J . Phys. Chem. 1987, 91, 4662-4671
4662
spectra for a ZnO/aqueous-solution interface under different conditions. In all these measurements, the sequence of the electrode potential was (1) +2.7 V vs. Ag in an equilibrium, (2) -0.6 V as a square pulse during 1.2 ms for electron accumulation, and (3) +2.7 V for electron depletion. Time was measured from the start of step 3. Curve a was taken for an electrode prepared with the ordinary procedures. The appearance of a concave, whose minimum (peak) is located at 132 ms, indicates the presence of an electron trap. Curve b was measured immediately after photoillumination of the same electrode with several pulses from a xenon lamp at an electrode potential of +2.7 V. The lack of the concave in curve b indicates the removal of the trap by photoexcitation. Curve c was taken after leaving this photoilluminated electrode in dark for 15 min. This curve shows that the trap was regenerated. No qualitative difference was observed when 2-propanol was added to the solution a t 50 vol %. The electron trap observed in the ICTS spectrum is easily removed by a photoillumination which induces photocorrosion of the electrode5 under the above anodic condition. Thus, this trap must be located a t the uppermost layer of the electrode and is most likely identified as an electron-accepting surface state. The origin of this surface state is probably a surface rearrangment of atoms induced by adsorbed water molecules, since the surface state is regenerated by a dark reaction after it is once erased by the photocorrosion. ( 5 ) Gerischer, H . J . Electrochem. SOC.1966, 113, 1174.
As to the energy level of a surface state of ZnO, Kohl and Bard6 estimated one in acetonitrile solution at 1.26 eV below the conduction band and Luth’ reported the first predominant one at 0.37 eV for zinc oxide crystal at low temperature. If 1.26 eV corresponds to the peak time (1 32 ms), v(E) must be 2 X 1020s-l, which seems unlikely because it indicates that detrapping from a deep 1.26-eV trap occurs with a high rate exceeding lattice vibration frequencies. Consequently, the surface state observed in our experiments is probably different from the 1.26-eV one. If 0.37 which is eV corresponds to 132 ms, v(E) will be 4 X IO6 SKI, possible. The width of the density-of-state distribution was calculated to be 0.07 eV in terms of eq 4 on the assumption that v(E) is practically independent of the energy in the energy region within this width. The present results demonstrate that ICTS is a potential method for the study of surface electronic states at semiconductor/liquid interfaces. Although this preliminary note is only concerned with the first observation of the ICTS spectrum for a semiconductor/liquid interface, extended experiments and analysis to reveal both the energy level and the attempt-to-escape frequency are in progress.
Acknowledgment. We are grateful to Dr. H. Okushi of Electrotechnical Laboratory for fruitful discussions. ( 6 ) Kohl, P. A,, Bard, A. J. J . Am. Chem. SOC.1977, 99, 7531. (7) Liith, H . Surf. Sci. 1973, 37, 90.
FEATURE ARTICLE Selection Rules for Vlbratlonal Energy Transfer: Vibrational Predissociation of van der Waals Molecules George E. Ewing Department of Chemistry, Indiana University, Bloomington, Indiana 47405 (Received: March 30, 1987)
Vibrational energy transfer under a wide variety of conditions-in van der Waals molecules, between colliding molecules in the gas phase, in the liquid and solid states, and on surfaces-is accomplished through coupling by intermolecular interactions. Since a common coupling mechanism underlies all these vibrational relaxation processes, we are able to extract a general selection rule: A relaxation channel of a vibrationally excited molecule is efficient only when the total change in effective quantum numbers for the process is small. The usefulness (and shortcomings)of this general rule for vibrational predissociation of van der Waals molecules will be illustrated by many examples from laboratory experiments and computational studies.
1. Introduction Selection rules are commonly associated with optical spectroscopy. We say, for example, that a molecule bathed in infrared radiation of the proper frequency can only undergo vibrational transitions when the quantum number changes by Au = *I. Rotational transitions are locked by the AJ = 0, f l selection rule. Likewise, other optical selection rules restrict changes in quantum numbers which identify electronic, nuclear, or electron spin states of atoms and molecules. These rules are not rigid, and exceptions uncovered by experiment make spectroscopic studies so important to understanding the coupling between electromagnetic radiation fields and molecules. It is this coupling that is responsible for the transfer of radiant energy by emission or absorption of photons. In a similar way, the selection rules we shall discuss here provide
a useful guide to understanding radiationless vibrational energy transfer processes induced by intermolecular coupling. That it is possible to offer a selection rule that applies over a wide variety of radiationless vibrational relaxation conditions is a consequence of the feature all these processes have in common. Whether we are discussing vibrational predissociation of van der Waals molecules or vibrational relaxation of colliding pairs in the gas phase, the energy flow is induced by intermolecular coupling. Likewise, vibrational flow in liquids, in solids, or on surfaces is accomplished through coupling of vibrations against chemical bonds within the molecule with motions against intermolecular potentials which connect the relaxing molecule to its neighbors. We begin with a quantitative statement of the general selection rule that guides vibrational energy transfer processes. Next we
0022-3654/87/209 1-4662$01 .50/0 0 1987 American Chemical Society
Feature Article present, in some detail, the specific semiquantitative selection rules which govern vibrational energy flow in van der Waals molecules. Examples that show the success and limitations of these selection rules are then considered. We end with a brief section of the future of the general selection rule for the interpretation of new experiments on van der Waal molecules. Caveat Lector. The selection rules we will provide here are not derived but rather have been extracted,'.2 through sometimes extreme assumptions, from theoretical models presented in a series of papers by Coulson and Robertson: Beswick and Jortner,"s and Ewing.61' The manner of this extraction is relegated to the Appendix in order to preserve, in the main body of the paper, the simplicity of the selection rules. We hope the reader will be satisfied by the physical arguments that go into constructing the selection rules and be convinced of their efficacy by the many demonstrations of their correspondence with vibrational predissociation data. As we shall see, lifetimes for vibrational predissociation into a variety of relaxation channels are observed to span 14 orders of magnitude. Our goal is not to list reliable numbers for these lifetimes. These numbers are being provided by detailed theoretical computations and laboratory measurements. But rather our contribution will be to offer a semiquantitative guide for understanding vibrational energy flow patterns.
2. The General Selection Rule The general selection rule is based on the reluctance of a vibrationally excited molecule to change quantum numbers during a relaxation process when the states are coupled by intermolecular interactions. In applying the selection rule, a careful accounting must be made of all the quantum numbers that can change in the process-vibrational, rotational, and translational. The following is a statement of the general selection rule: A relaxation channel of a vibrationally excited molecule is efficient only when the total change in the effective quantum numbers for the process is small. Specific selection rules will indicate how the quantum changes in translational, rotational, or vibrational levels accompany the vibrational predissociation process. We shall now show how the general and specific selection rules are applied to the relaxation of vibrationally excited van der Waals molecules.
3. Vibrational Energy Flow in van der Waals Molecules Here we examine relaxation of vibrationally excited van der Waals molecules such as 12*-.He,'2 HCI*-Ar,I3 H2*-Ar,'4 HF*--HF,'s*'6 C2H4*--Ar,'7-'9 FC6H4F*-Ar," and other^.^*^'-^^ (1) Ewing, G. E. Faraday Discuss. Chem. Soc. 1982, 73,402. (2) Ewing, G. E. In Intramolecualr Dynamics; Jortner, J., Pullman, B., Eds.; D. Reidel: Dordrecht, 1982; pp 269-285. (3) Coulson, C. A,; Robertson, G. N. Proc. R. SOC.London 1975, A342, 289. (4) Beswick, J. A.; Jortner, J. Chem. Phys. Lett. 1977, 49, 13. (5) Beswick, J. A.; Jortner, J. Adu. Chem. Phys. 1981, 47 (Part l), 263. (6) Ewing, G. E. Chem. Phys. 1978, 29, 253. (7) Ewing, G. E. J. Chem. Phys. 1979, 71, 3143. (8) Ewing, 0.E. Faraday Discuss. Chem. SOC.1982, 73, 325. (9) Ewing, G. E. In Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G., Ed.; Plenum: New York, 1981; p 75. (10) Morales, D. A.; Ewing, G. E. Chem. Phys. 1980, 53, 163. (11) Ewing, G. E. J. Chem. Phys. 1980, 7 2 , 2096. (12) Smalley, R. E.; Levy, D. H.; Wharton, L. J. Chem. Phys. 1976,64, 3266. (13) Hutson, J. M. J . Chem. Phys. 1984,81, 2357, 6413. (14) Hutson, J. M.; Ashton, J. A.; Le Roy, R. J. J. Phys. Chem. 1983,87, 2713. (15) Pine, A. S.; Lafferty, W. J.; Howard, B. J. J . Chem. Phys. 1984,81, 2939. (16) Huang, Z. S.;Jucks, K. W.; Miller, R. F.J. Chem. Phys. 1986,85, 3338. (17) Casassa, M. P.; Bomse, D. S.; Janda, K. C. J. Chem. Phys. 1981, 74, 5044. (18) Western, C. M.; Casassa, M. P.; Janda, K. C. J . Chem. Phys. 1984, 80, 4781. (19) Hutson, J. M.; Clary, D. C.; Beswick, J. A. J . Chem. Phys. 1984,81, 4474.
The Journal of Physical Chemistry, Vol. 91, No. 18, 1987
J
U
ut
4663
r/
Figure 1. Coordinates and quantum numbers for an idealized van der Waals molecule A-B--C. Adapted from ref 11.
We describe these complexes as A-B*-C where A-B* is a vibrationally excited chemical bonded molecule attached by a van der Waals bond to an atom C or a molecule C-D. Relaxation of the initially prepared complex can proceed by at least four
channel^:^.^^^^ A-B*...C A-B*-C A-B*--C-D A-B**...C
kwx
-
A-B
+ C + AEV-T
(1)
A-Bt
+ Ct + AEv-R
(2)
7-1
7-1
2A-B + C-D* + AEv-v
[A-B*4]*
i'
A-B
+ C + AEF-R
(3) (4)
In process 1, energy from the vibrationally excited chemical bond breaks the van der Waals bond with rate 7-1 and A-B (now relaxed) and C fly away with translational energy AEv-.I.. This is the vibration-translational (V-T) channel. If A-Bt and Ct contain rotational energy as in process 2, the relaxation is by the vibration-rotation (V-R) channel. W e define the energy remaining in the translational mode as AEv-R. It is also possible for these fragments to be translationally excited as well. In this event, we speak of the vibration-translation, rotation (V-T, R) channel and the energy left in translational motions is AEV-T,R. Vibrational excitation of one of the fragments, say C-D* as in eq 3, appears in the vibration-vibration (V-V) channel. Here the translation energy of the fragments A-B + C-D* is AEV+. These fragments might also contain rotational energy. Finally in process 4, energy initially localized in A-B** flows throughout the complex by exciting isoenergetic internal modes, against both chemical and van der Waals bonds, without predissociation. This is intramolecular vibrational relaxation (IVR) which produces [A-B*-C]* with rate kIVR.Alternatively, we can speak of the isoenergetic modes of [A-B*-.C]* being in Fermi resonance with A-B**-C?4 The vibrational predissociation of [A-B*.-C]* yields A-B C with AEF+ of translational energy. The fragments may contain rotational and possibly vibrational excitation as well. The algebraic statement of the general selection rule, expressed as the rate 7' for vibrational predissociation through the channels represented by processes 1-3, is given by
+
7-I i=
l O I 3 exp[-x(An,
+ Anr + An,)]
= lOI3 exp(-?rAnT) ( 5 )
The method of the extraction of eq 5 from theoretical models is provided elsewhere in the literature'2 and outlined in the Appendix. The selection rule expression, which loosely resembles an early method for estimating vibrational predissociation rates by Klemperer,2s has a satisfying physical explanation. The preexponential factor = l O I 3 s-' gives the typical collision frequency of A-B against C through the van der Waals bond which holds them together in the complex. The exponential term reflects the probability that the initial discrete state of the bound A-B*-C complex and final C will mix during the half-collision state of the fragments A-B and is thus a measure of the reluctance of the complex to change quantum numbers during vibrational predissociation.
+
(20) Butz, K. W.; Catlett, Jr., D. L.; Ewing, G. E., Krajnovich, D.; Parmenter, C. S. J . Phys. Chem. 1986, 90, 3533. (21) Janda, K. C. Adu. Chem. Phys. 1985,60, 201. (22) Miller, R. E. J. Phys. Chem. 1986, 90, 3301. (23) Levy, D. H. Ado. Chem. Phys. 1981, 47, 323. (24) Ewing, G. E. J . Phys. Chem. 1986, 90, 1790. (25) Klemperer, W. Ber. Bunsenges. Phys. Chem. 1974, 78, 128.
4664
The Journal of Physical Chemistry, Vol. 91, No. 18, 1987
The exponential expression contains the change in the total effective quantum number (An,) defined as the sum of the translational (Aq), rotational (&), and vibrational (An,,) quantum changes. The general selection rule thus states that relaxation will proceed most efficiently by that particular channel which corresponds to the smallest value of AnT, Le., the smallest change in the total quantum number. We will now offer an interpretation of these quantum numbers. The energy flow in van der Waals molecules requires a variety of coordinates and quantum states. Allow A-B to represent a chemically bonded diatomic molecule like I, or H F or even a polyatomic molecule such as C2H4 or p-difluorobenzene. This molecule is attached through a van der Waals bond to atom C (or another molecule C-D). The distance between the centers of mass of A-B and C is given by the radial coordinate r shown in Figure 1. Stretching vibration against the van der Waals bond along this coordinate is identified by the quantum number ut. Vibration against the chemical bond of A-B is given by the displacement coordinate x and the quantum number u. For polyatomic molecules we can use normal mode displacements and quantum numbers ul, u2. v3, etc. Internal rotatory motions of A-B against C through the van der Waals interaction is given by the relative displacement angle 0 and quantum number J. For simplicity, we imagine planar rotors. For cases where the barrier to internal rotation is very high, the J quantum number goes over into a vibrational motion of rotatory origin identified by quantum number ur.Il For polyatomic van der Waals moleculed, additional rotational and bending mode quantum numbers may be added.24 The quantum numbers for the fragments of vibrational predissociation are correlated through the same van der Waals molecule coordinates. For the radial coordinate r, the ut quantum number of the bound complex A-B.-C goes over into ql, which is proportional to the translational quantum number of the fragments A-B C. The free rotation of A-B relative to C about angle B is identified by the J quantum number. The quantum numbers of the vibrations involving chemical bonds have the same definitions in the fragments as in the complex. We illustrate in Figure 2 the vibrational predissociation of A-B*-C-D by the V-T channel. The example we have chosen is actually H-F*-sH-F.~~The van der Waals interaction potential between A-B and C-D is shown in the lower curve and has the analytical form of the Morse function
+
which can support a variety of bound states ut = 0, 1, .... The well depth De is measured from the bottom of the potential curve with centers of mass of A-B and C-D separated by distance re. A measure of the steepness of this curve is given by the range parameter a . The intermolecular parameters are imagined to remain unchanged for the A-B* and C-D interaction, and the potential curve is just shifted up by WAB, the vibrational energy in the A-B* chemical bond. The excited van der Waals molecule A-B*-C-D in our example has its chemical bond vibrationally excited to the LJ = 1 state. The wave function Iut) for the stretching motion against the van der Waals bond is shown in Figure 2 for the ut = 0 state. Its exact analytical form is obtained by solving the wave equation for the Morse potential function,26and it can be seen to resemble a harmonic oscillator wave function. Vibrational predissociation of A - B * - G D produces A-B + C-D flying away from each other over the lower potential curve of Figure 2 . The chemical bond vibration of A-B is now related to u = 0. Motion of the fragments against the van der Waals bond is described6*”by the translational wave function, lqt), shown in Figure 2 as a plane wave for large A-B + C-D separations but damped out rapidly near the repulsive wall of the potential curve. The translational wave function is characterized by the dimensionless q ~ a n t i t y ~ ~ ~ , ~ 4, = ( 2 ~ ~ E , ) ’ / ~ / a h ( 2 6 ) Vasan, V. S.; Cross, R. J. J . Chem. Phys. 1983, 78, 3869.
(7)
Ewing
A-BY
4000
-
+
C-D
2000
c
I
E
0
Y
h 0
v---
4%-T
I‘
L
al C
W 0
-2000
I
1i
A-B
+
C-D
re I
Figure 2. Potential surfaces, energy terms, and translational wave functions for vibrational predissociation. The example chosen is for the
complex HF-HF. Adapted from ref 11.
+
with pt = mABmC/(mAB mc) the reduced mass of the complex. For the complex A-B--C-D we replace mc by mcD. The translational energy of the fragments A-B C-D is given by E, = AEV-T for the V-T channel. For our example we have E, = AEv-r = WAB - Dosince the translational kinetic energy is the difference in the vibrational energy, W, between A-B* and A-B, and that lost in breaking the van der Waals bond from the ut = 0 zero point level, Do= De. We may interpret qt as proportional to the number of nodes of the (near) plane wave describing the motion of A-B C-D. For our purposes, what is important is the number of modes embraced by the intermolecular potential well since that is the region in which vibrational predissociation occurs. As we describe below, this number of nodes is approximately equal to 4 4 2 . Since quantum numbers are closely associated with nodal behavior, we may view q t / 2 as the effective translational quantum number of the fragments. In our example of Figure 2 we have q t / 2 = 11. We are now prepared to set down the effective quantum numbers for use of the selection rule expression of eq 5 . Application of the analytical expression for V-T vibrational predissociation rates of A-B*-C or A-B*-C-D for a wide variety of van der Waals molecules bound by Morse intermolecular potential functions like those shown in Figure 2 reveals, as described in the Appendix, the effective translational quantum number change
+
+
Ant = 14t/2 - LJtl
(8)
The quantum number change is essentially the difference between the effective number of nodes, q t / 2 , of the translational wave C, and the function of the predissociation fragments, A-B number of nodes, ut, in the van der Waals stretching vibrational wave function of A-B*--C. The exponential dependence on these quantum numbers in eq 5 is consistent with the poor FranckCondon type overlap expected for wave functions with widely differing numbers of nodes such as those shown for example by A-B*.-C-D predissociation in Figure 2. The influence of rotational degrees of freedom during the vibrational predissociation process is the most difficult to model.
+
The Journal of Physical Chemistry, Vol. 91, No. 18, 1987 4665
Feature Article
TABLE I: Quantum Numbers and Lifetimes for A-B*- .C Vibrational Energy Transfer
.
A-B*. .C A. Br2*".Ne B. N2****N2 C. HCI*...Ar D. HF*...Ar E. H2*...Ar F. D,*...Ar G. CzH4*-..Ne H. 12*".He I. Clz*...Ne J. HF***HF* K. p-DFB*. .Ar L. p-DFB*. .Ar M. (N0)2*
An, 0.4 10.8
A4 0 0
A4
AnT
1 1
1.4
3.1
2.4 3.8
1 1 1 1 2 1 1 1 1 2 1
2.5 6.0 7.0 1.o
1.3 5.4 4.1 1.4
-
8.5 5.4
0 0 0 0 0
3.7 0 0 0
channel 2 x 10-12
V-T V-T V-T, R V-T, R V-T V-T
11.8 6.5 7.3 7.0 8.0 3.0
>lo2 4 x 10" >3 x 10-4 3 x 10-2 1 x 10-0 3 x 10-10 1 x 10-10 >10-5 1 x 10-9 8X 4 x 10-8 4 x 10-1'
v-v
(u7 = 1 to V I 0 = 1) V-T V-T V-T, R V-T (u6 = 2 to Ug = 1) V-T (u6 = 2 to u6 = 0) V-T
2.3
6.4 8.8 2.4 10.5 6.4
'A, see text. B, see text. C, see text. D, using WAB= 3958 cm-I and Bo = 20.5 cm-' from ref 33, a = 2 X lo8 cm-l (a typical value),6Do= 92 cm-l [Douketis, C.; Hutson, J. M.; Orr, B. J.; Scoles, G. Mol. Phys. 1984, 52, 703, and a = 0.95. E, using a = 1.8 X lo8 cm-I, Do = 25 cm-I, and WAB= 4160 cm-' from ref 38. F, using a = 1.8 X 10" cm-I, Do= 30 cm-I, and WAB= 2990 cm-I from ref 38. G, see text. H, using relaxation of 12(u = 12)...He from the %(B) I2 state [Johnson, K.; Wharton, L.; Levy, D. J . Chem. Phys. 1978, 69, 27291, a = 2 X lo8 cm-I, and Do = 10 cm-I, see ref 2. I, relaxation of C12(u=l)...Ne from the XIZg+ C12 state, r > s, W A B = 560 cm-I [Brinza, D.; Swartz, B. A.; Western, C. M.; Janda, K. C. J. Chem. Phys. 1984, 84, 6272, Do= 55 cm-' (ref 30), a = 2 X lo8 cm-I. J, relaxation of the excited proton donor HF...HF*, using a = 1.6 X lo* cm-' and CY = 0.95," E, = (10)(11)(20.5) = 2255 cm-1,33WAB- Do = 2574 cm-1$77 = 1 X s.I6 K, see text and ref 20. L, see text and ref 20. M, using WAB= 1788 cm-I, Do = 550 cm-l, a = 3.1 X lo8 cm-I,l6 and r = 40 X s.~'
In the spirit of this presentation we will use the simplest possible treatment by replacing the reduced mass pLtfor the translational motion in eq 7 by Z/r: for the rotational motion. This substitution, first used by Moorez7to understand collision pair V-R vibrational relaxation, involves I, the vibrating molecule moment of inertia, and r,, the distance between its center of mass and the vibrating atom. With CH4* for example, r, becomes just the C-H bond length. For a diatomic molecule we have r, = w A B where rAB is the A-B bond length and
=m ~ / ( + m me) ~
(9)
gives the fraction of the vibrational displacement of B toward C in the A-B--C complex of Figure 1. With Z = &rAB2 and II, = mAmB/(mA + ms), the reduced mass of A-B, Moore's substitution yields the dimensionless quantity q, = ( 2 ~ , E , ) ~ / ~ / a a h
where E, is the rotational kinetic energy of the fragment A-Bt. Since E , = J(J l)h2/2Z, substitution into eq 10 reveals q, = J since for typical values of the molecular parameters arABa= 1. The rotational quantum number of A-Bt, J, is then quite close to 4,. By analogy to the effective translational quantum numbers and as discussed in the Appendix, we shall take
+
Anr
i=
lqr/2
- UrI
(11)
Overlap between the chemical bond vibrational wave functions is also given in terms of their effective quantum numbers. As described in the Appendix, we write Anv = yluf - uil = rlAul
(12)
where u, is the vibrational quantum number of A-B* within the C complex and uf labels the vibrational state of the A-B fragments. The factor y is the measure of the effectiveness of the coupling of vibrational motions of the chemical bonds in A-B* with the intermolecular or van der Waals bond which holds it to C in the complex. Without this specific vibrational coupling with the van der Waals bond there can be no predissociation. The value y = 1 for diatomic A-B is demonstrated in the Appendix. Equation 12 is then consistent with the Parmenter-Tang selection ruleszs and empirical observation^^^ that reveal each quantum change in an internal vibrational mode participating in vibrational energy transfer reduces the relaxation rate by a factor of lo-' to (= exp[-.lrluf - vi!]). In cases where the vibrations within
+
(27) Moore, C. B. J . Chem. Phys. 1965, 43, 2979. (28) Parmenter, C. S.; Tang, K. Y . Chem. Phys. 1978, 27, 127. (29) Lambert, J. D. Vibrational and Rotational Relaxation in Gases; Oxford University: London, 1977.
,
0
I
2
l
l
4
,
l
6
1
I
8
/
l
10
I
I
,
12
A* T Figure 3. The total quantum number change, AnT, and lifetimes, I , for vibrational predissociation. The line is the selection rule expression of eq 5. The data are from Table I. Data marked by ! result from a breakdown of our selection rule as discussed in the text. The upward arrows indicate that the experimental lifetimes are lower limits.
plyatomic A-B* do not effectively couple with the van der Waals stretching motions, y becomes large and exp(-xyluf - u,l) goes to small values. Predissociation through changes involving these vibrational modes is thus unlikely. We have now described the recipes for the specific selection rules of An,, An,, and An,, that go into the general selection rule of eq 5 . We will next examine how well these selection rules describe the experimental data. 4. Some Successful Applications of the Selection Rules Applications of our selection rules are presented in Figure 3. The line is the graphical representation of eq 5. Values of An, are the total effective quantum number change (An, An, An,,) according to the recipes of eq 7-12. The lifetimes, T , of vibrational predissociation are those observed either in a laboratory (experiments with lasers) or by detailed calculations (experiments with computers). Each letter in Figure 3 represents the vibrational predissociation of a particular van der Waals molecule identified in Table I. The coordinates of the letters are positioned according to the calculated value of AnT by our recipes and by the observed lifetime T . If the letter lies along or close to the line of eq 5 , then our simple selection rules are successful. If the letter is far removed from the line, then mechanisms are operating for the vibrational predissociation process that are outside the assumptions needed to obtain the simple selection rules. We can see from the examples in Figure 3 that the observed lifetimes span 14 orders of magnitude. The shortest lifetime, T =2X s, is from spectroscopic bandwidth measurements of
+
+
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The Journal of Physical Chemistry, Vol. 91, No. 18, 1987
Cline et al.30on Br2(u=27)-Ne. Vibrational predissociation produces BrZ(u= 26) N e within the B electronic state of Br, and is point A in Figure 3 and Table I. With Do = 62 cm-' and WAB = 64 cm-' we have E, = WAB- Do.= 2 cm-' (4 X lo-'' J). Taking a = 2 A-' (2 X 1O'O m-I) as a typical range parameter3-I0 kg), we obtain Ant = 0.4 using eq and p, = 17.8 amu (3 X 7 and 8. We have Anv = 1 and assume the V-T channel so An, = 0 and therefore AnT = 1.4. Point A in Figure 3 nearly lies on the theoretical curve. If we explore the V-R channel of eq 2, we must contend with the rather large moment of inertia of Br,. Let us compare the V-R and V-T channels for disposing of the same amount of kinetic energy by taking the ratio of eq 7 and 10. We find q,/q, = ( p r / p t ) ' / 2 / a= 3 using a = 0.5 and p, = 40 amu. The effective quantum number for the production of rotational energy is 3 times that for the production of translational energy. Thus, the V-T channel is favored over the V-R channel since it provides the smallest value for An,. This is consistent with the product distribution that shows little rotational excitation of the Br, fragment.M The selection rule also help understand the observed3O propensity for Au = -1 (or Anv = 1) relaxation on two counts. First, an increase in the Au change would increase &,and second, it would also increase Ant since E , would become larger. At the other lifetime extreme, consider vibrational predissociation of N,*-.N,. This lifetime has not been measured for the isolated van der Waals molecule, and its infrared spectrum under low resolution reveals no predissociation effect^.^' However, in a related experiment, the vibrational relaxation of N2(u=1) by collisions in the neat liquid occurs with 7 I10, s.32 If we imagine liquid nitrogen as a collection of van der Waals molecules, then this measurement provides a lower limit of 10, s for N2*-N2. This lifetime is indicated in Figure 3 by "B"with the arrow to indicate the lower limit of the measurement. This long lifetime for N2*-N2 vibrational predissociation is consistent with the calculated total change in quantum number of AnT = 11.8 for the V-T channel. This large value of An, is a consequence of the large product of reduced mass and vibrational frequency of N2*-.N2. Here we have used E, = WAB- Do = 2270 cm-' and a = 2 A-', a typical range parameter, to obtain An, = 10.8. After adding An, = 1 we obtain the value of AnT given in Figure 3 and in Table I. We favor V-T over the V-R channel because of the relatively large moment of inertia of N,. Thus, it would require J = 33 to place a rotating N, into near resonance with the energy released on vibrational predissociation: E, = J(J + l ) B = 2244 cm-' with B = 2 ~ 3 1 7 - l . ~Using ~ this value of E, into eq I O and 1 1 results in An, = 14.9 and AnT = 15.9. Possibly a more favorable V-R channel would divide angular momentum equally between the two N, fragments. In this case p, in eq 10 is replaced by the reduced mass of the two partners, pr/2.s4 As a consequence, An, = 10.5 and AnT = 1 1 . 5 which becomes competitive with the equally inefficient V-T channel. In any event the lifetime of N2*.-N2 is very long and consistent with our selection rules. With its large reduced mass and high vibrational frequency together with an inefficient V-R channel and closed V-V channel, N2*-N2 is likely the longest lived vibrationally excited van der Waals molecule. Vibrational predissociation of CO*.-CO, with its vibrational frequency and reduced mass similar to N2*.-N2, is likewise a slow process as indicated by liquid-state measurements which yield T L 1 s.35 Consider next vibrational predissociation of HCl*-Ar as explored in a thorough theoretical investigation by Hutson.13 His close coupling calculation result appears as point C in Figure 3 with T = 4 X s corresponding to the dominant V-T, R relaxation channel that produces vibrationally relaxed HClt in
+
(30) Cline, J. I.; Evard, D. D.; Reid, B. P.; Sivakumar, N.; Thommen, F.; Janda, K. C. Structure and Dynamics of Weakly Bound Molecular Complexes; Weber, A,, Ed.; Reidel: Dordrecht, 1987. (31) Long, C. A,; Henderson, G.; Ewing, G. E. Chem. Phys. 1973,2,485. (32) Chandler, D. W.; Ewing, G. E. J. Chem. Phys. 1980, 73, 4904. (33) Herzberg, G . Spectra of Diatomic Molecules; Van Nostrand: Princeton, 1950. (34) Townes, C. H.; Schawlow, A . 1_. Microwave Spectroscopy; McCraw-Hill: New York, 1955. ( 3 5 ) Chandler, D. W.; Ewing, G. E. Chem. Phys. 1981, 54, 241.
Ewing the J = 15 level. The V-T relaxation channel is calculated to be orders of magnitude less efficient. In applying our simple model, we first explore the V-T channel. We use WAB = 2886 cm-l and Do = 116 cm-' to obtain E, = 2770 cm-'. An exponential fit of the intermolecular potential surface36 gives as = 2.6 A-', and with = 18.9 amu, we obtain An, = 10.7. Since An,, = 1 , the total quantum number change becomes AnT = 11.7. It is this large quantum number change, due to both the high vibrational frequency and the large reduced mass, that is responsible for the sluggish vibrational predissociation by the V-T channel as found by Hutson.13 We can understand an efficient rotational relaxation by comparing the V-R and V-T channels if each accept the same kinetic energy. Taking the ratio of eq 7 and 10, we find q,/q, = (p,/ p , ) ' / 2 / a = 0.2, where p, = 0.97 amu and a = 0.97. Thus, rotational kinetic energy requires a much smaller effective quantum number than does translational kinetic energy. We now calculate the quantum numbers required by the V-T, R channel. Suppose that vibrational predissociation of HCl*-.Ar produces the nearresonant HClt fragment in its J = 15 level so that it carries away E, = J(J + l)B = 2540 cm-' of rotational kinetic energy (B = 10.6 cm-l 33). Use of eq 10 and 1 1 gives An, = 2.4. The remaining kinetic energy is translational, WAB - Do- E, = E, = 230 cm-I, giving An, = 3.1 by eq 7 and 8. With An,, = 1.0 we have AnT = 6.5 for this V-T, R channel together with T = 4 X s36to give the point C lying close to the line in Figure 3. Thus, our simple selection rule shows that the V-T, R channel is more efficient than the V-T channel in accord with the close coupling calc~lation.'~ In an analogous way, we predict that vibrational predissociation of HF(u=l)--Ar proceeds by a V-T, R channel to produce HFt in the J = 13 level. The selection rule yields a total quantum = 7.3 (Table I). The lifetime, change by this channel to be r>3 X s is identified by the letter D in Figure 3 with an upward arrow to indicate that the laboratory observation by Huang et al.37gives a lower limit. With its small moment of inertia, HFt rotation carries away much kinetic energy with little cost in angular momentum (Le., small J or An,). The V-R channel is therefore efficient. By contrast, the V-T channel is inefficient since relaxation involves translating the center of masses of HF and Ar at the expense of large linear momentum (Le., large An,). In comparing relaxation of H2(u=l)--Ar and D2(u=l)--Ar, data E and F in Figure 3, the V-T channel is efficient and the selection rule calculations are in reasonable accord with close coupling calculations of Hutson, Ashton, and Le Roy.'4 It should be noted that while the vibrational energy of H, is greater than that of D,, the lifetime for vibrational predissociation of H2(u=l)-Ar is less than that of D,(u=l)-Ar. Thus, the correlation of lifetimes is not with energy gap but rather with the effective quantum number change. Energy gap, E,, does appear in the relationship for translational quantum number, eq 7, but then so does the reduced mass, p,, and range parameter, a. It is all these molecular parameters taken together to produce the effective translational quantum number which guides vibrational predissociation. Our rules for H2*-.Ar or D,*-Ar relaxation by V-T, R channels suggest that rotational excitation of the fragments H, or D2 is not important. Close coupling calculations show,I4 on the contrary, that while the V-T channel is important, so are certain V-T, R channels. This demonstrates the quantitative shortcoming of our selection rules. They are not suitable for pinpointing which particular V-T, R channel is most efficient for relaxation, particularly in cases such as H,*.-Ar and D2*-Ar where q,/q, = ( p r / p t ) ' / 2 / a= 1 . Vibrationally and rotationally excited H2*--Ar has a relaxation lifetime 6 orders of magnitude shorter than the vibrational predissociation lifetime we have been d i s c ~ s s i n g .This ~ ~ occurs since (36) Holmgren, S. L.; Waldman, M.; Klemperer, W. J. Chem. Phys. 1978, 69, 1661.
(37) Huang, Z. S.;Jucks, K. W.; Miller, R. E. J. Chem. Phys. 1986,85, 6905. (38) Le Roy, R. J. In Resonances; D. G., Truhlar, Ed.; American Chemical Society: Washington, DC, 1984; ACS Symp. Ser. No. 263, p 23 1 ,
The Journal of Physical Chemistry, Vol. 91, No. 18, 1987 4667
Feature Article the rotational energy within H2*.-Ar is sufficient to break the van der Waals bond. The fragments Hz* Ar still contain vibrational energy. This process, termed rotational predissociation, has also been theoretically explored in HClt...Ar.3g While our selection rules are not designed to treat this channel, it is easy to see that the total effective change in quantum numbers, AnT, will be small. We have A h = 0, and since the fragments contain little translational energy An, will be small. Finally, the weak van der Waals bond requires only a small change in An, to provide dissociation. We turn now to the vibrational predissociation of C2H4(u,;; l)-.Ne with a 5 X 1O-Io s lifetime reported by Casassa et al. The excited vibration is an out-of-plane mode (v, = 949 cm-'), and the van der Waals bond has Do = 100 cm-'. The V-T channel puts E, = 950-100 = 850 cm-l into translation of the relatively massive CzH4and Ne fragments. Use of our selection rule expression (with a = 2 A-l and y = 1) yields AnT = 13, a value so large as to be incompatible with the observed lifetime. Likewise, considerations of the V-T, R channel reveal too large a value of AnT. The V-V channel, however, explains the experiment. We assume that a low-lying C2H4 mode (vl0 = 826 cm-I) absorbs energy, leaving only 23 cm-' into fragment translational energy corresponding to An, = 1.0. Now with An, = 2 (since two vibrational quanta are involved in this channel) we have AnT = 3.0 and with T = 5 X SI' we position point 6 in Figure 3 which we see lies near our theoretical curve of eq 5. Detailed close coupling calculations of Hutson et al.I9 bear out our simple arguments favoring the V-V channel. Since changes in vibrational level can absorb the greatest amount of energy with the least quantum change, we can expect V-V channels to be particularly efficient in vibrational predissociation processes. This has been illustrated elsewhere in other, more intricate, theoretical c o n s i d e r a t i ~ n s . ~ J ~ ~ ~ ~ In closing this section illustrating the successful applications of our selection rules, we need to reemphasize that quantitative predictions yielding numbers for T are unreliable. The error limits on T can span an order of magnitude or so, principally because of the uncertainty in the range parameter a and the approximate form of the specific selection rules expressed by eq 8, 11, and 12. The usefulness of the selection rules is in locating the most efficient relaxation channel and providing a crude range for the lifetime: picosecond, nanosecond, microsecond, and so on. However, even within this generous range of acceptable uncertainties several examples of vibrational predissociation lifetime measurements deviate dramatically from the selection rule predictions. These cases result from vibrational predissociation processes that proceed by mechanisms outside the assumptions required for the successful application of the selection rule expression. We will now explore the breakdown of these assumptions.
+
5. Failures of the Selection Rules Let us consider first the case of (HF),* vibrational predissociation. The V-T channel illustrated in Figure 2 corresponding to An, = 11 and An, = 1 yields a large total quantum number change anT= 12. The V-T, R channel using our recipes is clearly more efficient, and it places a HF fragment into the J = 10 state. The total quantum number change AnT = 8.8 as determined in Table I together with the observed lifetime T = 1 X lo4 s provides the location of point J! in Figure 3. The exclamation point by the letter is given to emphasize the dramatic deviation of its position from the theoretical curve of eq 5. Clearly the selection rules are not working. The difficulty is resolved by comparing the eAcectiue intermolecular potential surfaces for (HF),* predissociation by the V-T and V-T, R channels. The inefficiency of the V-T channel is understood by reviewing again Figure 2 which has been constructed for (HF),* with the fragments imagined to be in their nonrotating or J = 0 states. It is the severe mismatch between (39) Ashton, J. A.; Child, M. S.; Hutson, J. M. J. Chem. Phys. 1983, 78, 4025. (40) Beswick, J. A.: Jortner, J. J . Chem. Phys. 1981, 47, 6725.
18
Figure 4. The structure of HF-HF and the effective van der Waals radii of the rotating fragments HFt + HFt. Data from ref 41 and 42.
4000
1
-
I
I
E
2
2000 -
r L m
a,
c
Figure 5. Potential surfaces for HF + HF*, HF + HF, and HF + HFt. Translational wave functions of HF-.HF*, (ql,and HF + HFt, (qtl are also shown. Adapted from ref 43.
the ut = 0 wave function and the oscillations in the near plane wave with 4,/2 = 11 that leads to the large effective translational quantum change An, = 11 and the slow relaxation. Suppose, on the other hand, that each fragment is produced in a high rotational state. The effective potential surface for H F t + HFt then resembles Ne Ne since each rapidly spinning HFt molecule will have its strong anisotropic interactions averaged out to mimic those of isoelectronic Ne. This effective potential surface for spinning HFt HFt is dramatically more shallow than for the near-rigid HF-HF. Furthermore, the effective separation between HFt HFt is greater than that of HF-HF. This is illustrated schematically in Figure 4. In the lower part of this figure, the HF-HF equilibrium structure4' is drawn with Pauling's van der Waals radii.42 The upper part of the figure represents HFt + HFt. The spinning H F t molecules sweep out spheres defined by the outermost boundaries of their van der Waals radii. In order that the van der Waals radii of HFt HFt not overlap, the spinning molecules are shifted away from each other. In order to examine the consequences of the effective interaction of spinning HFt on vibrational relaxation, Halberstadt et al.43 performed close coupling calculations. In their model the proton donor was excited and the other monomer frozen at its equilibrium position: HF-.HF*. The products of vibrational predissociation
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(41) Dyke, T.; Howard, B. J.; Klemperer, W. J . Chem. Phys. 1972, 56, 2442. (42) Pauling, L. The Nature of the Chemical Bond, 3rd ed.; Cornell University: Ithaca, NY, 1960. (43) Halberstadt, N.; BrEchignac, Ph.; Beswick, J. A,: Shapiro, M. J . Chem. Phys. 1986, 84, 170.
4668 The Journal of Physical Chemistry, Vol. 91, No. 18, 1987
. -/---
Ewing
were taken to be HF(J=O) and HFt(J=lO). Surfaces for H F + u,= 2 I HF* and H F + HF from Figure 2 are reproduced in Figure 5. In addition, the effective HF + HF' (J = 10) surface is added as taken from the close coupling calculations. Not only is the HF H F curve more shallow than that of H F HF, consistent with u,, u, == 21 our qualitative considerations leading to Figure 4, but it is shifted up by the rotational energy J(J l)B of HFt. As a consequence, the H F HF* and H F H F t curves cross. The translational wave function (qtl of the fragments H F HFt, as illustrated in 1 Figure 5, now effectively overlaps the (uti wave function of HF.-HF* and accounts for the efficient vibrational predissociation by this channel. This curve crossing mechanism, first suggested by E ~ i n to g ~ ~ account for the efficient vibrational predissociation of both (CZH4)2* and (HF),*, has been studied in detailed close coupling calculation^^^*^^ that provide numerical values of 7 in reasonable accord with laboratory m e a s ~ r e m e n t s The . ~ ~ presentation ~~~ of the selection rule of eq 5 requires the effective intermolecular potential surfaces of initial and final states to be the same shape, merely displaced vertically by the energy in the vibrating chemical bonds as in Figure 2. For curve crossing cases, a different simple estimate of the efficiency of vibrational predissociation based on the Landau-Zener model has been provided elsewhere.44 The possibility of curve crossing is more likely for complexes with small r(A) moments of inertia and strong anisotropic van der Waals bonds. Thus, curve crossing is important to vibrational predissociation Figure 6. Potential surfaces for p-difluorobenzene-.Ar. Adapted from of (HF),* and (C2H4)2* but not for HCl*-Ar or 12*-.He. ref 20. Another class of anomalies in vibrational predissociation is exist. Because of this mixing, the original specification of the represented in certain relaxation channels of p-DFB*-Ar (pexcited state of p-DFB*-Ar as ut = 0, u6 = 2 has become blurred. difluorobenzene*-Ar) whose structure has Ar sitting above the The energy once localized, or so we imagined, into u6 = 2, ut = plane of the aromatic ring.20 Here it is vibrational predissociation 0 is now distributed among the variety of near isoenergetic ui, ut in the SI electronic state of the complex as measured by Butz et a1.20which we will explore. In one relaxation channel, p-DFBstates. We must now incorporate the presence of Fermi resonance into Ar with E, = AEv-T = 10 (u6=2).-Ar producesp-DFB(u,=l) our simple model for vibrational predissociation. Suppose we let cm-'. This ring vibration has a frequency v6 = 410 cm-' only slightly larger than the van der Waals bond strength of Do = 400 P112represent the admixture of, say, the ~ 1 =7 2, ut = 8 state wave cm-]. Using y = 1 so that An, = 1.0, we find AnT = 2.4. With function or other nearby state wave functions with the u6 = 2, ut = 0 state wave function. We may now rewrite our original sethe observed lifetime of 8 X loy9s the point K in Figure 3 is in reasonable accord with the theoretical curve. lection rule expression from eq 5 a~~~ Application of the selection rules for another observed relaxation T - ~= 1013P exp[-a(An, An, An,)] (13) channel, p-DFB(u6=2)-Ar p-DFB(u6=O) + Ar, with E, = AEv-T = 421 cm-I gives q t / 2 = An, = 8 . 5 , An, = 2, and consewhere the mixing coefficient converts into probability P . The quently AnT = 10.5. With the observed lifetime of 7 = 4 X lo-' interpretation of eq 13 for our example is that the collision fres, this point is L! in Figure 3 and deviates from the theoretical quency of Ar against the ring of a lOI3 s-' is now scaled by the curve by 6 orders of magnitude. Considerations of possible V-T, probability P of finding the initial state of the complex with, say, R channels also offer low predissociation efficiency because of u I 7 = 2, ut = 8 character. The effective quantum numbers retain the large moment of inertia of p-DFB. Again, the selection rules their usual significance except that now we refer to the relaxing have failed. state as u17 = 2, ut = 8 rather than u6 = 2, ut = 0. The final state The breakdown of the selection rules arises because the defiremains the same with p-DFB(u=O) + Ar and E, = = 421 nitions of the chemical bond vibrational quantum number vi (and cm-'. Now consider the change in effective translational quantum consequently &) and the van der Waals quantum number ut (and number. Since the energy gap is still large, we have 4 4 2 = 8.5 consequently An,) have been blurred by Fermi resonance. We and many nodes of the near plane wave of the final state of the are now considering the relaxation channel of eq 4 expressed early departing fragments. However, the initial state is now in the high in this paper. We present in Figure 6 potential surfaces associated van der Waals level of ut = 8 so the change in effective translational with some selected vibrational levels of p-DFB. The small energy quantum level is small, yielding Ant = Iq/2 - u,l = 18.5 - 81 = 0.5. gap between p-DFB**-Ar with ut = 0 and u6 = 2 and the disNow with An,, = 2 and assuming no rotational excitation so An, sociation limit corresponding to the fragments, p-DFB*(u6 = 1) = 0, we obtain An, = 2.5. We have suggested ways to estimate + Ar, is apparent in Figure 6 . It is consistent with the observed mixing probability elsewhere,24and a value for Fermi resonance efficient predissociation channel. Production of p-DFB(u6=O) and within p-DFB-Ar for this level is P = The net result in Ar on the other hand creates the fragments with large translational applying eq 13 yields T = lo-* s, a value within an order of energy or, what is more relevant to the discussion, a large magnitude of the observed lifetime. translational quantum number. Consider, however, the near Fermi resonance is responsible in increasing the efficiency of energy match between the ut = 0, u6 = 2 level (that is A-B**.-C vibrational predissociation because the effective change in in eq 4) and the ut = 8, uI7 = 2 level (that is [A-B*-C]* in eq translational quantum number can be dramatically reduced. We 4). These two levels can be mixed by Fermi resonance through have demonstrated the importance of Fermi resonance in vibrathe same type of coupling terms that can lead to vibrational tional predissociation of t e t r a ~ i n e - A r ~and ~ have suggested its predissociation. Since there are a variety of other ut, uI levels near possible role in (N,O), rela~ation.'.~We would suppose that as ut = 0, u6 = 2 numerous other possibilities for Fermi resonance molecules studied become more complicated the opportunities of Fermi resonance will increase' and the usefulness of the simple selection rule will diminish. It is for these larger molecules that (44) Ewing, G . E. Chem. Phys. 1981, 63, 411. we would expect the statistical predictions of RRKM theory to (45) Peet, A. C.; Clary, D. C.; Hutson, J. M. Faraday Discuss. Chem. become successful46in accounting for vibrational predissociation. SOC.,in press.
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The Journal of Physical Chemistry, Vol. 91, No. 18, 1987 4669
Feature Article
by mild extensions of the models for radiationless transitions found We turn next to assess the limitations of our selection rule for in the classic papers of Bixon and Jortner.” Until these calcuquantitative determination of the lifetime of vibrational predislations are performed, it is not clear that any simple selection rules sociation in the case of a highly forbidden relaxation channel. We exist to understand kIVRfor a wide variety of van der Waals consider the V-T channel of HCl*-Ar as explored in the thorough molecules. theoretical calculations of Hutson.13 He finds from a close couThe presence of the two rates, k,, and 7-l describing vibrational pling calculation the lifetime T = 1 X s. Using our estimate flow with eq 4, can have a complicating effect on both the apof AnT = 11.7, given earlier in this article, and eq 5, we find T pearance of the spectrum and the decay of the excited state. The = lo3 s. One cause of the discrepancy between our selection rule consequences of the interplay between klVRand 7-l is clearly laid predictions and the close coupling calculation may be in the out in the excellent review by McDonald53and applied to van der uncertainty in range parameter. A variation in the value selected, Waals systems by Gentry.54 a = 2.6 R-,-I,36by only -+IO% yields a range in T from 10 to lo4 s in eq 5. When the channel is highly forbidden, as in the case 6. The Future of the Selection Rules of V-T for HCl*-Ar, AnT is large and small changes in the range parameter make a big difference in the lifetime. A more serious The selection rules we have presented here have been shown cause of the discrepancy occurs because of the inadequacy of the to be useful in the interpretation of a wide collection of vibrational first-order perturbation Golden Rule expression. Huston shows predissociation data. This successful demonstration was illustrated that the lifetime for the V-T relaxation of HCl*-.Ar, using in Figure 3 by the many data that fall on the theoretical line that first-order perturbation theory, differs by orders of magnitude from expresses the selection rules. However, a number of examples, the close coupling calculation v a 1 ~ e . lLe ~ Roy has also criticized by their deviation from the theoretical line, indicate a breakdown the quantitative results of the Golden Rule c a l c u l a t i ~ n s .As ~ ~we of the selection rules. These cases can be rationalized by reademonstrate in the Appendix, our selection rule is extracted from sonable mechanisms for vibrational relaxation. Since the study the Golden Rule. Hence, when the Golden Rule is inadequate of vibrational predissociation of van der Waals molecules is for lifetime determinations, then so will our selection rule give currently an active area of research, we hope to see further tests poor quantitative results. It is important to recall, however, that of the usefulness of our general selection rule in the near future. while our selection rule gives poor numbers it is correct in preAs we remarked earlier in this article, the coupling of vibrational dicting and inefficient V-T channel and an efficient V-T, R motions through intermolecular interactions appears not only in channel. van der Waals molecules but in gaseous collisions and condensed Another class of anomolies is represented by the vibrational phases as well. The same general selection rule then applies to predissociation of (NO),*. It appears as point M! in Figure 3. vibrational relaxation processes in gases, liquids, and solids and The molecular parameters used in estimating AnT = 6.4 are to on surfaces. The demonstration of the usefulness of the general be found in Table I and the observed 40-ps lifetime is from the selection rule to these systems will be the subject of a future study. spectroscopic experiments of Casassa et a’!l The interpretation Acknowledgment. I thank Charles S . Parmenter for his many of the unexpectedly short lifetime for this vibrationally excited helpful suggestions during the preparation of this manuscript. The complex is not settled but is likely to be connected with the detailed comments of Jeremy M. Hutson have been particularly explanation for the unusually efficient collision pair vibrational relaxation of NO* NO NO NO.48 Here N i k i t i ~ ~ i ~ ~valuable. The financial support of the National Science Foundation is gratefully acknowledged. suggested that efficient relaxation occurs because of curve crossing among the variety of potential surfaces which correlate to the Appendix separated XzlI,,z or X2113/z electronic ground states of monomer NO. Somewhat analogous to the reasons we have given for We cannot provide a derivation of the general selection rule accelerated vibrational predissociation when curves cross because of eq 5 or the specific selection rules of eq 8, 11, and 12, but we of rotational effects (e.g., (C2H& and (HF),), we can expect a can show that their form reasonably follows from analytical exbreakdown of our selection rules when electronic surfaces cross. pressions of vibrational predissociation when the van der Waals Efficient overlap of the ( q J translational wave function of the molecule A-B*.-C is trapped in a Morse potential well. predissociation fragments with the bound state Iul) wave function We begin with the V-T channel of process 1 following early of the complex can occur even when the products have large kinetic treatments by Coulson and R ~ b e r t s o n Beswick ,~ and J ~ r t n e r , ~ . ’ energy provided the corresponding surfaces cross at an opportune and Ewing.6Jl We have the discrete state of the van der Waals region. The vibrationally excited (NO),* may then relax by this molecule, A-B*.-C, buried in the translational continuum of the sort of mechanism. Detailed theoretical calculations will have C fragment states. The calculation of the rate 7-l of A-B to establish this channel. vibrational predissociation suggests the application of the Golden It seems likely in any event that vibrational predissociation of Rule or its equivalent an excited electronic state of a complex will proceed at anomalous lifetimes when there are opportunities to cross over to nearby electronic surfaces. where Our selection rule is not adequate to interpret the IVR rate constant klVRof eq 4 where some experimental measurements are Vf = (2AEv-~/pt)’/~ (2A) now a ~ a i l a b l e . ~ Since ~ * ~ IVR ~ * ~is~ a near-resonant process within the van der Waals molecule, we can have large values of An, is the final velocity of the fragments, A-B C, in terms of their between the zero-order states of A-B**-C and [A-B*-C] *. translational kinetic energy The initial state wave function Consequently, the mixing probability P among the states will be for A-B*-C is identified by the quantum numbers ti) = Iui;u,,ut) small. However, the klVRrate is also proportional to p, the density = Iq)lu,)luI). Motions against the A-B chemical bond are given of nearby [A-B*-C]* states. Hence, a knowledge of both P and by the ]vi) harmonic oscillator wave function and are functions p is required to uncover kIVR.Estimations of kwR can be developed of vibrational displacement x in Figure 1. Bending motions of A-B against C through the angle 0 in Figure 1 are described by Iu,) which can be either free rotor or hindered rotor wave func(46) Kelley, D. F.; Bernstein, E. R. J. Chem. Phys. 1986, 90, 5164. tions” depending upon the extent of the anisotropy of the in(47) Casassa, M. P.; Woodward, A. M.; Stephenson, J. C.; King, D. S . J . termolecular potential holding the van der Waals molecule toChem. Phys. 1986, 85, 6235.
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(48) Stephenson, J. C. J . Chem. Phys. 1974, 60,4289. 1973, 59, 1523. (49) Nikitin, E. E. Opt. Specrrosc. 1960, 9, 8 . (50) Gentry, W. R. Resonances; D. G.Truhlar, Ed.; American Chemical Society: Washington, DC, 1984; ACS Symp. Ser. No. 263, p 289. (51) Ramaekers, J. J. F.; van Dijk, H. K.; Langelaar, J.; Rettschnick, R. P. H. Faraday Discuss. Chem. SOC.1983, 75, 183.
(52) Bixon, M.; Jortner, J. J . Chem. Phys. 1968, 48,715; 1969, 50, 4061. (53) McDonald, J. D. Annu. Rev, Phys. Chem. 1979, 30, 29. (54) Gentry, W. R. In Structure and Dynamics of Weakly Bound Molecular Complexes; Weber, A., Ed.; Reidel: Dordrecht, 1987.
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The Journal of Physical Chemistry, Vol. 91, No. 18, 1987
gether. Stretching vibrations against the van der Waals bond whose potential is given by the Morse function of eq 6 is expressed by the Iut) wave function as in Figure 2. The Morse well can support ut = 0, 1 , ..., d - 1 bound states" where
d = (2ptDe)'/2/~h
(3'4)
Since the parameters a and De of the Morse potential of eq 6 are usually only approximately known, we lose little useful information in rounding off d to an integer. The vibrational energy levels of the van der Waals stretching mode relative to the bottom of the V(r) well are expressed by D,! = -[ah(2d - 1
- 2u,)l2/8pt
+ D,
(4'4)
For the final state of A-B + C we have If) = (uf;J,qt) = Iuf)lJ)lq,). The vibrationally relaxed A-B is described by the final state harmonic oscillator wave function Iuf), Rotational motion of A-B and C becomes free as the fragments are removed a far distance from each other and is identified by the wave function IJ). The translational wave function, lqt), describes the fragments flying away from each other over the Morse potential as in Figure 2. The term V, of eq 1A couples the vibrational motion of A-B with motions involving changes in the van der Waals bond length. Following the Schwartz, Slawsky, and Herzfeld (SSH) theory of collision pair vibrational r e l a x a t i ~ n , we ~ ~ assume , ~ ~ that the interaction is modulated by the A-B vibration only through changes in the nearest-neighbor B--C distance. Because of the rapid decrease in van der Waals interaction with distance, vibrational displacement of A is ignored. The result is Vc(x,O,r) = V(r-asx) - V(r) = -asx[dV/dr]
(5A)
which is equivalent to saying that an increase in vibrational displacement, a s x , of B toward C produces the same energy change as a decrease by the same amount, dr, of the van der Waals bond length. The portion of the coupling term of eq 5A which is a function of r is the derivative of the Morse potential of eq 6 dV/dr = -2aDe[e-2a(Ne) - e-a(prc)]
(6.4)
and the fraction of the vibration of B toward C is weighted by the masses through a as given by eq 9. The angle A-B makes with the axis connecting the center of mass of A-B with C is 0 as shown in Figure 1 . The component of the vibrational displacement x of A-B pointing toward C depends on the orientation and is approximated by the function s = exp(-P cos
me)
(7'4)
borrowed from treatments which model vibrational relaxation of A-B* in a solid matrix.57 The parameter p gauges the extent to which higher order anisotropic terms influence the intermolecular potential function. The number of energetically equivalent orientations of A-B within the complex is given by m. The matrix element we need to evaluate is the three-dimensional integral (uf;J,qtlV,(ui;u,,ut)= o ( x ) ( s )(qtldV/drlut)
(8A)
where for harmonic oscillator vibrations of A-B we haves4 ( x ) = (uflxlq) = ui1/2h/(2prWAB)1'2
(9A)
where uf = vi - 1 and pr is the reduced mass of A-B. Fortunately, the radial portion of the integral of eq SA has been solved a n a l y t i ~ a l l y ~and - ~ *the ~ final form for the vibrational predissociation rate becomes (55) Schwartz, R. N.; Slawsky, Z. I.; Herzfeld, K. F. J . Chem. Phys. 1952, 20, 1591. ( 5 6 ) Herzfeld, K. F.; Litovitz, T. A. Adsorption and Dispersion of UItrasonic Waves; Academic: New York, 1959. ( 5 7 ) Freed, K . F.; Metiu, H . Chem. Phys. Lett. 1977, 48, 262.
Ewing 7-1
2 d - 2u, - 1 d ,(I1 [(n - X)' + ut! (2d - Ut - 1). n = l
= h-14n2(s)2a2a2(x)2De
with d an integer. This equation can be solved in a few minutes with a hand calculator. There are three intermolecular parameters needed to evaluate eq 10A the intermolecular well steepness given by a, its depth De, and the anisotropy parameter p. The only other physical quantities involve masses, the vibrational frequency of A-B*, and the reduced moment of inertia of the complex which is contained in (s) as we will demonstrate later. The rate expression given by eq 10A is fairly cluttered but has been evaluated for a variety of molecules for the V-T channel. What one finds for these calculations is that 7-1 z=
exp(-nq,/2)
(1 1'4)
Both eq 10A and 11A contain the exponential dependence on nq,. The exponential factor of determined empiri~ally~.~ for eq 1 1 A, arises because of the repeated appearances of q, elsewhere in the complete analytical expression of eq 10A. As we have noted earlier, we interpret 442 as the number of nodes in the translational wave function lqt) over the Morse potential well. The effective translational quantum number for the fragments then becomes q J 2 . Since ut counts directly the number of nodes of the van der Waals stretching vibrational wave function lu,), the change in effective translational quantum number Ant then becomes eq 8. Collapsed into the preexponential factor of 10l2 in eq 11A are the matrix elements relating changes in vibrational and rotational levels during vibrational predissociation. The matrix element associated with the vibrational change is a 2 u 2 ( x ) 2for uf - v, = -1 relaxation of A-B* to A-B. For extreme cases of diatomic molecules, say HC1 (ul=l) with W,, = 2880 cm-l and Br2(u,=27) with WAB= 64 cm-I, we find a2a2(x)2to range between 0.02 and 0.2 using known spectroscopic constant^^^^^^ and eq 9 and 9A. Thus, while the matrix element for the translational coordinate, e.g., exp(-nlq,/2 - uti), can vary over 20 orders of magnitude' depending on masses and energy gap, the vibrational matrix element only spans 1 order of magnitude. In order to cast the effect of changes in vibrational state into the form of eq 5 and 12, we set (a2a2(x)2)IAuI = exp(-nAn,)
( 12A)
where lAul = Iuf - uil. The appearance of IAul into the exponential of eq 12A is a consequence of higher order expansions of eq 5A which yield matrix element factors = = u ~ ~ ~for( xAu) = ~ -2, = a 6 a 6 ( x ) 6for Au = -3, and so 0n.11,s6358 Taking the logarithm of both sides of eq 12A and comparing with eq 12 reveal y = -(I/n) In ( a 2 u 2 ( ~ ) 2 )
( 1 3A)
In the extreme cases of HCl(u,= 1 ) and Br2(u,=27) we have just considered, we find y = 0.5 and 1.2. For simplicity, we usually take y = 1 in applying the selection rule of eq 5. This value of y is consistent with the empirical Parmenter-Tang selection rule.28 With this order of magnitude dependence on IAuI (or An.,) together with the An, relationship of eq 8, we modify eq 11A to 7-l
= 10'3(s)2exp(-r(An,
+ An,))
(1 4.4)
where we have introduced the effect of the ( s ) * rotational matrix element into the preexponential factor. Let us now deal with (s)*. When rotational changes introduce no effect on the vibrational predissociation process such as the V-T channel, we have ( s ) = ~ 1 . What we seek is an effective quantum number change An, for the matrix element ( s ) = ~ exp(-a(An,))
( 15A)
when considering the V-T, R channel. Earlier we have suggested this form by replacing p, in eq 7 with l/r: in order to exploit the (58) Wilson, Jr., E. B.; Decius, J. C.; Cross, P . C. Molecular Vibrations; McGraw-Hill: New York, 1955.
J. Phys. Chem. 1987,91,4671-4675 analogy between the effective translational quantum number qt and an effective rotational quantum number ql. Alternatively, a similar form for the rotational matrix element is obtained from Of A-B* in a 'lid matrix*57 It is found theories Of the to the that 7-1 OC exp(-aJnd where J m a ~ momentum state of A-Bt which has accepted the maximum vibrational energy from A-B*. The exponential factor u,near unity, depends on j3 and m of eq 7A. The theoretical model is in reasonable agreement with experimentally determined vibrational relaxation rates in matrices.59 We are reasonably assured
467 1
then of a comparable exponential Anr dependence for vibrational predissociation. Registry No. p-DFB, 540-36-3; Br,, 7726-95-6; Nz, 7727-37-9; HCI, 7647-01-0; HF, 7664-39-3; H2, 1333-74-0; D,,7782-39-0; C'H4, 74-85-1; I,, 7553-56-2; C12,7782-50-5; (NO),,16824-89-8; Ne, 7440-01-9; He, 7440-59-7; Ar, 7440-37-1,
(59) Legay, F.Chemical and Biological Applications of Lusers, 2nd ed.; Moore, C. B., Ed.; Academic: New York, 1977.
ARTICLES Interaction of Two .rr-Electron Systems: Spectroscopy of 9,lO-Dihydroanthracene Anunay Samanta, Kankan Bhattacharyya: and Mihir Chowdhury* Department of Physical Chemistry, Indian Association f o r the Cultivation of Science, Jadavpur, Calcutta-700 032, India (Received: September 16, 1985)
The absorption, fluorescence, and excitation spectra of 9,lO-dihydroanthracenein methylcyclohexaneisopentane (MCH-IP) glass and in n-heptane (Shpol'skii matrix) are reported. The fluorescence origin is found to be red-shifted from the excitation origin; this has been interpreted as due to exciton splitting of the 'B2"state of benzene with the forbidden state lying lower in energy. A comparison of the one-photon absorption with the two-photon fluorescence excitation spectrum supports this assignment. A theoretical estimate by the CNDO/S-CI method is in general agreement with the observations. Further, it is observed that in fluid solution the emission becomes blurred; this is tentatively ascribed to a conformational relaxation from a puckered to a nearly planar conformation. In the planar rigid n-heptane matrix at 7 7 K, the ground state is forced to a planar conformation.
Introduction
of through-space interaction is small. Nonetheless, there is a distinct possibility of through-bond interaction, especially in diWeak intermolecular interaction leading to the perturbation hydroanthracene where the two *-chromophoric units are joined of molecular energies of ground and excited states has been the by two monomethylene bridges. The through-bond interaction subject matter of theoretical and experimental investigation for a long time.'-I3 Polarized spectra of molecular crystals containing more than one unit in the unit cell have been studied thoroughly with a view to obtain information on excitonic interaction and (1) McClure, D. S. Solid State Phys. 1959, 8, 1. (2) Wolf, H.C. Solid State Phys. 1959, 9, 1. energy migration.'4.8-11 Apart from interests of its own, excimers (3) Schnepp, 0.Annu. Rev. Phys. Chem. 1963, 14, 35. serve as model systems for excitonic interaction, and because of (4) McClure, D. S.; Schnepp, 0. J. Chem. Phys. 1955, 23, 1575. the confinement of the molecular interaction to two units, they (5) Murrell, J. N.; Tanaka, J. Mol. Phys. 1963, 7, 363. lend themselves to tractable detailed theoretical a n a l y s i ~ . ~ ~ ~ J ~ J(6) ~ Vala, M. T.;Hillier, I. H.; Rice, S. A.; Jortner, J. J. Chem. Pfiys. 1966, 44, 23. However, "double molecule^",^^-^^ in which two identical a(7) Azumi, T.;McGlynn, S. P. J. Chem. Phys. 1965, 42, 1675. chromophores are held at close distances by insulating chemical (8) Ron, A.; Schnepp, 0. J. Chem. Phys. 1966, 44, 19. bonds, offer two advantages over excimers as model systems for (9) Birks, J. B.; Kazzaz, A. A. Proc. R . SOC.London, A 1968, 304, 291. studying weak interaction between identical molecular units. (10) Jones, P. F.;Nicol, M. J . Chem. Phys. 1965, 43, 3759. (1 1) Birks, J. B. In Exciplexes; Gordon, M., Ware, W. R., EMS.;Academic: Firstly, the geometry is precisely known, and secondly, the geNew York, 1975; p 39. ometry is variable to some extent by proper choice of insulating (12) Baiardo, J.; Spafferd, R.; Vala, M. J. Am. Chem. Soc. 1976, 98, 5225. bridges. Among bimolecules, the labile ones like benzil, mesitil, (13) Ohsaku, M.; Imamura, A,; Hirao, K. Bull. Chem. SOC.Jpn. 1978,51, biphenyl, etc., are not convenient systems for experimental study 3443. (14) McClure, D. S. Can. J. Chem. 1958, 36, 59. of small excitonic interaction, for the change in geometry in the (15) Bhattacharyya, K.; Ray, D. S.; Karmakar, B.; Chowdhury, M. Chem. excited state frequently causes blurring of the spectrum.17-20More Phys. Lett. 1981, 83, 259. defined are the rigid molecules like paracyclophanes where the (16) Samanta, A.; Bhattacharyya, K.; Chowdhury, M. Spectrochim. Acta, sandwich configuration offers maximum ?r-overlap and throughPart A 1986, 42A, 43. (17) Bhattacharyya, K.; Ray, D. S.; Chowdhury, M. J. Lumin. 1980, 22, space i n t e r a ~ t i o n . ~ *In * ,another ~~ set of rigid double molecules 95. like dihydroanthracene, dihydropentacene,21trans dimer of ace(18) Ray, D. S.; Bhattacharyya, K.; Bera, S. C.; Chowdhury, M. Chem. naphthylene,15J6 etc., the *-charge clouds of two halves are Phys. Lett. 1980, 69, 134. spatially displaced with respect to each other and, hence, the extent (19) Bhattacharyya, K.; Chowdhury, M. J . Photochem. 1986, 33, 61.
'Present address: Chemistry Department, Columbia University, New York. New York 10027. 0022-3654/87/2091-467 1$01.50/0
(20) Bera, S.C.; Mukherjee, R.; Chowdhury, M. Ind. J. Pure Appl. Phys. 1969, 7, 345.
(21) Samanta, A,; Bhattacharyya, K.; Chowdhury, M., unpublished work.
0 1987 American Chemical Society
