Binding energies of the ground triplet state a 3 u + of Rb2 and Cs2 in terms of thegeneralized Le Roy–Bernstein near-dissociation expansionV. B. Sovkov and V. S. Ivanov

Citation: The Journal of Chemical Physics 140, 134307 (2014); doi: 10.1063/1.4869981 View online: http://dx.doi.org/10.1063/1.4869981 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Updated potential energy function of the Rb2 a 3 u + state in the attractive and repulsive regions determinedfrom its joint analysis with the 230g state J. Chem. Phys. 139, 144303 (2013); 10.1063/1.4823496 Joint analysis of the Cs2 a 3 u + and 1 g (331g ) states J. Chem. Phys. 135, 024303 (2011); 10.1063/1.3606397 Experimental investigation of the R 85 b 2 a 3 u + triplet ground state: Multiparameter Morse long rangepotential analysis J. Chem. Phys. 131, 094505 (2009); 10.1063/1.3194290 ( ns 1/2 + np 1/2 )0 g Rb 2 and Cs 2 photoassociative spectroscopy of weakly bound levels: LuFano analysiscoupled to an improved LeRoyBernstein formula. AIP Conf. Proc. 935, 203 (2007); 10.1063/1.2795415 Calculation of accurate permanent dipole moments of the lowest + 1 , 3 states of heteronuclear alkali dimersusing extended basis sets J. Chem. Phys. 122, 204302 (2005); 10.1063/1.1903944

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

130.63.180.147 On: Wed, 13 Aug 2014 18:42:42

THE JOURNAL OF CHEMICAL PHYSICS 140, 134307 (2014)

Binding energies of the ground triplet state a3�+u of Rb2 and Cs2 in terms

of the generalized Le Roy–Bernstein near-dissociation expansionV. B. Sovkov and V. S. IvanovV.A. Fock Institute of Physics and Department of Physics of St. Petersburg State University,Ulyanovskaya Street 1, Petrodvoretz, St. Petersburg 198504, Russia

(Received 20 January 2014; accepted 19 March 2014; published online 7 April 2014)

Formulae of Le Roy–Bernstein near-dissociation theory are derived in a general isotope–invariantform, applicable to any term in the rotational expansion of a diatomic ro-vibrational term value.It is proposed to use the generalized Le Roy–Bernstein expansion to describe the binding ener-gies (ro-vibrational term values) of the ground triplet state a3�+

u of alkali metal dimers. The pa-rameters of this description are determined for Rb2 and Cs2 molecules. This approach gives arecipe to calculate the whole variety of the binding energies with characteristic accuracies from∼1 × 10−3 to 1 × 10−2 cm−1 using a relatively simple algebraic equation. © 2014 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4869981]

I. INTRODUCTION

The alkali metal dimers are interesting for many prospec-tive applications such as laser cooling, creating and exploringquantum condensates, cold atom (molecule) collisions, co-herent control, quantum computing, and many others (see,e.g., Refs. 1–17 and references therein). The lowest elec-tronic states X1�+

g and a3�+u of the dimers play an espe-

cially important role in those phenomena. For example, theX1�+

g ∼ a3�+u states of Cs2 were proposed to be a reference

system in a check of a possible variation of fundamental worldconstants.18–20

Within the series of works devoted to the spectroscopyof alkali metal dimers21–34 the ground triplet state a3�+

u hasalways been under our special attention. In part, the empiricalpotential energy function of this state was obtained in a formof a multi-parameter model function35–44 in Ref. 23 (see alsoRef. 24) for Cs2, and in Ref 21 (see also Refs. 13 and 25) forRb2. Recent observations18 of the Cs2 a3�+

u state agree withthe potential of Ref. 23 fairly well.

The potential energy function is, perhaps, the most im-portant characteristic of an electronic state of a moleculenecessary to simulate most of the spectroscopic and dynam-ical phenomena in which this state is employed.45 How-ever, those simulations usually require implementing rela-tively complicated numerical algorithms and computationalprograms, not always available at hand. Meanwhile, it is of-ten desirable to possess simpler equations suitable for a briskcalculation of observable molecular properties with com-monly available software. Such equations are very helpfulin interpreting experimental data, where they simplify ini-tial assignments of observed states and estimates of molecularparameters.

The well-known example of such an equation is the Dun-ham series,45–47 which can be treated as a correction to theharmonic oscillator and the rigid rotator approximations (seeEq. (A25) below). Despite the fact that the Dunham series wasproposed in the first half of XXth century, it is still widelyused up to now. The Dunham expansion is able to describe

the lower ro-vibrational levels fairly well. However, its well-known drawback is that it cannot be used to determine ener-gies of higher levels lying in the vicinity of the dissociationlimit.

An appropriate substitute for the Dunham series over-coming the mentioned difficulty is the near-dissociation ex-pansion based on the Le Roy–Bernstein theory. The princi-pal ideas of the theory were proposed in Refs. 48 and 49(see also Ref. 50) where an equation for the vibrational termvalue Gv was derived. In Ref. 51 an equation for the first ro-tational constant Bv , and in Ref. 52 (see also Ref. 53) analo-gous equations for the constants Dv , Hv , Lv were presented.Various aspects of the theory were studied in Refs. 54–61aimed to exploring properties of diatomic molecules, firstof all for the determination of the dissociation energies andthe long-range parameters—see also Refs. 62–73 and refer-ences therein. It was also proposed and proved to be effec-tive for an accurate description of large spectroscopic datasets.74

The primary aim of the present work was to find a rela-tively simple and compact description of all the binding ener-gies (ro-vibrational term values) of the Rb2 and Cs2 a3�+

u

states, which would be quantitatively equivalent in somesense to our potential functions of Refs. 21 and 23. The mostappropriate way to do it is the approach based on the Le Roy–Bernstein near-dissociation theory. Nonadiabatic couplings,fine and hyperfine interactions, etc. are beyond the scope ofthis work, while these effects may change the term valuesnoticeably on the scale of the uncertainties obtained in thepresent analysis.

There were two points we wish to highlight. First,the data for the rubidium dimer dealing with different iso-topomers had to be described with the same unique set ofparameters analogous to the unique potential energy func-tion. Second, we had supposed that contributions of higherterms of the rotational expansion beyond the ones consideredin the near-dissociation theory up to now may be substantialfor high rotational states. So, the other, no less important, aimof the work became a re-derivation of the Le Roy–Bernstein

0021-9606/2014/140(13)/134307/10/$30.00 © 2014 AIP Publishing LLC140, 134307-1

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

130.63.180.147 On: Wed, 13 Aug 2014 18:42:42

134307-2 V. B. Sovkov and V. S. Ivanov J. Chem. Phys. 140, 134307 (2014)

TABLE I. Parameters of Eq. (1) for the a3�+u state of Rb2. Missing values are all zero. The parameters a, d, γ D

were fitted, c6 and ED were taken from Ref. 21 and fixed. The energy ED of the dissociation limit is measuredrelative the bottom of the X1�+

g ground state.

Param. Value Param. Value Param. Value Param. Value

a10 − 0.1 a11 − 0.6597 a12 − 1.0407 d − 0.065

a20 3.914 a21 4.6 a32 1.878 γ D (√

cm−1Å) 25.5845a30 − 24.4788 a31 − 22.4316 a42 − 1.9503a40 59.049638 a41 53.7917 a52 0.63a50 − 77.344 a51 − 72.97 a13 − 0.63 c6 (cm−1Å6) 2.27 × 107

a60 58.46863 a61 57.59638 a23 0.24 ED (cm−1) 3993.664a70 − 24.08315 a71 − 24.7695 a14 − 1.56a80 4.2 a81 4.5

equations in the isotope-invariant form applicable to an ar-bitrary term in the rotational expansion of a diatomic ro-vibrational term value.

The part devoted to the generalization of the Le Roy–Bernstein formulae is placed in the Appendix, and applica-tions of the theory to Rb2 and Cs2 are presented in the mainbody of the article.

II. BINDING ENERGIES OF THE a3�+u STATE

OF Rb2 AND Cs2

We computed the sets of the term values E(v,N) of thea3�+

u state of Rb2 and Cs2 with the potential functions ofRefs. 21 and 23 using the renormalized Numerov method75–79

with the shooting algorithm to solve the eigenvalue problem.The Rb2 data spanned levels from v = 0 to v = 40 for lowerN and up to N = 154, v = 0 (the highest rotational statecontaining a bound level) for the two isotopomers 85Rb2 and87Rb2. The Cs2 data spanned levels from v = 0 to v = 55 forlower N and up to N = 215, v = 0 for the only natural iso-topomer 133Cs2. The designations v and N represent the vibra-tional and rotational quantum numbers.

We fitted these data sets to Eq. (A22) of the Appendixtruncated at j = 12 in a case of n = 6 (the leading power ofthe dissociation asymptote of the states under consideration).Namely,

E(v,N) = ED + (γD − γ )3

√c6

12∑j=0

Xj (6)

×[

λ

(γD − γ )2

]j

Qj (γD − γ ), (1)

where

γ = G(m)

(v + 1

2

), λ = G2(m)N (N + 1),

G(m) = ¯√2m

,

Qj(ζ ) is a correction function of the form

Qj (ζ )= 1 + a1jα(ζ ) + a2jα2(ζ ) + · · ·

1 + b1jα(ζ ) + b2jα2(ζ ) + · · · , α(ζ )= ζ

γD + dζ.

(2)The coefficients Xj(6) can be taken from the last column ofTable III in the Appendix, ED is the dissociation limit ofthe state, c6 is the dispersion coefficient, and m is the re-duced mass of the molecule (isotopomer). The relation G(m)= 4.1058045√

m(Å

√cm−1) is valid if m is measured in atomic

mass units.The resulting parameters are listed in Table I for Rb2 and

in Table II for Cs2. Fig. 1 presents the residuals of the final

TABLE II. Parameters of Eq. (1) for the a3�+u state of Cs2. Missing values are all zero. The parameters a, b, d, γ D were fitted, c6 and ED were taken the same

as in Ref. 23 and fixed. An information about the quantity c6 can be found in Refs. 16 and 18. The energy ED of the dissociation limit is measured relative thebottom of the X1�+

g ground state.

Param. Value Param. Value Param. Value Param. Value Param. Value

a10 − 3.28 a11 − 2.023 a12 − 9 b13 − 2.5902 b25 6.94a20 11 a21 − 12 a22 35.7358 b23 1.725a30 − 106.2 a31 114.262 a32 − 73.2 a14 − 49.2 d 0.4528

a40 687.28 a41 − 399.59 a42 85.1 a24 153.478 γ D (√

cm−1Å) 28.746a50 − 2074.33588 a51 724.05806 a52 − 37 a34 − 122a60 3276.69 a61 − 665.954 b12 − 6.403 b14 − 1.626a70 − 2651.6033 a71 246 b22 12.4 b24 0.61 c6 (cm−1Å6) 3.32055 × 107

a80 870 a13 1.3 a15 6.6 ED (cm−1) 3650.0321b10 − 3.8325 b11 − 2.35 a23 − 11.5017 a25 − 15.6b20 8.12 b21 3.11 a33 11 b15 − 4.865

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

130.63.180.147 On: Wed, 13 Aug 2014 18:42:42

134307-3 V. B. Sovkov and V. S. Ivanov J. Chem. Phys. 140, 134307 (2014)

-2

-1

0

1

2

3

4

5

Rb2 a3Σu+

δE, 1

0-3 c

m-1

0 20 40 60 80 100 120 140 N

0 20 40 60 80 100 120 140N

160 180 200-7-6-5-4-3-2-10123456

δE, 1

0-3 c

m-1

Cs2 a3Σu+

(a)

(b) σ = 1.3x10-3 cm-1

σ = 9.4x10-4 cm-1

FIG. 1. Residuals of the term values reproduced by Eq. (1) with the parame-ters of Tables I and II as a function of the rotational quantum number N: Thea3�+

u state of Rb2 (a) and Cs2 (b).

approximations (i.e., differences between values calculatedfrom Eq. (1) and the ones calculated from the solution of theSchrödinger equation with the potential functions of Refs. 21and 23). Notice that the sequential rounding and refitting pro-cedure described in Ref. 80 was applied to the values reportedin Tables I and II, so that these values exactly ensure the qual-ity of the reproduction seen in Fig. 1.

The standard deviations of the simulations are σ = 9.4× 10−4 cm−1, σ = 1.3 × 10−3 cm−1, and the maximum ab-solute deviations are 5.1 × 10−3 cm−1, 6.6 × 10−3 cm−1 forthe cases of Rb2 and Cs2, respectively. Such an accuracy per-fectly satisfies modern experimental requirements.

III. DISCUSSION

Notice that in fact we had started with fitting the 87Rb2

data alone. The resulting description had reproduced the otherisotopomer 85Rb2 data with almost the same quality as of thefinal computation, confirming the ability of the method to pre-dict data for various isotopomers from only one of them. Inthe final computation, whose results are presented above, thedata for both isotopomers were included.

It is known that the situation can be encountered, inwhich an approximant becomes physically unreliable, espe-cially near the borders of the interval, due to a poor stabilityof the problem. In fact, the statistical significance of the pa-rameters kept in an approximant is already a good confirma-tion of its stability. We performed a couple of additional teststo show that our result does not contain such an artifact.

The vicinity of the potential well bottom can be charac-terized by the Dunham coefficients, whose consistency canbe supported by the correlations45, 81, 82 discussed in the finalpart of the Appendix. We compared the values of the Dun-ham coefficient Y02 computed directly (Eq. (A29)) from theparameters of Tables I and II to their estimates Y ′

02 computedusing the relation, Eq. (A30). We found that Y02 = −2.646× 10−8 cm−1, Y ′

02 = −2.698 × 10−8 cm−1 for 85Rb2, Y02

= −2.526 × 10−8 cm−1, Y ′02 = −2.576 × 10−8 cm−1 for

87Rb2, and Y02 = −7.832 × 10−9 cm−1, Y ′02 = −8.051

× 10−9 cm−1 for Cs2. Those estimates are close enough toeach other, confirming our approximant stability, at least inthe lower part of the potential well.

The Le Roy–Bernstein theory was developed for thebound ro-vibrational states of a two-body system. However,Eq. (1) allows a smooth extrapolation to the rotationally pre-dissociated levels (shape resonances) above the dissociationlimit. We tested the ability of such an extrapolation to pre-dict the energies of those levels. We computed a syntheticspectrum of the transitions 85Rb2 23�0g (v′ = 5, J ′ = 100)→ a3�+

u (v, N = 100) with the split-operator algorithm83–88

using potential functions of Ref. 21, and compared positionsof the shape resonances in it with the ones predicted byEq. (1)—see Fig. 2. Although the accuracy of this predictionis lower than that for the bound levels, it can be considered asbeing reasonable. This result also confirms that our approxi-mation is stable in the vicinity of the dissociation limit.

As mentioned above, the highest rotational term ofEq. (1) computed was j = 12. We checked how the reduc-tion of it (i.e., the direct truncation of the rotational expan-sion to smaller j without a refit of the parameters) influencesthe quality of the reproduction of term values. The results ofthis test are shown in Fig. 3. It presents j-dependencies of the

FIG. 2. Fragment of the synthetic spectrum of the 85Rb2 23�0g

(v′ = 5, J ′ = 100) → a3�+u (v, N = 100) transitions (solid line) and pre-

dictions of the line positions with Eq. (1) and Table I (circles). Positions ofthe dissociation limit and the top of the rotational barrier are shown with thedashed vertical lines. The origin of the scale of the final state energy E is thebottom of the Rb2 X1�+

g ground state potential.

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

130.63.180.147 On: Wed, 13 Aug 2014 18:42:42

134307-4 V. B. Sovkov and V. S. Ivanov J. Chem. Phys. 140, 134307 (2014)

10-3

0.1

1

10

100

10-2

0 1 2 3 4 5 6 7 8 9 10 11 12

10-3

0.1

1

10

100

10-2

0 1 2 3 4 5 6 7 8 9 10 11 12j

j

σ an

d m

ax(|δ

E|),

cm

-1σ

and

max

(|δE

|), c

m-1

σ, all Nσ, N ≤ 10max(|δE|), all Nmax(|δE|), N ≤ 10

Rb2 a3Σu+

Cs2 a3Σu+

(a)

(b)

σ, all Nσ, N ≤ 10max(|δE|), all Nmax(|δE|), N ≤ 10

FIG. 3. Standard deviation σ and maximum absolute deviation max (|δE|) ofthe term value reproduction by Eq. (1) as a function of the number j of termsin Eq. (1): The a3�+

u state of Rb2 (a) and Cs2 (b).

standard deviations and the maximum absolute deviations forthe entire sets of data and for the subsets of the lowest rota-tional states N ≤ 10. It is seen that for truncation to j = 8for Rb2 and to j = 10 for Cs2, all residuals remain smallerthan 10−2 cm−1. For the subsets of N ≤ 10, the same bound-aries are maintained with j = 1 and j = 6 correspondingly.This means that the description of the Rb2 binding energiespresented here is somehow of a better quality than for Cs2. Itallows one to compute the low N subset of Rb2 data from onlya part of the parameters of Table I. On the other hand, this testdemonstrates that high terms of the rotational expansion areindeed able to make a noticeable contribution to the simulatedterm values, first of all, as expected, for high rotational states,but also, in some cases for relatively low rotational states aswell.

We have applied the proposed descriptions to the exper-imental data used in Refs. 21 and 23. The result for the mostaccurate set of data, namely, the deperturbed (in the samesense as in Ref. 21) binding energies of the 87Rb2 a3�+

u N= 0, 2, 4 states of Ref. 13, is shown in Fig. 4. Based on theconclusion of the previous test, we have truncated the rota-tional expansion to only the two lowest terms, i.e., to j = 1, inthis computation. Fig. 4 can be compared to Fig. 7(b) ofRef. 21, where an analogous result obtained by using the po-tential function directly was shown. The accuracy of the ex-perimental data reproduction has worsened (standard devia-tion σ = 1.0 × 10−3 cm−1 here versus σ = 5.1 × 10−4 cm−1

in Ref. 21). Nevertheless, it is still better than the one of therough experimental data reproduction in Ref. 13 (σ = 4.2

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

v

σ = 1.0x10-3 cm-1N = 0N = 2N = 4

87Rb2 a3Σu+

δE, 1

0-3cm

-1

0 5 10 15 20 25 30 35 40

FIG. 4. Residuals of the reproduction of the deperturbed experimental13

binding energies of the 87Rb2 a3�+u state with Eq. (1) and Table I truncated

to j = 1 (compare Fig. 7(b) of Ref. 21).

× 10−3 cm−1). The accuracy of the other experimental dataof Refs. 21 and 23 reproduction is almost indistinguishablefrom the one achieved in those works.

In some applications the value vD of the limiting vibra-tional number,48, 49, 51–60 showing the total number of levelsthe potential well contains, is of interest. According to theAppendix and the designations of Eq. (1) it can be computedfrom our parameter γ D with the equation

vD = γD

G(m)− 1

2.

This gives vD = 40.102 for 85Rb2, vD = 40.577 for87Rb2, and vD = 56.574 for Cs2. We have also checkedthe capability of the simple semi-empirical equation vD

= 1.78349De/ωe − 5/8 proposed in Ref. 61 to reproducethese values. This equation gives estimates of vD = 30.9 for85Rb2, vD = 31.3 for 87Rb2, and vD = 43.4 for Cs2, whichare clearly very far from the actual values. This means thatthis equation has a very limited applicability, mostly to thesystems studied in Ref. 61.

IV. CONCLUSIONS

We have presented the generalized equations of the LeRoy–Bernstein near-dissociation theory written in an isotope-invariant form and applicable to any term in the rotationalexpansion of a diatomic ro-vibrational term value. The near-dissociation expansion based on these equations has been pro-posed for a description of the binding energies (ro-vibrationalterm values) of the ground triplet state a3�+

u of Rb2 and Cs2.The accuracy of this description perfectly satisfies require-ments of modern experiments. The properties of it have beenstudied with a series of numerical tests. In part, the tests haveshown that high terms in the rotational expansion of the ro-vibrational term values are able to make a noticeable contri-bution, even for not very high rotational states.

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

130.63.180.147 On: Wed, 13 Aug 2014 18:42:42

134307-5 V. B. Sovkov and V. S. Ivanov J. Chem. Phys. 140, 134307 (2014)

APPENDIX: GENERALIZED FORMULAE OF THELE ROY–BERNSTEIN NEAR-DISSOCIATION THEORY

The purpose of the following discussion is the deriva-tion of the expression for the ro-vibrational energy E(v,N)of a diatomic molecule in terms of the vibrational v androtational N (sometimes J) quantum numbers implementingthe near-dissociation properties of the potential energy func-tion. This derivation mostly uses the same basic ideas of theLe Roy–Bernstein theory as of the original works cited in theIntroduction. The difference of our formulae from the originalone is that it is written in an isotope-invariant form and thatthe rotational expansion for the ro-vibrational term value isnot truncated (i.e., an expression for the arbitrary term of it isderived).

1. Mathematical note

In a case of an interdependence of three parametersX, Y, Z

Z = Z(X, Y )

their differentials and, consequently, partial derivatives areconnected via[

∂Z

∂X

]Y

dXZ +[

∂Z

∂X

]X

dYZ = 0,

[∂Y

∂X

]Z

= −[∂Z/∂X

]Y[

∂Z/∂Y]X

,

[∂

∂X

]Z

=[

∂

∂X

]Y

+[

∂Y

∂X

]Z

[∂

∂Y

]X

=[

∂

∂X

]Y

−[∂Z/∂X

]Y[

∂Z/∂Y]X

[∂

∂Y

]X

,

where a subscript designates which variable is fixed. Hence,[∂jY

∂Xj

]Z

= −[

∂

∂X

]j−1

Z

[∂Z/∂X

]Y[

∂Z/∂Y]X

= −([

∂

∂X

]Y

−[∂Z/∂X

]Y[

∂Z/∂Y]X

[∂

∂Y

]X

)j−1 [∂Z/∂X

]Y[

∂Z/∂Y]X

.

(A1)

2. Initial relations

Let us introduce new variables, convenient for a furtherderivation

γ = ¯√2m

(v + 1/2), λ = ¯2

2m[N (N + 1) − δ],

where m is the reduced mass and δ is specific for the the-ory of diatomic molecules; it depends on the coupling schemeand the symmetry of the state45 (δ = 0 for the � states). The

eigenenergy E is often expressed as

E(v,N) − ED = Gv + Bv[N (N + 1) − δ]

−Dv[N (N + 1) − δ]2 + · · ·

=∞∑

j=0

Y[v],j λj , (A2)

where ED is the dissociation limit, and −(E − ED) is the bind-ing energy. Obviously,

Y[v],j = 1

j !

([∂j (E(γ, λ) − ED)

∂λj

]γ

)λ=0

. (A3)

Substituting Eq. (A1) into Eq. (A3) results in

Y[v],j = − 1

j !

( ([∂

∂λ

]E

− [∂γ /∂λ]E[∂γ /∂E]λ

[∂

∂E

]λ

)j−1

× [∂γ /∂λ]E[∂γ /∂E]λ

)λ=0

.

Hence,

Y[v],j = − 1

j !

∑{jlkl}

S{jlkl}j y

{jlkl}[v],j ,

(A4)

y{jlkl}[v],j =

q∏l=1

∂jl γ

∂λkl ∂Ejl−kl

/ (∂γ

∂E

)q

,

where the braces {· · · } designate sets of integer indices,the summation is done over the sets {jlkl}ql=1 of the indiceskl ≥ 0 and jl ≥ kl satisfying

∑q

l=1 kl = j ,∑q

l=1(jl − kl) =q − 1, and S

{jlkl}j are dimensionless factors depending on

those sets. We have also omitted the subscripts indicatingwhat is held fixed in each differentiation as they are ratherobvious further on.

The computation of the values Y[v],j with Eq. (A4) re-quires the preliminary computation of the combinatorial fac-tors S

{jlkl}j and the values y

{jlkl}[v],j expressed via the derivatives

∂ iγ /∂λk∂Ei − k.

3. Combinatorial factor S{jl kl }j

A recurrent algorithm for the S{jlkl}j computations can be

proposed as follows. We have

[∂

∂λ−

(∂γ

∂λ

/∂γ

∂E

)∂

∂E

] q∏l=1

∂jl γ

∂λkl ∂Ejl−kl

/(∂γ

∂E

)q

=q∑

i=1

∂ji+1γ

∂λki+1∂Eji−ki

q∏l=1 (l =i)

∂jl γ

∂λkl ∂Ejl−kl

/(∂γ

∂E

)q

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

130.63.180.147 On: Wed, 13 Aug 2014 18:42:42

134307-6 V. B. Sovkov and V. S. Ivanov J. Chem. Phys. 140, 134307 (2014)

− q∂2γ

∂E∂λ

q∏l=1

∂jl γ

∂λkl ∂Ejl−kl

/(∂γ

∂E

)q+1

−∂γ

∂λ

q∑i=1

∂ji+1γ

∂λki ∂Eji−ki+1

q∏l=1 (l =i)

∂jl γ

∂λkl ∂Ejl−kl

/(∂γ

∂E

)q+1

+ q∂γ

∂λ

∂2γ

∂E2

q∏l=1

∂jl γ

∂λkl ∂Ejl−kl

/(∂γ

∂E

)q+2

.

Let us express the factor we are interested in as a sum

S{j ′

l k′l}

j =∑{jlkl}

sj ({j ′l k

′l}; {jlkl}) (A5)

of the terms originating from different factors S{jlkl}j−1 of the

previous set. Those terms can be found from

sj ({{jlkl(l = i)}, (ji + 1, ki + 1)} ; {jlkl}) = S{jlkl}j−1 ,

i = 1, 2, . . . , q,

sj ({{jlkl}, (2, 1)} ; {jlkl}) = −qS{jlkl}j−1 ,

(A6)sj ({{jlkl(l = i)}, (ji + 1, ki), (1, 1)} ; {jlkl}) = −S

{jlkl}j−1 ,

i = 1, 2, . . . , q,

sj ({{jlkl}, (1, 1), (2, 0)} ; {jlkl}) = qS{jlkl}j−1 .

Equations (A5) and (A6) make it possible to compute all thedesired values of S

{jlkl}j starting with S

{11}1 = 1. As those of

the factors S{jlkl}j , which are only different by a permutation

of the indices {jlkl}, are in fact equivalent, it is recommendedto reduce their number at every step of the computationalscheme by combining the equivalent factors into one. How-ever, even with the reduction, the total number of the factorsgrows rapidly—e.g., it equals 3247 at j = 10 and 13047 at j= 12. As these terms also have alternating signs, a loss of ac-curacy of Y[v],j can take place with finite–digit computations,so the computational accuracy should be controlled.

4. Derivatives ∂ iγ /∂λk∂Ei − k

The derivatives ∂ iγ /∂λk∂Ei − k can be estimated via a se-ries of approximations. Let us use the semi-classical quanti-zation rule of the WKB theory, which can be written in a form

γ = 1

π

∫ Rr

Rl

[E − U (R) − λ

R2

]1/2

dR, (A7)

where R is an inter-body separation, U(R) is an interactionpotential, Rl and Rr are the left and right boundaries of theclassically accessible region. An influence of the higher or-ders of the WKB theory on the near-dissociation propertieswas discussed in Ref. 56.

Further on we concentrate our attention on the near-dissociation energy levels, for which the system spends mostof the time in the long-distance region, making the major in-put into the integral, Eq. (A7). Within this assumption wecan use a well-known89 long-distance approximation for the

potential

U (R) = ED − cn

Rn, (A8)

where cn/Rn is the leading term of the dispersion asymptote,cn is the corresponding dispersion coefficient, and the index ndepends on the states of the atomic products of the dissocia-tion (n = 6 for the case of the electrically neutral S symmetryatoms). So

γ = 1

π

∫ Rr

Rl

[E − ED + cn

Rn− λ

R2

]1/2

dR. (A9)

For the derivatives in which we are interested, this gives[∂iγ

∂λk∂Ei−k

]λ=0

= (−1)i+k+1(2i − 3)!!

2iπ

×∫ Rr

Rl

[E − ED + cn

Rn

]−(2i−1)/2 1

R2kdR.

(A10)

This equation is valid for all nonnegative i if we as-sume (−3)!! = −1, (−1)!! = 1, and a common definitionof the double factorial i!! = 1 · 3 · · · · · i for positive oddindices i.

Let us restrict our consideration to those derivatives forwhich

in − 2k − n/2 + 1 > 0 (A11)

is satisfied. The case i = k = 0 for n ≥ 2 does not obeyEq. (A11) (that is why we could not apply the following ap-proximation to Eq. (A9) directly), while cases n > 2, i > 0as well as n = 1, k = 0 do not cause any problem. If the re-quirement, Eq. (A11), is satisfied, the integrand of Eq. (A10)either decreases or (in the less interesting cases −1 < in− 2k − n/2 ≤ 0) increases slowly enough with a decreaseof R. This makes it possible to introduce one more approxi-mation, namely, to change the left boundary Rl to zero[

∂iγ

∂λk∂Ei−k

]λ=0

= (−1)i+k+1(2i − 3)!!

2iπ

×∫ Rr

0

[ cn

Rn− (ED − E)

]−(2i−1)/2 1

R2kdR.

Various aspects of the approximations introduced abovealong with the ways to embed some corrections into the theorywere considered in Refs. 58 and 60.

As Rr = (cn/(ED − E))1/n, we have[∂iγ

∂λk∂Ei−k

]λ=0

= (−1)i+k+1(2i − 3)!!

2iπ

× (ED − E)(2k−1)/n−(2i−1)/2c−(2k−1)/nn

×∫ 1

0

[1

xn− 1

]−(2i−1)/2

x−2k dx,

where x = R/Rr. The latter integral is easily calculated.Finally,[

∂iγ

∂λk∂Ei−k

]λ=0

=Jik(n)(ED − E)(2k−1)/n−(2i−1)/2c−(2k−1)/nn ,

(A12)

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

130.63.180.147 On: Wed, 13 Aug 2014 18:42:42

134307-7 V. B. Sovkov and V. S. Ivanov J. Chem. Phys. 140, 134307 (2014)

where

Jik(n) = (−1)i+k+1(2i − 3)!!

2iπn

(

1−2i2 + 1

)

(1−2k

n− 1−2i

2

)

(1−2k

n+ 1

)(A13)

is a dimensionless coefficient depending only on the indicesn, i, k.

5. Coefficient Y[v],j as a function of the binding energyED − E(λ = 0)

Substituting Eq. (A12) into Eq. (A4) gives

y{jlkl}[v],j =

∏q

l=1 Jjlkl

(J10)q

×c−(2

∑l kl−q)/n−q/n

n

× (ED − E)(2∑

l kl−q)/n−(2∑

l jl−q)/2+q/n+(2∑

l 1−q)/2

=∏q

l=1 Jjlkl

(J10)qc−2j/nn (ED − E)−

(j−1)(n−2)−2n .

From the latter equation and Eq. (A4) we have

Y[v],j = AJ (n)c−2j/nn (ED − E)−

(j−1)(n−2)−2n , (A14)

where

Aj (n) = − 1

j !

∑{jlkl}

[S

{jlkl}j

∏q

l=1 Jjlkl

(J10)q

](A15)

is a dimensionless coefficient depending on the indices jand n.

6. Coefficient Y[v],j as a function of γ

(of the vibrational quantum number)

At the next step we should express the binding energyED − E of the non-rotating system (λ = 0) in terms of thequantum numbers. Practically interesting cases are the oneswith n > 2. The condition, Eq. (A11), makes it impossibleto find that expression via substituting i = 0. Instead, let usconsider Eq. (A12) with i = 1, k = 0[

∂γ

∂E

]λ=0

= 1

2√

πn

(

12 + 1

n

)

(1 + 1

n

) c1/nn (ED − E)−

n+22n , (A16)

integrate it over E and invert the resulting equation. We obtain

ED − E = Zn2n/(n−2)c−2/(n−2)

n (γD − γ )2n/(n−2) , (A17)

where

Zn = (n − 2)√

π

(1 + 1

n

)

(12 + 1

n

) (A18)

and γ D is the constant of integration. The limiting vibrationalquantum number vD of Refs. 48, 49, and 51–60, correspond-ing to the energy E = ED, can be expressed via γ D as

vD =√

2m

¯γD − 1

2.

However, contrary to vD , the value γ D is mass independent.In the special case n = 2, Eq. (A16) takes the form

∂E

∂γ

∣∣∣∣λ=0

= 2πc2−1/2 (ED − E) ,

whose solution is

ED − E = (ED − E0) exp[−2πc2

−1/2(γ − γ0)], (A19)

where γ 0 and E0 can be chosen arbitrarily, but in coordinationto each other. The potential well contains the infinite numberof levels for n ≤ 2.

It is interesting to note48 that, in spite of all the approx-imations, Eq. (A17) produces the well-known correct resultfor the Coulomb potential (n = 1) with vD = −1.

The parameters Y[v],j are calculated by substitutingEqs. (A17) (n = 2) or (A19) (n = 2) into Eq. (A14). For themost interesting case n = 2 this gives

Y[v],j = Xj (n)c−2/(n−2)n (γD − γ )4/(n−2)−2(j−1) , (A20)

where

Xj (n) = −Zn4/(n−2)−2(j−1)

j !

∑{jlkl}

[S

{jlkl}j

∏q

l=1 Jjlkl

(J10)q

](A21)

is a dimensionless coefficient depending on the indices jand n, including the case Yv0 = Gv = −(ED − E) with X0(n)= −Zn

2n/(n − 2).

7. Correction function and the final formof the generalized near-dissociation expansion

In practical applications it is customary to consider thegeneralized Le Roy–Bernstein near-dissociation expansion,which in our case takes a form

E(v,N) = ED +∞∑

j=0

Cjn(cn) (γD − γ )Kjn Qj (γD − γ )λj ,

(A22)where

Kjn = 4

n − 2− 2(j − 1), Cjn(cn) = Xj (n)c−2/(n−2)

n ,

γ = G(m)

(v + 1

2

), λ = G2(m) [N (N + 1) − δ] ,

G(m) = ¯√2m

,

and Qj(ζ ) is the correction function. The introduction ofan appropriate correction function, obeying the require-ment Qj(0) = 1, maintains the theoretically predicted near-dissociation behavior of Eq. (A22) at the asymptote E → ED

and expands the region of applicability of the theory to the en-tire set of the ro-vibrational term values, including those at thebottom of the potential well. This function is meant to factorinto the theory possible deviations from the approximationsimplemented above, including the higher terms of the disper-sion asymptote of the potential function, the exchange inter-action, and so on. Parameters of the function Qj(ζ ) are usu-ally fitted from the experimental data similar to the Dunhamcoefficients.

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

130.63.180.147 On: Wed, 13 Aug 2014 18:42:42

134307-8 V. B. Sovkov and V. S. Ivanov J. Chem. Phys. 140, 134307 (2014)

Various versions of the correction function were pro-posed and used in Refs. 57, 59, 63–65, 67–70, and 74.Reasonable choices are

Qj (ζ ) = a0j + a1jα(ζ ) + a2jα2(ζ ) + · · · (A23)

or

Qj (ζ ) = a0j + a1jα(ζ ) + a2jα2(ζ ) + · · ·

b0j + b1jα(ζ ) + b2jα2(ζ ) + · · · , (A24)

where

α(ζ ) = ζ

gD + dζ.

The reasonable assumptions a0j = b0j = 1 and gD = γ D en-sure that Qj(0) = 1, that the parameters a, b, and d are dimen-sionless, and that the un-physical degree of freedom (a possi-ble common factor of numerators and denominators) is elim-inated. However, we prefer to keep arbitrary values of thoseparameters for two reasons: (1) it allows us to equate the Dun-ham coefficients Yij (see Eq. (A25) below) to aij of Eq. (A23)with the implications G(m) = −1, γ D = 0, Cjn(cn) = 1, Kjn

= 0, gD = 1, d = 0; (2) it turns the functions given inEqs. (A23) and (A24) into standard polynomial and Padefunctions, and makes it possible to easily use standard librarycomputational routines to treat them.

The only mass-dependent factor of the representation,Eq. (A22), is G(m), which allows one to easily include var-ious isotopes in one analysis.

8. Numerical values of the coefficients Xj(n)

We have computed the coefficients Xj(n) given byEq. (A21), with the algorithm described above, forj = 0, 1, . . . ,12 and n = 3, 4, 5, 6 (all the cases occurring whena molecule dissociates to neutral atoms). The results are pre-sented in Table III. With the aim to control the accuracy, thecomputations were done several times with various numbersof significant digits. The table values are the ones obtainedwith 32 digits and rounded to 8 digits afterwards. The 16 digitvariant produced the same 8 digits up to j = 10 and differed

in 8th or 7th digit at j = 11, 12. The 8 digit variant producedcompletely different results starting with j = 8.

The coefficients X̄n(j ) of Ref. 53 can be com-puted from our dimensionless coefficients Xj(n) asX̄n(j ) = (¯/

√2)2n/(n−2)Xj (n) [where ¯/

√2 = 4.1058045

(Å−1√

cm−1amu)] up to the sign of X̄n(0), which should beswitched.

9. Connection with the Dunham series

The familiar Dunham coefficients Yij of the expansion

E = (ED − De) +∞∑

i,j=0

Yij (v + 1/2)i[N (N + 1) − δ]j ,

(A25)where De is the potential well depth so that Te = ED − De

is the energy of the potential well bottom, can be connectedwith the coefficients of Eq. (A22) by

Yij = [G(m)]2j+i

i!

[∂iY[v],j

∂γ i

]γ=0

, i + j > 0,

or

Yij = (−1)i

i![G(m)]2j+iCjn(cn)γ

Kjn

D

i∑k=0

(i

k

)γ −k

D

×k−1∏l=0

(Kjn − l)

[di−kQj (ζ )

dζ i−k

]ζ=γD

, (A26)

where computing derivatives of Qj(ζ ) (Eq. (A23) or (A24))causes no problem. In part

Y10 = ¯ωe = −G(m)X0(n)c−2/(n−2)n γ

(n+2)/(n−2)D

×(

2n

n − 2Q0(γD) + γD

[dQ0(ζ )

dζ

]ζ=γD

), (A27)

Y01 = Be = G2(m)X1(n)c−2/(n−2)n γ

4/(n−2)D Q1(γD), (A28)

Y02 = −D′e = G4(m)X2(n)c−2/(n−2)

n γ4/(n−2)−2D Q2(γD),

(A29)

TABLE III. Coefficients Xj(n) given by Eq. (A21) for use in Eqs. (A20) and (A22).

n

j 3 4 5 6

0 − 7.6002263 − 47.268180 − 82.746881 − 114.600141 12.570673 15.045929 10.631396 7.89779572 − 5.1979315 − 0.72365849 − 0.13874691 − 0.0397887363 0.28138183 − 0.013262912 − 0.0016008259 − 0.000277203744 − 0.015314692 − 0.0010228419 −4.8213006 × 10−5 −4.3259754 × 10−6

5 − 0.012665148 − 0.00010499359 −1.9561691 × 10−6 −8.9746184 × 10−8

6 − 0.0080770443 −1.2557421 × 10−5 −9.2760830 × 10−8 −2.1674979 × 10−9

7 − 0.0051897635 −1.6534625 × 10−6 −4.8479667 × 10−9 −5.7594805 × 10−11

8 − 0.0034437996 −2.3264235 × 10−7 −2.7086736 × 10−10 −1.6345480 × 10−12

9 − 0.0023592998 −3.4362446 × 10−8 −1.5890717 × 10−11 −4.8679456 × 10−14

10 − 0.0016618558 −5.2676613 × 10−9 −9.6762294 × 10−13 −1.5041679 × 10−15

11 − 0.0011985133 −8.3157177 × 10−10 −6.0677664 × 10−14 −4.7850032 × 10−17

12 − 0.00088180147 −1.3443385 × 10−10 −3.8965241 × 10−15 −1.5584759 × 10−18

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

130.63.180.147 On: Wed, 13 Aug 2014 18:42:42

134307-9 V. B. Sovkov and V. S. Ivanov J. Chem. Phys. 140, 134307 (2014)

where ωe is the principal vibrational frequency, Be is the equi-librium rotational constant, and D′

e is the centrifugal distor-tion coefficient. The latter designation D′

e is used in order notto confuse it with the potential well depth De.

This connection gives a way to check or imposecorrelations45, 81, 82 of Dunham coefficients in terms of the co-efficients of Eq. (A23) or (A24); e.g., the well-known relation

Y02 ≈ Y ′02 = −4

Y 301

Y 210

(A30)

along with the previous three equations transforms into thefollowing relation

Q2(γD) ≈ −4X3

1(n)

X20(n)X2(n)

Q31(γD)

/(

2n

n − 2Q0(γD) + γD

[dQ0(ζ )

dζ

]ζ=γD

)2

. (A31)

1I. Manai, R. Horchani, M. Hamamda, A. Fioretti, M. Allegrini, H. Lignier,P. Pillet, and D. Comparat, Mol. Phys. 111, 1844–1854 (2013).

2A. Jaouadi, E. Barrez, Y. Justum, and M. Desouter-Lecomte, J. Chem. Phys.139, 014310 (2013).

3J. Ma, J. Wu, G. Chen, Q. Fan, H. Feng, X. Dai, W. Sun, L. Xiao, and S.Jia, J. Phys. Chem. Lett. 4, 3612 (2013).

4Z. Wei, X. Ting, H. Yin, W. Gao-Ren, and C. Shu-Lin, Chin. Phys. B 22,013301 (2013).

5Y. Zhang, J. Ma, Y. Li, J. Wu, L. Zhang, G. Chen, L. Wang, Y. Zhao, L.Xiao, and S. Jia, Appl. Phys. Lett. 101, 131114 (2012).

6J. Li, Y. Liu, and S.-L. Cong, Mod. Phys. Lett. B 27, 1350109 (2013).7R. Rodriguez-Cantano, D. Lopez-Duran, T. Gonzalez-Lezana, G. Delgado-Barrio, P. Villarreal, E. Yurtsever, and F. A. Gianturco, J. Phys. Chem. A116, 2394 (2012).

8T. Takekoshi, C. Strauss, F. Lang, J. H. Denschlag, M. Lysebo, and L.Veseth, Phys. Rev. A 83, 062504 (2011).

9B. Gruener, M. Schlesinger, P. Heister, W. T. Strunz, F. Stienkemeier, andM. Mudrich, Phys. Chem. Chem. Phys. 13, 6816 (2011).

10P. S. Zuchowski and J. M. Hutson, Phys. Rev. A 81, 060703 (2010).11G. Auboeck, M. Aymar, O. Dulieu, and W. E. Ernst, J. Chem. Phys. 132,

054304 (2010).12P. Moroshkin, A. Hofer, V. Lebedev, and A. Weis, J. Chem. Phys. 133,

174510 (2010).13C. Strauss, T. Takekoshi, F. Lang, K. Winkler, R. Grimm, J. H. Denschlag,

and E. Tiemann, Phys. Rev. A 82, 052514 (2010).14C. Samuelis, E. Tiesinga, T. Laue, M. Elbs, H. Knöckel, and E. Tiemann,

Phys. Rev. A 63, 012710 (2000).15H. E. L. Martay, D. J. McCabe, D. G. England, M. E. Friedman, J. Petrovic,

and I. A. Walmsley, Phys. Rev. A 80, 033403 (2009).16N. Vanhaecke, C. Lisdat, B. T’Jampens, D. Comparat, A. Crubellier, and P.

Pillet, Eur. Phys. J. D 28, 351 (2004).17W. C. Stwalley and H. Wang, J. Mol. Spectrosc. 195, 194 (1999).18S. Sainis, J. Sage, E. Tiesinga, S. Kotochigova, T. Bergeman, and D.

DeMille, Phys. Rev. A 86, 022513 (2012).19K. Beloy, A. Borschevsky, V. V. Flambaum, and P. Schwerdtfeger, Phys.

Rev. A 84, 042117 (2011).20D. DeMille, S. Sainis, J. Sage, T. Bergeman, S. Kotochigova, and E.

Tiesinga, Phys. Rev. Lett. 100, 043202 (2008).21Y. Guan, X. Han, J. Yang, Z. Zhou, X. Dai, E. H. Ahmed, A. M. Lyyra, S.

Magnier, V. S. Ivanov, A. S. Skublov, and V. B. Sovkov, J. Chem. Phys.139, 144303 (2013).

22V. B. Sovkov, D. Li, V. S. Ivanov, A. S. Skublov, F. Xie, L. Li, and S.Magnier, Chem. Phys. Lett. 557, 66 (2013).

23F. Xie, L. Li, D. Li, V. B. Sovkov, K. V. Minaev, V. S. Ivanov, A. M. Lyyra,and S. Magnier, J. Chem. Phys. 135, 024303 (2011).

24F. Xie, V. B. Sovkov, A. M. Lyyra, D. Li, S. Ingram, J. Bai, V. S. Ivanov,S. Magnier, and L. Li, J. Chem. Phys. 130, 051102 (2009).

25B. Beser, V. B. Sovkov, J. Bai, E. H. Ahmed, C. C. Tsai, F. Xie, L. Li, V.S. Ivanov, and A. M. Lyyra, J. Chem. Phys. 131, 094505 (2009).

26F. Xie, K. V. Minaev, V. B. Sovkov, V. S. Ivanov, D. Li, and L. Li, Chem.Phys. Lett. 493, 238 (2010).

27D. Li, F. Xie, L. Li, V. B. Sovkov, V. S. Ivanov, E. Ahmed, A. M. Lyyra, J.Huennekens, and S. Magnier, J. Chem. Phys. 126, 194314 (2007).

28E. Ahmed, A. M. Lyyra, L. Li, V. S. Ivanov, V. B. Sovkov, and S. Magnier,J. Mol. Spectrosc. 229, 122 (2005).

29E. Ahmed, A. M. Lyyra, F. Xie, D. Li, Y. Chu, L. Li, V. S. Ivanov, V. B.Sovkov, and S. Magnier, J. Mol. Spectrosc. 234, 41 (2005).

30P. Yi, X. Dai, J. Li, Y. Liu, L. Li, V. B. Sovkov, and V. S. Ivanov, J. Mol.Spectrosc. 225, 33 (2004).

31V. S. Ivanov, V. B. Sovkov, and L. Li, J. Chem. Phys. 118, 8242 (2003).32V. S. Ivanov, V. B. Sovkov, N. Gallice, L. Li, Y. Liu, A. M. Lyyra, and S.

Magnier, J. Mol. Spectrosc. 209, 116 (2001).33V. S. Ivanov, V. B. Sovkov, L. Li, A. M. Lyyra, T.-J. Whang, and S. Mag-

nier, J. Chem. Phys. 114, 6077 (2001).34V. S. Ivanov and V. B. Sovkov, Chem. Phys. 213, 295 (1996).35J. S. Wright, J. Chem. Soc., Faraday Trans. 84, 219 (1988).36P. G. Hajigeorgiou and R. J. Le Roy, J. Chem. Phys. 112, 3949 (2000).37M. R. Manaa, A. J. Ross, F. Martin, P. Crozet, A. M. Lyyra, L. Li, C. Amiot,

and T. Bergeman, J. Chem. Phys. 117, 11208 (2002).38R. J. Le Roy, Y. Huang, and C. Jary, J. Chem. Phys. 125, 164310 (2006).39R. J. Le Roy and R. D. E. Henderson, Mol. Phys. 105, 663 (2007).40P. Qi, J. Bai, E. Ahmed, A. M. Lyyra, S. Kotochigova, A. J. Ross, C. Ef-

fantin, P. Zalicki, J. Vigué, G. Chawla, R. W. Field, T.-J. Whang, W. C.Stwalley, H. Knöckel, E. Tiemann, J. Shang, L. Li, and T. Bergeman, J.Chem. Phys. 127, 044301 (2007).

41H. Salami, T. Bergeman, B. Beser, J. Bai, E. H. Ahmed, S. Kotochigova, A.M. Lyyra, J. Huennekens, C. Lisdat, A. V. Stolyarov, O. Dulieu, P. Crozet,and A. J. Ross, Phys. Rev. A 80, 022515 (2009).

42J. Bai, E. H. Ahmed, B. Beser, Y. Guan, S. Kotochigova, A. M. Lyyra, S.Ashman, C. M. Wolfe, J. Huennekens, F. Xie, D. Li, L. Li, M. Tamanis, R.Ferber, A. Drozdova, E. Pazyuk, A. V. Stolyarov, J. G. Danzl, H.-C. Nägerl,N. Bouloufa, O. Dulieu, C. Amiot, H. Salami, and T. Bergeman, Phys. Rev.A 83, 032514 (2011).

43A. N. Drozdova, A. V. Stolyarov, M. Tamanis, R. Ferber, P. Crozet, and A.J. Ross, Phys. Rev. A 88, 022504 (2013).

44V. B. Sovkov, V. S. Ivanov, K. V. Minaev, and M. S. Aleksandrov, Opt.Spectrosc. 114, 167 (2013).

45H. Lefebvre-Brion and R. W. Field, The Spectra and Dynamics of DiatomicMolecules (Elsevier/Academic Press, Amsterdam/Boston/Heidelberg/NewYork/Oxford/Paris/San Diego/San Francisco/Singapore/Sydney/Tokio,2004), 766 pp.

46J. L. Dunham, Phys. Rev. 41, 713 (1932).47J. L. Dunham, Phys. Rev. 41, 721 (1932).48R. J. Le Roy and R. B. Bernstein, J. Chem. Phys. 52, 3869 (1970).49R. J. Le Roy and R. B. Bernsrein, Chem. Phys. Lett. 5, 42 (1970).50W. C. Stwalley, Chem. Phys. Lett. 6, 241 (1970).51R. J. Le Roy, Can. J. Phys. 50, 953 (1972).52M. S. Barwell, “Two-color photoassociation spectroscopy of the a3�+

u

state of Cs2,” M.S. thesis (University of Waterloo, Waterloo, Ontario,1975).

53R. J. Le Roy and M. G. Barwell, Can. J. Phys. 53, 1983 (1975).54W. C. Stwalley, J. Chem. Phys. 58, 3867 (1973).55R. J. Le Roy, in Molecular Spectroscopy (A Special Periodical Report of

the Chemical Society of London), edited by F. Barrow, D. A. Long, and D.J. Millen (Chemical Society, London, 1973), Vol. 1, pp. 113–176.

56S. M. Kirschner and R. J. Le Roy, J. Chem. Phys. 68, 3139 (1978).57R. J. Le Roy and W.-H. Lam, Chem. Phys. Lett. 71, 544 (1980).58R. J. Le Roy, J. Chem. Phys. 73, 6003 (1980).59J. W. Tromp and R. J. Le Roy, Can. J. Phys. 60, 26 (1982).60D. Comparat, J. Chem. Phys. 120, 1318 (2004).61P. G. Hajigeorgiou, J. Mol. Spectrosc. 286–287, 1 (2013).62R. J. Le Roy and R. B. Bernstein, J. Mol. Spectrosc. 37, 109 (1971).63E. J. Friedman-Hill and R. W. Field, J. Chem. Phys. 96, 2444 (1992).64J. Tellinghuisen, J. Chem. Phys. 78, 2374 (1983).65K. Viswanathan and J. Tellinghuisen, J. Mol. Spectrosc. 98, 185 (1983).66J. G. Ashmore and J. Tellinghuisen, J. Mol. Spectrosc. 119, 68 (1986).67K. J. Jordan, R. H. Lipson, N. A. McDonald, and R. J. Le Roy, J. Phys.

Chem. 96, 4778 (1992).68R. J. Le Roy, J. Chem. Phys. 101, 10217 (1994).69K. Ishikawa, N. Mukai, and M. Tanimura, J. Chem. Phys. 101, 876

(1994).

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

130.63.180.147 On: Wed, 13 Aug 2014 18:42:42

134307-10 V. B. Sovkov and V. S. Ivanov J. Chem. Phys. 140, 134307 (2014)

70D. R. T. Appadoo, R. J. Le Roy, P. F. Bernath, S. Gerstenkorn, P. Luc, J.Vergès, J. Sinzelle, J. Chevillard, and Y. DAignaux, J. Chem. Phys. 104,903 (1996).

71J. Koperski, J. Chem. Phys. 105, 4920 (1996).72J. Koperski, Sz. M. Kiełbasa, and M. Czajkowski, Spectrochim. Acta Part

A 56, 1613 (2000).73Q. Fan, W. Sun, and H. Feng, Spectrochim. Acta Part A 74, 911 (2009).74J. W. Tromp and R. J. Le Roy, J. Mol. Spectrosc. 109, 352 (1985).75B. R. Johnson, J. Chem. Phys. 67, 4086 (1977).76B. R. Johnson, J. Chem. Phys. 69, 4678 (1978).77J. P. Leroy and R. Wallace, J. Phys. Chem. 89, 1928 (1985).78J. P. Leroy and R. Wallace, J. Comput. Phys. 67, 239 (1986).79K. Davie and R. Wallace, Comput. Phys. Commun. 51, 217 (1988).80R. J. Le Roy, J. Mol. Spectrosc. 191, 223 (1998).

81J. K. G. Watson, J. Mol. Spectrosc. 80, 411 (1980).82V. G. Tyuterev and T. I. Velichko, Chem. Phys. Lett. 104, 596 (1984).83M. D. Feit, J. A. Fleck, Jr., and A. Steiger, J. Comput. Phys. 47, 412

(1982).84M. D. Feit and J. A. Fleck, Jr., J. Chem. Phys. 78, 301 (1983).85R. Heather and H. Metiu, J. Chem. Phys. 86, 5009 (1987).86V. Engel, Chem. Phys. Lett. 189, 76 (1992).87V. S. Ivanov, V. N. Serov, and V. B. Sovkov, in Abstracts of 31 Conference

of European Group on Atomic Spectroscopy, edited by F. Vedel (EuropeanPhysical Society, Marseille, France, 1999), pp. 267–268.

88V. N. Serov, V. B. Sovkov, V. S. Ivanov, and O. Atabek, J. Chem. Phys.115, 6450 (2001).

89J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gasesand Liquids (Wiley, New York, 1973).

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

130.63.180.147 On: Wed, 13 Aug 2014 18:42:42