The vibronically induced phosphorescence in benzene

Chemical Physics 175 ( 1993) 245-254 North-Holland

The vibronically induced phosphorescence in benzene

Boris F. Minaev Department of Chemistry, Cherkassy Engineering and Technological Institute, 257006 Cherkassy, Ukraine

Siiren Knuts Department of Quantum Chemistry University of Vppsala S-75120 Vppsala, Sweden

Hans &ren and Olav Vahtras Institute of Physics and Measurement Technology. University ofLinb”ping, S-58183 Linkilpng, Sweden

Received 30 March 1993

The phosphorescence spectnun of benzene is particularly rich in vibronic structure and provides the best-known example of a spectrum of electronic transitions that by symmetry is both spin and orbital forbidden. Such transitions, only allowed through the couplir~ of nuclear and electronic motions, are notoriously difficult to analyze both on theoretical and experimental grounds. We investigate the vibroniadly induced phosphorescence by means of m&Lconfiiumtion quadratic response theory akulations and expbre vibronic intensitic~, polarization directions, transition moments for benzene phospboresance and the radiative lifetime of triplet benzene. We find that the radiative decay of the 3BB,, state takes place predominantly through vibronic coupling among tbee%C-Cstretchingmodesvs=1601 cm-‘andu 9= 1178 cm-‘, with a close to complete out-of-plane polarization. The calculations predict relative intensities of different vibronic bands in good went with experiment. The oscillator stmngth for the 3Blu+1A,, absorption is predicted to 0.74~ 10-*“, to be compared with the experimental value of = 10-lo. The computed aver- aged radiative lifetime falls in the interval of 22 to 96 seconds, depending on the quality of the basis set, with the best prediction being 64 seconds.

1. Introduction

The characterization of singlet excited states and fluorescence spectra of benzene has been a popular target for semi-empirical and ab initio oriented quantum methods alike. The theoretical investiga- tions of triplet spectra and the phosphorescence of triplet benzene have, on the other hand, so far been conducted only by means of semi-empirical schemes [ 1-9 1. This mismatch of theory and calculation between singlet and triplet spectra is ironic from the point of view that the latter are the result of a much more complicated interaction, involving simultane- ous dipole and spin-orbit coupling, and in the particular case of benzene by the fact that it is necessary to account for vibronic coupling as well. This weak “third-order” nature of the benzene phosphorescence interaction leads to some computational and experimental diffkulties. Even the assignment of the

first excited triplet state in benzene was the object for some controversy, although the now established assignment of a 3Bl,, state [ lo,11 ] coincides with the original assignment, the one suggested by Goeppert- Mayer and Sklar [ 12 ] already in 1938. Basic to the benzene phosphorescence problem is the fact that the L 3Bi,-+~ ‘All transition is doubly forbidden by spin and orbital symmetry (O-O transition is missing). Perturbation due to spin-orbit coupling alone can therefore not provide electric dipole intensity for this transition without involving vibronic interactions. Results from semi-empirical calculations of benzene phosphorescence have not considered all channels nor all the interacting electronic states and have given rather different values for the radiative lifetimes. The experiments have been undertaken with different glassy media or with benzene isolated on crystal ma- trices and have not been completely unambiguous concerning the triplet state lifetime, quantum yield

0301-0104/93/$06.00 6 1993 Elsevier Science Publishers B.V. All rights reserved.

246 B.F. Minaev et al. /Chemical Physics I75 (1993) 245-254

and polarization of radiation. The question of the true radiative (phosphorescent) lifetime of benzene has therefore been considered open, although one seems to agree upon a “best experimental” value that focus on 30 s [ 9,13 1. Because of the obvious importance of benzene in theoretical organic chemistry it is desira- ble to investigate its singlet-triplet spectra from first principles. In the present work we address the lifetime of triplet benzene by means of ab initio calculations that employ a recently developed multi-configuration quadratic response theory method [ 141. We perform direct calculations of vibronic phosphorescence in order to explore the origin and importance of various mechanisms and vibronic channels for the lifetime of triplet benzene and for the polarization of its radiative decay.

2. The benzene phosphorescence problem

The vibrational analysis of the benzene phosphorescence bands indicate that the radiative activity is induced predominantly by eg vibrations [ 15,161. A weak but observable activity of b4 vibrations has also been found [ 15-l 7 1. By introducing spin-orbit and vibronic coupling through second order perturbation theory Albrecht [ 21 showed that the vibronic interaction within the triplet manifold is responsible for the major part of the phosphorescence intensity. This also follows from comparison of the vibrational structure in phosphorescence and fluorescence spectra [ 18 ] . The benzene phosphorescence spectrum in rigid glasses [ 15 ] reveals a dominant vibronic activity of the v8 (1601 cm-‘) and v9 (1178 cm-‘) ezs vibrations. The vg ( 608 cm- * ) mode is very active in fluorescence ( ‘B2,+ ‘A,,), but appears only weakly in the phosphorescence spectrum [ 10,15,17,18] (here and throughout the paper we use the Wilson numbering of benzene vibrations [ 19 ] ) . The key idea for the understanding of these features is connected with the pioneering work of Moffitt and Liehr on “pseudo-Jahn-Teller” interactions between rr-lr+ states induced by different eg vibrations [ 20,211. Moffitt noted from symmetry considerations that C- C stretching vibrations va (odd in Moflitt’s classifi- cation) are most effective for coupling between B,, and E,, states, whereas the benzene skeletal bending mode v6 (even) is most effective in B,,-E,, mixing.

Apart from the geometry distortion problem [ 111, these ideas explain well the observed vibronic fluorescence 1B2,+1Als and phosphorescence 3Bl,,+1Alg spectra. The first one borrows intensity from a very strong *El,- ‘A,, transition (lE,,- ‘BzU mixing by vg mode ) , the second one borrows from the spin-forbidden 3ElU-1All transition, which becomes allowed by the account of spin-orbit coupling (SOC) [ 2,451. The importance of 3E,U+3B1U mixing by vs vibrations is substantiated by the very characteristic 3BlUt1A10 absorption spectrum [22] and its Zee- man effect [ 231. An intense doublet at 239 and 252 cm-’ above the O-O transition in the absorption spectrum of low temperature crystal and also the 250 cm-’ vibronic band in O2 induced Tit&, absorption were attributed to the v8 vibration, the frequency of which is drastically reduced in the 3BlU state in comparison with the ground state value ( 1601 cm-’ ) [ 221. This is a consequence of the above- mentioned “pseudo-Jabn-Teller” interaction; the potential energy surface (PES) in the benzene lowest triplet state (T, ) is extremely flat and has few shallow minima, slightly distorted ( x0.02 A) from the Dsh conformation [ 11,24 1. We suppose that this dif- ference in PESs of S,-, and Ti states does not influence much the vibronic structure of phosphorescence spectrum described on the basis of vibrations of the So state alone. This follows from two arguments: (i) The vibrational wavefunction of the zero-point level demonstrates that the molecule in the Ti state is not confined to the minima but makes large excursions over the complete well [ 111. The free molecule thus appears hexagonal but even a weak crystal field can explain the miscellaneous host-dependent effects in ESR [25,26] and ODMR spectra [ 10,271 ‘of low temperature doped crystals and glasses: (ii) In spite of the large differences in vs vibrations in the upper and lower states (including the anharmonicity ) [ 111, the normal modes are qualitatively similar. We shall use the ground state vibrational analysis for the spin- vibronic treatment of the benzene phosphorescence. The Tr state is vibrationless during the phosphorescence decay at low temperature, so the ground state vibrations determine the frequencies of the phosphorescence vibronic bands and we suppose that they to a large extent determine the spin-vibronic perturbations, which govern the intensity.

Account of Moffitt’s idea together with an effec-

B.F. Minaev et al. /Chemical Physics I 75 (1993) 245-i-254 247

tive one-center SOC approximation leads in the framework of this approach to the consistent inter- pretation of the benzene phosphorescence spectra by the mechanisms represented in fig. 1. This is a summary of all schemes considered by Albrecht [ 2 1, and is supported by CNDO/S-CI calculations of vibronic and spin-orbit coupling [ 8 1. The major part of the phosphorescence intensity is determined here by channels I + II; this radiative activity induced by the ezs (mainly v8) modes, is determined by one-center SOC integrals and by intensity borrowing from high energy x-o excitation of ‘AI,-‘A2,, and ‘El,-‘Ei, types, which are out-of-plane polarized [ 7,8 1.

Albrecht’s analysis indicated that the strongest eze phosphorescence bands in rigid glass at 77 K are uni- formly polarized out-of-plane with approximately 60- 70% of the total intensity [ 2,28 1. The microwave induced delayed phosphorescence (MIDP ) experiments for a C6H6/CsDs doped crystal at 4.2 K shows that more than 90% of the ep phosphorescence must be out-of-plane polarized [ 10 1. This discrepancy re- lates to the relative importance of the channels I + II and III, see fig. 1. The last channel [ 51 as well as channel V [ 29 ] does not include one-center SOC in-

Mechanisms for phosphorescence of benzene:

tegrals and is supposed to be less effective [ 2 1. The estimations of SOC with one- and two-electron parts ofthe SOC operator [5,29] (3EI,]H,]‘E,,)=0.41 cm-‘, together with the account of o orbitals as sp2- hybrides for the simple construction of the o, ti ‘A2,, state wavefunction [ 51, leads to comparable intensity for the in-plane (channel III) and out-of-plane (channel II) polarization for the phosphorescence [ 2 1. This confers with the old photoselection experiment on benzene glass at 77 K [ 281, but contradicts the later MIDP experiment at 4.2 K [ 10 1, which must be considered more reliable (the glass results are af- fected by rotation of the excited molecules before de- caying [ lo] ). Up to now this contradiction has not been resolved theoretically. Recent semiempirical calculations within the n-approximation with an account of the complete form of the SOC operator [ 301 seem to support a large in-plane polarization charac- ter for aromatic phosphorescence. The other unre- solved question is connected with the role of the e2,v, mode in the benzene phosphorescence spectrum. Be- sides us it also bears considerable activity [ 15 1. The v9 vibration, however, is not expected to be active in the coupling between 3B1u and 3E1u states [ 8,201 and

Vib+c coupling

v: lAIItMMX.+ ‘E,+---H,@,~)-- +-1B2u- z

@Yv,5 H”-%

VI: lA a@-- ‘Blu

Fig. 1. Six main mechanisma of benzene phosphorescence.

248 B.F. Minaev et al. /Chemical Physics 175 (I 993) 245-254

its frequency does not change much upon &,-T, excitation [ 221. The other possible routes in the “pure vibronic + !!IOC” second-order perturbation theory, involving vibronic mixing in the singlet manifold (like scheme V) seem to be unimportant. The path which involves the intermediate ‘Ez,, states can only lead to in-plane polarized b, bands and are considered to have a negligible influence [ 8 ] (qualitatively it is very similar to path IV).

The b4 vibration v4 (707 cm-‘) appears weakly [ 15,161 and shows predominantly in-plane polarization of phosphorescence emission [ 21. CNJX/S- CI calculations predicted a rather weak SOC and vibronic coupling for the first 1 XAAzU state, involved in route IV, which corresponds to an order of magni- tude smaller intensity for the b, vibronic phosphorescence band in comparison with the e, activity [ 8 1. This result is proved to be artificial, because the electric dipole transition moment ( ‘AtiIMI ‘A,,) was underestimated, and vibronic coupling was overesti- mated by the CNDO approach as follows from recent calculations [ 3 11. The other bD vibronic (5: ) transition (us= 1005 cm-‘) is overlapped by the more intensive ( 8: ) band and was not analyzed before M.

These and the other mechanisms are computed and discussed below.

3. Colnpntations

We consider the electronic Hamiltonian in the zero- order Born-Oppenheimer (BO ) approximation,

H=I& +&,, (1)

and use a quadratic response theory approach [ 14 ] in which the SOC operator (H,), implemented in the complete Breit-Pauli form, it treated as a perturbation.

Both H,, and H,,, contain nuclear coordinates Q as parameters. The mixed spin multiplicity eigenfuno tions of H (as well as pure spin eigenfunctions of Ho) depend now on Q and this dependence is crucial for determination of the vibronic phosphorescence in-

*I We use the notation (?I$ ) of Calloman et al. [ 321 which in- dicates that for vibrational mode u.(a) the energy quantum is changing from v in the excited triplet state to v‘ in the ground state.

tensity. We can start with the approximate BO wavefunctions of the electronic Hamiltonian H,,

“@$,(r, Q> =‘%(r, QLG,(Qh d=w+ 1 , (2)

where A Yt( r, Q) and X,,(Q) are the electronic and the vibrational wavefunctions, respectively, at a given nuclear configuration (Q) , k is a spin sublevel and S is the total spin quantum number. We use the harmonic approximation and a standard Wilson treatment of vibrational normal coordinate analysis [ 19 1,

&p(Q) = ; xn,AQa) 3 (3)

where ~~((24) are the harmonic oscillator eigenfunctions, ,U is a vibrational quantum number, and (Y specifies the vibrational mode. For the BO wavefunction (2) the transition moment between the singlet ground state S, with v’th vibrational excitation ‘@J Q) and the first triplet Ti excited state with the vth vibrational level, 3@1,,(Q), is equal to

I’@:,&dQ

‘WC Q>

Here M and Mo are electronic and nuclear electric dipole operators, respectively. Accounting that the singlet and triplet eigenfunctions (2) of the zero-order Hamiltonian ( 1) are orthogonal to each other at any Q, the second term in the sum (4) is zero and we can rewrite (4) as

M<qk = I &A&(Q~&,, dQ 3 (5)

where M&r(Q) is the electronic !I,,-T, transition dipole moment for the kth spin sublevel.

M,,(Q)= j “yo(r,Q)M3’J’f(r,Q) dr. (6)

To the first order of the nuclear displacement we can expand this transition moment in a series

@J(Q) =~M2o) + C dX2a 9 (7) a

where

B.F. h4inaev et al. /Chemical Physics 175 (I 993) 245-254 249

6 G= ,[, ‘%uo(r, QW 3 Yf(r, Q) dr

1 a0

= &- WkdQ)lao-o. (8)

Finally from eq. (5) the intensity of the vibronic phosphorescence band is determined by

W$=&,T~QO) 1 &,v&,~dQ

+G I xs,, Q2G.v dQ . (9)

In a simple harmonic approximation (3) the first term gives the O-O band if the transition is allowed for non-distorted symmetrical structure. But in the case of benzene phosphorescence the 3B,U+‘Als transition is forbidden in Dsh symmetry even if the total Hamiltonian ( 1) is taken into account. In this case only vibronic phosphorescence bands ( U= 04 v’ = 1) appear through non-totally symmetrical b% and ep vibrations as follows from the scheme given in fig. I and as discussed in the previous sec- tion. Their transition moments are equal to (in au)

A4&=&2xv,)-“*. (10)

We do not consider the totally symmetrical a,, pro- gression in combination with eq modes nor other overtones because their intensities are the subject of Franck-Condon analysis, which is beyond the scope of the present paper.

To the first order of spin-orbit coupling perturbation ( 1) the S-Tk transition moment is equal to

MsT*(Q)= c (‘~~l~I”Y,)(‘~l~~13~~) s ‘~3 <Q> - 'Es(Q)

+ c (‘~l~~13~~)(3~~I~13Y”;) ‘C,(Q) - ‘G(Q) * (11) I

The normal coordinate dependence of this transition moment can be analyzed in different ways [ 21. The dependence of the electric dipole aIlowed S-S transition moments has often been accounted for by the Herzberg-Teller (HT ) expansion of vibronically mixed states [ 2,8 1. Little is known about the dependence on & of the SOC matrix elements in large molecules. In diatomics this dependence is often rather strong [ 8,331. But for polyatomic hydrocar-

bons one can suspect a somewhat smaller dependence. Albrecht [ 2 ] analyzed the spin-vibronic perturbation of the form (U-J,,&&) qualitatively and pointed out that its matrix elements are negligible. Instead of applying the HT perturbation theory we prefer to evaluate the nuclear coordinate dependence for the whole expression ( 1 I ) directly, and apply for that purpose the multi-configuration quadratic response (MCQR ) theory method of Vahtras et al. [ 14 ] which forms a generalization of the theory of Hettema et al. [ 341 when there is no longer permutation symmetry in the two-electron integrals and when the op erators may both have singlet and triplet symmetry. It goes back to the formalism for quadratic response functions presented in 1985 by Olsen and Jorgensen [ 351. The MCQR method is implemented for MCSCF (multi-configurational self-consistent ) ap- proximated reference states; the reference MCSCF wavefunctions are either of complete or restricted active space types [ 36 1. We refer to the original articles for different formal and computational aspects of MCQR theory. Here we only point out that with MCQR the full summations over states in eq. ( 11) is implicitly accounted for, which is an important fea- ture of the theory in particular when nuclear differentiation is carried out, since any truncation of these sums becomes size-inconsistent for different geometries. In the MCQR computations of the T,-S,-, transition moments ( 11) we consider the relevant es and bz, modes that by symmetry are active in the benzene phosphorescence. T1 represents here the first excited triplet state, 1 3B,U and So the ground state, 1 ‘A,,. The ground state force field, vibrational normal modes and frequencies have been obtained with MCSCF analytic gradient and Hessian calculations [ 371. Fre- quencies computed with the DZ basis set are compared with experimental ones in table 1. The T,-S, transition moments were obtained using distorted benzene geometries with atomic displacements along the normal modes, and the ones with the derivatives in eq. ( 8 ) obtained by numerical differentiation. The final formula for the radiative lifetime of the k spin sublevel produced by radiation in all ( VP) bands is (ZFS representation x, y, z is used [ 1 ] ) :

1 403 -=----c ]~::]z2nvLll, =k 3x1373 L1 (12)

where w= ‘E , - ‘~5, is the frequency of the transition.

250 B. F. Minaev et al. /Chemical Physics 175 (I 993) 245-254

Table 1 Relative intensities of different vibronic bands in benzene phosphorescence

Mode Wilson not.

Frequency (cm-‘)

v6.P. “c&

Intensity (%)

exp. [ 171 DZ DZR TZ TZR

bz, ~4 707 738 2.2 3.1 3.6 3.8 2.2 v5 1005 1064 5.0 4.8 7.5 9.7 6.4

eo ~6 606 602 2.1 0.3 0.1 0.3 0.1 v1 3057 3374 0.2 0.2 0.4 0.1 0.1 vs 1601 1726 65.8 64.5 61.9 54.3 58.4 vg 1178 1293 24.7 27.3 28.3 32.2 33.0

The average lifetime T,, at the fast lattice relaxation limit is given by 3/7,,= Ck 1 /rb

The basis sets employed in the MCQR calculations are the same as in previous calculations on singlet and triplet spectra of azabenzenes [ 38,391. These are the double Z (DZ ) and triple Z (TZ) Gaussian basis sets of Dunning [ 40,411 augmented with Rydberg-like functions (R) DZR and TZR, expanded at the center of symmetry #*. The quadratic response calculations were performed for various complete x active spaces, tested e.g. in ref. [ 3 11, see also table 2. Re- sults in tables 2-4 are shown for a complete 1c active space comprising l-2a2,, l-2e2”, l-2b2, and l-2e,, orbitals correlating all II electrons. The out-of-plane b, vibration breaks the u-x separability. The differ-

entiation along this mode has been carried out also with an active space that includes the la,,, lezr and 1 e is highest occupied and the 1 ezu, 1 e,, and 1 bzo lowest unoccupied orbitals.

All calculations have been carried out using the SIRIUS/ABACUS program package for multi-configuration self-consistent field (MCSCF) wavefunctions, linear and quadratic response functions [ 14,34,42,43] and geometry related properties [ 371.

4. Results and discussion

s* The Rydberg exponents are: s: 0.005885, 0.002040, p: 0.010265,0.003487; d: 0.014470.

The results of the present computations are presented through tables 1-5. The calculated and observed frequencies of the ezs and b, modes, which are relevant to the benzene phosphorescence problem are represented in table 1. This table gives also

Table 2 Spectral properties of states relevant for benzene phosphorescence obtained from response calculations at experimental equilibrium geometry for DZ, TZ, DZR and TZR basis sets (R=Rydberg functions). f; denote oscillator strengths (length form) for the ‘El, state. US-spaces: IL4= la,,, lo,,, lb%, l-2eis; IIB= l-3a2., lesv, lb* l-3eid IIC= l-2a,,,, I-Ze,, l-2bw l-2e,,

Level g.s. energy

(au)

Triplet states Singlet states ‘El. f;

‘B,” ‘El” ‘&U ‘Ak ‘El”

exp. 3.66 ‘) 4.54 b’ 6.93 c’ 6.93 c, 7.59 *’ 1.25 c) DZ-CAS(IL4) -230.7197 3.38 4.88 9.41 9.69 8.35 1.418 DZ-CAS(I-IC) -230.7255 3.46 4.89 9.42 9.69 8.20 1.432 DZR-CAS( IIB) -230.7255 3.45 4.90 7.25 7.28 8.19 1.360 TZCAS(IL4) -230.7504 3.32 4.84 8.85 9.07 8.21 1.352 TZR-CAS(IIB) -230.7558 3.39 4.86 7.25 7.28 8.06 1.256

a1 Ref. [I]. “)Ref. [44]. “Ref. [45]. d)Ref. [46]. e)Ref. [47].

B.F. Mnaev et al. /Chemical Physics 175 (1993) 245-254 251

Tabe13 Ttansition moments (au) calculated with direct vibronic coupling along e, modes. Results are displayed for different basis sets and for the active space described in text. Excitation energies (eV) are given for 0.2 au mass-weighted displacements form computed equilibirum along the respective normal mode

Level Excitation Transition moment Osc. strength % energy M” f;

mechanisms I + II e2s ug( 1601) DZ(IIB) 3.34 2.86x 1O-5 ~~(1178) G(606) v,(3057)

3.38 3.38 3.38

1.86x 10-S 1.90x 1o-6 1.49x 10-6

0.73x lo-‘0 0.31 x lo-‘0 0.03x lo-” 0.02x lo-”

~s( 1601) DZR 3.34 2.41 x 1O-5 0.52x lo-i0 ~~(1178) 3.38 1.63~10-~ 0.24x IO-‘O U6(606) 3.39 9.91x10-’ 0.01 x lo-” v,(3057) 3.39 2.04x 10-6 0.04x lo-”

~s(1601)TZ 3.28 1.27x 1O-5 0.15x10-‘0 vg(1178) 3.32 9.81 x 1O-6 0.09x lo-‘0 V6(606) 3.33 6.27x lo-’ 0.04x 10-12 u,(3057) 3.33 5.26x lo-’ 0.03x lo-‘2

vs( 1601) TZR 3.29 1.62x 1O-5 0.24x lo-lo vs(1178) 3.32 1.22x10-5 0.13x10-‘0 Q(606) 3.33 1.27x lo-’ 0.01 x lo-‘3 v,(3057) 3.33 5.94x 10-r 0.03x lo-‘*

mechanism III ew vs( 1601) DZ 3.34 1.48x 1O-6 0.39x 10-12 vs(1178) 3.38 1.86x lo-’ 0.06x lo-” v6(m) 3.38 1.64x lo-’ 0.04x 10-13 v,(3057) 3.38 1.53x 10-s 0.04x lo-‘5

~~(1601) DZR 3.34 1.46x 1O-6 0.38x lo-‘* vs(1178) 3.38 5.95x lo-’ 0.06x 1O-‘2 V6(606) 3.39 2.11 x 10-7 0.08x lo-” u,(3057) 3.39 9.48x 1O-8 0.02x lo-‘3

us(1601)TZ 3.28 1.15x10-6 0.24x 1O-‘2 ug(1178) 3.32 3.65x lo-’ 0.12x 10-i) vs(606) 3.33 4.07x lo-’ 0.03 x lo-‘2 v,(3057) 3.33 1.80x IO-’ O.O~XIO-‘~

IJ~( 1601) TZR 3.29 1.14x 10-6 0.23x 1O-‘2 vs(1178) 3.32 3.54x 10-7 0.02x lo-‘2 V6(606) 3.33 4.15x10-’ 0.03x lo-‘* vr(3057) 3.33 1.93x 10-7 0.07x lo-”

1.05x lo-‘0 ~

0.76x lo-lo

0.23x lo-‘O

0.37x lo-‘0

0.40x lo-‘2

0.46x 1O-‘2

0.29x lo-‘*

0.29x lo-‘*

Q I( 3Bx 1 zI ‘A,,) 1 for mechanisms I +II; I (‘Bf, lx(y) I ‘A,,) I for mechanism III.

the relative intensities for the vibronic bands of benzene phosphorescence as calculated with different

sets. Tables 3 and 4 give transition moments, M, M,, and AI’,, and oscillator strengths for S,,+c transi-

basis sets. The important vibrational modes are v,, v,, 06, v7, us and v9 with the standard Wilson classifi- cation. Excitation energies of states relevant for benzene phosphorescence obtained from response calculations are given in table 2 for the different basis

tions induced by the eD and b, modes, respectively. The radiative lifetimes for the different spin sublevels, 7k, and the average lifetimes, T,,, at the high-temperature limit are presented in table 5.

The most intensive phosphorescence vibronic band is the (88 ) band in complete agreement with experi-

252 B.F. Minaev et al. /Chemical Physics 175 (1993) 245-254

Table 4 Trarmition moments (au) cakxlated with direct spin-vibronic coupling along ba modea. Results am displayed for difkzwt basis sets and for the active space dcscxibed in text. FhAations energies (eV) are &II for 0.2 au mass-weighted dkpkements from computed equilibrium along the mspective normal mode

mechanism IV b,

Level

~~(707) DZ(lT) vs(990)

v,(707) DZR(i-I) vs(990)

v,(707) TZ(ZlI) rs(990)

v,(707) TZR(LlT) rs(990)

Excitation Transition moment A4 energy I (3WIx(~) I’&,) I

3.27 6.23x lO-6 3.27 7.75x 10-e

3.27 5.84x lO-6 3.27 7.99x lo-6

3.31 3.38x lO-6 3.31 5.38x lO-6

3.31 3.16x lO-‘j 3.31 5.35x 1o-6

Osc. strength I_& f;

3.48x lo-‘* 5.39x lo-‘* 8.87x lO-‘2

3.06x lO-‘2 5.72x lo-‘* 8.78x lO-‘2

1.02x lo-‘* 2.59x lo-” 3.62x lO-‘2

0.90x lo-‘* 2.57x lo-‘* 3.46x lO-‘2

Table 5 Lifetimes T(S) of the 3BB,, state for different basis sets and refer- ring to the four first mechanisms depicted in fig. 1. r, denotes the average lifetime at the high temperature limit. o and i denote out-of-plane and in-plane polarimtion, respectively

Level rZrt(o) rj$.(i) rz rY(i) r,,

DZ-CAS 16.4 194.2 15.1 4620 22.6 DZR-CAS 22.5 195.9 20.2 3779 30.3 T&CA!3 74.3 415.2 64.3 5803 95.9 TZR-CAS 46.6 497.0 42.6 5871 63.7

ments. It is interesting to note from table 2 that for all basis sets employed the movement along the vg normal mode produces a lowering of the excitation energy for the triplet state compared to the excitation energy at equilibrium ( 3.38 eV for the DZ basis set ) , while for all other e& modes the excitation energy is increased. This reflects the strong pseudo-Jahn-Teller effect along the VS mode, as predicted by Moffit [ 2 1 ]

and observed by van der Waals et al. [ 25 1, by Bur- land and Castro [ 22 1, and furthermore confirmed by ab initio calculations by Buma et al. [ 111. These results confirm that the lowest triplet state in benzene has few shallow minima on the potential energy surface around the hexagonal structure. The phosphorescence can be considered to start out from an effective Da symmetry of the T, state, because the nuclear motion in the zero-vibrational T, state is not confined to the trough. We have taken account of this fact in the present calculations. Correspondingly, the

ep C-C stretching mode, vg, appears to produce the most intensive phosphorescence band, with predominantly (92-98%) out-of-plane polarization in all basis sets. The other C-C stretching mode, 5, is also intensive in agreement with observations, see table 2. Wowever, it is not suspected to be active in pseudo- Jahn-Teller mixing (the v9 frequency is largely un- changed upon !$,+T, excitation, while the frequency of vg is changed drastically).

It is interesting to note that the strong intensity of the v9 mode is obtained by direct evaluation of the spin-vibronic activity along the ground state normal mode, without having to consider rotation of the modes (Duchinsky effect ) , as suggested by Buma et al. in ref. [ Ill. The latter authors also computed intensity ratios I( 89) /I( 8:) obtained in low-temperature “Shpolsky” experiments [ 17 1. Such ratios are, however, independent of spin-vibronic coupling, but depend on the proper form of the vibrational wavefunctions, and therefore on the Franck-Condon factors.

As seen in table 1 the relative intensities are very stable towards enlargements of the basis set, and they agree well with those experimentally measured. The strongest band is VS. In spite of the fact that the total intensity is rather basis set dependent, the intensity ratios for different vibronic bands are quite stable with respect to the refinement of the basis sets. The result for v6 deviates somewhat from experiment, which is not unexpected considering the very weak transition moments involved. The two b, vibronic

B. F. Minaev et al. /Chemical Physics I 75 (1993) 245-254 253

bands have comparable activity, v5 being somewhat more intensive than v, in all basis sets. These results are in good agreement with microphotometer record- ing of Shull [ 15 1. Albrecht [ 21 and other investiga- tors [lo] focused their attention only on v4 as the most prominent representative of bz, vibrations. This band (4: ) is well separated in frequency, while ( 5!’ ) is overlapped by the more intensive v8 ( 8’: ) band and appears only as a shoulder. However, the vs intensity, (59 ), is larger than that of v4 as documented by Shull [ 151 and by the %ru+‘AIL absorption measure- ments of Burland et al. [ 22 1.

From table 3 one notes that all ez, bands, except v6, are predominantly polarized out-of-plane. For v6 the in-plane and out-of-plane polarizations are approximately of equal strength in the TZ basis set, in the TZR basis set the in-plane polarization is even more intensive. This finding corresponds quite well with the microwave induced delayed phosphorescence (MIDP) experiment by van der Waals et al. [ lo] for C,jH,j/C,jDs doped crystals at 4.2 R.

As seen in table 4 the lifetimes for the spin sublevels rX and r,,, corresponding to out-of-plane polarization, are the shortest. In-plane polarized emission from the TTY spin sublevels, represented by the bz, vibrations, leads to approximately an order of mag- nitude longer radiative lifetime than for the eZll bands (mainly vg and v9). By contrast, the in-plane polarization from the the T; spin sublevel has negligible intensity, which means that this sublevel decays by very long radiative lifetimes (4000-6000 s for all basis sets). Because of these peculiarities the average lifetime, r,,, is approximately equal to 1.5r,.,,. All these features concord very well with results from the MIDP experiments.

The average phosphorescence radiative lifetime calculated with different basis sets varies between 23 (DZ) and 96 (TZ) seconds. The best quality basis set (TZR) gives 64 s. The so-called “best experimental” value is 30 s [ 91, but the proper estimation of non-radiative channels and the quantum yields still presents an open problem. From direct measure- ments of T1 +SO absorption in pure crystalline benzene by an impurity phosphorescence-photoexcita- tion method, Burland et al. [22] obtained an oscillator strength of x lo-‘O which leads to the estimated value of 52 s for the average lifetime. Al- though we find a lifetime of this order with the largest

of the wavefunctions employed we recall that the the oretical lifetime is the result of a delicate sum of different contributions from singlet-singlet and triplet- triplet o-x transition moments and excitation energies [ 8 1, which are supposed to be quite basis set dependent, expecially on the account of different Ryd- berg orbitals.

5. Summary

Using implementations of modem response theory we have made a detailed investigation of the vibra- tionally induced phosphorescence intensities in benzene. With the implicit state summations in perturbation series taken account of, the response calculations are capable of reproducing the relative phosphorescence vibronic band intensities quite well even for ground state wavefunctions optimized with moderate basis sets and correlating orbital spaces. We firmly establish that triplet state benzene depopu- lates through radiative emission that is polarized out- of-plane. This follows from the comparative strong vibronic coupling and thereby phosphorescence intensities induced by the e2# modes, in particular the vg and v9 modes. The in-plane polarization, represented mainly by b, vibronic coupling, is on the other hand weak, and contributes negligibly to the benzene phosphorescence lifetime. The intensity ratios for all vibronic bands are in very good agreement with experiment, and are not that sensitive to basis set. The radiation from the x, y spin sublevels (X and y are in- plane axes) are much more active than the z spin sublevel. Our calculations also manifest the pseudo- Jahn-Teller effect in the T, state induced by the ezs vg mode. Being spin and orbital forbidden the radiative decay of triplet benzene is very slow, the lifetime estimated from the best wavefunction employed amounts to over one minute.

Acknowledgement

This work was supported by a grant from CRAY Research Inc.

254 B.F. Minaev et al. /Chemical Physic 175 (1993) 245-254

[ 1 ] S.P. McGlyrm, T. Azumi and M. Kinoshita, Molecular spectmmopy of the triplet state (Prentice Hall, Engebvood Cliffs, 1969).

[2] A.C. Albrecht, J. Chem. Phys. 38 (1963) 354. [ 31 D.S. McClure, J. Chem. Phys. 20 (1952) 682. [4] M. M&r&ma and S. Koide, J. Chem. Phys. 20 (1952) 765. [ 51 E. Clementi, J. Mol. Spectry. 6 (1961) 497. [ 61 W.S. Veeman and J.H. van der Waals, Mol. Phys. 18 (1970)

[7]~F.Minaev,Izv.Vyssh.Uchebn.Zaved.Fii.8(1971) 118. [ 81 B.F. Minaev, Fiz. Mol. 7 ( 1979) 34. [9] J.W.Rabalais,H.J.MariaandS.P.McGlynn, J.Chem.Phys.

5 1 ( 1969) 2259. [lo] J. van Egmond and J.H. van der Waals, Mol. Phys. 26

(1973) 1147. [ 111 W.J, Buma, J.H. van der Waals and M.C. van Hemert, J.

Chem. Phys. 93 (1990) 3733. [12]M.Goe~MayerandA.L.Sklar,J.Chem.Phys.6(1938)

645. [ 131 E.C. Lim, J. Chem. Phys. 36 (1962) 3497. [ 1410. Vahtras, H. &pen, P. Jorgensen, H.J.Aa. Jensen, T.

Helgaker and J. Olsen, J. Chem. Phys. 97 ( 1992) 9 178. [IS] H.J. Shull, J. Chem. Phys. 17 (1949)295. [ 161 P.P. Dikun and B.Ya. Sveshnikov, Zh. Eksp. Teor. Fiz. 19

(1949) 1000. [ 171 G.C. Nieman, J. Chem. Phys. 50 (1969) 1660. [ 18 ] T.V. Ivanova and B.Ya. Sveshnikov, Opt. Spectry. 11 ( 196 1)

322. [19]E.B. Wilson, J.C. Decious and P.C. Cross, Molecular

vibrations (McGraw-Hill, New York, 1955). 1201 W. Moflitt, J. Chem. Phys. 22 (1954) 320. [2l]W.MofBttandA.D.Liehr,Phys.Rev. 106 (1957) 1195. [22] D.M. Burland, G. Castro and G.W. Robinson, J. Chem.

Phys. 52 (1970) 4100. [23] J. van Egmond, D.M. Burland and J.H. van der Waals,

Chem. Phys. Letters 12 (1971) 206. [24] Y. Gsamura, Chem. Phys. Letters 145 (1988) 541.

[25] J.vanEgmond,D.M. BurlandandJ.H.vanderWaals,Mol. Phys. 28 (1974) 457.

[26] S.R. Langhoff, E.R. Davidson and C.W. Kern, J. Chem. Phys. 63 (1975) 4800.

[27] M.A. El-Sayed, W.R. Mesman and J.B. Chodak, J. Chem. Phys. 57 (1972) 4061.

[28] P.G. Russell and A.C. Albrecht, J. C&em. Phys. 41 (1964) 2536.

[29]H.F.HamekaandJ.L.GosterhotT,Mol.Phys. l(l958) 364. [ 301 A.M. Pedash and A.V. Luzanov, Teor. Eksp. Khim. 27

(1991) 152. [31] S. Knuts, B. Minaev, H. &pen and 0. Vahtras, Theoret.

Chim. Acta (1993), in press. [ 321 J.H. Calloman, TM. Dunn and I.M. Mills. Phil. Trans. Roy.

Sot. A 259 (1966) 499. [ 331 H. Lefebvre-Brion and R.W. Field, Perturbations in the

spectra of diatom& molecules (Academic Press, New York, 1986).

[ 341 H. Hettema, HJ.Aa. Jensen, P. Jorgensen and J. Olsen, J. Cbem. Phys. 97 ( 1992) 1174.

[ 351 J. Olsen and P. Jorgensen, J. Cbem. Phys. 82 ( 1985) 3235. [36]H.J.Aa. Jensen, H. &r-en and J. Olsen, in: Modem

techniques in computational chemistry: MOTECC-90, ed. E. Clementi (Escom, I.&den, 1990) p. 435.

[ 371 T.U. Helgaker, J. AlmlsTf, H.J.Aa. Jensen and P. Jorgensen, J. Chem. Phys. 84 ( 1986) 6266.

[ 381 S. Knuts, 0. Vahtras and H. Agren, J. Mol. Struct. THEOCHEM 279 (1993) 249.

[ 391 H. &ren, 0. Vahtras and S. Knuts, to be published. [40] T.H. Dunning Jr., J. Chem. Phys. 53 (1970) 2833. [41]T.H. DunningJr., J. Chem.Phys. 55 (1971) 716. [42] H.J.Aa. Jensen and H. Agren, Chem. Phys. 104 (1986) 229. [43] P. Jorgensen, J. Olsen and H.J.Aa. Jensen, J. Chem. Phys.

74 (1988) 265. [44] S.D. Colson and E.R. Bernstein, J. Chem. Phys. 43 (1965)

2661. [45] P.M. Johnson and G.M. Koronowski, Chem. Phys. Letters

97 (1983) 53. [46] B.O. Roos, K. Andersson and M.P. Fiilscher, Chem. Phys.

Letters 192 (1992) 5. [47]M.W. Williams, R.A. MacRae, R.N. Hamm and E.T.

Arakawa, Phys. Rev. Letters 22 (1969) 1088.

The vibronically induced phosphorescence in benzene

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