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Spin-forbidden transitions in flavone
Spectrochimica Acta Part A 73 (2009) 1–5
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Spin-forbidden transitions in flavone Christel M. Marian ∗ Institute of Theoretical and Computational Chemistry, Heinrich Heine University Düsseldorf, Universitätsstr. 1, D-40225 Düsseldorf, Germany
a r t i c l e
i n f o
Article history: Received 4 August 2008 Accepted 5 January 2009 Keywords: Flavone Multi-reference configuration interaction Singlet and triplet excited states Intersystem crossing rate Phosphorescence lifetime
a b s t r a c t The ground and low-lying excited electronic states of flavone were investigated by means of quantum chemical methods including spin–orbit coupling. Minimum structures were determined employing (time-dependent) Kohn–Sham density functional theory. Spectral properties were computed utilizing a combined density functional and multi-reference configuration interaction (DFT/MRCI) method. Intersystem crossing (ISC) rate constants for the S1 T1 transition were computed using a discretized Fermi golden rule approach. For the evaluation of phosphorescence lifetimes a multi-reference spin–orbit configuration interaction procedure (DFT/MRSOCI) was invoked. According to the calculations the phenyl ring is twisted out of the benzopyrone plane by 28◦ in the electronic ground state whereas the nuclear frame is nearly planar in the lowest excited 1 (n∗ ) (S1 ) state and is slightly V-shaped in the 3 (∗ ) (T1 ) and 1 (∗ ) (S2 ) states. The calculations clearly show that the T1 state has mainly ∗ character. The large spin–orbit coupling of the S1 and T1 states and their small energy gap explain the high S1 T1 ISC rate for which a value of kISC ≈ 3 × 1011 s is computed, in good agreement with experimental build-up times of the Tn ← T1 absorption. In the absence of collisions and other nonradiative processes, the T1 state of flavone is prediced to be long-lived with a pure phosphorescence lifetime of P ≈ 4 s, in qualitative agreement with low-temperature measurements. The much faster decay of triplet flavone observed in fluid solutions is ascribed to nonradiative processes. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Flavone is known to be strongly fluorescent in acidic medium whereas it is non-fluorescent in neutral and alkaline media [1]. In the latter environments an almost quantitative triplet formation is observed upon UV radiation with quantum yields T close to unity [2,3]. Build-up times for Tn ← T1 absorption of the order of 30–40 ps [4] point towards very efficient inter-system crossing (ISC). In 50:50 ether and ethanol mixtures the lowest transition in the absorption spectrum was assigned to a 1 (n∗ ) state, followed by two strong 1 ( ∗ ) transitions [5]. There has been some controversy about the nature of the lowest-lying triplet state in chromones and related compounds [4,6–8]. While the fast ISC from the S1 (n∗ ) state to the triplet manifold is indicative of ∗ character of the T1 state [9], the short phosphorescence lifetimes of flavone compared to other ∗ triplets has been interpreted as being due to some admixture of n∗ character of a close-by n∗ triplet [4,5]. In this work, the low-lying electronic states of flavone are investigated by means of quantum chemical methods. In addition to vertical and adiabatic energies, ISC rates for the S1 T1
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transition as well as phosphorescence rates for T1 → S0 will be presented. 2. Methods At selected nuclear arrangements, separate DFT/MRCI calculations for singlet and triplet states are performed yielding LS-coupled MRCI wavefunctions. In the basis of these LS-coupled MRCI wavefunctions – augmented by appropriate spin factors to take account of different MS quantum numbers – electronic spin–orbit coupling matrix elements (SOCMEs) are evaluated. The inter-system crossing rate is evaluated in the Fermi Golden Rule approximation employing a discrete set of final vibrational states. Phosphorescence lifetimes are determined from spin–orbit multireference configuration interaction (MRSOCI) wave functions. 2.1. Nuclear geometries and vibrational frequencies All calculations were performed employing the valence triple zeta basis set with polarization functions (d,p) from the Turbomole library [10,11]. The equilibrium geometry of the electronic ground state was determined for a restricted closed-shell KS determinant using the B3-LYP density functional [12–14]. The minimum geometry of the lowest-lying triplet state was obtained by unrestricted density functional theory (UDFT). The geometries of the low-lying
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1 (n ∗ )
and 1 (∗ ) states were located by means of a TDDFT gradient [15]. All (U)DFT/TDDFT calculations were carried out utilizing by the Turbomole quantum chemistry program package [16]. Harmonic vibrational frequencies and normal modes were determined by a finite difference technique using the SNF tool [17]. 2.2. Spin–orbit free electronic structure calculations Electronic excitation energies and wave functions of 25 singlet and 25 triplet states were determined using the DFT/MRCI method by Grimme and Waletzke [18]. This method was shown to yield reliable electronic spectra of a variety of organic molecules at reasonable computational expense [19–25]. In the DFT/MRCI approach, an effective Hamiltonian is employed for the construction of the CI matrix. It entails BH-LYP [26,13] Kohn–Sham orbital energies of a closed-shell reference state and five empirical parameters (scaling of Coulomb and exchange integrals, damping of off-diagonal CI matrix elements, configuration selection threshold) that depend only on the multiplicity of the desired state, the number of open shells of a configuration, and the type of density functional employed, but not on the specific atom or molecule. For details concerning the selection criteria and the integration of DFT information into the MRCI procedure, we refer to the original publication [18]. In the MRCI runs, all valence electrons were correlated. For the RI approximation of the expensive four-index integrals of the two-electron Coulomb operator, RI-MP2-optimized auxiliary basis sets from the Turbomolelibrary [27,28] were employed. 2.3. Spin–orbit coupling matrix elements SOCMEs were evaluated using the Spock program package developed in this laboratory [29,30]. The spin–orbit mean-field (SOMF) Hamiltonian utilized in this work is an effective one-electron Hamiltonian derived from the one- and two-electron Breit-Pauli operators [31]. As an additional approximation, all multi-center spin–orbit integrals are neglected. In this way, the molecular mean field reduces to a sum of atomic mean fields. Mean-field orbitals are generated automatically for each atom from a restricted (openshell) Hartree–Fock atomic ground state calculation. The atomic mean-field integral program Amfi[32] makes use of spherical symmetry; spin–orbit integral evaluation is thus extremely fast.
second-order properties. It is common knowledge that sum-overstates expressions for these kind of properties converge very slowly with the number of intermediate states [37–39]. The MRSOCI method, recently developed in this laboratory [40], avoids this (0) ˆ and the problem by including the electrostatic Hamiltonian H ˆ spin–orbit Hamiltonian HSO on the same footing in a variational procedure. DFT/MRSOCI calculations were carried out for seven roots employing the merger of the configuration spaces of the preceding spin–orbit free DFT/MRCI. The convergence threshold for the iterative Davidson diagonalization of the DFT/MRSOCI wave functions was set to 5 × 10−9 EH . 3. Results and discussion 3.1. Ground and excited state geometries The nuclear frames at the ground and excited state minima are displayed in Fig. 1 where also the atom labels are shown. In the electronic ground state, the benzopyrone moiety of the flavone molecule is planar within the accuracy of the calculations. The phenyl ring is attached to the 2-position of the pyrone ring by a single bond, with the phenyl and pyrone planes twisted by ca. 28◦ . The twisting angle is somewhat larger than in older uncorrelated ab initio calculations [41,42]. In the S1 (n∗ )state, the molecule is nearly planar (see Fig. 1 and Table 1). The n∗ excitation leads to a shift of the carbonyl double bond to the adjacent C3 –C4 bond. The effects on the C2 –C3 and C2 –C1 bond lengths are smaller, but the partial conjugation effect is sufficient to induce a twist of the phenyl-pyrone torsion angle so that the rings are almost coplanar. The latter is also true for the T1 (∗ ) and S2 (∗ ) states. The nuclear arrangements of these ∗ states show a slight butterfly shape with the folding axis running through the two oxygen atoms. The 3 (∗ )
2.4. Inter-system crossing rates Rates of spin-forbidden nonradiative transitions were evaluated using a discretized form of the Fermi Golden Rule expression [33,34] ki{f } =
¯h
|Vif |2
(1)
{f for which |(Ef −Ei )|≤}
to which only those levels of the final state f contribute which lie in a chosen interval of width 2 centered around the energy of the initial state i. Herein the Vif constitute the coupling matrix elements between the intial and final Born–Oppenheimer states which are further approximated as ˆ SO |el |vib ≈ el |H ˆ SO |el vib |vib Vif = ivib |iel |H f i f f i f
(2)
(Condon approximation). The calculations of the ISC rate constants were performed using the Vibes program [35,36] which permits vibrational mode selection and a limitation of the number of vibrational quanta in each mode. 2.5. Phosphorescence rates ˆ SO induces direct coupling of the involved Unlike ISC where H electronic states, rates of radiative spin-forbidden transitions are
Fig. 1. Nuclear structure and atom labeling of flavone.
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Table 1 Minimum geometry parameters of flavone in various electronic states. Bond distances are given in pm, angles in ◦ . Parameter
State S0
T1
S1
S2
O1 –C2 C2 –C3 C3 –C4 C4 –O4 C4 –C4a C4a –C5 C5 –C6 C6 –C7 C7 –C8 C8 –C8a C8a –O1 C2 –C1
C1 –C2
C2 –C3
C3 –C4
C4 –C5
C5 –C6
C6 –C1
O1 –C2 –C3 –C4 C2 –C3 –C4 –O4
O1 –C2 –C1 –C2
C3 –C2 –C1 –C2
C2 –O1 –C8a –C4a C3 –C4 –C4a –C8a
136.2 135.3 145.4 122.6 148.1 140.1 138.2 140.1 138.4 139.5 137.2 147.3 140.0 138.9 139.2 139.2 138.8 140.0 −1.4 179.7 −28.2 151.2 −0.3 0.3
137.4 148.0 141.7 125.1 148.2 139.5 138.9 139.4 139.1 139.0 137.7 139.3 143.9 137.2 140.8 140.1 137.6 144.0 −6.0 179.2 −2.0 175.0 −6.6 3.7
137.7 139.9 137.8 131.0 145.3 140.3 138.8 139.4 139.1 139.0 136.0 143.0 141.8 138.4 139.5 139.7 138.2 141.8 −0.8 179.8 −2.8 176.8 −0.5 0.3
146.6 138.9 141.6 123.9 151.1 137.2 142.6 139.5 138.1 143.4 131.2 140.8 142.9 138.1 139.8 140.3 137.7 143.0 −8.6 −178.8 −3.5 174.6 −4.7 4.6
and 1 (∗ ) minimum structures differ, however, in the conjugation of the pyrone ring. In the T1 state, bond lengths indicative of single bonds are found for C2 –C3 and C4 –C4a while all other bonds exhibit partial double bond character (see Table 1). In the TDDFT optimized S2 minimum, the O1 –C2 and O1 –C8a bond lengths are unequal. The localization of the double bond character in the O1 –C8a bond has also consequences on the conjugation of the neighboring benzene ring. It is not clear at this moment whether the asymmetric distortion of the two pyrone bonds in the S2 state is an artifact of the geometry optimization method. The asymmetric structure is a true minimum at the TDDFT level. As we will see later, the DFT/MRCI method yields lower total energies of the S2 state at the T1 minimum than at the TDDFT optimized structure of the S2 state. 3.2. Electronic excitation energies At all geometries studied here, the order of electronic states is identical (Table 2). A situation like this is remarkable. For heterocycles many counterexamples can be found where curve crossings between (n∗ ), (∗ ), and ( ∗ ) excited states occur [43]. The most important molecular orbitals involved in transitions to low-lying electronic states of flavone are displayed in Fig. 2. The first excited state of flavone is the 3 (∗ )state. Its calculated adiabatic excitation energy of 2.59 eV (2.49 eV including zero-point vibrational energy (ZPVE) corrections) is only slightly lower than the experimental value for the 0–0 transition in 2methyltetrahydrofuran (2.69 eV) [2] or in 50:50 ether and ethanol solution (2.71 eV) [5]. The static dipole moment of the T1 state is about half the size of the ground state dipole moment so that we expect the transition to be blue shifted in polar solvents. Table 2 Excitation energies [eV] of singlet and triplet states in flavone at different nuclear geometries. State S0 S1 S2 T1 T2
Geometry 1 n ∗
1 ∗
1 ∗ 1 n ∗
S0
T1
S1
S2
0.00 3.30 4.21 3.04 3.17
0.49 3.21 3.90 2.59 3.20
0.44 2.93 3.97 2.74 2.88
0.64 3.43 3.98 3.03 3.37
Fig. 2. Molecular orbitals of flavone involved in the lowest electronic transitions.
At the S0 and S1 geometries, the 3 (n∗ ) state is near-degenerate with the 3 (∗ ) state, at least in isolated flavone. It can be assumed that the T2 minimum nuclear arrangement is very similar to the one of the S1 state so that the near-degeneracy will persist also in the T2 relaxed geometry. Slight out-of-plane motions will thus suffice to induce strong vibronic coupling between the two triplet states [44,45]. Polar solvents will increase the energy gap between the two triplets and thus reduce their coupling. The difference of their static dipole moments is smaller than in the corresponding singlets (see below) so that we can expect the T1 and T2 excitation energies to be less affected solvent by effects.
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At the ground state geometry, the static dipole moment of the S1 state is close to zero whereas the S0 and S2 states exhibit dipole moments around 4 Debye. The 1 (n∗ ) state will therefore be significantly blue shifted in polar solvents, thus decreasing the energy gap to the lowest 1 (∗ ) state. Pownall [5] reports that the first absorption band of flavone (S1 (n∗ ) ← S0 ) coalesces with the second band (S2 (∗ ) ← S0 ) in ethanol. This should, however, not be the case in unpolar solvents such as cyclohexane. Nevertheless, our calculated adiabatic excitation energy of the S1 (n∗ ) state of 2.93 eV (2.81 eV after ZPVE correction) is significantly lower than the maximum of the first broad peak of this weak absorption in cyclohexane solution (3.37 eV) [5]. Errors of this size are untypical of the DFT/MRCI method [19–25]. We rather assume that the 0–0 transition is not observed experimentally because of its small Franck-Condon factor (0.0016) in combination with its low electronic oscillator strength (0.0008). We compute the vertical S2 (∗ ) ← S0 absorption energy to be 4.21 eV, in good agreement with experimental maxima of absorption bands (4.19 eV in 50:50 ether and ethanol solution [5] and 4.08 eV in water at pH 7 [1]). As mentioned above, the excited state geometry optimized with TDDFT (B3-LYP) does not yield the lowest DFT/MRCI energy (see Table 2). Rather, the lowest DFT/MRCI excitation energy of the S2 state (3.90 eV, 3.80 eV including ZPVE correction) is obtained at the T1 minimum, in very good agreement with the redmost absorption peak (321.5 nm corresponding to 3.86 eV) in resonance enhanced two-photon ionization (R2PI) spectra of jet-cooled flavone [46]. Comparison with the measured value of 3.77 eV for the 0–0 transition in 50:50 ether and ethanol solution [5] shows that solvent effects on this transition are rather small. 3.3. Inter-system crossing rates ISC rates depend heavily on the size of the SOCMEs and the energy gap between the coupling states [47]. A small energy gap alone does not guarantee, however, the overlaps between the vibrational ground state of the initial state i and the vibrational levels of the final state f (Eq. (2)) to be sizable. For instance, a pair of nested states – i.e., states with nearly identically shaped potential energy wells that are vertically shifted with respect to each other – will in general have small vibrational overlaps and thus slow nonradiative transitions. Displacements of the nuclear geometry along the normal coordinates typically increase the vibrational overlap and thus enhance ISC. Our calculations place the zero vibrational level of the S1 state 2604 cm−1 above the T1 vibrational ground state. An interval of = 0.1 cm−1 above and below the initial state was chosen for integrating the transition probabilities. In this interval 57021 vibrational levels of the triplet state are located. The mode with the largest displacement is 1 , the phenyl torsion. More importantly, however, a considerable number of medium frequency modes, mostly asymmetric C–O–C and C–C–C stretches are among the modes with large displacements. For the S1 T1 transition 38 of the 75 vibrational modes of flavone show displacements larger than 0.1 in mass-weighted dimensionless reaction coordinates. Selecting only these 38 modes (3735 states) yields a transition rate within 10% of the full results. Furthermore, a combination of a few v = 0 → v = 1 excitations will suffice to bring the triplet levels into near degeneracy with the singlet vibrational ground state. A test calculation where the excitation level of all vibrational modes was set to at most 1 quantum per mode (852 vibrational levels in the interval) yields an ISC rate only one order of magnitude smaller than the full calculation. Limiting the excitation level to 3 quanta per mode (16,408 vibrational states) gives a value that is practically indistinguishable from the full result. These tests show that higher excitations of low-frequency modes
Table 3 Electronic spin–orbit coupling matrix elements [cm−1 ] and inter-system crossing rates [s−1 ] for the S1 T1 transition of flavone. ˆ SO |T1 | |S1 |H
kISC
x : 26.37 y : 29.28 z : 22.58 x+y+z
1.11 × 1011 1.37 × 1011 0.81 × 1011 3.29 × 1011
contribute only to a minor extent to the nonradiative transition. Matrix elements of the cartesian spin–orbit operator components and the corresponding ISC rates are shown in Table 3. The individual rates depend on the orientation of the molecule within the chosen coordinate system. Hence, comparison with experimental data is only meaningful for their sum. To the author’s knowledge, the ISC rate has not been determined experimentally. A rough estimate can be obtained, however, from the build-up times of the Tn ← T1 excitation spectra for which values of (30 ± 10) × 10−12 s in CCl4 up to (42 ± 8) × 10−12 s in ethanol were reported [4]. If the ISC is assumed to be the time-determining step, these buildup times correspond to ISC rates in the range of 2 to 3 × 1010 s−1 , somewhat smaller than our computed gas phase value of 3.3 × 1011 s−1 (Table 3). This difference is mainly attributed to solvent effects. Taking into account that polar protic solvents will increase the energy gap between the S1 (n∗ ) and T1 (∗ ) states leads us to conclude that the S1 T1 ISC rate will decrease with solvent polarity and proticity. Compared to the low fluorescence rate of the S1 state (kF ≈ 4.3 × 103 s−1 ) the ISC will remain efficient though. 3.4. Phosphorescence lifetimes The actual phosphorescence rates are calculated from variational wave functions. Nevertheless, it can be instructive to analyze the results in terms of the major contributions from perturbation theory. In the framework of the latter theory, the dipole transition moment of a spin-forbidden radiative transition is a sum of direct and indirect contributions [37,39]. The direct terms originate from the spin–orbit coupling of the zeroth-order wave functions, multiplied by the difference of their respective dipole moments and devided by their energy difference. The entries in Table 4 reveal that the three SOCMEs of the T1 and S0 wave functions are very small, as expected for a ∗ excitation [9]. The indirect terms involve spin–orbit coupling of the initial state to an intermediate state which in turn couples to the final state via the electric dipole operator and vice versa. We find large SOCMEs for the T1 and S1 pair of states (Table 4). The second-order coupling via S1 and other 1 (n∗ ) states is weak nevertheless because of their small electric dipole transition moments with S0 . In contrast, S2 and other 1 (∗ ) states exhibit strong dipole transitions to S0 , but their SOCMEs with T1 are typically below 1 cm−1 . Higher-order perturbation terms are usually neglected. In the present MRSOCI calculations electronic terms Table 4 Electronic spin–orbit coupling matrix elements [cm−1 ], dipole matrix elements [ea0 ], and phosphorescence lifetimes P [s] for the T1 → S0 transition of flavone. SOCME
x
y
ˆ SO |S0 | |T1 |H ˆ SO |S1 | |T1 |H ˆ SO |S2 | |T1 |H ˆ SO |T2 | |T1 |H ˆ SO |S0 | |T2 |H ˆ SO |S2 | |T2 |H P
0.88 18.40 0.28 17.28 23.41 12.92 T1a 5.26 3.59
0.06 22.69 0.26 21.32 26.58 17.44 T1b 1.56
High-temp. average
| |
z 0.11 19.60 0.12 18.42 23.24 14.88 T1c 383.9
|S0 | |S0 − T1 | |T1 | |S1 | |S0 | |S2 | |S0 |
0.9461 0.0347 2.2098
|S0 | |S0 − T2 | |T2 | |S2 | |S0 |
0.8977 2.2098
C.M. Marian / Spectrochimica Acta Part A 73 (2009) 1–5
are automatically included to all orders. Vibronic contributions which are expected to gain importance at elevated temperatures, have not been taken into consideration (see however below). At this level of approximation, phosphorescence lifetimes in the s range are obtained in the calculations. Qualitatively, this result is agreement with experiments conducted at 77 K in different glassy matrices. Under these conditions, decay times of the Tn ← T1 absorption of flavone range from 0.282 s in 9:1 methylcyclohexane-isopentane mixture [4] to 0.675 s in 1:1 ether-ethanol [5]. Taking into account that concurrent nonradiative decay processes are mediated by the interaction of the solute with the surrounding matrix, the agreement can be considered as good. The experimentally observed increase of the triplet lifetime with increasing solvent polarity [4] is consistent with the S1 state being the major source of indirect coupling. A blue shift of the latter state will increase the energy difference in the denominator of the second-order coupling term with T1 and thus diminish its contribution. How does the much shorter triplet lifetime measured at room temperature in degassed benzene (4.5 s after correction for selfquenching [2]) fit in? As the following discussion will show, such a short lifetime cannot be explained by phosphorescence alone but requires additional nonradiative decay channels. In Section 3.2 it was mentioned that the T1 and T2 states are near degenerate in some regions of the coordinate space. With decreasing energy gap, the (non-adiabatic) vibronic interaction between the two Born-Oppenheimer states will increase. Furthermore, higherorder spin–orbit coupling will gain importance. It should be kept in mind, however, that intensity borrowing from the nearby T2 3 (n∗ ) state via vibronic and spin–orbit interaction state can increase the phosphorescence rate to at most the value it has for the T2 → S0 transition. The calculated rate of a hypothetical phosphorescence originating from the T2 state amounts to 845 s−1 , i.e., it is about 3000 times higher than the value for the purely electronic T1 → S0 transition. The corresponding phosphorescence lifetime of 1.2 ms is, nevertheless, significantly longer than the above mentioned measured triplet lifetime in benzene solution. 4. Summary and conclusions Combined density functional and multireference configuration interaction methods were used to compute a variety of properties of flavone, including singlet and triplet excitation energies and rates of spin-forbidden transitions. While the phenyl ring is twisted out of the plane of the benzopyrone in the electronic ground state, a planar structure is preferred in the first excited singlet state which exhibits n∗ character. The lowest ∗ excited states take a butterfly-type shape. The calculations clearly show that the T1 state has ∗ character. T1 is populated at a rate of kISC ≈ 3 × 1011 s−1 from the S1 state. Among the most important accepting modes a variety of asymmetric C–O–C and C–C–C stretches are found. In the absence of nonradiative processes, T1 is long-lived with a calculated radiative decay rate of kP ≈ 0.3 s−1 . These values are in qualitative agreement with experimental build-up and decay times of transient Tn ← T1 absorption at 77 K. The calculations furthermore suggest that the redmost peak in the S1 ← S0 absorption spectrum (3.37 eV in cyclohexane solution [5]) does not correspond to the 0–0 transition because of its unfavorable Franck-Condon factor in combination with the weak electronic oscillator strength. Acknowledgment I would like to thank Karl Kleinermanns and Gernot Engler (Düsseldorf) for many valuable discussions. Financial support by
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Deutsche Forschungsgemeinschaft through SFB663/C1 is gratefully acknowledged. References [1] S. Kumar, S.K. Jain, N. Sharma, R.C. Rastogi, Spectrochim. Acta 57A (2001) 299–308. [2] K. Bhattacharyya, D. Ramaiah, P.K. Das, M.V. George, J. Phys. Chem. 90 (1986) 5984–5989. [3] V. Avila, C.M. Previtali, J. Chem. Soc., Perkin Trans. 2 (1995) 2281–2285. [4] K. Hamanoue, T. Nakayama, T. Miyake, H. Teranishi, Chem. Lett. 1981 (1981) 39–42. [5] H.J. Pownall, Spectrochim. Acta 30A (1974) 953–959. [6] R. Matsushima, T. Kishimoto, M. Suziki, Chem. Lett. (1976) 579. [7] Y. Sakagushi, H. Hayashi, H. Muray, Y.J. I’Haya, J. Phys. Chem. 90 (1986) 6416–6420. [8] T. Nakayama, T. Shimizu, Y. Torii, S. Miki, K. Hamanoue, J. Photochem. Photobiol. A: Chem. 111 (1997) 35–39. [9] M.A. El-Sayed, J. Chem. Phys. 38 (1963) 2834. [10] R. Ahlrichs, M. Bär, M. Häser, H. Horn, C. Kölmel, Chem. Phys. Lett. 162 (1989) 165. [11] A. Schäfer, C. Huber, R. Ahlrichs, J. Chem. Phys. 100 (1994) 5829. [12] A.D. Becke, J. Chem. Phys. 98 (1993) 5648–5652. [13] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [14] P.J. Stephens, F.J. Devlin, C.F. Chabalowski, M.J. Frisch, J. Phys. Chem. 98 (1994) 11623–11627. [15] F. Furche, R. Ahlrichs, J. Chem. Phys. 117 (2002) 7433–7447. [16] R. Ahlrichs, M. Bär, H.-P. Baron, R. Bauernschmitt, S. Böcker, P. Deglmann, M. Ehrig, K. Eichkorn, S. Elliott, F. Furche, F. Haase, M. Häser, C. Hättig, H. Horn, C. Huber, U. Huniar, M. Kattannek, A. Köhn, C. Kölmel, M. Kollwitz, K. May, C. Ochsenfeld, H. Öhm, A. Schäfer, U. Schneider, M. Sierka, O. Treutler, B. Unterreiner, M. von Arnim, F. Weigend, P. Weis, H. Weiss, Turbomole (vers. 5.6), Universität Karlsruhe (2002). [17] C. Kind, M. Reiher, J. Neugebauer, SNF Version 2.2.1: A Program Package for Numerical Frequency Analyses, Universität Erlangen (1999–2002). [18] S. Grimme, M. Waletzke, J. Chem. Phys. 111 (1999) 5645–5655. [19] C.M. Marian, J. Chem. Phys. 122 (2005) 104314. ´ J. Tatchen, C.M. Marian, J. Phys. Chem. A 109 (2005) 8410–8418. [20] K. Tomic, [21] C.M. Marian, J. Phys. Chem. A 111 (2007) 1545–1553. [22] J. Tatchen, C.M. Marian, Phys. Chem. Chem. Phys. 8 (2006) 2133–2144. [23] S. Salzmann, J. Tatchen, C.M. Marian, J. Photochem, Photobiol A: Chemistry 198 (2008) 221–231. [24] C.M. Marian, N. Gilka, J. Chem. Theor. Comp. 4 (2008) 1501–1515. [25] M. R. Silva-Junior, M. Schreiber, S. P. A. Sauer, W. Thiel, J. Chem. Phys. 129 (2008) 104–103. [26] A.D. Becke, J. Chem. Phys. 98 (1993) 1372–1377. [27] F. Weigend, M. Häser, H. Patzelt, R. Ahlrichs, Chem. Phys. Lett. 294 (1998) 143–152. [28] K. Eichkorn, O. Treutler, H. Öhm, M. Häser, R. Ahlrichs, Chem. Phys. Lett. 240 (1995) 283. [29] M. Kleinschmidt, J. Tatchen, C.M. Marian, J. Comp. Chem. 23 (2002) 824–833. [30] M. Kleinschmidt, C.M. Marian, Chem. Phys. 311 (2005) 71–79. [31] B.A. Heß, C.M. Marian, U. Wahlgren, O. Gropen, Chem. Phys. Lett. 251 (1996) 365. [32] AMFI is an atomic spin–orbit integral program written by B. Schimmelpfennig, University of Stockholm (1996). [33] A. Toniolo, M. Persico, J. Chem. Phys. 115 (2001) 1817–1827. [34] A. Toniolo, M. Persico, J. Comput. Chem. 22 (2001) 968–975. [35] J. Tatchen, Dissertation, Universität Düsseldorf (2006). [36] J. Tatchen, N. Gilka, C.M. Marian, Phys. Chem. Chem. Phys. 9 (2007) 5209. [37] S.R. Langhoff, C.W. Kern, Molecular Fine Structure, in: H.F. Schaefer III (Ed.), in: Modern Theoretical Chemistry, Vol. 4, Plenum, New York, 1977, pp. 381–437. [38] H. Ågren, O. Vahtras, B. Minaev, Advances in Quantum Chemistry 27 (1996) 71. [39] C. Marian, Spin–orbit coupling in molecules, in: K. Lipkowitz, D. Boyd (Eds.), in: Reviews In Computational Chemistry, Vol. 17, Wiley–VCH, Weinheim, 2001, pp. 99–204. [40] M. Kleinschmidt, J. Tatchen, C.M. Marian, J. Chem. Phys. 124 (2006) 124101. [41] L. Vrielynck, J.P. Cornard, J.C. Merlin, P. Bopp, J. Mol. Struct. 297 (1993) 227–234. [42] A. Mantas, E. Deretey, F.H. Ferretti, M. Estrada, I.G. Csizmadia, J. Mol. Struct. (Theochem) 504 (2000) 77–103. [43] W. Domcke, D.R. Yarkony, H. Köppel, Conical Intersections: Electronic Structure, Dynamics and Spectroscopy, World Scientific, Singapore, 2004. [44] Y.H. Li, E.C. Lim, Chem. Phys. Lett. 7 (1970) 15. [45] E.C. Lim, J. Phys. Chem. 90 (1986) 6770. [46] G. Engler, M. Nispel, C.M. Marian, K. Kleinermanns, submitted to Chem. Phys. Lett. [47] N.J. Turro, Modern Molecular Photochemistry, University Science Books, Sausalito, CA, 1991.
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Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy journal homepage: www.elsevier.com/locate/saa
Spin-forbidden transitions in flavone Christel M. Marian ∗ Institute of Theoretical and Computational Chemistry, Heinrich Heine University Düsseldorf, Universitätsstr. 1, D-40225 Düsseldorf, Germany
a r t i c l e
i n f o
Article history: Received 4 August 2008 Accepted 5 January 2009 Keywords: Flavone Multi-reference configuration interaction Singlet and triplet excited states Intersystem crossing rate Phosphorescence lifetime
a b s t r a c t The ground and low-lying excited electronic states of flavone were investigated by means of quantum chemical methods including spin–orbit coupling. Minimum structures were determined employing (time-dependent) Kohn–Sham density functional theory. Spectral properties were computed utilizing a combined density functional and multi-reference configuration interaction (DFT/MRCI) method. Intersystem crossing (ISC) rate constants for the S1 T1 transition were computed using a discretized Fermi golden rule approach. For the evaluation of phosphorescence lifetimes a multi-reference spin–orbit configuration interaction procedure (DFT/MRSOCI) was invoked. According to the calculations the phenyl ring is twisted out of the benzopyrone plane by 28◦ in the electronic ground state whereas the nuclear frame is nearly planar in the lowest excited 1 (n∗ ) (S1 ) state and is slightly V-shaped in the 3 (∗ ) (T1 ) and 1 (∗ ) (S2 ) states. The calculations clearly show that the T1 state has mainly ∗ character. The large spin–orbit coupling of the S1 and T1 states and their small energy gap explain the high S1 T1 ISC rate for which a value of kISC ≈ 3 × 1011 s is computed, in good agreement with experimental build-up times of the Tn ← T1 absorption. In the absence of collisions and other nonradiative processes, the T1 state of flavone is prediced to be long-lived with a pure phosphorescence lifetime of P ≈ 4 s, in qualitative agreement with low-temperature measurements. The much faster decay of triplet flavone observed in fluid solutions is ascribed to nonradiative processes. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Flavone is known to be strongly fluorescent in acidic medium whereas it is non-fluorescent in neutral and alkaline media [1]. In the latter environments an almost quantitative triplet formation is observed upon UV radiation with quantum yields T close to unity [2,3]. Build-up times for Tn ← T1 absorption of the order of 30–40 ps [4] point towards very efficient inter-system crossing (ISC). In 50:50 ether and ethanol mixtures the lowest transition in the absorption spectrum was assigned to a 1 (n∗ ) state, followed by two strong 1 ( ∗ ) transitions [5]. There has been some controversy about the nature of the lowest-lying triplet state in chromones and related compounds [4,6–8]. While the fast ISC from the S1 (n∗ ) state to the triplet manifold is indicative of ∗ character of the T1 state [9], the short phosphorescence lifetimes of flavone compared to other ∗ triplets has been interpreted as being due to some admixture of n∗ character of a close-by n∗ triplet [4,5]. In this work, the low-lying electronic states of flavone are investigated by means of quantum chemical methods. In addition to vertical and adiabatic energies, ISC rates for the S1 T1
∗ Tel.: +49 211 8113209; fax: +49 211 8113466. E-mail address: [email protected]. URL: http://www.theochem.uni-duesseldorf.de. 1386-1425/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.saa.2009.01.004
transition as well as phosphorescence rates for T1 → S0 will be presented. 2. Methods At selected nuclear arrangements, separate DFT/MRCI calculations for singlet and triplet states are performed yielding LS-coupled MRCI wavefunctions. In the basis of these LS-coupled MRCI wavefunctions – augmented by appropriate spin factors to take account of different MS quantum numbers – electronic spin–orbit coupling matrix elements (SOCMEs) are evaluated. The inter-system crossing rate is evaluated in the Fermi Golden Rule approximation employing a discrete set of final vibrational states. Phosphorescence lifetimes are determined from spin–orbit multireference configuration interaction (MRSOCI) wave functions. 2.1. Nuclear geometries and vibrational frequencies All calculations were performed employing the valence triple zeta basis set with polarization functions (d,p) from the Turbomole library [10,11]. The equilibrium geometry of the electronic ground state was determined for a restricted closed-shell KS determinant using the B3-LYP density functional [12–14]. The minimum geometry of the lowest-lying triplet state was obtained by unrestricted density functional theory (UDFT). The geometries of the low-lying
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1 (n ∗ )
and 1 (∗ ) states were located by means of a TDDFT gradient [15]. All (U)DFT/TDDFT calculations were carried out utilizing by the Turbomole quantum chemistry program package [16]. Harmonic vibrational frequencies and normal modes were determined by a finite difference technique using the SNF tool [17]. 2.2. Spin–orbit free electronic structure calculations Electronic excitation energies and wave functions of 25 singlet and 25 triplet states were determined using the DFT/MRCI method by Grimme and Waletzke [18]. This method was shown to yield reliable electronic spectra of a variety of organic molecules at reasonable computational expense [19–25]. In the DFT/MRCI approach, an effective Hamiltonian is employed for the construction of the CI matrix. It entails BH-LYP [26,13] Kohn–Sham orbital energies of a closed-shell reference state and five empirical parameters (scaling of Coulomb and exchange integrals, damping of off-diagonal CI matrix elements, configuration selection threshold) that depend only on the multiplicity of the desired state, the number of open shells of a configuration, and the type of density functional employed, but not on the specific atom or molecule. For details concerning the selection criteria and the integration of DFT information into the MRCI procedure, we refer to the original publication [18]. In the MRCI runs, all valence electrons were correlated. For the RI approximation of the expensive four-index integrals of the two-electron Coulomb operator, RI-MP2-optimized auxiliary basis sets from the Turbomolelibrary [27,28] were employed. 2.3. Spin–orbit coupling matrix elements SOCMEs were evaluated using the Spock program package developed in this laboratory [29,30]. The spin–orbit mean-field (SOMF) Hamiltonian utilized in this work is an effective one-electron Hamiltonian derived from the one- and two-electron Breit-Pauli operators [31]. As an additional approximation, all multi-center spin–orbit integrals are neglected. In this way, the molecular mean field reduces to a sum of atomic mean fields. Mean-field orbitals are generated automatically for each atom from a restricted (openshell) Hartree–Fock atomic ground state calculation. The atomic mean-field integral program Amfi[32] makes use of spherical symmetry; spin–orbit integral evaluation is thus extremely fast.
second-order properties. It is common knowledge that sum-overstates expressions for these kind of properties converge very slowly with the number of intermediate states [37–39]. The MRSOCI method, recently developed in this laboratory [40], avoids this (0) ˆ and the problem by including the electrostatic Hamiltonian H ˆ spin–orbit Hamiltonian HSO on the same footing in a variational procedure. DFT/MRSOCI calculations were carried out for seven roots employing the merger of the configuration spaces of the preceding spin–orbit free DFT/MRCI. The convergence threshold for the iterative Davidson diagonalization of the DFT/MRSOCI wave functions was set to 5 × 10−9 EH . 3. Results and discussion 3.1. Ground and excited state geometries The nuclear frames at the ground and excited state minima are displayed in Fig. 1 where also the atom labels are shown. In the electronic ground state, the benzopyrone moiety of the flavone molecule is planar within the accuracy of the calculations. The phenyl ring is attached to the 2-position of the pyrone ring by a single bond, with the phenyl and pyrone planes twisted by ca. 28◦ . The twisting angle is somewhat larger than in older uncorrelated ab initio calculations [41,42]. In the S1 (n∗ )state, the molecule is nearly planar (see Fig. 1 and Table 1). The n∗ excitation leads to a shift of the carbonyl double bond to the adjacent C3 –C4 bond. The effects on the C2 –C3 and C2 –C1 bond lengths are smaller, but the partial conjugation effect is sufficient to induce a twist of the phenyl-pyrone torsion angle so that the rings are almost coplanar. The latter is also true for the T1 (∗ ) and S2 (∗ ) states. The nuclear arrangements of these ∗ states show a slight butterfly shape with the folding axis running through the two oxygen atoms. The 3 (∗ )
2.4. Inter-system crossing rates Rates of spin-forbidden nonradiative transitions were evaluated using a discretized form of the Fermi Golden Rule expression [33,34] ki{f } =
¯h
|Vif |2
(1)
{f for which |(Ef −Ei )|≤}
to which only those levels of the final state f contribute which lie in a chosen interval of width 2 centered around the energy of the initial state i. Herein the Vif constitute the coupling matrix elements between the intial and final Born–Oppenheimer states which are further approximated as ˆ SO |el |vib ≈ el |H ˆ SO |el vib |vib Vif = ivib |iel |H f i f f i f
(2)
(Condon approximation). The calculations of the ISC rate constants were performed using the Vibes program [35,36] which permits vibrational mode selection and a limitation of the number of vibrational quanta in each mode. 2.5. Phosphorescence rates ˆ SO induces direct coupling of the involved Unlike ISC where H electronic states, rates of radiative spin-forbidden transitions are
Fig. 1. Nuclear structure and atom labeling of flavone.
C.M. Marian / Spectrochimica Acta Part A 73 (2009) 1–5
3
Table 1 Minimum geometry parameters of flavone in various electronic states. Bond distances are given in pm, angles in ◦ . Parameter
State S0
T1
S1
S2
O1 –C2 C2 –C3 C3 –C4 C4 –O4 C4 –C4a C4a –C5 C5 –C6 C6 –C7 C7 –C8 C8 –C8a C8a –O1 C2 –C1
C1 –C2
C2 –C3
C3 –C4
C4 –C5
C5 –C6
C6 –C1
O1 –C2 –C3 –C4 C2 –C3 –C4 –O4
O1 –C2 –C1 –C2
C3 –C2 –C1 –C2
C2 –O1 –C8a –C4a C3 –C4 –C4a –C8a
136.2 135.3 145.4 122.6 148.1 140.1 138.2 140.1 138.4 139.5 137.2 147.3 140.0 138.9 139.2 139.2 138.8 140.0 −1.4 179.7 −28.2 151.2 −0.3 0.3
137.4 148.0 141.7 125.1 148.2 139.5 138.9 139.4 139.1 139.0 137.7 139.3 143.9 137.2 140.8 140.1 137.6 144.0 −6.0 179.2 −2.0 175.0 −6.6 3.7
137.7 139.9 137.8 131.0 145.3 140.3 138.8 139.4 139.1 139.0 136.0 143.0 141.8 138.4 139.5 139.7 138.2 141.8 −0.8 179.8 −2.8 176.8 −0.5 0.3
146.6 138.9 141.6 123.9 151.1 137.2 142.6 139.5 138.1 143.4 131.2 140.8 142.9 138.1 139.8 140.3 137.7 143.0 −8.6 −178.8 −3.5 174.6 −4.7 4.6
and 1 (∗ ) minimum structures differ, however, in the conjugation of the pyrone ring. In the T1 state, bond lengths indicative of single bonds are found for C2 –C3 and C4 –C4a while all other bonds exhibit partial double bond character (see Table 1). In the TDDFT optimized S2 minimum, the O1 –C2 and O1 –C8a bond lengths are unequal. The localization of the double bond character in the O1 –C8a bond has also consequences on the conjugation of the neighboring benzene ring. It is not clear at this moment whether the asymmetric distortion of the two pyrone bonds in the S2 state is an artifact of the geometry optimization method. The asymmetric structure is a true minimum at the TDDFT level. As we will see later, the DFT/MRCI method yields lower total energies of the S2 state at the T1 minimum than at the TDDFT optimized structure of the S2 state. 3.2. Electronic excitation energies At all geometries studied here, the order of electronic states is identical (Table 2). A situation like this is remarkable. For heterocycles many counterexamples can be found where curve crossings between (n∗ ), (∗ ), and ( ∗ ) excited states occur [43]. The most important molecular orbitals involved in transitions to low-lying electronic states of flavone are displayed in Fig. 2. The first excited state of flavone is the 3 (∗ )state. Its calculated adiabatic excitation energy of 2.59 eV (2.49 eV including zero-point vibrational energy (ZPVE) corrections) is only slightly lower than the experimental value for the 0–0 transition in 2methyltetrahydrofuran (2.69 eV) [2] or in 50:50 ether and ethanol solution (2.71 eV) [5]. The static dipole moment of the T1 state is about half the size of the ground state dipole moment so that we expect the transition to be blue shifted in polar solvents. Table 2 Excitation energies [eV] of singlet and triplet states in flavone at different nuclear geometries. State S0 S1 S2 T1 T2
Geometry 1 n ∗
1 ∗
1 ∗ 1 n ∗
S0
T1
S1
S2
0.00 3.30 4.21 3.04 3.17
0.49 3.21 3.90 2.59 3.20
0.44 2.93 3.97 2.74 2.88
0.64 3.43 3.98 3.03 3.37
Fig. 2. Molecular orbitals of flavone involved in the lowest electronic transitions.
At the S0 and S1 geometries, the 3 (n∗ ) state is near-degenerate with the 3 (∗ ) state, at least in isolated flavone. It can be assumed that the T2 minimum nuclear arrangement is very similar to the one of the S1 state so that the near-degeneracy will persist also in the T2 relaxed geometry. Slight out-of-plane motions will thus suffice to induce strong vibronic coupling between the two triplet states [44,45]. Polar solvents will increase the energy gap between the two triplets and thus reduce their coupling. The difference of their static dipole moments is smaller than in the corresponding singlets (see below) so that we can expect the T1 and T2 excitation energies to be less affected solvent by effects.
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C.M. Marian / Spectrochimica Acta Part A 73 (2009) 1–5
At the ground state geometry, the static dipole moment of the S1 state is close to zero whereas the S0 and S2 states exhibit dipole moments around 4 Debye. The 1 (n∗ ) state will therefore be significantly blue shifted in polar solvents, thus decreasing the energy gap to the lowest 1 (∗ ) state. Pownall [5] reports that the first absorption band of flavone (S1 (n∗ ) ← S0 ) coalesces with the second band (S2 (∗ ) ← S0 ) in ethanol. This should, however, not be the case in unpolar solvents such as cyclohexane. Nevertheless, our calculated adiabatic excitation energy of the S1 (n∗ ) state of 2.93 eV (2.81 eV after ZPVE correction) is significantly lower than the maximum of the first broad peak of this weak absorption in cyclohexane solution (3.37 eV) [5]. Errors of this size are untypical of the DFT/MRCI method [19–25]. We rather assume that the 0–0 transition is not observed experimentally because of its small Franck-Condon factor (0.0016) in combination with its low electronic oscillator strength (0.0008). We compute the vertical S2 (∗ ) ← S0 absorption energy to be 4.21 eV, in good agreement with experimental maxima of absorption bands (4.19 eV in 50:50 ether and ethanol solution [5] and 4.08 eV in water at pH 7 [1]). As mentioned above, the excited state geometry optimized with TDDFT (B3-LYP) does not yield the lowest DFT/MRCI energy (see Table 2). Rather, the lowest DFT/MRCI excitation energy of the S2 state (3.90 eV, 3.80 eV including ZPVE correction) is obtained at the T1 minimum, in very good agreement with the redmost absorption peak (321.5 nm corresponding to 3.86 eV) in resonance enhanced two-photon ionization (R2PI) spectra of jet-cooled flavone [46]. Comparison with the measured value of 3.77 eV for the 0–0 transition in 50:50 ether and ethanol solution [5] shows that solvent effects on this transition are rather small. 3.3. Inter-system crossing rates ISC rates depend heavily on the size of the SOCMEs and the energy gap between the coupling states [47]. A small energy gap alone does not guarantee, however, the overlaps between the vibrational ground state of the initial state i and the vibrational levels of the final state f (Eq. (2)) to be sizable. For instance, a pair of nested states – i.e., states with nearly identically shaped potential energy wells that are vertically shifted with respect to each other – will in general have small vibrational overlaps and thus slow nonradiative transitions. Displacements of the nuclear geometry along the normal coordinates typically increase the vibrational overlap and thus enhance ISC. Our calculations place the zero vibrational level of the S1 state 2604 cm−1 above the T1 vibrational ground state. An interval of = 0.1 cm−1 above and below the initial state was chosen for integrating the transition probabilities. In this interval 57021 vibrational levels of the triplet state are located. The mode with the largest displacement is 1 , the phenyl torsion. More importantly, however, a considerable number of medium frequency modes, mostly asymmetric C–O–C and C–C–C stretches are among the modes with large displacements. For the S1 T1 transition 38 of the 75 vibrational modes of flavone show displacements larger than 0.1 in mass-weighted dimensionless reaction coordinates. Selecting only these 38 modes (3735 states) yields a transition rate within 10% of the full results. Furthermore, a combination of a few v = 0 → v = 1 excitations will suffice to bring the triplet levels into near degeneracy with the singlet vibrational ground state. A test calculation where the excitation level of all vibrational modes was set to at most 1 quantum per mode (852 vibrational levels in the interval) yields an ISC rate only one order of magnitude smaller than the full calculation. Limiting the excitation level to 3 quanta per mode (16,408 vibrational states) gives a value that is practically indistinguishable from the full result. These tests show that higher excitations of low-frequency modes
Table 3 Electronic spin–orbit coupling matrix elements [cm−1 ] and inter-system crossing rates [s−1 ] for the S1 T1 transition of flavone. ˆ SO |T1 | |S1 |H
kISC
x : 26.37 y : 29.28 z : 22.58 x+y+z
1.11 × 1011 1.37 × 1011 0.81 × 1011 3.29 × 1011
contribute only to a minor extent to the nonradiative transition. Matrix elements of the cartesian spin–orbit operator components and the corresponding ISC rates are shown in Table 3. The individual rates depend on the orientation of the molecule within the chosen coordinate system. Hence, comparison with experimental data is only meaningful for their sum. To the author’s knowledge, the ISC rate has not been determined experimentally. A rough estimate can be obtained, however, from the build-up times of the Tn ← T1 excitation spectra for which values of (30 ± 10) × 10−12 s in CCl4 up to (42 ± 8) × 10−12 s in ethanol were reported [4]. If the ISC is assumed to be the time-determining step, these buildup times correspond to ISC rates in the range of 2 to 3 × 1010 s−1 , somewhat smaller than our computed gas phase value of 3.3 × 1011 s−1 (Table 3). This difference is mainly attributed to solvent effects. Taking into account that polar protic solvents will increase the energy gap between the S1 (n∗ ) and T1 (∗ ) states leads us to conclude that the S1 T1 ISC rate will decrease with solvent polarity and proticity. Compared to the low fluorescence rate of the S1 state (kF ≈ 4.3 × 103 s−1 ) the ISC will remain efficient though. 3.4. Phosphorescence lifetimes The actual phosphorescence rates are calculated from variational wave functions. Nevertheless, it can be instructive to analyze the results in terms of the major contributions from perturbation theory. In the framework of the latter theory, the dipole transition moment of a spin-forbidden radiative transition is a sum of direct and indirect contributions [37,39]. The direct terms originate from the spin–orbit coupling of the zeroth-order wave functions, multiplied by the difference of their respective dipole moments and devided by their energy difference. The entries in Table 4 reveal that the three SOCMEs of the T1 and S0 wave functions are very small, as expected for a ∗ excitation [9]. The indirect terms involve spin–orbit coupling of the initial state to an intermediate state which in turn couples to the final state via the electric dipole operator and vice versa. We find large SOCMEs for the T1 and S1 pair of states (Table 4). The second-order coupling via S1 and other 1 (n∗ ) states is weak nevertheless because of their small electric dipole transition moments with S0 . In contrast, S2 and other 1 (∗ ) states exhibit strong dipole transitions to S0 , but their SOCMEs with T1 are typically below 1 cm−1 . Higher-order perturbation terms are usually neglected. In the present MRSOCI calculations electronic terms Table 4 Electronic spin–orbit coupling matrix elements [cm−1 ], dipole matrix elements [ea0 ], and phosphorescence lifetimes P [s] for the T1 → S0 transition of flavone. SOCME
x
y
ˆ SO |S0 | |T1 |H ˆ SO |S1 | |T1 |H ˆ SO |S2 | |T1 |H ˆ SO |T2 | |T1 |H ˆ SO |S0 | |T2 |H ˆ SO |S2 | |T2 |H P
0.88 18.40 0.28 17.28 23.41 12.92 T1a 5.26 3.59
0.06 22.69 0.26 21.32 26.58 17.44 T1b 1.56
High-temp. average
| |
z 0.11 19.60 0.12 18.42 23.24 14.88 T1c 383.9
|S0 | |S0 − T1 | |T1 | |S1 | |S0 | |S2 | |S0 |
0.9461 0.0347 2.2098
|S0 | |S0 − T2 | |T2 | |S2 | |S0 |
0.8977 2.2098
C.M. Marian / Spectrochimica Acta Part A 73 (2009) 1–5
are automatically included to all orders. Vibronic contributions which are expected to gain importance at elevated temperatures, have not been taken into consideration (see however below). At this level of approximation, phosphorescence lifetimes in the s range are obtained in the calculations. Qualitatively, this result is agreement with experiments conducted at 77 K in different glassy matrices. Under these conditions, decay times of the Tn ← T1 absorption of flavone range from 0.282 s in 9:1 methylcyclohexane-isopentane mixture [4] to 0.675 s in 1:1 ether-ethanol [5]. Taking into account that concurrent nonradiative decay processes are mediated by the interaction of the solute with the surrounding matrix, the agreement can be considered as good. The experimentally observed increase of the triplet lifetime with increasing solvent polarity [4] is consistent with the S1 state being the major source of indirect coupling. A blue shift of the latter state will increase the energy difference in the denominator of the second-order coupling term with T1 and thus diminish its contribution. How does the much shorter triplet lifetime measured at room temperature in degassed benzene (4.5 s after correction for selfquenching [2]) fit in? As the following discussion will show, such a short lifetime cannot be explained by phosphorescence alone but requires additional nonradiative decay channels. In Section 3.2 it was mentioned that the T1 and T2 states are near degenerate in some regions of the coordinate space. With decreasing energy gap, the (non-adiabatic) vibronic interaction between the two Born-Oppenheimer states will increase. Furthermore, higherorder spin–orbit coupling will gain importance. It should be kept in mind, however, that intensity borrowing from the nearby T2 3 (n∗ ) state via vibronic and spin–orbit interaction state can increase the phosphorescence rate to at most the value it has for the T2 → S0 transition. The calculated rate of a hypothetical phosphorescence originating from the T2 state amounts to 845 s−1 , i.e., it is about 3000 times higher than the value for the purely electronic T1 → S0 transition. The corresponding phosphorescence lifetime of 1.2 ms is, nevertheless, significantly longer than the above mentioned measured triplet lifetime in benzene solution. 4. Summary and conclusions Combined density functional and multireference configuration interaction methods were used to compute a variety of properties of flavone, including singlet and triplet excitation energies and rates of spin-forbidden transitions. While the phenyl ring is twisted out of the plane of the benzopyrone in the electronic ground state, a planar structure is preferred in the first excited singlet state which exhibits n∗ character. The lowest ∗ excited states take a butterfly-type shape. The calculations clearly show that the T1 state has ∗ character. T1 is populated at a rate of kISC ≈ 3 × 1011 s−1 from the S1 state. Among the most important accepting modes a variety of asymmetric C–O–C and C–C–C stretches are found. In the absence of nonradiative processes, T1 is long-lived with a calculated radiative decay rate of kP ≈ 0.3 s−1 . These values are in qualitative agreement with experimental build-up and decay times of transient Tn ← T1 absorption at 77 K. The calculations furthermore suggest that the redmost peak in the S1 ← S0 absorption spectrum (3.37 eV in cyclohexane solution [5]) does not correspond to the 0–0 transition because of its unfavorable Franck-Condon factor in combination with the weak electronic oscillator strength. Acknowledgment I would like to thank Karl Kleinermanns and Gernot Engler (Düsseldorf) for many valuable discussions. Financial support by
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