J. Phys. Chem. A 2010, 114, 9825–9831
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Bifurcation Phase Diagram for C2H2 Bending Dynamics Has a Tetracritical Point with Spectral Patterns† Vivian Tyng and Michael E. Kellman* Department of Chemistry and Institute of Theoretical Science, UniVersity of Oregon Eugene, Oregon 97403 ReceiVed: April 1, 2010; ReVised Manuscript ReceiVed: May 24, 2010
Critical points and bifurcations are considered for the acetylene effective Hamiltonian in the polyad space of total bend and vibrational angular momentum quantum numbers [Nb, l ]. A “phase diagram” is constructed for the surface of minimum energy critical points. The phases denote vibrational modes of different character, including new types of anharmonic modes born in bifurcations from the ordinary normal modes. A tetracritical point is the outstanding feature of the diagram. Patterns in the energy levels are considered. We start with a search for “defects” in the lattice of energy levels, similar to what is found with quantum monodromy in integrable systems. No such pattern is found. Instead, patterns are predicted in the first and second derivatives of the energy with respect to the quantum numbers, by close analogy to the theory of phase transitions. The tetracritical point finds striking manifestation in these patterns. This may be amenable to experimental test. Moreover, the minimum energy path, analogous to the reaction path, has derivative patterns that can already be compared with experiment. Agreement of theory and experiment is found within experimental error. I. Introduction It is clear after several decades of concentrated work by many investigators that there is an elaborate structure in the phase space of highly excited rotation-vibration dynamics of molecules. An accessible overview is presented in ref 1. The present article considers an example that arises in the bending dynamics of C2H2 with vibrational angular momentum l g 0: the bifurcation structure and “phase diagram” for certain states central to the acetylene-vinylidene isomerization process, namely, the “minimum energy surface of critical points in the polyad space”,2,3 reviewed in technical detail in Section II. This surface contains a minimum energy path that we have identified3 with the reaction path for the isomerization. These are the l ) 0 states, treated exhaustively in ref 4. A bifurcation takes place along the reaction path, a transition from trans to local bending motion. In moving along the reaction path, it therefore is necessary for the system to switch from normal to local mode motion, that is, to make the transition between phases. Extending to l g 0 gives the minimum energy reaction surface. This has a complex structure of bifurcations, as evidenced by a tetracritical point, where four kinds of modes or phases come together, as described in this article. The tetracritical point has a direct analogy to the theory of phase transitions,5 with a concrete occurrence in spin systems in complex magnetic materials first explored long ago6,7 and realized more recently in unconventional superconductors.8 The role of this kind of phase space structure in both spectroscopy and reaction dynamics is just beginning to be explored. In this article, we investigate patterns among the large numbers of levels in the lattice of states on the l g 0 phase diagram. One to check for is a pattern of “defects” in the lattice energy pattern, similar to that seen in the phenomenon of quantum monodromy, an interesting phenomenon in integrable systems associated with intramolecular potential barriers and also bifurcations in Fermi resonance. (Note that our system is †
Part of the “Reinhard Schinke Festschrift”. * Corresponding author. E-mail:
[email protected].
nonintegrable, so there is no question of actual monodromy in any usual sense.) We will see that no such monodromy-like pattern is associated with the C2H2 bending system and its tetracritical point. Instead, there are other more subtle patterns, potentially observable by experiment, in first and second derivatives of the energy with respect to the quantum numbers taken along natural directions of the lattice. We previously reported2 a moment of inertia backbending effect that involves certain functions of derivatives along one direction. However, at that time, we were not aware of the central organizing structure of the tetracritical point, which only became apparent subsequent to the work reported in ref 3. Now the pattern of derivatives is examined in the vicinity of the tetracritical point and also along the l ) 0 minimum energy reaction path. Discontinuities are found in the second derivatives. Such behavior is associated with bifurcations that occur in the Landau theory of second-order phase transitions.5,9 In fact, the molecular behavior we are discussing can be considered to be an example of quantum phase transitions in a molecular system. These are zero-temperature phenomena; that is, there is no entropy term in the potential, and consideration is taken of the critical points of the energy alone. This point of view has been emphasized especially by Iachello and coworkers.10,11 II. Minimum Energy Critical Points in the Polyad Space In a series of papers,1-4 we have explored acetylene (C2H2) pure bending dynamics through the method of critical points bifurcation analysis12 of the experimentally constructed13,14 spectroscopic fitting Hamiltonian. We use a classical version of the spectroscopic Hamiltonian, suitably reduced in dimension by taking into account conserved polyad numbers. The procedure first transforms, via the Heisenberg correspondence principle,15-17 the quantum effective Hamiltonian to a classical form in action-angle variables. A canonical transformation is then performed, which reduces the effective dimension of the phase space by projecting out the constants of motion corresponding to the polyad numbers Nb ) n4 + n5 and l ) l 4 + l 5. For a polyad effective Hamiltonian, the problem of finding the
10.1021/jp102957u 2010 American Chemical Society Published on Web 06/21/2010
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Figure 1. Phase diagram of minimum energy critical point families in the space of polyad numbers Nb, l . The four zones join at the tetracritical point.
anharmonic normal modes and new modes born in bifurcations reduces to that of finding the critical points of the reduced Hamiltonian: there is no need for numerical integration of Hamilton’s equations. The mathematical and numerical details of the critical points analysis are found in ref 3 and previous papers cited there. The critical points generally correspond to motions on invariant 2-tori (or periodic orbit 1-tori when l ) 0) and denote the key features in the nonlinear phase space flow; these are the normal modes and new types of anharmonic modes born in bifurcations. The families of critical points for C2H2 bending fall into six classes based on their physical character: normal (N); local (L); orthogonal (O); precessional (P); counterrotator (CR); and off-great circle (OGC). There is always one critical point that is the lowest energy state of its polyad3 corresponding to a stable vibration mode, and our present concern is with these. The lowest energy critical points belong to one of the character classes N, L, O, and OGC. The state space can be represented by a lattice of points, each characterized by its quantum numbers [Nb, l ] and one of these classes. These critical points form the minimum energy surface in the polyad space. As discussed in ref 3, the minimum energy path is an ascending trough of stable critical points along l ) 0. This is naturally related to the idea of an isomerization reaction path. III. Phase Diagram and Tetracritical Point The set of minimum energy critical points in the polyad space [Nb, l ] form a kind of phase diagram when divided into the four classes of different dynamical character. Figure 1 shows the phase diagram in the range Nb ∈ [0,38], l ∈ [0,20]. (Another related view for a wider range of quantum numbers will be considered later in relation to Figure 2.) Because of the physical constraint l e Nb + 2, a portion of the upper-left corner of the [Nb, l ] plane is not physical (indicated in white in Figure 1). The remaining area is partitioned into five zones with the two different normal mode families and three of the new families L, O, and OGC. The immediately evident outstanding feature of the phase diagram is the tetracritical point where the families N, L, O, and OGC meet. There is a parallel with tetracritical behavior in magnetic systems6,7 and unconventional superconductors,8 where there are two ordered regions and two transitional regions, the latter designated as disordered and “mixed” order; see the figures in ref 7. The boundaries between zones of the phase diagram Figure 1 are defined by bifurcations in which the character of the lowest-energy critical point changes. We will return to these bifurcations after an exploration of the possibility of a monodromy-like energy pattern of the phase diagram.
Figure 2. Energy-Momentum EM map of quantum states. Each arc of states corresponds to levels with the same Nb value.
IV. Absence of a Monodromy-Like Pattern In this section, we describe the phenomenon of quantum monodromy and its spectral pattern in integrable systems and indicate why it might be relevant to look for similar patterns in the phase diagram of the nonintegrable acetylene bends system. We find that it is not present, and consider in the following section what patterns we should instead look for. The phenomenon of monodromy in systems with several degrees of freedom has been an active topic of research in nonlinear dynamics.18-30,11 In quantum systems, monodromy appears as a visible “lattice defect” or “fracture” in a plot called an energy-momentum (EM) map of energy levels versus a quantum number, such as the vibrational angular momentum l considered in the present work. Underlying monodromy is a “topological obstruction” to constructing a consistent global set of action-angle variables. In molecular cases, monodromy often occurs in systems with multiple wells (“champagne bottle” potential19) including isomerization systems such as LiCN.26 The topological defect can also be associated with bifurcations. In this case, the barrier is dynamical rather than due to the molecular potential. The connection with bifurcations is seen very clearly in work of Cooper and Child,20,21 who examine the 2:1 Fermi resonance between stretch and bend. They relate monodromy and bifurcations in the following way. We had previously analyzed the standard Fermi resonance effective Hamiltonian for the 2:1 system with l ) 0 in terms of bifurcations31 and also a catastrophe map analysis32,33 of the bifurcations. Cooper and Child generalize the bifurcation and catastrophe map analysis to l g 0. They make EM maps of quantum energies versus l at fixed Nb. These show monodromy, and it is very clear how this is related to the bifurcation and catastrophe map analysis. See a recent review by Child21 for a comprehensive view of this work and many other topics in monodromy in molecules. The acetylene bends minimum energy surface with l g 0 is analogous in certain respects to the 2:1 Fermi resonance system. We have the same quantum numbers Nb and l . It can certainly be objected that monodromy, at least as presently understood, hinges on the existence of action-angle variables, which are defined for integrable systems, whereas the acetylene bends system is definitely not integrable, being a multiresonance system.3,4 Furthermore, we are concerned with the minimum energy states of each polyad; there is nothing like this in
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Figure 3. Schematic of the pitchfork-reverse pitchfork bifurcations in the L f OGC f O sequence.
previous studies of monodromy. The minimum energy surface is defined by restriction to the lowest energy critical point of each polyad. This in essence reduces the number of degrees of freedom to two, so that Nb and l constitute a complete set of quantum numbers. It thus seems reasonable to look for a monodromy-like pattern in our system, if only as a check on whether such an unexpected new pattern exists. Figure 2 plots as an EM map the energies of the quantum lattice points of the minimum energy surface as a function of l . It is quite evident in Figure 2 that no sign of a monodromylike fracture is visible around the tetracritical point, zone boundaries, or anywhere else. We will next examine the nature of the bifurcations that give rise to the phases of Figure 2 and then consider what this tells us about the patterns we should instead look for. V. Nature of the Bifurcations and the Analogy to Second-Order Phase Transitions Essential for what follows is the nature of the bifurcations involved in the transformations on the phase diagram. A boundary between zones is defined by a bifurcation in which the character of the critical point changes. At the zone boundaries in Figure 1, a pitchfork bifurcation occurs (or inverse pitchfork bifurcation in the reverse path), with the following linear stability change
(EE) T (EH) + 2(EE)
(1)
Such a bifurcation occurs if and only if one or both pairs of eigenvalues of the stability matrix vanish. At the point on the phase diagram where four segments converge, both pairs of the stability eigenvalues vanish. In the terminology of phase transitions, this is a tetracritical point.5 It is very illuminating to consider in detail the bifurcation behavior for a specific path in the vicinity of the tetracritical point. In a previous paper,2 we examined energy level patterns associated with particular sequences L f OGC f O of bifurcations. These sequences are vertical cuts with fixed Nb, crossing three of the zones on the right side of the tetracritical point in Figure 1. The bifurcation behavior is illustrated in the schematic diagram of Figure 3. At fixed Nb, increasing l from zero induces two bifurcations. At first, the L mode is the minimum energy critical point with (EE) linear stability. Then, the L mode undergoes a pitchfork bifurcation to give two OGC stable (EE) modes, as the L family becomes (EH), as in eq 1. The OGC are transitional modes, as seen in Figure 3. At even higher l values, there is an inverse-pitchfork bifurcation in which the transitional OGC modes merge and disappear into an O mode, which takes over as the minimum energy critical point with (EE) stability. The physical nature of this bifurcation
Figure 4. First-order derivative of energy ∂E/∂l for the EM map.
sequence L f OGC f O, including the transitional OGC modes in Figure 3, is examined in ref 2 in connection with modulation of the moment of inertia of the molecule along the bifurcation sequence. In connection with the tetracritical point, it is significant that the transitional purple branch of the bifurcation diagram in Figure 3 has the OGC character of the transitional region of the phase diagram, Figure 1. This is analogous to the magnetic systems with tetracritical point, where there are transitional regions between two regions of differently ordered magnetization; see the figures in ref 7. The nature of the bifurcations in Figure 3 enables us to understand the lack of any possibility, apart from the nonintegrability of our system, as discussed above, of a monodromylike spectral pattern. As becomes clear, especially from the work of Cooper and Child,20,21 monodromy in bifurcation systems is closely related to energy level patterns connected to spectral detection of a separatrix.34 However, Figure 3 shows that in the bifurcations of the present work, we are following a continuous evolution of stable fixed points, avoiding separatrices (except at the isolated points where the stable mode branches into the pitchfork). Therefore, there is no possibility of a monodromy-like pattern, which is consistent with our observations. We will instead pursue another direction suggested by the phase diagram of Figure 1. Bifurcation occurs at the crossing of a phase boundary, where the dynamical character of the vibration changes. A bifurcation is analogous to a phase transition, in which there is a change in the critical points of a free energy function. In the theory of phase transitions,5,9 a pitchfork bifurcation, the type we have here, is characteristic of a second-order transition. The derivatives of the free energy with respect to a control parameter, most often the temperature, show characteristic behavior: a crease in the first derivative ∂G/ ∂T and a jump discontinuity in the second derivative ∂2G/∂T2. Therefore, we will be led to look at derivatives of the energy in various directions along the lattice of energy levels. Rather than the temperature, the actions Nb, l are the control parameters with respect to which derivatives will be taken. We will be particularly interested in the pattern of derivatives in the
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Figure 5. Second-order derivatives of semiclassical energy with continuous energy (left) and second differences of quantum energies with ∆Nb, ∆l )1 (right). The continuous color shading encodes energy. The color contours of regions match the color coding of the phase diagram Figure 1.
vicinity of the tetracritical point, and also in the normal-to-local mode transition along the minimum energy isomerization path. VI. Derivatives on the Minimum Energy Surface Two evident natural directions in which to take derivatives of the energy are variation of l with Nb fixed and variation of Nb with l fixed. In the EM map Figure 2, these correspond, respectively, to vertical cuts with constant l and cuts along “arcs” of constant Nb. Taking the first derivative of the energy function with regard to l at fixed Nb results in the asymmetric “airplane” shown in Figure 4. Evident features are the creases where the first derivative curve remains continuous, but its direction undergoes an abrupt change, analogous to the change in the free energy function at a second-order phase transition. This prompts a look for jump discontinuities in the second derivatives.
Figure 5 shows the second derivatives ∂2E/∂Nb2, ∂2E/∂l 2, and ∂2E/∂Nb∂l in the vicinity of the tetracritical point. The lefthand set of Figures shows continuous derivatives for the semiclassical system. The right-hand set shows the finite difference approximation
∂2E/∂N2b ≈ E(Nb + 1, l ) + E(Nb - 1, l ) - 2E(Nb, l )
(2) for the potentially experimentally observable quantum spectra with p ) 1. Very distinctive are the “waterfall” patterns of discontinuity at the boundaries of the phase zones. The Figures with p ) 1 show the predicted experimental results as solid dots. Clearly, if these minimum energy critical point levels can be measured, then striking patterns in the second derivatives
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Figure 6. Second-order derivative ∂2E/∂φ2 with regard to the azimuthal angle φ.
Figure 7. Second-order derivative ∂2E/∂Nb with constant l .
should be found if the spectroscopic Hamiltonian has validity at these high Nb,l values. In Figure 6, we take the second derivative ∂2E/∂φ2 going around the tetracritical point in a cycle with angle φ, again with striking discontinuities at the phase boundaries. Figure 7 plots the second derivative ∂2E/∂Nb2 taken for various constant l , starting from l ) 0. Most relevant experimentally is l ) 0 to 2, where data already exist13 and in fact are one of the primary sources of the fitting Hamilton. This is especially significant because the l ) 0 series is the minimum energy path in the polyad space and hence corresponds to the reaction path for the isomerization in the bending space. The prediction is for a steep drop in the second derivative at the transition from
Figure 8. Collation of moment of inertia backbending curves from ref 2. The Figure plots a suitably defined moment of inertia versus square of rotational frequency for an asymmetric structure of the vibration C2H2 molecule.
normal to local modes; see the following section for a comparison with experiment. For higher values of l , yet more interesting discontinuities are evident, some of them already present in Figure 5 Finally, we wish to compare with the analysis here the moment of inertia backbending effect described in ref 2. This considers a suitably defined moment of inertia I as a function of angular momentum excitation l for sequences with fixed Nb. Figure 8 collects the backbending curves of ref 2. The closest parallel to what we have done in the present work is the second derivatives ∂E2/∂l 2 in Figure 5 (top panels). However, the two
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Tyng and Kellman Also note that one level (Nb ) 4) has not been observed experimentally but is calculated from the fitting Hamiltonian. The first two data points in Figure 9 using this level are therefore plotted in a distinct color. The experimental uncertainty is estimated in ref 13 to be 2σ ) 3 cm-1. We use this estimate for all of the levels in Table 1. In calculating the finite difference using eq 3, the uncertainty is propagated36 as 2σ ) 2(ΣiA2i σ2i )1/2 ) 1.84 cm-1. This number is plotted in Figure 9 as error bars. VIII. Conclusions
Figure 9. Comparison of predictions from Figure 7 with second differences from experimental data of Table 1 for l ) 0. The experimental second differences are shown as open squares with 2σ error bars calculated from experimental error propagation. Note that one level (Nb ) 4) has not been observed experimentally but is calculated from the fitting Hamiltonian. The first two data points in Figure 9 using this level are therefore plotted in a distinct (blue) color. The closed circles are predictions from the levels of the quantum fitting Hamiltonian.
TABLE 1: Lowest Energy Members of Some Bend Polyads and Their Second-Order Finite Differences, Used in Constructing Figure 9 Nb E (cm-1) ∆2E/∆Nb2 2 4 6 8 10 12
1230.4 2487.5 3770.3 5069.7 6387.8 7697.4
6.42 4.15 4.25 -1.28
reference obsd ref 13 unobserved, calculated from fit in ref 13 obsd ref 13 obsd ref 35 obsd ref 35 obsd ref 35
kinds of Figures are considerably different. The analysis here and the moment of inertia analysis of ref 2 are both tools that should be useful in future interpretation of experimental spectra. But now we turn to a presently feasible confrontation of the theory with existing experiments. VII. Comparison with Experiment Figures 5-7 for the second derivatives show striking predictions that would be very interesting to confront with experiment. For the time being at least, the levels involved in these predictions are mostly not amenable to experimental observation. The spectroscopic Hamiltonian from which they were obtained represents a huge extrapolation from experiment: it is based on data13 restricted to l ) 0, 1, 2, 3 for Nb values over a wide but still restricted range, but in fact, a first comparison with experiment is possible for some of the predictions of Figure 7. This is shown in Figure 9 for the second derivatives (actually second differences for finite Planck’s constant p ) 1) of l ) 0 states starting from the ground state. Within the given experimental error, theoretical predictions agree with experiment. It is interesting in Figure 9 how the second differences differ from the smooth second derivatives of the semiclassical limit. Figure 9 was obtained as follows. The reported lowest members in each of the bending polyads Nb ) 2, 4, 6, 8, 10, and 12 are listed in Table 1 as well as the second-order differences
In this article, we have explored the phase diagram for pure C2H2 bending states that are each the minimum energy state for given values [Nb, l ] of the total bend quantum number and vibrational angular momentum projection. These states form the minimum energy surface in the polyad space {Nb, l }. The phases are different classes of anharmonic modes, including normal modes and new modes born in bifurcations, first examined in previous work3,4 on the critical points bifurcation analysis of the C2H2 spectroscopic fitting Hamiltonian. The outstanding feature investigated here is the tetracritical point of the phase diagram. A previous report2 described a moment of inertia backbending effect along a particular bifurcation sequence on the minimum energy surface. The present work examines further kinds of energy level patterns associated with the global structure of the minimum energy critical points phase diagram, including the tetracritical point and the normal-to-local bending mode transition along the minimum energy isomerization reaction path. A search turns up not a trace of a monodromy-like spectral pattern. Consideration of the nature of the bifurcations involved in the minimum energy critical points states suggests instead an exploration of derivatives of the energy levels of the phase diagram based on a close analogy to the Landau theory of second-order phase transitions. There is a parallel between the bifurcations observed in the vicinity of the tetracritical point and the transition between regions in magnetic systems with a similar phase diagram. Second derivatives of the spectral energies taken in natural directions in the quantum number space show striking patterns that should be observable if the levels can be accessed by experiment. In fact, experiment and theory agree for the data available, within experimental error. The importance of the critical points analysis for the perspective here cannot be overemphasized: without the aid of the bifurcation analysis, one would never understand and probably not even suspect either the tetracritical point or the energy level patterns found here. This work adds to a growing body of exploration of energy patterns associated with phase space structure of large numbers of quantum molecular states, including the pattern of a dip in the energy level spacings at a separatrix,34 quantum monodromy, and the moment of inertia backbending effect.2 Acknowledgment. We would like to thank Dr. John Hardwick for comments on the propagation of experimental errors in the second difference calculations reported here. This work was supported by the U.S. Department of Energy Basic Energy Sciences program under contract DE-FG02-05ER15634. References and Notes
∆2E(Nb) ∆N2b
E(Nb + 2) + E(Nb - 2) - 2E(Nb) ) 4
(3)
The factor of four in the denominator is obtained from applying the interval ∆Nb ) 2 of the experimental data in eq 3.
(1) Kellman, M. E.; Tyng, V. Acc. Chem. Res. 2007, 40, 243–250. (2) Tyng, V.; Kellman, M. E. J. Chem. Phys. 2007, 127, 041101. (3) Tyng, V.; Kellman, M. E. J. Chem. Phys. 2009, 131, 244111. (4) Tyng, V.; Kellman, M. E. J. Phys. Chem. B 2006, 110, 18859– 18871. (5) Landau, L. D.; Lifshitz, E. M. Statistical Physics, 3rd ed.; Butterworth-Heinemann: Oxford U.K., 1984.
Bifurcation Phase Diagram for C2H2 Bending Dynamics (6) Fischer, M. E.; Nelson, D. R. Phys. ReV. Lett. 1974, 32, 1350– 1353. (7) Aharony, A. Phys. ReV. Lett. 1975, 34, 590–593. Aharony, A.; Fishman, S. Phys. ReV. Lett. 1976, 37, 1587–1590. (8) Park, T.; Ronning, F.; Yuan, H. Q.; Salamon, M. B.; Movshovich, R.; Sarrao, J. L.; Thompson, J. D. Nature 2006, 440, 65–68. (9) Heyde, K. L. G. Basic Ideas and Concepts in Nuclear Physics, 3rd ed.; Institute of Physics Publishing: Bristol, U.K., 2004. (10) Iachello, F.; Pe´rez-Bernal, F. J. Phys. Chem. A 2009, 113, 13273– 13286. (11) Caprio, M. A.; Cejnar, P.; Iachello, F. Ann. Phys. 2008, 323, 1106– 1135. (12) See ref 3 and sources cited therein. (13) Jacobson, M. P.; O’Brien, J. P.; Silbey, R. J.; Field, R. W. J. Chem. Phys. 1998, 109, 121–133. (14) Abbouti Temsamani, M.; Herman, M. J. Chem. Phys. 1995, 102, 6371–6384. (15) Heisenberg, W. Z. Phyz. 1925, 33, 879. Translated in Sources of Quantum Mechanics; van der Waerden, B. L., Ed.; Dover: New York, 1967. (16) Child, M. S. Semiclassical Mechanics with Molecular Applications; Clarendon Press: Oxford, U.K., 1991; Chapter 4.2. (17) Clark, A. P.; Dickinson, A. S.; Richards, D. AdV. Chem. Phys. 1977, 36, 63–139. (18) Cushman, R. H.; Bates, L. M. Global Aspects of Classical Integrable Systems; Birkha¨user: Basel, Switzerland, 1997. (19) Child, M. S. J. Phys. Chem. A 1998, 31, 657–670. (20) Cooper, C. D.; Child, M. S. Phys. Chem. Chem. Phys. 2005, 7, 2731–2739. (21) Child, M. S. AdV. Chem. Phys. 2007, 136, 39–94.
J. Phys. Chem. A, Vol. 114, No. 36, 2010 9831 (22) Sadovskii, D. A.; Zhilinskii, B. I. Phys. Lett. A 1999, 256, 235– 244. (23) Sadovskii, D. A.; Cushman, R. H. Physica D 2000, 142, 166–196. (24) Kozin, I. N.; Roberts, R. M. J. Chem. Phys. 2003, 118, 10523– 10533. (25) Waalkens, H.; Junge, A.; Dullin, H. R. J. Phys. A 2003, 36, L307– L314. (26) Joyeux, M.; Sadovskii, D. A.; Tennyson, J. Chem. Phys. Lett. 2003, 382, 439–442. (27) Efstathiou, K.; Joyeux, M.; Sadovskii, D. A. Phys. ReV. A 2004, 69, 032504. (28) Arangoa, C. A.; Kennerly, W. W.; Ezra, G. S. Chem. Phys. Lett. 2004, 392, 486–492. (29) Cushman, R. H.; Dullin, H. R.; Giacobbe, A.; Holm, D. D.; Joyeux, M.; Lynch, P.; Sadovskii, D. A.; Zhilinskii, B. I. Phys. ReV. Lett. 2004, 93, 024302. (30) Sadovskii, D. A.; Zhilinskii, B. I. Mol. Phys. 2006, 104, 2595– 2615. (31) Li, Z.; Xiao, L.; Kellman, M. E. J. Chem. Phys. 1990, 92, 2251– 2268. (32) Xiao, L.; Kellman, M. E. J. Chem. Phys. 1990, 93, 5805–5820. (33) Kellman, M. E.; Xiao, L. J. Chem. Phys. 1990, 93, 5821–5825. (34) Svitak, J.; Li, Z.; Rose, J.; Kellman, M. E. J. Chem. Phys. 1995, 102, 4340–4354. (35) Abbouti Temsamani, M.; Herman, M.; Solina, S. A. B.; O’Brien, J. P.; Field, R. W. J. Chem. Phys. 1996, 105, 11357–11359. (36) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing; Abramowitz, M., Stegun, I. A., Eds.; Dover: New York, 1970.
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