reaction paths and elementary bifurcations tracks: …€¦ · keywords: bifurcation theory;...

International Journal of Bifurcation and Chaos, Vol. 16, No. 7 (2006) 1913–1928c© World Scientific Publishing Company

REACTION PATHS AND ELEMENTARYBIFURCATIONS TRACKS: THE DIABATIC

1B2-STATE OF OZONE

S. C. FARANTOSInstitute of Electronic Structure and Laser,

Foundation for Research and Technology-Hellas, andDepartment of Chemistry, University of Crete,

Iraklion 711 10, Crete, Greece

ZHENG-WANG QU, H. ZHU and R. SCHINKEMax-Planck-Institut, Dynamik und Selbstorganisation,

D-37073 Gottingen, Germany

Received November 5, 2004; Revised March 15, 2005

Bifurcations of equilibrium points and periodic orbits are common in nonlinear dynamical sys-tems when some parameters change. The vibrational motions of a molecule are nonlinear, andthe bifurcation phenomena are seen in spectroscopy and chemical reactions. Bifurcations maylead to energy localization in specific bonds, and thus, they have important consequences for ele-mentary chemical reactions, such as isomerization and dissociation/association. In this article weinvestigate how elementary bifurcations, such as saddle-node and pitchfork bifurcations, appearin small molecules and show their manifestations in the quantum mechanical frequencies andin the topology of wave functions. We present the results of classical and quantum mechanicalcalculations on a new (diabatic) potential energy surface of ozone for the 1B2 state. This excitedelectronic state of ozone is pertinent for the absorption of the harmful UV radiation from thesun. We demonstrate that regular localized overtone states, which extend from the bottom ofthe well up to the dissociation or isomerization barrier, are associated with families of periodicorbits emanated from elementary bifurcations.

Keywords : Bifurcation theory; periodic orbits; classical and quantum molecular dynamics.

1. Introduction

Understanding the dynamics of molecules at ener-gies where chemical bonds break or are formedis not only of academic importance but basi-cally it is the foundation of chemistry. In theBorn–Oppenheimer approximation for the separa-tion of the electronic and nuclear motions in amolecule, Quantum Chemistry answers in part thisproblem by solving the electronic Schrodinger equa-tion to produce molecular potential energy surfaces(PES). The solution of the nuclear equations ofmotion is required to understand the elementary

chemical processes, such as isomerization and dis-sociation/ association. Since elementary chemicalreactions require the excitation of the molecule tohigh energy vibrational-rotational states, spectro-scopic methods which capture the molecule at theseextreme motions are applied, together with theo-retical methods to assign the spectral lines and toextract the dynamics.

In the second half of the twentieth centuryseveral vibration-rotation molecular spectroscopicmethods were developed which allowed one torecord high resolution spectra of highly excited

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1914 S. C. Farantos et al.

states. Among them the Stimulated EmissionPumping (SEP) and the Dispersed Fluorescence(DF) spectroscopy [Dai & Field, 1995] gave animpetus to the field of vibrationally excitedmolecules in the ground electronic state. SEP is adouble-resonance technique. A first laser promotesthe molecule to a single rovibrational level of anelectronically excited state, while a second laserstimulates the emission down to a highly excitedvibrational state of the ground electronic state.Due to the improved Franck–Condon access fortransitions corresponding to a significant change ofmolecular geometry, SEP gives access to a classof vibrational states which are not accessible bydirect overtone spectroscopy. Molecules studied bySEP and DF spectroscopy are C2H2 [Jacobsonet al., 1998a, 1998b, 1999], HCP [Ishikawa et al.,1998], SO2 [Yamanouchi et al., 1990, 1988; Sako& Yamanouchi, 1996], HFCO [Yamamoto & Kato,1998], HCO [Keller et al., 1996], DCO [Keller et al.,1997], HCN [Northrup et al., 1997] and NO2 [Delonet al., 2000]. Since the SEP and DF methods excitethe molecule at very high vibrational states, theyare ideal spectroscopic techniques to deduce thedynamics close to the isomerization or dissociationthreshold. As a matter of fact, the SEP spectra ofacetylene were the first which revealed vibrational(quantum) chaos at energies above the isomeriza-tion threshold of acetylene to vinylidene [Jonaset al., 1993].

Other high resolution spectroscopic methodswhich have contributed to the detailed study ofsmall polyatomic molecules are the High Reso-lution Fourier Transform and Laser Spectroscopyin several diversities: Frequency Modulation withDiode Lasers (FMDL), Cavity Ringdown Spec-troscopy (CRDS), Intracavity Laser AbsorptionSpectroscopy (ICLAS), Optoacoustic (OA) andOptothermal Spectroscopy, and PhotofragmentSpectroscopy. Several molecules have been stud-ied, such as H2O, HO2, CO2, HCN, HNO, N2O,OCS, HOCl, HCO, CH4, C6H6, NH3, HOOH,s-tetrazine, dimethyl-s-tetrazine pyridine, propyneto mention a few. A short description of these meth-ods are presented in [Herman et al., 1999]. Crimand coworkers [1999] have combined the photo-acoustic spectroscopy with a time of flight appa-ratus to control the products in unimolecular andbimolecular reactions by vibrationally exciting spe-cific chemical bonds of reactant molecules. Thisbond selective chemistry reveals energy localizationin specific bonds.

Laser femtosecond spectroscopy and molecu-lar beams have allowed to study isolated moleculesand follow a chemical reaction in real time. Thisrequires the simultaneous development of theoret-ical models to interpret the experimental results.The established theoretical methods based on anormal mode description of the molecular vibra-tions, applied at energies close to the equilib-rium point, are not valid for highly vibrationallyexcited molecules. The deviation from the harmonicapproximation of the potential energy surfaceimposes the need for the construction of accuratepotential functions that describe several and ener-getically accessible reaction channels. These arenonlinear functions and the application of non-linear mechanics to investigate the dynamics ofthe molecule is necessary. Polyatomic moleculesstimulate new computational challenges in solvingaccurately the Schrodinger equation and obtaininghundreds of vibrational states. Nowadays, triatomicmolecules can be treated with fully ab initio meth-ods, both in their electronic and nuclear parts.Tetratomic molecules are more difficult to deal with,in spite of the progress which has recently beenachieved. For example, six-dimensional calculationsup to energies of the isomerization of acetylene tovinylidene have been published [Zou & Bowman,2002].

Apart from the computational challenges smallpolyatomic molecules unravel conceptual and phys-ical interpretation problems. A result of thenonlinear mechanical behavior of a dynamical sys-tem at high energies is the simultaneous appear-ance of ordered motions and chaos, as well as thegenesis of new types of motions via bifurcation phe-nomena. What are the quantum mechanical coun-terparts of these classical behaviors? What are thespectroscopic fingerprints of the nonlinear dynam-ics in the molecules? In cases where the vibra-tional spectra depict isomerization and dissociationprocesses how can we identify them in the spec-tra? As a matter of fact, the progress of nonlinearmechanics forces us to re-examine the mechanismsof the breaking and/or forming of a single chemicalbond as it occurs in elementary chemical reactions.To answer the above questions new assignmentschemes which allow the classification of quantumstates in a meaningful and useful way are requiredand such novel methods have been developedthanks to the theory of periodic orbits (PO), theirbifurcations [Wiggins, 2003] and semiclassical the-ory [Gutzwiller, 1990].

Reaction Paths and Elementary Bifurcations Tracks 1915

The theory of periodic orbits plays a domi-nate role in the formulation of classical mechanics.However, what makes periodic orbits important formolecules are the advances obtained in semiclassi-cal theory in the past years. Gutzwiller [1990] usingFeynman’s path integral formulation has derived asemiclassical expression for the trace of the resol-vent (Green operator) of the quantum Hamilto-nian operator as a sum over all isolated periodicorbits, taking into account their linear stabilityindices. This formula provides approximate valuesfor the quantum eigenenergies of classically chaoticsystems. Berry and Tabor [1977] derived in a differ-ent way the trace formula based on the Einstein–Brillouin–Keller quantization rule. Their formulagives the quantum density of states as a coherentsummation over resonant tori, and therefore it isapplicable to integrable systems. Finally, a uniformresult bridging the Berry–Tabor and Gutzwillertrace formulas for the case of a resonant islandchain was derived by Ozorio de Almeida [1990].Last but not least, the importance of periodic orbitsfor a qualitative understanding of the localizationof quantum mechanical eigenfunctions in config-uration space came from the scarring theory ofHeller [1984]. It turns out, that for small polyatomicmolecules the probability density of eigenfunctionsis accumulated along short-period stable or the leastunstable periodic orbits. In some cases, such as over-tone states, we can associate isolated PO with spe-cific eigenfunctions.

Periodic orbits evolve with the energy of thesystem or any other parameter in the Hamilto-nian, bifurcate and produce new periodic orbitswhich portrait the resonances among the vibra-tional degrees of freedom. Generally, PO revealthe structure of phase space at different energies,i.e. the different types of motions. We can applythese methods with two different strategies. In thefirst one we follow the Global-approach, that is weemploy global potential energy surfaces valid overthe complete nuclear configuration space from theminima to the dissociation/isomerization channelsand compute families of periodic orbits from theminima of the potential up to and above the dis-sociation threshold or up to the energy where theyare annihilated.

The second strategy, named the patch-approach, is usually applied by spectroscopists.An effective Hamiltonian is fitted to spectroscopicresults reproducing part of the vibrational energyspectrum, and thus a patch of the global

Hamiltonian. The spectroscopic Hamiltonians,which are mainly of reduced type (less degreesof freedom) incorporate the resonance(s) amongthe frequencies of two different degrees of freedomobserved in experimental spectra. These Hamilto-nians can be extended to fit calculated spectra andthen a PO analysis is carried out in a similar fashionas in the Global-approach.

We have systematically studied the PO con-tinuation/bifurcation (C/B) diagrams for severaltypes of molecules; triatomic [Farantos, 1996]and tetratomic [Prosmiti & Farantos, 1995, 2003]molecules, van der Waals molecules [Guantes et al.,1999] and at energies below and above dissociationthreshold [Founargiotakis et al., 1997]. Similarly tothe landscape of the PES which reveals the relativestability/instability and reactivity of the molecularisomers, the families of periodic orbits portray thestructure of phase space at specific energies, andthey thus reveal the dynamics of the molecule atleast around the periodic orbits. It is now acceptedthat the regular motions and the chaotic motions inphase space consist of highly complicated entanglednetworks. Periodic orbits depicted by a C/B dia-gram may be considered as providing a higher orderof approximation to the dynamics of the molecule,after obtaining the zero order approximation withthe potential function.

The landscape of the PES may drasticallychange as energy increases in the molecule. Barriersand minima may disappear/appear. Similarly, thephase space changes with the total energy. Stable,quasiperiodic motions may turn to unstable chaoticones and vice versa. But most importantly, via thebifurcations of periodic orbits new types of motionmay emanate. The bifurcation theory of Hamilto-nian dynamical systems has mainly been developedin the last half of twentieth century. One impor-tant outcome of the theory is the identification ofthe elementary bifurcations which are described byvery simple Hamiltonians. In spite of their sim-plicity they can also describe a complex dynam-ical system close to particular critical energies.This makes elementary bifurcations generic. Formolecular Hamiltonian systems we have identifiedthe saddle-node, pitchfork and Hopf-like elementarybifurcations.

Bifurcation phenomena, i.e. the change ofthe structure of the orbits by varying one ormore parameters, are well known in vibrationalspectroscopy. For example, the transition from nor-mal to local mode oscillations, first discovered


in symmetric ABA molecules, is related to theelementary pitchfork bifurcation [Halonen, 1998;Lawton & Child, 1981; Prosmiti et al., 1999]. Inwhat follows we show that localization of energyis also obtained via another generic bifurcationof periodic orbits, the saddle-node (SN). Periodicorbits which emerge from SN bifurcations appearabruptly at some critical value of the energy, usu-ally in pairs, and change drastically the phase spacearound them. They penetrate in regions of nuclearphase space which the normal mode motions cannotpenetrate. Saddle-node bifurcations are of generictype, i.e. they are robust and remain for small(perturbative) changes of the potential function[Prosmiti & Farantos, 1995, 2003].

We initially determined the importance of SNbifurcations of periodic orbits in studies of the iso-merization dynamics in double well potential func-tions [Farantos, 1993]. These PO connect the twominima and scar the isomerizing wave functions,i.e. eigenfunctions with significant probability den-sity in both wells. Their birth is due to the unsta-ble periodic orbit which emanates from the saddlepoint of the potential energy surface. However, evenbelow the potential barrier a series of SN bifurca-tions of periodic orbits pave the way to the iso-merization process. The spectroscopic signature ofsuch SN bifurcations has been found in a num-ber of triatomic molecules [Ishikawa et al., 1999;Joyeux et al., 2002]. HCP was the first moleculewhere experimental evidence for SN bifurcationswas given. In the other extreme, infinite dimensionalsystems, such as periodic or random lattices, showspatially localized and periodic in time motions,called discrete breathers (DB), and it has beenshown that they can also be associated with saddle-node bifurcations [Flach & Willis, 1998; Kopidakis& Aubry, 1999]. Spectroscopic evidence for the exis-tence of discrete breathers can be found amongbiomolecules [Xie et al., 2000].

In the present article we apply PO analysis toan electronically excited state of ozone. In Sec. 2 weintroduce the concepts of elementary bifurcationsand show how saddle-node bifurcations emerge viasimple one-dimensional Hamiltonians, which how-ever, depict the dynamics on the complicated molec-ular potential energy surfaces. In Sec. 3 we presentour strategy to locate periodic orbits in multidi-mensional Hamiltonians and for general coordinatesystems. Section 4 describes the results on theexcited 1B2 electronic state of ozone. We concludewith Sec. 5.

2. Elementary Bifurcations

We call elementary those bifurcations which canappear in the simplest one- or two-dimensionalnonlinear systems by varying one or two parame-ters. Comparing these systems with the complicatedmultidimensional molecular Hamiltonians, we maythink that a simple one-dimensional Hamiltonian isonly of pedagogical use. This is not true, since wecan show that at the critical value of the parame-ter at which the bifurcation occurs, the system canbe reduced to one of lower dimension by using thecentral manifold theorem [Wiggins, 2003], and thenwe can describe it with a simple Hamiltonian bytransforming to normal forms. The mathematicaltheory of bifurcations in dynamical systems is welldeveloped and there are excellent books [Gucken-heimer & Holmes, 1983; Wiggins, 2003] and reviewarticles [Crawford, 1991] to introduce the subject.In this section we discuss how a saddle-node bifur-cation appears in the simplest nonlinear Hamiltoni-ans, that with a cubic and quartic potential.

2.1. Cubic potentials

For a system of one degree of freedom with aHamiltonian

H(q, p) =12p2 + V (q) (1)

the Hamilton’s equations of motion for a conserva-tive vector field are

q =∂H

∂p(2)

p = −∂H

∂q. (3)

q is the generalized coordinate, p its conjugatemomentum and V (q) the potential function. Thestationary points of the potential, i.e. dV (q)/dq = 0,are the equilibrium (or fixed) points of the vectorfield; q = p = 0. The above mentioned books [Guck-enheimer & Holmes, 1983] describe the elementarybifurcations of fixed points of vector fields. Here, wedo not attempt a complete cover of this subject butwe discuss mainly those cases which are related tothe saddle-node bifurcations.

We assume a general cubic potential

V (q) =13q3 − 1

2αq2 − βq − γ (4)

The equilibrium points of Hamilton equations arethen the roots of the second order polynomial


dV (q)dq

= q2 − αq − β = 0. (5)

The discriminant of the above equation is defined as

D = α2 + 4β. (6)

In order that Eq. (5) has two real roots (equilib-rium points) D ≥ 0 is required. Thus, the parabolaD = 0 defines the region in the two-parameterspace, (β, α), where these two roots exist. Figure 1exhibits this region and Fig. 2 shows the evolution

−1 −0.5 0 0.5 1 β

−2

−1

0

1

2

D < 0

D > 0

D > 0

Fig. 1. The sign of the discriminant D [Eq. (6)] in theparameter space (β, α) of a cubic potential.

−0.5 −0.1 0.3 0.7 β

−1

−0.5

0

0.5

1

Fig. 2. C/B diagram of a cubic potential. The coordinates qof the two equilibria are shown as function of the parameterβ and α = 0. Continuous line denotes the stable equilibriumpoint (minimum) and the dashed line the unstable equilib-rium point (maximum).

of the two equilibria of the potential by varyingthe parameter β assuming α = 0. This graph isa typical saddle-node continuation/bifurcation dia-gram. We notice, that there are no equilibriumpoints for negative values of β and at β = 0 thedouble root signals the genesis of the saddle-nodebifurcation. The two branches correspond to stable(solid line) and to unstable (dashed line) equilib-rium points. Stable means that trajectories closeto this point will remain in the nearby region,whereas unstable points mean that nearby trajecto-ries will deviate from it. Visualization of the equi-librium points of the corresponding potential func-tion explains better the stable and unstable terms.Such a plot is shown in Fig. 3 for several valuesof β.

In this figure we can distinguish three differ-ent regimes. For β < 0 there are no equilibriumpoints, for β > 0 there are two equilibrium points,one minimum and one maximum, and for β = 0(dashed line) there is one saddle point at q = 0. InFig. 4 several trajectories are plotted in the phaseplane, (q, p), and for β = 1. The dashed line denotesthe separatrix of the two types of motions allowedfor this dynamical system; closed stable orbits andunbound orbits. This phase space graph is typi-cal of a saddle-node bifurcation. It is important toemphasize that the structure of phase space doesnot change qualitatively by introducing a second

−2 −1 0 1 2q

−4

−2

0

2

4

Fig. 3. Plots of a cubic potential for several values of β.For β < 0 there are no equilibrium points, for β > 0 thereare two equilibrium points, one minimum and one maximum,and for β = 0 (dashed line) there is one saddle point atq = 0.


−2 −1 0 1 2q

−2

−1

0

1

2

Fig. 4. Trajectories which portray the phase space structurein the region of a saddle-node bifurcation (β = 1).

parameter. The C/B diagram remains the samewith two branches for values of α �= 0.

2.2. Quartic potentials

A general quartic potential is

V (q) =14q4 − 1

3αq3 − 1

2βq2 − γq − δ (7)

The equilibrium points are:

dV (q)dq

= q3 − αq2 − βq − γ = 0. (8)

This cubic equation can be reduced to a two param-eter equation with the transformations

x = q − α

3(9)

µ =α3

3+ β (10)

λ =2α3

27+

αβ

3+ γ. (11)

The reduced cubic polynomial is

x3 − µx − λ = 0. (12)

The discriminant is defined by

D = −µ3

27+

λ

4. (13)

The roots of Eq. (12) are:

(i) For D > 0, one real root and two imaginary.(ii) For D < 0, three different real roots.

−2 −1 0 1 2 λ

0

1

2

3

D < 0

D > 0 D > 0

Fig. 5. The sign of the discriminant D [Eq. (13)] in theparameter space (λ, µ) of a quartic potential.

(iii) For D = 0, three real roots two of thembeing equal. Figure 5 presents the sign of thediscriminant in the parameter space (λ, µ).The cusp curve defines the values of (λ, µ)where the discriminant is zero. Thus, crossingthis curve from positive to negative values of Dwe pass from one to three equilibrium points.A double degeneracy of equilibrium points isencountered at the cusp curve.

The C/B diagram for λ = 0 and varying theparameter µ is shown in Fig. 6. This is a typ-ical pitchfork bifurcation. The introduction of asecond parameter (λ �= 0) results in the C/B

−1 0 1 2 µ

−2

−1

0

1

2

λ = 0

Fig. 6. C/B diagram of a quartic potential and for λ = 0(pitchfork bifurcation). Continuous lines denote stable equi-libria and dashed line the unstable equilibrium point.


diagram shown in Fig. 7. Comparing the two con-tinuation/bifurcation diagrams for λ = 0 and λ �= 0(Figs. 6 and 7), we can see that the unstable branchin the pitchfork bifurcation (dashed line) becomesthe unstable branch of a saddle-node bifurcation,whereas one stable branch in the pitchfork bifurca-tion is the stable branch of the SN bifurcation. Wecan also see that the parent family in the pitchforkbifurcation turns to a hysteresis. We can think ofthis continuation/bifurcation diagram as a foldedsurface in the (λ, µ, x) space. The size of the gap inthe C/B curves with the parameter µ depends onthe value of the parameter λ. Increase of λ resultsin an increase of the gap (see Fig. 8).

The three equilibria in the quartic potentialare two minima and one maximum. In Fig. 9 we

−2 −1 0 1 2 µ

−2

−1

0

1

2

λ = 0.01

Fig. 7. As in Fig. 6 but for λ = 0.01. A saddle-node and ahysteresis bifurcation appear.

−2 −1 0 1 2 µ

−2

−1

0

1

2

λ = 0.1

Fig. 8. As in Fig. 6 but for λ = 0.1.

−3 −2 −1 0 1 2 3x

−3

−2

−1

0

1

2

3

−2.0

−1.5

−1.0

−0.5

0.0

λ µ = 1.0

Fig. 9. Potential curves of a quartic potential and for sev-eral values of λ. The dashed line is the symmetric double wellpotential (λ = 0).

Fig. 10. Phase space structure of a symmetric double wellquartic potential.

plot potential curves for several values of λ and forµ = 1. As λ approaches zero the kink in the poten-tial is transformed to a double well.

Trajectories plotted in the phase plane (x, p)are shown in Fig. 10 for the symmetric double wellpotential (λ = 0 and µ = 1). The separatrix (dashedline) emanates from the maximum of the potential,and it separates the two types of motions encoun-tered in this system.

3. Bifurcations of Periodic Orbits

The phase space structure of a dynamical system isdetermined by locating stationary objects, such as


(i) equilibrium points, (ii) periodic orbits, (iii) tori,(iv) reduced dimension tori, and (v) stable andunstable manifolds [Guckenheimer & Holmes, 1983;Wiggins, 2003].

Equilibrium points and periodic orbits locatedfor a range of energies or any other parameter,are a good starting point to portrait the struc-ture of phase space of small molecules. This hasbeen demonstrated by studying together the classi-cal and quantum mechanics for several triatomicmolecules [Farantos, 1996; Ishikawa et al., 1999;Joyeux et al., 2002; Joyeux et al., 2005]. If a globalPES is available in a curvilinear coordinate sys-tem for the molecule, we need general and robustalgorithms to locate periodic solutions of the equa-tions of motion. This problem is formulated asfollows.

If H(q,p) is the Hamiltonian of a system withN degrees of freedom, the equations of motion arewritten in Hamilton’s form as,

dqi

dt= qi(t) =

∂H

∂pi

dpi

dt= pi(t) = −∂H

∂qi, i = 1, . . . , N.

(14)

qi, i = 1, . . . , N, are the generalized coordinatesand pi, i = 1, . . . , N, their conjugate momenta. Thesymplectic properties [Arnold, 1980] of these equa-tions are better shown by considering coordinatesand momenta as the components of the generalizedcoordinate vector x,

x = (q1, . . . , qN , p1, . . . , pN )+, (15)

where + denotes the transpose of the 2N -D columnvector. The equations of motion are then written,

x(t) = J∂H(x)

∂x≡ J∂H(x) ≡ J∇H(x), (16)

where J is the symplectic matrix,

J =(

0N IN

−IN 0N

). (17)

0N and IN are the zero and unit N × N matrices,respectively. J∇H(x) is a vector field, and J satis-fies the relations,

J−1 = −J and J2 = −I2N . (18)

To locate in general stationary objects we solvea two-point boundary value problem, i.e. we wantto find those specific trajectories which satisfy at

two different times t = 0 and t = T a relation

B[x(0),x(T )] = 0. (19)

For example, to find periodic orbits we impose thefollowing two-point boundary conditions

x(T ) − x(0) = 0, (20)

where T is the period of time after which the trajec-tory by integrating the equations of motion returnsto its initial point in phase space, x0 = x(0).

The equilibrium points are defined by requiringx = 0, or

∇H(x) = 0. (21)

Since x(T ) = x(0), we can see that the equilibriumpoints are also solutions of Eq. (20).

To locate PO for a specific period T westudy the linearized system based on the Hartman–Grobman theorem [Guckenheimer & Holmes, 1983]which states that there is a continuous change ofcoordinates that relates the linearization to the vec-tor field (J∇H(x)). Thus, important conclusionsobtained from the linearized system apply to thenonlinear one as well.

If x(t) is a solution of Eq. (16) we want to knowthe behavior of a nearby trajectory

x′(t) = x(t) + ζ(t). (22)

From Eq. (16) we have,

x′(t) − x(t) = J∇H(x′) − J∇H(x). (23)

A Taylor expansion of the rhs of Eq. (23) up to thefirst order gives,

ζ(t) = J∂2H(x(t))ζ(t). (24)

∂2H(x(t)) denotes the matrix of second derivativesof the Hamiltonian evaluated at the original trajec-tory x(t) for time t.

If,

A(t) = J∂2H(x(t)),

then Eq. (24) is written as,

ζ(t) = A(t)ζ(t). (25)

These are 2N linear differential equations with timedependent coefficients, and they are called varia-tional equations.

The general solution of Eq. (25) can beexpressed by evaluating the fundamental matrixat time t, Z(t) [Yakubovich & Starzhinskii,1975]. This is the matrix with columns vectors


(1, 0, . . . , 0), (0, 1, 0, . . . , 0), . . . , (0, 0, . . . , 1) at t = 0,i.e.

Z(0) = I2N . (26)

The general solution of Eq. (24) is then given by,

ζ(t) = Z(t)ζ(0), (27)

where ζ(0) describes the initial displacement fromthe trajectory x0. The fundamental matrix satisfiesthe variational equations as can be easily proved;

Z(t) = A(t)Z(t). (28)

Thus, by solving Eqs. (16) and (28) we obtain thetrajectory with the chosen initial conditions andthe behavior of the nearby trajectories at the lin-ear approximation.

For periodic orbits the fundamental matrix att = T ,

M = Z(T ) =∂x(T )∂x0

, (29)

is called monodromy matrix, the eigenvalues ofwhich determine the stability of periodic orbit[Wiggins, 2003]. Because of the symplectic prop-erty of Hamiltonian systems if λi is an eigen-value of the monodromy matrix, then its complexconjugate λ∗

i , as well as the λ−1i and (λ−1

i )∗ arealso eigenvalues. The properties of the monodromymatrix and the different cases of stability have beendescribed before [Farantos, 1992]. For a system ofthree degrees of freedom the eigenvalues and thusthe kinds of stability are represented in Fig. 11.As energy varies, the eigenvalues of stable periodicorbits move on the unit complex circle. When theeigenvalues are out of the unit circle but on the realaxis the periodic orbit is single or double unstable,and finally four complex eigenvalues out of the unitcircle characterize a complex unstable PO.

Stationary points and periodic orbits arelocated numerically with a new version of POMULTprogram written in Fortran95 [Farantos, 1998].We apply multiple shooting methods which con-vert the two-point boundary value problem tom-initial value problems, i.e. we search for theappropriate initial values of coordinates andmomenta which satisfy the boundary condi-tions and the continuity equations [Farantos,1998] which guarantee a smooth orbit. Ana-lytic first and second derivatives of the poten-tial function required for the solution of theequations of motion and the variational equationsare computed by the AUTO–DERIV, a Fortran

Double Unstable

Stable Single Unstable

Complex Unstable

Real

λ

λ ∗

λ

λ ∗

1

1

λλ

1/λ2

2

λ 1

λ 1λ 2

∗

∗

1/λ1

1

λ 2

λ 2

1/λ∗

1/λ

1/λ2

i

−i

1−1

Imaginary

Fig. 11. Eigenvalues of the monodromy matrix for amolecule with three degrees of freedom. There are three pairsof complex conjugate eigenvalues one of which is always equalto one (not shown). The other two pairs move in the complexplane as energy varies. The positions of these eigenvalues withrespect to the complex unit circle determine the stability ofthe PO.

code for automatic differentiation of an analyticfunction of many variables written in Fortran[Stamatiadis et al., 2000].

4. Application to the 1B2-Stateof Ozone

Ozone is an important molecule for atmosphericchemistry because of its role in shielding earth fromthe harmful UV light. There have been numer-ous studies of its spectroscopy and photodissoci-ation [Houston, 2003]. The absorption of the UVradiation excites electronically the molecule to the(21A1) and the (11B2) states. Recently, we have car-ried out theoretical ab initio calculations and wehave shown that the two notorious absorption bandsobserved, Huggins and Hartley, are due to the exci-tation of the molecule from the ground electronicstate to the (11B2) state [Qu et al., 2004a, 2004b].The mechanism of the photodissociation of ozoneis complicated because of the involvement of sev-eral excited states. Nevertheless, the constructionof an analytical function for the diabatic (11B2)


state of ozone, as well as the calculation of highquality quantum mechanical vibrational eigenfunc-tions up to the dissociation threshold, as a resultof the previous studies, lead us to investigate fur-ther the dynamics of the molecule in this excitedstate. Here, we apply our periodic orbit techniquesto explore the structure of phase space, and thus,to trace reaction paths on this surface.

The potential energy for the diabatic 1B2-stateof ozone has been described in detail before [Quet al., 2004a, 2004b]. The equilibrium geometry ofozone in this excited state has a Cs symmetry, withone long-bond and one short-bond, in contrast tothe ground electronic state where the minimum is ofC2v symmetry with two equal bonds. Because of thepermutational symmetry of the molecule there arethree equivalent minima and several saddle pointsamong them. In Table 1 we tabulate the geometriesand the energies of one minimum and related saddlepoints. The nuclear configurations are described inJacobi coordinates, which are the distance R, of thelong-bond oxygen atom from the center of mass ofthe short-bond end atoms whose bond length is r,and the angle γ between the vectors R and r (seeFig. 12). This Jacobi coordinate system is appro-priate to describe the dissociation of a triatomicmolecule.

All quantum mechanical calculations are forzero total angular momentum. The discrete variablerepresentation [Light & Carrington, 2000] (DVR)is used to represent the Hamiltonian matrix. Thecalculations are done in symmetric-Jacobi coordi-nates: R′ is now the distance from the central oxy-gen atom to the center of mass of the two endatoms, r′ the distance between the two end atoms,and γ′ denotes the angle between the vectors R′and r′. The choice of this coordinate system sim-plifies the quantum mechanical calculations, sincethe Hamiltonian is symmetric with respect to theexchange of the two end atoms, i.e. it is symmetricwith respect to γ′ = 90◦. Thus, the eigenstates are

Table 1. Equilibrium points of the diabatic 1B2 PES ofozone.

Stationary Point Order r R γ Energy

minimum 0 2.258 3.751 2.198 3.429saddle-1 1 2.310 5.400 1.993 4.053saddle-2 1 2.703 3.401 2.294 4.108saddle-3 1 2.275 3.583 1.571 4.181saddle-4 2 2.301 4.126 1.571 4.242

O(1D)+O2(a1∆g) 2 2.316 ∞ — 4.046

Fig. 12. Potential energy contours for the diabatic 1B2-stateof ozone. Energies are in electronvolts, distances in Bohrs andthe angle in radians. The energies cover the interval of (4.0–4.5) eV in 0.033 eV steps. The points shown on the graphmark the saddle points 1 (dotted circle), 2 (stars), 3 (filledcircle) and 4 (filled square) (see Table 1). The periodic orbitsoverlayed are discussed in the text.

either symmetric or antisymmetric with respect toγ′ = 90◦ (see Fig. 15). Therefore, results related tothe quantum mechanical calculations will be pre-sented in the symmetric-Jacobi coordinate system,otherwise in the following with Jacobi coordinateswe mean the originally defined for the asymmetricgeometry.

The selected grid parameters are 0.5 a0 ≤ R′ ≤3.0 a0 with 64 potential optimized points [Echave &Clary, 1992] and 3.0 a0 ≤ r′ ≤ 7.0 a0 with 64 poten-tial optimized points. The angular coordinate is rep-resented by 64 Gauss–Legendre quadrature points[Bacic & Light, 1989] in the interval 0 ≤ γ′ ≤ 90◦.Only those points are retained in the grid whosepotential energy is smaller than 6.5 eV. Two types ofcalculations have been performed; Filter diagonal-ization [Grozdanov et al., 1995; Wall & Neuhauser,1995] and harmonic inversion [Mandelshtam &Taylor, 1997]. More details are given in [Qu et al.,2004a, 2004b].

In Fig. 12, isopotential curves are depictedin the (R, γ) plane. The symbols on this picturemark the positions of the saddle points tabulatedin Table 1. The saddle points which separate two


equivalent minima by exchanging their short andlong bonds are denoted by stars. The geometry ofthis saddle is better portrayed in projections of thepotential function in the plane with the two bondlengths of the molecule as coordinates shown inFig. 1 of [Qu et al., 2004a, 2004b]. Note the dif-ferences in the values of r for the saddle points.Except for saddle-4 which is of second order, i.e. ithas two unstable directions with imaginary frequen-cies, all the other saddles are of single order. Thestable degree of freedom for saddle-4 is the short-bond r as can be seen in Fig. 12 where this saddlepoint appears as a maximum.

The normal mode frequencies are estimated tobe ω3 = 1585 cm−1 for the short-bond local mode,ω1 = 743 cm−1 for the long-bond local mode, andω2 = 400 cm−1 for the bend. They are in an approx-imate ω2 : ω1 : ω3 = 1 : 2 : 4 resonance.

The evolution of the normal mode frequencieswith increasing energy in a nonlinear potential isbetter represented in a projection of the continua-tion/bifurcation (C/B) diagram [Allgower & Georg,1993] in the (E,ω) plane of those families of periodicorbits associated with the normal modes [Weinstein,1973; Moser, 1976], named fundamental or princi-pal families [Farantos, 1996; Ishikawa et al., 1999;Joyeux et al., 2002]. In Fig. 13 the three principal

3.5 4 4.5E / eV

200

400

600

800

ω (

cm-1

)

S3/2S1

SN1A

SN2A

SN3A

SN4A

B1

B1A

SN1C

SN2C

SN3C

S1A

SN5ASN1B

SN2B

s-1 s-2 s-3 s-4

Fig. 13. C/B diagram of the 1B2-state of ozone. S1 denotesthe long-bond mode, S1A its double period bifurcation. S3the short-bond mode, B1 the bend mode and B1A an earlysaddle-node bifurcation of B1. The symbol SN denotes cas-cades of saddle-node bifurcations. SNiA is related with theS1 family, SNiB with S1A family whereas the SNiC with theB1A. The arrows mark the energies of the saddle points (s-i)in the potential. PO families related with the s-2 have notbeen located.

families of periodic orbits are shown and they aredenoted as S1 for the long-bond stretch, B1 forthe bend and S3 for the short-bond stretch. B1exhibits an early saddle-node bifurcation, B1A, atabout 3.475 eV with a tiny energy gap, whereasa double period bifurcation is found for the S1(S1A) at about 3.78 eV. As we can see in this fig-ure the three most anharmonic families, S1, S1Aand B1A, develop cascades of saddle-node bifur-cations as energy approaches the dissociation limit(saddle-1) for the SNiB, the saddle-3 for the SNiCcascade, and saddle-4 for the SNiA, respectively.We have not searched for the PO which point tothe saddle-2. The saddle-node bifurcations are ofsimilar type as found in the quartic potential. Thefrequency of the parent family levels off as energyapproaches the critical energy where bifurcationoccurs, with the appearance of a saddle-node bifur-cation one branch of which inherits the character-istics of the parent family. The whole scenario isrepeated at higher energies resulting in a cascadeof such SN bifurcations. We do not distinguish thestability of the PO in this bifurcation diagram. Inmost cases the anharmonic branch is stable, at leastfor some energy, and the other branch is unstable,and therefore it is difficult to continue in energy.However, for a three-dimensional system, like ours,a SN bifurcation may show one single unstable andone double unstable branch instead, as is depictedin Fig. 11. For larger molecules with more degreesof freedom the number of combinations of courseincreases.

Representative PO at energies about 3.8 eV (ifthey exist) are depicted in Fig. 12. From this figureit becomes apparent that the double period bifur-cation (S1A) and the SNiB families of PO are thosewhich are directed towards the dissociation channel.

In Fig. 13 the point symbols denote the quan-tum mechanical frequencies obtained from theenergy differences of adjacent eigenstates whosewave functions can be clearly assigned. Theyare state overtone progressions of the short-bondstretch (S3), bend (B1A) and long-bond stretch(S1). The corresponding eigenfunctions have wellrecognized nodal structures, and therefore the num-ber of excitation quanta in each mode can easilybe assigned as (0, 0, v3) for the S3 states, (0, v2, 0)for B1A and (v1, 0, 0) for the S1 states. In orderto approximately account for the zero-point energy,the quantum results are shifted to lower energiesby the zero-point energy. The energy differences areplotted with respect to the upper level.


Representative eigenfunctions with their as-signment are shown in Fig. 14 and more in [Quet al., 2004a, 2004b]. The wave functions are plottedin symmetric Jacobi coordinates. The plots showthe |Ψ|2 for a specific iso-probability density surfaceviewed along γ′ axis, in the direction perpendicularto the (r′, R′) plane. To see their correspondenceto the periodic orbits which have been assigned, inFig. 15 we plot the periodic orbits in the symmetricJacobi (r′, R′) plane and for fixed angle γ′ = 111◦.The ranges of the coordinate intervals are the sameas in the wave functions.

In the symmetric Jacobi coordinates the S3 pro-gression (short-bond mode) has mainly excitationalong the perpendicular direction (γ′). The bend

Fig. 14. Representative wave functions which show the char-acteristic localization in configuration space of the main over-tone progressions. Symmetric Jacobi coordinates are used.The plots are viewed along the γ′ axis, in the direction per-pendicular to the (r′, R′) plane. The values of axes are forr′ [3.0, 7.0] a.u. and R′ [0.5, 3.0] a.u. Shading emphasizes the3D character of the wave functions.

Fig. 15. Projection of the potential function and periodicorbits in the (r′, R′) plane in symmetric Jacobi coordinates.The angle γ′ is kept fixed at 111◦.

progression from the beginning is associated withthe SN bifurcating family B1A and not with theprincipal bending family which is B1. As we can seein Fig. 13 there is a good agreement among the clas-sical and quantum mechanical B1A frequencies formost of the assigned energies. However, differencesare found for the S1 progression. These correspondto the long bond stretching states. The agreementamong S1 PO and quantum frequencies is satisfac-tory up to four quanta of excitation (4, 0, 0). At the(5, 0, 0) state systematic mixing with another typeof states sets in. Obviously, the (7, 0, 0) state shownin Fig. 14 is influenced from the double period bifur-cation of the S1 family, S1A. This explains the devi-ation of the quantum frequencies from the classicalcurve S1, in Fig. 13.

Unfortunately, for higher energies the mix-ing in the wave functions becomes significant andthat prohibits their visual assignment. However,we expect that wave functions which are directedtowards the dissociation channel are associated withSN bifurcations and particularly with the SNiBfamilies. Indeed, at energies close to the dissociationbarrier we found localized eigenfunctions similar tothe SNiB PO (see Fig. 14). It is difficult to assignadjacent levels for this family of states, and thus,the frequency of approximately 200 cm−1 given tothis state in Fig. 13 is only an estimate.

In previous studies, such as HOCl [Weishet al., 2000] and HOBr [Azzam et al., 2003], the


association of quantum mechanical overtone pro-gressions from the bottom of the well up to the dis-sociation threshold, with saddle-node bifurcationshas clearly been shown. Recently, in a study of theCH2(a1A1) molecule we have found unequivocallyassigned progressions of quantum states which leadabove the isomerization and dissociation barriers[Farantos et al., 2004; Lin et al., 2005], and theyalso follow the tracks of SN bifurcations. The agree-ment is not only in the frequencies but also amongperiodic orbits and eigenfunctions.

The saddle-node bifurcations which approachan equilibrium point have the characteristics of theSN bifurcation observed in the two parameter quar-tic potential, i.e. it springs off a main family of POwith a gap in the energy. In the case of molecules thesecond parameter is generated from the coupling ofthe vibrational modes. Indeed, we can show that thecascade of SN bifurcations is the result of high orderresonances between two vibrational modes. How-ever, SN bifurcations are also found above the bar-riers of isomerization/dissociation [Farantos, 1996].There is evidence that they result from the unstableperiodic orbits originated from the saddle point andthe Newhouse wild hyperbolic set [Newhouse, 1979;Borondo et al., 1996; Contopoulos et al., 1996].These periodic orbits seem to have no connectionto any main family and they appear abruptly likethe cubic polynomial type.

The cubic and quartic models seem to fit withour experience in locating SN bifurcations in molec-ular Hamiltonians. The quartic type, those whichare associated with a parent family, are foundby monitoring the curvature of the continuationcurve of the parent family. Continuation is car-ried out using the period as a parameter thanthe total energy. Then the derivatives dω(E)/dEand d2ω(E)/dE2 signal the approach to the criticalenergy of bifurcation. By giving a proper incrementto the period we overcome the gap and join thebranch of the SN family which is the continuationof the parent family. In the continuation schemethe starting initial conditions are those of a peri-odic orbit before the gap with period T0 and thenPOMULT converges to a new PO with a periodT0 + ∆T in one of the two branches of SN bifurca-tion. This is how we can find several members in acascade of SN bifurcations.

The cubic type is not associated with a par-ent family but it appears in the neighborhood ofstable/unstable entanglements of a principal unsta-ble periodic orbit which always exist above saddle

points of the potential [Moser, 1976]. Thus, we ini-tially propagate the stable and the unstable man-ifolds for some energies, and then search for loworder periodic orbits in their neighborhood.

5. Conclusions

Figure 4 shows the typical phase space structureclose to the SN bifurcation of a 1-D one param-eter cubic potential; a stable region surrounded byunstable trajectories. This reveals the importance ofsaddle-node bifurcations. Their sudden appearancein phase space with increasing energy may createstable regions even if the dynamics is chaotic atlower energies. If the stability region is compara-ble to the volume of �

N , then semiclassically weexpect the localization of an eigenfunction. The SNelementary bifurcation is generic and robust, per-turbations in the potential do not vanish a SN bifur-cation, as in the cubic potential the introduction ofone additional parameter (α �= 0) does not changethe structure of the phase space. For molecules wehave found two kinds of saddle-node bifurcations ofperiodic orbits. The first kind is associated to a par-ent family. The second seems to be related to thestable and unstable manifolds of unstable periodicorbits.

Another important elementary bifurcation isthe Hamiltonian–Hopf bifurcation [van der Meer,1985] which is encountered in three-dimensionalsystems and in systems with higher dimensionality.A Hopf-like bifurcation is expected when the twopairs of eigenvalues of the monodromy matrix areout of the unit complex circle (see Fig. 11). Acety-lene is one example for which such bifurcations havebeen found [Contopoulos et al., 1994; Prosmiti &Farantos, 1995]. In the future we plan to investi-gate further Hopf-like bifurcations in relation to thedynamics and spectroscopy of molecules.

Acknowledgments

Support from the Greek Ministry of Education andEuropean Union through the postgraduate programEPEAEK, “Applied Molecular Spectroscopy”, isgratefully acknowledged.

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