Abstract
It is unclear whether there is an intermediate dark state between the S_{2} and S_{1} states of carotenoids. Previous twodimensional electronic spectroscopy measurements support its existence and its involvement in the energy transfer from carotenoids to chlorophylls, but there is still considerable debate on the origin of this dark state and how it regulates the energy transfer process. Here we use ab initio calculations on excitedstate dynamics and simulated twodimensional electronic spectrum of carotenoids from purple bacteria to provide evidence supporting that the dark state may be assigned to a new A_{g} ^{+} state. Our calculations also indicate that groups on the conjugation backbone of carotenoids may substantially affect the excitedstate levels and the energy transfer process. These results contribute to a better understanding of carotenoid excited states.
Introduction
Carotenoids (Cars) take part in various processes in living organisms. In living animals and humans, they act as antioxidants preventing free radicals from destroying tissue cells^{1} and are relevant to the vision of retina^{2}. In the photosynthesis of plants and microorganisms, Cars are responsible for harvesting light, transferring energy to chlorophylls (Chls), and protecting against excessive light by quenching excited states of Chls^{3, 4}. Exploring the excitedstate dynamic properties of Cars is fundamental to understand the mechanism of photosynthesis and is also helpful for developing artificial lightharvesting systems. Despite the studies for several decades, our knowledge on the electronic structure and excitedstate properties of Cars, which determine the mechanism of energy decay in Cars and energy flow from Cars to Chls, is still limited. For example, the origin and nature of a dark state, which lies between the strongly onephoton allowed S_{2} (1B_{u} ^{+}) state and the forbidden S_{1} (2A_{g} ^{−}) state, has triggered intense research and has been under debate. This dark state would have a critical role in mediating the CartoChl energy transfer process and the depopulation of the S_{2} state^{4,5,6,7,8,9,10,11}. Discovery of this dark state would make the conventional carotenoid photophysics model S_{0}→S_{2}→S_{1}→S_{0} (Fig. 1a), where S_{0} is the ground state 1A_{g} ^{−}, no longer accurate. Therefore, a new model may be required to account for the excitedstate behavior of Cars^{4, 12, 13}.
A number of experimental and theoretical studies^{4, 8, 14, 15} have been carried out in the past 20 years to unravel this dark state. The 1B_{u} ^{−} state is the popular choice for the attribution of the dark state (Fig. 1b), and it also seems to be the only choice based on the present theory on the excitedstate structure of polyenes, which are closely related to the Cars. However, this assignment is controversial due to the contradiction between the properties of the dark state measured experimentally and the behavior of the 1B_{u} ^{−} state determined from both experiments and theory^{4, 9}. Especially, the conjugationlength dependence of the 1B_{u} ^{−} state is not in accord with that of the dark state as observed in the emission spectra and transient absorption spectra^{16,17,18,19,20}. Recent twodimensional electronic spectroscopy (2DES) and hightime resolution broadband pumpprobe spectroscopy measurements on Cars spheroidene (N = 10), rhodopin glucoside (RG) (N = 11) and spirilloxanthin (N = 13) demonstrate clearly the decay of the S_{2} state to the dark state and the energy transfer channel from the dark state to the Q_{ x } state of Chls^{6, 10, 21, 22}. From these experiments, emission energy of the dark state seems to lie a little bit (~0.1 eV) below that of the S_{2} state, irrespective of the conjugation length of Cars. Decrease of the 1B_{u} ^{−} energy with conjugation length is much steeper than this dark state. In addition, emission energy of the 1B_{u} ^{−} state might be smaller than 2 eV for Cars with N = 10–13 as predicted by Koyama et al.^{4, 18}, according to which energy transfer to the Chls Q_{ x } state, whose absorption energy is ~2.1 eV, cannot realize. Thus, there may exist some other, unknown excited state in the vicinity of the S_{2} state.
Another important issue is how much the dark state is involved in the CartoChl energy transfer. There is disagreement in the overall CartoChl energy transfer efficiency between theoretical prediction and experimental measurement, e.g., 20 vs. 50–60% for Rhodopseudomonas (Rps.) acidophila ^{23, 24}. As the spectral overlap of the dark state emission and Q_{ x } absorption is larger than that between the S_{2} and Q_{ x } states, the dark state, which is not considered in previous theoretical calculations, has been supposed to account for this disagreement^{6}. However, some experiments demonstrate that the amount of energy transferred from dark states to Chls is minor^{4, 25, 26}.
Here, we examine the excitedstate dynamics of two Cars from purple bacteria and their CartoChl energy transfer by virtue of manybody Green’s function theory and Förster–Dexter theory. 2D spectrum is also simulated by a density matrixbased dynamical method to compare with experiments. We provide evidence supporting a new dark state with A_{g} ^{+} symmetry, which is denoted by S_{ y }, in Cars. Its absorption energy would be higher than the S_{2} state, whereas its emission energy would be lower. The S_{ y } state is a singly excited state, in stark contrast to the wellknown dark states 2A_{g} ^{−} and 1B_{u} ^{−} which are doubly excited states constituted by the two triplet excitons. The excitedstate structure and the energy transfer appear to be highly sensitive to the methyl groups on the conjugated backbone of Cars.
Results
Excitation energy of S_{ y }
Extensive computational researches for the dark state in Cars have been performed using various firstprinciple methods^{8, 14, 27,28,29,30,31,32,33}. However, they mainly focus on the controversial 1B_{u} ^{−} state. With manybody Green’s function theory, we study the excitedstate dynamics of RG (N = 11) in Rps. acidophila (Fig. 2) and spirilloxanthin (N = 13) in Rhodospirillum rubrum. The calculated absorption energy of the S_{2} state for RG and spirilloxanthin are 2.65 and 2.42 eV. In experiments, the S_{2} energy is 2.48 eV for RG in methanol and 2.36 eV for spirilloxanthin in nhexane^{4}. In solution and the protein environment, absorption spectra of Cars redshift owing to the polarizability of the medium^{4, 22, 34, 35}. If taking this into account, our calculations agree well with experiments. We also prove that putting some protein fragments near Cars has limited effects on the excitation energies of Cars (Supplementary Fig. 2). Excitation energies of the S_{1} and 1B_{u} ^{−} states for RG (spirilloxanthin) are calculated to be 1.87 and 2.61 eV (1.68 and 2.32 eV), respectively, employing the scheme proposed by Tavan and Schulten^{36}. Above S_{2} by 0.55 eV for RG and 0.52 eV for spirilloxanthin, our results predict a new state (S_{ y }) of the A_{g} ^{+} symmetry. It is optically forbidden from the ground state and must be indiscernible in experimental optical absorption spectra. The S_{ y } state is a singly excited state, with the wave function represented dominantly by the transitions HOMO−1 → LUMO and HOMO → LUMO+1 (Figs. 3, 4; Supplementary Fig. 3 and Supplementary Note 2). The highlevel quantum chemistry approach EOMCCSD can also get this state and a similar S_{2}–S_{ y } energy gap (Supplementary Table 2). The transition dipole moment of the S_{ y } state is 0.7 and 1.6 Debye for RG and spirilloxanthin, respectively, much smaller than that of the S_{2} state (20.3 and 28.6 Debye) and comparable to the dark state measured in experiments^{18}. Moreover, the S_{ y } state remains optically forbidden when twisting the conjugated backbone of Cars from alltrans to cis configurations (Supplementary Fig. 4 and Supplementary Table 3).
Emission energy of S_{ y }
After optical absorption, Cars first relax in the S_{2} state. From the groundstate geometry (R_{0} in Fig. 3) to the excitedstate minimum geometry of the S_{2} state (R_{2} in Fig. 3), Stokes shift of the S_{2} state is 0.18 eV for both RG and spirilloxanthin. This is in accord with the shift of 0.15 eV for RG measured experimentally^{4, 23}. In this process, the S_{ y } state downshifts by 0.66 eV for RG and 0.68 eV for spirilloxanthin, respectively (Fig. 3). Now, at the potential minimum of the S_{2} state, the energy of the S_{ y } state is just a little bit higher (0.07 eV for RG and 0.02 eV for spirilloxanthin) than the S_{2} state. If extrapolating further along the R_{0}→R_{2} reaction coordinate, the energy of the S_{ y } state falls by an additional 0.1 eV (Fig. 3). The emission energy of the S_{ y } state (at R_{ y } in Fig. 3) is thus lower than that of the S_{2} state (see also Supplementary Figs. 8 and 9; Supplementary Note 4). Although the crossing point between the S_{2} and S_{ y } states is not in the R_{0}→R_{2} region, nonadiabatic transition from S_{2} to S_{ y } can happen with the aid of vibration. The S_{ y } state may also have a role in tuning the relaxation to lower states like S_{1} (Fig. 1c). The emission energy of the S_{ y } state for RG and spirilloxanthin is above the absorption energy of the chlorophyll Q_{ x } state, making the CartoChl energy transfer via the S_{ y } state realizable.
From the R_{0} to R_{2} geometries, the bond length alternation, i.e., the difference between the average lengths of C–C and C=C bonds, is reduced. As the S_{2} state is an ionic excited state, whereas the S_{ y } state can be considered to be a covalentlike one due to its weak transition dipole moment, the electronhole binding energy (E _{b}) in the S_{2} state should be much more influenced by structural variation than that in the S_{ y } state. We do find that from R_{0} to R_{2}, E _{b} in the S_{ y } state remains constant while that in the S_{2} state reduces by ∼0.5 eV. From R_{0} to R_{2}, the gap between the unoccupied and occupied molecular orbitals (E _{gap}) narrows by ∼0.7 eV. The excitation energy, which equals E _{gap}−E _{b} in physics, thus decreases faster for the S_{ y } state than the S_{2} state from R_{0} to R_{2} (Fig. 5c). Energies of the S_{1} and 1B_{u} ^{−} states are also found to exhibit much higher sensitivity on the bond length alternation than the S_{2} state^{8, 36}.
Dependence of the S_{ y } emission energy on conjugation length
Dependence of the emission energy of the dark state on the conjugation length of Cars is a crucial factor to determine the attribution of the dark state^{4}. Polyenes are ideal models to investigate this issue. We examine a series of polyenes H_{3}C–(C_{2}H_{2})_{ N }–CH_{3} with N = 6–15 (Fig. 5). Although the absorption energy of the S_{ y } state varies faster with the conjugation length than that of the S_{2} state (Fig. 5a), the S_{2}–S_{ y } energy gap at the potential minimum of the S_{2} state remains at ∼0.03 eV for N ≥ 10 with the S_{ y } below S_{2} (Fig. 5b). Taking into account the additional 0.1 eV downshift of the S_{ y } state to its own potential minimum as discussed above, the emission energy of the S_{ y } state is lower than that of the S_{2} state for N ≥ 9 and the gap between them remains at ∼ 0.1 eV. This agrees well with the transient absorption studies that the dark state lies below the S_{2} state for Cars with N ≥ 9^{11}, and also 2DES measurements that the potential minimum of the dark state is 0.1 eV below that of the S_{2} state for N = 10, 11, and 13^{6, 10, 21}.
2D spectrum of Rps. acidophila
In a recent study, Scholes and coworkers utilized the broadband 2DES to investigate carotenoid dark states in purple bacteria^{6}. The 2DES is a fourwave mixing technique that is extremely sensitive to electronic coherence and energy relaxation dynamics^{37, 38}. To verify our model, we calculate theoretical 2D spectrum of Rps. acidophila based on a model Hamiltonian that is consistent with our ab initio calculations. Note that the original 2D experiment is carried out using a broadband pulse that overlaps with the very red edge of the S_{2} band and the blue edge of the Q_{ x } band in a sample of Rps. acidophila. The experimentally observed 2D S_{2} peak at ~535 nm (2.34 eV) is dependent of the excitation laser spectrum, and is also in good agreement with the calculated S_{2} transition energy at the S_{2} minimum. This excitation energy corresponds to a highly displaced geometry along the bond lengthalternation coordinate. In this geometry, the fourstate model with the S_{ y } state energy lower than that of the S_{2} state applies, therefore, we place the S_{ y } energy minimum at 0.1 eV below the S_{2} energy minimum in our 2D simulation, in accordance with our calculations for RG. Figure 6 shows the simulated 2D spectrum at a delay time of 200 fs, and the simulated spectrum is in agreement with the experimental spectrum^{6}. Specifically, the diagonal peaks, S_{2}/S_{ y } crosspeak, and pronounced S_{2}/S_{1} excitedstate absorption peak that splits into two by the lowerdiagonal S_{2}/S_{ y } crosspeak are all correctly reproduced in our model simulations. Therefore, our theoretical calculations are consistent with the 2D experiments.
Influence of groups on the excitation energies of Cars
Comparing RG with the N = 11 polyene and spirilloxanthin with the N = 13 polyene, the S_{ y }−S_{2} energy gap in the Cars is the same as that in the polyene. However, the S_{2}−1B_{u} ^{−} and S_{2}−S_{1} energy gaps in the polyene are about 0.3 eV wider than those in the Cars of the same conjugation length. The structural difference between Cars and polyenes is reflected in two respects: (i) the conjugated backbone is symmetrical in polyenes but distorted in Cars, (ii) there are groups, e.g., methyl groups, on the conjugated backbone in Cars but not in polyenes. Through changing the shape of the polyene and substituting some hydrogen atoms on it by groups (Supplementary Fig. 5), we find that groups affect substantially the energy gap between the S_{2} state and the doubly excited states S_{1} and 1B_{u} ^{−}, whereas the gap between the S_{2} and S_{ y } states is independent of any modification to the conjugated backbone (Supplementary Table 4). This implies that energies of the S_{1} and 1B_{u} ^{−} states, and thus the competition between the S_{2}→S_{1} decay and the CartoChl energy transfer might be tuned by modifying the groups attached to the conjugated backbone.
Energy transfer from Cars to Chls
Contribution of the dark state to the CartoChl energy transfer is still an open question. It is recently proposed that the dark statemediated energy transfer rate is of the same magnitude as the S_{2}mediated one^{6, 21}. This is in contradiction with previous work^{4, 25, 26}. The nonadiabatic transition from the S_{2} to S_{ y } states is always present for longer Cars according to discussions in the previous sections. The S_{ y } state must be involved in the CartoChl energy transfer. We thus further investigate the energy transfer capability of the S_{ y } state, which is important for understanding the energy transfer mechanism in photosynthesis and resolving the controversies in this respect.
We calculate the energy transfer in the RGB850 and RGB800 pairs in Rps. acidophila as linked by dashed lines in Fig. 2b. Energy flow in these two kinds of pairs has been supposed to dominate the CartoChl energy transfer^{24}. Figure 7a shows the calculated absorption spectrum of the RGB800 pairs, which is comparable to the experimental spectrum of LH2 complex in Rps. acidophila strain 10050^{3, 6}. Positions of the Q_{ x } and Q_{ y } peaks deviate from the experimental ones by <0.1 eV. A chargetransfer state, with the electron excited from the Car to the Chl, appears between the S_{2} and Q_{ x } states, which can result in the formation of Car radicals as detected in experiments^{4}. The rate of energy transfer is evaluated via k = (2π/ℏ)V _{DA}^{2} J _{DA}, where V _{DA} and J _{DA} are the electronic coupling strength and the spectral overlap between donor (D) and acceptor (A) transitions, respectively. V _{DA} is calculated within the framework of manybody Green’s function theory.
We use the spectral overlap data from Krueger et al.^{24} and Ostroumov et al.^{21}, where J _{DA}(S_{2}−Q_{ x }) = 13J _{DA}(S_{2}−Q_{ y }) and J _{DA}(S_{2}−Q_{ x }) = J _{DA}(S_{ y }−Q_{ x }). Figure 7b compares the electronic coupling strength for each energy transfer passway from Car to Chl. We find that the energy transfer rate of the S_{2}B800 Q_{ y } channel is much higher than expected theoretically before^{24} and can reach 70% of that of the S_{2}B850 Q_{ x } channel which has been considered to be the dominating channel. This branching ratio is in good agreement with the experiment^{39}, supporting the experimental observations that both Q_{ x } and Q_{ y } have the role of energy acceptors and the amount of energy transferred to B800 is comparable to that to B850^{4, 39,40,41,42}. The S_{ y }mediated energy transfer is predominated by the S_{ y }B850 Q_{ x } channel. The overall S_{ y }mediated energy transfer rate is one order of magnitude smaller than that of the S_{2}mediated one. Thus, although the dark state participates in the energy transfer process, the CartoChl energy transfer is still governed by the S_{2} state (see Fig. 2c for the energy flow).
Discussion
Ever since the discovery of the intermediate dark state between the S_{2} and S_{1} states in Cars by Cerullo et al. in 2002, its role in photosynthesis becomes increasingly emphasized^{5}. On the basis of the conventional theoretical model for the electronic structures of polyenes developed by Tavan and Schulten^{36}, the 1B_{u} ^{−} state has long been regarded as the candidate for the dark state. The strongly dependence of the S_{2}–1B_{u} ^{−} energy gap on the conjugation length make this assignment questionable^{4, 9}. The S_{ y } state of the A_{g} ^{+} symmetry has a more moderate conjugation length dependence than the 1B_{u} ^{−} state. More importantly, variation of the S_{ y } emission energy with respect to the conjugation length is parallel to the S_{2} state, and their energy gap is kept at a small value (~ 0.1 eV) for Cars with N > 9. This is consistent with the experimental findings, such as those in 2DES, that the emission energy gap between the dark state and the S_{2} state is about 300 cm^{−1} and independent of the conjugation length^{4, 6, 10, 21}. This small energy gap ensures the extremely fast (~10 fs) internal conversion from the S_{2} state to the dark state as observed experimentally^{4, 5, 11}. The agreement between our theoretical 2D spectrum (Fig. 6) and the experimental data lends support to our model with a new A_{g} ^{+} state below S_{2} at a large bondlength alternation. Interestingly, the experimental 2D study resolves the diagonal peak due to the dark state. Furthermore, based on ultrafasttransient absorption and transientgrating experiments, some groups have suggested that the darkstate is due to a doubleminima structure on the S_{2} potential energy surface, not a distinct electronic state^{43, 44}. This doubleminima model hypothesizes that the lowest one of the S_{2} potential surface minima exists at the conformation where the carotenoid is twisted from the alltrans isomer by ~90° with respect to one C=C bond of the conjugated polyene backbone. This may be true for molecules with a short conjugated backbone such as the protonated Schiff bases^{45}, which is also demonstrated in our previous theoretical work^{46}. Nevertheless, this does not seem to be the case for molecules with a long conjugated backbone (Supplementary Fig. 10). The new state described in the present article, with strong energy shift along the bond length alternation coordinate that crossovers in energy with the S_{2} state (Fig. 3), does exhibit a doublewell like feature that may explain the spectral shifts observed in the recent experiments. Noticeably, the theoretical 2D spectrum based on our model correctly describes the S_{2}/S_{ y } crosspeak above the diagonal in the experiment, whereas in the doubleminima model one should expect to see extensive spectral diffusion and an elongated S_{2} peak along the detection wavelength (below the diagonal), which is inconsistent with the 2D experimental data. In addition, the clear diagonal S_{ y } peak is also not explained by the doubleminima model. On the basis of the above analysis, a carotenoid photophysics fourstate model is given in Fig. 1c, which involves the S_{1}, S_{ y }, and S_{2} excited states.
One important motive to study the dark state in experiments is to solve the obvious distinction in the estimated CartoChl energy transfer efficiency between previous theoretical calculations (20%) and experimental measurements (50–60%)^{6, 21, 24}. Above we have illustrated that the portion of energy transferred via the S_{ y } state is minor based on the assumption J _{DA}(S_{2}−Q_{ x }) = J _{DA}(S_{ y }−Q_{ x }) proposed by Ostroumov et al.^{21, 27}. Even if J _{DA}(S_{ y }−Q_{ x }) > J _{DA}(S_{2}−Q_{ x }), considering the smaller energy gap between the S_{ y } and Q_{ x } states than that between the S_{2} and Q_{ x } states, the contribution of the S_{ y } state cannot fill the gap between theoretical calculations and experiments. The electronic coupling strengths we calculate by manybody Green’s function theory are 1.5 times stronger than those from previous theoretical calculations by the transition densitycube approach^{24}. We suggest that the disagreement in energy transfer efficiency between previous theoretical work and experiments may be due not only to the absence of the dark state in the theoretical model, but also to the underestimation of electronic coupling strengths in previous calculations.
In conclusion, our study provides evidence for a new state S_{ y } of the A_{g} ^{+} symmetry in Cars, thus contributing to a better understanding of carotenoid excited states. Future experiments would be required to test the accuracy of our calculations.
Methods
Groundstate geometry
Densityfunctional theory (DFT) with the Coulombattenuating method variant of the Becke 3parameterLeeYangParr (CAMB3LYP) exchangecorrelation functional^{47} is used to optimize geometries of Cars and Chls by the Gaussian 09 program^{48}. CAMB3LYP has been shown to give more reasonable structures than other functionals^{14, 49}.
Excitation energy
Manybody Green’s function theory, which includes the combination of GW method and Bethe–Salpeter equation (BSE)^{50, 51}, is applied to compute the excitation energies with a Gaussian orbital based GWBSE package^{52, 53}. Calculations are performed at the level of full BSE, i.e., considering the mixing between resonant and antiresonant transitions, as Tamm–Dancoff approximation can cause large errors for organic molecules^{54,55,56}. This scheme has been applied for electronic excitations in many organic systems^{54, 57,58,59}. The S_{1} and 1B_{u} ^{−} states bear doubly excited character, involving two coupled triplet excitations. Their excitation energies cannot be obtained from BSE directly as BSE can only deal with one electron–hole pair excitation. Here, we estimate their energies according to Tavan and Schulten’s theory^{36}, i.e., E(S_{1}) = 2E(T_{1}) and \(E\left( {1{\rm{B}}_{\rm{u}}^  } \right) = E\left( {{{\rm{T}}_1}} \right) + E\left( {{{\rm{T}}_2}} \right)\) _{,} where T_{1} and T_{2} are the lowest two triplet states of Cars and are computed via GWBSE. This approach possesses high accuracy as proved by Tavan and Schulten. In the Supplementary Note 1, Supplementary Fig. 1, and Supplementary Table 1, a detailed discussion on the accuracy of our strategy to predict the S_{1} and 1B_{u} ^{−} energies is presented.
Potential minimum at the excited state
Equilibrium structure of Cars in the S_{2} state, which originates from the HOMO → LUMO transition (point R_{2} in Fig. 3) is optimized by the constrained densityfunctional theory (CDFT) where occupations in HOMO and LUMO are fixed at 1 during structural relaxation^{60}. As the S_{ y } state is composed predominantly by transitions HOMO1 → LUMO and HOMO → LUMO+1 with equal weight, its potential minimum cannot be predicted by structural optimization on its own energy surface using CDFT. We locate the potential minimum of the S_{ y } state (point R_{ y } in Fig. 3) approximately by extrapolating along the R_{0} → R_{2} reaction coordinate of the S_{2} state. We validate the reasonability of the S_{ y } potential minimum predicted by this scheme through relaxing Cars in the S_{ y } state with excitedstate forces from BSE. BSE forces are computed by the approach proposed by IsmailBeigi and Louie with the exception that we use finite difference method to evaluate derivatives of singleparticle energies and wave functions with respect to nuclear positions^{61, 62}. This technique to compute BSE forces has been well tested to give consistent results with those from IsmailBeigi and Louie on the excitedstate structures of CO and NH_{3} molecules^{61} (see Supplementary Note 3, Supplementary Table 5, and Supplementary Figs. 6 and 7 for details of the theory and test of the accuracy).
Theoretical 2D spectrum
We utilize a density matrixbased dynamical method to simulate 2D spectrum of Rps. acidophila. This method accounts for full densitymatrix dynamics and bath memory effects in the simulated 2D spectrum, and details of the theory are described in refs ^{63, 64}. The model adopted for the Rps. acidophila system consists of five carotenoid states (S_{0}, S_{1}, S_{ y }, S_{2}, and S_{ n }) and one chlorophyll state (Q_{ x }). The transition energies are set at 16,900, 17,810, 18,620, and 17,500 cm^{−1} for Q_{ x }, S_{ y }, S_{2}, and S_{1} → S_{ n }, respectively. This model places the S_{ y } energy at 0.1 eV below the S_{2} state to describe carotenoid electronic states at a geometry highly displaced along the bond lengthalternation coordinate, in accordance with our calculations for RG. The dynamics are described by a Lindblad master equation including five population relaxation terms: \({\tau _{{{\rm{S}}_2} \to {{\rm{S}}_y}}}\) = 0.3 ps, \({\tau _{{{\rm{S}}_y} \to {{\rm{S}}_1}}}\) = 1 ps, \({\tau _{{{\rm{S}}_y} \to {{\rm{Q}}_x}}}\)=2 ps, \({\tau _{{{\rm{S}}_1} \to {{\rm{S}}_0}}}\)=3 ps, \({\tau _{{{\rm{Q}}_x} \to {{\rm{S}}_0}}}\)=1 ps. The line broadenings are described by Gaussian static disorders with σ = 450 cm^{−1}, and couplings to a superOhmic bath with a spectral density \(J\left( \omega \right) = {\gamma _0}\frac{{{\omega ^3}}}{{\omega _{\rm{c}}^2}} {\rm {e}}^{  \omega /{\omega _{\rm{c}}}}\). The coupling strength γ _{0} and cutoff ω _{c} are set at 0.6 and 850 cm^{−1}, respectively. Note that in this work, we aim to demonstrate that our model produces 2D spectrum that is consistent with experimental results, therefore we do not perform full fittings to the experimental spectra, and the bath and dynamical parameters used in our model are only set tentatively. Additional calculations that provide nonadiabatic couplings and explore the potential surfaces more completely are required to fully describe the experimental 2D data.
Energy transfer
The electronic energy transfer rate, k, is calculated via k = (2π/ℏ)V _{DA}^{2} J _{DA} according to the Förster–Dexter theory. The electronic coupling strength V _{DA} is computed within the framework of manybody Green’s function theory via
where X _{D/A} is the BSE exciton wave function for donor (D) and acceptor (A), ω_{0} the energy transferred, S _{DA} the overlap matrix between donor and acceptor orbitals. K ^{x} and K ^{d} are the exchange and direct terms of the BSE electron–hole interaction kernel. Electronic coupling arising from these two terms resemble the Förster and Dexter coupling, respectively^{65}. V _{DA} computed by Eq. (1) consists excellently with that by the highlevel quantum chemistry approach CASSCF^{66}.
Data availability
All data supporting the findings of this study are available from the corresponding author on request.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grants Nos. 21433006, 21573131, 21173130, and 21603056) and the Natural Science Foundation of Shandong Province (Grant No. JQ201603). Y.C.C. thanks the Ministry of Science and Technology, Taiwan (Grant No. NSC 1052113M002012), National Taiwan University (Grant No. 103R891305), and Center for Quantum Science and Engineering (Subproject: 103R891401) for financial support. Computational resources have been provided by the National Supercomputing Centers in Jinan.
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Y.M. designed the calculations and J.F. carried out most of the calculations. C.W.T. and Y.C.C. performed the calculation of 2D spectra. T.C., X.L., and H.Y. took part in the optimization of configurations. Y.M. and M.R. performed coding on the manybody Green’s function theory and Förster–Dexter theory. Y.M., J.F., and Y.C.C. wrote the manuscript.
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Correspondence to Yuchen Ma.
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Feng, J., Tseng, C., Chen, T. et al. A new energy transfer channel from carotenoids to chlorophylls in purple bacteria. Nat Commun 8, 71 (2017) doi:10.1038/s41467017001207
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