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Regular Dynamics in Transition States with Flat Saddles J. C. Lorquet* Department of Chemistry, University of Liege, Sart-Tilman (B^atiment B6), B-4000 Liege 1, Belgium ABSTRACT: In the vicinity of a transition state, the dynamics is constrained by approximate local invariants of the motion even if the potential energy surface is anharmonic. The concept of local regularity near a saddle point is investigated in the framework of classical mechanics. The dynamics along the reaction coordinate decouples locally into a reactive mode and several bounded degrees of freedom. The partial energy stored in the unbounded mode is adiabatically invariant. Starting from a purely harmonic situation at the saddle point, anharmonicity coefficients are observed to come into play in a sequential way in the laws of motion. In most cases, each kind of anharmonic coefficient can be related to a particular feature of the potential energy surface or of the reaction path. These regularities account for previous classical trajectory calculations by Berry and co-workers, who observed that for flat saddles (i.e., those characterized by a low value of the modulus of the imaginary frequency), trajectories become temporarily collimated and less chaotic during passage through the transition state.
1. INTRODUCTION The classical theory of reaction dynamics is dominated by the concept of transition state.13 In the long history of the latter, saddle points play a leading role. Consider a region of a potential energy surface in the immediate neighborhood of a first-rank saddle point. For a system consisting of N harmonic oscillators in addition to the reactive (i.e., negatively curved) mode, the zeroth-order Hamiltonian1,2,4 H 0 ðp, qÞ ¼
1 2 1 ðps ω2s s2 Þ þ 2 2
N
∑j ðpj2 þ ωj 2qj2 Þ
ð1.1Þ
is separable and the dynamics is regular. In this equation, qj is the mass-weighted jth normal coordinate and pj its conjugate momentum; ωj is equal to the frequency of the jth mode (multiplied by 2π). The index s is reserved for the unbounded coordinate and ωs is the modulus of the pure imaginary frequency. At the saddle point, all of the coordinates are equal to zero and the potential energy vanishes. Of course, the actual potential energy surface is anharmonic and a correction has to be added to H0:58 H anh ðp, qÞ ¼ H 0 ðp, qÞ þ Csss s3 þ
Csjk sqj qk þ ∑ Cjkl qj qk ql þ ::: ∑j Cssj s2 qj þ ∑ j, k j, k , l
ð1.2Þ Decoupling of the reaction coordinate from a bath of oscillators for a nonseparable Hamiltonian was investigated early on by Miller.5,9,10 Marcus11 has suggested that vibrational adiabaticity might be local, i.e., good only in some limited region around the transition state. This idea has been substantiated by further research carried out by Wales, Berry, and Hinde on chaotic dynamics in rare gas r 2011 American Chemical Society
clusters.1214 These authors have analyzed in great detail the concept of local regularity in the vicinity of a saddle point. They were able to derive an interesting correlation between dynamics and a specific feature of the potential energy surface. They observed that for flat saddles (i.e., those characterized by a low value of the modulus of the imaginary frequency ωs), trajectories become temporarily collimated and less chaotic during passage through the transition state. By contrast, steep saddles (i.e., those that are sharply pinched) were found to be less efficient at decoupling vibrational modes from one another. In a second series of papers, Komatsuzaki and Berry68,15 further analyzed the concept of local regularity in the vicinity of a saddle point and developed additional convincing arguments in favor of the existence of local approximate invariants of motion associated with the reaction coordinate. Using Lie canonical perturbation theory, they constructed a nonlinear transformation to a hyperbolic coordinate system, which revealed the persistence of such invariants, at least in the region around a saddle point. Recently, a detailed mathematical analysis by Wiggins and coworkers1618 has provided a firm mathematical framework for the problem. The use of a special set of phase space coordinates, denoted normal form coordinates, was shown to greatly simplify the Hamiltonian in the neighborhood of a saddle. Three conditions are necessary for the dynamics to decouple into a reaction coordinate and bath modes. The first one is nonresonance between the frequencies of the bath modes. The second condition requires the anharmonic perturbation to be smooth, i.e., expressed as a differentiable function, e.g., as a polynomial of arbitrary order. Third, the perturbation should not be too strong. It then follows that approximate action integrals can be Received: February 7, 2011 Revised: April 6, 2011 Published: April 15, 2011 4610
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The Journal of Physical Chemistry A associated with each mode. It should be noted, however, that normal form coordinates are nonlinear functions of the original configuration space and associated momentum coordinates. The reverse of the medal, therefore, is that all calculations must be carried out in phase space, not in configuration space. We mention in passing an important but controversial matter showing that all problems concerning the local decoupling of the reaction coordinate from the bath at the saddle point are not solved. The RiceRamspergerKasselMarcus (RRKM) microcanonical transition state theory13 predicts that rate constants for unimolecular dissociation reactions should increase in a stepwise manner as the energy increases through each quantized transition-state energy level. Such steps have indeed been experimentally observed close to the energy threshold19,20 and interpreted as evidence of vibrational adiabaticity: “Energy is tied up in vibrational modes orthogonal to the reaction coordinate during the brief passage through the transition state region”.20 However, serious questions remain unanswered.2123 In particular, when examined in the light of quantum theory, a detailed analysis of these experiments reveals inconsistencies with quantum dynamical calculations of unimolecular dissociation, which predict fluctuations of decay rates over several orders of magnitude, especially at the dissociation threshold.21,22 Summing up, there is no denying that the general theoretical framework of reaction dynamics must be constructed in the full phase space. Yet most chemists have not given up the hope to infer dynamical phenomena from the topography of the potential energy surface, at least in some particular cases. The underlying hope is that in special cases some separation might be found between momenta and spatial coordinates. A well-known example is provided by the existence of adiabatic invariants,24 which are characterized by the fact that the translational momentum pr factorizes out of the expression of the Poisson bracket with the Hamiltonian. The present paper tries to find an additional instance. It is not inconsistent to relate the perturbation that results from anharmonicity to a specific feature of the potential energy surface because both are configuration space notions. With that purpose in mind, we come back to the observation that saddles that are relatively flat along the reactive mode s (i.e., that are characterized by a low value of the modulus of the imaginary frequency ωs) tend to regularize the dynamics close to the saddle point.1214 Wales, Hinde, and Berry have carried out extensive classical trajectory calculations on numerous potential energy surfaces obtained by summing Lennard-Jones or Morse pairwise potentials of various ranges. The dynamical effect was ascertained but the origin of the correlation between the frequency ωs and the regular nature of the dynamics could not be clearly determined by these calculations. We aim at shedding light on this problem. The present paper is organized as follows. In section 2, the basic equations that concern the dynamics along the reaction coordinate are derived. The specific role of each anharmonicity coefficient is studied in section 3. In section 4, we consider an analytical force field and show that the modulus of the imaginary frequency is directly connected to an important coupling coefficient and thus influences the regularity of the motion near the saddle. A general discussion follows in section 5.
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obeys closed-form equations: sharm ðtÞ ¼
qharm ðtÞ ¼ qj ð0Þ cosðωj tÞ þ j
pj ð0Þ sinðωj tÞ ωj
ð2.1aÞ ð2.1bÞ
The solution for the momenta can be obtained by differentiating these equations. Anharmonic trajectories can be obtained as a formal series solution by repeated application of the Poisson bracket equation24 •
X ¼ ½X, H
∑j
DH DX DH DX Dpj Dqj Dqj Dpj
!
! DH DX DH DX ð2.2Þ þ Dps Ds Ds Dps
which gives the rate of change of any dynamical variable X. For the reaction coordinate s, one has24 ! t2 sðtÞ ¼ s0 þ t½s, H0 þ ½½s, H, H0 2! ! t3 þ ð2.3Þ ½½½s, H, H, H0 þ ::: 3! When applied to the anharmonic Hamiltonian Hanh (eq 1.2), this formula leads to 1 1 sanh ðtÞ ¼ sharm ðtÞ As t 2 Bs t 3 þ ::: 2 6
ð2.4Þ
with As ¼ Bs ¼ 2ps ð0Þ
∑j k∑g j Csjk qjð0Þqk ð0Þ
ð2.5Þ
∑j Cssj qjð0Þ þ ∑j ∑k Csjk ½pjð0Þqkð0Þ þ pkð0Þqjð0Þ ð2.6Þ
In practice, when applied to eq 1.2, this expansion can be carried out up to the cubic term only. The analytical expression of the nest of Poisson brackets becomes extremely complex in the quartic term. The conjugate momentum can be derived from Hamilton’s first canonical equation applied to Hanh: panh s ðtÞ ¼
DH anh dsanh ¼ Dps dt
ð2.7Þ
Similar formulas can be derived in the same way for bounded motions. However, they are much more complicated and will not be reported here. The reliability of these analytic laws of motion has been confirmed by numerical integration of Hamilton’s canonical equations for some reasonable force fields. B. Partial Energies. The harmonic Hamiltonian H0 is separable and the partial energy stored in the unbounded mode
2. INFLUENCE OF ANHARMONICITY ON THE DYNAMICS A. Equations of Motion. The initial conditions are chosen so that s = 0 at time t = 0. Then, in the harmonic limit, the motion
ps ð0Þ sinhðωs tÞ ωs
Es ¼
1 2 ðps ωs 2 s2 Þ 2
ð2.8Þ
is related to an action integral in a simple way, even if the motion is unbounded,5,9,10 and is therefore an integral of the motion. 4611
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Invariance disappears when anharmonicity is taken into account, as shown by the nonzero value of the Poisson bracket evaluated at the saddle point ½H anh , Es ¼ ps ð0ÞAs
ð2.9Þ
The action associated with the reactive mode was first suspected1214 and then proved68,15 to be an approximate, but robust invariant of motion, whereas bath mode actions exhibit a lesser degree of invariance. The first factor in eq 2.9, namely ps(0), indicates that the partial energy stored in the unbounded mode is adiabatically invariant and that the dynamics is more regular in the saddle region simply because there is less kinetic energy there. This property is used to great advantage in the reaction path Hamiltonian model.25 The second factor, As, is directly related to anharmonicity and points out that regularity near the saddle point is determined by the magnitude of the coupling coefficients. Note, however, that the latter are weighting factors in a sum of products of amplitudes and that a high internal energy implies large values of qj(0) and qk(0). Thus, both factors are increasing functions of the internal energy. A more accurate appraisal of the constancy of Es can be obtained by substituting into eq 2.8 the laws of motion derived in eqs 2.4 to 2.7: 1 1 ps ð0Þ2 ps ð0ÞAs t ps ð0ÞBs t 2 þ ::: ð2.10Þ 2 2 The coefficient of the linear term coincides of course with the Poisson bracket [Hanh, Es] reported in eq 2.9. Equation 2.4 can be inverted, giving 2 3 !2 s 4 1 ωs s As s Bs s2 1 þ þ þ :::5 t ¼ ps ð0Þ 6 ps ð0Þ 2ps ð0Þ2 6ps ð0Þ3
time (eq 2.10) or in space (eq 2.12) and involving As (i.e., the Csjk coefficients). The next term, quadratic, contains Bs (i.e., the Cssj’s). The remaining anharmonicity coefficients (Csss and Cjkl) exert their influence at a later stage only. Note finally that if the study is limited to short-time dynamics, i.e., limited to terms cubic in time in the laws of motion, then it is unnecessary to include high-order anharmonicity terms (quartic, etc.) in the expansion of the potential energy (eq 1.2). We therefore examine in turn the properties of each kind of anharmonic coefficient and try to relate it to a particular feature of the potential energy surface or of the reaction path. A. Coefficient Csss and Influence of the Activation Barrier. Coefficient Csss is related to the asymptotic range of the reaction coordinate. The presence of a saddle point along the reaction coordinate implies the existence of a reverse activation energy barrier,13 which is an important parameter of the potential energy surface. A summary description of the asymptotic range of the potential energy can be given in terms of the function ωs2s2/2 þ Cssss3, which has its extrema at s = 0 and s = ωs2/3Csss. A positive value of Csss ensures that the second value corresponds to a minimum at positive values of s. The depth of this minimum is equal to the reverse activation energy barrier: jVsmin j ¼
Eanh s ðtÞ ¼
ð2.11Þ Substitution into eq 2.10 gives the expression of function of the unbounded coordinate: Eanh s ðsÞ ¼
1 Bs s2 ps ð0Þ2 As s þ ::: 2 2ps ð0Þ
Eanh s
as a
ωs 6 54Csss 2
ð3.1Þ
Hence ωs 3 Csss ¼ pffiffiffi 3 6jVsmin j1=2
ð3.2Þ
B. Coefficients Cssj and Curvature of the Reaction Path. Consider a two-dimensional (2D) subspace where all coordinates are set equal to zero except s and qj.
1 1 Vsj ðs, 0, :::, qj , :::; 0Þ ¼ ωs 2 s2 þ ωj 2 qj 2 2 2 þ Csss s3 þ Cssj s2 qj þ Csjj sqj 2 þ Cjjj qj 3 ð3.3Þ
ð2.12Þ
3. PROPERTIES OF THE ANHARMONIC COEFFICIENTS The previous results show that there is structure in the propagation of anharmonicity away from the saddle point. Starting from a strictly harmonic situation at t = 0, s = 0, anharmonicity progresses gradually as a function of time, both backward and forward, and more rapidly for momenta than for coordinates. That anharmonicity coefficients come into play in a sequential way can be seen by examining their appearance in the laws of motion (eqs 2.42.6). No correction appears as a linear term. For the reactive mode s, the Csjk coefficients are the only ones to be found in the t2 term, followed by the Cssj’s in the cubic term; Csss and Cjkl show up last in the quartic term (whose expression has not been reported). This has an influence on the dynamics, as will be seen later on. Momenta, which are time derivatives of the corresponding conjugate variables, are more sensitive to the influence of anharmonicity. As a result, the partial energy stored in the unbounded mode is now perturbed by a term linear either in
Since mass-weighted coordinates are used, the reaction path on this surface is the steepest descent path originating from the saddle point.25,26 It obeys the following equation ! Cssj 2 Csjj ζj ðsÞ ¼ 2 s 1 2 2 s þ ::: ð3.4Þ ωj ωj The curvature of the reaction path is equal to Cssj Csjj kj ðsÞ ¼ 2 2 1 6 2 s þ ::: ωj ωj
! ð3.5Þ
Equations 3.4 and 3.5 show that the combinations Cssj/ωj2 and Csjj/ωj2 determine an important feature of an anharmonic potential energy surface, namely, a drift of the reaction valley (toward positive values of qj if Cssj is chosen to be negative) and, as a consequence, a curvature of the reaction path. The major role is played by Cssj/ωj2, whereas Csjj/ωj2 provides a minor correction. C. The Elusive Csjk Coefficients. These coefficients play a particularly important role in the dynamics. In all of the equations that concern the reactive mode, namely the law of motion 4612
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sanh(t) (eq 2.4), the Poisson bracket [Hanh, Es] (eq 2.9), and the anh partial energies Eanh s (t) (eq 2.10) and Es (s) (eq 2.12), the anharmonicity coefficients Csjk are found to play a leading role via the quantity As. Coming back to the particular problem that is the subject of the present work, Hinde, Berry, and Wales have ascribed the regularity observed for flat saddles to the fact that the shape of the potential energy surface changes very slowly along the reaction coordinate.13,14 Translated into the present formalism, this remark suggests that vibrational decoupling can be traced back to a relative invariance of the force constants of vibrations orthogonal to the reaction coordinate, i.e., to the small magnitude of the third derivatives (∂/∂s)(∂2V/∂qj2) and (∂/∂s)(∂2V/∂qj∂qk). For Hamiltonian Hanh (eq 1.2), they are precisely equal to the anharmonicity coefficients Csjj and Csjk. Unfortunately, their physical significance is particularly elusive, as shown in what follows. A good idea of the shape of the 2D potential surface defined by eq 3.3 is given by two zero-energy equipotentials Vsj(s,0,...,qj, ...,0) = 0, which largely determine the shape of the dissociation valley. In the harmonic limit, these zero contours are two straight lines qj = ( (ωs/ωj)s. In an anharmonic situation, these straight lines become curves obeying the following equations ! Cssj ωs Csss ωs ωs 2 2 þ þ þ Csjj 3 þ Cjjj 4 s ð3.6Þ ζj ðsÞ ¼ s ωj ωj 2 ωs ωj ωj ωj ! 2 C ω C ω ω ssj s sss s s ζ s Csjj 3 þ Cjjj 4 s2 j ðsÞ ¼ ωj ωj 2 ωs ωj ωj ωj ð3.7Þ The separation between the zero contours is a conspicuous feature of the graph. It is given by ωs Δj ðsÞ ¼ 2 sð1 Aj s þ :::Þ ð3.8Þ ωj with Aj ¼
Csjj Csss þ ωj 2 ωs 2
ð3.9Þ
or, alternatively, after substituting eq 3.2 Aj ¼
Csjj ωs þ pffiffiffi ωj 2 3 6jVsmin j1=2
ð3.10Þ
In eq 3.8, the first factor 2(ωs/ωj)s represents the separation in the harmonic limit, whereas the expression in brackets is a corrective factor for anharmonicity. It determines a pinching of the zero contours for positive values of s (and a spreading out at negative values). The influence of Csjj on the separation between zero-energy equipotentials is minor because its contribution is obscured by 1/2 . This is somethe additional term Csss/ωs2 or ωs(54|Vmin s ||) what frustrating because one would have liked to relate the quantity As that appears in most of the equations of section 2 to a particularly conspicuous feature of the potential energy surface. As another deceptive conclusion, we note that the derived equations do not account for a channeling of reactive trajectories during passage through a transition state, at least if channeling is understood as restraining amplitudes. Although eq 3.8 leads to the conclusion that anharmonicity narrows the reaction valley for
positive values of s, it also predicts a widening out for negative values.
4. AN ANALYTIC POTENTIAL ENERGY SURFACE Uncovering correlations between dynamical effects (whose complete description requires the full phase space) and specific structural properties (which are restricted to configuration space) is a difficult matter. The most general method, namely running classical trajectories on a potential energy surface obtained by reliable ab initio calculations, supplies detailed information on many dynamical observables. However, correlation with a particular molecular parameter is indirect only. It results from a comparison of suitably averaged properties derived from a set of different surfaces where one particular feature is systematically varied.3 The present work has limited scope: we wish to understand the role played in the dynamics by the imaginary frequency ωs. Insight may be obtained via the use of model Hamiltonians containing parametrized force fields. The use of an analytic potential energy surface containing variable parameters is very appealing for our purposes, but caution has to be exercised: the force field must be realistic, i.e., flexible enough. A particularly awkward situation is encountered when the potential energy surface is complicated, with multiple minima connected to several dissociation asymptotes, and when the internal energy is large. Then, the reaction mechanism has sometimes little to do with the reaction path defined26 as the intrinsic reaction coordinate (IRC). These troublesome situations have been termed non-IRC dynamics22,27,28 or roaming atom kinetics.29,30 Hase and co-workers have studied reactions of this kind by direct dynamics simulation.22,27,28 Trajectories were integrated “on the fly”, with the potential energy and its derivatives obtained directly from quantum-chemistry programs, which requires cutting edge technology. Alternatively, Collins and co-workers have developed an astute interpolation method to construct potential energy surfaces, which gives good results even for a reaction that presents multiple reaction paths (either because there is more than one mechanism for the same reaction, or because there is more than one possible product).26,31 We now come back to our problem, and restrict it to reactions where knowledge of the IRC provides a good starting point for understanding the mechanism. The parametrized force field we propose here consists of a sum of Morse functions for the oscillatory modes, plus an inverted Lennard-Jones potential for the unbounded degree of freedom: V ðs, q1 , :::, qN Þ 0"
¼
4jVsmin j@
jVsmin j þ
N
#12 " #6 1 σ s ðq1 , :::, qN Þ σ s ðq1 , :::, qN Þ A s þ 21=6 σ s ðq1 , :::, qN Þ s þ 21=6 σ s ðq1 , :::, qN Þ
∑j Dj ð1 exp½βjðs, q1 , :::, qj 1 , qj þ 1, :::, qN Þqj 2 Þ
ð4.1Þ
(with all coordinates measured from the position of the saddle point). The reactive degree of freedom is parametrized as a LennardJones 6-12 potential, which is expected to be more realistic than a Morse function at large values of the reaction coordinate. Intermode coupling results from the fact that the parameter β of each anharmonic oscillator is assumed to depend on all degrees of freedom other than its own coordinate. In a similar 4613
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way, the value of parameter σs is assumed to depend on all of the vibrational coordinates. At the saddle point, the value of the Morse parameters is related to harmonic frequencies in a simple way: βj ð0, 0, :::, 0Þ ¼ ωj ð2Dj Þ1=2
ð4.2Þ
Similarly, the value of the function σs at the saddle point is related to the modulus of the imaginary frequency as follows σ s ð0, :::, 0Þ ¼ 621=3
jVsmin j1=2 ωs
ð4.3Þ
The value of the Morse parameters βj at an infinite value of the reaction coordinate is expected to be given by an equation similar to eq 4.2: βj ð¥, 0, :::, 0Þ ¼ ω¥j ð2Dj Þ1=2
ð4.4Þ
where ωj¥ denotes the vibrational frequency of a reaction product. At small intermediate values of s, the value of βj is empirically interpolated between these two extremes: βj ðs, q1 , :::, qj 1 , qj þ 1 , :::, qN Þ
¼ ððωj ω¥j Þ exp½γj ðq1 , :::, qj 1 , qj þ 1 , :::, qN Þs þ ω¥j Þð2Dj Þ1=2 ð4.5Þ where γj is a positive function of all coordinates except s and qj. Equations 4.14.5 form an ansatz that can be expected to be valid when the dynamical pathways followed by the trajectories do not deviate too much from the intrinsic reaction coordinate. Equation 4.1 is now expanded as a MacLaurin series and compared with eq 1.2. Equating coefficients of like terms leads to Csss ¼
Cssj
7ωs 3 pffiffiffi 12 2jVsmin j1=2
Dσs ¼ 621=3 jVsmin j1=2 Dqj
ð4.6Þ
ωs 3
ð4.7Þ 0
Csjj ¼ ωj ðω¥j ωj Þγj ð0, :::, 0Þ Csjk ¼ 0
ðj 6¼ kÞ
ð4.8Þ ð4.9Þ
The incidence of these formulas on the dynamics is examined in the next section.
5. DISCUSSION A. Flat Saddles. It has been seen repeatedly that anharmonicity coefficients come into play in a sequential way in the laws of motion of the reaction coordinate (eq 2.4) and that the Csjk and Csjj coefficients (i.e., the term As) determine the earliest stage in the dynamics. The coefficients Csjk (with j 6¼ k) vanish in the force field expressed by eq 4.1. They show up only if it contains three-body interaction terms. Furthermore, since initial conditions for coordinates are chosen randomly, statistical cancellation can be expected in the summation ∑j∑kCsjkqj(0)qk(0) δjk if the number of oscillators is large enough. As a result of the central limit theorem,32 the sum of N such random variables tends to zero
as N 1/2. Therefore, these coefficients can be disregarded in a first study. According to eq 4.8, coefficients Csjj are proportional to ωj(ω¥ j ωj). Attention should thus be paid to high-energy stretching modes. However, most of the latter have characteristic frequencies that should not change greatly along the reaction path. Thus, normally, when the first factor, ωj, is large, the second one, (ωj¥ ωj), is small. This would not be the case in more complicated situations, e.g., if a transition state containing a hydrogen bond fragments to a pure OH stretching mode. The second factor is then nonzero, but the magnitude of the frequency ωj is lower. Admittedly, such situations require care. Nevertheless, altogether, the short-time coupling between the oscillator and the reaction coordinate, measured by the term As, can be expected to remain weak in many cases. Afterward come terms cubic in time, which contain the Cssj coefficients in the term Bs. Equation 4.7 shows that these coefficients increase with the cube of the frequency ωs of the imaginary mode. The drift away from the straight line that characterizes the harmonic limit and the partial curvatures at the saddle point are proportional to ωs3/ωj2 (eqs 3.4 and 3.5). Thus, the lower the value of ωs, the better is the decoupling between the reaction coordinate and high-frequency vibrations. The cubic dependence on ωs should greatly contribute to the collimation of trajectories noted by Wales, Hinde, and Berry in their study of rare gas clusters.1214 The diagonal anharmonicity coefficient Csss is also proportional to ωs3. It is the last one to show up because it takes its origin in the asymptotic range of the potential energy surface, related as it is to the reverse activation energy barrier. The potential could be made more realistic by adding higher-order anharmonicity terms. However, the procedure described in paragraph 3A shows that quartic anharmonicity coefficients scale as ωs4, as ωs5 for quintic terms, etc. It is not hard to see that in a higher-order expansion such as c3s3 þ c4s4, the sensitivity to ωs would be higher than for a cubic term alone. Thus, inclusion of higher powers in the expansion of anharmonicity would reinforce our argumentation. As shown by eqs 2.10 and 2.12, regularity in the dynamics implies adiabatic constancy of the partial energy Eanh s . In short, the analysis in the framework of classical mechanics of a simple model describing standard cases accounts for the smallness of the first coefficients in the polynomial expansion of the anharmonic terms, at least in the case of flat saddles. B. Phase Space Structure. At the saddle point, the motion is governed by the integrable Hamiltonian H0 and is confined to a (N þ 1)-dimensional surface (or to a portion of it) in the constant energy (2N þ 1)-dimensional phase space. If all frequencies were real, the surface would be called a (N þ 1)torus.24 If one degree of freedom is unbounded, then the overall motion in phase space may be represented by considering the unbounded motion along a hyperbola in the (s, ps) plane and then plotting the trajectory of a bounded mode along a circle in a (qj, pj) plane perpendicular to the hyperbola. The overall picture might be termed a pseudotorus, although a small portion of it would be really needed. Mathematically minded readers would use the specific but awe-striking term NHIM (normally hyperbolic invariant manifold).1618,33,34 The motion remains confined to this surface in the immediate neighborhood of the saddle point as a result of the KolmogorovArnoldMoser (KAM) theorem concerning slightly perturbed integrable systems.24 4614
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The Journal of Physical Chemistry A Farther away, the perturbation increases and the KAM theorem no longer applies. However, the sequential appearance of anharmonicity coefficients in the laws of motion reported in section 3 provides additional information. Equations 2.4, 2.9, 2.10, and 2.12 teach us that a subset of nearly harmonic trajectories can be defined by imposing an additional condition As = 0; these trajectories do not deviate too much from eq 2.1a because the term quadratic in time then vanishes. A still better approximation to harmonic motion can be obtained by eliminating the perturbing cubic term via the additional condition Bs = 0; such trajectories sample a still more reduced phase space volume. The process can be continued. Thus, as predicted by the KAM theorem, perturbation of a simple integrable system leads to a division of phase space into regions of regular and irregular motion. However, as noted by a reviewer of this journal, the mechanism that guarantees stability is not the same for a KAM torus and for a NHIM.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
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