Photoexcited State Dynamics and Singlet Fission in Carotenoids

Abstract

We describe our simulations of the excited state dynamics of the carotenoid neurosporene, following its photoexcitation into the “bright” (nominally 1¹B_u⁺) state. To account for the experimental and theoretical uncertainty in the relative energetic ordering of the nominal 1¹B_u and 2¹A_g^– states at the Franck–Condon point, we consider two parameter sets. In both cases, there is ultrafast internal conversion from the “bright” state to a “dark” singlet triplet-pair state, i.e., to one member of the “2A_g” family of states. For one parameter set, internal conversion from the 1¹B_u to 2¹A_g^– states occurs via the dark, intermediate 1¹B_u state. In this case, there is a cross over of the 1¹B_u⁺ and 1¹B_u diabatic energies within 5 fs and an associated avoided crossing of the S₂ and S₃ adiabatic energies. After the adiabatic evolution of the S₂ state from predominately 1¹B_u⁺ character to predominately 1¹B_u character, there is a slower nonadiabatic transition from S₂ to S₁, accompanied by an increase in the population of the 2¹A_g^– state. For the other parameter set, the 2¹A_g energy lies higher than the 1¹B_u⁺ energy at the Franck–Condon point. In this case, there is cross over of the 2¹A_g and 1¹B_u⁺ energies and an avoided crossing of the S₁ and S₂ energies, as the S₁ state evolves adiabatically from being of 1¹B_u character to 2¹A_g^– character. We make a direct connection from our predictions to experimental observables by calculating the time-resolved excited state absorption. For the case of direct 1¹B_u to 2¹A_g^– internal conversion, we show that the dominant transition at ca. 2 eV, being close to but lower in energy than the T₁ to T₁ transition, can be attributed to the 2¹A_g^– component of S₁. Moreover, we show that it is the charge-transfer exciton component of the 2¹A_g state that is responsible for this transition (to a higher-lying exciton state), and not its triplet-pair component. These simulations are performed using the adaptive tDMRG method on the extended Hubbard model of π-conjugated electrons. The Ehrenfest equations of motion are used to simulate the coupled nuclei dynamics. We next discuss the microscopic mechanism of “bright” to “dark” state internal conversion and emphasize that this occurs via the exciton components of both states. Finally, we describe a mechanism relying on torsional relaxation whereby the strongly bound intrachain triplet-pairs of the “dark” state may undergo interchain exothermic dissociation.

1. Introduction

Carotenoids are a class of linear polyenes of high natural abundance. Carotenoids found in photosynthetic systems serve the dual functions of enhancing their light harvesting properties by absorbing photons in the visible spectrum not accessible to chlorophylls and protecting the light harvesting complexes from excess light.^1,2 The study of carotenoid photophysics is important for understanding their functions in photosynthetic systems.^3,4

The quasi-one-dimensional nature of polyenes enhances electron–electron interactions and electron–nuclear coupling, and gives rise to a complex excited state structure.⁵⁻¹² In 1972, it was observed that in polyenes there exists a symmetry-forbidden 2¹A_g^– “dark” excited state (usually labeled S₁) below the photoexcited 1¹B_u state (usually labeled S₂).^5,6 Then, in 1987, it was shown that there exist other dark states within the S₂–S₁ gap.⁸ Upon photoexcitation to the S₂ state, these dark excited states are involved in ultrafast internal conversion processes, giving rise to the exotic photophysical properties of polyenes, including singlet fission.

Singlet fission is a photophysical process by which a singlet photoexcited state dissociates into two spin uncorrelated triplets. In carotenoids, the first step of singlet fission is understood to be the internal conversion from the photoexcited 1¹B_u⁺ state to a correlated singlet triplet-pair state. The mechanism of this internal conversion process has been heavily debated.¹³ Spectroscopic studies of carotenoid excited states reveal that although the S₂–S₁ gap increases with conjugation length, the S₂ lifetime behaves nonmonotonically: initially increasing and then decreasing with conjugation length, in an apparent violation of the energy gap law. This implies for longer carotenoids, as for polyenes, that an intermediate dark state exists which might be involved in the internal conversion process.^14,15

Recent theoretical work using diabatic models continues to provide evidence for the importance of the low-lying dark excited states of polyenes to their photophysics, especially in the singlet fission process.¹⁶ However, ab initio calculations of polyene excited states argue that nuclear reorganization following photoexcitation can facilitate the internal conversion process, without needing to invoke intermediate dark states.^17,18

In a theoretical and computational study using the density matrix renormalization group (DMRG) method to solve the Pariser–Parr–Pople–Peierls (PPPP) model of π-conjugated electrons, Valentine et al.¹⁹ showed that the dark excited states, 2¹A_g^–, 1¹B_u, 3¹A_g^–, etc., belong to the same family of fundamental excitation with different center-of-mass kinetic energies. The triplet-pair nature of this 2A_g family (or band) of states was established by calculating the spin–spin correlation, bond dimerization, and triplet-pair overlaps.

In a recent paper we described our dynamical simulations of singlet triplet-pair production in photoexcited zeaxanthin using the adaptive time-dependent DMRG (tDMRG) method and Ehrenfest dynamics.²⁰ We chose a parameter regime where at the Franck–Condon point the diabatic energies satisfy E(2¹A_g^–) < E(1¹B_u) < E(¹B_u^–), while the adiabatic singlet states are S₁ ≈ 2¹A_g, S₂ ≈ 1¹B_u⁺, and S₃ ≈ 1¹B_u. Within 5 fs of photoexcitation to S₂, there is a diabatic crossover of the 1¹B_u⁺ and 1¹B_u energies but an avoided crossing of the S₂ and S₃ energies, such that S₂ evolves quasi-adiabatically from the 1¹B_u⁺ state to the 1¹B_u state. Since zeaxanthin possesses C₂ symmetry, there is no further interstate conversion from the 1¹B_u^– to the 2¹A_g.

In this paper, we extend that work to investigate internal conversion in neurosporene, a molecule of 18 conjugated carbon atoms that does not possess C₂ symmetry, thus permitting 1¹B_u⁺ to 2¹A_g state conversion. We consider two parameter sets, (a) E(2¹A_g^–) < E(1¹B_u) < E(¹B_u^–) at the Franck–Condon point, where internal conversion from the 1¹B_u to 2¹A_g^– states occurs via the intermediate 1¹B_u state and (b) E(1¹B_u⁺) < E(2¹A_g) < E(¹B_u^–) at the Franck–Condon point, where there is direct internal conversion from the 1¹B_u to 2¹A_g^– states. In both cases, we show that after 50 fs the yield of the singlet triplet-pair states is ca. 65%.

As well as describing the dynamical simulations of internal conversion, this paper has three further goals. First, we discuss our calculated transient absorption spectra from the evolving state, and we use our results to interpret experimental observations. In particular, we argue that a dominant absorption feature at ca. 2 eV, close to but lower in energy than a triplet state absorption, originates from the charge-transfer exciton component of the “dark” 2¹A_g^– state. Second, using the theory of the “dark” state of ref (21), where it was shown that this state contains both singlet triplet-pair and charge-transfer exciton character, we describe the microscopic mechanism of “bright” to “dark” state internal conversion in carotenoids. Finally, we examine the question of whether the bound intrachain triplet-pairs can undergo exothermic interchain dissociation. We show that this is possible if interchain transfer is accompanied by torsional relaxation, implying that the carotenoid dimers should have a twisted ground state geometry.

A companion paper²² describes our computational DMRG methodology for simulating the excited state dynamics of strongly correlated electron systems. It also analyses in greater detail the physics of the diabatic crossover and the adiabatic avoided crossing.

The contents of this paper are the following. After briefly introducing our model in section 2, section 3 describes our dynamical simulations of photoexcited state interconversion, as well as our calculation and interpretation of the transient excited state absorption. Section 3 concludes with a microscopic explanation of the ultrafast “bright” to “dark” interstate conversion observed in carotenoids. Section 4 shows that interchain singlet dissociation into triplets can be exothermic if accompanied by torsional relaxation. We conclude in section 5.

2. Computational Methods

The π-electron system is described by the extended Hubbard (or UV) model, defined by

1

where n labels the nth C atom, N is the number of conjugated C atoms, and N/2 is the number of double bonds.

is the bond order operator, ĉ_n,σ^† (ĉ_n,σ) creates (destroys) an electron with spin σ in the p_z orbital of the nth C atom, and N̂_n is the number operator. U and V correspond to Coulomb parameters, which describe interactions of two electrons in the same orbital and nearest neighbors, respectively, and β = 2.4 eV represents the electron hopping integral between neighboring C atoms.

The electron–nuclear coupling is described by

2

where α = 4.593 eV Å^–1 is the electron–nuclear coupling parameter and u_n is the displacement of the nth C atom from its undimerized geometry. The nuclear degrees of freedom are described by the classical Hamiltonian

3

where K = 46 eV Å^–2 is the nuclear spring constant.

To project out the high spin eigenstates of the Hamiltonian, we complement the Hamiltonian with

4

where Ŝ is the total spin operator and λ > 0.

The UV-Peierls (UVP) Hamiltonian, defined by

5

is invariant under a particle–hole transformation. For idealized carotenoid structures with 2-fold rotation symmetry, its eigenstates will have definite C₂ and particle–hole symmetries.¹² Therefore, its eigenstates are labeled either A_g^± or B_u. We define the eigenstates of (Ĥ_UVP + Ĥ_λ) as diabatic states. The ordering of these states is highly sensitive to the U and V parameters. It was shown in ref (20) that for the UVP model with U = 7.25 eV and V = 2.75 eV, the 1¹B_u^– relaxed energy is lower than the 1¹B_u relaxed energy for chain lengths N ≥ 10, while the 1¹B_u^– vertical energy is higher than the 1¹B_u vertical energy for N ≤ 22. Both the vertical and relaxed 2¹A_g^– energies are below the 1¹B_u energies for all relevant chain lengths. This implies that for these model parameters internal conversion from the 1¹B_u⁺ state to the 2¹A_g state is expected to proceed via the 1¹B_u^– state for 10 ≤ N ≤ 22.

Recent theoretical studies using highly accurate ab initio methods, however, suggest that internal conversion in carotenoids from the 1¹B_u⁺ state to the 2¹A_g state can proceed directly, without involving intermediate dark states.^17,18 In order to investigate this particular mechanism, we choose a second set of U and V parameters, namely U = 7.25 eV and V = 3.25 eV. The vertical and relaxed excitation energies for this parametrization are illustrated in Figure S1 of the Supporting Information. For this parameter regime, we find that for all relevant chain lengths the 2¹A_g^– relaxed energy is lower than the 1¹B_u relaxed energy, and that the 2¹A_g^– vertical energy is higher than the 1¹B_u vertical energy. Therefore, direct internal conversion from the 1¹B_u⁺ state to the 2¹A_g state is energetically possible for this parameter set.

Since Ĥ_UVP is invariant to particle–hole exchange, a symmetry breaking term Ĥ_ϵ is introduced into our model to facilitate internal conversion from the 1¹B_u⁺ state, which has positive particle–hole symmetry, to the triplet-pair states, which have negative particle–hole symmetry. This term is given by

6

where ϵ_n is the potential energy on the nth C atom.

We denote the ith singlet eigenstate of the full Hamiltonian Ĥ = (Ĥ_UVP + Ĥ_λ + Ĥ_ϵ) as S_i (where S₀ is the ground state). We define these to be adiabatic states. We note that for V = 2.75 eV (as in ref (20)), S₂ is taken to be the initial state, Ψ(t = 0), at time t = 0, as it has the largest 1¹B_u⁺ character. Similarly, for V = 3.25 eV, Ψ(t = 0) = S₁, as it has the largest 1¹B_u character.

As described in ref (22), Ĥ_ϵ is optimized under the constraint ϵ_n < ϵ_max such that the ground state π-electron density on the nth C atom reproduces the Mulliken charge densities of the π-system found via ab initio density functional theory (DFT) calculations. The optimized Ĥ_ϵ is given in Table S1 in the Supporting Information. The cutoff ϵ_max = 1.0 is chosen such that Ψ(t = 0) retains sufficient 1¹B_u⁺ character, while accurately reproducing the DFT densities.

The electronic states and the ground state equilibrium geometry are determined via the static DMRG method, while the time evolution of the initially prepared photoexcited singlet is determined via adaptive tDMRG.^23,24 The nuclear degrees of freedom are treated classically via the Ehrenfest equations of motion. The theoretical and computational techniques employed here are described in full detail in the accompanying methodology paper.²²

3. “Bright” to “Dark” State Internal Conversion

We begin our discussion of “bright” to “dark” state internal conversion by describing our dynamical simulations in section 3.1. These simulations predict that interstate conversion does happen within 10 fs. In section 3.2 we then explain how interstate conversion occurs and why it is so fast.

3.1. Computational Results

All of our calculations are performed on the carotenoid neurosporene, whose chemical formula is illustrated in Figure 1. As neurosporene does not possess C₂ symmetry, transitions from B_u to A_g states are not symmetry forbidden. This gives rise to a complex dynamical relaxation process involving the 1¹B_u⁺, 1¹B_u, and 2¹A_g^– diabatic states. This is in contrast to our previous work on zeathanxin,²⁰ which does possess C₂ symmetry, and therefore only exhibits 1¹B_u to 1¹B_u^– state interconversion.

Figure 1.

Structural formula of neurosporene, illustrating the 18 C atom (9 double-bond) π-conjugated system.

3.1.1. Internal Conversion from the 1¹B_u⁺ to 2¹A_g States via the 1¹B_u^– State

For the parameter set V = 2.75 eV the initial photoexcited system Ψ(t) is prepared in the adiabatic state S₂, as this has the largest overlap with the diabatic 1¹B_u⁺ state at the Franck–Condon point. The nuclei begin in the ground state geometry and experience resultant forces exerted by the electrons, which cause Ĥ_e–n to change and initiate the evolution of the electronic and nuclear degrees of freedom. Figure 2 shows the calculated adiabatic and diabatic excited energies as a function of time. A crossover of the diabatic 1¹B_u and 1¹B_u⁺ energies occurs at ∼5 fs, while the adiabatic S₂ and S₃ energies exhibit an avoided crossing, as the coupling between the diabatic states is nonzero, i.e., ⟨1¹B_u|Ĥ_ϵ|1¹B_u^–⟩ ≠ 0.

Figure 2.

Excitation energies as a function of time of the diabatic 2¹A_g^–, 1¹B_u, and 1¹B_u^– states and the adiabatic S₁, S₂, and S₃ states. These energies are found for neurosporene with V = 2.75 eV. The 1¹B_u and 1¹B_u^– energies exhibit a crossover at ∼5 fs, while the S₂ and S₃ energies (i.e., E₂ and E₃, respectively) exhibit an avoided crossing.

The triplet-pair nature of the system at time t is determined by calculating the probabilities that the system described by Ψ(t) occupies the triplet-pair states, 1¹B_u^– and 2¹A_g. Figure 3 shows the probabilities that Ψ(t) occupies the diabatic states 2¹A_g^–, 1¹B_u, and 1¹B_u^–, and the adiabatic states, S₁, S₂ and S₃. At the avoided crossing Ψ(t) exhibits an adiabatic transition from the 1¹B_u state to the 1¹B_u^– state. Within 10 fs, Ψ(t) predominantly occupies the 1¹B_u diabatic state.

Figure 3.

Probabilities as a function of time that the system described by Ψ(t) occupies the adiabatic states, S₁, S₂, and S₃, and the diabatic states, 2¹A_g^–, 1¹B_u, and 1¹B_u^–. The results are for neurosporene with V = 2.75 eV. Note that Ψ(t = 0) = S₂.

After the ultrafast adiabatic transition to the 1¹B_u^– state, the system continues to undergo a slow nonadiabatic transition to the 2¹A_g state. This is a consequence of the nonzero coupling between the 1¹B_u^– and 2¹A_g states, i.e., ⟨1¹B_u^–|Ĥ_ϵ|2¹A_g⟩ ≠ 0.

As both 2¹A_g^– and 1¹B_u are triplet-pair states and noting that P(S₃; Ψ(t)) ∼ 0 in the long-time limit, we extend the singlet triplet-pair yield calculation for a two level system^20,22 to calculate the total triplet-pair state probability, P_classical, as

7

After ∼50 fs this yield is ∼70%. (The probabilities, P(S_i, ϕ_j), that the adiabatic state, S_i, occupies the diabatic state, ϕ_j, are shown in Figure S2 of the Supporting Information.)

A scheme illustrating the internal conversion between the excited states is shown in Figure 4 for the case of the intermediate state.

Figure 4.

Schematic diagrams illustrating internal conversion between the excited states of neurosporene (using V = 2.75 eV). (a) Diabatic state representation: 1¹B_u⁺ (blue), 1¹B_u (red), and 2¹A_g^– (green). The dashed (solid) horizontal lines represent the vertical (relaxed) energies of the states. The approximate rate constants are also indicated. (b) Adiabatic state representation, showing S₂ evolve adiabatically from 1¹B_u to 1¹B_u^–, before the slower nonadiabatic transition from S₂ to S₁. S₂ (blue) and S₁ (green).

3.1.2. Direct Internal Conversion from the 1¹B_u⁺ State to the 2¹A_g State

As described in section 2, for the parameter set where V = 3.25 eV, the primary photoexcited state is S₁, which is predominately 1¹B_u⁺. The singlet adiabatic and diabatic energies as a function of time are illustrated in Figure 5. The 2¹A_g and 1¹B_u⁺ energies crossover within ∼3 fs. In contrast, as a consequence of the diabatic coupling of the 2¹A_g and 1¹B_u⁺ states, the S₁ and S₂ energies display an avoided crossing.

Figure 5.

Excitation energies as a function of time of the diabatic 1¹B_u⁺, 2¹A_g, and 1¹B_u^– states and the adiabatic S₁, S₂, and S₃ states. These energies are found for neurosporene with V = 3.25 eV. The 1¹B_u and 2¹A_g^– energies exhibit a crossover at ∼3 fs, while the S₁ and S₂ energies exhibit an avoided crossing.

The probabilities that Ψ(t) occupies the adiabatic and diabatic states, illustrated in Figure 6, show that the avoided crossing is accompanied by a transition of Ψ(t) from the 1¹B_u⁺ state to the 2¹A_g state, while predominantly remaining in the S₁ state. The oscillations of the diabatic populations can be understood by a quasistationary two-state approximation, as discussed in ref (22).

Figure 6.

Probabilities as a function of time that the system described by Ψ(t) occupies the adiabatic states, S₁ and S₂, and the diabatic states, 2¹A_g^– and 1¹B_u. These results are for neurosporene with V = 3.25 eV. Note that Ψ(t = 0) = S₁. (The dotted curve for the 1¹B_u⁺ occupation is an interpolation of the computed data from 8 to 15 fs, as during this time the 1¹B_u and 1¹B_u^– energies are quasidegenerate, making it difficult to numerically resolve these wave functions.).

Further evidence for the adiabatic transition is provided by the calculated probabilities that the adiabatic states occupy the diabatic states, shown in Figure 7. At t = 0, the photoexcited state, S₁, primarily occupies the exciton state, 1¹B_u⁺, while S₂ primarily occupies the triplet-pair state, 2¹A_g. After the avoided crossing, at which the diabatic states contribute equally to the adiabatic states, the S₁ state predominantly occupies the 2¹A_g^– state, while the S₂ state predominantly occupies the 1¹B_u state. The 2¹A_g^– yield after ∼50 fs is ∼60%.

Figure 7.

Probabilities as a function of time that the adiabatic states, S₁ and S₂, occupy the diabatic states, 2¹A_g^– and 1¹B_u. These results are for neurosporene with V = 3.25 eV. (The dotted curve for the 1¹B_u⁺ occupation is an interpolation of the computed data from 8 to 15 fs, as during this time the 1¹B_u and 1¹B_u^– energies are quasidegenerate, making it difficult to numerically resolve these wave functions.).

A scheme illustrating the internal conversion between the excited states is shown in Figure 8 for the case of no intermediate state.

Figure 8.

Schematic diagrams illustrating internal conversion between the excited states of neurosporene (using V = 3.25 eV). (a) Diabatic state representation: 1¹B_u⁺ (blue) and 2¹A_g (green). The dashed (solid) horizontal lines represent the vertical (relaxed) energies of the states. The approximate rate constant is also indicated. (b) Adiabatic state representation, showing S₁ evolve adiabatically from 1¹B_u⁺ to 2¹A_g.

3.1.3. Transient Excited State Absorption

Transient (i.e., time-resolved) spectroscopy is an important experimental technique used in the study of carotenoid photophysics. Seminal work which utilized transient spectroscopy experiments included the measurement of the S₁ lifetime of carotenoids,²⁵ direct observation of the S₁ dark state,²⁶ and the detection of dark intermediate states between the S₁ and S₂ states.^27,28

The theoretical transient absorption spectrum from state S_i at time t is given by the expression

8

where E_i is the energy of state S_i. In this section, we present our results for transient excited state absorption calculations using the Lanczos-DMRG method.^29,30 The computational methodology is described in ref (22).

We present results for V = 3.25 eV, when there is direct 1¹B_u⁺ to 2¹A_g state conversion. First, we consider an ideal system with both particle–hole and C₂ symmetries (i.e., we set Ĥ_ϵ = 0). Thus, at the Franck–Condon point (i.e., at t = 0 fs) S₁ ≡ 1¹B_u⁺ and S₂ ≡ 2¹A_g. The calculated initial absorption spectra from S₁ and S₂ and the triplet ground state, T₁, are shown in Figure 9. The three peaks arising from the S₁ state at 0.1, 1.3, and 1.4 eV are attributed to transitions from the 1¹B_u⁺ state to the 2¹A_g, 4¹A_g^–, and 5¹A_g states, respectively.

Figure 9.

Calculated transient absorption spectra at t = 0 from the S₁ ≡ 1¹B_u⁺, S₂ ≡ 2¹A_g, and T₁ states. Results are for a system with no broken-symmetry, i.e., Ĥ_ϵ = 0.

The 2¹A_g^– state is comprised of a singlet triplet-pair component and an odd-parity charge-transfer exciton component (see section 3.2 and refs (19 and 21)). If the absorption signal from the 2¹A_g state arose from a transition from its bound triplet-pair component to T₁T₁^*, we would expect this to be higher in energy than the T₁ to T₁ transition.³¹ We note, however, that this absorption peak is significantly lower in energy compared to the lowest absorption from the T₁ state. In contrast, if the absorption signal from the 2¹A_g^– state arose from a transition from its charge-transfer exciton component, the resulting n¹B_u state would be an even-parity exciton. We verify this latter hypothesis by calculating the exciton wave function of the n¹B_u⁺ state; this is shown in Figure 10.a By observing the nodal structure of the n¹B_u exciton wave function, we conclude that the 2¹A_g^– to n¹B_u transition does indeed arise from the charge-transfer exciton component of the 2¹A_g^– state to a Mott–Wannier exciton.¹²

Figure 10.

n¹B_u⁺ exciton wave function calculated using eq 18 of ref (19), using the PPP model with N = 54. This has two nodal surfaces along the relative (i.e., electron–hole) coordinate and is thus dipole-connected to the charge-transfer exciton component of the 2¹A_g state, which has one nodal surface along the relative coordinate. See ref (19) for the full parametrization.

Next, we consider the realistic Hamiltonian with the inclusion of the symmetry breaking term, Ĥ_ϵ. For this set of parameters, the 1¹B_u⁺ and 2¹A_g energies exhibit a crossover, while the S₁ and S₂ energies exhibit an avoided crossing (as shown in Figure 5).

The calculated absorption spectra at the Franck–Condon point from the S₁ and S₂ states and the triplet ground state, T₁, are shown in Figure 11a. At t = 0 fs, S₁ predominantly occupies the 1¹B_u⁺ state, while S₂ predominantly occupies the 2¹A_g state. The absorption at ∼0.3 eV corresponds to the S₁ to S₂ transition, while the absorption maxima observed around 1.4 eV is the transition from the 1¹B_u⁺ state to a high energy ¹A_g state. Due to the inclusion of the symmetry breaking term, the adiabatic state T₁ will have some diabatic triplet excited state character, and therefore two absorption peaks are observed from T₁. As for Figure 9, the transition from the charge-transfer exciton component of the 2¹A_g^– state to the n¹B_u state at ∼1.4 eV is lower in energy than the triplet absorption peaks.

Figure 11.

Calculated transient absorption spectra from the S₁, S₂, and T₁ states of neurosporene. (a) At t = 0, where S₁ ≈ 1¹B_u⁺ and S₂ ≈ 2¹A_g. Note the overlapping S₁ and S₂ absorption at ∼1.4 eV. (b) At t = 20 fs, where S₁ ≈ 2¹A_g^– and S₂ ≈ 1¹B_u. Here, the singlet state absorption spectra are weighted by P(Ψ(t); S_i).

As the evolving photoexcited system, Ψ(t), reaches equilibrium it occupies both the S₁ and S₂ states. Figure 11b shows the weighted absorption spectra at t = 20 fs of these states, as well as the absorption spectra of the T₁ state. After passing through the avoided crossing at ∼3 fs, the S₁ state now predominantly occupies the 2¹A_g^– state, while the S₂ state predominantly occupies the 1¹B_u state. This change of occupations is reflected in the calculated transient spectra, as the major signals originating from the S₁ state are now higher in energy compared to those originating from the S₂ state. At the relaxed geometry, the triplet peaks are blue-shifted to 2.1 and 2.3 eV, which are close to the relaxed excited state values previously calculated for polyenes.¹⁹

As the S₁ state at 20 fs predominantly occupies the 2¹A_g^– state, we attribute the absorption peak of the S₁ state at ∼2.1 eV to a transition from the charge-transfer exciton component of the 2¹A_g state. This energy is blue-shifted from the corresponding transition from the 2¹A_g^– component of the S₂ state at t = 0, because of the large reorganization energy of the 2¹A_g state. This energy is close to the experimentally observed 2¹A_g^– to n¹B_u transition of carotenoids.^13,25,31 We also note that the energy of the major absorption from S₂ around 0.9 eV agrees with the experimentally observed 1¹B_u⁺ to m¹A_g transition.³²

In summary, our transient excited state absorption calculations predict the following. At the Franck–Condon point there will be a photoexcited absorption at ca. 1.4 eV resulting from the 1¹B_u⁺ to n¹A_g transition (and a weak transition at ca. 0.3 eV resulting from the 1¹B_u⁺ to 2¹A_g transition). Within 20 fs, however, the adiabatic evolution of S₁ from 1¹B_u⁺ to 2¹A_g character results in a new transition from the charge-transfer exciton component of the 2¹A_g^– state at ca. 2.0 eV, while a weaker transition at ca. 0.9 eV arises from the residual 1¹B_u component of the evolving wave function.

3.2. How and Why Does “Bright” To “Dark” State Internal Conversion Occur?

As we have seen in the previous two sections, as a consequence of diabatic energy-level crossings, internal conversion from the optically “bright” state to the “dark” state occurs within 10 fs. In this section, we address the questions of how and why this process occurs.

As described in ref (21), the 2¹A_g^– state is a linear combination of a singlet triplet-pair and an odd-parity charge-transfer exciton. Triplet-pair binding occurs because when a pair of triplets occupy neighboring ethylene dimers a one-electron transfer converts them to the odd-parity charge-transfer exciton. This hybridization causes a nearest neighbor triplet–triplet attraction. Similarly, the 1¹B_u state is predominately a Frenkel exciton (i.e., an electron–hole bound on the same dimer). However, the 1¹B_u⁺ state also consists of some even-parity charge-transfer exciton components.b

In practice, because of substituent side groups, carotenoids do not possess definite particle–hole symmetry, and so neither do their electronic states. This means that the 2¹A_g state possesses some charge-transfer exciton components of even-parity, while the 1¹B_u state possesses some charge-transfer exciton components of odd-parity. As illustrated in Figure 12, it is these odd-parity charge-transfer components of the 1¹B_u state which readily interconvert to singlet triplet-pairs, and thus cause the “bright” to “dark” state internal conversion. In spirit, this is the same mechanism proposed to explain singlet triplet-pair production in acene dimers,³³ the difference being that acene molecules replace the ethylene dimers of carotenoid chains.

Figure 12.

Schematic diagram illustrating the internal conversion of the odd-parity charge-transfer component of the 1¹B_u state (a) to the nearest neighbor singlet triplet-pair component of the 2¹A_g state (b). Note that (a) is not a spin-symmetrized state; see ref (21) for an illustration of correctly spin-symmetrized singlet charge-transfer states.

Evidently, the greater the deviation from perfect particle–hole symmetry, the greater the amount of covalent and ionic mixing in the electronic eigenstates. This then implies a larger coupling and a higher internal conversion yield between the bright and dark states. In our model, the deviation from particle–hole symmetry is represented by Ĥ_ϵ (eq 6) and in particular by the value of {ϵ_n}. In our companion paper, ref (22), we quantify this statement by computing the probabilities of adiabatic and diabatic transitions as a function of {ϵ_n}. As ϵ_n → 0 the probability of a 1¹B_u⁺ to 2¹A_g transition vanishes, while the probability of a S₁ to S₂ transition becomes unity. Conversely, for large ϵ_n the probability of a S₁ to S₂ transition vanishes and the probability of a 1¹B_u⁺ to 2¹A_g transition increases.

Next, we explain why internal conversion happens within 10 fs, i.e., we address the equivalent question of why after photoexcitation there is a diabatic energy-level crossing of the Frenkel exciton and singlet triplet-pair state. The answer to this question lies at the heart of what makes the electronic properties of linear polyenes so fascinating. It involves the roles of both electron–electron interactions and electron–nuclear coupling. We approach this question in three ways.

We first consider the electronic states of linear polyenes in the absence of electron–nuclear coupling, i.e., when electronic interactions dominate. In this case polyenes are Mott–Hubbard insulators: there is a charge-correlation gap between the ground state (i.e., 1¹A_g^–) and the lowest-lying ionic state (i.e., 1¹B_u).^12,34 The ground state is a quantum Heisenberg antiferromagnet with, in the limit of infinitely long chains, a gapless spin density wave or triplet excitation (i.e., the 1³B_u^– state). These triplets weakly bind to form gapless singlet triplet-pairs (i.e., the 2¹A_g state). Thus, in this limit there is a large spin-correlation gap between the 1³B_u^– and 1¹B_u states, and the “dark” state lies energetically below the “bright” state.

Next, consider electronic states in the absence of electronic interactions, but when electron–nuclear coupling dominates. In this case polyenes are Peierls insulators, i.e., there is a band gap between the filled valence band and the empty conduction band as a result of the incipient bond order wave causing bond dimerization.¹² Now the 1³B_u^– and 1¹B_u states both have an excitation gap and are degenerate. The 2¹A_g^– state lies higher in energy. As already stated, the ground state is dimerized. The 1³B_u and 1¹B_u⁺ states, however, exhibit solitonic structures and a reversal of the bond dimerization in the middle of the chain from the ground state dimerization. Crucially, the solitons of the triplet (1³B_u state) are associated with spin-1/2 particles, i.e., spin radicals or spinons, while solitons of the singlet (1¹B_u⁺ state) are associated with an electron or hole.¹²

Finally, we consider the intermediate case, relevant for polyenes, when electronic interactions and electron–nuclear coupling are both important. Now, the ground state dimerization is enhanced.^12,35 The 1³B_u^– and 1¹B_u states exhibit a large spin-correlation gap. As a consequence, the 2¹A_g^– state has significant triplet-pair character, causing the 1¹B_u and 2¹A_g^– states to be quasidegenerate in the ground state geometry (i.e., at the Franck–Condon point).²¹ The relaxed geometries of the 1³B_u and 1¹B_u⁺ states are now quite different. The relaxed geometry of the 1³B_u state is similar to the noninteracting limit: there are soliton–antisoliton structures associated with the spin-radicals and a reversal of bond dimerization from the ground state. However, the relaxed 1¹B_u⁺ geometry is quite different from the noninteracting limit: as a consequence of the electron–hole interaction, the soliton and antisoliton attract forming an exciton-polaron whose bond dimerization is only slightly different from the ground state. Thus, the 1³B_u state exhibits a large reorganization energy, while the 1¹B_u⁺ state does not. Similarly, the 2¹A_g state, being composed of a triplet-pair, also exhibits a large reorganization energy.

In summary, the reasons that there is an energy level crossing between the diabatic bright and dark states are the following: First, because of the large spin-correlation gap, the dark state has a large triplet-pair component and thus the bright and dark states are quasidegenerate at the Franck–Condon point. Second, the triplet state (1³B_u^–) has a larger reorganization energy than the 1¹B_u state and consequently, because of its triplet-pair character, the reorganization of the dark state is much larger than for the bright state. Consequently, after photoexcitation to the bright state, nuclear forces cause the level crossing, and hence ultrafast internal conversion to the dark state.

4. Exothermic Intermolecular Singlet Fission

As shown in Figure S1 of the Supporting Information, and refs (19 and 20), the intramolecular triplet-pair binding energy varies from ca. 1 eV in short carotenoids to ca. 0.3 eV in long polyene chains. Thus, intramolecular singlet fission is a strongly endothermic process, consistent with the absence of experimental evidence for free triplets on isolated carotenoids generated via a singlet fission mechanism.^36,37

In this section we address the question of how intermolecular singlet fission can in principle be an exothermic process. Clearly, triplets on single molecules must be energetically stabilized to overcome the intramolecular triplet-pair binding. There are two causes for this. First, there is quantum deconfinement: a single triplet delocalized on a whole molecule has a smaller kinetic (or zero point) energy than a single triplet confined to half a molecule, which is the relevant comparison for two unbound triplets on the same chain. That is, it costs less energy to unbind a pair of triplets on separate molecules than on the same molecule. Second, self-localized triplets on two separate molecules experience a larger (negative) reorganization energy than two bound triplets on the same molecule. Crucially, there are two components to the reorganization energy: a term arising from C–C bond stretches and an additional term arising from torsional relaxation. Therefore, for exothermic intermolecular singlet fission to occur, the molecules must exist in an environment so that they have twisted ground state conformations.

To quantify these statements, we supplement the UVP model (introduced in section 2) by terms that model electron-torsional coupling. The π-electron transfer integral is β(θ) = β₀ cos θ, where θ is the dihedral angle between neighboring C–H groups. Assuming a small planarization in the excited state, i.e., assuming that δθ ≪ θ⁰, we may write

9

where θ⁰ is the ground state equilibrium dihedral angle. Thus, −β₀ sin θ⁰ is the electron-torsional coupling parameter that couples the bond-order operator, T̂, to the variation in dihedral angle, δθ. We also assume that there is a harmonic elastic energy

10

Applying the Hellmann–Feynman theorem implies that the equilibrium dihedral angle for the nth bond is θ_n = θ_n⁰ + δθ_n, where

11

and δu_n = (u_n+1 – u_n) is the change of bond length caused by electronic coupling to C–C bond vibrations.

We define the intermolecular triplet-pair binding energy, ΔE_TT, as twice the energy of triplets on separate molecules relative to the intramolecular triplet-pair state (i.e., 2A_g), namely,

12

where N is the number of conjugated carbon-atoms in each molecule. A negative value of ΔE_TT implies exothermic intermolecular singlet fission.

Our results are presented in Tables 1–3. Table 1 shows the results for a pair of carotenoid chains of 22 conjugated carbon-atoms each, for different ground state twists, θ⁰, and torsional force constants, K_rot. Evidently, in the absence of torsional relaxation (e.g., if θ⁰ = 0 or K_rot → ∞) intermolecular singlet fission is endothermic. For a fixed K_rot singlet fission becomes more exothermic for a more twisted molecule in the ground state. Similarly, for a fixed ground state twist, singlet fission becomes more exothermic as K_rot is reduced.

Table 1. Triplet-Pair Binding Energies, ΔE_TT, in meV Defined in eq 12, for a Pair of Carotenoid Chains, Both of 22 Conjugated Carbon Atoms^a.

	Krot (eV rad–2)
θ0 (deg)	6	8	10	∞
0	+102	+102	+102	+102
5	+75	+88	+93	+102
10	–5	+46	+68	+102
15	–134	–22	+27	+102
20		–111	–27	+102

^aA negative value implies exothermic intermolecular singlet fission.

Table 3. Triplet-Pair Binding Energies in meV for a Pair of Carotenoid Chains of 22 Conjugated Carbon Atoms^a.

intramolecular vertical	intermolecular vertical	bond relaxation	bond and torsional relaxation
781	226	102	–22

^aθ⁰ = 15° and K_rot = 8 eV rad^–2.

Table 2 indicates that for the same values of θ⁰ and K_rot intermolecular singlet fission becomes less favorable as the number of conjugated carbon atoms increases. This is because the energy gained by quantum deconfinement reduces with increasing chain length.

Table 2. Triplet-Pair Binding Energies in meV for a Pair of Carotenoid Chains of N Conjugated Carbon Atoms^a.

N	binding energy (meV)
18	–54
22	–22
26	+2

^aθ⁰ = 15° and K_rot = 8 eV rad^–2.

Finally, Table 3 lists the various contributions that favor exothermic intermolecular singlet fission. Quantum deconfinement onto two molecules reduces the intramolecular binding energy on 22-site chains from 781 to 226 meV; bond relaxation causes a 124 meV decrease in binding energy; while additional torsional relaxation causes another 124 meV decrease, rendering the process exothermic. Evidently, quantum deconfinement causes the largest reduction in binding energy, but all three components are necessary to enable exothermic singlet fission. In particular, it is necessary that the molecules are twisted in their ground states for exothermic singlet fission to occur.

5. Discussion and Conclusions

This paper has described our simulations of the excited state dynamics of the carotenoid neurosporene following its photoexcitation into the “bright” (nominally 1¹B_u⁺) state. We employed the adaptive tDMRG method on the UV model of π-conjugated electrons and used the Ehrenfest equations of motion to simulate the coupled nuclei dynamics.

To account for the experimental and theoretical uncertainty in the relative energetic ordering of the nominal 1¹B_u⁺ and 2¹A_g states at the Franck–Condon point, we considered two sets of parameters. In both cases there is ultrafast internal conversion from the “bright” state to a “dark” singlet triplet-pair state, i.e., to one member of the “2A_g” family of states.

For one parameter set, internal conversion from the 1¹B_u⁺ to 2¹A_g states occurs via the dark intermediate 1¹B_u^– state. In this case there is a cross over of the 1¹B_u and 1¹B_u^– diabatic energies within 5 fs and an associated avoided crossing of the S₂ and S₃ adiabatic energies. Such an intermediate state has been postulated to explain the violation of the S₂–S₁ energy-gap law in carotenoids.^14,15 Following the adiabatic evolution of the S₂ state from predominately 1¹B_u character to predominately 1¹B_u^– character, there is a slower nonadiabatic transition from S₂ to S₁, accompanied by an increase in the population of the 2¹A_g state. This scheme is illustrated in Figure 4.

For the other parameter set, the 2¹A_g^– energy lies higher than the 1¹B_u energy at the Franck–Condon point. In this case there is cross over of the 2¹A_g^– and 1¹B_u energies and an avoided crossing of the S₁ and S₂ energies, as the S₁ state evolves adiabatically from being of 1¹B_u⁺ character to 2¹A_g character. This scheme is illustrated in Figure 8.

We make a direct connection from our predictions to experimental observables by calculating the time-resolved excited state absorption. For the case of direct 1¹B_u⁺ to 2¹A_g internal conversion, we showed that the dominant excited-state transition at ca. 2 eV, being close to but lower in energy than the T₁ to T₁^* transition, can be attributed to the 2¹A_g component of S₁. Moreover, we show that it is the charge-transfer exciton component of the 2¹A_g^– state that is responsible for this transition (to a higher-lying exciton state), and not its triplet-pair component. This transition is blue-shifted from the Franck–Condon point, because of the large reorganization energy of the 2¹A_g state.

We next discussed the microscopic mechanism of “bright” to “dark” state internal conversion, emphasizing that this occurs via the exciton components of both states. It is also a fast and efficient process, because the strongly correlated nature of the dark 2¹A_g^– state means that it has a much larger reorganization energy than the bright 1¹B_u state.

Finally, we described a mechanism whereby the strongly bound intrachain triplet-pairs of the “dark” state may undergo interchain exothermic dissociation. This mechanism relies on the possibility of the unbound interchain triplets being energetically stabilized by quantum deconfinement and larger bond and torsional reorganization energies. We predict that this is only possible if the molecules are twisted in their ground states.

Irrespective of the ordering of the 1¹B_u⁺ and 2¹A_g states at photoexcitation, our simulations indicate that after 50 fs the yield of the “dark” predominately singlet triplet-pair states is ca. 65%. This implies, however, that the evolving state still has some “bright” (i.e., 1¹B_u⁺) component, which explains the weak emissive character of photoexcited carotenoids.

In an earlier paper we explained the origin of the intrachain triplet-pair binding,²¹ while in this paper we argue that exothermic interchain triplet-pair dissociation is possible if it is accompanied by torsional relaxation. Work is now in progress to build a full model of singlet triplet-pair dissociation and spin decoherence to understand the full kinetic process of singlet fission in carotenoid dimers. Future work will also investigate our model with fully quantized phonons. This will enable us to calculate the vibronic line shape of the photoinduced absorption spectra, in particular allowing for a comparison of the S₁ and T₁ transient absorption.³¹ We will also investigate the validity of the Ehrenfest approximation for the nonadiabatic S₂ to S₁ transition, discussed in section 3.1.1.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.2c07781.

• Parametrization of the UV-Peierls Hamiltonian, parametrization of the symmetry breaking Hamiltonian, Ĥ_ϵ, and probabilities that the adiabatic states, S₁, S₂, and S₃ occupy the diabatic states 2¹A_g^–, 1¹B_u, and 1¹B_u^– for case a (PDF)

The authors declare no competing financial interest.