molecular dynamics beyond the adiabatic approximation: new experiments and … · 2012-12-05 ·...

Ann. Rev. Phys. Chem. 1985. 36:277-320Copyright © 1985 by Annual Reviews Inc. All rights reserved

MOLECULAR DYNAMICSBEYOND THE ADIABATICAPPROXIMATION:.New Experiments and Theory

Robert L. Whetten, Gre~7ory S. Ezra, and

Edward R. Grant

Department of Chemistry, Baker Laboratory, Cornell University, Ithaca,New York 14853

INTRODUCTION

At the heart of the quantum mechanical description of molecular structureand dynamics lies the adiabatic or Born-Oppenheimer approximation (1,2). Starting with an assumption of separability of time scales for nuclear andelectronic motion, a familiar picture emerges of nuclei subject to well-defined forces corresponding to the potential energy surface for a particularelectronic state. Although the well-established success of these ideas in theareas of molecular spectroscopy (3-5) and reaction dynamics (6-9) is likelyto secure the adiabatic approximation as a continuing foundation ofmolecular science, the range of dynamical processes that lies within itsscope is far from complete. Recent experimental and theoretical advances inparticular are beginning to yield a coherent understanding of severalphenomena that, far from requiring minor corrections to the adiabaticapproximation for their explanation, by their very nature exist entirelyoutside its framework. Examples of importance in diverse areas ofchemistry and physics range from the dynamics of radiationless decay (10-15) and nonadiabatic processes in chemical reactions (16-19) to spectroscopy of excited (20-23) and ionized (24-26) states of isolatedmolecules, from single collision electronic energy transfer (27-41) to exeiton(42) and soliton (43) dynamics and spin-lattice relaxation (44). Specializedreviews on these topics are available.

2770066-426X/85/1101-0277502.00

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278 WHETTEN, EZRA & GRANT

In this article we adopt a much narrower view, concentrating onnonadiabatic bound states. Considerable recent progress has been made inthis area, with a strong and very profitable interaction between theory andexperiment. By emphasizing some theoretically tractable models thatthough simple are capable of quantitatively interpreting state-of-the-artspectroscopic measurements, we hope to provide the basis for an appreci-ation of the role of nonadiabatic coupling in the wider contexts notedabove. We endeavor in particular to use the treatment of nonadiabaticstates to illuminate the nature of the simpler or less rich dynamics obtainedin the limit that the adiabatic approximation is valid.

In the past few years a number of monographs and review articles havebeen devoted to subjects closely related to the material discussed here.Among these, special attention is drawn to the following :

1. First is the recent text of Bersuker (19). Aside from being a sound reviewof the principles of molecular dynamics, this work is also extremelyimportant because it extends beyond the formal chemical physicist’sviewpoint of eigenstates and energy level patterns and makes contactwith qualitative molecular orbital theories that are of wide-spread use inorganic and inorganic chemistry. It is accompanied by an exhaustivebibliography of work up to 1979 (45).

2. The recent reviews by K6ppel et al (46, 47) draw special attention to thenovel effects encountered when more than one vibrational, modeinteracts with electronic states. From this starting point K6ppel et alhave been led to apply the statistical theory of energy level spacings tocharacterize complex vibrational manifolds, and to identify importantapplications in the field of irreversible radiationless decay.

3. Judd (48) and Carrington (49) have reviewed the qualitative form adiabatic potential surfaces in the vicinity of intersections. (See also 50-56.) This work generally uses group theory to deduce the form of thesurface, but leaves off at the point at which an explicit connection withnonadiabatic dynamics must be made.

4. Nonadiabatic processes in the solid state are covered in the classic workof Davydov (42) (see also 43), in Englman’s comprehensive text (44), in Bersuker & Polinger’s review of cooperative effects and phasetransitions (57). Reviews in more peripheral areas are cited below.

Our review of nonadiabatic bound states begins with a discussion of aquantum formalism first developed over 25 years ago. This formalism hasrecently been remarkably successful in providing a complete quantitativeanalysis of a molecular vibronic spectrum, as discussed below. However,the intractability of the quantum approach for treating larger systemsremains an outstanding problem. Experimental manifestations of bound


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NONADIABATIC DYNAMICS 279

state nonadiabaticity are reviewed next, with an emphasis on isolatedmolecules. Two exceptionally promising developments are highlighted,both relating to Rydberg states. Finally, several semiclassical theories ofnonadiabatic processes are briefly reviewed, and vibronic dynamics in thevicinity of a conical intersection are explored within the framework of theclassical electron model.

QUANTUM THEORY OF NONADIABATICMOLECULAR STATES

The quantum theory of molecular bound states beyond the adiabaticapproximation has until recently remained well ahead of experiment in theinvestigation of spectroscopic and dynamical consequences of nonadiaba-ticity. This situation is now rapidly changing, as shown by advancesdescribed in the next section. In view of the nature of the experimental workdiscussed there, attention is restricted here to effects that occur in isolatedpolyatomic molecules; this leaves ample scope for treatment of aninteresting range of new and unexpected phenomena.

A long-standing difficulty in the theory has been the lack of a simplemeans for visualizing the effects of nonadiabaticity. This has been partiallyremedied by several recent calculations (58-61a) of nuclear probabilitydensities corresponding to exact vibronic states; comparison of thesedensities with those derived from purely adiabatic functions, as discussedbelow for triazine ; and by the development of classical analog hamiltoniansfor studying nonadiabatic dynamics in multimode systems.

The discussion to follow is a brief account of theoretical work directedtoward (a) understanding the implications of electronic degeneracy and theconsequent structural instabilities in molecules such as X3 (X = Li, Cu, Na,Ag, K) or planar C5H5 and C6H~- ; (b) the nature of vibronic states in thevicinity of degeneracies, and the most useful representations for describingelectronic state evolution; (c) problems of localization due to effects higher-order couplings; (d) interpretation of complex spectra in non-adiabatic systems and extraction of nonadiabaticity parameters fromexperiment; and (e) extension to multimode problems and related pheno-mena such as radiationless electronic decay.

Reviews of the theory of molecular bound states beyond the adiabaticapproximation have been given by Longuet-Higgins (62), Kolos (63),~)zkan & Goodman (64), Bishop & Cheung (65), and Lathouwers Leuven (66). Various aspects of fully quantum mechanical approaches nonadiabatic transitions in molecular collisions have been reviewed byBaer (38, 40), Garrett & Truhlar (35), Rebentrost (36), Tully (17), Child 32), Macias & Riera (41), and Kleyn, Los & Gislason (39). Several


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definitions of the term "vibronic coupling" have been discussed in the usefularticle by Azumi & Matsuzaki (67).

General Formalism

Most of the important manifestations of nonadiabaticity are present in thesimplest case of a symmetry-based electronic degeneracy; hence weconsider this class of problem in some detail [the results are extendable totreat less symmetrical cases (68)].

In the absence of electronic degeneracy, a good approximation to themolecular wavefunction is the familiar Born-Oppenheimer or adiabaticproduct of electronic and nuclear functions. In the vicinity of an n-folddegeneracy, known to be ubiquitous in the space spanned by polyatomicvibrational coordinates (54, 69-72), it is therefore natural to invoke expansion in some n-dimensional electronic basis. These general consider-ations can be formalized as follows: The time-independent spin-freeSchr6dinger equation for an isolated molecule is written :

[TN + H,(Q)]tI’,(q, Q) = E,~P.(q, 1.

where TN is the nuclear kinetic energy operator, and H~(Q) is the remainder

of the full molecular hamiltonian:

H~(Q) = T~ + U~ + U~ + U~. 2.

Note that this electronic hamiltonian, He(Q), depends parametrically on thenuclear coordinates, denoted collectively by Q. Terms in Eq. 2 arc theelectronic kinetic energy, the intcrelectronic repulsion potential, electron-nuclear attraction, and internuclear repulsion, respectively. Separation intocoupled vibrational equations can bc cffcctcd using either of two distinctstrategies :

One involves the use of diabatic states (62, 73). The exact molecularwavcfunction is expanded as follows :

u,,.(q, Q) = ~, ~’k(q ; Qo)zk(Q), 3.

where the orthonormal electronic states ~’k arc defined by solving theelectronic Schr6dingcr equation at a chosen reference configuration Q0 :

He(Qo)~’,(q) = E°~ d/~(q).

The vibrational wavcfunctions Zk arc determined by a set of coupledequations with hamiltonian matrix elements

Hk.k" = T~bk.k" + <~kIHe(Q)l~k’>. 5.


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Note that the nuclear kinetic energy is diagonal in the diabatic basis, andthe matrix elements of He(Q) can be further expanded to give:

Hk,k, = (T~v + E~ +AUmv) ~k~, + (~,~IA Ue~v[~%,). 6.

Each diagonal element therefore defines an effective vibrational hamil-tonian, consisting of the nuclear kinetic energy operator plus the Hellmann-Feynman (74, 75) potential for nuclear motion given by the internuclearrepulsions and the attraction to the electronic charge distribution [~ff~k].These vibrational hamiltonians neglect, however, all response of theelectronic state to changing nuclear configuration. The neglected off-diagonal coupling terms derive from the AU~s term, and it is therefore thechange in the electron-nuclear attractive potential with nuclear configur-ation that induces mixing of diabati¢ basis states, ~ving rise to bothadiabatic and nonadiabatic correlation of electronic and nuclear motion.We note that greatly improved fo~s of diabatic bases for practical solutionof Eq. 1 are possible (46, 47, 7~78).

An alternative strategy is to expand W(q, Q) using a basis of adiabaticelectronic states:

V,(q, Q) = E ~m(q ; Q)Zm(Q), 7.

where the ~,(q; Q) are solutions of the electronic Schr6dinger equation

He(Q)~m(q ; Q) = Em(Q)Om(q 8.

and depend parametrically on the nuclear coordinates {Q}. Expansion 7 isthe Born-Huang series (2), and the electronic eigenvalues E~(~) define usual adiabatic potential surNces. A set of coupled equations can beobtained for the adiabatic vibrational amplitudes, ~, where the coupling isnow induced by off-diagonal matrix elements of the nuclear kinetic energy(61a,b).

Diabatic states are in many ways more convenient for practicalcalculations, because they correspond to fixed electronic configurations,whereas adiabatic electronic states and associated potential energy surNcesemerge naturally from quantum che~stry computations (41).

Two-Sla~e Syslems

The simplest case of a symmetry-induced two-fold electronic degeneracy(n = 2) interacting with a sinNe doubly-degenerate vibrational mode, suchas the trigonal-molecule E x ~ problem, is important in the interpretationof sNctra discussed in the next section, and, additionally, it ~ustrates severalgeneral and unusual consequen~s of nonadiabaticity. To first order, thecorresponding adiabatic surNces show a linear divergence from the


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degeneracy point, characteristic of a conical intersection (69); addition of harmonic restoring force leads to the familiar cylindrically symmetricsombrero picture shown in Figure 1. Addition of higher-order couplingterms breaks the cylindrical symmetry by putting three-fold symmetricripples in the surfaces. Forms of adiabatic potential sheets in the vicinity ofcrossings have been studied in the comprehensive papers of Liehr (50-52)and others (49, 53-56).

The exact vibronic levels of the model E x e problem are easily obtainednumerically (62, 79) by employing adiabatic basis of functions that aredegenerate at the point of symmetry. Denoting the pair of diabaticfunctions ~, ±, where 0 ÷ = 0*_, the coupled vibrational equations are (inreduced units)

where p and ~b are polar vibrational coordinates and k is the lineardistortion constant. The energy levels W are functions of D = ½kz (thedistortion energy) as shown in Figure 2.

The eigenstates of this system are characterized by a good half-integerquantum number j (79). This is a remarkable result that merits somediscussion.

Diagonalizing the electronic hamiltonian in the basis of the two diabatic

V

Figure 1 Adiabatic potential energy sur-faces for a degenerate electronic statelinearly coupled by a doubly-degeneratevibrational mode. Coupling splits the iso-tropic paraboloid to give the familiar con-ical intersection of the linear Jahn-TellerE x e problem.


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states results in adiabatic functions:

i~er,1 .... : [ ~- e-i¢’/2~b + + ei*/2~b_]. 10.

It is easily seen that these functions chanye sixth upon encircling the conicalintersection in nuclear configuration space, i.e. q5 -, q5 + 2~. This is a generalresult (72, 80-86). Since the complete vibronic wavefunction ~P must single-valued, this implies that the associated vibrational amplitudeszupper.l .... must also change sign upon circling the point of degeneracy. As aconsequence of this sign change, in the linear limit the internal rotationquantum number j can take on only half-integer values (1/2, 3/2,...) stated above. The distinctive character of the level spectrum that results hasrecently been confirmed for the first time by direct observation (87).

The occurrence of half-integer quantum numbers in systems without spinimplies a topological or Kramers-type degeneracy, and is currently ofinterest in a wide variety of fields. For example, we note the growingliterature on quantization (and fractional quantization) of the Hallconductance in two-dimensional solids (85). Wilczek & Zee (84) have givena general discussion of the origin of fractional quantization, and derive aformula predicting half-integer quantum numbers for two-level systems(when adiabatic energies diverge linearly from the degenerate point). Theappearance of half-integer quantum numbers is important for the im-plementation of semiclassical descriptions of nonadiabatic phenomena.

2.8-

~--~ 2.4-

~ -

0.4-

I I I I I I I I I I I I0.4 0.8 1.2 1.6 2.0 2.4

Linear Coupling Parameter, D

Fi#ure 2 Correlation diagram giving therelative energies of the exact vibronic levelsof the linear E × e hamiltonian (Eq. 9) a function of the distortion parameter,D = 1/2k2.


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In series of papers, Mead (88-90) and Mead & Truhlar (80, 91) examined the consequences of these general results in scattering andbound-state problems. Mead noted that the form of the adiabati~ functions(Eq. 10) corresponds to a particular choice of yauye in the vibrationalproblem. It is possible to eliminate the sign change of vibrationalamplitudes, but at the expense of introducing a vector potential term intothe vibrational hamiltonian. This result has been termed the MolecularAharonov-Bohm Effect because of its formal similarity to the well knowninterference effect discussed by Aharonov & Bohn (92). The work of Meadand others shows that there are subtleties and surprises in the theory ofvibronic interactions, and that care must be taken when assessing thevalidity of any approximation scheme (61).

Higher-Order Effects and Localization

Several characteristic features of the linear case just described disappearwhen quadratic coupling or other symmetry lowering is allowed. Theseadditional coupling terms are important for the E x e problem in vibronicstates for which there are large vibrational displacements. There is both aloss of the free adiabatic pseudorotation of Figure 1, and a splitting of thedegeneracies of states in which j (taken customarily to be positive unlessotherwise noted) is 3/2 mod 3. These effects have now been characterizedexperimentally by means of observedj = 3/2 splittings in trifluorobenzenecation (93) and the ion cores of benzene (94) and cyclohexane (95) Rydbergstates. A quantitative account of the effects of higher-order coupling on thepositions and intensities of optical transitions has been given for the 3slE’

Rydberg state of sym-triazine (87, 96).Inclusion of quadratic coupling terms in the E x e system leaves the

diagonal matrix unchanged but alters the upper off-diagonal element toread:

kpe-i¢, + 1/2gp2e2~¢,, 11.

where the parameter # measures the strength of the quadratic coupling. Thecorresponding adiabatic surfaces are

o~p~,~i/E(p, ~b) = 1/2p2 -~ kp 1 +~l¢ cos (3~b) + -~-~-j 12.

For small ratios, gp/k, the energies reduce to the simple form :

E = 1/2[p2 T-gp2 cos (34~)] -T- kp 13.

to good approximation, and show directly the three-fold symmetry.The splitting of degeneracies forj = 3/2 mod 3 for small g is due to above-

barrier reflection in the three-fold symmetric angular potential.


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Asymptotically for large g the lowest state derived from j = 3/2 becomesdegenerate with the ground j = 1/2 pair, corresponding to localization inthree deep minima. A small energy gap between the near three-folddegenerate levels, due to tunneling, implies strong localization and near-static distortions from the symmetrical configuration, in which casenonadiabatic effects are relatively unimportant. The magnitude of thetunneling splitting determines the time scale on which a measurement willshow the molecule to be symmetrical. The concept of localization is ofcentral importance in the interpretation of temperature-dependent ESRspectra of the alkali trimers X3 (97-99). A uniform semiclassical treatmentof tunneling splittings for deep states in the E x e problem has been givenby Karkach & Osherov (100).

Cone-States or Resonances in Two-Level Systems

The successful treatment of low-lying states of the conical intersections interms of adiabatic approximations with correct boundary conditionssuggests that a similar model might work for some higher states. Inparticular, there has been speculation dating from the work of Longuet-Higgins et al (79, 101-105) that some stationary states exist on the uppercone. [Thompson et al (61) refer to these as "cone states" whereas lower-surface states are called "hat-states."] However, a thorough analysis bySloane & Silbey (106) showed that no exact wavefunctions are accuratelydescribed by states of the upper conical surface, and K6ppel & co-workers(107-110) have extensively described the role of simple conical intersectionsin ultrafast electronic decay. Nevertheless, this idea remains attractivebecause it is known that in order to account for the facts of molecularspectroscopy, for some energy gap the adiabatic Born-Oppenheimerapproximation must work well for excited states. Indeed, these very factsmotivated Born & Heisenberg (111) and later Born & Oppenheimer (1) attempt to separate electronic and nuclear motion.

The absence of simple resonant or cone states in the Sloane-Silbeytreatment has therefore not deterred further searching. Very recently,Thompson et al (61) have found a close correspondence between exact andadiabatic energies for large distortions and higher angular momentumstates. They give, in addition, one example of an exact wavefunction of thej = 5/2 type showing remarkable localization within the classical turningpoints of the upper conical surface. Such a discovery, if shown to be general,has wide implications, because these states would be regarded as precursorsto the type of excited electronic states observed in conventional spectro-scopy, and they should be semiclassically quantizable.

This observation leads us to the final topic of this section, that ofmultimode nonadiabatic states and their relation to spectral complexityand irreversible electronic decay.


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Multimode Vibronic Interactions and Radiationless

Electronic Decay

It is not possible to generalize very easily the above discussion tomultimode systems (systems of two degenerate or degenerate plus non-degenerate vibrations) because of the occurrence of highly complex levelspacings and coupled nuclear dynamics even in deep adiabatic states. Someof the simplest unusual features were first described for trigonal moleculesby Sloane & Silbey (106) and by Scharf & Jortner (112), but a full expositionawaited the work of Cederbaum, K6ppel and co-workers (46, 47).

To see the source of the complexity, note first that the major change in thediabatic formulation above is in the off-diagonal elements, which formultiple modes become

~, k,p~ exp (-- i~b~), 14.

where {a} is the set of modes. The adiabatic energies are coupled to give thelower adiabatic surface :

2 ,L-- k,p,22-t-2 ~ k,k#pap # COS((~at--~fl)1112

where 2 are the ha~onic force constants of the modes. This evidently leadsto highly correlated dynamics even at lowest energies (motion in one mode"tunes" the conical intersection in another). For this linear-distortionapproximation the surviving quantum number J is a sum of those fromindividual modes and takes on the half-integer values characteristic of thetwo-level electronic system. The di~culty of this problem is reflected by thefacts that (a) even for only three modes, to obtain exact eigenfunctionspushes the limit of present computational capabilities, and (b) no solutionsof the adiabatic approximation, Eq. 15, have been given. It is thereforenecessary to seek statistical approaches such as that suggested by Hailer etal (113, 114) or to look to semiclassical methods. Work in the latterdirection is reviewed in the final section below.

More generally, Cederbaum and co-workers have demonstrated successin multimode problems using the diabatic representation with potentialsfitted to ab initio quantum chemical calculations (47, 76). Using theLanczos method with simplifying assumptions derived from the precisespectroscopy of the situation at hand, they have been able to treat thepreviously intractable optical spectrum of NO: (114) and the photoelectronspectrum of C2H~ (109). Their work in this area represents an importantadvance for the interpretation of complicated molecular spectra.

In C2H~, two spin-doublet states are coupled by vibrations of the D2a


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(conical-intersection) type, The surfaces in the spectroscopically observablezone are so widely split that it had previously been believed that the twophotoelectron bands wcrc due to independent adiabatic states. Krppcl &co-workers’ reassignment (109) establishes clearly that this is not the case.Moreover, model calculations show (107) that starting a nuclear wave-packet on the upper surface yields decay on a time scale of a single vibra-tional period, and offers no evidence for stationary states.

The spectral line pattern in C2H~ is very complex, and in the analogouscase for neutral NO2 a statistical analysis has been carried out to show thatnearest neighbor level spacing statistics are described quite well by theWigner distribution (115) [or, better, by a Brody distribution (116)], found in the spectra of complex nuclei and nonlinear quantum systemswhose classical limit is chaotic (117-125). For these small-molecule casesthere is thus a strong connection between spectral complexity due toconical intersections and the "decay" or radiation-emission behaviorobserved. K6ppel & Meyer (110) have pursued this point further in a recentpaper exploring the connection among fluorescence yields, ultrafast decay,and nonadiabaticity as it appears most simply in doublet molecules (wherespin interconversions are unimportant). Other than the apparent conestates uncovered by Thompson et al (61), as yet little evidence has emergedfrom theoretical work for resonance spectra in nonadiabatic modelproblems. Clearly, this will be a major line of future work. In a similar vein,conical intersections are important in mediating chemical reactions ordecompositions, and when many modes are present a statistical theory isappropriate. Toward this end, Lorquet and co-workers (126-128a) havegeneralized transition state theory to nonadiabatic states and applied it tothe decomposition of C2H~+.

EXPERIMENTAL OBSERVATIONS ON THESTRUCTURE AND DYNAMICS OFNONADIABATIC STATES

The section above outlines how one can model potential energy surfaces,and, more importantly, calculate quantum states in the vicinity of conicalintersections. The formulation is general, but the specific case most easilytreated is one in which the degenerate electronic term is well isolatedenergetically, with strong restoring forces preventing very large amplitudenuclear motions from the intersection. Such systems are most oftenassociated with single-hole or single-electron degeneracies.

Quantum states derived from this simplest electronic configuration offeran excellent opportunity to probe the coupling of electronic and nuclearmotion. The pattern of energy levels characteristic of a given system can be


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understood with the help of correlation diagrams and a knowledge of likelydeviations from simple limiting cases. It now remains to ask whatexperimental methods can provide detailed information for a range ofvibronic systems. Considering typical energy spacings between vibronicstates, it is clear that optical spectroscopy is a prime candidate, with themost direct and unambiguous information coming from isolated moleculesin a single specified initial quantum state. A sketch of the range ofexperimental systems for which nonadiabatic effects are found confirms thisexpectation.

Survey of Nonadiabatic Effects: Rationale for an

Experimental Approach

The extreme sensitivity of vibronic states to perturbations has become oneof the best known consequences of an electronic degeneracy. Both therelatively large-amplitude nuclear motions and electronic mixings areknown to be easily influenced by environment (129), as underlined by thesustained interest in the cooperative Jahn-Teller effect (130). This impliesthat if we hope to study the individual vibronic states, as opposed to bulkresponse properties of nonadiabatic bound state systems, then it is essentialthat the molecule be isolated.

The ground states of stable molecules contain an even number ofelectrons and are usually totally symmetric electronic terms. Open-shellpolyatomics of sufficient symmetry often have electronic degeneracies, butas illustrated by work on metal trimers (60, 131), the reactivities of suchsystems generally require extraordinary measures in preparation andisolation. Many transition metal complexes have degenerate ground terms,but most of these are observed only in the solid state and solution.Consequently, strong nonadiabatic effects are rare in the lower vibroniclevels of the ground electronic terms of most molecules that can be readilyisolated. At higher energies, near thresholds for chemical reactions orunimolecular dissociation, strong vibronic effects can indeed be expected.However, the associated surface intersections, although often criticallyimportant in the reaction dynamics of such systems (132), usually do not

¯ have high symmetry and involve complex and very large amplitude nuclearvibrations. Thus, one cannot look to ground terms of stable molecules as asource of few-state nonadiabatic effects.

Among the valence excited states of many molecules having somesymmetry, a great number involve degenerate or near-degenerate electronicterms. It is perhaps noteworthy that few of these known excited states havea settled-upon interpretation (see for example 11). By definition, valencestates involve the strong interaction of at least two electrons or electronsand holes. In the spectroscopy of such systems it is also common for both


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the initial and final molecular orbitals to be degenerate, so that the excitedstate contains four closely related terms, two of which are degenerate. Insuch cases a description of the consequent vibronic coupling requires atleast a four-state treatment. Similar considerations apply to the groundstates of cyclobutadiene and eyclo-oetatetraene, where the two highestelectrons derive from the same degenerate molecular orbital (133).

The extraordinary complexity of such (valence) multielectron degenerateconfigurations is reflected in the long history (134-136) of effort on thelowest electronic transition in cyclopentadienyl radical CsH5. The groundstate Of C5H5 has the simple one-hole configuration, (e~)3, but its firstexcited terms derive from the configuration (e~)Z(e~)1, which produces noless than eight coupled states from the three-electron interaction. Thenumerous attempts to assign this spectrum serve as an indication of thecomplexity of degenerate valence states in general.

The easiest way to introduce a single hole in a set of degenerate orbitals issimply to remove an electron. The resulting ionic term has no inherentreason to be strongly coupled to any other, and recent experiments confirmthat the spectroscopy of ionized states indeed offers one of the bestopportunities for finding essentially pure two-state nonadiabatic problemsin molecular bound states (25, 26, 137, 138). Thus, it is not surprising thatthe most substantial experimental developments in the understanding ofthe two-state dynamic Jahn-Teller problem have come from advances inthe spectroscopy of ions: vibrationally-resolved photoelectron spectro-scopy (138-140) and, more .recently, laser-induced fluorescence. Theformer has until very recently been limited by electron velocity resolutionto offer only marginal separation of vibronic bands. The latter has greatpotential but most ion-excited states do not fluoresce.

One of the major exceptions is the family of halobenzene cations that hasbeen studied in great detail from this point of view over the past five years byMiller, Bondybey, and co-workers (25, 26, 137, 141-144) and by Leach,Cossart-Magos, and co-workers (145-148). Hexafluorobenzene cation hasprovided one of the most comprehensive tests of the two-state Jahn-Tellermodel to date (149), although its transition system still presents somedifficulties, as one between valence states (in this case of the ion).

On balance, valence complexities, together with the related problem oflow fluorescence quantum yields, must be viewed as a fundamentallimitation of optical spectroscopy as a general means for determining thevibronic structure of ions. Thus, the greatest long-term potential for theuniversal study of ion ground states appears to lie with photoelectrontechniques, especially in their most recent form as laser multiphotonphotoelectron spectroscopies (150-152). It is now believed that a resolutionnear 10 cm- 1 will soon be possible.


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A simpler, entirely optical method of studying ions is through thespectroscopy of molecular Rydberg states, in which one electron is excitedto high-lying orbitals of the Coulomb potential (153). To a goodapproximation, the excited electron has a negligible effect on the vibronicstructure and dynamics of the ion core (see below), so that the level patternsaccessed, and even the vibronic intensities, match those of the ionic state ofthe core. Until about eight years ago, however, the study of the vibronicstructure of molecular Rydberg states was extremely limited in scope. Withthe development of high peak power tunable dye lasers in the mid-1970scame the means to perform nonlinear spectroscopy on the high-lying statesof atoms and molecules. In this realm, particularly for multiphotonionization spectroscopy, Rydberg state resonances play a featured role(154). It is therefore easy to see that the excitation to Rydberg levels molecules whose highest-lying molecular orbitals are degenerate or near-degenerate offers a most promising means of studying two-state non-adiabatic phenomena.

A major advantage of this approach is that, like photoelectron spectro-scopy, the molecule is excited from the ground vibronic states of the neutralabout which there is much prior information. Thus, the nonadiabatic effectsobserved can be straightforwardly associated with the ionized state. Themajor advantage compared with the current practice of photoelectronspectroscopy is resolution; structure of the Rydberg state is probed at the n-photon resolution of the laser (typically m I cm-1). To complete theconnection between the Rydberg spectroscopy of the molecule and thevibronic structure of the ion, it remains only to justify the above assertionconcerning the separability of the ion-Rydberg electron subsystems.

Strong Vibronic Coupling in Molecular Rydberg States

An exceedingly simple and useful way to look at vibronically activeRydberg states is in terms of a strong coupling limit for the degenerate ioncore (155). In this limit the dominant vibronic coupling is that within thecore, so that, neglecting rotation, the molecular ion and Rydberg electronform noninteracting subsystems, except for the fact that the Rydbergelectron is governed by the symmetric average field of the ion and isorthogonal with respect to core orbitals. This is clearly an idealized limit,and such a separation is only appropriate when the magnitude of thevibronic coupling in the core is much greater than both the dependence ofthe Rydberg electronic energy on nuclear positions (for small amplitudevibrations) and Rydberg-valence interactions at the configuration level.

For a given Rydberg state, the zeroth-order vibronic energy levelsappropriate to this separation are simply those of the cationic subsystem


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shifted down by the Rydbcrg term value. Band intensities in optical spectraof Rydberg states also find a match with corresponding photoelectronspectra: Within this limit electronic and vibrational moments separate, andthe vibrational part corresponds precisely to the Franck-Condon factorexpression for the set of transitions from the ground state of the molecule tothe vibronic levels of the ground statc ion.

Symmetries of these individual vibronic levels, though, arc a composite,determined by the direct product of the ion core vibronic state symmetryand the orbital symmetry of the Rydberg electron. One implication of this isthat nondeoenerate Rydberg states of a degenerate ion core will arise only aspart of an electronic quartet, which remains four-fold degenerate in theideal-Rydberg limit. This results in a substantial reduction in the number ofdistinct Rydberg series that can be expected, as degeneracies and splittingsintroduced by the Rydberg electron are considered in a coupling hierarchythat proceeds on a eore-vibronic, term-by-term basis. Recognition of thishierarchy, with its symmetry implications for optical transitions, is criticalfor the proper assignment of Rydberg states and associated vibronicstructure in symmetric molecules.

This spectral simplification is one consequence of the idealized Rydberglimit and its approximate sum hamiltonian (155, 156). Other predictionsinclude quantum defects that carry for given series from convergence withone ionization potential to all others, and an excited state lifetimedetermined solely by radiative decay. Deviation from this limit with respectto these latter observables, which in itself often constitutes a form of strongnonadiabaticity, is reviewed below. For the present, we turn to anexamination of the completeness of this separation as a window on the coreand the nonadiabaticity derived from its conical intersection.

Structure of Nonadiabatic Bound States: Benzene andSym-triazineThere are two general sets of criteria against which one can easily judge theideal Rydberg-limit as an accurate predictor of the vibronic structure ofdegenerate core Rydberg states. With an extensive data set spanning manyseries, the degree to which vibronic structure conforms, series to series, withthe highest available resolution photoelectron spectrum, together with themagnitudes of quartet splittings, will give one indicator of how far Rydbergelectron-core interactions carry the system from simple separability.Another test, for a vibronic system that is simple enough, is to checkspectral positions and intensities within a single electronic state for internalconsistency against the best available theoretical calculation. Benzene andtriazine in turn offer cases of each of these tests. At the same time these


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important systems provide interesting examples of dynamical distortions inhighly symmetrical rigid molecules, which by classical rules of bondingwould be expected to retain high symmetry (133).

BENZENE By analogy to the halobenzene cations, we can expect benzene topresent a multimode vibronic coupling problem. This is confirmed both bytwo-photon Rydberg spectra (94, 156-158) and moderately well-resolvedconventional (159) as well as laser photoelectron spectra (150). This makes a definitive confirmation and interpretation of observed vibronicstructure by direct calculation difficult. However, the size of the data setcombined with the availability of the photoelectron spectrum provides themeans for a test of the first kind.

In conformity with ideal expectations, degenerate Rydberg states of thissystem that arise in zeroth order as products of the degenerate corewith nondegenerate Rydberg orbitals (for example nsa~g), show readilyassignable structure in Jahn-Teller active and symmetric modes, and thisstructure is virtually identical to that of the photoelectron spectrum.Rydberg states derived from degenerate hydrogenic orbitals provide aneven more sensitive test. As noted above, in the limit of vanishing Rydberg-core interaction, the simple product of orbital- with core-degeneracy willdouble the degeneracy of all such vibronic levels. A great many of these areaccidental degeneracies, and any perturbation (configuration interaction orvibration-Rydberg coupling) will strongly mix zeroth-order states andproduce characteristic splittings.

Such splittings are observed, but with one exception all are exceedinglysmall (156). Typical is the 15 cm-~ splitting of the E2g-Alg components ofthe 3d~ origin. The exception is the 3px origin, for which the measuredsplitting is 160 cm- ~ (94, 157, 160), a value that perhaps reflects the specialsignificance of this state as derived from the lowest degenerate Rydbergorbital.

This generally favorable picture of the ideal-Rydberg limit for thevibronic structure of benzene Rydberg states is supported by the smallmagnitudes of other deviations, such as origin perdeuteration shifts andtrends in vibrational frequencies. Thus, it is with reasonable confidence thatthe Rydberg data can be combined with the specific though much lowerresolution photoelectron spectrum to provide a qualitative picture ofvibronic coupling in the 2Elo C6H~- ion matching in completeness that forsubstituted benzene cations. Main conclusions at the current level ofunderstanding are as follows :

1. The dynamical Jahn-Teller effect in the 2Elg benzene cation is inherentlymultimode, involving at least Jahn-Teller modes v6 and v9. In benzene-h6, v6 dominates, with a total stabilization energy of about 250 cm-


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Greater apparent v9 activity is seen for -d6, a result consistent withmultimode coupling expectations (106). In both eases the number active modes is far smaller than that for the halobenzene cations.

2. Higher order Jahn-Teller effects are quite small, leaving single-modevibronic angular momentum largely unquenched.

3. A Renner effect in the out-of-plane mode v16 has been observed andcharacterized for the first time (94, 161). The observation of interferencesin combination states with Jahn-Teller active v6, as predicted by theory(162), sheds further light on multimode quenching and coupling vibronie angular momentum in nonadiabatic states.

SYM-TRIAZINE The two-photon absorption spectrum of sym-C3NaH3shows a set of sharp vibronic bands associated with a clear origin (To 55,791 cm- 1 ; 55,882 cm-1, _ d3) (87). These bands, pictured for triazine-hain Figure 3, are readily assigned to the lowest 3s~E’, Rydberg state. With itstotally symmetric Rydberg orbital and degenerate ion core, the limitingpicture just confirmed for benzene predicts this state to be Jahn-Telleractive. Analysis shows that it is, with coupling dominated by strong activityin a single normal coordinate, the lowest frequency e’ mode, Vr, a ring-bending vibration. This simplicity presents a unique opportunity. All otherknown examples of nonadiabatic states for which resolved vibronic spectrahave been recorded for isolated molecules are multimode systems. With itsstrong single-mode conical intersection, the 3slE’ state of triazine offers achance to (a) visualize the dynamics of nonadiabatic bound state vibronicmotion in an active vibrational space of manageable dimensionality, and (b)quantitatively test the few-parameter numerical coupling theories reviewedabove, from which can also be obtained accurate vibronic wavefunctions.

Comparison of the spectrum in Figure 3 with the correlation diagram inFigure 2 shows that a description in terms of simple linear coupling withk ~ 2.2 gives a good account of level positions through the conicalintersection and above, thus allowing the confident first-order assignmentof all the bands of the spectrum. Linear coupling alone, however, fails topredict the observed intensities (the spectrum shows strong bands thatwould be forbidden for strictly linear coupling) and certain distinctivesplittings. Inclusion of quadratic terms with g = 0.046 and optimization ofk to 2.14 gives quantitative and sensitive agreement with both positions andintensities of all measured bands (96, 163).

With this coupling strength, the zero-point and first few excited levels ofthe active mode, v6, lie substantially below the conical intersection. Simplecalculation shows that the exact level structure for such conditions isreasonably well described by the eigenvalues of the adiabatic lower surfaceBorn-Oppenheimer (or more correctly Born-Huang) hamiltonian. Motion


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in these lower states can thus be viewed in very simple terms, asapproximately that of a free internal rotor built on a radial harmonicoscillator with a level structure (79)

E(v,j) = o~(v 1/2) + Aj2 16.

where ~o is the fundamental frequency and A is the rotational constant. Thelatter term is determined by the amplitude of the trough, Po = k/2, viaA = hz/2rnp2o, where for the present case Po = 0.26/~ and A is calculated tobe 80 cm-a. Importantly, it is found that this rotational constant givesa precise account of the deepest few rotor states, so long as the angularmomentum quantum number j is taken to be a half-odd integer. Thisobservation constitutes the first direct confirmation of the appropriateness

55400 -55700 56000 56.300 56600 56900 57200 57500 57800TWO-PHOTON ENERGY (CM-1)

Figure 3 Two-photon absorption spectrum of the 3sXE’ Rydberg state of sym-triazine. Thestick fioure indicates theoretical positions and intensities for bands assigned in terms ofapproximatej quantum number as : 1/2 (origin), 5/2, 1/2, 7/2, 5/2, 1/2, 11/2, reading from theleft. (Cf Figure 2; 3/2 bands are forbidden. These and symmetric fundamentals andcombinations are suppressed.)


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of the choice of phase implied by the electronic basis functions of Eq. 10 informing the proper adiabatic limiting wavefunctions. It is interesting tonote that the energy of the first excited state of this adiabatic-limitpseudorotation is 160 cm-’, which corresponds to an extraordinary lowfrequency of 5 x 1012 sec-1 for internal motion in a molecule as rigid astriazine. The level structures predicted by each of the models discussed hereare summarized, together, with experiment, in Figure 4 (96, 163).

6OOAFR LJT L+Q Expt.

4OOI1~_~

200.

0

-zoo .......I "~""E ~/-~-- "’-...~ -400

~ -600

z .%_~-800

-~ooo % .... ~/z -~.

-1400

-1600

Figure 4 Comparison of vaNous predictions for the level structure of 3s~’ sym-t~azine withexperiment. From the left are: (a) the adiabatic fr~-rmor model (Eq. 16), (b) linear J~n-Teller,and (c) linear plus quadratic Jahn-Teller coupling. Dashed lines connect states of (ap-proximately) like vibrational angular momentum.


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296 WHETTEN~ EZRA & GRANT

This essentially two-parameter fit of the entire spectrum is significant :

1. It offers the most direct and dramatic confirmation yet of the classictheory formulated over 25 years ago by Longuet-Higgins and co-workers (62, 79).

2. It decidedly confirms the ideal-Rydberg limit (155) for the interpretationof vibronic coupling in molecular Rydberg states; vibration-Rydberginteractions, if present, are beneath the resolution of the correspondencebetween theory and experiment.

The adiabatic potentials derivable from this calculation must beregarded as very nearly exact to an energy at least as high as the highestmeasured eigenvalue. It is interesting to compare the potential in thepseudorotation coordinate around the bottom of the trough with theinternal rotor levels built on the first radial oscillator state. As Shown inFigure 5, the quadratic barriers to pseudorotation only weakly alter theshape of the potential surface. Despite this, and reasonably close corre-spondence between linear and linear-plus-quadratic eigenvalues, evidentsplittings and transfers of intensity show that, in the vibronic hamiltonian,these quadratic terms have a significant effect on the wavefunctions.

With near quantitative correspondence between experimental and

j=7/2

j=5/2

j-3/2

j-~/2

~- 0.00 0.78 1.57 2..35 3.14 3.92 4.71 5.49 6.28PSUEDOROTATION PHASE. RADIANS

Figure 5 Adiabatic potential in the pseudorotation coordinate ~b as calculated from Eq. 13.

The horizontal lines give the lowest five levels in the Jahn-Teller pseudorotation mode.


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theoretical intensities, wavefunctions can be derived with reasonableconfidence from the present calculations. These are conveniently visualizedas nuclear probability densitics (58-61). Figures 6 through 8 show plots nuclear probability densities (for complex vibronic wavefunctions) for thefirst four levels in thcj = (+ 1/2) mod 3 manifold. For the first two states scc that quadratic coupling modulates the nuclear probability densities(which are cylindrically symmetric for linear coupling alone). Nucleardensity accumulates over surface depressions or saddle points. The stateswith the unusual density pattern shown in Figure 8 are a close-lying pairlabeled byj = 1/2 and 7/2 in the linear limit, which repel by mixing due toquadratic terms.

To summarize, sym-triazinc presents unprecedented simplicity in anisolated nonadiabatic system. With its single active mode and large linear

Figure 6 Nuclear probability density for the ground vibronic state of 3s~E’ sym-tria~inecalculated as I Z +12+ IZ-I2 (see Eqs. 9 and 11) for coupling parameters corresponding to thetheoretical level structure of Figures 4 and 5 and intensities of Figure 3.


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Figure 7 Nuclear probability density for the second excited vibronic state of 3a~E’ aym-tdazine [the lowest ~xcited state of thej = 1/2 rood (3) block], as in Figure

stabilization, it is possible to visualize easily the dynamics of its vibronicpseudorotation in the adiabatic limit. From the precise correspondencebetween high resolution jet spectroscopy and theory, when carried tosecond order in coupling terms, there emerges a quantitative picture of therole of quadratic effects and the loss of cylindrical symmetry in splitting andmixing vibronie levels.

Slow-Electron Systems: Nonadiabaticity in the Failureof the Ideal-Rydber~l Limit and Related Systems

ACCIDENTAL RESONANT MIXING IN MOLECULAR RYDBERG STATES Rydbergstates of symmetric molecules provide a convenient view of the vibronicstructure of degenerate ion gores. More universally, Rydberg-moleeularcore assemblies also constitute a class of systems for which commensurateelectronic and nuclear frequencies are readily attainable. Such states are


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very important, dominating the resolved absorption spectrum of mostmolecules beginning at. energies above two-thirds of the first ionizationpotential. As mentioned above, the novel feature of such states is that oneelectron has escaped from the molecule and now "sees" it predominantly asa rigid ion or even a point charge. The level spacing for this electron reflectsits delocalized, hydrogenic character, and can be represented by theformula

E(n) = -R(n-~)-2 17.where R is the Rydberg constant, n a principal quantum number, and 6 thequantum defect, which is correlated with the orbital angular momentumand nearly independent ofn (164). Adherence to this zeroth order picture forthe case of a degenerate core reveals the nonadiabatic bound state structureof open-shell molecular ions. Deviations constitute a second importantclass of nonadiabatic interaction.

Three kinds of nonadiabatic interactions can be envisioned :

1. Levels of the same n but different 6 (different series) can be coupled vibration or, for very high n, by rotation. For vibration, Rydberg-nuclearinteractions decrease as n-a, so that significant splitting should beexpected only for lower Rydberg states. It is also necessary thatappropriate excited vibrational levels of the lower energy series overlaplower levels of the upper state. These two factors tend to confine strongvibrational couplings to levels of intermediate n. For example, considerthe s and d Rydberg orbitals of benzene (156), l(6), which are s(0.76),do(0.05), and dr(-0.12). At n = 3, 3s-3do-3dt spacings are 9000and 2000 cm- 1. Only high vibrational overtones have such energies, andthese should be only inefficiently scattered by the Rydberg electron, soone might expect the adiabatic approximation to be fairly successful. Atthe n = 5 level, however, the spacings are 1900 and 400 cm-1.

Fundamental vibrations of the correct symmetry are in this range andreadily mix the two hydrogenic states.

2. More generally, strong interaction can occur whenever fundamentalvibrational frequencies of proper symmetry-type match the gap betweenRydberg states of different principal quantum number n, for examplens ~-* (n- 1)do as observed in benzene (156).

3. As another form ofnonadiabatic coupling in many molecules, the lowestRydberg states having electronically excited cores (excited ionic states)lie overlapped with higher Rydberg states converging to the firstthreshold.

As with 1, examples of 2 and 3 have been observed (156). Such occurrencescan help confirm symmetry assignments of highly-excited molecular states.


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Of more interest from the present point of view, they also provide directinformation on the mixing of fast internuclear and slow electronicoscillations.

The first molecule to be considered from this viewpoint was H2, by Berry& Nielsen (165). These authors gave a general theoretical formulation in theadiabatic representation in which they constructed electronic wavefunc-tions and potential energy surfaces, derived the corresponding vibrationalstates, and finally evaluated the matrix elements of the kinetic energyoperator that couples the Born-Oppenheimer states. This procedure, inprinciple, yields exact energy levels (exclusive of rotation). It is, however, exhaustive theoretical treatment that is difficult to reproduce for morecomplex molecules. Among the important results is the identification ofzones of strong mixing (166). As discussed in the next subsection, thesystematics of these zones can be understood in a very natural way.

It is instructive to look at the interactions between Rydberg electrons andcore vibrations from a dynamic point of view. Consider the picture in whicha short-pulsed radiation field prepares a coherent state consisting of aparticular Rydberg orbital and core-vibration. For a two-state interactionthis state would begin to mix with one consisting of the other Rydbergorbital and core vibration pair. The Rydberg electron in effect inelasticallyscatters off the core. In a strictly two-state picture this mixing leads to a’simple pattern of quantum beats, and a higher resolution, longer timeexperiment would reveal the two mixed eigenstates of the interactionseparated by the beat frequency. Weaker interactions dephase these beatsand produce an apparent broadening of the individual eigenstates of thetwo-state interaction. Dynamically the relaxation becomes more complex,with the resulting states not well described in terms of any zeroth-orderseparation into ion normal modes and hydrogenic orbitals. Rather, in time-dependent language, the system evolves to a vibronically mixed (rando-mized) core in which complicated, large amplitude nuclear motion stronglycouples the possible final electronic states.

In recent nonlinear absorption experiments on ultracold molecules,Whetten et al (156) observed several pronounced examples of this type nonadiabatic interaction in benzene. These include at least one of each ofthe three types of interaction described above, including apparent couplingbetween the core-excited 3s Rydberg state converging to the second IP, andneighboring (zeroth-order) 1 and 5dl st ates ontheground state core. Indynamical language this latter case provides an example of core-electronicinelastic scattering of the Rydberg electron.

An equally dramatic example is found in the high angular momentumstates of H2 (167, 168). In the 50 orbitals of 2, molecular r otation i ssufficiently fast compared to evolution among the electronic states that the


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latter uncouple from the axis, producing a smaller splitting on eachrotational state of the ion core. These were detected by the 5g-4f infraredemission, and explained by an angular momentum coupling model. [Highangular momentum states of benzene have also been observed (156, 157),but it does not appear that this inversion of term spacings occurs, since therotational spacing in benzene is much smaller.] This is an example of aninverse adiabatic situation where it is now the vibrational spacings that arelargest. Such a situation is highly counterintuitive, but is to be expected formany systems, particularly those involving electronic fine structure (inphotodissociation, for example). Starting from this point of view is essentialto understand the term splittings in the Rydberg spectrum of benzene,where, as discussed above, it is best to construct electronic levels on avibrational state of the ion; the opposite approach leads to qualitativelyincorrect expectations (155). In these cases it is essential to avoid associatinga distinct adiabatic surface with each electronic term.

More generally, the occurrences of regions of strong mixing are readilypredicted in terms of known level spacings (vibrations and rotations of theion, Rydberg electronic spacings) (169). Furthermore, the information highly redundant because the wavefunctions producing the zeroth-orderpattern of levels are assumed known. Thus, for example, the knowledge ofthe strength of interaction coupling the v16 vibration of benzene to the 5fand 5d hydrogen-like electronic orbitals predicts the strength of thisinteraction for all other n (156). The correct means of taking this knowledgeinto account is via the multichannel quantum defect theory (MQDT) (166,170). This results in an enormous simplification of the description of theseinteractions. It is based on the electron-ion scattering problem, facilitatedby use of the asymptotically correct Coulomb potential, so posed as toallow the experimental data to solve the configuration interaction andnonadiabatic problem at the same time, yielding directly the nonadiabaticinteraction matrix elements with minimal assumptions.

Application to molecules has been concerned primarily with NO and H2,especially in the latter case with regard to rotational channels of interaction,but is readily extended in principle (171, 171a, 172). Very difficult regions the spectra of these molecules have yielded to this approach, which alsooffers the simple scattering interpretation of nonadiabatic dynamicsdiscussed above.

RADIATIONLESS DECAY AND AUTOIONIZATION An understanding of thenonadiabatic interaction of weakly bound electrons with vibration androtation is extremely important in the decay and ionization properties ofmolecular systems. To see this, let us consider again the interaction in whichan electron is scattered into a lower-lying orbital via excitation of a core-vibration. Imagine, as above, preparation of the coherent state consisting of


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the zeroth-order n! orbital; vibrational interaction mixes it with anotherstate (n-1)1’, which in effect serves as a doorway to a more dense set ofovertones associated with lower Rydberg states. At some stage anessentially continuous manifold of lower levels takes over, and one has aLorentzian linewidth corresponding to exponential decay.

The Rydberg states of benzene (156-158) taken at rotational tempera-tures near 0 K provide an example of precisely this coupling hierarchy.Pairwise mixings have been discussed above. Moreover, where severalstates of proper symmetry nearly coincide, one has higher-order mixings.Finally the homogeneous linewidth of each identifiable band reflects strongcoupling to the quasicontinuum structure beneath. This latter aspect isconfirmed by direct time-resolved experiments using femtosecond laserpulse-probe techniques carried out by Wiesenfeld & Greene (173). bandwidth of 100 cm-1 excites a superposition, which decays exponen-tially, as probed by time-resolved ionization. Significantly, for both benzeneand toluene, the agreement between dynamical (173) and spectroscopicmeasurements (156, 157, 174) is quite good, thus giving credence to the ideathat the final irreversible decay product is a set of states with energy in largeamplitude vibrations, states that cannot be ionized by a visible laser. Thusthe width of the homogeneous line is that corresponding to the net processby which the electron spirals inward, transferring energy to the core.

The widths or decay times measure the strength of interaction betweenthe background states and the pure electronic excitation, and as might beexpected this coupling strength decreases rapidly with n. This idea can becarried to an extreme to predict that Rydberg states of very high nl numbershave extreme stability; they are effectively metastable because of the rarityof a collision between the Rydberg electron and the ion core. With very lowtransition frequencies, radiative lifetimes are also very long. Some evidencefor this behavior has been gathered by Freund (175). Multiphotonexcitation with circularly polarized light could well offer an effectivespectroscopic means to produce these states.

Similar arguments apply to the phenomenon of autoionization, but theseare resonances in which the reverse flow (energy from vibrotation toelectronic) occurs. For the case of diatomics this manifestly nonadiabaticprocess has been studied within the MQDT formalism (176), but explicitly time-dependent experiments have yet been attempted.

NEGATIVE IONS In principle a similar treatment could apply to negative ionstates, but all the nice features of the Coulomb potential applicable aboveare not in force here. Space limitations preclude consideration of the highlyinteresting topic of resonant scattering here. Nonetheless, a point regardingautodetachment is worthy of mention.

The recent experimental observation of electron-dipole bound states


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gives a picture of autodetachment that is evidently related to rotationalpredissociation (177, 178). One can view these states in terms of the electroncycling about the positive end of the dipole, so weakly attached as to haveonly one bound state for most if not all angular momentum states of thecore. With the detached neutral molecule as a core, resonances abovethreshold are analogous to the case of a Rydberg state excited above theadiabatic IP. Here the core gives up very little rotational energy to detachan electron, yet the resonances are fairly long-lived.

CLASSICAL AND SEMICLASSICALAPPROACHES TO NONADIABATICPHENOMENA

As is apparent from the preceding discussion, the interaction betweentheory and experiment has played an important role in developing anunderstanding of vibronic states of molecules in which a doubly-degenerateelectronic state is coupled to one or a very small number of active modes.For such systems a full quantum variational treatment of the vibronichamiltonian is feasible, so that model parameters can be determined toprovide the best fit to experimental data (e.g. sym-triazine). The size of theassociated secular determinant, however, grows extremely rapidly with thenumber of active modes, and there are no useful decoupling approxima-tions that can be applied. Hence, the limits of current quantum-mechanicaltechnology are reached very quickly for problems involving only a modestnumber of modes. Although procedures such as the Lanczos algorithm(179) or the related Recursive-Residue method (180) have great potential solving very large eigenvalue problems, and although a statistical approachmay be appropriate for describing vibronic level structure in highly-exciteddense manifolds (113, 114), the situation described above points to thedesirability as an alternative of a semiclassical (asymptotically valid as -~ 0) theory of vibronic level spectra. Such a theory would ideally provideboth a quantitative treatment ofvibronic spectra in the multimode case anda conceptual framework for obtaining qualitative insight into manifes-tations of vibronic coupling; it would therefore be an extension tomultisurface phenomena of the various semiclassical approaches that havebeen successfully applied to single surface scattering and bound stateproblems.

A semiclassical theory of multimode vibronic phenomena as justdescribed is not available at present. Despite an enormous amount of workon the generalization of semiclassical methods to treat nonadiabaticprocesses occurring in collisions, the general problem involving multiple(> 2) interacting surfaces, strong and delocalized nonadiabatic couplings,


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and many nuclear degrees of freedom remains essentially unsolved. It is ourpurpose here to review a small body of work that is in our view .likely toprovide a foundation for future theoretical treatments of problems relevantto the recent experimental advances described in previous sections. Thus,we consider some recent classical and semiclassical approaches toradiationless transitions and bound states in Jahn-Teller systems. Most ofthe work we describe is within the framework of the "classical electronmodel" of Miller, Meyer & McCurdy; this model is therefore described insome detail.

In order to place this work in a proper complete context, it would benecessary to give a thorough discussion of all the major lines ofdevelopment in the semiclassical theory of nonadiabatic phenomena (ahuge literature). We stop short of this, presenting instead a few pertinentreferences. Reviews of many aspects of the subject can be found in Miller(181), Lain & George (33), Tully (17), Child (30--32), Nikitin (28), Ziilicke (29), Kleyn, Los & (3islason (39), Rebentrost (36), Ranfasni (182), Crothers (34), Ovchinnikov & Ovchinnikova (15), Heller & (183), Nakamura (184).

Background

A central issue in the theory of nonadiabatie processes is the extent to which"classical path" descriptions are meaningful or useful. That is, to whatextent can the heavy-particle (nuclear) degrees of freedom be taken undergo classical motion in an effective potential determined in somefashion by the quantal dynamics of the coupled electronic degrees offreedom? A path-integral formulation (181, 185-187) of the nonadiabaticscattering amplitude enables a formal answer to be given to this question:The exact (semiclassical) effective "potential" is of necessity nonloeal in timeand must be determined iteratively. This result clearly implies insuperablepractical difficulties unless other, perhaps uncontrollable, approximationsare introduced. Further work in this direction has led to the widely usedMiller-George (188) and Tully-Preston ("surface-hopping") (189, approaches, whereas the self-consistent matrix propagation method ofFreed & Laing (191-193), further developed by Herman & Freed (194), closer to the original path-integral approach.

At a simpler level, the classical motion of the nuclei can be taken to bedetermined self-consistently through interaction with the electronic sub-system. The nuclei then provide, through vibronic coupling terms in thecomplete molecular hamiltonian, an effective, time-dependent electronichamiltonian, which induces transitions in the manifold of electronic statesof interest. The time-dependence of the electronic state in turn gives rise to atime-dependent, effective (Ehrenfest) potential for the nuclei through the


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expectation value of the nuclear-coordinate dependent molecular hamil-tonian in the instantaneous electronic state. A specific example of this self-consistent coupling is treated in the next section.

Considerable effort has been devoted to assessing the validity of classicalpath methods in general (195, 196). For a system with a single nucleardegree of freedom and a localized crossing of two adiabatic curves, it isnecessary that (a) the form (i.e. slope) of the potentials be similar in region of the crossing, and (b) the nuclear kinetic energy be large comparedto differences in the two potential surfaces in the interaction region. Inaddition, the usual conditions for semiclassical treatment of nuclear motionshould hold. Despite such stringent requirements, the self-consistentcoupling approach underlies a substantial amount of work in the area. See,for example: Ehrenfest equations (200-202); first-order self-consistentEikonal method (197-199); Dynamical Hellman-Feynman theorem (203);time-dependent Hartree-Fock theory (204, 205); dynamical vibration-electronic coupling (206). In particular, the classical electron model Miller, Meyer & McCurdy (207-211) has the self-consistent formulation its starting point.

Nevertheless, severe conceptual difficulties and inconsistencies areinherent in such a mixing (albeit self-consistent) of classical and quantaldynamics. For example, in a system that is asymptotically in a super-position of two electronic states, the nuclei execute some averageddynamics, as opposed ~o motion determined by one or the other adiabaticsurface, as might be expected on physical grounds. The classical electronpicture is intended to surmount such problems, and to facilitate treatmentof resonant processes in electronic-vibrational and electronic-rotationalenergy transfer (212).

Review of the Classical Electron Picture

The classical electron model of Miller, Meyer & McCurdy, which is anattempt to overcome some of the difficulties of mixed dassical/quantaldynamics, works in the following way [we do not discuss the related butapparently less general Spin Matrix Mapping (210) and ClassicalPseudopotential (208) approaches]: For a system in which there are coupled electronic states of interest, the electronic hamiltonian can berepresented as an N × N matrix whose elements depend on the nuclearcoordinates {Q~} :

Hkk,(Q) k,k’= 1 ..... N. 18.

Adiabatic electronic representation is used for simplicity, but it is alsopossible to formulate the theory in the adiabatic basis (209). If the nuclearcoordinates Q(t) are taken to be given functions of time, the time-dependent


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Schr6dinger equation for the electronic wavefunction gives rise to a set ofcoupled equations for (a~(t)}, the coefficients in the diabatic basis

ihdk = ~ Hkk,(Q(t))ak,. 19.k"

A key step is now the change of variables [of a type originally due to Dirac(213)3

ak =-- n~/2ei°k, 20.

giving the polar decomposition of the complex coefficients (ak} into (real)amplitudes (nk) and phases (Ok}. Note:

0<nk<l , ~nk----1. 21.

Defining the "hamiltonian’

h(n, 0; Q) = a*kI-Ikk,ak, 22.kk’

= E ~1/2~1/2t-/ °--i(Ok-O~’) 23.kk"

as the expectation value of H in the electronic state, the remarkable result isobtained that the quantum mechanical set of coupled equations is entirelyequivalent to "hamilton’s equations" for the canonically conjugate pairs(n~, 0~):

Oh 0=--Ohk = 1,..., U. 24.fik = 00~’

[This equivalency obtains because mappings of type, Eq. 22, for arbitrarymatrix operators transform commutators into Poisson brackets (209).]Adding the classical nuclear kinetic energy then gives a completely classical,time-independent hamiltonian ~,~ that determines the self-consistentlycoupled nuclear/electronic time evolution through the classical equationsof motion for the nuclear degrees of freedom, together with Eq. 24 :

p29~(P,Q,n,O) Y’,~--~ +h(n,O;Q) 25.

zm~

/~, = ---- ~,=-- 26.oO,’

This time evolution preserves both total energy and electronic probability~, nk.

For the limit of a single-surface problem, [nxl = 1 and Eq. 26 simplyreduce to the classical nuclear equations of motion for the potential surface


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V(Q) =-h(nl, OI;Q). In the general two-state case, the classical electronhamiltonian is

~J~ = ~ P2~/Em~ + (1 - n)H~ + nil22

+ 2In(1 -- n)] 1/2H12 (cos 0) 27.

where a canonical transformation has been made to new variables n’ = nl+n2 ---- 1, n ~ n2, 0’ = 01, 0 = 02--01, so that only the relative phase 0appears. For the simplest two-state linear E x e Jahn-Teller problem (cfprevious section), the classical analog hamiltonian in the diabatic represen-tation is :

2 P~ +p2/2+2kp cos (O-qb)[n(1-n)] ~/2. 28.

At this point all that has been achieved is an elegant reformation of theEhrenfest force classical path equations. The major innovation in the workof Miller, Meyer & McCurdy is the realization that the classical analoghamiltonian can be used within the context of the classical S-matrix orquasiclassical trajectory approaches to determine nonadiabatic transitionamplitudes, provided the following "Langer modification" is made :

n~ ~ n~ + 1/2, 29.

together with a change in the form of ~ that maintains equality betweenthe semiclassical electronic eigenvalues and adiabatic potential surfaceenergies. The substitution of Eq. 29 ensures that there is a nontrivialdependence of the final electronic action upon the initial electronic phase.[For a justification of the Langer modification, see the discussion byHerman & Currier (209a).] The resulting modified hamiltonian for general two-state system is :

~/g’ = ~ P~/2rn~ + (1 - n)I-I1 ~ + nIt 22

+2[(n+ 1/2)(3/2--n)]I/2H12 cos 0. 30.

Applications of the general two-state classical analog hamiltonians, Eqs. 27and 30, and the linear Jahn-Teller hamiltonian, Eq. 28, in particular, arereviewed in the next section.

Applications of Classical Electron and Related Models

NONADIABATIC PROCESSES IN COLLISIONS The classical electron (Ehrenfestwith Langer modification) and related classical pseudopotential and spinmatrix mapping methods have been applied by Miller and co-workers to avariety of electronically nonadiabatic collision processes : quenching of an


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excited fine-structure state of the fluorine atom F*(EP1/2) by collision withH + or Xe (208); quenching of F*(2P1/2) by collision with H2 (E-R energytransfer) (210); quenching of Br*(2P1/2) by collision with n2(v = 0) transfer), for collinear (214) and three-dimensional (215) systems; charge transfer in Na + I collisions (216).

In all cases where corresponding close-coupled quantum calculations areavailable for comparison, there is encouraging agreement between the exactresults and quasidassical trajectory calculations of cross sections andtransition probabilities. In particular, resonant features are correctlydescribed using the classical analog methods. There is clearly tremendousscope for application of these techniques to nonadiabatic processes alreadytackled using the popular trajectory surface-hopping method (189, 190),and for quantization of the theory using the classical S-matrix formalism.

BOUND STATES In line with the emphasis laid in preceding sections of thisreview on nonadiabatic cffects in molecular bound states, wc proceed toconsider now in some detail the few applications made to date of theclassical electron picture to these problems. Such an approach is potentiallyvery fruitful, and much remains to bc done in this area.

Ultrafast nonradiative decay via conical intersections In a pioneeringquantum-mechanical calculation, K/Sppel (107) modeled the ultrafastinternal conversion between .~ ~ .~ states of C2H~-. The model system wastaken to be two near-degenerate states in the diabatic representationcoupled by a single nontotally symmetric mode, together with two totallysymmetric modes. Such a system has a multidimensional conical intersec-tion, since the totally symmetric modes "tune" the electronic energies. Thetime-dependent probability of being in the upper (diabatic) state (summedover all vibrational states) after sudden excitation from the neutralmolecule ground vibrational state was computed by direct integration ofthe time-dependent Schr6dinger equation.

The key result to emerge from this calculation is the extremely short timescale (~ 10-1’~ s) characterizing the radiationless decay. The necessity forcoupling of the electronic states via more than one vibrational mode, hencethe existence of a conical intersection, is borne out by a single modecalculation having revealed no rapid internal conversion. The dispositionof energy among vibrational modes in the electronic ~ vibrational energytransfer from the ground vibrational state of the upper diabatic electronicstate was also calculated.

It is natural to consider the applicability of the classical electron model tothis type of problem. In fact, Meyer (217) has recently investigated the samesystem as K6ppel by using an Ehrenfest classical analog hamiltonian, set upin the way outlined above. Using the classical histogram method for


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3 l0 WHETTEN, EZRA & GRANT

assigning electronic states, and taking the initial state used in (107) correspond to an ensemble of trajectories, remarkably good agreementbetween classical and quantum mechanical probabilities is obtained for thethree-mode problem. Reasonable but "less beautiful" agreement is ob-tained for model two- and one-mode systems. This is in accord with thegeneral principle that classical calculations are most reliable when someaveraging (in this case, over vibrational states) occurs.

The enormous potential of the classical electron model in this context isapparent when it is realized that anharmonic modes or several additionalvibrational degrees of freedom are easily included, while quantum mechani-cally such systems remain intractable. Meyer’s calculations show that modeanharmonicity has little effect on the short-time (t < 40 fs) decrease of theprobability of remaining in the upper state. Quenching of regularoscillations of P(t) occurs, however, at longer times. Increasing the numberof modes to ten, for example, has very little effect on the time-dependentprobability (a ten-mode quantal calculation is of course well beyondpresent-day capabilities), Finally, a recent comparison (110) of exactquantum and classical analog results for the time-dependent photonemission rate from a two-level two-mode system shows reasonable overallagreement with quantum calculations.

The classical electron model thus represents a computationally con-venient and tractable approach to multimode vibronic bound stateproblems, and considerable development along these lines can be expectedin the future. One advantage of a completely classical formulation of thevibronic problem is that examination of trajectories can provide insightinto complicated vibronic dynamics. In the next subsection we describesome recent calculations in which the classical electron picture is used tostudy bound states of linear JT systems.

Bound states of linear Jahn-Teller systems We now briefly survey theresults of our investigations of the classical analog hamiltonian for thelinear E × e Jahn-Teller bound state problem; details are given elsewhere(J. W. Zwanziger et al, submitted for publication). The hamiltonian Eq. 28 that for a multidimensional nonlinear classical dynamical system. In linewith much of the recent research in this area (218), we emphasize thequalitative appearance of trajectories, Poincar6 surface of section plots, andthe possibility of both quasiperiodic and chaotic motions. The relationbetween quantum and classical integrability for a two-state ("electronic")system interacting with a boson ("vibrational") mode has been studiedusing several approximations to the exact dynamics (219), including theself-consistent coupling formulation.

Judd (48, 220) has recently described Jahn-Teller trajectories for severalsystems. It should be noted that Judd’s work goes only as far as the


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determination of nuclcar trajectories on particular adiabatic potentialsheets, so that treatment of the self-consistently coupled dynamics de-scribed by Eq. 28 represents a considerable extension of this approach.

The hamiltonian Eq. 28 leads to six coupled equations for the variables(pp, p, pC, ~b, n, 0). Initial values of the pair (n, 0) can be chosen to correspondto starting on either adiabatic sheet, or diabatic state, etc. 0Note that wc dealhere only with the Langcr unmodified hamiltonian Eq. 28.) The complexityof the problem can bc reduced by noting that ~ is a 1. l resonancehamiltonian, since the angles 0 and ~b appear only in the combination (0-~b).Transforming to new canonical variables (Pp, p, It, ~1,12, ~b2) wc obtain hamiltonian ~ independent of ~, so that Ix is a constant of the motion:

~=P~ p~ 1~ + ~ + ~ E~ +~/~

+kp[(-I2+I~/2)(1+I2-I1/2)] ~/~ cos ~z" 31.

We therefore have effectively a two degree-of-freedom problem, and so canuse the standard surface-of-section method to characterize the dynamics.

Typical surfaces of section for a low energy trajectory (corresponding tothe lowest quantum state of this system, k = 2.0 andj = 1/2) are shown inFigures 9 and 10. The (Pp, p) section, Figure 9, shows straightforward radial

1.00 1.27 1.55 1.82 2.10 2 37 2.BB 2.g2 3.20RHO

~ Surface of section for a low energy trajectory for ~a~ilto~i~ ~q. ~1, ~th k = 2.0,


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librational motion for the most part, except for a prominent high orderresonance visible as a chain of eight islands. The (I2, t#2) section, Figure 10,shows relatively small fluctuations of the phase around a value of $2 near n,together with a central resonant zone corresponding to the island chain inFigure 9. In the limit of large k and low energy, which is the realm of"deepstates" With negligible nonadiabatic coupling, the phase $2 has the constantvalue n. Constancy of ~b2 implies perfect correlation between electronic andvibrational degrees of freedom. Nonadiabaticity is then manifest in theclassical electron picture in deviations of ~2 from the value r~, suchdeviations corresponding to a loss of correlation between electronic andnuclear variables.

Examination of nuclear configuration space projections of classicalanalog Jahn-Teller trajectories provides additional insight into the effectsof nonadiabaticity on nuclear dynamics within this model. In Figures 11 weshow two low-energy trajectories. One is obtained by solving the coupledequations of motion for hamiltonian Eq. 28; the other corresponds tomotion at the same energy on the lower adiabatic potential surface [wherethe latter includes the Born-Huang correction 1/8p2 (76, 105)]. The twotrajectories are remarkably similar! For motion on the lower, cylindricallysymmetric adiabatic surface, the nuclear angular momentum is a conserved

2.60 2.74 2.87 3.01 3.15 3.29 3.42PHI2

3.56 3.70

Figure I 0 (12, ~b2) surface of section at constant p for the trajectory of Figure


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0’g



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quantity, whereas for the low energy exact trajectory it is very nearly so.Increasing the energy reveals the effects of nonadiabaticity: In the slightlyhigher energy trajectory shown on the left in Figure 12, the nuclear radialturning points are beginning to develop a slight asymmetry, while on theright, at higher energy still, the projection of the nuclear motion developsloops and cusps, indicative of a nonadiabatic exchange of angularmomentum with the electronic degrees of freedom. At very high energies,we obtain for some initial conditions irregular-looking trajectories that canonly be described as "chaotic."

We have provided a brief overview of studies of nonadiabatic boundstates using the classical electron model. There is clearly much more to belearned about vibronic interactions using this approach. Particularlyintriguing is the possibility of calculating bound-state energies or energydifferences using some scmiclassical quantization of the classical analoghamiltonian, i.e. via a "requantization" of the original problem (221).

Other Semiclassical Approaches

In the preceding subsection we stress the potential utility of the classicalelectron picture as a basis for a semiclassical description of nonadiabaticbound states. Another natural framework for such a theory is themultichannel generalization of the WKB method (222). We therefore brieflyreview some recent progress in this direction.

The general theory of multichannel WKB approaches to molecularscattering phenomena has been reviewed by Eu (223). Herman has recentlyinvestigated uniform multichannel semiclassical wavefunctions for non-adiabatic scattering, for both one- (224) and multi-dimensional (225, 226)systems, and has also provided a helpful discussion and comparison ofprevious approaches to nonadiabatie scattering (225). Herman’s approachseems particularly promising, and is complementary to the classicalelectron method discussed above.

In the past few years some work has been done on the multichannelWKB quantization of vibronic bound states. Voronin et al (227) andO’Brien (228) have considered the high energy limit of the linear E x ~ Jahn-Teller problem, where the conical intersection represents a negligible pertur-bation of essentially harmonic motion on either adiabatic surface. In thislimit, good agreement is obtained (228) with the appropriate asymp-totic energy level expressions of Longuet-Higgins et al (79); in fact,impressive agreement with variational energy levels extends over a wideparameter range (227). At the other extreme, Karkach & Osherov haveinvestigated the semiclassical quantization of deep states in a quadraticE x e Jahn-Teller system (100). In this case the nonadiabatic couplingterms were ignored, and a uniform quantization of the angular nuclear


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NONADIABATIC DYNAMICS


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motion carried out after an adiabatic separation of the radial motion. Anotable feature of this calculation is the imposition of nonperiodicboundary conditions on the semiclassical angular wavefunction, thusreflecting the requirement of a change of sign of the nuclear wavefunctionupon traversing a loop in nuclear configuration space enclosing a conicalintersection (72, 80-85). The general problem of uniform quantization in multiwell potential with nonperiodic boundary conditions has recentlybeen discussed in some detail by Connor et al (229).

A semiclassical theory for scattering resonances in the upper sheet of atwo-dimensional conical intersection has been developed by Osherov andco-workers (230, 231), and a semiclassical treatment of the absorptionlineshape for an optical transition to a doubly degenerate electronic state isgiven in (232) (cf also 233). The multichannel WKB (or its generalization EBK) approach to bound states for general matrix hamiltonians has beendiscussed in the context of the Dirac electron (234) and quark dynamics(235).

ACKNOWLEDGMENTS

We thank co-workers Kenneth I-Iaber and Josef Zwanziger for their

invaluable contributions to much of the original research discussed in ouraccount. This work was supported by grants from the National ScienceFoundation, the Petroleum Research Fund and the US Army ResearchOffice. During the course of this writing, R.L.W. held a post-doctoralfellowship at the Corporate Research Science Laboratories, ExxonResearch and Engineering Company.

Literature Cited

1. Born, M., Oppenheimer, R. 1927. Ann.Phys. 84: 457-84

2. Born, M., Huang, K. 1956. DynamicalTheory of Crystal Lattices. Oxford:Clarendon

3. Herzberg, G. 1966. Molecular Spectraand Molecular Structure, Vol. 3. NewYork: Van Nostrand

4. Bunker, P. R. 1979. Molecular Sym-metry and Spectroscopy. New York:Academic

5. Papousec, D., Aliev, M. R. 1982.Molecular Vibrational-Rotational Spec-tra. New York : Elsevier

6. Miller, W. H., ed. 1976. Dynamics ofMolecular Collisions, Pt. A. New York:Plenum

7. Miller, W. H., ed. 1976. Dynamics ofMolecular Collisions, Pt. B. New York :Plenum

8. Truhlar, D. G., ed. 1981. Potential

Energy Surfaces and Dynamics Calcu-lations. New York : Plenum

9. Bernstein, R. B., ed. 1979. Atom-Molecule Collision Theory. New York :Plenum

10. Freed, K. F. 1976. Top. Appl. Phys. 15:23-168

11. Lim, E. C. 1977. Excited States 3 : 305-37

12. Freed, K. F. 1980. Adv. Chem. Phys.42 : 207-69

13. Di Bartolo, B., ed. 1980. RadiationlessProcesses. New York: Plenum

14. Jortner, J., Levine, R. D. 1981. Adv.Chem. Phys. 47(Pt. 1): 1-114

15. Ovehinnikov, A. A., Ovehinnikova, M.Ya. 1982. Ado. Quantum Chem. 16:161-227

16. Pearson, R. G. 1971. Acct. Chem. Res.4:152q50

17. Tully, J. C. 1976..See Ref. 7, pp. 217-67


Ann

u. R

ev. P

hys.

Che

m. 1

985.

36:2

77-3

20. D

ownl

oade

d fr

om a

rjou

rnal

s.an

nual

revi

ews.

org

by C

OR

NE

LL

UN

IVE

RSI

TY

on

05/2

7/06

. For

per

sona

l use

onl

y.



18. Salem, L. 1982. Electrons in ChemicalReactions. New York : Wiley

19. Bersuker, I. B. 1984. The Jahn-TellerEffect and Vibronic Interactions inModern Chemistry. New York : Plenum

20. Siebrand, W. 1976. See Ref. 6, Chapt. 621. Siebrand, W., Zgierski, M. Z. 1979.

Excited States 4 : 2-13222. Champion, P. M., Albrecht, A. C. 1982.

Ann. Rev. Phys. Chem. 33 : 353-7623. Ziegler, L. D., Hudson, B. S. 1982.

Excited States 5 : 41-14024. Eland, J. H. D. 1974. Photoelectron

Spectroscopy. New York : Wiley25. Miller, T. A., Bondybey, V. E. 1982.

Appl. Spectrosc. Rev. 18 : 105~i926. Miller, T. A. 1982. Ann. Rev. Phys.

Chem. 33 : 257-8227. Nikitin, E. E. 1968. In Chemische

Elementarprozesse, ed. H. Hartmann, p.43-77. New York : Springer

28. Nikitin, E. E. 1970. Adv. Quant. Chem.5 : 135-99

29. Nikitin, E. E., Ziilicke, L. 1975. Lect.Notes Chem. 8:1-175

30. Child, M. S. 1974. Molecular CollisionTheory. New York : Academic

31. Child, M. S. 1978. Adv. At. Mol. Phys.14: 225-80

32. Child, M. S. 1979. See Ref. 9, pp. 427-6533. Lam, K. S., George, T. F. 1980. In

Semiclassical Methods in MolecularScatterin# and Spectroscopy, ed. M. S.Child, pp. 179-261. Dordrecht : Reidel

34. Crothers, D. S. 1981. Adv. At. Mol.Phys. 17 : 55-98

35. Garrett, B. C., Truhlar, D. G. 1981.Theor. Chem. 6A : 215-89

36. Rebentrost, F. 1981. Theor. Chem. 6B:1-77

37. Hertel, I. V. 1982. Adv. Chem. Phys. 50:475-515

38. Baer, M. 1982. Adv. Chem. Phys. 49:191-309

39. Kleyn, A. W., Los, J., Gislason, E. A.1982. Phys. Rep. 90:1-71

40. Baer, M. 1982. Ber. Bunsenoes. Phys.Chem. 86: 448-58

41. Macias, A., Riera, A. 1982. Phys. Rep.90: 299-376

42. Davydov, A. S. 1971. Theory of Mol-ecular Excitons. New York : Plenum

43. Collins, M. A. 1983. Adv. Chem. Phys.53 : 225-337

44. Englman, R. 1972. The Jahn-TellerEffect in Molecules and Crystals. NewYork : Wiley-Interscience

45. Bersuker, I. B. 1984. The Jahn-TellerEffect. A Bibliographic Review. NewYork : Plenum

46. K6ppel, H., Cederbaum, L. S., Domcke,W., Shaik, S. S. 1983. An#ew. Chem. Int.Ed. 22 : 210-24

47. K6ppel, H., Domcke, W., Cederbaum,L. S. 1984. Adv. Chem. Phys. 57 : 59-246

48. Judd, B. R. 1984. Adv. Chem. Phys.57 : 247-309

49. Carrington, T. 1974. Acct. Chem. Res.7 : 20-25

50. Liehr, A.D. 1962.Ann.Rev.Phys. Chem.13:41-76

51. Liehr, A. D. 1963. J. Phys. Chem. 67:389-471

52. Liehr, A. D. 1963. J. Phys. Chem. 67:471-94

53. Carrington, T. 1972. Discuss. FaradaySoc. 53 : 27-34

54. Davidson, E. R. 1977. J. Am. Chem. Soc.99 : 397-402

55. Davidson, E. R., McMurchie, L. E.1982. Excited States 5 : 1-39

56. Davidson, E. R., Borden, W. 1983. J.Phys. Chem. 87 : 4783-90

57. Bersuker, I. B., Polinger, V. Z. 1982.Adv. Quant. Chem. 15:85-160

58. Gerber, W. H., Schumacher, E. 1978. J.Chem. Phys. 69:1692-1703

59. Meyer, R., Graf, F., Ha, T.-K.,Giinthard, Hs. H. 1979. Chem. Phys.Lett. 66:65 71

60. Morse, M. D., Hopkins, J. B.,Landridge-Smith, P. R. R., Smalley, R.E. 1983. J. Chem. Phys. 79:5316-28

61. Thompson, T. C., Truhlar, D. G., Mead,C. A. 1985. J. Chem. Phys. 82:2392-2407

61a. Mead, C. A. 1983. J. Chem. Phys. 78:807-14

61b. Thompson, T. C., Mead, C. A. 1985. J.Chem. Phys. 82: 2408-17

62. Longuet-Higgins, H. C. 1961. Adv. Spec.2: 429-72

63. Kolos, W. 1970. Adv. Quant. Chem. 5:99-133

64. Ozkan, I., Goodman, L. 1979. Chem.Rev. 79 : 275-85

65. Bishop, D. M., Cheung, L. M. 1980.Adv. Quant. Chem. 12 : 1-42

66. Lathouwers, L. L., Van Leuven, P.1982. Adv. Chem. Phys. 49:115-89

67. Azumi, T., Matsuzaki, K. 1977.Photochem. Photobiol. 25: 315-26

68. Gregory, A. R., Silbey, R. 1976. J. Chem.Phys. 65 : 4141-46

69. Teller, E. 1937. J. Phys. Chem. 41 : 109-16

70. Jahn, H. A., Teller, E. 1937. Proc. R. Soc.London Set. A 161 : 220-35

71. Jahn, H. A. 1938. Proc. R. Soc. LondonSet. A 164:117-31

72. Longuet-Higgins, H. C. 1975. Proc. R.Soc. London Set. A 344:147-56

73. Smith, F. T. 1969. Phys. Reo. 179: 111-23

74. Feynman, R. P. 1939. Phys. Rev.56: 340-43


Ann

u. R

ev. P

hys.

Che

m. 1

985.

36:2

77-3

20. D

ownl

oade

d fr

om a

rjou

rnal

s.an

nual

revi

ews.

org

by C

OR

NE

LL

UN

IVE

RSI

TY

on

05/2

7/06

. For

per

sona

l use

onl

y.



75. Deb, B. M. 1981. The Force Concept inChemistry. New York : Van Nostrand

76. K6ppel, H., Domcke, W., Cederbaum,L. S., yon Niessen, W. 1978. J. Chem.Phys. 69 : 4252-63

77. Cederbaum, L. S., K6ppel, H., Domcke,W. 1981. Int. J. Quant. Chem. 515:251-67

78. Cederbaum, L. S. 1983. J. Chem. Phys.78: 5714-28

79. Longuet-Higgins, H. C., Opik, U.,Pryce, M. H. L., Sack, R. A. 1958. Prec.R. Soc. London Set. A 244:1-16

80. Mead, C. A., Truhlar, D. G. 1979. J.Chem. Phys. 70: 2284-96

81. Berry, M. V. 1981. Ann. Phys. 131 : 163-216

82. Berry, M. V. 1984. Prec. R. Soc. LondonSet. A 392: 45-57

83. Simon, B. 1983. Phys. Rev. Lett. 51:2167-70

84. Wilczek, F., Zee, A. 1984. Phys. Rev.Lett. 52: 2111-14

85. Thouless, D. 1984. Phys. Rep. 110: 279-91

86. Berry, M.V. 1985. J. Phys. A 18:15-2787. Whetten, R. L., Grant, E. R. 1984, J.

Chem. Phys. 81 : 691-9788. Mead, C. A. 1980. J. Chem. Phys. 72:

3839~089. Mead, C. A. 1980. Chem. Phys. 49 : 23-

3290. Mead, C. A. 1980. Chem. Phys. 49 : 33-

3891. Mead, C. A., Truhlar, D. G. 1982. J.

Chem. Phys. 77 : 6090-9892. Aharonov, Y., Bohm, D. 1959. Phys.

Rev. 115 : 485-9193. Bondybey, V. E., Miller, T. A., English,

J. H. 1980. Phys. Rev. Lett. 44:1344-4794. Whetten, R. L., Fu, K. J., Grant, E. R.

1983. J. Chem. Phys. 79 : 2626-4095. Whetten, R. L., Grant, E. R. 1984. J.

Chem. Phys. 80:1711-2896. Whetten, R. L., Haber, K. S., Grant, E.

R. 1985. J. Chem. Phys. 82 : In press97. Lindsay, D. M., Herschbach, D. R.,

Kwiram, A. L. 1976. Mol. Phys. 32:1199-1213

98. Thompson, G. A., Lindsay, D. M. 1981.J. Chem. Phys. 74:959-68

99. Thompson, G. A., Tischler, F., Garland,D., Lindsay, D. M. 1981. Surf. Sci.106:408-14

100. Karkach, S. P., Osherov, V. I. 1978.Mol. Phys. 36:1069-84

101. Slonczewski, J. C. 1963. Phys. Rev. 131 :1596-1610

102. Slonczewski, J. C., Moruzzi, V. L. 1967.Physics 3 : 237-54

103. Volger, M. 1979. Phys. Rev. A 19:!-5104. Watson, J. K. G. 1980. Phys. Rev. A

22 : 2278-79

105. K6ppel, H., Hailer, E., Cederbaum, L.S., Domeke, W. 1980. Mol. Phys. 41:669-77

106. Sloane, C. S., Silbey, R. 1972. J. Chem.Phys. 56:6031-43

107. K6ppel, H. 1983. Chem. Phys. 77: 359-75

108. K6ppel, H., Meyer, H. D. 1984. Chem.Phys. Lett. 107 : 149-54

109. K6ppel, H., Cederbaum, L. S., Domcke,W. 1984. Chem. Phys. Lett. 110: 469-73

110. Meyer, H. D., K6ppel, H. 1984. J. Chem.Phys. 81 : 2605-19

111. Born, M., Heisenberg, W. 1924. Ann.Phys. 74:1-31

112. Schaff, B., Jortner, J. 1968. Chem. Phys.Lett. 2: 68-70

113. Haller, E., K6ppel, H., Cederbaum, L.S. 1983. Chem. Phys. Lett. 101:215-20

114. Hailer, E., K6ppel, H., Cederbaum, L.S. 1985. J. Chem. Phys. In press

115. Porter, C. E. 1965. Statistical Theoriesof Spectra. New York: Academic

116. Brody, T. A., Flores, J., French, J. B.,Mello, P. A., Pandey, A., Wong, S. S. M.1981. Rev. Mod. Phys. 53 : 385-479

117. Buch, V., Gerber, R. B., Rather, M. A.1982. J. Chem. Phys. 76:5387-5404

118. Pechukas, P. 1983, Phys. Rev. Lett. 51:943-46

119. Bohigas, O., Giannoni, M. J., Schmit, C.1984. Phys. Rev. Lett. 52:1-4

120. Hailer, E., K6ppel, H., Cederbaum, L.S. 1984. Phys. Rev. Lett. 52:1665-68

121. Seligman, T. H., Verbaarschot, J. J. M.,Zirnbauer, M. R. 1984. Phys. Rev. Lett.53:215-17

122. Bohigas, O., Giannoni, M. J., Schmit, C.1984. J. Phys. Lett. 45 : L1015-L1022

123. Berry, M. V., Robnik, M. 1984. J. Phys.A 17:2413-21

124. Pechukas, P. 1984. J. Phys. Chem. 88:4823-29

125. Meyer, H.-D., Hailer, E., K6ppel, H.,Cederbaum, L. S. 1984. J. Phys. A17: L831-L836

126. Vaz Pires, M., Galley, C., Lorquet, J. C.1978. J. Chem. Phys. 69 : 3242-49

127. Desouter-Lecomte, M., Galley, C., Lor-quet, J. C., Pires, M. V. 1979. J. Chem.Phys. 71 : 3661-72

128. Dehareng, D., Chapuisat, X., Lorquet,J. C., Galley, C., Raseev, G. 1983. J.Chem. Phys. 78:1246-64

128a. Desouter-Lecomte, M., Dehareng, D.,Leyh-Nihant, B., Praet, M. Th., Lor-quet, A. J., Lorquet, J. C. 1985. J. Phys.Chem. 89 : 214-22

129. Ammeter, J. 1980. Nouv. J. Chim. 4:631-37

130. Bersuker, I. B. 1984. See Ref. 19, pp.203-49

131. Gole, J. L., Stwalley, W. C., eds. 1982.


Ann

u. R

ev. P

hys.

Che

m. 1

985.

36:2

77-3

20. D

ownl

oade

d fr

om a

rjou

rnal

s.an

nual

revi

ews.

org

by C

OR

NE

LL

UN

IVE

RSI

TY

on

05/2

7/06

. For

per

sona

l use

onl

y.


Metal Bonding and Interactions in HighTemperature Systems: With Emphasison Alkali Metals. Washington, DC:Am. Chem. Soc.

132. Bersuker, I. B. 1984. See Ref. 19, pp.251-90

133. Salem, L. 1966. Molecular OrbitalTheory of Conjugated Systems. NewYork : Benjamin

134. Porter, G., Ward, B. 1968. Proc. R. Soc.London Set. A 303:139-56

135. Engelking, P. C., Lineberger, W. C.1977. J. Chem. Phys. 67 : 1412

136. Heaven, M., Dimauro, L., Miller, T. A.1983. Chem. Phys. Lett. 95 : 347

137. Miller, T. A., Bondybey, V. E. 1982.Philos. Trans. R. Soc. London Set. A307 : 617-31

138. Berkowitz~ J. 1979. Photoabsorption,Photoionization and PhotoelectronSpectroscopy. New York : Academic

139. Bersukcr, I. B. 1984. See Rcf. 19, pp.204-7

140. Rabalais, J. W. 1977. Principles ofUltraviolet Photoelectron Spectroscopy.New York : Wiley

141. Bondybey, V. E., Miller, T. A. 1980. J.Chem. Phys. 73 : 3053-59

142. Bondybey, V. E., Sears, T. J., Miller, T.A., Vaughn, C., English, J. H., Shiley, R.S. 1981. Chem. Phys. 61:9-16

143. Bondybey, V. E., Vaughn, C., Miller, T.A., English, J. H., Shiley, R. H. 1981. J.Am. Chem. Soc. 103 : 6303-7

144. Bondybey, V. E., English, J. H., Miller,T. A. 1983. J. Phys. Chem. 87:1300-5

145. Cossart-Magos, C., Cossart, D., Leach,S. 1979. Mol. Phys. 37 : 793-830

146. Cossart-Magos, C., Cossart, D., Leach,S. 1979. Chem. Phys. 41 : 345~2

147. Cossart-Magos, C., Cossart, D., Leach,S. 1979. Chem. Phys. 41 : 363-72

148. Cossart-Magos, C., Cossart, D., Leach,S., Maier, J. P., Misev, L. 1983. J. Chem.Phys. 78 : 3673-87

149. Raghavachari, K., Haddon, R. C.,Miller, T. A., Bondybey, V. E. 1983. J.Chem. Phys. 79:1387-95

150. Long, S. R., Meek, J. T., Reilly, J. P.1983. J. Chem. Phys. 79 ." 3206

151. Reilly, J. P. 1984. Israel J. Chem. 24:266-72

152. Wilson, W. G., Viswanathan, K. S.,Sekreta, E., Reilly, J. P. 1984. J. Phys.Chem. 88 : 672-73

153. Dunning, F. B., Stebbings, R. F., eds.1983. Rydberg States of Atoms andMolecules. London: Cambridge Univ.Press

154. Johnson, P. M., Otis, C. E. 1981. Ann.Rev. Phys. Chem. 32:139-57

155. Whetten, R. L., Grant, E. R. 1984. J.Chem. Phys. 80: 5999-6005


156. Whetten, R. L., Grubb, S. G., Otis, C.E., Albrecht, A. C., Grant, E. R. 1985. J.Chem. Phys. 82:1115-34

157. Whetten, R. L., Fu, K. J., Grant, E. R.1983. J. Chem. Phys. 79 : 2626-40

158. Grubb, S. G., Otis, C. E., Whetten, R. L.,Grant, E. R., Albrecht, A. C. 1985. J.Chem. Phys. 82 : 1135-46

159. Asbrink, L., Lindholm, E., Lingquist, O.1970. Chem. Phys. Lett. 5 : 609

160. Johnson, P. M., Korenowski, G. M.1983. Chem. Phys. Lett. 97:53

161. Whetten, R. L., Grant, E. R. 1985. J.Chem. Phys. 82 : In press

162. Scharf, B., Vitenberg, R., Katz, B., Band,Y. B. 1982. J. Chem. Phys. 77:4903-11

163. Whetten, R. L. 1984. Nonadiabaticbound states of isolated molecules. PhDthesis. Corncll Univ., Ithaca, NY. 603PP.

164. Robin, M. B. 1974. Higher ExcitedStates of Polyatomic Molecules. NewYork : Academic

165. Berry, R. S., Nielsen, S. E. 1970. Phys.Rev. A 1 : 383-94

166. Fano, U. 1983. Commun. At. Mol. Phys.13 : 157-69

167. Dressier, K., Jungen, Ch., Miescher, E.1981. J. Phys. B 14:L701-L704

168. Herzberg, G., Jungen, Ch. 1982. J.Chem. Phys. 77 : 5876-84

169. McDiarmid, R. 1978, J. Chem. Phys.69 : 2043-53

170. Wynne, J. J., Armstrong, J. A. 1979.Comm. At. Mol. Phys. 8:155-71

171. Jungen, Ch., Dill, D. 1980. J. Chem.Phys. 73 : 3338-45

171a. Jungen, Ch. 1984. Phys. Rev. Lett. 53:2394-97

172. Dagata, J. A., Findley, G. L., McGlynn,S. P., Connerade, J.-P., Baig, M. A.1981. Phys. Rev. A 24:2485-90

173. Wiesenfeld, J. R., Greene, B. I. 1983.Phys. Rev. Lett. 51 : 1745-48

174. Whetten, R. L, Fu, K. J., Grant, E. R.1984. Chem. Phys. 90:155-65

175. Freund, R. S. 1983. See Ref. 153, pp.355-92

176. Dill, D., Jungen, Ch. 1980. J. Phys.Chem. 84:2116-22

177. Lykke, K. R., Mead, R. D.., Lineberger,W. C. 1984. Phys. Rev. Lett. 52:2221-24

178. Mead, R. D., Lykke, K. R., Lineberger,W. C., Marks, J., Brauman, J. I. 1984. J.Chem. Phys. 81:4883-92

179. Golub, G. H., Van Loan, C. F. 1983.Matrix Computations. Baltimore: JohnsHopkins Univ. Press

180. Nauts, A., Wyatt, R. E. 1983. Phys. Rev.Lett. 51 : 2238-41

181. Miller, W. H. 1974. Adv. Chem. Phys.25 : 69-177


Ann

u. R

ev. P

hys.

Che

m. 1

985.

36:2

77-3

20. D

ownl

oade

d fr

om a

rjou

rnal

s.an

nual

revi

ews.

org

by C

OR

NE

LL

UN

IVE

RSI

TY

on

05/2

7/06

. For

per

sona

l use

onl

y.



182. Ranfagni, A., Mugnai, D., Englman, R.1984. Rev. Rep. 108 : 165-216

183. Heller, E. J., Brown, R. C. 1983. J.Chem. Phys. 79 : 3336-51

184. Nakamura, H. 1984. J. Phys. Chem. 88:4812-23

185. Feynman, R. P., Hibbs, A. R. 1965.Quantum Mechanics and Path Inte#rals.New York : McGraw-Hill

186. Pechukas, P. 1969. Phys. Rev. 181 : 166-74

187. Pechukas, P. 1969. Phys. Reo. 181 : 174-85

188. Miller, W. H., George, T. F. 1972. J.Chem. Phys. 56 : 5637-52

189. Preston, R. K., Tully, J. C. 1971. J.Chem. Phys. 54 : 4297-431M

190. Tully, J. C., Preston, R. K. 1971. J.Chem. Phys. 55 : 562-72

191. Freed, K. F. 1975. Chem. Phys. 10: 393-402

192. Laing, J. R., Freed, K. F. 1975. Phys.Rev. Lett. 34: 84%52

193. Laing, J. R., Freed, K. F. 1977. Chem.Phys. 19:91-117

194. Herman, M. F., Freed, K. F. 1983. J.Chem. Phys. 78 : 6010-20

195. Delos, J. B., Thorson, W. R., Knudson,S. K. 1972. Phys. Rev. A 6:709-20

196. Delos, J. B., Thorson, W. R. 1972. Phys.Rev. A 6 : 720-27

197. Billing, G. D. 1983. Chem. Phys. 100:535-39

198. Billing, G. D., Jolicard, G. 1984. J. Phys.Chem. 88 : 1820-25

199. Billing, G. D. 1984. Chem. Phys. 89:19%218

200. Swaminathan, P. K., Micha, D. A. 1982.Int. J. Quant. Chem. 516:377-90

201. Micha, D. A. 1983. J. Chem. Phys. 78:7138-45

202. Olson, J. A., Micha, D. A. 1984. J.Chem. Phys. 80: 2602-14

203. Kerner, E. H. 1959. Phys. Rev. Lett.2:152-53

204. Kerman, A. K., Koonin, S. E. 1976. Ann.Phys. 100:332-58

205. Tiszauer, D., Kulander, K. C. 1984.Phys. Rev. A 29 : 2902-11

206. Joachim, C. 1984. Mol. Phys. 52:1191-1208

207. Miller, W. H., McCurdy, C. W. 1978. J.Chem. Phys. 69 : 5163-73

208. MeCurdy, C. W., Meyer, H.-D., Miller,W. H. 1979. J. Chem. Phys. 70:3177-87

209. Meyer, H.-D., Miller, W. H. 1979. J.Chem. Phys. 70: 3214-23

209a. Herman, M. F., Currier, R. 1985.

Chem. Phys. Lett. 114 : 411-14210. Meyer, H.-D., Miller, W. H. 1979. J.

Chem. Phys. 71:2156~9211. Meyer, H.-D., Miller, W. H. 1980. J.

Chem. Phys. 72: 2272-81212. Miller, W. H. 1978. J. Chem. Phys.

68 : 4431-34213. Dirac, P. A. M. 1927. Proc. R. $oc.

London Set. A 114 : 243-65214. Orel, A. E., Ali, D. P., Miller, W. H.

1981. Chem. Phys. Lett. 79:137-41215. Ali, D. P., Miller, W. H. 1984. Chem.

Phys. Lett. 103 : 470-74216. Gray, S. K., Miller, W. H. 1982. Chem.

Phys. Lett. 93 : 341-44217. Meyer, H. D. 1983. Chem. Phys. 82:

197-205218. Noid, D. W., Koszykowski, M. L.,

Marcus, R. A. 1981. Ann. Reo. Phys.Chem. 32: 267-309

219. Graham, R., H6hnerbach, M. 1984. Z.Phys. B 57 : 233-48

220. Judd, B. R. 1978. J. Chem. Phys. 68:5643-46

221. Williams, R. D., Koonin, S. E. 1982.Nucl. Phys. A 391 : 72-92

222. Johnson, B. R. 1973. Chem. Phys. 2:381-99

223. Eu, B. C. 1984. Semiclassical Theoriesof Molecular Scattering. New York:Springer

224. Herman, M. F. 1983. J. Chem. Phys. 79 :2771-78



227. Voronin, A. I., Karkach, S. P.,Oscherov, V. I., Ushakov, V. G. 1976.Soy. Phys. JETP 44: 465-70

228. O’Brien, M. C. M. 1976. J. Phys. C 9:2375-82

229. Connor, J. N. L., Uzer, T., Marcus, R.A., Smith, A. D. 1984. J. Chem. Phys.80: 5095-5105

230. Voronin, A. I., Osherov, V. I. 1974. Soy.Phys. JETP 39 : 62-66

231. Karkach, S. P., Osherov, V. I., Usha-kov, V. G. 1975. Soo. Phys. JETP 41 :241-46

232. Karkach, S. P., Osherov, V. I. 1979.Mol. Phys. 38 : 1955-71

233. Ballhausen, C. J., Avery, J. 1981. Mol.Phys. 43:1199-1203

234. Bjorken, J. D., Orbach, H. S. 1981. Phys.Rev. D 23:2243-51

235. Bjorken, J. D., Orbach, H. S. 1982. Ann.Phys. 141 : 50-82


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