_{u}

^{+}and Optically-Forbidden 1B

_{u}

^{-}or 3A

_{g}

^{-}Vibronic Levels of Carotenoids: Possible Roles in the Light-Harvesting Function

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The unique excited-state properties of the overlapped (diabatic) optically-allowed 1B_{u}^{+} and the optically-forbidden 1B_{u}^{−} or 3A_{g}^{−} vibronic levels close to conical intersection (‘the diabatic pair’) are summarized: Pump-probe spectroscopy after _{u}^{+}-to-1B_{u}^{−} is allowed but 1B_{u}^{+}-to-3A_{g}^{−} is forbidden’. On the other hand, pump-probe spectroscopy after

_{u}

^{+}, 3A

_{g}

^{−}, 1B

_{u}

^{−}and 2A

_{g}

^{−}states

_{2h} symmetry gives rise to low-lying singlet states, including the _{u}^{+} and the _{g}^{−}, 1B_{u}^{−} and 3A_{g}^{−} states, concerning transitions from/to the ground 1A_{g}^{−} state [_{u}^{−} and 3A_{g}^{−} states of Cars were first identified by the measurement of resonance-Raman excitation profiles (RREP) [

As shown in _{g}^{−}, 1B_{u}^{−} and 3A_{g}^{−} states, as functions of 1/(2_{u}^{+} state is the 1B_{u}^{−} state for the shorter-chain Cars (_{g}^{−} state, for the longer-chain Cars (

_{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 (_{g}^{−} state, _{u}^{+} → 3A_{g}^{−} → 1B_{u}^{−} → 2A_{g}^{−} → 1A_{g}^{−}, has been identified in Cars (

The low-lying singlet states of all-_{u}^{+}, 1B_{u}^{−} and then 2A_{g}^{−}. This is the reason for the natural selection of the shorter-chain Cars in the all-_{u}^{+}-to-Q_{x}_{u}^{−}-to-Q_{x}_{g}^{−}-to-Q_{y}_{u}^{−}-to-T_{1} singlet-to-triplet fission reactions were determined [_{u}^{−} and 2A_{g}^{−} energies shown in _{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 light-harvesting function are left to be determined.

_{u}^{+}(0), 1B_{u}^{−}(0) and 3A_{g}^{−}(0) vibrational origins as shown in _{u}^{−}(0) level, for example, completely or approximately overlaps with the 1B_{u}^{−}(1) and 1B_{u}^{−}(2) levels in the shorter-chain Cars (_{g}^{−}(1), 3A_{g}^{−}(2) and 3A_{g}^{−}(3) levels in the longer-chain Cars (

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 [_{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 (^{−1}, inclusively taking into account the C=C (ν_{1}) and C–C (ν_{2}) stretching modes.

The definition of the diabatic pair includes _{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 (_{g}^{−} potential has been determined by pump-probe stimulated-emission spectroscopy of Cars (_{u}^{+}(0) + 1B_{u}^{−}(1) and 1B_{u}^{+}(0) + 1B_{u}^{−}(2) in Cars (_{u}^{+}(0) + 3A_{g}^{−}(1), 1B_{u}^{+}(0) + 3A_{g}^{−}(2) and 1B_{u}^{+}(0) + 3A_{g}^{−}(3) in Cars (

_{2h} symmetry in the ground state and, as a result, the singlet electronic states can be classified by symmetry into _{g}^{−}, _{u}^{−}, _{u}^{+} and _{g}^{+}, where the + and – signs are called Pariser’s labels [_{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.

The unique characteristics of the low-lying singlet-excited states of Cars in the all-_{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 _{g}^{−}, 1B_{u}^{−}, 1B_{u}^{+} and 2A_{g}^{−} potentials are located, in this order, in a small region of mass-adjusted normal coordinate of

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-_{2h} symmetry, the vibronic-coupling term between a pair of electronic states (_{i}_{j}_{u}^{+} state, for example, there is a good chance for the conjugated chain to take a twisted conformation (degrading the _{2h} symmetry) and to give rise to a certain value of the vibronic coupling. Then, in the vicinity of conical intersection where _{ii}_{jj}_{ii}_{jj}

Therefore, it becomes absolutely necessary to use diabatic expression, instead, setting the derivative coupling to be zero,

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 (_{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, _{u}^{+}-to-1B_{u}^{−} is allowed but 1B_{u}^{+}-to-3A_{g}^{−} is forbidden’. When a 1B_{u}^{+} + 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.

In comparison to the energy gap between the diabatic pair of Car (^{−1}) shown in _{σ} of ∼200 cm^{−1} still tend to _{σ} of ∼700 cm^{−1} can excite the optically-allowed and optically-forbidden diabatic counterparts

Visible-pump and near infrared-probe spectroscopy _{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.

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-_{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-

Section 3: We used ∼30 fs pulses for coherent excitation of Cars (_{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 (_{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 (_{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 (_{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 (_{u}^{+}(3) or 1B_{u}^{+}(4) level and probed fluorescence, by Kerr-gate fluorescence spectroscopy, to examine the slowest two steps of vibrational relaxation, _{u}^{+}(2) → 1B_{u}^{+}(1) → 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.

As shown in _{u}^{+} state is the 1B_{u}^{−} state in Cars (_{g}^{−} state in Cars (_{u}^{+} state is overlapped with the 3A_{g}^{−} state. Since the set of Cars is dissolved in nonpolar solvent (_{2h} symmetry in the ground state, before selective excitation with ∼100 fs pulses (the same symmetry should be conserved immediately after excitation).

_{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 (_{g}^{−} state in the longer-chain Cars (_{g}^{−} transient absorption is also seen in the second component of Car (

The decay time constants listed in _{u}^{+} lifetimes of Cars (_{u}^{−} lifetimes are around 0.25 ps, whereas the 3A_{g}^{−} lifetimes are around 0.10 ps.

_{u}^{+}(0) level (called ‘0 ← 0 excitation’). Here, we can see the singlet-state internal-conversion processes of 1B_{u}^{+} → 1B_{u}^{−} → 2A_{g}^{−} → 1A_{g}^{−} (ground). In this subsection, we focus on the _{u}^{+}(0) and the iso-energetic 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 (_{u}^{−} stimulated emission (

_{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

_{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).

The results of simulation can be summarized as follows: _{u}^{+}(0) + 1B_{u}^{−}(1) and 1B_{u}^{+}(0) + 1B_{u}^{−}(2), respectively. After the 1 ← 0 excitation, the vibrational relaxation of 1B_{u}^{+}(1) + 1B_{u}^{−}(2) → 1B_{u}^{+}(0) + 1B_{u}^{−}(1) and 1B_{u}^{+}(1) + 1B_{u}^{−}(3) → 1B_{u}^{+}(0) + 1B_{u}^{−}(2), respectively, take place. _{u}^{+}(0) stimulated emission is observed, whereas after the 1 ← 0 excitation, only the 1B_{u}^{+}(1) stimulated emission, instead. _{g}^{−} counterpart _{u}^{+}(1) → 1B_{u}^{+}(0) is observed at all in the set of SADS.

_{u}^{+} + 1B_{u}^{−} diabatic pair, and the bent and short arrows, internal conversion and vibrational relaxation, respectively. (a) _{u}^{+} and 1B_{u}^{−} diabatic pair and the vibrational relaxation of the diabatic pair, _{u}^{+} counterpart, are seen. (b) _{g}^{−} optically-forbidden counterpart _{u}^{+} manifold is seen in Cars (_{u}^{+} + 3A_{g}^{−} diabatic electronic mixing never takes place, and the 1B_{u}^{+} → 1B_{u}^{−} diabatic internal conversion, instead, takes place very efficiently.

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 (_{u}^{+} stimulated emission directly transforms into the 1B_{u}^{−} transient absorption in Cars (_{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 (

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. [_{g}^{−} signal in the pump-probe time-resolved spectra of Cars (

The above set of results lead us to the following symmetry selection rule, concerning the diabatic electronic mixing and diabatic internal conversion: ‘1B_{u}^{+}-to-1B_{u}^{−} is allowed but 1B_{u}^{+}-to-3A_{g}^{−} is forbidden’. This selection rule has been theoretically explained [

_{i} and _{2v} symmetries and (b) an energy diagram for _{g}^{−}(0) energy of 18,700 cm^{−1} in _{g}^{−}(0) regression line reproduced from ^{−1} including the 2A_{g}^{−}(0), 1B_{u}^{−}(0), 3A_{g}^{−}(0) and 1B_{u}^{+}(0) origins can be formed, a unique property of this particular Car.

_{u}^{+}(1) level (the absorption maximum) of all-

_{u}^{+} → 1A_{g}^{−} stimulated emission (red broken line) and the bleaching of the 1B_{u}^{+} ← 1A_{g}^{−} absorption (black dotted line) as well as the Gaussian profiles from the 1B_{u}^{−}(1) (blue broken line) and 3A_{g}^{−}(0), 3A_{g}^{−}(1), 3A_{g}^{−}(2) and 3A_{g}^{−}(3) (green broken line) vibronic levels. In the calculation of the Frank-Condon simulation, we analyzed the fluorescence data of _{u}^{−} potential treating collectively the C=C and C–C stretching modes. Here, the Gaussian profiles of the 1B_{u}^{−}(1) level is an approximation (actually it has a pair of wings on both sides; see _{g}^{−}(0) – 3A_{g}^{−}(3) 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-_{g}^{−}(0) and 3A_{g}^{−}(1), whereas 15-_{u}^{−}(1), 3A_{g}^{−}(0), 3A_{g}^{−}(1), 3A_{g}^{−}(2) and 3A_{g}^{−}(3), both in addition to a weak Franck-Condon profile from the 1B_{u}^{+}(1) level. This spectral change, on going from all-_{i} to _{2v}. _{g}^{−} Gaussian peaks seem to be enhanced after addition of IL.

_{u}^{+} → 1A_{g}^{−} stimulated emission (Franck-Condon) components and the 3A_{g}^{−} → 1A_{g}^{−} plus 1B_{u}^{−} → 1A_{g}^{−} stimulated emission (Gaussian) components. The overlapped 1B_{u}^{+} + 3A_{g}^{−} vibronic levels as well as the isolated 3A_{g}^{−}(0) and 1B_{u}^{−}(1) levels can give rise to a progression of peaks due to the multiple 1B_{u}^{+}(_{g}^{−}(_{u}^{−}(1) radiative resonance transitions to facilitate the mutual transfer of phonons (vibrational energy) between a pair of molecules in aggregates.

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 [_{g}^{−} potential being left to be determined. However, we saw a progression of the Gaussian-type 3A_{g}^{−} stimulated emission in isomeric _{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.

_{u}^{+}(0) + 3A_{g}^{−} (υ = 1–3) diabatic levels of Car (_{u}^{+}(0) stimulated emission, a set of peaks assignable to the 3A_{g}^{−} _{g}^{−}(0) energy obtained by the measurement of resonance-Raman excitation profiles (_{g}^{−}

The time-dependent changes in the fluorescence and absorption patterns can be characterized as follows: (1) The 3A_{g}^{−} stimulated emission appears first as a single peak having the 3A_{g}^{−}(0) energy; no vibrational structures ascribable to the Franck-Condon factors are seen at all. (2) The 3A_{g}^{−} transient absorption appears next also as a single peak having the 3A_{g}^{−}(0) energy. No vibrational structures are seen in the 3A_{g}^{−} transient absorption either. (3) The 3A_{g}^{−} transient absorption is longer-lived and gets stronger in intensity than the 3A_{g}^{−} stimulated emission at later delay times, the former becomes overlapped with the 1B_{u}^{−} transient absorption having a vibrational structure and less clear.

_{g}^{−} potential in reference to the 1A_{g}^{−} potential. Under this condition, all the vibrational wavefunctions in both the upper 3A_{g}^{−} and the lower 1A_{g}^{−} states become orthogonal, and _{g}^{−}(2) ↔ 1A_{g}^{−}(2), 3A_{g}^{−}(1) ↔ 1A_{g}^{−}(1) and 3A_{g}^{−}(0) ↔ 1A_{g}^{−}(0) emissive or absorptive transitions become

(a) The Car (_{u}^{+}(0) + 3A_{g}^{−}(2) diabatic pair (see Section 3.2 for the details of this state). Following the processes of vibrational relaxation in the 3A_{g}^{−} manifold, the 3A_{g}^{−}(2) → 1A_{g}^{−}(2), 3A_{g}^{−}(1) → 1A_{g}^{−}(1) and 3A_{g}^{−}(0) → 1A_{g}^{−}(0) stimulated emission, having exactly the same energy as that of the 3A_{g}^{−}(0) ← 1A_{g}^{−}(0) transition (_{g}^{−}(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}^{−}(0) ← 1A_{g}^{−}(0), 3A_{g}^{−}(1) ← 1A_{g}^{−}(1) and 3A_{g}^{−}(2) ← 1A_{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 (

The unique excited-state dynamics after coherent excitation with ∼30 fs pulses to the diabatic pair of Cars (_{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 _{2h} to _{s}. (b) The spectrally-broad ∼30 fs pulses enable the

To facilitate the spectral comparison, the initial stimulated-emission patterns that have been presented in

The pump-probe time-resolved spectra of Cars (_{u}^{+}(0) level._{u}^{+}(0) stimulated emission appears. _{u}^{+}(0) + X^{−}^{−}(υ) indicates the 1B_{u}^{−}(1), 1B_{u}^{−}(2), 3A_{g}^{−}(1), 3A_{g}^{−}(2) and 3A_{g}^{−}(3) levels for Cars (_{u}^{+} + X^{−}(υ) diabatic pair appears. This particular peak even tends to split into two in Car (_{u}^{+}(0) ← 1A_{g}^{−}(0) absorption and the 2A_{g}^{−} transient-absorption remain. _{u}^{−}(0)_{g}^{−}(0) and 1B_{u}^{+}(0) stimulated emission peaks._{u}^{−}(0) stimulated emission, appears in Cars (_{u}^{+}(0) stimulated emission ascribable to the 1B_{u}^{+}(0) → 1A_{g}^{−}(1) transition appears, while stimulated emission from the 1B_{u}^{+}(0) + X^{−}(υ) diabatic pair predominates. Later, both the 1B_{u}^{+}(0) and 1B_{u}^{−}(0) stimulated emission peaks are replaced by the 1B_{u}^{−} transient absorption peaks.

The above spectral characteristics can be explained in terms of the mechanism of quantum beat (see _{u}^{+}(0) + 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 _{u}^{+} and optically-forbidden 1B_{u}^{−} states; for the rigorous theoretical description of this quantum beat mechanism, see Eq. 48 in Ref. [_{u}^{−}(0) level that is responsible for stimulated emission is the result of vibrational relaxation from the 1B_{u}^{−}(2) optically-forbidden counterpart of the diabatic pair; this is attributed to one of the incoherent terms. (iii) The 1B_{u}^{+}(0) level exhibits strong 1B_{u}^{+}(0) → 1A_{g}^{−}(1) stimulated emission in this particular Car; this is ascribable to the other incoherent term.

Next, we proceed to the case of Car (_{u}^{−}(2) is replaced by 3A_{g}^{−}(1) as shown in _{u}^{+}(0) + 3A_{g}^{−}(1) pair (which follows the initial weak stimulated emission from the pure 1B_{u}^{+}(0) level) exhibits a strong, broad and long-lived peak. (ii) The 3A_{g}^{−}(0) stimulated emission is weakly observed. (iii) The pure 1B_{u}^{+}(0) → 1A_{g}^{−}(0) stimulated emission is probably hidden in the diabatic 1B_{u}^{+}(0) + 3A_{g}^{−}(1) profile. The stimulated emission due to the 1B_{u}^{+}(0) → 1A_{g}^{−}(1) transition is not clearly seen at all, most probably it is overlapped with the 3A_{g}^{−}(0) → 1A_{g}^{−}(0) stimulated emission.

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: _{u}^{+}(0) and 3A_{g}^{−}(1) pair of Car (^{−1}, whereas that between the 1B_{u}^{+}(0) and 1B_{u}^{−}(2) levels of Car (_{u}^{+}(0) + 3A_{g}^{−}(1) diabatic pair in Car (

As shown in Section 3.2, the coherent excitation of Cars (_{u}^{+}(0) + X^{−}(υ) diabatic pair _{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 (

_{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 clearly-split two peaks. Simultaneously, the 1B_{u}^{+}(0) → 1A_{g}^{−}(1) and the 1B_{u}^{−}(0) → 1A_{g}^{−}(0) weak stimulated emission peaks emerge. The set of stimulated emission components can be explained in terms of the quantum beat mechanism, as in the case of these Cars in THF solution. The latter pair of stimulated emission is replaced by the 1B_{u}^{−} transient absorption, and then by the combination of the _{u}^{+} ← 2A_{g}^{−} and T_{n} ← T_{1} transient absorption peaks.

The above sequence of events can be proven by the SVD and global-fitting analysis of bound Car (_{u}^{+} + 1B_{u}^{−}, 1B_{u}^{+} and 1B_{u}^{−} stimulated emission → the 1B_{u}^{−} transient absorption → the 2A_{g}^{−} and T_{1} transient absorption. The transformation is schematically presented in

_{u}^{+} emission (whose structure is not clear), the strong and extremely-broad pair of stimulated-emission signals ascribable to the 1B_{u}^{+}(0) + 3A_{g}^{−}(1) and 1B_{u}^{+}(1) + 3A_{g}^{−}(2) diabatic pairs appear. Simultaneously, a weak stimulated emission signal ascribable to the 3A_{g}^{−}(0) → 1A_{g}^{−}(0) transition appears and becomes replaced by the positive 1B_{u}^{−} transient-absorption signal. Eventually, it transforms into the 2A_{g}^{−} and T_{1} transient-absorption signals.

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 _{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

_{u}^{+}(0), 1B_{u}^{−}(0) and 3A_{g}^{−}(0) energies of Cars (_{u}^{−}(0) and 3A_{g}^{−}(0) levels than in the ionic 1B_{u}^{+}(0) level. The results strongly suggest the polarization of the conjugated chain, which must enhance the electronic mixing of the diabatic pair.

_{u}^{−}(0) and 3A_{g}^{−}(0) levels as well as the much broader stimulated emission from the 1B_{u}^{+}(0) + X^{−}(υ) diabatic pair strongly support the idea of enhanced polarization of the Car conjugated chain and the resultant stronger diabatic interaction. (b) 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 [

Our preliminary Kerr-gate fluorescence spectroscopy was found contradictory in our series of attempts to determine the 1B_{u}^{−} lifetime of neurosporene, Car (_{u}^{+}(0) level and probing by the use of the NIR white continuum. By subpicosecond time-resolved Raman spectroscopy, we obtained a value of 250 fs [_{u}^{−} lifetime after excitation to the 1B_{u}^{+}(0) level [_{u}^{−} lifetime has been consistently determined to be in the range of 240–270 fs.

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 [^{−1}, which is just below and above the 1B_{u}^{+}(3) level in Car (_{u}^{+}(4) level in Cars (

_{u}^{+} manifold (

_{u}^{+}(2) and 1B_{u}^{+}(1) starting levels of vibrational relaxation in terms of the optically-allowed 1B_{u}^{+} counterpart. (b) The lifetimes of component III are 250 and 240 fs in Cars (_{u}^{−} and 3A_{g}^{−} lifetimes of the corresponding the shorter-chain and the longer-chain Cars listed in

_{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 (_{g}^{−} transitions is also seen as expected from the energy diagram (_{u}^{+}(2), 1B_{u}^{+}(1) and 1B_{u}^{+}(0) counterparts (shown in red) and by a pair of the 3A_{g}^{−} progressions designated as 3A_{g}^{−}(_{g}^{−}(_{g}^{−} progression in all-^{−4} M (facilitating aggregate formation) to record very weak fluorescence. Here, the contribution of the 3A_{g}^{−} fluorescence predominates not only in fluorescence profiles I and II, but also in fluorescence profile III.

Thus, the fluorescence patterns can be clearly classified into two groups: one, in the shorter-chain Cars (_{u}^{+}-to-1B_{u}^{−} diabatic electronic mixing in the shorter-chain Cars _{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 _{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 (_{g}^{−} lifetime in Cars (_{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 (_{u}^{+}(2) + 1B_{u}^{−}(3) → 1B_{u}^{+}(1) + 1B_{u}^{−}(2) → 1B_{u}^{+}(0) + 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 _{u}^{−} vibronic levels _{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 (_{u}^{+}(2) + 3A_{g}^{−}(3) → 1B_{u}^{+}(1) + 3A_{g}^{−}(2) → 1B_{u}^{+}(0) + 3A_{g}^{−}(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.

Pump-probe stimulated-emission and transient-absorption spectroscopy _{u}^{+}(0) and 1B_{u}^{+}(1) levels of Cars (_{u}^{+}-to-1B_{u}^{−} is allowed but 1B_{u}^{+}-to-3A_{g}^{−} is forbidden’. On the other hand, Kerr-gate fluorescence spectroscopy after selective excitation to the 1B_{u}^{+}(3) or 1B_{u}^{+}(4) level of Cars (_{u}^{+}(2) → 1B_{u}^{+}(1) → 1B_{u}^{+}(0) vibrational relaxation processes, _{u}^{+}-to-3A_{g}^{−} diabatic electronic mixing and diabatic internal conversion become allowed. 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 _{2h} symmetry of the conjugated chain.

The above results demonstrate that _{u}^{−} state _{g}^{−} state can play important roles in the light-harvesting function: While the Cars (_{g}^{−} energy’ to BChl as far as the pair of pigment molecules are in close contact.

Pump-probe stimulated-emission and transient-absorption spectroscopy _{u}^{+}(0) level of Cars (_{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 (

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 _{g}^{−}(0) → 1B_{u}^{+}(0) to the 1A_{g}^{−}(0) → 3A_{g}^{−}(0) transition as well as from the 1A_{g}^{−}(0) → 1B_{u}^{−}(1) to the 1A_{g}^{−}(0) → 1B_{u}^{+}(0) transition. Further, FWM spectroscopy of Car (_{g}^{−}(0) → 1B_{u}^{+}(0) and the 1A_{g}^{−}(0) → 3A_{g}^{−}(1) transitions, which decayed with a coherence lifetime of as long as 1.06 ps.

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,

Pump-probe spectroscopy after _{u}^{+}-to-1B_{u}^{−} is allowed but 1B_{u}^{+}-to-3A_{g}^{−} is 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 _{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.

After our proposal of the 1B_{u}^{−} and 3A_{g}^{−} states, a variety of hidden states have been proposed between the optically-allowed 1B_{u}^{+}(S_{2}) and the optically-forbidden 2A_{g}^{−} (S_{1}) states. They include the S^{*} [_{x}^{‡} [

A comprehensive and elaborate review has been published by Polivka and Sundström [_{u}^{−} and S^{*} states is given. Despite of self-sacrificing and pains-taking effort, these authors had to conclude that the present proposals are still controversial.

An interesting review, based on femtosecond pump-probe electronic-absorption and subpicosecond stimulated-Raman spectroscopy, was published by Hashimoto, Yoshizawa, De Silvestri and Cogdell [

The present readers also need to understand what are written in this article is just along an attempt how far the present authors can proceed based on the following simple and systematic picture:

As described in Section 1 (Introduction), the present authors introduced the symmetry notation, 1B_{u}^{−} and 3A_{g}^{−}, based on the Pariser-Parr-Pople calculations done by Tavan and Schulten, including multi-reference double configurational interaction (PPP-MRDCI) [_{g}^{−}, 1B_{u}^{−} and 3A_{g}^{−} states’, in the ratio of 2:3.1:3.8 determined by the measurement of resonance-Raman excitation profiles for Cars (_{u}^{+} and the optically-forbidden 1B_{u}^{−} or 3A_{g}^{−} vibronic levels. They have found that just introducing the concept of diabatic interaction to the potential functions of the four low-lying singlet states (1B_{u}^{+}, 2A_{g}^{−}, 1B_{u}^{−} and 3A_{g}^{−}), they determined spectroscopically, was enough to explain the excited-state properties of the overlapped vibronic pairs.

To the best of the present authors’ knowledge, no corresponding work has been published. Therefore, it is still immature to write ‘a review article’. However, the motivation to publish ‘a summary’ of our most recent work is as follows: (1) The overlap of the vibrational ladders between the optically-allowed 1B_{u}^{+} and optically-forbidden 1B_{u}^{−} or 3A_{g}^{−} state is an intrinsic property of all-_{u}^{+} + 1B_{u}^{−} or 1B_{u}^{+} + 3A_{g}^{−}, exhibit sometimes the 1B_{u}^{−} or 3A_{g}^{−} property, in addition to the 1B_{u}^{+} property, depending on (a) the way of pulsed excitation (selective excitation with subpicosecond pulses or coherent excitation with femtosecond pulses), (b) 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 [_{u}^{+} + 1B_{u}^{−} pair, after selective excitation with 100 fs pulses in nonpolar solvent, as well as _{u}^{+} + 1B_{u}^{−} pair, after coherent excitation with 30 fs pulses in polar solvent. They must help to understand the subtle 1B_{u}^{+} and 1B_{u}^{−} transition dipole moments. Rather confusing ‘1B_{u}^{+}’ and ‘1B_{u}^{−}’ lifetimes may also depend on such experimental conditions. (ii) _{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) _{u}^{+} lifetime depends on the excited vibrational level in the 1B_{u}^{+} manifold (bottom or higher).

Here, in the rest of this section, we will describe the results of _{u}^{+} state of lutein,

In this theoretical paper, multi-reference Møller-Plesset perturbation theory with complete active-space configurational interaction (CASCI-MRMP) was applied to calculate the energies of the vertical π → π* transitions of all-_{u}^{+} state is the lowest optically-allowed excited state, while the covalent 2A_{g}^{−}, 1B_{u}^{−} and 3A_{g}^{−} states are the optically-forbidden states increasing in energy in this order. The calculations predict that the 1B_{u}^{−} state becomes lower than the 1B_{u}^{+} state at _{g}^{−} state becomes lower than the 1B_{u}^{+} state at

It was a challenge for the theoreticians to carry out highly accurate

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 π electrons were treated as active electrons. The effect of σ 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}^{−}, 1B_{u}^{+}, 2A_{g}^{−}, 1B_{u}^{−} and 3A_{g}^{−} states could be characterized by the use of it.

The calculated vertical-excitation energies did not exhibit a simple linear dependence on 1/(2_{0},

_{u}^{+}, 2A_{g}^{−}, 1B_{u}^{−} and 3A_{g}^{−}, determined by the measurement of resonance-Raman excitation profiles (

Following a preliminary report on _{u}^{+} state and ascribed it to quantum beat due to coherence coupling between the 1B_{u}^{+} and 1B_{u}^{−} vibronic levels [_{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 _{u}^{+} and 1B_{u}^{−} energies and potentials have been _{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 _{u}^{+} + 1B_{u}^{−} diabatic pair’ in 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.

The authors thank Heiko Lokstein for reading the manuscript. All the work described here was supported by the Open Research Center Project (entitled “Research Center of Photo-Energy Conversion”) of MEXT (Ministry of Education, Culture, Sports, Science and Technology, Japan).

The authors also thank reviewers who encouraged us to include other proposals of the hidden states.

The authors acknowledge permission by Prof. Alfred Holzwarth, RSC Publishing and Elsevier for use in this article of tables and figures that appeared in previous publications.

_{u}

^{+}→ 3A

_{g}

^{−}→ 1B

_{u}

^{−}→ 2A

_{g}

^{−}internal conversion in carotenoids following the energy-gap law identified by 5 fs spectroscopy

_{u}

^{+}→ 3A

_{g}

^{−}→ 1B

_{u}

^{−}→ 2A

_{g}

^{−}→ 1A

_{g}

^{−}in all-

^{1}B

_{u}

^{−}state in carotenoid-to-bacteriochlorophyll singlet-energy transfer in the LH2 antenna complexes from

_{u}

^{+}and optically-forbidden 1B

_{u}

^{−}or 3A

_{g}

^{−}vibronic levels of carotenoids: Effects of diabatic mixing as determined by Kerr-gate fluorescence spectroscopy

_{g}

^{−}potential in longer-chain carotenoids as revealed by a single persistent peak of 3A

_{g}

^{−}→ 1A

_{g}

^{−}stimulated emission followed by 3A

_{g}

^{−}← 1A

_{g}

^{−}transient-absorption

_{u}

^{−}and 3A

_{g}

^{−}states located just below the 1B

_{u}

^{+}state

_{g}

^{−}state in all-

_{u}

^{−}(0) level and the 1B

_{u}

^{+}(0) + 1B

_{u}

^{−}(1 and 2) diabatic levels upon excitation to the 1B

_{u}

^{+}(0) level in neurosporene and spheroidene

_{u}

^{+}, 1B

_{u}

^{−}and 3A

_{g}

^{−}states of carotenoids bound to LH2 antenna complexes from purple photosynthetic bacteria

_{u}

^{+}, 1B

_{u}

^{−}and 2A

_{g}

^{−}states of all-

_{u}

^{+}, 1B

_{u}

^{−}and 2A

_{g}

^{−}states of all-

_{u}

^{+}→ 1B

_{u}

^{−}→ 2A

_{g}

^{−}and fluorescence from the 1B

_{u}

^{−}state in all-

_{u}

^{+}state of carotenoids as determined by Kerr-gate fluorescence spectroscopy

_{2}→S

_{1}internal conversion in

Energies of the 1B_{u}^{+}(0), 3A_{g}^{−}(0), 1B_{u}^{−}(0) and 2A_{g}^{−}(0) vibronic levels of Cars (_{1} state determined by emission spectroscopy of Cars (_{x}_{y}_{1} state of BChls in LH2 and LH1 complexes are also shown (Reproduced from Ref. [

Chemical structures of typical Cars (

The vibrational ladder of the 1B_{u}^{+} state (labeled on the right-hand-side) overlapped with those of the 1B_{u}^{−} state in Cars (_{g}^{−} state in Cars (^{−1} (Reproduced from Ref. [

The 1B_{u}^{+}, 1B_{u}^{−} and 3A_{g}^{−} potentials and conical intersections between the 1B_{u}^{+} and 1B_{u}^{−} potentials in Cars (_{u}^{+} and 3A_{g}^{−} or 1B_{u}^{−} potentials in Cars (

Intensity profiles as well as numerical correlations (inset) between the time duration (FWHM_{t}) and the spectral width (FWHM_{σ}) for the 30, 60 and 100 fs pulses.

Subpicosecond time-resolved spectra after excitation with ∼100 fs pulses to the 1B_{u}^{+}(0) level of Cars (

Species-associated difference spectra (SADS) and time-dependent changes in population for the 1B_{u}^{+} and 1B_{u}^{−} states of Cars (_{u}^{+} and 3A_{g}^{−} states for Cars (

Subpicosecond time-resolved spectra after excitation with ∼100 fs pulses to the 1B_{u}^{+}(0) level of Cars (

After excitation to the 1B_{u}^{+}(1) level, instead; see the caption of

Fitting to the initial stimulated-emission profiles by the use of Franck-Condon factors for Cars (_{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. [

Diabatic electronic mixing between the 1B_{u}^{+} and 1B_{u}^{−} vibronic levels accompanying simultaneous stimulated emission in Cars (_{u}^{+} to 1B_{u}^{−} vibronic level in Cars (_{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. [

_{2h} and _{2v} symmetries. _{u}^{+}(0) level in _{g}^{−}, 1B_{u}^{−} and 2A_{g}^{−} levels determined by measurement of resonance-Raman excitation profiles (taken from _{g}^{−}(0) energy of 18,700 cm^{−}^{1} and the 3A_{g}^{−} regression line shown in _{u}^{+}, 3A_{g}^{−}, 1B_{u}^{−} and 2A_{g}^{−} states assuming a spacing of 1,300 cm^{−}^{1} (Reproduced from Ref. [

Subpicosecond time-resolved spectra after excitation with ∼100 fs pulses to the 1B_{u}^{+}(1) level of all-

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 _{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

Vibronic levels of the 1B_{u}^{+}, 3A_{g}^{−} and 1B_{u}^{−} states giving rise to the stimulated emission profiles shown in

Time-resolved stimulated-emission and transient-absorption spectra after _{u}^{+}(0) level of Cars (_{g}^{−} stimulated-emission and transient-absorption peaks as well as the 1B_{u}^{−} transient-absorption profile with a vibrational structure are indicated (Reproduced from Ref. [

The 1B_{u}^{+}(0) energies of Cars (_{g}^{−}(0) and 1B_{u}^{−}(0) energies determined by the stimulated-emission peaks in

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 stimulated-emission peak to a the single transient-absorption peak shown in _{g}^{−} and 1A_{g}^{−} states become orthogonal, and only the downward and upward transitions indicated by vertical arrows become allowed (Reproduced from Ref. [

Time-resolved stimulated-emission and transient-absorption spectra after coherent excitation with ∼30 fs pulses aiming at the 1B_{u}^{+}(0) level of Cars (_{u}^{+}(0) emission, the strong and broad persistent peak from the 1B_{u}^{+}(0) + X^{−}(υ) diabatic pair appears in each Car; here, the X^{−}(υ) level corresponds to the 1B_{u}^{−}(1), 1B_{u}^{−}(2) and 3A_{g}^{−}(1) levels in Cars (_{u}^{−}(0) → 1A_{g}^{−}(0), 3A_{g}^{−}(0) → 1A_{g}^{−}(0) and 1B_{u}^{+}(0) → 1A_{g}^{−}(1) stimulated emission peaks are also seen together with the 1B_{u}^{−} and 2A_{g}^{−} transient absorption peaks (Reproduced from Ref. [

A set of stimulated-emission patterns obtained by _{u}^{+}(0) level of Cars (_{u}^{+}(0) level of the same set of Cars in THF solution (

Schemes of relaxation processes when the set of Cars (_{u}^{+}(0) + X^{−}(υ) diabatic pairs (for the notation of X^{−}(υ), see the caption of _{u}^{+}(0) + 1B_{u}^{−}(2) diabatic pair as the coherent cross term, (ii) the short-lived 1B_{u}^{+}(0) stimulated emission as one of the split incoherent terms and (iii) the short-lived 1B_{u}^{−}(0) stimulated emission as the other split incoherent term. Also, the 1B_{u}^{+}(0) species internally-converts to 1B_{u}^{−}(2) and, then, vibrationally-relaxes to the 1B_{u}^{−}(0) species, which gives rise to transient absorption. In Car (_{u}^{−}(2) is replaced by 3A_{g}^{−}(1) in (i) and (iii).

Time-dependent changes in the integrated intensity of stimulated emission at the position of the 1B_{u}^{+}(0) energy in Car (_{u}^{+}(0) stimulated emission and the later stationary phase, stimulated emission from the 1B_{u}^{+}(0) + 3A_{g}^{−}(1) diabatic pair. After the subtraction of the background time profile, an oscillatory change emerges. The result supports the presence of the persistent stimulated emission generated by the quantum beat mechanism (Reproduced from Ref. [

Time-resolved stimulated-emission and transient-absorption spectra after coherent excitation with ∼30 fs pulses aiming at the 1B_{u}^{+}(0) levels of Cars (

The results of SVD followed by global fitting of time-resolved data matrices of Cars (_{u}^{+}(0) level. A set of SADS (upper panels) and time-dependent changes in population (lower panels) are shown. The sequential changes in the stimulated-emission and transient-absorption patterns (SADS) extracted from time-resolved spectra (

Relaxation schemes after coherent excitation to the 1B_{u}^{+}(0) level of Cars (

Comparison of the 1B_{u}^{+}(0), 1B_{u}^{−}(0) and 3A_{g}^{−}(0) energies of Cars (_{u}^{+}(0) energies were determined by conventional absorption spectroscopy, whereas the 1B_{u}^{−}(0) and 3A_{g}^{−}(0) energies, by the stimulated emission from these states obtained by coherent excitation of the diabatic pair, 1B_{u}^{+}(0) + X^{−}(υ) (Reproduced from Ref. [

Time-resolved Kerr-gate fluorescence spectra after selective excitation with ∼100 fs pulses to the 1B_{u}^{+}(3) level of the shorter-chain Cars (_{u}^{+}(4) level of the longer-chain Cars (

Results of SVD followed by global fitting of fluorescence data matrices of Cars (

Simulation of fluorescence patterns obtained as SAFS (taken from _{u}^{+} and 1B_{u}^{−} vibronic levels and a pair of progressions of the Gaussian-type downward transitions from the 3A_{g}^{−} vibronic levels (Reproduced from Ref. [

Mechanisms why the apparent 1B_{u}^{+}(0) lifetime agrees with the 1B_{u}^{−} lifetime in Cars (_{g}^{−} lifetime in Cars (_{u}^{+} potential, the 1B_{u}^{+}(0) diabatic counterpart relaxes through stimulated emission, whereas the 1B_{u}^{−} (3A_{g}^{−}) diabatic counterpart in the former (in the latter) set of Cars through internal conversion to the iso-energetic 2A_{g}^{−} vibronic level with the 1B_{u}^{−} (3A_{g}^{−}) lifetime (Reproduced from Ref. [

The energies of the optically-allowed 1B_{u}^{+} and the optically-forbidden 2A_{g}^{−}, 1B_{u}^{−} and 3A_{g}^{−} states (for vertical transition from the ground 1A_{g}^{−} state) that have been obtained by

Quantum beat due to coherence coupling between the 1B_{u}^{+} and 1B_{u}^{−} vibronic levels in lutein,

_{u}^{+} and 1B_{u}^{−} fluorescence and _{u}^{+} and 1B_{u}^{−} potential curves obtained by theoretical calculation based on (a). ESA and SE stand for excited-state absorption and stimulated emission (Reproduced from Ref. [

Dependence of the 1B_{u}^{+}, 1B_{u}^{−}, 3A_{g}^{−} and 2A_{g}^{−} lifetimes (in ps) on the number of conjugated double bonds (

Neurosporene ( |
Spheroidene ( |
Lycopene ( |
Anhydrorhodovibrin ( |
Spirilloxanthin ( | |
---|---|---|---|---|---|

1B_{u}^{+} |
0.10 | 0.10 | 0.02 | 0.01 | 0.01 |

1B_{u}^{−} |
0.24 | 0.23 | – | – | – |

3A_{g}^{−} |
– | – | 0.15 | 0.10 | 0.10 |

2A_{g}^{−} |
24.0 | 8.9 | 3.9 | 2.2 | 1.4 |