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Assigning vibrational spectra of chaotic molecules
22 December 1995
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 247 (1995) 195-202
Assigning vibrational spectra of chaotic molecules Zi-Min
Lu, Michael E. Kellman 1
Department of Chemistry, University of Oregon, Eugene, OR 97403, USA Received 10 July 1995; in final form 16 October 1995
Abstract A method is presented for detailed level-by-level approximate quantum number assignment and spectral pattern organization for a molecular Hamiltonian with a mixture of regular and chaotic classical dynamics. The large-scale bifurcation structure is determined for a spectroscopic Hamiltonian for H20. A Husimi transform method is used to assign approximate quantum numbers to the levels, and the spectrum is grouped into regular patterns. Pronounced spectral reorganization is observed in the region of most highly chaotic dynamics.
I. Introduction
The normal modes description of molecular vibrations is a cornerstone of molecular spectroscopy, and in fact of much of chemical science, including reaction dynamics. However, it is clear that the normal modes description must break down, especially at high vibrational energies, on account of anharmonicity and coupling between the modes. There are now many experiments that probe the 'post-normal modes' regime. A good deal of insight has come from methods of classical nonlinear dynamics, especially by way of understanding of the transition from regular to chaotic dynamics. However, it is an open question whether anything like a rational description of quantum spectra is possible for chaotic molecules, at least at the accustomed level of detail afforded by the normal modes description. This is a very important question in molecular science because spectroscopy (in the broadest sense, including both fre-
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quency and time domains) is the experimental window on the world of molecular potential surfaces, intramolecular dynamics, energy transfer processes, and transition states. From the even broader perspective of fundamental issues in physical systems, highly excited spectra of small molecules are interesting dynamical systems in which to investigate the correspondence between classical and quantum dynamics. Molecules are important experimental systems, the number of degrees of freedom is small but > 2, and they have interestingly complex Hamiltonians in that molecular potential energy surfaces have varied topographies with hills, valleys, and saddles. Perhaps most important, highly excited molecules tend to fall in the mixed regime characterized by a combination of regular and chaotic classical dynamics. Important experimental systems exist with classical dynamics in the mixed regime. Examples other than molecules include a variety of atomic systems in electric and magnetic fields [1,2]. In mixed systems, an important question is whether it is possible to order the spectrum by means
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0009-261 4(95)01220-6
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Z.-M. Lu, M.E. Kellman / Chemical Physics Letters 247 (1995) 195-202
of approximate quantum number assignments, thereby attaining, or at least approaching, the detailed spectral organization of regular systems, where one is accustomed to level-by-level analysis, for example, using normal modes assignments [3]. Or perhaps one is forced to rely on statistical analyses [4,5]. A complete answer to this question will probably await satisfactory understanding of the quantum-classical correspondence and semi-classical methods of quantization. These old topics once again are exciting great interest, because of the profusion of beautiful experiments on spectra and dynamics of 'simple' systems which nonetheless display complex classical dynamics. Most semi-classical methods are best-suited for either non-chaotic, integrable systems, e.g. Einstein-Brillouin-Keller (EBK) quantization of invariant tori [6-8], or highly chaotic systems, e.g. methods based on sums over periodic orbits [9]. Despite recent progress in the semi-classical quantization of mixed systems [10], fundamental issues still await complete understanding, and the feasibility of spectral assignment is an unanswered question at this time. In this Letter, lacking a final semi-classical theory, we explore assignment of a mixed system using an approach which is heuristic in nature, but which offers the immediate prospect of a practical method, especially for molecules.
2. Normal m o d e bifurcations to new anharmonic modes
The specific system considered is an effective Hamiltonian which has been used [11] to fit the experimental spectrum of H20. We analyze levels of this model at energies where there is a mixed classical phase space, with a significant degree of chaos. There are three basic steps to the procedure. First, we solve semi-analytically, i.e. without recourse to numerical integration of Hamilton's equations, for the large-scale bifurcation structure of the Hamiltonian. By solving for the large-scale structure we mean the principal periodic motions or nonlinear modes of the system. These are the periodic orbits that are 'born' in bifurcations from the N periodic motions, i.e. the original normal modes, that define the phase space structure at the lowest energy. It is possible to solve for the principal motions at any
given energy because [12-14] the spectroscopic Hamiltonian has a conserved quantum number, corresponding to an extra classical constant in addition to the energy. The next step is to assign the levels with approximate quantum numbers corresponding to the nonlinear modes determined from the bifurcation analysis. The values of the quantum numbers assigned to each level are obtained from a Husimi transform of each state to an action-angle phase space representation, using an SU(3) coherent states approach. Finally, the approximate assignments are used to arrange the spectrum into organized energy level patterns, including the levels which undergo strong perturbation due to the influence of classically chaotic dynamics. Our Hamiltonian is quantum spectroscopic fitting Hamiltonian used by Baggott [11] in a fit of the available data for H20: H = H 0 + VStretChl:l"b =2:lVstr-bend •
(l)
H 0 is diagonal in the basis I(n 1, n 2, rib)), where n 1, n 2 are quantum numbers for anharmonic local O - H stretch modes, and n b is the quantum number for the I/stretch includes 1:1 and bend. The interaction term -1: 2 : 2 resonant couplings between the stretches; the term V~t:rl-b~"d is a 2:1 resonant interaction between the stretches and the bend. The presence together of the stretch-stretch and 2:1 stretch-bend coupling makes the system non-integrable, and leads to a considerable degree of chaos at some energies. The quantum Hamiltonian (1) can be converted to a classical form via the Heisenberg correspondence principle [15,16], as we have done in a variety of cases for analysis spectroscopic Hamiltonians [17-22]. Despite the nonintegrability of the classical version of Hamiltonian (1), the generalized polyad 1 p = n 1+ n2 + ~n b is a good quantum number adapted [23,24] to the resonances of the Hamiltonian. Classically, the action 13 = P + 1 is a conserved quantity commuting under the Poisson bracket with the classical limit of H obtained through the Heisenberg correspondence principle [15,16]: HCl= HCl('l'l, "r2, "l'b; ~1, ~b2) where ~'i, thi, i = l, 2, b are the classical action and angle variables for the three modes, and the angles qJl = thl - thE, ~02 = thl - 2thb are adapted to the
Z.-M. Lu, M.E. Kellman / Chemical Physics Letters 247 (1995) 195 -202
resonance couplings. The polyad number conservation is due to the fact that H c~ depends only on the two angles ff~, ~b2. This suggests a more economical description in which we use the new set of action-angle variables ( I i, ~i, i = 1-3) in which I i are adapted to ~bi. Because of the conserved action 13, the angle ~O3 conjugate to the conserved polyad action 13 plays the role merely of a fast cyclic variable [14]. It turns out that even for the chaotic multiresonant Hamiltonian, as long as there is a polyad constant of motion, an enormous simplification is possible, in that one can solve analytically for the l a r g e - s c a l e b i f u r c a t i o n s t r u c t u r e [ 14], without actually performing numerical integration of Hamilton's equations. By the large-scale structure we mean the principal periodic motions or nonlinear modes that are born from the N normal modes that exist at low energy. These nonlinear modes are the shortest period orbits, which are solved for as the critical points (maxima, minima, and points of inflection) of the Hamiltonian in the reduced phase space Ii, ~bi, i = 1,2, without numerical solution of Hamilton's equations. The nonlinear normal modes of the system obtained in this way give the 'names' of the approximate quantum numbers that we will attempt to assign to the individual quantum levels. At low excitation, i.e. low P, there are three critical points, corresponding to the nonlinear normal modes proven mathematically to exist at low energy by Weinstein [25]. These are familiar in molecular spectroscopy as the symmetric stretch, antisymmetric stretch, and bend modes. As P increases, the number and the stability of the critical points changes in bifurcations, as summarized in the bifurcation diagram shown in Fig. 1. The details of obtaining this diagram are quite involved, and are presented in a separate paper [26]. Here we outline the method and present the most salient results. The procedure consists of obtaining analytical formulas for the values of the zeroth-order actions and angles at which the critical orbits and their bifurcations occur, as enabled by the classical polyad Hamiltonian described above. The dependence on the angles is almost trivial (again, a consequence of the polyad Hamiltonian), so this is left out of the bifurcation diagram, by this means reducing the number of dimensions to a tractable number. The remainder of the problem is to compute the critical points as a function of the polyad number
o
a0
197
I"11
Fig. 1. Bifurcation diagram for the H 2 0 molecule. The critical points are plotted against the polyad number P (vertical axis) and the number of quanta n~, n 2 in the zeroth-order modes.
(vertical axis) and the number of quanta nj, n 2 in the zeroth-order modes. The results have a fairly straightforward physical interpretation in terms of notions such as normal and local modes. In Fig. 1, the Weinstein normal modes exist at low polyad number P, with equal numbers of local stretch quanta nl, n2, in the stretch normal modes, on account of symmetry. (We leave the nearly pure-bend critical orbit out of the picture, because it does not undergo bifurcations. If shown, it would be nearly vertical, i.e. not contaminated much by the stretches, as expected for a 'bend'.) The two normal stretch modes are coincident in the lower right-hand comer of Fig. 1. At the first bifurcation A, the symmetric stretch becomes unstable and two local stretch modes are created. The local modes follow close to the nl = 0 and n 2 = 0 'walls' of the diagram, as expected for stretches of local character, where most of the energy is in one bond or another. The normal modes continue along nl = n 2 with ascending P. The change from solid to gray normal mode critical orbit in the diagram indicates that the symmetric stretch has become unstable in the bifurcation. (The antiysmmetric stretch remains stable.) The occurrence of both local and normal modes in the three-mode system can be understood qualitatively in terms of two-mode dynamics [17-20] as due to the 1:1 and 2 : 2 couplings between local stretch modes. The qualitative features of the normal-local bifurcation just described are discussed in detail (for two-mode systems) in Refs. [17-20].
Z.-M. Lu, M.E. Kellman / Chemical Physics Letters 247 (1995) 195 -202
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At B, a bifurcation takes place which roughly speaking is due to the 2:1 resonance between local stretch and bend. One of the local stretches and the bend coordinate combine to form 'resonant collective modes'; this is called zone II dynamics in a classification of the 2:1 system [19-22]. (The zones refer to a classification of the different kinds of vibrational dynamics of the 2:1 Fermi resonance Hamiltonian as described in a classical phase space analysis. In this system, unlike coupled stretches, the language of 'normal' and 'local' modes does not suffice. Detailed explanations are found in Refs. [19-22].) By symmetry, the other local stretch mode undergoes the same bifurcation at B'. At C, the stretch-bend system undergoes another bifurcation, to zone III dynamics in the 2:1 classification. It is possible as well for the normal stretch modes and the bend to have Fermi resonance bifurcations. At D, the symmetric normal mode and bend undergo a bifurcation to zone II resonant collective behavior; at E, this system undergoes a further bifurcation, to zone III dynamics. The stable critical orbits in Fig. 1 are the 'skeleton' around which the phase space is organized. The structures in phase space of least complexity are the stable critical orbits, or lowest-period orbits. Topologically, these are circles, i.e. one-dimensional tori in the multidimensional phase space. The 1-tori are nested at the center of sets of 2-tori, i.e. doughnutshaped surfaces. The 2-tori in turn are nested within sets of 3-tori. Outside of these regions of regular
"C2
2 V, Fig. 2. Poincare surface o f section for P = 8, e n e r g y = 25591 c m - t.
motion are chaotic regions. Within the chaotic regions there can sometimes be identified multidimensional analogs of island chains, which have higherperiod regular regions in the centers of the islands. The influence of the critical orbits and the chaotic nature of the phase space are seen in Poincare surfaces of section, taken at different energies for a given polyad number by integrating Hamilton's equations numerically for the classical version of Hamiltonian (1). A classical surface of section so obtained is illustrated in Fig. 2 for energy 25591 c m - I and P = 8 .
3. Assigning quantum numbers The three types of stable periodic orbits for P = 8 define three sets of quantum numbers that we will attempt to assign, both for overtone states with all the energy concentrated (apart from zero-point motion) along the periodic orbit, and combination states with the energy distributed among the three modes. The assignments are for the quantum energy levels obtained by diagonalizing the quantum matrix Hamiltonian (1) for H20. The first type of assignment is denoted (n l, n2, nb)+ indicating n 1 quanta in local stretch 1,n 2 quanta in local stretch 2, and n b quanta in the bend, with + denoting symmetrization with respect to 1 and 2. The second type is denoted In s, n a, n b] indicating n s quanta in the symmetric stretch, n a quanta in the antisymmetric stretch, and n b quanta in the bend. The third type is denoted {hi, nRt, nR2} indicating n] quanta in a local stretch mode, na] quanta in resonant collective mode 1, and nR2 quanta in resonant collective mode 2. We use the Husimi dislribution to assign these quantum numbers to individual levels, focusing here on key illustrative cases. The Husimi distribution is the squared overlap between an eigenstate ~b and a coherent state I z) [27,28]. The coherent state used in the analysis to follow is defined [29-31] with respect to the SU(3) algebra associated with the three-mode H20 oscillator system. Coherent states were first defined as non-spreading wave packets in a harmonic potential, hence the best quantum analogs of classical particle motion. Non-spreading states are not known for general quantum systems. The SU(3) coherent states of Refs. [28-31] are an attempt to
Z.-M. Lu, M.E. Kellman / Chemical Physics Letters 247 (1995) 195-202
devise, as closely as possible, analogs of harmonic oscillator coherent states for anharmonic systems. The fact that a Lie algebra such as SU(3) is used is not surprising because the harmonic coherent states can themselves be defined algebraically. For wave functions of the Hamiltonian (1), projections of the Husimi distribution in action space show pronounced peaks, and projections in angle space often have distinct oscillatory structure similar to nodal patterns. This suggests that the Husimi distribution is locating quantizing structures such as invariant tori, or in the chaotic region, remnants of tori. We start with a level which will be assigned with normal mode stretch and bend numbers as [ns, r/a, r/b] = [1, 5, 4]. Fig. 3b shows the projection of the Husimi distribution into the angle space ~0~, qJ2- There is a clear double peak along ~0I. In terms of a 'dynamical barrier' or 'dynamical tunneling' model [18,32,33], this is an indicator of normal mode stretch dynamics, with a separate bend mode. The double peak indicates n s = 1 quantum in the symmetric stretch. Fig. 3a shows the projection into the action space of the two local stretches. This distribution peaks close to ~'l = ~'2 = 3. We have found in assigning pure two-mode stretch systems that equal actions in the local stretches are a further marker of normal mode stretch dynamics. There are 6 total stretch quanta, so n a = 5 quanta in the antisymmetric stretch. Since P = n s + n a -t- lr/b, this leaves 4 quanta in the bend, and the assignment is [n s, r/a, rib] = [1, 5, 41. Next, we assign a level which turns out to be very near the stretch-bend resonant collective mode critical orbit. Fig. 3c shows the projection of the Husimi distribution onto the action space. There is pronounced peaking at (6, 0) and (0, 6) quanta in stretches 1 and 2, respectively. This indicates stretches which are local in character. The projection into the angle space contains further information. Fig. 3d shows a single pronounced peak in if2. In the dynamical barrier picture of a 2 : 1 resonance, this indicates that one of the local stretches, that with 6 quanta, is strongly coupled to the bend, resulting in a resonant collective mode. The single peak in ~P2 indicates zero quanta in this resonant mode: n R , = 0. The other stretch mode is uncoupled from the bend and so is assigned as a local stretch quantum number
199
(a)
2
~. .: ;i~:::!:i~
(b)
1
2
8
)
(f)
0
~
n~
~
0
Fig. 3. (a) Projection of the Husimi distribution for the eigenstate at 26714.96 cm- ~ onto the action space of the two local stretches. (b) Projection onto the angle space. (c) Projection for the eigenstate at 25573.93 cm- ~ onto the action space. (d) Projection onto the angle space. (e) Projection for the eigenstate at 26351.11 cm-~ onto the action space. (f) Projection onto the angle space. with n~ = 0. This leaves the quantum number nR2 to assign for the second resonant mode coordinate. With P = 8 and the values nR, = 0, n I = 0, there are 16 quanta in the second resonant mode. The complete assignment is then {hi, n R , , nR~} = {0, 0, 16}. A final example shows a state that cannot very plausibly be assigned like the first three states. The action projection in Fig. 3e indicates local stretches with (6, I) and (1, 6) quanta. However, the angle projection in Fig. 3f shows no clear pattern that leads to an assignment with one of the sets (n l, n 2, rib) ÷ or {r/l, r/R,, r/R2}" We classify this state as 'chaotic' and give it the 'nominal' assignment (r/l, n2, nb)+ chaotic= (6, 1, 2) + chaotic. The nominal assignment of 'chaotic states' will be of use for grouping the spectrum into ordered sequences of levels.
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4. 'User-friendly' assignment: reconstructing sequences and progressions All 45 levels of the P = 8 polyad have been assigned with one of the four types of quantum numbers (n l, n 2, nb )+, [n s, Oa, /7b] , {/'/1, nR,, nR2}' (121 , /72, rib)+ chaotic. W e n o w use these assignments to organize the spectrum into ordered energy level sets. As far as possible, we want to devise these in analogy to the groupings recognized by molecular spectroscopists in less highly excited systems, while remaining cognizant of the greater complexity inherent in the fourfold assignment demanded by the bifurcated phase space. The usual organizational scheme for lower energy classifies sequences in the number of quanta of the normal modes. A typical procedure is to group the spectrum into sub-polyads of constant n b and hence total stretch quantum number g s t r = n s + ha, with sequences in the stretch, so that states in the subpolyad are labeled [n s, ha, rib] : [Nstr, 0, rib], [Nst r - 1, 1, /Tb]' etc. When the dynamics are perturbed by a single resonance coupling between stretches, this scheme can be extended [34] so that within the polyad, if there is a transition between local and normal modes, the stretch labels change from (n l, n 2, nh )+, to [n s, /Ta, Y/b]" We want to extend this scheme further to the nonintegrable system (1) including the 2:1 coupling, maintaining the pattern of the subpolyad arrangement as much as possible. We need to place the resonant states {nl, r/R] , RR2} and the chaotic states (nl, n2 ' rib)_+ chaotic into sub-polyads with fixed values of the number of bend quanta n b, or at least nominal values of n b. For the chaotic states we use the nominal assignments (n I , n 2, n h) ± chaotic. For the states assigned with resonant collective modes as {n 1, n R t , nR2}, the nominal n I and n 2 are obtained from the H u s i m i p r o j e c t i o n onto the action space described above. Then, the nominal n b is obtained from P. For example, the case /TR } = {0, 0, 16} discussed above becomes {/71' /Ta I' nbiz ± nominal .~_ (n 1, n2, (6, 0, 4) 5 nominal The entire spectrum can now be organized into sub-polyads, using the dynamical assignments ( h i , rt2, nh )±, [ns, /Ta, /Tb] for local and normal mode states, and the nominal assignments (n l, rt2, rib)± nominal and (n 1, n 2 , nb )± chaotic for resonant and chaotic states. The local and normal mode states are
found to be only slightly displaced from the levels of the sub-polyads of the integrable Hamiltonian H 0 + vstretch with only stretch-stretch coupling. The situa1:1 tion regarding the resonant states and chaotic states is more complicated. Fig. 4 shows these levels placed into sub-polyads according to the nominal values of their bend quantum number n b, as well as some levels of the subpolyad which have local or normal assignments. Fig. 4 also shows the corresponding members of the sub-polyads of H 0 + VlSt:r~tch. The resonant and chaotic states (labeled 'nom(inal)' and 'cha(otic)' in the assignments) fit into the basic subpolyad pattern of H 0 +--l:lS/stretch, but are substantially displaced. This shows the effects of the resonant and chaotic dynamics on the spectrum, as interpreted through the quantum number assignments that we have made. The sequences devised here for polyad P = 8 can easily be generalized to include progressions by including different polyads.
5. Summary and prospects In summary, in this Letter we have attempted a level-by-level analysis of a realistic molecular system, as a prototype of what we hope will be a useful procedure for spectral analysis of systems with mixed dynamics. We have shown the feasibility in a molecular system with a mixture of regular and chaotic dynamics of using the classical bifurcation structure to make approximate quantum number assignments, by means of examination of Husimi phase space representations of the wave functions. We have used these assignments to arrange the levels into regular patterns, analogous to sequences customary at low energy in the normal modes regime. Pronounced shifts occur for individual levels assigned as resonant or chaotic. The Hamiltonian (1) analyzed here is the spectroscopic fitting Hamiltonian optimized in Ref. [11] to give the best possible fit to the known energy levels of H20. It gives a good fit (for a Hamiltonian with such a modest number of parameters) to the known experimental levels. However, one may ask whether it is likely to be accurate for the very high levels contained in the P = 8 polyad analyzed here. This question is especially acute given the paucity of
Z.-M. Lu, M.E. Kellman / Chemical Physics Letters 247 (1995) 195-202
experimental information generally available for states of symmetric triatomics with significant bend excitation. Another important consideration is that H 2 0 is virtually certain to be above the barrier to the linear configuration when the molecule contains 16 quanta nominally in the bend. However, this is not likely to be a factor for the levels of primary interest here: those in Fig. 4 with 2, 4, and 6 nominal bend quanta, in the region strongly affected by interacting 1:1 stretch and 2:1 Fermi resonance and chaotic dynamics, with marked deviation from the pure stretch-stretch Hamiltonian. With this confidence in
201
the applicability of our most interesting results to H20, we believe the conclusions obtained for the effective spectroscopic Hamiltonian (I) give impetus to experimental efforts to access the bending region of phase space in symmetric triatomics. The work presented here for the H 2 0 fitting Hamiltonian shows that it is possible both in principle and in practice to achieve a rational level-by-level analysis and organization of spectra of systems with a significant mixture of regular and chaotic dynamics. A future application to molecules will focus on larger systems, e.g. acetylene C2H 2. Another
28000
E (cm-1) • --,-•
[080] [170]
•
[072]
+
[162]
,
(522)+-nom
[260]
27000
•
[064]
(620) +-
'
[154] [244]
•
(612)'+ cha • •
(514)*cha
•
[056]
"
[146]
:
(416) ÷
( 7 1 0 ) +-
26000
--(604)+-nom •
(702)+" nom _ _
25000
•
( 5 0 6 ) ÷ - nom
(800)+-
I
I
I
i
0
2
4
6
nb Fig. 4. Portion of the spectrum of H 2 0 for polyad P = 8, including the region corresponding to the most chaotic dynamics. Lines correspond to levels of the nonintegrable Hamiltonian of Eq. (1) with l : l stretch-stretch and 2 : ! stretch-bend resonance interaction. Diamonds correspond to levels calculated of an integrable Hamiltonian H 0 + V~t:r[tch, identical to Eq. (l) except that it contains just the l : 1 stretch-stretch interaction. Levels labeled by a + - correspond to pair of states, which may be too nearly degenerate to resolve in the figure. The levels are assigned as in the text and arranged into sub-polyads labeled by the value of the 'nominal' quantum number n b.
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untested application is to systems which cross potential barriers, i.e. undergo isomerization, with abovebarrier H20 the prototype.
Acknowledgement Support of the Department of Energy under Basic Energy Sciences Grant DE-FG06-92ER14236 is gratefully acknowledged. We would like to thank Dr. John P. Rose for helpful discussions and for assistance in preparation of the figures.
References [1] R.V. Jensen, Nature 355 (1992) 311. [2] M. Courtney, H. Jiao, N. Spellmeyer and D. Kleppner, Phys. Rev. Letters 73 (1994) 1340. [3] G. Herzberg, Infrared and Raman spectra (Van Nostrand, New York, 1950). [4] R.L. Sundberg, E. Abramson, J.L. Kinsey and R.W. Field, J. Chem. Phys. 83 (1985) 466. [5] G. Sitja and J.P. Pique, Phys. Rev. Letters 73 (1994) 232. [6] A. Einstein, Verh. Dtsch. Physik. Ges. 19 (1917) 82. [7] J.B. Keller, Ann. Phys. 4 (1958) 180. [8] M. Tabor, Chaos and integrability in nonlinear dynamics (Wiley, New York, 1989). [9] M.C. Gutzwiller, Chaos in classical and quantum mechanics (Springer, Berlin, 1990). [10] S. Tomsovic and E.J. Heller, Phys. Rev. Letters 70 (1993) 1405. [11] J.E. Baggott, Mol. Phys. 65 (1988) 739.
[12] E. van der Aa, Celestial Mechanics 31 (1983) 163. [13] J.F. Sanders and F. Verhulst, in: Applied mathematical sciences series, Vol. 59. Averaging methods in nonlinear dynamical systems (Springer, Berlin, 1985). [14] M.E. Kellman, Dynamical analysis of highly excited vibrational spectra: progress and prospects, Molecular dynamics and spectroscopy by stimulated emission pumping, eds. H.-L. Dai and R.W. Field (World Scientific, Singapore, 1995). [15] W. Heisenberg, Z. Physik 33 (1925) 879, translated in Sources of quantum mechanics, ed. B.L. van der Waerden (Dover, New York, 1967). [16] A.P. Clark, A.S. Dickinson and D. Richards, Advan. Chem. Phys. 36 (1977) 63. [17] L. Xiao and M.E. Kellman, J. Chem. Phys. 90 (1989) 6086. [18] M.E. Kellman, J. Chem. Phys. 83 (1985) 3843. [19] Z. Li, L. Xiao and M.E. Kellman, J. Chem. Phys. 92 (1990) 225 I. [20] L. Xiao and M.E. Kellman, J. Chem. Phys. 93 (1990) 5805. [21] M.E. Kellman and L. Xiao, J. Chem. Phys. 93 (1990) 5821. [22] J. Svitak, Z. Li, J. Rose and M.E. Kellman, J. Chem. Phys. 102 (4340) 1995. [23] M.E. Kellman, J. Chem. Phys. 93 (1990) 6630. [24] L.E. Fried and G.S. Ezra, J. Chem. Phys. 86 (1987) 6270. [25] A. Weinstein, Inv. Math. 20 (1973) 47. [26] Z. Lu and M.E. Kellman, submitted for publication. [27] A. Perelomov, Generalized coherent states and their applications (Springer, Berlin, 1986). [28] W.-M. Zhang, D.H. Feng and R. Gilmore, Rev. Mod. Phys. 62 (1990) 867. [29] P. Leboeuf and A. Voros, J. Phys. A 23 (1990) 1765. [30] G. Wu, Chem. Phys. Letters 179 (1991) 29. [31] M.B. Cibils, Y. Cuche, P. Leboeuf and W.F. Wreszinski, Phys. Rev. A 46 (1992) 4560. [32] M.J. Davis and E.J. Heller, J. Chem. Phys. 75 (1981) 246. [33] E.L. Sibert III, J.T. Hynes and W.P. Reinhardt, J. Chem. Phys. 77 (1982) 3583. [34] M.E. Kellman and L. Xiao, Chem. Phys. Letters 162 (1989) 486.
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 247 (1995) 195-202
Assigning vibrational spectra of chaotic molecules Zi-Min
Lu, Michael E. Kellman 1
Department of Chemistry, University of Oregon, Eugene, OR 97403, USA Received 10 July 1995; in final form 16 October 1995
Abstract A method is presented for detailed level-by-level approximate quantum number assignment and spectral pattern organization for a molecular Hamiltonian with a mixture of regular and chaotic classical dynamics. The large-scale bifurcation structure is determined for a spectroscopic Hamiltonian for H20. A Husimi transform method is used to assign approximate quantum numbers to the levels, and the spectrum is grouped into regular patterns. Pronounced spectral reorganization is observed in the region of most highly chaotic dynamics.
I. Introduction
The normal modes description of molecular vibrations is a cornerstone of molecular spectroscopy, and in fact of much of chemical science, including reaction dynamics. However, it is clear that the normal modes description must break down, especially at high vibrational energies, on account of anharmonicity and coupling between the modes. There are now many experiments that probe the 'post-normal modes' regime. A good deal of insight has come from methods of classical nonlinear dynamics, especially by way of understanding of the transition from regular to chaotic dynamics. However, it is an open question whether anything like a rational description of quantum spectra is possible for chaotic molecules, at least at the accustomed level of detail afforded by the normal modes description. This is a very important question in molecular science because spectroscopy (in the broadest sense, including both fre-
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quency and time domains) is the experimental window on the world of molecular potential surfaces, intramolecular dynamics, energy transfer processes, and transition states. From the even broader perspective of fundamental issues in physical systems, highly excited spectra of small molecules are interesting dynamical systems in which to investigate the correspondence between classical and quantum dynamics. Molecules are important experimental systems, the number of degrees of freedom is small but > 2, and they have interestingly complex Hamiltonians in that molecular potential energy surfaces have varied topographies with hills, valleys, and saddles. Perhaps most important, highly excited molecules tend to fall in the mixed regime characterized by a combination of regular and chaotic classical dynamics. Important experimental systems exist with classical dynamics in the mixed regime. Examples other than molecules include a variety of atomic systems in electric and magnetic fields [1,2]. In mixed systems, an important question is whether it is possible to order the spectrum by means
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0009-261 4(95)01220-6
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of approximate quantum number assignments, thereby attaining, or at least approaching, the detailed spectral organization of regular systems, where one is accustomed to level-by-level analysis, for example, using normal modes assignments [3]. Or perhaps one is forced to rely on statistical analyses [4,5]. A complete answer to this question will probably await satisfactory understanding of the quantum-classical correspondence and semi-classical methods of quantization. These old topics once again are exciting great interest, because of the profusion of beautiful experiments on spectra and dynamics of 'simple' systems which nonetheless display complex classical dynamics. Most semi-classical methods are best-suited for either non-chaotic, integrable systems, e.g. Einstein-Brillouin-Keller (EBK) quantization of invariant tori [6-8], or highly chaotic systems, e.g. methods based on sums over periodic orbits [9]. Despite recent progress in the semi-classical quantization of mixed systems [10], fundamental issues still await complete understanding, and the feasibility of spectral assignment is an unanswered question at this time. In this Letter, lacking a final semi-classical theory, we explore assignment of a mixed system using an approach which is heuristic in nature, but which offers the immediate prospect of a practical method, especially for molecules.
2. Normal m o d e bifurcations to new anharmonic modes
The specific system considered is an effective Hamiltonian which has been used [11] to fit the experimental spectrum of H20. We analyze levels of this model at energies where there is a mixed classical phase space, with a significant degree of chaos. There are three basic steps to the procedure. First, we solve semi-analytically, i.e. without recourse to numerical integration of Hamilton's equations, for the large-scale bifurcation structure of the Hamiltonian. By solving for the large-scale structure we mean the principal periodic motions or nonlinear modes of the system. These are the periodic orbits that are 'born' in bifurcations from the N periodic motions, i.e. the original normal modes, that define the phase space structure at the lowest energy. It is possible to solve for the principal motions at any
given energy because [12-14] the spectroscopic Hamiltonian has a conserved quantum number, corresponding to an extra classical constant in addition to the energy. The next step is to assign the levels with approximate quantum numbers corresponding to the nonlinear modes determined from the bifurcation analysis. The values of the quantum numbers assigned to each level are obtained from a Husimi transform of each state to an action-angle phase space representation, using an SU(3) coherent states approach. Finally, the approximate assignments are used to arrange the spectrum into organized energy level patterns, including the levels which undergo strong perturbation due to the influence of classically chaotic dynamics. Our Hamiltonian is quantum spectroscopic fitting Hamiltonian used by Baggott [11] in a fit of the available data for H20: H = H 0 + VStretChl:l"b =2:lVstr-bend •
(l)
H 0 is diagonal in the basis I(n 1, n 2, rib)), where n 1, n 2 are quantum numbers for anharmonic local O - H stretch modes, and n b is the quantum number for the I/stretch includes 1:1 and bend. The interaction term -1: 2 : 2 resonant couplings between the stretches; the term V~t:rl-b~"d is a 2:1 resonant interaction between the stretches and the bend. The presence together of the stretch-stretch and 2:1 stretch-bend coupling makes the system non-integrable, and leads to a considerable degree of chaos at some energies. The quantum Hamiltonian (1) can be converted to a classical form via the Heisenberg correspondence principle [15,16], as we have done in a variety of cases for analysis spectroscopic Hamiltonians [17-22]. Despite the nonintegrability of the classical version of Hamiltonian (1), the generalized polyad 1 p = n 1+ n2 + ~n b is a good quantum number adapted [23,24] to the resonances of the Hamiltonian. Classically, the action 13 = P + 1 is a conserved quantity commuting under the Poisson bracket with the classical limit of H obtained through the Heisenberg correspondence principle [15,16]: HCl= HCl('l'l, "r2, "l'b; ~1, ~b2) where ~'i, thi, i = l, 2, b are the classical action and angle variables for the three modes, and the angles qJl = thl - thE, ~02 = thl - 2thb are adapted to the
Z.-M. Lu, M.E. Kellman / Chemical Physics Letters 247 (1995) 195 -202
resonance couplings. The polyad number conservation is due to the fact that H c~ depends only on the two angles ff~, ~b2. This suggests a more economical description in which we use the new set of action-angle variables ( I i, ~i, i = 1-3) in which I i are adapted to ~bi. Because of the conserved action 13, the angle ~O3 conjugate to the conserved polyad action 13 plays the role merely of a fast cyclic variable [14]. It turns out that even for the chaotic multiresonant Hamiltonian, as long as there is a polyad constant of motion, an enormous simplification is possible, in that one can solve analytically for the l a r g e - s c a l e b i f u r c a t i o n s t r u c t u r e [ 14], without actually performing numerical integration of Hamilton's equations. By the large-scale structure we mean the principal periodic motions or nonlinear modes that are born from the N normal modes that exist at low energy. These nonlinear modes are the shortest period orbits, which are solved for as the critical points (maxima, minima, and points of inflection) of the Hamiltonian in the reduced phase space Ii, ~bi, i = 1,2, without numerical solution of Hamilton's equations. The nonlinear normal modes of the system obtained in this way give the 'names' of the approximate quantum numbers that we will attempt to assign to the individual quantum levels. At low excitation, i.e. low P, there are three critical points, corresponding to the nonlinear normal modes proven mathematically to exist at low energy by Weinstein [25]. These are familiar in molecular spectroscopy as the symmetric stretch, antisymmetric stretch, and bend modes. As P increases, the number and the stability of the critical points changes in bifurcations, as summarized in the bifurcation diagram shown in Fig. 1. The details of obtaining this diagram are quite involved, and are presented in a separate paper [26]. Here we outline the method and present the most salient results. The procedure consists of obtaining analytical formulas for the values of the zeroth-order actions and angles at which the critical orbits and their bifurcations occur, as enabled by the classical polyad Hamiltonian described above. The dependence on the angles is almost trivial (again, a consequence of the polyad Hamiltonian), so this is left out of the bifurcation diagram, by this means reducing the number of dimensions to a tractable number. The remainder of the problem is to compute the critical points as a function of the polyad number
o
a0
197
I"11
Fig. 1. Bifurcation diagram for the H 2 0 molecule. The critical points are plotted against the polyad number P (vertical axis) and the number of quanta n~, n 2 in the zeroth-order modes.
(vertical axis) and the number of quanta nj, n 2 in the zeroth-order modes. The results have a fairly straightforward physical interpretation in terms of notions such as normal and local modes. In Fig. 1, the Weinstein normal modes exist at low polyad number P, with equal numbers of local stretch quanta nl, n2, in the stretch normal modes, on account of symmetry. (We leave the nearly pure-bend critical orbit out of the picture, because it does not undergo bifurcations. If shown, it would be nearly vertical, i.e. not contaminated much by the stretches, as expected for a 'bend'.) The two normal stretch modes are coincident in the lower right-hand comer of Fig. 1. At the first bifurcation A, the symmetric stretch becomes unstable and two local stretch modes are created. The local modes follow close to the nl = 0 and n 2 = 0 'walls' of the diagram, as expected for stretches of local character, where most of the energy is in one bond or another. The normal modes continue along nl = n 2 with ascending P. The change from solid to gray normal mode critical orbit in the diagram indicates that the symmetric stretch has become unstable in the bifurcation. (The antiysmmetric stretch remains stable.) The occurrence of both local and normal modes in the three-mode system can be understood qualitatively in terms of two-mode dynamics [17-20] as due to the 1:1 and 2 : 2 couplings between local stretch modes. The qualitative features of the normal-local bifurcation just described are discussed in detail (for two-mode systems) in Refs. [17-20].
Z.-M. Lu, M.E. Kellman / Chemical Physics Letters 247 (1995) 195 -202
198
At B, a bifurcation takes place which roughly speaking is due to the 2:1 resonance between local stretch and bend. One of the local stretches and the bend coordinate combine to form 'resonant collective modes'; this is called zone II dynamics in a classification of the 2:1 system [19-22]. (The zones refer to a classification of the different kinds of vibrational dynamics of the 2:1 Fermi resonance Hamiltonian as described in a classical phase space analysis. In this system, unlike coupled stretches, the language of 'normal' and 'local' modes does not suffice. Detailed explanations are found in Refs. [19-22].) By symmetry, the other local stretch mode undergoes the same bifurcation at B'. At C, the stretch-bend system undergoes another bifurcation, to zone III dynamics in the 2:1 classification. It is possible as well for the normal stretch modes and the bend to have Fermi resonance bifurcations. At D, the symmetric normal mode and bend undergo a bifurcation to zone II resonant collective behavior; at E, this system undergoes a further bifurcation, to zone III dynamics. The stable critical orbits in Fig. 1 are the 'skeleton' around which the phase space is organized. The structures in phase space of least complexity are the stable critical orbits, or lowest-period orbits. Topologically, these are circles, i.e. one-dimensional tori in the multidimensional phase space. The 1-tori are nested at the center of sets of 2-tori, i.e. doughnutshaped surfaces. The 2-tori in turn are nested within sets of 3-tori. Outside of these regions of regular
"C2
2 V, Fig. 2. Poincare surface o f section for P = 8, e n e r g y = 25591 c m - t.
motion are chaotic regions. Within the chaotic regions there can sometimes be identified multidimensional analogs of island chains, which have higherperiod regular regions in the centers of the islands. The influence of the critical orbits and the chaotic nature of the phase space are seen in Poincare surfaces of section, taken at different energies for a given polyad number by integrating Hamilton's equations numerically for the classical version of Hamiltonian (1). A classical surface of section so obtained is illustrated in Fig. 2 for energy 25591 c m - I and P = 8 .
3. Assigning quantum numbers The three types of stable periodic orbits for P = 8 define three sets of quantum numbers that we will attempt to assign, both for overtone states with all the energy concentrated (apart from zero-point motion) along the periodic orbit, and combination states with the energy distributed among the three modes. The assignments are for the quantum energy levels obtained by diagonalizing the quantum matrix Hamiltonian (1) for H20. The first type of assignment is denoted (n l, n2, nb)+ indicating n 1 quanta in local stretch 1,n 2 quanta in local stretch 2, and n b quanta in the bend, with + denoting symmetrization with respect to 1 and 2. The second type is denoted In s, n a, n b] indicating n s quanta in the symmetric stretch, n a quanta in the antisymmetric stretch, and n b quanta in the bend. The third type is denoted {hi, nRt, nR2} indicating n] quanta in a local stretch mode, na] quanta in resonant collective mode 1, and nR2 quanta in resonant collective mode 2. We use the Husimi dislribution to assign these quantum numbers to individual levels, focusing here on key illustrative cases. The Husimi distribution is the squared overlap between an eigenstate ~b and a coherent state I z) [27,28]. The coherent state used in the analysis to follow is defined [29-31] with respect to the SU(3) algebra associated with the three-mode H20 oscillator system. Coherent states were first defined as non-spreading wave packets in a harmonic potential, hence the best quantum analogs of classical particle motion. Non-spreading states are not known for general quantum systems. The SU(3) coherent states of Refs. [28-31] are an attempt to
Z.-M. Lu, M.E. Kellman / Chemical Physics Letters 247 (1995) 195-202
devise, as closely as possible, analogs of harmonic oscillator coherent states for anharmonic systems. The fact that a Lie algebra such as SU(3) is used is not surprising because the harmonic coherent states can themselves be defined algebraically. For wave functions of the Hamiltonian (1), projections of the Husimi distribution in action space show pronounced peaks, and projections in angle space often have distinct oscillatory structure similar to nodal patterns. This suggests that the Husimi distribution is locating quantizing structures such as invariant tori, or in the chaotic region, remnants of tori. We start with a level which will be assigned with normal mode stretch and bend numbers as [ns, r/a, r/b] = [1, 5, 4]. Fig. 3b shows the projection of the Husimi distribution into the angle space ~0~, qJ2- There is a clear double peak along ~0I. In terms of a 'dynamical barrier' or 'dynamical tunneling' model [18,32,33], this is an indicator of normal mode stretch dynamics, with a separate bend mode. The double peak indicates n s = 1 quantum in the symmetric stretch. Fig. 3a shows the projection into the action space of the two local stretches. This distribution peaks close to ~'l = ~'2 = 3. We have found in assigning pure two-mode stretch systems that equal actions in the local stretches are a further marker of normal mode stretch dynamics. There are 6 total stretch quanta, so n a = 5 quanta in the antisymmetric stretch. Since P = n s + n a -t- lr/b, this leaves 4 quanta in the bend, and the assignment is [n s, r/a, rib] = [1, 5, 41. Next, we assign a level which turns out to be very near the stretch-bend resonant collective mode critical orbit. Fig. 3c shows the projection of the Husimi distribution onto the action space. There is pronounced peaking at (6, 0) and (0, 6) quanta in stretches 1 and 2, respectively. This indicates stretches which are local in character. The projection into the angle space contains further information. Fig. 3d shows a single pronounced peak in if2. In the dynamical barrier picture of a 2 : 1 resonance, this indicates that one of the local stretches, that with 6 quanta, is strongly coupled to the bend, resulting in a resonant collective mode. The single peak in ~P2 indicates zero quanta in this resonant mode: n R , = 0. The other stretch mode is uncoupled from the bend and so is assigned as a local stretch quantum number
199
(a)
2
~. .: ;i~:::!:i~
(b)
1
2
8
)
(f)
0
~
n~
~
0
Fig. 3. (a) Projection of the Husimi distribution for the eigenstate at 26714.96 cm- ~ onto the action space of the two local stretches. (b) Projection onto the angle space. (c) Projection for the eigenstate at 25573.93 cm- ~ onto the action space. (d) Projection onto the angle space. (e) Projection for the eigenstate at 26351.11 cm-~ onto the action space. (f) Projection onto the angle space. with n~ = 0. This leaves the quantum number nR2 to assign for the second resonant mode coordinate. With P = 8 and the values nR, = 0, n I = 0, there are 16 quanta in the second resonant mode. The complete assignment is then {hi, n R , , nR~} = {0, 0, 16}. A final example shows a state that cannot very plausibly be assigned like the first three states. The action projection in Fig. 3e indicates local stretches with (6, I) and (1, 6) quanta. However, the angle projection in Fig. 3f shows no clear pattern that leads to an assignment with one of the sets (n l, n 2, rib) ÷ or {r/l, r/R,, r/R2}" We classify this state as 'chaotic' and give it the 'nominal' assignment (r/l, n2, nb)+ chaotic= (6, 1, 2) + chaotic. The nominal assignment of 'chaotic states' will be of use for grouping the spectrum into ordered sequences of levels.
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4. 'User-friendly' assignment: reconstructing sequences and progressions All 45 levels of the P = 8 polyad have been assigned with one of the four types of quantum numbers (n l, n 2, nb )+, [n s, Oa, /7b] , {/'/1, nR,, nR2}' (121 , /72, rib)+ chaotic. W e n o w use these assignments to organize the spectrum into ordered energy level sets. As far as possible, we want to devise these in analogy to the groupings recognized by molecular spectroscopists in less highly excited systems, while remaining cognizant of the greater complexity inherent in the fourfold assignment demanded by the bifurcated phase space. The usual organizational scheme for lower energy classifies sequences in the number of quanta of the normal modes. A typical procedure is to group the spectrum into sub-polyads of constant n b and hence total stretch quantum number g s t r = n s + ha, with sequences in the stretch, so that states in the subpolyad are labeled [n s, ha, rib] : [Nstr, 0, rib], [Nst r - 1, 1, /Tb]' etc. When the dynamics are perturbed by a single resonance coupling between stretches, this scheme can be extended [34] so that within the polyad, if there is a transition between local and normal modes, the stretch labels change from (n l, n 2, nh )+, to [n s, /Ta, Y/b]" We want to extend this scheme further to the nonintegrable system (1) including the 2:1 coupling, maintaining the pattern of the subpolyad arrangement as much as possible. We need to place the resonant states {nl, r/R] , RR2} and the chaotic states (nl, n2 ' rib)_+ chaotic into sub-polyads with fixed values of the number of bend quanta n b, or at least nominal values of n b. For the chaotic states we use the nominal assignments (n I , n 2, n h) ± chaotic. For the states assigned with resonant collective modes as {n 1, n R t , nR2}, the nominal n I and n 2 are obtained from the H u s i m i p r o j e c t i o n onto the action space described above. Then, the nominal n b is obtained from P. For example, the case /TR } = {0, 0, 16} discussed above becomes {/71' /Ta I' nbiz ± nominal .~_ (n 1, n2, (6, 0, 4) 5 nominal The entire spectrum can now be organized into sub-polyads, using the dynamical assignments ( h i , rt2, nh )±, [ns, /Ta, /Tb] for local and normal mode states, and the nominal assignments (n l, rt2, rib)± nominal and (n 1, n 2 , nb )± chaotic for resonant and chaotic states. The local and normal mode states are
found to be only slightly displaced from the levels of the sub-polyads of the integrable Hamiltonian H 0 + vstretch with only stretch-stretch coupling. The situa1:1 tion regarding the resonant states and chaotic states is more complicated. Fig. 4 shows these levels placed into sub-polyads according to the nominal values of their bend quantum number n b, as well as some levels of the subpolyad which have local or normal assignments. Fig. 4 also shows the corresponding members of the sub-polyads of H 0 + VlSt:r~tch. The resonant and chaotic states (labeled 'nom(inal)' and 'cha(otic)' in the assignments) fit into the basic subpolyad pattern of H 0 +--l:lS/stretch, but are substantially displaced. This shows the effects of the resonant and chaotic dynamics on the spectrum, as interpreted through the quantum number assignments that we have made. The sequences devised here for polyad P = 8 can easily be generalized to include progressions by including different polyads.
5. Summary and prospects In summary, in this Letter we have attempted a level-by-level analysis of a realistic molecular system, as a prototype of what we hope will be a useful procedure for spectral analysis of systems with mixed dynamics. We have shown the feasibility in a molecular system with a mixture of regular and chaotic dynamics of using the classical bifurcation structure to make approximate quantum number assignments, by means of examination of Husimi phase space representations of the wave functions. We have used these assignments to arrange the levels into regular patterns, analogous to sequences customary at low energy in the normal modes regime. Pronounced shifts occur for individual levels assigned as resonant or chaotic. The Hamiltonian (1) analyzed here is the spectroscopic fitting Hamiltonian optimized in Ref. [11] to give the best possible fit to the known energy levels of H20. It gives a good fit (for a Hamiltonian with such a modest number of parameters) to the known experimental levels. However, one may ask whether it is likely to be accurate for the very high levels contained in the P = 8 polyad analyzed here. This question is especially acute given the paucity of
Z.-M. Lu, M.E. Kellman / Chemical Physics Letters 247 (1995) 195-202
experimental information generally available for states of symmetric triatomics with significant bend excitation. Another important consideration is that H 2 0 is virtually certain to be above the barrier to the linear configuration when the molecule contains 16 quanta nominally in the bend. However, this is not likely to be a factor for the levels of primary interest here: those in Fig. 4 with 2, 4, and 6 nominal bend quanta, in the region strongly affected by interacting 1:1 stretch and 2:1 Fermi resonance and chaotic dynamics, with marked deviation from the pure stretch-stretch Hamiltonian. With this confidence in
201
the applicability of our most interesting results to H20, we believe the conclusions obtained for the effective spectroscopic Hamiltonian (I) give impetus to experimental efforts to access the bending region of phase space in symmetric triatomics. The work presented here for the H 2 0 fitting Hamiltonian shows that it is possible both in principle and in practice to achieve a rational level-by-level analysis and organization of spectra of systems with a significant mixture of regular and chaotic dynamics. A future application to molecules will focus on larger systems, e.g. acetylene C2H 2. Another
28000
E (cm-1) • --,-•
[080] [170]
•
[072]
+
[162]
,
(522)+-nom
[260]
27000
•
[064]
(620) +-
'
[154] [244]
•
(612)'+ cha • •
(514)*cha
•
[056]
"
[146]
:
(416) ÷
( 7 1 0 ) +-
26000
--(604)+-nom •
(702)+" nom _ _
25000
•
( 5 0 6 ) ÷ - nom
(800)+-
I
I
I
i
0
2
4
6
nb Fig. 4. Portion of the spectrum of H 2 0 for polyad P = 8, including the region corresponding to the most chaotic dynamics. Lines correspond to levels of the nonintegrable Hamiltonian of Eq. (1) with l : l stretch-stretch and 2 : ! stretch-bend resonance interaction. Diamonds correspond to levels calculated of an integrable Hamiltonian H 0 + V~t:r[tch, identical to Eq. (l) except that it contains just the l : 1 stretch-stretch interaction. Levels labeled by a + - correspond to pair of states, which may be too nearly degenerate to resolve in the figure. The levels are assigned as in the text and arranged into sub-polyads labeled by the value of the 'nominal' quantum number n b.
202
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untested application is to systems which cross potential barriers, i.e. undergo isomerization, with abovebarrier H20 the prototype.
Acknowledgement Support of the Department of Energy under Basic Energy Sciences Grant DE-FG06-92ER14236 is gratefully acknowledged. We would like to thank Dr. John P. Rose for helpful discussions and for assistance in preparation of the figures.
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[12] E. van der Aa, Celestial Mechanics 31 (1983) 163. [13] J.F. Sanders and F. Verhulst, in: Applied mathematical sciences series, Vol. 59. Averaging methods in nonlinear dynamical systems (Springer, Berlin, 1985). [14] M.E. Kellman, Dynamical analysis of highly excited vibrational spectra: progress and prospects, Molecular dynamics and spectroscopy by stimulated emission pumping, eds. H.-L. Dai and R.W. Field (World Scientific, Singapore, 1995). [15] W. Heisenberg, Z. Physik 33 (1925) 879, translated in Sources of quantum mechanics, ed. B.L. van der Waerden (Dover, New York, 1967). [16] A.P. Clark, A.S. Dickinson and D. Richards, Advan. Chem. Phys. 36 (1977) 63. [17] L. Xiao and M.E. Kellman, J. Chem. Phys. 90 (1989) 6086. [18] M.E. Kellman, J. Chem. Phys. 83 (1985) 3843. [19] Z. Li, L. Xiao and M.E. Kellman, J. Chem. Phys. 92 (1990) 225 I. [20] L. Xiao and M.E. Kellman, J. Chem. Phys. 93 (1990) 5805. [21] M.E. Kellman and L. Xiao, J. Chem. Phys. 93 (1990) 5821. [22] J. Svitak, Z. Li, J. Rose and M.E. Kellman, J. Chem. Phys. 102 (4340) 1995. [23] M.E. Kellman, J. Chem. Phys. 93 (1990) 6630. [24] L.E. Fried and G.S. Ezra, J. Chem. Phys. 86 (1987) 6270. [25] A. Weinstein, Inv. Math. 20 (1973) 47. [26] Z. Lu and M.E. Kellman, submitted for publication. [27] A. Perelomov, Generalized coherent states and their applications (Springer, Berlin, 1986). [28] W.-M. Zhang, D.H. Feng and R. Gilmore, Rev. Mod. Phys. 62 (1990) 867. [29] P. Leboeuf and A. Voros, J. Phys. A 23 (1990) 1765. [30] G. Wu, Chem. Phys. Letters 179 (1991) 29. [31] M.B. Cibils, Y. Cuche, P. Leboeuf and W.F. Wreszinski, Phys. Rev. A 46 (1992) 4560. [32] M.J. Davis and E.J. Heller, J. Chem. Phys. 75 (1981) 246. [33] E.L. Sibert III, J.T. Hynes and W.P. Reinhardt, J. Chem. Phys. 77 (1982) 3583. [34] M.E. Kellman and L. Xiao, Chem. Phys. Letters 162 (1989) 486.