J . Phys. Chem. 1989, 93, 5357-5365
5357
FEATURE ARTICLE I R Spectra of Highly Vibrationally Excited Large Polyatomic Molecules and Intramolecular Relaxation Aleksei Stuchebrukhov, Stanislav Ionov, and Vladilen Letokhov* Research Center f o r Laser Technology, institute of Spectroscopy, USSR Academy of Sciences, 142092 Troitsk, Moscow Region, USSR (Received: December 1 , 1988)
Theoretical and experimental aspects of the spectroscopy of highly vibrationally excited large polyatomic molecules and its relation to intramolecular vibrational relaxation (IVR) are discussed. The classification of various anharmonic interactions of high order according to their role in IVR and spectroscopy is done. The line shape, inhomogeneous and homogeneous T I and T2broadenings, and their relative contribution to the total line width are calculated. The theory presented is compared with recent spectroscopic results on the (CF3)JI molecule with vibrational energies E = (30-40) X lo3 cm-l.
1. Introduction Traditional molecular IR absorption spectroscopy deals with the study of individual vibrational-rotational levels near the bottom of the potential energy surface, which correspond to the excitation energy of one or a few vibrational quanta of some of the normal molecular modes. The progress in high-resolution experimental techniques has made it possible to study the finest details of vibrational-rotational interactions between individual levels. (See, for example, the work of Moore et al. on formaldehydela and Field and Kinsey et al. on acetylene and formaldehyde.*b) A huge number of such discrete levels have now been identified in various molecules. The impressive capabilities of linear and nonlinear IR high-resolution spectroscopy have been demonstrated in the studies of a low-lying energy spectrum ( E 6 4000 cm-') of the SF6 molecule,2a a classical object of studies on multiple-photon IR excitation.2b At the same time, the spectroscopy of isolated polyatomic molecules with high vibrational energies (of the order of or higher than the molecular dissociation threshold) is now really taking the first steps. The main difficulty in studying such molecules experimentally is that the ordinary excitation methods (heating or multiple-photon IR excitation whereby the molecule successively absorbs a large number of IR laser quanta) give rise to a fairly wide molecular energy distribution. Because of anharmonic red shifts, the spectra of different molecules are shifted with respect to each other, which leads to a high inhomogeneity of the spectrum of the entire ensemble. Such inhomogeneously broadened spectra contain almost no useful information about the spectrum of an individual molecule or its dynamics. To study molecules with a high and well-defined energy of E AE/2, AE E, (Figure l ) , the intramolecular relaxation times at E of large molecules are on a subpicosecond scale. To study intramolecular relaxation in this case, it is for now more convenient to use a spectroscopic approach, which will be used in the present paper. We will show that the line width and line shapes of absorption bands of excited molecules can be related to relaxation rates of mean energy and of polarization of the vibrational mode of a molecule in case they are taken from equilibrium. Note that in the low-energy region we can also say something about the re-
-
-
(13) Felker, P. M.; Zewail, A. H. J . Chem. Phys. 1985, 82, 2961, 2975, 2994. (14) Felker, P. M.; Lambert, W. R.; Zewail, A. H. J . Chem. Phys. 1985, 82, 3003. (IS) Blwmbergen, N.; Zewail, A. H.J . Phys. Chem. 1984, 88, 5459. (16) Bagratashvili,V. N.; Vainer, Yu. G.; Dolzhikov, V. S.; Kol'yakov, S. F.; Letokhov, V. S.; Makarov, A. A.; Malyavkin, L. P.; Ryabov, E. A.; Sil'kis, E. G.; Titov, V. D. Sou. Phys.-JETP (Engl. Trunsl.) 1981, 53, 512. (17) Mazur, E.; Burak, I.; Bloembergen, N. Chem. Phys. Lett. 1984, 105, 258. (18) Doljikov, V. S.;Doljikov, Yu. S.; Letokhov, V. S.; Makarov, A. A.; Malinovski, A . L.; Ryabov. E. A. Chem. Phys. 1986, 102, 155.
Stuchebrukhov et al. laxation of individual states, as in ref 13-15, but if we are interested in mode-selective laser chemistry,*Js then the relaxation rates of the mean energy of a whole mode, rather than that of individual levels, are pertinent. While the possibility of mode-selective dissociation is still in question because of the high rates of intramolecular redistribution, selectivity of excitation on a molecular level is nowadays a practically solved problem.2 The spectroscopy of excited particles is also of primary importance here. For choosing the optimum selective molecular excitation scheme the absorption bands of excited molecules must be known. We are going to demonstrate that some absorption bands of excited large polyatomic molecules are much narrower than it was customary to assume some time ago. (See, for example, ref 2b.) In this paper, we will discuss theoretical aspects of the IR absorption spectra of highly vibrationally excited isolated polyatomic molecules and the relationships between the spectra and intramolecular relaxation processes. The questions to be treated include the following. 1. What line shape can a highly excited molecule be expected to have, and how will the line shape change with varying molecular energy and size? 2. What are the anharmonic interactions responsible for line broadening, and what is the nature of this broadening in the quasicontinuum region? 3. What are the relative contributions of the processes involving no transfer of energy from the mode being excited (i.e., the T2 dephasing processes) and those involving such an energy transfer ( T , processes) to the homogeneous line broadening in the quasicontinuum region? In the concluding section, we will briefly discuss the spectra of the vtl mode of the (CF3)3CImolecule obtained recently in ref 4, 5, and 7 at energies of El = 37 600 cm-' and Ez = 43 000 cm-I, much in excess of the dissociation energy D = 19000 cm-I. Comparison between theory and experiment makes it possible to infer the intramolecular relaxation rate at such energies.
2. Absorption Spectra and the Pumped Mode-Heat Bath Model In a highly excited molecule with an energy of the order of or higher than the dissociation energy, the average vibrational mode occupation numbers tii become of the order of or greater than unity, 1. We are interested in the nonlinear interactions of the fil vibrational modes of the molecule under such circumstances. In our theoretical analysis below, we will disregard the fact that the molecule possesses rotational degrees of freedom. Such an idealization is more or less valid when the vibrational temperature, Tv, is much higher than the rotational one, TR. In that case, the absorption spectra of sufficiently large and excited molecules result mainly from purely vibrational anharmonic interactions between molecular modes. Such a situation can experimentally be realized in a supersonic molecular beam.I3J4 A nonrotating molecule is a system of coupled oscillators with the Hamiltonian
where woi are the frequencies of the normal modes whose total number is s, qi and pi are the normal dimensionless coordinates and momenta, respectively, and V(ql,...,qs) is the anharmonic interaction. The quantum and classical dynamics of an isolated molecule with Hamiltonian (1) become extremely involved even in the case of two degrees of freedom, s = 2.10-12*19As the molecular energy increases, vibrational motion becomes chaotic and the classical trajectories become unstable and start wandering randomly throughout the phase space. In the quantum domain, due to anharmonic interactions the zero-approximation wave functions In') = lnl,,,,,nJ)intermix strongly, thus forming the exact state of the type (19) Rice, S. A. In Photoselectiue Chemistry; Jortner, J., Levine, R. D., Rice, S . A,, Eds.;Wiley: New York, 1981; p 117.
The Journal of Physical Chemistry, Vol. 93, No. 14, 1989 5359
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In that case, the integrals of motion, i.e., the good quantum numbers n l , ..., n,, are, as a rule, destroyed so that the wave functions are marked by a single value of the whole molecular energy E. The classical and quantum properties of system (1) are now being intensively investigated.20-21 In the case of polyatomic molecules, Hamiltonian (l), widely used in various areas of physics, possesses a number of characteristic features. The number of degrees of freedom s here ranges between some ten to a few tens; i.e., the system is intermediate between the macroscopic one with s m and the small systems of the Henon-Heiles type10919.22 with s = 2. Another characteristic feature of system (1) in our case is the presence of chains of Fermi resonances with small detunings of the order of AR D at which the molecule can live a rather long period of time (7 >> w0r1) before it dissociates, the measure of such regions is small. For large molecules, this can be proved for energies that may be several times larger than the dissociation energy D.23*24 Let us assume that our molecule has a single mode active in IR absorption; this mode will be denoted vl. It will be referred to as the “pumped mode”; its frequency will be wol. The other molecular modes that are inactive in IR absorption form what is called the “heat bath”. Such subdivision of the molecule into two parts is widely accepted in the multiple-photon excitation (MPE) theory dealing with vibrationally excited molecule^.^^-^^ The anharmonic interaction between the pumped mode and the heat bath leads to the formation of an absorption band in the neighborhood of the frequency wO1, and our goal is to study the shape of this band and its evolution with the increase in the molecular energy. According to our approach, we express the total Hamiltonian of the molecule in the form
-
+
+
H =H
p
+ HB + VpB
(3) (4)
formed and its characteristic width are small compared to wol. As already noted, the behavior of individual trajectories in phase space and the properties of individual quantum states (3)-(6) are extremely complex. But in practice, we deal with molecular ensembles (e.g., molecules in a beam) and not with isolated molecules. For this reason, some averaging over neighboring levels or over initial conditions, in classical terms, inevitably exists. Such an averaging drastically simplifies the description of the system, for it eliminates all the finer details of molecular dynamics, as recently demonstrated in a series of numerical experiments. (See, for example, ref 29.) As stated, we will study an ensemble of molecules with Hamiltonians (3)-(6). The state of the ensemble should be specified by the density matrix
where g(E) is the density of the eigenstates IEi). We assume that the interval A contains a large number of the levels g ( E ) A >> 1 and that A 0
(10)
where a+ and a are the creation and annihilation operators in the Heisenberg representation, respectively, and q1 = (a+ + ~ ) / 2 l / ~ . The averaging in (10) is generally over the density matrix (7), but where it proves inconvenient, we will use the thermal state (8). In eq 10 the integrand is the standard response f ~ n c t i o n ~ ~ s ~ ~ for the vibrational mode v I . For a harmonic mode, a ( t ) = a exp(-welt), and taking into account that [a+, a] = 1, we have G(w) = l/(w01- w )
(1 1)
- WOI)
(12)
4 W L ) a *6(WL
The anharmonic interaction VpB includes all the cross terms containing both the pumped mode and heat bath coordinates. The weakness of the anharmonic interactions V and VpBmeans that the shift relative to the frequency wol of the absorption band being
+ io)
We are interested in G(o) and A(uL) for the molecule with anharmonic interactions. Because of these interactions, the exact Green functions of the various molecular modes prove to be linked so that the set of equations for these functions generally cannot be solved. Stuchebrukhov et al.32 have demonstrated that the calcualtion of the spectrum of the system can be reduced to the determination of the thermodynamic G F of the molecule and suggested a self-consistent theory for this purpose. In accordance with our intuitive beliefs, in the theory there arises the characteristic energy E,-the IVR threshold or onset above which the vibrational modes of the molecule are strongly mixed. Above E,, the effective anharmonic interaction, which is energy-dependent, is enough to overlap consecutive Fermi resonances and thus intermix the molecular modes; whereas below E,, the modes can (29) Sumper, B. G.; Thompson, D. L. J . Chem. Phys. 1985, 82, 4557. (30) Herzberg, G. Infrared and Raman Spectra of Polyatomic Molecules; Van Nostrand: New York, 1945. ( 3 1) Landau, L. D.; Lifshits, E. M. Statistical Physics; I. Nauka: Moscow, 1976. (32) Stuchebrukhov, A. A.; Kuzmin, M. V.; Bagratashvili, V. N.; Letokhov, V. S.Chem. Phys. 1986, 107, 429.
The Journal of Physical Chemistry, Vol. 93, No. 14, 1989
5360
a)
b)
Wsp
Figure 2. Radiative transitions in the quasicontinuum are assumed to be
from the narrow zone of crowded levels. The ensemble molecules occupy the lower zone with a width of AE X I I D ,the inhomogeneous component can be neglected altogether and the remaining broadening can be considered to be associated with the relaxation into the heat bath only. When y I >> 2XI1D,expression 21 yields
-
4 w ) a
(l/T)Tl/[(Bl - WL)* + TI2]
(25)
i.e., an ordinary Lorentzian contour. Thus, when the energy exchange rate with the bath is high, the anharmonicity of the pumped mode should only be included in the shift of the average transition frequency, the mode in other respects behaving like a harmonic oscillator of frequency QI(TV)with a damping of yl. This is the quasiharmonic pumped mode model discussed by us earlier.38-39As a rule, polyatomic molecules with high vibrational energy comply with the case y1 > 2XI1D,y I being of the order of a few tens of inverse centimeters, while 2XllD is of the order of a few inverse centimeters. Let us now consider another source of inhomogeneous broadening. Among the terms of the anharmonic interaction VPa,there will be those which are diagonal in the basis of the harmonic functions of the pumped mode and the heat bath. These are the type of qI2ql2,ql2qtqk2,and the like. In the first-order perturbation theory approximation, such anharmonic terms shift the mode frequency, while the wave functions remain the same. In the lower order anharmonic interaction, these shifts are linear in the occupation numbers of the heat bath modes: ~ I ( T v )=
w01
+ CXljn,
(26)
I# I
In this expression, X1, stand for the anharmonicity constants and n, the occupation numbers of the heat bath modes. The inhomogeneity that we would like to discuss here has the following origin. Let the total energy of a molecule be fixed at S
E
ZhwoI(n, +
'I
1
1/2)
c:
~ ~ a e i a~ e-f/Ti at
(28)
a
where the summation is over different ensemble molecules with different frequencies w, wl(nl,, ...,nsa). The inhomogeneous relaxation time T2/ depends on the dispersion, Az, of frequencies in the ensemble, which in turn depends on two factors-the dispersion of the occupation numbers n, and that of the anharmonicity constants XI,. Let Xl, =
WOI(X
+ t,)
(30)
[,
where x and are dimensionless constants, t, varying randomly among the heat bath modes to yield zero on the average, Ct, = 0. Consider the classical limit where the average energies in the modes are equal. The dispersion A2 is proportional to the dispersion of the constants and the dispersion 6 t of the mode energies:
[,
WI
- 001 = XE
Az2(o) = s ( t / 2 ) ( 6 t 2 )
(31) (32)
where s is the number of degrees of freedom. It was assumed that s >> 1. Introducing the average molecular frequency i3 and the average occupation number ( n )of a mode with such a frequency, we may represent the dispersion A2 in the somewhat different form Az2(0)= s(x2)(6nZ),where x = ([;)1/2i3. In the classical limit, this width is linear in the vibrational temperature of the molecule. As in the case of the intramode anharmonicity, we can analyze the effects of energy exchange among the heat bath modes. The calcualtions are much the same. Let y I denote the characteristic energy exchange rate. Then at y I 5 A2(0),A2(0)being given by formula 32, the characteristic spectral width A2 is given by the expression
Again we see that the inhomogeneous component A2(0) is strongly suppressed as y grows higher, so that at y > Az(0) the spectrum becomes homogeneous, its width being y2 = T2-l:
where y2 is the homogeneous phase relaxation rate or simply the pure dephasing rate.28,37,4W6It should be emphasized that the quantity y1which determines what type of broadening dominates is not a formal theoretical parameter but a definite characteristic of the molecule that we are going to calculate later. This rate depends itself on the molecular energy, and for one and the same molecule at various energies, both the extreme cases y >> A2(0)
(27)
This energy can be disposed among the molecular modes in a great with equality variety of ways according to the various sets lnl,,.,,ns), (38) Bagratashvili, V. N.; Kuzmin, M. V.; Letokhov, V. S.; Stuchebrukhov, A. A . Chem. Phys. 1985, 97, 13. (39)Stuchebrukhov, A. A. Sov. Phys.-JETP (Engl. Trunsl.) 1986,64, 1195.
(act))
(22)
If the relaxation rate y l is not very large (yl < XIID),integral (21) can be taken by expanding the exponent in powers o f t . Expanding to t3 and substituting t3 N (t)t2,where ( t ) is calculated self-consistently at y l = 0, we get A ~ L )a
27 remaining unchanged. Different occupation number sets correspond to different pumped mode frequencies wl(nl, ...,ns) (except for the case where XI, a.woj). If for some reason there is no energy exchange between vibrational modes, such an inhomogeneous component will contribute essentially to the broadening of the spectrum. This component is sometimes related to the inhomogeneous dephasing of the induced dipole moment in the molecular ensemble. Really, the induced dipole of different molecules in the inhomogeneous ensemble will oscillate with different frequencies w,, and due to dephasing of time factors the whole dipole will exponentially decay as follows:
(40)Stone, J.; Goodman, M. F. J. Phys. Chem. 1985, 89, 1250. (41) Makarov, A . A. Hyperfine Interucr. 1987, 38, 601. (42) Makarov, A. A,; Tyakht, V. V. Zh. Eksp. Teor. Pis. 1987,93, 17 (in Russian). (43) Fain, B. Physicu 1980, IOlA, 67. (44)Van Smaabn, S.;George, T. F. J . Chem. Phys. 1987, 87, 5504. (45)Arnoldus, H.F.;George, T. F. Phys. Rev. 1987, 836, 2987. (46)Narducci, L. M.; Mitra, S . S . ; Shatas, R. A.; Coultes, C. A . Phys. Rev. 1977, 116, 241.
5362 The Journal of Physical Chemistry. Vol. 93, No. 14, 1989
Stuchebrukhov et al.
and y > m*. Two qualitatively different possible situations are illustrated by Figure 4a,b which presents
-
+ +
7: = [ ~ @ ( ~ ) ~ / 8 ] (iiji i ~ l)p2rcs(wl,l=2)
(46)
where ni and nj are the characteristic values of the mean occupation numbers of those bath modes into which the pumped mode is decaying. When the molecular vibrational temperature Tv 0, y I does not vanish: y I # 0. When Tv 0, y1 determines the relaxation rate of the first excited level of the u1 mode. For resonances of the type of w1 w j - wi
-
-
YT = [d(3)2/8](iii- E,)pZlQ(ol,l=l)
-
-
(47)
The contribution from such processes vanishes when Tv 0. At high temperatures, 7: is linear in Tv in the region > 1 and is fairly accurately linear in the total molecular energy:
-
7:
Tv
N
E
(48)
5. Contribution to the Line Width from Pure Depbasing (yz) In the quasiharmonic approximation, the Hamiltonian of the mode being pumped has the form wla+a, and so the anharmonic potential terms of the type of a+aB2(q2,...,qs) lead in time representation of B2 to fluctuations of the pumped mode frequency. (Terms of higher orders in a+a have been discussed in ref 47.) Such fluctuations cause the pure dephasing of the pumped mode. The physical meaning of this statement is that the polarization d(0) induced in each molecule of the ensemble will decay with some rate y2 = T2-I: [e-i*l';i(o)le-'/T'
;i(t)
(49)
At the same time, the rate y2 determines the homogeneous absorption spectrum width of the molecule. (See eq 34.) We are interested in the rate y2 expressed in terms of molecular parameters. Let us proceed as we have done when calculating yl. The equation for the Green function G(o) has the standard form (37), and the self-energy, which we denote as Z2(w) in this case, is expressed as ZAw) = (B2) + G ~ ( o ) / G ( o ) S i e ( f ) ( [ B , (dt t) ) , BAO) a+(o)l)eiw' dt (50)
82 = B2 - (B2)
(51)
The first term gives a first-order correction to the frequency GI, while the second gives a second-order correction and the pure , ~ ~have dephasing rate y2.' In the lowest order a p p r ~ x i m a t i o n we y2 = ImSiB(t)(B2(t), B2(0))c dt = Im Z2(GI)
(52)
This is the general formula that we will make use of. We begin with the lowest, the fourth, order of anharmonicity. The anharmonic potential terms causing dephasing contribute to B2(q2,...,qs) in the form S
B2(42,...9qs) = (1/4)c@I:lid i-2
~~~~
~
~
(53)
~
(47) Stuchebrukhov, A. A.; Khromov, I. E. Chem. Phys. Lett. 1987,134,
251.
5364
The Journal of Physical Chemistry, Vol. 93, No. 14, 1989
We will further assume the anharmonicity constants of one and the same order to be equal: @\:Ii = From (52) one can find that the contribution from the terms of the above form is 72"' = (1 /32)@(4)2ii(ii
+ I)[(s
(54)
-
+
-
where the notation exactly corresponds to the case with y, (44). As distinct from (44), the resonance density is taken here at the zero frequency. It indicates the density of the resonant heat bath frequency combinations of the type wI ut + ... + wi ... - w, 0. If we introduce the rigidity parameter X < 1 of the molecule (see the preceding section), we find that the terms in sum ( 5 5 ) behave in qualitatively different fashions at different A's and different molecular energies. Two possibilities are illustrated in Figure 4a,b. If the product XZii is not small, the terms in the sum increase until m* is reached, where
+
m* = X2(s - 1 ) [ i i ( i i + 1)]1/2
(56)
This estimate also holds for y1 (44). In the neighborhood of the maximum m*, the terms of sum ( 5 5 ) may be represented in the form (Figure 4b)
y y=
Cy2*exp[-(m - m*lz/2m*1 m
where yz* is the contribution from m*. Putting m* evaluate the summation in (57) to get
(57)
>> 1, we can
+ 1)]'/4/X3a(s - 1 ) q x
y2 = 9(4)2{[ii(ii
yZw,with that from the lowest order nonresonant terms
One can show that
-
yZrCS/y2"' (YI/B)em*
- l)/yl]
where ii is the mean occupation number of the bath modes and y1 is the rate of energy exchange between them. The numerical factor here is due to the determination of the anharmonicity constants @ and has no special meaning. Note that this determination is not unambiguous, and so the numerical coefficient of y1,2may differ in various papers. When calculating the contributions from the individual heat bath modes, we have assumed the occupation numbers iii and relaxation rates y l i to be the same and equal to ii and yl, respectively. One can draw the following conclusions from the simplest formula (54). At low molecular energies, or when Tv 0, the contribution from y2 to the spectral width decreases rapidly as exp(-a/Tv), where B is the average heat bath frequency. The energy dependence of y2 is formally determined by the factor ii(ii l), but due consideration should be given here to the fact that the rate y I is itself energy dependent. If, for example, y I is determined by the third-order resonances, then at high energies y,a Tv so that (54)yields y2 cc Tv. It may happen that y1 will grow faster than Tv, and the contribution of y2 given by (54) to the spectral width will then grow slower or will even drop. Finally, consider the factor (s - 1) in (54). It points to a nonresonant character of the anharmonic interaction of type (54)and exerts substantial influence on the order of magnitude of the contribution from (54)because (s - 1) >> 1. Let us now analyze harmonic terms of a higher order. Earlier, we have considered the terms the type of d a q ? , there being (s - 1) terms in all. There may also be such terms as a+aq?q?, ' a+aq?q/2qk2,and so on. The number of such terms will grow sharply, but the corresponding anharmonicity constants will decrease at the same time. It is obviously impossible to tell a priori which terms will make the predominant contribution. As the molecular energy grows higher, the relationships between the terms change. To analyze this matter, let us consider the resonant contribution to dephasing from the terms of the type of a+aq,,,...,q,m. Such terms contribute to dephasing if among the frequencies wai, ..., a,, there exists a resonance such as w,,f wa2 f ... f warn 0 (i.e., -yl). The formula similar to (44) has the form39
-
Stuchebrukhov et al.
[ ( 2 ~ ) ' / ~ / 8 eexp(X2(s ] - l)[ii(ii + 1)]1/2)(58)
Let us now compare the contribution from the resonant terms ( 5 8 ) ,
(54),y2"'. (59)
Estimate (59) shows that the relationship betweeen the contributions from the different terms changes from molecule to molecule and depends on the molecular energy. As the energy grows higher, both y l and m* increase. In the classical region ii Z 1, m* = X2E/c,and so at high molecular energies yzre, Le., the contribution from the resonant terms, must predominate. In conclusion of this section, we would like to make a general note concerning the number of degrees of freedom, (s - l), of the heat bath. This number is of qualitative importance for the relaxation of the excited mode. Earlier, we have included all the modes except for the pumped one in the heat bath. Actually, the number of modes interacting directly with the pumped mode may be substantially smaller than the total number (s - 1). Those molecular modes that are not linked with the pumped mode by anharmonicity constants have no direct influence on the rates y, and y2. With the molecular energy specified, they only affect the vibrational temperature Tv of the molecule. On the other hand, it is exactly the temperature and not the total energy that determines the course of the intramolecular relaxation process. These considerations suggest that all the modes in the heat bath should be divided into two categories: active modes (those which directly interact with the pumped mode) and passive modes (those having no direct effect on the pumped mode). If the total number of the active heat bath modes plus the pumped mode active in IR absorption is denoted by sa, the symbol s in the preceding formulas should be taken to imply the active part of the molecule, i.e., sa. Such a division allows one to gain an insight into some specific features of absorption of similar molecules differing in the number of the degrees of f r e e d ~ m . ~ ~ . ~ ~
6. Comparison with Experiment for (CF3)&I In the preceding sections, we have separately analyzed the main sources of the absorption line broadening of a highly vibrationally excited molecule. Obviously, these sources in a real molecule are operative concurrently and may interfere with one another. Nevertheless, it will not be a great error if we consider the contributions from the different sources to be additive. In accord with this standpoint, the homogeneous broadening mechanisms will give a Lorentzian absorption line with a width of y = y, y2. As we have already seen, inhomogeneous broadening must in this case be strongly suppressed. Let us analyze the experimental results available. At present, there are only a few experiments reported in the l i t e r a t ~ r e ~ - ' , ~ ' ~ ~ on the measurement of absorption spectra of highly excited molecules in the quasicontinuum region. The results of most of them are difficult to interpret theoretically. The point is that vibrational excitation (e.g., multiple-photon excitation) gives rise to a wide energy distribution. Molecules possessing different vibrational energies have different absorption spectra which are shifted relative to one another on account of anharmonicity. This leads to a considerable inhomogeneous broadening (in addition to the one discussed earlier) which makes it impossible for one to obtain information about the spectrum of transitions in a molecule with a fixed energy E . Bagratashvili and co-workers4 have recently used a photodissociation technique to obtain absorption spectra for the v 2 ] mode (-950 cm-I) of the (CF,),CI molecule at energies of E , = 37 500 and E2 = 43 000 cm-' (Figure The technique consists of
+
.)'39
(48) Kuzmin, M. V.; Letokhov, V. S . ; Stuchebrukhov, A. A. Comments At. Mol. Phys. 1987, 20, 127, 139. (49) Ambartzumian, R. V.; Gorokhov, Yu.A.; Furzikov, N. P.; Letokhov, V. S . ; Makarov, G. N.; Purtetzky, A. A. Opt. Commun. 1976, 18, 517. (50) Borsella, E.; Fantoni, R.; Giadini-Guidoni, A,; Cantrell, C. D. Chem. Phys. Lett. 1982, 87, 281. Starkov, B. G.; Khokhlov, E. M. Zh. (51) Alimpiev, S . S . ; Zirkin, B. 0.; Eksp. Teor. Fiz. 1982, 83, 1634 (in Russian). (52) Boyarkin, 0. V.; Ionov, S . I.; Bagratashvili, V. N. Chem. Phys. Lett. 1988, 146, 106.
The Journal of Physical Chemistry, Vol. 93, No. 14, 1989 5365
Feature Article
v
900
920
960
940
W,, ,cm-’
Figure 5. Spectra of upward transitions, AY(uL),for the vZ1 mode of the (CF&CI molecule at two different molecular energy values: E l = 37 500 cm-I ( 0 )and E z = 43 000 cm-I (0). The line shapes are Lorentzian. The half-width at half-maximum, 7,is equal to 8.3 cm-’ for E , and 11 cm-I for E*.
measuring the spectrum of the dissociation yield increment, Sp(wL), caused by a weak probe laser pulse at the frequency wL in a beam of molecules highly excited preliminarily by another IR laser pulse.53 The authors have demonstrated that the contribution to 6p is made by molecules with a high energy of E > D that fall within a narrow energy interval of AE > y2,y1= y2, and y1> 1 and iii
