Estimation of Decay Rate Constants of 11Bu 21Ag and 21Ag 11A

ARTICLE pubs.acs.org/JPCA

Excited-State Dynamics of trans,trans-1,3,5,7-Octatetraene: Estimation of Decay Rate Constants of 11Bu ' 21Ag and 21Ag ' 11Ag Internal Conversions Reza Islampour,*,† Mahsasadat Miralinaghi,‡ and Azam Khavaninzadeh† † ‡

Department of Chemistry, Tarbiat Moallem University, 49 Mofateh Avenue, Tehran, Iran Department of Chemistry, Varamin-Pishva Branch, Islamic Azad University, Varamin, Iran ABSTRACT: The general expressions we previously derived for calculating internal conversion rate constants between two adiabatic displaced-distorted-rotated potential energy surfaces, by including all vibratinal modes, are applied to estimate the decay rate constants of 11Bu ' 21Ag and 21Ag ' 11Ag internal conversions in trans,trans-1,3,5,7-octatetraene molecule. The minimal models with respect to the number and types of vibrational modes are determined for these processes. Our calculations show that in the low temperature limit the 11Bu ' 21Ag internal conversion takes place on a 232290 fs time scale in the condensed phase and 2 ps in the gas phase, whereas 21Ag ' 11Ag internal conversion takes place on a 2 μs time scale under the isolated conditions.

1. INTRODUCTION Various spectroscopic observations, including absorption, fluorescence, and fluorescence excitation, have been performed to obtain information on the electronic structure and excitedstate dynamics of linear polyenes. It has been revealed that the linear (all-trans-) polyenes with four or more double bonds have two close-lying lowest singlet excited electronic states: the ionic 11Bu electronic state and the covalent 21Ag electronic state, which lies lower than the 11Bu state.13 The transition from the ground state 11Ag to the 11Bu state is dipole allowed, whereas to the 21Ag state is forbidden to first order. In the condensed phase, excitation to the 11Bu state is followed by fast relaxation to the 21Ag state, which is responsible for the observed Stokes-shifted fluorescence spectra. Unlike 1,3-butadiene and 1,3,5-hexatriene, which have immeasurable emissions, the longer polyenes with four through six double bonds exhibit fluorescence with a quantum yield that decreases as polyene length increases.4 The linear polyenes do not exhibit phosphorescence, and insensitivity of fluorescence quantum yields to an external heavy atom5 means that the intersystem crossing to the 13Bu or 23Ag state plays a minor role in the decay of the 11Bu or 21Ag state. Hereafter, we focus on all-trans-1,3,5,7-octatetraene and it is simply referred to as “octatetraene”. One- and two-photon fluorescence excitation spectra for the lowest energy singlet transition 11Ag f 21Ag of octatetraene in n-octane host crystal6 at 4.2 K and in He free-jet expansions,7 as well as its He free-jet fluorescence spectrum7 have been measured and vibrationally analyzed. The one-photon spectrum exhibits the features of a forbidden transition. The 000 band is absent and most of intensity is distributed among the bands that are progressions in FranckCondon active totally r 2011 American Chemical Society

symmetric ag modes built on several 010 false origins due to nontotally symmetric bu HerzbergTeller promoting modes. The two-photon spectrum is electronically allowed and the progressions in ag modes are built in the usual way on the 000 true origin. The 21Ag f 11Ag fluorescence spectrum is a good mirror image of the two-photon absorption. Conventional fluorescence measurements in solution have shown that excitation into either state 11Bu or 21Ag of octatetraene give exclusively 21Ag fluorescence. However, the 11Ag f 11Bu excitation in the gas phase leads to an unusual situation that is a violation of Kasha’s rule.8 Gavin et al.9 were the first to report the room temperature static gas emission, and the lack of a Stokes shift between absorption and emission implied a 11Bu f 11Ag assignment for the fluorescence. Heimbrook et al.10,11 later measured the 11Ag f 11Bu fluorescence excitation spectrum and a similar 11Bu f 11Ag fluorescence spectrum of octatetraene in a He supersonic jet. From the rough mirror image between the 000 excited emission and excitation spectra and the short lifetime, they concluded that the emission originates exclusively from the 11Bu state with no significant contribution from the lower lying 21Ag state. Bouwman et al.12 discovered that the vapors of room-temperature crystals of octatetraene (and other tetraenes and pentaenes) exhibit two distinct fluorescences: a strong, structured fluorescence identical to that previously reported for room-temperature vapors by Gavin et al., and ascribed to the fluorescence from 11Bu, and a broad, unstructured, weak fluorescence at longer wavelengths that, by comparison with the Received: April 27, 2011 Revised: June 20, 2011 Published: July 08, 2011 8860

dx.doi.org/10.1021/jp203895z | J. Phys. Chem. A 2011, 115, 8860–8869

The Journal of Physical Chemistry A

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Stokes-shifted fluorescence of octatetraene in the condensed phase, was assigned to the fluorescence from the 21Ag state. In addition, the ratio (11Bu f 11Ag)/(21Ag f 11Ag) of the integrated fluorescence intensity was found to be ∼16.6 for octatetraene. In a more recent work, the dual-fluorescence behavior of octatetraene has been also reported in acetonitrile as solvent13 at T = 293 K. Time-resolved fluorescence measurements on a picosecond time scale has demonstrated two distinct fluorescences after photoexcitation to the 11Bu state: a weak fluorescence originating from the 11Bu state, and a strong fluorescence assigned to the 21Ag state. In contrast to gas-phase results, the ratio (11Bu f 11Ag)/(21Ag f 11Ag) of integrated intensity was found to be 0.1 ( 0.05, which is much smaller than that found in the gas phase. This means that the internal conversion decay rate from the initially excited state 11Bu is larger in solution than in the gas phase. It has been estimated that the 11Bu ' 21Ag internal conversion in acetonitrile solution takes place on a 400 fs time scale. The temperature dependence of 21Ag f 11Ag fluorescence lifetime of octatetraene from 10 to 320 K in n-hexane, cyclohexane, and n-octane has been examined.14 The temperature dependence exhibits qualitatively similar features in all three solvents and has two distinct regions. In the lower-temperature region (below 178 K for octatetraene in n-hexane and cyclohexane and below 217 K for octatetraene in n-octane) the lifetime decreases slowly, while in the higher-temperature region it drops more rapidly with increasing temperature. The lower-temperature behavior has been attributed to adiabatic trans f cis isomerization around CdC double bond over a barrier of about 950 cm1 on the excited-state potential surface.13,14 The mechanism for the nonradiative process over a 1400 cm1 barrier in the higher-temperature region remains unexplained. On the other hand, there is both experimental7 and computational15 evidence that under isolated conditions, the trans f cis isomerization may occur nonadiabatically (at about 2100 cm1 above the ν48 false origin) by initiation on the first excited and completion on the ground electronic state potential surface. As an estimation of nonradiative lifetime, τnr, of the 21Ag state, we may use the fluorescence quantum yield9 0.58 (0.02) and fluorescence lifetime9,14 111 ns (4.4 ns) found in n-hexane at 77 K (295 K) to obtain τnr ∼ 264 ns (4.5 ns). It is our aim of this paper to calculate decay rate constants of 11Bu ' 21Ag and 21Ag ' 11Ag internal conversions of octatetraene using theoretical expressions developed in our previous works.16,17 In addition, the vibrational normal modes that have the highest contributions to these processes are investigated.

are some suitable vibrational basis sets. The eigenvalues Em(Q) of the electronic equation define the potential energy surfaces. ^ The matrix elements of H(r,Q) in the adiabatic vibronic states are given by16

2. THEORETICAL BACKGROUND

where FR(T)is the Boltzmann distribution function for the initial vibronic states and Ωab is the zerozero energy gap between the electronic states a and b. The calculation of the last relation is made easier by the generation function formalism first introduced by Kubo and Toyozawa.18 Lax19 introduced the method by applying the integral R representation of the delta function δ(x) = (1/2π) ∞ ∞dk exp(ikx) to eq 5 (see also refs 2025), whereby kIC(Ωab,T) can be defined as the Fourier transform of the generating function G(t): Z ∞ kIC ðΩab ;TÞ ¼ dt expðiΩab tÞ GðtÞ ð6aÞ

2.1. Internal Conversion Decay Rate Constant. The vibro-

^ nic (rotationless) nonrelativistic Hamiltonian operator H(r,Q) of a molecule consists of the electronic kinetic energy operator ^ N(Q), and the ^ e(r), the vibrational kinetic energy operator T T ^ total potential energy U(r,Q), where r and Q denote the set of the electronic and the set of the vibrational variables, respectively. In principle, the exact vibronic states ψj(r,Q) and the exact vibronic energies εj of a molecule can be obtained by diagonaliz^ ing the matrix of H(r,Q) in the complete set of adiabatic vibronic states {Φm(r,Q) χmμ(Q)}, where the adiabatic electronic states {Φm(r,Q)} are the solutions of the electronic Schr€odinger equation for the fixed nuclear configurations, and {χmμ(Q)}

b ðQ Þδmn ^ N ðQ Þ þ En ðQ Þ þ Λ Hmμ;nν ¼ Æχmμ j½T nn b ðQ Þjχ æ þ ð1 δmn ÞΛ nν mn

ð1Þ

where the nonadiabatic coupling operator, neglecting the terms involving second-order derivatives of the adiabatic electronic states with respect to the vibrational variables, is approximately given by Z ∂Φn ðr;Q Þ ∂ 2 b d3 r Φm ðr;Q Þ ð2Þ Λmn ðQ Þ= p ∂Q k ∂Q k k

∑

The vibrational basis sets are most conveniently chosen as the ^ N(Q) + En(Q) + eigenstates of the vibrational Hamiltonian T ^ nn(Q), thereby the Hamiltonian matrix elements reduce to Λ b ðQ Þjχ æ Hmμ;nν ¼ ε0nν δmn þ ð1 δmn ÞÆχmμ jΛ nν mn

ð3Þ

where ε0nν are defined as the zero-order adiabatic vibronic energy levels. Besides shifting the zero-order energies ε0mμ, the off-diagonal matrix elements Hmμ,nν can induce transitions among the adiabatic vibronic states. For the case in which only two electronic states a and b are involved in the vibronic coupling and that a single initially prepared adiabatic vibronic state |ΦaχaRæ couples to a large number of adiabatic vibronic states {|Φbχbβæ} via the off-diagonal Hamiltonian matrix elements HaR,bβ, it can be shown that the vibronic state |ΦaχaRæ undergoes irreversible first-order decay over time with a rate constant ΓaR(ε0aR) given by16 ΓaR ðε0aR Þ ¼

2π p

∑β jHaR;bβ j2δðε0aR ε0bβ Þ

ð4Þ

Equation 4 can be adapted to the statistical case of a complex molecule, wherein the vibrational relaxation takes place much faster than the electronic relaxation and leads to a Boltzmann distribution over the initial vibronic states. We may then define the thermally averaged internal conversion decay rate constant kIC(Ωab,T) as kIC ðΩab ;TÞ ¼

2π p

∑R ∑β FR ðTÞjHaR;bβ j2δðε0aR ε0bβ Þ

ð5Þ

∞

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dx.doi.org/10.1021/jp203895z |J. Phys. Chem. A 2011, 115, 8860–8869


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with GðtÞ ¼

1 p2

∑R ∑β FR ðTÞjHaR;bβ j2exp½iðER Eβ Þt=p

ð6bÞ

where ER and Eβ are the vibrational energies (excluding the zero-point energies) of the adiabatic electronic states a and b, respectively. The saddle-point method26 has been successfully applied for the approximate evaluation of integrals of the type appearing in eq 6a. The method states Rthat the major contribution from the integrand in the integral ∞ ∞dt exp[f(t)] comes from the vicinity of the saddle point ts where f(t) is maximum, so that ts is determined by the condition f 0 (ts) = 0. Expanding f(t) around ts to the second order as f(t) = f(ts) + 1/2f 00 (ts)(t ts)2 and introducing this expansion, with f(t) = iΩabt + ln G(t), into eq 6a and carrying out the integration we obtain " #1=2 2π expðiΩab ts Þ Gðts Þ kIC ðΩab ;TÞ= Ωab 2 þ G00 ðts Þ=Gðts Þ ð7Þ 16

In our previous work we derived an expression for the generating function G(t) for the general case in which the internal conversion takes place between two adiabatic displaced-distorted-rotated harmonic potential energy surfaces. It was assumed that a Duschinsky transformation27 of the type Q0 = JQ + D relates the vibrational normal modes Q0 of the upper electronic state to those Q of lower electronic state, where J is the Duschinsky rotation matrix that scrambles those normal modes belonging to the same irreducible representation Γj of the molecular symmetry group, so it has block-diagonal form, and D is a column vector whose components are the shift of the equilibrium nuclear configuration of the upper with respect to that of the lower electronic state. The derivation ended up with the following factorized expression for G(t) GðtÞ ¼ 2N ½hðtÞðdet Γ01 Γ1 T0 TW 1 W 2 Þ1=2 expðDT W 3 1 DÞΓp Y ½ðdet Γ01 Γ1 T0 TW 1 W 2 Þ1=2 expðDT W 3 1 DÞΓj

As an example, consider a two-mode system consisting of a nontotally symmetric undistorted promoting mode of angular frequency ωp plus a totally symmetric undistorted accepting mode of angular frequency ω. For this case we have ωp jRp j2 ½ð1 þ np Þ expð iωp tÞ þ np expðiωp tÞ GðtÞ ¼ 2p expfSð1 þ 2nÞ þ S½ð1 þ nÞ expð iωtÞ þ n expðiωtÞg

where S = ωD /2p is the HuangRhys coupling constant and n (np) is the mean occupation number of the states for the totally (nontotally) symmetric mode. We can now calculate the internal conversion decay rate constant for this case using eq 7. At low temperatures, the saddle point ts is determined by Sω exp(iωts) = Ωab ωp, and eq 7 gives " #1=2 ωp 2π 2 kIC ðΩab ;T ¼ 0Þ ¼ jRp j ωðΩab ωp Þ 2p ( ) ðΩab ωp Þ Ωab ωp ln exp S 1 Sω ω ð10Þ which illustrates the energy gap dependence of the rate constant. 2.2. Estimation of the Duschinsky Parameters J and D from the Intensities of Vibronic Bands. Following Herzberg and Teller,28 the intensity of a vibronic transition between the ground Φg(r,Q)χυ1υ2 3 3 3 υN(Q) and excited Φe(r,Q0 )χυ0 1υ0 2 3 3 3 υ0 N (Q0 ) adiabatic vibronic states is proportional to the square of the transition dipole moment defined by 0

μgv;ev0 =

ð8bÞ

where the first bracket on the right-hand side of eq 8a corresponds to the promoting modes, which induce the nonradiative transitions, and the next brackets to the accepting modes, which act as sinks for the electronic energy. The dimensions of the symmetric matrices W1, W2, W3, and W4 and diagonal matrices Γ0 , Γ, T0 , and T (all defined in ref 16) in each bracket are determined by the number of the vibrational modes that have the same symmetry species. In eq 8b R is the column vector of the nonadiabatic coupling matrix elements Rp = ipÆΦa|∂/ ∂Q0p|Φbæ0 in the Condon approximation, where Æ...æ0 means the value of the matrix elements in the vicinity of some reference configuration. It should be noted that for nontotally symmetric modes D is a null vector.

0

∑p ð∂μge ðQ Þ=∂QpÞ0 Æυ1υ2 3 3 3 jQpjυ1 υ2 3 3 3 æΓ 0

0

0

1

0

Æυ1 υ2 3 3 3 jυ1 υ2 3 3 3 æΓs 3 3 3

with

þ ðR † W 3 1 DÞðDT W 3 1 RÞ

0

for the dipole-allowed and by

ð8aÞ

1 † R ðW 3 1 þ W 4 1 ÞR 2

0

μgv;ev0 =μ0ge Æυ1 υ2 3 3 3 jυ1 υ2 3 3 3 æΓs Æυ1 υ2 3 3 3 jυ1 υ2 3 3 3 æΓ1 3 3 3 ð11aÞ

j6¼ p

hðtÞ ¼

ð9Þ

2

ð11bÞ

for the dipole-forbidden electronic transitions, where “0” refers to the evaluation of the electronic transition dipole moment μge(Q) and its derivatives with respect to the HerzbergTeller promoting modes Qp, ∂μge(Q)/∂Qp, at the equilibrium configuration of the ground electronic state. Factorization of the multidimensional vibrational integrals in eqs 11a and 11b as a product of vibrational integrals with lower dimensions is due to the fact that the Duschinsky rotation matrix, as pointed before, has block diagonal form, and each block corresponds to the vibrational modes that belong to the same symmetry species. In these equations Γs refers to the symmetry species of totally symmetric modes and Γ1, ... to the symmetry species of nontotally symmetric modes. The vibrational modes of each symmetry species are renumbered as 1, 2, .... Equation 11a shows that in all allowed electronic transitions the first band at low temperature is the 000 band, called the origin band, followed by progressions in totally symmetric modes and combination bands built on it. In contrast, it follows from 8862



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eq 11b that in the forbidden electronic transitions the first band at low temperature is one in which a nontotally symmetric mode is one quantum excited and the bands involving totally symmetric modes joins on it. The overall appearance of a forbidden band system is thus much like that of an allowed system except that it is built up on the false origins 010 (and 100 ) in the nontotally symmetric vibration instead of the 000 origin band. The occurrence of the progressions in polyatomic molecules depends on the extent of change in the equilibrium geometry as well as changes in vibrational frequencies upon the electronic transition. The relative contribution of these two factors depends on the symmetry of vibrational modes. For vibronic transitions involving totally symmetric vibrations the equilibrium geometry change is the dominating factor, while for the vibronic transitions involving nontotally symmetric vibrations any equilibrium geometry change is forbidden by symmetry, and frequency shifts may cause nontotally symmetric vibrations to gain some activities. Suppose the electronic transition occurs from the lowest vibrationless level of the ground electronic state, then the relative intensity along the progression of a totally symmetric, e.g., mode 1, relative to the (true or false) origin is given by 0

0

0

0

0

Iðυ1 Þ ¼ jÆ01 02 3 3 3 jυ1 02 3 3 3 æΓs j2 =jÆ01 02 3 3 3 j01 02 3 3 3 æΓs j2 ð12Þ and the relative intensity of the combination of, e.g., totally symmetric modes 1 and 2, relative to the origin by 0

0

0

0

0

2

0

0

Iðυ1 ;υ2 Þ ¼ jÆ01 02 3 3 3 jυ1 υ2 03 3 3 3 æΓs j =jÆ01 02 3 3 3 j01 02 3 3 3 æΓs j

Iðυi ;υj Þ ðβij Þk ðβii =2Þðυi kÞ=2 Hυi k ½βii 1=2 ðRδÞi ¼ ð1Þυi þ υj ðυi !υj !Þ1=2 k! ðυi kÞ! k¼0

∑

0

Iðυi ;υj Þ ð2Rij Þk ðβii =2Þðυi kÞ=2 0 Hυi k ½βii 1=2 ðRδÞi ¼ ð1Þυi ðυi !υj !Þ1=2 k! ðυi kÞ! k¼0

∑

2 0 ðRjj =2Þðυj kÞ=2 1=2 0 H fR ½ðE PÞδ g jj j υj k ðυ0j kÞ!

ð18Þ

The expressions for more-mode excitations can be derived similarly. Here Rij = (E 2P)ij, βij = (E 2Q)ij, where E is a unit matrix, δ = Γ0 1/2 D is the vector of the reduced displacements, and the symmetric matrices P and Q and the matrix R are defined by P ¼ Γ01=2 JG1 JT Γ01=2

ð19aÞ

Q ¼ Γ1=2 G1 Γ1=2

ð19bÞ

R ¼ Γ1=2 G1 JT Γ01=2

ð19cÞ

with G ¼ Γ þ JT Γ0 J

ð19dÞ

The dimensions of the matrices E, P, Q, R, G, and Γ and the vector δ are equal to the number of the vibrational modes entering in the related FranckCondon integral. It should be noted that the matrices P, Q, and R are not independent of one another. Knowing one can yield the other two from eqs 19a19d. To find the reduced displacement vector δ and the rotation matrix J of the Duschinsky transformation from the relative intensities of the vibronic bands measured experimentally, we may proceed as follows. Equations 19a19c with the help of eq 19d can be written, respectively, as

0

Iðυi ;υj Þ 0 0 0 ðRij Þk ðRii =2Þðυi kÞ=2 1=2 Hυ0 k fRii ½ðE PÞδi g ¼ ðυi !υj !Þ1=2 i k! ðυ0i kÞ! k¼0

∑

and in the emission spectra as 2 υi =2 υi ðβii =2Þ 1=2 Iðυi Þ ¼ ð1Þ H ½β ðRδÞ υi ii i ðυi !Þ1=2

ð17Þ

2

In the following we shall outline the procedure used for finding the parameters of the Duschinsky transformation from the relative intensities of a set of vibronic bands measured experimentally. The procedure assumes that the vibrational analysis has been made for the vibronic band system yielding vibrational frequencies of both the ground and excited electronic states. In a previous paper,29 we derived closed form expressions for the multidimensional FranckCondon integrals between displaced, distorted, and rotated potential energy surfaces. Making use of eqs 3539 of ref 29, we may write the relative intensities, with respect to the false (or true) origin, for one- and two-mode excitations in the absorption spectra as 2 ðR =2Þυ0i =2 0 ii 1=2 0 fRii H ½ðE PÞδ g ð14Þ Iðυi Þ ¼ i υ i ðυ0 !Þ1=2 i

2 0 ðRjj =2Þðυj kÞ=2 1=2 0 H fR ½ðE PÞδ g jj j υj k ðυ0j kÞ!

ðυj kÞ!

2

Hυj k ½β1=2 ðRδÞj jj

and for the case in which the ith mode is excited with υi quanta in the lower electronic state and the jth mode with υ0j quanta in the upper electronic state we have

ð13Þ

0

ðβjj =2Þðυj kÞ=2

JT Γ01=2 ðP1 EÞΓ01=2 J ¼ Γ

ð20aÞ

JΓ1=2 ðQ 1 EÞΓ1=2 JT ¼ Γ0

ð20bÞ

0

ðR 1 Þij ¼ Jij

ð15Þ

ð16Þ

ωi þ ωj ðω0i ωj Þ1=2

ð20cÞ

Equations 20a and 20b show that the rotation matrix J is found by diagonalization of either the matrix Γ0 1/2 (P1 E) Γ0 1/2 or Γ1/2 (Q1 E) Γ1/2, and eq 20c shows that the same matrix J is determined from the known matrix R. The matrix elements of P, Q, and R can be written in terms of the relative intensities given 8863



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in eqs 1418 as30,31 pffiffiffi 1 0 0 Pii ¼ ½1 Ið1i Þ ( 2I 1=2 ð2i Þ 2

ð21aÞ

1 0 0 0 0 Pij ¼ ( ½I 1=2 ð1i ;1j Þ I 1=2 ð1i ÞI 1=2 ð1j Þ 2

ð21bÞ

pffiffiffi 1 Q ii ¼ ½1 Ið1i Þ ( 2I 1=2 ð2i Þ 2

ð22aÞ

1 Q ij ¼ ( ½I 1=2 ð1i ;1j Þ I 1=2 ð1i ÞI 1=2 ð1j Þ 2 1 0 0 Rij ¼ ( ½I 1=2 ð1i ;1j Þ ( I 1=2 ð1i ÞI 1=2 ð1j Þ 2

pffiffiffi 2ðRδÞi

ð23Þ

ð24bÞ

So that the evaluation of the Duschinsky parameters (J, δ), apart from the sign of the elements of the column vector δ, using the experimentally measured relative intensities is completed. It follows that for displaced-distorted oscillator !2 ωi 0 0 S ð25aÞ Ið1i Þ ¼ 4 ωi þ ω0i i !2 0 ωi Ið1i Þ ¼ 4 Si ωi þ ω0i

Ωab

ð25bÞ

and for the displaced oscillator case I(10 i) = I(1i) = ωiDi2/2p = Si. Therefore, for the displaced oscillator case, the relative intensity of the transition in which only the ith totally symmetric mode undergoes one quantum excitation is a measure of the Huang Rhys factor Si of that mode.

3. RESULTS AND DISCUSSION In this section, we shall apply the low-temperature limit versions of eqs 7 and 8a,8b to calculate decay rate constants for the 21Ag ' 11Ag and 11Bu ' 21Ag internal conversions in octatetraene. In the low temperature limit the internal conversions take place from the ground vibrational level of the upper electronic state. The zerozero excitation energies to the covalent 21Ag and ionic 11Bu electronic states are collected in Table 1.6,7,9,32,33 The energy of the 11Bu state depends on the environment and is about 3500 cm1 lower in the matrix than in the gas phase. However, the energy of the 21Ag state is much less sensitive to the environment. The 48 vibrational modes of octatetraene molecule are distributed among the irreducible representations of the C2h

transition

(cm1)

2 Ag1 Ag

28948.7

supersonic jet

two-photon fluorescence excitationa

28560

n-octane at 4.2 K

one- and two-photon

1

1

phase

spectrum

fluorescence excitationb 1 Bu1 Ag 1

ð22bÞ

The ambiguity in sign in eqs 2123 can be removed by calculating the eigenvalues of the matrix Γ0 1/2 (P1 E) Γ0 1/2 in eq 20a, which have to be equal to the diagonal elements of the known matrix Γ or the eigenvalues of the matrix Γ1/2 (Q1 E) Γ1/2 in eq 20b, which have to be equal to the diagonal elements of the known matrix Γ0 . The reduced displacement δ can be determined from either eq 14 or eq 16 as pffiffiffi 0 ( I 1=2 ð1i Þ ¼ 2½ðE PÞδi ð24aÞ ( I 1=2 ð1i Þ ¼

Table 1. ZeroZero Excitation Energies (cm1) of the 21Ag and 11Bu Electronic States of Octatetraene

a

1

b

28621

n-hexane at 77 K

fluorescence emissionc

35553

supersonic jet

absorptiond

35523

static gas

absorptionc

32009

n-hexane at 4.2 K

fluorescence excitatione

32082

n-octane at 4.2 K

fluorescence excitatione

c

d

Reference 7. Reference 6. Reference 9. Reference 32. e Reference 33.

point group as 17ag + 16bu + 8au + 7bg; ag and bu are in-plane modes, while au and bg are out-of-plane vibrations. Table 2 displays vibrational frequencies of three lower electronic states of the molecule. Vibrational frequencies in the ground electronic state 11Ag have been experimentally determined by infrared and Raman spectroscopic measurements at room temperature.34 The experimental vibrational frequencies of ag and bu in-plane modes in the first excited electronic state 21Ag are provided by one- and twophoton 11Ag f 21Ag fluorescence excitation in He free-jet expansion7 and emission spectra of octatetraene in an n-octane host crystal at 4.2 K.9 The frequencies in 11Ag and 21Ag states, which were not determined experimentally, were obtained by scaling procedures as described in ref 35. The direct absorption spectra (in jet cooled and room temperature gas phase)9,32 and fluorescence excitation spectra (in n-octane and n-hexane at 4.2 K)35 of the 11Ag f 11Bu transition have been analyzed for determining the second excited-state vibrational frequencies of ag modes. Investigation of potential energy surfaces by an extended PoplePariserParr (PPP/CI) model has given adiabatic frequencies of bu modes in 11Bu electronic state.36 The displacement vectors D(S1) and D(S2), calculated from eq 24 using relative intensities of the first quanta of the progressions of ag modes in the 11Ag f 21Ag and 11Ag f 11Bu transitions,7,9,32,33,35 are presented in Table 3. In the present calculations we have neglected the rotation of normal modes upon excitation, so the Duschinsky matrix has been considered as a unit matrix. Displacement vector D between the normal modes of two excited electronic states 11Bu and 21Ag can be obtained from the relation D = D(S2) D(S1). Because the sign of each element of the column vectors D(S2) and D(S1) is undetermined, there are four possible values for each element of D that occur in pairs with opposite signs, thereby two possible sets of the HuangRhys factors corresponding to displacement D. The HuangRhys factors corresponding to displacements D(S1), D(S2), and D (S0j = ω0j Dj2/2p, where ω0j refers to the upper electronic state) are presented in Table 4. The 11Ag21Ag and 21Ag11Bu first-order nonadiabatic vibronic coupling constants induced by ag and bu vibrations, respectively, are collected in Table 5.35,37 Armed with Tables 15 we can now calculate the decay time constants for the 21Ag ' 11Ag and 11Bu ' 21Ag internal conversions. First we consider 21Ag ' 11Ag internal conversion wherein the 21Ag and 11Ag electronic states are vibronically coupled by the ag normal modes ν6 (consisting of in-phase CC double-bond stretches, CH2 scissors, and CH in-plane bends34) and ν12 (consisting of the in-phase CC single-bond stretches and CH 8864



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Table 2. Vibrational Frequencies (cm1) of 11Ag, 21Ag, and 11Bu Electronic States of Octatetraene ag modes 11Ag IR-Ramana

21Ag supersonic jetb

11Bu n-octaned

supersonic jete

static gas f

n-octaneg

n-hexaneg

1235

1246

1228

1228

1645

1666

1628

1646

v17

242

210

219

197

v16

342

336

340

348

v15

538

530

529

547

v14

956

965

v13

1136

1080

v12

1179

1225

1219

v11

1281

1276

1271

v10 v9

1291 1299

1280 1284

v8

1423

1445c

v7

1608

1509

v6

1617

1798

1006

1754

bu modes 11Ag

21Ag

IR-Ramana

supersonic jetb

11Bu n-octaned

theoryh

ν48

96

76

93

99

ν47

390

440

463

439

ν46

565

523

538

590

ν45 ν44

940 1139

952 1044

1054

935 1175

ν43

1229

1138c

1212

ν42

1280

1230c

1274

ν41

1303

1268c

1309

ν40

1405

1363c

1393

ν39

1569

1554

1478

ν38

1632

1591

1510 au modes

11Ag

21Ag

IR-Ramana

theoryc

ν25

55i

44

ν24

181

147

ν23

245

271

ν22

629

329

ν21

840

475

ν20

900

790

ν19

960

842

ν18

1011

928 bg modes

11Ag

21Ag

IR-Ramana

theoryc

ν32

164

177

ν31

343

271

ν30

722

297 8865



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Table 2. Continued bg modes 11Ag

21Ag

IR-Ramana ν29 ν28

theoryc

877 896

484 739

ν27

958

814

ν26

1080

984

Reference 34. Mid-infrared spectra (4000400 cm1) taken from a CCl4 solution and far-infrared spectra (50050 cm1) from an n-hexane solution. The Raman spectra in the region 18001000 cm1 taken from a CCl4 solution and in the region 1000300 cm1 from a cyclohexane solution at room temperature. The vibrational frequencies of ag(ν1ν5,ν17), au(ν25), bg(ν26,ν27,ν30,ν32), and bu(ν35, ν45) have not been determined from IR-Raman spectra. b Reference 7. c Reference 35. d Reference 6. e Reference 32. f Reference 9. g Reference 33. h Calculated (adiabatic) frequencies from ref 36. i Calculated frequency from ref 35. a

Table 3. Relative Intensities of the Totally Symmetric Fundamental Bands of the 11Ag 21Ag (Built on v48 False Origin) and 11Ag11Bu (Built on True Origin) Spectra of Octatetraene 11Ag21Ag supersonic jeta

11Ag11Bu supersonic jetb static gasc n-octaned n-hexaned

I(1017) I(1016)

0.035 0.559

0.11 0.05

I(1015)

0.04

0.02

I(1014)

0.061

I(1013)

0.143

0.03

I(1012)

0.659

0.23

0.36

0.416

0.572

I(1011)

0.141

I(1010)

0.083

I(109) I(108)

0.043 0.000

I(107)

0.178

I(106)

2.164

0.60

0.63

0.649

0.913

a

The experimental values of ref 7 scaled by the ratio of the lifetime of ν48 false origin to the lifetime of fundamental bands, as in ref 35. b Reference 32. c Reference 9. d Reference 33.

in-plane bends34). However, it is expected that ν6 will be more effective than ν12 in the internal conversion process due to its higher frequency (1798 cm1 vs 1225 cm1), significantly larger displacement parameter (2.413 vs 0.685), and larger nonadiabatic coupling constant (0.0533 vs 0.0306). To explore the vibrational modes that have the highest contributions in the process, the decay time constants τIC = 1/kIC have been calculated for different models with respect to the number and types of vibrational modes, and to investigate the effects of distortions, each model is subdivided to displaced and displaceddistorted submodels. For displaced submodels, the geometric averages of the vibrational frequencies ω0j and ωj, that is, (ω0j ωj)1/2, are used. The calculated time constants are presented in Table 6. The single mode model containing ν12 is absent from this table as it yields an extremely small decay rate constant. Table 6 shows that for the displaced models the time constant decreases monotonically with the increasing number of modes up to the 12-mode model and remains constant thereafter. A similar trend is observed for the displaced-distorted models; however, the decrease in time constant continues on to the

limiting value of about 2 μs determined by the 38-mode model. In addition, the ratio of the decay time constant for the displaceddistorted model to the decay time constant for the corresponding displaced model decreases with the number of modes from ∼56 for the 6-mode model to the limiting value of about 5 determined by the 38-mode model. Therefore, the effect of distortions of vibrational modes on 21Ag ' 11Ag internal conversion is to reduce its decay rate constant considerably. We may tentatively take the 3-displaced mode model consisting of ν6, ν12, and ν16 (an in-phase skeletal deformation mode34) as the minimal model for the process. The experimental quantum yield of 21Ag f 11Ag fluorescence for octatetraene in an n-hexane matrix at 77 K is estimated to be 0.58 ( 0.15 and the fluorescence lifetime to be 111 ns,38,14 so the lifetime of radiationless transition should be of the order of 1 μs. Dinur and Scharf40 calculated the nonradiative lifetime for this molecule in the range of energy gaps of 18 00029 000 cm1, which covers the 11Ag21Ag energy gap, and obtained τnr = 1.587, 0.462 μs for ΔE = 29 000, 28 000 cm1, respectively. So the present results are in reasonable agreement with the observed and calculated decay time constants. The temperature dependence of the decay time constant associated with 21Ag ' 11Ag internal conversion for the displaced-oscillator model has been also investigated using gasphase parameters. The decay time constant remains almost constant at a value of ∼0.3 μs up to 490 K. This may be considered as an indication that the decrease in 21Ag f 11Ag fluorescence lifetime with increasing temperature14,15 is due to another radiationless process, as trans f cis isomerization, in the 21Ag excited electronic state. Next we consider 11Bu ' 21Ag internal conversion in which the coupling between 11Bu and 21Ag electronic states occurs by the nontotally symmetric bu modes. Among bu modes, the mode ν38 (which is composed of CC double-bond stretches, CH2 scissors, and CH in-plane bends34) is the most effective promoting mode in that it has the highest frequency (Table 2) and the largest nonadiabatic coupling constant (Table 5). As can be seen from Tables 1, 2, and 4 for the 11Bu ' 21Ag transition, the zerozero energy gap, the frequencies of totally symmetric accepting modes, and HuangRhys factors depend on the phase and environment. Therefore, decay time constants associated with 11Bu ' 21Ag internal conversion have been calculated for octatetraene in free-jet expansions, static gas, and n-octane and n-hexane matrices. The results are presented in Tables 710. In these tables, the decay time constants given in parentheses 8866



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Table 4. HuangRhys Factors S0j for the 21Ag11Ag, 11Bu11Ag, and 11Bu21Ag Electronic States of Octatetraene 21Ag11Ag

11Bu11Ag

supersonic jet S017 S016 S015 S014 S013 S012 S011 S010 S09 S08 S07 S06 a

supersonic jet

static gas

11Bu21Aga

n-octane

supersonic jet

0.031

0.090

0.017 (0.221)

0.549

0.051

0.279 (0.960)

0.039

0.020

0.003 (0.119)

static gas

n-octane

n-hexane

0.062 0.136

0.027

0.685

0.241

0.381

0.433

0.596

0.037 (0.270) 0.116 (1.748)

0.047 (2.108)

0.030 (2.211)

0.003 (2.562)

0.610

0.649

0.653

0.929

0.496 (5.140)

0.475 (5.295)

0.474 (5.23)

0.273 (6.559)

0.140 0.082 0.043 0.000 0.167 2.413

The numbers in parentheses refer to the second possible set of the HuangRhys factors.

Table 5. Calculated First-Order Nonadiabatic Vibronic Coupling Matrix Elements ÆΦa|∂/∂Q0p|Φbæ0 transition 2 Ag ' 1 1

Aga

1

11Bu ' 21Agb

a

n-hexane

Table 6. Decay Time Constant Associated with 21Ag ' 11Ag Internal Conversion for Octatetraene in Free-Jet (Ωab = 28 949 cm1)

promoting modes

matrix elements (dimensionless)

Q6

0.0533

Q 12

0.0306

single mode model

Q48

0.02734

displaced

Q47

0.00416

2-mode model

Q46

0.00480

displaced

Q45 Q44

0.02037 0.02770

3-mode model

Q43

0.03293

6-mode model

Q42

0.02701

Q41

0.00819

displaced displaced-distorted

Q40

0.02877

12-mode model

Q39

0.01724

displaced

Q38

0.13335

displaced-distorted

21Ag ' 11Ag

displaced

Reference 35. b Reference 2.

correspond to the second possible set of the HuangRhys factors given in Table 4. It is seen that this second set leads to much smaller, and therefore unrealistic, internal conversion decay rates in the condensed phase. Taking mixed HuangRhys factors from the two possible sets given in Table 4 into the calculations is the alternative that is ignored here. Tables 710 indicate that mode ν6 plays a more important role than ν12, as an accepting mode, in the nonradiative process. This is due to the fact that ν6 has a higher frequency and, as Table 4 shows, it has a much larger displacement parameter (0.496 vs 0.116 in free-jet; 0.475 vs 0.047 in static gas; 0.474 vs 0.030 in n-octane; 0.273 vs 0.003 in n-hexane). Inclusion of more modes, beyond the two-mode model (ν6, ν38), does not cause a pronounced change in the rate of process. The time constant for both displaced and displaced-distorted models gradually decreases as the number of modes increases. However, the ratio decay time constant for the displaced-distorted model to the decay time constant for the corresponding displaced remains almost constant beyond the two-mode model (ν6, ν38) at values of about 0.55 and 0.52 for the free-jet and static gas, and from the

τIC = 1/kIC (μs)

types of modes 6

20.19 6, 12 2.72 6, 12, 16 2.13 6, 7, 11, 12, 13, 16 0.54 30.18 all ag modes 0.34 13.55

19-mode model

all g-types modes

displaced

0.34

displaced-distorted

3.25

38-mode model displaced

0.34

displaced-distorted

1.76

all modes

two-mode model thereafter at values of about 0.94 and 0.88 for n-octane and n-hexane, respectively. This allows us to conclude that the effect of distortions of vibrational modes on 11Bu ' 21Ag internal conversion is not as important as their effect on the 21Ag ' 11Ag process. Distortions increase the decay rate by a factor of almost 2 for the free-jet and static gas cases and have a minor effect on the decay rate for the n-octane and n-hexane cases. It is also concluded that the three-mode model consisting of promoting mode ν38 (bu) and two totally symmetric accepting modes ν6 and ν12, which have the highest displacement parameters (Table 4), may be considered as the minimal model for the 11Bu ' 21Ag process. In addition, Tables 710 show that 11Bu ' 21Ag internal conversion takes place more efficiently in the condensed phase (τIC = 232 fs in n-octane and 290 fs in n-hexane) than in the gas 8867



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Table 7. Decay Time Constant Associated with 11Bu ' 21Ag Internal Conversion for Octatetraene in Free-Jet (Ωab = 6604 cm1) 11Bu ' 21Ag

τIC = 1/kIC (ps)

single mode model distorted

displaced-distorted 2-mode model

38

distorted 12, 38

displaced displaced-distorted 2-mode model

6, 38

displaced-distorted

2.63 (0.73)

3-mode model

12, 6, 38

displaced

4.07 (0.97)

displaced-distorted

2.17 (1.42)

4-mode model

3.51 (1.06)

displaced-distorted

1.94 (1.54)

9-mode model 3.31 (0.97)

displaced-distorted

1.80 (1.39)

displaced-distorted

269.40 (2612.33)

3.24 (0.95)

displaced-distorted

1.75 (1.36)

17-mode model

276.26 (8045.41)

displaced-distorted

262.34 (11749.09)

17, 16, 15, 13, 12, 6 plus all bu modes

displaced-distorted

237.56 (9957.36) 12, 6, plus all bu modes


247.95 (6764.97)

displaced-distorted

232.22 (9682.80)

Table 10. Decay Time Constant Associated with 11Bu ' 21Ag Internal Conversion for Octatetraene in n-Hexane (Ωab = 3388 cm1) 11Bu ' 21Ag

displaced-distorted

1.65 (1.61)

single mode model distorted

τIC = 1/kIC (ps)

displaced displaced-distorted 2-mode model

types of modes 38

6458.50 328161.23 (0.54) 3601.49 (0.54)

2.69 (1.80)


4.95 (1.20)

displaced-distorted

2.61 (1.72)


4.66 (1.09)

6, 38 371.51 (4015.04)

displaced-distorted

329.18 (6105.65) 12, 6 ,38

displaced

369.01 (25072.63)

displaced-distorted

327.66 (38921.90) 12, 6, 43, 38 360.97 (23899.90) 319.13 (36800.22) 12, 6, 48, 44, 43, 42, 40, 38

displaced

340.35 (21602.29)

displaced-distorted

297.23 (32830.65) 2 ag plus 11 bu modes

13-mode model

12, 6, 43, 38

12,38

5155.12 (204.16)

8-mode model

12, 6, 38 5.06 (1.24)

336236.20 (188.83)

displaced


5.79 (0.55) 2.92 (0.77)

displaced-distorted

38

4-mode model

6, 38

displaced

types of modes

5361.95

3-mode model 12, 38

3-mode model

τIC =1/kIC(fs)

2-mode model

Table 8. Decay Time Constant Associated with 11Bu ' 21Ag Internal Conversion for Octatetraene in Static Gas (Ωab = 6574 cm1)

displaced-distorted 13-mode model displaced displaced-distorted

12, 6, 48, 44, 43, 42, 40, 38 253.06 (6962.92)

2.99 (1.11)

2-mode model displaced displaced-distorted

12, 6, 43, 38 269.58 (7683.22) 255.10 (11134.12)

displaced

displaced

single mode model distorted 2-mode model displaced displaced-distorted

12, 6, 38

displaced

16, 12, 6 plus all bumodes


11Bu ' 21Ag

6, 38 286.45 (1826.10)

8-mode model

16, 12, 6, 48, 44, 43, 42, 40, 38

displaced

5164.14 (175.98)

displaced

displaced displaced-distorted 16, 12, 6, 43, 38

displaced

12,38 21160.13 (166.21)

4-mode model

3.59 (1.10) 2.00 (1.61)

5-mode model

38

3-mode model

16, 12, 6, 38

types of modes

7234.74

2-mode model

1585.86 (0.82) 5.55 (0.52)

τIC = 1/kIC (fs)

single mode model

11507.30 (0.83)

displaced


11Bu ' 21Ag

types of modes

6905.68


Table 9. Decay Time Constants Associated with 11Bu ' 21Ag Internal Conversion for Octatetraene in n-Octane (Ωab = 3522 cm1)

displaced

333.75 (20971.81)

displaced-distorted

290.12 (31912.13)

12, 6, 48, 44, 43, 42, 40, 38 2.42 (1.56) 12, 6, plus all bu modes 4.57 (1.07) 2.36 (1.52)

phase (τIC = 2 ps). These results are in consistent with the ratio (11Bu f 11Ag)/(21Ag f 11Ag) of integrated fluorescence intensity that was experimentally estimated to be ∼16.6 in the gas phase12 and 0.1 ( 0.05 in acetonitrile solution.13 The internal conversion in acetonitrile solution at 293 K was estimated to be 400 fs.13 8868


The Journal of Physical Chemistry A The time constants calculated in this work are also comparable to time constants 570, 600, and 470 fs experimentally estimated for the 11Bu ' 21Ag internal conversion of trans,trans-1,8diphenyl-1,3,5,7-octatetraene in cyclohexane, dodecane, and octane, respectively, at room temperature.39

4. SUMMARY AND CONCLUSION The general expressions we derived in our previous work16 for calculating the internal conversion rate constants between two adiabatic electronic states, eqs 7 and 8a,8b, have been used to determine the decay rate constants for 11Bu ' 21Ag and 21Ag ' 11Ag internal conversions in trans,trans-1,3,5,7-octatetraene. A recipe that allows one to estimate the Duschinsky parameters J and D from the relative intensities of vibronic bands in absorptionemission spectra are also developed. The decay time constants obtained in the present work for the 21Ag ' 11Aginternal conversion under isolated conditions, Table 6, are in reasonable agreement with the observed and calculated decay time constants. The distortions of the normal modes considerably reduce the decay rate constant of this transition. It has been also explored that a 3-displaced mode model consisting of the three totally symmetric promoting modes ν6, ν12, and ν16, which have the highest HuangRhys factors (Table 4), can approximately produce the observed decay time constant for this process. The phase and environment dependency of the zerozero energy gap, frequencies of totally symmetric accepting modes, and HuangRhys factors (Tables 1, 2, and 4, respectively) for the 11Bu ' 21Ag internal conversion guided us to estimate the associated decay time in free-jet expansions, static gas, and n-octane and n-hexane matrices, Tables 710. We found that a three-mode model consisting of promoting mode ν38 (of symmetry bu), which has the highest vibronic coupling (Table 5), and two totally symmetric accepting modes ν6 and ν12 could be considered as the minimal model for the 11Bu ' 21Ag internal conversion both in the condensed and in the gas phase. Moreover, it has been found that distortions of vibrational modes to some extent increase the decay rate constant of this transition in the gas phase and have a minor effect on the rate in the condensed phase. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Tel: +982188848949. Fax: +982188820993.

’ REFERENCES

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Estimation of Decay Rate Constants of 11Bu 21Ag and 21Ag 11A

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