Excited-State Dynamics of Overlapped Optically-Allowed 1Bu+ and Optically-Forbidden 1Bu− or 3Ag− Vibronic Levels of Carotenoids: Possible Roles in the Light-Harvesting Function

The unique excited-state properties of the overlapped (diabatic) optically-allowed 1Bu+ and the optically-forbidden 1Bu− or 3Ag− vibronic levels close to conical intersection (‘the diabatic pair’) are summarized: Pump-probe spectroscopy after selective excitation with ∼100 fs pulses of all-trans carotenoids (Cars) in nonpolar solvent identified a symmetry selection rule in the diabatic electronic mixing and diabatic internal conversion, i.e., ‘1Bu+-to-1Bu− is allowed but 1Bu+-to-3Ag− is forbidden’. On the other hand, pump-probe spectroscopy after coherent excitation with ∼30 fs of all-trans Cars in THF generated stimulated emission with quantum beat, consisting of the long-lived coherent diabatic cross term and a pair of short-lived incoherent terms.


Low-Lying Singlet-Excited States and Their Light-Harvesting Function
Energetics. In bacterial photosynthetic systems having carotenoids (Cars) with n  913 conjugated double bonds in the all-trans configuration, shorter-chain Cars (n  9 and 10) are selectively bound to OPEN ACCESS LH2 antenna complexes for the light-harvesting function, which includes the absorption of light energy followed by singlet-energy transfer to bacteriochlorophyll (BChl) [1]. The all-trans conjugated chain having C 2h symmetry gives rise to low-lying singlet states, including the optically-allowed 1B u  and the optically-forbidden 2A g  , 1B u  and 3A g  states, concerning transitions from/to the ground 1A g  state [2,3]. The 1B u  and 3A g  states of Cars were first identified by the measurement of resonance-Raman excitation profiles (RREP) [1].  As shown in Figure 1, the observed slopes of the linear relations for the 2A g  , 1B u  and 3A g  states, as functions of 1/(2n  1), were in the ratio of 2:3.1:3.8, which is in excellent agreement with those theoretically predicted for shorter polyenes (n  58) in the ratio of 2:3.1:3.7 [4]. This is actually the basis for the assignment of these forbidden electronic-excited states. Because of the set of linear relations, the next low-lying singlet state below the 1B u  state is the 1B u  state for the shorter-chain Cars (n  9 and 10), whereas the 3A g  state, for the longer-chain Cars (n  1113). Figure 2 shows the chemical structures of the relevant Cars (n  913). Mini-9--carotene (n = 9) Dynamics. The singlet internal-conversion processes of 1B u  → 1B u  → 2A g  → 1A g  and the singlet-to-triplet fission followed by the triplet internal-conversion process of 1B u  → T 2 (A g ) → T 1 (B u ) have been identified in the set of Cars (n  913) in solution by pump-probe time-resolved spectroscopy using ~100 fs pulses. The singlet internal conversion including the 3A g  state, i.e., 1B u  has been identified in Cars (n  1113) by pump-probe spectroscopy using 5 fs pulses, and in Cars (n  10 and 11) by subpicosecond time-resolved Raman spectroscopy using ~100 fs pulses [1,5,6]. The low-lying singlet states of all-trans Cars have been found to give rise to plural channels of Carto-BChl singlet-energy transfer during the processes of internal conversion in the order, 1B u  , 1B u  and then 2A g  . This is the reason for the natural selection of the shorter-chain Cars in the all-trans configuration by antenna complexes. By transferring the rates of internal conversion within the Car and BChl a molecules in solution to those bound to the LH2 antenna complexes, the efficiencies of Car-to-BChl singlet-energy transfer through the 1B u  -to-Q x , 1B u  -to-Q x and 2A g  -to-Q y channels as well as the efficiencies of the 1B u  -to-T 1 singlet-to-triplet fission reactions were determined [1,7,8]. The sums of efficiencies through the three channels for Cars (n  9, 10, 11 and 11) were evaluated to be 88, 84, 51 and 54%, respectively, which nicely correlates to those determined by comparison of the electronic-absorption and fluorescence spectra, i.e., 92, 89, 53 and 55%. The sudden decrease in the Car-to-BChl singlet energy-transfer efficiency, on going from n  10 to n  11, was explained by the closing of the latter two channels due to the lowering of the 1B u  and 2A g  energies shown in Figure 1.
Thus, the important roles of the 1B u  state in the singlet-to-triplet transformation and the Car-to-BChl singlet-energy transfer have been determined. However, the roles of the 3A g  state in the lightharvesting function are left to be determined.

Diabatic Vibronic Levels
Energetics. Because of the unique linear relations among the 1B u  (0), 1B u  (0) and 3A g  (0) vibrational origins as shown in Figure 1, the 1B u  (0) level, for example, completely or approximately overlaps with the 1B u  (1) and 1B u  (2) levels in the shorter-chain Cars (n  9 and 10), whereas with the 3A g  (1), 3A g  (2) and 3A g  (3) levels in the longer-chain Cars (n  11, 12 and 13, respectively) as shown in Figure 3. overlapped with those of the 1B u  state in Cars (n  9 and 10) and the 3A g  state in Cars (n  1113) (both labeled on the left-hand-side). The spacing of all the vibrational ladders is set to be 1,400 cm 1 (Reproduced from Ref. [9]). We will call the pair of overlapped levels 'diabatic vibronic levels' or 'diabatic pair', because a diabatic basis set, instead of an adiabatic basis set, becomes necessary to theoretically describe their excited-state properties [10] (see Section 1.3). Thus, the shorter-chain Cars can form the 1B u   1B u  diabatic pairs, whereas the longer-chain Cars, the 1B u   3A g  diabatic pairs. It is to be noted that the energy gap between the diabatic pair is the largest in Car (n  11) and negligible in Car (n  10) (Figure 3), if we assume the interval of vibronic levels to be ~1,400 cm 1 , inclusively taking into account the C=C ( 1 ) and CC ( 2 ) stretching modes. The definition of the diabatic pair includes not only that the pair of vibronic levels is overlapped with each other, but also that the overlapped vibronic levels are located close to the conical intersection (see Figure 4). Concerning the set of potential functions, the shifts of the 1B u  , 1B u  and 2A g  potential minima, in reference to the ground 1A g  potential minimum, were determined by the Franck-Condon simulations of stationary-state fluorescence spectra from Cars (n  913) [9]. The 3A g  potential has been determined by pump-probe stimulated-emission spectroscopy of Cars (n  1113) after coherent excitation using 30 fs pulses [11], the details of which will be described in Section 3.1 of this article. Figure 4 clearly shows that the above-mentioned diabatic pairs, i.e., 1B u (2) in Cars (n  9 and 10) as well as 1B u (2) and 1B u  (0)  3A g  (3) in Cars (n  11, 12 and 13), are located close enough to the conical intersection, and satisfies the above condition of the diabatic pair. Another set of five pairs, which are located one vibrational-quantum higher, also fits this definition.
Spheroidene (n = 10) Lycopene (n = 11) Anhydrorhodovibrin (n = 12) Spirilloxanthin (n = 13) Dynamics. The all-trans conjugated chain of Cars has approximate C 2h symmetry in the ground state and, as a result, the singlet electronic states can be classified by symmetry into kA g  , lB u  , mB u  and nA g  , where the  and  signs are called Pariser's labels [2] and k, l, m and n indicate the ordering of the electronic states having the same symmetry (from the lowest to the higher energies). As shown in Figure 1, the low-lying singlet-excited states are in the order, 2A g  , 1B u  , 3A g  and 1B u  . Concerning Pariser's labels, optical transitions are allowed (forbidden) between electronic states with different signs (the same sign), whereas internal conversion is allowed (forbidden) between electronic states with the same sign (different signs).
As will be described in the next section, both the optically-allowed 1B u  counterpart and the optically-forbidden 1B u  or 3A g  counterpart keep their own symmetry properties, and behave as if they were totally independent symmetry-wise even after forming the diabatic pair.

Conical Intersection and Conservation of Symmetries in Singlet-Excited States
The unique characteristics of the low-lying singlet-excited states of Cars in the all-trans configuration can be summarized as follows: (a) Singlet-excited states having different symmetries, energetically in the order of 1B u   3A g − , 1B u  and 2A g   are closely located within a small energy difference in the region of 13 vibrational quanta (see Figure 1). (b) The minima of the 3A g  , 1B u  , 1B u  and 2A g  potentials are located, in this order, in a small region of mass-adjusted normal coordinate of q  02 (see Figure 4). (c) A pair of excited-state potentials with different symmetries can cross at the conical intersection keeping their own symmetry characteristics, the details of which are described below.
In the vicinity of a conical intersection, the Born-Oppenheimer approximation -on which the adiabatic description is based -breaks down. The reason for this is as follows: (1) The derivative coupling can be expressed as the vibronic coupling divided by the potential-energy difference like In the complete all-trans planar configuration in the ground state having the C 2h symmetry, the vibronic-coupling term between a pair of electronic states ( i and  j ) having different Pariser's  labels vanishes. However, when the Car molecule is excited to the 1B u  state, for example, there is a good chance for the conjugated chain to take a twisted conformation (degrading the C 2h symmetry) and to give rise to a certain value of the vibronic coupling. Then, in the vicinity of conical intersection where V ii (Q)  V jj (Q) approaches to 0, the derivative coupling diverges. (2) The time-dependent perturbation theory shows that the rate of change in any of the MO and CI coefficients is proportional to the value of V ii (Q)  V jj (Q) and, therefore, the reorganization of electronic wavefunction practically cannot take place in the vicinity of the conical intersection. Therefore, it becomes absolutely necessary to use diabatic expression, instead, setting the derivative coupling to be zero, i.e., and fix the MO and CI coefficients by the use of the orthogonal floating atomic orbitals. As a result, a pair of electronic wavefunctions, expressed by such a diabatic basis set, keeps the symmetry of each electronic state at any nuclear coordinate (Q), which makes the pair of nuclear potentials having different symmetries cross each other at the conical intersection, as shown in Figure 4. The above consideration has rationalized the apparently unique characteristics of the diabatic pair of electronic states: At the first glance, it looked strange and accidental that the symmetry properties of singlet-excited states conserve, as if they were totally independent from each other. However, it has turned out to be quite logical after we carefully consider the characteristics of the diabatic pair.
Working on the set of Cars (n  913) has been very fortunate, because the shorter-chain Cars (n  9 and 10) and the longer-chain Cars (n  1113) form completely different diabatic pairs in symmetries, i.e., 1B u   1B u  and 1B u   3A g  , respectively. This situation has enabled us to make a comparison between the two different combinations of symmetries in these diabatic pairs, and to establish the symmetry notation of the relevant singlet-excited states we have proposed.
In the electronic mixing of the diabatic pair ('diabatic electronic mixing') as well as in the internal conversion from a diabatic pair ('diabatic internal conversion'), we have found a common symmetry selection rule, i.e., '1B u diabatic pair vibrationally relaxes down to the bottom of the 1B u  potential, for example, the 1B u  (0) optically-allowed counterpart relaxes through radiative transition to the 2A g  or 1A g  state, whereas the 1B u  optically-forbidden counterpart relaxes through internal conversion to the iso-energetic 2A g  vibronic level followed by vibrational relaxation in the particular manifold. The above experimental results (to be described in detail in Section 2.2) evidence that the symmetry of each electronic state is totally conserved during the formation of the diabatic pair as well as in the splitting of the diabatic pair into the optically-allowed and the optically-forbidden counterparts.

Time-Resolved Spectroscopies Used
Pump-probe stimulated-emission and electronic-absorption spectroscopy. We used both ~100 fs and ~30 fs pulses for pump-probe stimulated-emission and electronic-absorption spectroscopy. Correlation between the time-duration and the spectral-width of these pulses is presented in Figure 5, showing the intensity profiles and the numerical values of FWHM.  In comparison to the energy gap between the diabatic pair of Car (n  11) (300 cm 1 ) shown in Figure 3, for example, the ~100 fs pulses with the FWHM  of ~200 cm 1 still tend to selectively excite one of the counterparts of the diabatic pair, whereas the ~30 fs pulses with the FWHM  of ~700 cm 1 can excite the optically-allowed and optically-forbidden diabatic counterparts simultaneously and coherently. Therefore, we call excitation with ~100 fs pulses 'selective excitation', whereas excitation with ~30 fs pulses 'coherent excitation'.
Visible-pump and near infrared-probe spectroscopy mainly probes transient-absorption spectra with some contribution of stimulated emission, while visible-pump and visible-probe spectroscopy probes the strong 1B u  stimulated emission as the optically-allowed counterpart. The latter spectroscopy is useful to identify stimulated emission from the optically-forbidden 1B u  or 3A g  counterpart of the diabatic pair, if any, and to conclude the presence or absence of the diabatic electronic mixing.
Kerr-gate fluorescence spectroscopy. We used this technique to probe fluorescence (mainly spontaneous emission) from the optically-forbidden vibronic levels of the Car molecules when they are being vibronically excited. This spectroscopic technique directly determines the energies of the emitting vibronic levels. Further, the excited-state molecules are free from disturbance by the probing radiation.

Contents of This Review
After the Introduction (Section 1), we are going to correlate the unique excited-state properties of the diabatic vibronic levels that we have found most recently. Those findings are classified into different categories in terms of the spectroscopic techniques used: Section 2: We used ~100 fs pulses for selective excitation: First, we excited to the 1B u  (0) level of a set of all-trans Cars (n  913) in nonpolar solvent, and mainly probed transient absorptions by the use of near-infrared (NIR) white continuum to determine the 1B u  , 1B u  and 3A g  lifetimes (Section 2.1).
Then, we excited the same set of Cars to the 1B u  (0) level and examined the presence or absence of stimulated emission from the optically-forbidden counterpart by the use of visible (VIS) white continuum. As a result, we found a symmetry selection rule in the diabatic mixing and diabatic internal conversion (Section 2.2). We also examined all-trans-and 15-cis--carotenes in nonpolar and polar solvents, and tried to find the effects of cistrans configurations and those of the polarization of the conjugated chain by polar solvents on the initial stimulated emission patterns. The effect of aggregation was also found (Section 2.3).
Section 3: We used ~30 fs pulses for coherent excitation of Cars (n  1113) in polar solvent, THF. Here, we observed stimulated emission followed by transient absorption from the 3A g  counterpart, exhibiting a single peak with the 3A g  (0) energy. The results lead us to conclude that the shift of the 3A g  potential, with respect to the ground 1A g  potential, is negligible (Section 3.1). The stimulated emission after coherent excitation of Cars (n  9 and 10) actually consisted of three components; one from the long-lived 1B u   1B u  diabatic pair, and the other two from the short-lived 1B u  and 1B u  counterparts. On the other hand, the stimulated emission after coherent excitation of Car (n  1113) consisted of one, from the long-lived 1B u   3A g  diabatic pair and the other two, from the short-lived 1B u  and 3A g  counterparts. The set of three components was explained by the mechanisms of quantum beat (Section 3.2). The same type of stimulated emission consisting of three components was observed after coherent excitation of Cars (n  911) bound to the LH2 antenna complexes, which accompanied the shift of the 1B u  (0) and 3A g  (0) levels to the lower energies and efficient triplet generation (Section 3.3).
Section 4: We used ~100 fs pulses to excite Cars (n  912) to the 1B u  (3) or 1B u  (4) level and probed fluorescence, by Kerr-gate fluorescence spectroscopy, to examine the slowest two steps of vibrational relaxation, i.e., 1B u 0). We found the breakdown of the above-mentioned symmetry selection rule in the diabatic electronic mixing and diabatic internal conversion, due to the degradation of molecular symmetry, while the Car molecules were being excited (Section 4.1). Section 5: We will summarize the results obtained (Section 2Section 4) and discuss the future trend of the present line of research.
After Section 6: Conclusion, we will introduce Section 7: Relevant work done by other investigators.  [12] As shown in Figure 1, the next low-lying singlet state below the 1B u  state is the 1B u  state in Cars (n  9 and 10), whereas it is the 3A g  state in Cars (n  1113); further, in Cars (n  10), the 1B u  state is overlapped with the 3A g  state. Since the set of Cars is dissolved in nonpolar solvent (n-hexane or a mixture of n-hexane and benzene), the conjugated chains are expected to keep the C 2h symmetry in the ground state, before selective excitation with ~100 fs pulses (the same symmetry should be conserved immediately after excitation).   Figure 7. Species-associated difference spectra (SADS) and time-dependent changes in population for the 1B u  and 1B u  states of Cars (n  9 and 10) and for the 1B u  and 3A g  states for Cars (n  1113) obtained by singular-value decomposition (SVD) followed by global fitting (Reproduced from Ref. [12]).  Figure 6 exhibits the pump-probe time-resolved spectra of Cars (n  913), and Figure 7 shows the species-associated difference spectra (SADS, top) and time-dependent changes in population (bottom) that have been obtained by singular-value decomposition (SVD) followed by global fitting of the spectral-data matrices by the use of a two-component sequential model. The first component, appearing immediately after excitation, is ascribable to the optically-allowed 1B u  state to which all the Car molecules were excited by the absorption of photons; each SADS consists of transient absorption and stimulated emission. The second component, appearing around 0.2 ps after excitation, is ascribable to the 1B u  state in the shorter-chain Cars (n  9 and 10), whereas to the 3A g  state in the longer-chain Cars (n  1113), according to the energy diagram ( Figure 1). The contribution of the 3A g  transient absorption is also seen in the second component of Car (n  10) as expected. The decay time constants listed in Table 1 show that the 1B u  lifetimes of Cars (n  9 and 10) are on the order of 0.1 ps, which is much longer than those of Cars (n  1113) on the order of 0.01 ps. On the other hand, the 1B u  lifetimes are around 0.25 ps, whereas the 3A g  lifetimes are around 0.10 ps.  Figure 8 shows the pump-probe time-resolved spectra of Cars (n  913) in nonpolar solvent after selective excitation with ~100 fs pulses to the 1B u

Excitation
In this subsection, we focus on the initial stimulated emission patterns that are useful in examining the presence or absence of diabatic electronic mixing between the optically-allowed 1B u  (0) and the isoenergetic optically-forbidden 1B u  or 3A g  vibronic levels. In this relation, we notice that the intensity of the strongest stimulated emission peak relative to the second-strongest one is much higher in the shorter-chain Cars (n  9 and 10) than in the longer-chain Cars (n  1113). The higher intensity in the former has turned out to be due to additional contribution of the 1B u  stimulated emission (vide infra).   Figure 9 shows the pump-probe time-resolved spectra when Cars (n  913) were excited to the 1B u  (1) level (called '1 ← 0 excitation') in terms of the 1B u  counterpart. Here, we need to remember the presence of the iso-energetic 1B u  or 3A g  diabatic counterpart; these diabatic pairs are shown on the top of Figure 3. We notice that the relative intensity of the strongest two stimulated-emission peaks changes with time, not in Car (n  12) but in Car (n  10), for example, the latter of which actually reflects vibrational relaxation (vide infra). Figure 10 exhibits the initial stimulated-emission patterns that have been extracted from the timeresolved spectra as SADS by means of the SVD and global-fitting analysis. In Cars (n  9 and 10), we succeeded in time-resolving the initial two components reflecting vibrational relaxation. We tried to simulate those negative signals in terms of the following components: the 1B u  stimulated emission (in red), the 1B u  stimulated emission (blue) and the bleaching of ground-state absorption (black) shown in broken or dotted line. Their sums (black dotted-broken line) are compared with the observed stimulated-emission patterns as SADS (black solid line).

Figure 10.
Fitting to the initial stimulated-emission profiles by the use of Franck-Condon factors for Cars (n  913). The stimulated-emission profiles were obtained as SADS by the SVD and global-fitting analysis of data matrices, the parts of which are presented in Figure 8 and Figure 9. Specification of lines: SADS (black solid lines), stimulated emission from 1B u  vibronic levels (red broken lines) and 1B u  vibronic levels (blue broken lines), the bleaching of the ground-state absorption (black dotted lines) and a sum of all the contributions (black dotted-broken lines) (Reproduced from Ref. [10]). Neurosporene (n = 9) The results of simulation can be summarized as follows: (a) Shorter-chain Cars (n  9 and 10): After the 0 ← 0 excitation, the observed stimulated emission can be explained by simultaneous stimulated emission from the diabatic pairs, i.e., 1B u After the 0 ← 0 excitation, only the 1B u  (0) stimulated emission is observed, whereas after the 1 ← 0 excitation, only the 1B u  (1) stimulated emission, instead. Neither the contribution of the 3A g  counterpart nor the vibrational relaxation of 1B u is observed at all in the set of SADS. Figure 11 summarizes the relaxation dynamics starting from the diabatic pair for all the set of Cars (n  913): The shadowed envelopes show simultaneous stimulated emission from the 1B u   1B u  diabatic pair, and the bent and short arrows, internal conversion and vibrational relaxation, respectively. (a) Shorter-chain Cars (n  9 and 10): As revealed by the simulation described above, the simultaneous stimulated emission from the 1B u  and 1B u  diabatic pair and the vibrational relaxation of the diabatic pair, i.e., υ  1 → υ  0 in terms of the 1B u  counterpart, are seen.
(b) Longer-chain Cars (n  11-13): Neither the stimulated emission from the 3A g  optically-forbidden counterpart nor the vibrational relaxation in the 1B u  manifold is seen in Cars (n  1113). This is because the 1B u   3A g  diabatic electronic mixing never takes place, and the 1B u  → 1B u  diabatic internal conversion, instead, takes place very efficiently. Figure 11. Diabatic electronic mixing between the 1B u  and 1B u  vibronic levels accompanying simultaneous stimulated emission in Cars (n  9 and 10) and diabatic internal conversion from the 1B u  to 1B u  vibronic level in Cars (n  1113). In the latter Cars, neither diabatic electronic mixing nor diabatic internal conversion between the 1B u  and 3A g  vibronic levels takes place. Diabatically-mixed states are shadowed, and internal conversion and vibrational relaxation are shown by long bent and short straight arrows, respectively (Reproduced from Ref. [10]).
Spheroidene (n = 10) Lycopene (n = 11) Anhydrorhodovibrin (n = 12) Spirilloxanthin (n = 13) As shown in the pump-probe time-resolved spectra and the simulation of the initial stimulated emission, the 1B u   1B u  stimulated emission transforms into the 1B u  transient absorption while taking vibrational relaxation in Cars (n  9 and 10), whereas the 1B u  stimulated emission directly transforms into the 1B u  transient absorption in Cars (n  1113). Although the resultant 1B u  transient absorption is the same, the mechanisms of its generation are different between the shorter-chain and the longer-chain Cars. It is continuous vibrational relaxation in the 1B u  manifold in the former, while it is the 1B u  to 1B u  internal conversion followed by vibrational relaxation in the 1B u  manifold in the latter ( Figure 11). In the latter case, we saw efficient 1B u  → 1B u  → 2A g  internal conversion in the time-resolved spectra and in the results of SVD and global-fitting analysis of the spectral-data matrices (see Figure S3 in Supporting Information of Ref. [13]). Thus, the reason for the absence of the 3A g  signal in the pump-probe time-resolved spectra of Cars (n  1113) (Figures 8 and 9) has been explained. The above set of results lead us to the following symmetry selection rule, concerning the diabatic electronic mixing and diabatic internal conversion: forbidden'. This selection rule has been theoretically explained [10].    Figure 13 shows a set of pump-probe time-resolved spectra after selective excitation with ~100 fs pulses to the 1B u  (1) level (the absorption maximum) of all-trans-and 15-cis--carotenes in nonpolar (n-hexane) and polar (DMF and DMF  IL) solvents; here, DMF indicates dimethylformamide and IL, an ionic liquid, i.e., methyl-3-octylimidazoluim tetrahydrofluoroborate. (1) level of all-trans-and 15-cis--carotenes in n-hexane, DMF and DMF  IL (see text) (Reproduced from Ref. [14]). (1) level is an approximation (actually it has a pair of wings on both sides; see Figure 10). Most importantly, the Gaussian peaks of the 3A g

 (1) Level of All-Trans-and 15-Cis--Carotenes in Nonpolar and Polar Solvents [14]
levels must originate from the negligible shift of the 3A g  potential in reference to the 1A g  potential (to be proven in Section 3.1). The stimulated-emission profiles of all-trans-and 15-cis--carotenes can be characterized as follows (see Figure 14): (a) Comparison between the two isomers: In n-hexane, all-trans--carotene exhibits a simplified stimulated-emission profile consisting of two main peaks, i.e., 3A g  (0) and 3A g  (1), whereas 15-cis--carotene exhibits a progression consisting of five peaks, i.e., 1B u , both in addition to a weak Franck-Condon profile from the 1B u  (1) level. This spectral change, on going from all-trans-to 15-cis--carotene, reflects the lowering of the symmetry of the conjugated chain from C i to C 2v . (b) Comparison among three solvents: In alltrans--carotene, the stimulated-emission peaks increase in number on going from n-hexane to DMF or DMF  IL. Some peaks are further enhanced in the latter polar solvent. In 15-cis--carotene, no changes in the number of peaks are observed on going from n-hexane to DMF or DMF  IL, although changes in the relative intensities are observed; 15-cis--carotene in DMF  IL gives rise to the clearest set of Gaussian peaks. We added IL to enhance the polarization of the conjugated chain; actually, some 3A g -Gaussian peaks seem to be enhanced after addition of IL. Figure 14. Simulation of the initial stimulated-emission patterns obtained as the first SADS by the SVD and global-fitting analysis of data matrices, parts of which are presented in Figure 13. The Franck-Condon profiles are used for the 1B u  (1) emission (red broken lines) and the bleaching of the 1B u  ← 1A g  absorption (black dotted lines). The Gaussian profiles are used for the 3A g  (green broken lines) and 1B u  (blue broken lines, as an approximation). The progression of stimulated emission peaks can be generated by resonance transfer of phonons (see text and Figure 15) (Reproduced from Ref. [14]).    Figure 14 (see text for the details) (Reproduced from Ref. [14]).  [11] We have been determining the 1B u  , 2A g  and 1B u  potentials of Cars by fluorescence spectroscopy: The shift of potential minimum, in reference to the ground-state 1A g  potential, was the largest in the 2A g  state, a middle in the 1B u  state, and the smallest in the 1B u  state [9], the 3A g  potential being left to be determined. However, we saw a progression of the Gaussian-type 3A g  stimulated emission in isomeric -carotenes (see Section 2.3), each of which strongly suggested the negligible shift of the 3A g  potential; here, the progression was ascribed to resonance transfer of phonons between a pair of Car molecules in aggregates as mentioned in Section 2.3. Figure 16 shows a set of pump-probe time-resolved stimulated-emission and transient-absorption spectra after coherent excitation with ~30 fs pulses to the 1B u   Figure 18. The negligible shift of the 3A g  potential, in reference to the 1A g  potential, a mechanism which gives rise to the transformation, with time, from a single stimulatedemission peak to a the single transient-absorption peak shown in Figure 16. Here, all the vibrational wavefunctions in the 3A g  and 1A g  states become orthogonal, and only the downward and upward transitions indicated by vertical arrows become allowed (Reproduced from Ref. [11]).  (2) level should be accumulated as thermal energy on the 1A g  vibronic levels after a while, when its dissipation is slow. This gives rise to the 3A g (2) absorptive transitions with the same transition energy as mentioned above.
Thus, the negligible shift of the 3A g  potential, in reference to the 1A g  potential, has been established by coherent excitation of the set of Cars (n  1113). Lycopene (n = 11)

Excitation to the 1B
The unique excited-state dynamics after coherent excitation with ~30 fs pulses to the diabatic pair of Cars (n  913) in THF solution stems from the following experimental conditions: (a) The interaction between the optically-allowed 1B u  (0) and the optically-forbidden iso-energetic 1B u  or 3A g  levels should become stronger in polar solvent than in nonpolar solvent due to the polarization of the conjugated chain and the resultant symmetry degradation from C 2h to C s . (b) The spectrally-broad ~30 fs pulses enable the simultaneous coherent excitation of the diabatic pair as mentioned above.
(c) The photon density (the number of photons per unit time) should be much higher in ~30 fs pulses than in ~100 fs pulses; therefore, there is a good chance that the diabatic pair becomes densely populated. These conditions gave rise to unique spectral patterns ( Figure 19) that are completely different from those obtained by selective excitation with ~100 fs pulses in nonpolar solvent ( Figure 8). Here, the Franck-Condon simulation of the stimulated emission is impossible.
To facilitate the spectral comparison, the initial stimulated-emission patterns that have been presented in Figure 8    The pump-probe time-resolved spectra of Cars (n  911) presented in Figure 19, that have been recorded under the above-mentioned experimental conditions (a), (b) and (c), can be characterized as follows: (i) An initial short-lived stimulated-emission peak from the 1B u  (0) level. Immediately after excitation, a weak and sharp peak ascribable to the pure 1B u (2) and 3A g  (3) levels for Cars (n  9, 10, 11, 12 and 13, respectively).) Subsequently, a much stronger, broad and long-lived stimulated-emission peak ascribable to the 1B u   X  (υ) diabatic pair appears. This particular peak even tends to split into two in Car (n  10) in (0) stimulated emission peaks. Around 0.060.10 ps, a peak systematically shifting to the lower energy with n, which is ascribable to the 1B u  (0) stimulated emission, appears in Cars (n  9 and 10). As shown in Figure 17, the energies of these stimulated emission peaks agree with those determined by RREP measurement. In the shorter-chain Cars (n  9 and 10), another 1B u The above spectral characteristics can be explained in terms of the mechanism of quantum beat (see Figure 21 referring to Figure 19). To be consistent to what will be described in Section 3.3, we will take the case of Car (n  10) as the first example: (i) The stimulated emission from the 1B u (2) diabatic pair, which follows the initial weak stimulated emission from the pure 1B u  (0) level, can be attributed to the coherent cross term. In phenomenological expression, the 1B u  (0) lifetime becomes substantially lengthened by back-and-forth exchanges between the optically-allowed 1B u  and optically-forbidden 1B u  states; for the rigorous theoretical description of this quantum beat mechanism, see Eq. 48 in Ref. [10].  (0)  X  (υ) diabatic pairs (for the notation of X  (υ), see the caption of Figure 19): The initial set of stimulated emissions can be explained in terms of quantum beat. In Car (n  10), for example, it consists of (i) the persistent stimulated emission from the 1B u  (1) transition is not clearly seen at all, most probably it is overlapped with the 3A g Thus, in terms of the quantum beat formalism, the 1B u  (0)  X  (υ) stimulated emission is ascribable to the coherent cross term, while the 1B u  (0) and the X  (0) stimulated emission, to the pair of split incoherent terms.
Finally, the key question is whether we can actually observe the real quantum beat: Figure 3 shows that the energy gap between the 1B u  (0) and 3A g  (1) pair of Car (n  11) is around 300 cm 1 , whereas that between the 1B u  (0) and 1B u  (2) levels of Car (n  10) is almost zero. Therefore, there is a good chance of observing the real quantum beat not in Car (n  10) but in Car (n  11). Figure 22 shows the time-dependent changes in the integrated intensity of stimulated emission from the 1B u  (0)  3A g  (1) diabatic pair in Car (n  11). After the subtraction of the background time profile, we can actually see the oscillatory changes in the fluorescence intensity. This result strongly supports our explanation of the observed spectral characteristics in terms of the quantum-beat mechanism.  As shown in Section 3.2, the coherent excitation of Cars (n  913) in a polar solvent, THF, exhibited not only the long-lived stimulated emission from the 1B u  (0)  X  (υ) diabatic pair but also the short-lived stimulated emission from the split 1B u  (0) and X  (0) levels, which have been explained in terms of the quantum-beat mechanism. Therefore, we have tried to find whether the excited-state dynamics, after the coherent excitation of Cars (n  911) that are bound to LH2 antenna complexes, is similar to, or different from, the excited-state dynamics free in THF solution.  Rsp. molischianum Lycopene (n = 11)  Figure 23 shows the pump-probe time-resolved stimulated-emission and transientabsorption spectra of Cars (n  911) bound to LH2 complexes from Rba. sphaeroides G1C, Rba. sphaeroides 2.4.1 and Rsp. molischianum. The time-resolved spectra can be characterized as follows:

Excited-state dynamics as revealed by time-resolved spectra and by the results of SVD and globalfitting analysis.
(a) Cars (n  9 and 10). Following the weak 1B u  (0) stimulated emission, strong stimulated emission from the 1B u  (0)  X  (υ) diabatic pair appears, giving rise to a broad peak or even clearlysplit two peaks. Simultaneously, the 1B u  The above sequence of events can be proven by the SVD and global-fitting analysis of bound Car (n  10), for example; the SADS and time-dependent changes in population are presented in Figure 24 (left-hand-side  Figure 25 (in the middle).   The above sequence of events has been proven by the SVD and global-fitting analysis of the spectral data matrix, the results of which are shown in Figure 24 (right-hand-side): The sequential transformation of Component 1 → Component 2 → Component 3 is ascribable to spectral transformation i.e., the pair of the 1B u   3A g  and 3A g  stimulated emission → the 1B u  transient absorption → the 2A g  and T 1 transient absorption. The transformation is schematically presented in Figure 25 (on the right end).
The singlet-state energies of Cars (n  911) when bound to the LH2 antenna complexes. Figure 26 compares the 1B u The much faster decay of the initial stimulated emission after the coherent excitation of the diabatic pair obviously reflects the branching pathway to the Car-to-BChl singlet-energy transfer in addition to internal conversion within Car. This pathway of Car-to-BChl singlet-energy transfer has been established as introduced in Section 1.1. (c) The most conspicuous change, upon the binding of Cars, is the efficient triplet generation. We have already shown that the triplet generation is due to singlet heterofission from the 1B u  state [1]. The high efficiency of triplet generation suggests a substantial twisting around the C=C bonds in the Car conjugated chain when bound to the apo-peptides and BChls.

Kerr-Gate Fluorescence Spectroscopy
Excitation to the 1B u 

(υ  3 or 4) Vibronic Level of Cars (n  912) in Nonpolar Solvent and Probing
Our preliminary Kerr-gate fluorescence spectroscopy was found contradictory in our series of attempts to determine the 1B u  lifetime of neurosporene, Car (n  9), for example: As described in In the previous Kerr-gate fluorescence spectroscopy, after excitation to the 1B u  (3) level, however, we obtained the 1B u  (0) lifetime of 260 fs, the value of which agreed with the 1B u  lifetime [21]. This result confused us. Therefore, we have decided to examine a set of Cars (n  912) after excitation at 12,500 cm 1 , which is just below and above the 1B u  (3) level in Car (n  9 and 10) and around the 1B u  (4) level in Cars (n  11 and 12). The results actually provided us with deeper insight into the unique excited-state properties of the diabatic pairs, which has turned out to be the reason for the apparent contradiction. Figure 27. Time-resolved Kerr-gate fluorescence spectra after selective excitation with ~100 fs pulses to the 1B u  (3) level of the shorter-chain Cars (n  9 and 10) and to the 1B u  (4) level of the longer-chain Cars (n  11 and 12) (Reproduced from Ref. [9]). 18 20 Energy / 10 3 cm -1 Figure 27 shows the time-resolved fluorescence spectra of Cars (n  912). The spectra of the longer-chain Cars (n  11 and 12) can be contrasted to those of the shorter-chain Cars (n  9 and 10) as follows: (a) Fluorescence decays faster in the longer-chain Cars than in the shorter-chain Cars. (b) The spectral profile is more stretched and the relative intensity in the higher-energy region is enhanced in the longer-chain Cars than in the shorter-chain Cars. (c) In each Car, the fluorescence maximum shifts to the higher energies with time, but this trend is less pronounced in the longer-chain Cars than in the shorter-chain Cars. The time-dependent high-energy shift is ascribable to the Franck-Condon factors and reflects the vibrational relaxation in the 1B u  manifold (vide infra). Since the rate of vibrational relaxation from υ  ℓ to υ  ℓ1 is proportional to the quantum number of the starting vibrational level, ℓ [22], hopefully, we would be able to time-resolve the last two steps of the slowest vibrational relaxations.  Table 1. In the comparison of those values, we need to take into account the fact that the present result is just on the edge of our spectroscopic and analytical techniques. Figure 29 shows the results of simulation for the fluorescence patterns obtained as a set of SAFS, which can be characterized as follows: (a) In Cars (n  9 and 10), fluorescence patterns I and II can be simulated in terms of the Franck-Condon factors for the downward transitions from each of the 1B u  and 1B u  diabatic pairs (shown in red and blue lines), but fluorescence pattern III is dominated by transition from the 1B u  (0) level. In the fluorescence pattern II of Car (n  10), the contribution of the 3A g  transitions is also seen as expected from the energy diagram ( Figure 1)  Thus, the fluorescence patterns can be clearly classified into two groups: one, in the shorter-chain Cars (n  9 and 10) and the other, in the longer-chain Cars (n  11 and 12). Most importantly, we have obtained not only the 1B u  -to-1B u  diabatic electronic mixing in the shorter-chain Cars but also the 1B u  -to-3A g  diabatic electronic mixing in the longer-chain Cars (11 and 12). The results indicate that symmetry degradation takes place while the Car molecules are being vibronically excited, presumably due to the twisting of the conjugated chain around the C=C bond(s). Then, the symmetry selection rule concerning the diabatic electronic mixing (see Section 2.2) breaks down, and the 1B u  -to-3A g  diabatic electronic mixing becomes allowed. Now, we will try to explain why the lifetime of 'the apparent 1B u  (0) level' agrees with the 1B u  lifetime in Cars (n  9 and 10), whereas with the 3A g  lifetime in Cars (n  11 and 12): Figure 30 presents the mechanisms: Here, each of the mixed diabatic pair is shown by a shadowed envelope, where the density of inclined lines is set proportional to the contribution of the optically-forbidden 1B u  or 3A g  counterpart. In considering the relaxation processes, we need to consider the selection rule in relation to the Pariser's  labels (see Section 1.2). (a) In the shorter-chain Car (n  9), for example, the vibrational relaxation, 1B u (1), takes place first. When the diabatic pair has reached to the bottom of the 1B u  (0) potential, the allowed relaxation for the 1B u  (0) counterpart is the instantaneous 1B u  (0) emission, whereas the allowed relaxation for the 1B u  (1) counterpart is the 1B u  (1) → 2A g  (4) internal conversion. Therefore, it is quite natural that the time constant of the latter process corresponds to the 1B u  lifetime. Actually, the allowed 1B u  -to-2A g  internal conversion is taking place so efficiently from the upper 1B u  vibronic levels that the 1B u  population has been almost exhausted before the diabatic pair reaches to the bottom of the 1B u  (0) potential. (b) In the longer-chain Car (n  11), for example, after the vibrational relaxation in the sequence of 1B u (1), the 1B u  (0) stimulated emission and the 3A g  (1) → 2A g  (4) internal conversion are to take place; the former must take place instantaneously, while the latter, with the 3A g  lifetime. Here, the 3A g  (1) level is still highly populated when the diabatic pair reaches to the bottom of the 1B u  potential.
This pair of observations reflects the unique excited-state properties of the diabatic pair consisting of the optically-allowed 1B u  counterpart and the optically-forbidden 1B u  or 3A g  counterpart. When the diabatic pair relaxes down to the bottom of the 1B u  potential, the 1B u  counterpart relaxes through emission, whereas the 1B u  or 3A g  counterpart relaxes through internal conversion. This is actually the splitting processes of the diabatic pair. The relative contribution of the optically-allowed and the optically-forbidden counterparts seems to vary in the processes of vibrational relaxation.

Summary and Future Trend
Pump-probe stimulated-emission and transient-absorption spectroscopy after selective excitation with ~100 fs pulses to the 1B u The results indicate that the symmetry selection rule holds immediately after excitation, but it breaks down while the Car molecules are being excited probably due to the degradation of the C 2h symmetry of the conjugated chain.
The above results demonstrate that not only the 1B u  state but also the 3A g  state can play important roles in the light-harvesting function: While the Cars (n = 1113) molecules are being excited they can efficiently transfer 'the 3A g  energy' to BChl as far as the pair of pigment molecules are in close contact.
Pump-probe stimulated-emission and transient-absorption spectroscopy after coherent excitation with ~30 fs pulses to the 1B u  (0) level of Cars (n = 913) in polar solvent gave rise to three stimulated-emission components, explained by the quantum beat mechanism, including the long-lived coherent cross term from the 1B u  + 1B u  or 1B u  + 3A g  diabatic pair and the short-lived 1B u  and 1B u  or 3A g  split incoherent terms. The lifetimes of the coherent terms from the diabatic pairs reach as long as ~2.5  10 2 fs. Basically the same type of stimulated-emission components were identified in Cars (n = 911) bound to LH2 complexes. Actually, the substantial shortening of the stimulated emission components were observed, supporting the idea of the Car-to-BChl singlet-energy transfer.
The results strongly suggest that the coherent excitation of the diabatic pairs strongly facilitate the light-harvesting function. A key question here is whether such coherent excitation of Cars can take place in Nature. Preliminary four-wave-mixing (FWM) spectroscopy of Car (n = 10) showed coherence transfer from the 1A g Therefore, there is a good chance that all the transitions eventually become coherently-coupled with one another and share the same phase while a set of Car molecules are being excited. (This reminds us the case where a pair of pendular hanging on the both ends of a bar eventually becomes synchronized.) Obviously, coherence dynamics beyond population dynamics is the key issue in the future in studying the dynamics of Car singlet-excited states and Car-to-BChl singlet-energy transfer.

Conclusions
Pump-probe spectroscopy after selective excitation of all-trans Cars (n  913) in nonpolar solvent, probing stimulated emission and transient absorption, identified a symmetry selection rule of diabatic electronic mixing and diabatic internal conversion, i.e., '1B u forbidden'. Kerr-gate fluorescence spectroscopy showed that this selection rule breaks down, due to the symmetry degradations when the Car molecules are being excited, and, as a result, the 1B u  -to-3A g  diabatic electronic mixing and internal conversion become allowed.
On the other hand, pump-probe spectroscopy after coherent excitation of the same set of Cars in polar solvent identified three stimulated-emission components, generated by the quantum-beat mechanism, consisting of the long-lived coherent cross term from the 1B u   1B u  or 1B u   3A g  diabatic pair and incoherent short-lived 1B u  and 1B u  or 3A g  split incoherent terms. The same type of stimulated-emission components were identified in Cars bound to LH2 complexes, their lifetimes being substantially shortened by the Car-to-BChl singlet-energy transfer. The low-energy shifts of the 1B u  (0), 1B u  (0) and 3A g  (0) levels and efficient triplet generation were also found. Therefore, there is a good chance that not only the 1B u state but also the 3A g state play the role of light-harvesting in bacterial photosynthesis. In all the above excited-state dynamics, the symmetry properties of the 1B u  , 1B u  and 3A g  counterparts are totally conserved during the formation of the diabatic pairs and also during their splitting and relaxation of the 1B u  counterpart through emission and the 1B u  or 3A g  counterpart through internal conversion. This is exactly what has been anticipated by the theoretical description (experimental condition) of the diabatic pairs. The observed energetics and excited-state dynamics of the diabatic pairs and their rigorous theoretical description using the diabatic basis set fully support the symmetry notations, the energy diagrams and the potential curves for all the 1B u  , 1B u  , 3A g  and 2A g  vibronic levels we have been proposing.

Relevant Works by Other Investigators
the 1B u  vibrational level of initial excitation (the lowest couple or much higher), and (c) the environment of the Car molecule (nonpolar or polar). (3) Therefore, if a laser spectroscopist of Cars were not aware of the phenomena and the mechanisms described in this article, there is a good chance he/she could become confused and introduce additional 'controversy' to this field. Therefore, the present authors really would like the above-mentioned leaders as well as relevant colleagues, in this particular field of Car excited states, to carefully read this summary. We suspect that the following figures, in this article, may be useful to solve the controversy already pointed out by Polivka and Sundström [28,29]: (i) Figures 10 and 11  ' lifetimes may also depend on such experimental conditions. (ii) Figure 24 (components 2 to 3 in the top-left panels) showing transformation from the 1B u  state into the 2A g  and T 1 states; the former is due to singlet internal conversion, while the latter is due to singlet-to-triplet fission. If the S * became time-resolved into the 1B u and T 1 (1 3 B u ) components, the contradiction between the 1B u  and S * states should be solved. (iii) Figure 10  Here, in the rest of this section, we will describe the results of ab initio calculations including both  and  electrons [32] (Section 5.1) and the observation of oscillatory intensity changes immediately after pulsed excitation to the 1B u  state of lutein, i.e., Car (n  10) [33,34] (Section 5.2). We think these two topics are relevant to our present findings.

Ab Initio Calculation of the  → * Excited States of Linear Polyenes [32]
In this theoretical paper, multi-reference Møller-Plesset perturbation theory with complete activespace configurational interaction (CASCI-MRMP) was applied to calculate the energies of the vertical  → * transitions of all-trans polyenes (n  314), focusing on the nature of the four lowest-lying singlet-excited states and their ordering: It has been determined that the ionic 1B u  state is the lowest optically-allowed excited state, while the covalent 2A g  1B u  and 3A g  states are the opticallyforbidden states increasing in energy in this order. The calculations predict that the 1B u  state becomes lower than the 1B u  state at n  7, while the 3A g  state becomes lower than the 1B u  state at n  11.
It was a challenge for the theoreticians to carry out highly accurate ab initio calculations of the longer polyenes to realize the state ordering determined by the use of resonance-Raman excitation profiles. Before starting the calculations, the authors had demonstrated (in other molecular systems) that CASCI-MRMP was more efficient than, and comparable in accuracy to MRMP based on CASSCF reference functions.
To calculate the vertical excitation energies from the ground state to the relevant singlet states, the ground-state equilibrium geometries were optimized at the MP2 level. A reference CASCI wave function was obtained by partitioning the SCF orbitals, and optimizing the expansion coefficients of all configurations that were generated by all the possible arrangement of the active electrons among the active orbitals. The 10 valence  electrons were treated as active electrons. The effect of  electrons was included through the perturbation calculation performed with MRMP, which was applied to each individual excited state.
In the present case of alternant hydrocarbons, the pairing properties are satisfied at the CASSCF and even the CAS-CI level. The 1A g   The calculated vertical-excitation energies did not exhibit a simple linear dependence on 1/(2n  1) but fit to an exponential function, for fixed n 0 , E n  E   (E n0  E  ) exp [a (n  n 0 )] Figure 31 shows an energy diagram for n = 913; approximate linear regression lines, as functions of 1/(2n  1), are drawn for comparison to those in Figure 1. The state ordering of the four lowestlying singlet states, i.e., 1B u  , 2A g  1B u  and 3A g  , determined by the measurement of resonance-Raman excitation profiles (Figure 1) is in general agreement with that predicted by the highly accurate ab initio calculations including the and -electrons ( Figure 31). The absolute values of the state energies and their dependence on n are still in poor agreement. When agreement between the observed and calculated state ordering becomes closer in the future, the results of calculations must provide us with a most reliable clue to confirm the symmetry notation of each singlet-excited state. It is much more straightforward to determine the symmetry of each excited state theoretically rather than spectroscopically, although the latter is ideal. [33,34] Following a preliminary report on -carotene [33], Alfred Holzwarth and his coworkers have most recently reported the intensity fluctuation of transient absorption immediately after the excitation of lutein, i.e., Car (n  10), to the 1B u  state and ascribed it to quantum beat due to coherence coupling between the 1B u  and 1B u  vibronic levels [34]. Oscillatory intensity changes were most pronounced in the 600700 nm region as shown in Figure 32a. The oscillatory changes in the electronic absorption were simulated by a scheme consisting of five components, whose time-dependent changes in population are shown in Figure 32b. The results of simulation are overlaid on the time profiles in Figure 32a. Interestingly, the profile and magnitude of intensity changes were strongly dependent on the solvent, which probably reflect the shift of the 1B u  counterpart in energy. Also, they are strongly dependent on the wavelengths of excitation and detection. To prove that the counterpart of the 1B u  state is the 1B u  state in the coherence coupling, the fluorescence pattern that does not exhibit the mirror image with respect to the electronic-absorption pattern (as shown in Figure 33a) was theoretically analyzed in lutein and -carotene. The 1B u  and 1B u  energies and potentials have been theoretically determined as shown in Figure 33b. Most importantly, the pair of the 1B u  and 1B u  energy levels is almost iso-energetic and located close to the conical intersection, showing the reliability of their theoretical calculations. The above results of analysis and interpretation in the case of -carotene and lutein seem to be closely correlated to our case of bacterial Cars. In our terminology, it may be called 'coherent excitation of the 1B u   1B u  diabatic pair' in lutein.

Oscillatory Intensity Changes in Electronic Absorption Immediately after Pulsed Excitation of Lutein
It is also encouraging, that these authors proved the presence of the 1B u  state, which is coherently coupled with the 1B u  state. Detailed discussion on the nature of the 1B u  state and the mechanism of quantum beat based on their sophisticated theoretical calculations are described in their original literature. Such theoretical analysis combined with spectroscopic studies will reveal the more detailed mechanisms of diabatic electronic mixing and quantum beat.