Molecular spectra, Fermi resonances, and classical motion

Molecular spectra, Fermi resonances, and classical motiona)

Eric J. Heller,b) Ellen B. Stechel, and Michael J. Davis Department of Chemistry, University of California, Los Angeles, California 90024 (Received 8 January 1980; accepted II April 1980)

Classically periodic molecular vibration (such as a totally symmetric stretch) can be unstable against the addition of small components of other modes, depending on anharmonic coupling strengths, near resonance of fundamental frequencies, and the total energy. We report here on some very strong correspondances between classical stability of the motion and quantum spectral features, wave functions, and energy transfer. The usual concept of a vibrational Fermi resonance turns out to apply best to the case where the transition to classical instability occurs at an energy below the first resonant quantum levels (this is the case for the famous Fermi resonance in CO2), In the (probably more common) event that resonant classical instability should set in above several quanta of energy in the mode of interest, the quantum spectrum shows tell-tale pre- and post-resonant signatures which include attraction of quantum levels (rather than the usual Fermi repulsion) and other features not normally associated with Fermi resonances. Evidence is presented which suggests that certain types of periodic motion in anharmonic molecules act as "traps", and are resistant to energy exchange with other types of motion. Numerical evidence linking the classical and quantum behavior, together with a new semiclassical theory presented here provides a very explicit connection between quantum and classical anharmonic motion.

I. INTRODUCTION

Much research has been focused recently on the ques-tion of intramolecular energy transfer. I The importance of this process to many areas of chemical physics is obvious. In this paper, we concentrate on a specific aspect of intramolecular energy redistribution which seems to have been passed by. Although impressive re-cent advances in the theory of the onset of stochastic motion in classical anharmonic systems have spurred studies of the parallel quantum mechanical problem,t,2 we emphasize that the present study does not concern the stochastic transition. Instead, we study a much simpler dynamical property, which is well understood classically, but which nonetheless has important impli-cations for energy redistribution. The parallel quantum problem seems to have been only obliquely examined. Since it is so much simpler than the question of stochastic motion, we are able to make a very specific connection between the classical and quantum mechaniCS, linked in this paper by numerical evidence and a firm semiclas-sical theory.

The dynamical property which is studied here involves the decay (or lack of it) of one mode of molecular motion into another. This sounds very much like the question of stochastic motion, but it is not. We are concerned here with individual mode-mode resonances in restricted regions of the phase space of the molecule. A beautiful picture of the onset of stochastic motion involves the overlap of two or more such resonant regions in phase space. a Isolated resonances may exist which strongly couple the modes, yet because of the isolation, the mo-t ion remains quasiperiodic.

The modes to which we refer are periodic motions of the molecule, which can always be found for certain

aWork supported by NSF Grant No. CHE77-13305. b)Alfred B. Sloan Foundation Fellow: Camille and Henry

Dreyfus teacher scholar.

initial conditions. For example, in a harmonic potential, all pure normal modes (energy totally in one mode) give rise to periodic orbits. When small anharmonicity is introduced, these closed orbits will not disappear, but will be found near by in phase space. In addition, new periodic orbits may appear which, however, have larg-er periods then the "fundamental" set of perturbed closed orbits. We shall for now focus on a special type of periodic orbit which is easiest to visualize and very relevant experimentally as well. This is totally sym-metric motion in a molecule which has exactly one such mode, including anharmonicities. Examples are CO2 ,

BFa, SFs, CH4, etc. Water will not do, because an-harmonic terms come in to mix the symmetric stretch and the symmetric bend, both of Al symmetry. The approach we describe is by no means limited to this case, and a future publication will deal with more gener-al types of motion. The experimental relevance comes from the fact that Franck-Condon transition often in-volve displacements only in the totally symmetric stretch coordinate. To find these periodic orbits numerically merely involves initiating the system with positions and momenta belonging to the totally symmetric representa-tion of the group of the molecule.

If the molecule could be represented by a point in phase space, with a totally symmetric initial condition as just described, then symmetry requires that it for-ever exhibit pure symmetric stretch motion, even in the presence of anharmonicities. Quantum mechanically, however, in a Franck-Condon transition involving a dis-placement in the totally symmetric coordinate, one has a wavepacket with a certain extent in position and mo-mentum surrounding the purely symmetric stretch phase space point. Therefore, it is reasonable that we should consider the history of trajectories which begin in some sense nearby the totally symmetric configuration. A confusion often arises in the quantum mechanical case: It seems paradoxial that energy could flow into unsym-metric modes, subsequent to the Franck-Condon transi-tion, even though the initial Franck-Condon wavepacket belonged only to the totally symmetric representation.

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Heller, Stechel, and Davis: Molecular spectra 4721

However, the system is allowed to couple into all over-tones and combinations involving unsymmetric modes whose direct product nonetheless contains a component in the totally symmetric representation, so there is no paradox. A complementary point of view, and one which is important for the' interpretation of the analogous clas-sical mechanics, is that a wave function may belong to the totally symmetric representation of the molecule, and yet possess amplitude in regions of configuration space which do not have this symmetry. (This simple fact accounts, for example, for vibronic allowedness in so called electronically forbidden transitions. )

If a periodic orbit is stable, then trajectories begin-ning in the vicinity of the periodic orbit are never able to wander very far. The "vicinity" surrounding a stable or elliptic fixed point can be quite large or arbitrarily small in classical mechanics. It seems fairly reason-able that for the stable nature of a periodic orbit to sig-nificantly affect the corresponding quantum mechanical Situation, the stable vicinity must compare favorably with a elementary quantum volume in phase space, i.e., Planck's constant h (or some appropriate power of it depending upon the dimension of the phase space). Classical details on a finer scale than this should some-how be washed out or averaged over.

If a periodic orbit is unstable, then initially close tra-jectories may wander quite far from each other, leading for example to energy exchange. If the symmetric stretch frequency (which is in general a function of the total energy in the symmetric stretch) satisfies low order "resonance" conditions with one or more other modes, then relatively small anharmonic coupling be-tween the modes can cause instability leading to flow from the symmetric stretch into the resonant mode. The resonance conditions are simply ws "" w. (1: 1 reso-nance), w.",,2w. (1: 2 resonance), 2ws ""w. (2: 1 reso-nance), etc. 4 If the resonances are not very close, then stronger anharmonic coupling is required to cause the motion to become unstable. Since in general the effec-tive frequencies change with total energy, and the effec-tive anharmonicity steadily increases with total energy there can be a complex interplay of these two effects, and it is possible for the symmetric stretch periodic motion, or for that matter, any periodic motion to con-tinuously evolve from stable to unstable and back again, and so on as total energy in the periodic mode is in-creased. For the systems we investigate here, there is a simple evolution from stable to unstable, and whether more complex behavior is important for realistic poten-tial surfaces is an interesting matter for future research.

Now consider the quantum mechanical situation. As pOinted out by Herzberg,5 the factors causing coupling and instability of two degrees of vibrational motion in the molecule are the same both classically and quantum mechanically. Using perturbation theory for example (see Sec. II. E) we see that the first feature mentioned above, namely, near resonance of fundamental frequen-cies, would lead to small energy denominators in the ex-preSSion for the perturbed wavefunction. Similarly, the second feature, namely the strength of the interac-

tion, would lead to large numerators. The prototypical behavior in the quantum mechanical situation is for a pair of nearly degenerate quantum levels arising from nearly resonant fundamental frequencies to interact strongly through the introduction of the anharmonic terms; this interaction is often noted in the spectrum through the mutual repulsion of the energy eigenlevels from their zero order positions, and from "intensity borrowing" which arises from the strong mixing of the zero order states. The repulsion and mixing are the hallmarks of a Fermi resonance. In general, the situa-tion is more complicated than this because there are many zero order levels which are approximately degen-erate. For example, in the 1: 1 resonance, between two modes "s" and "u," the levels (n .. 0), (n s -1,1), (n s -2,2), etc., may all be approximately degenerate. The relationship between the classical stability of the modes, and the fate of these zero order levels (i. e., their spacing and their appearance in the spectrum) under the influence of the anharmonic terms, is the main subject of this paper.

The above discussion is meant to provide a rough in-tuitive framework for the more general picture of Fermi resonances and their relationship to classical motion which is emerging from the more detailed considerations presented in the sections that follow. One of our main points is that it may just happen that the classical insta-bility, which our experience and explicit calculations have shown to very profoundly effect the quantum spec-trum, sets in above an energy corresponding to several symmetric stretch quanta. It turns out that this leads to a rich signature in the spectrum entailing "prereso-nant attractions, " "post-resonant broadening, " and other features which seem not to have been associated with Fermi resonances in the past. Even in the case that the classical instability occurs well before the first reso-nant levels in the absorption spectrum, (and this is the case that has traditionally been associated with Fermi resonance in the past) we can still extract some infor-mation about the degree of classical instability (in the vicinity of the symmetric stretch) from the width of the quantum absorption symmetric stretch bands, and vice versa.

In Sec. II we present results of classical and quantum calculations which provide strong empirical evidence for a close link between the stability of classical motion and corresponding quantum effects. In Sec. III, we provide a semiclassical theory which solidifies this relationship. In Sec. IV, we present a discussion of the Significance of these results, including some speculations on the pos-sible important role played by certain types of periodic stretch motion in trapping energy in molecules.

II. CLASSICAL AND QUANTUM EVIDENCE OF INSTABILITY

Before presenting a semiclassical theory (in Sec. III) linking quantum spectral features to classical dynamics, we expose here some evidence of the connection, mostly in the form of pictures, obtained from converged quan-tum, classical and semiclassical numerical calculations.

J. Chern. Phys., Vol. 73, No. 10, 15 November 1980

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4722 Heller, Stechel, and Davis: Molecular spectra

10

..... :, ..

6

4

2

ENERGY

FIG. 1. "Master diagram." The spectrum at the bottom arises out of the Hamiltonian! p; +! p;+ V(s ,u) [see Eq. (2.1) for V(s,u)] and the displaced wavepacket Eq. (2.3). The arrows point to the lines in the symmetric stretch Franck-Condon spectrum corre-sponding to the wavefunctions. Since the strongest lines were selected, the wave functions (plotted as contour diagrams for liJlI2 at the top) have maximum symmetric stretch character in their respective energy domains. Also shown are the time averages of trajectories and u, p. surfaces of section (second and third rows), both at the same energy as the corresponding wave function. See text for further details.

A. The master diagram

Figure 1 can be regarded as a "master diagram" for the major points we wish to make. In it will be found (i) the Franck-Condon spectrum for a displaced wave packet in the anharmonic potential

(2.1)

with Ws = 1. 0, w. = 1. 1, and'\ = - 0.11 (»is =m. = 1. 0); (ii) plots of IlJIn(s, u) 12 for selected eigenstates n which ap-pear with significant intensity in the spectrum; (iii) time averages of classical trajectories whose total energy equals that of the wavefunction beneath which they ap-

pear; and (iv) surfaces of section for these tra-jectories (and others "nearby" in their initial conditions) in the u,P. plane. As the energy increases in Fig. 1, changes take place which are directly reflected in all four features just mentioned. We now give a discussion of these effects.

First let us focus on the spectrum in Fig. 1. The wavepacket, which may be thought of as coming from another potential surface with a displaced equilibrium position in the symmetric coordinate s, generates a spectrum via the Franck-Condon factors in = I <n I ¢) 12 , where In) is the nth eigenstate and I ¢) the




TABLE 1. Pre resonant attraction of levels.

E a (n.,nu)

2.0450 (1,0) 0.0912 0.0921 0.094 2.1362 (0,1)

3.0415 (2,0) 0.0854 0.174 0.0869 0.087 3.1269 (1,ll 3.2157 (0,2)

4.0373 (3,0) 0.0785 0.164 0.0817 0.078 4.1158 (2,1) 4.2008 (1,2)

5.0323 (4,0) 0.0704 0.150 0.0765 0.068 5.1026 (3,1) 5.1827 (2,2)

6.0260 (5,0) 0.0608 0.135 0.0713 0.055 6.0868 (4,1) 6.1611 (3,2)

7.0182 (6,0) 0.0491 0.118 0.0661 0.036 7.0673 (5, II 7.1357 (4,2)

8.0082 (7,0) 0.0348 0.0979 0.0608 8.0430 (6,1) 8.1061 (5,2)

aH = +! +! w; S2 +Au2S, where Wu = 1.1, w.= 1. 0, >"=-0.11,

btl.jE= 1E(n.,0)-E(n.-1,lll, zerothorder=O.1. ctl.2E = 1 E(n., 0) -E(n. - 2, 2) 1 , zeroth order = O. 2. dt;E perturbative for b. et;E semiclassical, determined from trajectory and stability parameter at the quantum energy minus the zero point energy (1. 05 in this case). Instability occurs at E clao. = 6. 8, or Equant = 7.8.

displaced wavepacket. The spectral lines have been artificially broadened with a Gaussian line shape, lead-ing to

where T=57, and Wn is the nth eigenvalue. The wave packet cp is given by

cp(s, u) = (w. Wul1T2)1/4 exp[- (w./2)(s _4.0)2 - (w u/2)u2 J. (2.3)

This is the ground eigenfunction of a surface displaced 4.0 units in s, with the same fundamental frequencies w., wu' There are two notable features of the spectrum, aside from the fundamental symmetric stretch pro-gression and the overall breadth of the absorption band (which arise from periodic motion in the s coordinate and the steepness of the potential in the Franck-Condon region, respectively): The first is the presence of side peaks at low energy which grow in intensity as energy is increased. These represent combination overtones of the type (ns -2,2), (n.-4,4), etc., built upon the main (n., 0) progression seen at lower energy. [The (n. -1,1\ etc., lines are missing because they are odd functions of u while cp is an even function, i. e., they vanish by symmetry.J According to the fundamental frequencies, these combinations should be split by 0.2 units. In-stead, they are all split by less than this (see Table I).

Also, they get progressively closer together as energy increases and they begin to grow somewhat in intensity. Further, the combinations cannot be to disappear from this spectrum by taking a symmetrically displaced wave packet with different exponential parameters; in fact changing the frequencies in the ground and excited state [this would mean Wu appearing in cp would be dif-ferent from ,that in the potential Eq. (2.1)1 increases the intensity of the combination lines. If >.. were set to zero, and everything else were the same in our calculations, only a simple symmetric stretch progression (one line for each value of n.) would be seen. Thus we are forced to associate the residual combination intensity with an-harmonic coupling between the modes. The second notable feature is the dramatic growth in the number of combination peaks per n. band and a sharing of intensity as energy (thus n.) increases past E""S.O. The zero order labeling (n., 0), etc., begins to lose Significance as this mixing occurs and the n. bands begin to acquire an envelope with a width that increases as energy in-creases. The zero order state (n.,O) has been broad-ened and some sort of lifetime is associated with the width of the set of lines belonging to a given n.. Clearly, this lifetime is connected with the decay of the zero order state (n.,0), but more Significantly we shall be able (below) to relate it with the classical rate of exit from the symmetric stretch region. The lifetime has an indirect link with the lifetime of energy in the s mode,

Next we consider the wave functions6 associated with the peaks selected by the arrows in Fig. 1. Since the selected peaks have large Franck-Condon intensity and since cp is displaced along the symmetric stretch, we would expect the wave functions to have significant prob-ability along the symmetric stretch, and this is indeed the case. The outline surrounding each wave function is the eqUipotential contour for energy equal to that of the eigenfunction; thus, it forms the boundary where (aside from slight tunneling) the wave function can be found. We see that indeed the wave functions have much symmetric stretch character, but they become increas-ingly distorted as energy increases, until at the highest energy they have enough amplitude outside the symmet-ric stretch region to drastically reduce the Franck-Condon overlap, and indeed no wave function can be found which looks like a localized symmetric stretch in this energy region. (On the other hand, none is found which covers the allowed coordinate space more or less evenly, either. It can easily be shown that in two di-mensions' a "stochastic" eigenstate would possess an even distribution of probability over the allowed x-y region, after which it would fall off rapidly. The even distribution would be locally mOdified by nodal struc-ture.) At high energies (E > S) it is evident that many states now share the symmetric stretch character which belonged to the (n.,O) zero order state.

Associated with each wave function is the time average of a particular (single) trajectory, shown below the cor-responding wave function in each case in Fig. 1. Just as the wave functions split up into various classes in a given energy range, so too do the trajectories, and the trajectories shown are selected to give good corre-spondence with their associated wave functions. Still,




the detailed agreement is almost incredible, and it is not hard to find trajectories which correspond equally wei 1.7< a) The classical motion remains mostly integrable at all the energies shown in Fig. 1, except perhaps the highest, so one expects the trajectory-wave function cor-respondence to be close, given the success of semi-classically quantizing such systems. 7-10 For the pres-ent, this close correspondence between classical mo-tion and quantum wave functions provides a strong sug-gestion that the classical motion might tell us something about informative details of the spectrum, both qualita-tively (what does the mixing of intensity seen at higher energy have to do with stability of classical motion?) and quantitatively (can trajectories run on the potential surface predict spacings, intensities, and bandwidths?) (answers provided below!).

Lastly, Fig. 1 shows u,Pu Poincare surfaces of sec-tion for three trajectories at each of the four energies. These prove to be very informative and crucial to the semiclassical analysis of Sec. III. This surface of sec-tion4 amounts to plotting the value of u and Pu every time a running trajectory penetrates the s = ° line with Ps > 0. Since the trajectory with u = Pu = ° is necessarily stuck on the symmetric stretch line forever, this point on the surface of section represents a fixed point through which this special trajectory must pass each period in the s-motion. If either u or Pu are non vanishing, then successive passes through s =0 give rise to character-istic patterns in the vicinity of the fixed point. For example, at low energy the fixed point is surrounded with roughly elliptic curves each formed by the repeated passes of a single trajectory. If the trajectory passes very close to the fixed point in the u,Pu plane, it re-mains forever close in this low energy regime. This is tantamount to saying that a small amount of energy in the u mode remains forever small, and this is what happens at lower energy. However, at E =6.8, the central fixed point becomes hyperbolic, and now a small amount of energy in the u mode will grow in time at the expense of energy in the s mode; later it will eventually return to the s mode, etc. Thus above E = 6.8, the symmetric stretch motion has become unstable. Classically, a resonance has set in between the sand u modes above E = 6. 8 and these two modes are freely exchanging ener-gy. All four features shown in Fig. 1 herald the onset of the s-u resonant instability by switching gradually from preresonant behavior to post-resonant behavior in the vicinity of E"" 7.8, which is the classical reso-nance energy plus the zero point energy of 1. 05.

Although technically there is an abrupt change of the stability of the fixed point, from elliptic to hyperbolic, near E = 6.8, the volume of phase space which is affected by the hyperbola is at first quite small. That is to say, a technically unstable trajectory in the close vicinity of the fixed point follows a hyperbolic path in the surface of section which travels only a short distance from the symmetric stretch fixed point before returning, and tra-jectories further from the fixed point behave still as if they surround an elliptic point. At higher energy the volume of phase space occupied by the hyperbola is much larger. Thus even in the classical mechanics the turn-over to instability is smooth if viewed in this way. On

the other hand, the quantum wave function r/J, which oc-cupies a finite volume in phase space, is seriously af-fected by the instability (as seen in the spectrum) only after the hyperbolic region has grown to significant size. (The size of the wave packet is shown in one of the eigen-function plots in Fig. 1). Thus, as seen in Fig. 1, the real growth in the combination line intensities occurs only after the symmetric stretch has become unstable.

B. Relation to Fermi resonance

So far we have seen that the spectrum, wave functions, classical trajectory time averages, and classical sur-faces of section, all act in parallel and definitely show the effects of a transition to unstable symmetric stretch motion, with concomitant energy exchange between sand If. What has all this to do with the notion of Fermi resonance? As mentioned in the introduction, the pro-totype Fermi resonance 5 involves a pair of levels which mix strongly and repel one another. When this behavior is "chronic" in the sense that there are many states [(n., 0), (n s -1), 11, and possibly (n s - 2,2), (n s - 3,3), etc; = 1, 2, ... , where ws "" wu, then, as mentioned before, the resonance is expected to occur both classi-cally and quantally. For the system in Fig. 1, the reso-nance has waited until E = 7.8 to set in. Above this ener-gy, there is a classical resonance, and the new spectral features and qualitati ve changes in the wave function strongly suggest that the designation" Fermi resonance" be aSSigned to the E >8 region for this system, but not for E < 8. Clearly, this means we cannot always rely upon the classic intensity sharing and splitting of a pair of levels as a means of detecting a Fermi resonance. In this particular case, the system was not resonant at E "" 3.0, when only a pair of levels, namely (2,0) and (0,2) were available to interact with each other. [The (1,0), (0,1) levels give rise to a pair of levels, one of which is miSSing from the spectrum by symmetry, see Table I for the energies, however. 1 Indeed, the (2, 0) and (0,2) levels attract each other relative to zero order positions, and share intensity weakly. This is an exam-ple of what we term preresonant behavior. Later we shall see that pre resonant effects can also include re-pulsion of levels (but still very little mixing). This is another sign that to look only for the "classic" signs of Fermi resonance is somewhat misleading.

Under what circumstance, then, do we get the "clas-sic" Fermi resonance?l1 We maintain that the case of strong mixing and repulsion of the first pair of levels that can possibly interact (as in CO2 , for exam-ple11 ) simply corresponds to the event that the classical motion is unstable well below the energy of the inter-acting pair! Figure 2 shows a Franck-Condon spectrum for a case that was chosen to correspond to the CO2 symmetric stretch-to- bend interaction. As mentioned in Ref. 11, symmetric stretch classical trajectories run on the surface were unstable above about 20 cm-1 of energy, well below the first energy levels of the system. As expected, the first pair of levels are repelled and the spectrum (Fig. 2) shows also the mixing and broadening of all the higher overtone bands.

Note that for the case of 1 : 1 resonance between two




modes coupled by a >..su2 cubic anharmonicity, the matrix elements between the interacting levels (n.,O) (n. - 2,2), etc., vanish. This illustrates that it is certainly pos-sible to have Fermi resonance effects even though there is no first order coupling, although resonance is cer-tainly "easier" if there is such a coupling. This too has a direct classical counterpart, as we see below (Sec. lID and lIE).

C. Qualitative semiclassical picture: Spectra from classical trajectories

Considerable qualitative insight into the relationship of the quantum and classical mechanics of stable and un-stable periodic motion can be gained from a time de-pendent picture of molecular electronic spectra. 12 Fig-ure 3 shows a typical anharmonic two-dimensional po-tential surface, which might correspond to symmetric and unsymmetric stretch motion of a symmetric tri-atomic molecule. Supposing that an electronic transition has occurred, the Franck-Condon wave packet from the ground electronic state may find itself displaced from equilibrium in the excited electrOnic state, as shown in Fig. 3. The spectrum of the system is given by12

(2.4)

24.0

12.0

FIG. 2. Franck-Condon spectrum for CO2 Fermi resonance. v(s ,u) =! s2 +! (0. 49796)2u2 - O. 037su2, which mimicks CO2 (see Ref. Ill. The displaced wave packet is the ground state of the harmonic part of V(s,u) displaced to s =2. 0, where s is the symmetric stretch and u represents the bend coordinate. The entire spectrum is post-resonant (or Fermi resonant). In this system of units, the energies are multiplied by 1353 to obtain vibrational eigenvalues in cm-t • The FWHM bandwidth of the set of lines belonging to the zero order state (2,0) near E = 2.75 is 0.095 or 130 cm-I, or a lifetime of about ten symmetric stretch vibrational periods (0.3 psecl. The stability parameter r. (see Sec. uI> is 0.086 or 116 cm-t •

FIG. 3. Potential surface of Eq. (2. 1) with displaced wave packet of (2.3) shown. This situation generated the spec-trum of Fig. 1 via Eq. (2. 2).

where Eo is the energy of the ground state I X) of the lower potential surface; w is the frequency of the inci-dent radiation; 1<fl)=IJ.IX), where IJ. is the transition mo-ment between the two surfaces (a constant in the Condon approximation); and I <fl(t) is the solution of the time de-pendent Schrodinger equation on the new potential sur-face with 1<fl(0)= l<fl). Equation (2.4) says that every feature in the absorption spectrum can be obtained from knowledge of the evolution of the wavepacket I <fl(t) on the new potential surface. More specifically, only the overlap of this evolving wavepacket with itself at time t = 0 is required. The time -frequency uncertainty prin-ciple guarantees that short time dynamics of I <fl(t» will determine the broadest features of the absorption spec-trum, and long time dynamics of <fl(t) influences detailed features of Given the situation shown in Fig. 3, the wavepacket I <fl(t) will abruptly leave the Franck-Condon region where <fl resides, and the resulting decay of the overlap will determine the width of the absorption band. Of course, the steeper the potential surface in the direction of the symmetric stretch, the more rapid the decay of the overlap, and the broader the absorption band o This decay is visible as the first peak near t = 0 in Fig. 4. The second time of interest for the situation shown in Fig. 3 is the period of the symmetric stretch, since this is the time at which the wandering wave packet I <fl(t) returns to the initial Franck-Condon vicinity and again has significant overlap with <fl. However, upon

'-

0.2

0.0 .0

'- '-

TIME

FIG. 4. Magnitude of the correlation function I (¢ I ¢(t» I for the system of Figs. 1 and 3.

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this return and subsequent returns the evolving I q;(t) will in general have a reduced overlap with I q;(0). Nonetheless, this periodic function in time (see Fig. 4) leads in turn to the symmetric stretch vibrational pro-gression which is expected in frequency. Notice that our discussion of these first two characteristic times in the overlap has been couched in classical terms: the rate of the initial falling away of I q;(t», which is deter-mined by the steepness of the potential surface in the direction of steepest descent, and the periodicity of the motion in the symmetric stretch.

The third characteristic time affects the higher reso-lution details of the individual vibrational overtone fea-tures seen in the spectrum. These features, which are our main concern in this paper, may also be derived classically. In fact, the classical determinant of this higher resolution structure is indeed the stability of symmetric stretch motion. Accepting for the moment that the classical trajectories in the vicinity of the Franck-Condon wave packet evolve in much the same way as the wave packet itself, at least for moderately short times, it is apparent that if the trajectories suffer unstable falling away from the symmetric stretch region into remote regions of coordinate space, then the first few return visits of I q;(t) may result in progressively weaker overlap with q;. The decreasing peak values of I (q; I q;(t) I (for the first few vibrational periods) can be seen in Fig. 4. If the envelope of this decrease were to be continued (dashed line in Fig. 4), there would result a true, unresolvable broadening of the symmetric stretch peaks (this is in fact what occurs for certain un-bound potential energy surfaces12 ). In fact, because the potential surface is bound, new recurrences come in at longer times. These ultimately determine the structure which is present under the otherwise broadened peaks.

If the symmetric stretch motion is stable, then there will be structure in the time domain coming in after a time equal to the reciprocal of the frequency splitting of the combination-overtone pairs seen below resonance. Even though the wave packet which was used to generate I (q; I q;(t) I in Fig. 4 has an average energy above the threshold resonance energy, the spectrum still encom-passes some of the stable region and thus one does in-deed see this recurrence building up after t "'" 40. The semiclassical method described below for generating spectra from trajectories solidifies the trajectory -spec-t rum connection.

Some time ago, a method was developed in this lab-oratory for determining the evolution of the state q;(t) from classical trajectories. 13,14 Briefly, a cluster of five classical trajectories is run (2N + 1 in the general case, where N is the number of spatial dimensions) with initial conditions all near the average pOSition and mo-mentum of the wave packet; then the overlap and Fourier transform in Eq. (2.4) are performed, leading to a spec-trum. The method is approximate, and relies upon a local linearization of the classical dynamics. To the extent that it works, it provides a very explicit connec-tion between the classical mechanics and the spectrum,

3

l: 2

1

O.

20

l: 10

04 .75

l:

;1 15.5

4. E 8.

16.0 E

5.75

12. A

B

16.5 C

FIG. 5. Potential is (1.818181)2s2+0.1su2, the displaced Franck-Condon wave packet is the ground state of the harmonic part of V(s, u) displaced to s = 1. 54 in (a) and (b), and to s = - 2.52 in (c). In (a), the "low resolution" spectrum is . seen for both the exact [Eq. (2.2) with T = 7] and semiclassical [Eq. (2.4)]' with q,(t) determined by the trajectory cluster method of Ref. 13 with the argument of Eq. (2.4) multiplied by exp(- t2 /2T2), T = 7. The two spectra are virtually indistinguish-able. In (b), one of the overtone lines is examined at higher resolution, showing a difference in the semiclassical (dot-dash) and exact combination line. In (c), the displacement of q, puts it in the unstable region, and the semiclassical method gives the correct envelope. The areas under the two curves are identical. T = 57 in both.

because of the logical sequence

classical trajectories I

q;(t) I

(q; I q;(t) I

Z;(w)

It ;i.s a very successful semiclassical method for generat-ing spectra, because as Fig. 5 shows, not only are line positions available, but intensities as well. At a certain level of resolution, the exact quantum spectrum and the semiclassical one are indistinguishable [Fig. 5(a)]. At higher resolUtion, but below threshold for instability the combination-overtone peaks are seen [Fig. 5(b)1. The side peaks occur at an energy determined by periodici-ties in the trajectories which give rise to our semi-classical q;(t). Above instability, the linearized tra-jectories "think" the motion is permanently unstable, giving rise to broadened peaks in frequency. As Fig. 5(c) shows this broadened peak contains information




gg :.0 a 0.8

0.6 0.4 0.2

0.5

FIG. 6. Stability plot for the Mathieu equation. Shaded regions are unstable, ,f(i=2wjws' where So is the displace-ment of the s oscillator. Dashed lines show the stability parameter 11= I v I /rr in the stable regions, s in the unstable regions. From M. Abramowitz and A. Stegun, Handbook of Mathematral Functions (Dover, New York, 1965>'

about the envelope of the actual high resolution lines seen in the exact quantum calculation. The strong link between the classical mechanics and the spectrum pro-vided by the wavepackets is part of the motivation for the semiclassical theory presented in Sec. TIL

D. Mathieu equation model for stability

In 1961, Thiele and Wilson15 investigated the role of anharmonicity as a determining factor for a dissociating symmetric triatomic molecule. They recognized the stability of the symmetric stretch mode as a strong in-fluence affecting energy flow in the molecule. As they pointed out, much can be learned about the dynamics from a consideration of the stability of Mathieu's equa-tion. 16 Starting from Eq. (2.1), it is easy to see how the Mathieu equation arises. If nearly all the energy is initially in the s mode, then it is reasonable to assume that s behaves as

(2.5)

(assuming without loss of generality that Ps = 0 at t = 0), at least for relatively short time. Equation (2.5) is the exact solution for the pure s-mode periodic trajectory. Assuming this zero-order motion in the s mode, then the u mode may be considered separately, with a time dependent Hamiltonian given by

(2.6)

If we set T=%Wst, ,fa =2wulw., and q=4Asolw!, we have the classical equation of motion

d2 u y + (a -2qcos2T)U =0 , (2.7)

which is Mathieu's equation. The solution u(t) to this equation tells us about the early time evolution of energy in the u mode. The Mathieu equation is well studied, and its stability as a function of a and q is shown in Fig. 6. The shaded region corresponds to instability, which is more severe the deeper the penetration into the unstable region. Two conclusions are immediately ap-parent from Fig. 6: (i) The 1: 2 resonance is the broad-est in the sense of being easiest to reach with modest

values of q. Note that q is proportional to the anhar-monic coupling A times the square root of the energy. The higher resonances become increasingly narrow, at least for small values of q. (ii) The onset of instability is more sudden, severe, and occurs for smaller values of q ex..fE if a is on the "high Side" of exac t resonance, i. e., if Wu > Ws in the 1 : 1 case, etc. This is a strong effect and it gets stronger as the resonances get nar-rower. In Sec. III, we shall make use of the dashed lines appearing in Fig. 6; they will help predict spacings and bandwidths of the combination-overtone ensembles. For now, we restrict attention to the qualitative changes in the spectrum as resonance is approached and sur-passed. While Fig. 1 is fairly unequivocal about this relation, perhaps Fig. 7 will add further insight. In Fig. 7 the stability diagram is shown "sideways" and E ex q2 is plotted instead of q. Three different spectra

1.5 ----------- -

FIG. 7. 1: 1 resonance region of Fig. 6 (Yli = 2), plotted as E vs ra" 2wj w 8' Three spectra with baselines appearing at the value of ra appropriate for each of the three potentials generat-ing the spectra are shown. Note the qualitative change in the quantum spectrum as the classical resonance region is entered.

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are shown qualitatively, where the abscissa baseline has been drawn at the appropriate value of 2wuiw., and the spectrum is also plotted as a function of E. Note how the spectrum changes as energy increases into the resonance region, and how the resonance energy is very dependent upon the value of 2wu/w.. Note the decrease in spacing of the lines before resonance is reached for the two cases shown where Wu > w.. Compare the far greater stable region for the Wu < Ws case.

So far, our discussion has been in terms of the poten-tials of the form (2.1). One may well ask, what is the effect of higher order (and other cubic order) anhar-monicities? In general, the stability plot such as Fig. 6 will change, and will correspond to stability of Hill's equation,16 of which the Mathieu equation is a special case. The stability map can be determined by running trajectories for a short time (see Sec. Ill). Once the stability has been determined, the same strong correla-tions will follow between the classical and quantum be-havior. This will become much more evident below, as the semiclassical foundations of the correspondence are explained.

E. Perturbation theory

Second order nondegenerate perturbation theory can be applied to the eigenvalues (the first order shift vanishes). This predicts the correct trend toward attraction of levels in the case shown in Table I (other cases can show repulsion of levels in the preresonant regime, both in the exact and perturbative calculations), but the esti-mated magnitude of the pre resonant attraction is too small and contains no hint of the onset of the resonance.

Note that even the first pair, namely, (1,0) and (0,1), are "attracted" to a separation of 0.0912 from an un-coupled value of 0.1 in the 1: 1 resonance case of Table I. However, these levels are of different symmetry (even and odd about u = 0) and do not interact in any order. Still, they fit in to the total pattern of level splittings, and even perturbation theory predicts an attraction. One should not conclude that the levels must be of different symmetry for the attraction to occur. Examine the (2,0) and (0,2) levels, which do interact and are at-tracted to a splitting of 0.174 (0.2 in absence of coupling) and the (2,0) and (1,1) levels, which do not interact but 1I0netheless are split by almost exactly half that of the (2,0), (2,2) pair. One would expect this since the latter differ by two quanta and the former, by only one.

The semiclassical theory of Sec. III makes no dis-tinction between pairs of levels which do or do not inter-act; they must all fall into a regular pattern and, indeed, Table I shows they do.

In the previous section, we noted that the 1: 2 classi-cal resonance was the "broadest" in the sense that the unstable region is easiest to reach as a function of ener-gy or anharmonic parameter (A). Quantum mechanically this shows up in the existence of a first order coupling matrix element between, e.g., (2,0) and (1,2). Above, in the 1: 1 case, there are no first order couplings be-tween, e.g., (2,0) and (1,1).

III. SEMICLASSICAL THEORY OF ANHARMONIC SYMMETRIC VIBRATION SPECTRA

We now examine a simple semiclassical theory which is capable of describing and predicting all the major features seen in the preresonant and post-resonant Franck-Condon spectra shown in Fig. 1.

A. Quantization of the effective Hamiltonian

Some time ago, Gutzwiller l7a made a study of classical periodic orbits and their relation to the quantum density of states. He made clear the importance of considering the stability parameter of the orbit (see beloW), and pointed to certain difficulties of interpretation if the orbit is unstable. Gutzwiller's work and the work of Miller17b assumed one quantum condition per periodic orbit. Then Marcus, 7c Noid and Marcus, 7b and Milierl8 noted that, in general, N -1 additional quantum condi-tions were required. However, this too was shown by these workers to be ambiguous at best and wrong in sev-eral respects. Miller's work, 18 which assumed that the orbit was stable, concluded with the observation that the positions of the eigenvalues were incorrectly predicted for certain anharmonic potentials, which would include Eq. (2.1), for example. Berry and Taborl9 raise a re-lated objection in that the stability parameter analysis of periodic orbitsl7,lB breaks down if the closed orbits are not isolated; this is one reason for inaccuracy of the semiclassical eigenvalues. Also Noid and Marcus 7b showed that many completely spurious eigenvalues are found if periodic orbits are used to quantize the system.

Against this historical backdrop, we focus on our specific requirements in attempting to describe a spec-trum such as that seen in Fig. 1 in terms of classical theory. Below the classical resonance (or instability), we need to know the spacing of the combination-overtone lines, and for this the periodic orbit theory should be useful, 7a,17-19 notwithstanding the problem of incorrectly predicted energies mentioned above. We need also to be able to predict the intensities of the combinations, relative to the pure overtones, below resonance. Final-ly above resonance, we abandon the description of the individual lines, and instead we try to describe the band-width for the set of lines corresponding to what would have been the single line (n., 0) in the absence of coupling.

The periodic orbit methods, based as they are on the density of states, say nothing about the relative intensi-ties of the lines in a particular Franck-Condon spectrum. There is simply no information in the density of states about the initial state (what we call ¢) which generates the intensities via Franck-Condon factors or, equiva-lently, via Eq. (2.4). For the same reason, no explicit interpretation is given to the stability parameter (see below) in the above resonance case, where it is no longer simply related to the level spacings. More accurate semiclassical quantization methods based on a full ac-tion-angle variable treatment (instead of stability par-ameters) have been successfully applied to the Fermi resonant case. 7a However, intensities were again not available, so accuracy of eigenvalues was stressed, rather than the "spectral signature" of a Fermi reso-nance which we emphasize here.




Because of such deficiencies of existing periodic orbit theory, we adopt a different approach which provides in-formation about intensities and bandwidths. Simulta-neously, we shall learn that the eigenvalue positions which are most prominent in the spectrum are most ac-curately predicted.

Our approach is motivated by consideration of the time dependent overlap in Eq. (2.4) and the qualitative semi-classical picture of Sec. II. C, especially as applied to a symmetrical stretch. 12 The idea is as follows: A wave packet ¢, which is relatively localized, can be accu-rately propagated by assuming linearized dynamiCS in the vicinity of the center of the wave packet. This as-sumption, which implies that positions and momenta are linear functions of initial positions and momenta, is equivalent to the trajectory cluster method of propa-gating wave packets semiclassically, discussed in Ref. 13. For the present case of a totally symmetric stretch, linearized dynamics in turn implies separabil-ity of the s and the u degrees of freedom, i. e. , we may write

(3.1)

provided ¢ itself is separable at t =0. Given significant displacement (or energy) in the s mode, the overlap (¢ I ¢(t» will be nearly zero unless ¢.(t) is in the vicinity of ¢, i. e., once each period in the s motion, when (¢. I ¢.(t» is large. For the brief period that this overlap is large, the total overlap is modulated by (¢u I ¢u(t». It is evidently important to know this over-lap only for a short time once each period in the s mo-tion. This leads to the idea of looking at the u motion in "stroboscopic" fashion once each time (¢s I ¢.(t» is large, which in turn is closely related to the classical idea of the Poincare surface of section discussed in Sec. II. Given the Hamiltonian

H = t p; + t p; + V(s, u) , where

V(s, u) = V(s, -u)

(3.2a)

(3.2b)

(Le., s is a symmetric coordinate) the separable s mo-tion is clearly governed by

(3.3)

while the u motion in the separable approximation is governed by the time dependent Hamiltonian

(3.4)

The requirement (3. 2b) together with the linearization of the u motion leads to

(3.5)

where j is periodic with period 2rr /ws and time average zero, w. is the period of the s motion for the given ini-tial condition, and So is the maximum s displacement. The requirement thatj(t) time average to zero implies that Wu is in general a function of the energy in the s oscillator. Equation (3.5) is an example of Hill's equa-tion. 16

Since

(3.6)

the spectrum will be a convolution of the u and s spectra, L e.,

(3.7a)

where

etc., andE1u is the initial eigenvalue of the u oscillator. Thus, we may consider the sand u spectra separately. However, is just the Franck-Condon spectrum for the potential V(s, 0) and the initial wave packet ¢ •. Thus,

(3.8)

where

and

H. ¢ns =En• ¢n ••

Next the u spectra must be analyzed, and this is where the "stroboscopic" approximation mentioned above becomes useful. The strategy is to find an approximate time independent effective Hamiltonian:JC for u motion which can be quantized to give wave functions and ener-gies which generate an approximate from which the full spectrum can be derived via Eq. (3.7). Because (¢ul¢u(t» is needed only at and near t=O, ± T., ± 2T .. , ••• , the effect of :JC(t) on the history of the motion need be examined only at these times, and since .JC(t) has period T .. also, a great Simplification will result. We study first the classical motion induced by :JC(t) and its effect at time 0, ± T., .... The quantization will follow immediately because :JC(t) is a quadratic function of Pu andu.

Since JC(t) is a quadratic at all times, the values Pu(t) and u(t) will be linear functions of Pu(O), u(O), and in particular at t n; nTs' n '" 0, ± 1, ... , we have

(3.9)

where M n is the nth power of matrix M, which is a lin-ear mapping2° of the (u, Pu) plane into itself. It must also be area preserving. Since JC has period T .. the dynamics repeats itself each successive period with the result that the same mapping M applies each time a new multiple of T. is reached. The area preserving nature of the mapping implies det M = 1. Repeated application of M to a given (u, Pu) initial condition generates char-acteristic invariant curves in the (u, Pu) plane along which the mapped point must lie. In the case that the mapping M is stable, its eigenvalues iJ.% are complex and of unit modulus, i. e. ,

(3.10)

where v is real, and TrM is the trace of M. (Thus, in the stable case, I TrM I < 2). It is easy to show that the invariant curves in the stable case are ellipses in the (u, Pu) plane which in general, have major-minor axes




UNSTABLE Pi.!

. U actual . : . u' linearized

Veff

u' u'

FIG. 8. Typical surface of section invariant curves for the stable (left-hand side) and unstable cases. Below each is shown the corresponding effective potential.

not coinciding with the (u, Pu) axes. This is seen in Figs. 1 and 8 at the lowest energy, where the invariant curves are nearly elliptical (not quite exactly since the fully coupled, not the linearized, equations of motion were integrated). The parameters of the ellipses can be expressed in terms of the matrix elements of M, but the formulas are not especially illuminating.

The important point is that the invariant curves are phase space orbits of an effective (u, Pu) Hamiltonian JC. This Hamiltonian is a quadratic function of u and Pu, thus the elliptical paths. The phase space trajectories corresponding to JC not only lie on the invariant curves generated by M, they have a period such that in the time i. between successive mappings, JC carries the point (Pu(O),u(O» into the same point (Pu(i.),U(i.», as M does. However, the fact that the eigenvectors of M acquire phase of ± v after time i. necessarily implies

(3.11)

where k is an integer and where nu is the frequency of the periodic motion under JC. The stability parameter v is easy to obtain (see below), therefore the frequency nu is readily available.

It may seem that we have lost sight of the quantum mechanics of u motion, but this is not so, due to the per-fect correspondence of classical and quantum dynamics for harmonic Hamiltonians21 (even if time varying) such as that governing the u motion, Eq. (3.5). In Appendix A it is shown that the propagator U(i., 0) which propa-gates <p, i. e.,

(3. 12) ,. can be written

(3.13)

where Q is a quadratic form in u'Pu ' Evidently, at time i., we have

i. e., (3.14)

where JC is again the time independent effective Hamil-tonian. Finally, we invoke the stroboscopic approxima-

tion: for times near 0, ±i., ±2i., etc., we have

(3.15)

This is exact only at tn =ni •• but to the extent that the s overlap is peaked about these values, Eq. (3.15) is a valid approximation.

B. Stable case

Now we can take advantage of the time independent ef-fective Hamiltonian JC. Clearly, its quantized energy spectrum reads

(3.16)

Thus nut through v [Eq. (a.11)] and M [Eq. (3.10)], will determine the position of the combination bands relative to the pure symmetric stretch overtones.

Equally important are the Franck-Condon factors. These must be given by

(a.17)

where

JCcf>n =Enucf>n

In Appendix B some properties of the cf>n's and their Franck-Condon overlaps fnu are derived. It is evident that <Pu cannot be orthogonal to a typical cf>., for the latter are eigenfunctions of a Hamiltonian JC with invariant curves which are not symmetric with respect to the u axis, i. e., JC contains a term involving the product Pu' u. If <P0u is the ground state of the uncoupled u oscil-lator, then typically fnu = 1 < <P0u 1 cf>n> 12 will satisfy

fou:S 1 ,

while

o <flu «fou '

o <f2 u «flu' etc.

(3. 18a)

(3. 18b)

(a. 18c)

The Franck-Condon factors fnu' n 2: 1, give rise to com-bination lines in·the spectrum: we have

(3. 19)

and [see Eqs. (3.7) and (a.8)]

For example, suppose En =(n.+t)w., w.=2.0, nu =1.1, • E j = tw., and E j =tnu. Then a line (2,0) of strength 8 u

f2s fou corresponding to n. = 2, nu = 0, appears at w = 4.0, while the combination (1,2) appears with reduced strength f1. f2u at w =4.2. Sincefnu>O, n even, the combi-nations cannot have zero intensity, unless the s -u cou-pling is removed. By changing <Pu, the fnu can be mini-mized, but there will always be a residual intensity which is due to dynamiC, anharmonic coupling. This re-sidual intensity is expected to grow as the unstable re-gion is approached. This occurs because JC rotates its axes further away from the u,Pu axes and nu deviates further and further from the ull;coupled frequency wu' Both of these factors tend to increase Franck-Condon intensity in the combinations.




The spacing of the lines and the approach to instability is a most intriguing question. To answer it properly, we must address a slightly subtle point. The semiclas-sical approximations we have introduced (linearized dy-namics, stroboscopic effective JC) give rise to the entire Franck-Condon spectrum, via Eqs. (2.4), (3.6), (3.7), and (3.20). However, the question arises as to the choice of the "central" trajectory in phase space about which the dynamics is linearized, giving rise to the ef-fective JC. Often, this point would correspond to the average position and momentum of cP. However, the linearization is most accurate near the phase space do-main of the central trajectory, and thus it follows that the predicted spectrum is most accurate near the energy of the central trajectory. Therefore, to examine the question of the spacing of the combination overtone pairs, we determine the classical stability parameter v (and thus !1u) as a function of energy as we move through the Franck-Condon envelope. [The sign of v and value of the integer appearing in Eq. (3.11) are chosen so as to give the correct value for !1u in the uncoupled case. 1 The dashed lines in Fig. 6 show, in the stable region, the value of v'" vlrr as a function of position in the (2q la, Ja) plane. Although Fig. 6 is special to the Mathieu equation, the behavior as resonance is ap-proached is generic: v approaches 1 or 0, and as a re-sult !1u - pw., where p is an integer or half-integer, ac-cording to the order of the resonance. For example, if Ja = 1.1, then p = t. As a result, the combination lines get closer to their neighboring parent pure overtone lines as resonance is approached. This occurs because, for example, as !1u - tw., the line (n., 0) coincides with (n s -1, 2), whereas at lower energy [say the (2,0) and (1,2) pair 1 the splitting was - O. 2. Figure 6 shows that if the frequencies are such as to be on the "low side" of a resonance (for example, ra = 1. 9) then as E (thus q) is increased, v is not monotonic. This allows for some splitting apart of the combination-overtone pairs before they again approach each other closer to resonance. Moreover, on all the low sides (2w ucS mws)m=1,2, ••• resonance is harder to reach (see Figs. 6 and 7) as men-tioned earlier.

The trends predicted above are observed in practice, at least qualitatively. Table I and Fig. 1 show the "pre-resonant attraction" of quasidegenerate levels as pre-dicted above, although the spacing does not actually reach zero as resonance is approached. Also seen in Fig. 1 is the growth of the combination intensities as resonance is approached. Detailed calculations have confirmed the high side vs low side stability effects re-ferred to above and shown in Fig. 7 (see the conclusion). To recapitulate,

The pre resonant region is characterized by low combination band intensity, and combination-over-tone pairs which attract each other relative to their uncoupled positions. Some cases may show preresonant repulsion of levels. As resonance is approached, the combinations grow in intensity and the tendency toward attraction increases. It it easier to reach resonance from the "high side" (2w u?: mw.) than the "low side" (2wucS mws), es-pecially in the 1: 1 and 3: 2 cases.

Before examining the unstable case, we note that the predicted u-mode spacings [Eq. (3.16)1 are not correct even in the limiting case of an uncoupled anharmonic oscillator in u and s. 18 However, the higher overtones of u motion, where the error is worst, do not appear with measureable intensity in the spectrum. The effec-tive frequency !1u is most valid for the lowest u-mode overtones. Thus, when the factor of intensity is weighed in, which we have been able to do through the introduc-tion of JC, the physical meaning of the stability parameter analysis of periodic orbits 7c,d,17-19 becomes clearer.

c. Unstable case In the preresonant, stable case we learn to interpret

the elliptical invariant curves as those of an effective JC, whose frequency !1u and eigenfunctions <Pn determine the combination band spectrum. The same holds true above resonance, although JC corresponds to an inverted har-monic oscillator (see Fig. 8), and indeed the eigenvalues of M are now real and of the form

Jl±=e·iv , (3.21a)

where v is pure imaginary. Then

and !1u is also pure imaginary. The eigenvalues of JC thus form a continuum Eku and the eigenvectors <Pk sig-nificantly overlap cPu over a range of k values, giving rise to a broadened band for :6u(w). Since we are deal-ing with a bound potential, the true spectrum is of course discreet, and the broadened band results from linearization of the dynamics in our semiclassical ap-proximation: The semiclassical spectrum "thinks" the effective potential is globally an inverted barrier, where-as it is really only locally so, as suggested by the double well effective potential shown in Fig. 8: In prac-tice, it is found that the continuous band :6u(w) in the resonant region closely follows a true envelope of the corresponding overtone -combination set of discrete lines [belonging to a parent (n., 0) overtone 1 seen in the exact spectrum [see Fig. 5(c)1. This can be understood as follows: The:6 u is given by Eq. (3.7b), which shows that the lower resolution features come from shorter times in cPu(t). This short time dynamics is given cor-rectly by the inverted barrier approximation, so long as cPu does not initially extend into the double well region. At longer times, recurrences set in due to the double well potential, giving rise to the spectral lines under the envelope, but the linearized approximation misses this and settles for the envelope itself.

More quantitatively, the parameter ru =i!1u gives the bandwith of the envelope corresponding to each set of discrete lines belonging to a parent (n., 0) overtone. This is reasonable, since ru gives the exponential rate at which trajectories leave the u =0 region, according to Eqs. (3.21) and (3.9). Since the trajectories control the wave packet cPu(t), it can be shown that for long times I (cPul cPu(t) I goes as e-rut /2• One objection might be that there must be some dependence on the nature of the state cPu, and this is certainly true. However, calcula-




tions have shown that the width of the band is very close to ru over rather large variations in w when cPu takes the form (see Fig. 9)

(3.22)

Moreover, if w < r u, the band is skewed to the low ener-gy side, and if w > r u, the skewing is to the high e ne rgy side (Fig. 9).

Since the rate of escape, and thus r u, increases as the the unstable region (shaded areas in Figs. 6 and 7, for example) is penetrated further, the generic behavior will be increasing band widths with increasing energy. However, the bandwidths may not depend linearly on the energy or may not be a monotonic function of the en-I ergy, since the r u depends also on how w sand Wu vary with total energy in the s mode. It is possible, for strong "diagonal" anharmonicity, to carry the s-u modes out of resonance as energy is increased.

In practice the matrix M (and thus Ou or ru) is easily determined by the method of Greene. 22 This method re-quires running two trajectories in the vicinity of the central trajectory, for one period T •• These three tra-jectories are sufficient to determine M. The same tra-jectories are used in the cluster method. 13 Table II shows r u vs FWHM bandwidths for the resonant bands in Fig. 1.

The unstable region can be designated the Fermi

-5 -4 -3 -2 -I

FWHM=I.O

0·1.0 ·······w·3.0 ----w·0.25

,1'·. ----FWHM-I.I £ :\.

': \'. ·······FWHM .. I.I ': \'. f: \'.

//: \ .... 1-"""'1 .... / .' " •.....

-5 -4 -3 -2 -I 0 1 2 3 4 5 6 E

FIG. 9. Broadened bands due to decay on an inve rted barrier. The initial cp is given in Eq. (3.22), the potential is V(u) = -! n2u2 with a corresponding r w of 1. 0 W = 1. 0). Note how insensitive the FWHM is to the form of cpo

TABLE II. Fermi resonant band-widths.

EI'\>. a n b r e rd s u

8.0 7 0.049 0.091 9.0 8 0.066 0.101

10.0 9 0.082 0.110 11.0 10 0.092 0.116 12.0 11 O. 103 0.126

aEnergy of the maximum in the smoothed absorption band.

b ns is the symmetric stretch assign-ment of the band, i. e., the band arises out of the zero order (n s ' 0) level.

er"= I v I /Ts. where Ts=27r in this case and I v I is the stability param-eter. determined at (E ..... -1. 05).

FWHM measured bandwidth for the smooth fit to the spectral lines obtained from a full semiclassical wave bracket propagation and Fourier transform of < ¢ I cp(t). See Fig. 5(c) and the discussion in Sec. IID.

resonant region, for there is every reason to extend the definition of Fermi resonance to include the post-reso-nant, unstable behavior of the type we have just dis-cussed. In the event that there is a preresonant region in the spectrum, the Fermi resonance may set in without any repulsion of zero order levels either above or below resonance. Repulsion of levels seems to occur when there is no pre resonant domain, and the first resonant pair of levels are energetically in an unstable region; in which case, they interact strongly and split apart. An exception to this rule occurs for the "low side" pre-resonant cases mentioned above, where the stability parameter is not a monotonic function of energy. Another exception occurs for the first isolated pair of levels in the spectrum, if the levels are of the same symmetry. Even in the stable regime, a slight splitting but weak interaction will be noted for the first isolated pair.

IV. CONCLUSION

The connections between quantum spectral features and wave functions on the one hand, and classical dy-namics on the other, seem to be very close for the case of an isolated resonance examined here. A complete picture has emerged, cemented by numerical and semi-classical evidence. Further, the notion of a Fermi resonance has been augmented to include the possibility that two modes may not interact resonantly until midway in a spectrum, and the spectral signatures of this effect ha ve been identified. Earlier, 11 it was pointed out that CO2 is very deep in the post-resonant region, both clas-sically and quantally, even at the lowest energies in the spectrum, as regards the symmetric stretch-bend inter-action. However, the symmetric stretch-unsymmetric stretch interaction is pre-resonant until about 16500 cm-1 total energy. (A better potential surface will be needed to firm up this latter estimate.) It will be very




interesting if spectra can be found which show directly the transition from stable to unstable motion, not only from the point of view of this work but also since it will provide some evidence for the change from normal mode behavior at low energies to energy redistribution at high-er energies.

We believe the present work is the most explicit to date regarding the interpretation of homogeneous spec-tral bandwidths in terms of classical anharmonic motion and rates of classical flow. Pending results of calcula-tions on three or more dimensions now underway, the evidence seems to point to a large future role for clas-sical trajectory analysis of intramolecular energy flow and spectral features.

It is quite possible for two modes to remain stable and nonresonant until very high energies, due primarily to a poor match of the uncoupled frequencies to any low order resonant conditions. It remains possible in the absence of any evidence to the contrary that Significant stable do-mains can exist in the phase space of the molecule well up into the mostly stochastic regimes and perhaps above the dissociation energy of the molecule. Symmetric stretch motion of the type we have been considering is likely to be accessible experimentally. For example, many electronic transitions involve large displacement in the totally symmetric stretch coordinate. While stable domains may occupy a small fraction of the phase space of the molecule, their presence could significantly affect the energy flow in such totally symmetric stretch Franck-Condon spectra. The importance of stable periodic "traps" in the phase space of the molecule at high energy remains an important open question. It seems likely that the volume of such phase space traps must compare favorably with the appropriate power of Planck's constant h, lest its effect be "washed out."

Many possible extensions of the present work remain to be developed. A short list includes consideration of nonsymmetric periodic trajectories, extension to sev-eral dimensions, consideration of high order fixed points of the motion, and obtaining spectra in the vicinity, but not surrounding, periodic fixed points. It seems that the present approach, which attempts to relate spec-tral features at various levels of resolution with classi-cal and quantum dynamics for correspondingly short (low resolution) and long (high resolution) times, becomes in-valuable when the molecules considered contain more than three or four atoms. This is because individual spectral lines are becoming unresolvable (and irrele-vant) as densities of stat-es increase, and synthesizing the experimental spectra by smoothing over resolved ab initio calculations is no longer possible because far too many states are involved. Clearly, some sort of dynamic (time dependent) spectral analysis is required.

Even though the present work is couched mostly in the language of two dimenSions, there is a broad class of polyatomic systems for which the analysis carries over directly. This includes the possibility that a symmetric mode is excited, which then couples rapidly and strongly into one other mode; subsequent decay into other degrees of freedom will leave a spectrum very much like that seen in Fig. 1, still very describable in terms of two

dominant modes. Another case that can be synthesized from the presen t analysis is one symmetric mode cou-pled to two or more other modes which do not couple strongly with each other. Then we can write (in the case of three modes)

(4.1)

and the full. spectrum would simply be a convolution of three individual spectra, two of which arise from effec-tive Hamiltonia Je1

We present a final example which seems to raise the possibility of extreme sensitivity of molecular spectra, wave functions, and energy transfer to details of the po-tential surface. Figure 10 shows, at the top, two very similar potential surfaces VI and VII, which have the form

The uncoupled frequency WI is chosen to be 0.975, and wlI is 1. 025. This gives"; a I = 2WI/ Ws = 1. 95, and van =2WIJWs=2.05, where fa =2 is an exact 1:1 reso-nance. Figures 6 and 7 show that VI is expected to be more stable for a given energy than Vn, and the wave functions 1f!i and 1f!iI shown in Fig. 10 confirm this effect. The wave function 1f!I is the 109th eigenstate of VI' and 1f!1I is the 103rd eigenstate of Vn, and both have energies EI'" Ell '" 14. Both 1f!I and 1f!n are wave functions which possess the most symmetric stretch character near E = 14, as evidenced by Franck-Condon factors with a symmetrically displaced wave packet.

The classical trajectories (Mathieu equation in this case) show that the symmetric stretch is stable for Vr. but unstable for VII, near E = 14. There is a corre-sponding dramatic change in the wave functions, as seen in Fig. 10. The sensitivity of the wave functions to a small change in the potential surface evidenced here has direct implications for energy exchange and the spec-trum. The latter, of course, would show typical post-resonant broadening for VII and preresonant weak inter-actions for VI' A small (5%) change in a potential par-ameter has resulted in qualitatively different behavior for 1f!I and 1f!II' Such "bifurcations" are well studied in classical mechanics,23 and this example shows that they occur in quantum systems as well.

We have been able in this paper to provide a very specific link between the phenomenology of isolated classical resonances and the corresponding quantum phenomenology. Because of the relative Simplicity of isolated resonances, we have been much more specific about the quantum-classical correspondences than has been possible for the question of the onset of stochas-ticity.l,2

In the case of an isolated resonance we have achieved an understanding of certain spectral features in terms of the underlying dynamics, even to the point of pre-dicting bandwidths, overtone-combination spacings, and intensities using classical input. This has been accom-lished without any reference to quantization conditions on nonseparable potential surfaces. In abandoning the idea of predicting exact eigenvalues, we focus on dynam-ical properties of potential surfaces, and we extract di-




FIG. 10. Potential contours for VI (above left) and VII (above right), and their corresponding wave functions at E = 14, as de-scribed in text.

rectly the essential features leading to the observed spectra. This approach will become a necessity in dealing with "typical" polyatomics with vibrational densi-ties of states from 10 to 1014 per em-I. It has already proven very useful in the theory of polyatomic photodis-sociation spectral2 and Raman spectra. 24

ACKNOWLEDGMENTS This work was supported by NSF Grant No. CHE77-

13305.

APPENDIX A Suppose we approximate j(t) in Eq. (3.5) by piecewise

constant terms, of duration 1', i.e.,

(AI)

where n =0, ± 1, ...• Then

rjJ(t) "" exp[ -iHn+l(t -nl')l 0 •• e-iH2T e- iH1T rjJ(O) = UT(t)rjJ(O) • (A2)

Hn is the time independent quadratic Hamiltonian in the nth time interval given by replacingj(t) in Eq. (3.5) by

j(nl'). As 1'- 0, and n - 00, Eq. (A2) becomes exact. However, from the Baker-Hausdorff theorem, 25 i. e. ,

eA e B =exp {(A +B +HA, Bl +MA, [A, BlJ + ft{[A , Bl,BJ, + ••• }, (A3)

it is easily shown that

(A4)

where QIJ Qz, and Q3 are quadratic forms in Pu, u. Successive application of Eq. (A4) to Eq. (A2) shows

(A5)

where Q is again quadratic.

APPENDIX B Here we derive the overlap between the ground state

rjJ0u of a harmonic oscillator and the state <Pn of another harmonic oscillator whose major-minor axes u', are rotated by an angle () from the u,Pu axes (see Fig. 8). First we require the amplitude between the posi-tion vectors in the two systems




(u I u') = (- cscB /21Ti) 1 1 2

(B1)

which can easily be derived from the generating function for the (u,Pu)- (u', transformation. 26 Then we have

(<bOu!$n)= f dudu'(<bouiu)(ulu')(u' I $n) ,

where (<boul u ) = (W u/1T)1/4 e -<wu/2)u2 ,

(B2)

(B3a)

( 'I )-( -<ou/2)u'2H ("'" ') u $n - 2nn! n:rr e n V nu u . (B3b)

We can now perform the du integral in (B2) to give

(<boJ $n) = C1 f du ' e-C2U'2 Hn(ffiuu /) , (B4)

where

(.fw.If:. )112 ( 2 ) 1/2 C = ....:...=.JC.:.

1 2nn!n: wu+icotB

and

c = CSC20 i to nu

2 2wu + 2i cotO + 2" co +""2'

Next we employ the generating function for Hu, namely,

:!Hn(x-) , (B5)

to finally obtain

( (1T )1/2 ,..,,12 n! ( 1)n12 (<boJ$n>=I:1 C2 '-'3 (n!2)!-

where

C3 = (1 + nU/c2 ) •

(n even) ,

(n odd) , (B6)

Note that the odd states (n =odd) are thus missing in the spectrum, as must be true.

I For recent reviews, see S. A. Rice, inAdvances in Laser Chemistry, edited by A. Zewail (Springer, New York, 1978); and P. Brumer, in Advances in Chemical Physics, edited by S. A. Rice (John Wiley, New York), to be published.

2See P. Pechukas, J. Chern. Phys. 57, 5577 (1972); K. S. J. Nordholrn and S. A. Rice, ibid. 61, 203 (1974); 61, 768 (1974); I. C. Percival, J. Phys. A 7, 794 (1974); I. C. Percival and N. Pornpherey, J. Phys. B 31. 97 (1976); N. Pornpherey, J. Phys. B 7. 1909 (1974); W. H. Miller, J. Chern. Phys. (1976); (1976); N. Pornpherey, J. Phys. B 7. 1909 (1974); W. H. Miller, J. Chern. Phys. 64, 2880 (1976); M. V. Berry, J.

Phys. A 10, 2083(1977); E. J. Heller, Chern. Phys. Lett. 60, 338 (1979); R. M. Stratt, N. C. Handy, and W. H. Miller. J. Chern. Phys. 71, 3311 (1979); E. J. Heller. ibid. 72, 1337 (1980) .

3B . V. Chirikov, Research Concerning the Theory of Non-linear Resonance and Stochasticity, translated as CERN 71-40, Geneva (1971); D. W. Oxtoby and S. A. Rice, J. Chern. Phys. 65, 1676 (1976).

4See , e.g., J. Ford, Adv. Chern. Phys. 24, 155 (1973); in Fundamental Problems in Statistical Mechanics, edited by E. D. G. Cohen (North-Holland, Amsterdam, 1975), Vol. 3, p. 215.

5G. Herzberg, Infrared and Raman SPectra of Polyatomic Mole-cules (Van Nostrand, New York, 1945), p. 215.

6M. J. Davis and E. J. Heller, J. Chern. Phys. 71, 3383 (1979).

7(a) D. W. Noid, M. L. Koszykowski, and R. A. Marcus, J. Chern. Phys. 67. 2864 (1979); (b) D. W. Noid and R. A. Marcus, ibid. 62, 2119 (1975); (c) R. A. Marcus, Faraday Discuss. Chern. Soc. 55. 9 (1973); (d) W. Eastes and R. A. Marcus, J. Chern. Phys. 61, 430 (1974).

81. C. PerCival, J. Phys. A 7, 794 (1974); Adv. Chern. Phys. 36. 1 (1977).

9N. C. Handy, S. M. Colwell, and H. W. Miller, Faraday Discuss. Chern. Soc. 62. 29 (1977).

S. Sorbie and N. C. Handy, Mol. Phys. 33, 1319 (1977). "E. J. Heller, E. B. Stechel, and M. J. Davis, J. Chern.

Phys. 71, 4759 (1979). 12E. J. Heller, J. Chern. Phys. 68, 2066 (1978); 68, 38!n

(1978) . 13 E • J. Heller, J. Chern. Phys. 65, 4979 (1976); K. C. Kulan-

der and E. J. Heller, ibid. 69. 2439 (1978). 14See D. W. Noid, M. L. Koszykowski, and R. A. Marcus

[J. Chern. Phys. 67, 404 (1977)] for a method for obtaining "spectra" frorn trajectories. See also K. D. Hansel, Chern. Phys. 33. 35 (1978); Chern. Phys. Lett. 57, 619 (1978).

15E . Thiele and D. J. Wilson. J. Chern. Phys. 35, 1256 (1961). 16W. J. Cunningham, Introduction to Non-Linear Analysis

(McGraw-Hill, New York. 1958), Chaps. 9 and 10. 17(a) M. C. Gutzwiller, J. Math. Phys. 12, 343 (1971); (b) w.

H. Miller, J. Chern. Phys. 56, 38 (1972). 18W . H. Miller. J. Chern. Phys. 63, 996 (1975). 19M . V. Berry and M. Tabor, Proc. R. Soc. London Ser. A

349. 101 (1976). 20See M. V. Berry, N. L. Balazs, M. Tabor, and A. Voros

[Ann. Phys. (N. Y.) 122,26 (1979)] for a discussion of one-dirnensional quantum maps. Our quantum mapping is some-what different in that it arises out of a surface of section. See also G. Gasati, B. V. Chirikov. F. M. Izraelev, and J. Ford, in Lecture Notes in Physics. Vol. 93 (Springer, 1979).

21See , e.g., E. J. Heller, J. Chern. Phys. 62, 1544 (1975); 65, 1289 (1976).

22J. M. Greene, J. Math. Phys. 9, 760 (1969). 23A . Abraham and J. E. Marsden, Foundations of Mechanics,

2nd ed. (Benjamin Cummings, Reading, Mass .• 1979). 24Soo_Y. Lee and E. J. Heller, J. Chern. Phys. 71, 4777

(1979). 25See. e.g., R. M. Wilcox, J. Math. Phys. 8, 962 (1967). 26See , e.g., W. H. Miller, J. Chern. Phys. 53, 1949 (1970);

E. J. Heller, ibid. 66, 5777 (1977).



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Molecular spectra, Fermi resonances, and classical motion

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