J . Phys. Chem. 1991, 95, 2167-2175
2167
Rate Proportional to (Frequency Shift)* and Other “All Else Being Equal” Correlations in Vibrational Predissociatlon Robert J. Le Roy,* Mark R. Davies, and Mimi E. Lam Guelph- Waterloo Centre for Graduate Work in Chemistry, Uniuersity of Waterloo, Waterloo, Ontario N2L 3G1, Canada (Received: March 20, 1990; In Final Form: August 28, 1990)
It is shown that, within a simple ‘all else being equal” description, there should be a quadratic relationship between the vibrational frequency shift and vibrational predissociation rate of a van der Waals molecule. This occurs because both phenomena are driven by the same terms in the intermolecular potential. Model calculations used to demonstrate this behavior also delineate a special type of situation in which the widely quoted ‘momentum gap law” undergoes catastrophic failure. This emphasizes the fact that detailed discussions of trends in predissociation rates must always consider the nature of the intermode coupling function and its effect on the effective supporting potentials. Extension of these arguments to cases involving more than one internal degree of freedom suggests that Miller’s [Science 1988,240,4471 observation of (predissociation rate) 0: (frequency shift)2 correlations is implicit evidence for a two-step impulsive mechanism for vibrational predissociation.
1. Introduction
In the vibration1 predissociation of a van der Waals molecule, a quantum of energy initially localized in some internal mode of one of the component monomers leaks into the weak van der Waals bond, causing its rupture. The course of a process of this type is determined by the nature of the system’s potential energy surface. However, in spite of the wealth of theoretical and experimental studies performed over the past decade and a half,1-9 very few general principles concerning the nature of vibrational predissociation have been discerned. Noteworthy exceptions are the ‘momentum-gap law”, which predicts an exponential decrease in the predissociation rate with increasing relative momentum of the separating and the propensity or selection “rules” proposed by Ewing.6 However, the assumptions underlying these principles, and the conditions under which they may be expected to break down, do not seem to be fully appreciated. In a recent review, Miller9 noted an empirical correlation between vibrational predissociation lifetimes and the vibrational frequency shifts associated with formation of van der Waals dimers. At about the same time, in the context of a study of intramolecular vibrational redistribution (IVR) in tetrazeneargon complexes, Weber and Rice proposed qualitative rules for determining preferred IVR pathways.lO,” Although their origin in a discussion of a particular model for IVR intermode coupling may make their generality difficult to recognize, some of these rules should also apply to vibrational predissociation, and one of them echoes the type of behavior observed by Miller.9 The object of the present paper is to describe and demonstrate the interrelationship between two apparently distinct phenomena: vibrational predissociation and the vibrational frequency shift suffered by a monomer on forming a van der Waals molecule. As noted earlier by Ewing,’* they depend on different weighted averages of the same part of the potential energy surface. Pursuing this point further leads us to predict that, “all else being equal”, ~
( I ) Coulson, C. A.; Robertson, G. N. Proc. R. Soc. London 1974, A337, 167; 1975, A342, 289. (2) Child, M. S.Furuduy Discuss. Chem. Soc. 1977, 62, 307. (3) Beswick, J. A.; Jortner, J. Chem. Phys. Lett. 1977, 49, 13. Beswick, J. A.; Jortner, J . J. Chem. Phys. 1978, 68, 2277; 1978, 69, 512. (4) Ewing, G. E. Chem. Phys. 1978,29,253. Ewing, G . E. J. Chem. Phys. 1979, 71, 3143; 1980, 72, 2096. (5) Beswick, J. A,; Jortner. J. Adu. Chem. Phys. 1981, 47, 363. (6) (a) Ewing, G. E . Furuduy Discuss. Chem. Soc. 198t73.325.402. (b) Ewing, G. E. J. Phys. Chem. 1987, 91, 4662. (7) Le Roy, R. J. In Resonances in Electron-Molecule Scattering, Van der
Waals Complexes, und Reactive Chemical Dynamics; Truhlar, D.G . , Ed.; ACS Symposium Series No.263; American Chemical Society: Washington, DC, 1984; Chapter 13, pp 231-262. (8) Celli, F. G.; Janda, K. C. Chem. Reu. 1986, 86, 507. (9) Miller, R. E. Science 1988. 240, 447. (IO) Weber. P. M.: Rice, S.A. J. Chem. Phys. 1988,88, 6120. ( I 1 ) Weber, P. M.; Rice, S.A. J . Phys. Chem. 1988, 92, 5470. (12) Ewing, G. E. J. Phys. Chem. 1986, 90, 1790.
the vibrational predissociation rate should vary as the square of the level shift. The nature of the “all else being equal” conditions governing both this and a number of other correlations between the vibrational predissociation rate and parameters of the system will be delineated. This discussion also shows why vibrational frequency shifts should increase approximately linearly with the change in vibrational quantum number for the associated monomer transition. A description of the momentum-gap law which outlines and assesses its underlying assumptions is also included, and simple model calculations performed to illustrate the quadratic behavior mentioned above demonstrate that under certain circumstances the momentum-gap law can undergo catastrophic failure. The following section outlines the theory for a two-dimensional problem involving coupling between a single internal vibrational mode of the component monomer and the stretching or translational motion of separation along the van der Waals bond. The simplicity of this model system facilitates our drawing a number of conclusions regarding the nature of and interrelationship between vibrational level shifts and vibrational predissociation rates. Certain of these observations are then illustratred by the numerical calculations for simple yet realistic model systems, presented in section 111. This is followed by a more general discussion which examines the effect of additional internal degrees of freedom on the qualitative conclusions reached for the simple two-dimensional problem. 11. Outline of the Theory
A . Phenomenological Description. In vibrational predissociation, a metastable complex containing a monomer in vibrational state u‘ falls apart to yield the monomer fragment in vibrational state u”, where u” C 0’. For a model two-dimensional problem in which u’ = 1 and o f f = 0, the essential nature of this process is illustrated in Figure I . There, the monomer’s internal vibrational coordinate is denoted q and the intermolecular van der Waals bond coordinate denoted R; the vibrational states associated with the stretching of R are identified with the label n, while the continuum states corresponding to fragment dissociation are simply labeled by the total energy E . In the left-hand segment of Figure 1, the curve labeled U,(q) represents the potential energy governing the relevant internal vibrational mode of the isolated monomer, the functions Q,(q) are the associated wave functions for u = 0 and 1, and E,(u) are the corresponding free-molecule vibrational energies. In the right-hand segment, the curves r,!(R) and Vurp(R) represent the effective potentials governing motion along the van der Waals coordinate when the monomer is in the internal state identified by the subscript label. The differences between the shapes of these curves reflect the effect of vibrational excitation of the component monomer on the intermolecular potential energy function. Note that the asymptotes of the V J R ) potentials lie at the internal energies of the corresponding monomer species.
0022-365419112095-2167%02.50/0 , 0 1991 American Chemical Society ,
I
Le Roy et ai.
2168 The Journal of Physical Chemistry, Vol. 95, No. 6. 1991
the van der Waals bond, subject to a “frozen-monomer” potential V,(R) which does not depend on the internal state of the monomer; its eigenfunctions and eigenvalues are denoted $:(R) and l$(n), respectively. The interaction term AH(q,R) then couples these two types of motion, giving rise to both predissociation and the vibrational frequency shifts. For the vibrational predissociation process, AH(q,R) consists of a potential coupling term AV,(q,R) which may be expanded in terms of some chosen set of q-coordinate basis functions, {&(q)]:
AH(q,R) = AVc(qJ?)
0.5 1.0
4
I
2
9
3
4
5
6
R
Figure I . Schematic illustration of the potential energy curves and wave functions governing B vibrational predissociation process involving only one internal degree of freedom.
The n = 0 energies E,(v,n=O) and wave functions $,,n*o(R) shown in Figure 1 are associated with the ground state of the van der Waals stretching mode for complexes formed from groundstate ( u = 0’’ = 0) or excited-state (u = u’ = 1) monomer, while +dt&R) is the continuum wave function associated with the relative translational motion of the ground-state monomer and its partner at total energy E = E,(u’,n=O). Vibrational predissociation therefore corresponds to a metastable complex with total (zeroth order) wave function O,,(q) I / ~ , , , ~being ( R ) transformed into a dissociating state with total wave function @,49) $,Tt,E(R). The second phenomenon of interest here, the frequency shift associated with formation of the van der Waals bond, may also be readily described in terms of quantities shown in Figure 1. In general, the negative of the van der Waals binding energy, [E,(u) - E,(u,n=O)], may be thought of as the energy shift undergone by monomer vibrational level v upon forming the complex. The difference between two such shifts
Ad$’ = [E,(u’,O) - E,(u”,O)]- [E,(u’) - E,(u”)]
(5)
Thus, the frozen-monomer potential Vo(R) of eq 4 is simply the leading term in an expansion of the overall interaction potential for the complex: V,(q,R) = Vo(R)+ AV,(q,R). In most of the following, the expansion of eq 5 is truncated at a single term AV,(q,R) i= $l(q) Vl(R). Because of the large differences between the characteristic frequencies associated with the inter- and intramolecular vibrational motions, and the relatively small amplitude of the latter, it should usually be possible to choose a basis function + l ( q ) for which this is a good approximation. In a zeroth-order picture, the coupling term AVc(q,R) is completely neglected and the Schriidinger equation becomes exactly separable. Each total eigenvalue is then simply the sum of a free monomer level energy with an eigenvalues of HJR)
II
L
X$dq) VdR)
k=l
(1)
is then the shift of the free monomer transition frequency associated with complex formation. If the effective potentials V,(R) are identical for complexes formed from monomers in different vibrational states, the binding energies associated with motion along the van der Waals coordinate would,;ot depend on u, and the net vibrational frequency shifts AI$ would all be zero. However, such potentials are never exactly the same, since the bond lengths and electronic properties of a molecyle are always affected to some degree by the level of internal excitation. B. Vibrational Frequency Shift, In general, one may write the Hamiltonian for our two-dimensional model problem as a sum of three terms: Htot(q3) = Hm(q) + H J R ) + AH(q,R) (2) The term H,(q), which includes the potential U,,,(q),governs the intramolecular vibration of the free monomer and has eigenvalues (Em(u)]and eigenfunctions {@t,(9)]:
= Em(u) + E 3 n ) and the eigenfunctions are the product functions @ot(u,n)
%,,(q,R) = Q,(q) $%R)
(6) (7)
In this case, the vibrational frequency shifts are zero, since the e ( n ) values do not depend on v, and vibrational predissociation does not occur because of the absence of coupling to drive it. If the coupling term AVc(q,R) is not neglected, but the two degrees of freedom are adiabatically separated, the total SchrBdinger equation for the problem remains separable. However, the Hamiltonian governing motion along the van der Waals coordinate R is based on vibrationally averaged effective potentials such as those illustrated in Figure 1:
V J R ) = Vo(R)+ (O,(q)IAVc(q,R)IO,(q))
Vo(W + 41(u,u) Vi(R) where the general matrix elements 4,(U’P’?
(8)
= (@d(q)14I(q)l@u4q))
(9) are defined by the known monomer wave functions and the chosen expansion function $,(q). Note that, following Hutsor~,’~ we use the notation (I I) to denote an inner product which involves integration over only those variables associated with the functions in the outer brackets, yielding a result which is a function of the remaining variables; for the example in eq 8 the integration is performed over the variable q and the result is a function of R. The functions Y J R ) are the same diagonal radial channel potentials that appear in the close-coupling formulation of this p r ~ b l e m . ’ ~In * ’addition ~ to being vertically displaced by different values of Em(u),these curves differ in shape because the vibrational averages of the coupling function AV,(q,R) depend on u. It is this difference in shape which gives rise to the frequency shifts. The phenomenological eigenvalues E,(u,n) and eigenfunctions #,,“(I?) and IC;,E(R)shown in Figure 1 are clearly the eigenvalues of the vibrationally averaged effective adiabatic potentials of eq 8. The vibrational frequency shifts associated with complex formation are then due to the difference potentials AVddt(R) Vd(R)- Vd,(R) e [$~(u’,u’)- $,(u”,v”)]V~(R) (10)
Within first-order perturbation theory, these shifts may be written as
The second term in eq 2 (4) is the Hamiltonian for intermolecular vibration or translation along
(13) Hutson, J. M. In Advances in Molecular Vibrations and Collision Dynamics, in press. (14) Le Roy, R . J.; Carley, J. S . Adv. Chem. Phys. 1980, 42, 353.
Correlations in Vibrational Predissociation
The Journal of Physical Chemistry, Vol. 95, NO. 6, 1991 2169
The form of this expression suggests a third observation. Observation (iii): “All else being equal”, the predissociation where qu,,(R) is the eigenfunction of V J R ) corresponding to rate scales as the square of the strength of the coupling function eigenvalue E,(v,n), and u is either u’or 0”. AVc(q,R) or V l ( R ) .As with observation (i) this “all else being The form of eq 1 1 suggests the following observation: equal” condition ignores the effects of changes in the effective Observation ( i ) : “AN else being equal“, the magnitude of the radial wave functions I)~,,,,(R) and $,,,,,E(R)implied by changes vibrational frequency shijt will be directly proportional to the in AV,(q,R) or V l ( R ) . In the present case, this assumption is strength of the radial couplingfunction AV,(q,R) or Vl(R).Note somewhat more tenuous, but this third type of correlation should that this “all else being equal” condition neglects the fact that still be at least qualitatively valid for families of similar interactions changes in A V c ( ~ Ror) VI(/?) lead to changes in the vibrationally between similar partners. averaged potentials Vl,(R),and hence also in the effective radial The last line of eq 15 is also the starting point for derivation wave functions #,,(R) and the expectation values of V , ( R )apof the “momentum-gap law” for vibrational predissociation.16 pearing in eq 1 I . For the broad Gaussian-like wave functions Obtaining that result requires the introduction of three further characteristic of the ground van der Waals stretching state (see assumptions: (a) that the frozen-monomer supporting potential Figure I), this is not a severe constraint, so this correlation should Vo(R)is a Morse function, (b) that the coupling function AV,(q,R) be qualitatively valid for families of similar interactions. In is proportional to the product of a linear monomer stretching contrast, this dependence of # J R ) on v can massively affect coordinate times the first derivative of Vo(R),and (c) that the predicted vibrational predissociation rates (see section 111). effect of monomer vibrational averaging on the effective supporting A second type of correlation arises from the fact that, for potentials of eq 8 may be neglected. On combining these asvirtually any reasonable expansion function +i(q),its expectation sumptions with a few mathematical approximations, one obtains value over the monomer wave functions O,(q) will vary approxthe result that 7-I is directly proportional to exp(-rimately linearly with u. This is true for almost any reasonable (2pAE,)1/2//3h), where p is the reduced mass for motion along harmonic or anharmonic monomer potential. As a result, the the van der Waals bond, AE, = E,(u’) - E,(u‘? is the energy factor in square brackets in eq 1 1 should vary approximately released by the constituent monomer, and /3 is the exponent palinearly with Au. This leads to the following: rameter in the Morse potential model for Vo(R).The functionality Observation ( i i ) : The magnitudes of the frequency shijts of observation (iii) is also implicit in this r e s ~ l t but ~ - ~has tended associated with a given monomer vibrational series should into be overlooked, perhaps largely because relatively little is known crease approximately linearly with the change in the monomer about the coupling functions and because the remarkable success vibrational quantum number, bo. Note that this result is not of this “law” in explaining trends with respect to the factor contingent on a one-term truncation of the expansion for AV,(q,R), (2pAE,)1/2focused attention elsewhere. since additional terms in that expansion would merely contribute The conclusion that In ( F ’ )is proportional to the ”momentum additional terms of the same form as that seen in eq 1 I , each of gap” (2pAE,)1/2 has proved very useful. In particular, it showed which should scale approximately as Au. that the breadth of the range of observed predissociation rates C. The Vibrational Predissociation Rate. In the present is a straightforward manifestation of the effect of oscillatory discussion, the familiar golden rule formula15is used to estimate cancellation in bound -,continuum Franck-Condon factors, and vibrational predissociation rates. While not exact, the simplicity it explained the qualitative functional behavior of predissociation of this approach facilitates the development of physical intuition rates in a variety of practical cases.3-5,17-19However, the three regarding the essential nature of this process. Moreover, its underlying physical approximations deserve further examination. breakdown would most likely imply that no adequate effective The first approximation, the assumption that Vo(R)has the adiabatic separation of the intramolecular and van der Waals form of a Morse potential, is certainly not very restrictive, since modes of motion can be devised, in which case the types of simple the fact that the qualitative nature of bound and continuum correlations of interest here will not exist. functions are the same for any realistic potential means that its Within our adiabatic separation of the internal and van der breakdown should not significantly affect the functional behavior Waals stretching motions, at total energy E = Ec(u’,n?there exist predicted by the momentum-gap law. Assumption (b) requires up to (u’+ 1) independent eigenfunctions of the total Hamiltonian. the strength of the coupling function to vary linearly with that These are metastable state wave function of the supporting potential and causes the former to change sign precisely at the minimum of the latter. These are more serious W$&A = Q d q ) $d.n’(R) (12) constraints, since the strength of V , ( R )will certainly not always plus one continuum function associated with each vibrational scale with that of the effective supporting potential, while the fact channel for which E,(v”) < E,(u’,n’): that the bound -,continuum radial matrix element of eq 15 is dominated by oscillatory cancellation effects means that the precise %;:(%R) = Q d q ) $df.E(R) (13) position of an imposed zero in the integrand can have a significant effect on the resulting integral. Indeed, the model calculations Here, Or(q)and $l,3n(R) are as defined above, and $utj,E(R) is a reported below show that coupling functions Vl(R)having different continuum eigenfunction of Pd,(R). These product functions are shapes but the same “strength” (as characterized by the level obtained by solving the total SchrZKlinger equation while neglecting shifts) can yield markedly different predissociation rates. However, the off-diagonal coupling potentials these calculations also show that for a given type of coupling = (O,(q)lAVc(q,R)lO,f(q)) function an exponential dependence on the momentum gap is still obtained. ..Vt“-U”(R) = qq(v’,v’? V l ( R ) (14) The third assumption, neglecting the effects of the monomer Within the golden rule treatment, the total predissociation rate vibrational averaging implied by eq 8, is the most severe of these is given by approximations. Its requirement that the bound and continuum r-I = 2rc I ( S ~ ~ ~ ( S , R ) I A V , ( ~ , R ) I (15) ~ ~ ~ ~ ~ wave ( ~ , functions R ) ) ( ~ $,(R) and &,&(R) be eigenfunctions of the same c.”x.o effective radial potential makes them exactly orthogonal and 2*c [‘“10 C I ( $ ~ ~ , E ( R ) I ~ ’ ~ ~ ’ ~ ~ ) I $ ~ , ~ ( Rdefines ) ) I ~ their relative radial positions in a specific way. On the other hand, if the radial wave functions appearing in eq 15 are A4Y = [$I(v’P? - $ i ( ~ ” ~ ~ ’ ? l ( $ u ~ ( ~ ) l ~ i ( ~ ) l $ u ~(11) (R))
E””(R)
c
5*
2m2
c
u”=0
l ~ l ( V ’ ~ ~ ’ ~ 1 2 1 ( $ ~ ~ , E ( R ) l ~)IZ I(R)l$~~~~)
( 15) See,e.&: Perturbations in the Spectra of Diatomic Molecules; Lefebvre-Brion. H.. Field, R. W., Ed.; Academic Press: Toronto, 1986; Section 6.3.
(16) While first derived by Bawick and Jortner,’ the physical basis for this result was greatly clarified by the independent work of E ~ i n g . ~ (17) Levy, D. H. Ado. Chem. Phys. 1981,47, 323. ( 1 8 ) Cline, J. I.; Sivikumar, N.; Evard, D. D.; Bider, C. R.; Reid, B. P.; Halberstadt, N.; Hair, S . R.; Janda. K. C. J . Chem. Phys. 1989, 90. 2605. (19) Drobits, J . C.; Lester, M. I. J . Chem. Phys. 1988, 88, 120.
Le Roy et al.
2170 The Journal of Physical Chemistry, Vol. 95. No. 6, 1991
generated from the appropriate V.,(R) potentials, scaling or changing the shape of V , ( R ) while Vo(R)remains fixed will in general displace the effective bound and continuum functions relative to one another and can lead to massive changes in the predicted predissociation lifetimes. Indeed, the model calculations reported below show that this can give rise to a catastrophicfailure of the momentum-gap law, in which predissociation rates increase (rather than decreasing exponentially!) with increases in the momentum gap. This leads to the following: Observation (iu): The effect of monomer vibrational averaging can cause catastrophicfailure of the momentum-gap law. The form of the first factor inside the summation in the last line of eq 15 suggests an additional correlation which was recognized and discussed in ref 5. It is that, independent of momentum-gap considerations, the branching ratio for a given vibrational predissociation process will usually (for not too high v ) show a strong predilection to minimize Av and preferentially yield v” = 0’- 1 monomer fragments. However, this preference is based on the harmonic-oscillator-like behavior characteristicof wave functions and matrix elements for low vibrational levels, and the extreme anharmonicity of monomer wave functions for very high v may substantially weaken it. This indeed appears to be observed for predissociation from high vibrational levels of 12(B)-Ar, -Ne, and -HeZoand ICI(A)-Ne.19 Furthermore, the fact that Blazy et aL20 found evidence that the vibrational predissociation lifetime of 12(B,v’)-He may be significantly longer for v ’ = 61 than for lower levels, contradicting the predictions of the momentum-gap law, may partly reflect the increasing anharmonic behavior of this factor at very high 0’. This suggests a somewhat more speculative observation. Observation (0): For sufficiently high v, the effect of extreme anharmonicity on the intramolecular matrix elements may reverse both the momentum-gap law preference for small Av and the tendency for the predissociation rate to increase with 0’. D. Intercorrelations. Observations (i) and (iii) may be summarized as follows. Assuming an adiabatic separation of the motions along the internal and van der Waals coordinates, the Schriidinger equation for the system becomes separable. Within a first-order treatment, the frequency shifts associated with complex formation are diagonal expectation values of the difference potentials A VL,t,t,tJ(R) over the adiabatic initial or final bound-state wave functions. Such shifts will therefore scale linearly with the strength of the coupling function V l ( R ) . Similarly, within the golden rule treatment, the predissociation rate is proportional to the square of the off-diagonal matrix element of this same coupling function between the effective adiabatic metastable and continuum wave functions. This suggests the following. Observation (vi): “All else being equal”, the predissociation rate will LIary as the square of the level shift; thus,for a family of “similar” systems, the ratio ~ - ‘ / [ A v ; yshould ] ~ be approximately constant. A more quantitative basis for this observation is obtained on utilizing the fact that the sum in eq 15 is usually dominated by a single term, usually 0’’ = v ’ - 1. Combining eq 1 1 with the last version of eq 15 then yields
--
1-I e
[AlJp [$I
(v’,v9 - $1 (v”,v’?l
[ ( $ L v m l VI (R)l$”,n(R) ) 1
*
This expression shows that if all other terms remain unchanged, scaling the coupling function V l ( R )by a constant factor will not affect this ratio. The implied quadratic dependence of predissociation rate on frequency shift was discerned empirically by Miller9 and appears in another form in the IVR rules of Weber and Rice.Io The experimental data on which Miller’s conclusion was based are replotted in Figure 2;21in spite of the substantial (20) Blazy, J. A.; DeKoven, B. M.; Russell, T. D.; Levy, D. H. J . Chem. Phys. 1980, 72, 2439. (21) Miller, R. E. Private communication, 1989.
10.0 n
s . l
9.0
Q)
Y
E v
0 4
M
,“ 8.0
7.0 0.75
1.so
2.25
3.00
log*o(shift/cm-’)
Figure 2. For vibrational predissociation of a variety of complexes formed from H F (square points), HCN (triangular points), and C2H2 (round point), correlation between the experimental predissociation rates and frequency shift^.^,^'
scatter, the behavior implied by observation (vi) is clearly present. For a given type of monomer vibrational mode, such as the (anharmonic) stretching of H F or of the C-H bond in HCN, it seems reasonable to expect the leading term in eq 5 to involve the same type of expansion function &(q). This in turn implies that the expectation value differences [$l(v’,v’) - $l(v”,v”)] and off-diagonal matrix elements $l(u’,v’’)will be the same for the van der Waals complexes it forms with a range of different partners. For a family of such systems, the first ratio on the right-hand side of eq 16 may therefore be treated as a constant when discussing either variations of T - I / [Au;Yl2from one system to another or its functional dependence on the strength of the coupling or quantities such as the momentum gap. Momentum-gap law considerations lend additional credence to observation (vi). In particular, the energy gaps A&, appearing in the momentum-gap law exponent will be exactly the same for all complexes formed by a particular monomer, while the reduced mass p for its association with a wide range of partners will often be approximately constant.22 As a result, the degree of oscillatory cancellation in the bound continuum matrix elements in the numerator of eq 16 should be roughly constant for a family of complexes formed from that monomer and hence would not disrupt the qualitative cancellation of V l ( R )suggested by this expression. Of course, the “all else being equal” assumptions which would make observations (i), (iii), and (vi) exact are simply not true, and the effects of varying the coupling strength VI@) are actually fairly complex. In particular, the differences between the metastable- and continuum-state potentials increase with the strength of VI(R), displacing the metastable-state radial wave function &,JR) relative to the continuum function I ) + ~ ~ ( and R ) changing the entire nature of the integrand in the numerator of the last term in eq 16. Moreover, as the difference potential AVd@(R)grows, eventually the first-order approximation of eq 11 will not longer provide a very accurate estimate of the level shift (though this is expected to be a less serious source of error). The importance of these sources of breakdown of observation (vi) is examined in the model problem calculations presented in the following section. Observation (vi) will also tend to break down if the coupling function AV,(q,R) is not dominated by a single product term &(q) V l ( R ) . In such a case, the numerator of eq 16 would involve the square of a sum of terms, each affected differently by oscillatory cancellation effects, so the simple cancellation suggested by eq 16 would not occur. In conclusion, therefore, the “all else being equal” conditions underlying observations (i), (iii), and (vi) assume that the only quantity which changes from one case to another is a scaling parameter governing the strength of the coupling function Vl(R). In principle, this condition is internally inconsistent, since increasing the strength of V l ( R )simultaneously increases the dif-
-
(22) Indeed, for some 24 of the different species formed by HF and HCN with various partners?v2’ the values of p ’ / 2 had an rms spread of only 7%.
The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 2171
Correlations in Vibrational Predissociation ferences between the vibrationally averaged initial- and final-state supporting potentials and hence shifts the initial- and final-state wave functions relative to one another. Thus, while appealing for the insight they offer, and applicable for cases involving nottoo-large coupling strengths, these correlations will always eventually break down. Illustrations of such behavior will be seen below. 111. Illustrative Calculations for Two-Dimensional Model
Problems A. Description ofthe Model. The present section will focus attention of observations (iv) and (vi) and use numerical calculations for a model problem to illustrate the predicted (mis)behavior. In this study, the coupling function AV,(q,R) is represented by a single term from the expansion of eq 5, so the effective vibrationally averaged potentials have the form represented by the last version of eq 8. With no loss of generality, the basis function 4 , ( 9 )may be defined such that its expectation value is identically zero for the ground ut'= 0 state of the monomer. For infrared transitions originating in this state, or for vibrational predissociation yielding this product species, this simplifies the discussion by causing changes in the coupling function V l ( R )to affect only the eigenfunction $dCt.n.=o(R) and eigenvalue Ec(u',n'=O) of the metastable upper state and the integrand of the predissociation matrix elements (see eq 14). In the model calculations reported below, the radial potential Vo(R)was assumed to have the simple damped exponentiaL(6,b) form Vo(R) = A e - B R - D 8 ( R ) C 8 / R 8&(R)C,/R6 -
(17)
where m=O
(18)
is the damping function form introduced by Tang and ToenniesZ3 The potential parameters were based on those determined by LeRoy and Hutson for the H2-Ar system24but were scaled to yield alternative potential strengths. Three different functional forms were considered for th,e coupling strength function V , ( R ) : (a) the potential itself, v ) ( R ) = Vo(R),(b) th,e derivative of the repulsive exponential term in this potential, ubL(R) = and (c) the detivative of the overall potential, V ) ( R )= dVo(R)/dR. Since we are interested in how the frequency shift and predissociation rate depend on V l ( R )and the momentum gap, the scaling factors associated with the expectation values and matrix elements of 4,(q)were ignored, and the difference potential of eq 10 and the coupling function of eq 14 were both represented by the same expression where the parameter K controls the net strength of the coupling, and the superscript (Y (= a, b, or c) identifies the form chosen for the coupling function. For K = 0.1 and AE, = 1000 cm-I, these three functions are plotted in Figure 3 (dashed curves), together with the associated initial- and final-state potentials (solid curves). Results are presented for two families of model systems. In the first, the supporting potential well depth was set at 200 cm-I and the radial motion reduced mass at 20 amu; this case corresponds qualitatively to the system Ar-HCl. In the second, the well depth was 50 cm-I and the reduced mass p = 2 amu; this corresponds approximately to H2-Ar.25 Since we wish to test observation (vi) and not the first-order approximation for the level shifts (eq 1 I ) , the frequency shifts were calculated from eq 1 by using the exact ground-state eigenvalues of the appropriate vibrationally averaged potentials. The corresponding predissociation (23) Tang, K. T.; Toennies, J. P. J . Chem. Phys. 1984, 80, 3726. (24) Le Roy, R. J . ; Hutson, J. M. J . Chem. Phys. 1987, 86, 837. (25) The remainin parameters defining the supporting potential Vo(R) of eq 17 were @ = 3.6 and Re = 3.6 A, while C, = (30006) cm-I A6 scales with the well depth e, and C, and the preexponential factor A are defined in terms of the other parameters in the usual way (see eqs 5 and 6 of ref 24).
1-l
. S
o
f
-50
w
L
-100
-150 -200
3.5
3
I
I
I
4
4.5
5
R I Angstroms Figure 3. For K = 0.1 and AE, = 1000 cm-I, plot-of the three types of coupling functions considered (dashed curves): (a) = V,(R), (b) qb) = - @ A d R ,and (c) p-) = dV,(R)/dR, and of the associated vibrationally averaged potentials, V,(R) for u = 0 and 1 (solid curves).
w)
rates were approximated by using the golden rule result of eq 15. Introduction of eq 19 then yields the ratio used for our numerical tests of observation (vi): predissociation rate
n
D,(R) = 1 - e-oRC(@R)"/m!
2oo
7-1
=-= (frequency shift)2 [A&12
B. Resulrs. For a given potential, a chosen type of coupling function, and a selection of values of the free monomer level spacing AE,, the level shifts and vibrational predissociation rates were calculated for a wide range of values of the coupling strength scaling parameter K. The eigenvalues and eigenfunctions of the vibrationally averaged effective potentials V J R ) were generated by using a standard radial Schrodinger equation while the predissociation matrix elements were computed by using the general bound continuum program BCONT." For a range of cases in which the initial-state well depth is 200 cm-' and the reduced mass p = 20 amu, Figure 4 shows how the ratio of eq 20 varies with the magnitude of the frequency shift (which in turn varies approximately linearly with the coupling strength factor K ) . The three segments of the figure correspond to the three types of coupling function described above, while the curves for each case correspond to values of the energy gap AE, ranging from 2 to 50 times the depth of the supporting potential Vo(R). Figure 5 presents analogous results for the case in which Vo(R)has a depth of 50 cm-I, p = 2 amu, and the energy gaps AE, range from 2 to 200 times the depth of Vo(R). The key feature of the results in Figures 4 and 5 is the fact that the ratios calculated from eq 20 are indeed approximately constant over ranges of coupling strengths sufficient to increase the predissociation rate by many orders of magnitude. Irregularities appear in some of these plots when the increasing relative displacement of the wave functions in the integrand in the numerator of eq 20 causes the resulting overlap integral to change sign. Such irregularities-are most prominent for the case of coupling function (a) because q ) ( R )changes sign at relatively short distances where the product of the wave functions has a relatively large amplitude. ' ] ~amazingly In spite of this, the calculated values of ~ - l / [ A v ; : are
-
(26) Le Roy, R. J. University of Waterloo Chemical Physics Research Report CP-230R2, 1986. (27) Le Roy, R . J. Comput. Phys. Commun. 1989, 52, 383.
2172 The Journal of Physical Chemistry, Vol. 95, No. 6, 1991
I
Le Roy et al.
I
I
-4-
-0-
1o.m
-55
t
-I log,,l
-I2 -161 -2
-I 0 I loglo I (frequency s h i f t ) icm-'
I
I
0 I (frequency skift)/cm-'I
Figure 5. As in Figure 4, for the case in which V,,(R) has depth 50 cm-" and p = 2 amu.
2
Figure 4. For a case in which Vo(R)has depth 200 cm-' and p = 20 amu, plot of the ratio of eq 20 vs the associated frequency shift for a range of values of thc energygap A,!? ;segments a, b, and c correspond to use of coupling functions V)(R), b ) ( R ) ,and respectively.
V)(R),
constant in view of the many orders of magnitude range of lifetimes considered. Miller's empirical observation that the predissociation rate increases as the square of the frequency shift? which is qualitatively justified by the form of eq 16, is clearly further confirmed by these model calculations. However, these results also illustrate the sensitivity of this correlation to changes in the form of the coupling function. This is demonstrated in Figures 4 and 5 by the very different predissociation rates which may be associated with results obtained for the same AE, value and the same frequency shift but different types of coupling function (a = a or b or c). These differences highlight the danger of overinterpreting the results of approximate analytic treatments based on particular types of coupling functions. The results in Figures 4 and 5 also illustrate the sensitivity of the ratio of eq 20 to relative displacements of the initial- and final-state vibrationally averaged potentials. In particular, the deviations of the various curves from near-horizontal behavior shows that when the coupling strength and momentum gap are both large, the coupling strength does not cancel out of this ratio in the manner suggested by eq 16. Thus, while correlations such as observation (vi) may be expected to hold for families of similar systems for which the coupling is not too strong, they will certainly not be universally valid. Our model problem results may also be used to examine the momentum-gap law ~ r e d i c t i o n ~that -~ log
(T-')
-~(2pAE,,,)'/~/@h
2
(21 1
To this end, the results of Figure 5c have been replotted in a manner which tests this behavior?* For coupling strengths chosen to yield frequency shifts Av$ ranging from -0.01 to - t 00 cm-I, (28) Case c was chosen because it is the same typc of coupling function assumed in the momentum gap law
10
-
h
-5
-
AV = -0.009
The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 2173
Correlations in Vibrational Predissociation 250
0
-
-250
I
E .
u--___--
E ("'-1v')--m
a
*
K
-
Figure 8. Sample illustration of product fragment channels associated
with the different types of coupling functions driving vibrational predissociation of a metastable level of energy Ec(u,',u.l,n'=O). -1250
1
I
I
2.5
3
I
3.5
I
I
I
4
4.5
5
I
R IAngstroms Figure 7. Effcctive metastable-state potentials (solid curves), wave functions (dashed curves), and eigenvalues (short dotted lines) associated with a 200 cm-I deep potential, fi = 20 amu, and A& = 1000 cm-I, for a coupling function of type c and K = 0, 0.2, and 0.23, respectively.
the predissociation process also shifts the effective initial- and final-state potentials relative to one another. The resulting relative displacement of the initial- and final-state wave functions can completely change the integrand of the bound continuum overlap integral. For example, for the case of a 200 cm-' deep supporting potential, p = 20 amu, and a coupling function of type c, Figure 7 shows how increasing the strength of the coupling shifts the energy of the metastable level and displaces its wave function to smaller distances. Comparing these plots with the schematic wave functions seen in Figure 1 shows that such displacements will change both bound continuum matrix elements and the nature of their dependence on the internal energy release, AEm. While level shifts of a magnitude comparable to the supporting potential well depth may seem unlikely, another choice of coupling function could have effected the same wave function displacements seen in Figure 7 at the cost of much smaller level shifts. In any case, the object here is not to argue that such a complete breakdown of the momentum-gap law is highly probable. Rather, it is to point out that such effects can occur and hence to underline the fact that no serious discussion of trends in predissociation rates can neglect consideration of the nature of the coupling function and its effect on the effective supporting potentials.
-
-
IV. The Case of More Than Two Degrees of Freedom The preceding sections have considered only the simple twodimensional problem of the coupling between a single internal vibrational mode of the monomer and motion along the van der Waals bond. For a system that has more participating internal degrees of freedom, the same kinds of arguments apply, but the simultaneous coupling to additional degrees of freedom will tend to obscure the types of correlations discussed above. To illustrate this point, consider a system with two internal degrees of freedom, denoted cy and K , both of which couple to motion along the van der Waals bond. These may correspond to different monomer vibrations, such as a bend and a stretch, or to an internal rotation and a monomer stretch. These internal mode coordinates are denoted 9,, and qx,and the corresponding generalization of eq 5 is
an internal vibration, these @(q,) and #(q,) basis functions may be powers of the stretching coordinate itself, while for an internal rotation they might be, for example, Legendre polynomials. If the internal coordinate basis functions are chosen appropriately, the strengths of the radial functions Vk,/(R)should decrease monotonically with ( k l ) . In this case, the frequency shifts will be dominated by a sum of the two ( k I ) = 1 terms:
+
AU"~:~:" e [@(u,',u,') U.)&
- &'(O/,U,'')]
+
X
-
(J/U,,U",,(R)l~,,O(R)IJ/U~,~,~(R))
[a(u;P;)
- ?J!(U,",u;91
(J/",,"'.,(R)I~o,l(R)IJ/u.,,,"(R)) (23)
This result is distinctly more complicated than was eq 11, and there is no a priori reason for expecting the coupling strengths functions V1,,(R)and Vo,l(R)associated with different internal degrees of freedom to scale in exactly the same way from one system to another. The introduction of additional internal degrees of freedom affects the golden rule predissociation rates of eq 15 in two ways. The first is that there can be many more possible dissociation channels, since the single sum in eq 15 is replaced by a double sum over u / and u / . Momentum gap minimization considerations do simplify this situation by introducing a strong preference for the open channels corresponding to u,' - 1 and u,' - 1, where (u,',u;,n') are the quantum numbers identifying the metastable predissociating state. However, unless the metastable state corresponds to a very low frequency internal mode, predissociation may also be accompanied by increased excitation of some other internal mode. Thus, in addition to simple (internal vibration) (fragment translation) channels, this also allows cases in which (internal vibration) energy in one mode is converted into a sum of (fragment translation) plus (additional internal energy) in some other mode. This is schematically illustrated in Figure 8, where the Vk,/(R)labels associated with the three arrows indicate the component of the potential driving the production of fragments in that product channel (see below). Since the strengths of the V,,,(R) functions should decrease rapidly with ( k I), to a first approximation we can ignore terms in eq 22 for which k 2 2 or 1 2 2. However, the mixed term associated with V1,,(R) cannot be omitted so readily. This is shown by the form of the golden rule matrix element inside the sum in the first line of eq 15, as generalized to describe the predissociation of a metastable state characterized by quantum numbers (uk,u,,n') to yield fragments characterized by internal state quantum numbers 0", and u,":
-
+
&;,u/
( ua'14i'(qu) Iu,") 'U/,Um,!
('Jg,I$!
(91) ''6'(
(J/um~t,u;,~(R)l Vi.o(R)l A m * , ~ ; , )~ + (R) ) ( $'u.'p/,~(R) I0' , I ( R )I$U/,O;,~(') ) +
(u,'I4iYqa)lu/
) ( ux'14i(9x)lux") x
(J/u/*u/,€(R)I VI . I (R)lAJ/,u;,d(R) ) (24) The Kronecker delta's in this expression clearly prevent the terms ( k J ) = (1,O) and (0,l) from contributing to predissociation
2174 The Journal of Physical Chemistry, Vol. 95, No. 6, 1991
processes in which more than one of the monomer quantum ) usually numbers changes. At the same time, since V 1 , , ( Rwill be substantially weaker than the Vl,o(R)or Vo,,(R)functions, the latter terms should dominate fragmentation processes in which only one monomer quantum numbers change (see Figure 8). Thus, the predissociation channels in which only one, or the other, or both of the internal quantum numbers change will each be dominated by a different potential coupling term. Moreover, in spite of the relative weakness of Vl,l(R),the Franck-Condon overlap considerations underlying the momentum-gap law will strongly favor exit channels with the smallest possible energy release. Thus, in a predissociation process where the amount of internal energy converted to fragment translation may be decreased if a second internal mode gains energy, the relatively weak VIql(R)term may dominate the overall predissociation rate. Figure 8 illustrates a case in which this may occur. For the example shown in Figure 8, the calculated predissociation ratc involves a sum of terms associated with flux into the three possible product channels:
On combining eq 23 with eq 25, one obtains the generalization of eq 16 appropriate to this case:
The terms ( V k , , ) h h and ( Vk,r)bc respectively represent bound-state expectation values and bound-continuum off-diagonal matrix elements of the indicated function between appropriate radial wave functions, and the coefficients a - e represent matrix elements or expectation value differences involving the monomer vibrational coordinates (see eqs 23 and 25). In contrast, if written in the same form as eq 26, the one internal degree of freedom result of eq 16 is simply
7-'/[Ae:I2 = 2*,al( ~ 1 , o ) a ~ ~ / [ d ( ~ i , o ) b 6 (27) l~
Le Roy et al. Even so,the qualitative agreement with the prediction of our simple two-dimensional model seems better than should be expected, since the fact that product monomers are often produced in highly excited rotational statesg0implies that the predissociation process is driven by cross terms analogous to the Vll(R) function in the above example. In such a case, however, the above arguments suggest that the clear correlation seen in Figure 2 should be lost. This apparent contradiction is removed if the predissociation actually occurs as a two-step process, in which the initial energy release occurs impulsively within a relatively rigid complex with only one participating internal degree of freedom. With the overall rate decided in that initial step, the final product-state distribution would then be determined during the retreat along the fragment separation coordinate. The persistence of the correlation seen in Figure 2 appears to confirm the appropriateness of this interpretation. These results therefore lead to one additional conjecture. Observation (vii): When more than one internal degree of freedom participate in the predissociation process, correlations such as that predicted by observation (vi) should break down; their persistence is implicit euidence for a two-step impulsive mechanism for predissociation in these systems. This is a very nice conclusion, since it allows the simple reduced-dimension model of section 111 to facilitate the interpretation of what would otherwise be very complex multichannel processes. Note that while qualitatively similar, this mechanism differs from IVR in that the channel populated by the impulsive initial step is itself an unbound continuum state.
V. Concluding Remarks Examination of a simple two-dimensional model for vibrationally predissociating van der Waals dimers led to a number of observations regarding both the nature of this process, and its relationship to the vibrational frequency shift undergone by the monomer upon forming the dimer. These are the following: (i) "All else being equal", the magnitude of the vibrational frequency shift will be directly proportional to the strength of the same radial coupling function which drives the predissociation process. (ii) The magnitude of the frequency shifts associated with a given monomer vibrational series should increase approximately linearly with Po. (iii) "All else being equal", the predissociation rate varies as the square of the strength of the same radial coupling function which causes the vibrational frequency shift. (iv) The effect of monomer vibrational averaging can cause catastrophic failure of the momentum-gap law, yielding predissociation rates which increase (rather than decreasing exponentially) with the momentum gap. (v) For sufficiently high monomer vibrational excitation, anharmonicity effects may reverse both the momentum-gap law preference for Av = -1 predissociation and the tendency for the predissociation rate to increase with v . (vi) "All else being equal", the predissociation rate will vary as the square of the frequency shift. Of these observations, (ii) is perhaps the most obvious and (v) the most speculative. Numbers i and iii are of some interest in themselves, but are perhaps most important because they combine to give observation (vi). The latter three conjectures were presented previously in a different context by Weber and Rice,Io*l1 while (vi) was empirically discerned by Miller from examination of a range of experimental data? Finally, observation (iv) provides a dramatic warning that "all else is never equal" and that this fact can significantly affect even the most obvious expected trends. The message underlying many of these points is simply that when considering even qualitative trends and correlations in vibrational
If the same coupling term dominates both the numerator and denominator of eq 26, observation (vi) should still hold. If this is not the case, however, that correlation would occur only if from one system to another (a) the relative strengths of the three leading terms in eq 22 do not change and (b) the relative importance of the terms contributing to both eqs 23 and 25 does not change. In view of the complex subtleties of the oscillatory cancellation effects which afflict bound-continuum matrix elements, this last requirement is particularly stringent. For this reason, one would generally expect observation (vi) to break down for predissociation from a metastable state for which there exists more than one type of product channel. The above limitations on the region of validity of observation (vi) appear to be very restrictive. In particular, any van der Waals molecule that can undergo vibrational predissociation has at least two internal degrees of freedom, since in addition to the energized monomer vibration there will be a bending or internal rotation coordinate that correlates with fragment rotation. On the other hand, Miller did find an approximately quadratic relationship between frequency shifts and predissociation lifetimes for a wide range of systems (see Figure 2).9 Thus, in spite of the above objections, the kind of correlation suggested by observation (vi) does seem to be experimentally verified. (29) The exceptional case is H2-HF, which has a very small reduced mass When assessing the significance of this conclusion, it should and hence a much smaller momentum gap than is associated with the other HF complexes considered?.*' However, even the result for this system (the be noted that all of the cases considered in Figure 2 involve loss square point farthest to the left) still lies within the shaded band Seen in Figure of vibrational energy by an H-F diatom or H-C= monomer 2. fragment and that, with one exception, the momentum gaps as(30) Dayton, D. C.; Jucks, K . W.: Miller, R. E. J . Chem. Phys. 1989, 90, sociated with their vibrational predissociation are very ~ i m i l a r . ~ ~ . ~263 ~ I.
J . Phys. Chem. 1991, 95. 2175-2181 predissociation rates, one cannot ignore the effects of the strength and shape of the intermode coupling function. One final question regarding observation (iv) concerns exactly when one may expect this type of "catastrophic breakdown" to occur. Within the assumptions underlying the momentum-gap law, the metastable-state wave function only has significant amplitude in the region to the right (at larger R ) of the open-channel turning point, where the associated continuum wave function is purely oscillatory in nature. However, Figure 9 shows that this breakdown occurs when the relative displacement of the effective adiabatic radial-channel potentials moves the turning point of the open-channel potential into the classically allowed region between the turning points of the metastable state's closed-channel potential. Thus, it is not the magnitude or even the sign of the associated level shift that matters, but rather the change in the average value of R for the complex. As a result, if the effective rotational constant associated with the overall rotation of the complex increases by a even a few percent on internal excitation of the monomer, the momentum-gap law should be completely disregarded. Finally, extension of the arguments developed for the two-dimensional model problem to systems with more than one internal
2175
degree of freedom was seen to introduce substantial complications. In particular (vii) When significant couplings involving more than one internal degrees of freedom contribute to the predissociation process, correlations such as that predicted by observation (vi) should disappear; their observation would be implicit evidence for a two-step impulsive mechanism for predissociation. The fact that such correlations persist for the wide range of cases surveyed by Miller,9 for which a large number of bending or internal rotation states are accessible, is thus taken as evidence that these vibrational predissociation processes are fundamentally impulsive, with the distribution of fragment internal states being decided as the separating fragments retreat along the dissociation coordinate. Acknowledgment. We are grateful to Professor R. E. Miller for helpful discussions and for providing us with an unpublished data tabulation and to Dr. T. Slee, Mr. C. Chuaqui, and the two referees for their constructive comments. We are also pleased to acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada, through provision of both grant support to R.J.L. and an Undergraduate Summer Research Assistantship to M.E.L.
Spectroscopy of Indole van der Waals Complexes: Evidence for a Conformation-Dependent Excited State Michael J. Tubergen and Donald H. Levy* The James Franck Institute and The Department of Chemistry, The University of Chicago, Chicago, Illinois 60637 (Received: April 6, 1990: In Final Form: August 20, 1990)
Electronic excitation spectra of van der Waals complexes containing indole were obtained by using laser-induced fluorescence and resonantly enhanced, two-photon ionization. These complexes can be grouped into two classes. Acetamide, formamide, water, and methanol were found to shift the origin transition of the complex more than 400 cm-l to the red of the indole origin. Complexes of indole with methylated an+les, tetrahydrofuran, and 1,4-dioxane did not display transitions shifted as far to the red. Dispersed fluorescence spectra of the two groups of complexes were also found to differ. Complexes of indole with the amides, water, and methanol were found to fluoresce in a broad band centered 1500 cm-I to the red of excitation, while fluorescence from complexes of indole and the methylated amides, tetrahydrofuran, and 1 ,Cdioxane was sharp, with most of the intensity occurring at the excitation frequency. These differences are explained in terms of the ability of the complex partner to form a hydrogen bond with the ?r electron cloud of the indole. In such complexes, the interaction of the nearby polar solvent lowers the energy of the 'La state and allows it to mix effectively with the ILbstate. Complexes of water with I-methylindole and 3-methylindole are also shown to display the same two types of spectroscopic behavior.
Introduction The indole chromophore of the amino acid tryptophan has been the subject of extensive study because of its importance in protein spectroscopy.l In solution, the fluorescence behavior of indole is strongly affected by the environment of the chromophore. In polar solvents, the fluorescence from indole is shifted far to the red of the absorption band, but this large fluorescence shift is reduced in nonpolar solvents. One explanation of the fluorescence shift is that the excited indole molecule forms stable complexes, called exciplexes, with the solvent mole~ules.~JTwo different solvent binding sites have been identified for 3-methylindole dissolved in I - b ~ t a n o l . At ~ one site nucleophilic solvents are attracted to the indole nitrogen proton, while at the second site electrophilic groups are attracted to the T electron cloud around
the indole 3 position. Molecular beam studies of indole complexed with various solvents have shown that proton donation from the indole to the solvent is common in van der Waals complexes of indole and nucleophilic solvent^.^ Similar structures have been reported for van der Waals complexes of water with 7-azaindole5 and carbazole.6 Indole has two low-lying excited electronic states, the ILb and the 'La states, which complicate the interpretation of the fluorescence spectroscopy. The 'La state is thought to be strongly stabilized by polar solvents, while the ILb state is only slightly affected. The relative location of these two excited states has recently been determined for indole dissolved in cyclohexane and I-butanol.' The effect of the polar solvent on the IL, state is so strong that the 'La state becomes the lowest energy excited
( I ) Creed, D. Photochem. Photobiol. 1984,39, 537. (2) Hershberger, M.V.; Lumry, R.W.Photochem. Photobiol. 1976,23,
(4) Hager, J.; Wallace, S . C. J. Phys. Chem. 1984, 88, 5513. (5) Fuke, K.; Yoshiuchi, H.; Kaya, K . J . Phys. Chem. 1984, 88, 5840. (6) Bombach, R.; Honegger, E.; Leutwyler, S . Chem. Phys. Left. 1985, 118, 449. (7) Rehms, A. A.; Callis, P. R. Chem. Phys. Lett. 1987, 140, 83.
-19_..I
(3) Hershberger,
M.V.; Lumry, R.; Verrall. R. Photochem. Photobiol.
1981, 33, 609.
0022-365419112095-2175$02.50/0
0 1991 American Chemical Society
