Contents lists available at ScienceDirect

Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer 186 (2017) 167–178

http://d0022-40

E-m

journal homepage: www.elsevier.com/locate/jqsrt

LEVEL: A computer program for solving the radial Schrödingerequation for bound and quasibound levels

Robert J. Le RoyDepartment of Chemistry, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

a r t i c l e i n f o

Article history:Received 15 February 2016Received in revised form7 May 2016Accepted 26 May 2016Available online 1 June 2016

Keywords:Radial Schroedinger solverDiatomic vib-rotational energiesDiatomic Franck-Condon factorsQuasibound levelsQuasibound level widthsCentrifugal distortion constantsDiatomic matrix elementsDiatomic Einstein coefficients

x.doi.org/10.1016/j.jqsrt.2016.05.02873/& 2016 Elsevier Ltd. All rights reserved.

ail address: leroy@UWaterloo.ca

a b s t r a c t

This paper describes program LEVEL, which can solve the radial or one-dimensionalSchrödinger equation and automatically locate either all of, or a selected number of, thebound and/or quasibound levels of any smooth single- or double-minimum potential, andcalculate inertial rotation and centrifugal distortion constants and various expectationvalues for those levels. It can also calculate Franck–Condon factors and other off-diagonalmatrix elements, either between levels of a single potential or between levels of twodifferent potentials. The potential energy function may be defined by any one of a numberof analytic functions, or by a set of input potential function values which the code willinterpolate over and extrapolate beyond to span the desired range.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Determining the number, energies and properties ofvibration–rotational levels of a given one-dimensional oreffective radial potential, and calculating matrix elementsand transition intensities coupling levels of a single potentialor levels of two separate potentials, are ubiquitous problemsin chemical physics. The computer program LEVEL describedhere is a robust and flexible tool for performing such calcu-lations. Although the original version of this program wasbased upon the famous Franck–Condon intensity program ofZare [1–3], the present version has been considerably mod-ified over several decades, and now incorporates severalunique features. In particular, it can:

(i) automatically locate and calculate the widths of qua-sibound (orbiting resonance or tunneling predissocia-tion) levels;

(ii) calculate diatomic molecule inertial rotation andcentrifugal distortion constants for levels of a givenpotential;

(iii) readily locate levels with dominant wave functionamplitude over either well of an asymmetric doubleminimum potential;

(iv) automatically locate and calculate expectation valuesfor all vibration–rotation levels of any well-behavedsingle-minimum, “shelf state” or double minimumpotential;

(v) as an extension of (iv), it can automatically generate(for example) Franck–Condon factors and the radiativelifetimes for all possible discrete transitions allowedby specified (via the input data file) rotational selec-tion rules between the levels of a single potential, orbetween two different potentials. Although the pre-sent version only calculates Einstein A coefficientsusing Hönl–London factors for the case of singlet–singlet electronic transitions, it may readily be gen-eralized to treat other cases.

R.J. Le Roy / Journal of Quantitative Spectroscopy & Radiative Transfer 186 (2017) 167–178168

In the following, Section 2 presents the basic equationbeing solved, describes how the program functions, andoutlines some of its options. Section 3 then states theinput/output conventions, indicates the units assumed forthe physical parameters of interest, and presents a shell tofacilitate running the program on a UNIX or Linux system.Program operation is controlled by the contents of a datafile that is read (on Channel-5) during execution. Thestructure of this data file and the significance of the var-ious read-in parameters are described in Appendices A andB in a Supplementary Materials PDF document accom-panying this paper. Appendix C there then presents anddescribes sample data files and (truncated) output filesillustrating many of the capabilities of the code, whileAppendix D presents an outline the structure of the pro-gram and the roles of its various subroutines. The versionof the code described herein includes corrections andenhancements incorporated up to April 2016. Individualscurrently utilizing older versions [4] will likely find itdesirable to obtain the current version since it contains anumber of corrections and additional functionality. Theauthor would always appreciate having users inform himof any apparent errors or instabilities in the code, or ofadditional features that might be desirable for futureversions.

2. Outline of program operation and options

2.1. Solving the radial Schrödinger equation

The core of the program is concerned with determiningthe discrete eigenvalues and the associated eigenfunctionsof the radial or (effective) one-dimensional Schrödingerequation

� ℏ2

2μd2Ψ v;JðrÞ

dr2þVJ rð ÞΨ v;J rð Þ ¼ Ev;JΨ v;J rð Þ; ð1Þ

in which μ is the “effective” or reduced mass of the system,J the rotational quantum number, r the internuclear dis-tance, and the effective one-dimensional potential VJ(r) is asum of the rotationless (electronic) potential V(r) plus acentrifugal term. For the normal problem of a diatomicmolecule rotating in three dimensions, this centrifugalpotential has the form ½JðJþ1Þ�Ω2�ℏ2=2μr2, whereΩ� OMEGA is the projection of the electronic angularmomentum onto the internuclear axis. However, for thespecial case of a diatomic molecule rotating in twodimensions, a case invoked by setting the input parameterOMEGA499 (see the discussion of data input statement #6in Appendix B this term becomes ½J2�1=4�ℏ2=2μr2.Moreover, in order to allow for the special properties ofsmall molecular ions, the program normally defines thereduced mass appearing in Eq. (1) as Watson's “charge-modified reduced mass” [5],

μ¼ μW � MAMB

MAþMB�meQ; ð2Þ

in which MA and MB are the atomic masses of the twoatoms, me is the electron mass, and Q � CHARGE (see inputREAD statement #1 in Appendix B) is the (positive or

negative) integer net charge on the molecule. Alternately,it may use the usual two-body expression for the reducedmass, with the missing/added electron mass subtractedfrom/added to the mass of one atom or the other.

The core of the calculation is the solution of Eq. (1) todetermine the eigenvalues Ev;J and eigenfunctions Ψ v;JðrÞof the potential VJðrÞ. This is done in subroutine SCHRQ,which is based on the famous Cooley–Cashion–Zare rou-tine SCHR [1–3,6,7], but incorporates the ability to locateand calculate the widths of “quasibound” or tunneling-predissociation levels [8–11], metastable levels that lieabove the dissociation limit, but have their dissociationinhibited by a potential energy barrier.

The accuracy of the eigenvalues and eigenfunctionsobtained is largely determined by the size of the (fixed)radial mesh RH (see READ

#5 of the input data file) used inthe numerical integration of Eq. (1). For potentials that arenot too steep or too sharply curved, adequate accuracy istypically obtained using an RH value that yields a mini-mum of 30–50 mesh points between adjacent wavefunc-tion nodes in the classically allowed region. An appropriatemesh size may be estimated using the “particle-in-a-box”-based expression

RH¼ π NPN� μ�maxfE�VðrÞg=Cu� �1=2� �

:.

ð3Þ

in which NPN is the selected minimum number of meshpoints per wavefunction node (say 50), maxfE�VðrÞg is themaximum of the local kinetic energy (in cm�1) for the levelsunder consideration (in general it is ≲ the potential welldepth), and Cu ¼ 16:857629206 is the “inertial constant”, thevalue of ℏ2=2 in units ½amu cm�1 ˚A2� (see Section 3). Avalue of NPN that is too small will yield results that will nothave the desired accuracy, while too large a value mayrequire excessive computational effort or cause arraydimensions to be exceeded. Thus, while Eq. (3) is a usefulguide, a careful user should always perform test calculationsusing different RH values in order to verify that the calculatedresults achieve the accuracy desired for the specificapplication.

The numerical integration of Eq. (1) is performed in therange from RMIN to RMAX (see READ statement #5) using theNumerov algorithm [6,12]. This is a three-point propaga-tion scheme, so to initiate this integration, it is necessaryto specify initial values of the wave function at two adja-cent mesh points at each end of the range. For truly boundstates, the wave function at the outer end of the range,r¼ RMAX, is initialized at an arbitrary value (unity), whileits value at the adjacent inner mesh point is defined usingthe first-order semiclassical wavefunction [13]

Ψ v;JðrÞp ½VJðrÞ�Ev;J ��1=4exp �ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2μ=ℏ2

q Z r

½VJðr0Þ�Ev;J �1=2 dr0� �

ð4Þ

The Numerov procedure then propagates the wavefunc-tion inward.

At short range, most realistic intermolecular potentialsgrow very steeply, causing the wavefunctions in the clas-sically forbidden region to die off extremely rapidly withdecreasing r. As a result, the wave function at the innerend of the chosen range of integration is normally initi-alized by placing a node at the lower bound of this range,the read-in distance RMIN. This is effected by setting

R.J. Le Roy / Journal of Quantitative Spectroscopy & Radiative Transfer 186 (2017) 167–178 169

Ψ v;Jðr¼ RMINÞ ¼ 0, giving Ψ v;Jðr¼ RMINþRHÞ an arbitrary(non-zero) value, and allowing the Numerov scheme toproceed from there. This is the normal case for a diatomicmolecule problem. Note that one should normally setRMIN40, as the centrifugal contribution to the potentialbecomes singular at r¼ 0.

A special treatment of the inner boundary conditionmay be implemented if one is searching for eigenfunctionsof a precisely symmetric potential whose midpoint islocated at RMIN. For asymmetric levels that would have anode at that midpoint, the normal treatment describedabove will suffice. However, another approach must beimplemented for symmetric levels whose wave functionswould have zero slope at RMIN. This option is built intosubroutine SCHRQ, and is invoked by setting the controlparameter INNOD1r0. However, since this is an unusualcase, varying this parameter is not one of the normaloptions of the current version of the main program, and auser who wishes to deal with this case may chose either toadd parameter INNOD1 to READ

#24 (see lines #431-465 ofthe MAIN program), or to recompile the code with thevalue of INNOD1, defined on line #446 of the code, reset to0.

For special applications, if desired, a hard wall outerboundary condition may be imposed by setting the inputinteger IV ðiÞ (see READ

#25), which would otherwiserepresent the vibrational quantum number, equal to alarge negative number whose absolute value would spe-cify the radial mesh point at which this wall would beplaced. In particular, setting IVðiÞo�10 causes a hardwall (wave function node) to be placed at mesh pointnumber jIVðiÞj for level-i.

The Cooley procedure [6,7] for finding an eigenvalue ofEq. (1) is illustrated in Fig. 1. For any given trial energy thenumerical integration proceeds inward from RMAX andoutward from RMIN until the two solution segments meetat a chosen matching point rx. The discontinuity in theslopes of the inward and outward trial functions at rx isthen used to estimate the energy correction required toconverge on the eigenvalue closest to the given trialenergy [14,6], and this process is repeated until the energyimprovement is smaller than the chosen convergencecriterion (parameter EPS of Read #5). This procedureusually converges very rapidly, and for a single-minimumpotential it is insensitive to the choice of the matchingpoint rx as long as it lies in the classically allowed regionwhere the wavefunction amplitude is relatively large.However, to ensure high accuracy of calculated expecta-tion values or matrix elements, EPS should usually be set2 orders of magnitude smaller than the actual eigenvalueprecision required.

For an asymmetric double-well potential, wavefunc-tions usually have amplitudes of very different magnitudeover the two wells, and the eigenvalue correction algo-rithm [6,14] used by SCHRQ tends to become unstable ifthe matching distance rx lies in the well in which thewavefunction has very small amplitude. As a result, it isnormally necessary to require rx to lie in the well inwhich its amplitude is the largest. In the current versionof the program, this choice is set by the internal controlparameter INNER (� SINNER), which is given a default

value in Line #408 of the code, but as needed, is set to theappropriate value by the automatic vibrational level-finder subroutine ALF. As a result, calculations involvingvibrational levels of a double well or “shelf-state”potential are (normally) performed just as routinely asthose for a normal single-well potential.

In general, the outward and inward numerical inte-grations start at the input distances RMIN and RMAX (seeREAD

#5), respectively, which should lie sufficiently far intothe classically forbidden regions (where VJðrÞ4Ev;J) thatthe wavefunction has decayed by several orders of mag-nitude relative to its amplitude in the classically allowedregion. The present version of the code prints warningmessages if this decay is not by a factor of at least 10�9; ifsuch warnings are printed, a smaller RMIN or larger RMAXvalue may be needed in order to ensure that the desiredaccuracy is achieved. However, should RMIN or RMAX liesufficiently far into the classically forbidden regions that½VJðrÞ�E� becomes extremely large, the integration algo-rithm itself can become numerically unstable for the givenmesh size. For realistic diatomic molecule potential curves,this situation is only likely to occur near RMIN. If it does, awarning message is printed, and the beginning of theintegration range is automatically shifted outward untilthe problem disappears. However, use of a slightly largervalue of RMIN will cause these warning messages to dis-appear and (marginally) reduce the computational effort.For most diatomic molecules, a reasonable value of RMIN isca. 0.6–0.8 times the smallest inner turning pointencountered in the calculation, but for hydrides orother species of low reduced mass, smaller values may benecessary.

The program internally defines the upper bound on therange of numerical integration as the smaller of, the readin (READ #5) value of RMAX, and the largest distance con-sistent with the specified mesh and the internally defined(see Section 3) potential energy function and distancearray dimension NDIMR. As with RMIN, the choice of RMAXis not critical, provided (for truly bound states) that thewave function has decayed to an amplitude much smallerthan that in the classically allowed region, and the sameamplitude decay test of 10�9 mentioned above is usedhere. However, due to the anharmonicity of typical mole-cular potential curves, the requisite values of RMAX aremuch larger for highly excited vibrational levels than forthose lying near the potential minimum. In order to reducecomputational effort, an integration range upper boundrendðv; JÞ is therefore estimated for each level using thesemiclassical result of Eq. (4), which shows that thewavefunction dies off exponentially in the classically for-bidden region with an exponent of

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2μ=ℏ2

q Z rendðv;JÞ

r2ðv;JÞ½VJðrÞ�Ev;J �1=2dr ð5Þ

in which the turning point r2ðv; JÞ marks the outer end ofthe classically accessible region at energy Ev;J . For eachlevel it considers, SCHRQ first locates r2ðv; JÞ, and thendetermines a value of rendðv; JÞ that is sufficiently large toensure that this starting amplitude is smaller than that inthe classically allowed region by a factor of at least 10�9. Incalculations for levels spanning a wide range of energies,

Fig. 1. Illustration of the Löwdin/Cooley eigenvalue convergence proce-dure [14,6] showing wavefunctions at the first four trial energies thatconverge on the v¼ 1 eigenvalue of a model potential. This calculation setrx at the outer turning point for the given trial energy.

R.J. Le Roy / Journal of Quantitative Spectroscopy & Radiative Transfer 186 (2017) 167–178170

the employment of this procedure in the program canreduce the overall computation time by a factor of twoor more.

2.2. Locating quasibound levels and determining theirwidths

Quasibound levels, also known as orbiting resonances,shape resonances, or tunneling-predissociation levels, aremetastable eigenstates of Eq. (1) that lie at energies abovethe potential asymptote VJðr¼1Þ, but below the max-imum of a potential energy barrier that lies outside themain part of the potential well. Although they are part ofthe continuum of states with E4VJðr-1Þ, they are dis-tinct in that their wavefunction amplitude at small dis-tances, inside the potential barrier, is very much largerthat those for neighbouring continuum states, so that theymanifest themselves as distinct, sharp spectroscopic lines.The nature of the “inside-the-barrier” portion of thewavefunctions for such states is illustrated in Fig. 2, whichalso demonstrates the immense difference between thetunneling lifetimes of levels lying near the bottom of abarrier, and those lying near the top.

Following Refs. [8,9], program LEVEL locates quasi-bound levels by combining the standard Ψ v;Jðr¼ RMINÞ ¼ 0inner boundary condition with an outer boundary condi-tion defined by the slope of the inward-increasing Airyfunction at the third (outermost) turning point. This pro-cedure is the most accurate and efficient method forlocating quasibound levels that has been proposed to date[8–10,15,16]. It is nearly exact for narrow (long-lived)states, while for the very broadest levels lying marginallybelow barrier maxima (such as the v¼2 level in Fig. 2), thedifference with level energies obtained using alternativemethods are at most a small fraction (say, � 20%) of thewidth (FWHM) of that level. More accurate predictions forsuch short-lived states would require a detailed simulationof the actual observable property of interest, since differ-ent methods of observing a given quasibound level mayyield estimated level energies differing by a fraction of the

level width. For example, if such levels are being observedspectroscopically, the precise positions of the peaks in thebound -continuum spectrum should be calculated usinga photodissociation simulation code [17,18].

Calculation of the width or tunneling lifetime of aquasibound level by program LEVEL is based on Eq. (4.5) ofConnor and Smith [10]; a more transparent description ofthis procedure may be found in Section II.B of Ref. [11].This is a uniform semiclassical procedure in which thepredissociation rate may be thought of as being the pro-duct of the probability of tunneling past the barrier at thespecified energy, times the vibrational frequency (inverseof the vibrational period) for the system trapped in thewell behind the barrier. The actual calculation requires theevaluation of an integral of the type appearing in Eq. (5)across the classically forbidden interval between the twooutermost classical turning points (i.e., with the upperbound rendðv; JÞ replaced by the outermost turning pointr3ðv; JÞ), and of an analogous integral across the classicallyallowed interval between the two inner turning points:ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2μ=ℏ2

q Z r2ðv;JÞ

r1ðv;JÞ½Ev;J�VJðrÞ��1=2dr; ð6Þ

together with (the energy derivative of) a phase correctionfactor that takes account of the proximity to the barriermaximum [9,10]. This procedure yields widths that areexpected to be very reliable, particularly for narrow (long-lived) levels, but may have uncertainties of up to ca. 10% ormore for the very broadest levels [9,10]. To obtain moreaccurate widths for those broad levels would again requireone to perform a direct simulations of the process bywhich they are observed. The results in Fig. 2, in which thetunneling lifetimes are seen to range from ca. 1 fs to900 years, illustrate the extreme range of tunneling life-times that may arise.

On a practical note, if the outer end of the specifiednumerical integration range, RMAX, is smaller than theoutermost turning point, r3ðv; JÞ, of the metastable level ofinterest, the program attempts to generate a reasonableestimate of the width by completing the quadrature of Eq.(5) over the barrier analytically while approximating thepotential on the remainder of the interval by a centrifugal-type term C2=r2 attached to the potential function at RMAX.If this approximation is invoked, warning messages arewritten to the main Channel-6 output file (e.g., see theoutput for Case 3 in Appendix C).

2.3. Calculating diatomic molecule centrifugal distortionconstants

The rotational sublevels of a given vibrational level of amolecule have traditionally been represented by thepower series [19,20]

Ev;J ¼ GðvÞþBv½JðJþ1Þ�Ω2��Dv½JðJþ1Þ�Ω2�2þHv½JðJþ1Þ�Ω2�3þ⋯

¼Xm ¼ 0

KmðvÞ½JðJþ1Þ�Ω2�m ð7Þ

If desired, program LEVEL will compute values of theinertial rotational constant Bv ¼ ðℏ2=2μÞ⟨v; J ¼ 0j1=r2jv; J ¼0⟩ and of the first six centrifugal distortion constantsassociated with this expansion (�Dv, Hv, Lv, Mv, Nv and

Fig. 2. Quasibound level wavefunctions (red curves) and the associatedtunneling lifetimes for the three vibrational levels supported by theJ ¼ 32 centrifugally distorted potential (upper blue curve) for the (1Π)state of CHþ (adapted from Fig. 2 of Ref. [42]). (For interpretation of thereferences to color in this figure caption, the reader is referred to the webversion of this paper.)

R.J. Le Roy / Journal of Quantitative Spectroscopy & Radiative Transfer 186 (2017) 167–178 171

Ov). These constants have their normal significance whengenerated for levels of a rotationless (½JðJþ1Þ�Ω2� ¼ 0)potential [20], and are simply related to derivatives of theenergy with respect to ½JðJþ1Þ�Ω2� when calculated forvibration–rotational levels with J40. Calculation of theseconstants is invoked by setting input parameter LCDC40(see READ

#24), and is performed using a subroutine basedon Tellinghuisen's reformulation [21] of the exact quantummechanical method of Hutson [22], which is extendedherein to higher order to allow the calculation of Nv andOv. To ensure stable, fully converged evaluation of theseproperties, it is usually necessary to make the eigenvalueconvergence parameter EPS of READ

#5 and the value of theradial mesh RH distinctly smaller than would be necessaryto achieve the normal degree of convergence for theeigenvalue themselves.

2.4. Calculating expectation values, matrix elements, andEinstein A coefficients

If requested, the program will calculate expectationvalues or matrix elements of a function M(r), which maybe defined either by interpolating over an array of inputvalues, by a user-defined analytic function, or as a powerseries in a user-specified radial variable RFNðrÞ:

MðrÞ ¼XMORDRi ¼ 0

DMðiÞ � RFNðrÞi: ð8Þ

Parameters MORDR, IRFN and DREF defining the extent ofthe power series and the nature of the radial variableRFNðrÞ are input via READ

#26, and the power series coef-ficients DMðiÞ are input via READ

#27. In this last option, thenature of the radial variable RFNðrÞ is specified by thechoice of a value for input variable IRFN in the range�4rIRFNr9 (see comments regarding READ

#26 inAppendix B), while setting IRFNZ10 causes M(r) to bedefined by interpolating over and extrapolating beyond aset of read-in numerical values, and setting IRFN r�10

causes M(r) to be defined by a user-defined analytic radialfunction. For this last case, code for calculating the desiredfunction should be inserted into the program in themanner illustrated by the example in lines #593-614 of themain program. IfM(r) is to be defined by interpolating overan array of read-in points, the necessary information isinput via READs #28–31.

The conventional Franck–Condon factor FCF¼ j⟨Ψ v0 ;J0

jΨ v″ ;J″ ⟩j2 is the square of the matrix element of the zerothpower of RFNðrÞ, and will be generated whenever any off-diagonal matrix elements are calculated (i.e., wheneverinput parameter jLXPCTjZ3). In this case, the programalso assumes that M(r) is the transition dipole function (indebye), and uses its matrix element to calculate the Ein-stein A coefficient coupling the two levels in question. Forcases in which a pointwise (IRFN Z10) or analytic user-defined (IRFN r�10) matrix element argument functionM(r) is chosen, MORDR and DREF are dummy variables, andno DMðiÞ coefficients are read in. Note: to calculate onlyFranck–Condon factors, one should set MORDRo0, inwhich case IRFN and DREF are dummy variables and noDM ðiÞ values are read in.

The Einstein A coefficient for the rate of spontaneousemission from initial-state level ðv0; J0Þ into final-state levelðv″; J″Þ is defined by the expression [23]

A¼ 3:1361891� 10�7 SðJ0; J″Þ2J0 þ1

ν3 ⟨Ψ v0 ;J0 jMðrÞjΨ v″ ;J″ ⟩ 2 ð9Þ

Here: A has units s�1, M(r) is the dipole moment (ortransition dipole) function in units debye, ν the emissionfrequency in cm�1, SðJ0; J″Þ the Hönl–London rotationalintensity factor and Ψ v0 ;J0 and Ψ v″ ;J″ are the unit-normalized initial and final state radial wave functions.The present version of the code incorporates SðJ0; J″Þexpressions for singlet2singlet transitions obeying theparity selection rule, with ΔΛ¼ 0 or 71. Note that whileversions of this code prior to 7.7 used the SðJ0; J″Þ expres-sions of Herzberg [20], subsequent versions use the cor-rected Hönl–London factors recommended by Hanssonand Watson [24], which for Π–Σ or Σ–Π transitions are afactor of four larger than those listed by Herzberg [20]because of their use of a different definition of the per-pendicular transition dipole moment function (see alsoBernath [23]). To modify these selection rules and inten-sity factors, or to extend them to other cases, a user willneed to modify lines #60–106 of subroutine MATXEL.

2.5. Automatic location of any specified vibrational level(s) of a given potential

For each potential function considered, LEVEL beginsby examining the input instructions to determine thehighest vibrational level of interest, VMAX. It then usessubroutine ALF to locate all vibrational levels up to thatmaximum v, and calculates the first seven rotational con-stants (K ðvÞ for m¼1,7) for each. If a user wishes to havethe code find all bound (and quasibound, for potentialswith a barrier) levels, input parameter NLEV1 (see READ#24) should be set equal to some negative number whosemagnitude is much larger than the actual possible numberof levels (say, NLEV¼ �999). The actual level energy

R.J. Le Roy / Journal of Quantitative Spectroscopy & Radiative Transfer 186 (2017) 167–178172

search begins by locating the potential energy functionminimum (the lowest minimum for a double-well poten-tial). A simple three-point approximation is used to esti-mate the harmonic force constant there, and the harmonicoscillator formula is employed to estimate the zero-pointenergy of that well. This yields a trial energy for sub-routine SCHRQ that almost always leads to convergence onthe eigenvalue of the v¼ 0 level of that potential. ALF thencalls subroutine SCECOR, which evaluates the semiclassicalphase integral

v Eð Þþ1=2¼ 1π

ffiffiffiffiffiffi2μℏ2

s Z r2ðvÞ

r1ðvÞE�VðrÞð Þ1=2dr ð10Þ

and its energy derivative

d vðEÞþ1=2 �

dE¼ 12π

ffiffiffiffiffiffi2μℏ2

s Z r2ðvÞ

r1ðvÞE�VðrÞð Þ�1=2dr ð11Þ

over the well at the energy of the current level, and sincethe inverse of the latter provides a good estimate of thelocal vibrational level spacing ΔGvþ1=2, its addition to theenergy of the current level yields a good trial eigenvaluefor the next higher vibrational level. This procedure isnormally iterated until all levels have been found.

The above procedure is satisfactory for a single-wellpotential, or for levels lying above the barrier maximum ina double-well potential, but for levels in a double mini-mum potential lying below such a barrier maximum andabove the minimum of the shallower well, the level searchprocedure is more complicated. In practise, all but thelevels lying closest to the barrier maximum of an asym-metric double-minimum potential are predominantlyeither “inner-well” or “outer-well” levels, with largewavefunction amplitude over one well, and negligiblewavefunction amplitude over the other. Thus, the proce-dure begins by determining values of the phase integralsof Eqs. (10) and (11) over both wells, at the current energy.Whichever of the two fvþ1=2g integral values is closest tohaving precisely half-integer value identifies the currentlevel as being an “inner-well” or “outer-well” state [25].The inverse of the second integral for that well then pro-vides an estimate of the distance to the next higher levelassociated with that well. At the same time, the product ofthe inverse of the integral of Eqs. (11) for the other wellwith the difference between the value of the its fvþ1=2gintegral and the next larger half-integer number providesa semiclassical estimate of the distance to the next higherlevel centred over the second well. The lower of these twoenergies then defines our estimate for the trial eigenvalueand “well-identity” for the next higher level. That “well-identity” then allows the assignment of the subroutineSCHRQ control parameter INNER that assigns the inward/outward wavefunction matching point rx (see Fig. 1) to thewell over which the wavefunction is expected to havemaximum amplitude in the subsequent calculation. Ofcourse, at energies very near the maximum of that inter-well barrier, the levels often have no unique “well iden-tity”, and the above approach may break down. When thisoccurs, the code reverts to a “brute force” procedure ofsystematically changing the trial energy by small

increments until convergence on the desired “next” level isachieved.

Once the above process is complete, the stored valuesof the vibrational energies and rotational constants (seeSection 2.3) allow the code readily to generate an accuratetrial energy for any ro-vibrational levels of interest. If thisprocedure fails (e.g., if centrifugal distortion of a two-wellpotential causes the level of interest to shift to the otherwell), a call to subroutine ALF using the appropriate cen-trifugally distorted potential will normally allow the codeto climb up the well to locate and converge on the level ofinterest.

2.6. Defining the rotationless potential VðrÞ

The potential function package that reads requiredinput and returns the potential array and associatedparameters is controlled by subroutine PREPOT. It usessubroutine package GENINT for interpolation/extrapola-tion over a set of read-in turning points, and subroutinePOTGEN for generating analytic potential functions. Valuesof the necessary input parameters enter via READ

statements #6–23; for the 2-state case invoked by input-ting NUMPOT¼2, this block of input statements isread twice.

One may choose to define a potential either by a set ofNTP read-in turning points {XI ðiÞ, YI ðiÞ} input via READ

#9,or (by setting NTP r0) by an analytic function. In theformer case, interpolation over the input turning points toproduce the array with mesh size RH required for thenumerical integration of Eq. (1) is performed in a mannerspecified by the input parameter NUSE. For NUSE 40 thisinvolves the use of NUSE-point piecewise polynomials(typically NUSE¼8 or 10), while for NUSE r0 the inter-polation uses a cubic spline function. If the specified rangeof numerical integration [RMAX,RMIN] extends beyondthat of the input turning points, appropriate extrapolationprocedures are invoked. In particular, at distances smallerthan the second of the read-in turning points XI(2), thepotential is extrapolated inward with an exponentialfunction fitted to the first three turning points. Similarly, ifRMAX4XIðNTP�1Þ the potential for r4XIðNTP�1Þ isextrapolated outward either as an exponential-type func-tion or as a (sum of) inverse-power terms, as specified byparameters ILR, NCN and CNN of READ

#7 (see Appendix B).To define the potential by an analytic function, rather

than by an array of points, the integer input parameter NTPof READ

#6 should be set r0. The program then skips READs#7–9 and proceeds instead to #10–23 (see Appendix B),where it reads values of the parameters defining thechosen analytic potential. The present version of the codeallows use of the following eight families of analyticpotential energy functions.

2.6.1. Lennard–Jones(m,n) potentialsLennard–Jones(m,n) potentials have the form

VLJ rð Þ �Den

m�n

� � rer

� �m� m

m�n

� � rer

� �n� ; ð12Þ

in which De is the well depth and re the equilibriuminternuclear distance.

R.J. Le Roy / Journal of Quantitative Spectroscopy & Radiative Transfer 186 (2017) 167–178 173

2.6.2. “Generalized polynomial expansion function” (GPEF)potentials

The GPEF functional form [26] is

VGPEFðrÞ � β0z2 1þ

XNβ

i ¼ 1

βizi

0@

1A: ð13Þ

These functions are Dunham-like [19] polynomial expan-sions based on Seto's modification [27] of the dimension-less radial variable

z¼ rq�rqeaSrqþbSr

qe

ð14Þ

that was introduced by Šurkus et al. [26], and include thefamiliar Dunham [19], Simons–Parr–Finlan [28], and Ogil-vie–Tipping [29] expansions as special cases that areinvoked by particular choices for its parameters q, aS, andbS (see discussion of READ

#10 in Appendix B).

2.6.3. ‘Extended morse oscillator” or EMO potentialsEMO potentials have the functional form [27,30–32]:

VEMOðrÞ �De 1�eβðrÞ�ðr� reÞh i2

; ð15Þ

in which De and re are the well depth and equilibriuminternuclear distance, and

βðrÞ ¼ βEMOðrÞ �XNβ

i ¼ 0

βi ½yrefq ðrÞ�i ð16Þ

is a simple polynomial in the Šurkus-type variable [26]:

yrefq rð Þ ¼ yq r; rrefð Þ � rq�rqrefrqþrqref

; ð17Þ

in which q is a selected small positive integer and rref achosen expansion centre. That expansion centre has oftenbeen set equal to the equilibrium distance re, but there aresometimes substantial advantages associated with allow-ing it to have larger values [33,34]. Truncating the expo-nent expansion at the constant term yields the familiarsimple Morse potential [35].

Another special-case Morse-type function allowed byPOTGEN is Hua Wei's 4-parameter potential [36]:

VðrÞ ¼De ½1�e�bðr� reÞ�=½1�Ce�bðr� reÞ�� �2

: ð18Þ

2.6.4. “Morse/long-range” or MLR potentialsMLR potentials [37,33,34] are expressed as

VMLR rð Þ �De 1� uLRðrÞuLRðreÞ

e�βðrÞ�yrep ðrÞ� 2

: ð19Þ

These functions have the same Morse-like [35] structure asthe EMO potentials of Eq. (15), but the radial variable inthe exponent is now a dimensionless variable with thestructure of yrefq ðrÞ of Eq. (17), but with the expansioncentre fixed at the equilibrium distance rref ¼ re, and theexponential term has a pre-factor defined by an attractivelong-range potential tail function uLRðrÞ whose form is

given by theory [38–41] to be

uLR rð Þ ¼Xlasti ¼ 1

Dmi ρr �Cmi

rmi; ð20Þ

in which the Dmi ðrÞ are “damping functions” that approach1 as r-1, and (normally, see below) approach zero asr-0 sufficiently rapidly to suppress the singularities in theinverse-power terms [34], while ρ is a system-dependentrange-scaling parameter (see below).

The two defining features of the MLR potential formare: (i) since the pre-factor to the exponential term in Eq.(19) is equal to 1 at the equilibrium distance, re, andyrep ðreÞ ¼ 0, then VMLRðreÞ ¼ 0 and (ii) the requirement that

limr-1

β rð Þ � β1 ¼ ln2De

uLRðreÞ

� �; ð21Þ

which means that at large distances Eq. (19) becomes

VMLR rð ÞCDe�uLR rð Þþ 14De

½uLRðrÞ�2

CDe�Xlasti ¼ 1

Cmi

rmiþ 14De

Xlasti ¼ 1

Cmi

rmi

!2

: ð22Þ

The final form for Eq. (22) shows that although the MLRfunction does take on the theoretically predicted inverse-power-sum form at long range, unless mlasto2m1, thelimiting long-range behaviour of the quadratic term inEqs. (19) and (22) will change the long-range behaviourspecified by Eq. (20). This problem is most serious for casesin which m1 ¼ 3 [33], but can also cause difficulties forsystems with larger m1 values [42]. However, it is resolvedhere by the fact that, as necessary, the code internallymodifies the input Cmi coefficients, and/or adds additionalterms, so as to precisely cancel the effect of those spuriousnon-physical quadratic terms, as illustrated by the dis-cussion of the coefficients Cadj

6 ¼ C6þðC3Þ2=4De andCadj9 ¼ C3C

adj6 =2De for the m1 ¼ 3 case in Ref. [33].

The exponent coefficient function βðrÞ has normallybeen written as a constrained polynomial expansiondefined in terms of two Šurkus-type variables [26] thatboth have the form of Eq. (17), but are defined by differentintegers, p and q [33,34]:

βðrÞ ¼ βPE�MLRðrÞ � yrefp ðrÞβ1þ 1�yrefp ðrÞh iXNβ

i ¼ 0

βi½yrefq ðrÞ�i:

ð23ÞPotentials using this type of exponent coefficient functionare called polynomial-exponent MLR (PE-MLR) functions.

As an alternative to Eq. (23), βðrÞ may be represented asa natural cubic spline in the independent variable yrefq ðrÞ,passing through a specified set of input points, as descri-bed in Refs. [43,44],

βðrÞ ¼ βSE�MLRðrÞ �XNβ

k ¼ 1

Skðyrefq ðrÞÞβk; ð24Þ

in which the spline “basis functions” Sk yrefp ðrÞ� �

are com-pletely defined by the chosen mesh of values of yrefq ðrjÞ [45,46].In this case, the connection to the long-range behaviour of Eq.(22) is provided by the constraint of fixing βNβ

¼ β1.

R.J. Le Roy / Journal of Quantitative Spectroscopy & Radiative Transfer 186 (2017) 167–178174

Potentials using this second type of exponent coefficientfunction are called spline-exponent MLR (SE-MLR) functions.

Program LEVEL offers two choices regarding the formof the damping functions DmðrÞ in Eq. (20): one is based ona form proposed by Douketis et al. [47],

DDSðsÞm ρr

�¼ 1�exp �bdsðsÞ � ðρ rÞm

�cdsðsÞ � ðρrÞ2ffiffiffiffiffim

p( )" #mþ s

;

ð25Þand the other is based on a form proposed by Tang andToennies [48],

DTTðsÞm ρr

�¼ 1�e�bttðsÞ�ρr Xm�1þ s

k ¼ 0

ðbttðsÞ � ρrÞkk!

; ð26Þ

in both of which ρ is a system-dependent range-scalingparameter [47,34], while bdsðsÞ, cdsðsÞ and bttðsÞ are pre-viously optimized [34] system-independent parameterswhose values are built into the code.

Both of these generalized damping function forms [34]have the property that at very small r

DsmðrÞ=rmCrs ð27Þ

for all values of m and s. The original Tang–Toennies model[48] corresponds to the case s¼ þ1, and that would pre-sent no problem if uLRðrÞ were an additive contribution tothe potential (as it is for potential function forms ofSections 2.6.5–2.6.8, discussed below). However, it cannotbe used in the MLR potential form, as it requires sr0 toprevent the repulsive potential wall from turning over andapproaching zero as r-0 [34,44]. The original Douketiset al. [47] model corresponds to the case s¼ 0, and whilethat presents no formal problems for either MLR potentialsor the functions described below, the results of Ref. [34]suggest that either s¼ �1 or s¼ �1=2 might be a betterchoice, since they seem to lead to more realistic physicalbehaviour at very small r [34].

The long-range interaction energy in an MLR potentialis not restricted to having the form of Eq. (20), but couldtake on any form predicted by theory. In particular, in thepresent version of LEVEL it may be represented as one ofthe roots of a 2� 2 or 3� 3 diagonalization arising fromthe type of two-state [33,49] or three-state [50,51] cou-pling encountered for alkali dimers at the first nSþnPasymptote [44]. A more detailed discussion of variousfeatures of the MLR potential function forms may be foundin Ref. [44] and in Appendix B.

2.6.5. The “double-exponential long-range” or DELRpotential

DELR potentials [11] have the functional form

VDELRðrÞ � Ae�2βðrÞ�ðr� reÞ �Be�βðrÞ�ðr� reÞ þuLRðrÞ; ð28Þin which the exponent coefficient βðrÞ has exactly thesame form as that for the EMO potential (see Eq. (16)), anduLRðrÞ, the function chosen to represent the long-rangeregion, is assumed to have the form of Eq. (20) or any ofthe coupled-state functions mentioned at the end of pre-ceding subsection. The pre-exponential factors A and B aredefined in terms of the well depth and the properties of

uLRðrÞ at re, that is,A¼DeþVLRðreÞþV 0

LRðreÞ=β0; ð29Þ

B¼ 2Deþ2 VLRðreÞþV 0LRðreÞ=β0s; ð30Þ

in which V 0LRðreÞ � dVLRðrÞ=dr

r ¼ re

.An illustration of how a DELR function maybe used to

represent a potential energy function with a barrier lyingoutside the main well is provided by Fig. 3 of Ref. [44].

2.6.6. The generalized “HFD” potential functionThe present generalization of the “Hartree–Foch/Dis-

persion” or HFD potential function [52,53] is written as

VHFD rð Þ � AHFDrre

� �γ

ef�β1r�β2r2g �DHFD rð Þ

Xm

Cm

rm; ð31Þ

in which

DHFDðrÞ ¼ exp �α1 α2=r�1� �α3

n ofor roα2

¼ 1 for rZα2; ð32Þis a global damping function applied to all inverse-powerterms. The input parameters are the long-range coeffi-cients fCmg and damping function parameters fαig definingthe attractive inverse-power-sum term, the power γ andthe quadratic exponent coefficient β2, while the values ofexponential-term parameters AHDF and β1 are derivedinternally from the fact that the potential well has a depthof De at the equilibrium internuclear distance re.

2.6.7. The generalized Tang–Toennies (gTT) potential func-tion form

Generalized Tang–Toennies-type potentials consist ofthe sum of a repulsive exponential term, whose exponentand pre-factor may consist of multiple terms, plus a sum ofattractive inverse-power terms that are damped by thesame s¼ þ1 version of the generalized TT-type dampingfunction of Eq. (26) that is a key feature of the original TTform [48,54]. This structure has been used in a number ofpublished potentials [55–57], and the extended version ofthis form available in LEVEL incorporates most variantsfound in the recent literature:

VgTT rð Þ � β5þβ6rþβ7

rþβ8r

2þβ9 r3

� �

�e�fβ1rþβ2r2 þβ3=rþβ4=r

2g �uLRðrÞ; ð33Þin which the attractive outer-wall function uLRðrÞ has theform of Eq. (20) with the damping functions defined bys¼ 1 Eq. (26). However, while the “basic” Tang–Toenniesfunction [48,54] constrains the argument of the dampingfunction to be identical to the leading term in the expo-nent of Eq. (33), β1r, in the present implementation theseterms are independent; the damping function argument isρr with the shape parameter of Eq. (26) fixed at bttðsÞ ¼ 1.Moreover,while the original Tang–Toennies model onlyallowed for even powers mi in Eq. (20), with miZ6, andused the individual damping functions of Eq. (26) withs¼ þ1. LEVEL will accept any user-supplied set of powersfmig and either choice of damping function model. Theinput parameters for this model are then the set of powersfmig and coefficients fCmi g for the attractive outer branch of

R.J. Le Roy / Journal of Quantitative Spectroscopy & Radiative Transfer 186 (2017) 167–178 175

the potential, and the values of the parameters ρ andβ1�β9, while the well depth De and equilibrium distancere are properties, rather than defining parameters of thepotential.

2.6.8. The “X-expansion” or “Hannover polynomial poten-tial” (HPP) form

The HPP or “X-expansion” form that was introduced byTiemann and co-workers [58] represents the main part ofthe potential by a GPEF-like power series

VHPPðrÞ � β0þβ1ξþβ2ξ2þβ3ξ

3þβ4ξ4þ⋯ ð34Þ

in which ξ� ξðr;b; rmÞ ¼ ðr�rmÞ=ðrþbrmÞ. The possiblepresence of a non-zero linear term (β1a0) in Eq. (34)means that when β1a0 the parameter rm (which is inputas re ¼ REQ) only approximately corresponds to the equi-librium distance. At a specified small distance, this powerseries is smoothly joined to an exponential, and beyond aspecified large distance, it is attached to an inverse-powersum with the form of Eq. (20).

2.6.9. Other choicesNote that apart from the simple GPEF polynomial poten-

tials of Section 2.6.2, all of the above analytic potentials aredefined relative to the absolute energy at the asymptote,which is specified by input parameter VLIM (see READ

#6). Forthe HPP potential form this is effected by internally settinga0 ¼ VLIM�De, while the GPEF potentials of Eq. (13) aredefined with VLIM set at the potential minimum.

Users may also readily introduce their own analyticpotential functional forms, simply by replacing subroutinePOTGEN with their own potential routines and making inputparameter NTP of READ

#6 a negative integer. To retain con-sistency with the rest of the present code, such a user-prepared POTGEN subroutine should have the argument list:

POTGENðLNPT;NPP;IAN1;IAN2;IMN1;IMN2;VLIM;R;RM2;VV;NCN;CNNÞ:

The first argument, parameter LNPT, is an integer which isfixed in program LEVEL as LNPT¼ 1 [59]. The other inputquantities are the integer NPP specifying the size of the variousradial arrays, the integers specifying the atomic numbers(IAN1/IAN2) and mass numbers (MN1/MN2) of the iso-topologue of interest (required for calculating Born–Oppen-heimer breakdown terms, see below), the absolute energyVLIM (in cm�1) at the potential asymptote, the radial distancesarray Rðig (in Å) at which potential values are to be generated,and the squared inverse distance array RM2ðiÞ ¼ 1=RðiÞ2. Thesubroutine should return the desired NPP-point array ofpotential function values VVðiÞ (in units cm�1), as well as theinteger NCN and real positive coefficient CNN. Under the optionin which the program automatically searches for many or allvibrational levels of a given potential (when input parameterNLEV1 is large and negative, see READ

#24), the limiting long-range potential is assumed to have the formVðrÞCD�CNN=rNCN, and the parameters NCN and CNN

returned from POTGEN may be used in a near-dissociationtheory [60–62] algorithm to estimate the number and energiesof missing levels. If a user-specified analytic potential has abarrier maximum or dies off exponentially, rather than as aninverse power, NCN should be set at some large integer value(e.g., NCN¼99).

2.7. Born–Oppenheimer breakdown (BOB) radial strengthfunctions

In recent years, it has become increasingly common forcombined-isotopologue spectroscopic data analyses to requirethe inclusion of atomic-mass-dependent Born–Oppenheimerbreakdown (BOB) corrections to the rotationless and cen-trifugal potential energy functions. LEVEL will include suchterms if input parameter IBOB has a value greater than zero.These (optional) functions are defined as in Ref. [63]. In par-ticular, for each atom a¼ A or B, the additive correction to thepotential energy function is defined as a constrained poly-nomial in the Šurkus variable of Eq. (17), namely,

Ma�Mrefa

Ma

!yrepad ðrÞu

a1þ 1�yrepad ðrÞ

h iXNaad

j ¼ 0

uaj ½yreqad ðrÞ�

j

0@

1A;

ð35Þin which Ma is the mass of the specific isotope of atom-a, Mref

ais the mass of the chosen reference isotope of that species [63],yrepad ðrÞ and yreqad ðrÞ have the form of Eq. (17), but with rref ¼ re,and the integers pad and qad are chosen in the manner dis-cussed in Refs. [63,33,42].

The centrifugal BOB correction is a multiplicative factor1þgAðrÞþgBðrÞ� �

that is applied to the centrifugal con-tribution to the overall potential function VJ(r) of Eq. (1), inwhich the terms associated with the two atoms a¼ A or Bhave the same type of radial form as the “adiabatic”potential function corrections of Eq. (35):

ga rð Þ ¼Mrefa

Mayeqqna ðrÞt

a1þ 1�yeqqna ðrÞ

h iXNana

j ¼ 0

taj ½yeqqna ðrÞ�j

0@

1A: ð36Þ

Note, however, that this expression has no separate integerpna, since no general theoretical prediction regarding thelimiting long-range behaviour of these functions is avail-able. However, the algebraic structure of Eqs. (23) and (35)has been retained in order to allow for the treatment ofmolecular ions, for which ta1 would be non-zero [63].

3. Units, physical constants, array dimensions, input/output conventions, and program execution

Unless otherwise specified, the units of length andenergy used throughout this program, and assumed for allinput data, are Å and cm�1, respectively. One exception tothis rule is that the transition dipole function M(r) of Eqs.(8) used for calculating the Einstein coefficients of Eq. (9),as defined by the expansion coefficients DMðiÞ of READ

#27(see Appendix B) is assumed to be in debye (where1 debye¼ 3:335640952� 10�30 C m¼ 0:393430295 au).Note, however, that in the IRFNZ10 option that definesthe matrix element argument by numerically interpolatingover a set of read-in points (see READs #28–31), the main(Channel-6) output describing the read-in transitionmoment function values being interpolated over will(incorrectly) refer to their units as “cm�1” rather thandebye, since the interpolation is done by the subroutinepackage that was set up to deal with a pointwise inputpotential. Note too, that if a set of read-in points is used to

R.J. Le Roy / Journal of Quantitative Spectroscopy & Radiative Transfer 186 (2017) 167–178176

define the potential or the matrix element argument M(r),the values may be in any convenient units, as appropriateconversion factors are always also read in (see READs #8 and30) to convert them to the appropriate units.

Values of the physical constants appear in the programin two places. The first is the factor Cu � ℏ2=2¼ 16:857629206 amu ˚A2 cm�1 [64] appearing in theradial Schrödinger equation of Eq. (1) and in Eq. (3), whoseunits effectively define the units of the input/output vari-ables. The second is in the collections of terms defining thenumerical factor in Eq. (9) used in calculating the Einsteincoefficient for the rate of spontaneous emission, asdescribed above. Current values of these constants arebased upon the 2012 CODATA recommended values of Ref.[64]. In addition, the masses of all stable isotopes of allatoms taken from The AME2012 Atomic Mass Evaluationof Ref. [65] are stored in data subroutine MASSES.

The array dimension limits, which a user might wish tochange, are set in PARAMETER statements in the main driverroutine and in subroutines GENINT and SPLINT. In the former,NDIMR (currently 250 001) is the maximum dimension of theradial mesh array on which the potential, wave functions andradial expectation value/matrix element arguments aredefined. For systems of small reduced mass, it could safely beset considerably smaller than this. The second parameter set inthe main program is VIBMX (currently 400), which defines themaximum number of vibration/rotation levels for whichvibrational eigenvalues may be read and stored, and the upperbound on the number of rotational sublevels that may besaved when applying the NJM 40 option to generate auto-matically many J sublevels for a given v (see READ

#24). The twoother array-size parameters set inside the code are NTPMX

(currently 2000, set in PREPOT), which specifies the maximumnumber of potential function points (or radial matrix elementargument M(r) values, for IRFN410) that may be read in tobe interpolated over, and MAXSP (currently 8000� 4�NTPMX,set in SPLINT), the number of spline coefficients required forinterpolation over the read-in function values.

The program reads input data on Channel-5, writes stan-dard output to Channel-6, and optionally (controlled by para-meters LPPOT, LCDC and LXPCT of READs #6 & 24) writes acondensed output file to one or more of Channels 7–10. Thoseexecuting the program using a UNIX or Linux operating systemenvironment may wish to create and store in the system oruser's “bin” directory a shell named (say) “rlev”, such as thatshown here:

# UNIX shell “rlev” to execute the compiled version of

program LEVEL named

# lev.x, which is stored in the user's directory/upath/with input data

# file $1.5, and write output to $1.6, $1.7, etc. all in the

current

# directory.#

time �/upath/lev.x o $1.5 4 $1.6mv fort.7 $1.7 o& /dev/nullmv fort.8 $1.8 o& /dev/nullmv fort.9 $1.9 o& /dev/nullmv fort.10 $1.10 o& /dev/null

This shell allows the program to be executed with the simplecommand: rlev ⟨filename⟩, in which ⟨filename⟩:5 is the name

of the user-created input data file (note that ⟨filename⟩ maybe any name, and is normally chosen to identify a specificcase). For the above shell, the standard output from Channel-6 will be written to file ⟨filename⟩:6, and the Channel-7,Channel-8, Channel-9, and Channel-10 outputs will be writ-ten to files ⟨filename⟩:7, …, ⟨filename⟩:10, respectively.

4. Concluding remarks mea culpa

In conclusion, I should bring the reader's attention to acapability that is not incorporated into the present versionof LEVEL. In 2008 and 2011 Meshkov, Stolyarov and Ipublished two papers describing the use of an “adaptivemapping procedure” that replaces numerical integration tosolve Eq. (1) on the infinite domain rA ½0;1Þ by integra-tion using a dimensionless radial variable, such as yrefq ðrÞ ofEq. (17), on the finite domain yrefq A ½�1; þ1� [66,67]. Use ofthis procedure allows one to readily obtain accurate solu-tions for levels lying extremely close to dissociation, suchas sometimes observed for excited states of electronicallyexcited alkali dimers, or to calculate the s-wave scatteringlength associated with any given potential energy functionusing a single-pass finite-domain calculation. Unfortu-nately, the different array structures required by thisapproach means that it has had to be incorporated into aseparate, albeit parallel version of the code, which (up tillnow, anyway) has always lagged behind the most current“conventional” version of LEVEL, and has not yet beenposted for public use. However, until a properly docu-mented “adaptive mapping procedure” version LEVEL canbe posted for public use, I will continue to provide an(incompletely documented) version of such a code toindividuals who request it.

Acknowledgements

I am pleased to acknowledge the foresight and stimu-lation provided by the late Professor George Burns of theUniversity of Toronto who instigated my work on this typeof computational research tool back in 1966, and the grace,advice, and financial support provided by the late Pro-fessor Richard Bernstein, then of the University of Wis-consin, Madison, who patiently supported my mean-derings in this area. I must also acknowledge the inspira-tion and assistance provided by R.N. Zare's seminal 1964Franck–Condon intensity factor paper and the associatedreports [1–3]. I thank J.Y. Seto for assistance in developingthe initial version of the level-finder subroutine, ALF, and J.Y. Seto, G.T. Kraemer and K. Slaughter for assistance indeveloping and updating the atomic mass database sub-routine MASSES. I am also grateful to many users forcomments that identified errors and suggested usefulextensions of the code, with particular thanks to R.W.Field, P.F. Bernath, A.J. Ross, A.S. Dickinson, J.O. Hornkohland G.C. McBane. Finally, I am also grateful to my colleagueProfessor F.R.M. McCourt for his thorough and insightfulcomments on this paper. Finally, I am pleased to thankNatural Sciences and Engineering Research Council of

R.J. Le Roy / Journal of Quantitative Spectroscopy & Radiative Transfer 186 (2017) 167–178 177

Canada which has generously supported my work in thisarea for the past four decades.

Appendix A. Supplementary data

Supplementary material associated with this paperconsist of: a text file containing the full standalone FORTRANsource code for program LEVEL, a text file containing thesample data files that are presented in Appendix C, and aPDF document containing: (i) Appendix A, a structuredlisting of the READ statements in the program, (ii) AppendixB full definitions of all of the parameters and quantitiesread in the input data file, and descriptions of the asso-ciated program options, (iii) Appendix C, commentedlistings of illustrative sample input and output files, and(iv) Appendix D, a structured listing of the subroutinescomprising the code with brief descriptions of their func-tions. Note that the equation and reference numberingappearing in this Supplemental Material document refer tothose in this journal article, so for completeness we herebycite nine papers referenced there that are associated withparticular applications of this code, but had not beenreferred to above [68–76].

Anyone who wishes to be registered with the author asa user of this code, eligible to be sent any future bug fixesor updates, should fill in the online form at the wwwaddress ⟨http://scienide2.uwaterloo.ca/~rleroy/LEVEL16⟩”.

Supplementary data associated with this paper can befound in the online version at http://dx.doi.org/10.1016/j.jqsrt.2016.05.028.

References

[1] Zare RN, Cashion JK. The IBM share program D2 NU SCHR 1072 forsolution of the Schrödinger radial equation, by J.W. Cooley: Neces-sary and useful modifications for its use on an IBM 7090. Universityof California Lawrence Radiation Laboratory Report UCRL-10881;1963.

[2] Zare RN. Programs for Calculating the Relative Intensities in theVibrational Structure of Elecronic Band Systems, University of Cali-fornia Lawrence Radiation Laboratory Report UCRL-10925; 1963.

[3] Zare RN. J Chem Phys 1964;40:1934.[4] (a) RJ Le Roy. LEVEL 1.0. University of Wisconsin Theoretical Chem-

istry Institute Report WIS-TCI-429G; 1971; (b) Le Roy RJ. LEVEL 2.0.University of Waterloo Chemical Physics Research Report CP-58;1976; (c) Le Roy RJ. LEVEL 3.0. CP-110; 1979; (d) Le Roy RJ. LEVEL 4.0.CP-230; 1983; (e) Le Roy RJ. LEVEL 5.0. CP-330; 1991; (f) Le Roy RJ.LEVEL 5.1. CP-330R; 1992; (g) Le Roy RJ. LEVEL 5.2. CP-330 R2; 1993;(h) Le Roy RJ. LEVEL 5.3. CP-330 R3; 1994; (i) Le Roy RJ. LEVEL 6.0.CP-555; 1995; (j) Le Roy RJ. LEVEL 6.1. CP-555R; 1996; (k) Le Roy RJ.LEVEL 7.0-7.7. CP-642 – 642 R3, CP-655 & CP-661; 2000–2005; (l) LeRoy RJ. LEVEL 8.0. CP-663; 2007–2014; (m) Le Roy RJ. LEVEL 8.2. CP-668; May 2014 [see ⟨http://www.leroy.uwaterloo.ca/programs/⟩].

[5] Watson JKG. J Mol Spectrosc 1980;80:411.[6] Cooley JW. Math Comput 1961;15:363.[7] Cashion J. J Chem Phys 1963;39:1872.[8] Le Roy RJ, Bernstein RB. J Chem Phys 1971;54:5114.[9] Le Roy RJ, Liu W-K. J Chem Phys 1978;69:3622.[10] Connor JNL, Smith AD. Mol Phys 1981;43:397.[11] Huang Y, Le Roy RJ. J Chem Phys 2003:119;7398 [erratum: Huang Y,

Le Roy RJ. J Chem Phys 2007:126;169904].[12] Smith K. The calculation of atomic collision processes. New York:

Wiley-Interscience; 1971 [Chapter 4].[13] Child MS. Semiclassical mechanics with molecular applications.

Oxford: Clarendon Press; 1991.

[14] Löwdin P-O. An elementary iteration-variation procedure for solvingthe Schrödinger Equation. Technical note No. 11, Quantum Chem-istry Group, Uppsala University, Uppsala, Sweden; 1958.

[15] Hehenberger M, Froelich P, Brändas E. J Chem Phys 1976;65:4559.[16] Hehenberger M. J Chem Phys 1977;67:1710.[17] Le Roy RJ. Comput Phys Commun 1989;52:383.[18] Le Roy RJ. Kraemer GT. BCONT 2.2: a computer program for calcu-

lating bound -continuum transition intensities for diatomicmolecules. University of Waterloo Chemical Physics Research ReportCP-650 R2; 2004 [see ⟨http://www.leroy.uwaterloo.ca/programs/⟩].

[19] Dunham JL. Phys Rev 1932;41:721.[20] Herzberg G. Spectra of diatomic molecules. New York: van Nostrand;

1950.[21] Tellinghuisen J. J Mol Spectrosc 1987;122:455.[22] Hutson JM. J Phys At Mol Phys 1981;14:851.[23] Bernath PF. Spectra of atoms and molecules, 2nd ed. New York:

Oxford University Press; 2005.[24] Hansson A, Watson JKG. J Mol Spectrosc 2005;233:169.[25] An equally valid approach would be to calculate the expectation

value ⟨r⟩ and see whether it lies in the inner or outer well.[26] Šurkus AA, Rakauskas RJ, Bolotin AB. Chem Phys Lett 1984;105:291.[27] Seto JY. Direct fitting of analytic potential functions to diatomic

molecule spectroscopic data [M.Sc. thesis]. Department of Chem-istry, University of Waterloo; 2000.

[28] Simons G, Parr RG, Finlan JM. J Chem Phys 1973;59:3229.[29] Ogilvie JF. Proc R Soc Lond A 1981:378;287.[30] Lee EG, Seto JY, Hirao T, Bernath PF, Le Roy RJ. J Mol Spectrosc

1999;194:197.[31] Seto JY, Morbi Z, Charron F, Lee SK, Bernath PF, Le Roy RJ. J Chem

Phys 1999;110:11756.[32] Huang Y. Determining analytical potential energy functions of dia-

tomic molecules by direct fitting [M.Sc. thesis]. Department ofChemistry, University of Waterloo; 2001.

[33] Le Roy RJ, Dattani N, Coxon JA, Ross AJ, Crozet P, Linton C. J ChemPhys 2009;131:204309.

[34] Le Roy RJ, Haugen CC, Tao J, Li H. Mol Phys 2011;109:435.[35] Morse PM. Phys Rev 1929;34:57.[36] Hua W. Phys Rev A 1990;42:2524.[37] Le Roy RJ, Henderson RDE. Mol Phys 2007;105:663.[38] Margenau H. Rev Mod Phys 1939;11:1.[39] Hirschfelder JO, Curtiss CF, Bird RB. Molecular theory of gases and

liquids. New York: Wiley; 1964.[40] Hirschfelder JO. Meath WJ. Intermolecular forces. In: Hirschfelder

JO. editor. Advances in chemical physics, vol. 12. New York: Inter-science; 1967. p. 3–106 [Chapter 1].

[41] Le Roy RJ, Bernstein RB. J Chem Phys 1970;52:3869.[42] Cho Y-S, Le Roy RJ. J Chem Phys 2016;144:024311.[43] Tao J, Le Roy RJ. Pashov A. Merging of the spline-pointwise and MLR

potential forms. In: 65th Ohio State University international sym-posium on molecular spectroscopy, Columbus, Ohio; 2010. p. MG07.

[44] Le Roy RJ. Pashov A. betaFIT: a computer program to fit pointwisepotentials to selected analytic functions. J Quant Spectrosc RadiatTransf; 2016. http://dx.doi.org/10.1016/j.jqsrt.2016.03.036 (thisissue).

[45] Pashov A, Jastrzȩbski W, Kowalczyk P. Comput Phys Commun2000;1:622.

[46] Ivanova M, Stein A, Pashov A, Knöckel H, Tiemann E. J Chem Phys2011;134:024321.

[47] Douketis C, Scoles G, Marchetti S, Zen M, Thakkar AJ. J Chem Phys1982;76:3057.

[48] Tang KT, Toennies JP. J Chem Phys 1984;80:3726.[49] Gunton W, Semczuk M, Dattani NS, Madison KW. Phys Rev A

2013;88:062510.[50] Dattani N, Le Roy RJ. J Mol Spectrosc 2011;268:199.[51] Semczuk M, Li X, Gunton W, Haw M, Dattani NS, Witz J, et al. Phys

Rev A 2013;87:052505.[52] Hepburn J, Scoles G, Penco R. Chem Phys Lett 1975;36:451.[53] Aziz R, Nain V, Carley J, Taylor W, McConville G. J Chem Phys

1979;70:4330.[54] Tang KT, Toennies JP. J Chem Phys 2003;118:4976.[55] Hellmann R, Bich E, Vogel E. Mol Phys 2008;10:133.[56] Patkowski K, Szalewicz K. J Chem Phys 2010;133:094304.[57] Waldrop JM, Song B, Patkowski K, Wang X. J Chem Phys 2015;142:

204307.[58] Samuelis C, Tiesinga E, Laue T, Elbs M, Knöckel H, Tiemann E. Phys

Rev A 2000;63:012710.[59] While not used in LEVEL, LNPT is retained in the calling sequences

to facilitate the use of this subroutine package in other programs.

R.J. Le Roy / Journal of Quantitative Spectroscopy & Radiative Transfer 186 (2017) 167–178178

[60] Le Roy RJ. Molecular spectroscopy. In: Barrow RN, Long DA, MillenDJ, editor. Specialist periodical report 3, vol. 1. London: ChemicalSociety of London; 1973. p. 113–76.

[61] Le Roy RJ. J Chem Phys 1980;73:6003.[62] Le Roy RJ. J Chem Phys 1994;101:10217.[63] Le Roy RJ, Huang Y. J Mol Struct (Theochem) 2002;591:175.[64] Mohr P, Taylor B, Newell D. Rev Mod Phys 2012;84:1527.[65] Wang M, Audi G, Wapstra A, Kondev F, MacCormick M, Xu X, et al.

Chin Phys C 2012;36:1603.[66] Meshkov V, Stolyarov A, Le Roy RJ. Phys Rev A 2008;78:052510.[67] Meshkov VV, Stolyarov AV, Le Roy RJ. J Chem Phys 2011;135:154108.[68] Coxon JA, Hajigeorgiou PG. J Mol Spectrosc 1999;193:306.[69] Aubert-Frécon M, Hadinger G, Magnier S, Rousseau S. J Mol Spec-

trosc 1998;188:182.

[70] Le Roy RJ. J Mol Spectrosc 1999;194:189.[71] Peyerimhoff SD, Buenker RJ. Chem Phys 1981;57:279.[72] Le Roy RJ, Appadoo DRT, Anderson K, Shayesteh A, Gordon IE, Ber-

nath PF. J Chem Phys 2005;123:204304.[73] Henderson RDE, Shayesteh A, Tao J, Haugen CC, Bernath PF, Le Roy

RJ. J Phys Chem A 2013;117:13373.[74] Allard O, Pashov A, Knöckel H, Tiemann E. Phys Rev A 2002;66:

042503.[75] Barrow D, Slaman M, Aziz R. J Chem Phys 1991;91:6348.[76] Jäger B, Hellmann R, Bich E, Vogel E. Mol Phys 2009:107;2181

[erratum: Jäger B, Hellmann R, Bich E, Vogel E. Mol Phys2010:108;105].