Fermi resonance and vibrational energy flow in van der Waals

J . Phys. Chem. 1986, 90, 1790-1799

1790

Fermi Resonance and Vibrational Energy Flow In van der Waals Molecules: Aromatic Ring-Rare Gas Complexes George E. Ewing Department of Chemistry, Indiana University, Bloomington, Indiana 47405 (Received: September 9, 1985)

We offer a natural extension of a model used to discuss vibrational energy transfer in colliding molecules and van der Waals complexes. This extension allows an estimate of the matrix elements responsible for Fermi resonance between chemical bond and van der Waals bond vibrations in aromatic molecule-rare gas complexes. Together with an estimate of density of states for these complexes, we discuss the extent of Fermi resonance. We show how Fermi resonance can dramatically affect the vibrational predissociation channels and the spectroscopy of aromatic molecule-rare gas complexes. As an example of our model, we consider the experimental study of the s-tetrazine-argon van der Waals molecule.

1. Introduction Our present understanding of vibrational energy flow in van der Waals molecules has a historical parallel in the explanation of vibrational relaxation of colliding molecules. For colliding molecules, vibrational energy relaxation was first discussed 50 years ago by Landau and Teller, who assumed that energy was transferred via coupling through the intermolecular potential solely to the translation motions of the colliding partners.] The model showed that light molecules with low vibrational frequencies colliding at high velocities against a steep intermolecular repulsive wall would relax most efficiently. The propensity rules for vibrational relaxation by this model have been variously expressed as an energy a momentum gap: or an exponential gap (that is also associated with an adiabaticity parameter).Iq4” Over the years the form of the intermolecular potential function used to couple vibrational and translational motions of colliding molecules has become more realistic, but the same propensity rules remain. One of these intermolecular functions, the Morse potential, has yielded vibration-translation (V-T channel) efficiencies in reasonable agreement with experiment for a variety of colliding molecule^.^^^ The Morse potential has also been recruited into the discussion of vibrational predissociation of van der Waals molecules by Beswick and Jortner9 and by Ewing.Io The same energy gap,” momentum gap,I2 and exponential gapI3J4propensity rules have evolved from the theory of van der Waals energy flow that were found in the colliding-pair energy-transfer models. The model has been successful, for example, in quantitatively accounting for experimental measurements of vibrational predissociation in I,-He as reported in the pioneering investigation by Smalley, Levy, and Wharton.I5 A wealth of experimental data for relaxation of vibrationally excited colliding molecules have revealed energy transfer to ro(1) L. Landau and E . Teller, Fiz. Z h . 10, 34 (1936); D. Ter Haar, “Collected Papers of L. D. Landau”, Gordon and Breach, New York. 1965 (English translation). (2) J. T . Yardley, ‘Introduction to Molecular Energy Transfer”, Academic Press, New York, 1980. (3) J. D. Lambert, ‘Vibrational and Rotational Relaxation in Gases”, Oxford University Press, London, 1977. (4) K. F. Herzfeld and T. A. Litovitz, ‘Adsorption and Dispersion of Ultrasonic Waves”, Academic Press, New York, 1959. (5) R. N . Schwartz, Z . I. Slawsky, and K. F. Herzfeld, J. Chem. Phys., 20, 1591 (1952). (6) R. D. Levine and R. B. Bernstein, “Molecular Reaction Dynamics”, Oxford University Press, New York, 1974. (7) S. L. Thompson, J . Chem. Phys., 49, 3400 (1968). (8) H. Abdel-Halim and G. E. Ewing, J. Chem. Phys., 82, 5442 (1985). (9) J. A. Beswick and J . Jortner, Chem. Phys. Lett., 49, 13 (1977). (10) G. E . Ewing, Chem. Phys., 253 (1978). (11) J. A. Beswick and J. Jortner, J . Chem. Phys., 68, 2277 (1978). (12) G. E. Ewing, J. Chem. Phys. 71, 3143 (1979). (13) G. E. Ewing, Faraday Discuss. Chem. SOC.,73, 402 (1982). (14) G. E. Ewing in “Intramolecular Dynamics”, J. Jortner and B. Pullman, Eds., D. Reidel Publishing, Dordrecht, 1982, pp 269-285. (1 5) R. E. Smalley, D. H. Levy. and L. Wharton, J. Chem. Phys., 64,3266 (1976).

0022-3654/86/2090- 1790$01.50/0

tational levels or vibrational levels of the collision ~ a r t n e r . These ~.~ are termed the V-R or V-V relaxation channels, respectively. New propensity rules for these V-R and V-V channels have evolved from the theory of energy transfer in colliding molecule s y s t e m ~ . * - ~Analogously, ~’~ V-R and V-V channels have been applied to the discussion of vibrational predissociation of van der Waals molecule^.^^,'^ For collisional energy transfer in some sy~terns,’~ none of the V-T, V-R, or V-V propensity rules appear to work, and theoretical models have invoked Fermi resonance effects to account for relaxation e f f i c i e n c i e ~ . ’ ~We ~ ~are ~ now at this point in explaining energy flow in van der Waals molecules. Fermi resonance was introduced into discussions of energy transfer in van der Waals molecules before a clear need was to be found in explaining experimental data.17~18~21~22 Here Fermi resonance provides a mixing of a vibrational level in the chemically bonded molecule with near-resonant levels associated with vibrations against a van der Waals bond in the complex. There is now strong experimental evidence for Fermi resonance effects in two classes of van der Waals molecule energy-transfer studies. In one class are the s t ~ d i e s of ~ ~an- electronically ~~ and vibrationally excited aromatic molecule ( z g . C2H2N,, C&, C6H5NH2,or C6H4F2)complexed with a rare gas atom (e.g. He, Ne, or Ar). Relaxation patterns in these systems present two puzzling features. First, when vibrational predissociation occurs, its efficiency often shows no correlation with energy gap, momentum gap, or exponential gap. Second, emission from the complex often appears to be from a different vibronic level than that excited. While Fermi resonance has entered into the qualitative discussion of some of these results, we shall offer in this paper a theoretical model that can account for the effects in a semiquantitative way. We shall show that the energy levels involving vibrations against the van der Waals bond are coarsely spaced and Fermi resonance couples a small number of them to certain chemical bond vibrational levels. New relaxation channels are a natural consequence of this Fermi resonance. In another class of experiments by Janda et al.,* and by Gentry (16) C . S. Parmenter and K. Y. Tang, Chem. Phys., 27, 127 (1078). (17) G . E. Ewing, Faraday Discuss. Chem. SOC.,73, 325 (1982). (18) J . A. Beswick and J . Jortner, Adu. Chem. Phys., 47, 363 (1981). (19) J. Hager, W. Krieger, T. Ruegg, and W. Walther, J. Chem. Phys., 72, 4286 (1980). (20) W. Zinth, C. Kolmeder, B. Benna, A. Irgens-Defregger, S. F. Fisher, and W. Kaiser, J. Chem. Phys., 78, 3916 (1983). (21) J. A. Beswick and J. Jortner, J . Chem. Phys., 71, 7437 (1979). (22) G.E. Ewing, ‘Potential Energy Surfaces and Dynamics Calculations”, D. G. Truhlar, Ed., Plenum Press, New York, 1981, p 75. (23) D. V. Brumbaugh, J. E. Kenny, and D. H . Levy, J . Chem. Phys., 78, 3415 (1983). (24) J . J. F. Ramaekers, H. K. van Dijk, J. Langelaar, and R. P. H. Rettschnick, Faraday Discuss. Chem. Soc., 75, 183 (1983). (25) T. A. Stephensen, and S. A. Rice, J . Chem. Phys., 81, 1083 (1984). (26) E. R. Rernstein, K. Lawe, and M. Schauer, J . Chem. Phys., 80, 207 ( 1 984). (27) D. Catlett and C. S. Parmenter, private communication. (28) M . P. Casassa, D. S. Bomse, and K. C. Janda, J . Chem. Phys., 74, 5044 (1981).

0 1986 American Chemical Society

Fermi Resonance in Aromatic Ring-Rare Gas Complexes

f

The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 1791 ring (e.g. s-tetrazine) lies in the x-y plane, and the rare gas atom (e.g. Ar) lies above along the z axis as shown in Figure 1. The three normal modes of a rare gas atom and an aromatic ring against the van der Waals bond are shown qualitatively in the lower portion of Figure 1. The stretching mode, v,, then belongs to the a l irreducible representation, and the bending or tangential modes, vx, and vv, belong to b2 and bl, respectively. The stretching mode with displacement z is against the van der Waals bond whose potential we9Jotake to be of the Morse form: V(z) = De[exp(-2az) - 2 exp(-az)] (1)

t

The steepness of the well, with depth De, is given by the range parameter a. The resulting energy levels for this vibration are

I

m

a2h do G(u,) = -(u, 2V.h

I

+ 1/2)

a2h 4ncCc,

- -(u,

+ 1/2)2

(2)

where we have used u, to label the van der Waals stretching mode quantum number. (The energy units for G(u,) are in cm-I when we use cgs units for all the constants in this equation.) The reduced mass of the van der Waals molecule is

I 4 (a,) Figure 1. Normal modes for an aromatic molecule-rare gas atom van der Waals molecule. The example is taken from the qualitative dis-

(3)

placements in s-tetrazine-Ar. T h e displacements against the van der Waals bond are given in the lower portion of the figure. In the upper right portion, v, represents the 16a vibration of s-tetrazine.

where ma is the mass of the rare gas atom and m, is the mass of the ring. The parameter do is given by

et ethylene is complexed with another ethylene molecule to form a dimer. Vibrational excitation of the bending mode has resulted in predissociation and diffuse bands. This diffuseness was originally interpreted28as homogeneous Heisenberg broadening corresponding to a predissociation limited lifetime of s. An alternative explanation by GentryMinvolves Fermi resonance between the chemical bond bending vibrational level of ethylene and upper levels involving vibrations against the dimer van der Waals bond. In the absence of vibrational predissociation, coupling of these discrete levels would give rise to a multitude of closely spaced lines spread out over a region governed by Fermi resonance. On account of predissociation, each of these levels broadens and the lines blur together giving an overall width determined by Fermi resonance. We shall show in a later paper that this picture resembles the model developed by Bixon and Jortner3Iq3?for radiationless transitions in molecules with densely spaced vibrational levels. This paper is organized in the following way. Section I1 is entitled The Theoretical Model. The form of vibrations against the van der Waals bond for a rare gas atom complexed to an aromatic ring is first discussed in part A. The density of states of the van der Waals motions is explored in part B. The chemical bond and van der Waals bond vibrations are then classified in part C, and the coupling term that connects the chemical bond and van der Waals bond vibrational levels is given in part D. The vibrational shift, a consequence of the use of the coupling term in first-order perturbation of the vibrational energy levels, is discussed in part E. In part F we then describe Fermi resonance, using the coupling term and first-order perturbation of the vibrational wave functions. In part G, the qualitative appearance of the spectrum of the aromatic ring-rare gas atom complex is described. Finally, it is shown in part H how Fermi resonance can affect vibrational predissociation channels. In section 111, an example of the theoretical model is applied to the C2N4H2-Ar complex. In section IV, we conclude with a summary and a call for new experiments.

-

11. The Theoretical Model

A . van der Waals Bond Vibrations. We classify the van der Waals molecule according to the C2, point group. The aromatic (29) M. A. Hoffbauer, K. Liu, C. F. Giesse, and W. R. Gentry, J . Chem. Phys., 78, 5567 (1983). (30) W. R. Gentry in "Resonances", D. G. Truhlar, Ed., American Chemical Society, Washington, D.C., 1984, ACS Symp. Ser. No. 263, p 289. (31) M.Bixon and J. Jortner, J . Chem. Phys., 48, 715 (1968). (32) M.Bixon and J. Jortner, J . Chem. Phys., 50, 4061 (1969).

do =

(2p,De)'I2

ah

(4)

Since De and a are not accurately known, we are free to adjust either quantity slightly to make do an integer. The number of bound states supported by the Morse well is then equal to do. The vibrational wave functions (u,I are given e l s e w i ~ e r e . ~ ~ The bending modes are more difficult to model. An inspection of Figure 1 for v, shows that this vibration involves bending of the ring about the x axis, a dipping of the rare gas atom in an opposite sense, as well as tpngential motions of the atom and the ring along the y axis. We must resort to a normal-coordinate analysis34to see the connections among the vibrational frequency, the force constant, and the masses with their arrangement in the van der Waals molecule. When the atom (e.g. argon) is massive relative to the atoms making up the ring (e.g. C6H6or C2N4H2) and its distance from the ring is large relative to the radius of the ring, a consideration of the form of the normal mode shows that v, depends on the van der Waals molecule structure principally through the moment of inertia I, of the ring about the x axis. The normal mode displacement is then approximately the tilting of the ring against a relatively stationary atom. The form of the potential describing this motion can then be modeled by V(0,) = (V,/2)(1 - cos 28,) (5) where Ox is measured about the x axis (see Figure 1) with the factor of 2 appearing because the potential is symmetric to a rotation of the ring by ?r. The barrier hindering rotation is V, which is loosely related to De, the van der Waals bond strength. We have used this model for internal rotation in van der Waals complexes e l s e ~ h e r e . For ~ ~ levels near the bottom of the well the angular displacements are small, and the potential of eq 5 may be approximated which is that of a harmonic oscillator with bending force constant 2Vx. The corresponding energy levels appropriate to eq 6 are given (in cm-') by the first term in the equation G(%) =

V,O(U,

+ 1/2)

- gxx(% + 1 / a 2

(7)

where u, labels the bending mode quantum number with frequency (in cm-') (33) V. S. Vasan and R. J. Cross, J . Chem. Phys., 78, 3869 (1983). (34) G. Herzberg, "Molecular Spectra and Molecular Structure", Vol. 11, Van Nostrand, Princeton, NJ, 1945. (35) G. Henderson and G. E. Ewing, J . Chem. Phys., 59, 2280 (1973).

1792 The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 uXo

= 2(V,B,)'l2

Ewing

(8)

V, is the barrier height (in cm-I), and B, is the rotational constant (in cm-I) of the ring about its x axis. The corresponding harmonic oscillator wave functions (u,( are given by36

I y-1 ;0,0,1) The anharmonicity term in eq 7 with constant g,, will be discussed later. It accounts for the narrowing of the spacing between adjacent energy levels as shown by the solutions of the Matfiieu equation36for the upper regions of the cosinusoidal potential of eq 5 . For bending motion about they axis, there exists an identical set of equations like those in eq 5-10 but with the subscript x replaced by y. So far we have ignored coupling among the van der Waals vibrational modes. In fact, the coupling is important as can be appreciated by considering the effect on the bending modes when one of the highest stretching mode levels is populated. With high stretch excitation the atom has a high probability of being far from the ring and the van der Waals attraction is lowered. Consequently, the effective barrier to rotation V, in eq 8 is reduced and the bending frequency vxo is lowered. By an argument given elsewhere3' the effective well depth De for the stretching mode is reduced when high bending modes are excited. The natural consequence of these considerations is the necessity to include cross terms g,,, gy,, and gxyin the description of the energy levels of the van der Waals modes. We then write these term values as

Figure 2. Energy levels in Fermi resonance. The [ I ) state represents vibrations against chemical bonds only. The ..., Im + I ) , Im), Im - l ) , ... stack represents states involving both chemical bond and van der Waals bond vibrations. Fermi resonance occurs between I I ) and the ..., jnz - 1), Im), lm + I ) , ... stack.

function.39 We have made quantitative use of this potential function elsewhere.40 When the rare gas atom is far removed from the ring (e.g. the dissociated state), the bending or tangential motions go over into free rotation of the ring. The corresponding free rotor wave function may be approximated by36

(J,I = (1 / 2 a ) 1 ~ 2 e * i J ~ 8 ~

(13)

with J, labeling the rotational quantum number. There are corresponding wave functions for rotation about they axis labeled by Jy. B. Density of van der Waals Vibrational States. Simple expressions exist for estimating the density of vibrational states, p, for harmonic or Morse oscillator^.^^ However, since we believe that all the van der Waals vibrations are coupled, we shall use eq 11 to count the number of states directly. We shall find in a later example that the density of vibrational states is lowtypically p 1 to 10 states per cm-I in the upper reaches of the well. If we include rotational levels of the van der Waals molecule, the total state density increases greatly. However, we imagine here that all van der Waals molecules are in their lowest ( J = 0) rotational state. In any radiationless transition or Fermi resonance, the total angular momentum must be conserved so we do not include additional rotational states in our counting. The bending vibrations are associated with the double-minimum well of eq 5 , and each harmonic oscillator vibrational u, or uy level is twofold degenerate.34 However, since all the bending levels we will consider are well below the barrier, the degeneracy is not split appreciably by tunneling. We shall ignore these effects. C. van der Waals Molecule Vibrational Energy Levels. Vibrational energy levels for van der Waals molecules must reflect motions against both chemical bonds and the van der Waals bond. In order to provide a point of focus for this discussion, we can look at Figure 2 for the relative spacings for these levels. The notation Iui;ux, u,, u,) is used where ui labels the vibrational level of the ith normal mode against chemical bonds. The spacings of the low-lying van der Waals states, e.g. Iui - l;O,O,l) and Iui - 1;0,0,0) shown in the lower portion of Figure 2, are typically an order of magnitude less than chemical bond spacings Iui;O,O,O) and Iui - 1;0,0,0). It is convenient to call Iui;O,O,O) the 11) state which represents chemical bond vibrational excitation but no van der Waals bond excitation. In near resonance with the 11) state is a stack of states called ..., Irn - 1), Im), Im + 1) .... T,his stack represents vibrational excitation of the van der Waals bond to high ux, uy, and u, levels that are in combination with the lower ui- 1 level of the chemical bond vibration. In general, these states can be written Iui- 1;ux,u,,,u,) and are shown in the upper right portion of Figure

-

where vZo is taken from the first term in eq 2, v X o is given by eq 8, and vy0 is from an equation analogous to eq 8 but with the x subscript changed t o y . The anharmonicity term g,, is obtained from eq 2. There now remains the problem of choosing the values of the other g terms. For typical van der Waals parameters (to be discussed later) we can use eq 2 to discover that gz,/vzo This order of magnitude estimate of the ratio of anharmonic to harmonic constants for the van der Waals mode is consistent with that for ordinary molecules.34 We shall use this as a guide for estimating all the other anharmonicity constants. While the use of eq 11 may not be entirely satisfying for quantitative evaluation of the upper levels of van der Waals molecules, we accept it for estimating vibrational-state densities. When the rare gas atom is light (e.g. He) and the aromatic molecule massive (e.g. C14Hlo),the van der Waals bending motions go over into two tangential vibrations of the atom over the plane of the aromatic molecule along the x or y axis.38 This potential function can then be modeled by replacing eq 5 by the hyperbolic function

-

V(X) = V,(1 - cosh-2 bx)

(12)

where b is a range parameter determining the steepness of the potential well. This function, shaped like the cross section of a tulip, nicely mimics the shape of van der Waals potentials calculated for atom-aromatic ring interaction^.^^ Both the energy levels and wave functions associated with this hyperbolic potential function are known and resemble those for the Morse potential ~~

~

( 3 6 ) H. Eyring, J. Walter, and G. E. Kimbal, "Quantum Chemistry", Wiley, New York, 1944. (37) G. E. Ewing, Chem. Phys. 63,411 (1981). (38) M. J. Ondrechen, Z. Berkovitch-Yellin, and J. Jortner, J . Am. Chem. SOC.,103, 6586 (1981).

(39) 1. I. Gol'dman and V. D. Krivchenkov, "Problems in Quantum Mechanics", Pergamon Press, Reading, MA, 1961, (40) G. E. Ewing, unpublished work. (41) P. C. Haarhoff, Mol. Phys., 7, 101 (1963).

Fermi Resonance in Aromatic Ring-Rare Gas Complexes 2. Because u,, u,, and u, can take on many combinations of quantum numbers, there will be a number of states, ..., Im 1 ), Im),Im - l ) , ..., in near resonance with the 11) state. The energy levels shown in Figure 2 and their corresponding wave functions Iui;O,O,O) or I/), Iui - l;u,,uy,uz) or Im), etc., represent the uncoupled vibrations of the chemical or van der Waals bonds. They correspond to the zero-order vibrational wave functions for the van der Waals complex where Jui;ux,uy,u,) = ~u~)~u,)~u,)~u,) with I u i ) the harmonic oscillator chemical bond vibrational wave functio# and l u x ) , Iu,), and Iu,) as defined previously. There is, of course, coupling as we shall discuss below. First-order pertubation theory will result in a van der Waals shift of the unperturbed I/) state of energy E{Of to E,, E,(o) to E,, and so on. Second-order perturbation theory will cause a much smaller vibrational shift, but more important to our discussion, it will mix the zero-order Il) and Im)wave functions. This coupling will be discussed in the section on Fermi resonance. D. The Coupling Term. The form of the coupling terms for mixing between ring modes and van der Waals vibrations is analogous to that long used to understand collision-induced vibrational energy t r a n ~ f e r .It~ has also reappeared in discussions of energy flow in vibrationally excited van der Waals molec u l e ~ . ~ To ~ ' construct ~ ~ ' ~ the coupling term for our problem, the Morse intermolecular potential of eq 1 is expanded into a power series of the Q,, Qj, ..., out-of-plane chemical bond normal mode displacements of the ring molecule:

+

The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 1793 and consequently using the form of eq 14 for this mode gives Vf') = 0. However, we must realize that the ring polarizability, quadrupole moment, other electric moments, and even dimensions can change significantly with such an in-plane vibration. Consequently, the van der Waals interaction would be modulated and coupling between the ring mode and van der Waals modes would occur if they have the same symmetry. Likewise for out-of-plane ring modes, terms that concern vibrational modulation of polarizability, electric dipole moment, etc., may need to be added to eq 14. A more general form of fl')than that given in eq 14 will then be required to describe accurately coupling between chemical bond and van der Waals bond vibrations. We shall not be concerned about modeling these sorts of coupling terms for the present study although we have done so elsewhere.42 We do require, however, a test to show that eq 14 is a realistic description of the coupling interaction for the example we shall consider. This test is provided by calculating the vibrational shift. E. Vibrational S h i f t . The manifold of vibrational levels within the ring is expected to be altered when it becomes a van der Waals complex as a consequence of the coupling term, eq 14, we have just discussed. Indeed, shifts in vibrational levels in complexed molecules have been reported44and discussed.45 The magnitude of these shifts can be easily calculated by perturbation theory.46 The degree to which the calculated and observed shifts agree is then a measure of the success of the coupling terms of eq 14 to describe the van der Waals interaction. We might wish, for example, to calculate the energy El of the Iui;O,O,O) or 11) state of Figure 2. The energy of the ui level of the unperturbed harmonic oscillator in the ith mode in the uncomplexed ring as described by wave function Iui) is given by

Eto) = hcui(ui The term aI,which depends on the masses and positions of the ring atoms, picks out the z component of the displacements for the u, vibration. We obtain a, by transforming between normal and Cartesian coordinate^.'^^^^^^^ These displacements may be pointing either toward or away from the rare gas atom so the ring will need to tilt about the x or y axis in order to have a favorable interaction with the van der Waals bond. The Sr,, Srij,etc., terms in eq 14 accomplish this, and consequently they can couple the van der Waals bending modes with the vibrations in the ring. Since pi)is contained in the Hamiltonian for the system, it must be totally symmetric to all operations of the group (Le., C,, for our case). We therefore must have Sr,,Sr,j,etc., belong to the same irreducible representation r,,r,,,etc., as the corresponding Ql, Q,Q,, etc., normal modes. We shall use the simplest possible Sr,, ST,,, etc., terms here which contain the Ox and 0, bending coordinates. There will be four, each transforming as an irreducible representation of the C,, group: Sa, = cos 20, cos 20,

+ X)

(16)

This level is shifted to E, = E{o) + E /,I (1)

(17)

when the ring becomes a van der Waals molecule by complexing with an atom. We calculate the first-order correction by

E/,{')= (l1V')Il)= (v~;u~,u,,u,JV')~U,;U,,U~~U,~ (18) Using Vf') from eq 14, we find E,,;') =

(uzldV(z)/dzluz) + (ux~u,lSr,,lux~up)(uzld2V Z )/dz21uz) (19)

a i ( u i l Q i l u i ) (u,,u,lSr,lu,,u~)

Yla?(UilQ?lui)

Because (uilQilui) = 0, the first term in eq 19 vanishes. For the second term of eq 19 we use4' (uilQ?lui) = (vi + '/2)h/(2.rr~u,). Since Q? belongs to a,, we must have Sr,, = Sa,from eq 14, and standard integral tables48reveal (Olcos 20,lO) = (Olcos 20,lO) = 1. Use of the listed integral33shows (Old2V/dz210) = a2De(2do - l)/do. The result is

S,, = sin 20, sin 20, Sb, = sin 20, cos 20,

Sb2 = cos 20, sin 20,

(15)

These angularly dependent coupling terms are natural extensions of the expressions we have used elsewhere."j If the u, and u, van der Waals vibrations are tangential rather than bending and controlled by the potential of eq 12, the Sr,, Sr,,, etc., coupling terms of eq 15 must be modified by replacing the sin and cos terms by hyperbolic sinh and cosh terms.40 The coupling terms in eq 14 were generated with an important assumption which we must be mindful of. It requires that a component of the chemical bond displacement (through Sr, or a,, etc.) be directed along the z axis toward the rare gas atom. One can imagine couplings between the ring and atom to occur in different ways. For example, vibrations in the plane of the ring have no Cartesian displacements along the z axis (Le., a, = 0), (42) D. A. Morales and G. E. Ewing, Chem. Phys., 53, 141 (1980) (43) G. E. Ewing, J . Chem. Phys., 72, 2096 (1980).

The shift (in cm-I) of the vi

AIJ"i,o=

-

0 transition becomes

a?a2Deh(2do - 1)ui 4~c~jdo

when all constants in eq 21 are in cgs units with De and vi in cm-I. F. Fermi Resonance. When the vibrational states labeled ( I ) and ..., Im l ) , Im),Im - l ) , ... of Figure 2 are of the same symmetry, their corresponding wave functions will be mixed through a coupling term like that of eq 14. This mixing of the wave functions and concomitant shift of the energy levels is Fermi resonance. It means for our case that vibrational excitation of a chemical bond within the van der Waals molecule near energy

+

(44) K. E. Johnson, W. Sharfin, and D. H. Levy, J . Chem. Phys., 74, 163 (1981). (45) D. A. Morales, Thesis, Indiana University, 1983. (46) A. D. Buckingham, Proc. R. SOC.London, A, 248, 169 (1958); Trans. Faraday Soc., 56, 153 (1960).

1794 The Journal of Physical Chemistry, Vol. 90, No. 9, 1986

El scrambles motions against chemical and van der Waals bonds. Two types of energy terms enter into the first-order correction of the zero-order 11) or Im) wave f u r ~ c t i o n s . ~The ' ~ ~first ~ is the matrix E/,,(') = ( / " ( ' ) ~ w z )

(22)

with analogous ..., E/,wl('), ... terms. The second energy term depends on the separation between the vibrational levels

(23)

AE,,, = E, - E ,

with analogous terms ...AEl,,-,, AEl,,,,, .... Evaluation of E/,,(') in eq 22 follows directly from our definitions of the I/)and Im) states and the coupling term of eq 14. In the applications we shall use, and as shown in Figure 2 , ) l ) = Iui;O,O,O) and Im) = Iu, - l;u,,u,,u,). The result is E / , m ( ' ) = ~i(uilQilui- 1 ) (0,01Sr,l~,7~,) (OldV(z)/dzlu,)

Ewing L?

(0,Olsin 20, cos 26'yluz,u,.) = (Qlsin 20,lu,) (Olcos 20,lu, )

(26)

where (Olsin 2B,lu,) with

u,

= (-I)(~X

1)/*(u,!)-'/2(2/p,)u~~2 exp(-l / f i x ) (27)

odd and

exp(-l/&) (O(cos 20,lu,) = (-1)u~~2(u,!)-1~2(2/4,)"vi2

(28)

with uy even using listed integral^.^' The other (O,OJSr,lu,,u,) matrix elements are easily obtained from equations like eq 26 and the integrals above. The last brace in eq 24 involves the van der Waals stretching mode. This matrix element has been evaluated elsewhere33 to yield

At this point there are a variety of approaches we could take in estimating the mixing of the 11) and Im) states. In a classic series of papers, Bixon and J ~ r t n e r ~ ' ,treated ~ * , ~ ~some limiting cases depending on the density, p, of Im) states and the magnitude of the coupling matrix element. Following their a p p r ~ a c h , we ~' simplify the problem by imagining equal spacing between neighboring levels in the Im) stack: t

=

... Ern+>- E,+1 = E,+'

- E,, = E,,, -

Likewise, the matrix elements between an state are taken invariant:

Em-!= ...

(30)

I/) state and any Im)

(47) E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, 'Molecular Vibrations", McGraw-Hill, New York, 1955. (48) L. S. Gradsheteyn and I. M. Ryzhuk, "Tables of Integrals and Products", Academic Press, New York, 1965. (49) M. Bixon and J Jortner. J Chem Phys., 50, 3285 (1969).

=

P = ( n ~ pcosec ) ~ (apAE/,,)

...

(31)

(32)

If we further assume that the ( I ) state is midway between the nearest Im) and Im + 1 ) states, then AEl,, = c/2 and eq 32 reduces to P =

(sL?p)2

(33)

An examination of the detailed formulas3' shows that only the few Int) states nearest the 11) state undergo Fermi resonance for the range of p and u we have in our systems. To simplify the discussion, we shall imagine only a pairwise Fermi resonance between Il) and one of the states, Im) from the ..., Im - l ) , Im), Im l ) , .., stack. The first-order wave functions appropriate to the two perturbed levels become34

+

\k/ = (1 - P)1/*11)

{2a~uj(ujlQilvj - 1)) = 2aaj[2uih/4a~vj]~/* (25) The matrix elements for the angularly dependent coupling term depend on Sr,.For Sr,= Sb,we have from eq 15 and the harmonic oscillator wave functions of eq 9

= E I,m(1) = E /,,-'(I)

They go on to discuss molecular systems where t-lv = pu >> 1 , and significant mixing of I/) occurs with a multitude of Im) states which must be handled by a variational calculation. This limit applies to some van der Waals molecules containing ethylene as we shall show in a later paper. For the aromatic ring-rare gas complexes that we consider in this paper, we shall find that pu < 1 and wave function mixing may be discussed through first-order perturbation theory.50 A particularly simple result is obtained for P,the probability that 11) will take on character from the Im) stack:

= De(2aa,(uilQilui- 1 ) ) {(O,O~S~,~u,,u,)){ (Ole-"z - e-2arlu,)) ( 2 4 )

where we have made use of dV(z)/dz = 2 ~ D e ( e - ~-' e-2az). Each of the expressions in braces is dimensionless and analogous to a Franck-Condon overlap term between the 11) and Im) states for the corresponding displacement coordinate. The first term in braces in eq 24 is the matrix element for the chemical bond vibrational change. Standard listings of this matrix element47show that

=

+Pqm)

(34)

P)"*lm)

(35)

and 9, = P'/*11) - (1

-

-

As we shall show later in our example and confirmed by exper0.1. The perturbed levels described by Qland Q, iments, P will appear at nearly the same energies as [ I ) and Im), corresponding to a separation of 0.1 cm-I < e < 1 cm-' for the cases we shall consider. As we shall now show, the consequences of Fermi resonance will appear in the spectra and in the vibrational predissociation relaxation channels. G. Spectroscopy. In the spectra of the van der Waals molecules we shall consider, only a transition to the unperturbed 11) state carries oscillator strength. We call I/) the bright state. For the uncomplexed aromatic ring, this is a particular vibronic level of the excited electronic state. On formation of the complex, new zero-order ..., Im - l ) , Im), Im l ) , ... states are energetically accessible to the exciting light, but since they involve high vibrational levels against the van der Waals bond, they carry no oscillator strength and are therefore called dark states. Fermi resonance mixes oscillator strength from the [ I ) state into the ..., Im - l ) , Im), 1m + l ) , ... states. As we have just discussed, only a small number of states are affected by Fermi resonance. If we imagine only two levels in Fermi resonance as in the previous section, the transition that was initially to a single I!) state in the uncomplexed aromatic ring now can be a distributed over the 'k, and 'k, states. If one uses an estimate c given later, these lines will be separated by 0.1-1 cm-I. The two absorption lines will have a relative cross section given by 1 - P and P as eq 34 and 35 imply. Thus, two Fermi resonance vibronic transitions may then be observed in absorption measurements. Emission from the excited vibronic state will reflect the Fermi resonance since, as we shall show, the probability of Im) state character in the excited state is significant. The manifestation of Fermi resonance will depend on the nature of the experiment. During CW excitation, emission follows the selection rules for both the Im) or IuI - l;ux,uy,uz) state quantum numbers and the Il) or lu,;O,O,O) state quantum numbers. For narrow-band excitation, -0.1 c r ~ i - ~ an, individual Fermi resonance level, either PIor k ' , can be populated by tuning the source. If 9,is excited,

+

(50) P. A. Atkins, "Molecular Quantum Mechanics", Oxford University Press, New York, 1983.

The Journal of Physical Chemistry, Val. 90, No. 9, 1986 1795

Fermi Resonance in Aromatic Ring-Rare Gas Complexes then most emission (Le., with relative probability 1 - P ) , will follow the selection rules of the 11) state. However, if the lower cross section \k, state is excited, most emission will follow the selection rules of the Im) state. Pulsed laser experiments, in principle, would make possible direct measurements of intramolecular vibrational relaxation (IVR).5' While we have been talking about Fermi resonance in a twostate system, there may be the opportunity for mixing of several states from the ..., Im - l ) , Im), Im l ) , ... stack in excited aromatic ring-rare gas atom van der Waals molecules. Vibronic excitation to several levels may then be possible. The complexities in the spectroscopy of these van der Waals molecules will then mimic the spectroscopy of chemically bonded molecules where the Fermi resonance mixing of several levels and the time dependence of energy flow have recently been d e s ~ r i b e d . ~ ~ H . Vibrational Predissociation. Here we review vibrational predissociation (VP) of aromatic ring-rare gas atom van der Waals molecules like C2N4H2**-Ar. We describe these complexes as A-B**-C where A-R** is the vibrationally excited chemically bonded molecule attached by a van der Waals bond to atom C. Relaxation of the initially prepared vibrationally excited complex can proceed by a+ least four channel^:^'^^^

+

+ C + AEV-T A-B**-.C A-Bt + Ct + AEV-R A-B**..C A-B* + C + AEv-v A-B**.**C [A-B**.*C]* A-B + C + AEF-R A-B***-C

-

-+

-+

-+

A-B

-

(36) (37) (38) (39)

In the first channel, energy from the vibrationally excited chemical bond, A-B**, breaks the van der Waals bond and molecule A-B (now completely relaxed) and atom C fly away with translational energy labeled AEv-T. This is the vibrational-translational (V-T) channel. If the products contain rotational energy as in eq 37, they are written as A-Bt and Ct and the relaxation is by the vibrational-rotation (V-R) channel. The rotation manifests itself in the aromatic ring A-B in some J , and J , state together with the end-over-end rotation in the ring and the atom about their common center of mass. The energy released into the translational mode is now AEv-R. Some vibrational excitation in a lower level may remain in the ring after dissociation, which we indicate by A-B*. The relaxation is then by the vibration-vibrational (V-V) channel of sq 38. Here the translational energy of the fragments A-B* + C is labeled AEv-v. (It is likely that these fragments contain some rotational energy as well.) The last relaxation channel, eq 39, is represented conceptually as two steps although VP would occur as a single step. In the first step Fermi resonance occurs between the state 11) of A-B**-C corresponding to vibrational excitation localized solely into a chemical bond and state Im) of [A-B*-C] * involving excitation of vibrations against both chemical bonds and the van der Waals bond. The two states \kl and \k, describing [A-B*-C] * undergo vibrational predissociation (VP) to products A-B + C containing translational energy AEF-R. (These fragments may also contain some rotational energy.) We can calculate the rate constant T-] for VP by the V-T or V-R channels using the golden rule expressiong*l0 T-'

= (4/h2uf)l(fIV(1)li)12

(40)

dimensionless quantity, q, given by q = (2pvAE)1/2/ah

which roughly counts the number of nodes of the plane wave describing the translational motion of A-B C over the potential we11.13i14We will use q to label the translational quantum number of this plane wave. The rotatory motions of the fragments are difficult to model. We shall imagine as before43 that, for low rotatory states of the fragments near the turning point of the intermolecular potential well, the bending motions are crudely described by harmonic oscillations with wave functions given by eq 9 and the quantum numbers are u, and vu. As the fragments move away from the influence of the potential well, bending vibrations described by u, and vu go over into free rotation described by quantum numbers J , and J,. We shall imagine the initial state A-B**-C to have lost two vibrational quanta in its chemical bond to become A-B C by the V-T channel of eq 36 or A + Bt Ct by the V-R channel of eq 37 so the final state quantum number is vi - 2 and If) = lui-2;ux,uJ,,q). (We shall use this particular channel later.) With these definitions of li) and If, together with the second part of the coupling term pl) defined by eq 14 appropriate to Aui = 2, we have for the matrix element43

+

+

+

(flp')li) = a?(ui

- 21Qi21ui)( u x , u , l ~ r i j l O , O(q1d2Vz)/dz210) )

(43) The result for the VP rate using eq 43 into 40 is

q2)1[((do - u, -

1/A2 + do + q2)/d0I2exp(-rq)

is the final velocity of the fragments in terms of their translational kinetic energy A E by one of the relaxation channels of eq 36 or 37. The initial state in the absence of Fermi resonance is li) = 11) = ~ui;ux,u,,uz,) = Iui;O,O,O). The final state with the fragments flying apart over the intermolecular potential well is difficult to specify in general since it depends on the channel. The van der Waals stretching mode quantum number u, correlates to the (51) J. E. Baggott, M . 4 . Chuang, R. N. Zare, H. R. Dubal, and M. Quack, J . Chew. Phys., 82, 1186 (1985).

(44)

2a2a?(ui - 21Q;lui) = U ~ ~ ; ( ~ / ~ ~ C V ~-) 1)]1/2 [(U~)(U (45) ~ The matrix element for the angular coordinate is given in eq 26-28 or their like. We have noted el~ewhere'~.'~ that the matrix elements associated with the Q , z, 0,, and 0, coordinates all depend roughly on the exponential change in the respective quantum number. For the purpose of establishing propensity rules, the tedious formulas of eq 40-45 can then be collapsed to 7-l

-

+ Anr + An,)]

l O I 3 exp[-a(An,

We use13J4 Ant

-

lq/2 - %I

(46) (47)

for the translational quantum number change

where y

-

Anr

N

4IAuxI

+ IAuylI

(48)

1 for the rotational quantum change and

Anv (41)

1

where the matrix element involving the z coordinate has been evaluated elsewhere.43 The matrix element for the vibrational coordinate Qi shows4'

where uf = ( ~ A E / / . L ~ ) ' ~ ~

(42)

-

Juf -

uil

(49)

accounts for changes in the quantum numbers that label chemical bond vibrations. Equation 49 resembles the Parmenter-Tang propensity rule for collisional vibrational relaxation.16 Besides its algebraic simplicity, eq 46 has many advantages for our purposes. It has a satisfying physical explanation. The preexponential factor 1013s-l expresses the collision frequency of the aromatic ring molecule and its partner atom against the van der Waals bond. The exponential term reflects the reluctance of a molecule to change quantum numbers during a radiationless transition and gives the probability that the final and initial states of the process will mix during the collision. Equation 46 and the guides for determining selection rules in eq 47-49 easily show what

-

1796 The Journal of Physical Chemistry, Vol. 90, No. 9, 1986

factors can influence efficient VP. Any channel with a small total quantum change Ant An, An, will have a fast relaxation rate. This propensity rule based on quantum numbers corresponds to the exponential gap in other formulations of energy-transfer propensity r u I e s . ' ~ ~ - ~ Often it is the large change in the translational quantum number An, that limits the relaxation rate. In such cases Fermi resonances can make an enormous difference. Let us compare for example the V-T channel, eq 36, and the Fermi resonance channel, eq 39. For the V-T channel it is the I I ) state that is excited so we can use the propensity rule expression of eq 46 to estimate the VP rate constant. For the Fermi resonance channel we imagine that the \k, state has been excited which contains the Im) state with probability P as eq 34 implies. The propensity rule expression for VP from the Im) component of \kl then becomes

+

7-l

+

= lOI3P exp[-x(An,

+ Anr + An,)]

(50)

the factor P giving the probability that the initial state has i m ) character. If we imagine that in both channels the products A-B C have the same amount of translational kinetic energy, then 4/2 contributing to An, in eq 47 will be the same for both channels. In the V-T channel the initial state has u, = 0 so An, 14/21 is to be inserted into eq 46. However, for the Fermi resonance channel, the mixed state, Im), will occupy some high u, state in the van der Waals well. The translational quantum number change, An, lq/2 - u,I, that goes into eq 50 will then be considerably reduced. Since the VP rate depends on A 4 exponentially, the Fermi resonance channel can be orders of magnitude more efficient than the V-T channel. This increase in the VP rate constant by the Fermi resonance channel as a consequence of the small change in An, is, however, somewhat attenuated by the probability, P = 47.2 cm-I. The anharmonicity constant g,, = 1.6 cm-I was calculated from eq 2. The anharmonicity terms g, = gY, = 0.2 cm-' were chosen so that the bending mode energy levels (described by eq 7 for G(u,)) converge at the top of the potential barrier. The cross terms are g, = 0.2 cm-' and g,, = gy, = 0.6 cm-I. These spectroscopic constants yield by eq 11 the fundamental frequencies U, = 17 cm-I, uJ = 19 cm-], and uL = 43 cm-I, consistent with the experimental data. There is considerable uncertainty in the positions of the Irn) states in near resonance with the 11) state as shown in Figure 4. Some of this uncertainty is a consequence of the choice of the ~ ~eq ) 1 1 . A larger uncertainty anharmonicity terms for G ( u , , u , , L in

-

-

-

-

Fermi Resonance in Aromatic Ring-Rare Gas Complexes

The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 1797

c

-

have contributions from transitions such as, say T.-Ar(16a1;5,3,3)

400L I/ /

1 IV/ I

I

2

I

I

4

r I

I

I

I

I

6

8

Z + Z ~ (A) Figure 3. Energy levels and potential surfaces for s-tetrazine-Ar. The intermolecular potential surface for displacement of Ar against s-tetrazine is shown for several 16a vibrational states. The equilibrium separation is z , = 3.50 A, and z is the displacement from the minimum of the potential well. Only the u, van der Waals levels are illustrated. In Figure 4 all van der Waals levels in near resonance are shown by using a finer energy scale.

-

I16d ;1 1,5,0)

-

-

- -

-

-

-

-

I16a'; 1,9,2) 116d ;1,1,6) 116a1;3,1,5) 11 6 a2 ;O,O,O)

116a1;3,7,2) 116d;13,1,1$

'"I1 236

-

116a'; 5,3,3) 116a1;5,1,4)

Irn+l) '

Im) 1 rn- 1) -

:

116a'; 5,5,2) 116a'; 1 , l 1,l)

' 1

Figure 4. Estimate of the energy level spacings of all the ~16a';u,,uy,u,) states near 116a2,0,0,0) in s-tetrazine-Ar.

arises because the well depth we have selected as De = 350 cm-' may be in error by h50 cm-l. Referring again to Figure 3, it is uncertain whether u, = 0 of 16a2is in nearest resonance with u, = 6,7, or 8 of 16a'. The quantum numbers ux, uy, and u, associated with the 16aI vibration and the corresponding energy levels near the 116a2;0,0,0) state shown in Figure 4 therefore must not be taken literally, but rather they provide a rough idea of levels that make Fermi resonance possible. The density of states is rather low as we shall show later, with spacings o f t 0.5 cm-I. A wide range of quantum numbers are represented in those Im) states that can interact with the I I ) state. For the moment, we shall assume that the Fermi resonance between I I ) and the ..., Im - l ) , Im), Im + l ) , ... stack is significant and examine its consequence in directing the relaxation pathways from excited T-Ar. The first consequence is the appearance of additional fluorescence structure. The excited state is not simply the unperturbed 116a2;0,0,0) state, but rather with Fermi resonance it has admixtures of, for example, (16a';5,3,3), 116a1;1,1,6),or other states from the ..., Irn - 1 ), Im), )rn + 1 ), ... stack. As we shall show, this admixture is significant so that the fluorescence can

-

[T + Ar](16ao;Jx,J,,,q)

+ AEF-R

(54)

Let us now apply the propensity rule relationships we offered in section 1II.H. Since the initial state of eq 54 contains bending mode excitation (Le., u, = 5, uy = 3), the final state may contain some rotational excitation (Le., J , # 0, J y # 0) as well. (Both sets of quantum numbers are connected with angular displacements 0, and 0, as shown in Figure 1 .) In applying the propensity rule expressions, we shall suppose that the rotational excitation is modest, J, 0, Jy 0 so that AEF-R = AEV-T= 195 cm-I. We will likewise assume that the effects of rotational change are small and take An, 0. The chemical bond vibrational change is An, = 1 since deexcitation takes 16a' to 16a0. The most dramatic change is the translational quantum numbers. The value of q 11 is the same as in eq 52 since AE is essentially the same for both the V-T and Fermi resonance channels. However, since u, = 3, we have An, )4/2 - u,I 2.5. The total change in 3.5. If we assume quantum numbers is now Ant An, + An, that the \II, state has been excited, then eq 50 applies with the probability of Im) character of P 0.1 as we shall show later. s which we can compare with the observed The result is T lifetime of 11 n ~ . ~Clearly, ' the uncertainty in An, and An, makes the quantitative calculation of T meaningless. Had we chosen say (16a1;3,7,2) with u, = 2 as the Fermi resonance component, then we find Ant 4.5 and T s. Alternatively, the level s . The point we wish (16a2;1,1,6) with u, = 6 gives T to make is that any of these channels involving Fermi resonance can provide relaxation times 6 orders of magnitude or so more efficient than the V-T channel of eq 52. This efficiency occurs because Fermi resonance allows for a reduction in the change in translational quantum number An,, and this reduction is responsible for fast VP. We now attempt to justify the significant mixing of the 11) and ..., Im - l ) , Im), (m l ) , ... states that has been assumed for this example. Here we are faced with two problems-the determination of t and the determination of v. The determination of the spacing, t, between adjacent Im) states (Le., ~16a';ux,uy,uz)) near the I I ) state (Le., 116a2;0,0,0))is obtained by direct count from the energy level spacings given in eq 11 using the spectroscopic constants we have just given. Since each of the vibrational states must belong to one of the four (al, a2, b,, or b2) irreducible representations of the C, point group, we must multiply the calculated spacing by four to obtain t or, alternatively, divide by four to obtain the density of states p = e-!, for a particular irreducible representation. This result is shown in Figure 5 as a function of E, the total energy in vibrations against the van der Waals bond. We see an exponential dependence of E with E with 0.1 cm-I C t C 1 cm-' in the upper reaches of the potential well corresponding to the energy that the van der Waals modes must accept for Im) near I I ) as in Figure 4. Within the constraints of the observed frequencies for u,, v,,, and u, and with reasonable estimates of the anharmonicity terms g,, gxy,etc., the t separations for T-Ar are probably accurate to within a factor of 4. The spacings of these energy levels is clearly rather sparse. Next we estimate u. Here the problem is more difficult than locating t. The main difficulty is obtaining reliable estimates for the variety of acceptor Irn) states such as the ones illustrated in Figure 4. These states span extremes of quantum numbers from a state with the stretching mode highly excited, l16a1;l,l,6),to a state with only bending modes excited; 116a';l1,5,0). It is difficult to imagine that v is invariant to this variety of quantum states as we, following Bixon and J ~ r t n e r , are ~ ' forced to assume in eq 31. For the purposes of estimating the magnitude of Fermi

-

240

(53)

A spectral feature that correlates to T( 1sa') T( 16al) is indeed observed and has been discussed qualitatively in terms of Fermi resonan~e.~~.~~ With possible Fermi resonances described, we can now investigate their participation in VP. The consequences of Fermi resonance are dramatic. Consider for example the channel T-Ar(16a1;5,3,3)

-200

-

T-Ar(16a1;5,3,3)

+

- -

- -

+

1798 The Journal of Physical Chemistry, Vol. 90, No. 9, I986

Ewing order of magnitude with that of our model calculation. Relaxations from a number of vibrational excited states of T-Ar have been carefully s t ~ d i e d . Consider ~ ~ , ~ ~ the case where the 6b2 level of T-Ar is excited and two VP channels are observedz3with comparable efficiency even though their energy gaps are vastly different. These two channels may be written T-Ar(6b2;0,0,0)

-

-

[T + Ar](16bl;J,,Jy,q)

[T + Ar](16a';Jx,J,,q)

---

Figure 5. Matrix element u and separation e of van der Waals levels plotted against E , the total energy in motions against the van der Waals bond in s-tetrazine-Ar.

resonance in aromatic ring-rare gas van der Waals molecules, we will explore the matrix element between ( l ) = )16a2;0,0,0)and Im) = ~16a1;l,l,uz).We must therefore calculate u =

(Ilv')lm)= ( 16a2;0,0,0~V(')(16a';l,lu,)(55)

The matrix elements for u as a function of u, (and E ) are calculated from eq 24-29. From eq 25 with u, = 2 (Le., 16a2) and u, - 1 = 1 (Le., 16a') we obtain, using previous values of the molecular parameters, (2aa, (2lQ,ll)} = 0.3. For {(l,llSaJO,O)J= (llsin 20,lO) (llsin 28,lO) = 0.05 we have used eq 27 with 6, (or 6,) = 40 from eq 10 since :Y (or):v = 18 cm-I and B, (or By) = 0.22 cm--1.52The matrix elements for the variety of u, levels is given by eq 29. The result is plotted in Figure 5. We find throughout the range 100 cm-' < E < 350 cm-' that u 0.1~. We have of course not calculated all van der Waals ..., Im l ) , Im), Im l ) , ... states that can enter into Fermi resonance with I/) but only representative states. Calculation of u for the variety of (uxruy,uZ)van der Waals levels of symmetry a2 gives a wide range of answers for u. We must not accept these numbers quantitatively because the bending and stretching motions against the van der Waals bond are likely to be mixed among themselves at high ux, u,, and u, levels as we have already discussed in section 1I.A. The importance of this type of mixing has recently been demonstrated theoretically and experimental1 among C-H stretching and bending vibrations in chemically bonded fluorinated hydrocarbons.51 Here, as in our van der Waals case, the stretching fundamental frequency is roughly twice the fundamental bending frequencies and numerous opportunities for mixing of vibrational states are possible at high ux, u,, and u, levels. We are left with an estimate of u O.le for a representative matrix element between I/) and the ..., Im - I ) , Im), 1m + l ) , ... stack. If we take the result u 0 . 1 as ~ typical for the matrix element, then using eq 33 we find P 0.1, saying in effect that there is a 10% probability of the Im) stack being mixed in with 11) through Fermi resonance in forming the 9/ state. The results of Brumbaugh et al.23suggest a branching ratio following excitation to 16a2which corresponds to P = 0.26, a value within an

-

+

--

( 5 2 ) C . A. Haynam, D. V. Brumbaugh, and D. H. Levy. J . Chem. Phys., 80, 2256 (1984).

+ A E = 140 cm-'

(56)

In the upper channel of eq 56 a near-resonance V-V relaxation - Do = 724 - 403 - 330 = 0. is possible since AE = E6b2 Using the propensity expressions of eq 46-49, we obtain An, = 4 / 2 = 0, An, 0, and A h 3 (since 6b2 16b') so 7-l lo9 SKI.Now consider the lower channel without the intervention of Fermi resonance to assist the process. The energy gap is obtained from AE = E6b2 - E 1 6 a-~Do = 724 - 255 - 339 = 140 cm-' so by eq 42 q 9. Using eq 47, we have An, lq/2 - 01 = 4.5 0 and An, 3 (since 6b2 16a'). We obtain by and An, lo2 s-' or 6 orders of magnitude slower then the eq 46 T-' experimental m e a s ~ r e m e n t .We ~ ~ now introduce an Im) state, T.-Ar( 16a2;ux,uy,uz), in Fermi resonance with the initial I/) state T-Ar(6b2;0,0,0). Looking at the energy difference between 6b2 and 16a2,we can estimate the (16a2;0,0,u,)state in Fermi resonance with 16b2;0,0,0). The vibration energy in the van der Waals bond is E = Esb2 -- E16$ = 784 - 510 = 274 which would correspond 6, 7 , or 8. We then have An, to a van der Waals level of u, lq/2 - u,l 2 k 1 depending on the value of u, selected. We 1 (since 16a2 16aI) and An, 0 ignoring also have Anv rotational effects. Using the propensity rules of eq 50 together 0.1, we obtain T-' 10"' s-l. Thus, both the V-V with P and Fermi resonance channels with considerably different energy gaps give relaxation rates consistent with the experimental value of 2.6 X los We shall not go on to analyze the other channels so nicely reported in the experiment^.^^-^^ It would appear that the general features of our model can at least qualitatively account for the trends observed.

-

\

+ AE = 0

-

-

--

-

-

-

- -

-

-

-

-

-

IV. Summary As we have shown, Fermi resonance in aromatic ring-rare gas atom complexes can mix upper vibrational states. The zero-order I I ) state, A-B**-C, with vibrational energy concentrated in chemical bonds becomes mixed with a near-resonance zero-order Im) state, [A-B*--C]*, with vibrational energy in both chemical and van der Waals bonds. The wave functions resulting are q, and 9, as given in eq 34 and 35. Since 11) and Im) have different sets of quantum numbers, their spectroscopy and VP channels will be dramatically different. If the \kl state is excited in an experiment, then the probability of emission follows 11) or Im) states selection rules weighted by 1 - P and P , respectively, as eq 34 implies. The change in translational quantum numbers may be small for VP from an Im) state; consequently VP will be fast. The VP propensity rule is given by eq 50 and requires for its use a knowledge of the change in initial and final state quantum numbers, An, An, An,, as well as the probability, P, of Im) state mixing. The ( I ) state contribution to 9,is only slightly involved in VP since relaxation from I/) requires a larger change in translational quantum numbers. Excitation of the qrn state involves different mixtures of the zero-order 11) and I m ) states as shown in eq 35, and the spectroscopy is correspondingly altered. The propensity rule for VP, still favoring relaxation from the Im) state, has the factor 1 - P = 1 replacing P in eq 50. Consequently, VP following \k, state excitation will be much faster than VP following Q, state excitation. It would be convenient to be able to provide easily applied propensity rules for Fermi resonance as we have done for vibrational predissociation in the absence of Fermi resonance, but we do not believe this is possible. As we saw in Figure 5, both t and u track each other in a similar way as a function of the energy

+

+

1799

J . Phys. Chem. 1986, 90, 1799-1805

in Fermi resonance with 11). What values of u x , uy, and u, are responsible for the VP channel that involves Fermi resonance? Evidence for the dramatic changes in the spectroscopy and VP with \kl or \k, excitation that we predict will also be interesting. Future experiments may reveal patterns not obvious from the simple model we have offered.

content in van der Waals motions. Thus, large matrix elements, u, occur when the separation, e, of van der Waals bond vibrational levels is large. When the matrix elements become small, the separation of levels will be small, rather independent of the energy content, E , of the van der Waals bond; therefore, we have an lo-'. The mixing of essentially constant product, t-lu = pv the --, Im - l ) , Im), Im + l ) , ... stack with 11) as given by P = ( ~ p v is) then ~ roughly independent of E . There will, of course, be differences in the precise relative energy of 11) and a particular Im)state and in the value of u which will make the mixing change significantly different depending on the u, level. This will have to be estimated for each case. It may also be difficult to assign the u, and uy quantum numbers for the Im) state. However, crude estimates of relaxation channels should be possible following the examples we have given in this paper. It will be interesting for new experiments to explore which particular states from the --,Im - l ) , Im), Im l ) , ... stack occur

-

Acknowledgment. Dave Catlett is responsible for sparking my interest in Fermi resonance in aromatic molecule-rare gas complexes. I have benefited from my discussions with him and appreciate his contributing the program for counting the vibrational states shown in Figure 5. Charles Parmenter, Dave Catlett's thesis advisor, has helped me understand some of the subtleties of Fermi resonance and vibrational energy flow and has offered many good suggestions for making this manuscript readable. Finally, I acknowledge the support of the National Science Foundation. Registry No. Ar, 7440-37-1; s-tetrazine, 290-96-0.

+

Laser-Induced Excitation and Photodissociation of Potassium and Mixed Potassium-Na (1.5 % ) Clusters: Diverse Formation and Relaxation Processes and Bounds on Alkali Trimer (M-M,) Bond Energies from LIAF J. S. Hayden, R. Woodward, and J. L. Gole* Center for Atomic and Molecular Science and School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 (Received: September 9, 1985; I n Final Form: December 20, 1985) A supersonic expansion of a predominantly potassium + 1.5% sodium mixture has been used to form small potassium and mixed potassium-sodium clusters M,, n I2. We have observed and characterized laser-induced atomic fluorescence (LIAF) for potassium trimer (K,) and sodium-potassium dimer (NaK) over a range of experimental conditions. The observed LIAF spectrum for sodium-potassium dimer obtains in large part from the reactive encounter Na + K2 NaK + K in the late stages of expansion. The present studies lead to the determination of an upper bound (3610 cm-I) for the K2-K bond energy and provide further insight regarding the meaning of both the low- and high-frequency limits of the potassium and sodium dimer and trimer LIAF spectra. The onsets of the K3 and Na3 LIAF spectra appear to provide lower bounds for the K-K2 (2020 cm-I) and Na-Na2 (3050 cm-l) bond energies. The K2 excitation spectrum obtained under the most efficient expansion conditions is characterized by virtually equal effective rotational (80 f 5 K) and vibrational (80 10 K) temperatures in contrast to previous observations of sodium dimers produced in neat sodium expansionswhere T,,, (effective) > Trot(effective). Further, the characteristic cooling inherent in the NaK excitation spectrum for a relatively minor component in the supersonic expansion lags that for K2, a major component. Both effects are attributed to the nature of the collisional environment in the supersonic expansion.

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Introduction Recently, we reported a study of polyatomic metal cluster photodissociation.I In this study, specifically applied to the sodium trimer molecule, we employed the technique of laser-induced atomic fluorescence (LIAF) where the metal cluster M, is dissociated with a C W dye laser to yield the fragments M,-, and an electronically excited metal atom, M*. In specifically applying this technique to sodium trimer,' we monitored the emission from electronically excited M* in order to elucidate several of the features associated with the bonding and energy levels of sodium trimer, determine an upper bound for the Na2-Na cluster bond energy, and assess the behavior of this cluster under a variety of perturbing conditions. Here, we extend this technique to the potassium trimer (K3) and sodium-potassium dimer (NaK), formed from a pure metal supersonic expansion of a 98.5% potassium-1.5% sodium mixture, and studied over a range of experimental conditions. The present studies not only lead to the determination of an upper bound for the K2-K bond energy but also appear to provide further insight regarding the meaning of both the low- and high-frequency limits of potassium and sodium dimer and trimer LIAF spectra. The onsets of the K, and Na,

LIAF spectra would appear to provide lower bounds for the K-K2 and Na-Na2 bond energies. Because of the small sodium impurity in our potassium beam, we are afforded the opportunity to study the excitation and LIAF spectra of NaK and hence the characteristic cooling of a minor component in the supersonic expansion. While K2 is formed at very low internal temperature in our most efficient expansion, the heteronuclear NaK is characterized by a notably higher internal excitation. Upon perturbing the expansion we find that the LIAF spectrum is dominated overwhelmingly by K* 2P3/2emission.

Experimental Section The apparatus used in the present study is quite similar to that employed previously to study the LIAF spectra of sodium dimer and trimer.] The range of experiments involve modifications of the oven chamber depicted in Figure 1. Here, the output of an argon ion pumped dye laser is brought into a suitably equipped vacuum chamber and focused to a spot 400 pm in diameter. The dye laser beam intersects a supersonically expanded metal beam produced through use of an appropriate oven system. In this study the dye lasers, tuned by a home-built three-stage birefringent filter2 at a bandwidth of 0.5 cm-l, have been operative over the range

(1) Gole, J. L.; Green, G. J.; Pace, S. A,; Preuss, D. R. J . Chem. Phys. ( 2 ) Preuss, D. R.; Gole, J. L. Appl. Opt. 1980, 19, 702

1982, 76, 2241.

0022-3654/86/2090-1799$01.50/0

0 1986 American Chemical Society