angle-dependent cross-sections which Theory of Resonances are more or less structured. Sharp fea­ in Molecular Systems tures in a cross-section are usually asso­ ciated with the production of long-lived R. Lefebvre, Orsay intermediates. The characterization of (Laboratoire de Photophysique Moléculaire du CNRS) these unstable species also leads to the consideration of resonances. Thus we Resonances are met in many aspects of molecular dynamics such as may say that, whether obvious or unob­ photo- and predissociation of molecules and complexes, unimolecular vious, resonance identification is of con­ decay of polyatomic species and reactive scattering. Recent advances in cern in most atomic or molecular stu­ the theory of resonances has led to a better understanding of the nature dies. of resonant states in terms of associating to them quantum numbers reflecting the appropriate (sometimes unobvious) degrees of freedom. Which Definition for a Resonance? Whatever definition is adopted for a The resonance concept is present in physics but deals with recent advances resonance there is always the idea that a all fields of physics. An unstable particle in the theory which were initiated by complex energy is involved, which has is a manifestation of the resonance phe­ consideration of some molecular pro­ the form E = Er - iΓr/2. Since the fre­ nomenon. To take another example in a blems. In this varied and fast developing quency factor of a wave function is completely different area, a surface subject some selection has to be made exp(- (i/h)Et) such an energy produces plasmon in a solid can be described as a which is naturally guided by personal in­ in the density (squared modulus of the damped solution of Maxwell's equations terests. Thus no mention will be made of wave function) a factor exp(- Γrt/h). and the word resonance is again appro­ recent important advances which in­ We can therefore associate with a reso­ priate. An intense activity has been re­ volve electron motion (such as electron- nance a lifetime τr = h/Γr and a decay cently devoted to this concept in the molecule scattering or photoionization). rate Kr = Γr/h. However the preparation field of molecular physics. This is due We shall concentrate on molecular (or of a resonant state and the observation primarily to the use of experimental de­ heavy particle) motion in a few atom of an exponential time dependence of vices (essentially atomic or molecular system. the decay requires generally this reso­ beams and lasers) which allow for a con­ nance to be well isolated from others. siderable improvement in the quality of When are we finding resonance? Two resonances are said to be isolated the information obtained from a study of After centre of mass separation, the (or non-overlapping) when the diffe­ collisions or half-collisions involving wave equation of an atomic or molecular rence in the real parts of their complex atoms and molecules. The measured system is said to possess solutions cor­ energies largely exceeds their respective energy, time and angular distributions responding to either bound states or imaginary parts (to be called in short require often the consideration of the scattering states. In a bound state the their widths). resonances which mediate the proces­ particles (electrons and nuclei) are con­ Most of the traditional approaches to ses. There has been simultaneously con­ strained to remain in a finite region of the resonance problem consist in solv­ siderable progress toward a rigorous configuration space. The scattering ing the wave equation for real energies, theory of resonance states which are states allow for the description of colli­ this being supplemented by some proce­ described by solutions of the wave sion processes. The situation is in fact dure to determine the width. One merit equation under a special type of boun­ not as clear-cut as this. All excited of recent developments is the possibility dary conditions. Molecular physics with bound states are subject to radiative that is given of reaching directly their its large variety of interacting entities decay. It may also happen that the complex eigen-energies, even in condi­ and forms of potential gives many op­ bound character is the result of an ap­ tions where other approaches are bound portunities of applying these new tools. proximation and the real system is able to fail. We will start by recalling some of A favourable circumstance in this under­ to dissociate by the loss of electrons the real energy procedures. taking is the existence in quantum mole­ (ionization) or of atomic or molecular There is a formal tool which is almost cular physics and quantum chemistry of subunits (dissociation). Even the ground universally present in problems of mole­ a tradition of high numerical accuracy in state is not stable since collision with a cular dynamics : the coupled channel ap­ the manipulation of large basis sets and photon may lead to dissociation. Thus proach. The degrees of freedom of the the integration of coupled differential bound states are just a zeroth order view system are separated into two types: equations. This review is not centred on of decaying states, that is resonances. the coordinate (a distance) which des­ the experimental manifestations of the On the other hand, collision experiments cribes the relative motion of two entities resonance phenomenon in molecular lead to the determination of energy or and all other coordinates. The relative motion is usually the difficult part of the Fig. 1 — One and two- problem. For instance in a collision bet­ channel description of reso­ ween an atom and a diatomic molecule, nances. In both cases an in­ we can assume we know how to des­ cident wave associated with cribe vibration of the diatom and the an open channel can reach rotations (either of the diatom or of the the well, either by tunnelling system as a whole). This information is through the barrier, or be­ used to write the complete wave func­ cause of channel interaction tion in the form which splits the wave into ψ(q,r) = Σφn(q,r) Un(r) wavelets. The corresponding resonances are either called where r is the inter-fragment coordinate shape (case (a)) or Feshbach (from atom to centre of mass of the (case (b)). diatom in this example), q denotes col- 4 lectively all remaining coordinates and n is a label referring to the known informa­ tion. The dependence of the φn 's on r allows for the possibility of having the in­ ternal motion depending on the relative position (adiabaticity). One term in the Fig. 2 — Left: schematic spectrum of a one-channel Hamiltonian. E0 is the threshold energy and the thick segment symbolizes the continuum of scattering states. The resonance ener­ series may sometimes provide already a gies marked by crosses are "hidden". They cannot be calculated just by letting E be of the reasonable description of the dynamics. form Er - iΓr/2. The equation for the relevant function, Right: spectrum of the rotated Hamiltonian. The continuum is rotated clockwise and the say U(r), is found to have the form of the resonance energies are now visible. Two bound state energies marked by dots are unaffected radial Schrödinger equation. Retention by the operation. of two terms leads to two coupled dif­ ferential equations. One may in this way fore the possibility of defining a wave = k1/k0. After complex rotation, the build one-, two-, ... n-channel descrip­ function which has a purely outgoing wave function becomes tions of the dynamics. character, with an eigen-energy which is Uρ exp(iθ) ~ A(k) exp[- iKρ cos(θ -β )] Examples of potentials which may a complex number. This is the definition exp [Kρsin (θ- β)] correspond to one or two channels are of resonant states given by Siegert, + B(k) exp[- \Kp cos(θ —β)] shown in Fig. 1. Although a solution of which is of course not limited to our sim­ exp [- Kρsin (θ -β )] the wave equation exists for an arbitrary ple situation with only one open channel If sin(θ -β ) > 0 the unwanted (ingoing) energy above threshold energy E0, at (that is with only one channel with a component is now increasing so that its some particular energies the back and threshold below collision energy). The detection is made easier. The outgoing forth reflection of an incident wave wave function for large r behaves as component on the other hand is de­ reaching the well may lead to construc­ exp[ikr] = exp[ikor] exp[k1r]. creasing as ρ → + ∞. This situation is tive interference. The large amplitude The divergence at infinity is the beha­ the same as that met for a bound state. which results can be made the basis for viour expected for a decaying state: Beyond the relevant turning point, the a method for determining resonance observation of the density (or current) at bound state wave function for an ar­ energies (still real at this stage of the a distance r corresponds to breaking bitrary energy has both growing and work). From one resonance to the next events which occurred in the past at the decaying components. The quantized above in energy, one additional node will origin, when there was more 'activity' in energies are those energies which lead be present in the wave function in the the source. to disappearance of the growing term. region of the well. In the two cases de­ In some rare cases, S(k) is known This effect of using a complex variable picted in Fig. 1, the wave function U(r) analytically so that the resonance ener­ to solve second order linear differential behaves for large r as C(k)sin [kr + δ(E)] gies are immediately available, but the equations with outgoing boundary con­ with k2 ~ E. A change by π of the quan­ more common situation is to define ap­ ditions was already exploited by Hartree tity 5(E) (the phase shift) indicates a proximations yielding analytical forms during World War II in a study of radio change in nodal structure. Still another for S(k) (many of the semi-classical wave propagation. The method was re­ definition is based upon looking at the treatments are of this type). Finally there discovered years later in quantum me­ maxima in the time delay given by τd = is the task of determining as accurately chanics, first applied in a heuristic way 2h dδ/dE. This is the additional time it as possible the resonance energies once and then put on a firm mathematical takes for a wave packet to move across a form for the wave function has been basis for certain classes of potentials. the interaction region as compared to chosen. The complex rotation method The theory examines the spectral pro­ free motion. All these definitions are described in the next section provides a perties of the transformed Hamiltonian. easy to apply for isolated resonances very efficient and by now very popular Its spectrum is compared to that of the and in practice they give practically iden­ method for doing this. untransformed Hamiltonian in Fig. 2 in tical results. Each has its own recipe for Complex Rotation and Complex Eigen­ the single channel case. The important calculating the width. values point is that the continuum (the energies Turning now to the methods which Briefly speaking, the complex rotation of the scattering states) which starts at lead directly to the complex energies, we technique amounts to looking at the the threshold energy E0 is rotated clock­ start from the observation that in the consequences of substituting p exp(iθ) wise by angle 29, so that the resonant asymptotic region, U(r) is proportional to (with p real) instead of r into the wave energies which are below the real axis a combination of two exponentials equation. The effect of rotation can be are now discrete eigenenergies of the (describing the ingoing and outgoing appreciated by starting from the form of new Hamiltonian. The consequence is waves) : the wave function in the asymptotic that their eigenfunctions are localized. U(r) ~ [- exp(-ikr) + S(k) exp(ikr)] region when the coordinate is real. The This localization opens the route to two The scattering amplitude S(k) = exp function with a wave number of the form alternatives. The first procedure con­ (2i<5) is a number of modulus unity for a ko - ik1 can also be written as sists of developing the resonant wave real energy since <5 is real. However we U(r) ~ A(k) exp[- ik0r] exp[- k1r] function on a basis of localized functions can try to look for values of k (which + B(k) [exp ik0r] exp[k1r]. (harmonic oscillator functions, gaus- have to be complex) such that the Resonance states correspond to choos­ sians, etc.). The solution of the wave outgoing term dominates the ingoing ing k in such a way that only the out­ equation is then replaced by the diago- term. An extreme situation is a value kr going term is left. This means trying to nalization of the Hamiltonian matrix. of k such that S(kr) is infinite (kr is said to detect the vanishing of the ingoing com­ Because of truncation effects with a be a pole of S(k), that is S(k) has a factor ponent, which is normally dominated finite matrix the eigenvalues are found (k - kr)-1). A class of these poles is (because of the factor exp[-k 1r]) by the all to depend on 9. However it is observ­ found to be of the form kr = k0 - ik1 diverging outgoing component. Let us ed that some of them pause at some par­ with ko and k1 positive. We have there- write k in the form K exp ( —iβ) with tan β ticular values of 9 corresponding to the 5 dary condition. The energy is to be ad­ provide a good example. For the less ob­ justed until there is continuity of the vious cases semi-classical procedures wave function and of its derivative. This have been developed which extend to technique can be shown to be very the resonances an argument used for powerful and accurate. One may, for bound states. This consists in determin­ example, investigate the resonances of a ing approximate constants of the motion repulsive exponential potential A which allow for the application of quan­ exp ( - α r) with A, α > 0, a case hard tization rules similar to the Bohr-Som- to approach in terms of constructive in­ merfeld one. Finally advantage can be terference effects. One may prove also taken of the fact that there is an inherent that to a given potential there cor­ flexibility in the coupled channel ap­ responds generally a rich spectrum of proach. One may look for a system of resonant states. As an example, Fig. 4 coordinates which would decouple as gives a so-called 'string of resonances'. much as possible the channels so as to The resonance energies may have ima­ identify shape or Feshbach resonances. ginary parts which exceed considerably This idea has been successfully applied their real parts. Here again the strict ap­ to the analysis of resonances detected plication of a boundary condition pro­ in reactive scattering calculations (not duces results which go far beyond a sim­ to speak of experiments where the sub­ Fig. 3 — A θ-trajectory calculated for the ple intuitive picture. Such resonance ject is still in its infancy). The characte­ Ar-H2 van der Waals complex. Two poten­ states are however not as useful are rization of resonances in this case tials disposed as those of Fig. 1(b) are invol­ those of the more common type, al­ amounts to a study of the transition ved, with the diatom either in its lower or though they should be used in recons­ state of the theoretical chemists and it third rotational state. The energy is stabiliz­ ed for θ close to 0.07 rad. The width read on tructing the scattering amplitude from has even been suggested that this may the imaginary axis can be directly correlated its poles. lead some day to a 'spectroscopy of the with the spectral broadening mechanism transition state'. called rotational predissociation. Quantum Labelling the Resonant fulfilment of the condition ∂E/∂θ = 0. It States Stabilization Graphs and their Analyti­ is reasonable to identify these energies An interesting by-product of complex cal Continuation as being those of resonant states. Fig. 3 rotation is the possibility of associating We end this review, by pointing out gives an example of such a dependence quantum numbers in a rigorous way that the link between real and complex of the energy upon the rotation angle (a with the resonances of the one channel energy approaches can be made in a so-called θ-trajectory) for a van der problem. For bound states it is possible very suggestive way through the so- Waals complex (see J. Reuss and S. to introduce an auxiliary function W(r) called stabilization graphs. We have Stolte, EN, January (1985) p. 9). obeying a nonlinear differential equation seen that at a (real) resonance energy A different procedure consists in a (Milne's equation) in such a way that for the wave function has a large amplitude step by step numerical integration of the h → 0, one has the correspondence in the interior region. The matrix of the radial Schrödinger equation (or of the W -2(r) → k(r). The quantity W-2(r) has Hamiltonian can be diagonalized in a coupled channel equations) with intro­ sometimes been called quantum mo­ basis of integrable functions. If the basis duction of the appropriate boundary mentum. A rigorous quantization condi­ functions are made to depend on a sca­ conditions. In a radial problem these tion for motion on the r axis from - ∞ to ling factor (such as the parameter η of an conditions are vanishing at the origin + ∞ is exponential function exp(-ηr)) varia­ and an asymptotic behaviour which tion of this parameter is somewhat should be derived from the Siegert boun- The analogy with the (approximate) equivalent to changing the size of a box Bohr-Sommerfeld rule is striking; n+1 which encloses the system. The func­ Fig. 4 — The string of resonances associa­ instead of n + 1/2 accounts for motion tion approximating the resonance state ted with the potential 10 r2 exp(-0.5 r). The beyond the classical turning points. For is much less sensitive to this change vertical dashed line shows the position of the top of the barrier. Integer quantum num­ resonant states the theory can be simi­ than are the functions approximating bers can be associated rigorously with these larly developed after the wave equation the scattering states. Fig. 5 gives a typi­ resonances. has been written in terms of the variable cal diagram of energies versus the scal­ ρ exp(iθ). A function W[p exp(iθ)] is to be ing parameter. The resonant state calculated first and a rigorous quantiza­ shows up as the stable root (hence the tion condition of the same form with W expression stabilization graph), which in place of W is obtained which asso­ however is subjected to the non crossing ciates successive integers to the reso­ rule. The avoided crossings indicate pro­ nant states of a given potential (for in­ ximity of a complex value of η for which stance to the resonances belonging to ∂E/∂η = 0, with Ecomplex. Since a com­ the string displayed in Fig. 4). plex scaling factor η = ηo exp[i0] is No rigorous method exists for labell­ equivalent to the use of the variable η0r ing the resonances when several de­ exp[iθ], we are back to the complex rota­ grees of freedom are involved. A simple tion method (or at least a variant of it). case is that of moderately broadened This procedure has been successfully Feshbach resonances where it is natural applied to the determination of reso­ to label the resonances with the quan­ nance energies (and therefore of decay tum numbers of the bound states. rates) in systems of coupled oscillators Weakly bound van der Waals complexes which may simulate the behaviour of

6 small polyatomic molecules with a large excess of vibrational energy. An interes­ ting output of such calculations was the demonstration that even at these high energies there may exist resonant states with widely different life-times. This is important for the theory of unimolecular Fig. 5 — A stabiliza­ decay since this shows that we may tion graph for the po­ sometimes expect so-called mode-spe­ tential 7.5 r2 exp (- r). cificity in internal vibrational redistribu­ The eigenvalues of a tion of energy. matrix are plotted as a function of a (real) Suggested Reading scaling parameter. Resonances in Electron-Molecule Scatter­ The avoided cross­ ing, van der Waal's Complexes and Reactive ings can be exploited Chemical Dynamics, ed. D.G. Truhlar, ACS to determine the re­ Symposium Series, 263 (1984). sonance energy.

university of groningen the netherlands asks: professor/senior-scientist m/f in atomic physics (vac.nr. 850101/2493)

at the Department of Physics. The successful applicant is supposed to take on the daily responsibility for existing research in The appointed candidate will be associated with atomic physics and to initiate new programs in the Kernfysisch Versneller Instituut (K.V.I.) in atomic physics or related fields using the Groningen. research facilities of the K.V.I. He will participate in the teaching activities of the Department of The K.V.I. in Groningen is a national center for Physics. Nuclear Physics research that is jointly sponsored by the University of Groningen and the Foundation The appointment will carry a maximum salary of for Fundamental Research on Matter, F.O.M. The f 7.996,- per month (Dutch Civil Servants Code). regular staff of the K.V.I. consists of 75 persons, of which 35 are scientists. Applications, including a curriculum vitae, a list of publications and the names of three referees, In addition to the research in nuclear physics with should be addressed to the Director of Personnel, the light and heavy ion beams of the K = 160 MeV Rijksuniversiteit Groningen, P.O. Box 72, cyclotron of the K.V.I., there is an active program 9700 AB Groningen (The Netherlands), within a in atomic physics and related subjects. period of 4 weeks after date of publication, This research is based mainly on the use of quoting reference number 850101. beams of highly ionised atoms which are directly obtained from an ECR ion source. It is performed Further information on this position can be in close collaboration with groups from outside obtained from the chairman of the search the K.V.I. At this moment there are two committee: prof.dr. A. van der Woude, K.V.I., experimental set-ups available for studying telephone number The Netherlands, 50-115736. electron capture processes in ion-atom collisions, and one for studying the interaction of slow, Those wishing to draw attention to potential highly ionised atoms with surfaces. In addition candidates are welcome to contact the committee there is at the K.V.I. research on stepped surfaces chairman. by scattering of low-energy ions (LEIS).

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