Can resonances occur in the photodissociationcontinuum of a diatomic molecule? The role ofpotential discontinuities1

Joel Tellinghuisen

Abstract: Continuum resonances are standard fare in the instructional literature for quantum mechanics, where theyarise from the continuity conditions imposed on one-dimensional wavefunctions for piecewise-constant potential energyfunctions. Such resonance structure weakens progressively as the discontinuity in the potential is smoothed, showingthat the structure is specifically attributable to the discontinuity. Since diatomic molecular potential energy curvesseldom vary rapidly on the distance scale of the period of the wavefunction, such continuum resonances are not ex-pected in absorption continua. A historically interesting prediction of such structure in the Schumann–Runge continuum(B ← X) of O2 is attributed to the inadvertent incorporation of discontinuity in the B-state potential curve employed inthe computations.

Key words: quantum mechanics, continuum resonance, diatomic absorption, photodissociation continuum, numericalmethods.

Résumé : Les continuum de résonance sont des entités normales dans la littérature des étudiants de la mécaniquequantique où elles découlent des conditions de continuité qui sont imposées aux fonctions d’ondes unidimensionnellesdes fonctions d’énergie potentielle constantes. De telles structures de résonance perdent progressivement del’importance lorsqu’on étale la discontinuité du potentiel, ce qui démontre que la structure peut être attribuée spécifi-quement à la discontinuité. Comme les courbes d’énergie potentielle des molécules diatomiques varient rarement rapi-dement en fonction de l’échelle de distance de la période de la fonction d’onde, on ne s’attend pas à retrouver des telscontinuum de résonances dans les continuum d’absorption. Une prédiction historiquement intéressante d’une telle struc-ture dans le continuum de Schumann–Runge (B ← X) du dioxygène, O2, est attribuée à l’incorporation accidentelle dela discontinuité dans la courbe de potentiel de l’état B utilisée dans les calculs.

Mots clés : mécanique quantique, continuum de résonance, absorption diatomique, continuum de photodissociation,méthodes numériques.

[Traduit par la Rédaction] Tellinghuisen 830

Introduction

The Schumann–Runge (SR) absorption transition of O2(B3Σ−

u ← X3Σ−g) represents a prototypal diatomic photo-

dissociation spectrum having an adjoining discrete spectrum.From a practical standpoint, this system is arguably the mostimportant case of this type, since it largely accounts for the“vacuum” in “VUV”. As such it received significant atten-tion from Herzberg, especially in his landmark book ondiatomics (1–3). Although the nature of the continuous ab-sorption is now fairly well understood, it is only recentlythat it has arrived at this state (4–11). The absorption is es-sentially structureless in the region from the photodissoci-ation limit at 175 nm to the peak near 143 nm. Structuredoes occur at shorter wavelengths, because of a strong R-dependence in the transition strength and a “ledge” in the

adiabatic approximation of the B potential, resulting frominteractions between the valence and Rydberg 3Σ−

u states thatproduce the adiabatic B and E states. However, an early at-tempt at quantitative analysis of the SR continuum predictedthe occurrence of “wiggles” in the region of the strongestabsorption, which were attributed to continuum resonancesresulting from the presence of an attractive well in the po-tential (12). This result was never confirmed in subsequenttreatments, but it was also never explained. A reexaminationof the original treatment by Bixon et al. (12) has led me toconclude that the interesting structure in their computedspectrum was an artifact very likely stemming from anunphysical discontinuity inadvertently included in their B-state potential energy function. The present paper offers amore comprehensive look at the role of potential discontinu-ities in continuum resonances, with the conclusion that suchstructure must be very rare and weak in diatomic spectra,since realistic diatomic potentials are normally too smoothto support it.

To illustrate the last point first, consider the partial poten-tial energy diagram for O2 shown in Fig. 1 and the bound-free absorption spectra computed with these potentials inFig 2. The two spectra in the latter figure are virtually iden-tical, which implies that the B-state continuum wave-

Can. J. Chem. 82: 826–830 (2004) doi: 10.1139/V04-047 © 2004 NRC Canada

826

Received 17 December 2003. Published on the NRC ResearchPress Web site at http://canjchem.nrc.ca on 25 August 2004.

J. Tellinghuisen. Department of Chemistry, VanderbiltUniversity, Nashville, TN 37235, USA. (email:joel.tellinghuisen@vanderbilt.edu).

1This article is part of a Special Issue dedicated to thememory of Professor Gerhard Herzberg.

functions are identical where they overlap significantly withthe X-state wavefunction. The energy-normalized wave-functions shown in Fig. 1 indeed coincide at small R, in bothshape and amplitude, and this in spite of their quite differentnature where they diverge at larger R. This relationship hasbeen illustrated before for a system in I2 (14), and it is keyto the realization that the Franck–Condon properties of suchsimple absorption spectra are determined solely by the shapeof the final potential in the Franck–Condon region. This istrue for the discrete region, too, in the case of a potentialwith a bound well, where the attractive branch of the poten-tial serves just to “discretize” the spectrum (15, 16). Fromthe standpoint of the continuum wavefunctions (ψ) them-selves, which are obtained numerically in one pass by gener-ating outward to large R and then normalizing (17), in a veryreal sense the ψs “know where they are going” right fromthe outset. So we might then ask, what could cause them todeviate from this course?

Figures 3 and 4 provide an answer to the question justposed: the ψs deviate when they encounter a sharp “bump”,where sharp means abrupt on a distance scale comparable tothe period of the wavefunction. However, even the smallamount of smoothing illustrated by the dashed potential inFig. 3 suffices to eliminate most of the resonance structure,the exception being the region very near threshold.

The properties exhibited for the wavefunctions in Fig. 3can be seen also for the simple piecewise constant potentialsused to illustrate continuum resonances in quantum mechan-

ics textbooks. Such examples prove both instructive and use-ful for checking computational procedures, so I turn my at-tention to them before returning to the O2 historical puzzle.

Models and methods

The simplest model potential for a bound molecule is thepiecewise constant potential illustrated in Fig. 5. The wave-function in regions 1 and 2 can be taken as

[1] ψ1 = b1 sink1x and ψ2 = b2 sin(k2x + δ)

where k in each region is related to the kinetic energy ε inthat region by

[2] k22

2= µε�

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Tellinghuisen 827

Fig. 1. Potential curves and wavefunctions for the B state of O2.The solid-black B curve consists of RKR turning points (13), ex-tended at small R with an R–8 repulsion function. The magenta-coloured potential curve shares the same repulsive branch butdissociates to two ground state 3P O atoms at 42 046 cm–1 abovethe ground-state minimum (cf. 3P + 1D at 57 914 cm–1 for the Bstate, marked by the lower dashed line segment to right). Thedashed vertical lines denote the classical turning points for ν = 0in the X state. The wavefunctions are energy-normalized func-tions at an energy of 58 147 cm–1, marked by the horizontal dot-ted line.

Fig. 2. Franck–Condon density (FCD) computed for absorptionfrom ν = 0 in the X state of O2 to the two potentials illustratedin Fig. 1. The vertical line marks the photodissociation limit forthe “real” potential, having the dashed spectrum.

Fig. 3. Segment of RKR potential for B state (solid, lower right)and alternative extensions obtained by defining a Morse potentialby ωe and ωexe for the B state (giving De = 11 800 cm–1) andthen chopping the attractive branch of the potential off at thetrue dissociation limit (De = 8121 cm–1). The dashed potentialcurve was produced using an “avoided crossing” smoothing func-tion. The wavefunctions were obtained for an energy of58 147 cm–1 (dotted horizontal line), as in Fig. 1.

with µ the reduced mass and � Planck’s constant divided by2π. By making the wavefunction and its first derivative con-tinuous at x = a, we obtain

[3]bb k a

1

2

2

2

2 1 22

1

=

+ −ε

ε ε ε( ) cos

One of the two amplitudes is arbitrary until some normal-ization is chosen. In energy normalization, the amplitude isproportional to ε–1/4 (17). Thus, if our wavefunction is nor-malized at large x, b2

2 ∝ ε2–1/2, and

[4] bk a

12 2

1 2

2 1 22

1

∝+ −

εε ε ε

/

( ) cos

Alternatively, if ψ1 were energy normalized in region 1 (e.g.,for the dashed potential having V = 0 for all x > 0 in Fig. 5),it would have b1

′2 ∝ ε1–1/2. Any spectrum involving these

wavefunctions would sense the square of the amplitude, sothe ratio

[5] Qb

b k a=

′

=

+ −1

1

2

1 21 2

2 1 22

1

( )( ) cos

/ε εε ε ε

constitutes the continuum resonance spectrum. It shows spe-cifically the effect of the potential step at x = a, and it variesbetween a maximum of (ε1/ε2)

1/2 and a minimum of(ε2/ε1)

1/2.To investigate the effect of smoothing the discontinuity in

V(x) at x = a, we represent the potential by a hyperbolic tan-gent type function

[6] V xV y

y yx( ) ,=

+>−

01

0

where y = exp[p(x – a)]. A large value of the parameter pgives a sharp step, while a small value yields a soft step.

All of the computations employed numerical methods likethose described in ref. 17.

Results and discussion

Figure 6 illustrates continuum resonance spectra for themodified square well potential of eq. [6] with varying de-grees of smoothing, while Fig. 7 shows the potentials andrepresentative wavefunctions. The results for p = 1000 Å–1

confirm the validity of the numerical methods, since a p thislarge gives essentially the original box potential. With p =10 Å–1 (dashed potential in Fig. 7), the resonance structure isalmost gone. Correspondingly, in region 1, the energy-normalized ψ at the energy of the strongest resonance is onlyslightly larger than that for V0 = 0 (no well). (Of course, thedashed ψ matches the solid ψ in amplitude at large x, as itmust.)

I return now to the O2 puzzle that prompted this study.Bixon et al. (12) (BRJ) adjusted a linear repulsion potential

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828 Can. J. Chem. Vol. 82, 2004

Fig. 4. FCD just above the photodissociation threshold, forpotentials illustrated in Fig. 3. The continuum resonance at57 360 cm–1 (magenta) corresponds to the energy used to gener-ate the wavefunctions in Fig. 3. The three spectra are associatedwith the similarly depicted potential curves and wavefunctions inFig. 3.

Fig. 5. Potential diagram for square well model. The energygoes to ∞ at x = 0. The subscripts on the kinetic energy ε desig-nate regions 1 (0 < x < a) and 2 (x ≥ a).

Fig. 6. Continuum resonance spectra for modified square wellpotential of eq. [6] having V0 = 5000 cm–1, a = 1 Å, µ =10 amu, and p = 1000 Å–1 (red), 50 Å–1 (solid black), and10 Å–1 (fine dash). The broad dash lines delimit the extremevalues of Q for p = ∞.

in position and slope to achieve approximate agreement withthe experimental B ← X data. When they attached a boundwell to this potential at large R and repeated the computa-tion, they obtained oscillations in the spectrum. For the wellthey used RKR points from Singh and Jain (18) and Rich-ards and Barrow (19). Both of these show a great deal ofscatter at high energy (Fig. 8), and BRJ did not explain howthey dealt with this problem. Furthermore, the RKR curvesand the derived linear potential are disjointed, suggestingthat the BRJ working potential included an inadvertent dis-continuity. I have checked this possibility by using the po-tential of Fig. 1 (with which the points of Richards andBarrow are reasonably consistent) at large R, and the linearpotential of BRJ at small R, and repeating the computationof the spectrum for various assumed switchover distances.The two potential segments intersect near 1.29 Å with only a20% slope discontinuity; when this distance is taken forswitchover, the resulting spectrum agrees well with the spec-trum of BRJ for the linear potential (Fig. 9). The spectrumobtained by jumping from the linear potential to the RKRcurve at R = 1.326 Å (the minimum R of the Richards andBarrows points) agrees well in the positions of the wigglesbut undershoots in their amplitudes. Moving the jump pointto larger R increases the magnitude of the discontinuity andhence of the resonance structure; however, it also decreasesthe spacing of the peaks, so that for switchover at 1.365 Å,the structure more nearly matches that of BRJ in amplitudebut includes an extra shoulder at low frequency.

Without detailed knowledge of how the RKR data wereactually incorporated in the computations of BRJ, it is prob-ably not possible to reproduce their results any better thanshown in the solid spectrum of Fig. 9. For example, mostcommonly used algorithms for generating the wavefunctions(including the Numerov method used here and by BRJ) em-ploy a working potential of points evenly spaced in R, whichrequires interpolation when RKR data are used to define thepotential. The interpolators can perform badly on noisy datalike those near dissociation in Fig. 8. To illustrate, if just the

monotonically varying (in R) data from refs. 18 and 19 areused to define the curve, even a quadratic Lagrangian inter-polation can produce an anomalous bump in the potential atsmall R, which is sharp enough to act just like the otherforms of discontinuity considered here, giving pronouncedstructure in the spectrum (Fig. 10). Thus, it seems likely thatthe computations reported in (12) were afflicted with arti-factual potential errors of one or both of the types consid-

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Tellinghuisen 829

Fig. 7. Potential energy functions and wavefunctions relevant toFig. 6 for p = 1000 (solid black) and 10 (dashed) Å–1. Alsoshown (red) are V(x) and ψ(x) for V0 = 0. Wavefunctions arecentered at the energy ε1 = 5094 cm–1 where they were gener-ated (horizontal dashed line), corresponding to the first peak forp = 1000 Å–1 in Fig. 6.

Fig. 8. Repulsive branch of the O2 B-state potential, from RKRcalculations: ref. 18 (+), ref. 19 (�), ref. 20 (�) (cited by Rich-ards and Barrow (19) for ν levels lower than those reported).The solid curve is that shown in Fig. 1, and the dashed line is asreported by Bixon et al. (12). Energies are relative to the mini-mum of the B potential.

Fig. 9. O2B ← X FCD computed for absorption from ν′′ = 0 us-ing linear potential from ref. 12 and Fig. 8 merged smoothlywith potential of Fig. 1 at R = 1.29 Å (dashed), or discontinu-ously at 1.326 Å (solid black), and 1.365 Å (red).

ered here; as concluded by others long ago, the resonancestructure is not real.

A secondary moral of the present story is that in such mo-lecular applications, one must take care to devise model po-tential energy functions that are realistic. Another illustrationof this point comes from a study of the predissociation ofthe C state of N2

+ by the B state. The C state is embedded inthe continuum of the B state, with only very small Franck–Condon overlap between the two, making the computationsvery sensitive to the potentials (21, 22). When the B-statepotential was represented by a cut-off Morse curve like thatshown in Fig. 3 instead of by its RKR curve, the predictedC → B predissociation rates decreased by four orders ofmagnitude, causing earlier workers to dismiss this process asthe major mode of C-state predissociative decay (21).

Conclusion

I return now to the question posed in the title. Most realis-tic potentials do not include the kinds of discontinuities andabrupt changes considered here, so we expect any actualcontinuum resonance structure to be subtle to the point ofpossibly escaping experimental detection. Since the require-ment is a relatively rapid variation of the potential energyfunction on the distance scale of the wavefunction, the bestchances for observation would seem to be in light molecules(small µ) at small kinetic energies, for example just above abarrier. Le Roy and Bernstein (23) computed several suchresonances in the ground state of H2 but noted that they were“too diffuse to be spectroscopically observable”. Anotherqualifying example is the very narrow “hole” in the Franck–Condon density right at threshold, as noted by Allison and

Dalgarno (15). In this case, as the kinetic energy approacheszero, the period of the wavefunction becomes infinite, sothat even the very small variation in the potential energyfunction in the approach to dissociation might be viewed assignificant on the distance scale of the wavefunction.

With the unintentional exception of the lowest frequenciesin the highly structured spectrum in Fig. 10, I have specifi-cally excluded from consideration here cases where the con-tinuum levels are partially bound behind a potential barrier.Such shape resonances are certainly real and observable.The presence of a barrier can also affect the structure in thecontinuum. For example, if the model square well potentialtreated above and illustrated in Figs. 5 and 7 is altered to in-clude a square barrier V1 > V0 in a region a ≤ x ≤ a1, the con-tinuity conditions on the wavefunction lead to a much richerinterference-type structure in the pure continuum (ε > V1).By using smoothing functions like eq. [6], one can “turn off”the structure associated with either the discontinuity at x = aor that at x = a1, or both.

References

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830 Can. J. Chem. Vol. 82, 2004

Fig. 10. O2 B ← X FCD for ν′′ = 0 computed using linear po-tential from ref. 12 and selected, monotonically varying RKRpoints from refs. 18 and 19, with three-point (solid) and two-point (dashed) Lagrangian interpolation to produce the workingpotential. A segment of the latter is illustrated together with therelevant RKR points in the inset.