ComputerPhysicsCommunications52 (1989) 383—395 383North-Holland,Amsterdam

BOUND -, CONTINUUM INTENSITIES - A COMPUTERPROGRAMFOR CALCULATING ABSORPTION COEFFICIENTS, EMISSION INTENSITIESOR (GOLDEN RULE) PREDISSOCIATION RATES

RobertJ. Le ROY

Gue/ph-WaterlooCentrefor GraduateWork in Chemistry,Universityof Waterloo, Waterloo,Ontario N2L 3G1, Canada

Received17 April 1988

This paperdescribesa programfor calculatingthebound—~continuumtransitionintensitiesassociatedwith photodissocia-tion, spontaneousemissionor predissociation,of a diatomic molecule. If desired, it may alsoperform least-squaresfits toexperimentaldata to determineparameterscharacterizingthe transition moment (coupling function) and/or final-statepotential energyfunction(s).

PROGRAM SUMMARY

Title ofprogram: Bound—~Continuum Intensities(BCON1’) High speedstoragerequired: 752 kbytes

Catalognumber:ABHC No. of bits in a word: 32

Program obtainable from: CPC Program Library, Queen’s No. of lines in combinedprogram and testdeck: 2976Universityof Belfast, N. Ireland(seeapplicationform in thisissue), or from the author by electronic mall (LEROY@- Separatedocumentationavailable: program manual and de-WATDCS.BITNET) scription by RJ. Le Roy, University of Waterloo Chemical

PhysicsResearchReportCP-329(1988)

Computer: open-currentversion testedon a SUN-3 and aVAX-11/785 Keywords: photodissociation, absorption coefficient, pre-

dissociation,goldenrule, continuumemissionOperatingsystem:no operating system dependentconstruc-tions in the program; currently runs under Berkley UNIX Natureofthephysicalproblemversion4.2 on a SUN-3,andunderVMS version4.5 on a VAX This programcalculatesthebound—~continuumtransition

Programminglanguageused: FORTRAN

OO1O-4655/89/$03.50© ElsevierSciencePublishersB.V.(North-HollandPhysicsPublishingDivision)

384 Ri. I.e Roy/ Bound—~ ‘~ontinuumintensities

intensities associated with photodissociation, spontaneous Restrictionson complexityof theprogramemissionor predissociationfrom vibration—rotationlevelsof a Current dimensioningconsiderationspermit considerationofsingle (bound) initial-state potential into continuum levels of transitionsat up to 150 frequenciesand thecalculation and(ifoneor two distinct final electronicstates,and if desired,fits to desired)fitting of data for thermalinitial-state vibration—rota-experimentaldata to determineparameterscharacterizingthe tion populations corresponding to up to 4 temperatures.final-statepotential and/ortransitioncoupling functions. Simultaneoustransitionsto continuaassociatedwith up to two

(uncoupled)final electronicstatesmaybeconsidered.The finalMethodofsolution statepotential(s) and transition coupling function(s) are as-For arbitrary smoothpotentials,theradial or effectiveone-di- sumedto have particularfunctional forms(see long write up).mensionalSchrodingerequationis solvedusing the Numerovalgorithm to determinethe discreteinitial-state eigenvalue(s) Typical running time

andeigenfunction(s)and continuumfinal-statewavefunctions, The running time dependsentirely on the complexity of thewhich areusedwith the specifiedtransitioncoupling functions calculationbeingperformed.Forexample,a simplecalculationto calculatethedesiredmatrix elements, of the predissociationrates for thirteen levels required 5 s,

while threecyclesof an eight-parameterfit to thermal absorp-tion coefficients for 26 frequencies at two temperatures

required776 s on a SUN-3/260 with floating point accelerator.

LONG WRITE-UP

1. Introduction transitionintensitiesassociatedwith photodissoci-ation, spontaneousemission, or predissociation

Bound—* continuumtransitionsare widely en- from vibration—rotationlevels of a single(bound)counteredphenomenain chemical physics and initial-statepotential into continuumlevels of onemolecular spectroscopy.Quantitativesimulations or two distinct final electronicstates.If desired,itof photodissociationcross-sections(or absorption can also perform non-linear least-squaresfits tocoefficients) and continuum emission intensities experimentaldata to determinefinal-statepoten-are thereforevery useful for interpretingandpre- tial and/or coupling function parameters.Thedicting a wide variety of spectroscopicphenom- discussionbelow, and the sampledatafiles listedena, and for determining potential curves and with the program, are all concernedwith transi-transition moment functions from experimental tions of diatomic molecules,and the natureanddata[1,2].Similarly, studiesof radiationlessbound structureof the programretain strong memories

continuum transitions are increasingly being of this origin. However, such calculations may

usedto help improveour understandingof inter- also be used in approximatetreatmentsof vibra-statecoupling [3] andmolecularenergyredistribu- tional predissociationof Van derWaalsmoleculestion phenomena[4,5]. All of these phenomena [4,5] or polyatomicmoleculephotodissociation.dependon exactlythe sametype of radial matrix Thefundamentaltheoryunderlyingthesecalcu-element,and the presentpaperdescribesa com- lationsis nicely summarizedin a recentreview byputer programfor performing all three types of Tellinghuisen [2], while aspectsof the practicalcalculations. methodologyare describedin moredetail in refs.

The original version of the programdescribed [1,6,7]. In the following, section 2 presentstheherein was developed in order to address the basic equationsbeing used, while section 3 de-problem describedin ref. [1], simulation of the scribeshow the program functions and outlinesthermal absorptionspectrumof Br2, and the use some of its options.Section 4 then describestheof least squaresfits to experimentaldata to de- structure of the program and the roles of itstermine parameterscharacterizingthe potentials various subroutines,while section 5 states theand the transition momentfunctions for the two input/outputconventionsand indicatesthe unitsdifferent final electronicstatescontributingto this assumedfor the physical parametersof interest.spectrum.Subsequentgeneralizationsnow allow it The program’soperationis controlledby the con-to be used for calculatingthe bound—s continuum tentsof a data file which is read (on channel5)

R.J. Le Roy/ Bound-~ continuumintensities 3.85

during execution. The structureof this data file or by definingthe densityof statesto beunity andandthe significanceof the variousread-in param- requiring the asymptoticwavefunctionamplitudeetersaredescribedin detail in appendixA of the to be ~/p(E) [10]; the former convention isprogrammanualof ref. [8], while commentedlist- adoptedhere.ings of the varioussegmentsof the codeare found The three physical quantitieswhich may bein its appendicesB—F. Finally, section 6 below calculatedwith thisprogramare: (i) photodissoci-describesthe timings for three illustrative test ation cross-sectionsfor absorption of light byproblems,the first of which yields the test run moleculesin bound vibration—rotationlevels, (ii)ouput found at the end of this paper; complete Einstein emission coefficients for spontaneouslistings of the output obtained for all threecases bound— free emissionfrom givenvibration—rota-are found in appendixG of the programmanual tion levels, or (iii) the rate of a radiationless[8]. bound ....s free predissociationprocess,as given by

the Golden Rule approximation. In general,allthreeprocessesmay be accompaniedby a change

2. Swnmaryof the theory in the rotational quantum number, from J to J’.However, since the final states lie in the con-

The core of the programis concernedwith the tinuum, transitionscorrespondingto different val-evaluationof radial matrix elementsbetweenthe ues of (J‘-J) are superimposed.Since the corre-eigenfunctions“~v,J for discreteinitial-state levels spondingradial matrix elementstend to changewith vibrational and rotational quantumnumbers fairly slowly with (J’ — J), it is usually a veryv and J, respectively,and the energy-normalized good approximationto overlook this dependencefinal-statecontinuumwavefunctions“Ef’ for en- [1]. For the radiative processes(i) and (ii), theergyE andangularmomentumquantumnumber normalization propertiesof the rotational inten-

as mediatedby the transition coupling func- sity factorsallow the total transitionrateobtainedtion M( R): on summingoverthe rotationalbranches(J’ val-

M(v, J; E, .1’) = <‘I’v,J I M(R) I ~‘E.J’. (1) ues) associatedwith a given initial J-stateto becollapsedas:

The two wavefunctionsare solutionsof the radialSchrodingerequation ~[s~~~/(2J + 1)] V’~E,J’ I M(R) I ‘I’~,,/12

2~dR2 + v(R)+_J+1)l~=E~.~ [ 2~ R2 ] I <~~‘E,JI M(R) I ‘I’~) 12, (5)(2) whereS,~,is theusualrotational intensityfactor

This equationis implicitly solvedusingthe dimen- [2]. For most types of predissociation,exactly thesionlessdistancecoordinatex = R/R

0, where R0 sameapproximationwill apply, althoughthe cou-= 1 A. The boundinitial-statewavefunction‘~I’~J pling function may dependon J [3]. This “Q-

branch” approximationis built into the presentis unit normalized,program;however,ausermayremoveit by intro-

( ~I’,7,(x) ‘I’~(x) dx = 1 (3) ducing a loop over allowed values of (J’ — J)‘0 which would inclose lines 573—673of the main

while the final-statecontinuumwavefunction“E,J’ program (see appendixB of ref. [8]), and gener-is normalized over both distance and energy alizing the factor HONL appearingin line 577 of[1,9,10]. The latter condition may be imposed the presentcode.either by scaling the sinusoidalcontinuumwave- Utilizing the approximationof eq. (5), makingfunction to asymptoticallyhave unit amplitude the substitution ‘J’~’ = ~/p (E) “Ef’ whereand introducingthe densityof statesfactor: ~I’~J(x) is normalizedto asymptoticallyoscillate

with unit amplitude, and inserting appropriate1 1/2

p(E) = R0{8~tc/h[E— V(oo)j } (4) valuesfor the physical constants[11], the expres-

386 R.J. I.eRoy / Bound—. continuumintensities

sions for the threephysical quantitiesof interest Note that to facilitatecomparionswith experi-here are [2,3]: (i) for the photodissociationcross ment, the quantity actuallycalculatedfor case(i)section,in units [A2/molecule], is thedecadicmolarextinction coefficient, in units

[I,/mol cm],a~(v,J; E)

= [8~~PGei/(3hC)1I(’~EJI M(R) I ~v.i) 2 ç(v, J; E)

=3.2269566x10~~/~~v =NAa~(v,J)/1013 ln(10)

/ QG_ \2 =8.4397201~ii7~vx~f ~j(X)Me(X)~~j(X) dx)~ (6)

0 / r~—x(~J i(x)Me(x)’1~v.i(x)~) , (9)

(ii) for the Einstein emissioncoefficient, in units °

[s’/cmt], where NA is Avogadro’snumber. Eqs. (7)—(9) are

A,,(v, J; E) the fundamental expressionswhich the presentprogramsetsout to evaluate.

= [64iT~V~Gei/(3h)1IK~’E,JI M(R) I ~v.J) 12

= 2.4313849X 10_8~/ii7~v3

2 3. Outline of program operation and optionsX(f~i(X)Me(X)~J(X)dX) (7)

0 3.1. Determiningthe radial matrix elements

and(iii) for the Golden Rulepredissociationrate,in units [s~] (assumingeq.(5) holds), Oneof the first thingswhich mustbe specified

whenusingthis programis which type of physicalk(v, J) = [4TT2Gei/hJ I<~1’E,JI M(R) I “vi) 12 quantity is to be calculated: absorption coeffi-

= 9.17555390 < io~’~i7~ cients,spontaneousemissionratesor predissocia-tion rates.This choice is definedby the valuefor

I ~ \M ( ~ ~ d ~2 the integerparameterIFRPW read on the thirdX E,JkxJ e~XJ ~ X

1 line of the datafile (seeappendixA of ref. [8]). Its

(8) valueshouldcorrespondto the powerof r’ appear-ing in the correspondingtheoreticalexpressionof

Here, v[cm’] is the frequencyof the absorbedor eqs. (6)—(9): 1 for absorption,3 for spontaneousemitted light, the reducedmass~uhasunits amu, emissionand0 for predissociation.the final-statekinetic energyTf = [E — V~(R= c~c)] In the current version of this program, theis in cm

t, the function Me(x)= GeiM(R) has bound —s continuum radial matrix elementsmayunits debyefor absorptionor emissionandcm~ becalculatedin oneof two ways: (a) using“exact”for predissociation,and Gei = g~(0/ge (i) is the quantummechanicalwavefunctionsgeneratedbyratio of the final-state(f) to initial-state (i) elec- numerical solution of eq. (2), or (b) using Hunttronic degeneracyfactors[2]. Sinceit is not experi- andChild’s uniform harmonicapproximation[12].mentally separable,this last factor hasbeensim- The methodusedis determinedby the valueof theply incorporatedinto the effective transition cou- integerparameterlORD readin thesecondline ofpling function Me(x). Note that while Me(x) is the datafile; for lORD � 2 quantalwavefunctionssimply a function of internucleardistancefor most are always used, for lORD � 0 only the uniformoptical transitions,for predissociationit may also harmonicapproximationis used,while for lORDdependexplicitly on J. In the latter case, an = 1 the uniform harmonicapproximationis usedappropriateweighting should be introducedinto whenever possible,but the numerical approachthe factorHONL definedon line 577 of the main automaticallyreplacesit wheneverthe former isprogramlisted in appendixB of ref. [8]. inappropriateor breaksdown.

R.J. I.eRoy/ Bound—s continuumintensities 387

As describedin ref. [6,12],theuniformharmonic An appropriatemesh sizemay thus be calculatedapproximationis a very reliable procedurewhen usingthe expressionthe conditionsfor its validity are satisfied,and(atleastwith the presentcode) it requiressomewhat RH = ¶/ NH [( ~~/168576314)less computertime thana full quantalcalculation. xmax{E— V(R)}1”2] (10)However, in many applications,the savings in-volved (typically less thanca. a factorof 2) are not where NH is the desiredminimum number oflarge enoughto warrant the introduction of any points per wavefunctionnode, and the quantityapproximations.Calculations using this method max{ E — V(R)) is the maximum local kineticare performed in subroutines PREMSC and energy (in cm’1) encounteredin the initial orOVRMSC (seeappendixC of ref. [8]), anda user final state.SinceanNH valuewhich is too smalluninterestedin this option who wishes to shorten yields resultswhich are unreliable,while too largethe code could removetheseroutines and refer- avaluewill requireexcessivecomputationaleffort,encesto them andto the parameterlORD, without the user is advised to empirically determinetheprejudiceto the restof the program’sabilities. smallestvalueof NH which yieldsresultsof suffi-

The core of the quantalcalculationis the solu- cient accuracyfor the applicationof interest.tion of eq. (2) to determinefirst the eigenvalue(s) For both initial and final states,the numericaland the eigenfunction(s)‘I’0,,~( R) of the bound integrationstartsat adistanceRMIN (readstate-initial-statepotential V

1 (R), andthen the (asymp- ment #12) which must lie sufficiently far insidetotically unit amplitude)continuumwavefunction the inner turning points for both statesthat the‘I~EJ(R)of the final-state potential Vf(R). The wavefunctionamplitude has decayedto a valuefirst of these wavefunctionsis generatedin sub- severalordersof magnitudesmaller than the am-routineSCHRQ(seeappendixEof ref. [8]), which plitude in the classicallyallowedregion.Thepres-is based on the famous Cooley—Cashion—Zare entversion of the codeprints appropriatewarningroutines SCHR [13—17],but incorporatesspecial messagesif this decayis notby a factor of at leastfeaturessuch asthe ability to automaticallylocate ~o 8 and a smallerRMIN valueshouldperhapsquasibound levels and calculate their widths be adoptedin such cases.On the other hand,if[18,19]. Its capabilities are discussedelsewhere RMIN lies too far into the classically forbidden[13—19],andwill notbe consideredfurther here. region, where [V( R) — E] is extremelylarge, the

Eq. (2) is solvedfor thecontinuumlevels of the integrationalgorithm could becomeunstableforfinal-statepotential(s)in subroutineOVRLAP (see the given mesh size. If this situation is encoun-appendixC of ref. [8]). It is basedon the same tered,a warningmessageis printedandthe begin-Numerovnumericalintegrationprocedureusedin ning of the integration range is automaticallythe boundstatesubroutineSCHRQ,andthe same shiftedoutwardssothat no errorsarise. However,radial mesh is used for both wavefunctions.The useof a slightly larger valueof RMIN will causeoverlapintegralsof eqs.(6)—(9) are thereforeaccu- thesemessagesto disappearand (slightly) reducemulated as the final-statewavefunction is being the computationaleffort. For most cases,a rea-generated,so the latter neednotbe committedto sonablevalueof RMIN is ca. 0.6—0.8 times thememory. smallest inner turning point encounteredin the

The accuracyof both the initial- andfinal-state calculation.wavefunctionsis largelydeterminedby the sizeof While continuumwavefunctionsasymptoticallythe (fixed) radial meshRH (readstatement#12) become simple sine functions of constantampli-used in the numericalintegrationof eq. (2). For tude,conservationof computertime and minimi-potentialswhich arenot too steepor too sharply zation of accumulatingtruncationerror requirescurved, “adequate” accuracyis usually obtained theuseof aconvergencecriterionwhichallows theby useof a meshwhich allows for a minimumof outwardnumericalintegrationto be terminatedasfrom 10 to 30 meshpointsbetweenadjacentwave- early as possible. In the present subroutinefunction nodesin the classicallyallowed region. OVRLAP, this is accomplishedby fitting succes-

388 Ri. I.e Roy/ Bound—s continuumintensities

sive maxima of the oscillatory final-statewave- replacedby an NJ-point quadrature.The calcula-function to the semiclassicalor WKB form tion of the characteristicaverage J ‘s (in sub-

routine JAVGE, seeappendixD of ref. [8]) ne-‘~‘E.J(R) glects the effect of centrifugal distortion, but in

= f A/ [E — V (R)] 1/4 ‘~ view of the (normally) fairly gradualvariation ofradial matrix elementswith J, this should not

x sin1— (2pyh2)l/2

1R[ E — V,( R’)] 1/2 dR’ 1. introducesignificanterrors.Non-thermal rotational populationsmay also

(11) be treated in two ways. If the maximum numberof initial-state vibrational levels considered

The radialintegrationis assumedto be converged (parameterNV, readon dataline #2) is NV> 1,if the A valuesobtainedat threesuccessivemax- setting NJ = 0 causes the rotational quantumima agreeto within the relativeerror specifiedby numberto be fixed at J = 0. However,for NV = 1parameterOVRCRT (readon line #2 of the data andNJ = 0, J is fixed at the valueJFIX obtainedfile, appendixA of ref. [8]); typically, OVRCRT ~ via readstatement#11.1 x iO~. WhenNJ � 0, thethermal populationfactor for

each vibrational level is assumedto be propor-3.2. The initial statepopulationfactors tional to (kB T/B,,)e— Efl/k B~, whereE,, and B1, are,

respectively,the read-in vibrational energyEV( v

For absorption or emission from a thermal + 1) androtationalconstantBV(v + 1) for level v.

initial-statevibration—rotationpopulation,the ob- Valuesof thesemolecularconstantsentervia readservableproperty is a sum of terms of the type statements#9 and 10 (seeappendixA of ref. [8]).seen in eqs. (7) or (9), each weighted by the For NV> 1, theseread-in constantsare assumedappropriateBoltzmann populationfactor. In this to correspondto vibrationallevelsv = 0 to (NV —

case,the number of temperaturesto be consid- 1). While all NV of thesevibrational levels maybeered,NT, and their values,TMP(i), are read on included in the thermal vibrational populationlines #3 and 4 of the data file. However, if the (see data line #2), in the actualcalculation, theinitial-statevibrational androtationalpopulations sum over v is truncatedwhen fraction POPCRTare not thermal,onesetsNT = 0 and no valuesof of the total vibrational population is accountedTMP(i) are read in. for. Current array dimensionslimit NV to values

A sum over a thermal rotational population � 20.may be performedin one of two ways. First of all, Inside the program,both the trial initial-statefor negativevaluesof parametersNJ (readon data eigenvalues generated from the expressionline #2), a Boltzmann-weightedsumis performed E(v, J) = E1, + B)J(J + 1), and the thermalfor (integer)rotational quantumnumbersJ rang- population factors usedwhen NJ � 0, are calcu-ing from 0 to a maximum value of I NJ I. This lated using the read-in vibration—rotation con-direct sum is truncatedwhen the fraction of the stants.However, the discreteinitial-stateenergiesrotationalpopulationaccountedfor is greaterthan combined with the various transition frequenciesPOPCRT(read on data line #4). However, this to define the correspondingfinal-stateenergiesEapproachtends to be very timeconsuming. were the actualvaluesobtainedon solving eq. (2)

The secondway of summingover a thermal for that level.rotational population, invoked by settingNJ > 0,is the proceduredescribedin sectionII.B.2 of ref.[1]. In this case, the direct sum is replacedby a 3.3. Thetransition momentor couplingfuncttonsum over the contributions associatedwith theaverageJ valuesin eachof NJ equally-weighted SubroutinesOVRLAP or OVRMSC return val-segmentsof the rotational population for each ues of the overlap integral between ~ ( R) andvibrational level. In effect, the full sum overJ is ‘I’~,~(R),denoted FCA, plus the first two mo-

R.J. I.eRoy/ Bound—s continuumintensities 389

mentsof the property representedby the array 3.4 The initial-statepotential V5(R)TMF:

Values of the parametersdefining the bound

FCM = fs.1’;~(R)TMF(R)’I’0~(R)dR, initial-statepotentialare obtainedfrom readstate-0 ments #13—15. One may choose to define this

(12) potential eitherby a set of NTPI turning points

{RTPI(i), VTPI(i)} entering via read statementFCM2 = f ~‘~(R) TMF(R)

2~t’0~(R)dR. #14, or whenNTPI � 0, by an analyticfunction.

0 In the former case,NUSEI-point piecewisepoly-(13) nomials are used to interpolateover the read-in

turningpointsto producethe arraywith mesh sizeIt is often convenient to represent the transition RH required for the numerical integration, andcoupling function M(R) as a power series in a extrapolationto largeror smaller distancesis ef-shifteddistancecoordinate, fected by fitting exponentialsto the first and/or

TMF(R) = R— R,~, (14) last two or three read-in turning points. The

parameter VLIMI defines the energy at the

wherethe distanceR~ REX (seereadstatement asymptote of this potential, while the quantity#12) is the distanceabout which this function VSHFTI is an energy shift which (if non-zero)and the final-state potential(s) (see below) are must be addedto the read-in turning point en-expanded.In this case,the radial matrix element ergiesto make them consistentwith the value ofappearingin eqs. (6)—(9) is simply VLIMI.

If the user wishes to define the initial-stateM( v, J; E, J) = M0 X FCA + M1 X FCM + M2 potential by an analytic function rather than an

x FCM2 + ... (15) array of points, the integerparameterNTPI ofreadstatement#13 shouldbe set � 0, andwhile

whereM0, M1 and M2 are expansioncoefficients VLIMI retains the significance indicated above,whose (initial) values enter via read statement the othertwo parametersreadon this line become#16. If fits to experimentaldataare beingusedto dummyvariables.In this option, the programthendetermineor optimize theseexpansionparame- skips down to read statement#15 where it ob-ters, the requisitepartial derivatives {DFVB(j)} tains values of the parameterscharacterizingtheare readilydeterminedfrom theseFCA, FCM and analyticpotential of interest.As indicatedby theFCM2 values(see lines 649—660 of the listing in commentsand codein lines 323—347 of the mainappendixB of ref. [8]). program(seeappendixB of ref. [8]), the present

If desired, the user may change to a more version of this program allows for an analyticsophisticatedtype of the M(R) function by sim- initial-state potential with the generalizedHFDply redefining the TMF array generatedin line form [20,21]255 of the mainprogram;oneexampleof how this J~(R)= A, e1

3R — C6D6(R)/R

6— C8D8(R)/R

8maybe done,invoked by settingparameterNTPF (16)� 0 (see read statement#17), appearsin lines434—452 of the main program(see appendixB of definedby read-in values of the parameters$ =

ref. [8]). Of course,consistentvaluesmustbe used BETAI and C6 = C61, andof the well depthEPSI

for the expansioncoefficients M1 and for the and equilibrium distanceREQI which implicitlypartial derivativesrequiredfor the fits. Note,how- determineA1 andC8 [20], while D6(R) andD8( R)ever, that whateverfunctional form is used, the are “damping functions” of the form recom-programassumesthat a transition momentfunc- mendedby TangandToenmes[21]:tion will have the units debye while a couplingfunction driving predissociationwill have units D~(R)= 1 — e~ ~ (13R)

m/m!. (17)cm~. mO

390 Ri. Le Roy / Bound -.~ continuumintensities

However, a user may readily introduce his/her and D11 and A11 are definedby fitting eq. (18) toown favourite functional form by modifying this the two innermostread-in turningpoints.short segmentof code. A specialized“final-state-I” potential case is

invoked by settingNTPF= 0 (andNTPI = 0) andreading appropriate values for the parameters

3.5. Thefinal-statepotential(s) Vf (R) DVSHFT andCSCALE (readstatement#20). Inthis case,the coupling functionTMF(R) is defined

The program performs calculationsfor transi- as the constantCSCALE timesthe first derivativetions into continuumlevels of eitherone or two of the initial-statepotential of eqs. (16) and (17),final electronic states (see parameterIFREE on while the final-state-i potential is set equal to thedata line #2). Three options for specifying the initial-statepotential plus a sumof this TMF( R)final state(s)of thesetransitionsare possible:(i) it functionwith a constantenergyshift of DVSHFT.may be defined (at least in part) by a set of This option was introduced to facilitate certainread-in turningpoints,with additional parameters model studiesof vibrational predissociation[4,22].specifying its form at smaller distances (case A purely analytic final-statepotential is (cur-IFREE= 1), (ii) it may be a purely analytic func- rently) assumedto havethe functional form:tion (caseIFREE= 2), or (iii) simultaneoustransi-tions may occur to one state of each type (for fr~(R)= D12 + Af2 exp{ — [a1(R — R,)IFREE~ 1 or 2).

The parameterscharacterizing a final-state +a2(R — R,)2+ 1 }, (19)

potentialdefinedat least partly by read-inturningpoints (cases IFREE * 2) enter via read state- where R = REX is the usual reference length,ments #17—19 (seeappendixA of ref. [8]). This D

12 = VLIMF2 the energy at the potential asymp-form is appropriate, for example, for cases in tote, and the the factor A~2and the exponentwhich a final-state potential well has been de- parameters{ a1 } are definedby read-in valuesof

termined(perhapsby RKR inversion of discrete the potentialand its first threederivativesat R =

spectroscopicdata) but the short-rangerepulsive Rv (see readstatement#21).wall responsiblefor the continuum spectrumisunknown. As for a pointwise initial-state poten- 3.6. Fitting to experimentaldatatial, the read-inparametersconsistof the numberof turning points NTPF, the order of piecewise In addition to performing “forward” calcula-polynomial NUSEF, the energy at the potential tions of absorptionor emissionintensitiesor pre-asymptoteVLIMFI, and(if non-zero)the energy dissociation rates, the presentprogram may beshift VSHFTF to be addedto the read-in turning usedto perform least-squaresfits to experimentalpoints {RTPF(i), VTPF(i)} to make them C0fl data to determine fina/-state potential and/orsistent with VLIMF1. In addition, exponent transitioncoupling function parameters.Suchfitsparametersb1 and b2 characterizingthe short- are complementaryto direct inversion methodsrangeextrapolationof this potentialwith the func- for such properties [6,7], and may be usedtion form wheneverthe moredirect methodsare not suitable

[7].v~( R) In the presentversionof the program,suchfits

= D11 + A11 may simultaneously optimize: the exponentparametersof eq. (18) (D~1and A~1beingdefined

xexp( — [b1(R — R~)+ b2(R — R~)

2J } by the innermostturningpoints), the final-state-2(18) potential value and derivatives at R = REX

(which define parametersA~2and { aj } of eq.

entervia readstatement#19, whereR~= REX is (19)), andup to threecoefficientsin an expansionthe referencedistancereferred to in section 3.3, of the transition coupling function for each final-

R.J. LeRoy/ Bound—s continuumintensities 391

statepotential in powersof thevariable TMF(R) The least-squaresfits themselvesare performed(seesection3.3). The decisionregardingwhethera usinganiterativesteepestdescentsalgorithmbasedgiven variableis to be variedor held fixed in a fit on useof the multiple linear regressionsubroutineis governed by the values of the integer array packageMLREGR listed in appendix F of ref [8].(IVB( j) } enteringvia read statement#22. Note However,auserwho doesnot wish to makeuseofthat least-squaresfits of the type allowed by the the fitting capabilities of the programmay sim-presentprogramare fundamentallydifferent than plify his/hercodeby simply omitting this packagesimplemanualtrial-and-errormethods,in that all and replacingit by the dummy routine listed atparametersare varied simultaneouslyin a con- the end of appendixD of ref. [8].certed way, and that realistic estimatesof thecorrelatedparameteruncertaintiesare obtained.This allows much more sophisticated and 4. Programdescriptionmeaningfulfits to beperformedthanwould other-wise be possible. The presentsection lists the namesand mdi-

As in any least-squaresfit, values must be catesthe functionsof the varioussubroutinesusedgeneratedfor the partial derivativesof the (calcu- by this program,and indicatestheir heirarchy.Inlated) observableswith respectto the parameters particular,the level of indentationin the followingof the model. For transition coupling functions listing indicates which subroutines call whichexpressedas a simplepower seriespowerseriesin others;unlessstatedotherwise,eachsubroutineissome known function, the associatedpartial de- called exclusively by the immediately precedingrivativesmaybe obtainedby manipulationof the routinewith onelower level of indentation.overlapmomentsFCA, FCM, FCM2, . . . ,etc.(see

BCONTlines 649—660 of the programlisting, appendixBof ref. [8]). However, derivativeswith respectto the main program which reads all input data,the final-statepotential parametersmust be ob- preparespotentials, calls the eigenfunctionandtamednumericallyby taking differencesbetween overlapintegral routines,and(if desired)manages

the iterativeleast-squaresfitting procedurevalues calculated using incrementally differentparametervalues.The parameterincrementsused 11DAMPshould be small enough that the resulting first for the analytic initial-state potential of eq.differencesprovideaccurateestimatesof the cor- (16), TTDAMP calculatesthe valuesand firstrespondingderivatives,yet not so small that accu- derivativesof the dampingfunction of eq. (17)mulated numerical round-off errors affect their for n = 6 and8reliability. For secondandsubsequentcyclesof afit, appropriatevalues of theseparameterincre- PREJAVments are generatedinternally in a mannerde- for a thermal initial-state rotationalpopulation,fined by the value of parameterADACFT (see calculatesfactors determininghow to split theread statement# 6), while the fractionalparame- rotationalpopulationinto NJ segmentsof equalter changes{ FDVB( j )} defining theseincrements weight anddeterminethe averageJ for eachfor the first fitting cyclearereadon thelast line ofthe datafile. JAVGE

The data file for this casemust,of course,also for NJ > 0, it uses factors generatedin PRE-include the experimentalvalues of the quantity JAY to calculatetheNJ average-fvaluesrepre-beingfitted; theseentervia readstatements#6—8, sentingthe thermalrotationalpopulationfor awhich also read values of the convergencecrite- particularvibrationalleveltion for the fits, FITCRT, and of the constantADACFT characterizingthe potential parameter SCHRQincrementsusedto define the partial derivatives solves the Schrodingerequation to determinerequiredfor the fits. the eigenvalueand(unit normalized)eigenfunc-

392 Ri. I.e Roy/ Bound —s continuumintensities

tion of the vibrationallevel lying nearestto the MLREGRinput trial energy a self-containedpackageof subroutinesfor per-

forming multi-linear regressionand returningQBOUND fitted parametervaluesand their correlatedun-for quasiboundlevels (thoselying abovethe certainties.potential asymptote,but behind a potentialbarrier),definesthe Airy function boundary A detaileddescriptionof the data file contain-conditionat the third turningpoint required ing definitions for all input parametersand de-for the eigenvaluecalculation scribing the options they control is presentedin

appendixA of the programmanualof ref. [8].WIDTHa second entry to QBOUND, used whencalculating the tunneling predissociation 5. Units, physical constantsand input/outputcon-lifetime or width of a quasiboundlevel ventions

LEYQAD Unless otherwisespecified, the units of lengthcalled by QBOUND (from entry WIDTH) and energy used throughout this program, andand used to calculate integrals over the assumedfor all input data, are A and cm’,potentialwell andbarrier respectively. The only noteworthy exception is

that the transition dipole function governing ab-OVRLAP sorptionor emissionis assumedto be in debyefor a specified energy on a given final-state (althoughthe analogouspredissociationcouplingpotential, calculatesthe exact (numerical)con- function is in cm~).tinuum wavefunctionandreturnsvaluesof the The valuesof the physical constantsappearinoverlap integral (FCA) and overlap moments the programin threeforms.The first is the factor(FCM andFCM2) of eqs. (12) and(13) 2~t/h2= ~t[amu]/16.8576314[cm’A2] appearing

in the radial Schrodingerequationof eq. (2); thePREMSC secondis in the collectionsof termsdefining thewhen using the Hunt—Child [12] uniform numericalfactors in eqs. (6)—(9), and the third isharmonicapproximation to evaluatebound —s in the value of Boltzmann’sconstantkB in unitscontinuum matrix elements, begin by having cm~.Values of the requisiteconstantswere takenPREMSClocatethe initial-state turningpoints from the 1986 compilation[11].and vibrational energyderivativefor eachdis- The program reads input data on channel 5,creteinitial-state level writes standardoutput to channel 6, and op-

tionally (controlledby parameterIPNCH of readOVRMSC statement#2) writes a condensedoutput file tofor a specified energy on a given final-state channel 7. For computersnot running a SUNpotential,calculatesthe overlapintegral (FCA) Microsystems-typeUNIX operatingsystem, theand overlap moments(FCM and FCM2) im- call to I/O conditioning subroutine“ioinit” onplied by the uniform harmonicapproximation line 49 of the main program(see appendixB of[12] ref. [8]) shouldbe removedor replaced.

INTERPperformspiecewisepolynomialinterpolationon 6. Sampledataand output files, andtiminga given array of points and returns the func-tions value and (if desired)its derivativesat a The running time for thisprogramwill dependspecified point; called by PREMSC and entirely on the complexity of the calculationbeingOVRMSC, as well asby BCONT performed.Appendix G of the program manual

R.J. I.eRoy/ Bound —s continuumintensities 393

[8] presentssample data files and the resulting tance provided by R.N. Zare’s seminal 1964output for threecaseswhich illustrate someof the Franck—Condonintensity factor paper and thetypes of problemsto which this programmay be associatedreports [14,16,17]. This researchhasapplied.Whenrun on a SUN-3/260 workstation beensupportedby the Natural Sciencesand En-computerwith Floating Point Accelerator, they gineeringResearchCouncil of Canada.requiredthe computationtimeslisted below.

Case (i). Predissociationof levels v = 0—12 ofB(3H~~)-stateBr

2 onto the ~Hiu statepotential References(seefig. 26 of ref. [2]): 5 s.

Case(ii). Emission from level v = 13, J = 14 of [11R.J. Le Roy, R.G. Macdonaldand G. Burns, J. Chem.the d(

3111) stateof NaK into the a(

3~~)state Phys.65 (1976)1485.

continuumat 100 frequencies,calculatedusing [2] J. Tellinghuisen,Adv. Chem. Phys.60 (Photodissociation

the uniform harmonicapproximation(see ref. and Photoionization,ed. K.P. Lawley) (1985)299.

[7]): 7 ~. [3] H. Lefebvre-Brionand R.W. Field, Perturbationsin theSpectraof DiatomicMolecules(AcademicPress,Toronto,

Case (in). Three cycles of an eight parameter 1986)chap. 6.least-squaresfit of thermal absorptioncoeffi- [41J.A. Beswick and J. Jortner,Adv. Chem. Phys.47 (Pho-

cients for Br2, involving transitionsonto two toselectiveChemistry,part I, eds.J. Jortner,RD. Levine

final-statepotentialsat 26 frequenciesand two and S.A. Rice) (1981) 363.

temperatures(seeref. [1]): 776 ~. [5] K.C. Janda,Adv. Chem.Phys.60 (PhotodissociationandPhotoionization,ed. K.P. Lawley) (1985)201.

[6] M.S. Child, H. EssénandR.J. Le Roy, J. Chem. Phys.78(1983)6732.

The data file and resulting output for case (i) [71Ri. Le Roy, W.J. Keoghand M.S. Child, J. Chem. Phys.

comprisethe test run dataset and output listed 89 (1988) 4564.

below. This data file illustratesthe fact that the [8] RJ. Le Roy, University of Waterloo Chemical PhysicsResearchReportCP-329(1988); alsoavailablefrom theprogram can executecalculationsfor more that CPC ProgramLibrarian.

one case in a single run. In particular, at the [9] R.A. Buckingham,in: QuantumTheory I, Elements,ed.completionof eachcalculation,control returnsto DR. Bates(AcademicPress,New York, 1961)chap. 4.

the first readstatement,andif the input data file [101A. Messiah,QuantumMechanics,vol. 2 (North-Holland.

containsadditionaldata,executioncommencesfor Amsterdam,1962) section17.4.

that second case. In this example, the second [111 E.R. Cohenand B.N. Taylor, Phys.Today BGI1 (August1987).

calculationis for thesamephysicalproblemas the [12] P.M. Hunt and M.S. Child, Chem.Phys. Lett. 58 (1978)first, exceptthat thecalculationis performedusing 202.

the uniform harmonicapproximationratherthan [131J.W. Cooley,Math. Comput.15 (1961) 363.

exactnumericalwavefunctions. [141RN. Zare and J.K. Cashion, University of CaliforniaLawrence Radiation Laboratory Report UCRL-10881(1963).

[151J.K. Cashion,J. Chem. Phys.39 (1963) 1872.[161 R.N. Zare, University of California LawrenceRadiation

Acknowledgements LaboratoryReportUCRL-10925(1963).[17] RN. Zare,J. Chem.Phys.40 (1964)1934.[18] R.J. Le Roy andR.B. Bernstein,J. Chem. Phys. 54 (1971)

I am pleasedto acknowledgethe foresight and 5114.stimulation provided by ProfessorGeorge Burns [19] R.J. Le Roy, University of Waterloo Chemical Physicswho instigatedmy work on developingthe initial ResearchReportsCP-230(1985)or CP-330(1988).

version of this photodissociationcode in 1965, [20] R.J. Le Roy and J.M. Hutson. J. Chem. Phys.86 (1987)837.

long beforequantalcalculationsof this type were [21] K.T. Tang and J.P. Toennies,J. Chem. Phys. 80 (1984)

commonlyenvisioned.In the samehistorical vein, 3726.

I mustalsoacknowledgethe inspirationandassis- [22] R.J.Le Roy, unpublishedwork (1988).

394 Ri. I.e Roy/ Bound—~ continuumintensities

TESTRUN OUTPUT

Input data file read on channel

B——>l)PI)u State Predissociation Calculation for Br2o 2 2 13 0 1 1 —1 1 i.d—6 )NFIT 0050 IFREE NV NJ IRDAT NEWPOTIWR .. etc.O 1 0 0 (NT NFR IRWAVL IFRPW)

0.0 WAVL)i)15986. 16149. 16310. 16468. 16622. 16772. 16920. 17060. 17202. 17340. 17470.17600. 17722. EV)i).08093688 .08062135 .08030483 .07998644 .07966546 .07934242 .07901740 .07868937

.07835899 .07802770 .07769376 .07736 .07702 NV)!)39.95241 1.9 .0015 2.30 )ZMU RuIN RH REX)34 8 19742.15 15912.471.dO 1.dO

2.43880560481947 1436.436176 2.44847730819190 1300.5686762.45911757277757 1160. 941176 2.47092859694711 1017.6136762.48419656426633 870.646176 2.49934871959364 720.0986762.51707284721730 566.031176 2.53661036947724 408.5036762.56668908654296 247.576176 2.61075354611893 83.3086762.67780117351355 0.000000 2.75410909071243 83.308676

2.60713051707846 247.576176 2.86475092625963 408.5036762.90644235512562 566.031176 2.94496421077575 720.0986762.98159397214687 870.646176 3.01706031006518 1017.6136763.05183662324252 1160.941176 3.08626102163155 1300.5686763.12059398887302 1436.436176 3.15504947213098 1568.4836763.18981324084404 1696.651176 3.22505462789697 1820.8786763.26093464164737 1941.106176 3.29761204037464 2057.2736763.33524828934425 2169.321176 3.37401198351152 2277.1886763.41408314989862 2380.816176 3.45565776859132 2480.1436763.49895676083168 2575. 111176 3.54422889805589 2665.6586763.59175906524902 2751.726176 3.64187867166909 2833.2536761.0 0.0 0.016057.OOdO 7654. —35490. 151100. 0.0B——>i)pi(u State Predissociation Calculation for Bt2

0 1 2 13 0 0 0 —1 1 1.d—6

Resulting output file written to channel

B——>1(PI(u State Predissociatlon Calculation for Br2

For each of the 1 frequencies (cm—i) 0.0

Calculate Predissociation Rates in units: (sec—i)

Perform calculations using exact (numerical) final—state continuum wave functions

assuming asymptotic amplitude converged when constant to within 0.1OD—05 at 3 successive maximaFix J—0 rather than stun over a rotational distribution

Consider initial—state levels up to v—12 whose energies are 15986.000 16149.000 16310.000 16468.000 16622.00016772.000 16920.000 17060.000 17202.000 17340.00017470.000 17600.000 17722.000

and By—s are 0.0809369 0.0806213 0.0803048 0.0799864 0.07966550.0793424 0.0790174 0.0786894 0.0783590 0.07802770.0776938 0.0773600 0.0770200

Seduced mass ZMU — 39.952410(u) yields 92 — 2.3699895 and BFCT — 0.53324765E—05Integration starts at RNIN — 1.9000 (Angst.) and uses mesh RH — 0.0015000 (Angst.)

Add VSHFTI— 159,12.47)c,n—1( to make the NTPI— 34 initial—state potential points approach asymptote VLIMI— 19742.15

Multiply turning point distances by 1.000000000 and energies by 1.000000000 to convert units to Angstroms and cm—i

R(turn( E (turn) R)turn( E (turn) R(turn( E (turn) R)turn) S (turn) R)turn) S(turn)

2.43880560 1436.436 2.44847731 1300.569 2.45911757 1160.941 2.47092860 1017.614 2.48419656 870.6462.49934872 720.099 2.51707285 566.031 2.53861037 408.504 2.56668909 247.576 2.61075355 83.3092.67780117 0.000 2.75410909 83.309 2.81713052 247.576 2.86475093 408.504 2.90644236 566.0312.94496421 720.099 2.98159397 870.646 3.01706031 1017.614 3.05183662 1160.941 3.08626102 1300.5693.12059399 1436.436 3.15504947 1568.484 3.18981324 1696.651 3.22505463 1820.879 3.26093464 1941.1063.29761204 2057.274 3.33524829 2169.321 3.37401198 2277.169 3.41408315 2360.816 3.45565777 2480.1443.49895676 2575.111 3.54422890 2665.659 3.59175907 2751.726 3.64187867 2833.254

R.J. I.e Roy/ Bound—s continuumintensities 395

Use 8—point interpolation over the 34 initial—state turning points to generate 762 mesh points starting at the 366—thExtrapolate inward from S — 2.4475 with V — 15175.730 + 0.2747346D+i1~ exp( —6.7070344~R)Extrapolate outward from R — 3.5890 with V — 19742.150 — 0.28899210+06* exp( —1.554i412~R(

Expand transition moment functions about the point REX— 2.30000(Angstroms)Transition moment to final—state—2 ix: 1.000000 + 0.000000*(R_REX) + 0.OOt000*(R_REX)**2

Final—state—2 potential defined by value (relative to asymptote at VLIMF2— 16057.00 and derivatives at SEX — 2.3000 of:7654.000 —35490.000 051100.000 0.000

Final—state—2 potential curve (units 1/cm and Angst)

V — 16057.000 • 7654.000 • ssp(—( 4.6367912(R—R~) • 0.S792605(R—RZX)”2 + 0.000000(R—66X)~~3))

Pr.dlssociatiOfl Rate at E(v— 1, 2— 0)— 16148.54 is: 0.23i050D+05 (sec—i)Pr.dlssociation Rate at E(v— 2, 2— 0)— 16306.51 is: 0.5162110+08 )s.c—l(Predtssocistion Rate at E(v— 3, 2— 0)— 16465.12 is: 0.1155730+10 (sec—i)

Pr.diasocistion Rat, at E(v— 4, 2— 0)— 16616.32 is: 0.1792220+09 (s.c—i)Pr.dissocistion Rate at E(v— 5. J— 0)— 16766.04 is: 0.7242350+09 (s.c—i)Pr.diasociation Rats at E(v— 6, 2— 0)— 16914.23 is: 0.5538660.09 (s.c—i)pr.dissoctation Rats at E(v— 7, .3— 0)— 17056.63 is: 0.2392960*09 (s.c—i)Predissocistion Pats at E)v— 8, .3— 0)— 17195.76 is: 0.5550760+08 (sec—i)Pr.dis.ocistion Rats at E(v— 9, .2— 0)— 17331.03 ii: 0.7591430.06 (s.c—i)

Pr.dissociation Rat. at E(v iO, .2— 0)’ 17462.52 is: 0.1664070+06 (s.c—i)Pr.dissociation Rate at E(v— ii, 2— 0)— 17590.19 is: 0.5920710+06 (s.c—i)Prsdissociation Rate at E(v— 12. 3— 0)— 17714.00 is: 0.1039460+09 (sec—i)

Final calculated Transition Intensity Coefficients

TEMP — 0.0(K) TEMP —

WAVL FREQ State—i State—2 Total state—i Stste—2 Total

0.Ot 0. 0. + 0.3140265+10 — 0.3i4026E+i0

9——>i)pi(u State Predissociation Calculation for 3r2

Calculate exactly the same absorption/emission coefficients as in preceeding caseusing exactly the samepotentials and integration mesh

Calculate Prediasociation Rates in units: (sec—i)

Perform calculations using M.S. Child—s Uniform Harmonic Stationary Phase Approximation for overlap integralsexcept ... if no classically allowed stationary phase point found. us. exact (numerical> W.V. functions, whilsassuming asymptotic amplitude converged when constant to within 0.100—05 at 3 succeesivemaxima

Fix .2—0 rather than sum over a rotational distribution

Consider initial—state levels up to v—i2 whose energies are 15966.000 16149.000 16310.000 16468.000 16622.00016772.000 16920.000 17060.000 17202.000 17340.000

17470.000 17600.000 17722.000and 5W—s are 0.0809369 0.08062i3 0.0803048 0.0799864 0.0796655

0.0793424 0.0790174 0.0786894 0.0783590 0.07802770.0776938 0.0773600 0.0770200

:5. OVRMSCinappropriate at SF6— 16148.537 for v— 1 R(TP1,final) .GE. R)ITP2— 6i2,initial)

predissociation Rate at E)v 1, 3— 0)— i6i48.54 is: 0.2310500+05 (eec—i)OVPI4SC inappropriate at SF>>— 16308.508 for v— 2 R(TPi,final) .GE. R)ITP2 643,initial(

Pr.diasociation Rate at E)v 2, 3— 0)— 16308.5i is: 0.5i62ii0+08 (eec—i)Prediasociation Rate at E)v 3, 3— 0)— 16465.12 is: 0.1154420+10 (sec—i)Pr.disaociatton Rate at E(v— 4, 3— 0)— 16618.32 is: 0.1801940+09 (sec—i)Predissociation Rat, at E)v 5, J O( 16768.04 is: 0.7246990+09 (sec—i)Predissociation Rate at E(v— 6. 2— 0)— 16914.23 is: 0.5528440+09 (sec—i)Predisaociation Rate at E(v— 7, J— 0)— 17056.83 is: 0.2379640+09 (sec—i>Pr.diasociation Rate at E)v— 8, J 0) 17195.78 is: 0.5469010+08 (sec—i)Predissociation Rate at E(v 9, J 0> 17331.03 is: 0.6596060+06 (sec—i)Predissociation Rate at E)v— 10, .3— 0)— 17462.52 is: 0.1732640+08 (sec—i>Predissociation Rate at E)v— ii, .3— 0)— 17590.19 is: 0.6006670+08 (sec—i)Predissociation Rate at E(v 12, J 0) 17714.00 is: 0.1049830+09 (sec—i)

Final calculated Transition Intensity Coefficients

TEMP — 0.0(K( TEMP —WAVL FREQ State—i State—2 Total State—i State—2 Total

0.00 0. 0. + 0.3i3949E+lO — 0.3139495+10