continuum channel coupling of shape resonances in n2

Continuum channel coupling of shape resonances in N2E. D. Poliakoff, Sandeep Kakar, and R. A. Rosenberg Citation: The Journal of Chemical Physics 96, 2740 (1992); doi: 10.1063/1.462022 View online: http://dx.doi.org/10.1063/1.462022 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/96/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nuclear Astrophysics Studies with the Method of ContinuumDiscretized CoupledChannels AIP Conf. Proc. 1235, 228 (2010); 10.1063/1.3442599 ManyBody Dynamics Coupled to Continuum and “Pygmy”Resonances AIP Conf. Proc. 802, 296 (2005); 10.1063/1.2140671 Shaperesonanceenhanced continuum–continuum coupling in photoionization of CO2 J. Chem. Phys. 99, 1556 (1993); 10.1063/1.465324 Localization of a continuum shape resonance. Photoionization of CS2 J. Chem. Phys. 97, 4690 (1992); 10.1063/1.463870 Shape resonances in the photoionization of N2O J. Chem. Phys. 87, 224 (1987); 10.1063/1.453620

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

131.155.81.2 On: Sat, 22 Nov 2014 00:43:11

Continuum channel coupling of shape resonances in N2 E. D. Poliakoff Department o/Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803

Sandeep Kakar Department 0/ Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803

R. A. Rosenberg Synchrotron Radiation Center, University 0/ Wisconsin-Madison, Stoughton, Wisconsin 53589

(Received 21 August 1991; accepted 6 November 1991)

We have measured vibrational branching ratios for 2uu- 1 photoionization ofN2 in an effort to elucidate fundamental aspects of continuum channel coupling. Calculations have shown that photoejection of a 2u u electron from N 2 should be influenced by a shape resonance in the 3u g

..... EU u photoionization channel and that this continuum channel coupling can result in deviations from Franck-Condon behavior for the resulting N 2+ (B 2Lu+ ) ion. In the present study, the N2 molecules are ionized by monochromatic synchrotron radiation (25 < hv < 55 eV) and dispersed fluorescence is measured to determine the vibrational branching ratios v' = lIv' = 0 and v' = 2/v' = 0 for the N 2+ (B 2Lu+) state. The observed branching ratios are enhanced at hv::::::30 eV and we attribute this Franck-Condon breakdown to continuum coupling between the 2uu- 1 and 3ug- 1 ionization channels. However, our results exhibit significant discrepancies with theory. The areas of agreement and disagreement suggest useful avenues of further study to clarify the nature of continuum channel coupling in molecular photoionization.

I. INTRODUCTION

It is well known that resonances in one excitation chan­nel can influence ionization continua in other excitation channels. Indeed, autoionization, a process in which nomi­nally discrete states of one channel "decay" into alternative ionization continua, has been studied intensively for many years. 1 However, there has been very little study of how a continuum resonance in one ionization channel, such as a shape resonance, can influence alternative ionization chan­nels. Such phenomena clearly require additional study be­cause it is likely that effects of continuum channel coupling can be dramatic and widespread. In this paper, we report vibrational branching ratios for 2uu- 1 photoionization ofN2 in an effort to elucidate fundamental aspects of shape reso­nance mediated continuum channel coupling.

There are several reasons for choosing 2uu- 1 photoioni­zation ofN2 for the present study. First, a diatomic molecule serves as a useful prototype system because the independent particle behavior of shape resonant photoionization is well understood for diatomic samples.2-6 Second, theoretical studies7

- 9 have shown that the 2uu- 1 channel in N z does not support any shape resonances in the absence of continuum channel coupling. Third, there is a well-characterized shape resonance in the 3ug ..... EUu ionization channel, 10-12 so there is a possibility for the transfer of oscillator strength between the 3ug- 1 and 2u;; 1 ionization channels. Fourth, there have been high quality experimental 13. 14 and theoretical 15.16 stud­ies of channel coupling for this system. Thus, the results of the current study, in conjunction with the earlier work, are useful for elucidating how shape resonances in one ioniza­tion channel mediate the behavior of other ionization chan­nels. As shape resonances are ubiquitous for molecular sys-

tems,Z an understanding of the systematics of continuum channel coupling is required for developing predictive capa­bilities for fundamental molecular scattering phenomena.

In order to highlight the physical basis for continuum coupling of shape resonances, the process is described (qual­itatively) here and illustrated schematically in Fig. 1. The well-known 3ug ..... EUu shape resonancelO- IZ occurs in the en­ergy range studied and the quasibound resonant complex has significant amplitude within the ionic core. Thus, follow­ing shape resonant excitation of a 3ug electron into the con­tinuum, the probability of a "collision" between the quasi­bound continuum electron and a 2u u valence electron is enhanced, which can result in the ejection of the 2u u elec­tron, while the original continuum electron returns into the 3ug vacancy. Thus, the interaction between the shape reso­nant electron and an electron in the 2u u orbital can transfer shape resonant character from the 3ug- 1 channel to the 2u u- 1 channel. 13-16 Independent particle calculations have been performed using different calculational frameworks and they have consistently shown that the 2u ~- 1 channels do not support any shape resonances.7

-9 As a result, the

Franck-Condon approximation dictates that the vibrational branching ratios should be independent of excitation energy. Thus, we can infer that observed variations in vibrational branching ratios are due to interchannel coupling of reson­ances, either autoionization of discrete states or continuum coupling of shape resonances. In our measurements, the dis­persed fluorescence from the N z+ (B zLu+) ions (the state created upon ejection of the 2uu electron) is monitored to determine production of alternative vibrational levels.

As mentioned above, there have been previous studies on this system, both experimental 13,14 and theoretical. I5 ,I6 However, the experimental efforts have not been sufficient

2740 J. Chem. Phys. 96 (4), 15 February 1992 0021-9606/92/042740-05$06.00 © 1992 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

131.155.81.2 On: Sat, 22 Nov 2014 00:43:11

Poliakoff, Kakar, and Rosenberg: Shape resonances in Nz 2741

£XCIT£ So; £t£CTRON COHTIHUW COUPUHG

IHTD SHAPe It£SOHANCE So;-, - 211u- I

-H- -H- 111'u

2I1U-' STArr

FLUOResces

I Evac

FIG. 1. A schematic view of continuum channel coupling in N2 photoioni­zation.

to fully test and guide theory, and confusing gaps and discre­pancies remain. We briefly recount the past work to place the current study into context. The first investigation into continuum channel coupling was a theoretical study of N2 by Stephens and Dill.IS They predicted the energy depend­ence of the photoelectron asymmetry parameter «(3) of the 20',,- I continuum electron with the nuclei fixed and per­formed calculations using the mUltiple scattering model. 17

Their calculations predicted a large excursion in the (3 pa­rameter at hv:::::35 eV. This work was followed by an experi­mental study on the v = 0 level of the 2O'u- I channel, which indeed showed a variation in the(3parameter, 14 although the resonant feature occurred at hv:::::30 eV and the size of the dip in (3 was much less pronounced than predicted. IS These discrepancies were not viewed as significant, given that the calculations were performed at fixed internuclear separation and that a model molecular potential was used. The next advance emerged from a study by our group, which had suf­ficient sensitivity to measure the production of the v = 0 and v = 1 levels of Nt (B 2~u+)' The vibrational branching ra­tio v = lIv = 0 for the Nt (B 2~: ) state exhibited a reso­nant enhancement (::::: 10%) at hv::::: 30 eV and the energy position of this feature coincided precisely with the experi­mental curve for the photoelectron asymmetry parameter. 14

Technical constraints limited our previous study to photon energies below 32 eV, 13 which was unfortunate because the next theoretical study by Basden and Lucchese l6 predicted that the branching ratio would peak at higher energy, i.e., at 36 eV rather than at 30 eV. Equally serious was the discrep­ancy on the magnitude of the vibrational branching ratio peak. The experimental results 13 exhibited a small deviation from Franck-Condon behavior (::::: 10%), while the calcu­lated excursion from Franck-Condon behaviorl6 was dra­matically more pronounced ( ::::: 60%). The calculations em­ployed the Schwinger variational method, which is usually accurate for small systems at the independent particle level of approximation. 18 Basden and Lucchese l6 suggested that the experimental branching ratio curve terminated at too low a photon energy to observe the manifestations of the interchannel coupled shape resonance and, further, hypoth-

esized that the feature observed at hv::::: 30 eV was in fact an autoionization resonance. For completeness, we note that their predicted position of the resonance in the curve of (3(v = 0) vs energy also appeared at higher energy than the experimental result. 14

Thus, the present study provides new data to clarify these unresolved issues. First, this study extends over a much wider photon energy range than previous study, 13 en­abling us to test the predictions of Basden and Lucchese 16 for photon energies greater than 32 eV. Second, we have now performed measurements on the v = 2/v = 0 vibrational branching ratio, so that there will be additional information for comparison with theory. We have determined the vibra­tional branching ratios for the v' = 0, v' = 1, and v' = 2 lev­els of the Nt (B 2~u+ ) ion by measuring the relative intensi­ties of fluorescence transitions originating from these levels. 13,19,20 The excitation and fluorescence sequence used to monitor production of the vibrational levels of N2+ (B 2~u+ ) is summarized by Eq. I and is sketched in Fig. 2.

N 2(X I~g+) + hvexc ->N2+ (B 2~u+ ,v') + e-

22

20

,.... 18 >

CD '-"

>-(!) Q: W Z w

16 hI/axe 1 0

2

o

0.8 1.0 1.2 1.4

DISTANCE (A)

FIG. 2. Potential curves for N2 and Nt showing photoionization and flu­

orescence steps.

J. Chem. Phys., Vol. 96, No.4, 15 February 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

131.155.81.2 On: Sat, 22 Nov 2014 00:43:11

2742 Poliakoff, Kakar, and Rosenberg: Shape resonances in N2

The fluorescence intensity originating from level v' is a mea­sure of the rate of production of that level, i.e., its partial photoionization cross section (T~. 13

As we will show, these results underscore the impor­tance of obtaining vibrationally resolved data for investiga­tions into molecular ionization dynamics. This is a strong impetus for the dispersed fluorescence measurements that are employed in the preent study, as they provide highly resolved (induding rotationally resolved21

) data on the photoions over a wide range of incident photon energies. High resolution fluorescence data are accessible because the detection bandwidth in such experiments is not dictated by the energy bandwidth of the incident vacuum ultraviolet (VUV) radiation, as is the case for alternative methods, such as photoelectron spectroscopy. Thus, the convenient tunability of synchrotron radiation sources may be exploited to their full advantage. This capability allows us to probe molecular aspects of the problem (i.e., vibration and rota­tion), which are intrinsically sensitive to resonant phenome­na in molecular scattering.

II. EXPERIMENT

The experimental apparatus and method have been dis­cussed previously. 13,19 A brief description of the apparatus is given here and illustrated schematically in Fig. 3. The inci­dent radiation originates from a synchroton radiation source, and for the present study, we utilized the 6 m toroidal grating monochromator beam line22 at the Synchrotron Ra­diation Center at the University of Wisconsin at Madison. The monochromatized synchrotron radiation (I1E::::: 0.1 0 eV) was channeled to the interaction region in the experi­mental chamber by a two-stage differentially pumped capil­lary (2 mm i.d.). The exciting vacuum ultraviolet (VUV) radiation intersects an effusive molecular beam and the flu­orescence radiation is collimated by a planoconvex lens (op­tical aperture off /4). The fluorescence radiation exits the chamber through a window and passes through a shutter which is used to determine dark count contributions of the detector. Then the radiation is reflected into the horizontal plane and focused onto the entrance slit of the fluorescence

FIG. 3. A schematic of the experimental apparatus.

monochromator (Instruments SA model HR640; the opti­cal aperture is f /5.2). The dispersed radiation exiting the monochromator is refocused onto the photocathode of a cooled photomultiplier detector. For the present measure­ments, the fluorescence bandwidth was adjusted to I1A = 12 A. A vacuum ultraviolet photodiode measures the intensity of the incident VUV radiation. The fluorescence signal, pho­todiode signal, shutter control, and monochromator con­trols are interfaced to a computer via standard computer­aided measurement and control (CAMAC) electronics. One item that is not shown in Fig. 3 is a spherical mirror which is placed in the vacuum chamber to collect the radi­ation that is emitted in the opposite direction from the colli­mating lens. In practice, this nearly doubles the observed fluorescence signal.

In order to determine the contribution of higher-order radiation, we measured excitation spectra using an Al metal foil filter to reject higher-order radiation from the output of the excitation monochromator.23 This was useful at primary photon energies greater than 36 eV, as the metal foils elimi­nate radiation with hv> 72 eV, i.e., the higher-order compo­nents. As a result, we could estimate the contributions of higher-order excitation components to the observed fluores­cence signal and we found that the higher-order radiation did not appreciably affect the vibrational branching ratio, presumably because the photoionization cross section for the Nt (B 2"2.u+ ) state is monotonically decreasing.

Measurements were performed at several chamber pres­sures between 5 X 10-5 and 5 X 10-4 Torr. This was done to test for formation of Nt (B 2"2.: ) by secondary processes, such as electron impact ionization, rather than by photoioni­zation.

III. RESULTS Figure 4 shows a portion of the fluorescence spectrum

for N2+ (B 2"2. u+ ,v' -x 2"2.g+ ,v") and includes transitions originating from the v' = 0, 1, and 2 levels of Nt (B 2"2. u+). The intensity of fluorescence from a particular level is pro­portional to the rate of production of that level, i.e., its par­tial photoionization cross section. 13 By taking ratios of flu­orescence intensities for transitions with different upper vibrational levels, we can generate vibrational branching ra-

hv .. -32eV

0i-~~~~MM~~~~~~~~~~~~ 4150 4200 4250 4300

Fluorescence wavelength (A)

FIG. 4. The vibrationally resolved fluorescence spectrum ofN2+ photoions.

J. Chem. Phys., Vol. 96, No.4, 15 February 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

131.155.81.2 On: Sat, 22 Nov 2014 00:43:11

Poliakoff, Kakar, and Rosenberg: Shape resonances in N2 2743

tios, 13 which serve as useful indicators of resonant ionization pathways. The procedure for converting a fluorescence in­tensity ratio into a vibrational branching ratio has been dis­cussed previously and is briefly recounted here. Specifically, we must normalize the fluorescence intensity using the strength of the observed transition, as summarized by Eq. (2)24

au Iu//(qu/'~/)

au' Iu,/./(qu·[' .~./.} (2)

Here, u and u' represent alternative vibrational levels for the upper electronic state. Similarly, I and I' denote different vibrational levels for the lower electronic state. The Franck­Condon factor and the frequency of the fluorescence transi­tion from upper level i to lower level! are denoted by q if and vi/' respectively. Note that all quantities on the right-hand side of the Eq. (2) (Le., I if, qi/' and vif) pertain to the ionic fluorescence transitions and not to the initial ionization pro­cess. This equation converts ratios of fluorescence intensities into vibrational branching ratios. Thus, by measuring the fluorescence intensities as a function of the excitation ener­gy, we can generate a curve of vibrational branching ratios. Implicit in this equation is that different upper levels u and u' do not have different predissociation rates. 19,20 This assump­tion appears to be valid for this study as the present data agree well with the photoelectron data for the vibrational branching ratio. 2s The latter are unaffected by predissocia­tion. We may point out that even if our results were affected by predissociation, the relative behavior of vibrational branching ratio as a function of photon energy will not be altered.

Figure 5 shows vibrational branching ratio curves and a broad resonance is clearly visible in the v' = 2/v' = 0 curve at hVm ::::;30 eV. There is also a small peak in the branching ratio in the v' = l/v' = 0 curve, in good agreement with pre­vious data, 13 although the statistical scatter is greater in the

Nt (e2I!) branching ratios 0.015

0.010

0.005 0

:.;:; e CI 0.000 c 0.300 :c 0 c e 0.200 lD

0.100

0.000 20 30 40 50 60

Excitation energy (eV) FIG. 5. Vibrational branching ratios for N,' (B 0'2"+ ). The solid curve is the calculated prediction by Basden and Lucchese (Ref. 16).

current results. A pressure dependence was found for the v' = 2 curve,

although not for the v' = 1 or v' = 0 curve. We found that the behavior ofthis curve was reproducible at any pressure at or below 10-4 Torr and the data reported was obtained at pressures below 10-4 Torr for all scans. Even with this re­striction oflow sample densities, the experimental results are of sufficient quality to demonstrate the pronounced devi­ation from Franck-Condon behavior in the v' = 2/v' = 0 branching ratio curve.

IV. DISCUSSION

The basic description of continuum channel coupling was described qualitatively in the introduction and illustrat­ed in Fig. 1. We now extend this description to see how a shape resonance in one channel might affect the vibrational branching ratio for another continuum. More rigorous dis­cussions may be found in the theory papers by Stephens and Dill, IS and Basden and Lucchese. 16 When a 3ag electron in N2 is ejected into a Eau continuum channel with hv::::;30 eV, the continuum electron is quasibound due to the temporary trapping by a potential barrier (in this case, a centrifugal barrier for the 1=3 partial wave.3,10 The result is that the photoelectron has dramatically increased amplitUde in the region of the other valence electrons, so the transition ampli­tude between the bound electron wave function and the con­tinuum photoelectron is enhanced commensurately.2 The trapping of the photoelectron is sensitive to the shape of the potential sensed by the photoelectron, which depends on the internuclear separation. Another consequence of this elec­tron being in this quasibound continuum orbital 13

-16 is an

enhancement of the probability of a collision with an elec­tron in the 2uu orbital. Let us suppose that the target mole­cule is frozen at some fixed internuclear distance, e.g., R = Re of the neutral ground state. When a 3ag electron is excited into the quasibound EUu shape resonance at the ener­gy where the cross section is at a maximum, the continuum electron can "collide" with a 2a u electron and oscillator strength will be transferred to 2a;; I channels. This process will have different probabilities for creating alternative vi­brationallevels of the 2au- I hole state because the vibration­al wave functions for these levels will sample the internu­clear distance R = Re in varying amounts. So let us hypothetically shift the internuclear distance slightly, e.g., to R = Re + oR. In so doing, we have shifted the photon ener­gy to a lower value where the cross section for ejecting the 3ug electron is at a maximum. At this lower peak energy, we can again couple the 3ag- 1 and 2a;; I ionization. The vibra­tional levels being populated as a result of this transfer of oscillator strength at this lower energy will be different than at the higher energy because the new internuclear distance (R = Re + oR) is projected onto the vibrational wave func­tions of the N2+ (B 2~u+) in a different fashion than when R = Re. This qualitative reasoning leads to the key result, namely, that the interchannel coupled resonance will alter the vibrational branching ratios as a function of photon ener­gy for the nominally nonresonant 2a u- I channel.

The theoretical results confirm our expectation that vi­brational branching ratios for the N/ (B 2J.u+ ) state exhibit

J. Chern. Phys., Vol. 96, No.4, 15 February 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

131.155.81.2 On: Sat, 22 Nov 2014 00:43:11

2744 Poliakoff, Kakar, and Rosenberg: Shape resonances in N2

deviations from Franck-Condon behavior. 16 However, the agreement between experiment and theory is not satisfac­tory. The results shown in Fig. 5 for the v' = 1!v' = 0 branching ratio show only a small deviation at hv;::::30 eV, indicating that the 3a g -> Ea u shape resonance is not strongly affecting this vibrational branching ratio. This contrasts strongly with theory, shown as the solid line in Fig. 5, which predicts a large excursion ( > 60%) at hv;:::: 35 eV. On the other hand, the v' = 2/v' = 0 branching ratio shows signifi­cant non-Franck-Condon behavior, but there are not yet calculations available for comparison.

There are mechanisms other than shape resonances that might be responsible for the observed behavior in the vibra­tional branching ratios. For example there are several shake­up states in N 2+ that have been observed in x-ray photoemis­sion spectrocopy (XPS) .26 It is possible that autoioinization of Rydberg states converging to these excited ionic limits can appear as resonance features in the vibrational branching ratio spectra. Moreover, several of these multiply excited ionic states have binding energies in the range 25 < Eb < 45 eV.26 However, such effects would be limited to comparati­vely narrow energy ranges. The structure observed in the v' = 1!v' = 0 curve is comparatively sharp and may be due to autoionization. However, the excursion observed in the v' = 2/v' = 0 curve extends over a range of 15 eV and this large width is most consistent with the explanation of chan­nel coupling of a shape resonance presented in the introduc­tion.

If the peaks observed in the vibrational branching ratio profiles are indeed due to the interchannel coupled shape resonance, then we must conclude that there is something missing from the theoretical or calculational framework de­vised by Basden and Lucchese. 16 If, on the other hand, the features observed in Fig. 5 are due to an autoionizing reso­nance, a more puzzling question emerges-namely, why do we not see the excursion in the v' = 1!v' = 0 ratio predicted by theory?

The present results on the 2a; 1 channel provide a strin­gent test for theoretical methods aimed at understanding the continuum coupling. The cross section for direct ionization into the 2a u- 1 channel is small, so channel coupling effects can have significant and dramatic effects on the partial pho­toionization cross section for the 2a u- 1 channel. 27 Moreover, the cross section for the v' = 2 level is particularly weak, so it is not surprising that the v' = 2/v' = 0 ratio curve exhibits the most dramatic resonance effects in the present study.

For completeness, we note that the angularly resolved photoelectron results for this system 14 provide an indepen­dent test of theory, and that experimental photoelectron asymmetry parameters (i.e, f3 's) do not agree satisfactorily with theory either. 16

•27 On the other hand, recent work on

photoionization of CO2 has shown good agreement with the­ory for transfer of oscillator strength due to continuum channel coupling mediated by shape resonances.28 There is clearly something missing from the N2 studies and this needs to be addressed as N2 is one of the most thoroughly charac­terized systems, experimentally and theoretically. Collec­tively, these results underscore the need for further studies of continuum channel coupling in molecular photoionization

as the results cannot be interpreted quantitatively without a picture of how resonance character is transferred between ionization continua. We are currently attempting to study the valence isoelectronic system of CO photoionization and these results may shed some light on this issue. We hope that the current study on N2 will provide additional impetus for characterizing and clarifying the role of channel coupling in shape resonant photoionization.

ACKNOWLEDGMENTS

This research was carried out at the Synchrotron Radi­ation Center (SRC), at the University of Wisconsin at Madi­son. We thank the staff ofSRC, particularly Dr. Roger Han­sen and Dr. Michael Engelhart, for their cooperation and assistance. Two of the authors (E.D.P. and S.K.) also grate­fully acknowledge support of this work from the National Science Foundation (Grant No. NSF-CHE-9001590), the Louisiana Educational Quality and Support Fund, and the Louisiana State University Center for Advanced Micro­structures and Devices (CAMD).

'J. Berkowitz, Photoobsorption, Photoionization, and Photoelectron Spec­troscopy (Academic, New York, 1979).

2J. L. Dehmer, A. C. Parr, and S. H. Southworth, in Handbook on Synchro­tron Radiation, edited by Geoffrey V. Marr (North-Holland, Amsterdam, 1987), Vol. 2.

31. Nenner and A. Beswick, in Handbook on Synchrotron Radiation, edited by Geoffrey V. Marr (North-Holland, Amsterdam, 1987), Vol. 2.

4p. W. Langhoff, ACS Symp. Ser. 263,113 (1984). sB. 1. Schneider and L. A. Collins, ACS Symp. Ser. 263, 65 (1984). 6J. L. Dehmer, D. Dill, and A. C. Parr, in Photophysicsand Photochemistry in Vacuum Ultraviolet, edited by S. P. McGlynn, G. L. Findley, and R. H. Huebner (Reidel, Dordrecht, 1985), p. 341.

7S. Wallace, D. Dill, and J. L. Dehmer, J. Phys. B 12, L417 (1979). 's. Wallace, Ph. D. thesis, Boston University, 1979. 9R. R. Lucchese, G. Raseev, and V. McKoy, Phys. Rev. A 25, 2572 ( 1982). IOJ. L. Dehmer, D. Dill, and S. Wallace, Phys. Rev. Lett. 43, 1005 (1979). "R. R. Lucchese and V. McKoy, J. Phys. B 14, L629 (1981). 12J. B. West, A. C. Parr, B. E. Cole, D. L. Ederer, R. Stockbauer, and J. L.

Dehmer, J. Phys. B 13, L105 (1980). 13E. D. Poliakoff, M. H. Ho, G. E. Leroi, and M. G. White, J. Chern. Phys.

84,4779 (1986). 14S. H. Southworth, A. C. Parr, J. E. Hardis, and J. L. Dehmer, Phys. Rev.

A 33, 1020 (1986). ISJ. A. Stephens and D. Dill, Phys. Rev. A 31,1968 (1985). 16B. Basden and R. R. Lucchese, Phys. Rev. A 37,89 (1988). I7D. Dill and J. L. Dehmer, J. Chern. Phys. 61, 692 (1974). ISR. R. Lucchese, K. Takatsuka, and V. McKoy, Phys. Rep. 131, 147

(1986). 19L. A. Kelly, L. M. Duffy, B. Space, E. D. Poliakoff, P. Roy, S. H. South­

worth, and M. G. White, J. Chern. Phys. 90, 1544 (1989). 2"There is a typographical error in Eq. (2) of Ref. 19; the termsf.., andf..j

should be exchanged. 21E. D. Poliakoff, J. C. K. Chan, and M. G. White, J. Chern. Phys. 85, 6232

(1986). 22R. K. Cole, F. K. Perkins, E. L. Brodsky, A. Filipponi, E. Korpela, D. C.

Mancini, C. H. Pruett, D. J. Wallace, J. T. Welnak, and F. Zanini, Rev. Sci. Instrum. 60, 2093 (1989).

23F. R. Powell, P. W. Vedder, J. F. Lindblom, and S. F. Powell, Opt. Eng. 29,581 (1990).

24D. L. Judge and G. L. Weissler, J. Chern. Phys. 48, 4950 (1968), and references therein.

25J. L. Gardner and J. A. R. Samson, J. Chern. Phys. 60, 3711 (1974). 26J. A. Nichols, D. L. Yeager, and P. Jorgenson, J. Chern. Phys. 80, 293

( 1984), and references therein. 27R. R. Lucchese and R. W. Zurales, Phys. Rev. A 44, 291 (1991). 28R. R. Lucchese,J.Chem. Phys. 92,4203 (1990); P. Roy, I. Nenner, M. Y.

Adam, J. Delwiche, M. J. Hubin-Franskin, P. Lablanquie, and D. Roy, Chern. Phys. Lett. 109, 607 (1984).

J. Chern. Phys., Vol. 96, No.4, 15 February 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

131.155.81.2 On: Sat, 22 Nov 2014 00:43:11