J . Phys. Chem. 1990, 94, 1249-1267
1249
ARTICLES Unraveling of Vibronically Scrambled Electronic Spectra: The S2-S0 Transition in Azulene Warren D. Lawrancet and Alan E. W. Knight*-$ Division of Science and Technology, G r i f f h University, Brisbane, Queensland 41 11, Australia (Received: September 16, 1988; In Final Form: August 25, 1989)
Fluorescence excitation spectra, absorption spectra, and dispersed fluorescence spectra following excitation of a variety of absorption features associated with the S2('AI)-SO(IA1)electronic transition of azulene have been measured. Spectra have been obtained for low-pressure azulene vapor at 300 K and for a free-jet expansion of azulene seeded in argon. A theory involving strong vibronic coupling between the S2('AI) and S,( ,Al) electronic states, mediated by totally symmetric vibrational modes, is developed and implemented in an analysis of the strongly perturbed S2-So absorption system. The analysis shows that S2-S4 vibronic coupling in azulene involves significant participation by at least 6 of the set of 12 a, modes (excluding C-H stretches) in an interaction network that causes substantial scrambling of the identities of vibrational levels in the S2 manifold. The spectral evidence in combination with the theoretical analysis also indicates that there is further scrambling of vibrational identity in the S2 state as a result of weaker interactions (coupling constants of 10-20 cm-I) between two-quantum a, combinations that stem from second-order vibronic effects. The analysis serves to illustrate how vibronically scrambled spectra may be unraveled by using dispersed fluorescence spectroscopy to probe the vibrational identity of the mixed states.
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Introduction A basic function of molecular spectroscopy is the identification of the eigenstates of molecules. This is normally achieved within the framework instigated by Born and Oppenheimer,' in which the energy of a molecular state is viewed as the sum of electronic, vibrational, and rotational energies, and the molecular eigenstate is written as a product of separate electronic, vibrational, and rotational wave functions. In addition, the vibrational motion is normally treated in terms of harmonic oscillator wave functions. However, as one begins to probe the finer details of molecular spectra and intramolecular dynamics, it becomes increasingly apparent that an adequate description of molecular eigenstates is only possible if the interactions between the zero-order levels referred to above are taken into account. Indeed, many of the manifestations of the breakdown of this basis are very small in terms of energy shifts but appear rather more striking in the form of time-dependent phenomena, such as internal conversion, intersystem crossing,2-" and intramolecular vibrational redistribut i ~ n . ~ - ' ' Larger deviations, however, appear as perturbations to the energy levels and to the expected intensities of spectroscopic transitions. Vibronic coupling is one manifestation of the breakdown of the separability of electronic and vibrational motion and indeed is the dominant perturbation that effects transitions between electronic states. In this work we address the problem of calculating the eigenstates of a vibronically perturbed molecule, in particular, the energy levels and vibronic wave functions for the second excited singlet state of azulene. Our approach is not exclusively theoretical; rather it is based on a careful evaluation of the relevant spectral observations. The seeds of inspiration are to be found in Wessel and McClure's6 analysis of the S2 So absorption transition in naphthalene in the region of the S2 origin. The spectrum in this region is highly structured, due to near resonance vibronic coupling between the S2levels and high-lying SI vibrations. Wessel's aim was to reproduce the observed absorption structure by taking a set of zero-order vibrational states and mixing
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'Present address: School of Physical Sciences, The Hinders University of South Australia, Bedford Park, S.A. 5042, Australia. Deutsche Forschungsgemeinschaft Guest Professor, 1987, Institut fur Physikalische Chemie der Technischen Universitat Miinchen.
*
0022-3654/90/2094- 1249$02.50/0
them through arbitrary couplings. Wessel's work represented a bold, new direction in molecular spectroscopy: he demonstrated that complex spectra, which arise through couplings between a single state with oscillator strength and a large number of relatively dark levels, can be modeled successfully. The unraveling of Majewski, Kommandeur, and co-workers' molecular eigenstate spectrag using Green's function methodology,I0 although formally different from the trial and error approach used by Wessel, owes its genesis to Wessel's work. In the wake of Wessel's study of naphthalene, there have been a number of analyses of vibronically perturbed spectra using vibronic coupling calculations akin to those used to model the naphthalene S2origin band." However, these calculations have generally been restricted to treating only one or two inducing modes. Our present aim is to demonstrate that vibronic coupling calculations for a large set of inducing modes is also possible and indeed demostrably reliable if one makes use of the information generated through single vibronic level (SVL) fluorescence (1) Born, M.; Oppenheimer, R. Ann. Phys. 1927,84, 457. (2) Bixon, M.; Jortner, J. J . Chem. Phys. 1969, 50, 4061.
(3) Burland, D. M.; Robinson, G. W. Proc. Natl. Acad. Sci. U.S.A. 1970, 66, 257. (4) Avouris, P.; Gelbart, W. M.; El-Sayed, M. A. Chem. Reu. 1977, 77, 793. (5) Freed, K. F.; Nitzan, A. J . Chem. Phys. 1980, 73, 4765. (6) Parmenter, C. S. J . Phys. Chem. 1982, 86, 1735; Faraday Discuss. 1983, 75, 7. (7) McDonald, J. D. Annu. Reu. Phys. Chem. 1979, 30, 29. (8) Wessel, J. Ph.D. Thesis, University of Chicago, 1970. (9) Van der Meer, B. J.; Jonkman, H. Th.; Kommandeur, J.; Meerts, W. L.; Majewski, W. A. Chem. Phys. Lett. 1982,92,565. van der Meer, B. J.; Jonkman, H. Th.; Kommandeur, J. Laser Chem. 1983,2,77. Kommandeur, J. In Stochasticity and Intramolecular Redistribution of Energy; LeFebvre, R.,Mukamel, S., Eds.; Reidel: Dordrecht, 1987. (10) Lawrance, W. D.; Knight, A. E. W. J . Phys. Chem. 1985,89, 917. ( 1 1) Fischer, G. Vibronic Coupling Academic: New York, 1984. (12) Hunt, G. R.; Ross, I. G. J . Mol. Spectrosc. 1959, 3, 604. Hunt, G. R.; Ross, I. G. J . Mol. Spectrosc. 1962, 9, 50. (13) Lacey, A. R.; McCoy, E. F.; Ross, I. G. Chem. Phys. Lett. 1973.21, 223. (14) Barker, J. R. J . Phys. Chem. 1984,88, 11. ( 1 5 ) Shi, J.; Barker, J. R. J . Chem. Phys. 1988, 88, 6219. (16) Brouwer, L.; Hippler, H.; Lindemann, L.;Troe, J. J . Phys. Chem. 1985.89, 4608. (17) Kasha, M. Discuss. Faraday SOC.1950, 9, 4.
0 1990 American Chemical Society
Lawrance and Knight
1250 The Journal of Physical Chemistry, Vol. 94, No. 4, 1990
spectroscopy. We have chosen to study the second excited singlet state of azulene, because this state has long been recognized, though not analyzed, as an example of strong vibronic coupling involving a number of mode^.'^^'^ The vibrational analysis of the azulene S2 transition has persisted as one of the outstanding unsolved spectroscopic problems. In the past several years, azulene has been the focus of considerable attention by kineticists as a medium for exploring vibrational relaxation processes at very high vibrational energies."I6 The choice of azulene as a target molecule in these studies stems from some of its spectroscopic attributes, in particular the location of two optically accessible electronic states, SI and Sz, at 14000 and 28 000 cm-I, that serve as initiators via internal conversion for an energetically narrow population of vibrationally highly excited molecules in the So state. Azulene attracted early attention as a photophysical anomaly through its flaunting of Kasha's r ~ l e . ~ ' - 'It~ fluoresces strongly from its S2state rather than its SI state. It is also a renegade with respect to its spectroscopic behavior: we summarize some of the features of azulene spectroscopy. The S2 So transition (IAl ]Al) shows no hint of the mirror-image relationship that one normally observes between electronic absorption and fluorescence spectra in aromatics. The solution fluorescence shows a single strong feature at -1580 cm-l, while the absorption spectrum shows two strong and vibrationally complex regions, one at lo00 cm-I above the origin and the other at -0 1300 cm-'. The S2-So transition displays essentially no progressions. The lack of progressions in totally symmetric modes indicates that the S2 electronic state possesses a geometry similar to that of the ground electronic state; this is confirmed by the similarity between the rotational constants in the S2 and So states.20 The 4 K mixed-crystal absorption spectra reveal a rich structure, far in excess of that which can be accounted for in terms of a , fundamentals alone.z1 Indeed, the only conventional assignments involve combinations of a 661-cm-, mode, presumably the S2 counterpart of the 671-cm-' So fundamental. Hunt and RossI2 have suggested that the S2 Sotransition is dominated by strong vibronic interactions involving the S4( !A,) state, which lies some 8000 cm-l above Sz. The S4-SO transition is strongly allowed (f = 0.86). Polarized crystal absorption measurementsZ1show little evidence for b, activity and serve to confirm that the vibronic interactions are predominantly with S4 rather than with the S3 ( l B l ) electronic state. The first real breakthrough in arriving at an understanding of the azulene spectrum came from a vibronic interaction model proposed by Lacey, McCoy, and Ross.I3 On the basis of the host crystal dependence of intensities and spectral line positions in the 1000- and 1300-cm-I regions in absorption, Lacey et al.13 postulated that S2-S4 coupling proceeds through a strong inducing mode with unperturbed frequency 1050 cm-I. The interaction 1300 scheme predicts two bands in absorption, one at -0,O 1000 cm-I. These correlate with cm-I and the other at -0,O the observed strong absorption bands in the 1300- and 1000-cm-l regions. The calculated positions and intensity ratios are dependent upon the SrS4 energy gap. As this gap is reduced, the calculations predict that intensity should flow from the 1300-cm-l region to the 1000-cm-l region together with a significant shift of the IOOO-cm-' band toward lower energy. These predictions are correlated with the changes observed in the mixed-crystal absorption spectrum as different crystalline hosts are used to vary experimentally the S4-S2 electronic energy gap. These "medium-dependent" spectral changes extrapolate to the gas-phase observations also. While this model explains the broad features of the spectrum, it has not escaped criticism. In particular we mention the magnetic
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+
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-
-
-
+
circular dichroism (MCD) study by Dekkers and Westra.22 These authors support an earlier speculation advanced by Craig and SmallZ3that the spectral anomalies in azulene arise through transition moment interference and suggest that the 1000- and 1300-cm-l bands in the absorption spectrum are overtones, rather than perturbed fundamentals as suggested by Lacey, McCoy, and Ross.13 The analysis of the Sz So transition is thus not a settled issue. Even if we suppose for the moment that the two-interacting-states model proposed by Lacey et al. is correct in spirit, we must admit that the model accounts for only two regions in absorption, and there are a number of other features that require explanation. On the other hand, Dekkers and Westra's analysisz2 cannot account for any of the medium-dependent effects that pervade the mixed-crystal spectroscopy of azulene, or the extraordinarily rich vibrational structure, seen in the mixed-crystal and vapor absorption spectra of azulene, relative to the Sl-So transition in its isomer, naphthalene. In a more recent study, Fujii et alez4have measured the free-jet Sz-So excitation spectrum of azulene and have obtained some dispersed fluorescence spectra following excitation of some of the absorption features. However, their analysis of the spectroscopy is limited and little is added to the understanding of the Sz-So spectrum beyond that advanced by Lacey et al. Our work advances substantially the understanding of the Sz-So transition in azulene. We verify that extensive vibronic coupling, involving nearly all of the a, modes in S2,is responsible for most of the observed idiosyncracies of azulene's spectroscopy. Our model survives the critical tests that can be imposed using state-selected dispersed fluorescence spectroscopy to probe the characteristics of the vibrational eigenstate of the S2emitting level.
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Vibronic Coupling Theory The appropriate basis set for spectroscopic a n a l y ~ i often s ~ ~ used unwittingly, incorporates the so-called crude-adiabatic electronic wave functions +(q,Qo),with accompanying vibrational wave functions x(Q). The term crude adiabatic refers to wave functions for which the electronic motion is completely independent of the vibrational motion. To include vibronic coupling interactions we proceed from the crude-adiabatic basis to a nonadiabatic basis where we refer to the total wave function \k(q,Q). We first consider the solution of the SchrGdinger equation for molecular systems. The total molecular Hamiltonian can be expressed as HT(4,Q) = Te(q)+ T N ( Q )+ U(q9Q)+
v(Q)
(1)
where q represents the electronic coordinates, Q represents the normal coordinates for the nuclei, Te(q) and T N ( Q )are the electronic and nuclear kinetic energy operators, respectively, and U(q,Q)and V(Q)are the electronic and nuclear potential energies, respectively. The molecular eigenstates are defined according to
+
(18) Beer, M.; Longuet-Higgins, H. C. J . Chem. Phys. 1955, 23, 1390. Viswanath, G.;Kasha, M. J . Chem. Phys. 1956, 24, 574. (19) Sidman, J. W.; McClure, D. S. J . Chem. Phys. 1956, 24, 757. (20) McHugh, A. J.; Ramsay, D. A.; Ross,I. G.Aust. J . Chem. 1968,21, 2835. (21) Lacey, A. R. Ph.D. Thesis, Sydney University, 1972.
This equation cannot be solved directly because the term U(q,Q) prevents a separation of variables. The approximate solution to eq 2 given by the crude-adiabatic wave functions, +(q,Qo)x ( Q ) , is obtained as follows. The terms in the total Hamiltonian that refer to the electronic coordinates are separated out and defined as the electronic Hamiltonian: H k Q ) = TJq) + U(q,Q)
Q, can be any arbitrary configuration. However, since spectroscopic analyses more commonly reveal information concerning (22) Dekkers, H. P. J. M.; Westra, S.W. T. Mol. Phys. 1975, 30, 1795. (23) Craig, D. P.; Small, G. J. J . Chem. Phys. 1969, 50, 3827. (24) Fujii, M.; Ebata, T.; Mikami, N.; Ito, M. Chem. Phys. 1983, 77, 191. (25) Azumi, T.; Matsuzaki, K. Photochem. Photobiol. 1977, 25, 315.
The Journal of Physical Chemistry, Vol. 94, No. 4 , 1990 1251
S2-SoTransition in Azulene the equilibrium geometry, we take Qo to be the equilibrium configuration. HF(q,Qo)is then the electronic Hamiltonian for the crude-adiabatic electronic wave functions that are calculated independent of the nuclear motion:
[Hc(q3Qo)- Ec(Qo)l$(q7Qo)= 0
(4a)
[Te(q) + U(q,Qo) - E,(Qo)l$(q,Qo) = 0
(4b)
i.e. These crude-adiabatic wave functions form a complete set with which one can describe the total wave function:
$t(q,Q) = C$k(q,Qo) X k
dQ)
In proceeding from the exact wave functions to the crudeadiabatic wave functions, we have identified precisely the terms that are neglected in the Hamiltonian when arriving at the crude-adiabatic basis set. Thus, the determination of the exact wave functions from the crude-adiabatic basis set simply involves replacing the neglected terms in the Hamiltonian and operating on the crude-adiabatic basis functions. Diagonalization of the total Hamiltonian in the crude-adiabatic basis gives eigenfunctions that are the total wave functions expressed in terms of the crude-adiabatic wave functions. The relevant matrix elements are of the form
(5)
where Xkr(q)is the expansion coefficient. We may now (rigorously) identify the x k r ( Q )with the vibrational wave functions. The Schrodinger equation for the total molecule can be expressed as 0 = [HT(q,Q)- ET] C$dq,Qo) k
= [ T N ( Q )+ V(Q) + A U q , Q ) + U
Qo)
XdQ)
-
ET] C$k(q,Qo) X k
dQ)
(6)
Multiplying through by fij(q,Q0)and noting that the $k(q,Qo) form a complete orthonomal set, we have 0 = [ T N ( Q )+ V(Q) + ($jC;.(q,Qo)IAv(q,Qo)l$j.i(q,Qo))+
EAQo) - E T I X ~ ~ + Q)
k#i
($j(q,Qo)lAv(q,Q)I$k(q,Qo) )Xkr(q)
There is no restriction on t and r here. Note from eq 10 that there is no coupling between the vibrational levels within an electronic state. Thus the diagonal elements are just the energies of the crude-adiabatic states. Equation 11 gives the off-diagonal elements that correspond to the vibronic coupling between the two crude-adiabatic states $j(q,Qo) xjf(Q ) and $k(q,Qo)xkr(Q). It may be seen that in the crude-adiabatic framework the coupling between electronic and vibrational motion arises because of the dependence of the electronic potential on the vibrational coordinates. AU(q,Q) is not separable in terms of the variables q and Q. However, if we expand U(q,Q) in a Taylor series we have
The coefficients xkr(Q)are obtained as the solutions of the coupled differential equations defined by eq 7. If we set the term that involves the sum over k # j in eq 7 to zero, this equation for the coefficients reduces to the form of a Schrijdinger equation for vibrational wave functions:
[ T N ( Q )+ V(Q) + C$j(q,Qo)lAU(i(4,Q)l$j(q,Qo)) + Ee(Q0) - E T I X ~ ~ ( = Q )0 (8) and the total wave function is approximated by
'kt(q,Q) e 'kjt(q,Q)= $ j ( Q t Q o ) x j r ( Q )
(9)
Thus, in this crude-adiabatic approximation the total molecular wave function is given by the product of an electronic and a vibrational wave function defined by eq 4 and 8. We pause here to consider the vibrational Schrijdinger equation defined by eq 8. Of particular interest is the term
= U(q,Qo) + Au(q9Q) by definition (see eq 3). The subscript Qo indicates that the derivative is evaluated at Qo. Thus
Au(q,Q) =
Substituting this expression for AU(q,Q)into expression 11 gives
($j(q,Qo)lAU(q,Q)I$j(q,Qo) )
This term involves an integral over the electronic coordinates q but remains a function of the vibrational coordinates Q. The
effect of this term is to add to the potential energy term V(Q). In other words, the nuclear potential energy changes in a manner that depends upon the nature of the electronic state $j(q,Qo).This means that the vibrational wave functions need not be exactly the same in the different electronic states. As is witnessed by the mirror image intensity relationship that is often observed between absorption and fluorescence spectra for aromatics, the main effect of this term is usually only a first-order perturbation correction to the energy of the vibrations in the different electronic states. Except in cases where the changes bring different levels into close resonance, the influence of this term on vibrational wave functions normally goes undetected. Hence, spectral assignments remain conventional and we acknowledge merely a change in vibrational frequencies between the ground and excited electronic states. This term is exactly analogous to the so-called medium-independent Duschinsky effect, which has been the subject of investigation by Small and co-workers.26 (26) Burke, F. P.;Eslinger, D.R.;Small, G.J. J . Chem. Phys. 1975,63, 1309.
(13) Selection Rules for First-Order Vibronic Coupling. The couplings between the crude-adiabatic wave functions are defined by the terms in expression 13. The vibrational wave functions interact through terms of the form (xjt(Q)IQn...Qmlxki(Q)
)
(14)
The vibrational wave function x,,(Q) is approximated by the product of harmonic oscillator wave functions d(Qn): Xj,(Q)
nuwdj(Qn)
(15)
where u, is the number of quanta in the harmonic oscillator 4,(Qn) for the tth vibration, and the subscript j refers to the crudeadiabatic electronic state in which the vibrational motion occurs. From eq 13 the first-order vibronic coupling term is given by
For a given Qn,the vibrational component can be expressed in terms of the harmonic oscillator wave functions as
1252 The Journal of Physical Chemistry, Vol. 94, No. 4, 1990
fn(" ' ~ j ( Q ~ ) l ) Q n I ~ I ' ~ ~ , () Q1 p ) m
P
Lawrance and Knight azb'
(17)
t
1
Because the potential for the nuclei is in general different in the two coupled electronic states, {4j}is a different set from {&}. Indeed, the 4, may be expressed as a linear combination of the 4kand vice versa. Thus eq 17 need not readily simplify. However, if the two electronic states involve similar potentials, the and 4 k can be regarded as being essentially the same, permitting expression 17 to be reduced to ("WQn) IQnlU**'4(Qn))
(18)
which yields the well-known selection rule, Au, = f l for the coupling. For higher order terms in the vibronic coupling expansion (eq 13) there are similar vibrational selection rules. For example, the second-order term requires Au, = 0, f 2 or Au,, = f l , Aum = f l for it to remain nonvanishing. We turn now to the electronic component of the first-order vibronic coupling term:
This contribution may be interpreted as a measure of how much the electronic wave functions are changed from the crude-adiabatic wave functions when the molecule is distorted along the normal coordinate Q,. Thus expression 18 defines which of the vibrational levels of a particular mode are coupled through the first-order interaction, while expression 19 defines how strongly they are coupled. Expression 19 has the symmetry selection rule that in order for the integral over the electronic coordinates to be nonvanishing, the product symmetry of qj(q,Qo), Jlk(q,Q0), and Qn must contain an element of the totally symmetric representation. U(q,Q), as part of the Hamiltonian, is necessarily totally symmetric. For notational simplicity, we will label this electronic coupling integral 8,'such that
Noting that the vibrational integrals defined by expression 18 contain a dependence27 upon the vibrational frequency, v,; we define a new constant
where h is Planck's constant. Using the notation (Q,) as a shorthand expression for the vibrational integral,27the first-order vibronic coupling term becomes (see expression 16)
where
and Equation 22 summarizes the first-order vibronic coupling selection rules: 8, contains a symmetry restriction while (Q,) contains the restriction Au, = * I . /3, provides a measure of the intrinsic coupling strength produced by motion along Q,, while (Q,) is a measure of the variation of the coupling with the vibrational quanta. Our present calculations are restricted to include only first-order coupling. This is justifiable, given that there is experimental confirmation that the higher order vibronic coupling terms are
Figure 1. Schematic showing the levels coupled through first-order vibronic coupling with two active modes (vertical lines). The diagram is truncated at two quanta of the fundamentals and at two quanta in each mode for the combinations. The coupling strengths are indicated in the diagram: 0,is the intrinsic coupling strength of v, while Ob is that for pb.
generally less important. The most comprehensive data available are for benzene. An extensive analysis of the SVL fluorescence spectra from a number of levels of benzene indicate that second-order couplings are in general at least an order of magnitude smaller than first-order couplings.28 In the S2absorption spectrum of azulene, it was estimatedI3 that second-order couplings are of the order of 25 cm-I, relative to first-order couplings of 400 and 1300 cm-I. The situation in azulene thus appears to be akin to that of benzene. In another study, we discuss the effects of higher order couplings in a ~ u l e n e . ~ ~ It is important here to differentiate between first-order vibronic coupling calculations and perturbation theory treatments of the energies and wave functions resulting from the coupling. The truncation of the vibronic coupling expansion at the first term separates the strongly coupled fundamentals from the weakly coupled overtones and combinations. (Combinations and overtones built on a vibronically active fundamental do, of course, couple in first order through this fundamental.) The accuracy of a perturbation treatment in calculating the eigenvalues and eigenvectors of the first-order coupling matrix is determined by the properties of the matrix at hand and is totally independent of the fact that the coupling is truncated at first order. As shall become apparent, the higher order effects within the first-order coupling matrix are quite substantial when a number of modes are coupled strongly, and a full matrix diagonalization is necessary to obtain the wave functions and energies with sufficient accuracy to be useful in comparisons with experiment. Vibronic Coupling Calculations: Setting Up the Hamiltonian. The first-order vibronic coupling scheme for two active vibrations is shown in Figure 1. By active vibrations we mean those for which 8, # 0. This interaction scheme arises from the Au = *I selection rule governing the vibronic coupling in first order. In this diagram we have truncated the vibrational manifolds such that the total number of quanta in each vibrational mode is 1 2 . The vertical lines in Figure 1 indicate coupling between the vibrational levels in the two electronic states. For example, the level (28) Knight, A. E. W.; Parmenter, C. S.; Schuyler, M. W. J. Am. Chem.
(27) Wilson, E. B.; Decius, J. C.; Cross, P. C . Molecular Vibrations; McGraw-Hill: New York, 1955.
SOC.1975, 97, 1993, 2205. (29) Knight, A. E. W. In ExcitedStates; Lim, E. C . , Innes, K. K., Eds.; Academic: New York, 1988. Rock, A. B.; Knight, A. E. W. To be published.
S2-So Transition in Azulene
The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1253
a*
s 2 a’ 00
Unperturbed
T
1580 ell”
Vibronic Coupling Perturbatlon
Dmehinrky effect
-,
2 State LMR Model
Figure 3. Schematic of the two-level vibronic interaction model proposed by Lacey, McCoy, and Ross13 to account for absorption activity in the 1000- and 1300-~m-~ regions. Only the one-quantum S2vibrations are shown. The model is discussed in detail in the text.
s,
a1
00
, L
Figure 2. Schematic showing the first-order vibronic coupling scheme for one vibronically active mode. Vertical lines indicate coupled levels.
a, is coupled to a2 with a strength of 2II2fl,. Thus the Hamiltonian has a nonzero element of magnitude 21/2p,connecting a , and a2. The input required for vibronic coupling calculations are the unperturbed vibrational energies in the two crude-adiabatic electronic states (the diagonal matrix elements) and the inherent coupling strength for each active mode, fl, (used to calculate the off-diagonal elements). Given these quantities it is a simple matter to formulate the coupling matrix from the Au = f l vibrational selection rule and the vibrational quantum dependence illustrated in eq 22a and 22b. The selection rule of Av = f l between the vibrational manifolds of the coupled electronic states means that only vibrations separated by Au(tota1) = 2 , 4 are coupled through secondary interactions within a particular electronic state. Thus the matrix splits into two submatrices of equal size, one with only odd total vibrational quantum states in one electronic state and even in the other, and vice versa for the other matrix. Predicting Spectra from Vibronic Coupling Calculations. Diagonalization of the Hamiltonian yields (i) the eigenvalues, which give the energies of the total wave functions, and (ii) the eigenvectors, which define the wave functions themselves, expressed as linear combinations of the crude-adiabatic basis set. The optical selection rules are formulated in terms of these crude-adiabatic states. Thus by taking the eigenvectors, the optical transitions to and from the mixed state can be determined. Consider, for example, a hypothetical situation in which coupling between two electronic states S Iand S2occurs via a single active mode, u,. Transitions between the So and SI states are forbidden by symmetry; however, S2-So transitions are allowed. In this hypothetical situation we choose for simplicity to set the Franck-Condon factors such that (nlm) = ,6, where, ,a is the Kronecker delta. The coupling scheme for this simple situation is shown in Figure 2. It can be seen that the one-quantum u, level in S , is coupled to the origin of S2and the two-quantum u, state in S2. The mixed wave function \k, corresponding to the perturbed a’ level in SI is given by
where we have truncated the matrix at two quanta of v,. Here the superscripts on the refer to the number of quanta. The energy of $, is E,. In the absence of this coupling the electric dipole forbidden SI-Sotransition would not be observed. However, because of the coupling, an absorption transition from the zero-quantum level of the ground state to the SIstate will be seen at energy E,, because the state $, contains a component of the S2 origin. This is the only absorption transition allowed from the ground-state zeroquantum level: all other transitions vanish because the transition moment will not contain a component involving the S2 origin.
+
In fluorescence from +,, transitions to the So origin and the u/ = 2 level will be observed, because of the S2 component in the mixed wave function contributing to the transition. The above discussion gives a flavor for how one can predict the absorption and single vibronic level (SVL) fluorescence spectra from a knowledge of the optical selection rules, the eigenvalues, and the eigenvectors of the total Hamiltonian.
Review of the Lacey, McCoy, and Ross (LMR) Proposal for Vibronic Coupling in Azulene S2 Lacey, McCoy, and Ross (LMR),’ measured the S2 So absorption system of azulene at 4 K in a variety of crystal hosts. They observed a spectrum rich in structure, with two regions of prominent intensity, at ca. 1000 and 1300 cm-I above the origin. The position of the 1300-cm-I region is relatively immune to changes in host; however the 1000-cm-’ band appears to gradually shift to -900 cm-’ as the host is changed from durene through cyclododecane, biphenyl, and naphthalene. The shifts toward lower energy in the 1000-cm-’ region are accompanied by a loss of intensity from the 1300-cm-’ region and a corresponding increase in the 1000-cm-’ region. These authors further note the lack of mirror symmetry between the S2-So absorption and emission spectra measured in solution. Emission is characterized by a maximum at 1580 cm-I, while no prominent bands are observed in this region in absorption. In order to explain these observations, LMR proposed a vibronic coupling model in which two a , fundamentals induce coupling between the S2and S4 states. The S2and S4 states are separated by ca. 8000 cm-l in the vapor. On the basis of the observation that the SI frequencies of azulene are almost unchanged from those of the ground state, LMR maintain that a band would be present at ca. 1580 cm-l in the absence of vibronic coupling. LMR propose that the 1580-cm-’ (unperturbed) fundamental is responsible for strong vibronic coupling between S2and S4. Another a, fundamental is proposed to lie in the region of 1050 cm-l (unperturbed) and induce weak coupling between S2and S4. The identity of this latter state is left open to speculation. The consequence of the vibronic interaction is shown schematically in Figure 3. In a matrix diagonalization of the total Hamiltonian, the unperturbed states on the left of this diagram give rise to the perturbed states on the right of the diagram through the full subtlety of the matrix interactions: this is equivalent to an infinite-order perturbation correction to the wave functions. A physical feeling for the process can, however, be gleaned from perturbation theory. The second-order perturbation correction to the energy levels forces the perturbed 1580 level close to the perturbed 1050 level. This is the shift shown in the center of Figure 3. The vibrational wave functions for these levels now contain common elements due to the interaction with S4 and are no longer orthogonal. The interaction between them is analogous to the case of Fermi resonance; they are mutually repelled, causing the perturbed 1580 level to shift back to higher energy and the perturbed 1050 level to shift still lower. This additional mixing is referred to as the Duschinsky e f f e ~ t ~and ~ , ~produces ’ a flow
-
(30) Duschinsky, F. Acta Physicochim. U.R.S.S. 1937, 1 , 551.
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The Journal of Physical Chemistry, Vol. 94, No. 4, 1990
of intensity from the higher to the lower band. The intensity of an absorption transition terminating in these coupled levels comes from the component of the S4 origin which is mixed into the wave functions. The other S4 components are not seen because the molecular geometry is such that the Franck-Condon factors are essentially zero for all (n10) transitions. The relative size of the S4 origin coefficient in the 1000- and 1300-cm-' bands changes according to the coupling strength. Hence changes in the coupling strength are manifest as intensity variations. Thus, as a consequence of the coupling interactions the 1580cm-' level is pushed to the 1300-cm-I region, the 1050-cm-' level is pushed to 1000 an-', and there is a flow of intensity from the higher level to the low level in absorption. The resulting absorption spectrum (calculated) shows two prominent features, one at 1300 cm:l and the other at 1000 cm-I. In different hosts the S2-S4 energy gap changes, resulting in an effective change of the coupling strength. The S2-S4 gap in the vapor is -8000 cm-', while in naphthalene host it is reduced to -6000 cm-'. It will be recalled from perturbation theory that the state mixing the coupling has increased is approximately proportional to 1 by a factor of ca. 1.3 between the vapor and naphthalene host. At higher coupling strengths the lower level is shifted to still lower energies (ca. 900 cm-I), while the higher energy level remains essentially unchanged. An increased flow of intensity to the lower level accompanies this shift. This model explains quite well the gross behavior of the crystal spectra and is regarded by LMR as the primary vibronic effect. In order to achieve the shifts in position and intensity flow, LMR required coupling matrix elements of magnitude 1300 cm-I for the 1580 level and 400 cm-I for the 1050 level, Le., &sso = 1300 cm-' and @,050 = 400 cm-I. In an extension of this basic picture, LMR explain the proliferation of structure observed in absorption in terms of a mechanism whereby two-quantum levels of a l symmetry (i.e., combinations involving two b,, bz, or a2 modes) also couple to the S4 state, via second-order vibronic coupling (arising from the second term in eq 12). The coupling of these states is weak (LMR estimate -25 cm-I) and, by itself, would certainly not give rise to bands of the required intensity. However, in an effect analogous to the Duschinsky mixing which arises between the two strongly coupled a, modes, these levels can gain intensity from the perturbed 1580- and 1050-cm-' states; i.e., there is a flow of intensity from the 1300- and 1000-cm-l bands into the surrounding two-quantum combinations. This effect is labeled vibronic Fermi resonance by LMR. These authors demonstrate by a simple count of the two-quantum states of azulene that the number of bands in the spectrum appear to correlate very well with the density of twoquantum a , modes. In summary, the LMR model proposes that the S2-So transition in azulene is dominated by a primary vibronic interaction between the S2 and S4 states induced by a 1580-cm-l mode (strong) and a 1 0 5 0 - ~ m mode - ~ (weak). In addition, there are weak vibronic interactions induced by two-quantum states of a , symmetry; these states thus couple to the vibronically active a, fundamentals and there is a flow of intensity into the two-quantum states. However, as LMR emphasize, their model does not constitute an analysis of the azulene S2 So spectrum, and many features of the spectrum remain unexplained. Alternative Viewpoints Concerning the S2-So Transition. The LMR model for extensive vibronic coupling between S2and S, in azulene, giving rise to sweeping changes in the S2 So transition relative to that expected from a mirror symmetry relationship with fluorescence, is certainly the most detailed of the proposals suggested so far to account for the spectral observations. It is important to note that this model is based on the medium dependence of the absorption spectrum. We also note that the model 1300 cm-I, calculations require large coupling strengths, P I 5 8 0 primarily because of the large energy displacements that are necessary to mimic the correct band positions.
-
/e
+-
-
-
(31) Small, G . J . J . Chem. Phys. 1971, 54, 3300
Lawrance and Knight On the basis of a study of the SIstates of certain azaazulenes and azulene itself, Burke, Eslinger, and have suggested that the S2 state of azulene may be also subject to the so-called medium-independent Duschinsky effect (see the discussion following eq 9). In other words, the dependence of the vibrational potential on the electronic state may cause a change in the vibrational levels between the ground and S2 states. Since the excited-state vibrations x' can be expressed as a linear combination of the ground-state vibrations x": 3N-6
it is easily seen that the medium-independent Duschinsky effect can, if sufficiently strong, give rise to intensity variations between absorption and emission, independent of any vibronic coupling effects. In addition, the changes in the excited-state potential can act to reduce the excited-state frequencies somewhat. Burke et aLZ6propose this mechanism as an adjunct to the LMR model and suggest that the inclusion of this effect may reduce the size of the vibronic coupling matrix elements required to replicate the spectral patterns. A further comment on the LMR model is contained in the magnetic circular dichroism study by Dekkers and Westra.22 These authors confirm that the S2-So transition is subject to vibronic coupling between the S2and S4 states. In their opinion however, "the 1000 cm-I and 1300 cm-' frequencies are not fundamentals but overtones". This proposition appears not to be based on any direct experimental evidence. These authors also speculate that transition moment i n t e r f e r e n ~ emay ~ ~ account for the different vibrational activity in absorption and fluorescence. However, the medium-dependent shifts that prompted the LMR model are difficult to rationalize within a framework that allows only transition moment interference. A Spectral Probe of the LMR Model The above discussion demonstrates the need for further tests of the LMR model. In particular, experiments that probe the vibronically active modes are necessary, and an uncluttered view of the vapour absorption is essential. We have obtained the Sz Oofluorescence spectrum at high resolution (at room temperature) and the S2 So fluorescence excitation spectrum in a supersonic free jet expansion. The jet-cooled excitation spectrum provides a view free from room temperature congestion (vapor) or from phonon activity (crystal). The relative intensities of all the vibronic bands may therefore be measured with unprecedented precision and with essentially no ambiguities. @ Fluorescence: Expectations Based on the LMR Model. Before examining the Oo fluorescence spectrum, it is interesting to consider what we expect to see, based on the LMR model. The similarity between the rotational constants in the ground and second excited electronic statesz0indicates that the geometry change involved in excitation is very small and leads to the conclusion that Franck-Condon activity will be restricted. This is borne out in the vapor and crystal absorption spectra, in which there are essentially no progressions; the only clear combinations involve the 661-cm-' frequency.2' The lack of progressions has also been noted by Dekkers and Westra.22 These latter authors suggest on the basis of an analysis of their MCD spectrum of the S2 Sotransition that 80% of the absorption intensity comes from intensity borrowing from the S4 state via vibronic coupling. (It is interesting to note that in spite of this, these authors refute the model of LMR.) The Franck-Condon intensity is limited almost entirely to the 0; transition, although the analysis of the MCD data is consistent with the absorption measurements that limit Franck-Condon progression activity to a 660-cm-l mode. Thus, according to the LMR picture, the main absorption features arise from a vibronic interaction involving predominantly two modes, with unperturbed energies of 1580 and 1050 cm-I. As discussed earlier, the perturbed S2origin will correspondingly contain a component of the one-quantum 1580- and 1050-cm-l levels from the S4 state. Due to its stronger coupling, the
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-
Sz-SoTransition in Azulene
The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1255
0 (0 LD r
I
Oo Fluorescence Predicted (LMR)
0
i 0
z
c
.f
'p
active state in the LMR model. Only the levels 671 and 822 form significant combinations. Thus a contribution to the intensity from the S2 component, via a (110) type Franck-Condon factor, would appear to play a role for these two states. For other bands, the contribution, if any, to fluorescence intensity from the Sz Franck-Condon component relative to that from the S4 vibronic component is unclear. Nevertheless, it is safe to conclude that the activity observed in emission is indicative of many more a , levels being involved in the vibronic interactions than just the two suggested by LMR. It is not surprising to find evidence suggesting vibronic activity in a larger number of a, modes. The calculations of LMR were principally confined to modeling the 1OOO-and 1300-cm-I regions, these being the regions affected by changes in host. Vibronic activity outside these two areas is certainly possible. Nevertheless, the Oofluorescence spectrum is consistent with the basic hypothesis of LMR, that the 1580-cm-, mode is the dominant vibronic inducing mode. The extent to which the inclusion of other a, coupling modes in the 1000-1600-~m-~region changes the coupling calculations, and hence modifies the predicted absorption spectrum, is unclear from the LMR model. What is clear, however, is that the beginnings of a more detailed understanding of the absorption spectrum must be contained within this extra vibronic activity that is evident in the Oo spectrum. Ground-State Vibrational Assignments: Choice of the a, Fundamentals. Information contained in the Oo fluorescence spectrum concerning the a, fundamentals may be extracted provided that the So vibrational assignments are known with some security. Indeed, our present understanding of the Sz-Sotransition in azulene indicates that Oo fluorescence can assist in the confirmation of So a, vibrational assignments. Azulene has 12 a , modes below the C-H stretch region (17 a, modes in total). Lacey's compilationz1of Sovibrational frequencies and assignments for these a, modes is derived from several sources.12,32-35 More recently, Chao and K h a n ~ have ~ a ~assigned ~ a set of a, fundamentals on the basis of resonance effects in the Raman spectrum of azulene. A comparison between the assignments by Lacey and Chao and Khanna indicates that 10 of the 12 a, modes below the C-H stretch region can be assigned with security. Four bands vie for the two remaining assignments. Lacey's work suggests that one of these is almost certainly at 1058 cm-I, with the other fundamental being at 1630 cm-I. Chao and Khanna on the other hand do not observe any band in the 1060-cm-l region and assign the remaining fundamentals to be 1160 and 1457 cm-I. These authors draw specific attention to the 1630-cm-' band. In their Raman spectrum they observe a band at 1640 cm-' which they assign to be the first overtone of the 822-cm-l fundamental, and they conclude that the previous assignment of the 1630-cm-' band as a fundamental is incorrect. As discussed above, the Oo fluorescence spectrum contains a reasonably intense band at 1055 cm-I. We conclude that Lacey's assignment is correct in this respect and that the remaining a , fundamental is either 1160 cm-l or 1457 cm-I. Our choice is 1160 cm-I. We suggest that the band observed by Chao and Khanna at 1457 cm-I, and assigned as the fundamental u,, is in fact the combination of 401 1055. We propose that this combination derives intensity through Fermi resonance with the 1450-cm-I fundamental. High-resolution (-0.5 cm-I) scans of the Oo fluorescence spectrum in this region reveal structure consistent with this proposal. In the light of the structure revealed in the Oofluorescence spectrum and a comparison between the compilation of Laceyzl a , propose ~~ the and the Raman data of Chao and K h a n ~ ~we assignment of the a, So fundamentals given in Table I. Where possible we have used the frequencies observed in the Oo fluorescence spectrum in the vapor.
I Energy / cin-'
Figure 4. S2 Oo fluorescence spectrum predicted from the LaceyMcCoy-Ross model involving two coupled modes, one at 1050 cm-l (0 = 400 cm-I) and the other at 1580 cm-I (0= 1300 cm-I). Origin Fiuore8cence
f
26500
27500 Energy /
28500
cm-'
Figure 5. Oo fluorescence spectrum from the S2electronic state of azulene excited at 28 759 cm-l (vac). The excitation bandwidth was
