Reversible coupling of radical pair spin dynamics to a locally excited electronic singlet state

Abstract

Thermally assisted delayed fluorescence (TADF) in electron–donor–bridge–acceptor triads has recently been shown to provide a new way of observing the spin dynamics of charge-separated states (CSS) corresponding to linked radical pairs. In this work, we present a theoretical approach for extending standard quantum-dynamical models to describe this system. Using a representative example that combines four electronic radical-pair spin states with three nuclear spin states of a single nitrogen nucleus, we extend the Hilbert space from 12 to 15 dimensions by including the excited singlet state S₁. We derive Liouvillian operators that account for the kinetic coupling between S₁ and the CSS, including decay, charge separation, and recombination, and illustrate the resulting dynamics with numerical examples.

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Introduction

Radical pairs are chemical intermediates in which weak magnetic interactions – significantly smaller than thermal energy at room temperature – with both internal and external magnetic fields can exert remarkable influence over the rates of highly exergonic chemical reactions, such as those associated with covlalent bond formation or the recombination of charge-separated states.

These effects originate from the weak splitting of four spin sublevels – one singlet and three triplet – and their associated hyperfine states, which arise from the coupling between the unpaired electron spins in the radicals and nearby magnetic nuclei. Moreover, the chemical process is governed by strict spin selection rules that require conservation of spin multiplicity between the initial radical pair and the final product state. Experimental and theoretical investigations of these and related phenomena, which now constitute the scientific field of spin chemistry,^1–7 have significantly advanced our understanding of chemical reactivity.

Historically, the first discoveries in spin chemistry were the observations of non-Boltzmann spin state populations in EPR and NMR spectra – phenomena attributed to chemical origins and termed chemically induced dynamic electron polarization (CIDEP)^8–10 and chemically induced dynamic nuclear polarization (CIDNP).^11–14 Soon after, the first sizeable magnetic field effects were discovered in chemical reactions,^15,16 a phenomenon later dubbed MARY,¹⁷ followed by the discovery of magnetic isotope effects (MIE)¹⁸ and by magnetic resonance exploiting magnetic field dependent reactions yields (RYDMR).¹⁹ Magnetic field effects have also been discovered in radio luminescene, resulting from recombination fluorescence of radical ions, with oscillations indicating the coherent nature of spin processes involved.^20,21 Insights into the underlying spin mechanisms gained from all these discoveries have proven valuable in elucidating primary electron transfer processes in photosynthesis^22–24 and have led to the hypothesis that avian magnetic navigation may be based on the radical pair mechanism – an idea that has inspired extensive research and continues to gain experimental support to this day.^22–30

The energetics of a radical pair (RP) are depicted diagrammatically in Fig. 1. Depending on the multiplicity of the precursor, the RP is initially generated with either singlet or triplet electron-spin correlation. The principal energetic contributions are exchange interaction, which separates the singlet and triplet states even in the absence of an external magnetic field, and Zeeman interaction, which splits the triplet sublevels and may give rise to a condition in which one triplet substate becomes degenerate, i.e., resonant, with the singlet state. The electronic spin states are coupled through the hyperfine interaction (hfi). The isotropic, time independent component of the hfi gives rise to coherent singlet–triplet transitions, whereas the anisotropic, typically stochastically modulated hyperfine interaction (ahfi) induces relaxation. In many cases, a classical kinetic description^31,32 that incorporates coherent processes provides an adequate representation of the observed kinetics and their magnetic-field dependence. However, such approaches are insufficient for simulating quantum beats arising from time-independent interactions such as isotropic hfi or g-factor differences between singlet and triplet substates. These effects can be observed under specific experimental conditions at early times following RP formation. Situations of this kind require a full quantum-dynamical treatment.

Fig. 1 Magnetic field dependent state diagram of a radical pair, here represented by a charge separated state (CSS) in an electron donor-bridge-acceptor triad showing the energetic situations between singlet and triplet substates with exchange splitting 2J in different magnetic fields. The circular and linear arrows represent coherent and incoherent transitions, respectively, between the different substates. The rate constants of spin selective recombination processes to singlet and triplet product states are denoted k_S and k_T for singlet and triplet, respectively.

From an information science perspective, radical pairs (RPs) can be regarded as pairs of entangled qubits, a concept that has recently gained attention in the context of quantum information science.³³ In this framework, the maintenance and control of quantum coherence are of central importance.

Coherence in RPs is manifested through the observation of quantum beats, which arise from singlet–triplet (S/T) spin mixing. Although these spin substates cannot generally be distinguished spectroscopically, differences in their reaction rates toward spin-selective product channels may produce stepwise features in the overall reaction kinetics. Such effects, while difficult to detect, have been observed on the picosecond timescale under strong magnetic fields, where differences in Larmor precession frequencies – resulting from distinct g-factors of the radical partners – enhance S/T₀ transition rates.^34,35

Spin quantum beats are more readily accessible by spectroscopic techniques capable of distinguishing RP states of different spin multiplicities, notably electron paramagnetic resonance (EPR) spectroscopy.^36,37 In contrast, optical detection cannot directly discriminate between RP spin states. However, following photoexcitation by a second laser pulse after initial RP formation, the enhanced reactivity of the electronically excited radicals promotes rapid conversion into products of the corresponding multiplicity. This process enables optical readout of the spin state, allowing quantum beat detection via transient optical signals.³⁸

An alternative approach to differentiating RP spin states relies on singlet recombination to a fluorescent excited state, transferring singlet–triplet oscillations directly into the fluorescence signal. In non-polar solvents, radical ion pairs generated by high-energy radiation typically possess sufficient excess energy to form locally excited singlet states upon recombination, leading to the first observations of spin quantum beats in radioluminescence.^21,39,40 In polar solvents, by contrast, photoexcited RPs generally lie below the energy of fluorescent locally excited singlets. A notable exception occurs with exciplex formation, which, however, requires sufficient translational and rotational mobility for the radical partners to adopt the correct mutual orientation in an exciplex.^41–45 In rigid, covalently linked donor–bridge–acceptor (D–B–A) systems, exciplex formation is typically precluded, and the nearest fluorescent singlet state remains energetically inaccessible from the RP.

Recent studies by Mani and co-workers,^46,47 and subsequently by the Lambert group,⁴⁸ have demonstrated that structural modification of D–B–A triads can render the S₁ state energetically accessible to the charge-separated state (CSS) – which represents the RP in these systems. This enables the observation of delayed fluorescence arising from reverse charge recombination of the RP. The magnetic-field dependence of this delayed fluorescence provides compelling evidence identifying the RP as its precursor state. Although spin quantum beats have not yet been directly observed in thermally activated delayed fluorescence (TADF) from CSSs in D–B–A triads, these phenomena remain of significant interest. On one hand, they provide a basis for magnetic-field–sensitive imaging, and on the other, fluorescence detection offers a promising tool for single-molecule spin chemistry studies.

In this paper, we examine to what extent the TADF signal reflects the intrinsic dynamics of the CSS, and conversely, how the CSS dynamics are influenced by kinetic coupling to the S₁ state. We introduce a formal framework that integrates the reversible kinetic coupling of S₁ to the CSS with a quantum dynamical treatment of the CSS. This framework is then employed to address these two aspects through numerical model calculations.

Theoretical basis

The stochastic Liouville equation (SLE) for the time dependent spin density matrix (DM) [small rho, Greek, circumflex](t) has been well established as a theoretical framework to deal with the spin and reaction dynamics of radical pairs of all kinds^3,49,50 including the charge separated states (CSS) we are interested in here.^{32,38,51–57} For rigidly linked donor-bridge-acceptor triads the Liouville superoperator [script L] comprises the following contributions, [script L]_H describing the coherent spin motion due to the spin Hamiltonian Ĥ

[script L]_H[small rho, Greek, circumflex] = −i([Ĥ[small rho, Greek, circumflex] − [small rho, Greek, circumflex]Ĥ]) (2)

[script L]_K describing the stochastic decay into reaction channels selective for singlet and triplet multiplicity. The established standard form as introduced by Haberkorn⁵⁸ iswith Q̂_S and Q̂_T the projection operators on the electronic singlet and triplet spin substates, respectively, and k_S and k_T the rate constants for reaction into the pertinent reaction channels and [script L]_R describing relaxation and dephasing among the spin sublevels according to various time dependent interactions, such as anisotropies of hfc and g-tensor, electron spin–spin dipolar coupling, and fluctations of S/T splitting caused by stochastic fluctuations of rotational motion and molecular conformation. Its particular contributions will be specified in the application part below.

In solving the SLE it has become convenient to vectorize the DM by stacking the lines of the matrix into a vector, which we will denote by r. In this formalism, the Liouville superoperators are represented by square matrices applied to r. The pertinent conversion of the superoperators in eqn (2) and (3) into the matrices L_H, L_K, and L_R is based on the general matrix relation

Â[small rho, Greek, circumflex]B̂ → (Â ⊗ B̂^T)r (4)

where the symbol ⊗ denotes the Kronecker product and B̂^T is the transposed matrix of B̂.

Thereby the SLE is turned into a matrix equation

with the general solution

r(t) = e^Ltr(0) (6)

In the following, we consider the 4N-dimensional Hilbert space , spanned by the tensor product of a four-dimensional electronic spin space and an N-dimensional nuclear spin space. The corresponding Liouville space has dimension (4N)², and the Liouvillian superoperator is represented by a matrix denoted L_{(4N)²×(4N)²}. We then extend the Hilbert space to dimension 5N by including the excited precursor state S₁ alongside the four CSS electronic spin states. The associated Liouville space therefore has dimension (5N)², with Liouvillian matrix representation denoted by L_{(5N)²×(5N)²}.

For a general nuclear state defined by the nuclear quantum numbers {ν₁,ν₂,ν₃,…ν_n}, with ν_i ranging from 1 to k_i the dimension N of the nuclear Hilbert space is given by:

In the present treatment, we restrict the analysis to three nuclear states arising from a ¹⁴N nucleus, i.e. N = 3. Consequently, the dimensionality of the Hilbert space increases from 4 × 3 = 12 to 5 × 3 = 15.

Subsequently, we derive the formalism for incorporating the formation of the CSS from the S₁ state with rate constant k_CS as well as its regeneration from the singlet substate of the CSS with rate constant k_rCS. It is assumed that no nuclear spin dynamics occur in the S₁ state and that the nuclear spin DM remains unchanged during both, charge separation and regeneration processes.

Incorporation of S₁ into the electronic/nuclear spin space

The state ϕ(σ,ν) in Hilbert space represents the direct product of the electronic spin state σ in the Zeeman product basis and ν the nuclear spin state of ¹⁴N, where

σ = αα, αβ, βα, ββ, (S₁) (8)

is indexed by the numbers 1–4 (5) and

ν = 1,0,−1 (9)

is indexed by the numbers 1–3.

The ordered electronic/nuclear product states (1–12 or 1–15 if including S₁) are

αα1,αα0,αα−1,αβ1,αβ0,αβ−1,βα1,βα0,βα−1,ββ1,ββ0,ββ−1, (S₁1,S₁0,S₁−1) (10)

The quantities ρ_12×12(σ,ν;σ′,ν′) or ρ_15×15(σ,ν;σ′,ν′) represent density operators acting on a 12- or 15-dimensional Hilbert space. The space of such operators is a 144 (225)-dimensional Liouville space. Under vectorization,

ρ_12×12(σ,ν;σ′,ν′) → r₁₄₄(i) (11)

where

i(σ,ν;σ′,ν′) = ν′ + 3(σ′ − 1) + 4 × 3(ν − 1) + 3 × 4 × 3(σ − 1) (12)

or

ρ_15×15(σ,ν;σ′,ν′) → r₂₂₅(j) (13)

where

j(σ,ν;σ′,ν′) = ν′ + 3(σ′ − 1) + 5 × 3(ν − 1) + 3 × 5 × 3(σ − 1) (14)

The Liouvillian superoperators acting in the 12²- and 15²-dimensional Liouville spaces are represented by the matrices L_144×144 and L_225×225 with matrix elements defined in the Liouville basis as

(i|L_144×144|j) ≡ L_144×144(i,j) and (i|L_225×225|j) ≡ L_225×225(i,j) (15)

Equivalently, the same matrix elements may be written in their explicit Hilbert-space-index representation as

(σ,ν;σ′,ν′|L_144×144|s,n;s′,n′) ≡ L_{(12)²×(12)²}(σ,ν;σ′,ν′|s,n;s′,n′) (16)

and

(σ,ν;σ′,ν′|L_225×225|s,n;s′,n′) ≡ L_{(15)²×(15)²}(σ,ν;σ′,ν′|s,n;s′,n′) (17)

here, the notation L_{(12)²×(12)²} and L_{(15)²×(15)²} emphasizes the explicit Hilbert-space-index structure of the corresponding representations. The composite Liouville-space indices i and j are obtained from the index sets (σ,ν;σ′,ν′) and (s,n;s′,n′), respectively, according to the mapping defined above for the vectorization of the density operators r₁₄₄(i) and r₂₂₅(j).

The quantum dynamical calculation of the combined ^1,3CSS/S₁ system begins with the routine evaluation of the superoperator matrix for the isolated CSS system, as has been extensively documented in the literature (see also the Mathematica file in the SI).^{32,51,54–57,59} A detailed specification, together with representative model calculations, is provided below.

To construct the Liouville-space representation required for combining the electronic/nuclear S₁ state with the CSS states, we extend the Liouvillian matrix L_144×144 to L_225×225. Using the index mappings defined in eqn (10) and (11), L_144×144 is first expressed in its explicit Hilbert-space-index representation L_{(12)²×(12)²}(σ,ν;σ′,ν′|s,n;s′,n′). The resulting Hilbert-space-index representation is then embedded in the extended Liouvillian L_{(15)²×(15)²}. At this stage, all matrix elements involving the electronic index 5 are set to zero. The additional S₁-related matrix elements required to describe population transfer are introduced separately below. Thus, the present construction merely enlarges the state space and does not yet generate nuclear-spin evolution within S₁ or coherences between S₁ and the CSS states.

Finally, the extended Liouvillian is transformed back from the explicit Hilbert-space-index representation to its Liouville-space matrix representation according to eqn (12).

L_225×225(i(σ,ν;σ′,ν′), j(s,n;s′,n′)) = L_{(15)²×(15)²}(σ,ν;σ′,ν′|s,n;s′,n′) (18)

The kinetic processes to be represented are depicted in the scheme shown in Fig. 2. In the following their representation as operators in the Liouvillian are developed.

Fig. 2 Level scheme defining the stochastic rate processes: S₁-decay to ground state (”k_f” for simplicity, it encompasses also radiationless processes competing with fluorescence), S₁-transition to S-CSS (k_CS), regeneration of S₁ from S-CSS (k_rCS), S-CSS recombination to ground state (k_S), T-CSS recombination to local triplet state (k_T). Coherent and incoherent transitions among the CSS spin sublevels S-CSS and T-CSS (T₊-CSS, T₀-CSS, T₋-CSS) are indicated by a circular arrow.

Depletion kinetics of S₁

Through decay by k_f and k_CS, the S₁ part of the DM is diminished as follows:here the projection operator Q_S1,15 on the S₁ subspace is given by a 15 × 15 matrix with diagonal elements of 1 in positions 13–15 and zeros otherwise.

The DM operation given by eqn (19) is translated to the Liouvillian matrix by

Population of S-CSS from S₁

This contribution to the evolution of the DM (DM) is expressed ashere, P_ex is a 15 × 3 projection matrix with unit elements along the diagonal positions 13 to 15 and zeros elsewhere. The operation P^T_exρP_ex extracts the 3 × 3 submatrix of ρ corresponding to the S₁ state.

Similarly, P_in is a 15 × 3 projection matrix with unit elements along the diagonal positions 7 to 9 and zeros otherwise. The transformation P_inP^T_exρP_exP^T_in therefore inserts the extracted 3 × 3 submatrix representing the S₁ state into the position corresponding to the ¹CSS state within the coupled-basis representation of ρ. Finally, the transformation matrix T₁₅ converts the resulting matrix representation into the Zeeman basis. The matrix transformation given by eqn (21) translates to the Liouville space as

L_CSS,CS = k_CS(T₁₅P_inP^T_ex) ⊗ (P_exP^T_inT⁻¹₁₅)^T (22)

Effect of reverse charge separation on the density matrix (DM)

The effect of reverse charge separation (rCS) on the DM is expressed within the Haberkorn convention (cf. eqn (3)) aswhich corresponds to the Liouville space operatorhere Q_1CSS,15 denotes the projection operator onto the ¹CSS spin substate and U₁₅ denotes the 15 × 15 unit matrix.

Repopulation of S₁ by reverse charge separation from CSS

The corresponding time dependence of the DM due to the repopulation of the S₁state via reverse charge separation is given bywhere Q_in = P_ex and Q_ex = Q^T_in = P^T_in are projection operators related to P_ex and P_in. The combined operation of these matrices, when applied to the DM in the coupled basis T⁻¹₁₅ρT₁₅ yields the required contribution to the S₁ submatrix.

The corresponding Liouville-space operator is then given by

L_S1,rCS = k_rCS(Q_inQ^T_exT⁻¹₁₅) ⊗ (T₁₅Q_exQ^T_in)^T (26)

Finally, the complete Liouville operator, required for obtaining the time-dependent solution of the DM according to eqn (6), can be expressed as

L = L_225×225 + L_S1,CS + L_CSS,CS + L_CSS,rCS +L_S1,rCS (27)

Numerical applications

Parametrization

The various components of the Liouvillian in eqn (5) will be parametrized as follows:

Hamiltonian Besides Zeeman interaction with equal g-factors g_e of the two radicals, and exchange interaction J, we take into account isotropic hyperfine coupling with only one nitrogen atom specified by a_N. Thus

Ĥ = a_N(Î_N,xŜ_1,x + Î_N,yŜ_1,y + Î_N,zŜ_1,z) + ω₀(Ŝ_1,z + Ŝ_2,z) − 2J(Ŝ_1,xŜ_2,x + Ŝ_1,yŜ_2,y + Ŝ_1,zŜ_2,z) (28)

where Ŝ_i,x, Ŝ_i,y, Ŝ_i,z and Î_N,x, Î_N,y, Î_N,z denote the components of the respective spin operators and ω₀ = γ_eB₀ is the Larmor frequency.

Reaction operator. The form of the reaction operator has been given in eqn (3), the parameters k_S and k_T denoting the specific reaction constants for singlet and triplet recombination.

Relaxation operator. The relaxation operator involves the effect of anisotropic hyperfine interaction at the nitrogen center with a coupling constant ΔA and the rotational correlation time τ_c. The Nakashima–Zwanzig formalism as descibed by Fay et al.⁶⁰ was applied. Since the reaction constant k_S can compensate for general, field independent relaxation, we did not include such a contribuition explicitly in our calculations here. Furthermore S/T dephasing is taken into account by the super operator matrix

L_STD = −k_STD(Q̂_S ⊗ Q̂_T + Q̂_T ⊗ Q̂_S) (29)

where Q̂_S and Q̂_T are the projection operators on singlet and triplet CSS in Hilbert space, and k_STD is the singlet/triplet dephasing rate constant.

In the following we will apply the general formalism to calculate the S₁-coupled CSS dynamics for two typical scenarios, where the CSS evolution in the absence of S₁-coupling is

(a) purely coherent

(b) follows a realistic scenario, recently observed in an experimental case

Purely coherent spin evolution

The parameter set for this case was taken as given in Table 1 (time units are in ns):

Table 1 Parameter values for coherent spin evolution (time units: ns)

kCS	krCS	kf	kS	kT	kSTD	2J	ΔA, mT	a, mT
1	0.025	0.25	0	0	0	0	0	1.0

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In case of decoupling from S₁ it leaves the RP evolution completely coherent, with no decay and relaxation. The values of the rate constants involving S₁ represent efficient coupling to the CSS. In Fig. 3 the time dependences of the populations of several states of interest are shown. In all cases, the CSS state oscillates between S-CSS and T-CSS with an angular frequency of 3/2 a = 0.264 at zero field and of a = 0.166 at high field. These frequencies correspond to the energetic hyperfine splittings at the pertinent fields. The delayed fluorescence represented by the population of S₁ oscillates synchronously with the S-CSS population with about equal depth of modulation. For the phase relation cf. information below.

Fig. 3 Time evolution under conditions of coherent spin evolution (Table 1). Populations of S₁(×100, red), total CSS (black), cyan: exponential simulation, S-CSS in S₁-coupled CSS (blue), (a) and (c) at zero field, (b) and (d) at 200 mT. (c) and (d) show the simulations of S₁ decay including the simulation of their oscillations and their damping (purple).

As shown in the SI (cf. Section S1), the uncoupled CSS does not decay, which is trivial for k_S = k_T = 0, and the oscillations are free of damping. Thus it can be concluded that under kinetic coupling conditions it is the process of reverse CS that causes the damping of the oscillations in the CS-state. In the literature^58,61 it has been shown that spin-selective reaction described by the Haberkorn approach causes dephasing by a rate constant of ½ k_ssel, if k_ssel is the rate constant of the spin-selective reaction. One should note, however, that this result has been obtained under restricted conditions (e.g. S, T₀) model system. In the present case, with k_rCS as the rate constant of the spin-selective process, and consideration of the full four spin-state system (S, T₊,T₀, T₋) the associated dephasing rate constant is ¼ k_rCS corresponding to a value of 0.025/4 = 0.00625 in the above example. Although this aspect has not been further investigated, we note that for a purely electronic 5-state system comprising the 4 CSS spin states and the kinetically coupled S₁ state, with coherent oscillations driven by the exchange interaction, the rate constant of dephasing (decoherence) has been found to vary between k_rCS/2 and k_rCS/4.

Regarding the decay time of delayed fluorescence we refer to the classical rate description (cf. eqn (30), Here the symbol ³LE refers to a locally excited triplet state) described in the parallel paper.⁴⁸

In that simplified picture, the internal spin dynamics of the CS-state is not considered explicitly, but a stationary ratio of the four spin substates is assumed with a concomitant global decay constant k_CSS. The following relation has been derived for the slow component (delayed fluorescence) of S₁ decay.

In the limit of slow k_CSS it is approximated by

For the parameter values of the present example rate constants of 0.00245 and 0.00295 have been obtained in zero field and high field, respectively. A value of 0.0049 is obtained for k_slow,0 by eqn (32). In the simple model, however, the rate constant is not spin-dependent. Produced from S₁, the CSS always remains in the S-CSS substate. Taking into account the decrease of the average singlet character of the CSS under conditions of full spin dynamics, it is qualitatively clear that the actual recombination rate must be reduced. As shown in the SI (cf. Section S2) the reduction factors of 0.50 and 0.60 for zero field and high field, respectively, can be rationalized by the average singlet character of the CSS.

As shown in Fig. 3, the population of S₁, and thus the delayed fluorescence intensity, closely follows the population of the singlet component of the S-CSS, accurately reproducing its oscillation period, damping, and overall decay. On the other hand, Fig. S1 demonstrates that kinetic coupling between the CSS and S₁ modifies the decay behavior of the unperturbed CSS and further damps any oscillations peculiar to the CSS spin dynamics, with the rate constants k_f, k_CS and k_rCS exerting distinct influences. These effects are analyzed and rationalized in detail in Section S5 of the SI. In addition to the features illustrated in Fig. 3, Fig. S6 highlights a further effect, namely a phase shift between the oscillations of the S₁ and S-CSS populations. This phase shift is governed primarily by the sum of the rate constants, k_CS + k_rCS.

In principle, quantum beats can be observed in radical pairs when the spin dynamics are dominated by a small number of hyperfine couplings or by the Δg mechanism, the latter typically requiring high magnetic fields. If these conditions are satisfied for an isolated CSS state, the reversible kinetic coupling to the S₁ state, which is a prerequisite for TADF, does not in itself suppress the oscillations, provided that the associated rate constant k_rCS does not lead to efficient decoherence according to the criterion given in eqn (33).

To illustrate this point, we have added to the SI (cf. Section S4) a simulation of a CSS state previously shown to exhibit quantum beats³⁸ under a hypothetical reversible kinetic coupling to the S₁ state using the parameter values listed in Table S1. The simulations demonstrate that pronounced TADF quantum beats would still be observed.

Table 2 Generic set of parameters (time units: ns)

kCS	krCS	kf	kS	kT	kSTD	τ	2J, mT	ΔA, mT	a, mT
1	0.025	0.25	0.001	0.02	0.1	0.6	10	1.5	1.0

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Thus, if quantum beats are present in the isolated CSS, their observability by TADF depends on whether the reversible kinetic coupling to the S₁ state remains sufficiently slow to preserve that coherence.

More general parametrization

Secondly, we investigate a more general parametrization, loosely adapted to parameters corresponding to the experimentally studied triad (Cl)TAA–mB–PDI in ref. 48.

In Fig. 4, we represent the decays of S₁, the singlet part S-CSS of the CS-state, and the total CSS population as obtained quantum dynamically with the parameters in Table 2 for the S₁-CSS coupled system at zero field, the 2J-resonance field of 10 mT, and the effective high-field case at 200 mT. For the parameters chosen, a slight oscillating feature is still recognized for the singlet components S₁ and S-CSS at the resonance field B = 2J. The oscillations clearly depend on the value of k_STD. For k_STD = 0 they are more pronounced than for k_STD = 0.1 in the standard parameter set, and they have disappeared for k_STD = 0.5 (cf. inset in Fig. 4b)).

Fig. 4 Time evolution under conditions represented by the parameters in Table 2. Populations of S₁ (×100) (a and d), S-CSS (b and e) in S₁-coupled CSS, total CSS (c and f) in S₁-coupled CSS at fields of 0 mT (blue), 10 mT (red) and 200 mT (black). Lower row: corresponding log plots. Inset in b) represents the curves at 10 mT for various values of k_STD. For comparison with the signals resultig in the case of absent kinetic coupling to S₁ cf. SI, Section S3.

In our case, the Fay and Manolopoulos⁶² criterion for suppression of coherence in the S/T(±) transition at resonance (involving T₊ for positive J and T₋ for negative J) can be written as

with

For k_STD = 0 and the other parameter values given in Table 2, γ_ST± is dominated by the large rate constants k_T and k_rCS and has a value of 0.023. This value is well below the threshold in eqn (33a), indicating that coherence is not efficiently suppressed. In contrast, for k_STD = 0.5, γ_ST± = 0.123, exceeding the threshold; accordingly, the oscillations are expected to disappear. This prediction is confirmed by the inset in Fig. 4b.

At zero field and in high field the energy gap between S and T is too large, and the amplitude of the oscillations too small for them to be observed in the decay signals. As follows from the log plots in Fig. 4, after the short initial stage the signals seem to decay exponentially. However, a closer look at the decay constants evaluated as a function of delay time (cf. Fig. 5) reveals that the instationary stage lasts about 200 ns, after which the decay rates become stationary. In Fig. 5a, this behaviour is demonstrated for the total population of the CS-state. Fig. 5b complements this information by showing that the relative populations of S₁ and the S-CS state become constant after about 200 ns, meaning that their decay rates are exctly the same as that of the total CSS population.

Fig. 5 (a) Decay constant k_CSS (in absolute units) of the CS-state as function of delay time, calculated from the time derivative of the log of the CS-state population at 0, 10 and 200 mT. (b) time dependence of fractional populations of S₁ and S-CS-state. (numerical calculation of the time derivation was not feasible in those cases, due to S/T₀ oscillations that, albeit of very small amplitude, lead to large variations of the slope).

The dependence of k_slow on the magnetic field is shown in Fig. 6. Several representative cases are considered. With neglect of relaxation (ΔA = 0) and without explicit S/T dephasing (k_STD = 0) (cf. Fig. 6a) the resonance appears relatively sharp and the base line outside the resonance is completely flat. The value of 0.006 (6 × 10⁶ s⁻¹) corresponds to the sum of k_slow due to the kinetic coupling according to eqn (32) plus k_S = 0.001 (10⁶ s⁻¹⁾ according to Table 2. Switching on of relaxation (cf. Fig. 6b) leads to an increase of k_slow at zero field because now relaxation allows for some admixture of T-CSS which decays faster (k_T = 0.025 (2.5 × 10⁶ s⁻¹)) than S-CSS. At the same time, the resonance broadens. On the high field side, k_slow deceases below the zero field value, because now the transitions S/T₊ and S/T_- are switched off due to the Zeeman splitting. Nevertheless the S/T₀ transition rate due to relaxation is high enough to populate enough T₀ as to yield much faster decay than without relaxation in case of Fig. 6a. Upon additional switching on of S/T dephasing (Fig. 6c) the broadening of the resonance increases.

Fig. 6 Field dependence of k_slow for certain variations in the parameters. Red (solid line and datapoints): k_slow determined from exponential decay constant of total CSS population between 200 and 400 ns, calculated with full kinetic coupling to S₁. Blue, data points: quantum dynamically obtained decay constant k_CSS,qdwoS1 of CSS without kinetic coupling to S₁ but otherwise the same parameter values. Black, solid line: k_CSS,cl value calculated by the classical model (eqn (32)). Except for the resonance region, the classical model yields very good predictions of the CSS-decay constant kinetically uncoupled from S₁ In (a), the blue data points around 10 mT are problematic because the exponential decay approximation is poor. At earlier times, the actual decay constant for 10 mT is larger than the stationary values at the neighbouring fields.

In Fig. 6 are also shown the rate constants k_CSS,qdwoS1 describing CSS decay, calculated quantum dynamically with the present parameter set but with the kinetic coupling to S₁ switched off. Relative to the corresponding k_slow values, these data points exhibit an approximately constant downward shift of 4.9 × 10⁶ s⁻¹.

The solid lines represent the rate constants k_CSS,cl obtained by applying the classical model to the quantum-dynamical k_slow values calculated for the fully coupled S₁/CSS system. These values are obtained from eqn (34), which follows from eqn (31) by solving for k_CSS.

Remarkably, although k_CSS,cl is derived from the classical kinetic model of Scheme (30) using k_slow values obtained from a full quantum-dynamical calculation that includes the S₁ kinetics, the resulting “classical” values reproduce the quantum-dynamical rate constants k_CSS,qdwoS1 of the isolated CSS rather well. Significant deviations are observed only in the vicinity of the 2J-resonance, if the exponential approximation to the decay kinetics breaks down (cf. Fig. 6c).

The nearly field-independent parallel shift between the k_slow and k_CSS,cl curves—and hence also, to a good approximation, between k_slow and the quantum-dynamical rate constants k_CSS,qdwoS1 of the uncoupled CSS—follows directly from eqn (28):

where the final approximation is valid for k_slo ≪ k_f, which is typically satisfied under the present conditions.

Conclusions

In this work, we have extended the spin-chemical description of hyperfine-coupled, rigidly linked radical pairs–typically realized as charge-separated states (CSS) in electron-donor-bridge-acceptor triads – to the case of reversible kinetic coupling with an electronically excited, fluorescent singlet state (S₁), which forms the basis of thermally activated delayed fluorescence (TADF). We have presented a systematic method for expanding the (4 × n_N)-dimensional electron-spin × nuclear-spin Hilbert space by an additional (1 × n_N)-dimensional S₁ × nuclear subspace. Compact expressions for the corresponding Liouville superoperators have been derived to account for S₁ decay and regeneration via fluorescence (k_f, including nonradiative decay), charge separation forming the singlet CSS (k_CS), reverse charge separation regenerating S₁ (k_rCS), and the associated modifications of the CSS induced by reverse charge separation.

This formulation preserves full control over the nuclear-spin subspace during electronic transitions between S₁ and the CSS. Although hyperfine interactions are absent in S₁ and nuclear spin states are therefore conserved in this manifold, they continue to influence the overall spin dynamics in the CSS through the reversible kinetic exchange between S₁ and CSS. In this sense, the S₁ state acts as a shelving state for the nuclear spin degrees of freedom.

Numerical examples, ranging from minimal models to more realistic parameter sets, demonstrate that the present treatment provides a complete description of the sub-state information contained in the system. We show that the fluorescence signal from S₁ closely follows the population of the singlet CSS, including regimes exhibiting coherent oscillations. Comparison of the CSS dynamics under TADF conditions with those of a CSS not subject to reversible coupling to S₁ reveals an accelerated decay that is well described by the classical kinetic model (cf. eqn (32)). Although quantum beats in the unperturbed CSS are damped by reverse electron transfer at a rate determined by k_(rCS), they should remain observable under most realistic conditions.

These results establish a general theoretical framework for analyzing spin-dependent TADF and open new avenues for extracting radical-pair spin dynamics from time-resolved fluorescence measurements.

Conflicts of interest

There are no conflicts of interest to declare.

Data availability

The data supporting this article have been included in the main manuscript and as part of the supplementary information (SI). Supplementary information is available. See DOI: https://doi.org/10.1039/d6cp00916f.

Acknowledgements

The author gratefully acknowledges the fruitful cooperation with Prof. Christoph Lambert and his group at the University of Würzburg and helpful discussion with Prof. Nikita Lukzen, International Tomography Center and Novosibirsk State University, Novosibirsk, Russia.

References

1. K. M. Salikhov, Y. N. Molin, R. Z. Sagdeev and A. L. Buchachenko, Spin polarization and Magnetic Effects in Radical Reactions, Elsevier, Amsterdam, the Netherlands, 1984.
2. K. M. Salikhov, Bull. Russ. Acad. Sci.: Phys., 2025, 89, 323–333.
3. U. E. Steiner and T. Ulrich, Chem. Rev., 1989, 89, 51–147.
4. U. E. Steiner and H. J. Wolff, in Photochemistry and Photophysics, ed J. J. Rabek and G. W. Scott, CRC Press, Boca Raton, 1991, vol. IV, pp. 1–130.
5. Dynamic Spin Chemistry. Magnetic Controls and Spin Dynamics of Chemical Reactions, ed S. Nagakura, H. Hayashi and T. Azumi, Kodansha and Wiley, Tokyo and New York, 1998.
6. P. J. Hore, K. L. Ivanov and M. R. Wasielewski, J. Chem. Phys., 2020, 152, 120401.
7. A. M. Lewis, Spin Chemistry Community website, https://spin-chemistry-community.github.io/about/, (accessed 22.01., 2026).
8. R. W. Fessenden and R. H. Schuler, J. Chem. Phys., 1963, 39, 2147–2195.
9. B. Smaller, J. R. Remko and E. C. Avery, J. Chem. Phys., 1968, 48, 5174–5181.
10. R. Kaptein and J. L. Oosterhoff, Chem. Phys. Lett., 1969, 4, 195.
11. R. Kaptein and J. A. den Hollander, J. Am. Chem. Soc., 1972, 94, 6269.
12. Chemically induced magnetic polarization, ed. A. R. Lepley and G. L. Closs, Wiley-Interscience, New York, 1973.
13. R. Kaptein, in Biological Magnetic Resonance: Volume 4, ed. L. J. Berliner and J. Reuben, Springer US, Boston, MA, 1982 DOI:10.1007/978-1-4615-6540-6_3, pp. 145–191.
14. M. Goez, in Advances in Photochemistry, ed D. C. Neckers, D. H. Volman and G. von Bünau, Wiley, New York, 1997, vol. 23, pp. 63–163.
15. R. Z. Sagdeev, K. M. Salikhov, T. V. Leshina, M. A. Kamkha, S. M. Shein and Y. N. Molin, J. Exp. Theor. Phys., 1972, 16, 422–424.
16. K. M. Salikhov, F. S. Sarvarov, R. Z. Sagdeev, Y. N. Molin, T. V. Leshina, M. A. Kamkha and S. M. Shein, Proceedings of XI European Congress on Molecular Spectroscopy, Tallin, 1973, 363.
17. W. Lersch and M. E. Michel-Beyerle, Chem. Phys., 1983, 78, 115–126.
18. A. L. Buchachenko, Russ. Chem. Rev., 1976, 45, 761–792.
19. E. L. Frankevich, A. I. Pristupa and V. I. Lesin, Chem. Phys. Lett., 1977, 47, 304–308.
20. B. Brocklehurst, Chem. Phys. Lett., 1976, 44, 245–248.
21. J. Klein and R. Voltz, Phys. Rev. Lett., 1976, 36, 1214–1217.
22. H. J. Werner, K. Schulten and A. Weller, Biochem. Biophys. Acta, 1978, 502, 255–256.
23. R. Haberkorn and M. E. Michel-Beyerle, Biophys. J., 1979, 26, 489–498.
24. A. J. Hoff, Q. Rev. Biophys., 1981, 14, 599–665.
25. K. Schulten, C. E. Swenberg and A. Weller, Z. Physik. Chem. N. F., 1978, 111, 1–5.
26. K. Maeda, K. B. Henbest, F. Cintolesi, I. Kuprov, C. T. Rodgers, P. A. Lidell, D. Gust, C. R. Timmel and P. J. Hore, Nature, 2008, 453, 387–390.
27. M. Tiersch and H. J. Briegel, Phil. Trans. R. Soc. Lond. A, 2012, 370, 4517–4541.
28. I. A. Solov'yov, T. Ritz, K. Schulten and P. J. Hore, in Quantum effects in biology, ed M. Mohseni, Y. Omar, G. Engel and M. B. Plenio, Cambridge University Press, Cambridge, UK, 2014.
29. H. G. Hiscock, S. Bourne Worster, D. R. Kattnig, C. Steers, Y. Jin, D. E. Manolopoulos, H. Mouritsen and P. J. Hore, Proc. Natl. Acad. Sci. U. S. A., 2016, 113, 4636–4639.
30. T. P. Fay, L. P. Lindoy, D. E. Manolopoulos and P. J. Hore, Faraday Discuss., 2020, 221, 77–91.
31. H. Hayashi and S. Nagakura, Bull. Chem. Soc. Jap., 1984, 57, 322–328.
32. J. H. Klein, D. Schmidt, U. E. Steiner and C. Lambert, J. Am. Chem. Soc., 2015, 137, 11011–11021.
33. M. R. Wasielewski, M. D. E. Forbes, N. L. Frank, K. Kowalski, G. D. Scholes, J. Yuen-Zhou, M. A. Baldo, D. E. Freedman, R. H. Goldsmith, T. Goodson, M. L. Kirk, J. K. McCusker, J. P. Ogilvie, D. A. Shultz, S. Stoll and K. B. Whaley, Nat. Rev. Chem., 2020, 4, 490–504.
34. P. Gilch, F. Pöllinger-Dammer, C. Musewald, U. E. Steiner and M. E. Michel-Beyerle, Science, 1998, 281, 982–984.
35. M. Liu, J. Zhu, G. Zhao, Y. Li, Y. Yang, K. Gao and K. Wu, Nat. Mater., 2025, 24, 260–267.
36. G. Kothe, S. Weber, E. Ohmes, M. C. Thurnauer and J. R. Norris, J. Am. Chem. Soc., 1994, 116, 7729–7734.
37. S. Weber, E. Ohmes, M. C. Thurnauer, J. R. Norris and G. Kothe, Proc. Natl. Acad. Sci. U. S. A., 1995, 92, 7789–7793.
38. D. Mims, J. Herpich, N. N. Lukzen, U. E. Steiner and C. Lambert, Science, 2021, 374, 1470–1474.
39. B. Brocklehurst, Radiat. Phys. Chem., 1997, 50, 213–225.
40. O. A. Anisimov, V. L. Bizyaev, N. N. Lukzen, V. M. Grigoryants and Y. N. Molin, Chem. Phys. Lett., 1983, 101, 131–135.
41. A. Weller, Z. Physik. Chem. N. F., 1982, 133, 93–98.
42. H. Staerk, W. Kühnle, R. Treichel and A. Weller, Chem. Phys. Lett., 1985, 118, 19–24.
43. M. Chowdhury, R. Dutta, S. Basu and D. Nath, J. Mol. Liq., 1993, 57, 195–228.
44. D. R. Kattnig, A. Rosspeintner and G. Grampp, Angew. Chem., Int. Ed., 2008, 47, 960.
45. S. Richert, A. Rosspeintner, S. Landgraf, G. Grampp, E. Vauthey and D. R. Kattnig, J. Am. Chem. Soc., 2013, 135, 15144–15152.
46. J. T. Buck and T. Mani, J. Am. Chem. Soc., 2020, 142, 20691–20700.
47. N. Lin, M. Tsuji, I. Bruzzese, A. Chen, M. Vrionides, N. Jian, F. Kittur, T. P. Fay and T. Mani, J. Am. Chem. Soc., 2025, 147, 11062–11071.
48. T. Groß, M. Holzapfel, A. Schmiedel, B. Woodward, U. E. Steiner and C. Lambert, Chem. Sci., 2026, submitted.
49. K. Maeda, A. J. Robinson, K. B. Henbest, H. J. Hogben, T. Biskup, M. Ahmad, E. Schleicher, S. Weber, C. R. Timmel and P. J. Hore, Proc. Natl. Acad. Sci. U. S. A., 2012, 109, 4774–4779.
50. T. P. Fay, L. P. Lindoy and D. E. Manolopoulos, J. Chem. Phys., 2019, 151, 154117.
51. J. Schäfer, M. Holzapfel, A. Schmiedel, U. E. Steiner and C. Lambert, Phys. Chem. Chem. Phys., 2018, 20, 27093.
52. U. E. Steiner, J. Schäfer, N. N. Lukzen and C. Lambert, J. Phys. Chem. C, 2018, 122, 11701–11708.
53. D. Mims, A. Schmiedel, M. Holzapfel, N. N. Lukzen, C. Lambert and U. E. Steiner, J. Chem. Phys., 2019, 151, 244308.
54. S. Riese, L. Mungenast, A. Schmiedel, M. Holzapfel, N. N. Lukzen, U. E. Steiner and C. Lambert, Mol. Phys., 2019, 117, 2632–2644.
55. C. Lambert, C. Roger, A. Schmiedel, M. Holzapfel, N. Lukzen and U. E. Steiner, Phys. Chem. Chem. Phys., 2024, 26, 24983–24994.
56. C. Roger, A. Schmiedel, M. Holzapfel, N. N. Lukzen, U. E. Steiner and C. Lambert, Phys. Chem. Chem. Phys., 2024, 26, 4954–4967.
57. P. Mentzel, L. Gerhards, D. Koppenhöfer, A. Schmiedel, M. Holzapfel, N. Lukzen Nikita, A. Solov'yov Ilia, U. E. Steiner and C. Lambert, J. Am. Chem. Soc., 2025, 147, 23068–23078.
58. R. Haberkorn, Mol. Phys., 1976, 32, 1491–1493.
59. N. N. Lukzen, J. H. Klein, C. Lambert and U. E. Steiner, Z. Physik. Chem. N. F., 2017, 231, 197–223.
60. T. P. Fay, L. P. Lindoy and D. E. Manolopoulos, J. Chemi. Phys., 2018, 149, 064107.
61. J. A. Jones and P. J. Hore, Chem. Phys. Lett., 2010, 488, 90–93.
62. T. P. Fay and D. E. Manolopoulos, J. Chem. Phys., 2019, 150, 151102.

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