J . Phys. Chem. 1991, 95,4969-4976
4969
ARTICLES A Time-Dependent Study of the Fluorescence Spectrum of Ammonia Sandra L. Tang,**+Evan H. Abramson, and Dan C.Imre Department of Chemistry, University of Washington, Seattle, Washington 981 95 (Received: April 6, 1990; In Final Form: January 30, 1991)
We present the first A-2emission spectrum for v‘= 0 in ammonia. We use timedependent quantum mecha+al calculations of the emission spectrum of NH3 from v’ = 0 of the A state as? tool to investigate the minimum of the A state potential energy surface. We use the emission spectrum as a test of the A state surface that we generated in an earlier study-of the absorption spectrum. Comparison of the experim_entalemission spectrum with that calculated by using our modified A state shows that the equilibrium bond length for the A state is =1.08 A. We also show that various layers of resolution of the emission spectrum tell us about different regions on the ground-state potential energy surface.
1. Introduction The A-8transition in ammonia has been the subject of numerous theoretical and experimental studies. What makes this molecule attractive is the apparent simplicity of its electronic spectrum. The absorption spectrum exhibits a single progression with line spacings corresponding to the umbrella bending motion. This is to be expected-since the ground state is bent (pyramidal) whereas the excited A state is known to be planar. Activity in a single mode simplifies the problem at first glance, but when the entire volume of data is considered, the apparent simplicity turns into an interesting puzzle. Rotational analysis of some of the ND31J6.27.42.43 and NH3 bands27*42-46 shows that two geometry changes occur during the electronic transition. The rotational constants indicatz that the N-H bond lengths at the equilibrium geometry of the A state are substantially longer than those in the ground state. This geometry change should be large enough to produce a significant signature in the absorption spectrum in the form of a short symmetric stretch progression. Yet, as we mentioned above, in both N H 3 and ND3 only the bending mode appears active in the spectrum. The disappearing symmetric stretch rogression has been a subject of many studies in the past. 2-3P The A-8transition can also be studied by recording emission rather than absorption spectra. While absorption spectra probe the forces on the excited state at the ground-state equilibrium geometry, resonance emission spectra provide information about forces on the ground electronic surface at the excited-state equilibrium geometry and, thus, indirectly tell us about the excited-state equilibrium geometry. This is particularly true when the emitting state is the vibrationless v’ = 0. Emissions from higher vibrational levels are quite harder to interpret since the complex shape of the initial wave function needs to be taken into account. Emission from v’ = 0 for ND, has been previously reported by Gregory and Lipsky.I5 However, no experimental spectra were available for N H 3 emission from u ’ = 0. Emissions from v’= 1 and 2 for NH325941 and from u’ = 2Jor ND341are also available. The question of what is the true A state minimum has recently been rai-d in a series of theoretical Ab initio surfaces for the A state place the excited-state minimum at RN+ = 1.046 A, which is significantly shorter than the bond length as determined by rotational a n a l y ~ i s . ~For J ~ the ground-state equilibrium bond length, RN+ = 1.014 Thus, in one case AR = 0.07 A, while in the other case AR = 0.032 A. ‘Current address: Lockheed Missiles and Space Co., Orgn. 62-92, Bldg. 579, 1 I 1 1 Lockheed Way, Sunnyvale, CA 94089-3504.
0022-3654/91/2095-4969$02.50/0
There is always a question, as pointed out by Rosmus et al.,32 of how one goes about using rotational constants to obtain a rigid
( I ) Douglas, A. E. Discuss. Far. Soc. 1963, 35, 158. (2) Duncan, A. B. F. Phys. Rev. 1935,47, 822. (3) Tannenbaum, E.; Coffin, E. M.; Harrison, A. J. J . Chem. Phys. 1953, 21, 311. (4) Watanabe, K. J . Chem. Phys. 1954,22, 1564. ( 5 ) Bencdict, W. S.;Plyler, E. K. Can. J. Phys. 1957, 35, 1235. (6) Walsh, A. D.; Warsop, P. A. Trans. Faraday Soc. 1961, 57, 345. (7) Okabe, H.; Lcnzi, M. J . Chem. Phys. 1%7,47, 5241. (8) Smith, W. L.; Warsop, P. A. Trans. Faraday Soc. 1967, 64, 1165. (9) Kuchitsu, K.; Guillory, J. P.; Bartell, L. S. J . Chem. Phys. 1968.49, 2488. (10) Morino, Y.; Kuchitsu, K.; Yamamoto, S. Spectrochim. Acra 1968, 24A, 335. (1 1) Harshbarger, W. R. J . Chem. Phys. 1970, 53, 903. (12) Rosenstock, H. M.; Bottcr, R. Recenr Developments in Mass Spectrometry; Ogata, K., Hayakawa, T., Eds.;University of Tokyo: Tokyo, 1970; p 191.
(13) Harshbarger, W. R. J . Chem. Phys. 1972,56, 177. (14) Rabalais, J. W.; Karlson, L.; Werme, L. 0.;Bergmark, T.; Siegbahn, K. J . Chem. Phys. 1973.58, 3370. (15) Gregory, T. A,; Lipky, S.J . Chem. Phys. 1976,65, 5469. (16) Hackctt, P. A.; Back, R. A.; Koda, S. J. Chem. Phys. 1976,65,5103. (17) Back, R. A.; Koda, S. Can. J . Chem. 1977,55, 1397. (18) Koda, S.; Back, R. A. Can. J . Chem. 1977,55, 1380. (19) Runau, R.; Peycrimhoff, S.D.; Buenker, R. J. J . Mol. Specrrosc. 1977, 68, 253. (20) Donnelly, V. M.; Baronavski, A. P.; McDonald, J. R. Chem. Phys. 1979, 43, 271. (21) Avouris, P.; Rossi, A. R.; Albrecht, A. C. J . Chem. Phys. 1981, 74, 5516.
(22) Suto, M.; Lee. L. C. J . Chem. Phys. 1983, 78,4515. (23) Ashfold, M. N. R.; Dixon,R. N.;Strickland, R. J. Chem. Phys. 1984, 88, 463. (24) Vaida, V.; H a s , W.; Rocbber, J. L. J . Phys. Chem. 1984,88,3397. (25) Ziegler, L. D.; Hudson, B. J. Phys. Chem. 1984,88, 11 10. (26) Ashfold, M. N. R.; Bennett, C. L.; Dixon, R. N. Chem. Phys. 1985, 93, 293. (27) Ziegler, L. D. J . Chem. Phys. 1985,82, 664. (28) Xie, J.; Sha, G.; Zhang, X.; Zhang, C. Chem. Phys. Lerr. 1986, 124, 99. (29) Engelking, P. C.; Vaida, V. Inr. J . Quantum Chem. 1987, 29, 73. (30) Vaida, V.; McCarthy, M. I.; Engelking, P. C.; Rosmus, P.; Werner, H. J.; Botschwina, P. J . Chem. Phys. 1987.86, 6669. (31) McCarthy, M. I.; Rosmus, P.; Werner, H. J.; Botschwina. P.; Vaida. V. J . Chem. Phys. 1987,86,6693. (32) Rosmus, P.; Botschwina, P.; Werner, H. J.; Vaida, V.; Engelking, P. C.; McCarthy, M. I. J. Chem. Phys. 1987,86, 6677.
0 1991 American Chemical Society
Tang et al.
4970 The Journal of Physical Chemistry, Vol. 95, No. 13, 1991
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Figure 2. Experimental low-resolution emission spectrum from 0’ = 0 for NH3 as obtained by Koda et al.” Reprinted with permission from
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Figure 1. Jet-cooled absorption spectrum for (a) NH, and (b) ND, from Vaida et aI.% Reprinted with permission from ref 30. Copyright 1987 American Institute of Physics.
geometry in a polyatomic molecule. We felt the difference between 1.OS and 1.035 A should be large enough to produce significantly different emission spectra. As in any other spectroscopy study, one is faced with the fact \hat the_spectrum is determined by two surfaces, in this case, the A and X states. (We are assuming here that the transition moment surface does not Significantly affect the spectrum. Inspection of the transition moment surface from Rosmus et al. shows that the transition moment surface does not change much in the region of interest and thus, our aswmption is valid.) To absolutely pin down the minimum on the A state using the electronic spectra and a Franck-Condon type of calculation requires a precise knowledge of the k state, which is not available. The ab initio surface for the k state was shown to reproduce much of the data, and we will use it as p u b l i ~ h e d .pf ~ ~course, one could tweak this surface as we have done for the A state to better a ree with all the emission and IR data. We felt that the CEPA state in ref 32 is very close to reality. Our approach at the outset was not to set out to prove whose surface is the best but rather to provide a procedure by which one could take the spectra and learn about the two surfaces involved. Learning to isolate features in the spectra and correlate them with specific regions of one of the potential surfaces was the objective of this study. Despite what we state above about the spectrum being a product of two surfaces, some of the information in the spectrum depends entirely on one of the two surfaces. Generally, intensities depend on the difference between the two electronic states while eigenenergies (line positions) depend only on the final surface. We present here an experimental, as well as a theoretical study, of the emission from u’ = 0 for NH3. The paper is or anized as follows: section I1 is a qualitative discussion of the A- transition both in absorption and in emission; section 111 presents the technical details for both the theoretical as well as the experimental studies. In the next section we utilize a time-dependent formalism to calculate the emission spectrum. Our treatment provid_es,a pictorial presentation of the dynamics associated with the A-X
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Chem. Sa.1986.82, 163. (34) Dixon, R. N.Chem. Phys. Lett. 1988. 147, 377. (35) Tang, S.L.; Imre, D. G.;Tannor, D. J . Chem. Phys. 1984,93,5919.
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transition. Finally, we investigate emission spectra that were calculated assuming different excited-state potential energy surfaces, and we compare these results to the experimental spectrum. 11. Qualitative Treatment of the A-2 Transition
Figure 1 shows the absorption spectrum for jet-cooled NH3 and ND3.30 These are the highest quality spectra available.’O The diffuseness of the spectra is due to the rapid dissociation of NH3/ND3 to NH2 H/ND2 + D. It is evident that only a single progression is present. There is an alternation in intensity between even and odd levels due to nuclear spin statistics.r)*48The spacings are on the order of 900 cm-’ for NH, corresponding to bending type motion. As we mentioned earlier, this result is counterintuitive considering the rotational analysis which indicates a s u b stantial change in N-H bond lengths, in addition to the large change in bond angle for this bent to planar transition. For contrast, Figure 2 shows a low-resolution emission spectrum o b tained by Koda et a!l9 for ND3. What we find is that the lowresolution emission spectrum exhibits a single progression as well. This time, however, the progression is dominated by a frequency on the order of 2500 cm-’, clearly indicating stretching motion rather than bending, despite the fact that we have a planar-tebent transition. Thus, these two spectra present seemingly contradictory information. In a recent publication, we presented a theoretical study of the A-W absorption ~pectrum.’~The object of that study was to examine several models for this transition. Each of the models incorporates features that we feel are essential in order to comply with known data, e.g., a planar excited state and a change in bond lengths in agreement with the rotational data. The object was to test whether one can produce an absorption spectrum with a single progression despite the substantial change in equilibrium geometry for two modes. We find that two different phenomena can qualitatively account for the disappearance of the stretch progression. One model involves a potential energy surface shaped in such a way as to guide the motion of a wave packet originating in u” = 0 so that as it moves toward planarity, the bond lengths increase simultaneously and the two geometry changes are incorporated in a single motion. Such a classical trajectory is shown in Figure 3. It is perfectly periodic, and it returns to the Franck-Condon region with a frequency corresponding mostly to bending motion. The spectrum is expected then to exhibit only that frequency, i.e., a single progression spectrum. Figure 4 shows an absorption spectrum calculated quantum mechanically based on such a surface, and as we predicted from the classical mechanics, despite changes in both bend and stretch, only one frequency appears in the spectrum. In our study, we found that
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The Journal of Physical Chemistry, Vol. 95, No. 13, 1991 4971
Fluorescence Spectrum of Ammonia
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Figure 4. Calculated absorption spectrum corresponding to the same surface used to generate the classical trajectory in Figure 3.
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spectra based on this model. We separated the absorption spectrum into two spectra, one for the lines originating from d’ = O+ (Figure sa) and the other for the lines originating from promotion of d’=0- (Figure 5b). The observed spectrum is the sum of the two with appropriate weighting factors based on temperatureaB The two top spectra clearly show two progressions, one of the form (0,n)(where the first index denotes the quanta in the symmetric stretch and the second index denotes the quanta in the bend mode) and the other progression with one quantum excited in the stretch and of the form (1,n). We find that when the two spectra are added to produce the overall absorption spectrum (bottom part of figure) only a single progression is apparent. This is due to an accidental 3: 1 degeneracy between three quanta of the bend and one quantum of the stretch. Figure 6 shows a classical trajectory for this surface. It clearly demonstrates activity in two modes. The 3:l degeneracy is also exemplified in Figure 6. We concluded that it is possible to find models that can produce single-progression spectra despite large changes in two coordinates. Recent data based on the NH3 A emission33 suggest yet another mechanism to explain the discrepancies of the stretch mode. The authors find that states of the form (1,n) dissociate extremely rapidly producing very broad lines that would make spectral detection of this progression very difficult in NH3. If this were true we would expect a broad background under the blue end of the spectrum. Indeed, Figure l a shows such a feature in the spectrum. In ND3, however, lines are narrow enough to suggest that the stretch mode must be active in the model we present. We believe that in ammonia both of the earlier mentioned mechanisms are active simultaneously to some degree and the *truth” is somewhere between the two models. We were able to reproduce quantitatively the absorption data for both N H 3 and ND3 based on the above model. Now we are faced with a new challenge based on the lowresolution emission spectrum. We need to come up with a model that will explain the ‘disappearance” of a bending progression for a planar-to-bent transition. Figure 7 shows the ground state CEPA ab initio potential energy surface3* in the two coordinates we are treating here. The double
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Figure 5. Calculated absorption spectra for the model that accounts for
the observed line intensities. (a) Calculated spectrum for promotion of o” = O+ to the upper electronic surface. (b) Calculated spectrum for promotion of ut’= 0- to the upper electronic surface. (c) The sum of (a) and (b) with appropriate weighting factors. The spectrum has been multiplied by w as prescribed by eq 1 in ref 35. qualitatively this model fits the data as far as line positions and the overall appearance of a single progression. However, quantitatively we find that the line intensities do not match the experimental observations. It seemed as if another mechanism might be a t play in this case. We found that when we tried to account for the observed spectral intensities,.we introduced new bands into the spectrum with the stretch excitation seemingly destroying the major feature we were trying to account for. Figure 5 shows three
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4972 The Journal of Physical Chemistry, Vol. 95, No. 13, 1991
Tang et al. 111. Technical Details
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Figure 8. Experimental NH3 emission spectrum from v' = 0.
minimum in the bending coordinate is clearly visible. A transition originating in u' = 0 of the A state places a wave packet on the ground-state surface at the linear geometry, Le., the wave packet lands exactly on the dividing line between the two minima. This implies that the net forces in the bending coordinate are zero. On the other hand, if the excited state is displaced in the symmetric stretch coordinate as the rotational data suggest, and as shown in Figure 7, the wave packet will feel a strong force pulling it towards shorter bond lengths. If it does not spread too rapidly, it will return to its birth place. Since this motion is purely in the stretch coordinate, it will give rise to a spectrum dominated by stretch progressions as has been observed (ref 49 and Figure 2). This phenomenon is reminiscent of the spectrum of dissociative symmetric triatomic molecules such as SO2%and H20." In these molecules the two relevant coordinates are symmetric stretch and antisymmetric stretch. Motion on the saddle region of the potential produces a symmetric stretch progression, and spreading in the perpendicular direction results in dissociation, which is expressed in the spectrum as broad lines. In ammonia, spreading corresponds to bending activity that obviously is not dissociative. This bending activity should then produce additional structure indicative of bending motion. This spectral structure, however, is bound by the previously determined low-resolution spectrum carrying the stretch signature. Figure 8 shows our experimental NH, emission spectrum. We note that our qualitative prediction is in excellent accord with the observed data. The spectrum shows activity in both stretch and bend. The bend progressions are "confined" to remain within our low-resolution stretch features. The qualitative arguments seem to be consistent with the data. The question remains whether a quantitative quantum mechanical treatment will be in accord wiih our intuition. The other question is, which A state surface Kill be+ reproduce the experimental data? A recent study of the A X transition by Rosmus and c o - w o r k e r ~utilized ~~ potential energy surfaces with bond lengths differing from the ground state by only 0.033 A rather than the difference of 0.07 A predicted by the rotational analy~is.~J~ Since the overall extent of the low-resolution emission spectrum from u' = 0 arises from motion in the symmetric stretch (see Figure 2), we expect that this spectrum will provide very sensitive information as to the changes in bond lengths between the two states. As with the absorption spectrum, it is the intensity pattern that will provide the clues. The line position is determined entirely by the ground-state potential energy surface. A quantitative quantum mechanical time-dependent treatment of the emission spectrum in two dimensions follows this section. The potential energy surfaces are the same ones we have used in our study of the absorption spectrum. In this respect, this study is an independent test of the surfaces we used before.
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(36) Ebata, T.; Nakazawa, 0.;Ito, M.Chem. Phys. Leu. 1988,143, 118. (37) Zhang, J.; Imre, D. G. J . Chem. Phys. 1989, 90,1666.
Experimental Part. Light from the third harmonic (355 nm) of a Nd:YAG laser was used to pump a Coumarin 440 dye in a Lambda Physik FL2002 dye laser. The light from the dye laser was then frequency doubled with a BBO crystal, and the resulting UV light was focused into the stainless steel sample cell. The incident laser light had a frequency corresponding to 46065 cm-l. The fluorescence was imaged onto the monochromator slit by the combination of a plano-convex 2-in.-diameter, 2-in. focal length lens followed by a 2-in.-diameter, 541. focal length lens. We used a SPEX 1401 monochromator in the single monochromator configuration. The output from an RCA 4501/V4photomultiplier with a LeCroy 1OX amplifier was sent to a gated integrator. The output from the gated integrator was sent directly to a chart recorder. The pressure of the ammonia in the sample cell was 4 Torr. The ammonia (99.99%) was used without further purification as purchased from Matheson. Potential Energy Surfaces. Potential energy surfaces published by Rosmus and c o - ~ o r k e r swere ~ ~ used in t k s study. The ground-state CEPA surface and the CASSCF A state su_rfaces were used. In addition, modifications to the CASSCF A state were made to create a surface that better fit the experimental line positions and intensities. The full details of the modifications to the original a b initio surfaces can be found in ref 35. The published surfaces were written in terms of the N-H bond angle and N-H bond length (they assumed that all N-H bond angles and lengths are equal a t any given time). A nonlinear coordinate transformation was used to transform the potential surface from these internal coordinates to mass-weighted Cartesian coordinates. The full details of the coordinate transformation can be found in the appendix of ref 35. In the mass-weighted Cartesian coordinate system, the X coordinate represents motion mostly in the symmetric stretch coordinate, while the 2 coordinate represents motion mostly in the umbrella bending coordinate. The transition moment surface published in ref 32 was also transformed to mass-weighted Cartesian coordinates for the calculation of spectra. The emission spectrum is given by
f(u) = kU3J:d"f/h (+ 'nlpA-Xe"'"P'/hp,+xl+ ',,) dt
(1)
where ($',I is the initial wav: function (in our case, the initial wave function is u' = 0, the A state's lowest eigenstate), 7f, is the ground-state Hamiltonian with the potential being the ab initio CEPA surface, pA-x is the transition moment surface iven in ref 32, pA&',,) = I4(0)) is the initial wave packet, and e d/*14(0)) = 4(t)) is a moving wave packet evolving on the ground state surface. Aside from the kw3 frequency factor, the formula for the emission spectrum is identical with that for the absorption spectrum. To calculate the spectrum, we need u' = 0. This state is calculated by projecting it out of a moving wave packet on the excited state as we described in ref 35. We will use the same approach to calculate eigenstates on the ground-state surface. Numerical Methods. The quantum dynamics are obtained by an FFT grid method developed by Kosloff and c o - w ~ r k e r s . ~ ~ ~ ~ ~ The time-dependent SchriSdinger equation is solved on a twodimensional grid of size 32 X 64. We checked for convergence by halving the grid spacing and time step and noting no change in the dynamics nor the calculated spectrum.
4
IV. Emission from Y' = 0 We begin our study of the emission spectrurp from u' = 0 by first preparing th_eeigenstate for u' = 0 of the A state. First we use the modified A state surface from ref 35. We use the surface (38) Kosloff, D.; Kosloff, R. J . Compur. Phys. 1983, 52, 35. (39) Kosloff,R.; Kosloff, D. J . Chcm. Phys. 1983, 79, 1823. (40) Heller, E. J. Acc. Chem. Res. 1981, I4,368.
The Journal of Physical Chemistry, Vol. 95, No. 13, 1991 4913
Fluorescence Spectrum of Ammonia
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Figure 10. I ( # l $ ~ ( t ) ) lversus time.
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Figure 9. (a-h) Snapshots of I#(?))at various times: (a) time = 60 au (b) time = 120 au, (c) time = 180 au, (d) time = 360 au, (e) time = 600 au, (f) time = 3000 au, ( 9 ) time = 9000, (h) time = 15000 au ( I au = 0.024 17 fs). The gray dot on each frame indicates the maximum of I@(t=O)).
that gave the best agreement with the experimental absorption spectrum. We then multiply the u’= 0 eigenstate by the transition moment from ref 32 as prescribed in eq 1 to yield j # o ) . Next, the modified wave packet pI$+,) is placed on the ground state surface and propagated according to the time-dependent Schradinger equation to yield I d ( t ) ) . Finally, to obtain the emission spectrum we take the self-overlap, and Fourier transform this function. Figure 9 shows snapshots of the time evolution of I+(t)J. The gray dot represent! the center of I4(0)), which is the A state minimum, on the X state surface. Initially, the only force felt by the wave function is along the symmetric stretch coordinate, in the planar geometry. As a consequence, a t short times we see that the wave packet’s motion is mostly oscillatory in the symmetric stretch coordinate. Concurrently, the wave packet begins to spread in the umbrella bend coordinate. As portions of the spreading wave packet leave the saddle region, they feel the two minima at the pyramidal geometry and accelerate toward the bent geometry. At this point, it is no longer easy to identify the types of motion the wave packet is executing simply by observing pictures in coordinate space. Clearly, the amplitude which spread into the wells will return to the saddle region as the dynamics continue. Rather than observing the coordinate space representation, we can examine the observables such as the self-overlap, ( d o l ~ ( t ) ) , which is the quantity that determines the spectrum. Figure IO shows I(&&(t))l. Note that there are three different time scales involved in the autocorrelation function. T ~the , initial falloff of the auto correlation function, is the shortest time feature. It corresponds to the wave function’s initial move away lin momentum space) from the Franck-Condon region on the X state. Hence, T A is determined by the forces in the Franck-Condon region. For example, if the slope in the Franck-Condon region is steep, then the wave function will move away from its initial momentum very quickly and T A will be small. Note that in our specific case, the slope is determined by the displacement in the symmetric stretch. As the displacement in the symmetric stretch coordinate increases, the slope in this coordinate also increases. When we Fourier transform ( ~ $ ~ l C $ ( tto) ) yield the emission spectrum, we find that T A corresponds to the broadest ‘feature”
in the emission spectrum, that is, the entire envelope of the spectrum. Since the overall width is determined by the local slop in the potential, we can find the relative displacement of the X and the A states in the symmetric stretch coordinate by fitting the overall width of the spectrum to the experimental value. Thus, to find the relative displacement requires an experimental spectrum with resolution on the order of 5000 cm-’ only. Moreover, since the dynamics for short time is almost exclusively in the symmetric stretch coordinate, this calculation can be done in one dimension. When we consider the dynamics for slightly larger time, we find a new maximum in ( $ 1 ~ 1 $ ( t ) at ) T B . This is the time it takes I d ( t ) ) to complete one symmetric stretch vibration as shown in Figure 9. When we Fourier transform (dold(t))we find that the frequency wB is slightly larger than the fundamental symmetric stretch frequency for the X state as determined by IR spectroscopy. This is due to the fact Ihat wB is determined by motion along the saddle region of the X state, while the IR symmetric stretch frequency for the ground state is determined at the equilibrium geometry, Le., at the bottom of the two wells. Previously, when we discussed the spectrum we found two frequencies: a high one at about 3500 cm-’ and a lower frequency due to the umbrella bend. Correspondingly, the autocorrelation function shows recurrences on two time scales: the fast one with , bending period 78 modulated by a slower period labeled T ~ the frequency. As with the emission spectrum, we find that the autocorrelation exhibits two types of structure, high frequency, TB, or symmetric stretch motion and a lower frequency with a period T~ which is much slower than 78. This of course, is due to motion in the bending coordinate. The spectrum then will show the two frequencies, wB and wc.
V. Comparison between Theory and Experiment One of the goals of this study was to provide a test for the equilibrium geometry of the excited state. The study of Rosmus on the basis of a b initio results that and c o - ~ o r k e r concluded s~~ the difference between ground- and excited-state equilibrium bond lengths is only 0.033 A, which is almost a factor of 2 smaller than that predicted by the rotational analysis. We, on the other hand, showed that one can fit the observed absorption data while retaining a bond length displacement of 0.07 A. The emission spectrum from u’ = 0 is expected to be very sensitive to this parameter. The overall width of the entire spectrum is determined by where the u’ = 0 wave function lands and how far from the minimum in the stretch it is. Figure 11 shows a comparison between the experimental spectrum and the s trum calculated by using an excited state displaced by 0.07 r w h i l e Figure 12 shows a comparison with the original ab initio surface. Clearly what we predicted is observed: the surface with a larger displacement produces a broader spectrum, or in other words, more intensity into peaks of the form (1,n)and (2,n)as compared with the spectrum produced by the original ab initio surface. As a crosscheck, we calculate the emission spectrum for ND3 using the
Tang et al.
4974 The Journal of Physical Chemistry, Vol. 95, No. 13, 1991
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Figure 11. Experimental emission spectrum from u'= 0 (filled circles) versus the calculated spectrum with a modified ab initio upper surface
(solid line).
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Figure 12. Experimental emission spectrum from u' = 0 (filled circles) versus the calculated spectrum with the original ab initio upper surface (solid line).
same surface as in Figure 11 and compare with the experimental data published by Gregory and LipskyI5 (see Figure 13). We find excellent agreement indicating that the displacement of 0.07 A is more consistent with the data. We need to insert here a qualifying note for the statements above. As is the case for many spectra, the intensity pattern is determined by two surfaces and the spectrum is a signature of the differences between them. To this point, we have ignored the ground-state surface and assumed that it is completely known. We have been using the CEPA ab initio ground state which shows excellent agreement as far as equilibrium geometry and fundamental freq~encies.~~ That, of course does not guarantee that the surface is true to the molecule in the planar geometry region and one could have changed the spectra by modifying the ground state rather than the excited state. In this case we could have kept the original a b initio excited state and changed the ground-state minimum in the symmetric stretch to shorter bond lengths only in the planar region. However, we do not see any compelling reason to choose that route since shifting the excited state to larger displacement in the symmetric stretch produces agreement with one more piece of information, namely, the rotational analyses. Thus, we feel that the two emission spectra for NH, and ND, provide support to the rotational data and for a larger bond length in the excited state than that predicted by the a b initio study. What Remains To Be Fixed? When we thoroughly examine the emission spectrum, we find that one can isolate pieces of information in the spectrum and correlate them with specific regions of the potential energy surface, for either the excited or the ground state. A careful comparison of the calculated and the experimental spectrum, despite their good agreement, reveals areas that need to be changed.
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energy (cm .I)
Figure 13. Experimental ND3 emission spectrum from u' = 0 (unfilled circles)Isversus the calculated spectrum with a modified ab initio upper surface (solid line).
To make the proper changes, we need to understand which aspect of the surface produces a given feature in the spectrum. We start by examining the low-resolution spectrum, Le., when the individual bending levels are not resolved. The relative intensities are determined by the slope of the ground-state potential at the planar geometry corresponding to the excited state's minimum. This feature shows good agreement with experiment. Next, we examine the line positions, again for the low resolution peaks, ignoring the finer structure. These line positions are determined by the shape of the ground-state potential along the symmetric stretch at the planar geometry only. Thus, the spacings provide information about the symmetric stretch frequency for groundstate planar NH,. We find here that the calculated spectrum shows good agreement for the lower levels, (O,n),and it overestimates the energy of the second low-resolution peak by about 100 cm-I. This suggests that a local correction of the potential in the planar region made by increasing the anharmonicity slightly will produce a better fit. As expected, the same trend is found for ND3. The last feature in the low-resolution spectrum to be examined is the width of the individual broad band, or in other words, the relative intensities of the bending lines within each stretching band. This is determined by how long the wave packet survives on the saddle. The longer it remains there, the narrower the low-resolution peaks will appear or the more one of the bending lines will dominate in intensity. This width is determined then by the shape of the potential along the bending coordinate in the planar geometry. Due to its double minimum nature, the potential is a maximum at the planar geometry, and the steeper the potential is (the higher the imaginary frequency is) the faster the wave packet will spread and the broader will be the low-resolution peak, since the steeper the potential, the faster is the wave packet spread. Moreover, by examining the relative intensities within each of the individual broad peaks we can learn about the shape of the bamer at different bond lengths. For example, the (0,n) peaks are a probe of the region near the minimum of the symmetric stretch, while the (3,n) peaks probe a wider region in the symmetric stretch. We find good agreement in relative intensities between theory and experiment for both NH, and ND,. Finally, we turn to the high-resolution spectrum. The line positions for the (n,O) peaks are determined by the shape of the potential at the equilibrium angle of the ground state, i.e., a t the bent geometry. We find that for n = 0 and 1 agreement is fairly good, but for higher n lines are off by as much as 200 cm-'. This is similar to what we found for the low-resolution spectrum. This indicates that an increase in anharmonicity in the symmetric stretch (with the necessary correction in the planar geometry being about one-half that in the bent geometry) will shift the highresolution peaks to their positions as well as the low-resolution features.
The Journal of Physical Chemistry, Vol. 95, No. 13, 1991 4975
Fluorescence Spectrum of Ammonia A Slightly Different Interpretationof the Emission Spectrum. In the previous section, we have made connections between spectral features and specific regions of the potential. We used a dynamic point of view to make these connections. It is possible to view the spectrum from a different angle as well. For example, we have stated that the eigenenergies of the states (60)provide information about the potential in the bent geometry. That statement is fairly obvious. All states with zero quanta in the bend have amplitude only in the bent region and thus can provide information only about that region. We are used to thinking about eigenstates and their shape in coordinate space. It is then easy to associate the information with the region where the eigenstate resides. Most of the information we extracted from the spectrum is based on the low-resolution spectrum, and we were able to connect the lowresolution spectral features with the planar geometry. Somehow it must then be true that the low-resolution peaks correspond to states residing on the barrier. This is a slightly foreign concept since it is not clear what is meant by the states represented by the low-resolution peaks. These obviously are not eigenstates of 7 f . They are superpositions of states. The first peak is a superposition of eigenstates of the form (O,n),which includes the eigenstates (O,O), (O,l), (0,2), etc., with squared amplitudes given by the relative intensities of the individual spectral peaks in the high resolution spectrum. How then does one go about constructing such a state? We are still missing one very important piece of information that cannot be derived from the spectrum; we do not know the relative phases of the states. Thus, we cannot just create any superposition of eigenstates with the proper intensities. To construct these states, we return to the dynamic point of view. Low resolution always implies short time. Also, we remind the reader that the eigenstate (WEB)at energy E,, can be obtained by filtering out of the moving wave packet all frequency components except that corresponding to the desired state. This is done according to the equation
c,,pPER)= x - 2 Re[e-'E.'/*14(t))]dt
I
I
l
a
energy (1o3 cm-') 1
c
Figure 14. Low-resolution emission spectrum (frame a) with qR extracted from each of the three center peak positions (frames b-d).
(2)
and assuming I4(0)) is real. c,,, the coefficient is given by the overlap (3)
Now if the integral in eq 2 is damped by a decay function as in the equation
= ~ m e - r t e - i E J / h 1 4dt (t))
0
5 10 15 energy (1o3 cm-' )
20
(4)
and r is chosen large enough such that states with energies near E, do not get filtered out, then the resulting is a superposition of states with energies near E,. Thus, by choosing r one can create a variety of superposition states. Now if I4(t)) represents the moving wave packet in the emission experiment and we were to choose E, to match the low-resolution peak center and r was chosen to reproduce the low-resolution peak width, the states Q,, will be the superposition states representing the low resolution peaks. In the same sense, by choosing r one gets to observe the time integrated dynamics with times weighted according to the lifetime l/r. As we choose smaller values of r, longer time dynamics contribute to I*,), and obviously the closer the states will be to eigenstates. Figure 14 shows the low-resolution spectrum and the states extracted at the three center peak positions. I' was chosen to be large such that short time dynamics dominate the wavefunctions. Frames W show the three wave functions as extracted from I+(t)). We find that the state (0,n) produces a wave function with no nodes. It is localized around the minimum of the symmetric stretch and centered exactly at the planar geometry. The next state shows one node, and the last state shows two nodes. As we expected, these superposition states do reside on the barrier, and thus our interpretation of the low-resolution spectrum as providing information about the saddle region (planar geometry)
Figure 15. High-resolution emission spectrum (frame a) with qR extracted from the same energies as in Figure 14.
was justified. Figure 15 shows what happens when r is made smaller to the point where individual eigenstates can be resolved. The node count in the symmetric stretch remains as before. However, we find large amplitude in the bent geometry and these states all look like (42). These are the eigenstates nearest to the peak centers and the ones with the largest coefficients in the short time I*,,)'s. Obviously, we have lost an important feature by letting r get smaller; the resulting states are no longer localized. The long time/high resolution observables are determined by large sections of the surface and one can no longer point to specific regions. By analyzing the spectrum in layers of resolution, we are able to learn about the potential and the dynamics step by step. The overall width of the entire spectrum, determined by the shortest time dynamics, provides information about the Franck-Condon
4976
J . Phys. Chem. 1991, 95, 4976-4982
region only. The low-resolution spectrum showing the symmetric stretch progression is determined by the shape of the potential in the planar geometry and finally, as we let I$(t)) explore the entire potential, the fully resolved spectrum is a signature of large regions of the potential surface.
VI. Conclusion We presented here experimental results for the resonance emission from u’ = 0 of NH,. The spectrum is very similar to that obtained for ND31sand higher vibrational levels of NH3.25*41 All these spectra show structure that can be broken into two time scales: stretch oscillation and bending motion. This is true even for emission from u’> 0. This indicates that even states with zero (41) Ziegler, L. D.; Roebber. J. L. Chem. Phys. Lett. 1987, 136, 377. (42) Ziegler, L. D. J . Chem. Phys. 1986, 84, 6013. (43) Henck, S.; Yan, W. B.; Lehmann, K. K. The 44th Molecular Spectroscopy Symposium; Ohio State University: Columbus, 1989. (44) Chung, Y. C.; Ziegler, L. D. J . Chem. Phys. 1988,89,4692. (45) Ziegler, L. D.; Kelly, P. B.; Hudson, B. J . Chem. Phys. 1984, 81, 6399. (46) Ziegler, L. D. J . Chem. Phys. 1987,86, 1703. (47) Helminger, P.; DeLucia, F. C.; Gordy, W. J . Mol. Spectrosc. 1971, 39, 94. (48) Wilson, Jr., E. B. J . Chem. Phys. 1935, 3, 276. (49) Koda, S.; Hackett, P. A.; Back, R. A. Chem. Phys. Lett. 1974, 28, 532.
amplitude in the planar geometry, such as v’ = (0,l) vibrate along the symmetric stretch before spreading into the two minima. We have provided a fully quantum mechanical treatment of the emission spectrum in the two active dimensions. We find that an excited state with bond lengths of 1.08 A reproduces the emission spectra for both NH3 and NDp We have used the CEPA a b initio ground state from ref 32. The good agreement with experiment suggests that the ground-state potential energy surface in these two dimensions represents ammonia very well. The only significant modification we suggest is a slight increase in the anharmonicity in the symmetric stretch. This would not change any of our conclusions significantly. We were able to correlate specific regions of the potential with features in the spectrum, thus providing a direct route for construction or modification of potential energy surfaces. We have shown that low-resolution spectra can be correlated with superposition states. Unlike many eigenstates, these can be highly localized to specific regions of the potential and thus provide information about those regions. Acknowledgment. We gratefully acknowledge Dr. Stewart 0. Williams and Dr.Jinzhong Zhang for useful discussions. This work was supported by NSF Grant No. CHE-8707168, “Spectroscopy of the Transition State and Vibrational Structure of Polyatomic Molecules”.
Transition-State Energies from Experimental Spectra Sandra L. Tang*?+and Dan C.Imre Department of Chemistry, University of Washington, Seattle, Washington 981 95 (Received: April 6, 1990; In Final Form: February 13, 1991)
We present a simple method for extracting transition-state parameters for photodissociation and inversion reactions from very easily obtained experimental data. Our method uses electronic spectra to fairly accurately determine the potential energy at the minimum barrier to symmetric isomerization. Our method is inherently more sensitive to the barrier height than other methods that utilize a need to accurately measure splitting energies of the vibrational levels. The method we present derives the barrier height via the energy of a wave function sitting atop the barrier whose height is to be determined. Thus, it is suitable for dissociative, Le., reactive potentials, as well. We present the NH3tunneling barrier and the photodissociation of H20as examples.
I. Introduction Most unimolecular chemical transformations involve nuclear motion from one potential minimum to another over some type of barrier. The shape and height of the barrier play a crucial role in determining reaction rates. Being able to experimentally characterize this important region of the potential energy surface has been an important task of molecular dynamicists for many years. Many molecules have multiple equivalent minima with barriers between them. Every bent triatomic has a maximum at the linear geometry for example. Many symmetric triatomics of the type ABA have excited states of C, symmetry with true minima corresponding to the two AB bond lengths being unequal. There are multitudes of examples. The molecule we discuss in this paper is ammonia, which is pyramidal in the ground state. The barrier separates the two equivalent minima and is at the planar geometry. There is really no substantive difference between a barrier separating two stable minima, as in ammonia, and a barrier in a reaction path. The dynamics of crossing are exactly the same. The dynamics away from the barrier is the primary difference in the two cases. We will show this later when we use the same Current address: Lockheed Missiles and Space Co., Orgn. 62-92, Bldg.
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0022-3654/91/2095-4976$02.50/0
method to determine the NH3 inversion barrier and the H + OH exchange reaction barrier. A great deal of effort has been spent trying to determine the height of the inversion barrier for The usual ap(1) Dennison, D. M.; Uhlenbeck, G. E. Phys. Reu. 1932, 41, 313. (2) Wright, N.; Randall, H. M. Phys. Reu. 1933, 44, 391. (3) Cleeton, C. E.; Williams, N. H. Phys. Reu. 1934, 45, 234. (4) Manning, M. F. J. Chem. Phys. 1935, 3, 136. (5) Migeotte, M. V.; Barker, E. F. Phys. Rev. 1936, 50, 418. (6) Wall, F. T.; Glcckler, G. J . Chem. Phys. 1937, 5, 314. (7) Dcnnison, D. M. Rev. Mod.Phys. 1940, 12. 175. (8) Sheng, H.-Y.; Barker, E. F.; Dennison, D. M. Phys. Rev. 1941,60,786. (9) Bleaney, B.; Penrose, R. P. Nature 1946, 157. 339. (IO) Good,W. E. Phys. Reu. 1946, 70, 213. ( 1 1) Townes. C. H. Phys. Rev. 1946, 70, 665. (12) Newton, R. R.; Thomas, L. H. J . Chem. Phys. 1948. 16. 310. (13) Loubser, J. H. N.; Klein, J. A. Phys. Reo. 1950, 78, 348A. (14) Lyons, H.; Rueger, L. J.; Nuckolls. R. G.; Kcssler, M. Phys. Reo. 1951, 81, 630. (15) Weiss, M. T.; Strandberg, M. W. P. Phys. Reu. 1951. 83, 567. (16) Costain. C. C.; Sutherland, G. B. B. M. J. Phys. Chem. 1952.56.321. (17) Townes, C. H.; Schawlow, A. L. Microwooe Spectroscopy; McGraw-Hill: New York, 1955. (18) Bemdict. W. S.; Plyler, E. K. Can. 1. Phys. 1957, 35, 1235. (19) Swalen, J. D.; Ibers, J. A. J . Chem. Phys. 1962, 36, 1914. (20) Varandas, A. J. C.; Murrell, J. N . J . Chem. Soc., Faraday Trans. 2 1977, 73, 939. (21) Herzberg, G. Molecular Spectra and Molecular Structure, II. Infrared and Raman Spectra; Van Nostrand Reinhold Co.: New York, 1945.
0 1991 American Chemical Society
