THE JOURNAL OF CHEMICAL PHYSICS 129, 074302 共2008兲

S1 ] S0 transition of 2,3-benzofluorene at low temperatures in the gas phase A. Staicu,1,a兲 G. Rouillé,1 Th. Henning,1 F. Huisken,1,b兲 D. Pouladsaz,2 and R. Scholz3 1

Laboratory Astrophysics Group of the Max Planck Institute for Astronomy at the Friedrich Schiller University Jena, Institute of Solid State Physics, Helmholtzweg 3, D-07743 Jena, Germany 2 Institut für Physik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany and Walter Schottky Institute, Technical University of Munich, Am Coulombwall 3, D-85748 Garching, Germany 3 Walter Schottky Institute, Technical University of Munich, Am Coulombwall 3, D-85748 Garching, Germany

共Received 23 May 2008; accepted 11 July 2008; published online 18 August 2008兲 The S1共 1A⬘兲 ← S0共 1A⬘兲 absorption spectrum of jet-cooled 2,3-benzofluorene 共Bzf兲 has been measured by cavity ring-down spectroscopy. The potential energy surfaces of the Sn=0,1,2 states of Bzf have been investigated with calculations based on the time-dependent density functional theory 共TD-DFT兲. At the B3LYP/TZ level of theory, TD-DFT does not deliver a realistic difference between the excited S1 and S2 potential energy surfaces, a problem which can be avoided by introducing a reference geometry where this difference coincides with the observation. In this geometry, an expression for the Herzberg–Teller corrected intensities of the vibronic bands is proposed, allowing a straightforward assignment of the observed a⬘ modes below 900 cm−1, including realistic calculated intensities. For vibronic bands at higher energies, the agreement between calculated and observed modes is deteriorated by substantial Dushinsky rotations and nonparabolicities of the potential energy surface S1. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2967186兴 I. INTRODUCTION

Polycyclic aromatic hydrocarbons 共PAHs兲 are considered to present an important portion of the carbon-bearing molecules that exist in interstellar medium. They were proposed as carriers of the unidentified infrared emission bands of different nebulae1 and of the optical absorption bands observed in the spectra of reddened stars, i.e., the diffuse interstellar bands.2,3 In this context, we have studied the electronic spectroscopy of several PAHs in the gas phase at low temperatures and low densities by pulsed jet cavity ringdown laser absorption spectroscopy.4–8 The absorption features measured by this technique are most relevant for a comparison with astronomical observations. Other electronic CRDS studies on PAHs in supersonic jets were reported by Salama and co-workers.9–14 In this paper, we present a combined experimental and theoretical study of the jet-cooled S1 ← S0 transition of 2,3benzofluorene 共Bzf, C17H12兲. A preliminary account was given in an earlier overview article, which, however, was devoted to a more general subject.7 Prior to our work, this molecule was studied by matrix isolation spectroscopy.15–17 High-resolution matrix spectra of Bzf were first reported by Nakhimovsky et al.15 using an n-hexane matrix at 5 K. For each band, the spectra showed a multiplet structure caused by the various orientations of the matrix-embedded molecule. The newer high-resolution spectra of Bzf obtained by Geigle and Hohlneicher16 at 14 K in an Ar matrix were free a兲

Present address: National Institute for Laser, Plasma and Radiation Physics, Laser Department, 077125 Magurele, Bucharest, Romania. b兲 Electronic mail: [email protected]. 0021-9606/2008/129共7兲/074302/10/$23.00

of multiplet structures as they used site-selective laser excitation/fluorescence spectroscopy. The most recent Ar matrix isolation study carried out by Banisaukas et al.17 using a conventional UV spectrometer showed absorption bands that were much broader than those observed in the earlier site-selective matrix experiment.16 Interestingly, it was found that the Herzberg–Teller effect plays a significant role in the S1 ← S0 transition of Bzf.16 Whereas Geigle and Hohlneicher16 calculated the vibronic absorption and emission spectra of Bzf with a force field model, normal density functional theory 共DFT兲 and timedependent density functional theory 共TD-DFT兲 have been employed in the present study to determine the geometries and vibrational modes of Bzf in its S0–S2 states. From these results, the vibrational intensity pattern in the S1 ← S0 spectrum was calculated and discussed in comparison with the experimental data. II. EXPERIMENT

The pulsed jet laser spectrometer employing cavity ringdown spectroscopy 共CRDS兲 has been described in detail elsewhere.6 It consists of a pulsed supersonic jet nozzle incorporated into a small vacuum chamber equipped with two adjustable extension tubes on which two high-reflectivity mirrors are mounted, thus forming a high-quality optical cavity. The 1-mm-diameter nozzle is attached to an electromagnetically driven valve 共series 9, Parker Hannifin Corporation, General Valve Division兲 and can be heated up to 500 ° C in order to provide a sufficiently high vapor pressure of the PAH powder contained in a small reservoir inside the valve.

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As carrier gas, we used Ar or Ne with a backing pressure of 2.5 bars. Bzf was purchased from Fluka and had a purity of better than 98%. In the present experiment, the temperature of the nozzle was varied between 175 and 240 ° C. The gas pulse duration was 600 ␮s and the background pressure in the vacuum chamber was in the range of 共0.6– 1兲 ⫻ 10−2 mbar. The jet was probed at distances between 4.5 and 19 mm downstream from the nozzle exit. The light source of the CRD spectrometer was provided by a tunable pulsed dye laser pumped by the second harmonic 共␭ = 532 nm兲 of a neodymium doped yttrium aluminum garnet laser having a repetition rate of 20 Hz. The visible laser light, whose linewidth was approximately 0.1 cm−1,6 was frequency doubled by a potassium dihydrogen phosphate 共KDP兲 crystal and the UV laser beam thus generated was coupled into the optical cavity. For this study, the cavity was formed by planoconcave mirrors with a radius of curvature of 250 mm placed 420 mm apart. Several sets of mirrors were used to cover the wavelength range of interest.7 Photons transmitted by the cavity rear mirror were detected by a photomultiplier tube and the observed decay signal was averaged over 64 laser shots by a digital oscilloscope before being transferred to a computer to calculate the losses per pass. The wavelength calibration of the spectra was performed using an Fe–Ne hollow cathode lamp and measuring the absorption lines of neon when the laser was scanned through the Bzf spectrum. The accuracy of the band positions is ⫾1.5 cm−1. All wave numbers are given for vacuum. III. CALCULATIONS WITH TIME-DEPENDENT DENSITY FUNCTIONAL THEORY

The ground state geometry of Bzf was optimized with the B3LYP hybrid functional,18 and the minima of the potential energy surfaces 共PESs兲 in the lowest excited singlet states S1 and S2 were obtained with TD-DFT in the adiabatic approximation.19 Similar to previous TD-DFT calculations17 based on the BLYP functional, the ordering of the S1 and S2 configurations of Bzf was inverted. As shown in Fig. 1, in the ground state geometry, the origin of the strong S2 ← S0 transition dominated by the excitation from the highest occupied molecular orbital 共HOMO兲 to the lowest unoccupied molecular orbital 共LUMO兲 共1La ← 1A in the Platt notation20兲 was found at 31 041 cm−1 共3.848 eV兲, significantly below the weak S1 ← S0 transition 共 1Lb ← 1A兲 at 32 027 cm−1 共3.971 eV兲, which is mainly composed of the electronic excitations LUMO← 共HOMO-1兲 and 共LUMO+ 1兲 ← HOMO. For several aromatic compounds, TD-DFT shows similar deviations for the lowest two transition energies, and the situation for benz关a兴anthracene resembles that of Bzf.21 From a comparison of the TD-DFT transition energies of Bzf with the experimental data discussed below, we find that the calculation underestimates the origin of the strong S2 ← S0 共 1La ← 1A兲 transition by about 0.15 eV, whereas the origin of the weak S1 ← S0 共 1Lb ← 1A兲 transition is overestimated by about 0.26 eV. These systematic deviations follow rather precisely the overall trend observed in TD-DFT calculations for a series of polycyclic aromatic molecules, revealing that on average the B3LYP functional underestimates the

J. Chem. Phys. 129, 074302 共2008兲

FIG. 1. Calculated PESs for the Sn=0,1,2 electronic states of Bzf as a function of the molecular geometry q.

more ionic 1La ← 1A transition energies by 0.18 eV, whereas the more covalent 1Lb ← 1A transitions are overestimated by 0.24 eV.21 As all orbitals involved in the S1 ← S0 and S2 ← S0 transitions of Bzf are ␲ states transforming according to the representation A⬙ of the point group Cs, the transitions S1 ← S0 and S2 ← S0 have A⬘ symmetry, so that in the relaxed excited geometry of the molecule, only internal vibrations transforming according to the A⬘ representation can be elongated. For basis sets up to double ␨,22 the lowest two excited state PESs crossed between the ground state geometry and the relaxed excited geometry of the PES in the electronic configuration S1. Only a rather large triple-␨ 共TZ兲 basis23 gave PES of S1 and S2 that did not cross between the geometries of interest, which is the case described in Fig. 1. The calculation of the internal vibrations at the S1 and S2 minima, however, was useless because the interaction between the two lowest PESs had a strong influence on the vibrational frequencies. For the lower PES 共S2兲, this resulted in a saddle point with a large imaginary frequency, whereas the upper PES 共S1兲 had a vibrational frequency of around 2200 cm−1, clearly in an unphysical region. On selected points along the deformation pattern connecting q共S0兲 and q共S1兲 but covering both sides of the minimum of S1, we have performed further calculations of the vibrational properties. As the energetic difference between the two lowest PESs remained rather small, the artifacts resulting from the interaction between them were not removed. In order to come to a situation resembling the observed ordering, i.e., E共S2兲 − E共S1兲 = 0.295 eV, we have deformed the molecule along the direction defined by the minima of S2 and S1, as visualized on the right hand side of Fig. 1. In the reference geometry called qⴱ, we find the PES at energies of E共S0兲 = 0.400 eV, E共S1兲 = 4.175 eV, and E共S2兲 = 4.470 eV with respect to the minimum of S0. This defines clearly the lowest energy where we can obtain a splitting E共S2兲 − E共S1兲 = 0.295 eV, in accordance with the experimental situation,

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Electronic spectrum of 2,3-benzofluorene

J. Chem. Phys. 129, 074302 共2008兲

FIG. 2. 共Color online兲 Optimized geometry q共S0兲 of benzofluorene in the electronic ground state 共black兲 and modified geometry qⴱ 共gray/light blue兲. The corresponding potential energies are given in Fig. 1.

and the vertical transition energies are only about 0.07 eV above the observed values. As shown in Fig. 2, this geometry qⴱ is still rather similar to q共S0兲, so that we expect only a minor influence of the nonparabolicities of the three PESs on the vibrational properties of either singlet state. Nevertheless, as the intention behind the construction of the geometry qⴱ was a modification of the difference between the PESs relating to S2 and S1 as large as 0.35 eV, it is clear that the energetic cost of this geometry on the PES of S0 of 0.40 eV has to be significantly larger than the reorganization energies of 0.13 eV at the minimum of the PES S1 and of 0.20 eV at the minimum of the PES S2. The failure of DFT in describing 1La states with rather ionic character relates to the wrong asymptotics of the exchange-correlation functional, decaying much faster than −1 / r, as expected for the Coulomb potential.24 For the B3LYP hybrid functional in the present work, this deficiency is reduced but not removed. Obviously, our qⴱ geometry cannot cure this systematic shortcoming of DFT, but it combines two relevant ingredients for quantitative predictions: a reasonable difference between the lowest excited configurations S1 and S2 and a reliable gradient of this difference with respect to changes in the nuclear positions. In Sec. IV, we will discuss vibrational modes on the PES of S1, calculated from numerical differentiation of this PES around the reference geometry qⴱ defined above. The Huang–Rhys factors25 of these modes are obtained by projecting the deformation ⌬q = q共S1兲 − q共S0兲 between the minima of the PESs relating to S1 and S0 onto the vibrational eigenvectors. The dependence of the transition dipole on deformations around the reference geometry qⴱ induces substantial Herzberg–Teller corrections of the intensities related to the vibronic bands.

IV. RESULTS AND DISCUSSION A. Experimental results

Figure 3 presents a comparison of the Bzf spectra as measured in the gas phase and in Ar matrices in the wave number range between 29 600 and 31 500 cm−1. The supersonic jet gas phase spectrum displayed in the uppermost panel was obtained with Ar as carrier gas at 2.5 bars and a nozzle temperature of 200 ° C. The jet was probed at a distance of 4.5 mm from the nozzle exit. The strongest band at 29 894.3 cm−1, which is reduced in intensity due to

FIG. 3. 共Color online兲 S1 ← S0 absorption spectra of Bzf measured in a supersonic jet by CRDS 共a兲 and in Ar matrices by conventional absorption spectroscopy 共Ref. 17兲 共b兲 or by site-selective fluorescence 共Ref. 16兲 共c兲. Bands labeled with a star are presented in more detail in Fig. 4. Note that the lower frame has been shifted by 240 cm−1 to higher wave numbers to account for the matrix shift.

saturation effects, can be assigned to the origin of the S1共 1A⬘兲 ← S0共 1A⬘兲 transition. The features at higher frequencies belong to vibronic transitions from the v = 0 level of the electronic ground state to excited vibrational levels in the S1 state.7 In the investigated spectral range, i.e., between the origin band and 1600 cm−1 above the origin band, we observe vibronic transitions corresponding to the excitation of modes dominated by CCC in-plane bending motions 共ⱕ700 cm−1兲 and of modes involving mainly CC stretching as well as CH in-plane bending motions 共1000– 1650 cm−1兲. The molecular structure of Bzf, which is also depicted in Fig. 3, has the symmetry elements of the Cs point group and the S0 and S1 states transform according to its A⬘ irreducible representation. Although both types of vibrational transitions, i.e., a⬙ ← a⬘ and a⬘ ← a⬘, are allowed by symmetry selection rules, we should expect only transitions involving the totally symmetric modes if we consider that the Franck– Condon principle applies. In that case, all vibronic bands observed in the present spectrum should be assigned to a⬘ vibrational modes of the S1 state. Moreover, they can appear as bands with a mixed A- and B-type character. In the lower panels of Fig. 3, the spectra obtained by Ar matrix spectroscopy are shown 共adapted from Refs. 16 and 17兲. For a better comparison, the scale of the matrix spectra is shifted by 240 cm−1 to higher wave numbers. While the

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TABLE I. Vibrational frequencies of the S1 state of Bzf measured in the S1共 1A⬘兲 ← S0共 1A⬘兲 spectrum observed in a supersonic jet and in an Ar matrix. The results of the computational study are included for comparison. Observed

Calculated

Jeta 共cm−1兲

Ar matrixb 共cm−1兲

145

153 197

298 343 349 359 446 459 554 702 710 765 807 817 839 880 958 984c 1059 1098 1117

共cm−1兲 142.7 298.4

352

356.1

447

462.5

557 702 712 768

566.7

822

821.5

881 959 985

944.8

666.4 759.4

1098

Observed Jeta 共cm−1兲 1127 1155 1194 1264 1319 1336 1371 1375c 1381 1394 1417 1423 1437 1457 1465 1468 1476 1486 1524 1534

Calculated

Ar matrixb 共cm−1兲

共cm−1兲

1341 1376 1395

a

This work. Reference 16. c Pair of overlapping bands. b

conventional Ar matrix study of Banisaukas et al.,17 shown in Fig. 3共b兲, resulted in broad features with full widths at half maximum 共FWHMs兲 of the order of 70 cm−1, which left individual vibronic transitions unresolved, the site-selective fluorescence spectrum of Geigle and Hohlneicher16 in Fig. 3共c兲 is comparable to our gas phase data. The agreement between the vibrational frequencies obtained by us and Geigle and Hohlneicher16 can be validated in Table I. Still, the bands observed in the Ar matrix are three times broader, and a typical Ar matrix redshift of −240 cm−1 is present in their spectra. Moreover, the bands measured in an Ar matrix do not display rotational contours. In contrast, vibrational bands recorded in the gas phase may reflect a rotational structure, a fact which can be very helpful in determining the symmetry of the vibronic transition. In Fig. 4, the profiles of the origin band and the three most intense vibrational features observed at 710, 984, and 1375 cm−1 above the origin are shown. The corresponding bands are marked with a star in Fig. 3共a兲. All bands were recorded in the jet at the same probing distance of 4.5 mm, while the valve temperature was maintained at 175 ° C for the origin band and somewhat higher 共200 ° C兲 for the other features. For the origin band, a lower temperature was used to avoid saturation effects resulting from the strong absorption. As we will show below, the double-peak structure in the origin band corresponds to P, Q, and R rotational branches, with the P and Q branches being too close to show a dip between their maxima. When we introduced our preliminary

results,7 we proposed a calculated profile based on the rotational constants and the transition moment orientation obtained by ab initio calculations at the HF and RCIS/ 6-31 + G共d , p兲 frozen core levels for the electronic ground and excited states, respectively. As will be discussed below, we came to the conclusion that our earlier calculation did not reproduce the observations correctly because of inversion of the nature of the two lowest singlet-singlet electronic transitions. For comparison, we have carried out calculations with the Zerner intermediate neglect of differential overlap 共ZINDO兲 technique,26 which we estimate to be reliable in describing the lower electronic excited states of PAHs.8 The ZINDO technique was used as implemented in the GAUSSIAN package.27 The electric dipole transition moment for S1 ← S0 has been found to be essentially oriented like the A principal axis of inertia of the molecule. Therefore, the rotational profile is considered to be of the A type with a FWHM of 1.6 cm−1, the lower-energy peak corresponding to the overlapping P and Q branches and the higher-energy peak to the R branch. The simulated profile shown in Fig. 4共a兲 by the dotted line was obtained using the PGOPHER program.28 Equilibrium rotational constants determined in our DFT and TD-DFT studies of the S0 and S1 states, respectively, were employed, and the profile was considered to be of the A type. Values for the constants are A = 0.052 107 cm−1, B = 0.008 811 2 cm−1, and C = 0.007 547 3 cm−1 for S0 and A⬘ = 0.051 634 cm−1, B⬘ = 0.008 770 8 cm−1, and C⬘ = 0.007 507 7 cm−1 for S1. The constants calculated for S2, which corresponds to the

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Electronic spectrum of 2,3-benzofluorene

J. Chem. Phys. 129, 074302 共2008兲

FIG. 5. 共Color online兲 Temperature study of the bands at −22 cm−1 共left兲 and +145 cm−1 共right兲 relative to the origin band in the range between 200 and 240 ° C.

FIG. 4. 共Color online兲 Close-ups of the bands labeled with a star in Fig. 3. The relative positions with respect to the origin band are given in the figure. Except for the temperature 共175 ° C for the origin band and 200 ° C for the others兲, the experimental parameters were the same. Simulated 共dotted兲 and synthetic 共dashed兲 profiles are shown in 共a兲 共see text兲. In 共c兲 and 共d兲, vertical lines indicate the centers of overlapping bands as determined by the simulations 共dashed lines兲.

observed S1 state considering the electronic structure, are A⬘ = 0.051 587 cm−1, B⬘ = 0.008 762 8 cm−1, and C⬘ = 0.007 500 8 cm−1, and they yield a band contour very close to that determined with the other set. Rotational levels characterized by values of the quantum number J up to 140 were taken into account. From the comparison of the observed origin band with the present simulation, we derived a rotational temperature of 16 K at this sampling position 共4.5 mm兲. This is somewhat higher than the 12 K determined in our previous publication.7 For the band at 710 cm−1, we notice a rotational profile of the same type as for the origin band, with the overlapping of P and Q branches appearing more clearly. In contrast, the features at 984 and 1375 cm−1 appear to be composed of two close-lying bands of the same symmetry, as suggested by the simulations that are presented in panels 共c兲 and 共d兲. Each simulated feature represents the sum of two overlapping synthetic bands of identical contour, differing only by their positions and intensities. Their shape, shown as a dashed line in panel 共a兲, has been determined by fitting two Lorentzians to the observed profile of the origin band. Although we did not investigate all small vibrational bands at higher temperatures and higher resolutions in order to study their profile in detail, we found for most observed bands the same profile as for the origin band, i.e., an A type. Particularly interesting is the small feature observed on

the red side of the origin band at −22 cm−1 关see Fig. 3共a兲兴. Theoretically, this band can be assigned either to a hot or sequence band or to a van der Waals 共vdW兲 complex of Bzf with Ar.6 We observed that this band did not change in intensity when the valve temperature was varied, while other small bands located on the blue side of the origin band experienced a significant increase in intensity. For example, in Fig. 5, we compare the behavior of the −22 cm−1 band with that of the band at +145 cm−1 when the temperature was increased from 200 to 240 ° C. All spectra displayed in Fig. 5 were recorded with Ar as carrier gas at a pressure of 2.5 bars and a probing distance of 4.5 mm. The observed behavior can be explained if we assume that the −22 cm−1 band is due to a Bzf· Ar complex. The intensity of the vibrational band at +145 cm−1 increases as the temperature and Bzf vapor pressure are raised. In contrast, for a Bzf· Ar complex, we expect a compensation of this effect since a higher temperature makes dimer formation less probable. In another experiment, we kept the temperature constant at 200 ° C and varied the distance r from the nozzle exit from 4.5 to 19 mm. The results are displayed in Fig. 6. While the origin band and the 710 cm−1 vibrational band experience a reduction in intensity with roughly r−2, the intensity of the band at −22 cm−1 seems to be not affected by the distance variation. This behavior again suggests that the band on the red side can be assigned to a Bzf· Ar vdW complex. Further

FIG. 6. 共Color online兲 Study of the band profile as a function of the distance in the jet. All bands were recorded at 200 ° C. The distance between the probing volume and the nozzle exit was varied between 4.5 and 19 mm.

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the Ar atom on the planar Bzf molecule, either above the 5-C ring, like in the case of the fluorene complex with argon,31 or above the inner 6-C ring, like in Te· Ar.34 The maximum at −22 cm−1 is produced by the partial overlap of the two bands.

B. Theoretical analysis

FIG. 7. 共Color online兲 Comparison of the carrier gases argon and neon. The band at −22 cm−1 was absent when neon was used. Due to less efficient cooling, the origin band measured with neon at T = 175 ° C is somewhat broader than the respective band observed under otherwise identical conditions with argon.

collisions and the decreasing temperature favor the formation of complexes and compensate the general decrease in density. The final proof for this interpretation was provided by a third experiment, where we employed neon instead of argon as carrier gas. The spectra shown in Fig. 7 reveal that the Bzf· Ar feature at −22 cm−1 is absent when neon is used. In this experiment, the temperature was varied between 175 and 225 ° C, the Ne pressure was kept at 2.5 bars, and the probing distance was set to 4.5 mm. Only for the smallest temperature was the origin band measured; at higher temperatures, due to the strong absorption, this band is saturated. Taking into account the three experiments just discussed, we can derive for the origin band of the Bzf· Ar vdW complex a shift of −22 cm−1 from the monomer origin band. It is interesting to compare the present result with that obtained for tetracene 共Te兲, the polyacene formed with four linearly fused benzene rings, which has a structure rather similar to that of Bzf. For the Te· Ar vdW complex, a shift of −41 cm−1 from the origin band of the tetracene monomer was reported.29,30 Apparently, the change in the interaction between Ar and the PAH resulting from the electronic excitation is weaker in the case of Bzf than in the case of tetracene. This is somewhat remarkable given that, in a similar comparison, the shifts observed for the anthracene 共An兲 and fluorene complexes with argon are very close with −40 and −44 cm−1, respectively.6,31 We were also able to observe the peculiarity of the S1 ← S0 transition of Bzf relative to those of tetracene and anthracene when the molecules were embedded in helium nanodroplets. Whereas shifts of −103 and −61 cm−1 were observed for Te@ HeN and An@ HeN, respectively,32,33 a shift of +5 cm−1 was measured for Bzf@ HeN.7 In order to explain the peculiar behavior of Bzf, a detailed theoretical study of the variation in the dispersion interaction caused by the electronic excitation is necessary, which is beyond the scope of the present study. Finally, we would like to discuss the peculiar shape of the Bzf· Ar feature, which seems reproducible in all spectra shown in Fig. 5. The two minima observed at −21.6 and −23.2 cm−1 can possibly be assigned to the origin band positions of two Bzf· Ar complexes differing in the position of

In their study of Bzf, Geigle and Hohlneicher16 calculated the vibronic absorption and emission spectra by using a force field model. Their calculation revealed that, especially for the weak S1 ← S0 transition, an interference between Franck–Condon and Herzberg–Teller terms can have a significant influence on the intensities of the vibronic bands. In the following, we calculate the Herzberg–Teller corrections starting from vibrational modes on the S1 surface as obtained in the reference geometry qⴱ defined in Sec. III. As the CRD spectra have been obtained in a supersonic jet after expansion through a nozzle, the molecules have a very low internal temperature, so that we can assume that photon absorption starts from the lowest vibrational level for all internal vibrations. This state is denoted 0g, with the subscript g standing for the electronic ground state, while the subscript e will refer to the electronically excited state. We consider that only transitions to the vibrationless S1 state, 0e, and to states in which a single vibration is excited by one quantum of energy, 1e, can yield bands of significant intensities. The aim of the present section is to calculate these intensities, I共1e ← 0g兲, relative to that of the origin band, I共0e ← 0g兲. For Bzf, the lowest transition S1 ← S0 has a small oscillator strength corresponding to a small transition dipole, so that internal deformations may admix a substantial amount of the large transition dipole related to the strong transition S2 ← S0. In order to avoid tedious notations with indices for the various internal vibrations in the electronic state S1, the following notation for the transition dipoles toward different vibronic levels will be restricted to a single internal mode. If we are only interested in the ratio between the intensity of a vibronic band and the origin band, the influence of the other modes collapses to a common prefactor for both intensities, so that it does not affect their ratio. With this simplification, the intensity of a transition from the lowest vibronic level 兩0g典 in the electronic ground state to a level 兩␯e典 in an optically excited state is governed by the square of the transition dipole moment, ˆ 兩⌿g0 典. ␮e␯e,g0g = 具⌿e␯e兩␮ g

共1兲

Expanding the wave functions in terms of an electronic part ⌽ and a vibrational part ␹ as ⌿e␯e = ⌽e共q兲␹␯e共q兲 and ⌿g0e = ⌽g共q兲␹0g共q兲, the transition dipole reads35

␮e␯e,g0g = 具␹␯e共q兲兩␮eg共q兲兩␹0g共q兲典.

共2兲

The electronic transition dipole ␮eg共q兲 can be expanded around the reference geometry, in our case qⴱ, according to

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Electronic spectrum of 2,3-benzofluorene

␮eg共q兲 ⬇ ␮eg共qⴱ兲 + 共q − qⴱ兲ⵜq␮eg共qⴱ兲 + O共共q − qⴱ兲2兲. 共3兲 Truncating this Taylor series after the first order and inserting it into Eq. 共2兲, one obtains

␮e␯e,g0g ⬇ 具␹␯e兩␹0g典␮eg共qⴱ兲 + 具␹␯e共q兲兩共q − qⴱ兲 ⫻兩␹0g共q兲典ⵜq␮eg共qⴱ兲,

TABLE II. Vibrational eigenfrequencies of a⬘ modes obtained at the geometry qⴱ on the PES S1, Huang–Rhys factors S, Herzberg–Teller corrections f HT, and ratios of the intensities of the fundamental vibronic bands to the origin band, given by f HTS. Modes with Huang–Rhys factors lower than 0.006 or with frequencies higher than 1800 cm−1 have not been included. Mode indexa

Frequency 共cm−1兲

S

f HT

I共1e ← 0e兲 / I共0e ← 0g兲

54 53 52 51 49 46 44 43 41 39 36 35 32 30 29 28 27 26 25 24 23 17 15 14 13 12

142.7 298.4 356.1 462.5 566.7 666.4 759.4 821.5 944.8 1021.3 1156.3 1166.5 1209.7 1244.0 1267.5 1303.5 1320.3 1345.0 1354.5 1362.3 1386.1 1532.2 1568.6 1600.1 1611.4 1677.1

0.012 0.012 0.025 0.017 0.040 0.154 0.014 0.017 0.027 0.010 0.008 0.012 0.023 0.006 0.024 0.018 0.025 0.025 0.009 0.010 0.040 0.026 0.211 0.030 0.033 0.021

1.51 0.44 1.06 0.70 1.35 1.64 1.06 1.05 0.71 0.70 1.41 0.63 1.89 2.15 1.30 1.51 1.42 1.38 1.31 2.51 0.98 0.34 0.96 1.70 1.37 1.97

0.018 0.005 0.027 0.012 0.054 0.252 0.015 0.018 0.019 0.007 0.012 0.007 0.044 0.014 0.031 0.028 0.036 0.035 0.012 0.026 0.039 0.009 0.203 0.051 0.045 0.041

共4兲

where the lowest order corresponds to the Franck–Condon approximation and the first order gives the Herzberg–Teller correction. The entire expression yields the Franck–Condon– Herzberg–Teller 共FC-HT兲 approximation for the transition dipole, allowing the study of the influence of a vibrational deformation onto the size and direction of the transition dipole. As we are only interested in the vibronic levels 兩1e典 and 兩0e典 for each internal vibration, the general expressions for the FC-HT corrections36 can be simplified to these specific cases. Denoting the deformation of the excited Bzf molecule in the relaxed excited geometry of the S1 configuration as ⌬q = q共S1兲 − q共S0兲

共5兲

and taking into account that only the part of ⌬q along the eigenvector of a specific mode is required, a straightforward calculation gives the following expressions: 具0e兩␮eg共q兲兩0g典 = e−S/2

␮eg共qⴱ兲 + ␮eg共qⴱ + ⌬q兲 , 2

冋

共6兲

具1e兩␮eg共q兲兩0g典 = − 冑Se−S/2 ␮eg共qⴱ兲

册

␮eg共qⴱ + ⌬q兲 − ␮eg共qⴱ兲 1 − S , − 2 S 25

where S is the Huang–Rhys factor of the mode. native notation reads

−

冋

Same convention as in Ref. 16.

共7兲

An alter-

␮eg共qⴱ + ⌬q兲 + ␮eg共qⴱ兲 具0e兩␮eg共q兲兩0g典 = e−S/2 , 2 具1e兩␮eg共q兲兩0g典 = − 冑Se−S/2

a

共8兲

␮eg共qⴱ + ⌬q兲 + ␮eg共qⴱ兲 2

册

␮eg共qⴱ + ⌬q兲 − ␮eg共qⴱ兲 . 2S

共9兲

Only in the case when the transition dipole moment does not change along the deformation does the second term in Eq. 共9兲 vanish, so that the intensity ratio in the Franck– Condon approximation is recovered and I共1e ← 0g兲 = S. I共0e ← 0g兲

共10兲

Otherwise, I共1e ← 0g兲 = Sf HT , I共0e ← 0g兲

共11兲

where f HT represents the Herzberg–Teller correction factor, which can be expressed by

f HT =

冋

␮eg共qⴱ + ⌬q兲 + ␮eg共qⴱ兲 兩␮eg共qⴱ + ⌬q兲 + ␮eg共qⴱ兲兩

−

1 ␮eg共qⴱ + ⌬q兲 − ␮eg共qⴱ兲 S 兩␮eg共qⴱ + ⌬q兲 + ␮eg共qⴱ兲兩

册

2

.

共12兲

The above expressions shall only include the part of the deformation ⌬q along the vibrational eigenvector. They have been evaluated under the assumption that the vibrational eigenvectors on the potential surfaces S0 and S1 coincide. We have checked that for modes with frequencies up to 900 cm−1, the eigenvectors in ground and excited state PESs are close to each other. On the other hand, in the range above 1100 cm−1, the eigenvectors in the excited potential are rearranged by substantial Dushinsky rotations. Nevertheless, the curvature of the ground state PES along the vibrational eigenvectors in the excited potential is only weakly affected, so that the usual form of wave function ␹0g as used in the above equations is still a reasonable approximation. The calculated harmonic frequencies, the associated Huang–Rhys factors, and the Herzberg–Teller correction factors are summarized in Table II. Figure 8 compares the experimental spectrum with a stick spectrum based on the contents of Table II. Since the intensity of the origin band and that of the band at 710 cm−1 have been measured under the same conditions in Fig. 6, we

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Staicu et al.

FIG. 8. 共Color online兲 Calculated stick spectrum of the S1共 1A⬘兲 ← S0共 1A⬘兲 transition of Bzf taking the Herzberg–Teller correction into account 关panel 共a兲兴. The band intensities are given with respect to the origin band. Experimental spectrum of Fig. 3共a兲 scaled so as to show the band at 710 cm−1 with its relative intensity as determined in Fig. 6 关panel 共b兲兴. In either spectrum, the vibrational shift is relative to the position of the origin band at 29 894.3 cm−1.

know that the ratio of their intensities is 0.25⫾ 0.01. This has allowed us to normalize the observed spectrum in Fig. 8. The observed ratio of 0.25 is to be compared with our calculated value of 0.252. It can be also noted that, in the excitation spectrum of Geigle and Hohlneicher,16 the strong band at 710 cm−1 has an intensity between 20% and 25% of that of the origin band. Considering band intensities, the agreement between observation and calculation is very good in the range up to 900 cm−1 above the origin band. In that range, the calculated band positions, actually unscaled harmonic frequencies, are in reasonable agreement with measurements, although the result is not as precise as expected for DFT calculations based on the B3LYP functional. Nevertheless, this calculation has allowed for a straightforward band assignment given in Table I. On the other hand, at first glance, the observed and calculated spectra do not coincide in the range beyond 900 cm−1 above the origin band. As the strong band measured at 1375 cm−1 has an intensity close to that of the calculated band at 1569 cm−1, it would be tempting to make a one-to-one assignment. The difference between their positions is so large, however, that such a correspondence cannot be established. In the range below 1100 cm−1, several weak bands could not be assigned to fundamental excitations of a⬘ modes despite the good quality of the calculations for that region. Some of these bands could be harmonics or combination bands. For instance, the bands with vibrational shifts of 702 and 1059 cm−1 could be respectively assigned to the second

J. Chem. Phys. 129, 074302 共2008兲

and third harmonics of the band at 349 cm−1. Other bands in that range, however, cannot enter any of these categories and, therefore, cannot be attributed to a⬘ modes. They do not seem to correspond to bands of the Bzf· Ar complex either, which would be revealed by a shift of −22 cm−1 relative to a monomer band. These bands may arise from second harmonics of a⬙ modes with a substantial change in their frequency after excitation to the PES representing S1. As they should have a rotational profile of C type, an analysis of their contours would contribute to verify this fact, but the weak intensity of the bands prevents such analysis. As briefly mentioned above, the accuracy of the DFT and TD-DFT vibrational frequencies we have obtained is inferior to typical applications of the same functional. For example, a deviation like the one between the measured frequency of 710 cm−1 and the calculated value of 666 cm−1, which is as large as 44 cm−1, represents more than 6% of the observed value. Presumably, these relatively large deviations result from the use of the geometry qⴱ, so that nonparabolicities of the excited PES can reduce the curvature with respect to the harmonic approximation applicable close to a potential minimum. When comparing our TD-DFT-based theoretical spectrum with the one that Geigle and Hohlneicher16 calculated using a semiempirical approach, we find that our result is in much better agreement with measurements in the range from the origin band up to 900 cm−1. Indeed, we have calculated an intensity ratio of 0.252 for the strongest band at 710 cm−1 relative to the origin band, whereas Geigle and Hohlneicher16 obtained a ratio of 0.071, to be compared with the measured value of 0.25. Moreover, our calculated position for this band, 666 cm−1, if compared to that obtained by Geigle and Hohlneicher16 共778 cm−1兲, is also closer to the measured value. On the other hand, their approach was clearly more successful for the modes with higher frequencies. For instance, in the case of the strong band we have measured at 1375 cm−1 and for which they reported a relative intensity of approximately 0.200 in their excitation spectrum, they calculated a relative intensity of 0.216 and a position of 1409 cm−1. The usually excellent quality of the vibrational modes resulting from DFT was one of our reasons for using this approach. Moreover, we did not have access to more advanced procedures for an optimization of the geometry and a calculation of the vibrational modes in the excited state. This has led us to examine what alternative approaches may be used in the future. Concerning the calculation of electronic states, we have already mentioned that the CIS technique had failed in describing the energetic ordering of the lower electronically excited states of Bzf. On the other hand, the semiempirical INDO or ZINDO 共Ref. 26兲 techniques are more reliable in that respect for PAHs in general, as observed, e.g., in the case of benzo关g , h , i兴perylene.8 As demonstrated by Grimme and Izgorodina,37 other techniques that perform significantly better than TD-DFT-B3LYP are the CIS共D兲 共Ref. 38兲 ab initio technique and its spin component-scaled variant SCS-CIS共D兲. In order to fuel the discussion, we have calculated the Sn=1,2 ← S0 excitation energies with ZINDO, CIS共D兲, and RI-CC2.39,40 The results are given in Table III.

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J. Chem. Phys. 129, 074302 共2008兲

Electronic spectrum of 2,3-benzofluorene

TABLE III. Calculated excitation energies and oscillator strengths for the lowest singlet electronic states of Bzf. The calculations have been carried out at different geometries indicated in the second column. Measured values are included for comparison. Level of theory

CIS共D兲/TZ CIS共D兲 / 6-31G共d兲 CIS共D兲 / 6-31G共d兲 CIS共D兲 / 6-31G共d兲 RI-CC2/TZ ZINDO TD-DFT-B3LYP/TZ TD-DFT-B3LYP/TZ Experimentalb

1

Geometry

q共S0兲 q共S0兲 S1 关CIS/ 6-31G共d兲兴 S2 关CIS/ 6-31G共d兲兴 q共S0兲 q共S0兲 q共S0兲 qⴱ

L a ← 1A

1

L b ← 1A

Energy 共cm−1兲

Strength

Energy 共cm−1兲

Strength

41 222 40 238 36 916 39 018 38 196 31 346 32 632 32 815 32 031

-a 0.3172 0.6399 0.2434 -a 0.3560 0.2176 0.112 0.056

35 310 34 923 33 687 33 591 34 694 30 095 33 108 30 431 29 647

-a 0.0234 0.0101 0.0301 -a 0.0116 0.0117 0.038 0.024

a

Not determined. Measured in Ar matrix.17

b

All calculations have been made at the geometry of the electronic ground state determined at the DFT-B3LYP/TZ level. The CIS共D兲 calculations have also been carried out at the geometries of the S1 and S2 states optimized at the CIS/ 6-31G共d兲 level. The calculations making use of the TZ basis set were carried out with the TURBOMOLE 5.7 package,41 while the others were performed with the 27 GAUSSIAN software. The ZINDO technique gives S2 ← S0 as the transition with the dominant LUMO← HUMO character, i.e., 1La ← 1A. It also yields a weaker oscillator strength for S1 ← S0 than for S2 ← S0, in agreement with the measurements of Banisaukas et al.17 The CIS共D兲/TZ and RI-CC2/TZ approaches, applied at the DFT-B3LYP/TZ geometry of S0, also led to the correct ordering of the excited electronic states, although the excitation energies are rather high. Lower energies are obtained when they are calculated with CIS共D兲 / 6-31G共d兲 at the geometries of S1 and S2 optimized at the CIS/ 6-31G共d兲 level. The oscillator strengths were not determined in the CIS共D兲/TZ and RI-CC2/TZ calculations. As already mentioned, the TD-DFT-B3LYP/TZ calculation carried out at the ground state geometry gives contrary results. Moreover, as reported in Sec. III and shown in Fig. 1, the optimization of the geometries of the excited states with TD-DFT do not lead to improved results. Therefore, it appears that approaches based on the CIS共D兲 and RI-CC2 techniques are advantageous as far as the ordering of the lowest transitions of PAHs is concerned, but the calculation of the vibrational modes with these methods would require increased numerical effort. Finally, from the comparison of the oscillator strengths in the optimized ground state geometry and at the reference geometry qⴱ reported in Table III, it is clear that the ratio of the oscillator strengths of the S1 ← S0 and S2 ← S0 transitions is around one-third at qⴱ only, in reasonable agreement with the transition strengths observed on Bzf in solid Ar.17 This feature of the TD-DFT calculation at qⴱ, together with the correct spacing between the PESs of S2 and S1, contributes to the quantitatively meaningful Herzberg–Teller corrections found at this specific reference geometry.

V. CONCLUSIONS

In this paper, we have presented new results on the absorption spectroscopy of Bzf for the transition from the ground state, S0, to the first electronically excited singlet state, S1. The experimental studies were conducted in the gas phase using CRDS in combination with supersonic jets. The position of the S1共0兲 ← S0共0兲 transition was located at 29 894.3 cm−1 共334.5 nm兲. The matrix shift for argon was determined to be around −240 cm−1. Based on several measurements, a small feature redshifted by 22 cm−1 from the origin band could be assigned to the origin band of the Bzf· Ar complex. More precisely, we assigned this feature to the origin band of the S1 ← S0 transition of the complex. The measured shift is remarkable if compared with those determined for other PAHs presenting structural similarities with Bzf, namely, tetracene, fluorene, and anthracene, for which the redshifts are 41, 44, and 40 cm−1, respectively. In spite of the difficulties caused by the small energy difference between the S1 and S2 states, we have obtained a reasonable theoretical absorption spectrum based on a TDDFT approach applied to a slightly modified molecular geometry. The Herzberg–Teller contribution has been taken into account in the calculation of the intensities. Although the agreement between the theoretical and observed spectra is very good only for vibrational modes with frequencies lower than 900 cm−1, we consider our calculations to be the best approach to an ab initio study realized for Bzf until now since only parametrized force fields had been used before.16 Of course, the rearrangement of the lowest two transitions in accordance with the experiment cannot be considered to be fully ab initio because it would have been impossible without knowing the measured transition energies. On the other hand, in cases where the calculated difference between the lowest excited PESs S1 and S2 gives a fair representation of the experimental situation, the expression we have derived for calculating the Herzberg–Teller correction of the vibronic intensities can be applied directly to the deformation between the optimized geometries of the PES related to the excited state S1 and the ground state S0. Concerning Hartree–

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074302-10

Fock-based schemes such as CIS共D兲 and RI-CC2, we have found that they determine the correct ordering of the lowest transitions of Bzf, in agreement with previous studies for other PAHs.37 Our construction of a deformed geometry qⴱ with a reasonable separation of the lowest two excited states was a key ingredient for quantitatively meaningful Herzberg–Teller corrections of the vibronic intensities of the S1 ← S0 transition. This success indicates that adiabatic TD-DFT is quantitatively reliable for several quantities involved in this calculation, including the gradient of the separation between the PESs S2 and S1, together with the admixture of the stronger S2 ← S0 transition dipole to the smaller S1 ← S0 transition dipole for the deformation pattern arising between the S1 and S0 geometries. The principal failure of common functionals for the rather ionic 1La ← 1A transitions has been ascribed to the wrong asymptotic behavior of the most common exchangecorrelation functionals, and a promising remedy may be based on functionals such as LB94 with the correct Coulomb asymptotics,42 which lead to a significant amelioration of transition energies close to the ionization threshold.24 However, to the best of our knowledge, this functional has not yet been applied to an analysis of the ordering of 1La and 1Lb states in polyaromatic molecules. Instead, the BLYP and B3LYP exchange-correlation functionals were corrected in their asymptotic behavior to a −1 / r dependence, but the application of the resulting functionals to Bzf did not correct the wrong ordering of the lowest two calculated transitions.17 Two further strategies for possible improvements in TD-DFT are worth mentioning. First, time-dependent current-density-functional theory43 as applied to polymers44 might reduce the deviations of the calculated 1La ← 1A and 1 Lb ← 1A transition energies. Second, different variants of TD-DFT such as the Tamm–Dancoff approximation instead of the adiabatic approximation seem to reduce the systematic error of the 1La state of polyacenes, but the influence on the overestimated 1Lb energies remains rather small.45 Our results should motivate the testing and improvement in exchange-correlation functionals applied to TD-DFT-based calculations and the development of procedures to calculate vibrational modes in electronically excited states with these techniques, including Herzberg–Teller couplings. ACKNOWLEDGMENTS

This work was supported by a cooperation between the Max Planck Institute for Astronomy and the FriedrichSchiller-Universität Jena as well as by the Deutsche Forschungsgemeinschaft 共DFG兲 in the framework of the Forschergruppe Laborastrophysik. The authors are grateful to K. P. Geigle for fruitful discussions and for communicating calculated semiempirical intensities as well as to M. Vala for providing his computational results regarding the vibrational energies of Bzf in its electronic ground state. We wish to thank M. Schreiber for giving us the opportunity to use the computational facilities at Technische Universität Chemnitz. 1

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