q 0 + qnxr
(2) for pa ≫ 1 3 p, 4 t
(4.16)
[recall that pD ≤ 1 is the efficiency of transfer of CIDEP to R*radicals, defined in eq 3.20]. Later we will demonstrate quite good accuracy of this expression by numerical calculations with previously derived general matrix formulas. The expression 4.15 is conveniently represented in the form of (linear) dependence of Q = 1/Pn on xr = pt−1:
(4.11)
Pne ≈
t
(4.18)
(4.10)
Pne ≈
le = Pn0−1[Pn(pt )/pt ]| p ≪ 1
(4.17)
(1) for pa ≪ 1 2 aq p , 3 t
and
are the parameters independent of pt. In particular, in the case of rt ≈ rq analyzed in section IV.B one gets expression 4.15 with
Qualitative arguments, close to those applied in the discussed limit pa ≪ 1, allow one to easily find the estimations P̅enm = 1/6 and Pqm = 5/6, taking into account that values of these parameters are determined by five highly reactive states (|2⟩, | 3⟩, |4⟩, |5⟩, |6⟩) and one nonreactive state (|1⟩). These estimations completely agree with the prediction of formulas 4.9 3. Effect of Weak Reactivity (rq < rt). In sections IV.B1 and IV.B2 of our analysis we have assumed fairly strong reactivity, for which the quenching radius rq ≈ rt. For weak reactivity, corresponding to rq < rt, no simple analytical expressions for observables can be obtained in general. It is possible, however, to derive quite simple formulas in limits of weak and strong transitions in c- and a-regions. As applied to parameters Pμn = P̅μn /Pq and Pq, these formulas are written as Pne ≈
(4.15)
and
−1 ⎧ ⎞⎡ ⎛ 3 ⎞⎤⎫ 1⎛4 Q 0e ≈ 5⎨1 + ⎜ + pq + 3pq 2 ⎟⎢1 − ϕ⎜ pa ⎟⎥⎬ ⎠⎣ ⎝ 4 ⎠⎦⎭ 8⎝3 ⎩
(4.14)
Naturally, in the particular case rq = rt formulas 4.10−4.13 coincide with corresponding limiting variants of eqs 4.3 and 4.5,
(4.22) 11359
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Article
⎡ ⎛ 4 ⎞⎤ 3 7 ⎛ 3 ⎞⎤⎡ 1 (2 + pq )−1⎢1 + ϕ⎜ pa ⎟⎥⎢1 + pq ϕ⎜ pa ⎟⎥ ⎝ 3 ⎠⎦ ⎣ 2 9 ⎝ 4 ⎠⎦⎣ 2
Pq(xr ) = Pq0 + (Pq∞ − Pq0)ϕ(qr xr )
Unfortunately, expressions for parameters P0q(pq, pa), P∞ q (pq, pa), and qr(pq, pa) turn out to be somewhat cumbersome; that is why they are presented in Appendix A.
(4.23)
SR*-contribution terms Q 0q ≈ 8{[pq f1 (pa ) + pq 4 f2 (pa )][1 + φ(pa )]}−1 qnq ≈
⎛ 16 ⎞⎤ ⎛ 1 ⎞⎤⎡ 1 −1 ⎡ 1 pq ⎢1 + 15ϕ⎜ pa ⎟⎥⎢1 − pq ϕ⎜ pa ⎟⎥ ⎝ 11 ⎠⎦⎣ ⎝ 11 ⎠⎦ ⎣ 2 2
(4.27)
V. APPLICATION AND DISCUSSION A. Analysis of Experimental Data. The proposed rigorous numerical approach (section III) as well as simple analytical limiting and interpolation formulas are very useful for the analysis of experimental results on CIDEP generation in TR-quenching processes. The most informative are experimental investigations of the dependence of CIDEP Pn on the parameters, determining the value of pt, for example, diffusion coefficient Dr and the Zeeman frequency ωB. In this work we mainly analyze the experimental results on Dr-dependence of CIDEP, generated in quenching of the excited T-state of 1-chloronaphthalene by 2,2,6,6-tetramethylpiperidine-1-oxyl (TEMPO) radicals.10 Fitting of numerical and analytical dependences to experimental data10 is displayed in Figure 3a. Noteworthy is that in the calculation we use the Dr-
(4.24)
(4.25)
2 −1
with φ(x) = (16/3)[3 + 0.1(1 − x) ] , 1 f1 (x) = [3 − 2ϕ(x)] and f2 (x) = 1 − f1 (x) 5 (4.26)
High accuracy of the linear approximation 4.19 for the dependence Q(xr) with interpolation expressions 4.21−4.26 is demonstrated in Figure 2a,b for two values of the parameter pD
Figure 2. Dependence of Q = Pn−1 on xr = Dr/Dt 4.19, evaluated numerically with eqs 3.14−3.18 (dots) and analytically with eqs 4.19−4.26 (full lines) for two R*-contributions pD = 1 (a) and pD = 0 (b). Other parameters used are pa = 0, pq = 1.0 (1); pa = 0, pq = 0.4 (2); pa = 3, pq = 1.0 (3); pa = 3, pq = 0.4 (4).
Figure 3. Comparison of experimental (○) and theoretical functions Q(Dr) = Pn−1(Dr) (a) and Pq(Dr) (b). Theoretical inverse-CIDEP dependence Q(Dr) is calculated numerically (full line) and analytically (*) for pD = 1 and other parameters presented in section V.A. Dependence Pq(Dr) is evaluated numerically (eq 3.22) and analytically [with eq A1] (full and dashed lines, respectively) for the same values of parameters of the model.
(see section III.D): pD = 1 and pD = 0, respectively. Some deviation of the numerically calculated exact dependence Q(xr) = Pn−1(xr) from the linear one (4.19) is observed only for small xr < 1 (large pt ≫ 1), which can be realized at very small diffusion coefficients Dr ≳ 10−7 cm2/s for realistic values of parameters of TR-systems (see subsequent text). Noteworthy is that this deviation manifests itself, in particular, in the discrepancy between predictions of exact and approximate formulas for P̅qn(pt ≫ 1) at pq ≪ 1 [eqs 4.11, 4.13, and 4.24]: P̅qn(pt ≫ 1) = 0 and P̅qn(pt ≫ 1) = 1/Qq0, respectively. Analysis shows, however, that this discrepancy is fairly strong only in cases when values of Qqn(xr) are large (i.e., the corresponding CIDEP Pqn is small) and the manifestation of R*-contribution in CIDEP Pn (thus in Qn) is negligibly small (see Figure 2a,b). The xr-dependent reduced TR-quenching rate Pq(xr) = Kq (xr )/K diff (section III.D) can also be described by interpolation formula, which is represented in the form
dependent quenching probability pq(Dr) = rq(Dr)/rt, obtained by fitting of the experimental dependence of the dimensionless TR-quenching rate Pq(Dr) = Kq(Dr)/Kdiff (see section III.D), shown in Figure 3b. The theoretical Q(Dr) = Pn−1 (Dr)-dependences are calculated assuming the case of large S0R*-contribution, pD = 1, which is believed to be the most realistic [though the case pD < 1 will also be briefly discussed (in section V.B), for comparison]. As for other parameters of the model, in the calculation we have used the following values: d = 7 × 10−8 cm, rt = ra = 7.7 × 10−8 cm, D = 2.1 × 1010 s−1, and ωB = 6.0 × 1010 s−1. Note that the observables under study do not depend on d itself, depending, however, on length parameters rν, (ν = t, a, q), 11360
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cm−1. This value of α agrees quite well with that α ≈ 2.85 × 108 cm−1, which is calculated from the magnitude Dt = 0.27 × 10−6 cm2/s, estimated earlier by fitting experimental data with derived formulas (assuming pD = 1). With the use of the obtained value of α one can estimate the exchange interaction |J0| at a contact (r = d): |J0| ≈ ωB eα(rt−d) ≈ 4 × 1011 s−1. (4) Very good agreement was obtained between experimental and theoretical dependences Q(Dr) and confirmed the high accuracy of the proposed theory and counted in favor of a suggested large contribution of R*-radicals to CIDEP; i.e., pD ≈ 1. Fairly good agreement, however, can been obtained for small pD < 1 as well, but at the cost of using unrealistic values of some other important parameters of the model. For example, for pD = 0 the agreement is gained only for too large value α ≈ 3.3 × 108 cm−1 and under the assumption of weakness of transitions in aregions, pa ≈ 0, which looks completely unrealistic for the system under study, taking into account the significant strength of transition-inducing ZFS-interaction. (5) The validity of the assumption pD = 1, demonstrated as applied to the process under study, does not, in principle, mean that the more general case pD < 1 cannot be realized in other processes. The reduction of R*-contribution (∼pD < 1) is possible in the presence of the spin-dependent-interaction induced CIDEP relaxation during TR-quenching processes accompanied by capture of TR-pairs into long-lived intermediate state. The proposed theory can be quite helpful for studying such processes. (6) In our work the main attention has been paid to the analysis of CIDEP generation in TR-quenching for intermediate values of relative diffusion coefficients Dr ≳ Dt, corresponding to pt ≲ 1, although the proposed theory is, of course, applicable for fairly large diffusion coefficients Dr ≫ Dt as well (for which pt ≪ 1 or xr ≫ 1). According to eq 4.19, in the limit Dr ≫ Dt one can evaluate CIDEP using the simplified expression Q ≈ qnxr, i.e., Pn ≈ qn−1pt. Even more simple analytical formulas 4.10−4.13 can be applied for weak and strong transitions in a-regions. These formulas enable one to easily evaluate the effect of small quenching rate rq < rt (small probability pq = rq/rt < 1). In particular, they show that Pqn/Pen ∼ pq, and, therefore, the CIDEP contribution of R*-radicals (S0R*-pairs) becomes very small in the limit of small pq ≪ 1, which is quite expected for large Dr, as it is demonstrated by the above analysis. (7) In the limit of very large diffusion coefficients Dr the previously mentioned expressions 4.23 and 4.10-4.13 must be corrected to take into account the effect of fast VZFSfluctuations, which strongly manifest themselves in generated CIDEP, when ωBτV(Dr) ≲ 1 where τV(Dr) ∼ d2/Dr is the correlation time of fluctuations. The correction, however, reduces to the simple term in these expressions, resulting in the final formulas for CIDEP Pnc of the form9,11,12 Pnc ≈ PnΦ(ωBτV), where Φ(x) = (1/2)[ϕ(x2) + ϕ(4x2)] with ϕ(x) defined by eq 4.7.
whose values are determined by d (section II), but only formally (because of applied definitions of parameters rν). The Dr-dependent parameter pa = κara (eqs 3.18, 4.6), characterizing the efficiency of transitions in a-regions, is quite accurately estimated within the slowly fluctuating interaction approximation (section III.B).9,11,12 The important parameter Dt (eq 3.5) is considered as adjustable. Theoretical functions Q(Dr), presented in Figure 3a, are obtained for Dt = 0.27 × 10−6 cm2/s. Figure 3a demonstrates quite good agreement between the experimental dependence Q(Dr) and theoretical ones, calculated for chosen realistic values of parameters of the model. B. Discussion of Results. Here we discuss the basic assumptions and most important results of the proposed treatment. (1) First, some comments are needed on the accuracy of the applied approximation r1 ≈ r2,3 = rt (see eq 3.2). The difference of exact results from those, obtained in this approximation, is controlled by the parameter ξt = (r2,3 − r1)/rt = ln(2)/αrt. For realistic parameters of the model, discussed in section V.A (see also later text), the estimation yields ξt ∼ 4 × 10−2, which means that the estimated relative difference of predictions of exact theory and applied approximation is about 4%. Comparison of results of numerical calculations with the exact11,12 and proposed approximate approaches shows that, in reality, the difference is usually even smaller (∼1−2%). Similar estimation can be obtained for the accuracy of the considered approximation ra ≈ rt (section IV.B). The fact is that for values of parameters, used in preceding fitting of experimental data, we get the estimation δa = (ra − rt)/rt ≲ 0.1. Numerical calculations show that, for δa ≲ 0.1, in the wide region of parameters of the model, the relative difference between the approximate CIDEP values Pn and the exact ones is about 5% or less. (2) It is worth recalling that the definition 4.1 predicts about two times larger CIDEP value Pn than that obtained with the usually applied formula for CIDEP in geminate TR-quenching of TR-pairs in the equilibrated quartet state.1 Nevertheless, we have found good agreement between experimental and theoretical CIDEP, predicted by the proposed (correct) theory (as is seen from Figure 3a), although for values of parameters of the model somewhat different from those applied in previous interpretations with formula for geminate TR-quenching (see later text). (3) CIDEP is essentially determined by the transition efficiency pt (3.5). The value of pt is controlled by a number of parameters, one of which is the rate α of exponential decrease of the exchange interaction with distance (eq 2.3): pt ∼ α−1. In some earlier-mentioned previous investigations the value α ≈ 1.4 × 108cm was used,1 which, however, does not seem to be quite realistic. The fact is that a fairly accurate formula for α is well-known (see, for example, ref 19): α ≈ αR + αT ,
where αX = αH(EX /E H)1/2
(5.1) −1
with X = R,T. In the relation 5.1 αH ≈ 1.9 × 10 cm and EH ≈ 13.5 eV are the inverse size of the wave function and the ionization energy of the ground state of the hydrogen atom,19 and EX is the ionization energy of the particle X (X = T, R). For experimental values of ionization energies of analyzed Tand R-particles, ET ≈ 6 eV,15,20 for 1-chloronaphthalene and ER ≈ 8.2 eV21,22 for TEMPO-radical, we estimate: αT ≈ 1.3 × 108 cm−1 and αR ≈ 1.4 × 108 cm−1 and, therefore, α ≈ 2.7 × 108 8
VI. CONCLUSION In this work we have theoretically analyzed specific features of the CIDEP generation in liquid-phase TR-quenching processes. The approximate SLE is proposed, which allows one to easily (but quite accurately) describe the experimental results. 11361
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The Journal of Physical Chemistry A With this SLE we have derived simple analytical formulas, accurately describing the CIDEP and quenching rates in the wide region of parameters of the model. In particular, they describe specific features of CIDEP generation at relatively small diffusion coefficients Dr, for which the effect of nonadiabatic transitions in c- and a-regions are fairly strong; i.e., pt > 1, and pa > 1. Especially simple expressions are found in the limit of large Dr, when pt ≪ 1 (but pa ≲ 1) and CIDEP pn(pt) ∼ pt. These expressions are shown to be very useful for treating a number of recent experimental results.1 Concluding our consideration, it is worth emphasizing the importance of the proposed theory in connection with recent studies of the RTPM-like CIDEP-generation mechanisms in some spin-selective processes. Here we mention two of them: (1) One of examples of such processes is quenching of the lowest excited singlet state of molecular oxygen O2(1Δg) by TEMPO radicals,1,5,23 [represented by the kinetic scheme O2(1Δg) + R → O2(3∑−g ) + R*]. This reaction is found to result in fairly large net CIDEP of radicals, determined by spin evolution of [O2(3∑−g )R*]-pairs. The results of our work can be very helpful for the analysis of CIDEP, generated in O2(1Δg) quenching, for which fairly large pt > 1 is expected because of very strong ZFS-interaction in the T-state O2(3∑−g ) (D ≈ 7.5 × 1011 s−1). (2) Another example is solid-state TD-quenching, i.e., quenching of T-excitations by doublet (D) particles, for instance, charge carriers (polarons). This kind of process leads to CIDEP (i.e., electron spin polarization) of carriers,24 which can be very important for spintronic applications.25
■
The analysis of the numerically calculated dependence of quenching probability Pq = Kq/Kdiff (section III.D) on pt enables us to propose the approximate expression 4.27 for Pq(xr) with parameters P0q(pq, pa), P∞ q (pq, pa), and qr(pq, pa) obtained by simple interpolation formulas:
qr ≈ q1 +
(ν = 0, ∞)
(A1)
1 (q − q1)(pq + 3pq 2 ) 4 2
where α0 = 0.5 + 0.07ϕ(pa ),
α∞ = 0.53ϕ(pa )
q1 = 0.19 + 0.12ϕ(2pa )
■
■
REFERENCES
(1) Kawai, A.; Shibuya, K. Electron Spin Dynamics in a Pair Interaction between Radical and Electronically-Excited Molecule as Studied by a Time-Resolved ESR Method. J. Photochem. Photobiol., C 2006, 7, 89−103. (2) Blättler, C.; Jent, F.; Paul, H. A. Novel Radical-Triplet Pair Mechanism for Chemically Induced Electron Polarization (CIDEP) of Free Radicals in Solution. Chem. Phys. Lett. 1990, 166, 375−380. (3) Kawai, A.; Okutsu, T.; Obi, K. Spin Polarization Generated in the Triplet-Doublet Interaction: Hyperfine-Dependent Chemically Induced Dynamic Electron Polarization. J. Phys. Chem. 1991, 95, 9130− 9134. (4) Kawai, A.; Shibuya, K. Charge-Transfer Controlled Exchange Interaction in Radical-Triplet Encounter Pairs as Studied by FT-EPR Spectroscopy. J. Phys. Chem. A 2007, 111, 4890−4901. (5) Moscatelli, A.; Sartori, E.; Ruzzi, M.; Jockusch, S.; Lei, X.; Khudyakov, I.; Turro, N. Photolysis of Endoperoxides in the Presence of Nitroxides: A Laser Flash Photolysis Study with Optical and ESR Detection. Photochem. Photobiol. Sci. 2014, 13, 205−210. (6) Takahashi, H.; Iwama, M.; Akai, N.; Shibuya, K.; Kawai, A. Pulsed EPR Study on Large Dynamic Electron Polarisation Created in the Quenching of Photo-Excited Xanthene Dyes by Nitroxide Radicals in Aqueous Solutions. Mol. Phys. 2014, 112, 1012−1020. (7) Goudsmit, G.-H.; Paul, H.; Shushin, A. I. Electron Spin Polarization in Radical-Triplet Pairs. Size and Dependence on Diffusion. J. Phys. Chem. A 1993, 97, 13243−13249. (8) Shushin, A. I. The Relaxational Mechanism of Net CIDEP Generation in Triplet-Radical Quenching. Chem. Phys. Lett. 1993, 208, 173−178. (9) Shushin, A. I. Nonadiabatic Transitions in Liquid Phase Reactions. Net Electron Polarization in Radical Pair Recombination and Triplet-Radical Quenching. J. Chem. Phys. 1993, 99, 8723−8732. (10) Kobori, Y.; Takeda, K.; Tsuji, K.; Kawai, A.; Obi, K. Exchange Interaction in Radical-Triplet Pairs: Evidences for CIDEP Generation by Level Crossings in Triplet-Doublet Interactions. J. Phys. Chem. A 1998, 102, 5160−5170. (11) Shushin, A. I. Detailed Analysis of the Mechanism of Net Electron Spin Polarization in Liquid Phase Triplet-Radical Quenching. Chem. Phys. Lett. 1999, 313, 246−254. (12) Shushin, A. I. The Fast and Slow Rotation Approximations for the Description of the Kinetics of Triplet Radical Quenching and Electron Spin Polarization. Mol. Phys. 2002, 100, 1303−1310. (13) Blank, A.; Levanon, H. Optimal Magnetization in Liquids, Generated by Triplet-Doublet Interaction. Mol. Phys. 2002, 100, 1477−1488. (14) Stavitski, E.; Wagnert, L.; Levanon, H. Magnetic Field Dependence of Electron Spin Polarization Generated through Radical-Triplet Interactions. J. Phys. Chem. A 2005, 109, 976−980. (15) Swenberg, C. E.; Geacintov, N. E. In Organic Molecular Photophysics, Vol. 1; Birks, J. B., Ed.; Wiley: New York, 1973. (16) Blank, A.; Levanon, H. Interaction between Polarized Triplets and Stable Radicals in Liquid Solutions. J. Phys. Chem. A 2001, 105, 4799−4807. (17) Shushin, A. I. Magnetic Field Effects in Radical Pair Recombination. I. CIDNP and CIDEP in Geminate Recombination. Chem. Phys. 1990, 144, 201−222. (18) Shushin, A. I. Magnetic Field Effects in Radical Pair Recombination. II. Spin Relaxation and CIDN(E)P in Bulk Recombination. Chem. Phys. 1990, 144, 223−239. (19) Nikitin, E. E.; Umanskii, S. Ya. Theory of Slow Atomic Collisions; Springer: Berlin, 1984. (20) Gotkis, Y.; Oleinikova, M.; Naor, M.; Lifshitz, C. TimeDependent Mass Spectra and Breakdown Graphs. 17. Naphthalene and Phenanthrene. J. Phys. Chem. 1993, 97, 12282−12290.
Interpolation Formula for the Effect of Transitions in a-Region on Quenching Rate
⎛1 ⎞ ⎜ p ⎟ /(1 − α p ) νq ⎝ 3 q⎠
ACKNOWLEDGMENTS
The work was supported by the Russian Foundation for Basic Research (Projects 13-03-00388 and 14-03-00546).
APPENDIX A
Pqν ≈
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Article
q2 = [0.35 + 0.15ϕ(1.5pa )][1 − 0.3ϕ(0.2pa )]
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Present Address
Moscow Institute of Physics and Technology, State University, 141700, Institutskii per. 9, Dolgoprudny, Moscow Region, Russia Notes
The authors declare no competing financial interest. 11362
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(21) Cockett, M. C. R.; Ozeki, H.; Okuyama, K.; Kimura, K. Vibronic Coupling in the Ground Cationic State of Naphthalene: A Laser Threshold Photoelectron [Zero Kinetic Energy (ZEKE) Photoelectron] Spectroscopic Study. J. Chem. Phys. 1993, 98, 7763−7772. (22) Kubala, D.; Regeta, K.; Janečková, R.; Fedor, J.; Grimme, S.; Hansen, A.; Nesvadba, P.; Allan, M. The Electronic Structure of TEMPO, Its Cation and Anion. Mol. Phys. 2013, 111, 2033−2040. (23) Mitsui, M.; Takeda, K.; Kobori, Y.; Kawai, A.; Obi, K. Unusually Large Dynamic Electron Polarization in an O2(1Δg)-2,2,6,6-Tetramethylpiperidine-1-oxyl Radical System. J. Phys. Chem. A 2004, 108, 1120−1126. (24) Shushin, A. I. Generation of Electron Spin Polarization in Disordered Organic Semiconductors. Phys. Rev. B 2012, 86, 0352061−035206-10. (25) Ž utić, I.; Fabian, J.; Das Sarma, S. Spintronics: Fundamentals and Applications. Rev. Mod. Phys. 2004, 76, 323−410.
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