The Jahn–Teller effect in triptyceneAlan Furlan, Mark J. Riley, and Samuel Leutwyler Citation: The Journal of Chemical Physics 96, 7306 (1992); doi: 10.1063/1.462434 View online: http://dx.doi.org/10.1063/1.462434 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/96/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Relativistic E × T Jahn–Teller effect in tetrahedral systems J. Chem. Phys. 129, 224102 (2008); 10.1063/1.3035189 Jahn–Teller effects in the coronene anions and cations J. Chem. Phys. 110, 249 (1999); 10.1063/1.478100 The Jahn–Teller effect: An introduction and current review Am. J. Phys. 61, 688 (1993); 10.1119/1.17197 A trimer vibronic coupling model for triptycene: The Jahn–Teller and Barnett effects J. Chem. Phys. 98, 3803 (1993); 10.1063/1.464009 On the Jahn–Teller and Pseudo‐Jahn–Teller Effect J. Chem. Phys. 51, 3129 (1969); 10.1063/1.1672466

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The Jahn-Teller effect in triptycene Alan Furlan, Mark J. Riley, and Samuel Leutwyler Institutfur anorganische, analytische und physikalische Chemie, Universitat Bern, Freiestr. 3, CH-3000 Bern 9, Switzerland

(Received 7 January 1992; accepted 21 January 1992)

The irregular vibronic structure resolved in the SI <-So resonant two-photon ionization (R2PI) spectrum of supersonically cooled triptycene (9, 1O-dihydro-9, 10 [ 1 '2'] benzenoanthracene) is assigned in terms of a single-mode E' ® e' Jahn-Teller vibronic Hamiltonian for the excited state, with linear and quadratic coupling terms. The Jahn-Teller active vibrational mode is a benzene wagging framework mode. To fit to the observed vibronic levels yields a very low frequency V e, = 47.83 cm- I and linear and quadratic terms are k = 1.65 andg= 0.426. This fit accounts for :::::98% of the observed absorption band intensities over the observable range 0--350 cm - I. The quadratic term is unusually large, leading to localization of the lowest vibronic levels in the three symmetry-equivalent minima. Emission spectra from 13 vibronic levels in the excited E' state show extended vibrational progressions with up to 25 members in the analogous e' ground state vibration, which is highly harmonic in the electronic ground state. The Franck-Condon factors of the fluorescence emission spectra calculated with the E' state Jahn-Teller parameters fitted to the absorption spectrum also yield a quantitative fit to observed emission intensities. The eigenvectors of the E' state vibronic levels are hence determined to great precision; the lowest five can be classified as radial oscillator and/or hindered rotor states, while higher levels have mixed character. Several eigenvectors are strongly localized in the upper sheet of the adiabatic Jahn-Teller surface, corresponding to "cone" states.

I. INTRODUCTION

In degenerate molecular electronic states, nonadiabatic (or non-Born-Oppenheimer) coupling arises due to the nonseparability of the electronic and nuclear motions. This coupling of degenerate electronic and vibrational wave func­tions is known as the Jahn-Teller effect, and much of the relevant theory was worked out more than 30 years ago. 1-3 Spectroscopic manifestations of the Jahn-Teller effect were quite rare for a long time, but are now well known in metal trimer molecules [Li3 (Refs. 4-6), Na3 (Refs. 7-10), CU3 (Ref. 11), Ag3 (Ref. 12), Al3 (Ref. 13), and Mn3 (Ref. 14)]. Manifestations of the Jahn-Teller effect in large mole­cules are still comparatively sparse; the experimentally best studied examples are the halobenzene cations C6H3F 3+ , C3H3CI3+' C6F3Br3+' and C6F/, 15 and Rydberg states of sym-triazine. 16

We have recently investigated the electronic spectrosco­py of jet-cooled triptycene (9,10-dihydro-9,1O[1',2']ben­zenoanthracene, for the structure, see Fig. 1) by resonant two-photon ionization (R2PI) and fluorescence emission spectroscopy. This molecule consists of three benzene (or strictly speaking, o-xylene) "monomer" chromophores fixed at 120· to each other by two bridgehead C-H groups. Together with molecules such as barrelene and p-cyclophane it is a fundamentally important system for the study ofinter­actions between equivalent chromophores. The first uv ab­sorption spectra measured by Wilcox 17 were compared to those of o-xylene, and the changes in transition frequency and oscillator strength were discussed in terms of the exciton model. The excitonic interaction of monomer excited states

results in A ; and E' excited states, an assignment followed by other authors l8.19 and which can also be derived from semiempirical calculations.20.21 The exciton model was also employed by Bersohn, Even, and Jortner (BEJ), who stud­ied the laser fluorescence excitation spectrum of jet-cooled triptycene. 18 On the basis of symmetry arguments they also concluded that the first spin-allowed electronic transition is I A ; -+ IE'. They also identified four different low-energy vi­brations with fundamental or overtone frequencies in the range 50--125 cm - t, and assigned three of these as Jahn­Teller-active e modes. Unfortunately, their reported spec­trum is not correct (see Sec. III B), invalidating their vi­bronic assignments.

As we show below, the E ' S I state oftriptycene turns out to be a textbook example for the study of the Jahn-Teller

Triptycene

FIG. L Chemical formula of triptycene,

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effect, for several reasons: (1) The observed absorption (R2PI) spectrum up to;:::::350 cm- l excess energy can be almost perfectly fitted with respect to frequencies and band intensities in terms of a single Jahn-Teller active mode, tak­ing only linear and quadratic Jahn-Teller coupling terms into account. Although there are several other e' modes which could couple to the E' electronic state, only one of these is active. (2) The electronic A ; ground state is nonde­generate and is highly harmonic along the isolated Jahn­Teller active e' coordinate, which is a benzene wagging mode. The ground state vibrational states are two-dimen­sional harmonic oscillator wave functions to an excellent approximation, simplifying the analysis considerably. (3) Van der Waals complexes triptycene' Rn with R = Ne, Ar, Kr, and Xe, and n = 1-3 can be produced in the supersonic expansion. These ada toms induce weak deformations of the intramolecular electronic surfaces. The Jahn-Teller coupled E I state is especially susceptible to external perturbations. For both n = I and n = 2, the deformations are oriented mainly along one component of the Jahn-Teller active pair of vibrational coordinates. Also, by increasing the size and polarizability of the rare-gas atom along the series Ne to Xe, the perturbation strength can be changed in a well-defined way. The results and analysis of these studies are presented elsewhere. n

In the following, we report the R2PI spectrum and sin­gle vibronic level fluorescence spectra from 13 excited-state levels (Sec. III). The theoretical analysis ofthe spectra (Sec. IV) was performed in terms of a single-mode E' ® e' surface, including linear and quadratic couplings. This surface is used to derive vibronic frequencies and intensities, i.e., an absorption spectrum from the A I vibrationless state, as well as a set of calculated fluorescence emission spectra from each of 13 E I state vibronic levels (Sec. III B). The agree­ment with the experimental absorption and emission spec­tral data is quantitative with respect to both frequencies (eigenvalues) and intensities (wave functions) for the low­est ;::::: 20 levels in the E I state, showing that the approxima­tions in the theoretical treatment are justified. The photoion­ization efficiency curves from three E' state levels are given in Sec. V; these confirm the expectation from semiempirical MO calculations~0.21 that the non degenerate ion ground state has similar geometry to the neutral ground state. Final­ly, \ve show the derived adiabatic surfaces and vibronic prob­ability densities, and discuss the extent of Jahn-Teller split­ting and localization in the 51 state oftriptycene as compared to other systems. 4

-lh

II. EXPERIMENT

Electronic absorption spectroscopy of jet-cooled tripty­cene was performed by two-color resonant two-photon ioni­zation spectroscopy (2C-R2PI). The molecular beam appa­ratus has been described elsewhere. 23 The conical nozzle has a 0.4 mm opening diameter, 600 inner angle and 2 mm length. Triptycene (Aldrich, 98%) was placed inside the nozzle, and heated to 150- I 90 °C. The carrier gases used were pure He, Ne, and Ar, as well as mixtures of 10% and 30% Ar in Ne, at backing pressures of 1-2 bar. Argon acts not only as a

coolant but also as a complexing agent, and the vdW com­plexestriptycene'Arn (n = 1-6) were also observed;22 com­pared to other substrates of similar size and weight, tripty­cene forms complexes very poorly. Neon is the best coolant, but a much weaker complexing agent, and only the vdW complexes with n = 1,2 were observed22 ). No helium com­plexes with triptycene were observed.

Two independently tuneable dye lasers provided light for the resonant 50 ..... 5 1 excitation in the region VI ;:::::36300 cm - 1 (frequency-doubled Fluorescein 27 and Coumarin 153) and for ionization in the range V 2 = 28 000 ± 1000 em -1 (DMQ). The excitation dye laser pumped by the sec­ond or third harmonic output of a Nd:Y AG laser Quanta­Ray DCR2A, whereas the ionization dye laser was pumped by a XeCI excimer laser Questek 2220. The bandwidths were 0.3 cm- l (FWHM) and the pulsewidths ;:::::6 and;::::: 10 ns, respectively. The laser frequencies were calibrated using op­togalvanic transitions in Ne* and Ar*, and converted to vacuum cm - 1. A Questek 9200 synchronization unit re­duced the relative time jitter between the Nd:Y AG and ex­cimer laser pulses to ± 1 ns, using optical feedback signals from both lasers. R2PI spectra could be recorded with the excitation laser alone, but the two-color experiment proved to be much more sensitive: since the first electronic transi­tion of triptycene is intense, with an osciIIator strength /;:::::0.05,17 excitation laser intensities> 10 kW Icm2

(;::::: 10 ,uJ Ipulse) already lead to saturation of the optical transition. The excitation laser beam was expanded 5 X along the mo­lecular beam direction using anamorphic prisms, and pulse energies were < 5 ,uJ. Due to the low cross section for the ionization step, high ionization energies could be used ( 1000 kW /cm2, 2 mJ/pulse). No ion signal was obtained with the excitation or the ionization laser alone. The ionization ener­gy hV2 was adjusted so that the excess energy was < 500 cm - I. The two laser beams were aligned almost parallel and created an excitation/ionization volume of 1 X 5 X 5 mm\ with 1 mm in the direction of the ion flight tube. The tempo­ral delay between hVI and hV2 was adjusted to < 2 ns using fast photodiodes.

For photoionization efficiency (PIE) measurements the energy hVI was fixed toa resonance in the51 state and the ionization frequency scanned over the ionization threshold to higher frequencies. The PIE measurements are not cor­rected for the Stark shift induced by the source electric field of 158 V Icm; the true adiabatic IPs have ;:::::80 cm- l higher energy.

The 12C-triptycene mass peak (254amu) was well sepa­rated from the isotopes containing a 13C atom. Since the isotope shifts of the various possible 13C species are notice­able ( 1-2 cm - 1 ), R2PI spectra of pure 12C-triptycene were collected, resulted in narrower spectral bands. The recorded data were corrected for variations in the baseline (caused essentially by pickup of excimer laser electromagnetic noise during the first,us after each laser shot) and laser energies.

For dispersed fluorescence measurements, triptycene was excited on selected narrow bands at xl D;::::: 20 in front of the nozzle. Fluorescence was collected with a quartz lens, dispersed with a 1.0 m monochromator (SPEX 1407), and detected with a Hamamatsu R-928 PM. The monochroma-

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7308 Furlan, Riley, and Leutwyler: Jahn-Teller effect in triptycene

tor bandpass was 4-7 cm-' (FWHM). Decay curves were recorded with the same digitizer. The averaged data were transferred to a computer, where the integral over a chosen time was stored. All emission spectra were corrected for scattered light appearing at the excitation frequency. Excita­tion spectra were also recorded. A RCA 7265 photomulti­plier collected the undispersed emission. The fluorescence lifetime oftriptycene was determined as 'Tn = 20 ± 1 ns.

III. EXPERIMENTAL RESULTS

A. R2PI spectra

The R2PI S, ,,-So spectrum oftriptycene in the spectral region of the first electronic transition is shown in Fig. 2. Only weak spectra features were observed beyond 36 700 cm - " and none below 36 300 cm - '. The same spectral re­gions were also recorded with pure neon and helium as car­rier gases. The neon spectra have slightly sharper lines and exhibit some weak features that we attributed to tripty­cene' Nen complexes which dissociate to triptycene ions after ionization. The spectrum shown in Fig. 2 exhibits only bands which can be attributed to the bare triptycene mole­cule and not to van der Waals complexes with Ar. The R2PI spectra of the complexes triptycene· Ar n' with n = 1-6 are published elsewhere.22 To avoid cluster fragmentation and appearance in the triptycene mass channel, the ionization lase energy was adjusted to excess energies in the ion ground state Eexc < 500 cm - '. Further discrimination of Ar com­plexes was achieved by probing the gas pulses within the first 50 ps, where the highest concentration of noncomplexed triptycene was found. The spectrum hardly changed when the backing pressure was reduced from 2 to 1 bar.

All observed R2PI bands have widths of 1.0-1.4 cm - " essentially due to the rotational envelope, which causes a short tail on the high energy side of the peaks. The longest tails appear in the spectra recorded with helium, reflecting the relatively high temperature in the helium seeded molecu­lar beam.

d e

b

Ci) :t::: c: :::l

.ci .... ~

(ij c: en "w a f c: J2

TABLE I. Measured and calculated vibronic band positions and intensities in the SI - So absorption spectrum, relative to the SI - So electronic origin a at 36 319 em-I. The IT parameters are given in Table III.

Measured Calculated Assignments

Band AV ReI. Av ReI. label (em-I) intensity (em-I) intensity (nj) a (np,n,,) b

a 0.0 1.00 0.0" 1.00 (0,112) (O,Q)

b 32.8 3.96 33.1c 4.59 (0,5/2) ( 1,0) c 56.6 0.59 56.3c 0.18 (I, 112) (0,1) d 66.3 4.74 67.lc 5.57 (0,7/2) (2,0)

92.4 0.02 93.1 0.04 (1,5/2) (1,1) e \05.3 4.78 \06.3c 5.05 (2, 112) (3,0)

\08.3 0.Q2 (0, 11/2) (0,2) f 131.7 0.92 131.9c 0.89 (1,7/2) g 140.9 1.65 142,5< 0.98 (0, 13/2) h 149.7 1.66 150.3c 2.30 (3, 1/2)

169.7 0.21 169.4c 0.05 (0,3) j 175.5 1.22 174.3c 1.08 k 186.4 0.90 187.7c 0.38 I 193.6 1.61 192.7c 2.51 m 202.7 0.15 205.3c 0.07 n 213.5 0.26 0 218.0 0.27 216.9c 0.51 p 223.4 1.03 222.3c 0.79 q 231.4 0.71 231.3c 1.63

236.2 IE-3 r 238.1 0.30

249.2 0.05 s 257.1 0.66 257.0" 0.63

263.3 0.01 277.8 0.06 274.3 0.14

282.9 0.01 u 286.3 0.13 287.2 3E-4

294.6 0.Q3 301.2 0.Q3

y 303.7 0.14 w 322.0 0.19 x 361.5 0.\0

a Assignment with vibrational and pseudorotational quantum numbers n andj (see the text).

b Assignment with approximate radial and angular quantum numbers (np,n,,).

c Vibronie levels included for the fit of the parameters hv', k, and g.

k p

q s

x

36300 36400 36500 Frequency [cm -1]

36600 36700

FIG. 2. R2PI spectrum oftriptyeene in the region of the S,-So electronic origin. The carrier gas was 30% Ar!70% Ne at 2 bar backing pressure. The vibronie band labels a to x are referred to in the text, tables, and following figures.

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Furlan, Riley, and Leutwyler: Jahn-Teller effect in triptycene 7309

Bersohn et ai. (BEl) have previously measured the flu­orescence excitation spectrum of jet-cooled triptycene and analyzed this spectra range. 18 They assigned the SI +-So elec­tronic origin to the peak marked b in Fig. 2, and feature a as a hot band. We find clear evidence that the origin must be assigned to the band a. This follows from both the Jahn­Teller analysis presented in the next section, which shows excellent agreement with the experimental spectra if a is cho­sen as origin but fails if b is chosen, and the PIE curves shown and discussed in Sec. VI.

The bands numbered a, b, d, e, h, I seem to form an excited-state progression with a spacing of approximately 35 cm - I; the maximum Franck-Condon factor (FCF) is ob­served for v' = 3, indicating a sizable displacement of the potential surfaces. However, the spacings are quite irregular, ranging between 32.8 and 44.4 cm - I. Also, the positions and intensities of the remaining R2PI bands c, f, g, i, j, k, m, etc., do not allow a simple vibrational assignment. The band posi­tions are collected in Table I. We note that the band positions reported by BEl (Table I in Ref. 18) are incorrect.

Since there is considerable theoretical and experimental evidence for the assignment of the lowest electronic state of triptycene as E', an obvious explanation of the irregular vi­bronic structure is due to an E I ® e' Jahn-Teller effect, and this will be our basic hypothesis in the following. However, before discussing further features of the R2PI spectra we present the dispersed emission spectra which give the key to the understanding of the R2PI spectra.

:§" 'c ::>

€ ~

.i!-'u; c: .l!l .£

-1200 -800 -400 Displacement [cm -1]

FIG. 3. Single vibronic level ftuorescence emission spectra of jet-cooled triptycene ( 1.8 bar Ne) excited to levels a, b, c, and d. The frequency is given as displacements from the excitation frequency (see Table I). Note that the dominant ground state vibrational progression appearing in all spectra is in the same mode of 64 cm - 1 frequency. Other weaker ground state vibrations appear at 348 and 486 cm -I (cr. Table VI).

-1200 -800 -400 0 Displacement [cm -']

FIG. 4. Single vibronic level ftuorescence emission spectra of jet-cooled triptycene (1.8 bar Ne) excited to levels e, f, g, and h.

~ 'E ::>

.ci ~ c '00 c Q)

c

FIG. 5. Single vibronic level ftuorescence emission spectra of jet-cooled triptycene (1.8 bar Ne) excited to levels j, k, I, q, and s.

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7310 Furlan, Riley, and Leutwyler: Jahn-Teller effect in triptycene

TABLE II. Measured band positions and intensities of the dispersed flu-orescence spectra with the excitation energy fixed on a, b, c, and d. All ener-gies and intensities are relative to the corresponding excitation bands.

-Av ReI. intensityI/I(v" = 0) v' (em-I) a b c d

0 0.0 1.00 1.00 1.00 1.00 1 64.2 1.64 1.21 1.70 0.62 2 128.4 3.02 1.01 4.09 0.11 3 192.4 3.61 0.61 6.16 0.14 4 256.4 4.09 0.19 6.14 0.33 5 320.0 4.10 0.03 7.19 0.48 6 385.0 3.94 0.09 5.89 0.38 7 449.1 3.26 0.27 4.48 0.28 8 513.0 2.72 0.42 4.28 0.12 9 576.4 2.05 0.53 3.45 0.11

10 639.6 1.62 0.51 2.44 0.17 11 702.8 1.06 0.50 2.04 0.27 12 766.8 0.80 0.40 1.42 0.33 13 831.0 0.51 0.37 1.27 0.40 14 895.2 0.25 0.23 1.07 0.37 15 959.2 0.19 0.14 0.74 0.26 16 1023.0 0.13 17 1087.0 0.21 18 1149.5 0.14

B. Emission spectra

The emission spectrum shown in Fig. 3 (a) was obtained when exciting band a at 36 319 cm - I. Analogously, emis­sion spectra obtained upon excitation of subsequent vibronic levels are given in Fig. 3 (levels b-d), Fig. 4 (levels e-h), and Fig. 5 (levelsj,k,l,q,s). Since in all emission spectra the emis­sion starts at the excitation wavelength, we conclude that little or no relaxation takes place in the excited electronic state within Tie ~ 20 ns. Table II lists the band positions and intensities of the emission spectra from levels a-cl.

We discuss the emission spectrum from the SI origin first [Fig. 3 (a) ]. The following facts are unusual about this spectrum: (a) A progression with a low-frequency vibra­tional spacing of 64.2 cm - I is observed. This frequency is almost twice the average vibrational frequency in the excited state. (b) The spectrum is not mirror symmetric to the R2PI spectrum: the maximum intensity of the progression is at v" = 5, compared to Vi = 3, and the progression is exceed­ingly long, extending out to v" = 15. (c) The progression is extremely harmonic, the change of frequency' between the first and the tenth spacing being 0.2 cm- I

, equivalent to a 0.03% decrease per consecutive spacing. (d) The FC enve­lope is a simple Poissonian, corresponding to the expected emission pattern from a Vi = 0 level. (e) Transitions are ob­served predominantly to one ground state mode. A second ground state vibration appears weakly at 348 cm - I, and an extensive progression of the 64 cm - I mode ~ccurs in combi­nation with this mode.

The emission spectra from all subsequent excited-state vibronic levels from h up to s also show a dominant progres­sion in the same 64 cm - I ground state mode. The progres­sions are even longer than from level a, in some cases extend­ing out to v" > 30. The emission spectra from the strong

levels h, d, and e show an FC envelope which reflects one, two, and three nodes in the respective wave functions of these levels. Conversely, the emission spectrum from the weak excitation h shows an envelope which is very similar to that from level a. We also note that the emission spectra from the close-lying levels f, g, h are closely related to each other.

These observations imply that (a) the;::::: 35 cm -I excit­ed-state vibration corresponds to the 64 cm - I ground state vibration; (b) since the emission from all of the excited-state levels show related progressions in the same ground state mode, they must belong to the same excited-state vibrational mode; (c) the wave functions of the lowest five vibrational levels a-e are closely related with respect to their nodal pat­terns. The combination of closely related vibrational wave functions with irregular frequency spacings and intensity patterns is a typical attribute of vibronic coupling.

IV. THEORETICAL BACKGROUND AND CALCULATIONS

The E ® e Jahn-Teller system has been extensively stud­ied theoreticallyl-4.24-29 and in this section we merely em­phasize the features which are relevant to the present work. The original literature should be referred to for a more de­tailed description.

A. Symmetry considerations

In the D3h molecular point group of triptycene vibra­tions of e' symmetry will be Jahn-Teller active in electronic states transforming as either E I or E ". By symmetry argu­ments alone,25,26 it can be inferred that the vibronic wave functions will transform as irreducible representations of the C3v point group which is the homomorphic image of D3h in the e' vibrational space.26 This Jahn-Teller system can then be represented by a vibronic Hamiltonian which is identical to that for all molecular point groups which have C3v as a subgroup (Le., Td , 0h''')' By the epikernal principle2s the Jahn-Teller potential energy surface will have minima at C2v symmetry which, by making the association O'h (D3h )

-+ O'yz (C2v ), corresponds to minima along the following three directions of the two dimensional e' vibrational space:

QEx,

1 !3 -2"QEx +TQEY'

1 !3 -2"QEX -TQEY'

The dynamic vibronic wave functions, however, retain the full C3v vibronic symmetry. It is interesting to consider the form of the potential energy surface because of the effects that small perturbations, such as rare-gas complexation may have on it. Transitions from the vibronic ground state of the molecule (low temperature limit) transform as totally sym­metric and only A I -+ E vibronic transitions are allowed. Here capital italic letters are used to denote the irreducible representations of vibronic states. In addition, ifthe vibronic coupling is confined to first order only, then the vibronic states can be classified by a half-odd integral pseudorota-

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tional quantum number, j, and only the transitions AI-Eu1 = 1/2 are allowed.2

When second order vibronic coupling is included, only 2j(mod3) remains a good quantum number. The value 2j( mod3) = 0 corresponds to vibronic states of A I or A2 symmetry, while the vibronic states of E symmetry have 2j(mod3) = ± 1. The much weaker selection rule of the allowed vibronic transitions A)-+E means that many more lines will become allowed with second order vibronic cou­pling. Similar arguments hold for the case where an anhar­monic term is included into the vibronic Hamiltonian,26 and the same selection rules hold.

B. The vibronic Hamiltonian

With the advent of efficient diagonalization routines, the usual method of solving vibronic problems is the vari­ational procedure where the wave functions are expanded in a vibronic basis set and a secular matrix is diagonalized. The vibrational part of the basis is usually simple harmonic oscil­lator functions for which matrix elements are easily evaluat­ed. The eigenvalues are monitored at different levels of basis set truncation and can be obtained to arbitrary accuracy. For all molecules whose point group is a factor group ofC3", the general vibronic Hamiltonian to second order can be written for a diabatic electronic basis t/lx, t/ly as

H=Ho+H)T'

Ho =~(_~_~+X2+ Y2)I (2) hv 2 ax2 ayZ ' H)T J;; = k [ - Xu. + YUx ]

The symbols I, ux ' U z are the 2 X 2 unit and Pauli spin matri­ces, X, Yare the coordinates of the e' vibration Qx' Qy made dimensionless and k, g are the first and second order Jahn­Teller coupling constants, respectively. The Hamiltonian is in units of the harmonic frequency hv of the basis functions. The vibronic wave functions'll k are expanded in a basis set which is a product of two one-dimensional harmonic oscilla­tor functions ¢i> ¢j based at the eqUilibrium undistorted D3h geometry of the ground state

n

'11k = 2: [axij.kt/lAi¢j +ayij.kt/lY¢i¢j]· (3) i+j=O

Here t/l, ¢, and'll denote electronic, vibrational and vibronic states, respectively. For n levels of the vibrational basis in­cluded, the total basis size is N = (n + 1)( n + 2) and an N X N matrix must be diagonalized. Appropriate linear com­binations can symmetry block this matrix into smaller ma­trices of size

n(mod3) ~O n(mod3) = 0,

A IA2 : I,(n + l)(n + 2) I,n(n + 3), (4)

E: !(n + l)(n + 2) j(nz + 3n + 3),

where A I' A 2' and E are the irreducible representations of the C),. vibronic group. If the ground state is assumed harmonic,

then the basis functions in Eq. (3) are eigenfunctions for the ground state vibrational functions. The energies Ek and wave functions'll k of the excited state obtained from evalu­ating Eq. (2) in the basis of Eq. (3) are then used to con­struct a theoretical spectrum

N

I(E) = 2: [laxOO•k 12 + layOO,k IZ]8(E - Ek ), (5) k=1

Since the basis functions are centered at the ground state geometry, only the lower values of k contribute intensity to the spectrum, as only these levels have non-negligible a xOO or ayOO coefficients in their wave functions. This means that only a subset of the N eigenvalues need to have converged in the diagonalization of the N X N matrix. Such a problem is then ideally suited for diagonalization with the Lanczos rou­tine which is efficient at calculating a few eigenvalues/vec­tors of large matrices. 27 A basis size of n = 52 (N = 2862) was found to be sufficient for the parameter values used in the present study.

c. Unequal frequencies in the ground and excited states

The experimental results indicate that the vibrational frequencies in the ground and excited states are very differ­ent. There are two approaches that can be used to take this into account: The ground state wave functions, although harmonic, can be expanded in the different harmonic oscilla­tor basis which has the excited state harmonic frequency hv' and solved in a separate matrix diagonalization. The har­monically spaced eigenfunctions will be linear combinations of the basis functions. The spectrum can then be calculated from a (more complicated) expression analogous to Eq. (4) summing over both ground and excited state coefficients.

Alternatively, and the approach used here, is to keep the harmonic ground state with frequency hv" as a basis set, and therefore retain Eq. (4), and to scale the coordinates of the excited state Hamiltonian so that it is in terms ofthe coordi­nates of the ground state basis. 28

X' = ~ hv" X, r = ~ hv" Y. (6) h~ h~

The vibronic Hamiltonian is then given by

H~ _ I ( a2 a2 X'z Y'Z) (7 )

hv" -"2 - ax ,2 - ar2 + + , a

HE =Ho +Hh, (7b)

Ho = [~(_~_~)+ K;(X,2+ y'2)]I, hv" 2 ax'2 ar2 2

H' ~=k'[ -X'u + Y'u ] hv" Z x

+ g' [ - (X,2 - y,z)uz - 2X'Y'ux ]'

2 In the present case we wish to have the excited state with a harmonic frequency of hv' in the absence of a Jahn-Teller effect, but expressed in terms of the ground state hamonic basis which has a frequency of hv" - 64 cm - I. Both ground and excited state Hamiltonians in Eqs. 7(a) and 7(b) are in

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7312 Furlan, Riley, and Leutwyler: Jahn-Teller effect in triptycene

units of the ground state harmonic frequency hv". This is achieved by rescaling the coordinates to a larger size in the excited state, which has the effect of decreasing the potential energy and increasing the kinetic energy. Note that the ki­netic energy operator is the same in both tPA and tPE elec­tronic states.

The parameters of the scaled and unscaled Hamiltonian are simply related

k-- k g---g , _ ( hv' )3/2 , _ ( hv' )2 hv'" hv'"

K' =(~)2 2 hv"

(8)

By substitution, it can be seen that the vibronic Hamilto­nians in Eqs. (2) and (7b) are identical, however, ifEq. (7b) is used, only a single diagonalization is required, together with the simple Eq. (4) for calculating the spectrum. It should be stressed however that Eqs. (2) and (7b) give iden­tical answers, only the vibronic basis functions are different. Care must be taken, however, when relating the coordinates X', Y' to the original coordinates X, Yin Eq. (2).

D. Search strategy for best fit parameter

From the fluorescence spectra, which are almost per­fectly harmonic, the ground state vibrational frequency can be fixed at 64.2 cm -I. The most intense peak of the emission spectrum from the origin, shown in Fig. 3(a), is the fifth member of the progression. From simple semiclassical argu­ments, this gives the displacement of the minima of the excit­ed state potential energy surface with respect to that of the

ground state as Po = ~2' 5.5 - 3.3 in terms of the ground state coordinates. This can then be used as a first approxima­tion to relate the magnitude of the first and second order coupling constants.29

Ikl =Po[i-Igl]· (9)

The key to the present interpretation is recognizing that the weak excitation - 56 cm - \ above the origin "belongs" to the origin in the sense that approximately the same emission pattern is observed from both peaks. The levels forming the main excited progression, a, b, d, e, show nodes in the Franck-Condon envelopes of their emission spectra. These spectra can be interpreted as a projection of the excited state wave functions onto the ground state potential energy sur­face. The vibrational levels a, b, d, e can be assigned approxi­mate radial quantum numbers np = 0, 1,2,3 which are re­flected as the 0, 1, 2, 3 node pattern of their respective emission spectra. The small peak c at 56 cm - 1 is then as­sumed to have the same np ' but a higher angular quantum number n", than that of the origin (band a).

Experience with many spectra simulations showed that the second order vibronic coupling must be either weak or strong, the intermediate case producing many more spectra lines than observed experimentally. In the weak coupling limit, only lil = 5/2, 7/2 will mix into the ground vibronic state (lil = 1/2) to first order, as only /lj = ± 2 states will be directly connected by the g matrix elements. This was confirmed by calculating the expectation values of

[ - (X 2 - y2)uz - 2XYux ] (10)

in eigenfunctions of the linear J ahn-Teller E ® e case. The energy levels as a function of the first order Jahn-Teller cou­pling constant only are discussed below (Sec. V A). The only cases which reproduce the small 56.3 cm -\ peak at a position 10 cm- I below the np = 2 peak is when k:::;:0.5 (lil = 5/2) or k - 1.0 (lil = 7/2). Both these cases are un­tenable because (a) the fluorescence spectra, although hav­ing the correct node structure, are far too short; (b) the energy spacings and intensities of the calculated main excit­ed state progression is very different to that observed experi­mentally; (c) in each case both lines should be observed; (d) the higher energy spectral features beyond band d are not reproduced.

At intermediate values of g, states of different j values become mixed in higher order and many spectra lines be­come allowed, as there is a high density of "rotational" states due to the soft motion in the angular direction. At higher values of g, the spectrum becomes simple again as the barrier height between the three minima increases and vibrational levels become quantized within these minima. The frequen­cies within these minima are approximately given by29

hvp = hv'(1-lgl),

hv~ = 199hVp I. V' 2 -Igl (11)

These expressions are only very approximate as the potential is extremely anharmonic, especially in the angular direction. However it provides a starting point for further optimi­zation. With hvp - 33 cm -\, hv", - 56 cm -\ and the condi­tion given by Eg. (9) gives k = 2.3, g = 0.3, hv' = 48 cm - \. These parameters were then scaled using Eqs. (8) to the basis used in the Hamiltonian given in Eg. (7b). The spec­trum was then simulated using Eq. (4). It was found that by increasing g the main spectral features could be well repro­duced. A one-to-one correspondence between observed and calculated peaks allowed a least-squares refinement of the parameters hv', k, and g, which are given in Table III. The calculated stick vibronic absorption spectrum is shown in Fig. 6. Comparison with the measured R2PI spectrum, giv­en in the top half of Fig. 6, shows that the correspondence is excellent, with a few slight exceptions (cf. Fig. 2 and Table I), which will be discussed below. Figure 7 shows a perspec­tive plot of the adiabatic potential energy surface as a func­tion of the two Jahn-Teller-active coordinates Qx and Qy,

TABLE III. lahn-Teller parameters obtained from the experimental fit.

Equation (2) "Unsealed"

hv' = 47.83 em -Ia

k = 1.65b

g= 0.426"

Equation (7) "Scaled"

hv" = 64.2 em-I K; =0.555 k' = 1.061 g' = 0.236

a Frequency of unperturbed JT active mode e' in the IE' excited state. b Linear coupling constant, related to the Moffitt-Thorson parameter D by

k =,ffi5. <Quadratic coupling constant, is equal to b given in Ref. 24.

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Furlan, Riley, and Leutwyler: Jahn-Telier effect in triptycene 7313

e -"!. f--_-.J'_~ n;

~ e o

o 300 100 200 -1 Relative frequency [em ]

FIG. 6. Calculated stick absorption spectrum, compared to the measured R2PI spectrum. All observed bands a-s, except nand r, are assigned with quantitative agreement to the same Jahn-Teller active vibration e' with hI" = 47.83 cm- I, k = 1.65 and g = 0.426.

using the parameters given in Table III. Figure 8 is a cut through the adiabatic surface of Fig. 7 along one of the dis­tortion coordinates Qx' showing the Jahn-Teller stabiliza­tion energy, the localization energy, and the calculated posi­tions oflevels a-I. The corresponding quantities, which were derived from the parameters of Table III, are given in Table IV.

V. DISCUSSION

A. Comparison of calculated and measured spectra

The measured R2PI spectrum is compared to the calcu­lated absorption spectrum in Fig. 6. A complete listing of the experimental band positions and intensities is given in Table I. Table I also includes all 27 calculated E' symmetry vi­bronic levels with energies up to 300 cm - I above the origin. For higher levels the harmonic basis should be further ex­panded to obtain reliable level positions. All bands experi-

FIG. 7. Plot of the adiabatic potential energy as a function of the Jahn­Teller active coordinates Q, and Q" using the potential energy terms ofEq. (2) and the parameters of Table III. The energy intervals are 10 em-I, where the energy is relative to the conical intersection at Qx ,Q, = 0,0.

200

E .£100

-100

-4 0 4 displacement along Q x

FIG. 8. Cut through the adiabatic surface shown in Fig. 7 along the distor­tion coordinate Qx' The calculated energy levels (with reI. intensity> 15%, see Table I) are shown. The Jahn-Teller stabilization energy is the differ­ence between the conical intersection and the global minimum in this plot; the localization energy is the difference between the two minima.

mentally observed are reproduced by the calculation, with the exception of the five weak excitations labeled as n, r, v, W,

and x. They probably belong to other e' or a; modes and combinations thereof. The emission spectra from these bands should be completely different from emission spectra from the e' vibrations presented in this work. Unfortunately, these bands are weak, and emission difficult to measure. The 17 calculated vibronic levels with intensities > 3% relative to the origin, which are marked in Table I, were used for the least square refinement of the parameters hv', k, and g. We note that the band intensities as determined by R2PI are accurate to ± 20%, depending not only on the FC overlap between So and S, but also on the ionization efficiency. As will be shown in Sec. VI the ionization efficiency in the case oftriptycene is not constant over the range of the spectrum; there is a stepwise increase of efficiency at higher excitation energies, the step position depending on the ionization ener­gies, which is held fixed during the R2PI spectrum. A com­parison of R2PI spectra recorded with different ionization energies and an fluorescence excitation spectrum showed that the intensity differences are < 20%.

The calculated emission spectra for emission from levels a-d (Fig. 9), levels e-h (Fig. 10), and levels j-l, q, and s (Fig. 11), are in very good agreement with the measure­ments, even for the highest measured vibronic levels. The calculated band positions and intensities for emission spec­tra from levels a-d are given in Table V, to be compared to

TABLE IV. Quantities derived from the parameters of Table III.

EJT = 113.4 em-I Eloc = 67.77 em-I

Po = 2.87 p, = 1.16 3a=0.93cm- 1

Jahn-Teller stabilization energy Barrier height, localization energy Radial position of the minima Radial position of the saddle points Tunneling splitting

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7314 Furlan, Riley, and Leutwyler: Jahn-Teller effect in triptycene

a 1I11II111

II1III 11II

11111111

d 111111 I I III II I I I I

-1200 -800 -400 0 Displacement [cm -1]

FIG. 9. Calculated stick fluorescence emission spectra from levels a-d in the 'E' state, using the parameter set given in Table III, and assuming a single active e' mode in the I A I state. Note the excellent agreement between these calculated and the experimental emission intensities and positions in Fig. 3 for all four emission spectra.

e

h

FIG. 10. Calculated stick emission spectra from levels e, C, g, and h, calcu­lated as for Fig. 9. Note again the excellent agreement between theoretical and experimental e' mode intensities (shown in Fig. 4) for all four emission spectra.

", "111111111111111111111111

k " , 1111111111111111111111 111111

!B

: 1~.<...L.1.' I 1-LLJ1 I 1.-L-1-1 I L.J....LI I I-LLJII 1--,-,-1 1 II....LJ..I 1-'--'--11 I '..J...J...J,I 11....LJ..1 I LL.J.I I '...L...I...JIII

c Q)

1;;

q

.:, "111111'1",,1,,, 1111111111111111,

-2000 -1600 -1200 -800 -400 0 Displacement [cm "1

FIG. II. Calculated stick emission spectra from levelsj, k, I, q, and s, calcu­lated as for Fig. 9, to be compared with the experimental data in Fig. 5.

TABLE V. Calculated band positions and intensities of the fluorescence emission spectra with the excitation energy fixed on levels a, b, c, and d. All energies and intensities are relative to the corresponding excitation bands. The energies are multiples of the harmonic ground state frequency 64.2 em-I.

-~v ReI. intensity III(u" = 0) u' (em-I) a b c d

0 0.0 1.00 1.00 1.00 1.00 1 64.2 1.64 1.06 1.77 0.52 2 128.4 2.99 0.74 10.3 0.05 3 192.6 3.40 0.35 19.2 0.\3 4 256_8 3.84 0.07 18.0 0.30 5 321.0 3.54 om 24.0 OAO 6 385.2 3.20 0.09 17.7 0.27 7 449A 2.58 0.22 18.6 0.16 8 513.6 2.02 0.35 12.9 0.06 9 577.8 1.47 OA1 1\.3 0.07

10 642.0 1.04 OA2 7.85 0.12 11 706.2 0.70 0.38 6.17 0.20 12 770A 0.46 0.32 4.26 0.26 \3 834.6 0.29 0.25 3.11 0.29 14 898.8 0.18 0_19 2.18 0.28 15 963.0 0.11 0.13 1A5 0.25 16 1027.2 0.07 0.09 0.96 0.21 17 \091.4 0.04 0.06 0.63 0.17 18 1155.6 0.02 0.04 0.40 0.12

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Furlan, Riley, and Leutwyler: Jahn-Teller effect in triptycene 7315

the experimental data in Table II. The intensity variations of the calculated emission spectra from the levels 1, q, and s in Fig. II exhibit somewhat larger deviations from the mea­sured data. The deviations could be caused by the following effects: (a) Anharmonicity which was not included in the calculation. Although the JT -quadratic coupling has a simi­lar effect on a linearly JT-perturbed potential surface as an­harmonicity, namely, warping of the sombrero potential, the surface could still be improved by including a cubic term in the Hamiltonian. (b) OtherJT-active (e' ) and inactive (a;) modes in the vicinity of the JT vibronic bands can have a dramatic impact on position and intensity by mode-mixing, as has been shown by Meiswickel and Koppel. 30 Possible candidates for mixing are the levels corresponding to bands nand r, which lie at 213.4 and 238.1 cm - I. Band n is the first unexplained peak in the R2PI spectrum, and the calculated intensities of the close-lying levels 0, p, and q are less good than the others. Also, the second vibrational mode in the electronic ground state is observed at 211 cm - I, very close to the frequency value oflevel n in the excited state, indicating that both may be due to the same vibration.

In the last column of Table I we give assignments for a number ofvibronic levels in terms of approximate radial and angular quantum numbers np ' n",. These numbers are easily found from the probability density functions of the different excited state levels, shown in Fig. 12. The radial quantum number counts the radial nodes between p = ° and p = 00,

and nq> counts the angular nodes between, e.g., - 60° and 60°. Consequently, band b corresponds essentially to a vibra­tion where the amount of static distortion is increased and reduced. Band c corresponds to a hindered pseudo rotation. If the localization barrier separating the three potential minima were very high these would be the two normal modes resulting from the static symmetry lowering D3h -.C2v (e' -.a l + hi)' The intensities of the vibronic bands depend on the probability density near p = ° in the excited state. For example, the wave function of the level (np,nq» = (1,1) has almost no density near p = 0, thus a low FC overlap with the v" = ° wave function. Therefore, the transition So(v" = 0) --,SI (1,1) has very low intensity. The level d, on the other hand, corresponds to the second radial excitation (0,2), and has large excited state density nearp = 0.

Besides np and n", a second assignment is given in Table I in terms of the pseudorotational quantum number j; j is a good quantum number only if the quadratic coupling con­stant g is zero. The number n( = 0,1,2, ... ) labels levels with the same value of j in ascending order. It becomes a good quantum number in the free rotor limit: g = ° and k --. 00.

The energy spacings of a linearly JT-coupled potential sur­face is shown in Fig. 13(a). The right hand side of the dia­gram approaches the radial vibrator/free rotor limit. All sol­id lines converge to rotational levels belonging to n = 0, the dash-dotted lines belong to n = 1, and so on. The calculated k value for triptycene is marked by a vertical bar. With this k value fixed the energy levels are represented as a function of the quadratic coupling constant g in Fig. 13(b). Again the value found for triptycene is marked by a vertical bar. The pseudorotational quantum numbers given in Table I were

determined by tracking back the energy levels from g = 0.42 to O. For such strong quadratic coupling as in our case, though,j does not remain a good quantum number, because wave functions belonging to different (nJ) values mix too strongly.

Note that the sign of the coupling constants are undeter­mined in fitting the present experimental data and both here are taken as positive using the phase conventions given by Eq. (2). To determine their absolute signs, the form of the vibrational coordinates must also be defined. These signs do not effect the present treatment except that if the product k· g is negative, the contour plots in Fig. 7 should be reflected in the Qy axis and the AI' A2 symmetry labels for the vibronic energy levels in Fig. 13 should be reversed.

The splitting due to tunneling between the three equiva­lent minima is the frequency difference between the lowest e' and a I vibronic states calculated from the above Hamilto­nian as av = 0.93 cm -I, equivalent to a pseudorotation time of 36 ps. This internal rotation time is commensurate with external rotation frequencies, and strong Coriolis couplings may be expected.

For the illustration of the coupling constants found in the calculation the adiabatic surface is shown in Fig. 8 as a cut through the surface along the Qx distortion coordinate (the component of symmetry type a I in C2v )' The localiza­tion energy is the difference between the energies of the two minima in the cut surface, and the JT stabilization energy is the energy difference between the global potential energy minima and the conical intersection.

B. Shape of Jahn-Teller active vibration

While the analysis presented above is highly consistent, it does not yield any information on the shape, i.e., the vibra­tional displacements, of the J ahn-Teller active vibration. As a first approach, a normal coordinate calculation was made for the ground IA ; state. This is not trivial, due to the large size of the molecule (96 normal modes); first attempts to model the intramolecular force field via known force con­stants of structurally similar but smaller hydrocarbons gave ambiguous results. Thereafter the semiempirical MOPAe 6.0

method using the AM I Hamiltonian34 was used: A full structure optimization was followed by a normal coordinate analysis. A complete discussion of the results and compari­son with ab initio results together with the analysis ofIR and Raman spectra will be given elsewhere. 35 For the interpreta­tion of the present results, we note the following: (a) The lowest-frequency normal mode V96 indeed turns out to be­long to the e' representation, with a calculated frequency V<)6 = 75 cm -I. This is :::: 15% higher than the observed fre­quency, which seems acceptable in view of the known range of deviations of AMI frequencies from experiment by ± 15%.36 The semi empirical calculation thus also provides

a qualitative identification of the JT-susceptible vibration. (b) The vibrational coordinates < and e; of V<)6 are shown in Fig. 14. The displacements are large-amplitude angular or wagging motions of the benzene rings around the innermost ring C-C bond. This coordinate is qualitatively similar to the e' vibrational coordinate of the archetypal M 3 (M = Li, N a,

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7316 Furlan. Riley. and Leutwyler: Jahn-Teller effect in triptycene

(al (bl

(el (d)

FIG. 12. Calculated probability density functions of the five lowest energy levels a--e. The node structure clearly shows the suitability of the __ -.1!_I_ I ___ ~ •• - ~ Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.14 On: Fri, 04 Nov

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(e)

FIG. 12. (Continued.)

(a)

4

3 ,.....,

B

o 2 3

k

Furlan, Riley, and Leutwyler: Jahn-Teller effect in triptycene 7317

1/2

1/2

4 5

-1 75 em

FIG. 14. Calculated pair of normal vibrations ex (left) and ey (right) of the lowest-frequency e' mode. The calculated frequency V<)o = 75 cm - '.

K, etc.) molecules. (c) The next-lowest normal modes be­long to thea;', e", a~, and a; representations. None of these modes is allowed as a single-quantum excitation in the emis­sion spectrum. (d) The next allowed vibration is an e' mode calculated to lie at V91 = 331 cm - I, well separated from the lowest-frequency mode. We tentatively assign the ground­state excitation experimentally observed at 348 cm - I to this mode (cf. Fig. 3 and Table VI). (e) From the shape of the vibration we conclude that a high sensitivity of both ground­and excited-state Vw} vibration frequencies to external sol­vent perturbations can be expected.

(b)

4

3

-, 2

'j,-----

a

.. · ........ A, A2

- E

0

0.0 0.1 0.2 0.3 0.4 0.5 0.6

g

FIG. 13. The calculated energy levels of the E ® e Jahn-Teller problem. The vertical lines show the values obtained from fitting the experimental spectrum, k = 1.65 andg = 0.426. Only transitions given by heavy lines are allowed from the ground electronic state. (a) The energy levels as a function of k. The levels can be given the exact quantum number j. Levels with the same n are shown by the same line pattern. Only transitions into iii = 1/2 levels are allowed from the ground state. (b) The energy levels as a function of g for fixed k = 1.65. The levels with 2j(mod3) = 0 are split into states of A, and A2 symmetry. Only transitions into the E levels (solid lines) are allowed from the ground state.

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7318 Furlan, Riley, and Leutwyler: Jahn-Teller effect in triptycene

TABLE VI. Listing of weak ground state vibrations measured in emission spectra different from the 64.2 em - , progression.

Ground state energy (em -')

211 276 348 486 655

C. Cone states

Observed in emission spectrum

s q a,b,d,e,g,h d b,d-h

Speculation about the existence of long lived states lo­calized within the upper "cone" of the adiabatic potential energy surface dates back to early Jahn-Teller studies. 37.38

More recently, an interesting comparison has been made24 of plots of eigenvalues vs the linear coupling constant obtained from both exact-nonadiabatic and approximate-adiabatic calculations. In these plots, regions were identified where the levels could be associated mainly with either one or the other adiabatic potential energy surface. In the case of those levels mainly associated with the upper adiabatic potential energy surface, the calculated wave functions show a high degree of localization within the upper cone. This method of finding cone states in the linear E ® e problem was found to work better for high values of the half-integral angular momen­tum quantum number j (see Fig. 8 in Ref. 24). This is under­stood as the states being centrifugally stabilized37 by avoid­ing the inner region of the cone where the two potential energy surfaces, in the Born-Huang adiabatic approxima­tion, become very close. Recently, 16 it has been stressed that the localization of the wave functions is never complete in

these cone states and this is important in model studies of nonradiative decay.39,4o

A well localized cone state has been calculated for the E ® e Jahn-Teller system in a Rydberg state of sym-tria­zine16 where the parameters were obtained from fitting the lower levels to an experimental spectrum. The cone state itself was not observed, however. These workers also con­cluded that second order vibronic coupling would in general disrupt such localizations. It comes as some surprise then to find in the present system that the calculated wave functions for several energy levels observed in the experimental R2PI spectrum show a high degree oflocalization. Notwithstand­ing the large second order vibronic coupling required to sim­ulate the spectral features, the wave functions of bands 1, s, and to a lesser extent q, are largely localized within the upper cone. The second order coupling may actually promote the experimental observability of these states, by mixing wave­functions with higher values of j into the observed states. These higherjlevels are normally forbidden for E ..... A transi­tions in the linear E ® e case.

The absorption spectrum is actually an ideal method of detecting the cone states as their maximum probability den­sity lies directly above the ground vibronic state density maximum. There is some experimental confirmation for the localization of these states as there is a large intensity maxi­mum in the Franck-Condon envelope near the origins of the LIF spectra, i.e., for Llu = 0, - 1, - 2 from bands I and s. To our knowledge, this appears to be the first direct experimen­tal evidence of cone states in Jahn-Teller systems.

D. Comparison to other systems

Table VII compares different molecular systems exhi­biting JT distortions. For the halobenzene cations, studied

TABLE VII. A comparison of Iahn-Teller parameters for different molecules (the ground state values ofLi and Na trimers are calculated).

Electronic Ref., hv(e') System state method" (cm-') k g EJT/hv E,,,,jhv

s-C6H3C13+ X 2E" 15b 417 1.11 0.005 0.62 0.006 C6F 6+ X 2E,g 15b 425 1.17 0.005 0.68 0.007 s-C6F3Br,+ X 2E" 15b 260 0.75 0.75 S-C6H 3F ,+ X 2E" 15b 475 0.80 0.017 0.80 0.011 s-triazine 3s 'E' 16b 661 2.14 0.046 2.40 0.21

CU3 X2E' l1(f)b 132 1.56 0.27 1.68 0.72 11 (g)b 137 1.86 0.22 2.23 0.81

Li3 X2E' 4" 278 1.96 0.22 2.46 0.89 41" 250 2.25 0.14 2.94 0.72

Na3 X2E' 42" 86 3.72 0.11 7.78 1.52 41" 98 3.89 0.21 9.57 3.32 43" 50 19.2 5.50

B 2E'(+2A;) 8b 127 4.04 om 8.26 0.20 42(a)a 86 4.90 0.03 12.5 0.84

Mn3 X 14b 130 0.21 0.02

Triptycene 'E' b 47.8 1.65 0.43 2.37 1.42

" Parameters from ab initio calculations. b Parameters fitted to spectra.

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Furlan, Riley, and Leutwyler: Jahn-Teller effect in triptycene 7319

by Sears et ai,,15 all four e' vibrations were analyzed, and non-negligible mode mixing was found. We reproduce only the parameters for the vibration V6 which causes the largest JT distortion. The triptycene IE' state exhibits the lowest Jahn-Teller mode frequency and the largest localization en­ergy (in units of hv) of all the systems measured so far. It is also remarkable that in all other organic molecules the qua­dratic coupling does not have a strong influence on the spec­tra, in contrast to triptycene. On the other hand the ground states of metal trimers seem to exhibit much stronger linear and quadratic coupling. Li3, Na3 as well as triptycene are examples of statically distorted JT-potential surfaces, i.e., Eloc/hv> 1. We note that the large second order coupling constant, g, found in the present study, may be partially due tothe 1 E' electronic state coupling with a higher IA ; state. A first order pseudo-Jahn-Teller coupling between E and A states can lead to an effectively larger value of g than if the E state is considered in isolation. This possibility is currently under investigation. We note that an 1 A ; state is predicted to be close lying by the exciton model,17-19 but quite far re­moved (1-2 eV) by semiempirical calculations.2o,21 An analogous (A \B E) ® e vibronic coupling case has also re­cently been postulated for Na3•

41-43

VI. PHOTOIONIZATION EFFICIENCY CURVES

The PIE curves shown in Fig. 15 were recorded with the excitation laser fixed on the bands a, b, c, and d in Fig. 2. The energies in the PIE curves are the sum of the excitation and the ionization energies. All PIE curves show a remarkably sluggish onset, especially when compared to the onsets of

~ a 'r: :J

.e ~

ro c Ol 'iii b c .2

c

d

FIG. 15, Photoionization efficiency curves obtained via excitation at the 1 E' levels a, b. c, and d.

planar aromatic molecules: the increase of the ion signal from 0% to 90% takes place within 500-1000 em - I, as com­pared to 5 em - I in naphthalene32 and 25 em - I in phenanth­rene33 (b) and 33(c) (excitation at the og band in both cases). This slow onset in triptycene is an indication for a large geo­metric change between the excited SI state and ionic state. The onset is, however, much steeper in the curves band d than in the curves a and c. Another conspicuous feature in all curves is the stepwise increase of the ion current after every 30-40 em - I, and large steps after every 200 em - 1. This re­flects the vibrational structure in the ion state of triptycene. The increase at each step is given by the Franck-Condon overlap between the vibrational levels in the excited and the ion state. We suppose that the 30-40 em - I steps belong to a progression of the analogous e' mode in the ion state. In curves a and c the steps 3-5 are the largest, whereas in curves band c the first two steps are the largest. This behavior is a mirror image of the corresponding emission curves: the lar­gest FCF from levels a and c are at v" = 5, from levels band d at v" = 0 and 1. This means that the first ion state oftripty­cene is not degenerate and therefore not Jahn-Teller distort­ed. From simple MO calculations one expects that the first ion state has symmetry A ;, since one electron is removed from an a; molecular orbital,Z°·21

As has been mentioned in Sec. V, the fact that the curves a and b have no relative shift proves that the level a at 36 319 cm- I in the R2PI spectrum cannot represent a hot band but can definitively be assigned to the electronic origin of the SI excited state.

VII. CONCLUSIONS

The main results of this work can be summarized as follows: The first singlet excited state of triptycene exhibits a strong Jahn-Teller effect of the E' ® e' type. Both the linear and quadratic coupling terms are relatively large, being k = 1.65 and g = 0.426. The observed levels can be fitted with a single-mode Jahn-Teller Hamiltonian including only linear and quadratic coupling terms. Due to the large qua­dratic coupling, there are three marked minima on the lower sheet of the adiabatic Jahn-Teller surface. Five vibronic lev­els lie below the conical intersection, which is 113.5 cm- I

above the potential minimum, and the lowest two levels even lie below the localization energy, which is 68 em - I. By the usual definition,29 the 1 E' state of triptycene exhibits a static Jahn-Teller effect.

The first vibronic excitation of 32.8 em - I (band a) cor­responds essentially to a radial vibration within the wells of the lower adiabatic sheet of the JT surface; other almost purely radial vibrations, with corresponding radial quantum numbers up to np = 4 can also be assigned. The second exci­tation at 56.6 em - I can be viewed as the first angular or hindered pseudorotation level, and analogous angular vibra­tion levels are assignable up to n", = 4.

Using the JT potential parameters fitted to the absorp­tion spectrum, the IE' state eigenvectors were calculated, as well as the Franck-Condon factors for fluorescence emis­sion. The calculated spectra were compared to the set of experimental emission spectra from 13 e' vibronic levels, which extend over a ~ 1200-2000 em - I frequency range.

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7320 Furlan, Riley, and Leutwyler: Jahn-Teller effect in triptycene

Quantitative agreement was found for the emission spectra from the first nine levels (bands a-j) and good agreement was observed for the next four levels (bands k,l,q,s), proving the very high quality of the derived JT parameters.

Based on the calculation of the eigenvectors, several of the higher excited levels were identified as cone states, i.e., states whose wave functions are strongly localized within the upper sheet of the adiabatic surface. The experimental prop­erties associated with the cone states are ( 1) relatively high intensity in absorption from u" = 0, since the wave functions of these state lie vertically above the ground state wave func­tion, and (2) high intensity for transitions near the elec­tronic origin in emission.

Using the semiempirical MOPAC 6.0/ AMI method to calculate the intramolecular force field in the 1 A; ground state, followed by a normal mode analysis, the lowest-fre­quency mode of this molecule was identified as an e' mode, calculated to be 75 em -1, in qualitative agreement with the observed value of 64.2 em - 1. The mode is essentially a wag­ging motion of the benzene rings. The coupling of this mode to the electronic excitation can be viewed in terms of an ex­cimer-type approach of two of the benzene rings in the IE' state.

ACKNOWLEDGMENTS

We thank P. Fluckiger and Professor J. Weber for per­forming the MOPAC calculation, and Dr. R. Knochenmuss, Professor H. B. Burgi, and Professor H. U. Gudel for stimu­lating discussions. This work was supported by the Schweiz. Nationalfonds under Grants Nos. 20-28995.90 and 20-29585.90.

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