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Strong vibronic coupling in the first excited singlet state of diphenylhexatriene by an asymmetric low-frequency mode
2 October 1998
Chemical Physics Letters 295 Ž1998. 56–62
Strong vibronic coupling in the first excited singlet state of diphenylhexatriene by an asymmetric low-frequency mode M. Pfeiffer ) , W. Werncke, S. Hogiu, A. Kummrow, A. Lau Max-Born-Institut fur ¨ Nichtlineare Optik und Kurzzeitspektroskopie, D-12489 Berlin, Germany Received 17 June 1998; revised 27 July 1998
Abstract Raman studies of diphenylhexatriene in the first excited singlet state evidence strong vibronic coupling between two close 1B ur2A g states. The geometrical change after photoexcitation contains a dominant contribution of a low-frequency mode with b u symmetry near 40 cmy1. This mode is identified as a relatively strong line in the Raman spectrum. Vibronic coupling mainly affects two C5C stretching Raman bands near 1700 cmy1. A two-dimensional effective potential for the lowest excited singlet state is derived which models the coupling between the coordinate of the 40 cmy1 mode and the C5C stretching coordinate, thereby explaining the occurrence of the two bands and their strong solvent shift. The model gives, in the zero gap limit, a double-well potential for the C5C stretching coordinate, due to the pseudo-Jahn–Teller effect. Modulation of this potential by the low-frequency cycle explains the spectral broadening observed in the Raman spectra. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Upon electronic excitation, polyenic hydrocarbons show considerable alterations in the geometrical parameters of their nuclear skeleton. During the transition to a stable excited state, the C–C bond lengths along the conjugated chain can change by up to 5 pm. Compared to the ground state, the sequence of bond-order alternation may be switched or there occurs bond-equalization — both are found in energetically neighboured excited singlet states. As the bond length variations are comparable with the zero-point amplitudes of molecular vibrational modes
the polyenes are candidates for strong vibronic coupling effects, especially in case of a narrow energetic gap between the coupled states. In fact, a relatively large strength for the vibronic coupling was determined by Orlandi and Zerbetto w1x for the adiabatic coupling operator between the electronic ground state Ž1A g . und the lowest excited singlet state Ž2A g . of the hexatriene molecule Vab . This parameter derives from the conventional Herzberg–Teller expansion of electronic wavefunctions about a nuclear equilibrium configuration linear with respect to the normal coordinate Q according to
E Va b Ž Q . s ²Fa )
Corresponding author. Fax: q49-30-6392-1429.
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 9 3 5 - X
EQ
H Ž Q . F b :Q s aab Q .
Ž 1.
M. Pfeiffer et al.r Chemical Physics Letters 295 (1998) 56–62
Until recently vibronic coupling in polyenes was discussed with respect to one C5C stretching mode around 1600 cmy1 , which belongs to the a g species in molecules with an inversion centre and which is strongly Raman active. It couples both the 1A g ground state and the 2A g excited singlet state, thus leading to an upshift of the mode frequency of ; 50 cmy1 in the excited state and a downshift of the same order in the ground state w1x. Vibronic coupling between the two lowest excited singlet states belonging to the 2A g and 1B u species in analogous molecules was discussed recently. The same C5C stretching vibration is assumed to act as the active mode w5x, but to allow coupling by this mode the action of external symmetry breaking effects had to be assumed. Recently, we published experimental results giving evidence of strong vibronic coupling effects in the Raman spectrum of the lowest excited singlet state of diphenylhexatriene ŽDPH. w2x. The molecular structure of DPH and the term scheme relevant for the spectroscopic investigations is given in Fig. 1. DPH belongs to the diphenylpolyenes and there is a well-known level crossing of the two lowest excited singlet states on going from
Fig. 1. Molecular structure and schematic energy level diagram of DPH containing the singlet states involved Ž ´ excitation channel, - -) ground state Raman probe, — — ) Raman probe of the excited singlet level xxx..
57
the stilbene and diphenylbutadiene to molecules with a longer polyenic chain w3x. In diphenylbutadiene, the calculated 0–0 energy gap E2Ag y E1Bu s q350 cmy1 , while for diphenyloctatetraene E2Ag y E1Bu s y2100 cmy1 holds in a vacuum. For DPH the gap is close to zero. On applying solvents with different polarizabilities, the energy of the 1B u state can be shifted relative to that of the 2A g state, leading to drastic Raman shifts of two prominent bands between 1620 and 1850 cmy1 belonging to the abovementioned C5C stretching motion of the polyenic chain w4x. The frequencies of both bands are downshifted by more than 50 cmy1 on varying the solvent from hexane to dimethylsulfoxide Žvide infra.. Compared to the other Raman modes in the excited state which show practically no shifts, the two prominent bands are extremely broad with a FWHM of ; 70 cmy1 , in contrast to other modes of normal width Ž- 10 cmy1 .. The strong solvent dependence of the two vibrational frequencies is directly related to the gap variation between the two excited states and gives evidence for the action of strong vibronic coupling. Applying time-resolved measurements on the picosecond time scale we derive in Ref. w4x the result that all observed Raman bands are to be assigned to a single excited vibronic state. The two intense Raman bands result from one electronic state of the molecule. With respect to their frequencies they nearly coincide with either the strongest Raman band found in the 1B u state and that found in the 2A g state for related polyenes in which strong vibronic coupling is missing. As both these bands come from the same normal coordinate in the two separated electronic states they are ascribed to result from one vibrational coordinate in the case of DPH. Taking into consideration the presence of the inversion centre in the excited state, in this Letter we propose a model of vibronic coupling between the diabatic 1B u and 2A g states mediated by one lowfrequency mode of b u species. Besides this coupling has to be taken into consideration, the effect of the well-known 1A gr2A g coupling by the CC stretching mode of a g species as discussed in Ref. w1x. The model allows the qualitative explanation of the experimental findings about the two Raman bands: their downshift on applying solvents with growing refractive indices as well as their high spectral broadening. The experimental resonance Raman
58
M. Pfeiffer et al.r Chemical Physics Letters 295 (1998) 56–62
spectrum of DPH shows a low-frequency mode that plays a crucial role in this model.
3.1. Quantum chemical description of the lowest excited singlet states
2. Experiment The resonance Raman spectrum of DPH was measured in the transverse geometry using a double monochromator, spectral resolution set at 2.5 cmy1 . The solution in cyclohexane Ž0.2 mM. was pumped through a flow cuvette. The spectrum was recorded using the 334.5 nm line of an Ar 2q laser for excitation Ž50 mW.. The result shown in Fig. 2 is corrected by the usual Bose–Einstein factor nŽ v , T . q 1 where n Ž v , T . s exp Ž h vr2 p k B T . y 1
3. Modeling of the observed vibronic coupling
y1
,
Ž 2.
where nŽ v , T . the thermal population of the v th vibrational mode and k B the Boltzmann constant. The slowly variant fluorescence background is subtracted. For comparison the spectrum of the neat solvent is shown to scale. The solvent contribution to the spectrum of the solution of DPH was determined by normalizing to the intensity of the solvent lines at 385 cmy1 and the very strong line at 802 cmy1 . The solvent has only a single broad peak at 29 cmy1 in the low-frequency range. DPH gives the dominant contribution with a broad low-frequency mode at 35 cmy1 . Additional narrower lines below 300 cmy1 appear at 64, 97, 135, 187 and 257 cmy1 .
Fig. 2. The resonance Raman spectrum of DPH in the lowfrequency range Ž lexcit s 334.5 nm. after correction by the thermal mode population applying the Einstein–Bose formula. For comparison the spectrum of the pure solvent is given. ŽTriangles mark the dominant low-frequency modes for DPH and solvent respectively, and the point of spectra normalization..
Quantum chemical calculations show a distinct planar structure of C 2h symmetry for DPH in the two lowest excited singlet states. A semiempirical routine using the PM3 Hamiltonian gives the correct ordering of excited states by energy if configuration interaction is applied including double excitation. The calculation can be confined to the two highest occupied levels of the ground state configuration and to the two following lowest unoccupied molecular orbitals. These four molecular orbitals are practically determined by the p-electron system of the hexatriene backbone. They belong — with increasing energy — to the b u , a g , b u and a g symmetry species, respectively. Compared to the closed-shell configuration for the 1A g state, the second excited singlet of 1B u type is nearly perfectly a configuration with singly excited H ™ L transition ŽH denotes the HOMO, L the LUMO orbital.. The first excited singlet of 2A g type can be described as a combination of two singly excited configurations ŽH ™ L q 1, H y 1 ™ L. and one doubly excited closed-shell structure ŽH, H ™ L, L.. 3.2. Determination of the modes responsible for Õibronic coupling Symmetry conditions require that the vibronic coupling between the 2A g and 1 B u states can be mediated by vibrational modes of b u type only. For the 1A g –2A g coupling only a g-type modes are admitted and it can be expected that the modes with the largest origin shift, i.e. with the strongest Raman activity will contribute substantially. The equilibrium geometries of the 2A g and 1 B u states differ markedly from that of the ground state. By decomposing the nuclear displacements in normal coordinates the modes contributing dominantly can be identified. Atomic positions are calculated optimizing the geometry for the respective electronic state on the quantum chemical level characterized in Section 3 applying the MOPAC 93 package w6x. In taking the normal coordinates calculated for the ground state
M. Pfeiffer et al.r Chemical Physics Letters 295 (1998) 56–62
the relative contributions of the in-plane modes are given in Fig. 3. The modes are numbered by increasing wavenumbers. For both final geometries, Fig. 3 shows that from the b u manifold there is a dominating contribution from just one low-frequency mode calculated at 40 cmy1 in the ground state. The same normal mode appears in both excited states with slightly shifted frequency at 42 cmy1 in the 1B u and 49 cmy1 in the 2A g state. For the a g manifold there is one dominant contribution from the high-frequency totally symmetric C5C stretching mode. The corresponding normal coordinate — named ja-coordinate by Zerbi et al. w7x — represents a vibration with antiphased elongations in all neighbouring CC bonds of the polyenic chain. It is known to give the largest origin shift value along the p-chain and generally results in the strongest bands in the resonance Raman spectra of the polyenes. The vibrational animation for both modes is given in the upper part of Fig. 3. The b u mode is the zero-node string mode of the polyenic chain. The applied semiempirical calculation of vibrational frequencies reproduces the experimental fre-
59
quencies in the low-frequency range with an accuracy better than 10% and we take the broad intense band observed at 35 cmy1 in the ground state spectrum as the candidate for the b u coupling mode. Based on this decomposition procedure we chose a 3-state model with two active coupling modes Žnormal coordinates Q1 for the a g mode at 1595 cmy1 , Q2 for the b u mode at 40 cmy1 . to describe the vibronic coupling in the case of DPH. It is the generalization of the two-state, one-mode model of coupling between 1A g and 2A g states which is parametrized to reproduce the upshift of the 1600 cmy1 mode in the excited state relative to its ground state frequency. Within the 3-state model we estimate the effective two-dimensional potential energy surface EŽ Q1 , Q2 . by diagonalization of the diabatic Hamiltonian
H0
Hdiab s Ž a 01 Q1 . ) 0
a 01 Q1 H1
Ž a 12 Q2 . )
0
0
a 12 Q2 . H2
Ž 3.
The indices denote the electronic singlet states involved Žindex 0 for the 1A g ground state; 1 and 2
Fig. 3. Decomposition of geometrical displacements in the two lowest excited singlet states according to normal coordinates of inplane modes. Modes number 3 and 78 are those taken as involved in the vibronic coupling. Vibrational animation of these modes Ža: b u mode at 40 cmy1 , b: a g mode at 1595 cmy1 . is shown, connected by arrows to the corresponding mode contribution.
M. Pfeiffer et al.r Chemical Physics Letters 295 (1998) 56–62
60
for the lowest excited 2A g and 1B u states, respectively.. The vibronic coupling constant a 01 is set equal to the value derived by Orlandi and Zerbetto ˚ y1 amuy1r2 .. for hexatriene w1x Ž< Vge < s 1.65 eV A Our estimates within the PPP method give for a 12 a value approximately one order of magnitude lower ˚ y1 amuy1r2 . than that and it is set equal 0.2 eV A Within our approximation there is no vibronic coupling between the 1A g –1B u states through the b u mode. The harmonic approximation is chosen for the diabatic potentials. Thus the potentials in the two excited singlet states take the form
Hi Ž Q 1 , Q 2 . s
1 2
Ž v 12 Q12 q v 22 Q22 . q Bi1 Q1 q Di , Ž 4.
with normal frequencies being taken as equal in the 3 states. An origin shift B i appears for the totally symmetrical mode Q1 , Di are the electronic gaps relative to the ground state. In transparent solvents, the effective vibronic coupling according to the first approximation varies with the Lorenz and Lorentz molar refraction according to the relation 2 a 21
D21
s
2 a 21
D21
vacuum
ž
1qb
n2 y 1 n2 q 2
/
,
Ž 5.
The equation is derived applying Hudson’s rule w8x for the gap dependence on the solvent refractive index n. Here D21 s D2 y D1 , b is a positively valued parameter related to the transition dipole and polarizability properties of the solute molecule, here chosen as a fit parameter. The inclusion of the coupling to the Q2 mode gives a dependence of the adiabatic potentials in the 3 states on the Q1 coordinate as presented in Fig. 4a. To reproduce the two-band appearance of the C5C stretching motion as derived in Ref. w2x from the experimental findings, a large relative origin shift between the two diabatic potential minima has to be assumed. Fig. 4b shows details of the potential in the range of avoided level crossing. Here changes in the
Fig. 4. Ža. Effective potentials for the totally symmetric C5C stretching coordinate around 1700 cmy1 in the 3 vibronically coupled states. In the 2A g state the pseudo-Jahn–Teller effect leads to a double-well potential. ŽAbscissa values of elongation normalized to the zero-point oscillation for this mode Q10 .. Žb. Enlarged segment of the range of avoided level crossing between the 1B u and 2A g singlet states. Potential curves for two different strengths of vibronic coupling. Žc. Contour diagram of the effective 2-d vibrational potential for the first excited singlet state of DPH resulting from strong vibronic coupling Žthe cut at Q2 s Q20 r2. represents the lower potential in Žb..
potential curvatures by solvent-dependent variation of the effective vibronic coupling according to Eq. Ž5. are scheduled. Fig. 4c is the contourplot of the potential derived from diagonalization of expression Ž3., modelling the lowest excited singlet state. Fig. 4b represents the cut taken at the value of the mean amplitude for the low-frequency mode at zero-energy excitation, given by the double-arrow line in Fig. 4c.
M. Pfeiffer et al.r Chemical Physics Letters 295 (1998) 56–62
3.3. The double-minimum potential for the Q1 coordinate Under the condition that the origin shift between the two diabatic minima for the 1600 cmy1 mode is of the order of about 4 times the zero-energy amplitude of this mode, a double minimum appears for the 2A g state, in which two quasiharmonic oscillations can be excited. It results from the removal of level crossing for the 1B u and the 2A g potentials. This double minimum appears in the case of DPH through vibronic coupling by the b u mode under the condition of nearly zero gap between the two lowest excited singlet states. From this model it follows that one band of this vibration is close to the frequency of the 1650 cmy1 vibration in the diphenylbutadiene where the lowest excited singlet state is an 1B u state. The other band Žat 1750 cmy1 in DPH. lies close to the C5C stretching band found in the excited state spectrum in the diphenyl-octatetraene, which originates from the 2A g state. Based on this model we interpret the anomalous character of the two bands in the C5C stretching range with respect to solvent shift and high spectral broadening as the evidence for the action of a pseudo-Jahn–Teller effect in the excited state of DPH.
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4. Consequences of the model 4.1. SolÕent shift of the two C 5C stretching frequencies The frequencies of analog vibrational modes in two closely lying electronic states under vibronic coupling normally will drift in the opposite direction if the effective vibronic coupling is varied by solvent alteration. This results from the upshift of the potential curve in the higher state near to the avoided crossing which gives a stronger force constant while in the lower state the force constant is weakened by enhancing vibronic coupling strength. The fact that the two frequencies of both CC stretching bands of DPH shift in the same direction on changing the solvent within a single molecule model can be explained only assuming that both bands originate from the same electronic state. The model proposed here, assuming a double-well potential for the CC stretching vibration for the molecule, is able to explain the experimentally observed solvent-induced frequency shifts upon the variation of the effective vibronic coupling. The observed dependence is quantitatively reproduced by Eq. Ž4., as shown in Fig. 5. Enhanced repulsion between the
Fig. 5. Frequency shifts for the two C5C stretching bands on solvent dependent vibronic coupling Ža: model calculation assuming a vacuum 0 gap D21 s y0.01 eV; assumed shift between 1B ur2A g minima is 4)Q10 , b: measured dependence on solvent polarizability ; Ž n 2 y 1.rŽ n2 q 2... The abscissa values in Ža. and Žb. are related by Eq. Ž5..
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M. Pfeiffer et al.r Chemical Physics Letters 295 (1998) 56–62
two coupled adiabatic potentials ŽFig. 4a. on enhancing the refractive index of the solvent leads to a lowering of the curvature near to the minima of the effective 2A g potential. 4.2. Broadening by Õibrational coupling under the condition of the pseudo-Jahn–Teller effect Within the proposed model there is a simple explanation of the broadening of the two highfrequency stretching bands. It results from the excited state dynamics proceeding predominantly within the two-dimensional subspace of molecular vibrational coordinates as given by the contour-diagram in Fig. 4c. Fig. 4a and b gives the cut through the potential at Q2 s Q 20r2 with a double well along the Q1 coordinate. Vibronic coupling of the 2A g and 1B u states by the low-frequency b u mode, according to the coupling scheme of Eq. Ž3. results in an anharmonic coupling between the Q1 and the Q2 coordinates. The periodic variation of the Q1 potential during the oscillation in the Q 2 mode modifies the Q1 curvature and leads to a broadening of the bands for the high-frequency oscillation. The lowering of the saddle point at maximum elongation of the Q2 mode may lead to a cross over of the primarily excited wavepacket from one valley of the Q1 potential to the other, thus periodically varying the probability of finding the molecule at each region of the potential. The modulation of the Q1 potential with the cycle of the low-frequency oscillation yields a Fourier transform representing a spectral broadening D v1 f v2. As spectral broadening and solvent effects for vibrational modes are absent for all other modes, one can assume the essential coupling in the excited state to be confined to the two coordinates Q1 and Q2 . The other modes shall be described by independent normal coordinates.
5. Conclusions Applying the parameters of the given model derived in combining recent experimental results on vibrational spectra in the lowest excited singlet electronic state and quantum chemical estimates on a semiempirical basis, the essential features of the anomalous spectra found in the excited singlet state of DPH can be explained in a uniform manner. It is shown that from one vibrational degree of freedom two bands may arise as a consequence of the pseudo-Jahn–Teller effect. The experimental findings are related to a form of excited state coherent dynamics appearing under the condition of an avoided level crossing. A modulation of the high-frequency C5C stretching vibration by the low-frequency mode proceeds which should be directly observable by femtosecond pump–probe experiments. Acknowledgements We like to thank Dr. F. Zerbetto, University of Bologna, Italy, for valuable discussion about the role of vibronic coupling in the excited state Raman spectra from polyenes. We also thank Professor T. Elsaesser for critically reading the manuscript. References w1x G. Orlandi, F. Zerbetto, Chem. Phys. 108 Ž1986. 187. w2x S. Hogiu, W. Werncke, M. Pfeiffer, A. Lau, T. Steinke, Chem. Phys. Lett. 287 Ž1998. 8. w3x B.E. Kohler, J. Chem. Phys. 93 Ž1990. 5838. w4x W. Werncke, S. Hogiu, M. Pfeiffer, A. Lau, to be published. w5x T. Noguchi, H. Hayashi, M. Tasumi, G.H.J. Atkinson, J. Phys. Chem. 95 Ž1991. 3167. w6x J.J. Stewart, MOPAC 93 Code, Fujitsu, Tokyo, 1993. w7x M. Gussoni, C. Castiglioni, G. Zerbi, Adv. Spectrosc. 19 Ž1991. 251. w8x J.R. Andrews, B.S. Hudson, J. Chem. Phys. 68 Ž1978. 4587.
Chemical Physics Letters 295 Ž1998. 56–62
Strong vibronic coupling in the first excited singlet state of diphenylhexatriene by an asymmetric low-frequency mode M. Pfeiffer ) , W. Werncke, S. Hogiu, A. Kummrow, A. Lau Max-Born-Institut fur ¨ Nichtlineare Optik und Kurzzeitspektroskopie, D-12489 Berlin, Germany Received 17 June 1998; revised 27 July 1998
Abstract Raman studies of diphenylhexatriene in the first excited singlet state evidence strong vibronic coupling between two close 1B ur2A g states. The geometrical change after photoexcitation contains a dominant contribution of a low-frequency mode with b u symmetry near 40 cmy1. This mode is identified as a relatively strong line in the Raman spectrum. Vibronic coupling mainly affects two C5C stretching Raman bands near 1700 cmy1. A two-dimensional effective potential for the lowest excited singlet state is derived which models the coupling between the coordinate of the 40 cmy1 mode and the C5C stretching coordinate, thereby explaining the occurrence of the two bands and their strong solvent shift. The model gives, in the zero gap limit, a double-well potential for the C5C stretching coordinate, due to the pseudo-Jahn–Teller effect. Modulation of this potential by the low-frequency cycle explains the spectral broadening observed in the Raman spectra. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Upon electronic excitation, polyenic hydrocarbons show considerable alterations in the geometrical parameters of their nuclear skeleton. During the transition to a stable excited state, the C–C bond lengths along the conjugated chain can change by up to 5 pm. Compared to the ground state, the sequence of bond-order alternation may be switched or there occurs bond-equalization — both are found in energetically neighboured excited singlet states. As the bond length variations are comparable with the zero-point amplitudes of molecular vibrational modes
the polyenes are candidates for strong vibronic coupling effects, especially in case of a narrow energetic gap between the coupled states. In fact, a relatively large strength for the vibronic coupling was determined by Orlandi and Zerbetto w1x for the adiabatic coupling operator between the electronic ground state Ž1A g . und the lowest excited singlet state Ž2A g . of the hexatriene molecule Vab . This parameter derives from the conventional Herzberg–Teller expansion of electronic wavefunctions about a nuclear equilibrium configuration linear with respect to the normal coordinate Q according to
E Va b Ž Q . s ²Fa )
Corresponding author. Fax: q49-30-6392-1429.
0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 9 3 5 - X
EQ
H Ž Q . F b :Q s aab Q .
Ž 1.
M. Pfeiffer et al.r Chemical Physics Letters 295 (1998) 56–62
Until recently vibronic coupling in polyenes was discussed with respect to one C5C stretching mode around 1600 cmy1 , which belongs to the a g species in molecules with an inversion centre and which is strongly Raman active. It couples both the 1A g ground state and the 2A g excited singlet state, thus leading to an upshift of the mode frequency of ; 50 cmy1 in the excited state and a downshift of the same order in the ground state w1x. Vibronic coupling between the two lowest excited singlet states belonging to the 2A g and 1B u species in analogous molecules was discussed recently. The same C5C stretching vibration is assumed to act as the active mode w5x, but to allow coupling by this mode the action of external symmetry breaking effects had to be assumed. Recently, we published experimental results giving evidence of strong vibronic coupling effects in the Raman spectrum of the lowest excited singlet state of diphenylhexatriene ŽDPH. w2x. The molecular structure of DPH and the term scheme relevant for the spectroscopic investigations is given in Fig. 1. DPH belongs to the diphenylpolyenes and there is a well-known level crossing of the two lowest excited singlet states on going from
Fig. 1. Molecular structure and schematic energy level diagram of DPH containing the singlet states involved Ž ´ excitation channel, - -) ground state Raman probe, — — ) Raman probe of the excited singlet level xxx..
57
the stilbene and diphenylbutadiene to molecules with a longer polyenic chain w3x. In diphenylbutadiene, the calculated 0–0 energy gap E2Ag y E1Bu s q350 cmy1 , while for diphenyloctatetraene E2Ag y E1Bu s y2100 cmy1 holds in a vacuum. For DPH the gap is close to zero. On applying solvents with different polarizabilities, the energy of the 1B u state can be shifted relative to that of the 2A g state, leading to drastic Raman shifts of two prominent bands between 1620 and 1850 cmy1 belonging to the abovementioned C5C stretching motion of the polyenic chain w4x. The frequencies of both bands are downshifted by more than 50 cmy1 on varying the solvent from hexane to dimethylsulfoxide Žvide infra.. Compared to the other Raman modes in the excited state which show practically no shifts, the two prominent bands are extremely broad with a FWHM of ; 70 cmy1 , in contrast to other modes of normal width Ž- 10 cmy1 .. The strong solvent dependence of the two vibrational frequencies is directly related to the gap variation between the two excited states and gives evidence for the action of strong vibronic coupling. Applying time-resolved measurements on the picosecond time scale we derive in Ref. w4x the result that all observed Raman bands are to be assigned to a single excited vibronic state. The two intense Raman bands result from one electronic state of the molecule. With respect to their frequencies they nearly coincide with either the strongest Raman band found in the 1B u state and that found in the 2A g state for related polyenes in which strong vibronic coupling is missing. As both these bands come from the same normal coordinate in the two separated electronic states they are ascribed to result from one vibrational coordinate in the case of DPH. Taking into consideration the presence of the inversion centre in the excited state, in this Letter we propose a model of vibronic coupling between the diabatic 1B u and 2A g states mediated by one lowfrequency mode of b u species. Besides this coupling has to be taken into consideration, the effect of the well-known 1A gr2A g coupling by the CC stretching mode of a g species as discussed in Ref. w1x. The model allows the qualitative explanation of the experimental findings about the two Raman bands: their downshift on applying solvents with growing refractive indices as well as their high spectral broadening. The experimental resonance Raman
58
M. Pfeiffer et al.r Chemical Physics Letters 295 (1998) 56–62
spectrum of DPH shows a low-frequency mode that plays a crucial role in this model.
3.1. Quantum chemical description of the lowest excited singlet states
2. Experiment The resonance Raman spectrum of DPH was measured in the transverse geometry using a double monochromator, spectral resolution set at 2.5 cmy1 . The solution in cyclohexane Ž0.2 mM. was pumped through a flow cuvette. The spectrum was recorded using the 334.5 nm line of an Ar 2q laser for excitation Ž50 mW.. The result shown in Fig. 2 is corrected by the usual Bose–Einstein factor nŽ v , T . q 1 where n Ž v , T . s exp Ž h vr2 p k B T . y 1
3. Modeling of the observed vibronic coupling
y1
,
Ž 2.
where nŽ v , T . the thermal population of the v th vibrational mode and k B the Boltzmann constant. The slowly variant fluorescence background is subtracted. For comparison the spectrum of the neat solvent is shown to scale. The solvent contribution to the spectrum of the solution of DPH was determined by normalizing to the intensity of the solvent lines at 385 cmy1 and the very strong line at 802 cmy1 . The solvent has only a single broad peak at 29 cmy1 in the low-frequency range. DPH gives the dominant contribution with a broad low-frequency mode at 35 cmy1 . Additional narrower lines below 300 cmy1 appear at 64, 97, 135, 187 and 257 cmy1 .
Fig. 2. The resonance Raman spectrum of DPH in the lowfrequency range Ž lexcit s 334.5 nm. after correction by the thermal mode population applying the Einstein–Bose formula. For comparison the spectrum of the pure solvent is given. ŽTriangles mark the dominant low-frequency modes for DPH and solvent respectively, and the point of spectra normalization..
Quantum chemical calculations show a distinct planar structure of C 2h symmetry for DPH in the two lowest excited singlet states. A semiempirical routine using the PM3 Hamiltonian gives the correct ordering of excited states by energy if configuration interaction is applied including double excitation. The calculation can be confined to the two highest occupied levels of the ground state configuration and to the two following lowest unoccupied molecular orbitals. These four molecular orbitals are practically determined by the p-electron system of the hexatriene backbone. They belong — with increasing energy — to the b u , a g , b u and a g symmetry species, respectively. Compared to the closed-shell configuration for the 1A g state, the second excited singlet of 1B u type is nearly perfectly a configuration with singly excited H ™ L transition ŽH denotes the HOMO, L the LUMO orbital.. The first excited singlet of 2A g type can be described as a combination of two singly excited configurations ŽH ™ L q 1, H y 1 ™ L. and one doubly excited closed-shell structure ŽH, H ™ L, L.. 3.2. Determination of the modes responsible for Õibronic coupling Symmetry conditions require that the vibronic coupling between the 2A g and 1 B u states can be mediated by vibrational modes of b u type only. For the 1A g –2A g coupling only a g-type modes are admitted and it can be expected that the modes with the largest origin shift, i.e. with the strongest Raman activity will contribute substantially. The equilibrium geometries of the 2A g and 1 B u states differ markedly from that of the ground state. By decomposing the nuclear displacements in normal coordinates the modes contributing dominantly can be identified. Atomic positions are calculated optimizing the geometry for the respective electronic state on the quantum chemical level characterized in Section 3 applying the MOPAC 93 package w6x. In taking the normal coordinates calculated for the ground state
M. Pfeiffer et al.r Chemical Physics Letters 295 (1998) 56–62
the relative contributions of the in-plane modes are given in Fig. 3. The modes are numbered by increasing wavenumbers. For both final geometries, Fig. 3 shows that from the b u manifold there is a dominating contribution from just one low-frequency mode calculated at 40 cmy1 in the ground state. The same normal mode appears in both excited states with slightly shifted frequency at 42 cmy1 in the 1B u and 49 cmy1 in the 2A g state. For the a g manifold there is one dominant contribution from the high-frequency totally symmetric C5C stretching mode. The corresponding normal coordinate — named ja-coordinate by Zerbi et al. w7x — represents a vibration with antiphased elongations in all neighbouring CC bonds of the polyenic chain. It is known to give the largest origin shift value along the p-chain and generally results in the strongest bands in the resonance Raman spectra of the polyenes. The vibrational animation for both modes is given in the upper part of Fig. 3. The b u mode is the zero-node string mode of the polyenic chain. The applied semiempirical calculation of vibrational frequencies reproduces the experimental fre-
59
quencies in the low-frequency range with an accuracy better than 10% and we take the broad intense band observed at 35 cmy1 in the ground state spectrum as the candidate for the b u coupling mode. Based on this decomposition procedure we chose a 3-state model with two active coupling modes Žnormal coordinates Q1 for the a g mode at 1595 cmy1 , Q2 for the b u mode at 40 cmy1 . to describe the vibronic coupling in the case of DPH. It is the generalization of the two-state, one-mode model of coupling between 1A g and 2A g states which is parametrized to reproduce the upshift of the 1600 cmy1 mode in the excited state relative to its ground state frequency. Within the 3-state model we estimate the effective two-dimensional potential energy surface EŽ Q1 , Q2 . by diagonalization of the diabatic Hamiltonian
H0
Hdiab s Ž a 01 Q1 . ) 0
a 01 Q1 H1
Ž a 12 Q2 . )
0
0
a 12 Q2 . H2
Ž 3.
The indices denote the electronic singlet states involved Žindex 0 for the 1A g ground state; 1 and 2
Fig. 3. Decomposition of geometrical displacements in the two lowest excited singlet states according to normal coordinates of inplane modes. Modes number 3 and 78 are those taken as involved in the vibronic coupling. Vibrational animation of these modes Ža: b u mode at 40 cmy1 , b: a g mode at 1595 cmy1 . is shown, connected by arrows to the corresponding mode contribution.
M. Pfeiffer et al.r Chemical Physics Letters 295 (1998) 56–62
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for the lowest excited 2A g and 1B u states, respectively.. The vibronic coupling constant a 01 is set equal to the value derived by Orlandi and Zerbetto ˚ y1 amuy1r2 .. for hexatriene w1x Ž< Vge < s 1.65 eV A Our estimates within the PPP method give for a 12 a value approximately one order of magnitude lower ˚ y1 amuy1r2 . than that and it is set equal 0.2 eV A Within our approximation there is no vibronic coupling between the 1A g –1B u states through the b u mode. The harmonic approximation is chosen for the diabatic potentials. Thus the potentials in the two excited singlet states take the form
Hi Ž Q 1 , Q 2 . s
1 2
Ž v 12 Q12 q v 22 Q22 . q Bi1 Q1 q Di , Ž 4.
with normal frequencies being taken as equal in the 3 states. An origin shift B i appears for the totally symmetrical mode Q1 , Di are the electronic gaps relative to the ground state. In transparent solvents, the effective vibronic coupling according to the first approximation varies with the Lorenz and Lorentz molar refraction according to the relation 2 a 21
D21
s
2 a 21
D21
vacuum
ž
1qb
n2 y 1 n2 q 2
/
,
Ž 5.
The equation is derived applying Hudson’s rule w8x for the gap dependence on the solvent refractive index n. Here D21 s D2 y D1 , b is a positively valued parameter related to the transition dipole and polarizability properties of the solute molecule, here chosen as a fit parameter. The inclusion of the coupling to the Q2 mode gives a dependence of the adiabatic potentials in the 3 states on the Q1 coordinate as presented in Fig. 4a. To reproduce the two-band appearance of the C5C stretching motion as derived in Ref. w2x from the experimental findings, a large relative origin shift between the two diabatic potential minima has to be assumed. Fig. 4b shows details of the potential in the range of avoided level crossing. Here changes in the
Fig. 4. Ža. Effective potentials for the totally symmetric C5C stretching coordinate around 1700 cmy1 in the 3 vibronically coupled states. In the 2A g state the pseudo-Jahn–Teller effect leads to a double-well potential. ŽAbscissa values of elongation normalized to the zero-point oscillation for this mode Q10 .. Žb. Enlarged segment of the range of avoided level crossing between the 1B u and 2A g singlet states. Potential curves for two different strengths of vibronic coupling. Žc. Contour diagram of the effective 2-d vibrational potential for the first excited singlet state of DPH resulting from strong vibronic coupling Žthe cut at Q2 s Q20 r2. represents the lower potential in Žb..
potential curvatures by solvent-dependent variation of the effective vibronic coupling according to Eq. Ž5. are scheduled. Fig. 4c is the contourplot of the potential derived from diagonalization of expression Ž3., modelling the lowest excited singlet state. Fig. 4b represents the cut taken at the value of the mean amplitude for the low-frequency mode at zero-energy excitation, given by the double-arrow line in Fig. 4c.
M. Pfeiffer et al.r Chemical Physics Letters 295 (1998) 56–62
3.3. The double-minimum potential for the Q1 coordinate Under the condition that the origin shift between the two diabatic minima for the 1600 cmy1 mode is of the order of about 4 times the zero-energy amplitude of this mode, a double minimum appears for the 2A g state, in which two quasiharmonic oscillations can be excited. It results from the removal of level crossing for the 1B u and the 2A g potentials. This double minimum appears in the case of DPH through vibronic coupling by the b u mode under the condition of nearly zero gap between the two lowest excited singlet states. From this model it follows that one band of this vibration is close to the frequency of the 1650 cmy1 vibration in the diphenylbutadiene where the lowest excited singlet state is an 1B u state. The other band Žat 1750 cmy1 in DPH. lies close to the C5C stretching band found in the excited state spectrum in the diphenyl-octatetraene, which originates from the 2A g state. Based on this model we interpret the anomalous character of the two bands in the C5C stretching range with respect to solvent shift and high spectral broadening as the evidence for the action of a pseudo-Jahn–Teller effect in the excited state of DPH.
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4. Consequences of the model 4.1. SolÕent shift of the two C 5C stretching frequencies The frequencies of analog vibrational modes in two closely lying electronic states under vibronic coupling normally will drift in the opposite direction if the effective vibronic coupling is varied by solvent alteration. This results from the upshift of the potential curve in the higher state near to the avoided crossing which gives a stronger force constant while in the lower state the force constant is weakened by enhancing vibronic coupling strength. The fact that the two frequencies of both CC stretching bands of DPH shift in the same direction on changing the solvent within a single molecule model can be explained only assuming that both bands originate from the same electronic state. The model proposed here, assuming a double-well potential for the CC stretching vibration for the molecule, is able to explain the experimentally observed solvent-induced frequency shifts upon the variation of the effective vibronic coupling. The observed dependence is quantitatively reproduced by Eq. Ž4., as shown in Fig. 5. Enhanced repulsion between the
Fig. 5. Frequency shifts for the two C5C stretching bands on solvent dependent vibronic coupling Ža: model calculation assuming a vacuum 0 gap D21 s y0.01 eV; assumed shift between 1B ur2A g minima is 4)Q10 , b: measured dependence on solvent polarizability ; Ž n 2 y 1.rŽ n2 q 2... The abscissa values in Ža. and Žb. are related by Eq. Ž5..
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M. Pfeiffer et al.r Chemical Physics Letters 295 (1998) 56–62
two coupled adiabatic potentials ŽFig. 4a. on enhancing the refractive index of the solvent leads to a lowering of the curvature near to the minima of the effective 2A g potential. 4.2. Broadening by Õibrational coupling under the condition of the pseudo-Jahn–Teller effect Within the proposed model there is a simple explanation of the broadening of the two highfrequency stretching bands. It results from the excited state dynamics proceeding predominantly within the two-dimensional subspace of molecular vibrational coordinates as given by the contour-diagram in Fig. 4c. Fig. 4a and b gives the cut through the potential at Q2 s Q 20r2 with a double well along the Q1 coordinate. Vibronic coupling of the 2A g and 1B u states by the low-frequency b u mode, according to the coupling scheme of Eq. Ž3. results in an anharmonic coupling between the Q1 and the Q2 coordinates. The periodic variation of the Q1 potential during the oscillation in the Q 2 mode modifies the Q1 curvature and leads to a broadening of the bands for the high-frequency oscillation. The lowering of the saddle point at maximum elongation of the Q2 mode may lead to a cross over of the primarily excited wavepacket from one valley of the Q1 potential to the other, thus periodically varying the probability of finding the molecule at each region of the potential. The modulation of the Q1 potential with the cycle of the low-frequency oscillation yields a Fourier transform representing a spectral broadening D v1 f v2. As spectral broadening and solvent effects for vibrational modes are absent for all other modes, one can assume the essential coupling in the excited state to be confined to the two coordinates Q1 and Q2 . The other modes shall be described by independent normal coordinates.
5. Conclusions Applying the parameters of the given model derived in combining recent experimental results on vibrational spectra in the lowest excited singlet electronic state and quantum chemical estimates on a semiempirical basis, the essential features of the anomalous spectra found in the excited singlet state of DPH can be explained in a uniform manner. It is shown that from one vibrational degree of freedom two bands may arise as a consequence of the pseudo-Jahn–Teller effect. The experimental findings are related to a form of excited state coherent dynamics appearing under the condition of an avoided level crossing. A modulation of the high-frequency C5C stretching vibration by the low-frequency mode proceeds which should be directly observable by femtosecond pump–probe experiments. Acknowledgements We like to thank Dr. F. Zerbetto, University of Bologna, Italy, for valuable discussion about the role of vibronic coupling in the excited state Raman spectra from polyenes. We also thank Professor T. Elsaesser for critically reading the manuscript. References w1x G. Orlandi, F. Zerbetto, Chem. Phys. 108 Ž1986. 187. w2x S. Hogiu, W. Werncke, M. Pfeiffer, A. Lau, T. Steinke, Chem. Phys. Lett. 287 Ž1998. 8. w3x B.E. Kohler, J. Chem. Phys. 93 Ž1990. 5838. w4x W. Werncke, S. Hogiu, M. Pfeiffer, A. Lau, to be published. w5x T. Noguchi, H. Hayashi, M. Tasumi, G.H.J. Atkinson, J. Phys. Chem. 95 Ž1991. 3167. w6x J.J. Stewart, MOPAC 93 Code, Fujitsu, Tokyo, 1993. w7x M. Gussoni, C. Castiglioni, G. Zerbi, Adv. Spectrosc. 19 Ž1991. 251. w8x J.R. Andrews, B.S. Hudson, J. Chem. Phys. 68 Ž1978. 4587.