Faraday Discuss. Chem. SOC., 1991, 91, 31-46 Diffuse Structures and Periodic Orbits in the Photodissociation of Small Polyatomic Molecules Reinhard Schinke,* Klaus Weide and Bernd Heumann Max-Planck-Institut f u r Stromungsforschung, 0-3400 Gottingen, Germany Volker Engel Fa kulta t f u r Ph ysik, Herman n - Herder3 tr, 3, Albert - Lud w igs- Un iversi ta t Freibu rg, D- 7800 Freiburg, Germany The relation of diffuse vibrational structures in UV-absorption spectra of (small) polyatomic molecules and internal vibrational motion in excited electronic states are investigated. The method of choice is the propagation of time-dependent wavepackets with the autocorrelation function serving as the link between the energy dependence of the spectrum and the time dependence of the molecular motion in the excited electronic state.For the purpose of this paper we characterize diffuse structures as very short-lived resonances with 'lifetimes' of the order of at most one internal vibrational period. In particular, we study a model system for the photodissociation of symmetric triatomic molecules ABA such as H20, C 0 2 and 03, the photodis- sociation of H 2 0 in the second continuum, and the fragmentation of H2S. The existence of unstable periodic orbits and their influence on the dissoci- ation dynamics is especially elucidated. In the case of H2S we demonstrate that the diffuse absorption structures are caused by symmetric stretch motion in a binding state which is strongly coupled to a dissociative state. Diffuse structures can be regarded as very broad resonances in excited electronic states.They manifest transition-state spectroscopy in the original sense of the word. 1. Introduction Many UV-absorption spectra of polyatomic molecules exhibit so-called diffuse vibra- tional structures, i.e. structures which are relatively broad and not well resolved. Fig. 1 shows two typical examples, namely the absorption-spectrum-of water in the two lowest bands. The corresponding excited electronic states, A 'B1 and B 'Al, are both dissociative and correlate with H('S) + OH(211) and H( 'S) + OH(*Z), respectively. Both spectra are composed of a broad background with superimposed weak undulations. The background indicates fast and direct dissociation on a timescale much shorter than an internal vibrational period, whereas the undulations manifest temporary excitation of an internal mode in the excited complex.The books of Robin' and Okabe2 contain many similar examples of diffuse structures in absorption spectra of polyatomic molecules. The central spectroscopical questions are: 1, what type of internal molecular motion is concealed by diffuse structures? 2, What is the timescale for internal trapping? 3, What is the appropriate 'zeroth-order' picture? Despite their simplicity, absorption spectra like those shown in Fig. 1 are very difficult to reveal and the likelihood to incorrectly assign the diffuse structures by using an oversimplistic model is actually high. The very weak undulations superimposed to the first abs9rption band of water were originally ascribed to excitation of the bending mode in the A state.3 Detailed theoretical studies employing an accurate potential-energy surface (PES), however, have undoubtedly proved this assignment to be wrong.4 The 3132 Difuse Structures and Periodic Orbits 1 I I I I I I ( a ) base \in( I I I I 1 I I I I 150 160 17 0 180 190 120 130 140 h /nm Fig.1 ( a ) Comparison of the measured3 (- - -) and the calculated4 (-) absorption spectrum of H20 in the first continuum. Theory and experiment are normalized at the maximum. ( b ) Comparison of the measured (- - -) and the calculated’ (-) absorption spectrum of H 2 0 in the second continuum. The arrow indicates the threshold energy for the production of H + OH(’X) structures superimposed to the second continuum of water were attributed to bending excitation as well.3 Although this interpretation is in principle correct, the picture which evolves from a recent dynamical analysis5 substantially differs from the conventional spectroscopic conception.Usually one interprets distinct structures in absorption or emission spectra in terms of normal modes and assumes that the coupling between the various degrees of freedom is weak. Any assignment presumes, in one way or another, that the full multidimensional Schrodinger equation describing the nuclear motion can be approximatel? cast into separate one-dimensional equations (adiabatic separation); this allows one to label each peak in the spectrum by a set of quantum numbers ( u , , v2, u3, . . . ) which specify the degree of excitation in each mode, e.g.stretching, bending etc. This procedure is undoubtedly correct and extremely prolific in extracting information on the structure of the excited electronic state from the absorption spectrum, provided the underlying assumptions, namely small displacements from equilibrium and/ or adiabatic separabil- ity, are indeed valid? The spectra for bound-bound transitions with total energies well below the dissociation threshold of the upper-state PES can be analysed in this way. However, as the excitation energy increases, the molecule performs oscillations of larger and larger amplitudes and the assumption of separability breaks down, i.e. theR. Schinke et al. 33 coupling between the various modes becomes significant and cannot be neglected. The conventional spectroscopic picture gradually loses its applicability and at last, when the classical motion is chaotic, it becomes useless.As the photon excites the molecule to still higher energies, above the dissociation threshold, at least one of the coordinates becomes unbound and the complex ultimately dissociates. Nevertheless, the excited molecule may live long enough to allow the development of broad structures in the spectrum, even for energies high above the dissociation threshold. Since the motion is unbound, normal coordinates are completely impractical and an exact solution of the multidimensional Schrodinger equation using scattering, i. e. Jacobi coordinates, is compulsory. The aim of this article is to demonstrate that diffuse absorption structures, which from the ordinary spectroscopic point of view look rather uninteresting, may hide very rich molecular motion in the excited electronic state.However, in order to reveal the beauty and the complexity of this internal motion it is absolutely necessary to have a good deal of information about the PES on which the dissociation proceeds and to perform exact dynamical calculations. Simple models with a few adjustable parameters may satisfactorily reproduce the spectrum, but that does not guarantee that these models are realistic and that we have learnt anything about the actual motion in the excited state. 2. Theory Absorption spectra can be calculated in the time-inde~endent”~ as well as the time- dependent’,’’ approach of photodissociation. In the time-independent picture we solve the stationary Schrodinger equation ( H - E)*ex(E,f) = O (1) in the excited electronic state (index ex) for fixed energy E = Ei+ ho, where Ei is the energy in the ground electronic state and Aw is the energy of the photon.The index f distinguishes via the appropriate boundary conditions the possible product channels. Assuming a weak light-matter interaction the various partial absorption cross-sections for absorbing a photon with energy ho and generating the products in channel f are calculated by the Golden rule expression’ ‘,12 where qgr(Ei) is the nuclear wavefunction in the ground electronic state (index gr); the transition dipole function p is usually assumed to be independent of the coordinates and therefore it is in most cases omitted. Summing over all final product channels, yields the total absorption cross-section, i.e.the spectrum. alternatively the time-dependent Schrodinger equation In the time-dependent approach of spectroscopy and photodissociation’~l’ we solve where a,,,( t ) represents the wavepacket evolving in the excited electronic state. Eqn. (4) is solved subject to the initial condition a,,,( t = 0) = pqgr( Ei). The total absorption cross section follows then from the Fourier-transformation34 Difuse Structures and Periodic Orbits Fig. 2 Schematic illustration of the two-dimensional PES for a symmetric triatomic molecule ABA for fixed bending angle. R , and Rz denote the two A-B bond distances. The shaded area at short distances indicates the Franck-Condon region and the two arrows illustrate the main dissociation path of classical trajectories starting in the FC region with S( t ) being the autocorrelation function The autocorrelation function contains the entire history of the wavepacket, for example, how fast it escapes from and how often it recurs to its place of birth. Eqn.( 5 ) relates the time-dependent dynamics in the upper state to the energy dependence of the absorption spectrum and is therefore ideally suited to analysing vibrational structures in the spectrum. The time-independent and time-dependent approaches are completely equivalent and comprise the same basic postulates (weak light-matter interaction); they merely provide different views and methods to calculate absorption cross-sections. 3. Photodissociation of Symmetric Triatomic Molecules As a first example of diffuse structures we consider the photofragmentation of a symmetric triatomic molecule ABA such as C02, O3 and H20.Owing to the symmetry, the excited molecule can dissociate in two identical ways: A+ BA and AB+ A. Fig. 2 depicts a typical PES of the LEPS type, for fixed ABA bending angle. It has a sadd!e at short A-B bond distances and two identical product channels. The PES of H,O(A 'Bl), for example, has for each bending angle an overall behaviour of this type.4 The standard coordinates used to describe the bound motion in the ground electronic state are the symmetric and asymmetric stretch coordinates as illustrated in Fig. 2. In the course of the fragmentation the asymmetric stretch mode turns into the dissociation mode and the symmetric stretch coordinate becomes the vibrational coordinate of the AB fragment.Upon excitation in the Franck-Condon region (indicated by the shaded area) the wavepacket o r alternatively a swarm of trajectories, if we treat the photodissociation classically, will immediately slide down the potential slope at the inner wall of theR. Schinke et al. 35 2.0 3.0 4.0 5.0 EIeV Fig. 3 Calculated absorption spectra for the model C 0 2 system as a function of the energy in the excited state. The vertical lines in (a) indicate the eigenenergies ( nss = 0, 1,2, . . .) of the symmetric stretch (ss) motion in the excited electronic state. The spectra in (b) and (c) are calculated with the same excited-state PES but using different equilibrium bond distances for the ground electronic state in order to ‘magnify’ the low- and high-energy branches of the spectrum.Adapted from Ref. 4 saddle and disappear in the two product channels as indicated by the heavy arrows in Fig. 2. The dissociation is fast and direct and leads to a broad absorption spectrum. The time-independent calculations of Kulander and Light,13 considered as a model for the photodissociation of C 0 2 , confirmed this view. Fig. 3( a ) depicts the absorption spectrum obtained from a time-dependent wavepacket cal~ulation;’~ it agrees essentially with the original spectrum reported by Kulander and Light. The spectrum is rather broad, but superimposed with diffuse structures which, especially on the blue side of the spectrum, look quite irregular. ‘Magnification’ of the high-energy side by shifting the ground-state equilibrium inward to smaller bond distances [Fig.3( c)] actually reveals a quite erratic looking spectrum. We emphasize that these structures are real and not caused by a numerical artifact! Several attempts have been made in the past to analyse this spectrum, however, only with modest S U C C ~ S S . ‘ ~ ~ ’ ~ - ” Following the models of Pack’’ and Heller,”36 Difuse Structures and Periodic Orbits 0 40 80 120 t/fs Fig. 4 Autocorrelation function S ( t ) corresponding to the spectrum shown in Fig. 3(a). T , , T2 and T3 denote the periods of the unstable periodic orbits shown in Fig. 5 ( a ) , ( c ) and ( d ) , respectively which were actually established before the calculations of Kulander and Light, the main progression, indicated by the vertical lines in Fig.3 ( a ) , have been ascribed to excitation of symmetric stretch motion on top of the barrier of the excited-state PES. This assignment is correct, as the following discussion will elucidate, but it explains only part of the diffuse structures. What type of internal motion causes the additional structures, especially in the spectrum shown in Fig. 3(c)? The picture becomes substantially simpler in the time-dependent approach. Fig. 4 depicts the autocorrelation function corresponding to the spectrum of Fig. 3( a ) . The rapid drop from the original value S ( 0 ) = 1 to zero within a few femtoseconds reflects the immediate ‘dephasing’ of the wavepacket due to the strong acceleration towards the saddle point.The main part of the wavepacket actually follows the route indicated by the arrows in Fig. 2. After ca. 35 fs, however, the autocorrelation function shows three well resolved recurrences with small amplitudes; they show that a small portion of the evolving wavepacket does not dissociate directly but recurs to its place of birth. A certain fraction of the quantum-mechanical wavepacket or of the swarm of classical trajectories is trapped in the inner region for at least one internal period. Fourier- transformation of each recurrence separately yields Gaussian-type ‘spectra’ modulated by cos ( 2 n / T.). Since the three recurrence times TI, T2, and T3 are incommensurable the addition of these partial ‘spectra’ explains the complicated structures superimposed on the broad background, which stems from the major peak of the autocorrelation function at t = 0.In this way we can rationalize the diffuse structures in terms of a series of well resolved but unrelated recurrences with periods T,. However, we really understand this generic fragmentation process only if we know what kind of molecular motion causes the recurrences. The time-dependent wavepacket itself is not prolific in this case because the portion that dissociates directly obscures the fraction that is temporarily trapped. Moreover, the wavepacket contains simultaneously all energies as well as all different types of internal motion and this superposition additionally complicates the analysis. The incommensurability of TI, T2 and T3 suggests that three different forms of short-time trapping are actually involved in the dissociation process.Classical trajectories provide the real understanding of the molecular motion. Launching trajectories randomly in the transition region and following their evolution reveals that, as expected, the majority dissociate directly with a high degree of vibrational excitation of the CO fragments. However, a small fraction do not immediately escape but return once to the FC region. A more careful analysis brings to light three genericR. Schinke et al. 37 2 3 2 3 R,l% R,l% Fig. 5 Unstable periodic orbits for the model C02 system. The energy is 2.5 eV for each trajectory. The heavy dot at short bond distances indicates the Franck-Condon point. For a more detailed discussion see the text unstable periodic orbits which, loosely speaking, guide some of the randomly started trajectories.Periodic orbits are unique features of the upper-state Hamiltonian; they live for ever, despite the fact that their total energy is high above the dissociation threshold. On the other hand, the relevant periodic orbits are very unstable and fragile; the slightest distortion rapidly destroys the perfect periodicity and initiates rapid dissoci- ation. Fig. 3( a), ( c ) and (d) display the three special trajectories relevant for the present model system; the energy corresponds to the maximum of the absorption spectrum in Fig. 3(a). The first periodic orbit is actually very simple and known for a long time from the work of Heller;" it represents symmetric stretch motion on top of the saddle between the two product channels. Its period T, agrees exactly with the time of the first recurrence.The analysis of scattering resonances in exchange reactions like H + H2 -P H,+ H has laid bare the corresponding 'asymmetric stretch' or 'hyperspherical' periodic orbit shown in Fig. 5(6);*' it has motion 'perpendicular' to the symmetric stretch periodic orbit. Since it is well separated from the FC region, where the motion in the upper state begins, this orbit by itself, however, cannot explain the two remaining recurrences. For a recurrence to occur the wavepacket must return to its place of birth at the inner wall of the saddle region, and this is possible only if symmetric stretch motion is involved as well. Thus, the asymmetric stretch periodic orbit can support a recurrence only in combination with symmetric stretch motion.Fig. 5 ( c ) and (d) depict the two simplest types of periodic motion that combine symmetric and asymmetric stretch motion. The corresponding times elapsed between the start at the inner slope of the PES and the first return to the FC region, T2 and T,, agree exactly with the two later recurrences observed in Fig. 4. The periodic orbits influence the dissociation in the excited electronic state in the following way. If we launch randomly a large number of trajectories in the FC region some of them might begin their journey very close to one of the periodic orbits. If the38 Difuse Structures and Periodic Orbits 'displacement' (in four-dimensional phase-space) is sufficiently small these trajectories stay for at least one period in the proximity of the periodic orbit and manage to return to their place of birth. There they start again with new initial conditions and most likely they succeed to dissociate.The periodic orbits act like guidelines for the trajectories rushing down the hill towards the product channels. Since they are highly unstable only trajectories that start extremely close to them have a chance to become temporarily trapped. According to the semiclassical theories of Gutzwiller,21 Balian and Bloch,22 and Berry and Tabor,23 based on the path-integral formulation of quantum mechanics due to Feynman and H i b b ~ , ~ ~ the classical periodic orbits guide the quantum-mechanical wavepacket in the same way. The extremely good agreement between the recurrence times of the quantum-mechanical wavepacket on one hand and the periods of the classical periodic orbits on the other hand provides sufficient evidence that the periodic orbits are indeed the real cause for the diffuse structures superimposed to the broad absorption spectrum.The amplitudes of the recurrences of S( t ) and consequently the amplitudes of the diffuse structures reflect the stability and the robustness of the underlying classical periodic orbit. Excitation of symmetric stretch motion on top of the barrier between the two identical product channels as predicted by Pack" and Heller," rather than bending motion as assumed by Wang et aZ.,3 causes the very weak diffuse structures in the first continuum of water [Fig. 1 ( a ) ] .The calculated autocorrelation function25 has a small recurrence after ca. 20 fs, which clearly stems from symmetric stretch motion of the quantum- mechanical wavepacket. The bending angle was fixed in this calculation. Actually, including the bending motion additionally blurs the diffuse structures because the barrier height and therefore the energies of the symmetric stretch mode depend slightly but noticeably on the HOH bending angle.4 The absorption spectrum of ozone in the Hartley band resembles qualitatively the model spectrum shown in Fig. 3( c ) . * ~ Using the a6 initio PES of Sheppard and Walker,27 Johnson and Kinsey2' related some of the recurrences of the autocorrelation function, obtained by Fourier-transformation of the measured spectrum, to various types of unstable periodic trajectories.Neither is the experimental spectrum fully resolved (thermal broadening), nor is the calculated PES one hundred per cent correct, which makes the direct relation between the recurrences on one hand and the periodic orbits on the other hand disputable. In this regard see also the exact wavepacket calculations of Le Qu6r6 and L e f o r e ~ t i e r ~ ~ using the PES of Sheppard and Walker. To make the analysis most rigorous one needs a full three-dimensional PES of high quality3' and exact dynamical calculations. 4. Large-amplitude Bending Motion in the Photodissociation of H,O(fi) The photodissociation of water in the second continuum, % 'A, --+ fi 'A, provides a second example for the usefulness of unstable periodic orbits. The experimental spec- trum shown in Fig.1 (6) exhibits a regular undulating structure superimposed on a broad background, wkich has been ascribed to excitation of high overtones of the bending motion in the B state.3 The explanation goes as follows: the ground state is bent with an equilibrium angle of a,= 104". Since the excited state is linear, a,= 180°, the one-fiimensional Franck-Condon overlap with the zero-point bending wavefunction in the X siate is appreciable only if the bending motion is higbly excited_. Strong quenching of the B state, caused by non-adiabatic coupling with the A and/or X state, significantly broadens each bending band and the result is a broad overall absorption spectrum with a regular progression of spikes sticking out of the background. The diffuse structures represent the remnants of the individual bending lines. This conventional spectroscopic picture rests on the adiabatic separation of the bending motion from the stretching degrees of freedom.In accordance with this picture, Wang et aZ.3 uniquelyR. Schinke et al. 39 60L I I I I \ \ I I \ I I I 1 2 3 4 5 6 Fig. 6 Two-dimensional PES for the fi ‘A, state of water. a is the HOH bending angle and RH--OH is the distance from one H atom to the OH fragment. The other 0-H bond distance is fixed at the equilibrium separation in the ground electronic state. Energy normalization is such that E = O corresponds to H+OH(’Z). The potential is based on the ab initio calculations of Theodorakopoulos, Petsalakis and Buenker, Chem.Phys., 1985, 96, 217. The cross marks the equilibrium in the ground state and the ellipse indicates the corresponding zero-point wavefunction. The heavy arrow illustrates the main dissociation path for the quantum-mechanical wavepacket or a swarm of classical trajectories. The dashed curve represents the unstable periodic orbit discussed in the text for an energy of 0.5 eV above the dissociation threshold labelled each peak of the spectrum by a bending quantum with the maximum correspond- ing to v 2 = 11-12. This picture is essentially wrong! The equilibrium configurations in the k and in the fi state are significantly displaced not only in the angular coordinate but also-in the H-OH stretch coordinate. Fig. 6 shows a two-dimensional representation of the B-state PES as a function of the H-OH bond and the HOH bending angle a ; the other 0-H bond is fixed at its value in the ground electronic state.The deep well in the linear configuration arises from a conical intersection with the ground electronic st_ate; it dominates the entire dissociaticn dynamics. The equilibrium separation in the X state is 1.8a0 compared to 3a, in the B state. As a consequence of this immense displacement in both directions the absorption of the photon is confined to en_ergie_s slightly below and mainly above the H+OH(*C) dissociation threshold. The X -+ B transition is a bound-continuum rather than a bound-bound transition! In this high-energy region, however, the bending motion does not adiabatically separate from the stretching motion and any decoupling scheme is meaningless.Below the threshold the (bound) motion is chaotic (in terms of classical mechanics) and above the threshold H20 dissociates40 Difluse Structures and Periodic Orbits rapidly into products H + OH( ’Z). The two-dimensional bound-state wavefunctions for energies in the re ion of the dissociation energy clearly prove the adiabatic separation Fig. 1 (6) compares the experimental spectrum with an exact quantum-mechanical, time-dependent wavepacket calculation using the two-dimensional PES of Fig. 6.5 The calculation includes two degrees of freedom: one of the H-0 bonds, which becomes the dissociation coordinate, and the HOH bending angle. It reproduces the overall shape of the measured spectrum remarkably well, especially the broad background and the superimposed diffuse structures. The amplitude of the undulatips at the onset of the spectrum is artificially large because strong quenching of the B state, caused by non-adiabatic coupling to the lower electronic states, is not taken into account in the model. The corresponding autocorrelation function (Fig.4 in Ref. 5 ) reveals a weak recurrence after ca. 40 fs whose amplitude is of the order of only 5% of the initial value for t = 0. This single recurrence causes the diffuse structures on top of the broad background. As in the foregoing case discussed in section 3, the exact quantum-mechanical wavepacket does not bring to light in a clear way what kind of internal motion is actually responsible for the recurrence. Running a few classical trajectories again yields real insiKht.The majority of trajectories, ca. 95%, starting randomly at the steep slope of the B-state PES above the ground-state equilibrium immediately slide down into the deep potential well, cross the linear H -0- H configuration, and dissociate directly, as indicated schematically by the heavy arrow in Fig. 6.32 These trajectories lead to extremely high rotational excitation of the OH(2Z) products, as known from the earliest experiments on the photodissociation of water in the second continuum.33 The remaining trajectories do not dissociate directly but perform, on average, one large-amplitude bending and stretching oscillation in the potential well. They swing to the other side of the well and return, like a boomerang, to their starting position.After the first recurrence they make a second try with different initial conditions and dissociate. This type of motion evinces strong coupling between the bending and the stretching degree of freedom. Taking into consideration only the bending mode is inadequate! In view of the foregoing section it is no surprise that an unstable periodic orbit ‘guides’ those trajectories which are ‘trapped’ for one period in the well before they manage to e ~ c a p e . ~ One example is depicted in Fig. 6. Although this periodic orbit has a total energy of 0.5 eV above the dissociation threshold it does not break apart but lives for ever. Its curvature clearly manifests the coupling between bending and stretching motion. As the ejected trajectories are drawn into the deep potential well the periodic orbit, pictorially speaking, captures a minor fraction of them in its proximity.Since the periodic orbit is very fragile, however, the captured trajectories do not survive for more than one period before they escape and dissociate. Deep inside the well at low energies the periodic orbit describes pure bending motion with a small amplitude. Owing to the coupling with the stretching coordinate the curvature of the orbit increases with energy and at the same time the amplitude grows as well. This type of periodic orbit persists from low energies, where the motion is regular, to very high energies in the region of the dissociation threshold, where the classical motion is chaotic, and even up into the continuum. Its stability or robustness gradually decreases as the energy rises.There are several other types of periodic orbits, which, however, do not take part in the overall dissociation dynamics. The simplest one is certainly pure H-OH stretching motion for a fixed bending angle of a = 180”. However, the overall dissociation path runs roughly perpendicular to this orbit such that the probability for capturing a trajectory that rushes down the potential slope is negligibly small. It must be emphasized that the extremely short ‘lifetime’ of the excited complex prohibits labelling of the individual absorption peaks by quantum numbers ! The stationary wavefunctions do not exhibit any clear nodal structure which would allow such an a~signrnent.~’ to be inadequate. 5R. Schinke et al. 41 Segev and S h a p i r ~ ~ ~ were the first to attempt ajynamical interpretation of the diffuse structures in the absorption spectrum of H20( B).They performed two-dimensional calculations in the time-independent approach using the PES of Flouquet and H ~ r s l e y . ~ ~ The spectrum of Segev and Shapiro exhibits a mixture of broad and very narrow resonances which were attributed to pure bending motion supported by a shallow rim at short H-OH distances. This interpretation is essentially different from ours; Segev and Shapiro did not consider the possibility of trapping inside the deep potential well. 5. Non-adiabatic Effects in the Photodissociation of H2S The absorption spectrum of H2S in the first continuum provides another interesting example of the relation between molecular motion in excited electronic states and diffuse structures.Like the spectrum of H20(A), the spectrum of hydrogen sulphide shows a progression of weak undulations on top of a broad backgr~und'~~'' which, as in the case of water, had been ascribed to excitation of internal bending motion in the upper electronic state.36 This assignment, which essentially rests on the approximate agreement of the energy spacing (1118 cm-') with the bending frequency in the ground electronic state (1290 cm-I), was still used in the late eighties to explain some unexpected findings for the rotational state distributions of the e S fragment.38 After the detailed theoretical analysis of the dissociation of water in the A state4'*' it also appears believable to relate the diffuse structures to excitation of the symmetric stretch motion on top of the saddle between the two identical product channels.This conjecture has been put forward by Xie et aZ.39 Both the interpretation in terms of bending excitation and in terms of symmetric stretch motion on the barrier of a dissociative PES are wrong in the light of recent a6 initio calculations. The actual dynamics of the photodissociation of H2S are much more involved. Recent a6 initio calculation^^^ unambiguously show that, unlike the photodissociation of water, the dissociation of H2S involves two excited electronic states. In C, symmetry they both have 'A" symmetry and therefore crossing of the corresponding PES is forbidden in the adiabatic representation. In CZv symmetry, however, they have different electronic symmetries, B1 and 'A2 , and the resulting potential curves actually can cross.The lower adiabatic PES is dissociative if one H-S bond is cleaved and the upper adiabatic PES is bound. For a fixed bending angle of 92" the two surfaces cross twice on the CZv symmetry h e , at ca. 2.5 a, and at 3.4 a,. The crossing in the inner region occurs right at the transition point and therefore it makes a rigorous dynamical treatment extremely difficult (and challenging, of course). In water the binding excited state is much higher in energy and therefore it does not at all take part in the excitation in the first continuum. Our earlier dynamical calculations40 were performed in the adiabatic representation with the HSH angle fixed at 92".The coupling strength between the two Born-Oppen- heimer states was adjusted to reproduce roughly the diffuseness of the spectral features. The coordinate dependence of the coupling was completely unknown at that time. In the meantime we extended the previous study with respect to both the a6 initio and the dynamical calculation^.^^ Although these new calculations are still regarded as pre- liminary, we present at this stage our present result for the absorption spectrum in order to illustrate the complexity of molecular motion that can be concealed by diffuse vibrational structures. The new ab initio calculations are performed at the MRDCI level" at ca. 2000 nuclear geometries, varying both H-S bonds as well as the bending angle In this way we construct two adiabatic PES and the corresponding transition dipole surfaces with the ground electronic state.The lower PES, shown in Fig. 7(6) foz a =92", is dissociative with an overall shape similar to the analogous surface for H20(A). Inciden- tally we note that the saddle between the two H + HS channels is significantly narrower42 Difuse Structures and Periodic Orbits a" 2 Q 2 3 4 5 'HS/ 2 3 4 5 R H S I Fig. 7 The binding ( a ) and the dissociative ( b ) adiabatic PES surfaces for H2S for a bending angle of 92". The energy spacing between the contours is 0.5 eV. The closed circles indicate the equilibrium in the ground electronic state. The arrow in (a) illustrates the initial motion of abound( t ) and the arrow in ( b ) indicates the dissociative motion of t ) than for H,O with the consequence that trapping in the symmetric stretch coordinate as described in section 3 is negligibly weak. Taking into account only this PES and neglecting the other electronic state does not yield diffuse structures! The other PES depicted in Fig.7 ( a ) is binding in all directions. In addition we extract from the MRDCI calculations the so-called mixing angle 0 which describes the mixing of the two leading electronic configurations; it changes from 0 to n / 2 across an avoided cro~sing.~' The mixing angle allows us to transform fromR. Schinke et al. 43 -2.0 -1.5 -1.0 -0.5 0 EIeV Fig. 8 Calculated absorption spectrum for H2S in the first continuum. us, denotes the quantum numbers of the symmetric stretch mode in the binding electronic state.E = 0 corresponds to three ground-state atoms, i.e. H + H + S the adiabatic to the non-adiabatic repre~entation~~ which, in contrast to Ref. 40, is employed in our new dynamical calculations. The coupling strength between the two non-adiabatic states is thus completely determined by the ab initio calculations. The relatively small energy separation between the two adiabatic potentials in the FC region indicates strong non-adiabatic coupling. The nuclear motion is treated in the time-dependent approach with two two- dimensional wavepackets, @bound(f) and @diss,(t), one evolving in each of the non- adiabatic states. The coupling between them is controlled by the off-diagonal element of the Hamiltonian matrix. It is completely determined by the ab initio calculations.The angular motion is described in the rotational sudden approximation, i e . we perform separate calculations for fixed bending angles a! and finally we average over CY with the square of the bending wavefunction in the ground electronic state serving as weighting function. Incidentally we note that the depcndence on the bending angle is crucial and, in contrast to the photodissociation of H20( A), it cannot be neglected. These calculations yield the absorption spectrum as well as the final vibrational state distribution of the HS fragment. They include two three-dimensional PES two three-dimensional dipole moment functions and the mixing angle, which is also a function of all three coordinates. They are parameter free! All aspects of the calculations will be published in full detail later.Fig. 8 depicts the calculated total absorption cross-section utOt( E ) with energy- normalization such that H + H + S corresponds to E = 0. The calculated spectrum con- sists, like the experimental one, of a broad background with superimposed, very diffuse structures. The spacing of ca. 1050 cm-I agrees reasonably well with the average spacing of 1118 cm-' of the measured spectrum. In contrast to the experiment, the theoretical spectrum is slightly asymmetric. In full accord with the results of van Vecn et al.,45 Xu et aZ.46 and Xie et ~ 1 . ~ ~ [but in contrast to the photodissociation of H,O(A)] HS( n = 0 ) is by far the dominant channel over the entire spectrum. In this paper we consider only the origin of the diffuse structures and defer the discussion of the final vibrational distributions to a forthcoming publication.According to our ab initio calculations the dissociation process proceeds in the following way: first, the photon excites almost exclusively the binding non-adiabatic state whereas the probability for exciting the dissociative state is negligibly small, i.e. I@bound(0))2 = 1 and I@diss.(0)l' = 0. The motion in the non-adiabatic states begins at t = 0 with the wavefunction in the bound state starting to perform an oscillation along the44 Difluse Structures and Periodic Orbits symmetric stretch coordinate as indicated by the arrow in Fig. 7 ( a ) . However, the very strong coupling to the dissociative non-adiabatic state rapidly reduces the probability in the bound state and simultaneously increases the probability for finding the system in the dissociative non-adiabatic state.At each instant an appreciable portion of @bound passes over to the dissociative PES where it immediately escapes as indicated by the arrows in Fig. 7 ( b ) . The quenching is so efficient that the wavepacket in the bound state has time for only one return to its birthplace before the entire wavepacket propagates on the dissociative PES. This leads to a single recurrence of the autocorrelation function with comparably small amplitude. This recurrence in turn induces the weak diffuse structures observed in the spectrum. According to this picture, the diffuse structures in the spectrum of H2S ar the consequence of symmetric stretching motion in a bound rather than a dissociative PES.The quenching rate, i.e. the non-adiabatic coupling strength controls the lifetime in the bound state and therefore the width as well as the amplitudes of the structures. A weaker coupling leads to sharper resonances while an even stronger coupling has the effect to additionally blur the structures. The diffuse structures reflect the eigenenergies of the symmetric stretch motion in the bond non- adiabatic state. In conclusion, the photodissociation of H2S in the first continuum can be classified as Herzberg’s‘ type I predissociation with strong coupling between the binding state, which carries almost 100% of the oscillator strength, and a dissociative state, as van Veen et u Z . ~ ~ originally surmised. Incidentally, note that a modification of the two dipole moment functions (in the non-adiabatic picture) in such a way that (a)bound(0)12/(~diss(o)12 = 0.8 : 0.2 instead of 1 : o tends to increase the spectrum at the low-energy side with the consequence that the agreement with the measured spectrum improves.It is important to note that the overall degree of vibrational excitation of HS as well as the amplitude of the diffuse structures depend sensitively on the coupling strength, the \@bound( O)l’/l@diSS(o)l* ratio, and the bending angle. The dissociation of H2S reveals a wealth of interesting internal motion in two electronic states. The picture evolving from our ab initio calculations is, in our opinion, essentially different from the model suggested recently by Dixon et d4’ These authors assume that the photon excites preferentially a dissociative state which is weakly coupled to a bound state.The vibrational eigenstates of the latter give rise to narrow resonances superimposed on the broad spectrum that originates from the dissociative state. In order to mimic the diffuseness revealed by the measured spectrum Dixon et al. introduce a phenomenological damping term. Our calculations yield automatically the right degree of diffuseness without any artificial fit parameter! Furthermore, Dixon et al. neglect the dependence on the bending angle which is not justified according to our experience. The present calculations are preliminary in so far as the rotational degree of freedom is treated in the rotational sudden ap roximation. In order to compare with the measured rotational state distributions of HS3 and with Raman spectra, which in contrast to H20 show substantial excitation of the bending the rotational degree of freedom should be treated exactly.Such calculations are in progress. The transformation from the adiabatic to the non-adiabatic representation is very crucial. The method we employed in the present calculations is one possibility. In the future we will examine other possibilities and their influence on the various observables. P 6. Discussion Diffuse structures can be viewed as very broad or short-lived resonances; they manifest short-time trapping in the intermediate region between the excited complex and the dissociation products. Diffuse structures in absorption spectra illustrate transition-state spectroscopy in the most general ~ e n s e .~ ” ~ ’R. Schinke et al. 45 Because of the extremely short ‘lifetime’ the time-independent approach is not useful for analysing and interpreting diffuse structures, except when an adiabatic separation is obvious.52 The time is too short to allow the development of a clear nodal structure which would facilitate one to classify the underlying molecular motion. The time- dependent picture, with the autocorrelation function as the link between the molecular dynamics and the absorption spectrum, is more generative. A single recurrence immedi- ately shows that some fraction of the quantum-mechanical wavepacket or the swarm of classical trajectories is temporarly trapped and returns to its starting position.Running classical trajectories provides the deepest insight into the molecular motion. They elucidate unambiguously the kind of molecular motion that the recurrences and therefore the diffuse structures conceal. Once the general behaviour of the trapped trajectories is uncovered, it is relatively easy to find the corresponding unstable periodic orbits which actually influence and guide the dissociation dynamics. Usually, symmetry considerations are very helpful to find the appropriate periodic orbits. For a given system there are often many periodic orbits, but only one or at most several of them actively take part in the dissociation dynamics. Other periodic orbits may be relevant for full collisions, for example. The instability of the orbits controls the ‘lifetime’ of the complex and therefore the diffuseness of the spectral features.Unstable periodic orbits are exceedingly important for the analysis of spectra in energy regimes where conventional spectroscopic means are not applicable anymore (large-amplitude motion of strongly coupled oscillators). A beautiful example is the hydrogen atom in a strong magnetic field (see Ref. 53 for a recent review and an extensive list of references). Molecular examples include Na354 and H f.55 Periodic orbits provide the real understanding of diffuse vibrational structures and their dynamical message. Can we attribute a specific lifetime to the diffuse structures? Our answer is negative! Let us consider, for example, the photodissociation of water in the second continuum.The vast majority of trajectories dissociate immediately on a timescale of cu. 10 fs. Very few trajectories are trapped for at most one vibrational period before they dissociate as well. Under such circumstances we find it meaningless to extract a lifetime. Diffuse structures superimposed on a broad background, like the examples in Fig. 1, look very simple and probably can be reproduced by several more or less sophisticated models, but only one explanation can be correct ! The spectrum of H2S clearly illustrates this ambiguity. 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Paper 01055726; Received 11 th December, 1990