- Email: [email protected]
Theoretical description of diabatic mixing and coherent excitation in singlet-excited states of carotenoids
Chemical Physics Letters 474 (2009) 342–351
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Theoretical description of diabatic mixing and coherent excitation in singlet-excited states of carotenoids Hiroyoshi Nagae a, Yoshinori Kakitani b, Yasushi Koyama b,* a b
Kobe City University of Foreign Studies, Gakuen-Higashimachi, Nishi-ku, Kobe 651-2187, Japan Faculty of Science and Technology, Kwansei Gakuin University, Gakuen, Sanda 669-1337, Japan
a r t i c l e
i n f o
Article history: Received 7 January 2009 In final form 15 April 2009 Available online 18 April 2009
a b s t r a c t This Letter discusses the following characteristic excited-state properties of closely-overlapped ‘diabatic’ states of carotenoids, including 1Bþ u ; 1Bu and 3Ag , on which we have reported systematic experimental studies. A rigorous general formulation is developed and applied for the adiabatic and diabatic descriptions in which the breakdown of the Born–Oppenheimer approximation is used as a criterion to differentiate them: (i) the observed selection rules concerning electronic mixing and internal conversion between the 1Bþ u and 1Bu or 3Ag diabatic levels after incoherent excitation, and (ii) a long lifetime of the apparent þ 1Bu stimulated emission and the oscillatory change in the fluorescence intensity after coherent excitation of both pairs of diabatic levels. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction The 1B u and 3Ag states of carotenoids (Cars) were first identi fied in 2002, in addition to the well-known 1Bþ u and 2Ag states, by the measurement of resonance-Raman excitation profiles [1]. When the conjugated chain is in the all-trans configuration having C2h symmetry, the optical transition from and to the ground 1A g state is allowed for the 1Bþ u state, but it is forbidden for the 1Bu ; 3Ag and 2Ag states; the former and latter states are hereafter denoted as ‘optically-allowed and -forbidden states’, respectively. Fig. 1a shows that the energies of the low-lying singlet states decrease in proportion to 1/(2n + 1), where n is the number of conjugated double bonds, while Fig. 1b shows that the vibrational levels of the optically-allowed 1Bþ u state approximately overlap with those of the next low-lying 1B u and 3Ag states in the shorterand longer-chain Cars, respectively, due to the unique arrangement of the origins of these singlet states, as shown in Fig. 1a. If we assume the same vibrational energy gaps of 1400 cm1 for all the electronic states, the 1Bþ u ð0Þ vibronic level (labeled on the right hand side), for example, can overlap with the 1B u ð1Þ and 1Bu ð2Þ levels (labeled on the left-hand side) in neurosporene (n = 9) and spheroidene (n = 10), but with the 3A g ð1Þ; 3Ag ð2Þ and 3Ag ð3Þ levels in lycopene (n = 11), anhydrorhodovibrin (n = 12) and spirilloxanthin (n = 13). Spectroscopically, we observed that excitation to the optically-allowed 1Bþ u ð0Þ level can cause excitation to each isoenergetic, optically-forbidden level. We call the pair of overlapped levels ‘diabatic levels’ because the energy gap between each pair of vibronic levels is so small that the observed excited-state
* Corresponding author. Fax: +81 79 565 8408. E-mail address: [email protected] (Y. Koyama). 0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2009.04.039
dynamics need to be described on the diabatic basis instead of the adiabatic basis (vide infra). We have found that the excitedstate dynamics depends on the symmetries of the pair of the diabatic levels, incoherent or coherent excitation (by the use of 100 and 30 fs pulses, respectively), and the polarity of the solvent [2–6]. We will first summarize below, as the basis of our discussion in this Letter, the crucial features of our experimental observations published in sequence and then present the rigorous diabatic theory formulated in the present study, by which we will put forth our consistent interpretation of the observed phenomena. 2. Survey of experimental results 2.1. Excitation with 100 fs pulses of all-trans Cars (n = 9–13) in nonpolar solvents: selection rules for the electronic mixing and internal conversion of a pair of diabatic vibronic levels Fig. 2 shows the pump–probe time-resolved spectra of a set of 1A Cars (n = 9–13) after (a) the 1Bþ u ð0Þ g ð0Þ and (b) the þ 1Ag ð0Þ excitations (abbreviated as 0 0 and 1 0 exci1Bu ð1Þ tations). The relaxation scheme includes singlet internal conver or 1Bþ sion: 1Bþ u þ 1Bu u ðv ide infraÞ ! 1Bu ! 2Ag ! 1Ag ; we focus our attention here on the initial stimulated-emission components. Fig. 3 shows the species-associated difference spectra (SADS) after the 0 0 and 1 0 excitations that were obtained by singular-value decomposition (SVD) followed by global fitting [2,3]. The shorter-chain Cars (n = 9 and 10) and longer-chain Cars (n = 11–13) exhibited completely different compositions in the stimulatedemission patterns.
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
n
(a)
13
12
11
spheroidene (n = 10), on the other hand, the vibrational relaxation þ 1Bþ u ð1Þ þ 1Bu ð3Þ ! 1Bu ð0Þ þ 1Bu ð2Þ took place in 40 fs, and the ð0Þ þ 1B internal conversion 1Bþ u u ð2Þ ! 1Bu ð0Þ occurred in 100 fs. The pairs of these time constants, observed in both Cars, approximately reflect the rule that the rate of vibrational relaxation from t = ‘ to t = ‘ 1 is proportional to ‘ [7]. Thus, the presence of electronic mixing causing simultaneous stimulated emission and internal conversion between the pair of 1Bþ u and 1Bu diabatic vibronic levels has been corroborated for these short-chain Cars.
9
1Bu+
20
–1 3 Energy / 10 cm
10
3A g–
16
1Bu–
2A g–
12
0.04
0.05 1 / (2n + 1) n
(b)
13
12
11
10
Energy / 103 cm–1
1Bu+
20
4
13
1 2
3
02
0 1
2
1
0
1
0
0
0
9
3
1
2
0
1
2
1
1
0
0
0 1 0
1Bu–
3A g–
16
0
0
12
0.04
343
0.05
2.1.2. Cars (n = 11–13) The stimulated-emission patterns of this set of longer-chain Cars can be characterized as follows (see the three panels of Fig. 3 on the right-hand side): (i) The patterns can be simply simulated by the stimulated emission from the 1Bþ u state and the bleaching of the ground-state absorption; no additional components are needed in the fitting. (ii) Each of the stimulated-emission patterns in the 1 0 excitation completely differs from that in the 0 0 excitation; no signs of vibrational relaxation appear at all in the 1 0 excitation. In this set of Cars, the stimulated-emission patterns in the 0 0 excitation can be simulated simply by the 0 excitation are ex1Bþ u ð0Þ emission, whereas those in the 1 ð1Þ emission in addition to the bleaching of pressed by the 1Bþ u the ground-state absorption. No electronic mixing has been found either between the 1Bþ u ð0Þ level and the 3Ag ð1Þ; 3Ag ð2Þ or 3Ag ð3Þ þ levels, or between the 1Bu ð1Þ level and the 3Ag ð2Þ; 3Ag ð3Þ or 3A g ð4Þ levels in the Cars including lycopene (n = 11), anhydrorhodovibrin (n = 12) and spirilloxanthin (n = 13), respectively. Fig. 4 (the three panels on the right-hand side) shows the relaxation dynamics for the set of these Cars. Neither electronic mixing (simultaneous stimulated emission) between the 1Bþ u and 3Ag vib þ ronic levels nor internal conversion from the 1Bu to the 3Ag vib ronic level is observed for the longer-chain Cars. The 1Bþ u to 1Bu internal conversion is observed, instead. In summary, the same selection rule applicable for the electronic mixing and internal conversion of the diabatic levels of Cars has been established; i.e., Bu-to-Bu is allowed whereas Bu-to-Ag is forbidden.
1 / (2n + 1) Fig. 1. (a) Energies of the 1Bþ u ð0Þ (d), 3Ag ð0Þ (h), 1Bu ð0Þ (s) and 2Ag ð0Þ (j) levels determined by the measurement of resonance-Raman excitation profiles for crystalline Cars, including mini-9-b-carotene, spheroidene, lycopene, anhydrorhod ovibrin and spirilloxanthin (n = 9–13, respectively) [1]. The 1B u ð0Þ (N) and 3Ag ð0Þ (D) levels determined by stimulated emission are also shown (see Section 2.2.1; here, for Car (n = 9), neurosporene was used instead). Solid regression lines are drawn, for the set of energy values, as the linear functions of 1/(2n + 1). (b) Arrangement of the 1Bþ u ; 1Bu and 3Ag vibronic levels shown in solid, dotted-broken and broken lines, respectively. The 1Bþ u ð0Þ and 1Bu ð0Þ levels were determined by ð0Þ level was taken from (a). The fluorescence spectroscopy [22,23], while the 3A g spacing of the vibrational levels is set to be 1400 cm1.
2.1.1. Cars (n = 9 and 10) The stimulated-emission patterns of this pair of Cars can be characterized as follows (see the two panels on the left-hand side): (i) Each pattern can be simulated as a sum of stimulated emission from the 1Bþ u vibronic level and that from the 1Bu level, in addition to the bleaching of the ground-state absorption. (ii) The 0 0 excitation exhibits only a single stimulated-emission pattern, whereas the 1 0 excitation exhibits two time-resolved stimulated-emission patterns, reflecting a step of vibrational relaxation. Fig. 4 schematically shows the relaxation dynamics for a pair of Cars (two panels on the left): In neurosporene (n = 9), the vibra þ tional relaxation 1Bþ u ð1Þ þ 1Bu ð2Þ ! 1Bu ð0Þ þ 1Bu ð1Þ took place in ð0Þ þ 1B 60 fs, and the internal conversion 1Bþ u u ð1Þ ! 1Bu ð0Þ (or in another viewpoint, the 1Bu vibrational relaxation), in 120 fs. In
2.2. Excitation with 30 fs pulses of all-trans Cars (n = 9–13) in a polar solvent: coherent excitation of a pair of diabatic vibronic levels Fig. 5 shows the 30 fs pump–probe time-resolved spectra of Cars (n = 9–13) in THF, a polar solvent [5,6]. A spectral width of 1000 cm1 for the 30 fs pulse can coherently excite the optically-active 1Bþ u ð0Þ level and the isoenergetic optically-forbidden 1B u or 3Ag diabatic level. The time-resolved spectral patterns are substantially different from the case of incoherent excitation with 100 fs pulses in non-polar solvents (Fig. 2), reflecting completely different excited-state dynamics. As for the 1Bþ u stimulated emission, for example, we do not observe at all either the strong vibrational structures that are ascribable to the Franck–Condon downward transitions as stimulated emission or the bleaching of the ground-state absorption. In the longest-chain Cars (n = 12 and 13), we do observe þ the process of 1Bþ u ð1Þ ! 1Bu ð0Þ vibrational relaxation, indicating þ the simultaneous excitation of the 1Bþ u ð1Þ and 1Bu ð0Þ levels by the timely-short and spectrally-broad 30 fs pulses. After this vibrational relaxation, we observe a single persistent peak with the 1Bþ u ð0Þ energy as in the case of the shorter-chain Cars (n = 9 and 10). We will report in the following the three pieces of information obtained from the particular pump–probe time-resolved spectra shown in Fig. 5, which include (a) stimulated emission from the þ 1B u ð0Þ or 3Ag ð0Þ level, (b) vibrational relaxation in the 1Bu ; 1Bu þ and 2Ag manifolds, and (c) a long-lived emission with the 1Bu ð0Þ
344
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
Neurosporene (n = 9) ← (a) 0 0 excitation
ΔOD = 0.19
Spheroidene (n = 10)
ΔOD = 0.12
Lycopene (n = 11)
Anhydrorhodovibrin (n = 12)
ΔOD = 0.10
ΔOD = 0.09
Spirilloxanthin (n = 13)
–0.40 ps
ΔOD = 0.09
–0.10 –0.06
1B+u + 1B–u
1B+u + 1B–u
1B+u
1B+u
1B+u
1B–u
1B–u
–0.02 0.00 0.02 0.06
1B–u
0.10 0.12
1B–u
0.16
1B–u
0.20 2A–g 2A–g
0.30
2A–g
0.50
2A–g
1.00
2A–g
3.00 5.00 10.00 15.00
16
20 24 Energy / 103 cm–1
20 24 Energy / 103 cm–1
20 24 Energy / 103 cm–1
ΔOD = 0.10
ΔOD = 0.10
16 20 Energy / 103 cm–1
16 20 24 Energy / 103 cm–1
(b) 1 ← 0 excitation
ΔOD = 0.44
ΔOD = 0.12
– 0.40 ps
ΔOD = 0.05
– 0.10 – 0.06
1B+u + 1B–u
1B+u + 1B–u
1B+u
1B+u
1B+u
1B–u
1B–u
– 0.02 0.00 0.02 0.06
1B–u
0.10 0.12
1B–u
0.16
1B–u
0.20 2A–g 2A–g
0.30
2A–g
0.50
2A–g
1.00
2A–g
3.00 5.00 10.00 15.00 16
20 24 Energy / 103 cm–1
20 24 Energy / 103 cm–1
20 24 Energy / 103 cm–1
16 20 Energy / 103 cm–1
16 20 24 Energy / 103 cm–1
Fig. 2. Pump–probe time-resolved stimulated-emission and transient-absorption spectra of Cars (n = 9–13) in non-polar solvents (n-hexane for n = 9 and 10 and mixtures of n-hexane and benzene for n = 11–13) after (a) the ‘0 0’ and (b) the ‘1 0’ excitations with 100 fs pulses.
345
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
Neurosporene (n = 9) (a) 0 ← 0 excitation
Spheroidene (n = 10)
1Bu+(0) + 1Bu–(2)
1Bu+(0) + 1Bu–(1)
16
Lycopene (n = 11)
20 24 Energy / 103 cm–1
Anhydrorhodovibrin (n = 12)
1Bu+(0)
20 24 16 Energy / 103 cm–1
Spirilloxanthin (n = 13)
1Bu+(0)
20 24 Energy / 103 cm–1
16
1Bu+(0)
20 Energy / 103 cm–1
16
20 Energy / 103 cm–1
(b) 1 ← 0 excitation
1Bu+(1) + 1Bu–(2)
1Bu+(1) + 1Bu–(3)
1Bu+(1)
1Bu+(1)
20 24 Energy / 103 cm–1
20 Energy / 103 cm–1
24 16
20 Energy / 103 cm–1
24
1Bu+(0) + 1Bu–(2)
1Bu+(0) + 1Bu–(1)
16
16
1Bu+(1)
20 24 Energy / 103 cm–1
20 24 Energy / 103 cm–1
Fig. 3. Fitting to the initial stimulated-emission components (SADS) by the use of Franck–Condon factors for Cars (n = 9–13). Specifications of the lines used are as follows: þ For all Cars, SADS (black solid line), bleaching of the ground-state absorption (black dotted line), emission from the 1Bþ u ð0Þ level (blue broken line), emission from the 1Bu ð1Þ ð1Þ and 1B ð2Þ levels, respectively (green level (red broken line), and a sum of all the contributions (black dotted-broken line). For Cars (n = 9 and 10), emission from the 1B u u broken line) and emission from the 1B u ð2Þ and 1Bu ð3Þ levels, respectively (magenta broken line).
Neurosporene (n = 9)
Spheroidene (n = 10)
Lycopene (n = 11)
1Bu+
30
1Bu+
1Bu– 1B v' = 3
v' = 1
v'' = 2
Energy / 103 cm–1
20
v' = 1
1Bu+
3A g– v' = 3
1Bu– v'' = 3
v'' = 1
v' = 0
v' = 1
v' = 2
1B
v' = 0
v'' = 3
v' = 1 v' = 0
v'' = 3 v'' = 2
v'' = 1
v'' = 0
– u
v'' = 4
v'' = 2
v'' = 1
v' = 3
v' = 2
1Bu–
v'' = 2
v'' = 0
3A g–
v' = 3
v' = 2 v' = 1
v' = 0
v'' = 2
v'' = 0
1Bu+
v' = 3 v' = 2 v'' = 3
v' = 0
v'' = 1
1Bu+
Spirilloxanthin (n = 13)
3A g–
– u
v' = 2
v'' = 3
Anhydrorhodovibrin (n = 12)
v'' = 1
v'' = 0
v'' = 0
10 1A g–
1A g–
v=3
v=1
–5
0 q
5 –5
0 q
0 q
v=2
v=1
v=0
5 –5
v=3
v=2
v=1
v=1 v=0
1A g–
v=3
v=2
v=2
v=0
1A g–
v=3
v=3
v=2
0
1A g–
v=1
v=0
5 –5
0 q
v=0
5 –5
0 q
5
Fig. 4. Relaxation dynamics showing diabatic electronic mixing between the 1Bþ u and 1Bu vibronic levels accompanying simultaneous stimulated emission for Cars (n = 9 and 10) (the two panels on the left-hand side), and those showing diabatic internal conversion from the 1Bþ u to 1Bu vibronic level for Cars (n = 11–13) (the three panels on the right-hand side). In the latter, neither diabatic electronic mixing generating simultaneous stimulated emission nor diabatic internal conversion between the 1Bþ u and 3Ag vibronic levels took place. Diabatically-mixed states are shadowed, and vibrational relaxations and internal conversions are shown by short and bent arrows, respectively.
energy, which exhibit an oscillatory change in the fluorescence intensity (quantum beat). Finally, we will present the scheme that
can consistently observations.
explain
all
the
above-mentioned
set
of
346
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
Neurosporene (n = 9)
Spheroidene (n = 10)
Lycopene (n = 11)
Anhydrorhodovibrin (n = 12)
Spirilloxanthin (n = 13)
– 0.04 ps ΔOD = 0.18
ΔOD = 0.03
ΔOD = 0.02
ΔOD = 0.01
ΔOD = 0.01 – 0.02 0.00
1B
1B u+
+ u
1B u+
1B u+
1B u+ 0.02 0.04
1B
+ u
1B u–
1B
+ u
3A
– g
3A
– g
3A
– g
0.06
1B u– 0.08 0.10 0.15
1B u–
1B u– 0.20
1B u– 0.25
1B u–
1B u– 0.30 0.60 0.90 1.20 1.50
2A g–
2A g–
2A g– 3.00
2A g– 18 20 Energy / 103 cm–1
5.00
2A g– 18 20 Energy / 103 cm–1
16 18 Energy / 103 cm–1
16 18 Energy / 103 cm–1
Fig. 5. Pump–probe time-resolved spectra of Cars (n = 9–13) in THF, a polar solvent, after the 0 larger labels and downward arrows, whereas transient-absorption peaks, by smaller labels.
2.2.1. Stimulated emission from the 1B u ð0Þ and 3Ag ð0Þ levels þ Following the very first 1Bu emission, each of the shorter- and longer-chain Cars exhibits a weak stimulated-emission peak ascribable to either the 1B u or the 3Ag state. Each peak systematically shifts to lower energy with increasing n. Fig. 1a also shows a plot of the 1B u (N) and 3Ag (D) energies, as functions of 1/(2n + 1), observed here as stimulated emission, in reference to the linear dependence of the 1B u ð0Þ and 3Ag ð0Þ energies that have been determined by the measurement of resonance-Raman excitation profiles. Both linear relations practically agree completely. Since each Car is excited to the 1Bþ u ð0Þ level as well as the isoenergetic ð1Þ and 1B vibronic levels, i.e., the 1B u u ð2Þ levels in Cars (n = 9 and 10) and the 3Ag ð1Þ; 3Ag ð2Þ and 3A g ð3Þ levels in Cars (n = 11, 12 and 13), respectively, there must be some mechanism of rapid vibrational relaxation after excitation to these diabatic levels.
16 18 Energy / 103 cm–1
20
0 excitation with 30 fs pulses. Stimulated-emission peaks are indicated by
2.2.2. Vibrational relaxation in the 1Bþ u ; 1Bu and 2Ag manifolds In Cars (n = 9), transient absorptions showing vibrational relax ation from the 1B u ð1Þ to the 1Bu ð0Þ vibronic level followed by that ð1Þ to the 2A ð0Þ level are observed; the 1B from the 2A u ð1Þ and g g 2Ag ð1Þ peaks appear on the lower-energy side of the 1B u ð0Þ and 2A g ð0Þ peaks, respectively. These electronic states must be generated, through internal conversion, from the 1Bþ u ð0Þ level as one of the incoherent components (see Scheme 1 and Eq. (48), vide infra). In Cars (n = 10 and 11), however, only the 2A g ð1Þ and 2Ag ð0Þ transient absorptions are observed.
2.2.3. A long-lived stimulated emission with the 1Bþ u ð0Þ energy showing oscillatory change in intensity As shown in Fig. 5, the stimulated-emission peak with the 1Bþ u ð0Þ energy persists even longer than 5 ps in all of the Cars
347
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
3. Theory explaining the experimental observations
Neurosporene (n = 9) 1Bu– (1)
IC
1Bu+(0)
1Bu+(0) + 1Bu– (1)
hν
VR
hν
(b)
1Bu– (0)
1Bu– (1)
(c)
VR
1Bu– (0)
transient absorption
hν
3.1. Theoretical background Let us first define the adiabatic and diabatic electronic states and their properties, focusing our attention on the ground and singlet-excited states: The wavefunction of a molecular system can be generally written as
X
Wðr; Q Þ ¼
vk ðQ Þ/k ðr; Q Þ;
ð1Þ
k
(a)
where r denotes a set of electronic coordinates, Q, the nuclear normal coordinates, and {uk(r, Q)}, an orthonormal complete basis in the electronic Hilbert space; here, Q is a parameter. Substituting Eq. (1) into the time-independent Schrödinger equation leads to
Lycopene (n = 11) 1Bu– (2) VR
IC
1B u+(0)
1B u+(0) + 3Ag– (1)
hν
3A g–(1)
hν
VR
X
Hij ðQ Þvj ðQ Þ ¼ Evj ðQ Þ;
ð2Þ
l
with –
1Bu (0)
(b)
(c)
transient absorption
–
3A g (0) hν
(a) Scheme 1.
(n = 9–13). By means of near-infrared pump–probe spectroscopy of the same set of Cars, when they were incoherently excited to the þ 1Bþ u ð0Þ level by the use of 100 fs pulses, the 1Bu lifetimes were determined to be 100, 100, 20, 10 and 10 fs for Cars with n = 9, 10, 11, 12 and 13, respectively [8]. Therefore, the lifetimes of the presently-observed stimulated emissions are far beyond the lifetime of the pure 1Bþ u state. We have further observed oscillatory changes in the integrated intensity of the above-mentioned persistent peak in Car (n = 11) with an interval of 150 fs, which corresponds to an energy gap of 220 cm1 [6]. By means of spectrally-resolved photon-echo spectroscopy, Davis et al. [9] also observed an oscillation with a 120 fs time interval corresponding to the energy gap of 275 cm1 for the same Car in n-hexane. Therefore, the oscillatory changes in the fluorescence intensity can be ascribed to the ‘quantum beat’ between the pair of diabatic levels, i.e., 1Bþ u ð0Þ and 3A g ð1Þ.
Hij ðQ Þ ¼ T ij ðQ Þ þ V ij ðQ Þ þ Dij ðQ Þ þ Gij ðQ Þ; 2 2 h X @ 2 h @ 2 dij ¼ ; T ij ðQ Þ ¼ 2 2 a @Q a 2 @Q 2
ð3Þ
V ij ðQ Þ ¼ h/i jHe ðr; Q Þj/j i; @ @ 2 Dij ðQ Þ ¼ h /i /j ; @Q @Q + +! * * 2 2 @2 @2 h h @ /i 2 /j Gij ðQ Þ ¼ /i 2 /j ¼ @Q @Q 2 2 @Q 2 @ @ h X /k /i /k /j : @Q @Q 2 k
ð5Þ
ð4Þ
ð6Þ
ð7Þ
Here, He(r, Q) is the electronic Hamiltonian and the brackets indicate integration over the coordinates r. If uk(r, Q) is chosen as the eigenfunctions of He(r, Q), noting that Vij(Q) = Vii(Q)dij with Kronecker’s delta dij, the matrix Hamiltonian in Eq. (3) takes the form 2
@2 h þ V ii ðQ Þ 2 @Q 2
Evi ðQ Þ ¼
!
vi ðQ Þ h2
XX j
a
ðaÞ 2 X @F ij ðQ Þ X ðaÞ h ða Þ þ F ki ðQ ÞF kj ðQ Þ 2 a @Q a k
ðaÞ
F ij ðQ Þ
@ v ðQ Þ @Q a j
!
vj ðQ Þ;
ð8Þ
ðaÞ
2.2.4. Possible mechanisms Scheme 1 provides a possible mechanism for a pair of Cars (n = 9 and 11) based on the quantum beat generated by coherent excitation to each pair of diabatic levels, which can explain all the observations (a), (b) and (c): In lycopene (n = 11), for example, the emission after coherent excitation of the 1Bþ u ð0Þ and 3Ag ð1Þ diabatic levels can be described by the following three terms: (a) the incoherent 3A g ð1Þ term (right-hand side) generating the 3A g ð0Þ emission after rapid vibrational relaxation, (b) the incoherent 1Bþ u ð0Þ emission (left-hand side), which can subsequently generate the 1B u ð0Þ transient absorption after the 1Bþ u ð0Þ to 1Bu ð2Þ internal conversion followed by the 1B u ð2Þ ! 1Bu ð1Þ ! 1Bu ð0Þ vibrational relaxation, and (c) þ the coherent 1Bu ð0Þ and 3Ag ð1Þ cross term generating oscillatory emission reflecting the quantum beat (center). (Subsequently, the 1B u ! 2Ag internal conversion takes place as shown in Fig. 5.) Basically the same mechanism should hold in neurosporene, which is expected to generate a quantum beat between the 1Bþ u ð0Þ and 1B u ð1Þ states.
where F ij ðQ Þ is the derivative coupling, ðaÞ
F ij ðQ Þ ¼
@ / : /i @Q j
ð9Þ
a
In the adiabatic or Born–Oppenheimer (BO) approximation [10], the derivative couplings in Eq. (8) are neglected. Then, the BO equation for the nuclear motion reads 2
@2 h þ V ii ðQ Þ 2 @Q 2
!
viv ðQ Þ ¼ Eiv viv ðQ Þ;
ð10Þ
and the overall BO wave function is simply the product of /Ai ðr; Q Þ and viv(Q). Hereafter, superscripts ‘A’ and ‘D’ denote ‘adiabatic’ and ‘diabatic’ bases, respectively. By the use of the commutation relation, [He, @/@Qa] = @He/@Qa, the derivative coupling can be expressed as
@ A / ¼ /A @He /A /Ai ðV ii ðQ Þ V jj ðQ ÞÞ: j i @Q a @Q a j
ð11Þ
348
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
This equation shows that the derivative coupling becomes significant and even diverges as the energy gap reD in the denominator E duces to zero, if the vibronic coupling, /Ai j@He =@Q a j/Aj , does not vanish. The BO approximation breaks down in such a case. Then, a new orthonormal basis, in which the derivative couplings vanish and the basis functions exhibit a weak dependence on the nuclear coordinates, becomes necessary [11–13]. Such a basis, called ‘the diabatic basis’ f/Di ðr; Q Þg, can be obtained by the unitary transformation of the adiabatic BO basis as
/Di ðr; Q Þ ¼
X
/Aj ðr; Q ÞU ji ðQ Þ:
ð12Þ
j
Here, Uji(Q) can be determined so that /Di ðr; Q Þ satisfy
@ D / ¼ 0; /Di @Q a j
ð13Þ
and obey
X ða Þ @ U ij ðQ Þ ¼ F ik ðQ ÞU kj ðQ Þ: @Q a k
ð14Þ
The unitary property of the matrix Uij(Q) is due to the anti-HermicðaÞ ðaÞ ity of the derivative coupling matrix, i.e., F ij ðQ Þ ¼ F ji ðQ Þ. It has been shown that diabatic levels do not strictly exist if a truncated set of adiabatic levels is used [14]. However, if we assume the existence of an approximate diabatic basis for a truncated set of adiabatic levels, the formal solution can be written in a matrix form as
UðQ Þ ¼ exp
Z
!
Q
FðQ Þ dQ ;
ð15Þ
3.2. Electronic mixing and internal conversion between the diabatic levels of carotenoids When a Car molecule is excited to the 1Bþ u state and its conjugated chain loses perfect C2h symmetry, the diabatic description becomes necessary. (See Adiabatic 3.) We note that the diabatic levels (3A g or 1Bu ) approximately keep Ag or Bu symmetry for an arbitrary value of Q even under the conditions of so-called ‘avoided crossing’ or ‘conical crossing’ [20], due to the properties of diabatic wavefunctions as described in Section 3.1. (If the diabatic wavefunction has Ag (Bu) symmetry at Q0, as shown in Eq. (16), it should keep the same symmetry at an arbitrary value of Q shifted from Q0.) Then, we consider the eigenstate of the matrix Hamiltonian of Eq. (17) described on the diabatic basis. Assuming that the diabatic potential V Dii ðQ Þ can be approximated by a harmonic potential, we get
! 2 @2 h 1 i i i Evi ðQ Þ ¼ þ ðQ DQ Þx ðQ DQ Þ vi ðQ Þ 2 @Q 2 2 X V ij ðQ Þvj ðQ Þ; þ
where DQi stands for the shift of the potential minimum in the ith diabatic electronic pffiffiffiffiffiffiffiffiffi state. Using the dimensionless normal coordinates, qð¼ x= hQ Þ, and taking energy in the unit of angular frequency, we have
evi ðqÞ ¼ hi ðqÞvi ðqÞ þ
Q0
/Di ðr; Q 0 Þ ¼ /Ai ðr; Q 0 Þ:
ð16Þ
If we adopt a basis set so determined, the matrix Hamiltonian in Eq. (3) takes the form 2
Evi ðQ Þ ¼
!
vi ðQ Þ þ
X
V Dij ðQ Þvj ðQ Þ
ð17Þ
j
D E V Dij ðQ Þ ¼ /Di jHe ðr; Q Þj/Dj :
ð18Þ
Here, Dij(Q) and Gij(Q) becomes negligible, since they are expressed only by the derivative couplings (see Eqs. (6) and (7)). The physical meaning of the diabatic basis can be exemplified by the orthogonalized floating atomic orbitals [16,17]. (See Adiabatic 1 in Supplementary material.) Thus, we can use a set of wavefunctions with constant MO and CI coefficients, based on the orthogonalized floating atomic orbitals as a diabatic basis. According to Fermi’s golden rule, the rate of internal conversion between the ith and jth electronic states can be given as [18,19]
2p X ¼ Bu jT ij j2 dðEiu Ejv Þ; h uv
Z
ð22Þ
viu ðQ ÞV Dij ðQ Þvjv ðQ ÞdQ :
with
! 2 1 i i i @ i h ðqÞ ¼ þ ðq D Þx ðq D Þ þ ei ; x 2 @q2 i
ð23Þ
(hereafter, it is assumed that xi does not depend on the value of i, i.e., xi = x). Introducing eigenfunctions of hi(q) such that
1 i h ðqÞtin ðq Di Þ ¼ xi n þ þ ei tin ðq Di Þ; 2
vi ðqÞ ¼
X
ð24Þ
ð19Þ
ð20Þ
C i;n tin ðq Di Þ:
ð25Þ
n
Substituting this into Eq. (22), multiplying the resultant equation by tin ðq Di Þ from the left, and integrating over q, we get
1 þ ei C i;n 2 E X XD þ tin ðq Di ÞV Dij ðqÞtjm ðq Dj Þ C j;m :
eC i;n ¼ xi n þ j–i
ð26Þ
m
Solving this simultaneous equations for Ci,n, we obtain energy eigenfunctions
WðpÞ ðr; qÞ ¼
where Bu is the Boltzmann factor of the initial state, Tij is the matrix element of the perturbation between the initial and final states, and Eiu (Ejv) is the vibronic energy of the initial (final) state. (See Adiabatic 2.) In the diabatic basis, Tij can be expressed, from Eq. (17), as
T ij ¼
V Dij ðqÞvj ðqÞ;
where n represents a set of quantum numbers of vibrational modes, we can expand vi(q) in the form
with
ki!j
X j–i
where Q0 is a reference point, which can be chosen arbitrary. The diabatic basis coincides with the adiabatic basis at this reference point Q0 [15] as
@2 h þ V Dii ðQ Þ 2 @Q 2
ð21Þ
j–i
X
ðpÞ
C i;ni /Di ðr; qÞtini ðq Di Þ:
ð27Þ
i;ni
(See Adiabatic 4.) Eq. (26) shows that the condition for /Di and /Dj to mix with each other is that V Dij ðQ Þ, defined by Eq. (18), has a non-vanishing value. This guarantees the presence of the diabatic mixing between the elec D and /D1Bu ) and tronic wavefunctions of the 1Bþ u and 1Bu states (/1Bþ u the absence of the diabatic mixing between those of the 1Bþ u and D D ) since He(r, Q) has approximately the Ag and / 3A 3A g states (/1Bþ g u symmetry. Thus, the observed selection rule for diabatic mixing has been theoretically proven.
349
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
Then, we examine emission from N-degenerate diabaticallyP ðpÞ mixed states expressed by Ni¼1 C i /Di ðr; qÞtini ðq Di Þ; p ¼ 1; . . . ; N. ðpÞ Since fC i g is a unitary matrix, it follows that N X
ðpÞ
Ci
ðqÞ
C i ¼ dp;q ;
N X
ðpÞ
ðpÞ
Ci
C j ¼ di;j :
ð28Þ
p¼1
i¼1
The transition-dipole moment for the emission from these excited states to the ground state, i.e., /0 ðrÞt0O ðq D0 Þ, invoking the Condon-approximation, is given by
D
E
W0 ðr; qÞjljWðpÞ e ðr; qÞ ¼
N X
ðpÞ
Ci
i¼1
/0 ðrÞjlj/Di ðrÞ
D
E
t0O ðq D0 Þtini ðq Di Þ :
ð29Þ
E Abbreviating t D Þtini ðq Di Þ as F i;ni , we have the ensemble average of emission spectra as
D
/0 ðrÞjlj/Di ðrÞ
hW0 ðr; qÞjljWe ðr; qÞi2 ¼
N X
0 O ðq
0
2 jhW0 ðr; qÞjljWðpÞ e ðr; qÞij
p¼1
¼
N X N X
¼
dij F i;ni F j;nj
i;j¼1
N X F i;n 2 : ¼
ð30Þ
i
i
Then, we obtain
hW0 ðr;qÞjljWe ðr;qÞi2 ¼
N D E2 X /0 ðrÞjlj/Di ðrÞiht0O ðq D0 Þtini ðq Di Þ : i¼1
ð31Þ Thus, the observed simultaneous stimulated emission from the electronically-mixed 1Bþ u and 1Bu states is theoretically explained (when, N = 2, /D1 ¼ /D1Bþu ; /D2 ¼ /D1Bu ). Now, we consider internal conversion between a pair of diabatic levels. Let the initial and final states be /Di ðr; qÞtimi ðq Di Þ and /Dj ðr; qÞtjnj ðq Dj Þ, respectively. The internal conversion rate from /Di timi to /Dj tjnj can be given, from Eqs. (19) and (20), as
kimi !jnj ¼
E2 2p D j D t ðq Dj ÞjV ji ðqÞjtimi ðq Di Þ dðhe0i h Xnj X þh xia mi a he0j hxja nja Þ; a
ih
WðtÞ
¼ ðH lEðtÞÞWðtÞ;
dt
ð32Þ
1 t2 EðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi exp 2 Eðeixt þ eixt Þ; 2s 2ps2
1
s ¼ pffiffiffiffiffiffiffiffi sp :
Regarding the term lE(t) as perturbation to H and setting that W(1) = Ug, the Schrödinger equation, Eq. (34), can be solved by the use of the successive approximation method. Denoting the solution by We(t), we obtain, up to the first order in the radiation field,
h/Di jHe
Thus, the rate is proportional to the square of ðqÞð ðr; qÞj/Dj iÞ. This leads to the same selection rule for internal conversion as that already proven for electronic mixing, i.e., 1Bþ u to 1Bu is al þ lowed, but 1Bu to 3Ag is forbidden. This selection rule also explains internal conversion but the why not the 1Bþ u to 3Ag þ 1Bu to 1Bu internal conversion was observed for the longer-chain Cars (n = 11–13). 3.3. Coherent excitation of a pair of diabatic levels In a polar solvent, an electric field can be generated by the fluctuation of the solvent permanent dipoles. This electric field induces the polarization of the conjugated chain of the solute Car molecule. As a result, the conjugated chain loses both the C2h and particle hole symmetries, which may facilitate the 1B u and 3Ag states with certain transition-dipole moments. If a short laser pulse is applied to the Car molecule in this situation, the coherent excitation of the totally optically-allowed 1Bþ u state and the isoenergetic, partially optically-allowed 1B u or 3Ag state can be anticipated.
Z
i h
1
dt 1 Gðt1 ÞlGðt t1 ÞUg Eðt t 1 Þ;
ð37Þ
0
where GðtÞ ¼ exp hi Ht . Expressing by Ca(t) the probability amplitude of finding Ua at time t, we get
C a ðtÞ hUa jWe ðtÞi Z 1 dt 1 hUa jGðt1 ÞjUa ilag Ug jGðt t1 ÞjUg Eðt t 1 Þ; ¼i
ð38Þ
0
where lag = hUa|l|Ugi. Applying the Wigner–Weisskopf approximation [21] to the Green functions for the excited state, i.e.,
hUa jGðtÞjUa i ¼ eca t=2ixa t
ðt > 0Þ;
ð39Þ
we have
C a ðtÞ ¼
i l h ag
Z
1
dt 1 eca t1 =2ixag t1 Eðt t 1 Þ with dxag ¼ xa xg ;
0
ð40Þ where h xa and ca are the energy and the lifetime of state a. Substitution of Eq. (35) into this equation yields
C a ðtÞ ¼
ð33Þ V Dij
ð36Þ
2 ln 2
a
1X i xa : 2 a
ð35Þ
where x and s are the mean frequency and the duration of the pulsed radiation, respectively; s is related to the FWHM duration time sp of the pulse intensity E2(t) (exp(t2/s2)) as
where
e0i ¼ ei þ
ð34Þ
where H is the Hamiltonian, l, the dipole moment of the molecule, and E(t), the radiation field expressed by
We ðtÞ ¼
ðpÞ ðpÞ C i C j F i;ni F j;nj
p¼1 i;j¼1 N X
The Schrödinger equation for such a system reads
i Elag exp i x ðxag ica =2Þ t 2 " 2 # ðx xag Þ þ ica =2Þ s2 exp 2
ðx xag Þs ca s t pffiffiffi erfc i þ pffiffiffi pffiffiffi ; 2 2 2 2s
ð41Þ
pffiffiffiffi R 1 where erfcðzÞ ð2= pÞ z expðs2 Þds. (Note that erfc(1) = 2 and erfc(0) = 1.) Denoting the modulus and phase of
" 2 exp
ðx xag Þ þ ica =2Þ 2
2
s2
# ;
ð42Þ
p by Aa(x) and da(x), we obtain for t > 2 2s
C a ðtÞ ¼
i Elag exp i x ðxag ica =2Þ t Aa ðxÞeida : 2
ð43Þ
In terms of Ca(t), We(t) can be written as
We ðtÞ ¼
X
C a ðtÞUa :
ð44Þ
a
The fluorescence intensity for the transition from the excited state We(t) down to the ground state Ug can be expressed by
2 Ifl ðtÞ ¼ Ug jljWe ðtÞ :
ð45Þ
350
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
Combining Eqs. (43)–(45), we obtain
2 X i Ifl ðtÞ ¼ Ejlag j2 exp½iðxag ica =2ÞtAa ðxÞeida a 2
pffiffiffi ðt > 2 2sÞ: ð46Þ
Taking into account inhomogeneous broadening due to the statistical distribution of transition frequencies, we finally obtain
Here, Q is defined in such a way that viu(Q)@ vjv(Q)/@Qa takes the maximum value at Q , and /Di ðr; Q Þ is related to /Ai ðr; Q Þ by the unitary matrix Uij(Q), as already shown in Eq. (16),
/Di ðr; Q Þ ¼
N X
/Aj ðr; Q ÞU ji ðQ Þ:
ð52Þ
j¼1
Substituting this into Eq. (51) results in
2
X i 1 Ifl ðtÞ ¼ Ejlag j2 exp iðxag ica =2Þt D2a t 2 Aa ðxÞeida a 2 2 pffiffiffi ðt > 2 2sÞ; ð47Þ
! N N X X @ ðpÞ ðpÞ ða Þ /0 ðr; Q Þ W ðr; Q Þ ¼ Ci F oj ðQ ÞUji ðQ Þ : @Q a Q ¼Q i¼1 j¼1
where Da is the root-mean-square amplitude of the frequency fluctuations. Now, let us assume that only a pair of electronic excited states have appreciable values of Aa(x). Denoting the two states by a and b, we get
If the right-hand side of this equation approaches to zero for certain ðpÞ ðpÞ values of C 1 ; . . . ; C N , then, the lifetime of the state described by W(p)(r, Q) can become very long. Thus, the origin of the persistent peak can be proposed as an over lap of the 1Bþ u ð0Þ level and the isoenergetic 1Bu or 3Ag vibronic level.
Ifl ðtÞ ¼
E2 2 2 2 2 jlag j4 A2a ðxÞeca tDa t þ jlbg j4 A2b ðxÞecb tDb t 4 2 2 2 þ 2jlag j2 jlbg j2 Aa ðxÞAb ðxÞeðca þcb Þt=2ðDa þDb Þt =2 cos½ðxag xbg Þt ðda db Þ :
4. Conclusion
ð48Þ
Here, the first two terms present the incoherent decays of the two states, and the last cross term, the coherent decay (with the averaged rate) which is modulated at the angular frequency of xag xbg (‘quantum beat’). The conditions for the coherent quantum beat to be observed is that (i) both states have appreciable transition-dipole moments and that (ii) both state-energies must be close enough to the pulse frequency x so that Aa(x) and Ab(x) have non-vanishing values, which requires the relation that (x xag) s 6 1 and (x xbg)s 6 1. From this pair of inequalities combined with Eq. (36), we can find the sufficient conditions for the energies of the two states to be coherently excited,
pffiffiffiffiffiffiffiffi jxag xbg jsp 2 ln 2;
ð49Þ
where sp is the FWHM pulse duration. The above expressions can be easily extended to incorporate the vibrational levels, by replacing subscript a by a pair of subscripts a and u, representing the electronic states and vibrational levels. As shown in Section 1, the 1Bþ u ð0Þ level is in close proximity with the 1B u ð1Þ level in Car (n = 9) and with the 3Ag ð1Þ level in Car (n = 11). As described in this subsection, when these carotenoid molecules are dissolved in polar solvent, these ‘isoenergetic’ levels become partially optically-allowed, and the mixing of the pairs of diabatic levels should take place due to such extremely-small energy differences. Thus, Eq. (48) verifies the essential part of Scheme 1, and Eq. (49) shows the experimental conditions we actually achieved. Let us consider a set of diabatically-mixed states,
WðpÞ ðr; Q Þ ¼
N X
ðpÞ
C i /Di ðr; Q Þ:
ð50Þ
i¼1
We assume a large energy gap between the constituent diabatic states f/Di ðr; Q Þg and the ground state u0(r, Q). Then, from Eqs. (19) and (S2) in Supplementary material, the internal conversion rate from the diabatic states to the ground state, through the ath vibrational mode, is roughly proportional to the square of the modulus,
@ ðpÞ W ðr; Q Þ /0 ðr; Q Þ @Q a Q ¼Q N X @ D ðpÞ U ðr; Q Þ ¼ C i /0 ðr; Q Þ : @Q a Q ¼Q i¼1
ð53Þ
ð51Þ
The experimentally observed selection rules concerning electronic mixing and internal conversion between 1Bþ u ð0Þ level and the isoenergetic 1B u or 3Ag vibronic level as well as a long fluorescence lifetime and oscillatory changes in fluorescence intensity originating from the above pairs of vibronic levels (showing the apparent 1Bþ u ð0Þ stimulated emission) have been explained by the use of a theory based on the diabatic description. The experimental observations and their rigorous explanation by the diabatic theory support the energy diagram and the symme try notations concerning the 1Bþ u ; 1Bu and 3Ag vibronic levels shown in Fig. 1. Acknowledgements This work has been supported by a grant, Open Research Center Project ‘The Research Center for Photo-Energy Conversion’ to Y. Koyama. Y. Kakitani has been supported by a Grant-in-Aid for Scientific Research (C), 19579005, from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.cplett.2009.04.039. References [1] K. Furuichi, T. Sashima, Y. Koyama, Chem. Phys. Lett. 356 (2002) 547. [2] P. Zuo, A. Sutresno, C. Li, Y. Koyama, H. Nagae, Chem. Phys. Lett. 440 (2007) 360. [3] A. Sutresno, Y. Kakitani, P. Zuo, C. Li, Y. Koyama, H. Nagae, Chem. Phys. Lett. 447 (2007) 127. [4] G.E. Bialek-Bylka, Y. Kakitani, C. Li, Y. Koyama, M. Kuki, Y. Yamano, H. Nagae, Chem. Phys. Lett. 454 (2008) 367. [5] C. Li, T. Miki, Y. Kakitani, Y. Koyama, H. Nagae, Chem. Phys. Lett. 450 (2007) 112. [6] T. Miki, Y. Kakitani, Y. Koyama, H. Nagae, Chem. Phys. Lett. 457 (2008) 222. [7] J.T. Fourkas, H. Kawashima, K.A. Nelson, J. Chem. Phys. 103 (1995) 4393. [8] R. Fujii, T. Inaba, Y. Watanabe, Y. Koyama, J.-P. Zhang, Chem. Phys. Lett. 369 (2003) 165. [9] J.A. Davis, L.V. Dao, M.T. Do, P. Hannaford, K.A. Nugent, H.M. Quiney, Phys. Rev. Lett. 100 (2008) 227401. [10] M. Born, R. Oppenheimer, Ann. Phys.-Berlin 84 (1927) 457. [11] F.T. Smith, Phys. Rev. 179 (1969) 111. [12] T. Pacher, L.S. Cederbaum, H. Köppel, Adv. Chem. Phys. 84 (1993) 293. [13] A. Troisi, G. Orlandi, J. Chem. Phys. 118 (2003) 5356. [14] C.A. Mead, D.G. Truhlar, J. Chem. Phys. 77 (1982) 6090. [15] M. Baer, Chem. Phys. Lett. 35 (1975) 112. [16] P.-O. Löwdin, J. Chem. Phys. 18 (1950) 365. [17] W. Weber, W. Thiel, Theoret. Chem. Acc. 103 (2000) 495.
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351 [18] S.H. Lin, J. Chem. Phys. 44 (1966) 3759. [19] M. Bixon, J. Jortner, J. Chem. Phys. 48 (1968) 715. [20] W. Domcke, D.R. Yarkony, H. Koppel (Eds.), Conical Intersections: Electronic Structure, Dynamics and Spectroscopy, World Scientific, New Jersey, 2004. [21] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford University Press, New York, Oxford, 1995.
351
[22] R. Fujii, T. Ishikawa, Y. Koyama, M. Taguchi, Y. Isobe, H. Nagae, Y. Watanabe, J. Phys. Chem. A 105 (2001) 5348. [23] R. Nakamura, R. Fujii, H. Nagae, Y. Koyama, Y. Kanematsu, Chem. Phys. Lett. 400 (2004) 7.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Theoretical description of diabatic mixing and coherent excitation in singlet-excited states of carotenoids Hiroyoshi Nagae a, Yoshinori Kakitani b, Yasushi Koyama b,* a b
Kobe City University of Foreign Studies, Gakuen-Higashimachi, Nishi-ku, Kobe 651-2187, Japan Faculty of Science and Technology, Kwansei Gakuin University, Gakuen, Sanda 669-1337, Japan
a r t i c l e
i n f o
Article history: Received 7 January 2009 In final form 15 April 2009 Available online 18 April 2009
a b s t r a c t This Letter discusses the following characteristic excited-state properties of closely-overlapped ‘diabatic’ states of carotenoids, including 1Bþ u ; 1Bu and 3Ag , on which we have reported systematic experimental studies. A rigorous general formulation is developed and applied for the adiabatic and diabatic descriptions in which the breakdown of the Born–Oppenheimer approximation is used as a criterion to differentiate them: (i) the observed selection rules concerning electronic mixing and internal conversion between the 1Bþ u and 1Bu or 3Ag diabatic levels after incoherent excitation, and (ii) a long lifetime of the apparent þ 1Bu stimulated emission and the oscillatory change in the fluorescence intensity after coherent excitation of both pairs of diabatic levels. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction The 1B u and 3Ag states of carotenoids (Cars) were first identi fied in 2002, in addition to the well-known 1Bþ u and 2Ag states, by the measurement of resonance-Raman excitation profiles [1]. When the conjugated chain is in the all-trans configuration having C2h symmetry, the optical transition from and to the ground 1A g state is allowed for the 1Bþ u state, but it is forbidden for the 1Bu ; 3Ag and 2Ag states; the former and latter states are hereafter denoted as ‘optically-allowed and -forbidden states’, respectively. Fig. 1a shows that the energies of the low-lying singlet states decrease in proportion to 1/(2n + 1), where n is the number of conjugated double bonds, while Fig. 1b shows that the vibrational levels of the optically-allowed 1Bþ u state approximately overlap with those of the next low-lying 1B u and 3Ag states in the shorterand longer-chain Cars, respectively, due to the unique arrangement of the origins of these singlet states, as shown in Fig. 1a. If we assume the same vibrational energy gaps of 1400 cm1 for all the electronic states, the 1Bþ u ð0Þ vibronic level (labeled on the right hand side), for example, can overlap with the 1B u ð1Þ and 1Bu ð2Þ levels (labeled on the left-hand side) in neurosporene (n = 9) and spheroidene (n = 10), but with the 3A g ð1Þ; 3Ag ð2Þ and 3Ag ð3Þ levels in lycopene (n = 11), anhydrorhodovibrin (n = 12) and spirilloxanthin (n = 13). Spectroscopically, we observed that excitation to the optically-allowed 1Bþ u ð0Þ level can cause excitation to each isoenergetic, optically-forbidden level. We call the pair of overlapped levels ‘diabatic levels’ because the energy gap between each pair of vibronic levels is so small that the observed excited-state
* Corresponding author. Fax: +81 79 565 8408. E-mail address: [email protected] (Y. Koyama). 0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2009.04.039
dynamics need to be described on the diabatic basis instead of the adiabatic basis (vide infra). We have found that the excitedstate dynamics depends on the symmetries of the pair of the diabatic levels, incoherent or coherent excitation (by the use of 100 and 30 fs pulses, respectively), and the polarity of the solvent [2–6]. We will first summarize below, as the basis of our discussion in this Letter, the crucial features of our experimental observations published in sequence and then present the rigorous diabatic theory formulated in the present study, by which we will put forth our consistent interpretation of the observed phenomena. 2. Survey of experimental results 2.1. Excitation with 100 fs pulses of all-trans Cars (n = 9–13) in nonpolar solvents: selection rules for the electronic mixing and internal conversion of a pair of diabatic vibronic levels Fig. 2 shows the pump–probe time-resolved spectra of a set of 1A Cars (n = 9–13) after (a) the 1Bþ u ð0Þ g ð0Þ and (b) the þ 1Ag ð0Þ excitations (abbreviated as 0 0 and 1 0 exci1Bu ð1Þ tations). The relaxation scheme includes singlet internal conver or 1Bþ sion: 1Bþ u þ 1Bu u ðv ide infraÞ ! 1Bu ! 2Ag ! 1Ag ; we focus our attention here on the initial stimulated-emission components. Fig. 3 shows the species-associated difference spectra (SADS) after the 0 0 and 1 0 excitations that were obtained by singular-value decomposition (SVD) followed by global fitting [2,3]. The shorter-chain Cars (n = 9 and 10) and longer-chain Cars (n = 11–13) exhibited completely different compositions in the stimulatedemission patterns.
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
n
(a)
13
12
11
spheroidene (n = 10), on the other hand, the vibrational relaxation þ 1Bþ u ð1Þ þ 1Bu ð3Þ ! 1Bu ð0Þ þ 1Bu ð2Þ took place in 40 fs, and the ð0Þ þ 1B internal conversion 1Bþ u u ð2Þ ! 1Bu ð0Þ occurred in 100 fs. The pairs of these time constants, observed in both Cars, approximately reflect the rule that the rate of vibrational relaxation from t = ‘ to t = ‘ 1 is proportional to ‘ [7]. Thus, the presence of electronic mixing causing simultaneous stimulated emission and internal conversion between the pair of 1Bþ u and 1Bu diabatic vibronic levels has been corroborated for these short-chain Cars.
9
1Bu+
20
–1 3 Energy / 10 cm
10
3A g–
16
1Bu–
2A g–
12
0.04
0.05 1 / (2n + 1) n
(b)
13
12
11
10
Energy / 103 cm–1
1Bu+
20
4
13
1 2
3
02
0 1
2
1
0
1
0
0
0
9
3
1
2
0
1
2
1
1
0
0
0 1 0
1Bu–
3A g–
16
0
0
12
0.04
343
0.05
2.1.2. Cars (n = 11–13) The stimulated-emission patterns of this set of longer-chain Cars can be characterized as follows (see the three panels of Fig. 3 on the right-hand side): (i) The patterns can be simply simulated by the stimulated emission from the 1Bþ u state and the bleaching of the ground-state absorption; no additional components are needed in the fitting. (ii) Each of the stimulated-emission patterns in the 1 0 excitation completely differs from that in the 0 0 excitation; no signs of vibrational relaxation appear at all in the 1 0 excitation. In this set of Cars, the stimulated-emission patterns in the 0 0 excitation can be simulated simply by the 0 excitation are ex1Bþ u ð0Þ emission, whereas those in the 1 ð1Þ emission in addition to the bleaching of pressed by the 1Bþ u the ground-state absorption. No electronic mixing has been found either between the 1Bþ u ð0Þ level and the 3Ag ð1Þ; 3Ag ð2Þ or 3Ag ð3Þ þ levels, or between the 1Bu ð1Þ level and the 3Ag ð2Þ; 3Ag ð3Þ or 3A g ð4Þ levels in the Cars including lycopene (n = 11), anhydrorhodovibrin (n = 12) and spirilloxanthin (n = 13), respectively. Fig. 4 (the three panels on the right-hand side) shows the relaxation dynamics for the set of these Cars. Neither electronic mixing (simultaneous stimulated emission) between the 1Bþ u and 3Ag vib þ ronic levels nor internal conversion from the 1Bu to the 3Ag vib ronic level is observed for the longer-chain Cars. The 1Bþ u to 1Bu internal conversion is observed, instead. In summary, the same selection rule applicable for the electronic mixing and internal conversion of the diabatic levels of Cars has been established; i.e., Bu-to-Bu is allowed whereas Bu-to-Ag is forbidden.
1 / (2n + 1) Fig. 1. (a) Energies of the 1Bþ u ð0Þ (d), 3Ag ð0Þ (h), 1Bu ð0Þ (s) and 2Ag ð0Þ (j) levels determined by the measurement of resonance-Raman excitation profiles for crystalline Cars, including mini-9-b-carotene, spheroidene, lycopene, anhydrorhod ovibrin and spirilloxanthin (n = 9–13, respectively) [1]. The 1B u ð0Þ (N) and 3Ag ð0Þ (D) levels determined by stimulated emission are also shown (see Section 2.2.1; here, for Car (n = 9), neurosporene was used instead). Solid regression lines are drawn, for the set of energy values, as the linear functions of 1/(2n + 1). (b) Arrangement of the 1Bþ u ; 1Bu and 3Ag vibronic levels shown in solid, dotted-broken and broken lines, respectively. The 1Bþ u ð0Þ and 1Bu ð0Þ levels were determined by ð0Þ level was taken from (a). The fluorescence spectroscopy [22,23], while the 3A g spacing of the vibrational levels is set to be 1400 cm1.
2.1.1. Cars (n = 9 and 10) The stimulated-emission patterns of this pair of Cars can be characterized as follows (see the two panels on the left-hand side): (i) Each pattern can be simulated as a sum of stimulated emission from the 1Bþ u vibronic level and that from the 1Bu level, in addition to the bleaching of the ground-state absorption. (ii) The 0 0 excitation exhibits only a single stimulated-emission pattern, whereas the 1 0 excitation exhibits two time-resolved stimulated-emission patterns, reflecting a step of vibrational relaxation. Fig. 4 schematically shows the relaxation dynamics for a pair of Cars (two panels on the left): In neurosporene (n = 9), the vibra þ tional relaxation 1Bþ u ð1Þ þ 1Bu ð2Þ ! 1Bu ð0Þ þ 1Bu ð1Þ took place in ð0Þ þ 1B 60 fs, and the internal conversion 1Bþ u u ð1Þ ! 1Bu ð0Þ (or in another viewpoint, the 1Bu vibrational relaxation), in 120 fs. In
2.2. Excitation with 30 fs pulses of all-trans Cars (n = 9–13) in a polar solvent: coherent excitation of a pair of diabatic vibronic levels Fig. 5 shows the 30 fs pump–probe time-resolved spectra of Cars (n = 9–13) in THF, a polar solvent [5,6]. A spectral width of 1000 cm1 for the 30 fs pulse can coherently excite the optically-active 1Bþ u ð0Þ level and the isoenergetic optically-forbidden 1B u or 3Ag diabatic level. The time-resolved spectral patterns are substantially different from the case of incoherent excitation with 100 fs pulses in non-polar solvents (Fig. 2), reflecting completely different excited-state dynamics. As for the 1Bþ u stimulated emission, for example, we do not observe at all either the strong vibrational structures that are ascribable to the Franck–Condon downward transitions as stimulated emission or the bleaching of the ground-state absorption. In the longest-chain Cars (n = 12 and 13), we do observe þ the process of 1Bþ u ð1Þ ! 1Bu ð0Þ vibrational relaxation, indicating þ the simultaneous excitation of the 1Bþ u ð1Þ and 1Bu ð0Þ levels by the timely-short and spectrally-broad 30 fs pulses. After this vibrational relaxation, we observe a single persistent peak with the 1Bþ u ð0Þ energy as in the case of the shorter-chain Cars (n = 9 and 10). We will report in the following the three pieces of information obtained from the particular pump–probe time-resolved spectra shown in Fig. 5, which include (a) stimulated emission from the þ 1B u ð0Þ or 3Ag ð0Þ level, (b) vibrational relaxation in the 1Bu ; 1Bu þ and 2Ag manifolds, and (c) a long-lived emission with the 1Bu ð0Þ
344
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
Neurosporene (n = 9) ← (a) 0 0 excitation
ΔOD = 0.19
Spheroidene (n = 10)
ΔOD = 0.12
Lycopene (n = 11)
Anhydrorhodovibrin (n = 12)
ΔOD = 0.10
ΔOD = 0.09
Spirilloxanthin (n = 13)
–0.40 ps
ΔOD = 0.09
–0.10 –0.06
1B+u + 1B–u
1B+u + 1B–u
1B+u
1B+u
1B+u
1B–u
1B–u
–0.02 0.00 0.02 0.06
1B–u
0.10 0.12
1B–u
0.16
1B–u
0.20 2A–g 2A–g
0.30
2A–g
0.50
2A–g
1.00
2A–g
3.00 5.00 10.00 15.00
16
20 24 Energy / 103 cm–1
20 24 Energy / 103 cm–1
20 24 Energy / 103 cm–1
ΔOD = 0.10
ΔOD = 0.10
16 20 Energy / 103 cm–1
16 20 24 Energy / 103 cm–1
(b) 1 ← 0 excitation
ΔOD = 0.44
ΔOD = 0.12
– 0.40 ps
ΔOD = 0.05
– 0.10 – 0.06
1B+u + 1B–u
1B+u + 1B–u
1B+u
1B+u
1B+u
1B–u
1B–u
– 0.02 0.00 0.02 0.06
1B–u
0.10 0.12
1B–u
0.16
1B–u
0.20 2A–g 2A–g
0.30
2A–g
0.50
2A–g
1.00
2A–g
3.00 5.00 10.00 15.00 16
20 24 Energy / 103 cm–1
20 24 Energy / 103 cm–1
20 24 Energy / 103 cm–1
16 20 Energy / 103 cm–1
16 20 24 Energy / 103 cm–1
Fig. 2. Pump–probe time-resolved stimulated-emission and transient-absorption spectra of Cars (n = 9–13) in non-polar solvents (n-hexane for n = 9 and 10 and mixtures of n-hexane and benzene for n = 11–13) after (a) the ‘0 0’ and (b) the ‘1 0’ excitations with 100 fs pulses.
345
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
Neurosporene (n = 9) (a) 0 ← 0 excitation
Spheroidene (n = 10)
1Bu+(0) + 1Bu–(2)
1Bu+(0) + 1Bu–(1)
16
Lycopene (n = 11)
20 24 Energy / 103 cm–1
Anhydrorhodovibrin (n = 12)
1Bu+(0)
20 24 16 Energy / 103 cm–1
Spirilloxanthin (n = 13)
1Bu+(0)
20 24 Energy / 103 cm–1
16
1Bu+(0)
20 Energy / 103 cm–1
16
20 Energy / 103 cm–1
(b) 1 ← 0 excitation
1Bu+(1) + 1Bu–(2)
1Bu+(1) + 1Bu–(3)
1Bu+(1)
1Bu+(1)
20 24 Energy / 103 cm–1
20 Energy / 103 cm–1
24 16
20 Energy / 103 cm–1
24
1Bu+(0) + 1Bu–(2)
1Bu+(0) + 1Bu–(1)
16
16
1Bu+(1)
20 24 Energy / 103 cm–1
20 24 Energy / 103 cm–1
Fig. 3. Fitting to the initial stimulated-emission components (SADS) by the use of Franck–Condon factors for Cars (n = 9–13). Specifications of the lines used are as follows: þ For all Cars, SADS (black solid line), bleaching of the ground-state absorption (black dotted line), emission from the 1Bþ u ð0Þ level (blue broken line), emission from the 1Bu ð1Þ ð1Þ and 1B ð2Þ levels, respectively (green level (red broken line), and a sum of all the contributions (black dotted-broken line). For Cars (n = 9 and 10), emission from the 1B u u broken line) and emission from the 1B u ð2Þ and 1Bu ð3Þ levels, respectively (magenta broken line).
Neurosporene (n = 9)
Spheroidene (n = 10)
Lycopene (n = 11)
1Bu+
30
1Bu+
1Bu– 1B v' = 3
v' = 1
v'' = 2
Energy / 103 cm–1
20
v' = 1
1Bu+
3A g– v' = 3
1Bu– v'' = 3
v'' = 1
v' = 0
v' = 1
v' = 2
1B
v' = 0
v'' = 3
v' = 1 v' = 0
v'' = 3 v'' = 2
v'' = 1
v'' = 0
– u
v'' = 4
v'' = 2
v'' = 1
v' = 3
v' = 2
1Bu–
v'' = 2
v'' = 0
3A g–
v' = 3
v' = 2 v' = 1
v' = 0
v'' = 2
v'' = 0
1Bu+
v' = 3 v' = 2 v'' = 3
v' = 0
v'' = 1
1Bu+
Spirilloxanthin (n = 13)
3A g–
– u
v' = 2
v'' = 3
Anhydrorhodovibrin (n = 12)
v'' = 1
v'' = 0
v'' = 0
10 1A g–
1A g–
v=3
v=1
–5
0 q
5 –5
0 q
0 q
v=2
v=1
v=0
5 –5
v=3
v=2
v=1
v=1 v=0
1A g–
v=3
v=2
v=2
v=0
1A g–
v=3
v=3
v=2
0
1A g–
v=1
v=0
5 –5
0 q
v=0
5 –5
0 q
5
Fig. 4. Relaxation dynamics showing diabatic electronic mixing between the 1Bþ u and 1Bu vibronic levels accompanying simultaneous stimulated emission for Cars (n = 9 and 10) (the two panels on the left-hand side), and those showing diabatic internal conversion from the 1Bþ u to 1Bu vibronic level for Cars (n = 11–13) (the three panels on the right-hand side). In the latter, neither diabatic electronic mixing generating simultaneous stimulated emission nor diabatic internal conversion between the 1Bþ u and 3Ag vibronic levels took place. Diabatically-mixed states are shadowed, and vibrational relaxations and internal conversions are shown by short and bent arrows, respectively.
energy, which exhibit an oscillatory change in the fluorescence intensity (quantum beat). Finally, we will present the scheme that
can consistently observations.
explain
all
the
above-mentioned
set
of
346
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
Neurosporene (n = 9)
Spheroidene (n = 10)
Lycopene (n = 11)
Anhydrorhodovibrin (n = 12)
Spirilloxanthin (n = 13)
– 0.04 ps ΔOD = 0.18
ΔOD = 0.03
ΔOD = 0.02
ΔOD = 0.01
ΔOD = 0.01 – 0.02 0.00
1B
1B u+
+ u
1B u+
1B u+
1B u+ 0.02 0.04
1B
+ u
1B u–
1B
+ u
3A
– g
3A
– g
3A
– g
0.06
1B u– 0.08 0.10 0.15
1B u–
1B u– 0.20
1B u– 0.25
1B u–
1B u– 0.30 0.60 0.90 1.20 1.50
2A g–
2A g–
2A g– 3.00
2A g– 18 20 Energy / 103 cm–1
5.00
2A g– 18 20 Energy / 103 cm–1
16 18 Energy / 103 cm–1
16 18 Energy / 103 cm–1
Fig. 5. Pump–probe time-resolved spectra of Cars (n = 9–13) in THF, a polar solvent, after the 0 larger labels and downward arrows, whereas transient-absorption peaks, by smaller labels.
2.2.1. Stimulated emission from the 1B u ð0Þ and 3Ag ð0Þ levels þ Following the very first 1Bu emission, each of the shorter- and longer-chain Cars exhibits a weak stimulated-emission peak ascribable to either the 1B u or the 3Ag state. Each peak systematically shifts to lower energy with increasing n. Fig. 1a also shows a plot of the 1B u (N) and 3Ag (D) energies, as functions of 1/(2n + 1), observed here as stimulated emission, in reference to the linear dependence of the 1B u ð0Þ and 3Ag ð0Þ energies that have been determined by the measurement of resonance-Raman excitation profiles. Both linear relations practically agree completely. Since each Car is excited to the 1Bþ u ð0Þ level as well as the isoenergetic ð1Þ and 1B vibronic levels, i.e., the 1B u u ð2Þ levels in Cars (n = 9 and 10) and the 3Ag ð1Þ; 3Ag ð2Þ and 3A g ð3Þ levels in Cars (n = 11, 12 and 13), respectively, there must be some mechanism of rapid vibrational relaxation after excitation to these diabatic levels.
16 18 Energy / 103 cm–1
20
0 excitation with 30 fs pulses. Stimulated-emission peaks are indicated by
2.2.2. Vibrational relaxation in the 1Bþ u ; 1Bu and 2Ag manifolds In Cars (n = 9), transient absorptions showing vibrational relax ation from the 1B u ð1Þ to the 1Bu ð0Þ vibronic level followed by that ð1Þ to the 2A ð0Þ level are observed; the 1B from the 2A u ð1Þ and g g 2Ag ð1Þ peaks appear on the lower-energy side of the 1B u ð0Þ and 2A g ð0Þ peaks, respectively. These electronic states must be generated, through internal conversion, from the 1Bþ u ð0Þ level as one of the incoherent components (see Scheme 1 and Eq. (48), vide infra). In Cars (n = 10 and 11), however, only the 2A g ð1Þ and 2Ag ð0Þ transient absorptions are observed.
2.2.3. A long-lived stimulated emission with the 1Bþ u ð0Þ energy showing oscillatory change in intensity As shown in Fig. 5, the stimulated-emission peak with the 1Bþ u ð0Þ energy persists even longer than 5 ps in all of the Cars
347
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
3. Theory explaining the experimental observations
Neurosporene (n = 9) 1Bu– (1)
IC
1Bu+(0)
1Bu+(0) + 1Bu– (1)
hν
VR
hν
(b)
1Bu– (0)
1Bu– (1)
(c)
VR
1Bu– (0)
transient absorption
hν
3.1. Theoretical background Let us first define the adiabatic and diabatic electronic states and their properties, focusing our attention on the ground and singlet-excited states: The wavefunction of a molecular system can be generally written as
X
Wðr; Q Þ ¼
vk ðQ Þ/k ðr; Q Þ;
ð1Þ
k
(a)
where r denotes a set of electronic coordinates, Q, the nuclear normal coordinates, and {uk(r, Q)}, an orthonormal complete basis in the electronic Hilbert space; here, Q is a parameter. Substituting Eq. (1) into the time-independent Schrödinger equation leads to
Lycopene (n = 11) 1Bu– (2) VR
IC
1B u+(0)
1B u+(0) + 3Ag– (1)
hν
3A g–(1)
hν
VR
X
Hij ðQ Þvj ðQ Þ ¼ Evj ðQ Þ;
ð2Þ
l
with –
1Bu (0)
(b)
(c)
transient absorption
–
3A g (0) hν
(a) Scheme 1.
(n = 9–13). By means of near-infrared pump–probe spectroscopy of the same set of Cars, when they were incoherently excited to the þ 1Bþ u ð0Þ level by the use of 100 fs pulses, the 1Bu lifetimes were determined to be 100, 100, 20, 10 and 10 fs for Cars with n = 9, 10, 11, 12 and 13, respectively [8]. Therefore, the lifetimes of the presently-observed stimulated emissions are far beyond the lifetime of the pure 1Bþ u state. We have further observed oscillatory changes in the integrated intensity of the above-mentioned persistent peak in Car (n = 11) with an interval of 150 fs, which corresponds to an energy gap of 220 cm1 [6]. By means of spectrally-resolved photon-echo spectroscopy, Davis et al. [9] also observed an oscillation with a 120 fs time interval corresponding to the energy gap of 275 cm1 for the same Car in n-hexane. Therefore, the oscillatory changes in the fluorescence intensity can be ascribed to the ‘quantum beat’ between the pair of diabatic levels, i.e., 1Bþ u ð0Þ and 3A g ð1Þ.
Hij ðQ Þ ¼ T ij ðQ Þ þ V ij ðQ Þ þ Dij ðQ Þ þ Gij ðQ Þ; 2 2 h X @ 2 h @ 2 dij ¼ ; T ij ðQ Þ ¼ 2 2 a @Q a 2 @Q 2
ð3Þ
V ij ðQ Þ ¼ h/i jHe ðr; Q Þj/j i; @ @ 2 Dij ðQ Þ ¼ h /i /j ; @Q @Q + +! * * 2 2 @2 @2 h h @ /i 2 /j Gij ðQ Þ ¼ /i 2 /j ¼ @Q @Q 2 2 @Q 2 @ @ h X /k /i /k /j : @Q @Q 2 k
ð5Þ
ð4Þ
ð6Þ
ð7Þ
Here, He(r, Q) is the electronic Hamiltonian and the brackets indicate integration over the coordinates r. If uk(r, Q) is chosen as the eigenfunctions of He(r, Q), noting that Vij(Q) = Vii(Q)dij with Kronecker’s delta dij, the matrix Hamiltonian in Eq. (3) takes the form 2
@2 h þ V ii ðQ Þ 2 @Q 2
Evi ðQ Þ ¼
!
vi ðQ Þ h2
XX j
a
ðaÞ 2 X @F ij ðQ Þ X ðaÞ h ða Þ þ F ki ðQ ÞF kj ðQ Þ 2 a @Q a k
ðaÞ
F ij ðQ Þ
@ v ðQ Þ @Q a j
!
vj ðQ Þ;
ð8Þ
ðaÞ
2.2.4. Possible mechanisms Scheme 1 provides a possible mechanism for a pair of Cars (n = 9 and 11) based on the quantum beat generated by coherent excitation to each pair of diabatic levels, which can explain all the observations (a), (b) and (c): In lycopene (n = 11), for example, the emission after coherent excitation of the 1Bþ u ð0Þ and 3Ag ð1Þ diabatic levels can be described by the following three terms: (a) the incoherent 3A g ð1Þ term (right-hand side) generating the 3A g ð0Þ emission after rapid vibrational relaxation, (b) the incoherent 1Bþ u ð0Þ emission (left-hand side), which can subsequently generate the 1B u ð0Þ transient absorption after the 1Bþ u ð0Þ to 1Bu ð2Þ internal conversion followed by the 1B u ð2Þ ! 1Bu ð1Þ ! 1Bu ð0Þ vibrational relaxation, and (c) þ the coherent 1Bu ð0Þ and 3Ag ð1Þ cross term generating oscillatory emission reflecting the quantum beat (center). (Subsequently, the 1B u ! 2Ag internal conversion takes place as shown in Fig. 5.) Basically the same mechanism should hold in neurosporene, which is expected to generate a quantum beat between the 1Bþ u ð0Þ and 1B u ð1Þ states.
where F ij ðQ Þ is the derivative coupling, ðaÞ
F ij ðQ Þ ¼
@ / : /i @Q j
ð9Þ
a
In the adiabatic or Born–Oppenheimer (BO) approximation [10], the derivative couplings in Eq. (8) are neglected. Then, the BO equation for the nuclear motion reads 2
@2 h þ V ii ðQ Þ 2 @Q 2
!
viv ðQ Þ ¼ Eiv viv ðQ Þ;
ð10Þ
and the overall BO wave function is simply the product of /Ai ðr; Q Þ and viv(Q). Hereafter, superscripts ‘A’ and ‘D’ denote ‘adiabatic’ and ‘diabatic’ bases, respectively. By the use of the commutation relation, [He, @/@Qa] = @He/@Qa, the derivative coupling can be expressed as
@ A / ¼ /A @He /A /Ai ðV ii ðQ Þ V jj ðQ ÞÞ: j i @Q a @Q a j
ð11Þ
348
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
This equation shows that the derivative coupling becomes significant and even diverges as the energy gap reD in the denominator E duces to zero, if the vibronic coupling, /Ai j@He =@Q a j/Aj , does not vanish. The BO approximation breaks down in such a case. Then, a new orthonormal basis, in which the derivative couplings vanish and the basis functions exhibit a weak dependence on the nuclear coordinates, becomes necessary [11–13]. Such a basis, called ‘the diabatic basis’ f/Di ðr; Q Þg, can be obtained by the unitary transformation of the adiabatic BO basis as
/Di ðr; Q Þ ¼
X
/Aj ðr; Q ÞU ji ðQ Þ:
ð12Þ
j
Here, Uji(Q) can be determined so that /Di ðr; Q Þ satisfy
@ D / ¼ 0; /Di @Q a j
ð13Þ
and obey
X ða Þ @ U ij ðQ Þ ¼ F ik ðQ ÞU kj ðQ Þ: @Q a k
ð14Þ
The unitary property of the matrix Uij(Q) is due to the anti-HermicðaÞ ðaÞ ity of the derivative coupling matrix, i.e., F ij ðQ Þ ¼ F ji ðQ Þ. It has been shown that diabatic levels do not strictly exist if a truncated set of adiabatic levels is used [14]. However, if we assume the existence of an approximate diabatic basis for a truncated set of adiabatic levels, the formal solution can be written in a matrix form as
UðQ Þ ¼ exp
Z
!
Q
FðQ Þ dQ ;
ð15Þ
3.2. Electronic mixing and internal conversion between the diabatic levels of carotenoids When a Car molecule is excited to the 1Bþ u state and its conjugated chain loses perfect C2h symmetry, the diabatic description becomes necessary. (See Adiabatic 3.) We note that the diabatic levels (3A g or 1Bu ) approximately keep Ag or Bu symmetry for an arbitrary value of Q even under the conditions of so-called ‘avoided crossing’ or ‘conical crossing’ [20], due to the properties of diabatic wavefunctions as described in Section 3.1. (If the diabatic wavefunction has Ag (Bu) symmetry at Q0, as shown in Eq. (16), it should keep the same symmetry at an arbitrary value of Q shifted from Q0.) Then, we consider the eigenstate of the matrix Hamiltonian of Eq. (17) described on the diabatic basis. Assuming that the diabatic potential V Dii ðQ Þ can be approximated by a harmonic potential, we get
! 2 @2 h 1 i i i Evi ðQ Þ ¼ þ ðQ DQ Þx ðQ DQ Þ vi ðQ Þ 2 @Q 2 2 X V ij ðQ Þvj ðQ Þ; þ
where DQi stands for the shift of the potential minimum in the ith diabatic electronic pffiffiffiffiffiffiffiffiffi state. Using the dimensionless normal coordinates, qð¼ x= hQ Þ, and taking energy in the unit of angular frequency, we have
evi ðqÞ ¼ hi ðqÞvi ðqÞ þ
Q0
/Di ðr; Q 0 Þ ¼ /Ai ðr; Q 0 Þ:
ð16Þ
If we adopt a basis set so determined, the matrix Hamiltonian in Eq. (3) takes the form 2
Evi ðQ Þ ¼
!
vi ðQ Þ þ
X
V Dij ðQ Þvj ðQ Þ
ð17Þ
j
D E V Dij ðQ Þ ¼ /Di jHe ðr; Q Þj/Dj :
ð18Þ
Here, Dij(Q) and Gij(Q) becomes negligible, since they are expressed only by the derivative couplings (see Eqs. (6) and (7)). The physical meaning of the diabatic basis can be exemplified by the orthogonalized floating atomic orbitals [16,17]. (See Adiabatic 1 in Supplementary material.) Thus, we can use a set of wavefunctions with constant MO and CI coefficients, based on the orthogonalized floating atomic orbitals as a diabatic basis. According to Fermi’s golden rule, the rate of internal conversion between the ith and jth electronic states can be given as [18,19]
2p X ¼ Bu jT ij j2 dðEiu Ejv Þ; h uv
Z
ð22Þ
viu ðQ ÞV Dij ðQ Þvjv ðQ ÞdQ :
with
! 2 1 i i i @ i h ðqÞ ¼ þ ðq D Þx ðq D Þ þ ei ; x 2 @q2 i
ð23Þ
(hereafter, it is assumed that xi does not depend on the value of i, i.e., xi = x). Introducing eigenfunctions of hi(q) such that
1 i h ðqÞtin ðq Di Þ ¼ xi n þ þ ei tin ðq Di Þ; 2
vi ðqÞ ¼
X
ð24Þ
ð19Þ
ð20Þ
C i;n tin ðq Di Þ:
ð25Þ
n
Substituting this into Eq. (22), multiplying the resultant equation by tin ðq Di Þ from the left, and integrating over q, we get
1 þ ei C i;n 2 E X XD þ tin ðq Di ÞV Dij ðqÞtjm ðq Dj Þ C j;m :
eC i;n ¼ xi n þ j–i
ð26Þ
m
Solving this simultaneous equations for Ci,n, we obtain energy eigenfunctions
WðpÞ ðr; qÞ ¼
where Bu is the Boltzmann factor of the initial state, Tij is the matrix element of the perturbation between the initial and final states, and Eiu (Ejv) is the vibronic energy of the initial (final) state. (See Adiabatic 2.) In the diabatic basis, Tij can be expressed, from Eq. (17), as
T ij ¼
V Dij ðqÞvj ðqÞ;
where n represents a set of quantum numbers of vibrational modes, we can expand vi(q) in the form
with
ki!j
X j–i
where Q0 is a reference point, which can be chosen arbitrary. The diabatic basis coincides with the adiabatic basis at this reference point Q0 [15] as
@2 h þ V Dii ðQ Þ 2 @Q 2
ð21Þ
j–i
X
ðpÞ
C i;ni /Di ðr; qÞtini ðq Di Þ:
ð27Þ
i;ni
(See Adiabatic 4.) Eq. (26) shows that the condition for /Di and /Dj to mix with each other is that V Dij ðQ Þ, defined by Eq. (18), has a non-vanishing value. This guarantees the presence of the diabatic mixing between the elec D and /D1Bu ) and tronic wavefunctions of the 1Bþ u and 1Bu states (/1Bþ u the absence of the diabatic mixing between those of the 1Bþ u and D D ) since He(r, Q) has approximately the Ag and / 3A 3A g states (/1Bþ g u symmetry. Thus, the observed selection rule for diabatic mixing has been theoretically proven.
349
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
Then, we examine emission from N-degenerate diabaticallyP ðpÞ mixed states expressed by Ni¼1 C i /Di ðr; qÞtini ðq Di Þ; p ¼ 1; . . . ; N. ðpÞ Since fC i g is a unitary matrix, it follows that N X
ðpÞ
Ci
ðqÞ
C i ¼ dp;q ;
N X
ðpÞ
ðpÞ
Ci
C j ¼ di;j :
ð28Þ
p¼1
i¼1
The transition-dipole moment for the emission from these excited states to the ground state, i.e., /0 ðrÞt0O ðq D0 Þ, invoking the Condon-approximation, is given by
D
E
W0 ðr; qÞjljWðpÞ e ðr; qÞ ¼
N X
ðpÞ
Ci
i¼1
/0 ðrÞjlj/Di ðrÞ
D
E
t0O ðq D0 Þtini ðq Di Þ :
ð29Þ
E Abbreviating t D Þtini ðq Di Þ as F i;ni , we have the ensemble average of emission spectra as
D
/0 ðrÞjlj/Di ðrÞ
hW0 ðr; qÞjljWe ðr; qÞi2 ¼
N X
0 O ðq
0
2 jhW0 ðr; qÞjljWðpÞ e ðr; qÞij
p¼1
¼
N X N X
¼
dij F i;ni F j;nj
i;j¼1
N X F i;n 2 : ¼
ð30Þ
i
i
Then, we obtain
hW0 ðr;qÞjljWe ðr;qÞi2 ¼
N D E2 X /0 ðrÞjlj/Di ðrÞiht0O ðq D0 Þtini ðq Di Þ : i¼1
ð31Þ Thus, the observed simultaneous stimulated emission from the electronically-mixed 1Bþ u and 1Bu states is theoretically explained (when, N = 2, /D1 ¼ /D1Bþu ; /D2 ¼ /D1Bu ). Now, we consider internal conversion between a pair of diabatic levels. Let the initial and final states be /Di ðr; qÞtimi ðq Di Þ and /Dj ðr; qÞtjnj ðq Dj Þ, respectively. The internal conversion rate from /Di timi to /Dj tjnj can be given, from Eqs. (19) and (20), as
kimi !jnj ¼
E2 2p D j D t ðq Dj ÞjV ji ðqÞjtimi ðq Di Þ dðhe0i h Xnj X þh xia mi a he0j hxja nja Þ; a
ih
WðtÞ
¼ ðH lEðtÞÞWðtÞ;
dt
ð32Þ
1 t2 EðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi exp 2 Eðeixt þ eixt Þ; 2s 2ps2
1
s ¼ pffiffiffiffiffiffiffiffi sp :
Regarding the term lE(t) as perturbation to H and setting that W(1) = Ug, the Schrödinger equation, Eq. (34), can be solved by the use of the successive approximation method. Denoting the solution by We(t), we obtain, up to the first order in the radiation field,
h/Di jHe
Thus, the rate is proportional to the square of ðqÞð ðr; qÞj/Dj iÞ. This leads to the same selection rule for internal conversion as that already proven for electronic mixing, i.e., 1Bþ u to 1Bu is al þ lowed, but 1Bu to 3Ag is forbidden. This selection rule also explains internal conversion but the why not the 1Bþ u to 3Ag þ 1Bu to 1Bu internal conversion was observed for the longer-chain Cars (n = 11–13). 3.3. Coherent excitation of a pair of diabatic levels In a polar solvent, an electric field can be generated by the fluctuation of the solvent permanent dipoles. This electric field induces the polarization of the conjugated chain of the solute Car molecule. As a result, the conjugated chain loses both the C2h and particle hole symmetries, which may facilitate the 1B u and 3Ag states with certain transition-dipole moments. If a short laser pulse is applied to the Car molecule in this situation, the coherent excitation of the totally optically-allowed 1Bþ u state and the isoenergetic, partially optically-allowed 1B u or 3Ag state can be anticipated.
Z
i h
1
dt 1 Gðt1 ÞlGðt t1 ÞUg Eðt t 1 Þ;
ð37Þ
0
where GðtÞ ¼ exp hi Ht . Expressing by Ca(t) the probability amplitude of finding Ua at time t, we get
C a ðtÞ hUa jWe ðtÞi Z 1 dt 1 hUa jGðt1 ÞjUa ilag Ug jGðt t1 ÞjUg Eðt t 1 Þ; ¼i
ð38Þ
0
where lag = hUa|l|Ugi. Applying the Wigner–Weisskopf approximation [21] to the Green functions for the excited state, i.e.,
hUa jGðtÞjUa i ¼ eca t=2ixa t
ðt > 0Þ;
ð39Þ
we have
C a ðtÞ ¼
i l h ag
Z
1
dt 1 eca t1 =2ixag t1 Eðt t 1 Þ with dxag ¼ xa xg ;
0
ð40Þ where h xa and ca are the energy and the lifetime of state a. Substitution of Eq. (35) into this equation yields
C a ðtÞ ¼
ð33Þ V Dij
ð36Þ
2 ln 2
a
1X i xa : 2 a
ð35Þ
where x and s are the mean frequency and the duration of the pulsed radiation, respectively; s is related to the FWHM duration time sp of the pulse intensity E2(t) (exp(t2/s2)) as
where
e0i ¼ ei þ
ð34Þ
where H is the Hamiltonian, l, the dipole moment of the molecule, and E(t), the radiation field expressed by
We ðtÞ ¼
ðpÞ ðpÞ C i C j F i;ni F j;nj
p¼1 i;j¼1 N X
The Schrödinger equation for such a system reads
i Elag exp i x ðxag ica =2Þ t 2 " 2 # ðx xag Þ þ ica =2Þ s2 exp 2
ðx xag Þs ca s t pffiffiffi erfc i þ pffiffiffi pffiffiffi ; 2 2 2 2s
ð41Þ
pffiffiffiffi R 1 where erfcðzÞ ð2= pÞ z expðs2 Þds. (Note that erfc(1) = 2 and erfc(0) = 1.) Denoting the modulus and phase of
" 2 exp
ðx xag Þ þ ica =2Þ 2
2
s2
# ;
ð42Þ
p by Aa(x) and da(x), we obtain for t > 2 2s
C a ðtÞ ¼
i Elag exp i x ðxag ica =2Þ t Aa ðxÞeida : 2
ð43Þ
In terms of Ca(t), We(t) can be written as
We ðtÞ ¼
X
C a ðtÞUa :
ð44Þ
a
The fluorescence intensity for the transition from the excited state We(t) down to the ground state Ug can be expressed by
2 Ifl ðtÞ ¼ Ug jljWe ðtÞ :
ð45Þ
350
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351
Combining Eqs. (43)–(45), we obtain
2 X i Ifl ðtÞ ¼ Ejlag j2 exp½iðxag ica =2ÞtAa ðxÞeida a 2
pffiffiffi ðt > 2 2sÞ: ð46Þ
Taking into account inhomogeneous broadening due to the statistical distribution of transition frequencies, we finally obtain
Here, Q is defined in such a way that viu(Q)@ vjv(Q)/@Qa takes the maximum value at Q , and /Di ðr; Q Þ is related to /Ai ðr; Q Þ by the unitary matrix Uij(Q), as already shown in Eq. (16),
/Di ðr; Q Þ ¼
N X
/Aj ðr; Q ÞU ji ðQ Þ:
ð52Þ
j¼1
Substituting this into Eq. (51) results in
2
X i 1 Ifl ðtÞ ¼ Ejlag j2 exp iðxag ica =2Þt D2a t 2 Aa ðxÞeida a 2 2 pffiffiffi ðt > 2 2sÞ; ð47Þ
! N N X X @ ðpÞ ðpÞ ða Þ /0 ðr; Q Þ W ðr; Q Þ ¼ Ci F oj ðQ ÞUji ðQ Þ : @Q a Q ¼Q i¼1 j¼1
where Da is the root-mean-square amplitude of the frequency fluctuations. Now, let us assume that only a pair of electronic excited states have appreciable values of Aa(x). Denoting the two states by a and b, we get
If the right-hand side of this equation approaches to zero for certain ðpÞ ðpÞ values of C 1 ; . . . ; C N , then, the lifetime of the state described by W(p)(r, Q) can become very long. Thus, the origin of the persistent peak can be proposed as an over lap of the 1Bþ u ð0Þ level and the isoenergetic 1Bu or 3Ag vibronic level.
Ifl ðtÞ ¼
E2 2 2 2 2 jlag j4 A2a ðxÞeca tDa t þ jlbg j4 A2b ðxÞecb tDb t 4 2 2 2 þ 2jlag j2 jlbg j2 Aa ðxÞAb ðxÞeðca þcb Þt=2ðDa þDb Þt =2 cos½ðxag xbg Þt ðda db Þ :
4. Conclusion
ð48Þ
Here, the first two terms present the incoherent decays of the two states, and the last cross term, the coherent decay (with the averaged rate) which is modulated at the angular frequency of xag xbg (‘quantum beat’). The conditions for the coherent quantum beat to be observed is that (i) both states have appreciable transition-dipole moments and that (ii) both state-energies must be close enough to the pulse frequency x so that Aa(x) and Ab(x) have non-vanishing values, which requires the relation that (x xag) s 6 1 and (x xbg)s 6 1. From this pair of inequalities combined with Eq. (36), we can find the sufficient conditions for the energies of the two states to be coherently excited,
pffiffiffiffiffiffiffiffi jxag xbg jsp 2 ln 2;
ð49Þ
where sp is the FWHM pulse duration. The above expressions can be easily extended to incorporate the vibrational levels, by replacing subscript a by a pair of subscripts a and u, representing the electronic states and vibrational levels. As shown in Section 1, the 1Bþ u ð0Þ level is in close proximity with the 1B u ð1Þ level in Car (n = 9) and with the 3Ag ð1Þ level in Car (n = 11). As described in this subsection, when these carotenoid molecules are dissolved in polar solvent, these ‘isoenergetic’ levels become partially optically-allowed, and the mixing of the pairs of diabatic levels should take place due to such extremely-small energy differences. Thus, Eq. (48) verifies the essential part of Scheme 1, and Eq. (49) shows the experimental conditions we actually achieved. Let us consider a set of diabatically-mixed states,
WðpÞ ðr; Q Þ ¼
N X
ðpÞ
C i /Di ðr; Q Þ:
ð50Þ
i¼1
We assume a large energy gap between the constituent diabatic states f/Di ðr; Q Þg and the ground state u0(r, Q). Then, from Eqs. (19) and (S2) in Supplementary material, the internal conversion rate from the diabatic states to the ground state, through the ath vibrational mode, is roughly proportional to the square of the modulus,
@ ðpÞ W ðr; Q Þ /0 ðr; Q Þ @Q a Q ¼Q N X @ D ðpÞ U ðr; Q Þ ¼ C i /0 ðr; Q Þ : @Q a Q ¼Q i¼1
ð53Þ
ð51Þ
The experimentally observed selection rules concerning electronic mixing and internal conversion between 1Bþ u ð0Þ level and the isoenergetic 1B u or 3Ag vibronic level as well as a long fluorescence lifetime and oscillatory changes in fluorescence intensity originating from the above pairs of vibronic levels (showing the apparent 1Bþ u ð0Þ stimulated emission) have been explained by the use of a theory based on the diabatic description. The experimental observations and their rigorous explanation by the diabatic theory support the energy diagram and the symme try notations concerning the 1Bþ u ; 1Bu and 3Ag vibronic levels shown in Fig. 1. Acknowledgements This work has been supported by a grant, Open Research Center Project ‘The Research Center for Photo-Energy Conversion’ to Y. Koyama. Y. Kakitani has been supported by a Grant-in-Aid for Scientific Research (C), 19579005, from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.cplett.2009.04.039. References [1] K. Furuichi, T. Sashima, Y. Koyama, Chem. Phys. Lett. 356 (2002) 547. [2] P. Zuo, A. Sutresno, C. Li, Y. Koyama, H. Nagae, Chem. Phys. Lett. 440 (2007) 360. [3] A. Sutresno, Y. Kakitani, P. Zuo, C. Li, Y. Koyama, H. Nagae, Chem. Phys. Lett. 447 (2007) 127. [4] G.E. Bialek-Bylka, Y. Kakitani, C. Li, Y. Koyama, M. Kuki, Y. Yamano, H. Nagae, Chem. Phys. Lett. 454 (2008) 367. [5] C. Li, T. Miki, Y. Kakitani, Y. Koyama, H. Nagae, Chem. Phys. Lett. 450 (2007) 112. [6] T. Miki, Y. Kakitani, Y. Koyama, H. Nagae, Chem. Phys. Lett. 457 (2008) 222. [7] J.T. Fourkas, H. Kawashima, K.A. Nelson, J. Chem. Phys. 103 (1995) 4393. [8] R. Fujii, T. Inaba, Y. Watanabe, Y. Koyama, J.-P. Zhang, Chem. Phys. Lett. 369 (2003) 165. [9] J.A. Davis, L.V. Dao, M.T. Do, P. Hannaford, K.A. Nugent, H.M. Quiney, Phys. Rev. Lett. 100 (2008) 227401. [10] M. Born, R. Oppenheimer, Ann. Phys.-Berlin 84 (1927) 457. [11] F.T. Smith, Phys. Rev. 179 (1969) 111. [12] T. Pacher, L.S. Cederbaum, H. Köppel, Adv. Chem. Phys. 84 (1993) 293. [13] A. Troisi, G. Orlandi, J. Chem. Phys. 118 (2003) 5356. [14] C.A. Mead, D.G. Truhlar, J. Chem. Phys. 77 (1982) 6090. [15] M. Baer, Chem. Phys. Lett. 35 (1975) 112. [16] P.-O. Löwdin, J. Chem. Phys. 18 (1950) 365. [17] W. Weber, W. Thiel, Theoret. Chem. Acc. 103 (2000) 495.
H. Nagae et al. / Chemical Physics Letters 474 (2009) 342–351 [18] S.H. Lin, J. Chem. Phys. 44 (1966) 3759. [19] M. Bixon, J. Jortner, J. Chem. Phys. 48 (1968) 715. [20] W. Domcke, D.R. Yarkony, H. Koppel (Eds.), Conical Intersections: Electronic Structure, Dynamics and Spectroscopy, World Scientific, New Jersey, 2004. [21] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford University Press, New York, Oxford, 1995.
351
[22] R. Fujii, T. Ishikawa, Y. Koyama, M. Taguchi, Y. Isobe, H. Nagae, Y. Watanabe, J. Phys. Chem. A 105 (2001) 5348. [23] R. Nakamura, R. Fujii, H. Nagae, Y. Koyama, Y. Kanematsu, Chem. Phys. Lett. 400 (2004) 7.