A Periodic Orbit Bifurcation Analysis of Vibrationally Excited

Jul 16, 2010 - molecules by bifurcation analysis are those of Kellman and co-workers.2-4 ... tion theory in studying the nonlinear mechanical behavior...
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A Periodic Orbit Bifurcation Analysis of Vibrationally Excited Isotopologues of Sulfur Dioxide and Water Molecules: Symmetry Breaking Substitutions† Frederic Mauguiere,*,‡ Michael Rey,‡ Vladimir Tyuterev,‡ Jaime Suarez,§ and Stavros C. Farantos§,| UniVersity of Reims, GSMA, Moulin de la Housse, B.P. 1039, 51067 Reims, France, Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas (FORTH), P.O. Box 1527, Vasilika Vouton, Heraklion 71110, Crete, Greece, and Department of Chemistry, UniVersity of Crete, P.O. Box 2208, Vasilika Vouton, Heraklion 71305, Crete, Greece ReceiVed: April 5, 2010; ReVised Manuscript ReceiVed: June 15, 2010

Theoretical predictions and assignment of highly excited vibrational states and their organization is one of the most important challenges in molecular spectroscopy. A systematic procedure to investigate such problems is locating the principal families of periodic orbits that emanate from the stationary points of the molecule and then following their evolution with the total energy. This results in constructing continuation/bifurcation diagrams that assist in locating the critical bifurcation energies and to discover new types of vibrational modes. Another parameter that may influence the dynamics of a molecule is isotopic mass substitution. In this article, we investigate the effect of symmetry breaking by isotopic mass substitution of triatomic molecules with C2V symmetry in classical and quantum dynamics. Sulfur dioxide and water molecules in their ground electronic state are studied by employing accurate potential energy surfaces. Continuation/bifurcation diagrams of periodic orbits are constructed by varying the energy and the mass of one oxygen atom of sulfur dioxide and one hydrogen atom of a water molecule. The transition from normal-to-local mode vibrations is studied in terms of a pitchfork to a center-saddle elementary bifurcation of periodic orbits. The results presented in this article aim to help the assignment of experimentally obtained spectra. Introduction Bifurcation (branching) phenomena, i.e., the appearance of new motions and the change of geometry of the old orbits by varying the energy or other parameters of the system, are wellknown in vibrational spectroscopy. For example, the transition from normal-to-local mode oscillations, first discovered in symmetric ABA molecules, can be understood in classical mechanical phase space as an elementary pitchfork bifurcation.1 Among the first studies of the phase space structures of triatomic molecules by bifurcation analysis are those of Kellman and co-workers.2-4 Employing mainly effective two-mode Hamiltonians with one or two resonance terms, they have analyzed the bifurcation diagrams of several small molecules. Most importantly, using the spectroscopic constants in the Hamiltonian as control parameters Xiao and Kellman3 have constructed the catastrophe map with which the dynamical behaviors of several types of coupled vibrations can be understood in terms of elementary bifurcations. The significance of introducing bifurcation theory in studying the nonlinear mechanical behaviors of the molecules is the generalization of phenomena such as normal-to-local mode transitions and the comprehension of the spectroscopic resonances. As a matter of fact, the well-studied Darling-Dennison and Fermi resonances are particular cases †

Part of the “Reinhard Schinke Festschrift”. * Corresponding author. Address: Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas (FORTH), P.O. Box 1527, Vasilika Vouton, Heraklion 71110, Crete, Greece. Tel: +30 2810 545058. Fax: +30 2810 391305. E-mail: [email protected]. ‡ University of Reims. § Foundation for Research and Technology-Hellas (FORTH). | University of Crete.

of nonlinear resonances being inevitably introduced in nonlinear mechanical systems. Elementary bifurcations5 are very common in excited polyatomic molecules with the simplest one, the center-saddle (CS), being ubiquitous. HCP was the first molecule where CS bifurcations were identified spectroscopically (ref 6 and references therein). Further studies on HOCl, HOBr, and HCN showed that cascades of CS bifurcations pave the way to dissociation or isomerization, as the molecule is excited along the reaction coordinate.7 In this article, we investigate the implications on molecular spectroscopy and dynamics of the substitution of an atom with its isotope by constructing continuation/bifurcation (C/B) diagrams of families of periodic orbits (POs). Particularly, we examine the impact of symmetry breaking by isotopic substitution in classical and quantum dynamics of triatomic C2V symmetry molecules. Vibrational frequencies, as well as the corresponding eigenfunctions, are calculated for water and sulfur dioxide in the ground electronic state using accurate potential energy surfaces (PESs). C/B diagrams are constructed by varying the energy and the mass of one hydrogen atom in water and one oxygen atom of sulfur dioxide molecule. The latter is treated as a continuous mathematical parameter, thus producing alchemical species, which help us reveal how mass variation affects the PO structure of the system. Isotopic substitution is a well-known method in studying reaction mechanisms and assigning spectral lines to specific chemical bonds. With respect to chemical reactions, isotope effects have mainly been attributed to the change of the zero point energy. A well-known example is the unusual dependence of ozone formation rate constants on oxygen isotopes.8 Despite considerable progress in recent works,9-11 the role of mass

10.1021/jp1030569  2010 American Chemical Society Published on Web 07/16/2010

J. Phys. Chem. A, Vol. 114, No. 36, 2010 9837 substitution in the qualitative changes of the dynamics of vibrationally excited molecular states yet remains less investigated. Vibrationally excited molecules are expected to show nonlinear mechanical behaviors, such as various types of resonances, localization of wave functions in specific bonds, chaotic unassignable states, and so forth. These drastic changes in nuclear motions can be understood in terms of bifurcation phenomena, which generate new type of vibrational modes emanated from the fundamental ones.12-15 Nonlinear classical mechanics offer a systematic way to study complex systems. The hierarchical detailed exploration of the molecular phase space structure requires first the location of the equilibrium points of the PES, and then the location of POs that emanate from the equilibria, the tori around stable POs, stable and unstable manifolds for the unstable POs,5 and even transition-state objects such as the normally hyperbolic invariant manifolds (NHIMs).16-19 We infer that POs can be considered as the first approximation to the dynamics of the molecule, with the equilibria of the Hamiltonian being the zero order approximation. Over the past years, we have constructed C/B diagrams of POs by plotting the frequency of the vibrational motions as a function of the total energy, for a plethora of triatomic molecules.6,7,13,20 All these studies demonstrate the alignment of overtone quantum wave functions along principal (fundamental) and bifurcating POs and the universality of elementary bifurcations such as pitchfork, CS, and period doubling.5 However, changes in other parameters of the Hamiltonian are expected to affect the dynamics of the molecule as well. The masses involved in the kinetic part of the Hamiltonian as well as in the PES beyond the Born-Oppenheimer approximation are among the most important parameters. The present study is distinguished from most of the previous studies on the nuclear dynamics of sulfur dioxide and water molecules by the employment of accurate PESs and the effect of mass substitution, and thus symmetry breaking, on the normal-to-local mode transition. We compare the C/B diagrams with new experimental data and also with nonperturbational quantum mechanical calculations using a large basis set. For better understanding, we construct a new type of C/B diagram based on calculating “averaged tangent angles” along the POs. Also, the changes in the phase space are compared with changes in wave functions directly computed from the molecular PES. Computational Methods POs have been located with the POMULT FORTRAN code,21 which is based on two-point boundary value multiple shooting solvers.22-24 The code has mainly been developed for locating POs in molecular Hamiltonian systems with many degrees of freedom, and it utilizes damped Newton-Raphson and secant methods. With the multiple shooting algorithms we convert the two-point boundary value problem to p-initial value problems, i.e., we search for the appropriate initial values of coordinates and momenta that satisfy the boundary conditions and the continuity equations.21 The stability of POs, i.e., how fast nearby to PO trajectories escape from it, is determined from the eigenvalues (λ) of the so-called monodromy matrix.25 For conservative Hamiltonian systems, one pair of eigenvalues of the monodromy matrix is always equal to 1. For a system of N degrees of freedom, the remaining N - 1 pairs of eigenvalues move on the complex plane by varying the control parameters, collide, and produce new POs. For systems with three degrees of freedom and higher, as energy varies, for example, the eigenvalues of stable POs

move on the unit complex circle. When the eigenvalues are out of the unit circle but on the real axis, the PO is single or double unstable, and finally four complex eigenvalues may come out of the unit circle and the PO is characterized as being complex unstable. When the eigenvalues collide, new POs emerge. Collisions at +1 signal the birth of CS or pitchfork bifurcations. Collisions at -1 give a period doubling bifurcation. Generally, every time the phase of the stability parameter of a stable PO, σ, satisfies the relation T/(2π/σ) ) m/n, with m and n being integers, then new POs are born with a period nT. The quantum mechanical calculations were carried out by solving the time independent Schro¨dinger equation with discrete variable representation (DVR)26 or variational methods. Assignment of the levels is obtained by observing the nodal structure of the corresponding eigenfunctions. Results and Discussion The normal-to-local mode transition for SO2 and H2O molecules presents two quite different cases. For sulfur dioxide, the transition occurs at high excitation energies, whereas for water an early transition has been observed. There are already several studies of the normal-to-local bifurcation of these molecules by Lawton and Child,1 Sibert et al.,27 Kellman,4 Sako and co-workers,28,29 and Prosmiti et al.30 Results of quasiclassical calculations have been reviewed by Child.31 Three-mode studies including the bend mode have been presented by Lu and Kellman32 and Keshavamurthy and Ezra.33 The latter article describes a detailed study for the symmetric water molecule, H2O. The investigators used a three-mode effective Hamiltonian, which, however, can be reduced to a twomode one because of the existence of a constant of motion, the superpolyad number. By employing the standard Chirikov resonance analysis as well as locating POs and 2-tori, they have proposed an assignment of the quantum mechanical vibration levels for the polyad number equal to 16. The focus of this work was mainly to extract the phase space structure and to study the classical-quantum correspondence using the simple model of Baggot for an effective Hamiltonian of the symmetric water isotopologue. However, Keshavamurthy and Ezra33 clearly state that “they extend the Baggot Hamiltonian beyond the limit of strict applicability to H2O (in particular well above the energy at which the molecule can become linear) in order to study the dynamically interesting case”. The SO2 is known to be a typical normal mode molecule.1 Because of a relatively weak resonance coupling between the symmetric stretch (ss) and antisymmetric stretch (as) modes, they keep their normal mode character up to high vibrational excitations. For this reason, most of the assignments of experimental spectra of SO2 have been carried out using normal mode quantum numbers. Yamanouchi et al.34 have carried out spectroscopic studies and calculations on the fluorescence spectra of SO2 that have allowed them to determine experimentally high vibrational levels and to find characteristic signatures of normal-to-local mode transition. Sako and coworkers29 have fitted observed vibrational energies with an algebraic effective Hamiltonian and have deduced the corresponding effective wave functions. The extrapolation of the (ss) overtone series to quantum numbers V1 ) 18-22 has shown the trend of the localization of the (ss) normal mode wave functions along the valence bonds. Prosmiti and co-workers30 have assigned this transition to local modes using the classical bifurcation analysis with the empirical potentials of Murrell and Sorbie35 and of Kauppi and Halonen.36 In the present work we use the Martin-Zuniga37 PES of SO2, which provides considerably more accurate spectroscopic results

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compared to previously available potentials. The initial ab initio surface had been computed by Martin at the CCSD(T) level with the AVQZ+1 basis. Zuniga et al.38 converted the PES to a 3D Morse-cosine expansion and optimized a few lower order parameters in order to correct the potential energy function with respect to the (as) normal mode. The final PES has a consistent behavior at sufficiently large nuclear displacements, and it is thus suitable for both spectroscopic and dynamical studies. An accurate PES for the water isotopologues constructed by Partridge and Schwenke39 has been proved to be very successful for spectroscopic applications at various spectral ranges and temperatures. The Partridge and Schwenke PES is based on extensive ab initio calculations at the ICMRCI+Q/5Z level of theory with the account of core-valence corrections and with subsequent optimization using spectroscopic data. The validity of this surface for very highly excited vibration-rotation states of the water isotopologues has been confirmed by many experiments40-44 on the emission of high-temperature media assignment of water lines in sunspots. Note that, for water, some other accurate potential functions have been recently published with different isotopologues treated separately (ref 45 and references therein). An advantage of the Partridge and Schwenke PES for the present study, is that this surface includes mass-dependent adiabatic contributions beyond the Born-Oppenheimer approximation, usually referred to as DBOC terms.46 All potential functions of various water isotopologues are thus represented by a unique self-consistent set of the potential parameters with a good accuracy. This allows studying the effects of isotopic symmetry breaking, H2O f HOD, and substitution on the vibrational dynamics. The potential of Partridge and Schwenke is expressed in terms of the bending angle and Morse radial variables, and in its final form was empirically optimized to reproduce experimentally observed rovibrational levels. Figure 1a,c gives a comparison of the classical POs projected in the valence bonds coordinate space that correspond to normalto-local mode bifurcation for the C2V isotopologues of sulfur dioxide (a) and water (c) molecules, respectively. For both molecules this bifurcation occurs in the (ss) vibration mode. Panels a and c show trajectories in the (r1,r2) bond length plane matched on the map of the PES. For energies below the critical bifurcation energy (Eb), 0 < E < Eb, all configuration space trajectories of Figure 1a for SO2 and Figure 1c for H2O are superimposed and correspond to the (ss) normal mode family. In a given molecule, the (ss) trajectories only differ by their amplitudes. This is symbolized by the thick bisector line in Figure 1a,c. For E > Eb, the (ss) family becomes unstable, and two new POs are born symmetric with respect to the bisector line: one tends to be located along r1 and the other along the r2 bond length. This localization allows defining a characteristic angle for a PO. For each orbit we fit the curve by a polynomial in the configuration space, and we calculate the average tangent to this curve. By taking the inverse of this tangent, we get an angle that characterizes the average slope of the orbit. Then for a given PO family, we can plot the variation of this deviation angle of the trajectories with the energy of the molecule (Figure 1b,d). To our knowledge, this type of diagram, which represents the classical viewpoint on the normal-to-local mode transition in such a simple and clear way, has not been used in previous PO studies. Until Eb, the angle of the trajectory slope is 45°, which corresponds to (ss) normal mode vibration. After Eb, these POs become unstable, and the angles corresponding to the two new

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Figure 1. Localization of POs along the valence bonds as energy increases after the pitchfork bifurcation, which describes a normal-tolocal mode transition in C2V symmetry molecules. Dashed curves correspond to equipotential contours of the PES. (a) POs for SO2 at several energies before and after the pitchfork bifurcation. (b) Variation of the averaged tangent angle with the energy (see text) during the pitchfork bifurcation of the (ss) PO family of SO2. (c) Plot similar to that in panel a, but for the H2O molecule. (d) Plot similar to that in panel b, but for H2O.

stable POs sharply bifurcate, one branch tending to 0° (loc1) and the other to 90° (loc2). For the H2O molecule (Figure 1d) the critical bifurcation energy, Eb, of the normal-to-local mode transition occurs at much lower energy than for SO2 (Figure 1b), which is consistent with previous studies of C2V isotopologues.1,28,29,47 Furthermore, we can see that localization in water molecule is more abrupt than in sulfur dioxide. Comparing the Eb, which has been determined by MartinZuniga PES (17046 cm-1), to those predicted by Prosmiti et al.30 (16150 cm-1 with the Murrell-Sorbie35 PES and 16639 cm-1 with the Halonen et al.36 PES), as well as the value of Weston and Child48 based on a model of kinetically coupled Morse oscillators (18229 cm-1), we can see that the MartinZuniga PES gives an intermediate critical energy. The C/B energy-frequency diagrams for the three fundamental vibration modes provide a useful overview of the classical phasespace structure of a triatomic molecule. The case of SO2 and its isotopologue 16OS18O calculated with Martin-Zuniga PES are shown in Figure 2. The continuous curves represent the principal and bifurcating families of POs. Each point of a curve denotes a PO of the nuclear vibrations with fixed amplitude corresponding to the total energy that is given at the horizontal axis. It is seen that the frequency of POs decreases, and thus the period increases with the excitation energy due to the anharmonicity of the potential. From the eigenvalues of the monodromy matrix, we deduce that the (as) principal family remains stable up to the dissociation limit, whereas the (ss) normal mode family undergoes a pitchfork bifurcation at the energy Eb/hc ) 17046 cm-1. At this energy, the phase space around the (ss) family turns from stable to unstable, and two new local mode PO families (loc1, loc2) with the same frequency are born. With increasing energy, the frequency of the two local modes families deviates to lower values with a larger negative slope. For the bending mode family in Figure 2a, we find an early change of the slope followed by a cascade of CS bifurcations

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Figure 3. Representative POs of the 16OS16O isotopologue projected on (a) the bond length coordinate plane, (b) POs of the bend family at 10 000 cm-1 and 20 000 cm-1 before the CS bifurcation and two families, CS1 and CS2, after the bifurcation projected onto the bond length, bond angle plane. Panels c and d show POs of the 16OS18O isotopologue. (c) Representative POs before CS bifurcation ((t) and (bi) families) at energies of 5000 cm-1, 10000 cm-1, and 15000 cm-1. (d) Representative POs after CS bifurcation all taken at an energy of 35000 cm-1. Labels are explained in the text. Dashed curves correspond to equipotential contours in the range [0, 45 000] cm-1 with a step of 5000 cm-1. Figure 2. C/B diagrams of the principal families of POs for (a) the C2V symmetry SO2 molecule, and (b) the Cs symmetry isotopic substituted 16OS18O molecule. The pitchfork bifurcation, which describes the transition from normal-to-local vibrational modes in the C2V symmetry, and the CS bifurcation for the asymmetric Cs species are shown. The labels are explained in the text. Points marked by triangles, diamonds, and so forth are quantum energy differences of adjacent overtones states assigned to the three vibrational quantum numbers (V1,V2,V3), which denote the number of quanta in the (ss), bend, and (as) normal modes, respectively.

(CS1s, CS2s). As it has been observed in other molecules,13 this is due to the linearization of the molecule by increasing the total energy and the existence of a potential saddle point. The points denoted by triangles, diamonds, and so forth represent quantum mechanical energy differences of adjacent energy overtone states. In Figure 2a we plot the energy differences (EV+1,0,0 - EV,0,0)/hc and (EV,0,1 - EV-1,0,1)/hc of successive quantum energy levels computed from the same PES that was used for the classical analyses. Quantum mechanical energies and wave functions of stationary vibrational states of SO2 were obtained by global variational calculations in normal coordinates. As primitive basis set for the variational calculations, we used a direct product of harmonic oscillator wave functions for (as) and bending vibrations and Morse oscillator wave functions for the (ss) mode. This allowed calculating all matrix elements of the vibrational Hamiltonian analytically without a loss of accuracy. To ensure convergence of highly excited vibrational states, more than 40 000 primitive threedimensional (3D) wave functions were included in the variational calculations. It is seen that the classical curves from the PO families smoothly interpolate these series of quantum states, thus confirming previous results30 for the C2V species. Above Eb, the quantum-mechanical energies do not follow the classically unstable (ss) normal mode family, but they are aligned along local-mode POs.

Breaking the symmetry of the molecule by isotopically substituting one of the oxygen atoms (16O f 18O), we can see in Figure 2b the transformation of the pitchfork bifurcation of the (ss) family to a CS elementary bifurcation. This bifurcation abruptly gives birth to two new families of POs: one stable (l) and one unstable (bil). The gap in energy between the pitchfork bifurcation of the C2V species and CS bifurcation of the 16OS18O isotopologue makes it difficult to connect these two bifurcations without slowly varying the mass of the oxygen. The bending fundamental in the 16OS18O molecule shows behavior similar to that of the symmetric molecule with the characteristic cascade of CS bifurcations as energy approaches the barrier to linearization. The labeling of PO families for asymmetric ABA′ molecules belonging to the Cs point group is given after careful analyses of the PO alignment in the bond length coordinate space. One can no more use (ss) and (as) labels, which are specific for a C2V symmetry group. In general, the label (bi) stands for “bisector” orientation in the (r1,r2) plane, somewhat similar to the (ss) of the parent ABA molecule, and “t” stands for “transversal” vibration, somewhat similar to the (as) of ABA. For a symmetry-breaking isotopic substitution with mA′ > mA, the label “l” stands for the motion mainly aligned along the “light” bond BA, and “h” stands for the motion mainly oriented along the “heavy” bond BA′. A comparison of coordinate space projections of POs corresponding to the C2V isotopologue of the sulfur dioxide molecule (16OS16O) with those of the Cs isotopologue (16OS18O) is given in Figure 3. Panels a and b correspond to the stretching and bending vibrations of 16OS16O, respectively, and panels c and d display the shapes of 16OS18O POs for valence bond vibrations. Equipotential energy curves are also shown in the energy range [0, 45 000 cm-1]. Figure 3c,d shows that, in the

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case of 16OS18O, the CS bifurcation induces the following changes: the (ss) is transformed to (bi) PO, which continuously turns into the heavy mode h vibration. In addition, two new families of POs are suddenly born: one stable (l) and one unstable (bil). The focus of this work is to study the effect of isotopic symmetry breaking substitution on the vibrational dynamics. It is well known that such changes may drastically alter the phase space structure and the nature of quantum states and transitions. In order to trace these changes in detail, we introduce a “virtual nucleus” by a continuous variation of the mass of one of the oxygens for sulfur dioxide or one hydrogen nucleus in water molecule.49 In other words, we consider mathematically a sequence of ABA′ “virtual molecules” having the masses mA,mB,mA + ∆m. We adopt a mass variation step ∆m ) 0.5 amu in the mass range of [16,18] amu for the sulfur dioxide molecule and a step ∆m ) 0.2 amu in the range of [1,2] amu for the water molecule. Figure 4a shows the two highest frequency principal families of POs for OSO′ virtual species, whereas Figure 4b,c shows the low and high stretch frequency principal families, respectively, for HOH′ virtual species. In both panels a and b, we can see how the initial pitchfork bifurcation of the symmetric molecules is transformed into a CS bifurcation and its evolution by varying the mass of the chosen atom. Black curves and labels in Figure 4 correspond to C2V symmetry molecules, while other colors correspond to PO families of virtual isotopologues of the Cs symmetry group computed at various ∆m values. The family emanated from (as) POs of the parent 16OS16O molecule remain stable (Figure 4a), and no bifurcation is observed. For the family emanated from (ss) POs of the parent C2V molecule, the pitchfork bifurcation turns to CS bifurcation (Figure 4a), which is also often called a saddle-node one in the literature. In this bifurcation, two families of POs are born: one that tends to the two stable (loc1) and (loc2) POs of the parent molecule, and another one that runs parallel to the unstable parent (ss) PO. There appears a gap between the pitchfork bifurcation of the parent C2V molecule and the CS bifurcations of Cs isotopologues, which are shifted to higher energies with increasing ∆m. For the case of isotopic substitutions on water molecules, we have split the results into two diagrams (Figure 4b,c) for simplicity. For this molecule, the relative mass variation ∆m/m is much larger than that in the previously considered case of 16 O f 18O substitution in sulfur dioxide, and the change in the phase space is more dramatic. The gap in energy between the pitchfork bifurcation of C2V water species and CS bifurcation of the HOD isotopologue makes it difficult to connect these two bifurcations without slowly varying the mass of the hydrogen. By introducing mass variations with increments of 0.2 amu, we break the symmetry in the Hamiltonian changing the point group from C2V to Cs. In order to clearly understand the trends in the PO geometry variation, the mass of one peripheric hydrogen atom is changed “quasi-continuously” from 1 to 2 amu as shown in Figures 4b,c. For the physically realizable HOD deuterated water isotopologue (m′ ) 2 amu), the CS bifurcation appears at a considerably higher energy, 27 500 cm-1 (Figure 4b). It is also worth observing in the same figure the leveling off of the frequency slope h1 families at high energies. This change in the energy/frequency behavior, which signals a new CS bifurcation, is shifted to lower energies as the mass parameter increases. Thus, the gap of the first CS bifurcation increases with the mass, whereas it decreases for the CS bifurcation of the h1 family. On the other hand, the effect

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Figure 4. Effect of symmetry breaking isotopic substitutions on C/B diagrams in the energy-frequency domain. A mass of one “end atom” is formally considered as a mathematical parameter in the Hamiltonian that varies quasi-continuously. (a) Stretching POs of the sulfur dioxide molecule with a mass of one “oxygen atom” in the [16,18] amu range; (b) changes in the parent (ss) family of the water molecule with a mass of one “hydrogen atom” in the [1,2] amu range; (c) transformation of the pitchfork bifurcation to center-saddle ones in water “isotopologues”.

of isotopic substitution on the (as) PO family is quite different from the case of sulfur dioxide, as can be seen in Figure 4c. As soon as the symmetry of the molecule is broken, the families (colored curves) corresponding to the (as) family follow the old (loc1, loc2) line at low energy, which constitutes the l1 part of this family (Figure 4c). As energy increases, the slope of the curve changes to make this line parallel to the old (as) family at high energies, which we call the (t) part of this family. The effect of the mass variation, (mA,mB,mA′) f (mA,mB,mA + ∆m), on the space orientation of bifurcating vibration modes in ABA′ sequences is shown in Figure 5. The bifurcation diagrams in the energy/tangent angle plane are shown in the

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Figure 5. Effect of symmetry breaking isotopic substitutions on energy-average angle C/B diagrams with the same mass variations as in Figure 4. (a) Quasi-continuous changes from 16OS16O to 16 OS18O with steps of 0.5 amu; (b) quasi-continuous changes from HOH to HOD (see text).

panel a for sulfur dioxide and panel b for the water molecule and for several masses. Tangent angles for a coordinate space projection of POs are plotted versus total energy. These angles were calculated in the same way as was discussed for the symmetric SO2 and H2O molecules at the beginning of the section. Figures 5a and 5b show the impact of symmetry breaking isotopic substitutions on the normal-to-local mode transition in these two different cases. The pitchfork bifurcation in the PO family corresponding to C2V isotopologues (the black curves) turns to the CS bifurcations for Cs isotopologues. It is clearly seen that the orientation toward the heavy bond PO (h/h1) changes continuously, whereas the light bond PO (l/l2) can appear after a large energy gap only. The bisector-oriented POs (bil) remain unstable after the critical bifurcation energy, E b. These conclusions are confirmed by quantum mechanical calculations. A sample of vibrational wave functions that we have computed with the Martin-Zuniga PES by a variational method is given in Figure 6 for 16OS16O (a) and 16OS18O (b) species. For both isotopologues the nodal lines of wave functions for these overtone series match very well the coordinate space projections of corresponding classical POs in Figures 1 and 3. More details on these series of wave functions and comparisons with POs are given in Figures SI-1 and SI-2 of the Supporting Information. In the case of the C2V isotopologue, the normal-to-local mode transition for the (ss) progression occurs in the range V1 ) 19-21 at energies of about 20 000 cm-1, in agreement with previous studies.28,30 The shapes of wave functions as can be

Figure 6. Wave functions of two isotopologues of the sulfur dioxide molecule in the bond lengths (r1,r2) plane computed variationally from the Martin-Zuniga PES.38 (a) Example of overtone (V,0,0) and combination (V - 1,0,1) states for the C2V symmetry 16OS16O isotopologue near the pitchfork bifurcation corresponding to the normalto-local mode transition; (b) 16OS18O isotopologue: example of (0,0,V) states in the (t) progression and of (V,0,0) states in the (bi/h) progressions in Figure 2b; the latter one aligns along the heavy bond mode. The (0,0,12) state is perturbed by an accidental resonance. (EV - ZPE) in cm-1 is given at the bottom for each state.

seen in Figure 6a are quite similar to those obtained by Sako et al.29 using an algebraic model. In the case of Cs isotopologues, convergence of variational calculations is more difficult to be achieved because all vibration states belong to the same A′ symmetry type. This requires larger dimension Hamiltonian matrices built on primitive functions. The increasing density of anharmonically coupled states results more often in accidental resonances involving highly excited bending states. These perturbations also make the assignment of the wave functions more complicated than in the C2V case. To our knowledge, no spectroscopic assignment for V > 3 has been given in the literature for this molecule. Our identification to classical PO series was based on the analyses of 3D space distributions of wave functions that was rather demanding in terms of the density of grid points. We were able to obtain a clear-cut assignment of the stretching state series up to ∼22 000 cm-1. For the (0,0,V3) progression, both POs and wave functions behave nearly as antisymmetric overtones of the 16OS16O molecule. For some wave functions of this series, i.e., (0,0,11), (0,0,14), and (0,0,16) (see Figure SI-2 of the Supporting Information), there appear pronounced manifestations of accidental resonance perturbations. The conclusions of the classical PO analysis are also very well confirmed for the other stretching progression, which is labeled as (V1,0,0) in Figure 6b. Up to the (6,0,0) state, the nodal lines of wave functions are nearly rectilinear in (r1,r2) projection

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Figure 7. Energy-frequency C/B diagram, quantum energy differences of adjacent overtone states and representative wave functions for the C2V isotopologue HOH of the water molecule. Upper PO family (black curves and labels): (as) PO family. The (ss) PO family exhibits the pitchfork bifurcation at about 3000 cm-1 corresponding to the transition to local modes, which are shown with pink curves and labels. The bending PO family and its CS bifurcation is shown in green (see the text for details).

plane and fit the bisector direction of (bi) POs. Then, the curvatures of the nodal line follow the classical POs, which are gradually transformed form (bi) to h type: in the energy range approaching the CS bifurcation, the slope of the wave function tends to be along the r1 bond (see Supporting Information, Figure SI-2 for details). In Figure 7 we plot the energy difference of successive levels in the (as), (ss) (converted to local modes) and bending progressions of the C2V water isotopologue as a function of the vibration energy. These plots are superimposed with the C/B diagram of POs. For a comparison of classical and quantum mechanical results, we have shifted the quantum vibration levels by the zero-point energy of 4638 cm-1. As in the case of SO2, the curves for the classical POs smoothly interpolate the quantum calculations. The bending progression (b) shown with green labels undergoes a cascade of CS bifurcations when approaching the linearization barrier estimated at about 11400 cm-1. For the (as) progression, the quantum points given by black triangles are slightly shifted down with respect to the classical PO curve at the top of the diagram. No bifurcations occur in this series both in classical and in quantum analyses as is confirmed by (0,0,V3) wave functions. As a typical example, wave function (0,0,9) belonging to the (as) progression is given in the upper-right corner of Figure 7. In contrast, the quantum mechanical (ss) progression does not follow the unstable branch of the (ss) PO family, but bifurcates along the local mode POs as is expected from the C/B diagram. This bifurcation is clearly shown by the shape of quantum local mode wave functions, with a typical example shown in the lower-right corner of Figure 7. A sample of wave functions corresponding to normal-to-local mode transition (given in the Supporting Information, Figure SI-3) confirms the results of Xiao and Kellman50 obtained with empirical algebraic models. The local mode notations [n1,n2](,V2 are more appropriate than the normal mode (V1,V2,V3) labels for vibration excitation above four quanta in this progression. Here n1 and n2 denote

the number of vibrational quanta in OH bonds along r1 and r2 coordinates, respectively. Note, that the bifurcation in quantum states is not as sharp as in the classical picture (SI Figure SI3). In virtue of the Wigner theorem, the stationary quantum states are either symmetric [n,0]+,0 or antisymmetric [n,0]-,0 under a C2 permutation of equivalent O-H bonds, contrary to the classical (loc1,2) POs. In agreement with Li and Guo results,51 the calculated doublets of quantum energies dEn ) E([n,0]-,0) - E([n,0]+,0) decrease very rapidly with n, and this is usually considered as a quantum mechanical signature of the normalto-local mode transition.27,47,52-54 These local mode doublets in the HOH isotopologue have been recently observed for very highly excited overtones by Maksyutenko et al.55 and Grechko et al.45 using sophisticated triple-resonance quantum stateselective laser spectroscopy experiments up to the dissociation limit, which is estimated at D0 ) 41 146 cm-1.55 However, contrary to the predictions of simple two-dimensional effective models, this decreasing of energy differences, dEn, in quasidegenerate local mode doublets is not monotonous. In Figure 8a we give a histogram of |dEn| versus the overtone quantum number, n, in a logarithmic scale. On the scale of the picture, there are no noticeable differences between calculations and observations. A regular decrease is perturbed at n ) 6, n ) 9, and n ) 13. As can be seen in the classical C/B diagram of H2O (Figure 7), these perturbations coincide with successive period doubling bifurcations corresponding to local-mode/ bending resonances. New POs born in these bifurcations are denoted in Figure 7 as loc-b1, loc-b2, loc-b3 and loc-b4. Representative POs are shown in Figure 8b displaying mixed stretching and bending vibrations. Examples of quantum counterparts of these local-mode/ bending resonances are shown in Figure 9. The vibrational wave function in panel b of Figure 9 has a clear assignment as the antisymmetric local mode [6,0]-,0 state at 19 780 cm-1 with six nodes along each O-H bond at the right-hand side projection.

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Figure 8. (a) Energy difference dEn ) E([n,0]-) - E([n,0]+) between local mode quasi-degenerate doublets versus the local mode quantum number (n) in logarithmic scale. The [6,0](, [9,0](, and [13,0]( doublets are perturbed by accidental resonances involving the bending mode, as shown in Figure 7. This explains a deviation from a monotonous decrease in this progression. Each time a perioddoubling bifurcation occurs, the quasi-degeneracy of the doublets is destroyed. This not only affects the energy degeneracy but also the sign of the energy difference. Colors have been used in order to emphasize the sign of the energy difference: red for positive value and blue for negative. (b) Representative POs of period-doubling bifurcations in the C/B diagram of Figure 7 resulting from a stretching local mode/bending 1:2 resonance.

This is a 1:2 resonance with the state at 19720 cm-1 which has a normal mode character and it is easily assigned as (V1,V2,V3) ) (0,6,3) by counting nodes in the angular direction (vertical axis at the left-hand side of Figure 9b), and in the (as) direction (left-hand side of Figure 9b). In spectroscopic notations, this resonance is usually referred to as ωs = 2ωb, where ωs stands for a stretching vibration frequency, and ωb stands for a bending frequency. Figure 9a shows wave functions of a very similar shape that suggests that these states are strongly mixed by an anharmonic resonance. By counting nodes, these states are assigned as symmetric local mode [6,0]+,0 state and normal mode (0,8,2) state coupled by the same ωs = 2ωb resonance. Panels c and d show the wave functions that correspond to another strongly perturbed local mode doublet [9,0]+,0, [9,0]-,0 and of their resonance partners. Classical dynamics of bending and stretching vibrations in a HOD molecule have been studied by Kim et al.56 in an approximation were the potential coupling has been completely ignored. In our analyses, a full PES including mass dependent Born-Oppenheimer breakdown corrections39 was taken into account. The C/B diagram of POs for the deuterated Cs water isotopomer together with quantum energy differences of adjacent overtone states in the corresponding progressions and characteristic examples of wave functions are given in Figure 10. The changes in classical phase space structure for HOH f HOD isotopic substitution are quite different from those of previously discussed 16OS16O f 16OS18O symmetry breaking substitution. Figure 11a,b demonstrates that the POs are rapidly oriented along the O-H and O-D bonds. The initial (as) mode of the parent HOH molecule continuously tends to the light O-H bond (denoted as l1 in Figure 11a), and the (ss) mode continuously tends to the heavy O-H′ bond (denoted as h1 in Figure 11a). For low energies, no bifurcations occur with the mass variation, mH′, in the range mH′ ∈ [1,2] amu. At mH′ ) mD, the normal mode vibrations have local mode characters near the bottom of the PES (Figure 11a). This means that the normal mode quantum number (V1, V2, V3) for labeling local mode states of HOD is justified at least up to the bifurcation energies. For energies above the bifurcation points, we use “primed” quantum numbers

Figure 9. Wave functions in quasi-degenerate local mode doublets perturbed by (loc1,2)/bend resonances: (a,b) the doublet [6,0](,0 together with their partner, with which they are supposed to be in resonance. (c,d) The same but for the doublet [9,0](,0 (see text for details).

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Figure 10. Energy-frequency C/B diagram, quantum energy differences of adjacent overtones states and representative wave functions for the Cs isotopologue HOD. Upper frequency light (l1) mode corresponds to OH vibration, and the lower frequency heavy (h1) mode overtones correspond to OD vibration. The bending mode (b) and stretching local mode families (l1, h1) undergo center-saddle bifurcations. Progressions of quantum states after the bifurcations are identified corresponding to “transversal” (t), “curved” (c), and (h2) POs (see text for details).

(V′) in order to emphasize the changes that occur in the shape of both POs and the corresponding wave functions. For the bending progression of HOD (green curves and labels in Figure 10), several CS bifurcations occur, earlier than in HOH. In the case of the high-frequency stretching PO family (blue curves and labels in Figure 10), the light bond O-H vibration (l1) turns to a “transversal” vibration (t). By increasing energy the (t) PO family behaves more and more like the antisymmetric vibration of the C2V isotopologue. This is supported by comparing their coordinate space projections in Figure 11 and quantum mechanical wave functions. Examples of such wavefuctions are given on the top of Figure 10: the (0,0,4) state at 13854 cm-1 corresponding to the (l1) PO family and the (0,0, 11′) state at 33048 cm-1, corresponding to the (t) PO family. More wave functions of (l1 /t) progressions are presented in Figure SI-4 of the Supporting Information. Two new PO families are abruptly born in a CS bifurcation. One is denoted by bil, tending to align along a bisector line on the (r1,r2) plane, and it is initially double unstable. The initially single unstable branch is denoted by l2, and it is oriented along the light O-H bond. Examples taken at an energy of 35000 cm-1 are given in Figure 11b. Similarly, for the lower frequency PO family (red curves and labels in Figure 10), the heavy bond O-D stretching vibration (h1) undergoes a CS bifurcation at nearly the same energy; however, the stability and the shapes are quite different as shown in Figure 11a,b. The rectilinear (h1) PO family turns continuously to the (c) family (“curved” trajectories in (r1,r2) plane). Among the two newly born POs, the one denoted as h2 is again stable, nearly rectilinear in the (r1,r2) plane and is aligned along the heavy bond. The other one (bih) is unstable. An example of an overtone state in the (h1) progression strongly perturbed by a stretching-bending resonance is given in Figure 12. Table 1 tabulates the energies of quantum overtone progressions of HOD assigned according to the C/B diagram of POs of Figure 10. The convergence error of vibration levels is estimated to be ∼0.1 cm-1 or better. Note that the comparison

Figure 11. Coordinate space projections of representativePOs for HOD in the bond length plane. (a) Three representative POs for local mode stretching families at energies of 5000, 10 000, and 20 000 cm-1 and before the two CS bifurcations. (b) Representative POs for all families after CS bifurcations at 35 000 cm-1. The labels of the different families originated from the trajectory geometries in configuration space.

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Figure 12. Wave function for the (10,0,0) state of HOD, which shows a strong resonance with the bending mode.

with the results of triple-resonance laser spectroscopy experiments45,55 confirm the excellent accuracy of the potential of Partridge and Schwenke,39 at least up to 30000 cm-1. For higher levels, this PES underestimates the vibrational energies typically by about 10 cm-1 in the range of 35000 cm-1. The calculated uncertainties gradually increase with energy, but a qualitative trend remains correct. The importance of knowing the molecular classical phase space structure is demonstrated in cases where physically meaningful assignments of vibrational bands are sought in spectroscopic experiments. Examples are the discussions by Theule et al.57 and Voronin et al.43 on the assignment of OH overtone states for HOD near 25 000 cm-1. Voronin et al.43 have attempted to fit the energy levels from (0,0,1) to (0,0,8) and concluded that a simple local mode model based on a twoparameter Morse oscillator “is not appropriate for describing highly excited O-H overtones” in the HOD molecule. Although, our sequence of quantum numbers in Table 1 formally coincides with that of Voronin et al.,43 one has to keep in mind that after ∼20 000 cm-1 the CS bifurcation changes the shape of quantum states, and the quantum number V3 has a different physical meaning. This is why we use the label V3′. In the energy range

influenced by the bifurcation and after it, the “overtone” states are no longer pure O-H vibration of local mode character. In classical mechanical sense the (t) POs form a new family of “transversal” collective vibrations where the heavy bond motion also contributes (Figure 11b). This is confirmed from the shape of wave functions in Figures 10 and SI-4. Furthermore, examination of the coefficients in the basis set expansion of the eigenvectors reveals such a mixing. For example, in the beginning of the overtone state series (in the range from (0,0,1) to (0,0,5)), the main contribution comes from the primitive |0,0, V3〉0 function with a coefficient of about 0.93. For (0,0,6) and (0,0,7) states, these coefficients drop to 0.77 and 0.73, respectively, whereas contributions from the basis set functions |1,0,5〉0 and |1,0,6〉0 with coefficients of 0.25 and 0.51, respectively, appear. Near the bifurcation critical energy the expansion of the (0,0,8′) state has the largest contribution from the |1,0,7〉0 basis function with a coefficient of 0.56. This explains the difficulties of assigning states of rectilinear (l1) and “transversal” (t) character by using a simple local mode model. Conclusions A PO analysis has been carried out for sulfur dioxide and water molecules. The behavior of these molecules in isotopic substitutions has been investigated by varying the mass of one oxygen atom and one hydrogen atom in small increments for each species. PESs of high accuracy for the ground electronic state have been employed. We have analyzed and compared the C/B diagrams of the principal families of POs projected in the (energy-frequency) domain. A new scheme to project C/B diagrams in the nuclear configuration space is presented by plotting the average tangent angles along the PO. The two emerging families born after the pitchfork bifurcation of the (ss) family are clearly shown in these plots, as well as the pace of divergence from the parent PO. Effective Hamiltonians provide a local description of the dynamics by incorporating those resonances among the degrees of freedom active at a certain energy range. The method we follow may be called “global” since the Hamiltonian includes a global PES which describes the total configuration space of the molecule: all bound-state isotopomers and several dissocia-

TABLE 1: Assignment of HOD Overtone States with Respect to POs Bifurcation Diagrama energy (cm-1)

assignment

Obs.

energy (cm-1)

assignment

Obs.

energy (cm-1)

assignment

Obs.

2723.41 5363.29 7917.47 10378.13 12765.94 15064.29 17280.14 19412.30 21461.47 23427.89 25312.00 27120.66 28816.47 30448.61 31992.43 33444.83 34804.66

(1,0,0) (2,0,0) (3,0,0) (4,0,0) (5,0,0) (6,0,0) (7,0,0) (8,0,0) (9,0,0) (10,0,0) (11,0,0) (12,0,0) (13′,0,0) (14′,0,0) (15′,0,0) (16′,0,0) (17′,0,0)

2723.68 5363.82 7918.17 10378.95 12767.14 15065.71

3707.91 7251.35 10632.74 13854.68 16920.68 19836.57 22623.73 25329.61 27966.12 30538.71 33048.71

(0,0,1) (0,0,2) (0,0,3) (0,0,4) (0,0,5) (0,0,6) (0,0,7) (0,0,8′) (0,0,9′) (0,0,10′) (0,0,11′)

3707.47 7250.52 10631.68 13853.63 16920.02 19836.88 22625.53

1403.52 2782.02 4145.47 5419.95 6690.37 7914.36 9086.42 10119.37 11111.92

(0,1,0) (0,2,0) (0,3′,0) (0,4′,0) (0,5′,0) (0,6′,0) (0,7′′,0) (0,8′′,0) (0,9′′,0)

1403.48 2782.01 4145.47 5420.04 6690.41 7914.32

28950.07 30743.02 32467.85

(13′′,0,0) (14′′,0,0) (15′′,0,0)

a Vibration energies E/hc are calculated from Partridge-Schwenke PES39 using 80 DVR points in Radau radial coordinates, 120 associated-Gauss-Legendre angular points, and a maximum dimension secular problem of 15 000. (V,0,0) and (0,0, V) are local mode states corresponding to heavy h1 and light l1 bond vibrations. Primed and double primed quantum numbers correspond to series after the bifurcations given in Figure 10. States (V′,0,0) and (V′′,0,0) are assigned to the (h2) and (c) families; states (0,0, V′) are assigned to the (t) family (see text for details).

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tion channels. Usually, global PESs are analytical functions constructed from ab initio electronic structure calculations and optimized to accurately reproduce experimental measurements: dissociation energies and spectroscopic transition frequencies. There are no limitations in the type of coordinate systems used to describe the PES. Although global Hamiltonians can be cast in some reduced algebraic forms, and therefore analysis of the resonances similar to that of Kellman4 can be carried out (see also refs 12-15 and 58-60 and references therein), investigation of the phase space structure based on POs and higher dimension objects is usually performed numerically revealing the rich structure of phase space and the entanglement of motions. Local effective Hamiltonians have the advantage of semiclassically quantazing the energy, and thus establishing the correspondence among overtone and combination quantum states and classical tori. This allows focusing on certain essential phenomena, which can have quite general applicability, but it is well-known that these simple models have limitations in terms of molecular dissociation properties, number of resonances included, and extrapolating them to very high vibration states and to other isotopologues. In this sense, these two approaches are complementary ones. Applications with both global and local methods13 demonstrate that their synergy provides the best means for assigning complex vibrational spectra. The constructed C/B diagrams unravel the correlation of the pitchfork bifurcation, which appears at 3000 cm-1 for the water molecule and at 17 046 cm-1 for SO2, with that of the CS bifurcation, which appears at rather high frequencies, 27 500 cm-1 for HOD and 24 860 cm-1 for 16OS18O. The frequency gap, characteristic of the CS bifurcations, quickly increases with the mass for water and in smaller pace for sulfur dioxide. Exact quantum mechanical DVR and variational calculations for solving the time-independent Schro¨dinger equation produce regular and highly localized eigenstates for the overtone states of both species and their isotopologues, from low to high energies. A good correspondence in the alignment of eigenfunctions along the POs is clearly demonstrated. This together with the frequency bifurcation diagrams unravel the importance of bifurcations as well as the complexity in assigning highly excited vibrational molecular states. Acknowledgment. The present work was financially supported by the European Union Transfer of Knowledge grant (MTKD-CT-2005-029583). The support from IDRIS CNRS and the Regional Champagne-Ardenne computer centers is also acknowledged. Supporting Information Available: Vibrational overtone and combination states of SO2, vibrational overtone stretching states of 16OS18O, vibrational doublet states of H2O, and vibrational overtone stretching states of HOD. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Lawton, R. T.; Child, M. S. Mol. Phys. 1981, 44, 709–723. (2) Li, Z.; Xiao, L.; Kellman, M. E. J. Chem. Phys. 1990, 92, 2251– 2268. (3) Xiao, L.; Kellman, M. E. J. Chem. Phys. 1990, 93, 5805–5820. (4) Kellman, M. E. Annu. ReV. Phys. Chem. 1995, 46, 395–422. (5) Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 2nd ed.; Springer-Verlag: New York, 1990; Vol. 42. (6) Ishikawa, H.; Field, R. W.; Farantos, S. C.; Joyeux, M.; Koput, J.; Beck, C.; Schinke, R. Annu. ReV. Phys. Chem. 1999, 50, 443–484. (7) Joyeux, M.; Grebenshchikov, S. Y.; Bredenbeck, J.; Schinke, R.; Farantos, S. C. AdV. Chem. Phys. 2005, 130, 267–303.

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