Relying on the quantization rule of Raab and Friedrich [Phys. Rev. A (2009)
in press], we derive simple and accurate formulae for the number of rotational
states supported by a weakly bound vibrational level of a diatomic molecular
ion. We also provide analytic estimates of the rotational constants of any such
levels up to threshold for dissociation and obtain a criterion for determining
whether a given weakly bound vibrational level is rotationless. The results
depend solely on the long-range part of the molecular potential.Comment: 4 pages, 1 figur

Friedrich, Bretislav

Lemeshko, Mikhail

English

arXiv

Relying on the quantization rule of Raab and Friedrich [Phys. Rev. A (2009) in press], we derive simple and accurate formulae for the number of rotational states supported by a weakly bound vibrational level of a diatomic molecular ion. We also provide analytic estimates of the rotational constants of any such levels up to threshold for dissociation and obtain a criterion for determining whether a given weakly bound vibrational level is rotationless. The results depend solely on the long-range part of the molecular potential

Lemeshko, M.

Friedrich, B.

MPG.PuRe

arXiv:0910.5743v1  [physics.atom-ph]  30 Oct 2009Rotational structure of weakly bound molecular ionsMikhail Lemeshko and Bretislav FriedrichFritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany(Dated: October 30, 2009)Relying on the quantization rule of Raab and Friedrich [Phys. Rev. A (2009) in press], we derivesimple and accurate formulae for the number of rotational states supported by a weakly boundvibrational level of a diatomic molecular ion. We also provide analytic estimates of the rotationalconstants of any such levels up to threshold for dissociation and obtain a criterion for determiningwhether a given weakly bound vibrational level is rotationless. The results depend solely on thelong-range part of the molecular potential.PACS numbers: 33.15.-e, 33.15.Mt, 34.20.Cf, 03.65.Ge, 03.65.NkKeywords: molecular ions, cold collisions, rotational states, near-threshold quantization, WKB, analyticmodelsI. INTRODUCTIONMolecular systems bound by a potential which variesasymptotically with the inverse power of the distance, r,between their constituentsV (r)r→∞∼ −Cnrnwith n > 2 (1)are amenable to an accurate analytic semiclassical(WKB) treatment, as long as the system’s states are suffi-ciently ensconced within the potential energy well. How-ever, for states near threshold for dissociation, the WKBapproximation fails, as the system’s classical action, pro-portional to momentum, no longer exceeds Planck’s con-stant. And yet, it is the near-threshold states that havecome to the fore recently, through the work in cold-atomphysics where such states arise in photo- and magneto-association [1, 2, 3] or other types of “assisted” collisionsof (ultra)cold atoms [4] or atomic ions [5, 6, 7, 8]. There-fore, a considerable effort has been expended at amendingthe WKB approximation to also allow for tackling near-threshold states analytically. A leading approach is thatof H. Friedrich et al., who showed that, firstly, the bound-state eiegenenergies, Eb, can be expressed in terms of thequantization function, F (Eb) [9], which relates the state’sintegral quantum number v to the generally non-intergralquantum number, vth, of a state exactly at threshold, viaF (Eb) = vth − v (2)Secondly, they were able to find an explicit analytic formof the quantization function for attractive inverse-powerpotentials with n = 6 [10, 11], and, most recently, withn = 4 [12]. Note that the binding energy, Eb = D − Ev,with D the dissociation energy and Ev the energy of thevibrational level v, is thus positive, Eb > 0.In our previous work [13], we have shown that for eachvibrational level, v, the rotational angular momentum,J , can take a critical value, J∗, such that the vibrationallevel is pushed up to threshold, thereby causing the level’sbinding energy to vanish. Hence the angular momentumJ in excess of J∗, J > J∗, dissociates the molecule, cf.Figure 1. Furthermore, we have shown that the criticalangular momentum is related to the quantization func-tion byJ∗ = F (Eb)(n − 2) (3)The corollary of Eq. (3) is that the number of rotationalstates supported by a weakly bound vibrational level of adimer is given by the integer part of the critical angularmomentum, Jmax = Int[J∗]. By making use of the ex-plicit form of the quantization function of refs. [10, 11]for n = 6, we were able to evaluate J∗ and estimate therotational constant B for near-threshold states of 85Rb2.Here we take advantage of the recently derived explicitform of the quantization function for the n = 4 long-range potential and analyze the rotational structure ofhighly-excited H+2 and133Cs+2 molecular ions. The accu-racy of the approach based on the quantization functionis compared with that of an exact numerical solution ofthe corresponding Schrödinger equation.II. ROTATIONAL STATES OF WEAKLYBOUND MOLECULAR IONSThe quantization function for the n = 4 case takes theform:F (Eb) = Fth(κ) + Fip(κ)[Fcr(κ) + FWKB(κ)](4)where the individual terms, herein introduced for conve-nience, comprise a near-threshold dependence,Fth(κ) =2bκ− (pκ)22π [1 + (G6κ)6 + (G7κ)7]; (5)an “interpolation” term, which gives a smooth transitionbetween low-κ and high-κ behavior,Fip(κ) =(G6κ)6 + (G7κ)71 + (G6κ)6 + (G7κ)7; (6)a term which corrects the reflection phase due to thepotential of Eq. (1),Fcr(κ) = −14+u12πκ1/2+u32πκ3/2+u52πκ5/2+u72πκ7/2; (7)2TABLE I: The un coefficients of Eq. (7). The coefficients ofEqs. (5) and (6) are: b ≡ 1, p2 ≡ 2π3, G6 = 1.622576 andG7 = 1.338059.n1 3 5 75√π Γ(14)48 Γ(34)−35√π Γ(34)384 Γ(14)475√π Γ(54)3584 Γ(−14)−63305√π Γ(74)221184 Γ(−34)and a pure WKB contribution,FWKB(κ) =κ1/22√πΓ(34)Γ(54)(8)cf. Eq. (38) of ref. [12]. In Eqs. (4)–(8), the dimensionlesswavenumber κ is defined byκ ≡ k(C42m~2)12= E12b C1242m~2(9)with k =√2mEb/~ the wave vector and m the diatomic’sreduced mass. The parameters b, p and u in Eq. (4) arelisted in Table I (cf. Tables I, II of ref. [12]). Note thatin order to avoid confusion with the rotational constant(defined below), we changed the symbols B6,7, used inref. [12] for the adjustable-length parameters, to G6,7.The quantization function (4) can be simplified by ne-glecting the Fth(κ) and Fcr(κ) terms and setting Fip(κ)to ≈ 1, which results in the Leroy-Bernstein (LB) ap-proximation [14]. Neglecting only the Fth(κ) and Fip(κ)terms yields the improved Leroy-Berstein approximation(iLB), which accounts for short-range deviations of thetrue potential from V (r) of Eq. (1), see refs. [10, 15]. Theassumptions about the various terms of Eq. (4) inherentto the approximations are listed in Table II.Neglecting any coupling of the molecular rotation, wecan estimate the rotational constant, B, from the ro-tational energy, BJ∗(J∗ + 1), required to promote thevibrational level bound by Eb to thresholdB =EbJ∗(J∗ + 1)(10)The values of the rotational constant B obtained fromEq. (10) for 133Cs+2 and H+2 are listed in Tables III andIV together with the essentially exact values, Bexact. Thelatter were calculated fromBexact = 〈v|~22mr2|v〉 (11)with the vibrational wavefunctions obtained from a nu-merical solution of the Schrödinger equation for the po-tentials of refs. [8] and [16].The value of the binding energy Eb for which a vibra-tional state is unable to support molecular rotation canTABLE II: Terms of the quantization function of Raab andFriedrich (RF), Eq. (4), inherent to the LB and the iLBapproximations. Also shown are the ranges of the reducedwavenumber κ wherein the approximations apply. See text.RF iLB LBAll termsFth = 0Fip = 1Fth = 0Fip = 1Fcr = 0All κ κ ≈ 1 κ ≫ 1be derived from Eq. (3) with J∗ = 1. Hence the criterionfor a level to be rotationless isEb < d4~m1/2 C1/44(12)The parameter d4 has the same value for all potentialwells with an 1/r4 tail, namely d4 = 2.9105. Withinthe LB and iLB approximations, the coeeficients diLB4 =2.9096 and dLB4 = 0.7386. These values come close tothose given in Table III of ref. [13].III. RESULTS AND DISCUSSIONA. 133Cs+2 molecular ionAs an example, we analyzed the rotational structureof the 133Cs+2 ion, for which an accurate potential energycurve was published recently [8]. Table III compares, forthe last three vibrational levels of 133Cs+2 , the values ofthe critical rotational angular momentum J∗ and rota-tional constant B, calculated from Eqs. (3) and (10), withexact results. The table also lists the values obtained bythe Leroy-Bernstein and improved LeRoy-Bernstein ap-proximations. One can see that for all the states consid-ered, the predicted values of J∗ come close to the exactvalues, as do the values of J∗iLB . However, J∗LB, given bythe purely semiclassical LB theory, are quite off the exactvalues. One can also see from Table III that the model’sestimate of the rotational constants B is within 25% ofthe exact value. In the case of the LB approximation, thedeviation of J∗LB from the exact value happens to be inthe direction such that it improves the agreement of theLB rotational constant with the exact one; however, thisimproved agreement has to be regarded as serendipitous.B. H+2 molecular ionWe have also looked at the other end of the mass scaleand tackled the rotational structure of the H+2 ion. Theexact values of J∗ and B were obtained by numerically3rV   (r)effJ = 0J > J *v(J = J *) = v   (J = 0)thv(J = 0)v(J > J *)FIG. 1: (Color online) A schematic illustrating the role of the centrifugal term in the effective potential, Veff(r) = −C4r4+~22mJ(J+1)r2. The energy splittings have been exaggerated for clarity. Shown is the position of a rotationless vibrational level,v(J = 0) (dashed line), as well as the position of the same level when pushed up by the centrifugal term to threshold,v(J = J∗) = vth(J = 0) (full line, at threshold). When the rotational angular momentum J exceeds the critical value J∗, thecentrifugal term pushes the vibrational level above threshold, v(J > J∗) (full line, above threshold), thus leading to dissociation.TABLE III: Critical angular momenta J∗ and rotational con-stants B obtained for the last three vibrational states of the133Cs+2 molecular ion in different approximations; Eb and Bare given in 10−8 cm−1. See also Table II and text.v Eb J∗ J∗exact J∗iLB J∗LB B Bexact BiLB BLB371 5.7 0.63 0.63 0.63 1.01 5.53 4.02 5.51 2.81370 499.4 2.63 2.63 2.63 3.08 52.4 35.5 52.4 39.7369 3731.1 4.62 4.64 4.62 5.10 143.5 96.4 143.5 120.0TABLE IV: Critical angular momenta J∗ and rotational con-stants B obtained for the three least-bound states of the H+2molecular ion in different approximations; Eb and B are givenin cm−1. See also Table II and text.v Eb J∗ J∗exact J∗iLB J∗LB B Bexact BiLB BLB19 0.707 1.27 1.33 1.27 1.70 0.24 0.15 0.24 0.1518 23.411 3.60 3.70 3.60 4.07 1.41 0.92 1.41 1.1417 153.67 6.03 6.56 6.03 6.51 3.62 2.10 3.62 3.14solving the the Schrödinger equation for the potential ofref. [16]. Table IV compares the exact critical angularmomenta and rotational constants with those obtainedfrom the models. Although the potential well of the hy-drogen molecular ion is rather shallow, the near-thresholdrotational structure is still governed by the long-rangepotential tail, as attested by a reasonable agreement ofboth J∗ and B with the exact results. Since none of thelast vibrational states lies in a “pure near-threshold re-gion,” the LB and iLB approximations are not too far offeither.IV. CONCLUSIONSThe advent of the physics of translationally cold atomsand molecules has secured a new prominence for longrange interactions [17]. The r−4 attraction features mostnotably in the ion-induced dipole interaction betweenan ion and and an atom or molecule, as well as in theCasimir-Polder potential of an atom interacting with asurface [18]. The bound states near threshold are ofparticular interest. This is because such states are non-classical, often spending most of their lifetime in the far-out, classically forbidden region of the potential [4]. Also,molecular species, whether ionic or neutral, in such statescan be probed using nonresonant laser light, by “shak-ing” [19]. On the other hand, threshold states deter-mine low-energy scattering behavior, subject to studiesin traps [5].Within this context (and beyond), it is helpful to pos-sess the means the assess the rotational structure nearthreshold. Herein, we provide simple formulae that en-able such an assessment to be carried out. These for-mulae yield the critical value of the rotational angularmomentum needed for dissociation; the rotational con-stant; and a criterion for determining whether a givenvibrational state can support molecular rotation.AcknowledgmentsWe are grateful to Harald Friedrich and Patrick Raabfor discussions and to Gerard Meijer for encouragementand support.[1] T. Köhler, K. Goral, P. S. Julienne, Rev. Mod. Phys. 78,1311 (2006).[2] C. Chin, R. Grimm, P. Julienne, E. Tiesinga,arXiv:0812.1496 (2008).4[3] K. M. Jones, E. Tiesinga, P. D. Lett, P. S. Julienne,Rev. Mod. Phys. 78, 483 (2006).[4] A. S. Jensen, K. Riisager, D. V. Fedorov, E. Garrido,Rev. Mod. Phys. 76, 215 (2004).[5] A. T. Grier, M. Cetina, F. Oručević, V. Vuletić,Phys. Rev. Lett. 102, 223201 (2009).[6] Z. Idziaszek, T. Calarco, P. S. Julienne, A. Simoni,Phys. Rev. A 79, 010702(R) (2009).[7] P. Zhang, A. Dalgarno, R. Côté, Phys. Rev. A 80,030703(R) (2009).[8] M. J. Jamieson, A. Dalgarno, J. Tharamel, J. Phys. B.42, 095203 (2009).[9] H. Friedrich and J. Trost, Phys. Rep. 397, 359(2004).[10] H. Friedrich, P. Raab, Phys. Rev. A 77, 012703 (2008).[11] P. Raab, H. Friedrich, Phys. Rev. A 78, 022707 (2008).[12] P. Raab, H. Friedrich, Phys. Rev. A, in press (2009).[13] M. Lemeshko, B. Friedrich, Phys. Rev. A 79, 050501(R)(2009).[14] R. J. LeRoy, R. B. Bernstein, J. Chem. Phys. 52, 3869(1970).[15] H. Jelassi, B. Viaris de Lesegno, L. Pruvost, Phys. Rev. A73, 032501 (2006); ibid 77, 062515 (2008).[16] L. Hilico, N. Billy, B. Grémaud, D. Delande,Eur. Phys. J. D 12, 449 (2000).[17] in Cold Molecules: Theory, Experiment, Applications(eds. R.V. Krems, W.C. Stwalley and B. Friedrich) pp.3-38 (CRC Press, 2009).[18] J. Phys: Conf. Ser. 19, 1 (2005).[19] M. Lemeshko, B. Friedrich, Phys. Rev. Lett. 103, 053003(2009).

Rotational structure of weakly bound molecular ions

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