Photodissociation Dynamics of Triatomic van der Waals Molecules Citation Cline, Joseph Isaac, III (1988) Photodissociation Dynamics of Triatomic van der Waals Molecules. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/f38f-hb60. https://resolver.caltech.edu/CaltechTHESIS:01182013-155753371 Abstract The spectroscopy and vibrational predissociation dynamics of triatomic van der Waals molecules are studied experimentally and theoretically at the state-to-state level of detail. Laser-induced fluorescence spectra of HeCl 2 are measured and the geometry and vibrational predissociation rate are obtained by a rotational analysis of the spectra and the determination of homogenous linewidths. A laser pump-probe technique is used to measure the vibrational and rotational state population distribution of the product Cl 2 . Although the Cl 2 fragment has little rotational energy, its rotational distribution is bimodal. A symmetry selection rule for the dissociation results from the symmetry of the He • • • Cl 2 van der Waals potential. Quantum mechanical calculations on a realistic potential energy surface are successful in modeling the experimental spectroscopy and dynamics. The vibrational predissociation dynamics of NeCl 2 is also measured using the pump-probe technique. In this case the Cl 2 fragment shows significantly more rotational excitation than in the dissociation of HeCl 2 . The rotational distributions are bimodal and are relatively independent of the energy of the prepared state. The NeCl 2 binding energy is estimated from thresholds for the populations of fragment rotational levels. The vibrational predissociation of NeBr 2 is studied by dispersing the fluorescence of the Br 2 fragment. The product Br 2 is rotationally cold. The closure of vibrational product channels is used to determine the binding energy of the molecule. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Chemsitry Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Thesis Availability: Public (worldwide access) Research Advisor(s): Janda, Kenneth C. Thesis Committee: Kuppermann, Aron (chair) Janda, Kenneth C. Beauchamp, Jesse L. Dougherty, Dennis A. Defense Date: 18 January 1988 Funders: Funding Agency Grant Number NSF UNSPECIFIED Record Number: CaltechTHESIS:01182013-155753371 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:01182013-155753371 DOI: 10.7907/f38f-hb60 Related URLs: URL URL Type Description https://doi.org/10.1063/1.450507 DOI Article adapted for Chapter 2. https://doi.org/10.1103/physreva.36.1944 DOI Article adapted for Chapter 3. https://doi.org/10.1063/1.452204 DOI Article adapted for Chapter 4. https://doi.org/10.1007/978-94-009-3969-1_39 DOI Book chapter adapted for Chapter 5. Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 7412 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 22 Jan 2013 17:30 Last Modified: 05 Nov 2021 20:04 Thesis Files Preview PDF - Final Version See Usage Policy. 27MB Repository Staff Only: item control page

Photodissociation Dynamics of Triatomic van der Waals Molecules Thesis by Joseph Isaac Cline III In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1988 {Submitted January 18, 1988) i i Acknowledgements I acknowledge leadership, advice, my advisor, Ken and friendship career. His interest chemistry has guided in my the own his for Janda, throughout my graduate fundamental thinking physics about of scientific problems, and his approach to science as a discipline, not as a business, has helped me to keep my perspective. fostered a his spirit research of mutual group that Ken support and sacrifice within is refreshing in an in tensely competitive world. I thank my fellow Techers Dwight Evard, Sally Hair, and Brian Reid for their many contributions to the work in this thesis and for many good times when we weren't working. I will always remember the camaraderie we shared that made the dreaded move adventure. Dwight had a in this kept me greatly to Pittsburgh something of an hand in most of the experiments thesis, while Sally's bright personality and wit smiling through all appreciate scientific many discussions but the cloudiest wide-ranging, with Brian. times. always Without I fruitful his help I would have never completed the theoretical calculations on I also getting me thank Fritz Thommen and Francis Celii started in the lab at Caltech, and for for great adventures outside the lab. N. Sivakumar deserves much of the credit for the iii We spent many long, early pump-probe experiments at Pitt. often exciting nights constitute the in the lab together , which easily of the work this thesis . highlights in Siva ' s endurance and persistence as an experimentalist and his easy-going personality made him an ideal collaborator . also acknowledge I Alberto Beswick and theoretical aspects of Pittsburgh . Nadine backbone of the HeC1 thank I keeping me Craig Mike helpful Nadine this conversations with about the Halberstadt work during supplied the their programs visits to form the and for that calculations in Chapter 3 . 2 Sherman physically Bieler many deserves fit for and credit his friendship sane while for in assisting the me "Boig." with the last of the experiments I did in Pittsburgh. Craig, Sally Brian, thanks and Jeff Tomer deserve special proofreading and helping to assemble go to Rob Coalson for this thesis. generously allowing me for Thanks the use of his laser printer over the last year . I thank my parents and family for their unwavering support and confidence, which played an indispensable role in seeing me through the crises of the last few years. Finally, Science I Foundation financial support in this thesis. gratefully for that a acknowledge predoctoral the fellowship made possible all of the National and the research iv Abstract The spectroscopy dynamics of and predissociation vibrational triatomic van der Waals molecules are studied experimentally and theoretically at the Laser-induced fluorescence spectra state-to-state level of detail . measured and and technique rotational product Cl . 2 symmetry is state used to the Cl the spectra A laser linewidths. measure population Although rotational energy, A rotational analysis of the determination of homogenous pump-probe are 2 the geometry and vibrational predissociation rate are obtained by a and HeC1 of the vi brat i anal fragment 2 the of distribution has little its rotational distribution is bimodal. selection rule for from the symmetry of the He•••Cl Quantum mechanical calculations the 2 dissociation results van der Waals potential. on a realistic potential energy surface are successful in modeling the experimental spectroscopy and dynamics . The vibrational predissociation dynamics of NeC1 also measured case the Cl 2 using the than rotational distributions NeC1 2 technique. In is this fragment shows significantly more rotational excitation independent pump-probe 2 of in the the dissociation are energy of bimodal the of and HeC1 are prepared binding energy is estimated from 2 . The relatively state. thresholds for The the v populations of fragment rotational levels. The vibrational predissociation of NeBr by dispersing the fluorescence of product is rotationally cold . Br vibrational 2 product binding energy of channels the molecule . is the Br used 2 is studied fragment. The to 2 The closure of determine the vi Table of Contents Acknowledgements Abstract . List of Figures. List of Tables Chapter 1. Introduction Chapter 2. The Laser-Induced Fluorescence Spectrum of the HeCl 2 van der Waals Molecule . ii iv vii viii 1 10 Chapter 3. State-to-State Vibrational Predissociation Dynamics and Spectroscopy of He•••Cl 2 : Experiment and Theory . 33 Chapter 4 . Product State Distributions for the Vibrational Predissociation of NeCl 2 . 117 Chapter 5 . Vibrational Predissociation Dynamics of the NeBr 2 van der Waals Molecule . 157 Appendix A. Classical Trajectory Program '. 186 Appendix B. Probe Laser Spectrum Fitting . 194 vii List of Figures Chapter 1. Introduction Figure 1 Triatomic vibrational modes Chapter 2. LIF Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 of HeC1 2 Excitation spectrum of HeC1 2 Excitation spectrum of HeC1 2 Excitation spectrum of HeC1 2 • Test of energy gap law Momentum gap model 3 17 18 19 26 29 Chapter 3. Dynamics of HeC1 2 Figure 1 Laser pump-probe scheme . Figure 2 Experimental apparatus Figure 3 Rigid, T-shaped model for HeC1 2 Figure 4 Parity-selected excitation spectra Figure 5 Parity-selected excitation spectra Figure 6 Homogeneous broadening of 12~ band Figure 7 Cl 2 fragment probe laser spectra Figure 8 Product rotational distributions Figure 9 Av=-1.-2.-3 distributions for u'=12 Figure 10 Coordinate system for calculations Figure 11 Contour plot of HeC1 2 vdw potential Figure 12 vdw vibratiorlal wavefunction . Figure 13 Calculated excitation spectra Figure 14 Calculated rotational distributions Figure 15 Classical model for dissociation . Figure 16 Franck-Condon model for dissociation 41 43 45 48 51 53 56 65 69 72 78 84 87 95 103 105 Chapter 4. Dynamics of NeC1 2 Figure 1 Cl 2 fragment probe laser spectrum . Figure 2 Spectrum of Au=-1 channel for u'=ll Figure 3 Spectrum of Au=-2 channel for u'=ll Figure 4 Fragment rotational distributions . Figure 5 35 Cl 37 Cl fragment probe spectrum Figure 6 Parity-selected excitation spectra Figure 7 Energy level diagram for dissociation Figure 8 Centrifugal barrier to dissociation . 124 126 127 129 135 137 139 140 Chapter 5. Dynamics of NeBr 2 Figure 1 Total fluorescence spectra of NeBr 2 Figure 2 Br 2 fragment emission spectra . Figure 3 Threshold for population of Au=-1 163 166 175 viii List of Tables Chapter 2. LIF of HeCl 2 Table 1 HeCl 2 spectral shifts and linewidths . . 21 Chapter 3. Dynamics of HeCl 2 Table 1 Rotational distributions for u=8 level Table 2 Rotational distributions for u=6,12 Table 3 Summary of distributions in Fig. 8 Table 4 Potential parameters for HeC1 2 . Table 5 vibrational levels for HeC1 2 Table 6 Calculated lifetimes and blue shifts Table 7 Rotational wave function symmetries Table 8 Rotational energy levels of HeCl 2 60 61 64 76 81 83 90 Chapter 4. Dynamics of NeCl 2 Table 1 Binding energy determination for NeC1 2 Table 2 Summary Au=-1 rotational distributions Table 3 Summary Au=-2 rotational distributions 144 147 148 91 Chapter 5. Dynamics of NeBr 2 Table 1 Vibrational product-state distributions 172 1 Chapter 1 Introduction 2 The research in dynamical problem. Consider depicted in Fig. If 1. ABC a concerns thesis this the triatomic has a basic molecule non-linear ABC geometry, then the molecule has 3 vibrational degrees of freedom: an A-B stretching motion, and a bending vibration, A-B potential stretching energy frequency a R; C-(AB) stretching motion, Furthermore, ~- sur face is is high very very assume deep relative that the and the A-B to shallow a C-(AB) potential and low C-(AB) vibrational frequency. vibrational energy is stretch excess of in initially the deposited well potential. then any coupling eventually lead to dissociation depth among of the into of the r; If the the A-B C-{AB) vibrations will Since molecule. the frequency mismatch of the two vibrations is large, the coupling is is inefficient relatively slow. and the dissociation The quantum numbers of process the dissociating state can then be well defined. This fundamental central variety problem, importance topic of of vibrational this other in predissociation, chemical physics, and is of is the thesis. It is closely related to a chemical phenomena, for example, intramolecular vibrational energy redistribution (IVR) and vibrationally and rotationally inelastic collisions. formal theory predissociation, often that of used to the decay describe a The vibrational zero-order, bound, 3 R FIG . 1 . Vibrational degrees of freedom for molecule ABC. 4 1 2 is analogous to that for · state embedded in a continuum, many processes other and radiative the as such 3,4 non-radiative decay of electronica 11 y excite d states. The given questions potential explored energy in this surface and thesis are, for a atomic masses, how rapidly the initially prepared state decays, and what the vibrational and rotational quantum states of the recoiling fragments are. changing the alters the succeeded Furthermore, masses in the and dynamics we of want to potential the molecule. experimentally measuring understand energy This and how surface work has theoretically calculating all the relevant degrees of freedom for such a process at the state-to-state level of detail. van der Waals molecules 5 are excellent candidates for studies of this type because the dissociation dynamics generally occurs on a single electronic potential surface. The research in this class van of der complexes. thesis concerns a Waals molecules: These molecules have made the characteristics that experimental and theoretical studies: have optical transitions electronic states by tunable well have the that lasers. characterized, them to very in the fall rare-gas topic halogen of many First,the halogens long-lived, range excited easily accessed these transitions so the understanding closely related spectroscopy of specific additional Second, that energy are very of the the van der Waals species 5 Third, is simplified. many cases, prepared sufficiently in a electronic, Finally, rich, vibrational predissociation is, slow and the dynamics of ranging from translational rotational state quasibound vibrational, also been because they systems for potential intensively constitute which energy is be known numbers. surprisingly to vibrational to in rare-gas ICl complexes, effects in HeC1 .9 2 . 8 to 11 y, 10,13 . t h eoret1ca invitingly simple dynamical on realistic calculations can be 2 7 The rare-gas halogens studied, detailed surfaces precisely quantum can process in multistep IVR processes in ArC1 have with pure transfer interference of molecule complexes nearly rainbow phenomena observation the rotational these a energy that in quantitatively compared to experimental data. This thesis presents predissociation of HeC1 order of 2 . NeC1 increasing mass of quasibound vibrational research 2 , on the and NeBr the complex. level the of 2 . pu 1 se. following The quantum decomposition is state of B probe laser pulse or by dispersion of the halogen fragment the larger rotational to perform truly the determined following in electronically tunable dye halogen fragment either by a second the fluorescence of dissociation. constants of HeC1 state-selected organized In each case a excited state of the molecule is prepared by a laser vibrational 2 • it Because of· is possible experiments that 6 are compared to detailed theoretical analysis, small rotational constants of NeBr whereas the prevent 2 rotational state resolution of the dynamics. Chapter 2 of this thesis describes the laser-induced fluorescence spectrum The rotational is analyzed of the structure to obtain of B+-X transition individual information of HeC1 2 . vibrational about the bands molecular geometry, and the homogeneous broadening of the spectra is measured to obtain the decay rate of the quasibound state. The experiment described in Chapter 3 uses the laser pump-probe technique to examine the dynamics of HeC1 2. It is possible to measure the product state distribution for decomposition of individual complex. Although only released during dissociation excitation of products, unusual bimodal realistic He•••Cl 2 the a rotational small rotational potential levels fraction of the the the energy rotational into goes of shows an population distribution. A surface constructed for energy fragment is for which three-dimensional quantum calculations agree with the measured spectroscopy. Use of the golden rule approximation in a quantum mechanical calculation of the agreement dynamics gives good This work represents experimental and theoretical data. one with of the the studies experimental most of camp 1 e te vibrational predissociation yet performed for any molecule. Chapter 4 discusses the vibrational predissociation 7 of NeCl The Cl 2 . thus increasing fragment 2 shows considerably more the dissociation of excitation than in rotational levels allowed populated . A observed. The c atom the mass of bimodal bond by conservation rotational energy of in Fig. rotational HeC1 of is 2 all ; 2 energy are is also population NeC1 1. obtained by the measurement of thresholds for the production of particular fragment rotational levels . The effect of is explored in Chapter vibrational Br 2 i ncreasing 5, is obtained following dissociation . the Br of which presents predissociation of fragment the mass by the diatom AB a study of the the NeBr . The state dispersing its fluorescence 2 The larger rotational of inertia of 2 fragment results in significantly less rotational excitation binding than energy partitioning in of of the NeBr the 2 decompos i tion of is analysis obtained fragments by among NeC1 the 2 . The of the available vibrational product channels . This research shows that , while the gross vibrational predissociation dynamics of these molecules is mostly consistent with the previous understanding of process, rotationally resolved, reveal new and exc i ting phenomena . state-to-state It is shown this studies that the symmetry of the van der Waals potential leads to dynamical selection rules in photodissociation analogous to those of inelastic scattering. Despite the fact that 3-dimensional 8 quantum calculations reproduce the experimentally observed rotational distributions be explained in of terms of a process. These further semiclassical and binding energy contributes to results He- and Ne- Cl intermolecular forces. . they cannot simple physical model for the for are exciting candidates studies. Finally, the information the 2 general obtained in s true tural this understanding research of weak 9 REFERENCES 1 A good introduction to this problem is given in C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Vol. 2 {Wiley, New York. 1977). p. 1343. 2u. Fano. Phys. Rev. 124, 1866 {1961). 3M. Bixon and J. Jortner, J. Chern. Phys. 48, 715 ( 1968). 4M. Bixon and J. Jortner, J. Chern. Phys. 50, 4061 {1969). 5 6 7 8 9 An introduction to the properties of van der Waals molecules is given in B.L. Blaney and G.E. Ewing, Ann. Rev. Phys. Chern. 27, 533 {1976). D.H. Levy, Adv. Chern. Phys. 47, 323 {1981). J.C. Drobits, J.M. Skene, and M.I. Lester, unpublished results presented at the Conference on the Dynamics of Molecular Collisions. July 1987, Wheeling, WV. Chapter 3 of this thesis. D. D. Evard, J. I. Cline, and K. C. preparation; D.D. Evard, Ph.D. Institute of Technology, 1988. 10 J.A. Beswick and {1981). 11 J. Jortner, N. Halberstadt, J.A. Beswick. Phys. 87, 3966 {1987). Janda, manuscript in Thesis, California Adv. Chern. and K.C. Phys. Janda, 47, J. 363 Chern. 12 S.K. Gray, S.A. Rice, and M.J. Davis, J. Phys. Chern. 90, 3470 ( 1986). 13 S.B. Woodruff and D.L. Thompson, ( 1979). J. Chern. Phys. 71, 376 10 Chapter 2 The Laser-Induced Fluorescence Spectrum of the HeCl 2 van der Waals Moleculet tPublished ( 1986) . in a similar form in J . Chern. Phys. 84, 1165 11 The Laser-Induced Fluorescence Spectrum of the HeCl 2 van der Waals Molecule Joseph I . Cline , a) Dwight D. Evard, Fritz Thommen,b} and Kenneth C . Jandac) The Arthur Amos Noyes Laboratory of Chemical Physics , d} California Institute of Technology, Pasadena, California 91125 ABSTRACT Laser-induced fluorescence excitation spectra of free-jet expansions of He and Cl 2 have been recorded . 1 Spectrosco~ic features observed ~ 3 . 5 cmto the blue of 5 the free Cl 2 vi broni c origins of the 7-0 through the . 12-0 bands of the B3 IT{O + )~X 1 L+ system were attr1buted to u g the He 35 Cl 2 molecule. The observed spectra were well simulated by aT-shaped, rigid, asymmetric top model. The spectra were observed to display homogeneous broadening dependent on the vibrational band excited . Predissociation lifetimes were calculated and structural parameters determined by fitting the observed contours . The distance from the He atom to the Cl 2 center of mass is 3.8 ± 0.4 and 3 . 9 ± 0 . 4 A for the ground and excited states, respectively. a)National Science Foundation Predoctoral Fellow . b)Present address : CIBA-Geigy, Basel , Switzerland. c)Camille and Henry Dreyfus Teacher-Scholar. d}Contribution No. 7283. 12 I . INTRODUCTION van der Waals molecules atom weakly bound to a system for dynamical which consisting of a rare diatomic molecule provide a fundamental processes may be tested . state vibrationally excited in of theories simple molecular The decay the diatomic gas rate stretch of a is a strong function of how efficiently that state couples to a continuum of dissociative final system. Spectral 1 ine states of broadening the atom-diatom provides a means of measuring this rate . The suited rare gas for such dihalogens (AX ) provide systems well 2 The B3 Z1{0+)+-X 1 .I+ electronic studies . g u transition has been carefully studied for which greatly aids in the analysis molecule The triatomic spectra . can be prepared in a few wavenumbers vibrational perturbs component. are vibrational "' 10 -9 s. to the blue The electronic van der of rare Waals the gas structure Fluorescence lifetimes whereas homogeneous predissociation Thus, the van der Waals molecule free-jet expansion and is detected a bandheads. the of the dihalogens , for diatomic a tom only of the Cl 2 weakly halogen Br , , and 2 linewidths lifetimes dissociation free of r2 indicate less than occurs on the potential Levy and co-workers surface of one electronic state. The pioneering work of 1-5 13 characterized the these systems in that Hei general features of the dissociates, prepared in a vibrational state u 2 yielding an r fragment in the u'-1 vibrational state with 2 little rotational excitation. The structure determined triangular, the He to be and 2 Nei 2 . 1-5 NeC1 2 , 6 7 · atom NeBr , 2 8 10 ' of Hei being r 2 bond axis. perpendicular bisector of the Ari of They determined studies of Hei 2 . their dynamics was 2 on the Subsequently, HeBr and 11 have 2 been studied . Here we report molecule . At this the excitation spectrum of the HeC1 2 time exists a large allowing body of molecules, reveal the nature of the dissociation dynamics and van der T-shaped geometry and to lighter have vibrational In addition, diatomic fragments there may s tate-dependent, = -1 propensity Numerous theoretical predissociation of Hei reproduce the rotational and Jortner 12 reproduce 2 have been made. vibrational the Ewing 15 trends that the significantly r2 . approaches experimentally and is evidence exhibit greater rotational excitation than that that provided one quantum is sufficient to break the van der Waals bond. and made All of these molecules share a dissociation rates consistent with the Au' rule, be on these Waals molecular structure. comparisons data to the 12-16 vibrational The results £ragmen t' s observed state distributions. have given in dissociation general Beswick equations rates as a 14 function of v for AX Waals potentials order, and quanta in energy along diatomic the The diatomic and van der species. 2 are assumed to be separable to zero spectroscopically prepared state the diatomic mode and a relative translational the van der vibrational dissociation energy Waals coordinate spacing, E the der of van v ,-E has equal . v - 1 to the the less . Waals v bond. The dissociation rate is roughly proportional to the square of the matrix element between the initial van der Waals wave function coupled by the potential perturbation operator to a rapidly oscillating translational energy "energy-gap" accessible plane of the "momentum-gap" or states with excitation will have the relative dissociation. This wave with predicts model rotational that vibrational) (or larger matrix elements because of a reduced relative translational energy. In this paper we show that the observed trend in the HeC1 2 predissociation rate is consistent with this model. Structural information analysis the of is spectra. obtained The results by a are rotational compared to previous studies of related complexes. II. EXPERIMENTAL The experimental previously described, expansion was used 6 to apparatus used was except form similar to one a continuous nozzle jet. Liquid that the Cl 2 was 15 maintained at 233-243 K under The typically 600 psi . vapor expanded xyz was He at gaseous through a a total mixture pressure of He Cl 2 of and 25 JJ.m nozzle moun ted on an translatable stage into a vacuum chamber maintained at pressures below 10 -4 Torr . A Lumonics 861-S pulsed excimer XeCl was used to pump a laser operating Lambda Physik FL2002E dye laser beam was focused light baffles angle. to linewidth of 0 . 04 em intersect to the light f i 1t e r s was The the molecular beam at a plane of the filtered ( OG-515 , 530) . by molecular appropriate Spectra were right collected was fluorescence and laser Scattered beams and detected by an S-20 photomultiplier. laser -1 through brewster angle windows and Laser-induced perpendicular laser This laser equipped with a Fabry-Perot air-spaced etalon. produces 1-3 mJ pulses with a on Schott normalized glass to laser pulse energy by a Gen-Tec energy meter, which measured the beam energy it left the apparatus. fluorescence signal from the photomultiplier signal the from homebuilt as energy MCSA integrators. or Both interfaced to Labmaster interface. scanning averaging. of an the meter Stanford the IBM-PC dye were recorded Research MCSA and microcomputer by microcomputer laser and the either a a boxcar were boxcars through The signal and Systmes the total The Tecmar controlled the acquisition and 16 III. RESULTS The jet laser-induced fluorescence expansion (11--o) in band is approximately disappears the -1 to reduced of Fig. the blue backing to the HeC1 2 van der bands 2 B~x of band head the This 2 peak, . Hei 2 . and HeBr 2 . is Similar Waals molecule. transition. Excitation spectra were to fit nuclear spin for rotational the feature the 7--<) through 12--o bands of spectra were recorded for the Cl The 1. pressures . analogous to those observed for NeC1 attributed the free the in shown 3.7 em at region spectrum of a of the rigid statistics . corresponding rotor model, Literature constants of the free which values X and 17 B Cl 2 included were used electronic states, while the rotational temperature and inhomogeneous linewidth were optimized The inhomogeneous the laser rotational by linewidth bandwidth and temperatures a least-squares includes contributions Doppler broadening. were K 1 technique. and from Typical inhomogeneous linewidths were 0.07 cm- 1 . Averaged He 35 Cl are shown in Figs. HeC1 2 spectra The bond Cl 2 assumed to 2 spectra and computer-generated fits 2 and 3 were fit, length in be for the 11--o and s--o bands . assuming the identical a T-shaped van der Waals to that reported structure. molecule for was the 17 8-X (11-0) Vi 1- z :::::> >a: ~ 1- 35CI 2 iD ~ 19814 cm- 1 FIG. 1. Excitation spectrum about the a 5 Cl 2 11~ bandhead at 0.04 cm- 1 laser resolution. The more intense lines of free Cl 2 are not to scale because of saturation of the detection electronics. 18 19814 19815 19816 19817 cm-1 . 36 FIG. 2. Excitation spectrum of the He Cl 2 11-o band at Upper trace is the 0.04 cm- 1 laser resolution. experimental spectrum. The lower trace is the computergenerated fit, assuming a T - shaped structure . 19 19357 19358 315 FIG. 3. Excitation spectrum of the He Cl 2 8+-0 band at 1 Upper trace is the 0 . 04 cmlaser resolution . experimental spectrum . The lower t ·race is the computergenerated fit, assuming a T-shaped structure. 20 17 inhomogeneous 1 inewid th and rotational temperature determined in the Cl fit were used uncomplexed molecule. as fixed values in The the analysis of The distance from center of mass, electronic RX states , and the RB ' for band 2 the corresponding the He atom the origin, to the Cl 2 ground and excited and homogeneous the linewidth was varied in the fit to a rigid, asymmetric top Hamiltonian. Band origins of HeC1 to origins the were band set these to their simulations of 2 were computed relative 35 adjacent the literature values . show the that 17 band results contour is quite somewhat less the sensitive to the absolute values of RX and RB. and bands be simulated yielded similar values for Rx. did not have a the of the 3.4-4.0 A. Since bands. average, the and closely the v' Rx = 3. 8 is common A, for bands was taken as the optimum value. the 8+-0 summarized values of RB calculated by systematically with v .. 0 . 024 A, the range measured and standard 9+-0 in The results Table procedure did 1. The not vary The average value of RB over measured bands was 3.937 A as this in All bands were then of are the to all fixed parameter. calculations fits the fit over fit again with this RX as a these the The absolute value of Rx = 10-12 bands same RX of The 8+-0 and clear effect on the quality of simulations which The to could (RX-RB) peaks, 2 sensitive 9+-0 difference c1 the The error in (RB-RX) is taken deviation average. of this 19179.307 19353.976 19517.746 19670.509 19812.366 19943.386 7-0 8-0 9-0 10-0 11-0 12-0 • From Reference 1 7 RB - Rx = 0.127 ± 0.024A Rx = 3.8±0.4A Voo(Ch )• Band 19182.71 19357.47 19521.30 19674.16 19816.09 19947.16 Voo(HeCh ) 3.40 3.49 3.55 3.65 3.72 3.77 shift cm- 1 0.0105±0.0016 0.0193±0.0032 0.0296±0.0035 0.0546±0.0019 0.1014±0.0014 506±77 275±45 179±21 97.2±3.4 52.37±0.72 r(FWHM)~!ll- 1 _r___1E§) Table 1 : He35 C}:z spectral shifts and estimated linewidths "'..... 22 Assigning error limits to the absolute determination of RX is not straightforward. asterisk in Fig. unblended line, The is 3 feature attributed assigned to the 0 indicated to a by single, an nearly The 00 ~1 01 transition. position of this line with respect to the band origin is a function only of RX . by the fit was relatively value of RX over a relative to Since the band origin as determined insensitive wide range, the band to the the position of origin allows an this of Estimating the error determination of transition to be the error in RX is ± 1 A. overly conservative estimate. is more sensitve This is probably an than the fit these to the overall error bars Our experience in varying RX over a indicates that increase laser the indicate . reasonable . the in Fitting the entire spectrum over a wide range of RX shows that contour line absolute determination linewidth, absolute ± 0.4 A would wide range is more Changing RX by this amount corresponds to an an uncertainty in the sum of of the squares of the error by 25 to 100 % and a qualitatively poor fit to the data. It should also this be noted that centrifugal bond could be significant and lines of appreciable distortion that intensity these with J in spectra contain as high Quoted errors in the fitted homogeneous linewidths, the 95 % confidence limits. are taken to be the laser weak as 4. r. are The errors in the band shifts linewidth . Weak Franck-Condon 23 and reliable determination of predissociation van der Waals radii for HeC1 2 prevented linewidths homogeneous narrow factors lifetimes and u'~7. bands with IV. DISCUSSION Three molecule, quantities which can gas have be been measured compared halogen to for previous These other rare complexes. shift, the van der Waals bond length, the HeC1 2 studies are the of band and the vibrational predissociation lifetime. The blue to the free 35 Cl that of respect the He 35 c1 shift of 2 to the B state This result is analogous blue shift observed significantly from 2 band origin relative increases roughly linearly with vibrational to in the Hei the NeC1 2 quantum smooth and 2 data, 7 increase HeBr where number, 2 u'. in the differs but the band shifts fluctuate with u' . A commonly used mode 1 for these sys terns is one in which the rare gas atom feels the "effective potential" of the halogen atoms, This approach is separation of model existence the which move in a deep potential analogous to the Born-Oppenheimer electronic of a well. and nuclear motion. In this blue shift indicates that the effective van der Waals potential in the X state is deeper than that in the B state. Hence an increasing blue shift indicates simply that the B state van der Waals well depth 24 The physical basis of such decreases with increasing u an increasing Waals shift potential would surface expectation value of alternative model that increase is to be the change because the Cl 2 the an bond suggests consider of in the van der increase in the length an with explanation perturbation of for the the Cl-Cl bond is stiffened in HeC1 effect lead spacing. Simple qualitatively to an increase calculations reproduce blue shift with u the These in the Cl based on trend of an Cl-Cl {described more fully below). would this Using a potential because of the presence of the He atom. simple atom-atom pairwise potential model An u 2 This 2. vibrational such a increase model in superimpose~ increases are the on an initial blue shift at u' = 0 because of a more shallow van der Waals potential surface for the B state electronic distribution. Both mechanisms probably take part in producing the observed blue shift. Vibrational predissociation lifetimes for HeC1 2 follow the trends predicted by the theories of Ewing 15 and Beswick and Jortner. Beswick and Jortner 12 have proposed the expression r to describe with u' . a u !h !h ] e xp __-...;c;;..~;;;._..t;;;!-L;.___ [ the variation of the predissociation lifetime In this equation w is the diatomic vibrational 25 ~ energy spacing, is the van der Waals reduced mass, the Morse range parameter, and C is a a is Figure 4 constant. (w~)~ within shows that ln{r/u') is linear with respect to our experimental error. Comparison more insight of these data into the dissociation dissociation of Hei type process. The existence of energy and increase modes Ewing hence the exist 15 the to nearly pure V-T the that the channel v-v small serves to channel is closed. in is nearly large rotational fragments gap to accept frequency vibrational products, only The dynamics . energy the can accept gives 2 V-T in the Hei other channels lower V-R and 2 be a Since no Dissociation inducing a thought reduce closed because of inertia 2 HeBr rate. shows fragment. is to Hei 2 inertia of with amounts large of the 1 moments of energy large change in angular momentum. 2 without This results in a rapidly angularly oscillating rotational wavefunction that has small coupling matrix elements with the prepared state. This corresponds classically to impact parameters Br 2 and Cl so the the dissociation large half-collision. have successively larger rotational constants, 2 V-R channel accepting energy. complexes with the for unrea~onably opening of should become Comparison similar energy this channel. of more available bands spacings The (w for for the He show evidence of Hei 12--D 2 = 108 cm- 1 ). band and 26 -4 • -5 - • -> t:::; - .£ -6 -722~--------~~---------L----------~----------~--------~. ~ M ~ ~ V FIG . 4. Plot of ln(r/u') vs (~w) ~ as a test of the energy 95 % gap law for the HeC1 2 van der Waals molecule. confidence limits are shown. 27 the band 14+-0 221.2. was ps, respectively . from a 53. extrapolated plot in Fig. 4.) vary weakly with probably dramatic to a linear He-rare the Approximating the HeX is = 110 em -1 ) (w nearly 2 der Waals potentials number of the constant in cannot vibration to translation is to rare some this be attributed energy gap rotational 18 AEV-T Thus, series. the gas. potential as described below, lifetimes there lifetime 2 van in that (The HeCl of fit change implies lifetimes least-squares gas atomic have the simply problem, excitation and of the lighter diatomic. The HeC1 measured lifetimes are surprisingly close to 2 for NeC1 2 . 7 The dissociation those dynamics are closely related to the shape of the van der Waals B state potential model surface. A crude surfaces of He- and Ne- Cl pairwise atom-atom 2 for the potential may be constructed by using a The Cl-Ne and Cl-He the Ar-Ne and Ar-He scattering data of Lee et al. 18. 19 The Cl-Cl bond length potentials may be potential. estimated by is fixed at its vibrationally averaged value in the B(v ' ) state and the rare gas atom is constrained the perpendicular bisector of the c1 depths and of 34 em such a -1 previously, van der . 6 Waals potential respectively. this method is molecule well for bond axis. 2 NeC1 As to move along 2 and HeC1 has been The well 2 are 100 discussed likely to overestimate depths. Integration these of the 28 Schrodinger equation for gives a these one-dimensional potentials zero point energy of Making the 15 em 18 and simplification that -1 Ne- and for the dissociation is a pure Y-T process, the relative translational for the the prepared state fragments 48 cm- 1 for NeCl of and 112 em 2 -1 for HeCl 2 in one might expect u to longer a significantly is 12 = u Based on this . simple energy gap argument, have energy lifetime = 12 HeCl 2 than = 12 u The energy gap is somewhat compensated for by the lower reduced mass potentials lead significantly effect is of HeC1 to more 2 Also, . the diffuse than illustrated in Fig. that for Note that 5. atom-atom function wave a model that NeC1 2 is This . the prepared vibrational wave function has nearly the same overlap with the final state plane wave for both HeC1 2 and NeC1 2 . In conclusion, we report here the observation of the van der fluorescence. rigid, der The molecule spectra asymmetric structure. van Waals top model, Homogeneous Waals homogeneous bond can using be linewidths simulated assuming 1 inewid ths, lengths laser band are a well calculated. rapid dependent on by a triangular shifts, indicate induced and the Broad vibrational predissociation that quantum number of state. Comparison of these results to previous studies of Cl is 2 in the HeC1 2 the vibrational excited electronic related molecules gives evidence of rotational excitation 29 ISO(a) ~---=----:-::---::---:: ' ,-, ,. V {R), V =12 1 ~, \ I \ I \..' V I E u 1 \ I \ I \ I \ I '-' I \.I = 12 90 1 - - - - - - - - - V{R}, V =II 01 3 240 (b) 5 7 9 He- em (CI 2 ) distance (.I\) II NeCI2 V{R}, vI = 12 I v = 12 ,, I \ \ I E u 120 I \ I "'''/!f "' V(R}, v = II 1 Ne-cm(CI 2 ) distance (J~) FIG . 5. Momentum gap model for Heand Ne-Cl 2 photodissociation constrained to perpendicular dynamics. ~t is the wave function of the spectroscopically prepared state . ~f is the final state wave function. 30 of the Cl 2 dissociation fragment. V. ACKNOWLEDGEMENTS This Foundation. Petroleum work was supported Acknowledgement Research Fund the is made to administered Chemical Society and the W.M. support of this research . by National Science the donors of by the the American Keck Foundation for partial 31 REFERENCES 1 2 3 4 5 6 7 8 9 R.E. Smalley, D. H. 64, 3266 (1976). K.E. Johnson, L. 69, 2719 ( 1978). Levy, Wharton, and L.Wharton, and D.H. w. Sharfin, K.E. Johnson, L . Chern . Phys. 71,1292 (1979). Levy, Wharton, J. Chern. Phys . J. Chern. Phys. and D.H. Levy, J. J . A. Blazy, B . M. DeKoven, T.D. Russell, and D.H. Levy, J. Chern . Phys. 72, 2439 (1980) . K.E. Johnson, W. Sharfin, 74, 163 ( 1981). and D . H. Levy, J. Chern . Phys. D.E. Brinza, C.M . Western, D. D. Evard, F. Thommen, B.A . Swartz, and K . C . Janda, J. Phys . Chern. 88, 2004 (1984). D.D . Evard, F . Thommen, J . I. Phys. Chern . 91, 2508 (1987). Cline, B.A. Swartz, D. E. Brinza, C.M. J. Phys. Chern. 88, 6272 (1984). F . Thommen, D.D . 82. 5295 ( 1985) . Evard, and K.C. Western, and K. C. Janda, J. and K.C. Janda, J. Phys. Janda, Chern. 10 J.I . Cline, D.D. Evard, B . P. Reid, N. Sivakumar, F. Thommen, and K.C. Janda, in Structure and Dynamics of Weakly Bound MoLecuLar CompLexes, A. Weber, ed . (D. Reidel Publishing Co. , Dordrecht, 1987), p. 533; Chap. 5 of this thesis. llL.J . van de Burgt, J.-P. Nicolai, and M.C. Heaven, J. Chern . Phys. 81, 5514 ( 1984). 12J . A. Beswick and J. Jortner, J. Chern. Phys. 68, 2277 (1978) . 13J.A. Beswick (1978) . 14 and J. Jortner, J. Chern. S.B . Woodruff and D.L . Thompson, J. Chern. (1979). Phys. 69, 512 Phys. 71, 376 32 15 16 17 18 G.E. Ewing, Faraday Discuss . Chern . Soc. 73, 325 {1982). E. Segev and M. Shapiro, J. Chern. Phys. 78, 4969 {1983) . J . A. Coxon, J. Mol. Spectrosc. 82, 264 {1980) . C . H. Chen, 601 {1973). 19 P.E. C . Y. Ng, Y . T. 1996 {1974). Siska, and Y . T. Lee, and J . A. Lee , J . Chern . Barker, J. Chern. Phys. 59, Phys. 61, 33 Chapter 3 State-to-State Vibrational Predissociation Dynamics and t Spectroscopy of He•••Cl : Experiment and Theory 2 t Part of this work has been published as a brief report in Phys . Rev . A 36, 1944 (1987) . 34 State-to-State Vibrational Predissociation Dynamics and Spectroscopy of He•••Cl 2 : Experiment and Theory Joseph I. Cline,a} Brian P. Reid,b} Dwight D. Evard.c} N. Sivakumar, Nadine Halberstadt,d} and Kenneth C . Janda Department of Chemistry University of Pittsburgh Pittsburgh, PA 15260 ABSTRACT State-to-state measurements are reported for the vibrational predissociation of HeC1 2 • Single rovibronic quasibound levels are prepared so that dynamical effects not previously observed in photodissociation experiments are measured. Because of the symmetry of the He• • •Cl 2 potential, specific initial rotational levels of the quasibound state decay to either even or odd j fragment rotational levels. The Cl 2 fragment appears predominantly with the loss of one Cl-Cl vibrational quantum and with rotational quantum numbers j ~ 16. Fragment rotational distributions show two distinct peaks. Three-dimensional quantum calculations are performed using an atom-atom potential energy surfac~. which is adjusted to bring both the calculated spectroscopy and dynamics into good agreement with the experimental data . The bimodal rotational distributions are not consistent with any simple physical picture for the dynamics. a}Present address : Joint Institute for Laboratory Astrophysics, National Bureau of Standards and University of Colorado, Boulder , CO 80309-0440. b}Present address: Department College, Marietta, OH 45750. of Chemistry, Marietta c}Present address : Physikalisch Chemisches Institut der Universitat Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland. d}Present address: Laboratoire de Photophysique Moleculaire, Universite de Paris Sud, 91405 Orsay, France. 35 I. INTRODUCTION The photo-fragmentation dynamics of has been the subject In favorable cases, theoretical study. obtain a nearly complete experimental quantum state distribution of that results from the dissociation it is possible to description of the the decomposition fragments excitation at threshold. and experimental intense of small molecules a 1 2 · specific One energy difficulty above in the analysis of these data has been the inability to construct quantitatively surfaces. surface, accurate Often, which molecule involves states for which these cases it optically a excited the is prepared one is or state excited more to is to describe state the system of a excited difficult 2 energy repulsive electronic unknown. In precisely the even the supersonic expansion. in the photodissociation of HeCl initial potential detailed potential molecules are cooled in a the state when However, , reported in this paper, can be described in remarkable detail; there is no averaging over the electronic, vibrational, or even rotational quantum state-to-state dynamics numbers of is measured the molecule. that can be True compared directly to quantum mechanical calculations. The vibrational molecule is a predissociation of process dynamically a van der analogous to Waals the indirect dissociation of a covalently bound molecule on an electronically adiabatic potential energy surface. 3 An 36 initial state that is quasibound (i..e . , long lived enough for vibrational and rotational quantum numbers defined) to be well is produced by optical excitation and decays via coupling among the vibrational modes of the molecule. this study, for example, energy in excess of Waals binding energy of HeC1 the Cl-Cl stretching mode In the van der is optically deposited into 2 of the complex. The van der Waals molecule undergoes vibrational predissociation when this excess frequency vibrational vibrational of the He and the Cl here is quite 2 modes that involve many from low into relative motion The process discussed constituents. distinct coupled is energy other pump-probe photodissociation experiments that measure the dynamics of the direct dissociation covalently bound molecules excited, electronically of on purely repulsive potential energy surfaces. van der Waals molecules have characteristics that make them particularly attractive systems for the detailed study of fragmentation vibrational these systems surfaces often energy is flow. the 4 The simplicity involved. very dynamics well-defined The construct a principal of dynamics electronic the covalently bonded fragments. intramolecular and the advantage potential occurs on potential a of energy single, surface of It is relatively easy to realistic overall potential that accounts for weak van der Waals forces between the constituents of the 37 Theoretical complex. naturally to a motions from analogous treatment this problem the covalent bonds Born-Oppenheimer Zero-order vibrational formed wave products of in a manner of nuclear separation and electronic motion. as leads zero-order decoupling of the van der Waals the vibration of to of functions that states are describe the vibration of covalent bonds and vibrational wavefunctions for the van der Waals stretches and bends. is then evaluating among described these by zero-order been developed experimental states . The dynamics higher-order Theoretical couplings models have that can quantitatively reproduce detailed results . Still, there are few systems for which potential energy surfaces have been constructed that have been rigorously tested experimentally. The rare gas-halogen 5-17 and interhalogen 18-21 van der Waals molecules have been intensively studied because they appear to provide the best hope for a complete description of state-to-state dynamics by both experiment and theory. fragment Fragmentation rates simplicity of such in state distributions are known for (with the notable exception of ClF). variety and of as vibrational these molecules, fundamentally inelastic energy these cases systems Despite the relative they provide examples of a important dynamical half-collisions. redistribution, and quantum interferences. some A second phemomena intramolecular rotational rainbows, important aspect of 38 this research is aimed at using the rotational structure of the transitions to aid the complete potential energy vibronic determination of the In surface. the past the spectroscopy has been analyzed in terms of simple rigid-top models to give effective molecular In these highly non-rigid molecules, of the molecular geometry obtained only surface for by the measurable considering molecule; even at this simple treatment surprisingly is Precise agreement with measured spectra, the true deviations the geometries. effective. however, can be potential energy from relatively rigidity are rotational low resolution of these experiments. In a recent brief report, pump-probe experiments on HeC1 completely class of state-resolved molecules. 2 . work HeC1 is 2 5 we laser pres en ted which represent the most yet for a performed particularly this favorable candidate for state-resolved experiments for the following reasons . First, excitation the molecule has a relatively uncongested spectr?m sufficiently long that (large, rotational lifetime, thus broadening). so it individual quasi bound is possible states constants and a homogeneous small to optically prepare with precisely known rotational and vibrational quantum numbers. potential energy between He and Cl certainly determined simple van der Waals forces. the relatively light He can impart limited Third, because surface by Second, 2 is the almost 39 angular momentum to is easily the Cl incorporated 2 . into the number of a final channels Fourth, calculation. the final momentum of the products results in a He wavelength commensurate molecular with effects are enhanced . well known so that dimensions so that quantum the spectroscopy of Cl 2 is Finally, the quantum state of the Cl 2 fragment can be measured following decay of the prepared quasibound state. This and paper presents a rather theoretical description of complete experimental the decomposition dynamics The energies of the bound and quasibound levels of the this molecule are extracted from resolved structure in excitation different spectrum levels and the dynamics of is measured by the pump-probe several technique. These results are used in combination with data from other sources to construct a potential energy surface for HeC1 Three-dimensional quantum pres en ted sur face, for this 2. mechanical calculations are which are in agreement with both the spectroscopic and dynamical experimental results. HeC1 2 can thus be regarded as a the of study demonstrated molecular how prepared state of vibrational modes the interactions between state-to-state fragmentation dynamics the molecule, of "solved" model the the character depends It dynamics. on the is initially on the coupling among the molecule, recoiling of system in and upon fragments. these data the the The truly permits the 40 measurement typically effects of scattering experiments, in observed only dynamical symmetry in particular, constraints and possibly interference phenomena. II. EXPERIMENTAL DESCRIPTION A. Pump-Probe Scheme The scheme for preparing quasibound HeCl and probing 2 tuned Cl a vibronically with u quanta of stretching mode and no initial excitation of of HeCl 2 Waals vibrational modes. correlates to the radiative lifetime previously study Cl 2 6 as features excited, vibration 2 3 + B II(Ou) the HeCl in through specific the 2 a B~X rotational The He•••Cl vibrational 2 levels w . 1 the Cl-Cl the van der to the structure of 22 ~305 2 (The J.LS. 23 ) transition have been laser-excited shifted 3-4 cm-l rotational in state. 1 B 3 II(O+)~x ~+g (u'~) band origin. 22 u laser shown quasibound is of identified is electronic state of HeCl This long-lived Cl Vibrational bands of excited, fragment 2 prepare state to the vibronically The pump laser frequency, schematically in Figure 1. is the fluorescence blue of each By scanning the of HeC1 (B,u') 2 these are bands, prepared. bond is so weak that one quantum of the Cl-Cl stretch decomposition. has sufficient Intramolecular energy vibrational to cause energy redistribution results in decay of the prepared state with a lifetime, T, which depends upon the initially prepared 41 Cl CE,v,j±l) 2 probe HeC1 CB,v~J) 2 I pump I wl HeCl 2cx,v':::0,J) fluor Vibrational ~He+ Cl CB,v~n,j) 2 Predissociation T"v100ps 1 fluor \"vffiS Cl 2 CX) FIG. 1. Laser pump-probe scheme for preparation of quasibound levels of HeC1 2 (B,v',J) and measurement of the vibrational and rotational state distributions of the Cl 2 (B,v,j) fragment. 42 vibrational (T- 10°-10 3 ps). state The quantum state of through the the Cl (B) fragment is determined by tuning w 2 2 E(O+)+-B 3 IT(O+) . . trans1t1on. u g 24 For both excitation spectra and product distribution measurements, undispersed fluorescence from function of w the c1 state E 2 is detected as a and w . 2 1 B. Apparatus A schematic diagram of the experimental apparatus is shown in Fig. 2. described. expansion Cl 8 The vacuum apparatus has been previously HeC1 was 2 (150 J.l.m prepared diameter in nozzle) a of pulsed, a free jet of 0 . 6% mixture 2 in He at a total pressure of 350 psi. The mixture was obtained by passing He over a vessel containing liquid Cl The rotational temperature of van der 2 Waals molecules formed in the jet was approximately 0.6 K. Two provided which dye tunable traveled perpendicular providing 0.05 em -1 pump to 1 the -1 etalon bandwidth, -10 ns from the jet XeCl probe through to excimer laser, laser pulses, (w ) 2 the vacuum expansion. produce 1 -20 ns The output of to produce and a fundamental. the w an apparatus, The dye laser pulse was operated with an angle-tuned, band width. attenuate by and (w ) 1 free frequency-doubled -0 . 2 em pumped colinearly the w intracavity was lasers, pulse. The w 2 with the probe dye -20 ns cutoff pulses pulses a laser with a to filter was used pulse was delayed Laser power was maintained such 43 Computer ~------~Boxcar Grating tn Grating tt2 Etalon _ __.. Crystal Laser Energy Valve Trig Boxcar Trig Laser Trig 0. 6% Cl2 in He 350 psi Dye tt2 Excimer Dye ttl Pump FIG. 2. Schematic diagram of experimental apparatus. Two dye lasers are pumped by a XeCl excimer laser. Dye laser #2 receives 60 % of the excimer pulse energy; dye laser #1 receives the remaining 40 % . The probe beam (w 2 ) is generated as the second harmonic of dye #2. while the pump beam (w 1 ) is the fundamental of dye #1. The pump and probe beams are combined by a dichroic optic and travel collinearly through light baffles to intersect the pulsed jet. Fluorescence is collected by a lens (not shown) and imaged through defining slits onto a photomultiplier. See text for more details. 44 that the experimental signals fluorescence was scaled linearly with pulse energies. UV photomultiplier, and a reflecting a where integrators signals. of the sensitive UV The w pulse energy was monitored was averaged a filter was used portion fluorescence onto UV pass visible laser scatter. by imaged of 2 the UV beam detected by a into to a reject dye cell Gated photodiode. the photodiode and photomultiplier A microcomputer controlled the synchronization pulsed electronics, valve, scanned laser the dye trigger, laser detection and gratings, etalon, and doubling crystal, and recorded the data. III. EXPERIMENTAL RESULTS A. Excitation spectra: dynamical selection rule In a previous paper by 6 we studied the He 35 c1 laser-induced fluorescence visible fluorescence from fragment was as through (see the Fig. recorded He 35 1) . Cl 2 spectrum the LIF simple (LIF}. the wl frequency absorption; no probe laser individual lines in could not experiment, rotational a T-shaped geometry for Fig. 3. The structure, rotational be the distinct rigid, dissociation the Although excitation molecule 2 In that experiment, completely band was tuned was the He used 35 resolved contours Cl 2 in displayed which was modeled assuming the molecule, structure of the as shown in vibronic bands 45 z (b) R x (a) Y (c) FIG. 3. Rigid. T-shaped model for HeC1 2 • R and r are vibrationally averaged bond lengths. a. b. and c indicate the inertial axes for the asymmetric top analysis of the HeC1 2 excitation spectra. 46 was best fit using a He to c1 RX = 3.8±0.4 A when the in the difference electronic states, exhibited center-of - mass distance of 2 ground in R electronic between RB-RX' measurable (X) the ground and was 0.127±0 . 024 homogeneous state, A. and excited The spectra broadening that was a function of the number of vibrational quanta deposited in the stretch. Cl-Cl progressions modes : in the B~X all There was no excited van der evidence Waals any of vibrationa l transitions occur between ground van der Waals vibrational levels . The pump-probe technique util i zed in the present work offers significant advantages over the LIF method of recording The excitation spectra. is significantly improved . signal-to-noise ratio In a one laser LIF experiment, the exceedingly long lifetime of the B electronic state of Cl 2 severely most of limits the Cl (B) 2 signal collection efficiency because fragments in the jet fluoresce they have moved out of the detection region . is eliminated transition in fully allowed . in the pump-probe outweighs the absorbed reach E advantage measure of how detection Efficient collection o f fluorescence to the the This problem the present experiment because used the fact state. that technique excitation spectrum the E~B scheme is the prompt E state two photons must However, present after is of the be principal the ability to HeC1 depends on 2 what fragment state is being monitored by the w 2 pulse. 47 Typical Fig . 4. on pump-probe the 35 c1 region of the probe superposition of the R( 0) -R( 10) in particular case no the probe laser, in Fig. shows excitation of rotational state the Since 1 ines . is a selected by the LIF experiments spectrum of Ref. 6 . obtained by again 2 to He 35 cl 35 fragment Cl 2 (B,u'=8) levels that decay to give (B,u=7,j=l0) state are observed. The excitation spectrum positioned on the 2 obtained P(9) line when is the shown spectra of Figs. 4b and 4c result from of which, transitions, when added we first reported in Ref. probe in Fig. 5, laser 4c. is The two different sets together, all the transitions observed in Fig. which contains but in this case w is positioned on the P(lO) 2 the 35 c1 E~B 0~7 transition so that only transitions the B~X 8~ band. 2 the spectrum is similar to the excitation observed tuning w , 1 in is positioned transition fragment spectra 4b shown E~B 0~7 band and the pump 2 1 bandhead this w , 2 w , is tuned across the He 35 c1 frequency, 1 ine spectra are In Fig . 4a the probe frequency, the bandhead of The excitation 4a. is account for This separation, the result of a symmetry selection rule in the fragmentation dynamics due to the symmetry of the He-Cl By applying a asymmetric critical symmetry spectra, the This symmetry, parity, HeC1 2 van der Waals potential. rigid, these initial 2 is asymmetric given top by the rotational top analysis becomes behavior to evident . of the wavefunction and 48 l (a) (c) 19356.5 19357.5 Pump Energy/em- 1 FIG. 4 . Solid curves are spectra of He 36 Cl 2 obtained by scanning w 1 through the He 36 Cl 2 B~X (8~) band and (a) positioning w2 on the bandhead of the 36 Cl 2 E~B (0~7) transition, (b) positioning w2 on the P(lO) line of the 36 Cl 2 E~B (0~7) band, and (c) positioning w2 on the P(9) line. Dotted curves are calculated asymmetric top spectra for transitions to onl~ (b) even- or (c) odd-parity rotational states of He 3 Cl 2 (B,v'=8). The rigid, T-shaped model has He to Cl 2 center-of-mass distances of 3.67 and 3 . 79 A and Cl - Cl distances of 1.99 and 2.64 A in the X and B electronic states, respectively. The linewidth is set to that of Ref . 6 and the Boltzmann rotational temperature is 0.6 K. 49 the final exchange HeCl fragment of the Cl asymmetric 2 pulse). upon First, consider the initial wavefunction (prepared by the a toms. top wavefunction rotational The asymmetric top rotational states, ]k k • of a c HeCl can be assigned as having either even or odd parity, 2 depending upon whether they are symmetric or antisymmetric upon atoms operation 25 26 ' as of operator seen in Fig. fragment rotational whether j is even or odd. that initially decay only The parity of 3. wavefunction Cl the Cl {B) 2 simply given is by The observed selection rule is prepared rotational fragment rotational to the exchanges that states of states one of parity the same parity. The transition dipole for the HeCl 2 B~X transition lies parallel to the a inertial axis (see Fig. 3), so that "a-type" asymmetric top expected in the excitation spectrum. The calculated spectra in Fig. 4 asymmetric top rotational odd- {Fig. transitions parity to are either levels for even- 6 (Fig. of were used. transitions 4b) or He (B,u=8). center-of-mass distances, Ref. rotational RX and RB' Considering very non-rigid molecule, the similar fact to 27 the 4c) to those of that HeCl the agreement with are 2 is a observed line positions is remarkable. For reported practical in this reasons paper were most of the performed using experiments the He 35 c1 2 50 isotopic higher The homonuclear He species. natural abundance than the and the spectroscopy of 35 Cl 37 c1. than that of 35 c1 However, for the homonuclear 2 nuc 1 ear para-forms. of He 35 cl species, is known in much more detail species, it should be noted nuclear exchange 35 The that, symmetry c1 nucleus is a = 3/2) and 35 c1 35c1 exists in ortho- and The nuclear spin state of the wou ld b e 2 species has a 2 heteronuclear plays a potentially important role. fermion (I 35 Cl the decomposition. expec te d to be 35 c1 2 conserve d component t h roug h ou t However, as shown in Fig. 5, parity is also conserved in the decomposition of He 35 c1 37 c1 in which nuclear exchange symmetry cannot affect the dynamics. The observed parity selection rule not only conserves nuclear spin; rather, Waals potential He 35 Cl 37 it results from the symmetry of (or symmetry in the case of Cl) with respect to interchange of Cl atoms. Quantitative fits excitation spectra of using rigid, The near the van der the line the to these "parity-selected" type in Fig. 4 asymmetric positions described above. for rovibronic asymmetric top rotational states of a particular parity were calculated. Each line was a to then convoluted with a homogeneous linewidth prepared state and an into line model strengths transitions and top were generated · account the laser Voigt profile due to the 28 decay which includes rate inhomogeneous linewidth and doppler linewidths. of that the takes Typical 51 ~ c .(f) $ c Q) u c (a) Q) u (f) ~ 0 => [L > :J (b) 19340 19341 19342 19343 Pump Energy/em_, FIG. 5. Parity-selected excitation spectra of the He 36 Cl 37 Cl B~X (8~) band exactly analogous to that shown in Fig. 4 for the He 36 Cl 2 isotopic species. Spectrum (a) shows transitions to odd parity levels; (b) shows transitions to even parity levels. Note that there is no measurable breakdown of the parity selection rule. 52 linewidths 0 . 07 em B-x -1 of the the measurement (FWHM) from transitions . In model the frequency center-of-mass fits to the of the uncomplexed Cl and 2 asymmetric top the He Cl the rotational rigid, origin, band distances of were profiles inhomogeneous Gaussian to 2 temperature, and the homogeneous linewidths were optimized by a least-squares method. The inhomogeneous 1 inewid th Cl-Cl was fixed to the value measured as described above. distances of were assumed to be unperturbed by the He atom and were the particular extracted spectra from of set to their average 22 vibrational state. the the present the fits HeC1 of 6-o 2 the presence and 8-o The values bond for lengths "parity-selected" bands agree within experimental error with those previously determined by the asymmetric top analysis of of Ref. The 6. rigid, the total asymmetric fluorescence top model of cannot perfectly simulate the excitation spectra: to the spectra of Fig . deviations . more A spectroscopy of HeC1 2 4 spectra HeC1 2 the fits show experimentally significant sophisticated treatment will be presented in Sec . of the IV(C). B. Homogeneous Broadening of High Vibrational Bands There is no detectable homogeneous broadening of the He 35 but c1 2 {B,u=6) higher broade~ing. level within our experimental resolution, bands homogeneous u This do effect show is measurable illustrated in Fig. 6. 53 (a) 19946 19948 19947 Pump Energy/em- 1 FIG. 6. Homogeneously broadened parity-selected excitation spectra of the Hea 6 Cl 2 B+-X ( 12+-0) band. Spectrum (a) shows transitions to even parity levels; (b) shows transitions to odd parity levels. 54 ~ Homogeneous linewidths for the bands with 8 reported in linewidths measured the (FWHM) respectively, 12 were study . 6 The homogeneous the u' = 20 and 24 levels study to be 1.0 and 2.0 of this in LIF previous ~ u corresponding em -1 of 5 .4 and widths of the complicated by the lifetimes to were 2.5 ps . Interpretation = 20 of the and 24 bands possibility of electronic u u' = 12, of homogeneous HeC1 to ground-state average lifetime of the 13 -305 ~s to electronic predissociation of slower 2 that , that the 2 Cl for 2 high atoms vibrational vibrational Although the electronic predissociation with a is predissociaton levels III(D) of rate of vibrational predissociation . still rate of that suggests HeC1 rate the 29 30 · Cl {B,u')13) 2 free and is 671 ns ~ 12 23 ) . u evidence is presented in Sec. very P / 3 2 ~ u' ~ 25 levels (compared HeC1 35 c1 (B) predissociation . > 21 and all u' > 12 levels undergo electronic j' predissociation much is 2 2 may comparable undergo to the It is then no longer possible to identify our experimental lifetime as a purely vibrational predissocition lifetime. u The lifetimes of the = 20 and 24 levels of HeC1 2 are longer than what would have been expected based on an extrapolation of the energy gap analysis of Ref. 6 applied is work clear that more is to the u required decomposition dynamics of the high u' to ~ 12 levels; understand it the levels of HeC1 {B) . 2 55 c. Fragment vibrational population rotational and distributions Having transitions individual there is considered the of i t is clear levels of the rotational little homogeneous vibrational levels (u' ~ be positioned spectroscopy 8) of HeC1 on an isolated vibrational band to prepare vibrational level . The probe 2 (B) B state For the 2 broadening. line specific lowest a rotationalto measure the rotational and vibrational populations of the fragment resulting from the can within then be used Cl 2 laser when the pump laser can rotational a B+--X prepare to how HeC1 the of decomposition of this specific state. Fig. An example 7, which of such shows an p~obe experiment laser is spectra presented of in the (B,u=5) fragment resulting from decomposition of different rotational Fi g. 7a levels was B+--X , 6~. 35 c1 2 E+--B, 0 He o bt a i ne d by +--1 00 01 (B,u=6). 2 pos1t · i on i ng transition and 1 ~- 1 ~ w 1 The 35 spectrum on tuning w the 2 He 35 through in c1 2 the The spectrum of Fig . 7b was c1 2 E+--B, 2 B+--X, 6~. 0+--5 spectra appear type rigid rotor spectra consisting of an R-branch head and a to the red of The was positioned on the He 35 c1 1 10 +--1 11 transition. 1 c1 0+--5 transition. obtained when w as simple 35 of well-resolved P-branch, the bandhead. of the spectra in Fig. 7. which extends There are two unique aspects First, because of the symmetry 56 (a) (b) 38856 38859 38862 38865 38868 Probe Energy/em- 38871 1 FIG. 7. Pump-~robe spectra obtained by scanning w 2 through 5 the fragment Cl 2 E~B (0~) band and (a) positioning w 1 36 on the He Cl 2 B~X (6~. 1 10 ~1 11 ) transition and (b) positioning Wt on the n~x (6~. Ooo+--lot) transition. These spectra reveal the rotational state distribution of the (a) 1 10 and (b) 0 00 rotational levels of He 36 Cl 2 (B.u·=6). 57 selection rule He 35 c1 states . states The of other spectra is in P-branch the this the the even-parity state decays only to product. The odd parity 35 the Cl 2 at j unobservable unusual fragment Cl 2 distribution rotational lines, which are resolved has The intensity distribution to 3 to levels for only predissociation of 2 the fragment intensity =2 the of j bimodal population of fragment = 5; feature even distinctly spectrum . maxima Cl III(A), initial rotational level decays to only odd j 10 u in Sec . (B , u'=6,Jh. h. =0 ) 00 a c 2 rotational 1 discussed and > 14 . j 35 Cl 2 and 10 There is local decays no to measurable vibrational levels other than 2 observed HeC1 9 in for channel is (B,u'=6) the vibrational transfer of one vibrational quantum to the van der Waals modes . Fragment rotational state populations were extracted from the line intensities fitting the intensity, in the Cl E~B 2 spectra by I(w) , of the observed spectra to an equation of the form I(w) = 2 [r 1 _ 1 • J L[w-wp(J) .r] j + I. 1 . L[w-wR(j) J+ • J where the rotational level . summation quantum is states, over j, all and C .r]] + C . observed is the {1) fragment background A Gaussian instrumental lineshape function, L, was 58 used with a r. of typically 0.2-0.3 em -1 (FWHM). linewidth. The frequencies of individual R- and P-branch lines, wR(j) and were wp(i). calculated using spectroscopic constants of Refs. line intensities are given by 1 .• 1 . •J the band origins and The relative 22 and 24. 31 = N(j) S J.. . J./(2j+l) s where the 1 (2) . , j are the Honl-London factors. 32 and N(j) is the relative population of the fragment state that appears with rotational dependence spectral of quantum number Einstein coefficients the range spanned by the band is parameters optimized in the fits were In general, one of the two fragment E--B intensities of individual, lines measured, can be bandhead. constant with v', v' bands, in most the the small negligible. The r. C, and the N(j). branches so of the that the completely resolved rotational the the P-branch the other change branch in the is extended forms cases a of few branch the of fall other taken in obtaining the lowest within branch so j for the the congestion that care two or three rotational levels. lines of of the needs of lower However, rotational the correct populations a rotational the R-branch for the higher v' bands.) extended bandhead over extended is while of frequency rotational bands (Because The j. the to be lowest The N(j) were obtained by 59 fitting, one at a time, the intensities of the lines the extended branch, beginning with the highest j working successively towards the lowest j is known, Thus, (2). the extended branch their line and line. Once N(j) the intensities of lines in the other branch are fixed by Eq. of in populations intensities Excellent when fitting the lowest j lines that are not completely resolved, can still of lines in the fits were obtained bandhead are by since the accounted for. extracted be this method to the Cl rotational state population 2 fragment spectra. fragment (B) distributions were extracted from the fragment spectra for = 6, the u' u·~ 8, 12, transition. 20, and 24 bands of In all cases the He 35 c1 2 B~x. the dominant vibrational predissociation product vibrational channel is the one in which one quantum of transferred to the Cl-Cl stretch is decompose the molecule (termed the "Au= -1 channel"). Tables 1 and 2 35 tabulate the fragment c1 2 (B,u'-1) = 0 and rotational state population distributions of the 1 1 rotational channel c1 2 (B,u'=6,8,12). of of the u'=6 and 8 accounts for all of the observable population, the fragment decomposition vibrational state population decays rotational minimum at to unobservable distributions j He 35 levels = 6 and 7, are states, levels bimodal the For Au = -1 product rotational for j with a > 15, the distinct and parity is strictly conserved 60 Rotational 35 P(j), of quantum state Table 1: fractional population distributions, c1 (B,u'=7) following dissociation of the 1=0 and 1 2 35c1 levels of He (B,u'=8) 2 Prepared He 35 c1 (B,u'=8) rotational state, Jk'k.' 2 a c Oao 0.17 i=O <E E 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -1 11 1 0.35 0.31 0.34 0.34 0.34 0.20 0.21 0.11 0.11 0.02 0.03 0.00 0.03 0.07 0.07 0.11 0.08 0.11 0.11 0.09 0.06 0.08 0.06 0.03 0.02 5.3 -1 -1 Era t ( j max= 14) = 29.7 em lto 0.15 0.32 ro t)/cm avail. =165.3 em lot 0.04 0.02 4.8 5.5 5.1 61 Rotational distributions, quantum N(j) • Table 2: fractional state 35Cl (B,u'-1) of 2 population following dissociation of He 35 c1 2 (B,u',J'k.'k.') a. c 35 Prepared He c1 state: u 2 (B) 6,0oo J=O <E E ro t)/cm au a i.l./cm E(j max 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -1 -1 )/em -1 6,110 0 . 19 . . Jk_.k.· 12,0 00 0.23 0.26 0.32 0.31 0.33 0.22 0.18 0.12 0.02 0.04 0 . 01 0.08 0.06 0.11 0.11 0.09 0.10 0.06 0.05 0.04 0.02 0.01 0.01 5.5 6.3 4.1 186 . 7 187.1 121.8 31.1 35.5 26 . 5 a c 62 in the decomposition decomposition of The dynamics. ] = 0 the and 1 distributions levels of the u for = 6,7 states are identical to each other within the experimental error. The primarily a fragment rotational E auai.l' is quantum, Eu , complex, D0 . He 35 type process . v~T less the For value highest kinetic the Cl-Cl c D (B) 0 measurable the energy, vibrational decomposition ) releases in rotation 00 energy of for energy of fragment is <E ro t> = 5.3 em [the The energy available translational example, cl 2 (B,u'=8.Jk k =0 average is the the B state dissociation energy of a The and given by of predissociation vibrational is -1 the Eauai.l = 165.3 em -1 . the of , which is only 3% of E auaL. 1 calculated fragment of in Sec. rotational IV(B)]. The J. max = 14. level, accounts for only 18 % of the available energy. For HeC1 the higher maximum contour. the levels (u' ) 8) of the 2 B state, homogeneous broadening precludes exciting a single rotational the vibrational intensity Thus, prepared by In these cases w position on the 1 was tuned vibrational to band the measured probe laser spectra represent fragment contributions leve 1. state distribution averaged of the initial set of the pump laser. In order over rotational to the states compare the fragment population distributions for decomposition of the high vibrational states to those of the lower vibrational 63 states, c1 fragment 2 spectra for the lower vibrational levels were also recorded with w maximum intensity position on the HeC1 distribution of substantial contributions with 1 = 1.2. distributions Fig. 8. Since both even- and 3 extracted A summary of and 2 tuned to the 1 odd-parity HeC1 were prepared. these states with levels rotational 2 Population spectra are shown in the results is tabulated in Table 3. the even- and odd-parity states undergo dynamics, (B) bandhead so that a from from HeC1 2 the even and odd j independent distributions are normalized and tabulated independently in Fig. 8 and Table 3. Fig. The "initial 8 and rotational-state Table differ 3 averaged" from results of rotationally the state-selected distributions in Tables 1 and 2 in that the distinct population minimum and that at clearly apparent higher. The slightly increased j j max j = 6,7 is no longer is one or two quanta max probably results from the contributions of higher total angular momentum 1 states of HeC1 2 . The disappearence of the node = 2,3 in the distribution clearly results from averaging over initially prepared rotational rotational state population distributions for states. Nodes in the fragment the 1 = 2,3 rotational levels may appear at different positions or may disappear altogether obscures nodes in so the The anharmonicity of that averaging over measured rotational these levels distributions. the Cl-Cl potential causes E ava .... l. to 64 Table 3: Summary of distributions the for fragment rotational state population dissociation of HeC1 (B,u') for averaged 2 initial rotational state given in Fig. 8. Prepared He 35 c1 (B) state: u' 2 u 6 8 + <E ro t)/cm E(j max -1 + 6.2 5.9 6.3 )/em -1 40.3 + 4.5 2.2 5.8 38.5 35.5 parity 12 + + 1.9 2.1 5.1 18.4 34.3 34.0 24 20 30.3 2.4 10.4 16.0 12.1 E auai.l./cm -1 186.7 165.3 121.8 -46.0 -19.5 Probe Trans. u(E)+--u(B) 0.-5 0+--7 4+--11 18+--19 13+--23 65 36 FIG. 8. Cl 2 fragment rotational state population distributions for the Au=-1 vibrational channel. These values were extracted from the fragment probe laser spectra when the pump laser was tuned to the maximum of the He 35 Cl 2 B~X (u'~} bands (a superposition of initial rotational levels is prepared}. (a}, (b), (c), (d), and (e) show the distributions for u' = 6, 8, 12, 20, and 24, respectively. The even j distributions (filled circles} and odd j distributions (open circles) are independently normalized to 100 %. 66 ~ 5~ 0 t: 0 • 0 • • 0 a...~ (b) s--7 • (a) 6__.5 ~ 0 ~~ 0 ~PI •0 • 0 • 0 • cf.co '<t • 0 o 4 6 ~ 8 o ~ • 2 lO I") 0 5~ 0 ~= 0 •0 • 0 • 0 '<t 2 4 w ~ m 0 • ~~ o ~ (d) 20--19 ~~ ~ cf.co u 8 oN • 0 6 • .5~ co }~rt 4 Fragment q Rotationd State, j ~ (c) 12--11 ~ 0 04--,--.-~--.--r--.--.~~~ w ~ m Fragment ~ Rotationd State, j 2 u '<t 6 Fragment 8 u 0 ~PI •0 0 m ~ q Rotationd State, j ~ ~ 0 q- w ~ •0 • cf.co o w ~ m Fragment q Rotationd State, j 2 4 6 8 (e) 24--23 •0 .5 ~ ......,N 0 i~ • ~: 0. 0 rfco '<t 0 • • 0 • 0 0,_-,--.--r--.-~~~~~~~ o 2 4 s Fragment 8 u • ~ w ~ m q Rotationd State, j u ~ 67 change by more than a vibrational states population in in the factor of six Fig. and Table 3. peak of the 8 high-j distribution for dissociation of over the range of The loss of rotational =20 and 24 levels the v is probably due to kinematic constraints associated with the of For j small value E . . avat. 1 example, classical for dissociation classical in the impact parameter of the parameter relative velocity of the level A. impact parameter of 3.6 impact high dependence of the results Cl Note corresponds The large the = 10 j to a increase much lower in the dissociation of because of the constant, the average that, rotational 2 whereas from the fragments levels. v 2.6 X. of = 20 v of = 8 level corresponds = 14 in the dissociation of the v to a observation v rotational energy of the fragments, <E ro t>. decreases with increasing still v so relatively small that rotation fraction of Eavail accounts even for for a the high v ' levels . Although only the the decomposition of Av = -1 <1% of and Av the = -3 fragment channels, cu 1 significantly so that excited. observed in = 12 state approximately 5% population appear respectively. homogeneous broadening of position was = 6 and 8 states of HeC1 2 {B), the v in the decomposition of the v and channel the only v the = 12, 0 in Despite the Av = -2 the large it is possible to 00 ~1 01 transition is Rotational distributions for 68 decomposition of shown in Fig. the ] = o00 level of HeC1 It is apparent that 9. 2 available energy increases by more than a from channels, the Au = -1 to constant within our the minimum at the Au = -3 experimental j=6 also are although the factor of three remains J. max The position of error. remains (B,u=12) unchanged, while the relative population in the high j peak of the distribution increases. = -1 channel D. Attempts to observe the closing of the Au As u' is increased, a point is reached at which the energy in a single vibrational quantum of Cl sufficient to dissociate that the Au = -2,-3, only allowed . Levy and coworkers measure D 's for Attempts to measure such a 0 direct by nonadiabatic an channels are 16 first van used der The of so energetically this effect Waals to molecules. of D onset for 0 of HeC1 D 0 2 were electronically observation of the in the decomposition of HeC1 limit < D0 ) Eu threshold and thereby obtain a apparent relaxation. upper {i.e. measurement the relaxation channel complex r2 rare-gas experimental prevented fixes the is no longer 2 Au = -1 (B,u=26) 2 < E(u=26) = 19 em -1 . Even though this is likely to be near the threshold for closure of Au = -1, Because of estimate the Au low signal of the = -1 channel levels for partitioning u is ~ among still 20, dominant. a quantitative the vibrational 69 (a) .du o 2 4 6 8 n ~ ~ = -1 m ~ Fragment Cl2 Rotational State, j (b) .du = -2 o 2 4 6 8 n ~ ~ m ~ Fragment q Rotational State, j (c) .du = -3 o 2 4 6 8 n ~ ~ ~ ~ Fragment C~ Rotationd State, j FIG. 9. Rotational state population distributions for dissociation of the 1 = 0 level of He 36 Cl 2 (B,u'=12) . The .du = -1 vibrational channel (a) accounts for most of the fragment population. The .du = -2 channel (b) accounts for -5 X and the .du = 3 channel (c) for less than 1 % of the total fragment population . Experimental data are the filled circles, while open circles are calculated distributions. 70 The experimental signal became channels is not possible. i ncreasingly HeCl weaker 2 (B,u')26) channel at higher could be remained no channel for signal However, observed . dominant the and v Au = -1 all the the for vibrational bands studied. A possible explanation intensity of that perturbation the enhances [see the Sec . successively below the electronic state Cl 2 (B,u=27) 6 A! 22 ) . in resonance signals were threshold outer This the free Cl (B) 2 scale of prove is still obtained responsible for this 2 point Our Strong double the level the -10 ns time result does not predissociation however, for prevent 2 to B would this vibrational electronic effect ; Cl Cl (B,u ' =27); for slow relative that the with turning effect the pump-probe experiment . conclusively is the of experiment . electronic predissociation rate of of levels These vibrational levels lie classical detection competitive level a fragment the 2 dissoc i ation is vibrational in caused by the He atom 2 predissociation rate of Cl (B) to {the degradation Cl vibrational predissociation . just the higher of electronic III(B)] for we can is offer no is the other simple explanation for these observations. IV . CALCULATIONS A. Potential Energy Function One of the primary goals of this work 71 construction of a quantitatively accurate potential energy surface for spectroscopic functional hand, data reported form for the above the potential time-consuming dynamical the Modeling nature a flexible On the other demands surface. of and the calculations described below places practical constraints on the number of parameters have the that can be adjusted. potential containing a It is described by adjustable parameters surface small number of desirable a to function that alter the potential in a physically obvious way. Reid. Cl 2 van der surfaces. a.L. 33 Waals That parameter initial et calculated bound molecules study expansion) attempts dynamics of HeC1 used a potential at 2 for modeling levels model of rare-gas potential energy highly flexible (Legendre form. We that the found experimentally our observed were quite successful using the simpler atom-atom potential form described below and that further refinement gave good agreement with the spectroscopy as of to well. In this paper we use the notation describe the degrees of freedom of HeC1 scheme is depicted in Fig. 10. The 2 . Ref. 33 This coordinate internal coordinates of HeC1 2 are the He to Cl center-of-mass distance, R. 2 Cl-Cl bond length, the vectors R and r use a modified form r, and 'Y, which is along which R and r of the potential the the angle between are measured. energy function We of 72 FIG. 10. (a) shows the relationship of the molecule-fixed frame (x , y,z) to the space-fixed frame (X,Y,Z) for a generalized atom-diatom system A-BC . The molecule-fixed frame is rotated from the space-fixed frame by the Euler angles c<t>. 9,0). (b) shows the internal coordinates in the molecule fixed frame . R is the distance from the atom, A . to the center-of-mass of the diatom, BC . r is the B to C distance and ~ is the angle between the lines along which R and r are measured. x is the angle between the xz plane and the plane of the molecule . For clarity, ~ has been chosen as ~/2 . Both coordinate frames have their origins at the center-of-mass of the A-BC system . This figure and caption are courtesy of Brian Reid. 73 z a) 8 ·. ... z --- ...·· X X z b) R y 74 Halberstadt, et al.., 34 given by (3a) X * (3b) where 2 VM(r,R.~) = cCl l [{1-exp[-a(xi-xm)]} 2 -1] (4) i=l and (5) This functional (B) excited form is used for both electronic states of HeC1 2 the ground (X) . sum of VM is a and two Morse potentials between the He atom and each Cl atom, and describes He•••Cl 2 the well potential. and The hard wall regions are the He to of Cl the atom distances given by and = R2 + r xm is 2 4 ± rR the potentials. cos~ position is (6) , of the minimum smoothed into of the the Morse long-range anisotropic van der Waals potential, VvdW' appropriate for HeC1 of 2 . the The smoothing is initiated at the inflection point atom-atom Morse potentials , x * =x m+ln(2)/a. The 75 parameter p function. Th e are controls expressed the shape relatively x fitting rn as two of the van term rn Waa l s Legendre potential adjustable x d er smoothing the of . . 35 coe ff 1c1ents expansions; e . g .• form places rather rigid constraints the few (X), strength anisotropic This functional on the but has the advantage of Only parameters . and a were varied systematically in (B), spectroscopic and dynamic data. (The same value for a was used for both electronic states . ) The set of optimal potential paramters that provided the simultaneous agreement with all the available experimental data are g i ven in 4. Table The sensitivity of the spectroscopy and dynamics calculations to these parameters is discussed in more detail in the following sections . course. Of the optimum choice of a value for one parameter is determined by the values of the other parameters . With the exception of eCl' the parameters were optimized to fit experimental for III(B) . we energy, D 0 were • electronic scaling systems results unable of HeC1 state 2 . . As discussed in Sec . 2 measure directly the binding HeC1 to The value of D can be estimated 0 for HeC1 to 2 be in the B em -1 by the experimentally measured D ' s for the related 0 16 36 Hei . Nei . and NeC1 . The values of a and 2 2 2 were calculation adjusted of the to give vibrational good agreement predissociation with the dynamics 76 Table 4 : Potential Parameters for HeC1 = 16.5 em -1 X (X) = 3.55 A m e.Cl (B) = 14.0 em E.Cl {X) p X = 4 a= 1.6 ,-1 J\ c6,0 = 13 . 3 X 10 3 em- 1 A6 C8,0 = 10 . 95 X 10 5 em -1 A,8 3.2 x 10 5 em -1 2 A8 m (B) = 3 . 60 A -1 77 while simultaneously giving a calculated value of D close 0 to A contour plot of the estimated value. the X,u' '=0 state of HeC1 V(r is shown in Fig . 2 0 .R.~) for 11 . B. Calculation of Bound and Quasibound States The the secular equation method used quantum quasibound, has been Reid, et continuum levels fully described by al...; of Fig . 10, 2 is 33 therefore, a Using paper for the bound, atom-diatom systems Halberstadt, only is given here. of this of calculation and calculations HeC1 mechanical in et rough al.... 34 outline and the of the coordinate scheme the vibrational and rotational Hamiltonian for 34 (7) ~Cl is the vibrational Hamiltonian for free Cl 2 ;,:Cl The and m 2. (8) 2 eigenfunctions ECl(u). 2 and eigenvalues of In Eqs. and = (mHemC1 )/(mHe+mC1 ). 2 (7) j is the Eq. (8) are )( (r) u (8) angular rotational 2 momentum associated with the rotation of r, and l is the 78 HeCI2 X, v + 0 + + + + -6 -4-20 2 4 6 + + Chlorine Axis (A) FIG. 11. Equipotential contour plot of the He•••Cl 2 potential for the ground electronic state, using the potential parameters in Table 4. The Cl-Cl bond length is set to its vibrationally averaged value of 1.998 A for u = 0. Contours are drawn at -30, -20, -10, -5, 0 {dashed contour), +100, and +200 cm- 1 • 79 orbital R. angular The V(r,R.~) Sec. total angular is the IV(A). vibrational Coxon. momentum associated with is thus rotation of given by j+l.. intermolecular van der Waals potential of and the intramolecular the RKR is VCl ( r) Cl-Cl 2 potential 22 I momentum the Both from VCl (r) 2 V(r,R.~) and calculations of depend upon the partitioned to electronic state of the complex. The Hamiltonian decouple the He•••Cl that the van effective der He•••Cl vibrational 2 2 of is motion from the Cl-Cl vibration Waals levels potential, motion. (7) Eq. Following are calculated averaged Ref. 37 for over the we define 34, so an Cl-Cl the zero-order Hamiltonian jf v,v + V (R.~) + Eel (u) + v,v 2 where Vv,v (R.~) intermolecular = <)( v IV(r,R.~) I'< v > i s potential vibrational wavefunctions, constant Cl of 2 (Eq. and in vibrational the (3)] B n (9) u2 j2 average over of the the Bv is the rotational state u. The zero-order wavefunctions are given by 0 l]!vk.(r.R) " = '<u(r) ~vk.(R,r) {10) 80 A is an eigenfunction of :Jt The -~'uk.(R,r) U,U are diagonalization of the zero-order set of basis functions calculated by Hamiltonian matrix for a constructed as products of harmonic oscillator functions. <f> n (R). Waals which bond Appendix describe and angular which 1), the radial basis motion of the internal the the complex . J. angular momentum, is the projection pi = ±1 is the function. It is I of on on ]Mpi In the Sjn (R,r). M is the A the projection of spaced the fixed Z-axis; the body inversion parity of the shown Ref. 33 that in (see and quantum number associated with total der bending A overall rotation of van ]Mpi A A SjO (R , r) functions, describe the fixed z-axis; angular :Jt u,u n and basis does not couple angular basis functions having different values of 1. M. basis and p .. Furthermore, functions with 1. (even). :Jt even j U,U (odd) The eigenfunctions of :Jt ]Mpi -~'p. . J where parity of p. = +1 J the (even) state. U,U or -1 results rotationless 1=0 of the bound levels of to those with j odd can thus be labeled as (odd) is indicates the the same Cl state calculation for the = 0 = 8 the state are given in Table 5 for not couple angular parity The exchange parity discussed in Sec. The will III(A) . X,u'' and B,u' the potential determined by the parameters of Table 4 . The energy of the lowest bound state gives of the binding energy, 81 Table 5: Bound .state energy levels of the X, u' 0, I 0 and the B. u' = 8, I = 0 states of HeC1 2 • b indicates the number of quanta in the van der Waals bending mode along the coordinate "Y . = Energy/cmBending State b X.u'' 0 1 2 3 -12.82 -7.43 -5.74 -3.42 1 = relative to dissociation threshold =0 B,v' =8 -9.39 -4.65 -3.61 -0.31 82 n0 = -9.39 em -1 three The higher der van Waals vibrational bound states involve excitation of the bending mode about the angle There ~. are no bound levels involving the van der Waals stretch along the R coordinate in either the X or B electronic states. The blue shift of the HeC1 from the corresponding Cl 2 B~X. 2 u · ~ u ·~ band origins is a From Table 5 the measure of the difference D (X)-D (B). calculated blue-shift 3.43 em -1 of 0 0 the HeC1 . close Calculated and experimental vibrational bands are shift increases binding energy, to the with of the Cl 2 V (R.~). U,U probability (the the 12 density Note that to give the ..p shows for rotationless, level). of shifts in Table means This intermolecular wavefunction Fig. value band 3.49 em for 6 . that the is -1 other The blue B state n0 (B). decreases with the number of quanta in the Cl-Cl stretching mode. averaging 8~ blue which u', B~X 2 experimental presented band origins B~X. is the potential the result of V(r,R.~) effective the over potential, 001 wavefunction and associated 1 the ground u' = 8, van der electronic B Waals state vibrational there is significant delocalization of the He atom even in the lowest van der Waals vibrational state. The continuum, coupling of which has been negle.cted the zero-order in states this to the calculation. 83 Table 6: Summary of 1=0 bound state and dynamics calculations for HeC1 2 (B,v') and comparison with experimental results of Ref. 6. ~ 6 7 8 9 10 11 12 blue shift/em - 1 Y.P. Lifetime/ps expt. calc. expt. calc. - 3.37 3.40 3.43 3.47 3.51 3.55 3.60 - 612 436 311 231 170 128 95 3 . 40 3.49 3.55 3.65 3.72 3.77 - 506 275 179 97 52 84 HeC~ (a) c 0 ~ L2 a> ~ ~ 4 2 D) -2 ACJidre ~(~ 0 (b) FIG. 12. (a) shows the vibrational wave functton of the lowest2 van der Waals quasibound Vibrational level for density. He'"C1 (B.u·=8.J=O). (b) shows the associated probability 85 shifts the energies The magnitude of of the golden IV(D) Sec . of the line this quasibound shift is a vibrational 34 which calculation and found 0 . 0425 cm-l the coupling of Ne-Cl 2 continuum is (He- Ne-Cl and will of the be 2 have predissociation lifetimes), the used treatment to 2 in dynamics. an center shifted was same NeCl that resonance the the accuracy compared the golden rule NeCl 2 {B . v=l1,]=0) to slightly . dissociation predissociation of Since states measure of the calculate et ai ., shape the rule approximation, to Halberstadt , of exact of the only by the zero-order states order very for both He- similar and vibrational the golden rule approximation should also be valid for HeCl 2 . Further technical details of the calculation are given in Appendix 2 . C. Spectroscopy The spectroscopy of simulated if transition line initial this bound rotational section extended to bands of HeCl quasi bound state among states, and strengths B~X the can be energies, and the population distribution are known. In calculation states is X B the levels with of >0 J these 2 the in bound both the and electronic states. An initial X,v' '=0,]' 'k ' ' k ', a transition dipole to the level is coupled by the c quasibound B , v' ,J'k'k' a Selection rules and line strengths for these levels. c transitions 86 are by determined strength transitions between the two factors is the analysis of Appendix 1 is that the = 0 or ±1, Api = ±2 and for Calculation levels. factors 1 ine the of the line strength described in Appendix 1. A result of transition selection rules are AJ = ±2. Ap. J B~X. The Fig. 8~ 13 shows calculated spectra for the HeC1 2 band using the potential parameters of Table 4. simulated rotational spectrum was population calculated for distribution, a Boltzmann and the intensities were calculated using Eq . (A7) of The lines are convoluted with profile done for the previously described The quality of the the simulation Voigt found dependent on the potential parameters x be expected since of the van der secular better equation fit to obtained described in Sec. and fits. be most (X) and x the was Ill (B), to the position effective bond simulated spectrum obtained from gives experimental from the III(A), levels, hence calculation the simulation rotational well The lengths RX and RB . the Ill as 1. top to these parameters determine Waals Appendix asymmetric was line a than spectrum asymmetric particularly although quantitatively top for qualitatively the treatment the the higher results appear quite similar. It is interesting is lengths obtained in the rigid, to determine asymmetric how the bond top analysis of 87 FIG . 13. Solid curves are exferimental. parity-selected excitation spectra of He 3 Cl 2 (B,u ' =8) obtained as described in Fig . 4 . Stick spectra are line positions i nvolving transitions among levels with 1 ~ 3 calculated as described in Sec . IV(C); more important transitions in this study are labeled with the asymmetric top designations for ease of identification . Table 8 shows the symmetry correspondence of these calculated levels and the asymmetric top functions . Dotted curves are simulated spectra convoluted with the proper Voigt lineshape for this vibrational transition. Note the subtle improvement of these fits over those obtained using the asymmetric top model of Fig . 4. (a) shows transitions to even-parity levels ; (b) shows transitions to odd-parity levels. 88 .... 0 ...... ! N 0 N (a) ... 0 ... ...... l... 0 ...-1 ... ! T"l 0 0 0 ... ... N N N ..0 • (f) .., l... (I') T""f l ... 0 c(l) -+--' c C\1 l (l) uc 2 0 C\1 l... l.., (l) u (f) N 0 0 0 (b) 0 ~ ...... 0 ::J T""f > :=J T""f u: l... 0 0 (I') ...... ~ l... N C\1 N 0 C\1 ...... !... N 0 ~ l ... 0 T""f 19356.5 o- N C\1 l... N C\1 19357.5 Pump Energy/em- 1 89 HeC1 and are related to the potential surface of the molecule 2 to the average geometry of In order the complex . to compare directly the asymmetric top energy levels to those obtained by diagonalization of the Hamiltonian of Eq. the correlation of the the >JI (9). asymmetric top rotational must be established. ]Mp. wavefunctions and L first pi The asymmetric for the D the B top rotational point group . 2 wavefunctions respect to c b2 the as seen in Fig. operator 3), Ba. and B c n2 are antisymmetric with (which exchanges the Cl while those belonging to 25 representations are symmetric. of Thus, the A and Bb the wavefunctions have even parity. The rotational rules for a.-type asymmetric top transitions are A~Ba. 4b and Bb~Bc contains a.L., 33 spectrum Fig. transitions, only B showed how of a. ~A 4a while or then the A and Bb selection A~B and a. contains only spectrum of Fig. transitions . to correlate atoms the wavefunctions symmetry have odd parity and The bas i s Those wavefunctions belonging to and Be representations of a. form a the symmetry of Reid, et ]Mpi the ..Pp. J to the asymmetric correlation for HeC1 Table 8 top 2 wavefunctions. some of symmetry is shown in Table 7 . shows how the energy levels obtained in the asymmetric top analysis of Fig. levels The calculated using Eqn. 4 compare with the energy (9) . the highest rotational With the exception of levels of the X state, the 1 -1 -1 -1 a Bb B c B -1 1 I A pj 1 I Asymmetric Top Symmetry 1 Pt Wavefunction Symmetry Table 7: Symmetry Correlation of Calculated Wavefunctions with Asymmetric Top Functions (£) 0 91 Table 8: Comparison of the rotational energy levels of the ground van der Waals vibrational state of HeCl 2 for the asymmetric top (AT) model (calculated for the bond lengths given in Fig. 4) or the secular equation calculation (SE) of Sec ·. IV(C) . The symmetry correlation of the levels is also given. Energy levels are calculated for the u'' 0 level of the X state and the u' 8 level of the B electronic state. = Ro t ational Level AT D2 Rep. 1k k a. c A Ooo Energy/em -1 x. u=O Energy/em -1 B. U=8 EAT(X) ESE(X) EAT(B) 0.0 0.0 0.0 0 .0 0.383 0 . 400 0.233 0.239 0.470 0.460 0.405 0 . 407 0.573 0.597 0.448 0.461 1 . 098 1.113 0.693 0.708 1.134 1.110 0.829 0.826 1.443 1.517 0 . 957 0 . 990 SE ] pi. pj B a. lot Bb B 11 1 c 11o A 2o2 B c 2t:z 0 1 1 l_t - 1 1 1 -1 1 1-1 2 1 1 2_1 1 Bb Ba. 211 -1 A 22o B a. 3o3 Bb B 313 c 312 A 322 B a. 321 2 1 2_1 - 1 2 1 1 3_1 -1 3 1 -1 3_1 1 3 1 1 3_1 -1 Bb B c 331 3 1 -1 330 1_1 1 221 = ESE(B) 1.704 1.691 1.472 1.486 1.757 1.773 1.479 1. 497 2 . 091 2.062 1 . 367 1.383 2 . 102 2.058 1. 461 1 . 452 2 . 701 2.819 1.716 1.775 2.854 2.830 2.172 2.179 3.063 3.139 2.206 2 . 225 3.574 3.484 3. 141 3.132 3.593 3.525 3.141 3.133 92 agreement the between asymmetric levels top secular equation calculation is within a a wavenumber. significant shows in a microwave experiment, experimental top model can simulate of highly especially rotational levels , in the spectrum. s~x As for resolution the agreement the rigid , the rovibronic spectrum with surprising transitions among the which account for expected, this be molecule non-rigid accuracy, would deviations such asymmetric this few hundredths of Although that within our the and rotational lower the intense features levels equation calculation given in Table 8 of the secular cannot be produced by any single set of asymmetric top rotational constants . The 3.8 A He•••Cl 2 distance from the asymmetric top fit to the experimental spectrum is the same as the average value of R for the probability density of Fig. The bond length derived from 12 along the asymmetric ~ = 90 0 . top analysis for a T-shaped model thus gives an accurate description of the effective geometry of the complex . ~he average bond length is significantly longer than the 3.35 A position of the well minimum the ~ = 90 0 the van der along section of the potential surface. No progressions in Waals modes could be measured experimentally . has been attributed excited state to potential the Previously, similarity of surfaces. vibrational the ground However, the this and line 93 strength factors that transition intensity to the calculated for our potential indicate the ground van der Waals of the B state mode accounts for roughly 80 % of the total line strength from the ground van der Waals mode of the X state. Most of excited bending the remaining mode, and strength the is 1 ine to the first strengths for transitions to successively higher excited levels decrease rapidly. it is Assuming our potential unclear observed. why no van Our experimental surface der to Waals be accurate, progression signal-to-noise should enable It may us to measure at least the first combination band. be that the excited van is der Waals levels are severely homogeneously broadened and are difficult to detect. D. Dynamics The neglects zero-order the weak Hamiltonian, coupling due jf to of U,U the Eqn. intermolecular potential, V(r,R,"Y), among the wavefunctions of Eq. The rate the decay of initially prepared quasibound state, (9) (10). zero-order is given within the golden rule approximation by the half-width, 34 ( 11) where the are the off-diagonal 94 matrix the elements Cl of the vibrational 2 continuum intermolecular The that are described elsewhere. and 34 j uf f the describes methodology continuum wavefunctions r. ,p wavefunctions. wavefunction products . potential of the the integration between "' (R,r) is dissociated calculation of of the Eq. (11) The partial width, . (12) 1.---+UfJf is a proportional to the population fragments of appearing in the vibrational state uf with the rotational jf . quantum number given by T The respect to The lifetime of the initial state is n . = 2r t coupling v potential, rotations of utuf of Eq . and be either antisymmetric (pj = -1). is = 7r in Fig. "Y the matrix elements must . (12) to symmetric 10. In order be non-zero, symmetric (p. J with for both ,Pi. = 1) or This requirement leads to parity conservation in the dynamics calculations. Dynamics dissociation calculations of the J =0 Waals vibrational state. were level of the The results of performed for ground der van the calculations are compared with experiment in Table 6 and in Figs . 9 and 14. The calculated vibrational predissociation lifetimes fall within a factor of two of the experimental values; 95 (a) 6-+5 o m Q w m Fragment Cl2 Rotational State, j 2 4 6 B ~ (b) 8-+7 o 2 4 6 e n ~ w m ~ Fragment q Rotational State, j FIG. 14. Comparison of experimental and calculated fragment rotational population distributions for vibrational predissociation of the J = 0 level of 5 Hea Cl 2 (B,u'). (a) is for u' = 6: (b) is for u' = 8. Filled circles represent experimental data; open circles are calculated rotational distributions. 96 however, the with than do and vibrational u rotational calculated lifetimes population fragment and the rotational second fractional Calculated are in evident, the also Both the minimum in population distribution at maximum population slowly distributions show good agreement with experiment. the more lifetimes . actual the decrease but second the j = 6 calculated maximum is smaller than was observed experimentally. The population of the Au' in the decomposition of for 1 % = -2 vibrational HeC1 (B,u'=8,]=0) 2 channel accounts of total the population. Experimentally, we estimate the population of this channel to be experimental and calculated the decomposition calculated and less -3 again in of population channels good than is 1 %. Fig. rotational distribution 96.6, agreement the population number of fragmentation; Cl 2 of 3.3, with large calculated The as low overall j for among and level . the 0.1 %, higher j quanta The Au= -1.-2. respectively, both the distributions, the peak In increases transferred in with the this effect is not as dramatic in In all is distributions experiment. vibrational in the second peak of as the however, the calculation. the HeC1 (B,u'=12,]=0) 2 experimental and calculated rotational relative compares 9 cases the rotational distribution observed experimentally, populations agreement the calculated population of is not whereas are larger than the calculation the observed. with the 97 experiment remarkable , however, is . the considering simplicity of our potential energy surface. rotational distributions and relatively insensitive to the of potential . However, both The the value lifetime distributions are extremely sensitive For example, x decreasing a lifetimes are used the m and to constant, rotational the values ,-1 by 0.4 ~ increases -1 increases lifetime by approximately four hundred picoseconds. the correct it is possible experimental a Holding other parameters decreasing eel from 16 to 10 em given value of eCl' of The lifetime the lifetime by several hundred picoseconds. also varies inversely with eel· in to adjust a lifetime for a the For a to obtain given initial vibrational level. Both the position of the intervening minimum and the high tail j of the shifted to larger j of a fragment rotational values as a distribution is increased: are high values lead to greater rotational excitation in addition to shorter lifetimes. Decreasing enhances the value of considerably the minimum in the rotational distribution. It is possible to adjust the potential parameters to bring the quantitative eel T = 8 em -1 dynamics calculation into essentially agreement with data. Choosing a= 1 . 8 y-1. ~ = 680 ps and an excellent the and X m = 3.65 A reproduction of the gives observed 98 Such a potential. however. has a rotational distribution. 16 em -1 . which supports only one well minimum of only bound state with a These values are unrealistically binding energy, the parameters -1 to The values of IV(A)]. in Table 4 were selected 2.8 em compared when small data from related systems [See Sec . of D (B). 0 to give the best simultaneous agreement with all the available data. V. DISCUSSION The these success of calculations the atom-atom additive is particularly exciting potential in because it suggests that simple potential energy surfaces constructed in this manner for closely related van der Waals molecules may be reasonably accurate. It is interesting to compare our He•••Cl Morse potential to the experimentally measured potentials for the nearly isoelectronic He•••Ar system . Our value of eCl (X) 20.4 em -1 for = 16.5 em -1 e(He•••Ar). = 3 . 55 A. well minimum, X 3 . 50 A value for the atom-atom use of scattering Ill. data (X) The close to the value of position of the is is slightly longer the He• • •Ar potential. potentials provides a good intermolecular potentials of 39 than It appears derived initial simple, He•••Cl from the that rare-gas estimate of the rare-gas halogen van der Waals systems. O ' Loughlin scattering 35 experiment has performed a on He•••Cl (X,u"=0); 2 crossed-beams however, the 99 results did potential. anisotropy, the range not lead to Both the a unique determination well depth, D (X) AR (X)= 1.14 e of values e of = 33 cm- 1 a and A. of our potential fall within spanned by the two best potentials proposed in that study . It is important to distinguish the origins of parity conservation in this experiment from an effect observed in another et at . , have recent 40 photodissociation Valentini. experiment. have studied the photodissociation of ozone and shown that formation the of 16 0 homonuclear 2 (1A ) g fragment occurs with preferential population of the even j rotational levels. the odd j population by a give This is due to selective depletion of the second decomposition channel fragment rotational levels because Heteronuclear population of which of nuclear fragments either even or in due exchange parity distinct to nuclear conservation from the decomposition. of the der He • • •Cl 2 van HeC1 observed for both He 35 c1 2 Waals 2 2 from potential; and He 35 c1 37 Cl. j symmetry . levels; ozone effects. is rules the the the is The completely ozone non-adiabatic selection result odd preferential no symmetry in symmetry HeC1 exchange rotational electronically photodecomposition only the decomposition of observed The have show odd symmetry selection rule solely can to in the symmetry of effect is This effect is precisely the half-collision analogue of the Aj = ±2,4, ... 100 selection rule 37 c1 the 35 c1 Since the center-of-mass of lie precisely half the distance parity conservation should not, scattering. inelastic rotationally in species does not between the Cl nuclei. in principle, be complete . However, breakdown of the selection rule for He 35 Cl 37 Cl is too small to be accurately measured in this experiment. The general features predissociation dynamics of vibrational rule propensity HeCl and rotational excitation of the Cl the now predictions rules for molecules . of the vibrational 41 that • 2 is, the small the fragment , 2 vibrational the of = -1 Au of degree follow closely well-established propensity transfer der energy in van Waals The details of the rotational distributions, however, merit further discussion. The unusual rotational bimodal distributions explained as a result complete rotational selection has been structure of of the c1 averaging and fragment 2 vibrational initial state The reasonable representation of to a simple physical analysis of the that quantum surface is a lead results, our they picture dynamical be states; provide assurance experimental cannot initial cal cu lations the the over achieved. like in observed data potential do for in mechanical energy the forces, not the but immediately dynamics. terms of An other theoretical approaches is beyond the scope of the present paper, briefly: but we consider two simple treatments a 101 classical impulsive dissociation and the Franck-Condon approximation for dissociation. In a First, we consider the classical dissociation. theoretical study van of der calculated bimodal the vibrational Waals molecule, rotational predissociation Hutson, et distributions and of a.1. . 42 suggested that they were the result of oscillatory rainbow structure associated recent with a classical theoretical studies rainbow features of diatomic rotational rotational explored have More rainbow . observed in the rotational distributions fragments covalently bound impulsive dynamics of the direct By molecules. on photodissociation of simply treating electronic dissociative the the potential surface classically, and weighting the classical trajectories by agreement has been obtained with experimentally fragment rotational the photodissociation of initial distributions several qualitative wavefunction, small for observed direct the molecules. 43 It is possible to draw an analogy between the latter treatments and vibrational predissociation. The optical transition that couples a bound state to the continuum on a repulsive electronic the potential surface is analogous to small potential couplings that lead to the decay of a zero-order quasibound der Waals continuum of unbound vibrational states. classical level of a trajectories van are computed molecule to the In this method, for different 102 initial angular system to j(~ 0 ). obtain which system a following j(~ singularity Fig . 15 the final the classical the results for 2 (B.u ' =12,]=0) In "classical of the atom-diatom excitation function, " angular momentum of rainbow rotational A the 0 ) passes through a maximum, resulting in shows surfaces . 0. dissociation. in calculations ~eC1 the gives results when ~ configurations. our these B,v ' distribution. classical of = 11. 1 0. calculations trajectory dissociation impulsive the on rotational the and of 9 potent i al initial relative velocity of the fragments is set to zero. and trajectories were to run for obtain j(~ a variety of 0) . The initial angular configurations initial He•••Cl fixed by conservation of energy; the various product channels of vibrational decomposition . the j ~ transferred in the predict that trajectory calculations 8 Au = -1 channel rotational energy . For rainbow) this rainbow to j ~ 13 the second maximum of distribution rotational the available energy for quanta classical However , (which is reason is number the the R. the of for distance, is determined by position channel. the The 2 would be should for the from Au = -3 the experimental associated insensitive to we that believe move the with available the second maximum of the distribution cannot be attributed simply to a rotational rainbow effect. A limiting case in photodissociation theory is the 103 ,, ;------................ , ......,,_ .. ,,'' ,'' _,,-' ,,,'' \ \\ \ ... ····························... 1\ ,~' ~ ·.\ ,' ,, , I I I \ \ \ \ \ \ / • I ·.... '' \\ ,' \\ \~ \\ ".. \ ,' / "". . a :. ".,, "~·.,,,, :\ .. "~ ') 0 10 20 30 40 50 60 70 80 90 Initial angular configuration, -y(t == 0) FIG. 15. Classical trajectory calculations for impulsive dissociation of HeC1 2 (B,u'=12,J=0) on the He•••Cl 2 (B,u'=11) potential surface. Solid curve shows the final angular momentum of the Au = -1 channel Cl 2 fragment, j, as a function of "Y(t=O), the initial angular geometry of the complex. The dotted curve is calculated for the = -2 channel and the dashed curve is for the Au= -3 channel. See text for initial conditions. The method used to compute the classical trajectories is discussed in Appendix A to this thesis. Au 104 Franck-Condon approximation, rotational populations are 3 in which the relative final given by the overlap of the initially prepared state with the free rotor states, YjO' N(j) = I<"'.IY. l. J 0 >1 The result calculated of 2 {13) this in Sec. treatment, using the is presented IV(B). wavefunctions in Fig. Franck-Condon distribution does show weak structure, it much more rapidly the but falls experimental off sophisticated version successfully model of the the oscillatory than does However, distribution. more a theory Franck-Condon observed The 16. 44 may This dynamics. is suggested by the insensitivity of the distributions to the available energy and the degree low of rotational excitation. Yet distribution effect. third possibility is result a the of a that from different rise to The near two or more configurations the same final total node in result from the decay suggests of the of the molecule state of the rotational a single that give fragment. distributions initial the distributions are interferences, and that oscillatory j interference this results from that large bimodal trajectories originating rotational or the quanta! Within a semiclassical picture, the superposition of {which is truly the level) result of features are 105 0 (.() o 2 4 s 6 n w Q Fragment Cl2 Rotational State, j FIG. 16. Decomposition of the wave function of the level in terms of the 0 = 0 spherical harmonics as a test of the Franck-Condon theory of photodissociation. Open circles show the overlaps given by Eq. (13}. Filled circles show the experimental rotationa~ population distribution. He ~Cl 2 (B,v'=8,]=0) ground van der Waals vibrational 3 106 missing because parameters they distribution for third maximum of HeC1 (B,u ' =12,]=0) 2 that appears of experimental the this be vibrational measurement success of Au in shows a of the Fig. 15 continuation Unfortunately, oscillatory structure . population to the channel did we are of a this low fractional not third maximum, these calculations, = -3 calculated The require. impact large unphysically the permit but given confident that it exists . VI. SUMMARY The dynamics been studied of at unprecedented the state HeC1 the photodissociation of state-to-state resolution allows 2 This level. the has experimental observation of dynamical symmetry constraints arising from the symmetry of the He•••Cl major vibrational transfer of one fraction of the rotation; 2 van der Waals potential. predissociation product Cl 2 vibrational available however, the channel quantum. energy observed The is the Only a as fragment appears fragment small rotational population distribution is distinctly bimodal . The bond analysis provide of a lengths the extracted from an asymmetric spectra of this molecule representation of the effective excitation reasonable top geometry of the complex. A potential energy surface was constructed for this 107 molecule on which calculations were with both fragmentation The potential experiment -1 and These dynamics that gave theory has mechanical calculations the complex and measured in the agreement a best binding agree the energy laboratory. of between 12.8 and and He-Cl atom-atom equilibrium distances of 3.55 and 3 . 60 A in states, performed . quantum the excitation spectroscopy of the 9.4 em three-dimensional the ground and (X) excited (B) electronic respectively. The rotational distributions are not consistent with any simple bimodal dynamical rotational model for the dissociation. distribution may arise from The quanta! interference effects in the photodissociation dynamics. VII. ACKNOWLEDGEMENTS The authors acknowledge the financial support of the National Science Foundation and NATO for an international collaboration grant. We appreciate he 1 pfu l discussions with Prof. J.A. Beswick . VIII. APPENDICES 1 . Line Strength Factors This intensities appendix and outlines selection the rules for calculation of transitions between two energy levels (J.{a}) and (J' ,{a'}). electronic of state HeCl 2 is denoted the The excited (B) with primes, while 108 the ground (X) momentum of J is the total angular state is unprimed. the level and {a} represents the set of quantum numbers and labels required to completely specify the M-states, particular which comprise absence of level. The the (J, {a}), level external fields . 2]+1 The IJ.M,{a}>. are degenerate line strength in 31,45 the for transitions between these levels is S(]',{a'};J.{a}) = 2 l<J' ,M' .{a'} lt·J.LIJ.M.{a}>l 2 (A1) . M' .M The sum field is over vector, r Defining assuming a all and to M and J.t is lie states, M' the along transition the r is dipole spaced-fixed random distribution of the molecular electric operator. Z-axis and orientations gives S ( J' . {a' } ; J. {a}) = 3 2 I< J' . M' • {a' } I GIJ.z IJ. M. {a}> 12 . (A2) M' ,M J.Lz is the projection of HeC1 2 , (the J.t on the spaced-fixed Z-axis. In J.t for the B-X transition lies along the vector r Cl-Cl . no t at1on, 45 axis). t he Using spherical tensor operator z-projection o f J.t is given by 1 J.Lz = IJ.tl (4~/3)~ 2 n6:<¢.e.o) Y 1 q(~.~) q=-1 (A3) 109 1 are the Wigner functions and the YjO are the the DMO where spherical harmonics. combinations of I1.M.{a}> are The formed as linear the basis set described in Sec.-IV(B), 1Mp. a 1 • { a }~ (R)9 . IJ.M.{a}> = Opn n JO A A t. {R. r) (A4) O~O.j,n where 33 {A5) The same used for a 1 ·{a} Opn ~ n (R) both are the harmonic the X and elements oscillator the of B the {A2). the Wigner functions be shown that and the are states. The eigenvectors obtained in IV(B). substituting Eqs. After considerable effort, functions electronic the bound state calculation of Sec. We proceed by basis (A3) using spherical and (A5) into the properties of harmonics, it can 110 = IJ.LI2 [2~+1] S{J',{a'}:J.{a}) \ I I \L 0 n,n'aO'p'n'aOpn L (j0, 10 j'O) n,n j,j' X x {1-pi.pi_)2 l [{1+o 0 . 0 ){1+o 0 .. 0 )J-!I.! l [c<J0.1qiJ'O'><J0,1qiJ'O'> o~o.o·~o q J] 2 . ( A6) (rn,j'rn'lj''rn'') are + pi. ( -1) 1 < 1-0. 1 q I 1. 0' > < j -0. 1 q I j '0' > In [ 2j+1]!1.! 2j'+1 Eqn. (A6) expressions such Clebsh-Gordon coefficients and as 1 onrn are Kronecker deltas. The line strength, S, determines the selection rules and the relative intensities between levels. rise AJ basis j The functions in Sec. IV(B). J allowed for transitions to be non-zero give the optical transition. The 1 have couple only coefficients involving (j,0,10IJ' ,0) will factor which j = j'±1. Since, as discussed the wavefunctions contain only even or odd basis functions rule Ap . for Clebsh-Gordon = 0,±1. the The conditions for S to selection rules Non-zero of (parity pj = ±1). we have the selection = ±2. Assuming that the rotational population of HeC1 formed in the distribution, jet can be described the relative intensity of a as a 2 (X) Boltzmann line is given by I ( ] ' .{a'}:J.{a}) ~ w1 S(]' .{a'}:J.{a})•exp[-E(J.{a})/kT] (A7) 111 Where is the frequency, laser temperature, and k is T rotational the is the Boltzmann constant. 2. Technical Details of the Calculations Bound-state calculations were performed 1 more =0 bound-state spherical energy harmonics levels were two In the calculation of different sets of basis functions. the using of Tables included in and 5 the 6, angular basis set than in the 1 > 0 spectroscopy calculations such as The that in Tab 1 e 8. computational type of calculation are discussed aspects of in more detail this in Ref. 33. the 1 consisted of the In =0 calculations, lowest 17 harmonic for an equilibrium position of R 16 em 0 =0 -1 . Since spherical functions of Eq. 1 = 0, radial have 0 = 0, harmonics are needed in prediagonalizing at = 4.0 A for the The angular the angular first set functions = 4 A and a frequency of e must (A5). basis oscillator we first R the 20 0 basis part =0 of so that the was only angular formed by the potential spherical harmonics. The 12 lowest energy angularly diagonalized functions were then used in the diagonalization of the full Hamiltonian. Because of computer memory constraints, the 1 >0 bound-state calculations to obtain the spectroscopy used a smaller number of spherical harmonics in the angular basis 112 set . In this case, must be angular basis included. prediagonalizing independently the for all 0 highest of > 0 value The 6 functions these diagonalization of lol ~ 1 part of the potential of for the first =0 functions angular each with formed by first The angular basis was spherical harmonics. and functions 0, lowest energy 0 with energies lower 6 were retained for the full Hamiltonian . the than 12 the overall Energies are estimated to be converged to within a few thousandths of a wavenumber of the ground the energies of excited van der for rotational Waals vibrational level ; levels van der Waals levels are less well converged . In the Au = -1 dynamics calculations, continuum eigenfunctions corresponding to dissociated states with j up to 28 were included . 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Herzberg, MoLecuLar Spectra and MoLecuLar Structure (van Nostrand Reinhold, New York, 1945}, Vol . II, p. 55. 39 R . A. Az i z, P. W. R i 1 ey. U. Schleusener, G. Scoles, and U. 71, 2637 (1979). Buck, G. Valbusa, J. 40 Maneke, J. Chern. Phys. J . J. Valentini, D.P. Gerrity, D.L. Phillips, J . C. and K.D. Tabor, J. Chern. Phys. 86, 6745 (1987). 41 G.E . Ewing, J . Phys . Chern. 91, 4662 (1987). Nieh, 116 42 J . M. Hutson, C . J . Ashton, and R . J. LeRoy , j . Phys . Chern. 87 , 2713 (1983} . 43 R . Shinke, j . Chern. Phys. 85 , 5049 (1986}. 44 See , for example, R . W. Heather and j . C . Light, Phys. 78, 5513 (1983). 45 R.N . Zare , AnguLar press, Chap.VI. Momentum in Quantum j. Chern. Mechanics, in 117 Chapter 4 Product State Distributions for the Vibrational Predissociation of NeC1 2 t Part of this work has been published as a in J. Chern. Phys . 86, 1636 (1987) . t communication 118 I. INTRODUCTION coworkers, 1 Levy and studies of that the general predissociation dynamics explained in selection rules . have recently study of recent work terms 2 has of complexes, ICl, show 2 , that wide intramolecular redistribution prior evidence rotational in distributions. Because of ideal the energy candidates the vibrational dynamical simple and the gross are similar the of of the This dynamical the to product effects state such as energy vibrational 12 6-10 to complexes. 2 while rainbows group this techniques Br be can molecules 3- 5 and fragment surfaces for of . i on 10,11 d ecompos1t to structures 7 ' 9 potential al.. variety preliminary for molecules, these measurements a Waals der pump-probe molecules detailed distributions et Cl revealed these experimental relatively of laser rare-gas seminal features of Lester, used their van rare-gas established features in the and interference and population rotational the relative simplicity of the of they are studies of these fundamental molecular photofragmentation. molecules, theoretical 13 In a previous communication 6 we reported preliminary results of measurements of the product-state distributions for the vibrational paper we present a predissociation full account of of NeC1 2 . In this these laser pump-probe 119 In measurements. prepares NeCl technique, laser level vibrational excited electronic state. stretching mode of the well energy the Cl-Cl molecule is Ne•••Cl van der Waals potential. greatly stretch to molecule as soc ia ted leads pump quasi bound B 2 a in a 2 long-lived in this the in low excess of frequency with to decomposition. the quasibound depth of the the Cl-Cl modes motion 2 the vibrational Coupling of Ne•••Cl of The vibrational the pulse of the eventually The van der Waals well depth is so small that the transfer of even one vibrational quantum is sufficient because of to the break large the Ne•••Cl frequency mismatch between covalent and van der Waals vibrational modes, is inefficient so that the However. bond. 2 vibrational the this process predissociation rate is relatively slow. The spectroscopy of the B(u' )-X(u' ') transitions of has been Homogeneous previously linewidth predissociation state function a vibrational =9 quanta, in the B electronic Cl-Cl stretching of u • and ranges from u electronic state of dynamics potential energy surface. the lifetime number for geometry . that the 33±2 ps decomposition show of to u' detail. 14 - 16 in measurements vibrational is studied = 13 . occurs NeCl The lifetime of -305 J.LS; 17 thus. on 2 258±42 ps a single for the B the electronic has a T-shaped effective In the ground (X) electronic state the Ne atom 120 A from the Cl 2 center-of-mass . lies 3.57±0 . 04 Recently, and we presented a theoretical dynamics of revealed v~T study of the comprehensive experimental vibrational predissociation 9 the closely related HeC1 that the decomposition system. That work 2 of HeCl is primarily a 2 process and is governed by a dynamical selection rule resulting from the potential. which allows parity with only respect fragment bimodal for symmetry to initial the population fragment. rotational the energy levels states of of Cl der even a toms same parity. was terminated forbidden by 2 . or odd decay to measured below well conservation As expected, NeC1 same symmetry selection rule as HeC1 Waals An unusual distribution which energy and angular momentum . van the interchange states having rotational of 2 obeys the The larger mass of the Ne atom leads to greater rotational excitation of Cl 2 product; with conservation populated all in fragment of the distributions show rotational dissociation. The structures is that, independent level and from the symmetry and observed the initially thus are independent those these studies conservation excited of to . rotational of can be rotational similar the most surprising result of aside constraints, 2 the consistent momentum observed in the decomposition of HeC1 Perhaps states energy and angular bimodal of the of energy distributions NeC1 2 amount are vibrational of energy 121 released to is different quite simple, the product degrees from what of This freedom. would be result predicted by a impulsive model of vibrational energy transfer. II. EXPERIMENTAL A. Pump-Probe Scheme The scheme for preparing the initial, excited quasibound NeC1 Cl 2 fragment technique 2 following we have molecules and decomposition described in for is detai 1 vibrationally detecting analogous the to 9 elsewhere. a The scheme is outlined as follows: NeC1 (X,u' '=0.]' ') + nw1 2 NeC1 (B.u' ,]') 2 ~ pump (1) Ne + Cl (B,u'-z,j) 2 vibr. pred. (2) Cl (E,u,j±1) 2 probe (3) detection (4) Cl (B,u'-z,j) + nw2 2 Cl (E) 2 ~ ~ respectively. In Eq. quasi bound of w . 1 B(u')~X(u' '=0) band. laser is state, 19 Eq. Cl 2 (E) of positioned pu 1 se NeC1 2 of correlate to the uncomplexed (1). a pump laser pulse prepares the level frequency, probe states states 18 and a NeC1 (B,u' ,]') 2 Cl (B) + nw 2 fluor The X and B electronic initial, ~ NeC1 (B). 2 on the The maximum pump laser of NeC1 a Following decomposition in Eq. excites (3). the Finally, is detected as Cl (B) 2 fragment to 2 (2). the the prompt fluorescence the experimental signal. By 122 w , 2 tuning the probe laser frequency, Cl 2 E+--B transition, the through the fragment population distribution of the products is measured. In the previous position w levels the of smaller to 1 initial in positioning w the cleanly rotational congestion band . study of HeCl it was possible of NeC1 single, The measured 2 isolated fragment state . leads bands to rotational quasibound vibrational B(u ' )+--X(u ' '=0) on a 1 , individual prepare constants the 2 to that The spectral precludes rotational line in population distribution of the Cl {B) fragment thus represents an average over the 2 set of initial by the w rotational levels of NeC1 (B,u ' ) 2 prepared pulse. 1 B. Apparatus NeC1 pulsed, van 2 free passing Ne and der jet held at -77 C at depending upon molecules expansion He 0 Waals in a {General pulsed Valve, pulsed nozzle was used. system, 14 optics, and nozzle 300 ~m ~m formed in prepared a by ratio over liquid Cl from 150 to 450 psi. Either source. 2 a nozzle diameter) or a nozzle diameter) type The vacuum apparatus and pumping laser detail elsewhere . Briefly, 201MSC) pumped laser dye mixture ranging piezo-ceramic (Lasertechnics , 150 solenoid a 9 : 1 Ne:He pressures the of were se t up 9. 10 are described an excimer (Lambda (Lambda Physik 2002£) in Physik provided 123 tunable pump (w ) pulses, with a 0 . 2 cm-l bandwidth over a 1 wavelength range centered about 500 nm. An angle-tuned etalon was sometimes installed in this dye laser to attain 0.04 cm-l resolution. 2002), pumped doubled to by the provide wavelength range followed the width of the A second dye laser (Lambda Physik same laser, was excimer 0.02 em centered pulse pulses by are -1 probe about ~10 pulses 250 nm. ns; frequency w the temporal so that wider two pulses are somewhat overlapped in time. a The however, significantly over 2 pulse the The two laser pulses travelled collinearly through the vacuum apparatus. The intersection volume of the laser and nozzle pulses was defined by lens a photomultiplier eliminate detector. scattered fluorescence. fluorescence by a gated wl of Cl (E) 2 and Optical fi 1 ters laser light dye laser pulse. pulses onto a were used to and visible resulting were preamplified and integrator. the energy of imaged slits Photomultiplier UV monitored and A the second gated the recorded integrator second harmonic of Laser tuning, from the probe sychronization, and signal averaging were computer-controlled . II. RESULTS 1 shows fragment the probe resulting laser spectrum of the from decomposition of 124 --0 .(J) c ~ c Q) u c Q) u (J) ~ 0 ::J [L > :::) 38670 38730 Probe Energy/em- 38790 1 FIG. 1. Laser pump-probe spectrum obtained when w 1 is tuned to the maximum intensity position on the Ne:l 5 Cl 2 E+--B {11+--0} transition and w 2 is scanned through the :l 6 Cl 2 E+--B 2~ and 3+--10 bands. The relative populations of the two vibrational channels are obtained from the integrated intensities of the two bands (after a small correction for the Franck-Condon factors) . 125 Ne 35 cl {B,u'=11) 2 (u' is the number of vibrational quanta in the Cl-Cl stretch and the complex has no excitation of the vibrational modes involving Ne•••Cl 1, the pump frequency, of the Ne 35 the c1 35 c1 B~x. 2 E~B 2 w . 1 In Fig . motion). 2 was positioned on the maximum 11~ band, and w2 was tuned through transition. the Comparing relative intensity of the feature due to the 35 c1 {B) fragment with 2 10 35 vibrational Cl 2 (B,u=9) quanta fragment, remaining it is to apparent decomposition mechanism for NeCl is 2 that that the dominant the loss of a Cl 2 vibrational quantum {the Au= -1 channel). no measurable population signal of the for the Au = -3 35 Cl (B,u=8) 2 channel is the of single There was fragment : negligible . the Using Eqn . {A1) of Chap . 5 to correct the integrated intensities of the fragment Cl 2 E~B. 20 Franck-Condon factors, 3~10 2~9 and bands for the the Au = -2 channel accounts for 6 % of the total population. E~B . Figs . 2 and 3 3~10 and 2~9 bands, rotational structure of branches. The R the 35 c1 (B) fragment 2 respectively, of Fig. 1. The show in detail these branch bands forms a consists bandhead of to and R the P blue , whereas the P branch extends to the red so that the higher j rotational resolved. lines (j fragment Cl spectra lies 2 .) in is of the P branch rotational Although the the most rotational completely number of the the intensity in both lines resulting from Cl of quantum are 2 126 -0 .(f) cQ) -+-' c Q) 0 cQ) 0 (f) ~ 0 ::J [L X10 > ~ 38725 38735 Probe Energy/em- 38745 1 FIG . 2. Signal-averaged spectrum of the fragment 36 Cl 2 E+-B {3+-10) band of Fig . 1 when w 1 is positioned on the 36 Ne Cl 2 B+-X {11+-0} band. The fragment rotational population distribution of the Au = - 1 channel is extracted from this spectrum. 127 ~ . (f) c ~ c G) u c X10 Q) u (f) ~ 0 ::::J [L > ~ 38620 38630 38640 Probe Energy/em- 38650 1 FIG. 3. Signal-averaged spectrum of the fragment 36 Cl 2 E+--B {2+--9) band of Fig . 1 when w 1 is positioned on the 35 Ne Cl 2 B+--X {11~) band. The fragment rotational population distribution of the Av = -2 channel is extracted from this spectrum. 128 fragments in the lower they differ dramatically in the intensities of the high j lines of Fig. 3, the Cl the P(24) fragment 2 however, decreases (j In Fig. 2 there is an abrupt termination the P-branch between spectrum of levels ~ 12). of the P branch. rotational j the gradually lines in the Au= -1 intensity to and P(25) of the unobservable P in the channel. In branch levels lines around the P(30) line. In Fig . 2 (the the P = -1 Au the channe 1) . = 24 termination of restrictions imposed by the conservation of energy; is a limited vibrational products. branch at amount quantum of for In contrast, the P branch kinematics of would the energy in Fig . is available rotational the gradual decline in the of j 3 result of there in the Cl excitation of the 2 (the Au= -2 channel), intensity of appear the abrupt to be decomposition; the high j lines the result of the there still remains plenty of energy to populate yet higher rotational levels . Cl and 3 was NeC1 2 fragment were initial w1 2 spectra analogous recorded quasibound tuned to B~x. u·~ Fig. 4 distributions for the to those decomposition levels of NeC1 (B,u'). 2 the maximum intensity in Figs. of 2 several In each case, position of the band. shows fragment extracted from rotational the line state population intensities in the 129 36 FIG. 4. Cl 2 (B,u'-z) fragment rotational population distributions for the vibrational predissociation of 36 Ne Cl 2 (B,u'). z is the number of Cl 2 vibrational quanta transferred in the dissociation. Since the parity of the initial level is conserved in the dynamics, the populations of the even j fragment levels (filled circles) and odd j levels (open circles) are each independently normalized to 100 %. (a) u' = 6, z = 1: (b) u' = 7, z = 1; (c) u' = 8, z = 1; (d) u' = 9, z = 1: (e) u' = 11, z = 1; (f) u' = 11, z = 2: (g) u' = 12, z = 1: (h) u' = 13, z = 1; (i) u' = 13, z = 2 130 (a) 6--+5 0 • 0 • 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Fragment Cl2 Rotational State, j • (b) 7--+6 0 • 0 • 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Fragment Ck Rotational State, j ~ 0 ~ c~ :8~ _Q •0 :::JN &; ceo 0 (c) 8--+7 • 0 Q) ~tO &.-or • 0 0 2 4 6 8 10 12 14 16 18 202224262830 Fragment C~ Rotational State, j 131 0 N {d) 9---+8 • 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30. Fragment Cl2 Rotational State, j 0 •0 •0 11---+10 (e) • 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Fragment Ck Rotational State, j ~ SQ c ~ .Q :t =§_~ &~ 0 0 • 0 ••• 0 {f) • 0 • 11---+9 0 • 0 2 4 6 8 1012141618202224262830 Fragment q .Rotationd State, j 132 0 N ~ • c ~ ........... Q ~ 0 0 • 0 :=lN •0 0 (g) • o_..- 0 &~ c • CX) 0 •0 Q) ~ cf 12~11 '¢ N 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Fragment Cl2 Rotational State, j 0 • •o 0 (h) 13~12 • 0 • 0 • 0 •o. 0 • 0 0 2 4 6 8 1012141618202224262830 Fragment Cl2 Rotational State, j • • 0 (i) • 13~11 0 •oe eo o • 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Fragment Cl2 Rotational State, j 133 experimentally simulation. observed Cl The details of described elsewhere. high branch P j, simulation of In lines overlap the computer simulation procedure are were individually band head congested this manner the low region of j resolved, fit prior to region of the the R branch the intensities of the accounted for while extracting the low j low j by In the computer fitting, lines the spectra. that 9 spectra E~B 2 branch are P populations. populations extracted from the spectra exhibit large variations for two uncertainty is larger result of reasons. for the First, the low populations j the spectral congestion in the probe laser spectrum, the experimental low j populations are the specific rotational levels 21 If coupling can theoretical be sensitive of the complex. neglected study of NeC1 to as of Halberstadt, 2 the low j the initial coriolis suggested is J ) 0 levels of the complex that have a Ne-Cl a so the intensities of individual Second, very as region of lines cannot be measured as accurately. the The et by a l. • 21 the then ]-projection onto bond axis of 0 cannot decay to fragment rotational levels with j < 0. J the total angular momentum of the result, low j where 2 the the pump initial quasibound distribution laser is is is level of NeCl sensitive positioned on 2 . As a to precisely the NeC1 2 band contour. In Fig. 4 the odd j and even j rotational 134 independently normalized to 100 % because populations are they undergo independent showed initial that dynamics. rotational In Refs. levels of 7 and 9 we HeC1 (B,u') can 2 decay only to Cl (B) fragment rotational 2 levels having the same a parity as the initial symmetry of the He·· •Cl Precisely the same effect level, van 2 is as result observed for the potential . Waals der of NeC1 2 The . parity of the initially prepared quasibound level of NeC1 is given by the behavior upon exchange of the Cl Cl (B) atoms. rotational w is positioned on 11~ band. The 1 B~x. wavefunction The parity of j the final even and odd E~B. 3~10 band of 35 c1 37 c1 Fig . 5 shows the odd. j when the rotational wavefunction is even for 2 for of 2 the maximum weak of the even/odd Ne 35 c1 37 Cl intensity j alternation observed in the resolved P branch is a result of larger the w number of 1 rotational pointed frequency being positioned such molecules are even-parity levels out conservation 35 c1 37 c1 inversion potential. that is prepared than odd-parity for Ne not completely center-of-mass center of However, 35 c1 37 c1 does the in not Ne••• the in that It levels. the initial should dynamical rigorous lie a parity because precisely 35 c1 37 c1 van decomposition of the at der He be 35 the Waals c1 37 c1 (for which it is possible to prepare individual rotational levels of definite parity), there was no experimentally measurable violation of parity conservation and the same 135 ~ ·cn c ~ c (1) u c Q) u (J) ~ 0 :::; [L > :::::) 38730 38740 38750 38760 Probe Energy/em_, FIG. 5. Laser pump-probe spectrum of the fragment :~s C l37Cl E+--B (3+--10) band obtained when w 1 is positioned on the maximum of the Ne 36 Cl 37 Cl B+--X (11+--0) band. The weak even/odd alternation is a result of w 1 ~reparing a larger 37 6 fraction of even parity levels of Ne Cl Cl than odd parity levels. 136 is likely to be true for NeCl The 35 c1 more dramatic 2 . even/odd alternation j in the E~B bands for the decomposition of Ne 35 c1 2 is the 2 result of nuclear exchange symmetry. species exists degeneracy should in ortho and ratio . Since the not dynamical parity of result change the the forms para vibrational spin nuclear conservation symmetry of that is the have predissociation state of mandatory Ne 35 c1 3 :5 a van 2 both as a der Waals potential and nuclear exchange symmetry. As a result possible to obtain consist of either even of B~X or detect individual an the fragment Cl Fig . 2 B~X 2 to odd spectra by line is . which levels of parity-selected positioning in Spectra of this spectra, the obtained in one-laser LIF the "normal" Fig. rotational 2 it the w 2 P-branch to of transition and tuning w through the 1 By in phenomenon, These rotational absorption. sh6wn state recorded 6. Also B parity . are E~B adding symmetry excitation spectra of NeCl transitions excitation NeC1 this 6 even- type are shown in and odd-parity excitation excitation spectrum previously studies of NeCl are calculated 2 is recovered . asymmetric top rotational spectra for transitions to either even- or oddparity levels. The structural parameters of Ref. used in the asymmetric top calculations . 16 were 137 . ,.,. .....- .... .. . ..-·-- ' - -0 .(f) c 2c Q) u c (a) Q) u (f) ~ 0 ::::::s [L > ::J I -~ ~ (b) ·.- ~~• 19357 19358 19359 19360 Pump Energy/em_, FIG . 6. Solid curves are parity selected excitation spectra of Ne 36 Cl 2 obtained by scanning w 1 through the Ne 36 Cl 2 B-X (B+-0} band and (a) positioning w 2 on the P(20) line of the 36 Cl 2 E-B (0-7) transition, or (b) positioning w 2 on the P(19} line . Dotted curves are calculated asymmetric top spectra for transitions to only (al evenor (b) odd- parity rotational states of Ne 6 Cl 2 (B,u'=B). The rigid, T-shaped model has Ne to Cl 2 center-of-mass distances of 3.57 and 3.51 A in the X and B electronic states, respectively. The computed lines are convoluted with the experimental lineshape, and the Boltzmann temperature is 1.0 K. 138 IV. DISCUSSION The decomposition channel terminated at a contrast, the for the Au = -1 NeC1 (B.9LU'L13) are abruptly distributions rotational of 2 well-defined rotational rotational distributions . for 1 max the 9 ~ ~ u' all levels can be used 13 In max· Au = -2 ~ 8 The abrupt termination appear to decay gradually to zero. t for J. = -1 channels for u' decomposition channels and the Au a level, to obtain the van der Waals bond energy in the B electronic state, We first rotationless. for the consider ] = 0. moment the Au = -1 of NeC1 (B. u' =12). level the diagram energy of the the state and fragment in Fig. initial Cl (B.u=11) 2 the of Assuming 2 that represents channel 7 accurately NeC1 (B.u'=12.]=0) 2 rotational levels, we see that it is not energetically possible to populate fragment Cl 2 (B,u=1l.j) of angular possible Cl levels with momentum still fragment 2 ]. is angular rotational and the fragments. curves for the t is further decreases the orbital Fig. decomposition of than level. momentum the the reaction coordinate, the Ne atom conservation Since is zero. we must have j fragmentation. recoiling Imposing rotational angular momentum, the > 24. j the Cl R, 2 8 of the the Cl 2 following momentum potential NeC1 (B,u'=12,]=0) which is to tal = -t, where j angular shows maximum 2 of energy along the distance between center-of-mass. The isotropic, 139 Cl 2 (8 v=121 j=O) I -- Nee 1 2 ( 8 1v=12 j=25 ~ .. --------------------- --..:..::..=··-r=~"----- --~~ • I J =0 ) - j=23 j =2 2 E avail .,,. .,, Cl 2 (8~v=111 j=O) FIG. 7. This energy level diagram illustrates the energetic constraints in the decomposition of NeC1 2 {B.u'=l2). The energy available for rotational and translational kinetic energy of the fragments is 1 E ava "1. = <.~{Cl 2 ) - D0 = 77 cm- • <.~{Cl 2 ) is the energy in .. the Cl 2 vibrational quantum f131 cm- 1 ) and D 0 is the binding energy of NeC1 2 (54 em- ). Only rotational levels of Cl 2 {B.u=ll) with j ~ 24 can be populated energetically; conservation of angular momentum imposes further constraints as described in the text. 140 FIG. 8. Schematic diagram illustrating the effect of the centrifugal barrier in the vibrational predissociation of NeC1 2 (B,v'=12). The centrifugal barriers in the effective radial potentials for fragment levels with j > 22 lie above the energy of the prepared state. This prevents population of the j = 23 and 24 fragment levels , even though they are energetically allowed, as shown in Fig . 7 , 3 NeC1 2 (B, u=l2, 1=0)----+···· 4 R (Ne•••Cl 2 )t.l. Erot ----+ ~1 ~1 5 6 -----V~ff{u=ll,L;,o) V~ff(u=l2,l::O) 2 (B,u=ll,J=0) 2 {B,u=ll,J=22) ..... ,.j). ..... 142 long-range part of the NeC1 potential is given by 2 22 (5) V(R) Where C 6 has the value 1 . 33x1Q centrifugal potential due gives the effective to 5 em -1 0 ,6 A Addition of . the the orbital angular momentum potential along reaction the coordinate, (6) where J.l is the reduced mass given by mNemC12/(mNe+mC12). The height of the centrifugal barrier is then V barrier It can (1..) -- (54 c )-~ 6 be centrifugal seen in barrier energetically (7) Fig. 8 prevents allowed limits on the value of energy that the population levels. presence of the Upper of the highest and lower n0 (B) can now be obtained from the experimental determination of jmax and the known energy of the Cl 2 vibrational w(12) = E[Cl (B,u=12 , j=0)] 2 bracketed by quantum given by 143 w( 12 ) - Erot(u= 11 ·imax+ 1 } - Vbarrier(jmax+ 1 } ~ DO(B) ~ w( 12 ) - E rot (u ' J.) is Erot(B.u= 11 ·imax) - Vbarrier(jmax). rotational of the energy the Cl 2 (B,u,j) where B are u and D u the spectroscopic rotational and distortion constants Table the shows 1 the results these of By considering the range of n (B) obtained 0 calculations. in {S) analysis of several rotational bands, D (B) 0 is 1 bracketed to lie between 52.0 and 56.0 cm- . In Table 1 note that for the u' chosen as even 23. e~perimentally. In though Fig. 2 j it = 11 state, j = 24 was observed seen that can be population of j = 24 is substantially less than j if the j = 24 channel were only behavior has also been observed result of the fact that distribution rotational higher includes in the = 23, as open. This and observed rotational contributions levels the ArC1 10 ' 11 2 the complex with 1 levels of 1 rotational partly was max Some of of NeC1 (B,u'=11} 2 a initial from > 0. is have the energy such that they lie just above the threshold for production of the Cl {B,u=10,j=24) 2 of the comp 1 ex 1 i e fragments only temperature of be 1 ow with the fragment, j jet is whereas lower the threshold and < 24 . Since 1 levels can decay the to rotational 1.0 K (measured in the fits of 144 Table 1: Bracketing of the ·binding energy threshold in the decomposition of u jmax 13 12 11 9 21 22 23 25 c.u(u' )/em 120.29 131.02 141 . 89 163.77 52.0 ~ D (B) 0 ~ 56.0 em 57.7 ~ D (X) 0 ~ 61.7 em -1 -1 of NeC1 NeC1 2 by the 2 (B,9~v'~ll). -1 49 . 5 - 56.0 51.0- 58.0 51.9 - 59.5 52.0 - 60 . 6 jmax 145 Fig. 6), lying there more complex. lies is than This closer little population 2 em -1 to the the above suggests that the =0 1 value This range for n (B) 0 is revised from in earlier commun1cat1on chose j = 24 for max . preliminary . the dissociation of the of D (B) 0 -1 52.0-56.0 em range . our levels of level the of 1 initial actual bound lower of the estimate 6 in which we = 11 state the u' of NeC1 {B) . 2 The NeC1 B+--X, 2 the associated Cl binding energy than the in average 14 bands B+--X, 2 in the shift to D (B). anisotropy and state. 5.66 em are isotropic long-range part of of NeC1 (B) 2 For this manner was assumed 2 is for the n0 (X) ~ ~ the observed -1 calculation, 13 Both the of actual and the the van der Waals potential to be the same as for NeCl {X) . 2 determined be considered as an This estimate of somewhat smaller than 23 larger through 9 61 . 7 em parts this is energy should the in state binding measurement. NeC1 neglected reason -1 from This means that the By adding short-range potential blue-shifted electronic we have 57 . 7 0 the ground of are bands . u'.-...Q B electronic b 1 ue bands u'.-...Q only the binding the in this approximate energy measured for binding . 24 which have values of D (X) 0 2 1 1 between 68.5-72 . 6 cm- and 72.4-74 . 7 cm- . respectively. energies of NeBr 2 and Nei The maximum classical photodissociation impact half-collision parameter, b, for is given the by 146 b = h(j rnax (j rnax +1)]~/~u. where u is the relative velocity of large R. the fragments at from the given translational by em -1 the dissociation of kinetic = w(u') - Etrans 2 2 rotational of fragments, Assuming = 22 jrnax gives a maximum distributions the complex for u ~ initial determined by constraints which prematurely 8. in the classical > 8. these other level, 4 are of the u', levels appears than to be conservation terminates of rotational the These constraints do not appear to be simply kinematic in nature; for Fig. in vibrational for jrnax distributions for u' picture the . of classical is obtained A, much larger than the molecular independent energy, of u Erot(jrnax) -DO. NeC1 (B.u'=12) dimensions of NeC1 energy observation impact parameter of 6.4 The The value of in a simple, the dissociation, impulsive, one would expect more rotational excitation of the fragments in the Au = -2 = -1 the much half-collision. No channel than in larger amount the of Au energy population of rotational channel in the because levels with j of > 30 was observed, regardless of the energy of the half-collision. Tables 2 and 3 present a the Cl 2 fragment experimentally appearing in rotational measured rotation, summary of the analysis of distributions spectra. <E ro t >, for The average for the all the energy Au = -2 distributions is not significantly different than that for I Eauai.t Probe Trnsl v(E)+-v(B) em -1 0+-5 142 . 1 120 . 2 112.0 I E(jmax) -1 em em -1 18 . 0 17 . 5 - <Erot> + 6 I I I - 0+-6 131.5 101.7 109.5 15 . 5 16 . 1 + 1 - 2 15.9 15 . 7 - 13.2 13.2 + 12 (B,u'), Eauait 14.9 15 . 4 + 11 (B) state: u u' parity 9 + Cl 2 10 . 8 10.8 + 13 I I 0+-7 120.7 I I 2+-9 109 . 76 I I 3+-11 87 . 8 I I 4+-12 77.2 I I 5+-12 66 . 3 I 99.392 . 0 I 82.889 . 7 I 78.271.9 I 63 . 698 . 6 I 51. 561 . 3 13.9 14.6 + 8 35 = 54 em -1 . Prepared Ne is calculated assuming D (B) 0 of the Au= -1 channel for decomposition of NeC1 Summary of the fragment rotational state population distributions Table 2 : --l ,l>. 1-' 148 Table 3: Summary of · the distributions of NeCl 2 fragment rotational the Au = -2 channel (B,u ' ), = 54 em . Prepared Ne 35 c1 2 (B) state: v v parity 11 13 + <E ro t)/cm -1 Eava i.l./cm -1 Probe Trans. u(E)+-u(B) population for decomposition of calculated is -1 state + 16.7 16.8 16.6 16 . 6 124 . 8 116 . 7 117.3 109.8 240.6 197 . 3 2+-9 4+-11 assuming 149 the ~ v = -1 Au 8, large the despite distributions. change in EavaiL . The Au = -1 rotational bimodal for v' ~ = 20 and 21 is j termination of For 8. distributions are distinctly v ) apparently the 8, obscured the Au by distribution the Au = -2 is at energetic The bimodal to one observed in the predissociation of HeC1 = -2 maximum the the rotational distributions . rotational distribution is similar vibrational second also However. . 2 bimodal. in HeC1 whereas 2 for rotational population decreases kinematic limit rotational monotonically with j. Apparently. excitation in that for HeC1 the the dissociation of NeC1 2 is = j 14. for is 2 Assuming = 30, j that the while maximal impact parameters (the origin of the kinematic constraint) for the · He- and Ne-Cl the classical predicts half-collisions are 2 relation that the of ( ll(Ne) l/ll(He) I limits should be given by reduced masses collision energies. the then = JJ.Ub = b(2JJ.E avat..l.)!h correctly ll.l ratio similar, the excitation = [30(30+_1)/14(14+1)]!h = 2.1) square root ([JJ.(Ne)/JJ.(He)J E avaiL' rotational !h are of the of provided = 2 . 0). the ratio same . However. the the as mentioned above, this very simple picture does not account correctly the for excitation toE insensitivity of the rotational ~ l. · avac.. Halberstadt, et al. . 21 have performed 3-dimensional 150 quantum mechanical on calculations a potential model surface for NeC1 (B,v'=ll) and obtained product rotational 2 distributions calculated using the golden distributions rule showed approximation. The significantly less rotational excitation than is observed experimentally, and there was no distribution. evidence for peak second a the in This is probably due to deficiencies in the potential surface used . An analogous calculation for HeC1 on 2 a potential surface optimized to reproduce experimental data gave good agreement with experimental the distribution, and a should bring the full with experiment. HeC1 2 study the origin refined of this potential it was not simple physical phenomenon. The rotational surface quantum calculation However, to draw a bimodal into for NeC1 2 agreement possible in the picture to explain results of such a large basis set diagonalization calculation are difficult to interpret . The dynamics of NeC1 is rationalized not Here we easily explore some in possible 2 , terms 1 ike that of HeC1 of a 2 , simple model. approaches to future performed close-coupled the further analysis of this data. Hutson, calculations et for the al.. 25 vibrational predissociation that predicted a bimodal distribution of They speculated that from oscillatory the bimodal structure ArHD the HD fragment. distribution associated of with a originates rotational 151 rainbow singularity in rotational classical the distribution. Interference patterns arise because of superposition of initial angular rise to the The position distribution energy , trajectories originating configurations of the same final rotational of the minimum appears whereas to the be of of the that give the products . second independent position different molecule level and from the peak of the of our available classical rainbow feature would be expected to scale with approximately the square root E of a.ua.il." 26 The phenomenon has been used successfully to model distributions for a particularly who distr i bution an of rotational variety of photodissociations relevant reported rainbow rotational result is that unusual, NO of Sato, "inverted" fragments 27 One a. L • , et 28 rotational the from direct photodissociation (not vibrational predissociation) of the Ne- and Ar-NO van der Waals molecules . In that study, the position of the maximum in the rotational distribution was found to be proportional half-collision energy and mass , compe l ling rotational of rainbow to the the both square root evidence for effect . Because the rotational distribution of not possible to vibrational predissociation square the of the reduced presence of NeC1 of root 2 the of the the insensitivity to Ea.ua.il.' it is identify the rotational distributions for patterns associated with a with quantum rotational rainbow . interference 152 Another method distributions mo 1 ecu 1 es In this state In is is of the the wavefunction in of terms the of a the weighting of bending that the avai lab 1 e a free that rotational distribution energy the model sophisticated 29 to NeC1 application would 2 observed distributions: be of function is is in model independent in agreement It may be that a Franck-Condon the successful such work . level this products, set . predicts rotational Thus, 2 initial basis model rotor basis wavefunction. for the rotor fragment with the experimental results for NeC1 more of Franck-Condon given triatomics of . . 21 su dd en approx1mat1on. bending form, population initial predicts photodissociation F ranc k - C on d on or simplest the the decomposed is given by the the picture its that for rotational predicting for in describing beyond the the scope of the present paper. In conclusion, rotational and vibrational state distributions are measured in this work fragment of several initial vibrational of one modes. the vibrational channel Cl-Cl The abruptly the vibrational rotational terminated constraints. production for In of these the for predissociation of quasi bound levels. is the van distributions are, in because energy cases, highest of the measured allowed 2 from 2 dominant the to the Cl NeC1 The dissociation quantum product- transfer der Waals some cases, conservation thresholds rotational level for are 153 used to estimate the binding between 57.7 and 61 . 7 cmstates for dynamics, j of which 30, constraints regardless and fragments. Rotational do It the not to affect available excitation of distributions seems the observed distributions the of rotational are bimodal. for NeC1 2 (X) lie For dissociation from initial energy translational model of the the rotational distributions extend to a maximum about channel 1 energy in decomposition; terms it is the for impossible of a energy recoiling the to for Au = -1 rationalize simple physical clear that further theoretical work is required to understand the dynamics of this molecule . V. ACKNOWLEDGEMENTS We thank Nadine Halberstadt for helpful discussions and gratefully acknowledge the National Science Foundation for the financial support of this research . 154 REFERENCES 1 2 3 4 5 6 7 8 9 D.H . Levy, Adv. Chern. Phys. 47, 323 (1981). G. E . Ewing, J . Phys. Chern. 91, 4662 (1987). J . C . Drobits, J . M. Skene, and M. I. Lester, J. Chern. Phys . 84, 2896 (1986). J . M. Skene, J.C. Drobits, and M.I. Lester, J. Chern. Phys . 85 . 2329 ( 1986) . J.C . Drobits (1981). and M. I. Lester, J . Chern. Phys. J . I. Cline, N. Sivakumar , D.D . Evard, and K . C . Chern . Phys . 86 , 1636 (1987) . J.I. Cline, N. Sivakumar, D. D. Phys . Rev. A 36 , 1944 ( 1987) . 74, 163 Janda, J. Evard , and K.C. Janda, N . Sivakumar, D.D . Evard, J.I. Cl i ne, Chern . Phys . Lett . 137, 403 (1987) . and K.C . Janda, J.I. Cline, B.P. Reid, D.D. Evard, N. Sivakumar , N. Halberstadt, and K. C . Janda, manuscript in preparation; Chap. 3 of this thesis. 10 D . D. E var d , Ph .D. Technology, 1988 . 11 D.D . Evard, preparation. J . I. Thesis, Cline, and California Institute of K.C. manuscript in Janda, 12 J . C. Drobits , J . M. Skene, and M.I. Lester, poster presentation at the 1987 Conference on the Dynamics of Molecular Collisions, July, 1987, Wheeling, WY. 13 N. Halberstadt and J.A. Soc . 73, 357 (1982) . Beswick, Faraday Discuss. Chern. 155 1 4 D . E . Brinza. C . M. Western . D . D. Evard, F. Thommen, B . A. Swartz, and K.C. J anda, J . Phys. Chern. 88, 2004 (1984). 15 D.D . Evard. F. Thommen, 84 , 3630 (1986) . and K.C. 16 D.D . Evard, F . Thommen, J . I . Phys. Chern. 91, 2508 (1987) . Janda, Cline, J. Chern. a nd K . C. Phys. Janda, 17 M. A . A . Clyne and E . Martinez. J . C.S. Faraday II 76, ( 1980) . 18 Ishiwata, T. Shinzawa , T. Kusayanagi, J . Chern. Phys . 82, 1788 (1985) . °Franck-Condon factors for the E-B and I. Tanaka. transition of were calculated using the RKR potentials of Refs . 19. 21 N. Halberstadt, J . A . Beswick , Phys . 87. 3966 (1987) . 22 1275 J . A . Coxon. J . Mol . Spectrosc . 82 , 264 (1980). 19 T. 2 J . M. J. O'Loughlin, Technology, 1986 . Ph . D . Thesis, and K . C . Janda, California J. 35 c1 2 18 and Chern . Institute of 23 J.I. Cline, D.D. Evard. B . P . Reid, N. Sivakumar, F. Thommen, and K . C . Janda, in Structure and Dynamics of WeakLy Bound MoLecuLar Complexes, A. Weber, ed . (D . Reidel Publishing Co . . Dordrecht, 1987). p . 533. 24 J.A. Blazy. B . M. DeKoven, T . D. J. Chern . Phys . 72 . 2439 (1980) . Russell, and D.H. Levy , 25 J . M. Hutson. C . J . Ashton , and R . J . LeRoy, J . Phys . Chern . 87, 2713 (1983). 26 27 28 R. Shinke and V . Engel, J. Chern. Phys. 83, 5068 (1985). R . Shinke, J. Chern. Phys . 85, 5049 (1986) . K. Sato , Y . Achiba, H . Nakamura. and K. Kimura, J. Chern. Phys. 85, 1418 (1986) . 156 29 R.W. Heather (1983). and J.C. Light, J. Chern. Phys. 78, 5513 157 Chapter 5 Vibrational Predissociation Dynamics of the Ne•••Br 2 van der Waals Molecule tThis work was published in a Structure CompLexes, Dordrech t, and Dynam t c s A . Weber, ed. 1987) . p. 533. t slightly different of (D . Vfeak.l.y Reidel Bound form in .Mol.ecul.ar Publishing Co .. 158 I. INTRODUCTION Levy and coworkers molecular dynamics of their Nei Hei 2 the vibrational and Ari in the 2 results 2 because have theory perform. 2 It was of for that found ~u a excited deal great mo 1 ecu 1 e is the of levels state. of up Hei . 2 of 1 The attention relatively that vibrational vibrational dominated by the predissociation B electronic received for study the the rare-gas halogen complexes with on work . initiated easy to predissociation to u = 27. is = -1 vibrational channel, and that the vibrational predissociation lifetime of the complex ranged from 221 ps for = 12 u' to 38 ps for u = 27 . The mechanism of this decomposition has been interpreted as a nearly V-T pure type process decomposition rate upon and the dependence of the number of vibrational quanta initially in the I-I stretch in terms of a energy gap momentum 3 4 gap • energy-gap law model. is at An early indication the 2 that or the least qualitatively accurate over a very large range of vibrational predissociation rates was the (T observation > 10- 5 s) of quasi bound vibrational for the ground electronic state of NeC1 levels 2 .5 The V-T mechanism for vibrational predissociation of Hei 2 and Nei 2 was supported by dispersed experiments, which revealed the product I rotationally cold. For there 2 fluorescence molecules to be is considerable 159 rotational energy the ability of Ari mechanism, the in products . 1 This to dissociate via a Au 2 allowing the first two may be due to = -3 sequential quanta energy of r 2 stretch to be randomized over the van released from the der Waals modes before the third quantum dissociates the molecule. In this experiment we show that dynamics of NeBr is different from predissociation too 2 not expected on basis of the However, measurements of vibrational dynamics for rare-gas related what Hei that rotational excitation of have that and NeiCl have consistent with an shown distributions some HeiCl cases the there significant is a configuration. two molecules it with same the results mechanism that that consistent applies to Hei 2 . Nei the bent structures of HeiCl probability to a are of large peak in the relatively large laboratory 7 ' 8 ICl 2 . and Y-T.R geometry appears that impulsive NeBr . 2 and NeiCl 6 rotational molecular the al.. et impulsive from these have systems predissociation mechanism For have predissociation product bent Y-T results. 2 Skene, products. a might Nei halogen indicated in one and 2 vibrational primarily process, the the except increase the classical impact paramters, fragment rotational distribution at values indicate of j. Results that the mechanism leading from for our NeC1 2 vibrational predissociation is also different from that of 160 NeBr NeC1 2 . The (B,v' 2 there is primary = 11) vibrational significantly more would have been predicted collision mechanism. such a model = -1 Av remains dissociation as for the basis predict of a the classical 2 ; for however, excitation Interpretation of would NeBr rotational on channel than simple halfresults using impact parameters > 6 A. The been spectroscopy previously of studied excitation spectrum of the in NeBr B+--X 2 A detail. the NeBr B+--X, 2 transition high resolution 10+--0 band has been analyzed to obtain the effective molecular geometry of complex. 9 The structure in molecule both was found electronic state and the two 3.65±0.01 bond bond excitation. two electronic fact involving modes the predissociation range from from by The very of the A 0.6 electronic lifetimes 355±100 ps to for to that Br 2 was the electronic potentials der for the by the shown combination bands Waals vibrational spectrum. NeBr Ne similarity of upon similar van a T-shaped considering intermolecular are a the A in the X electronic surprising, homogeneous respectively. 10 with searching revealed no excitation in obtained the states that careful is increases That have A in the B state. lengths length to states, center-of-mass distance of 3.67±0 . 01 has Vibrational (B,v') have been broadening measurements, which 1.7±0.3 ps for 2 v = 10 and 27, 161 Linebroadening molecule excitation decay rate of the energy spectra give is distributed electronically excited exploited well-characterized halogen information among Studies the 11 Waals der van about the the prepared state but do not describe how channels . halogens . in measurements By of rare-gas of the available product of predissociation the dispersing fragment the halogen complexes have B-X transitions of the B---+X fluorescence of the rotational and the dissociation. its vibrational quantum state distribution is obtained. we present data on Br 2 Here the product-state distribution of the fragment from the photodissociation of NeBr . 2 II. EXPERIMENTAL over A mixture a trap maintained grade, naturally reagent of 9:1 Ne : He {Spectra Gases) 0 at -40 C occurring NeBr 2 was prepared in a pulsed, mixture through a 500 100-140 psig. concentration atom. bandhead. 0.7 K of as 12 as The rotational lines. was clusters to containing (MCB , 2 abundances). can the be the than one Ne in LIF detected from temperature relative minimize more blue-shifted rotational by Br free jet expansion of this taken clusters features measured isotopic passed nozzle at stagnation pressures of Care Higher-order experiments ~m containing was of the NeBr the jet intensity of 2 was 162 The beam from an excimer pumped dye (Lambda laser bandwidth) crossed the (Lambda Physik EMG 201MSC) Physik FL 2002, 0.2 cm- 1 > 80 nozzle diameters expansion The fluorescence from emitting species in the jet was either dispersed or, in some cases, downstream of to tal the nozzle . f 1 uorescence was measured. The optical system for collecting total fluorescence from emitting species in the jet, the vacuum chamber, previously described. 13 tube used was to at A measure Franck-Condon factors emission and for wavelengths excitation wavelength; 14 pumping system fluorescence total therefore, redder was used to collected by an optical favor 2 than scattered laser was rejected by appropriate red-pass filters . monochrometer spectra. the B-X transition of Br significantly been photomultiplier S-20 cooled has disperse the light A Spex 1269 product emission system described elsewhere. tube measure dispersed fluorescence spectra. The preamplified integrators signals (Stanford used A red-sensitive RCA 31034 photomultiplier photomultiplier was 15 to were recorded by boxcar Research Systems). An IBM-PC microcomputer controlled the synchronization of the pulsed valve, lasers, and boxcar, scanned the dye laser and monochrometer, and performed signal averaging. III. DATA AND RESULTS Fig. 1 shows a to tal f 1 uorescence spectrum of the 163 79,81 Br2 I Total Fluorescence Spectrum Br2 (10- 0) Band .€ Cl ...."'u ..9 u 7llBr2 81Br2 v Cl u ..u v J 1 0 ::I ~ 1. 7 9 1.73 Total Fluorescence Spectrum Br2(21- 0) Band FIG. 1. Total fluorescence species in the jet. spectra of the Br 2 and NeBr2 164 jet obtained by excited Br species. band 2 have molecule. 12 Product NeBr 2 detecting all The weak been the fluorescence features assigned to In Fig. 1. 27+--0 transition to the the of each der Waals the b 1 ue NeBr van 2 the homogeneous broadening of state distributions for the Ne 79 Br 2 B~X. observable . clearly the photodissociation of the decomposition . Only the Ne 79 Br studied since it species in detail isotopic The experiment is to '=0 van der Waals u·~u· and recording the emission spectrum of the Br in the is were obtained by tuning the excitation laser maximum of other from band fragment of 2 isotopic species was 2 best the the separated from the spectrum . excitation in the excitation (1) vibrational predissociation (2) product fluorescence (3) can be best described following steps: NeBr (X.u' '=0,]' ') + hv 2 1 a.ser NeBr (B,u' ,]') 2 ~ ~ NeBr (B,u' ,]') 2 Ne + Br (B,u'-z,j) 2 The vibrational predissociation lifetime in Eq. from 355 ps radiative is for u lifetime -s J.Ls, 11 so = 10 for that a to 1.7 ps fluorescence for u of Br (B) 2 negligible amount of (2) ranges = 27. in Eq. The (3) fluorescence 165 occurs from the van der Waals complex its to prior fragmentation. Fig. the shows 2 Br emission product 2 spectra obtained by positioning the pump laser on various Ne B~X. band u'~ through the maxima fragment and Br monochrometer slits were obtain acceptable contours, emission B~X 2 than (-5 em required were resolved of For molecule. excited, are Br stretch As higher 2 is so is used vibrational of 2 B~X Relative populations fragment spectra complicate of to dissociate states of NeBr Particularly One is Br the Br of the > 28. vibrational states of that extraction collisions of quanta 2. the the are 2 from Fig. of 2 the the decomposition are extracted of distributions. the is the dramatic cutoff of intensity in the one quantum loss channel for u ' lifetime the that in which one lost become increasingly more probable. the Br 2 band (B,u'(26) 2 to that the product channels in which multiple Br striking in Fig. the NeBr 2 Wide resolution) transitions obtained. the -1 Br monochometer bands. ratios dominant product vibrational channel quantum the emission signal-to-noise rather bands, scanning 79 2 2 There of the B are three phenomena quantitative relatively electronic product-state long fluorescence state. Inelastic fragment with other species in the expansion prior to fluorescence will make the observed Br fragment-state distribution different from that 2 initially 166 FIG. 2. Br 2 fragment emission spectra observed when the 79 laser is positioned on the maximum of the Ne Br 2 B--.X. u·~ vibrational band. The transition labels on the far left of each row of spectra indicate the number of quanta 79 deposited in the Br-Br stretch of Ne Br 2 (B). and the Br 2 fragment emission spectra on that row show the relative fluorescence intensity of the Br 2 (B) fragments that appear in different product channels. Each column of spectra correspond to a specific Au relaxation channel: left column, Au= -1: center column, Au = -2: right column, Au = -3. The identity of each emission band is indicated by a transition label, u'---+u' '. The Au = -1 channel is dominant while i t is energetically open: however, the population of other vibrational channels increases with 79 u'. In the decomposition of Ne Br 2 (B,u'=28), the Au= -1 channel has completely closed. {The weak emission to the red of the 79 Br 2 B---+X, 27---+1 origin is due to isotopic contamination.) See the text for further discussion. 167 Band Excited v' ~ v" 0 0 Energy Relative to Origin of 79 0 8 r2 8 -+X ·v'-+ v" Transition ( cm-1 ) 168 In such collisions there is produced by the dissociation. the possibility for electronic relaxation of the fragment Br . 2 nozzle, the number density of molecules varies as nozzle source. the inverse square By 11 in Far from the the the distance from the increasing the distance of the number of the expense of collisions is intensity. The relative amount of vibrational from reduced nascent Br at in 2 fluorescence . downstream of -1o 16 /cm 3 . the relaxed where measured relaxation of fragment of NeBr determined The Appendix to of measurements Br 2 are density (B) the nascent rotational is vi brat i o na 11 y of 8 % the this chapter shows of the vibrational used to correct the of Br population distributions 2 by diameters number exceeded decomposition relaxation of the nascent fragment. for the 2 vibrational Collisions also alter state distribution to some degree . The rotational temperature from the same vibrational is identical to jet within the relaxation nozzle population uncomplexed measured vibrational 80 the never experimental was fluorescence the expansion and dispersing position uncomplexed these (B) a nozzle. unrelaxed population. how experiment in 2 At the the this exciting uncomplexed Br the the nozzle, region the expansion of excitation of and vibrational. rotational. of state uncomplexed error. 2 emitting in which it was prepared 0.7 K rotational experimental Br temperature However. of the vibrationally 169 inelastic the collisions rotational rotational also state appear to alter significantly For of temperature of Br example, the which has (B.u'=21). 2 undergone an inelastic collision in which Au'=-1 Br 2 produced by the photodissociation of NeBr , 2 significantly more Russell, energy et at . energy that very vibrational relaxation in low-energy the section collision there energy. relaxation of Br 2 actually This Br 2 translationally was not the laser the low-energy They collisions implies for section cross and that with increasing that vibrational (B) produced by van der Waals molecule smaller than (B). In order excited Br (B) significantly collisions, uncomplexed in decreases result predissociation should be uncomplexed high K. will have (B) and He in a free jet expansion. 2 a cross ~8 careful study of occurs is that than 16 have performed a redistribution collisions of r found translational is 2 altered 10+-0 B+--X pump-probe to measured confirm rotational in NeBr 2 discussed for that the distribution vibrationally band of technique that elastic was studied by in Chap. in 3 which the probe laser is delayed by only 10 ns so that the product state collisions. 17 distributions Within are experimental rotational distribution of the u'=9 Br affected by uncertainty, the fragment the not 2 was same in both measurements. Another complication in obtaining product-state 170 distributions undergoes results electronic depends upon reduces significantly Br 2 from i ts fragments for of the fluorescence the the was that lifetime of states j to spectra averaged for the electronic predissociation laser effect excited state in the potential to this effect, the first 400 ns Since rate This the due only pulse. which populated minimize emission following the B state 2 rate a . 11. 18 J. higher the Br at state, order In fact predissociation rotationa 1 dissociation . distortion the the constants j-dependent are not known for each band studied, no attempt was made to correct the distributions the values for obtained for the by technique this rotational dispersed (which electronic effect. The temperature fluorescence should agreement be of and Br practically predissociation) indicate (B,v'=10) 2 the of pump-probe unaffected that by electronic predissociation does not qualitatively change the results obtained. Finally, presence of expansion. there other The bands of Waals molecules species. NeBr 2 Also, Br are interferences isotopic 2 species broad Lorentzian prevent selective associated with because of the monochrometer, it is fluorescence from the difficult 79 Br 2 arising present profiles of excitation only one from the in the higher of Br 2 van v der isotopic limited resolution of our to the and separate the totally 79 sr 81 sr species. 171 Among the bands completely free emission by the studied, only the from contamination of 79 mind, the good qualitative Br 81 A description quantitative rotational quantum determined in detailed are 79 Br 2 in should be accepted as a of the photodissociation of description laser 22-o these caveats distribution state and the dispersed With Br species . results presented here dynamics. 10-o the remains pump-probe final to experiments be such as those described in Chap . 3 . Table 1 presents the product state distributions for the vibrational predissociation of Ne the spectra of Fig. Because 1. 79 Br of extracted from 2 the uncertainties described above, only one significant figure is reported . Populations are extracted intensities, corrected for collisional fluorescence Appendix. from Franck-Condon Rotational the observed vibrational relaxation and factors as described in the temperatures are reported for the Au= -1 channels of u'=10 and u'=22 states of NeBr (B). 2 IV. DISCUSSION Ne 79 Br Comparison of the (B.u=27) and Ne 2 79 product Br 2 emission (B,u'=28) shows spectra a for dramatic decrease in the apparent product population in the Au = -1 channel. the 79 Br That the weak emission near where emission from (B,u'=27) fragment of Ne 79 (B,u'=28) should 2 occur is actually due entirely to the 79 Br 81 Br species, is 2 Br 172 Table 1. Vibrational Product-State Distributions for 79 Photodissociation of Ne Br 2 the Relative Population of Product Channel Au Band Excited u '+--u' ' 10+--{) 22+--{) 27+--{) 28+--{) ( 1 l ( 2 l ( 3 l Au=-1 Au=-2 Au=-3 Au=-4 Au=-5 1(1 ) 1 ( 2) 0.7 0 ( :l> 0 0.2 1 1 0. 1 0.6 0.6 - - 0. 1 0.3 0.4 0 0. 1 0.2 =6 K Trot = 5 K T rot The weak fluorescence observed in this channel is entirely . . d ue to t h e d ecompos1t1on o f Ne 79B r 81B r ( see text ) . 173 s h own b y t he a b sence Because of 27----+1 band origin. the 79 Br · any em i ss1on at of B----+X. 2 the very broad homogeneous linewidth of the van der Waals features in this region, is no longer isotopic species product the selectively The = -1 Br Au channel vibrational D (B) 0 bond, kinetic energy u. When The amount of Ne 79 Br stretch, bond is the (&)27 = 63.5 em energy Ne 79 bracketed to be Shl. ft 12 a Ne 79 B r origin is a potential the Ne 79 the u Because of the in excited (B) energy 2 Br 2 ~ the Au = -1 the B channel for Au = -1 channel closes at Thus, the der the B van electronic n0 (B) ~ 63.5 em -1 . origin (X) from the 79 the Br 2 X the Br state 79 Br is blue 2 band the van der Waals state the blue shift bandhead to state 2 Waals The the electronic By adding state, close. The the ground from must quantum of b an d state. decreases in a in 2 u energy -1 59.7 bandhead of in channe 1 measure of how much deeper is Br Au = -1 = 59.7 em -1 . where of the th of energy (B). the the of is u of < n0 , trans w energy released (B,u'=27) 2 u'=28, wu E the dissociation energy of anharmonicity of the Br-Br B state potential, w with by is and quantum in then negligible, is translational one has. where (B) 2 only Assuming that the Br 2 (B,u'=28). 2 energy fragments, Ne• • •Br excite of relative recoiling the 79 rotational conservation to NeBr 2 . of closed for Ne fact, is possible it bond than in of the the bond energy is 174 ~ 68.5 is ~ D (X) 0 close Swartz, to e t for NeBr 72.6 em an a 1. 12 on 79 of Br . 2 = -1 Au channel 79Br81Br of of a band For NeBr that cannot u' = 28, 2 the the is (B,u'=28) positioned band, Lorentz ian of 79 the the origin fluorescence of of detectable) and -1 half- When the of the it also excites the blue tail of Br of 81 the Ne 79 Br (B,u'=27) 79B the other r2 81 the isotopic Br 2 (B,u'=28), . Br (B,u =28) threshold for product. f 1 uorescence the excitation laser is 79 Br the monochrometer the Lorentzian profile of Ne at maximum separate experiment where at . = 27 for width full absorption at an energy that lies above population u upon the profile on fluorescence populate laser 2 lifetime detected be should excitation Br by for the Lorentz ian lineshape is -3 em 79 made · · 1ncreas1ng an maximum of Ne -1 the question of why weak fluorescence the in = 27 excitation Ne b as1s · It may appear contradictory that (B,u'=28). u the em of (B,u')27). 2 observed from This bond energy determination estimate There remains is -1 In a is positioned band (so that species is not tuned through the a weak threshold occurs when the energy of the prepared state is sufficient to predissocate via the one-quantum mechanism. This phenomena is illustrated in Fig. 3. The fragment rotational can be distributions closely obtained simulated by for a the Br 2 Boltzmann 175 FIG. 3 . Threshold for population of the Au = -1 channel. The top figure schematically illustrates how the energy of NeBr 2 (B,u'=28) level overlaps with the continuum product states. The bottom figure shows the observed emission intensity from 79 Br 2 (B,u'=26) and 79 Br 2 (B.u'=27). when the laser is tuned across the Ne 79 Br 2 B+--X, 28+--0 band. Since excitation anywhere on the band profile gives rise 79 to a state that can decay into the Br 2 (B,u'=26) continuum {the Au= -2 channel), the full Lorentzian profile of the Ne 79 Br 2 B+--X, 28+--0 absorption is observed when the emission from 79 Br 2 (B,u'=26) is monitored . 79 Detection of fluorescence from Br 2 (B,u'=27) {the Au = -1 channel) is not observed until the excitation ener9~ exceeds the threshold for production of products in the Br 2 (B,u'=27) set of continuum product states. 176 Prepared State Product States 20 co C\J flw27 + Erot + E trans i i 10 II - - - - - - - - - - - -=~ E(Br2 ,B,v'= 27) > w - -20 0 1 Q) > c Q) a: I E u 10 / Emission from Br2 (B,v'=27) 0 Fluorescence Intensity 177 distribution, preclude somewhat distribution maximum largest of -1 our limited non-Boltzmann must have rotational a does not In fact, the termination at some that is the behavior . definite quantum resolution number, jmax' that can be populated according to conservation j energy. 70 em although For the Ne is available for 79 Br 2 Au = -1 channel, rotational and relative (B,u'=10) product ~ translational kinetic energy and jmax rotational The observed 35 . temperature of 6 K has an average of 4 em -1 of energy in rotation and gives practically no population of rotational levels approaching strongly with the NeC1 in which allowed 2 state has significant -1 ) an d rotation . 7 • 8 of NeBr 2 (B.u'=27) is kinematically very the due to 2 distribution energetically population (available 15 Cm {~ (B,u'=26) the contrasts and cold Br This rotational an average Of The distribution = 35 . (B.u'=11) every energy = 87 em j small -1 of energy in rotational 1 K) product amount of from energy available for rotation near the closure of the one-quantum channel. These results are in complete dynamical propensity vibrational predissociation of Because the of rules relatively agreement proposed van der large by with the Ewing for Waals molecules. rotational constant 4 of Cl 2 , a Cl 2 fragment c~n accept much more rotational energy than a Br 2 fragment in the same rotational state. This 178 means that Br must have a 2 high-j levels than c1 of rotational values of j in order 2 of the larger population of to accept the same amount According energy. are much to Ewing's model. low probability. so the large fragment Br 2 should be rotationally cold . For high s tate s u of NeB r (B) . 2 the vi brat i o na 1 product state distributions are qualitatively quite broad, with appreciable excitation of that the Nei 2 is NeBr number significant result intensity (B) 2 of u' . ' where for present = 27 and vibrational population observed still Nei 2 Au = -5 It is apparent 28. product increases in at with channels A u'. Nei 2 (B,u'=33), the vibrational predissociation of (B,u'=21) that more 90% of conclusion, the upon dissociation of NeBr is 19 and is 20 distribution 2 the population significant where observed in several product channels. In with similar product population is present in the Au= -1 channel for for much as of product would be energy expected from consideration of the momentum gap theories of Beswick and Jortner is 2 imparted and Ewing, 3 in that little rotational energy to fragment ( <E the Br 2 ro t > ,., 4 K). and the highest open vibrational product channel is favored in the dissociation. The binding energy of NeBr (X) state electronic is bracketed dynamical threshold to be 68.5 ~ n0 by ~ 2 in the ground measurement 76.2 em -1 . of a 179 V. ACKNOWLEDGEMENT We gratefully the acknowlege Science National Foundation for the financial support of this research. VI. APPENDIX This appendix populations of how the photodissociation integrated describes of intensities of can 2 the experimental measurements of be were extracted from from calculated fragment the from the fluorescence and the population relaxation of Vibrational 2 vibrational fragment NeBr uncomplexed Br (B). relative state population the integrated rotational each vibrational emission band of the Br 2 ratios contours of B~X transition using the equation [Br (u 1-z)] 2 = 1 [Br (u )] 2 where I U u~~u~ I vvl .v~ 1 1 1 1 band, f I vI tv quI I u I I I v 3 u I • I quI-z' u I I I v u -z, u I It is the observed integra ted intensity of the 14 and quI' u I I is the Franck-Condon factor, derivation of relaxation of the the kinetic initial equations distribution laser pulse follows closely that of Sharf in, cw (A1) 3 is the frequency of the transition. The the ,U I u -z. VI laser experiment. 21 uncomplexed Br 2 are The relaxation describing following et a.L. processes the for a for 180 A(u : ,u " ) t. fluorescence (A2) 2Br + X quenching (A3) Br 2 (uj) + X . vibrational relaxation (A4) Here we adopt the conventional notation where the primed u is the vibrational electronic state, and electronic state. neg 1 ig i b 1 e with quantum the in number double primed u the in excited the ground The duration of the excitation pulse is respect to the rate of these processes. The population of the initially prepared vibrational level u: of the B state, t. [Br (u:)]. decays as t. 2 [Br (u:)] t. 0 2 [Br 2 (ui_)] 0 is following laser the e -ft (A5) concentration excitation, t is (u{_) of Br the elapsed 2 immediately time since the laser pulse, and r 2 R(ui_, uf )[X] - Q(ut_)[X] . = uj<ut_ u The decay (A6) rate, f, is assumed to be independent of the 181 initial vibrational state, vi_. over a narrow range of initial states . The population of Br (ui_-l} varies in time as 2 d[Br 2 (ui_-l}] dt Applying the initial condition of [Br (ui-l}]t=O = 0 gives 2 [Br 2 (ui-2)] + At a = e [Br (u:)] R(u:,u:-l}[X]t t 0 t t 2 = R(ui,ui-2)[X][Br 2 (ui)] 0 t e -ft (AS} -ft ~R(ui-l,ui-2}R(ui,ui-l}[X] 2 [Br 2 (ui)J 0 t 2 e -ft g i v en o b s e r v a t i on t i me . T, the vi bra t i o na 1 . (A9} sta te population ratios are [Br 2 (ui-l}]l [ [Br (u:)] 2 t = (AlO} t=T R(u: .u:-2}[X]T t t (All} Now consider distribution that the arises relaxation from the of the population photodissociation of 182 Again, subsequent duration fragmentation with respect time t=O, the to the laser lifetime of NeBr the rate of 2 pulse and the are negligible the relaxation processes. At the initial population of each vibrational state is [Br 2 (ui-z)] 0 , where z quanta the lost initial of in to be discussed in Sec. the number of Br dissociation. Applying and the conditions constants is equal III. assuming to those the for vibrational 2 this set relaxation uncomplexed time-dependent of rate Br 2 as vibrational state populations are e -rt e -rt (A12) e -rt (A13) -rt [Br 2 (u:-3)] = [Br (u:-3)] e L 0 2 L At a + R(u{-l,u{-3)[X][Br 2 (u{-1)] 0 t e -rt + ~R(u{-2.u{-3)R(u{-l.ui-2)[X] 2 [Br 2 (u{-1)] 0 t 2 e -rt + [Br 2 (u{-2)] 0 R(ui-2,u{-3)t e-rt . given ratios are observation time, T, the (A14) product population 183 (A15) (A16) where f and f 1 2 are the values obtained in Eqs. (AlO) and {All) for an experiment on uncomplexed Br 2 . The branching ratios [Br 2 (v.£.-z)] 0 /[Br 2 (v.£.-1)] 0 are obtained by experimental measurements of the relaxation of uncomplexed Br 2 and then by using Eqs. (A15) and (A16). In our experiment the emission is observed for a short but finite length boxcar gate. (A12-14) equations of the (AS-9), and to be integrated over the observation time. The are of time, This then determined requires somewhat by Eqs. more but the final ratios are identical. the (A5), width algebraically complex, 184 REFERENCES 1 D.H. Levy, Adv. Chern. Phys. 47, 323 (1981}. 2 3 4 5 6 7 8 9 J.A . Beswick and (1981}. J . Jortner, Adv. Chern . Phys. 47, 363 G . E. Ewing, Faraday Discuss. Chern. Soc . 73 , 325 (1982). G . E . Ewing, J. Phys . Chern . 91, 4662 (1987). D.E. Brinza, B.A. Swartz. C.M. J . Chern. Phys . 79 , 1541 (1983) . Western. and K.C. Janda , J.M. Skene, J.C . Drobits, and M.I. Lester, J. Chern . Phys. 85 , 2329 (1986) . J.I. Cline, N. Sivakumar. D.D . Evard, and K.C. Chern. Phys. 86, 1636 (1987). Janda, J . Chap . 4 of this thesis. F. Thommen, D . D . 82, 5295 ( 1985) . Evard, and K. C . Janda, J . Chern. Phys . 10 J.I. Cline, D . D. Evard, B.P. Reid, N. Sivakumar, F. Thommen, and K . C . Janda, in Structure and Dynamics of Weak.l.y Bound Mol.ecul.ar Compl.exes, in A. Weber, ed. (D. Reidel Publishing Co., Dordrecht, 1987), p. 533. 11 M. C. Heaven, Molecular Spectroscopy, Chern. Soc. Per. Rep . , The Chemical Society, London, in press. 12 B.A. Swartz, D.E . Brinza. C.M. J . Phys. Chern. 84, 6272 (1984) . Western, and K.C. Spec. Janda, 13 D . E. Brinza , C.M. Western, D . D. Evard, F. Thommen, B . A . Swartz, and K . C. Janda, J. Phys. Chern . 88, 2004 (1984). 14 J.A . Coxon, 639 (1972). 15 J. Quant. B.A. Swartz, Ph.D. Technology, 1984. Spectrosc . Thesis, Radiat. Transfer 12, California Institute of 185 16 T . D . Russell , B . M. DeKoven, J . A . J . Chern . Phys. 72, 3001 {1980) . 17 N. Sivakumar , preparation . 18 J . I. Cline , and K . C . M. Siese, E . Tiemann, 117, 208 {1985). 19 and J . E . Kenny, K.E . Johnson, W. Chern . Phys . 72 , 1109 {1980). 20 U. and D.H. Janda, Wul f, Levy , manuscript in Chern . Phys . Lett . Sharfin, and D.H . Levy , J . J . A. Blazy , B . M. DeKoven, T . D . J . Chern. Phys . 72, 2439 {1980). 21 Blazy, Russell, w. Sharfin, K . E . Johnson, L . Wharton, Chern. Phys. 71, 1292 {1979). and D . H . Levy, and D . H. Levy, J. 186 APPENDIX A Classical Trajectory Program classical generates following FORTRAN computer code trajectories for atom-rigid diatom The an It is designed to be the starting point for system. semiclassical analyses of direct photodissociations using the theory of Schinke. The classical Hamiltonian of the system is = H(R,-,,P,j) where R,-,,j,L, 3, P is and~ the rotational have the same definitions as in Chap. radial momentum constant atom-diatom potential as described Defining zero, the of conjugate to diatom, and over the diatom vibration following Eq. (9) the averaged immediately total angular momentum of requires j = -L. the R, dR p crt = 2j [B + d-r the is the Chap. system, ], 3. as The classical equations of motion dP dt 2~R 2 ] v of are then crt = ~ is B + L = _av BR ~R3 av ~ dt = -a..., 187 The initial conditions are chosen according given the in R . Schinke , j. semiclassical Chern . Phys . treatment photodissociation of of allows li 0 l to vary from 0 to 3fi. It is assumed subroutine POTEYAL , is that the recipe (1986) for direct the namely, P0 = 0 , is chosen such that the total energy sys tern 5049 of triatomic molecules; ...,0 is varied , and R0 the 85, to E. However, a this potential which returns wavenumbers as a function of R in is program also given in the potential a energy in A and ..., in radians . 188 c PROGRAM TRAJ C C C C C C C C C C PROGRAM TO CALCULATE CLASSICAL TRAJECTORIES FOR ATOM-DIATOM DISSOCIATIONS. THE DIATOMIC BOND LENGTH IS FIXED. GENERALIZED COORDS FOR THE PROBLEM: R -> ATOM-DIATIOMIC CENTER-OF-MASS DISTANCE P -> CONJUGATE MOMENTUM FOR R GAMMA -> ANGLE OF Rvector WITH RESPECT TO DIATOMIC AXIS J -> CONJUGATE MOMENTUM FOR GAMMA, I.E. ANG. MOMENTUM C C FOR A DESCRIPTION OF THE CLASSICAL HAMILTONIAN FOR THIS SYSTEM SEER. SCHINKE, J. CHEM. PHYS .. VOL. 85, 5049 {1986). c c c c IMPLICIT REAL*4 (J) COMMON FMU,SR,JCONV,ECONV,B C******************************************************** c c c c c c c c c PHYSICAL CONSTANTS FOR MOLECULE B. ROTATIONAL CONSTANT OF DIATOMIC, IN CM~{-1) B=.12616 FMU, REDUCED MASS OF ATOM-DIATOMIC SYSTEM, IN AMU FMU=3.784 SR. THE BOND LENGTH OF THE DIATOMIC, IN ANGSTROMS SR=2.7637 SR IS NOT USED IN TRAJ, BUT MAY BE NECESSARY IN THE SUBROUTINE POTEVAL THAT COMPUTES THE POTENTIAL. EO, THE TOTAL ENERGY OF THE SYSTEM, IN CM~(-1) E0=121.809 C******************************************************** c c c c c c c c c PARAMETERS FOR TRAJECTORIES NANG, NUMBER OF ANGLES TO CALCULATE TRAJECTORIES NANG=19 AMAX, GAMMA WILL RANGE FROM 0. TO AMAX. IN DEGREES AMAX=90. MAXIT, MAXIMUM NUMBER OF ITERATIONS MAXIT=5000 RMAX. MAXIMUM RADIAL DISTANCE TO CALCULATE TRAJECTORIES, ANGSTROMS RMAX=25. DELTAT, THE TIME INCREMENT, IN PICOSECONDS DELTAT=0.01 C******************************************************** c PI=3.14159 189 c c C******************************************************** c c c c c c c c c c c c c c c c c C c INITIAL CONDITIONS: THE INITIAL CONDITIONS ARE SELECTED WITH INTEGRAL VALUES OF J UP TO 6 Hbar , PO=O. AND RO SELECTED FOR ENERGY CONSERVATION . GAMMAO IS VARIED . P. THE RADIAL MOM . OF THE REDUCED MASS, IN UNITS OF AMU*ANGSTROM/PS PO=O.O J. THE ANGULAR MOM. OF THE REDUCED MASS, IN UNITS OF HBAR GAMMA=VARIED . R=SELECTED TO GIVE THE CORRECT VALUE OF EO-EZPT GAMMA IS DEFINED IN DEGREES AS : c \. \.R \. B IS THE (A-B) ROTATIONAL CONT.(CM-1) FMU IS THE C-(A-B) REDUCED MASS (AMU) R,SR IN ANGSTROMS C GAMMA \. C A----------------- - -------B C SR c c C******************************************************** c c c c c c c c c c c c c c c c c c c INTERNALLY . THIS PROGRAM OPERATES IN THE FOLLOWING UNITS : R -> ANGSTROM P -> AMU*ANGSTROM/PS GAMMA -> UNITLESS J -> AMU*(ANGSTROM**2)/PS V (ENERGY) - > AMU*(ANGSTROM**2)/(PS**2) TO GET ENERGY IN REDUCED UNITS. MULTIPLY E IN CM~(-1) BY THE FACTOR ECONV ECONV=1 . 196259 B IS GIVEN AS THE SPECTROSCOPIC BIN CMA(-1), IT MUST BE CONVERTED TO THE CLASSICAL VALUE OF 1/2I, IN UNITS OF 1/(AMU*(ANGSTROM**2)) TO GET CLASSICAL B. MULTIPLY SPECTROSCOPIC B BY BCONV BCONV= . 029660 B=B*BCONV TO GET ANG. MOM. IN REDUCED UNITS. MULTIPLY J IN UNITS OF HBAR BY THE FACTOR JCONV JCONV=6 . 350772 J=J*JCONV 190 c C******************************************************** c C LOOP OVER THE ANGLES. COMPUTE TRAJECTORIES C DIRECT OUTPUT TO UNIT 9 OPEN(9.STATUS='NEW',FORM= ' FORMATTED'} c c DO 300 I=1.NANG GAMMA=(I-1}*(AMAX/(NANG-1)} c c CONVERT GAMMA TO RADIANS GAMMA=GAMMA*2*PI/360 GAMMAO=GAMMA c c c c ASSUME INITIAL ANGULAR MOMENTUM RANGES FROM JO=O TO 3 LOOP OVER INITIAL ANGULAR MOMENTUM DO 250 II=1.4 JO=(II-1}*2 . *JCONV GAMMAO=GAMMA PO=O . c c c c c COMPUTE INITIAL VALUE OF RO, SO THAT Erot(R}+V(R)=EO: I . E . INNER TURNING POINT OF THE COLLISION ETOLER=0.01 ENERGY TOLERANCE c c RGUESS=2.0 LOWER LIMIT FOR RO IS ASSUMED TO BE 2 ANGSTROMS c c 40 50 c c c c c 60 DR=0.05 FORMAT(A) CALL POTEVAL{COS(GAMMAO),RGUESS,SR.VGUESS,DVDR) EROT=B*J0**2+(J0**2}/(2*FMU*RGUESS**2} FOLD=EROT/ECONV+VGUESS-EO RGUESS=RGUESS+DR CALL POTEVAL{COS{GAMMAO),RGUESS,SR,VGUESS . DVDR) EROT=B*J0**2+{J0**2)/{2*FMU*RGUESS**2) IF{(FOLD*(EROT/ECONV+VGUESS-EO)) .GT. 0.) GOTO 60 RGUESS=RGUESS-DR DR=DR/2 . IF(ABS{EROT/ECONV+VGUESS-EO) .GT. ETOLER) GOTO 50 THE VALUE OF RO IS NOW KNOWN IF JO>O THEN CALCULATE TRAJECTORY FOR +JO AND -JO DO 225 IJ=1 , 2 IF ((II .EQ . 1} . AND. (IJ . EQ. 2}} GOTO 225 IF {IJ .EQ . 2) JO=-(II-1}*2.*JCONV 191 c c C c RO=RGUESS GAMMAO=GAMMA PO=O. CALL POTEVAL(COS{GAMMAO).RO,SR.V.DVDR) E=({P0**2)/(2*FMU)+B*J0**2+{J0**2) & /{2*FMU*R0**2)+V*ECONV)/ECONV COMPUTE TRAJECTORY IT=O T=O. C OUTPUT TO SCREEN AND TO UNIT 9 C INITIAL CONDITIONS WRITE ( * . ' (A) ' ) ***************************************' WRITE{*. '(A)') ' T, R, P, GAMMA. J,E' 100 WRITE{*. '{6F12.5)') & T,RO , PO , GAMMA0*360/{2*PI).JO/JCONV,E WRITE{9. '(A)') ' T. R, P. GAMMA , J.E' WRITE(9. '(6F12 . 5) ') & T.RO.PO.GAMMA0*360/{2*PI).JO/JCONV.E 110 CALL PROPAGATE{RO,PO,GAMMAO.JO,DELTAT) IF {GAMMAO .LT. (2*PI)) GOTO 120 GAMMAO=GAMMA0-2*PI*INT(GAMMA0/{2*PI)) c IF ALL POINTS ON THE TRAJECTORY ARE REQUIRED. C REMOVE C'S C C WRITE{*. '(A)') ***************************************' C WRITE(*. '(A)') 'T, R, P, GAMMA, J,E' C WRITE(*. '(6F12.5)') T.RO,PO.GAMMA0*360/(2*PI).JO/JCONV,E C ENDIF c 120 c C C 200 225 250 IT=IT+1 T=T+DELTAT IF (RO . GT. RMAX) GOTO 200 IF (IT .GT . MAXIT) GOTO 200 GOTO 110 TRAJECTORY COMPLETED. WRITE OUT FINAL STATE OF SYSTEM CALL POTEVAL(COS(GAMMAO),RO,SR,V,DVDR) E=((P0**2)/(2*FMU)+B*J0**2+(J0**2) & /(2*FMU*R0**2)+Y*ECONV)/ECONV WRITE(*. '(6F12 . 5)') & T,RO,PO,GAMMA0*360/(2*PI).JO/JCONV.E WRITE(9. '(A)') 'T, R. P. GAMMA. J,E' WRITE(9, '{6F12.5)') & T.RO.PO,GAMMA0*360/(2*PI).JO/JCONV,E CONTINUE CONTINUE 192 300 c c C C C C C C C c c c CONTINUE CLOSE(9) STOP END SUBROUTINE PROPAGATE(RO.PO.GAMMAO.JO,DELTAT) THIS SUBROUTINE PROPAGATES THE TRAJECTORY THROUGH ONE TIME STEP, DELTAT USES RUNGE-KUTTA METHOD FOR SYSTEMS OF DIFFER. EQUATIONS DESCRIBED IN G.A. KORN AND T.M. KORN, MATHEMATICAL HANDBOOK FOR SCIENTISTS AND ENGINEERS. 2ND ED. (McGRAW-HILL, NEW YORK. 1968), P. 783. IMPLICIT REAL*4 (I.J.K.M,N) COMMON FMU.SR.JCONV,ECONV.B K1=DELTAT*FFUN(RO,PO,GAMMAO.JO) M1=DELTAT*GFUN(RO.PO,GAMMAO.JO) N1=DELTAT*HFUN(RO.PO.GAMMAO.JO) 01=DELTAT*IFUN(RO,PO,GAMMAO.JO) K2=DELTAT*FFUN(RO+K1/2,PO+M1/2,GAMMAO+N1/2,J0+01/2) M2=DELTAT*GFUN(RO+K1/2,PO+M1/2,GAMMAO+N1/2,J0+01/2) N2=DELTAT*HFUN(RO+K1/2,PO+M1/2,GAMMAO+N1/2,J0+01/2) 02=DELTAT*IFUN(RO+K1/2,PO+M1/2,GAMMAO+N1/2.J0+01/2) K3=DELTAT*FFUN(RO+K2/2,PO+M2/2,GAMMAO+N2/2,J0+02/2) M3=DELTAT*GFUN(RO+K2/2,PO+M2/2,GAMMAO+N2/2,J0+02/2) N3=DELTAT*HFUN(RO+K2/2,PO+M2/2,GAMMAO+N2/2,J0+02/2) 03=DELTAT*IFUN(RO+K2/2,PO+M2/2,GAMMAO+N2/2.J0+02/2) K4=DELTAT*FFUN(RO+K3,PO+M3,GAMMAO+N3,J0+03) M4=DELTAT*GFUN(RO+K3,PO+M3.GAMMAO+N3.J0+03) N4=DELTAT*HFUN(RO+K3.PO+M3.GAMMAO+N3.J0+03) 04=DELTAT*IFUN(RO+K3,PO+M3,GAMMAO+N3.J0+03) RO=RO+(K1+2*K2+2*K3+K4)/6 PO=PO+(M1+2*M2+2*M3+M4)/6 GAMMAO=GAMMAO+(N1+2*N2+2*N3+N4)/6 JO=J0+(01+2*02+2*03+04)/6 RETURN END C C THESE ARE THE FOUR CLASSICAL EQUATIONS OF MOTION FOR THE SYSTEM: C DR/DT REAL*4 FUNCTION FFUN(R.P,GAMMA.J) IMPLICIT REAL*4 (J.K.M,N) COMMON FMU,SR.JCONV.ECONV,B FFUN=P/FMU RETURN END c c C DP/DT 193 c C c C REAL*4 FUNCTION GFUN(R,P,GAMMA.J) IMPLICIT REAL*4 (J,K,M,N) COMMON FMU,SR,JCONV,ECONV.B DR=.001 CALL POTEVAL(COS{GAMMA),R+DR,SR,POTl,DVDR) CALL POTEVAL(COS(GAMMA}.R-DR.SR.POT2,DVDR) DVDR={POT1-POT2}/(2.*DR) DVDR=DVDR*ECONV GFUN=J**2/(FMU*R**3) - DVDR RETURN END DGAMMA/DT REAL*4 FUNCTION HFUN(R,P,GAMMA,J) IMPLICIT REAL*4 (J.K.M.N) COMMON FMU,SR.JCONV,ECONV,B HFUN=2*B*J+J/(FMU*R**2} RETURN END DJ/DT REAL*4 FUNCTION IFUN(R,P,GAMMA,J) IMPLICIT REAL*4 (I.J.K.M.N) COMMON FMU,SR,JCONV.ECONV,B DG=O.OOOl CALL POTEVAL{COS{GAMMA+DG}.R,SR,POTl,DVDR) CALL POTEVAL{COS{GAMMA-DG),R,SR,POT2,DVDR) DVDG=(POT1-POT2}*ECONV/{2*DG} IFUN=-DVDG RETURN END 194 APPENDIX B Program for Extraction of Rotational Populations from Probe Laser Spectrum "POP" is a VAX-FORTRAN program to extract rotational populations diatomic by simulating vibronic interactively on a the experimental compared . line intensities is designed bands. It terminal with graphics and Except calculated for the spectra 1 in to be run capability, can be program should be transportable to other computers . the Since graphics capabilities are typically system-dependent, graphics subroutines are not graphics calls listed, are included which here . our Generic be should so directly subroutines, graphics 1 L- L easily implemented on any system . The calculated spectrum is determined by two sets of parameters, documented in the function FUNCTN. The first set contains the spectroscopic constants (e.g., rotational constants experimental conditions (e.g., laser lineshape and background level). The second set parameters of and rotational constants are known, are origin) gives levels. populations spectrum . band and the relative Usually, they optimized are to if set populations the first , fit the of the spectroscopic and then the experimental 195 All and is with a given the program I/0 is contained in the main program documented list in the of parameters the opportunity to be listing. or are otherwise, input a new list, is presented fitting options and to alter settings The user the retained, list. respond If is then the current with separated by commas. a "/"; 196 c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c PROGRAM POP PROGRAM TO EXTRACT ROTATIONAL POPULATIONS FROM THE LASER-INDUCED FLUORESCENCE SPECTRA OF DIATOMIC MOLECULES . WRITTEN BY JOE I . CLINE SET UP FOR SINGLET-SIGMA----- >SINGLET SIGMA TRANSITIONS ONLY. BUT SUBROUTINE FUNCTN COULD BE MODIFIED FOR THE SELECTION RULES OF OTHER TYPES OF TRANSITIONS . A GAUSSIAN LASER LINESHAPE (FWHM) , GAMMA. IS ASSUMED MAGNETIC SUBLEVELS ARE ASSUMED TO BE STATISTICALLY POPULATED , I.E., AN ISOTROPIC ORIENTATION OF THE PROBED MOLECULES . FOR NASCENT DIATOMIC FRAGMENTS FROM PHOTODISSOCIATIONS THIS MEANS THAT NO ALIGNMENT IS ASSUMED-THE METASTABLE STATE OF THE PARENT MOLECULE IS LONG-LIVED . REFERENCES: E.U . CONDON AND G. H. SHORTLEY , THE THEORY OF ATOMIC SPECTRA (CAMBRIDGE U. PRESS, CAMBRIDGE , 1935), CHAP . IV . G. HERZBERG MOLEC . SPECTRA AND MOLEC. STRUCTURE (VAN NOSTRAND REINHOLD, N. Y. , 1950) , VOL . 1 . MARQUARDT NONLINEAR LEAST SQUARES OPTIMIZATION IS USED . THE MARQUARDT PROGRAM USED HERE IS A SUBSTANTIALLY MODIFIED VERSION OF THE CODE GIVEN IN : P . R. BEVINGTON DATA REDUCTION AND ERROR ANALYSIS FOR THE PHYSICAL SCIENCES (McGRAW-HILL , N. Y., 1969), CHAP . 11. IMPLICIT REALMS (A-H , O-Z) CHARACTERM50 DNAME CHARACTERM50 PNAME CHARACTER*50 DUMMY DIMENSION XX(1200) , YY(1200) COMMON/DATA/X(1200),Y(1200) , YFIT(1200),NPTS COMMON/FIT/SIGMAY(60),NTERMS,MODE.DELTAA(60), &ASIGMA(60),FLAMDA,CHISQR,QDELTAA(60) COMMON/PARMS/A(60),QA(60),JUGGLE(15) INITIALIZE FITTING PARAMETERS 197 c c c c c c c c c c c c c c c c c c c c c c c C c c C c DO 5 I=l. 10 JUGGLE{I)=O DELTAA{I)=O. ASIGMA(I)=O . A(I)=O. QDELTAA(I)=O. QA(I)=O . 5 CONTINUE GET FILENAMES: THE .SPC FILE CONTAINS THE EXPERIMENTAL SPECTRUM IN THE FORMAT : NPTS -NUMBER OF -poiNTS IN SPECTRUM XXI , YYl -LASER FREQ .. INTENS. AT POINT 1 XX2. YY2 AT POINT 2 THE .DAT FILE CONTAINS THE FITTING DATA IN THE FORMAT: NVAR, MODE -NO. PARAMETERS, MODE NOT USED NFIRST, NLAST -FIRST AND LAST POINTS TO FIT CONY -CONVERGENCE CRITERION -PARAMETER NO. 1 QA{l) -PARAMETER NO. 2 QA{2) QA{NVAR) QDELTAA{l) -INCREMENT FOR PARAMETER NO. QDELTA(NVAR) WRITE ( * , 1 0 ) 10 FORMAT{' INPUT NAME OF SPC FILE:') READ (*.20) DNAME 20 FORMAT (A) OPEN {l,FILE=DNAME,STATUS='OLD') WRITE (*,30) 30 FORMAT(' INPUT NAME OF DAT FILE:') READ (*,20) PNAME OPEN {2,FILE=PNAME,STATUS='OLD') READ IN FIT PARAMETERS READ {2, '(2I6)') NVAR,MODE MODE=O READ (2, '{2I6)') NFIRST,NLAST READ {2. '{F20.0)') CONY IF {CONV.EQ.O.) CONV=l.E-6 IF {CONV.LT.O.) CONV=l.E-6 READ IN DATA SET READ {1, '{I6)') NPTSX DO 40 I=l,NPTSX 1 198 c c c c C C C c c c c C c C C c C c C c C c C c C C READ (1, '(3F20.0)') XX(I).YY(I),YFIT(I) 40 CONTINUE CLOSE{!) DO 50 I=l,NVAR 50 READ (2, '(F20 . 0)') QA(I) DO 60 I=1,NVAR 60 READ (2, '{F20 . 0)') QDELTAA(I) 70 CLOSE{2) ASSIGN X,Y MATRICES FOR DATA RANGE SELECTED ONLY DATA IN THE SELECTED RANGE WILL BE FIT THESE DATA ARE SUPPLIED TO CURFIT IN THE X,Y ARRAYS. CALL RANGE{NPTS,NFIRST,NLAST,X,Y,XX,YY) IFLAG=-1 NTERMS=O WRITE SPECTROSCOPIC PARAMETERS TO SCREEN 400 WRITE{*.410) {QA{J),J=1,15) 410 FORMAT ((3F20.10)) READ IN ANY CHANGES TO PARAMETERS IF PARAMETER NO. 1 .LT. 0 THEN TERMINATE PROGRAM. READ(*,*) (QA(J).J=1,15) IF (QA{1) .LT. 0.) GOTO 2020 WRITE POPULATION PARAMETERS TO SCREEN WRITE(*,410) (QA(J).J=20,60) READ IN ANY CHANGES READ(*,*) (QA(J).J=20,60) WRITE{*,410) (QDELTAA(J).J=l,15) READ(*,*) (QDELTAA{J).J=1,15) OPPORTUNITY TO ALTER CONVERGENCE CRITERION WRITE(*, '{F20.10)') CONY READ(*.*) CONY INPUT FIRST AND LAST POINTS TO FIT WRITE(*, '(2I6)') NFIRST,NLAST READ(*,*) NFIRST,NLAST CALL RANGE(NPTS,NFIRST,NLAST,X,Y,XX,YY) READ IN NTERMS, THE NUMBER OF PARAMETERS TO FIT FOLLOWED BY THE LIST OF NTERMS PARAMETER NUMBERS. WRITE(*.420) NTERMS,(JUGGLE(J).J=l,15) READ(*,*) NTERMS,(JUGGLE(J),J=1,15) 420 FORMAT (I4,(10I4)) 199 c IF {NTERMS . EQ. 0) THEN IF NOT FITTING ANYTHING, JUST CALCULATE THE ERROR DO 427 K=1,NPTS 427 YFIT{K)=FUNCTN(X,K,QA) NFREE=NPTS-NTERMS CHISQR=FCHISQ(Y,SIGMAY.NPTS,NFREE,MODE,YFIT) WRITE(*.217) CHISQR GOTO 300 C NOW JUMP TO PLOT ROUTINE END IF C c C C C c c C c c THE FIRST NTERMS PARAMETERS IN JUGGLE{) ARE TO BE FIT, TRANSFER THE PROPER VALUES FROM QA{) TO A() . CURFIT WILL OPTIMIZE THE PARAMETERS IN A(). CALL ORDERA{JUGGLE,A,QA,DELTAA,QDELTAA,NTERMS) INITIALIZE COUNTERS AND MARQUARDT PARAMETER FLAMDA=.001 I=O IFLAG=O WRITE PROGRESS OF ITERATION TO SCREEN 200 WRITE (*,205) I 205 FORMAT(' ITERATION NO ',I6) DO 210 J=1,NTERMS c WRITE PARAMETER, VALUE, ESTIMATED UNCERTAINTY 210 WRITE (*,215) J.A(J).ASIGMA(J) 215 FORMAT(' A(',I3 , ')= ', E20.10,E20.10) IF(I . EQ.O) GOTO 100 WRITE (*,217) CHISQR 217 FORMAT(' CHISQR=',E20.10) IF (IFLAG.EQ . 1) GOTO 300 c 100 CALL CURFIT CURFIT DOES THE NON-LINEAR OPTIMIZATION IF {I . GT . 0) GOTO 105 ACHISQR=CHISQR I=I+1 GOTO 200 105 DIFF={ACHISQR-CHISQR)/ACHISQR I=I+1 ACHISQR=CHISQR C CHECK FOR CONVERGENCE TOLERANCE IF {DIFF.GT . CONV) GOTO 200 IFLAG=1 GOTO 200 300 CONTINUE c c C PLOT ROUTINE C CALCULATE LIMITS FOR PLOTTING c 200 XMIN=X{1) XMAX=X(1) YMIN=Y{1) YMAX=Y{1) DO 510 I=1.NPTS IF (X(I) .GT . XMAX) XMAX=X(I) IF (X(I) . LT . XMIN) XMIN=X(I) IF (Y(I) . GT . YMAX) YMAX=Y(I) IF (YFIT(I) .GT. YMAX) YMAX=YFIT{I) IF (Y(I) .LT. YMIN) YMIN=Y(I) 510 IF {YFIT{I) .LT. YMIN) YMIN=YFIT(I) WRITE(*. '(4F20 . 10)') XMIN,XMAX.YMIN,YMAX c!!! 1!!!!!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! C C C THESE ARE GRAPHICS CALLS TO SYSTEM-DEPENDENT SUBROUTINES, ONLY THEIR GENERIC FUNCTION IS DESCRIBED. THEY ARE ASSUMED SINGLE PRECISION. C INITIALIZE GRAPHICS MODE CALL GSTART c c c SET UP PLOT LIMITS ON PLOT DEVICE CALL MAP(SNGL{XMIN). SNGL(XMAX). SNGL(YMIN). & SNGL(YMAX)) c DRAW AXES. NOT CURRENTLY IMPLEMENTED c CALL AXIS{SNGL{XMIN), SNGL(XMAX), SNGL{YMIN). c & SNGL(YMAX) . . TRUE . ) c PLOT INITIAL POINT OF EXPERIMENTAL SPECTRUM CALL POINT{SNGL(X(1)),SNGL{Y{1))) c JOIN SUBSEQUENT DATA POINTS TO THE FIRST DO 515 I=2,NPTS 515 CALL JOIN{SNGL(X(I)).SNGL(Y(I))) c PLOT THE CALCULATED SPECTRUM CALL POINT(SNGL(X(1)),SNGL(YFIT(1))) DO 517 I=2,NPTS 517 CALL JOIN(SNGL(X(I)),SNGL(YFIT(I))) READ(*,20) DUMMY C!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! c c c c c C C 2020 PLOTTING IS COMPLETE. NOW JUMP BACK TO INPUT SECTION TO DO ANOTHER FIT . GOTO 400 FITTING IS COMPLETED. BEFORE TERMINATING, WRITE CALCULATED SPECTRUM TO SPC FILE OPEN (1,FILE=DNAME,STATUS='OLD') WRITE ( 1, ' ( I6) ') NPTSX DO 2010 I=1,NPTSX IF (I . GE. NFIRST) THEN IF (I .LE . NLAST) THEN WRITE (1,'(3E20.10)') & XX(I),YY(I).YFIT(I-NFIRST+1) ELSE 201 c c WRITE (1,'{3E20.10)') XX(I),YY(I) END IF ELSE WRITE {1, ' {3E20 . 10) ' ) XX(I).YY(I) END IF 2010 CONTINUE c c WRITE UPDATED PARAMETERS TO DAT FILE. OPEN {2,FILE=PNAME,STATUS='OLD ' ) WRITE {2. ' {2I6) ' ) NVAR,MODE WRITE (2. '(2I6)') NFIRST, NLAST WRITE {2. '{E20 . 10)') CONV DO 1000 I=1,NVAR 1000 WRITE {2. '{E20.10)') QA(I) DO 1010 I=1,NVAR 1010 WRITE (2. '(E20.10)') QDELTAA(I) 1020 CLOSE {2) END c c c C c SUBROUTINE RANGE(NPTS,NFIRST,NLAST.X.Y . XX,YY) IMPLICIT REALMS (A-H.O-Z) DIMENSION X(1200),Y{1200),XX{1200),YY{1200) ASSIGN X.Y MATRICES FOR DATA RANGE SELECTED NPTS=NLAST-NFIRST+1 41 DO 43 I=1, NPTS X(I)=XX(NFIRST+I-1) Y(I)=YY(NFIRST+I-1) 43 CONTINUE RETURN END c c CMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM c c c SUBROUTINE CURFIT: ADAPTED FROM P . R . BEVINGTON c DATA REDUCTION AND ERROR ANALYSIS FOR THE c PHYSICAL SCIENCES c {McGRAW-HILL, N.Y .. 1969), P . 237 . c THE MARQUARDT ALGORITHM FOR LEAST-SQUARES FIT OF c AN ARBITRARY FUNCTION TO DATA IS DESCRIBED THEREIN . c SUBROUTINE CURFIT IMPLICIT REALMS (A-H,O-Z) DIMENSION WEIGHT{1200),ALPHA{15,15),BETA(15), 1 DERIV(15),ARRAY{15,15).B{15) COMMON/DATA/X{1200),Y{1200),YFIT{1200),NPTS 202 c C C C c c C c c C c c C c COMMON/FIT/SIGMAY(60),NTERMS,MODE.DELTAA(60). 1 ASIGMA(60),FLAMDA,CHISQR,QDELTAA(60) COMMON/PARMS/A(60),QA(60),JUGGLE(15) 11 NFREE=NPTS-NTERMS IF (NFREE) 13,13,20 13 CHISQR=O. GOTO 110 EVALUATE WEIGHTS IN THIS PROGRAM MODE IS FIXED AT ZERO. (EQUAL WEIGHTING OF ALL POINTS) 20 DO 30 I=l, NPTS 21 IF (MODE) 22.27,29 22 IF (Y(I)) 25,27 , 23 23 WEIGHT(I)=l . /Y(I) GOTO 30 25 WEIGHT(I)=l./(-Y(I)) GOTO 30 27 WEIGHT(I)=l. GOTO 30 29 WEIGHT(I)=l . /SIGMAY(I)**2 30 CONTINUE EVALUATE ALPHA AND BETA MATRICES 31 DO 34 J=l,NTERMS BETA(J)=O. DO 34 K=l.J 34 ALPHA(J.K)=O. CALL ORDERA(JUGGLE,A,QA,DELTAA,QDELTAA,NTERMS) 41 DO 50 I=l,NPTS CALL FDERIV(X.I.A.DELTAA.NTERMS,DERIV.JUGGLE,QA) DO 46 J=l,NTERMS BETA(J)=BETA(J)+WEIGHT(I)*(Y(I) & -FUNCTN(X,I,QA))*DERIV(J) DO 46 K=l,J 46 ALPHA(J.K)=ALPHA(J.K)+WEIGHT(I)*DERIV(J)*DERIV(K) 50 CONTINUE 51 DO 53 J=l,NTERMS DO 53 K=l,J 53 ALPHA(K,J)=ALPHA(J.K) EVALUATE CHI SQUARE AT STARTING POINT 61 DO 62 I=l,NPTS 62 YFIT(I)=FUNCTN(X,I.QA) 63 ACHISQR = FCHISQ (Y.SIGMAY,NPTS,NFREE,MODE.YFIT) INVERT MODIFIED CURVATURE MATRIX TO FIND NEW PARAMETERS 71 DO 74 J=l,NTERMS DO 73 K=l.NTERMS 203 c c c c c c c 73 ARRAY(J.K)=ALPHA(J.K)/SQRT(ALPHA(J.J)*ALPHA(K.K)) 74 ARRAY(J.J)=1 . +FLAMDA 80 CALL MATINV(ARRAY,NTERMS,DET) 81 DO 84 J=1,NTERMS B(J)=A(J) DO 84 K=1,NTERMS 84 B(J)=B(J)+BETA(K)*ARRAY(J.K) & /SQRT(ALPHA(J.J)*ALPHA(K,K)) IF CHI SQUARE INCREASED, INCREASE FLAMDA AND TRY AGAIN CALL ORDERQA(JUGGLE.B,QA,NTERMS) 91 DO 92 I=1,NPTS 92 YFIT(I)=FUNCTN(X.I,QA) 93 CHISQR=FCHISQ(Y,SIGMAY.NPTS,NFREE.MODE.YFIT) IF (ACHISQR-CHISQR) 95,101.101 95 FLAMDA=10 . MFLAMDA GOTO 71 EVALUATE PARAMETERS AND UNCERTAINTIES 101 DO 103 J=1,NTERMS A(J)=B(J) 103 ASIGMA(J)=SQRT(ARRAY(J.J)/ALPHA(J.J)) FLAMDA=FLAMDA/10. 110 RETURN END c c C******************************************************** c c C C c c c SUBROUTINE ORDERA(JUGGLE.A,QA.DELTAA.QDELTAA.NTERMS) STUFF PROPER VALUES FROM QA() PARAMETER ARRAY INTO A() FOR OPTIMIZATION BY CURFIT IMPLICIT REAL*8 (A-H,O-Z) DIMENSION JUGGLE(15),A(60),QA(60), & DELTAA(60),QDELTAA(60) DO 100 I=1.NTERMS A(I)=QA(JUGGLE(I)) DELTAA(I)=QDELTAA(JUGGLE(I)) 100 CONTINUE RETURN END C******************************************************** c c SUBROUTINE ORDERQA(JUGGLE,A,QA,NTERMS) 204 C C c c c FOLLOWING THE FITTING, UPDATE QA() WITH THE OPTIMIZED PARAMETERS IMPLICIT REAL*8 (A-H,O-Z) DIMENSION JUGGLE(15),A(60),QA(60) DO 100 I=l,NTERMS QA(JUGGLE(I))=A(I) 100 CONTINUE RETURN END C******************************************************** c c C C c C C C c c c c c c C c c c FUNCTION FCHISQ: ADAPTED FROM P.R. BEVINGTON DATA REDUCTION AND ERROR ANALYSIS p. 194. CALCULATE ERROR: GIVEN BY THE SUM OF THE SQUARES OF THE DIFFERENCES OF THE EXPERIMENTAL AND CALCULATED INTENSITIES. FUNCTION FCHISQ (Y,SIGMAY,NPTS,NFREE,MODE,YFIT) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION Y(1200),SIGMAY(1200),YFIT(1200) 11 CHISQ=O. 12 IF (NFREE) 13,13,20 13 FCHISQ = 0. GOTO 40 ACCUMULATE CHI SQUARE 20 DO 30 I=l, NPTS 21 IF (MODE) 22,27,29 IN THIS PROGRAM, MODE IS FIXED AT ZERO. 22 IF (Y(I)) 25,27,23 23 WEIGHT=l./Y(I) GOTO 30 27 WEIGHT= I. GOTO 30 25 WEIGHT=l./(-Y(I)) GOTO 30 29 WEIGHT=l./SIGMAY(I)**2 30 CHISQ=CHISQ+WEIGHT*(Y(I)-YFIT(I))**2 DIVIDE BY NUMBER OF DEGREES OF FREEDOM 31 FREE=NFREE 32 FCHISQ:CHISQ/FREE 40 RETURN END 205 C******************************************************** c c c c c c c c c c c c c c c c c SUBROUTINE MATINV: ADAPTED FROM P.R. BEVINGTON DATA REDUCTION AND ERROR ANALYSIS INVERT A SYMMETRIC MATRIX, ALSO CALCULATES ITS DETERMINANT SUBROUTINE MATINV (ARRAY,NORDER,DET) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION ARRAY{l5,15).IK{l5).JK{15) 10 DET=l. 11 DO 100 K=l,NORDER FIND LARGEST ELEMENT ARRAY(I.J) IN REST OF MATRIX AMAX=O. 21 DO 30 I=K.NORDER DO 30 J=K.NORDER 23 IF (ABS{AMAX)-ABS{ARRAY(I.J))) 24,24,30 24 AMAX=ARRAY(I.J) IK{K)=I JK(K)=J 30 CONTINUE INTERCHANGE ROWS AND COLUMNS TO PUT AMAX IN ARRAY(K.K) 31 IF (AMAX) 41,32,41 32 DET=O. GOTO 140 41 I=IK(K) IF (I-K) 21,51,43 43 DO 50 J=l.NORDER SAVE=ARRAY(K,J) ARRAY(K.J)=ARRAY(I,J) 50 ARRAY(I,J)=-SAVE 51 J=JK(K) IF (J-K) 21,61,53 53 DO 60 I=l,NORDER SAVE=ARRAY(I,K) ARRAY(I,K)=ARRAY(I.J) 60 ARRAY(I.J)=-SAVE ACCUMULATE ELEMENTS OF INVERSE MATRIX 61 DO 70 I=l,NORDER IF (I-K) 63,70,63 63 ARRAY(I,K)=-ARRAY(I,K)/AMAX 70 CONTINUE 71 DO 80 I=l,NORDER DO 80 J=l,NORDER IF (I-K) 74,80,74 206 c c c c c 74 IF (J-K) 75,80,75 75 ARRAY(I.J)=ARRAY(I.J)+ARRAY(I.K)*ARRAY(K.J) 80 CONTINUE 81 DO 90 J=1 , NORDER IF (J-K) 83,90,83 83 ARRAY(K,J)=ARRAY(K,J)/AMAX 90 CONTINUE ARRAY(K.K)=1./AMAX 100 DET=DET*AMAX RESTORE ORDERING OF MATRIX 101 DO 130 L=1,NORDER K=NORDER-L+1 J=IK(K) IF (J-K) 111,111 , 105 105 DO 110 I=1.NORDER SAVE=ARRAY(I,K) ARRAY(I.K)=-ARRAY(I.J) 110 ARRAY(I , J)=SAVE 111 I=JK(K) IF (I-K) 130,130,113 113 DO 120 J=1 , NORDER SAVE=ARRAY(K,J) ARRAY(K,J)=-ARRAY(I.J) 120 ARRAY(I,J)=SAVE 130 CONTINUE 140 RETURN END C******************************************************** c c ~ ' - c c c c c c c c c c c c c c c c c c FUNCTION FUNCTN (X,I,A) THIS SUBROUTINE CALCULATES THE INTENSITY AT THE POINT X(I) FOR THE PARAMETER ARRAY A() . THE PARAMETERS HAVE FOLLOWING SIGNIFICANCE TO FUNCTN : A(2) IS THE GAUSSIAN LASER LINEWIDTH CM-1, FWHM A(3) IS THE LASER FREQUENCY CORRECTION, IN CM-1 A(5) IS THE FREQUENCY OF THE BAND ORIGIN , CM-1 A(6) BY ROT . CONSTANT FOR LOWER ELECTRONIC STATE A(7) BY ROT . CONSTANT FOR UPPER ELEC . STATE A(8) JMAX , THE HIGHEST POPULATED ROTATIONAL LEVEL A(9) THE BACKGROUND NOISE LEVEL: THE BASELINE A(10) DV DISTORTION CONSTANT FOR LOWER ELEC. STATE A(11) DV FOR UPPER ELEC. STATE A(20)-A(20+JMAX) RELATIVE POPULATIONS OF ROTATIONAL LEVELS , E.G . A(20) IS THE POPULATION OF J=O A(21) IS THE POPULATION OF J=1 A(20+JMAX) IS POPN . OF J=JMAX . PARAMETERS 4 AND 11-19 ARE UNUSED IN THE CURRENT 207 C c c c c c c c c PROGRAM. IMPLICIT REAL*S (A-H.O-Z) COMMON/LINES/ R(50),P(50),RINTENS(50), & PINTENS(50),JMAX DIMENSION X(1200),A(60) 20 FORMAT(A) JMAX=IDINT(A(S)) PUT FREQUENCY POSITIONS OF R- AND P- BRANCH LINES INTO R() AND P() CALL LINEPOS(JMAX,R,P,A) PUT INTENSITIES OF LINES INTO RINTENS AND PINTENS CALL INTENS(JMAX.R.P . A.RINTENS,PINTENS) FUNCTN=O . CALCULATE INTENSITY AT X(I) SUM UP CONTRIBUTIONS OF EACH CONVOLUTED LINE IN IN THE SPECTRUM : DO 300 J=1,JMAX+1 c R-BRANCH FUNCTN:FUNCTN+RINTENS(J)*GAUSS(X(I),R(J),A(2)) 300 CONTINUE c P-BRANCH DO 400 J=1.JMAX FUNCTN:FUNCTN+PINTENS(J)*GAUSS(X(I),P(J).A(2)) 400 CONTINUE c c ADD IN THE BACKGROUND TO THE INTENSITY FUNCTN:FUNCTN+A(9) RETURN END c c C******************************************************** c c FUNCTION GAUSS (W,WO,FWHM) IMPLICIT REAL*S (A-H,O-Z) C COMPUTES GAUSSIAN CONVOLUTION FUNCTION C WO IS CENTER OF GAUSSIAN C FWHM IS FWHM OF GAUSSIAN C 1 . 0 IS PEAK HEIGHT C W IS POINT TO CALC . GAUSSIAN Z=(W-WO)/FWHM IF (Z**2 .GT . 10 . 0D+OO) THEN GAUSS:O . GOTO 100 END IF C=4 . 0*DLOG(2 . 0DO) GAUSS=DEXP(-C*Z**2) 100 RETURN END ' c 208 c C******************************************************** c c C C C c c SUBROUTINE LINEPOS(JMAX,R,P,A} CALCULATES THE FREQUENCIES OF THE P- AND R- BRANCH LINES. USING THE ROTATIONAL CONSTANTS AND DISTORTION CONSTANTS FOR EACH ELECTRONIC STATE. IMPLICIT REAL*8 (A-H,O-Z} DIMENSION R(50) , P(50) DIMENSION A(60) DO 100 J=O, JMAX EB=A(6}*J*(J+1}-A(10}*(J*(J+1}}**2 K=J+1 EEPLUS=A(7)*K*(K+1}-A(11}*(J*(J+1}}**2 R(J+1}=EEPLUS-EB+A(5)+A(3} IF (J .GT. 0) THEN K=J-1 EEMINUS=A(7)*K*(K+1}-A(11)*(J*(J+1}}**2 P(J)=EEMINUS-EB+A(5)+A(3) END IF 100 CONTINUE RETURN END C******************************************************** c c C C C C C C c c c SUBROUTINE INTENS(JMAX,R,P , A,RINTENS,PINTENS) COMPUTE INTENSITY OF EACH LINE USING THE POPULATIONS, A(20} ... A(20+JMAX). MUST BE CORRECTED FOR HONL-LONDON FACTORS SEE, FOR EXAMPLE , HERZBERG, MOLECULAR SPECTRA AND MOLEC. STRUCTURE, VOL . 1., P. 208, AND EQ. 2 OF CHAPTER 3 OF JIC CALTECH THESIS. IMPLICIT REAL*S (A-H,O-Z) DIMENSION R(50).P(50),A(60),RINTENS(50),PINTENS(50} RINTENS(1}=A(20} DO 100 J=1.JMAX RINTENS(J+1}=A(J+20)*(J+1}/(2*J+1} PINTENS(J}=A(J+20}*J/(2*J+1} 100 CONTINUE RETURN END C******************************************************** c c SUBROUTINE FDERIY (X,I.A.DELTAA,NTERMS, 209 C C C C & DERIV,JUGGLE,QA) CALCULATES DERIVATIVES OF FUNCTION FUNCTN WITH RESPECT TO THE FITTING PARAMETERS. QA() DERIVATIVES ARE CALCULATED NUMERICALLY USING THE INCREMENTS IN DELTAA(). IMPLICIT REAL*8 (A-H, 0-Z) DIMENSION X(1200),A(60).DELTAA(60),DERIV(60). & JUGGLE(15),QA(60) 11 DO 18 J=1,NTERMS AJ=A(J) DELTA=DELTAA(J) A(J)=AJ+DELTA CALL ORDERQA(JUGGLE,A,QA,NTERMS) YFIT=FUNCTN(X,I,QA) A(J)=AJ-DELTA CALL ORDERQA(JUGGLE.A,QA,NTERMS) DERIV(J)=(YFIT-FUNCTN(X,I,QA))/(2.*DELTA) A(J)=AJ 18 CALL ORDERQA(JUGGLE.A,QA,NTERMS) RETURN END