Analysis of the a 3S+ state of KRb

Warren T. Zemke,1 Robin Côté,2 and William C. Stwalley21Department of Chemistry, Wartburg College, Waverly, Iowa 50677, USA

2Physics Department, University of Connecticut, 2152 Hillside Road, Storrs, Connecticut 06269-3046, USAsReceived 23 December 2004; published 20 June 2005d

We report analyses and comparison of three high qualityab initio KRb a 3S+ state potentialsfRousseauetal., J. Mol. Spectrosc.203, 235 s2000d; Parket al., Chem. Phys.257, 135 s2000d; Kotochigovaet al., Phys.Rev. A 68, 011501s2003dg. Eigenvalues for all vibrational levels and precision binding energies for high-lyinglevelssand corresponding triplet scattering lengthsd are presented. By adjusting the inner wall of a potential, weshow that we can reproduce measured scattering lengths of two isotopomers and predict scattering lengths ofall isotopomers.

DOI: 10.1103/PhysRevA.71.062706 PACS numberssd: 34.20.2b, 31.50.2x, 34.50.2s

I. INTRODUCTION

Even though the lowest triplet states of the alkali diatomicmolecules are not directly accessible from the lowest singletstatessthey are dark statesd and conventional molecular spec-troscopy is limited in studying them, interest in these states isstill very high. For example, the critical parameter for thestability of a Bose-Einstein condensatesthe scattering lengthdis extremely sensitive to the triplet state potential at largenear-dissociation distances. With the new experimental tech-nique of photoassociationsPAd of molecules from ultracoldatoms and the multiple resonance technique of perturbationfacilitated optical-optical double resonancesPFOODRd,these triplet states have received considerable experimentalattention. The review by Wang and Stwalleyf1g describes PAexperiments and analyses in general while focusing on K2.The review by Li and Lyyraf2g examines work on the manytriplet states of Li2 and Na2, with special emphasis onPFOODR. In addition, coupled with the experimental results,the availability of both high quality theoretical potential en-ergy curves and long range dispersion and exchange coeffi-cients provide an accurate description of the potential overall internuclear distances. In fact, studies of the lowest tripletstates necessarily involve a strong interplay between theoryand experiment.

Recent studies on thea 3Su+ states in sodium and potas-

sium illustrate the current interestsboth theoretical and ex-perimentald in the lowest triplet state and its potential energycurve. In the determination of collisional cross sections forK2, Côtéet al. f3g performed scattering length calculationson several modela 3Su

+ state potentials, all of which at-tempted to characterize properly the three regions of the po-tential: the inner repulsive wall, the attractive shallow well,and the outer long range region of the well. Theyf3g showedthe sensitivity of the scattering length both to slight adjust-ments in the inner wall and the long-range tail of the poten-tial; they noted that a 0.7% shifts1.78 cm−1d in the dissocia-tion energyDe caused a significant shift in the last boundenergy level, resulting in a change in the sign of the scatter-ing length. Wanget al. f4g performed double resonance PAexperiments on K2 and observed binding energies of levelswithin 0.2 cm−1 of the 4S1/2+4S1/2 asymptote; theyf4g cal-

culated an improved precise scattering length for thea 3Su+

statesat=−33±5a0, wherea0 is a Bohr radiusd.In Na2, Ivanovet al. f5g constructed a new accuratea 3Su

+

state potential energy function that included spin-rotation in-teraction to match continuouss23Sg

+→a 3Su+d and discrete

s33Pg→a 3Su+d spectra; their goal was to resolve disagree-

ments with previous empiricala 3Su+ potential curves and

high quality ab initio curves. Araujoet al. f6g reported im-pressive two-color PA experiments on the very high-lyingvibrational levels of thea 3Su

+ potential, two of which arewithin 0.2 cm−1 of the dissociation limit.

There is very limited experimental information on theKRb a 3S+ state potential curve. Inouyeet al. f7g report ob-serving Feshbach resonances in collisions between ultracold40K and 87Rb atoms. Theyf7g used theab initio a 3S+ statepotential of Rousseauet al. f9g smoothly matched to disper-sion and exchange contributions to form a complete potentialcurve and calculate a scattering length; to fit the data theyvaried the scattering length and determined a valueat=−281±15a0. The same group has also published anothervalue for the same isotopomer practically at the same timesuatu=250±30a0, see Goldwinet al. f8gd; this valuesif takento be negatived, is close to their more recent value, althoughoutside their error bars. This illustrates the difficulty in ob-taining accurate experimental values. In their study ofFeshbach resonances, Simoniet al. f10g predicted at=−410±80a0. Earlier, based on elastic cross-section mea-surements, Ferrariet al. f11g determined scattering lengthsfor all six isotopomers of KRb. Skenderović et al. f12g ob-served satellite absorption bands for KRb in a mixture of K2and Rb2. They interpreted the bands as arising from absorp-tion from both X 1S+ and a 3S+ states; they also used thecurves from Rousseauet al. f9g in their spectral simulations.

Along with the Rousseauet al. f9g work, in the literaturethere are two other high qualityab initio calculationsf13,14gon theX 1S+ anda 3S+ states of KRbsalong with many otherexcited singlet and triplet state potentialsf9,13gd. The abinitio calculations on thea 3S+ state are the focus of thiswork and provide an opportunity to systematically analyzehow small differences in the three important regions of thepotential curve affect the number and energies of the vibra-tional levels in this relatively shallow and broad potential.

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Our purpose is to identify those differences as well as helpdetermine which potential might be most useful in the inter-pretation of experimental measurements. Another goal of thiswork is to find out if these potentials can be optimized sothat we recover scattering lengths consistent with experimen-tal values.

II. POTENTIALS

Earlier we used the Parket al. f13g a 3S+ state pointwisepotential sVParkd in the construction of a hybridX 1S+ statepotential curvef15g for KRb. The Rousseauet al. f9g pointssVRousd are available electronically and have been used asnoted above. The Kotochigovaet al. f14g points sVKotd arealso available.

From the minimum of the curve we refer to the potentialasVsRd whereVù0. From the separated atom limit we referto the potential asDVsRd; DVø0 as long as it is attractiveand DVsRd=VsRd−De. The latter convention is particularlyuseful in describing the long range portion of the potential,i.e., the dispersionDVdisp and exchangeDVex contributions tothe potential.

A comparison of the three theoretical potentials is pre-sented in Table I. The dissociation energyDe, equilibriuminternuclear distanceRe, and number of vibrational levelswithin the well describe the attractive portion of the well.Energies«v sv is the vibrational quantum numberd of theuppermost near dissociation vibrational levels are presentedas binding energies and are negative in sign. The uppermostlevels show exactly how close the molecule is to dissociationand provide a description of the long range portion of thepotential. The crossing pointRc salso called classical turningpointd is the distance at which the inner wall of the potentialDVsRcd turns positiveswhere the potential is no longer at-tractived.

Results of calculations on a fourth potential curve are in-cluded in Table Isright-hand columnd. This “X-based” curveis defined asVasRd=VXsRd−2DVexsRd, whereVXsRd is theAmiot and Vergèsf16g Rydberg-Klein-ReessRKRd curve forthe X 1S+ state and where the exchange energyDVexsRd=−CRae−bR, based on the formula of Smirnov and Chibisovf17g. The exponentsa and b are expressed in term of theatomic ionization energiesIK and IRb of potassium and ru-bidium sin atomic unitsd: rX;Î2IX, so thatb=rK +rRb anda=2/rK +2/rRb−1/srK +rRbd−1. In this work, we usedb=1.11892 a.u., anda=5.25603 a.u. The constantC wasdetermined from the Parket al. f13g X and a statepotentials: DVexstheoryd=fVXsRd−VasRdg /2 and C=−DVexstheoryd /Rae−bR; see Refs.f15,18g for details f19g.The X state RKR points extended out to only 19.7a0. Toextend the potential to longer range, the potential was setequal to the dispersion and exchange contributionsVLRsRd=De+DVdispsRd−DVexsRd for all R.19.7a0. DVdispsRd=−C6/R6−C8/R8−C10/R10 where the dispersion coefficientsCn are the accurate values of Derevianko and co-workerssRef. f20g for C6; Ref. f21g for C8 and C10d. Values for theconstants inDVdisp and DVex are given in the footnotes ofTable II.

The VKot and VRous pointwise potentials were alreadylisted as energies with respect to the asymptotefi.e., DVsRdg.We dropped a point atR=12.8a0 in theab initio data ofVKotbecause it was leading to energy levels with unphysical be-havior. TheVPark points were reported as absolute energiesEsRd and their potential points were determined fromDVsRd=EsRd−Easym, whereEasymwas the asymptotic energyat R=132a0. The final potential functionsVsRd were cubicspline fits to allab initio points plus the long range analyticalform VLR; the match at long range was such that each poten-tial V and its slopedV/dR were continuous. The match forthe VKot potential was at 50.0a0 and for theVPark potential,56.7a0. Note that at such large separations, the contributionof Vex is negligible for those two potentials. The lack ofsmoothness and the limitations due to round-off at longrange in theVRous potential pointsssee Table IId caused the

TABLE I. Comparison of potential energy curves for the39K85Rb a 3S+ state.a The binding energies«v are with respect tothe asymptotic energy and in cm−1.

VKot

f14gVPark

f13gVRous

f9g VX−2DVexb

De scm−1d 299.1064 244.5150 238.3556 258.4627

Re sa0d 11.19 11.22 11.09 11.64

Rc sa0dc 9.16 9.33 9.28 10.12

vmaxd 33 32 31 29

−«vmax−1 0.0361 0.0816 0.0315 0.0383

−«vmax0.00075 0.00356 0.00043 0.00094

aValues forDe, Re, and Rc were determined by a spline fit to thepoints of the potential.bVasRd=VXsRd−2DVexsRd, whereVXsRd is from the RKRX statepotentialf16g andDVexsRd=−CRae−bR. Values forC, a, andb arelisted in the footnotes of Table II.cThe crossing pointRc is the distance on the inner wall of thepotential whereDVasRd changes sign.dThe maximum vibrational quantum number; the number of levelsequalsvmax+1.

TABLE II. Long-range KRba 3S+ state potentialssin cm−1d.

Rsa0d

DVRous

f9gDVPark

f13gDVKot

f14g DVLRa −DVex

16.0 −77.0 −76.8781 −75.2710 −75.5010 18.1783

20.0 −25.0 −22.4653 −19.4370 −19.5651 0.6686

24.0 −10.0 −7.4532 −6.0574 −6.0897 0.0198

28.0 −5.0 −2.7286 −2.2696 −2.2791 0.0005

32.0 −3.0 −1.2112 −0.9810 −0.9843

36.0 −2.0 −0.6752 −0.4718 −0.4731

40.0 −1.0 −0.4354 −0.2463 −0.2468

50.0 −0.0 −0.1035 −0.0629 −0.0629

aDVLR;VLR−De=DVdisp−DVex, with DVdisp=−C6/R6−C8/R8

−C10/R10, where C6=4274 hartreessa0d6, C8=493 000 hartreessa0d8, and C10=63 000 000 hartreessa0d10, and with DVex=−CRae−bR, whereC=0.00 231 382 hartrees,a=5.25603 withR ina0 andb=1.11892sa0d−1.

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match of the long range potential with theirf9g potential tooccur at 16.0a0. It should be noted that values forDe, Re, andRc in Table I were determined by the fit to the points for eachpotentialfi.e., De=−DVsRedg, and the given precision inDe

values is somewhat artificial.Table II compares the long range behavior of the threeab

initio potentials with the analytical long range potentialDVLRsRd=DVdispsRd−DVexsRd. At long range all three poten-tials show at least semiquantitative agreement withVLR andeach other. The long range behavior of theVa=VX−2DVexpotential is identical to that of the long range potentialVLRfor Rù20a0 and hence the potential is not included in TableII.

III. ANALYSIS AND CONCLUSIONS

Using theEIGEN code of Sando and Dalgarnof22g eigen-values for the three potentials were calculated; see Table III.The mesh of grid points for integration was analyzed care-fully; integration over the region of the wells5,R,25a0dused a “tight” grid with enough closely spaced pointsfs0.001–0.004da0g to thoroughly cover the highly oscillatorybehavior of the wave functions. The outer limit of integrationwas alwaysRù460a0, but sometimes was extended to7000a0 to cover the broad tails of wave functions with ener-gies very close to dissociations,0.000001 cm−1d.

The breadth and depth of the potential well have a signifi-cant impact on the number of vibrational levels supportedwithin the well. As the depthsDe valued increases for thethree ab initio potentials, the number of levels increases.However, theVa=VX−2DVex potential is deeper than eitherthe VRous or the VPark potential but supports only 30 levelsbecause it is not as broadsRc=10.1 rather than 9.3a0d; seeTable I.

Direct eigenvalue calculations yield 34, 33, and 32 levelsfor the VKot, VPark, andVRous potentials, respectively, for allisotopic combinations except the heavier isotopomers for theVPark and VKot potentials; see Table IV. The limits of ourbinding energiess«bd are ,0.000001 cm−1. To more care-fully examine the long range behavior of the potentialswhereDVex is very smallsfor Rù24a0d, we plotted the cuberoot of the uppermost binding energies against the vibra-tional quantum numberv. Sinceu«vu1/3 for levels near disso-ciation are proportional tosvD−vd f23,24g, we obtained val-ues forvD, the effective vibrational quantum number at thedissociation limit, for all six isotopomersssee Table IVd. ThevD values indicate that for theVPark potential there should bea v=33 level for40K87Rb, 41K85Rb, and41K87Rb and for theVKot potential av=34 level for40K87Rb sconsistent with ac-tual eigenvalue calculationsd. However, we do not obtain for40K85Rb av=33 level for theVParkpotential, even though forthat isotopomervD=33.03 ssee Table IVd. This inconsis-tency, as well as the irregular behavior ofvD with increasingreduced masssTable IVd, seems due to the “roughness” intheVPark potential curve forRù28a0, demonstrated in TableII with the comparison ofDVPark with the smoothDVdisp atlong range. For theVRous potential the highest vibrationallevel is v=31 si.e., 32 levels totald. This is consistent withSimoni et al. f10g who report 31±1 vibrational levels based

on the Rousseauet al. f9g potential. However, using theRousseauet al. f9g potential, Skenderović et al. f12g in theirsimulations calculate 38 levels, in significant disagreementwith our calculationssregardless of the potentiald.

From 32 to 50a0, where DVex is vanishingly small,DVLR.−C6/R6−C8/R8 and a plot ofR6s−DVab initiod against1/R2 should be linear at largeR and yield a slope close to theC8 value and an intercept close to theC6 value. The plot ofR6s−DVKotd yielded values forC6 andC8 in excellent agree-ment f25g with the values based on the atomic calculationsf20,21g, whereas the plot ofR6s−DVParkd deviated signifi-cantly from being linear. This behavior is consistent with ourconclusion about theVPark potential in the preceding para-graph. Based on the remarkable fit of theDVKot potential tothe long range potentialDVLR=DVdisp−DVex ssee also Table

TABLE III. EigenvaluesEv sin cm−1d for three39K85Rb a 3S+

stateab initio potentials.

v VKot f14g VPark f13g VRous f9g

0 8.8803 8.6713 8.5149

1 26.1000 25.5545 25.2940

2 43.2304 41.8072 41.4141

3 60.2934 57.4192 56.6192

4 77.2540 72.3858 71.2603

5 93.9907 86.7037 85.2711

6 110.4225 100.3739 98.5705

7 126.4609 113.3980 111.2305

8 142.0125 125.7784 123.2321

9 156.9786 137.5174 134.4902

10 171.2659 148.6157 145.0731

11 184.8155 159.0728 155.1752

12 197.5849 168.8871 164.8142

13 209.5250 178.0557 173.9891

14 220.6195 186.5762 182.5957

15 230.8766 194.4456 190.5489

16 240.3098 201.6637 197.8585

17 248.9359 208.2337 204.5066

18 256.7750 214.1626 210.5012

19 263.8481 219.4647 215.8405

20 270.1759 224.1611 220.5382

21 275.7799 228.2783 224.6060

22 280.6827 231.8478 228.0606

23 284.9083 234.9039 230.9267

24 288.4841 237.4947 233.2362

25 291.4421 239.6463 235.0298

26 293.8206 241.3408 236.3578

27 295.6656 242.5808 237.2801

28 297.0317 243.4406 237.8651

29 297.9823 243.9565 238.1871

30 298.5878 244.2439 238.3241

31 298.9242 244.4334 238.3552

32 299.0703 244.5114

33 299.1056

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II d, we would normally favor theirf14g ab initio potentialover the other two. However, theVPark potential seems toyield significantly better agreement with measured scatteringlengths than doVKot and VRous potentials. The long rangebehavior of the potential is not the only criterion by which toevaluate the potential curve.

Table IV lists scattering lengthsat and the uppermost vi-brationalsbindingd energies for each of theab initio poten-tials; for comparison the table also lists published valuesf7,8,10,11g sbased on the Rousseauet al. f9g potentialsmoothly matched to the long range potential, except for Ref.f11gd. Our calculations were performed using the codes ofCôtéet al. f26g. Based on both sign and magnitude ofat, nopotential agrees well with earlier experimental resultsf7,8,10,11g. The best agreement with measured scatteringlengths comes with theVPark potential. Agreement is verygood for all isotopomers except for40K87Rb ssee Table IVd.However, because the scattering length for40K87Rb is largeand positivesi.e., thev=33 level is just barely boundd, a verysmall change in the potential could shiftat from a large posi-

tive value to a large negative value. Ferrariet al. f11g notethat their analysis is based on only 32 bound statessenergylevelsd.

In order to match the calculated scattering lengths withthe experimental values, we modified bothVParkandVKot. Weachieved this by simply shifting smoothly the inner wall “in”or “out” for R,Re;

Rshifted= R+ sS R− Re

Rc − ReD , s1d

wheres corresponds to a shift “in”sif negatived or “out” sifpositived of the classical turning pointRc, all other pointsbeing shifted proportionally. All points forRùRe are notmodified. We illustrate the effect of such a shift in Fig. 1.The modified potentials are then cubic splined and used tocompute the scattering lengths for all isotopomers. Figure 2illustrates the change in the scattering length for theVKotpotential as a function of the shifts. We found that for a shifts=−0.069a0, we could obtain scattering lengths for both

TABLE IV. Effective vibrational quantum numbers at dissociationsvDd, binding energiess«vd, and scat-tering lengthssatd of the a 3S+ state of KRb for three different potentials.

Isotopomer 39K85Rb 39K87Rb 40K85Rb 40K87Rb 41K85Rb 41K87Rb

vD

VKot 33.39 33.49 33.69 33.82 33.98 34.11

VPark 32.65 32.79 33.03 33.23 33.24 33.17

VRous 31.32 31.44 31.61 31.73 31.88 32.00

−«vmaxs0.001 cm−1d

VKot 0.75 1.73 4.21 6.95 11.78 0.0023a

VPark 3.56 6.16 12.31 0.0011b 0.19b 0.62b

VRousc 0.43 1.13 2.89 4.93 8.58 12.55

at sa0dTheory

VKot 119 86 45 13 −100 3223

VPark 65 33 −95 1085 186 130

VRous 143 103 66 39 −15 −152

Shiftedd

VKot 68 40 −25 −225 310 153

VPark 73 45 −23 −243 319 164

Exp.e

Ferrari 58−6+14 31−6

+16 −38−17+37 −261−159

+170 329−55+1000 163−12

+57

et al. f11gSimoni −410−80

+80 156−5+10

et al. f10gInouye −281−15

+15

et al. f7gGoldwin −250−30

+30

et al. f8gaFor the Kotochigovaet al. f14g potential, this binding energy is forvmax=34 snot 33d.bFor the Parket al. f13g potential, these binding energies are forvmax=33 snot 32d.cFor the Rousseauet al. f9g potential,vmax=31.dThe shift values ares=−0.069a0 ands= +0.032a0 for VKot andVPark, respectively.eInferred values from a single measurement of a different isotopomer are underlined.

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40K87Rb and41K87Rb that are simultaneously inside the errorbars of the measured values of Goldwinet al. f8g and Simoniet al. f10g. This was the onlys value for which agreementwith experimentalat could be achieved in the range −0.1,s,0.1a0: largers would lead to unphysical features in thepotential. Because of the uncertainty between the values ofInouyeet al. f7g and Goldwinet al. f8g, we decided to com-pare with the latter. Theat values for the shifted potential arelisted in Table IV.

For VPark, an agreement is obtained fors=0.032a0; TableIV again lists theat values for this shifted potential. In thiscase, however, the scattering length for41K87Rb is less sen-

sitive to s than for VKot, while it changes drastically for40K87Rb, and hence we can find a value ofs that wouldaccommodate all measured scattering lengths for those twoisotopomers. Therefore no clear conclusion forat can bedrawn fromVPark. Because of the highly accurate long rangeportion of VKot and the matching of measured scatteringlengths, we favor the shiftedswith s=−0.069a0d VKot poten-tial for the a 3S+ state of KRb.

We observe that the scattering length is quite sensitive tothe complete potential energy curve, including the breadth ofthe curve; noteRc=9.33a0 for VPark, compared to 9.28a0 and9.16a0 for VRousandVKot, respectively. For the shifted poten-tials VParkss=0.032d and VKotss=−0.069d, we find Rc

=9.36a0 and 9.09a0, respectively. We also observe that asexpectedf27g the reduced mass of the KRb isotopomer af-fects very significantly the scattering length; see Fig. 3.Rather than calculating the scattering length directly, for veryweakly bound states the approximationf28g relating at in-versely to the square root of the binding energyfat

=" /Î2m«v, wherem is the reduced massg can be tested forKRb. For thev=34 level in theVKot potential for 41K87Rband thev=33 level in theVPark potential for 40K87Rb, theapproximation is very satisfactory; but it works less well fordeeper bound levelsf29g. Based on the scattering lengthcomparison for all six isotopomers, we favor the two shiftedpotentials for thea 3S+ state of KRb. The decision as towhich is better should be postponed until further experimentshave been carried out.

The theoretical framework for thea 3S+ state of KRb is inplace and should be a helpful guide for further experimentalefforts. Photoassociation experiments may be capable ofcharacterizing a number of the high-lying energy levels in-cluding the last one just below dissociationf30g. The veryrecent observationf31g of the photoassociation, moleculeformation, and trapping of ultracold39K85Rb molecules inthe a 3S+ state is very encouraging. However, more spectro-scopic investigations are needed to experimentally observethe vibrational levels in thea 3S+ state of KRb.

FIG. 1. Potential curves as a function of the internuclear sepa-ration R: VKot slong-dashed lined, VPark ssolidd, and VRous sdashedlined. The inset shows the effect of shifting the inner wall for threevalues ofs for VKot.

FIG. 2. Scattering length as a function of the shifts for VKot.The left panel showsat for both isotopomers40K87Rb and41K87Rbover a larger range ofs values: the sharp variation ofat ass growsindicates an additional vibrational level appearing. The open andclosed circles denote the values ofat for s=0 ands=−0.069a0 forboth isotopomers. The right panels showat for both isotopomersand the range of experimental values shown by the shaded area. Fors in the vicinity of −0.069a0, both calculated values ofat are withinthe error bars of the measured values.

FIG. 3. Scattering lengths as a function of the reduced mass forthe shiftedVKot with s=−0.069a0.

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ACKNOWLEDGMENTS

Support received from the National Science Foundation isgratefully acknowledged. We are particularly grateful to Pro-

fessor Monique Aubert-Frécon, Professor Gwang-Hi Jeung,and to Dr. Svetlana Kotochigova for sharing unpublished nu-merical results with us.

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based on theVXsRd andVasRd potentials of Ref.f13g over therange 13øRø21a0. Based on potentials from Ref.f14g overthe range 15øRø25a0, the calculated constantCKot wasfound to be in good agreementswithin 1.0%d with the CPark

value. One standard deviation in the fit of either valueCPark orCKot is 1.4%.

f20g A. Dervianko, J. F. Babb, and A. Dalgarno, Phys. Rev. A63,052704s2001d.

f21g S. G. Porsev and A. Dervianko, J. Chem. Phys.119, 844s2003d.

f22g K. M. Sando and A. Dalgarno, Mol. Phys.20, 103 s1971d.f23g R. J. Le Roy, inMolecular Spectroscopy 1, edited by R. F.

Barrowet al. sChemical Society of London, London, 1973d, p.113.

f24g W. C. Stwalley, Contemp. Phys.19, 65 s1978d.f25g In the case of theDVKot potential,C6 andC8 values agreed to

within 0.1% and 4.0% of those from Refs.f20,21g, respec-tively sthe DVRous potential is identical toDVLR=DVdisp

−DVex for Rù16a0d.f26g R. Côté, M. J. Jamieson, Z.-C. Yan, N. Geum, G.-H. Jeung,

and A. Dalgarno, Phys. Rev. Lett.84, 2806s2000d.f27g G. F. Gribakin and V. V. Flambaum, Phys. Rev. A48, 546

s1993d.f28g R. Côté, M. J. Jamieson, Z.-C. Yan, N. Geum, G. H. Jeung,

and A. Dalgarno, Phys. Rev. Lett.84, 2806s2000d.f29g In 41K87Rb for theVKot potential, the directly calculated scat-

tering lengthatsdirectd=3223a0 while using the approxima-tion, atsapproxd=3066a0 sa 5% differenced. In 40K87Rb for theVPark potential, atsdirectd=1085a0 while atsapproxd=1410a0

s30%d. In 39K85Rb, where the binding energy of the last levelis more than 100 times largersu«bu,0.0004–0.0008 cm−1dthan in the case of the two preceding examples, forVKot,atsdirectd=119a0 while atsapproxd=55a0, and for VRous,atsdirectd=143a0 while atsapproxd=72a0.

f30g As in the a 3S+ state of RbCs: Thomas Bergemansprivatecommunicationd.

f31g D. Wang, J. Qi, M. F. Stone, O. Nikolayeva, H. Wang, B.Hattaway, S. D. Gensemer, P. L. Gould, E. E. Eyler, and W. C.Stwalley, Phys. Rev. Lett.93, 243005s2004d.

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