Models for Low-Temperature Helium Dimers and Trimers

LAWRENCE L. LOHR, S. M. BLINDER Department of Chemistry, University of Michigan, Ann Arbor, Michigan, 48109-1055

Received 10 April 2001; revised 19 June 2001; accepted 2 July 2001

DOI 10.1002/qua.1807

ABSTRACT: We have shown that a simple model based on pairwise deltafunction potentials can, with an appropriate choice of parameters, represent the general behavior of ultra-low temperature helium dimers and trimers, including all possible isotopomers. 4 4 4 3 The species He2, He3,and He2 He are stable, with binding energies of approximately 4 3 1.3, 100, and 10 mK, respectively. An Eﬁmov state for He2 He is also predicted, bound by 4 3 3 4 3 3 2.4 mK. The remaining dimers and trimers, He He, He2, He He2,and He3,are evidently unstable. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem 90: 419–423, 2002

Key words: helium dimer; helium trimer; deltafunction potential; Eﬁmov state

average internuclear separation to expand to 98.1 b Introduction (51.9 Å). No bound states exist for the isotopomers 4 3 3 He He or He2. he stability of small clusters of helium atoms, The helium trimer has been the subject of many T with binding energies in the millikelvin range, computational studies [10 – 22]. Among the more has been confirmed in recent years using highly re- recent of these, Nielsen et al. [20] used Faddeev fined experimental techniques including mass spec- methods [23, 24] to obtain binding energies of 125.2 troscopy and diffraction by nanoscale sieves [1 – 5]. and 2.269 mK, respectively, for the ground and 4 Concurrent theoretical work has determined a excited states of He3 and a binding energy of highly accurate He–He interatomic potential [6 – 8] 4 3 13.06 mK for the ground state of He2 He. Also us- and confirmed the stability of certain helium dimers ing Faddeev methods, Roudnev and Yukovlev [21] and trimers. A highly accurate quantum Monte obtained binding energies of 117.1 and 1.665 mK, re- Carlo computation by Anderson et al. [9] predicted 4 4 spectively, for the ground and excited states of He3. the stability of the He2 dimer. The computed mini- 4 In both studies the energy of tne He3 excited state mum of the potential well is at 5.61 b (2.97 Å), with 4 a well depth corresponding to /k = 10.970 K. This lies slightly below that of the He2 ground state. The potential can support only a single bound state with results of Blume et al. [22] obtained using hyper- 4 v = 0, J = 0 and a remarkably minute binding en- spherical coordinates for He3 agree to within 1% ergy of 1.310 mK. Such weak bonding allows the with those of Nielsen et al. The various studies are in 4 3 3 agreement that the isotopomers He He2 and He3 4 3 3 Correspondence to: L. L. Lohr; e-mail: [email protected]. are unbound, as are the dimers He He and He2.

We have this leeway since we are constructing an Deltafunction Models empirical approximation to the accurate potential. The larger value of r0 also takes account of the exten- The purpose of this article is to present a highly sive delocalization of the wave function. With this idealized model which reproduces the essential fea- parameter, we obtain a spectacular overlap integral tures of helium dimers and trimers without the need of 0.99942 between the LM2M2 radial distribution 4 for elaborate computations. The fact that the He function and that computed from Eq. (3). Using dimer supports but a single bound state is highly Eq. (4), we can now assign the value λ =−1.07434. suggestive of a deltafunction potential, which has In fact, a bound state does not exist for Eq. (2) the same property. Moreover a deltafunction can unless λ<−1. For other isotopic variants, λ is ap- be very weakly bonding, allowing for highly delo- proximately proportional to the reduced mass. This 4 3 calized wave functions. Indeed Jee et al. [17] have would imply for He He that λ43 ≈−0.9207, con- employed ﬁtted spherical delta-shell potentials in sistent with a nonexistent bound state. As we will 4 their study of He3, obtaining a binding energy of see later, an even smaller negative value for λ43 is 183 mK, about 50% higher than the accepted value appropriate. of around 120 mK. Their value, like ours given below, is very much dependent on the choice of parameters in the potential. Helium Trimers In an earlier contribution to this journal [25], the authors proposed a “Dirac bubble potential” in The Schrödinger equation for a trimer with arbi- which the interatomic interaction is idealized as an trary masses and pairwise deltafunction interations attractive deltafunction on a sphere of radius r ,viz. 0 can be written (in atomic units h¯ = m = 1) V(r) =−const δ(r − r0). (1) − 1 ∇2 − 1 ∇2 − 1 ∇2 1 2 3 The Schrödinger equation for the relative motion of 2M1 2M2 2M3 the two helium atoms can be written λ12 λ23 + δ(r12 − r0) + δ(r23 − r0) 1 2 2µ12r0 2µ23r0 −ψ (r12) − ψ (r12) 2µ r λ31 12 + δ(r − r ) (r , r , r ) µ 31 0 1 2 3 + λ − = 2 31r0 δ(r12 r0)ψ(r12) Eψ(r12), (2) = r0 E (r1, r2, r3), (5) assuming a state of zero angular momentum. Here where µij ≡ MiMj/(Mi + Mj). Assuming zero trans- µ = M/2, where M is the mass of a 4He atom, lational and rotational energies, this can be reduced 7296.3 au. There exists just one bound state with to an equation in the three interatomic coordi- (unnormalized) wave function given by nates [27]: −κr> 2 2 sinh κr< e 1 ∂ 2 ∂ 1 ∂ 2 ∂ ψk(r) = ,(3)− + − + κr< κr> µ 2 ∂ µ 2 ∂ 2 12 ∂r12 r12 r12 2 23 ∂r23 r23 r23 { } where r> and r< are the greater and lesser of r12, r0 ∂2 ∂ 2 − 2 − 2 ∂2 − 1 + 2 + r12 r23 r31 and the parameters are related by µ 2 ∂ ∂ ∂ 2 31 ∂r31 r31 r31 2M1r23r31 r23 r31 2κ λ + = r2 − r2 − r2 ∂2 r2 − r2 − r2 ∂2 −2κr 0. (4) + 23 31 12 + 31 12 23 1 − e 0 r0 2M2r31r12 ∂r31∂r12 2M3r12r23 ∂r12∂r23 The ground-state energy is equal to E =−κ2/2µ = + λ12 − + λ23 − −κ2/M. We equate this to the LM2M2 computed δ(r12 r0) δ(r23 r0) 2µ12r0 2µ23r0 value [26], −4.148×109 hartree or −1.310 mK, which = −1 λ31 determines the parameter κ 0.005520 b . + δ(r31 − r0) (r12, r23, r31) The ﬁrst obvious candidate for the bubble ra- 2µ31r0 = dius r0 would be the computed potential minimum, E (r12, r23, r31). (6) 5.61 b. This would be an appropriate choice for Lacking exact solutions to Eq. (6), we consider very heavy atoms. However, we obtain better re- variational approximations of the form sults by using instead the maximum of the LM2M2 radial distribution function, namely r0 = 13.15 b. (r12, r23, r31) = φ1(r12)φ2(r23)φ3(r31), (7)

420 VOL. 90, NO. 1 LOW-TEMPERATURE HELIUM DIMERS AND TRIMERS where the “orbital functions” φi(rij) have the func- With the choice of parameters r0 = 13.15 b and tional form of the dimer solutions Eq. (3), with the λ44 =−1.07434, the wave function φ(r12)φ(r23)φ(r31) κij now treated as variational parameters. Note that is optimized for κ = 0.0099, giving a binding en- this function is totally symmetrical with respect to ergy of 96.1 mK. This compares well with the best interchange of any two particles, as is appropriate Faddeev result of 117 mK. The average internuclear for a Bose–Einstein system. One must not fail to distance r12 is computed to be 24.3 b. An indication account for the deltafunction contributions result- of the spread of the interatomic distribution is given = = 2 − 2 1/2 = ing from second derivatives at the cusps rij r0, by the standard deviation σ [ r12 r12 ] whereby 17.2 b. + − An augmented variational computation verified φ (r ) = φ (r ) − φ (r ) δ(r − r ) 0 0 0 ij 0 the existence of an excited bound state. Following 2κ = φ(r )δ(r − r ). (8) the method of Hylleraas, we consider the linear − −2κr 0 ij 0 1 e 0 variational function Homonuclear Trimer (r12, r23, r31) = φ(r12)φ(r23)φ(r31) We have recently done a preliminary computa- × c0 + c1(r12 + r23 + r31) tion on the 4He trimer, with the objective of study- 3 + c2(r12r23 + r23r31 + r31r12) ing the origin of three-body forces [28]. The three + c r2 + r2 + r2 + c r r r . (15) orbital functions can then be chosen with the same 3 12 23 31 4 12 23 31 parameter κ, now treated as variational parameter, This function, like Eq. (7), is symmetrical with re- such as to minimize spect to interchange of any two particles. The five H dτ roots of the secular equation give two bound states E(κ) = .(9) − − 2 dτ with energies 99.75 and 2.412 mK (plus three positive roots corresponding to unbound excited 4 We set the Mi in Eqs. (5) and (6) all equal to M( He) states). The Hylleraas computation thus gives a = and the µij M/2. Exploiting the equivalence of slightly improved ground state, with binding en- the three interatomic coordinates, the variational energy 99.75 mK. More significant, however, is the ergy can be reduced to prediction of a weakly bound excited state. This ev- 3 T 2κ λ D idently represents the Efimov state in the vicinity of κ = −κ2 + + + E( ) −2κr . (10) 1–2 mK, as conjectured by several workers. M N 1 − e 0 r0 N We must evaluate the normalization integral 2 Heteronuclear Trimers N = φ(r12)φ(r23)φ(r31) dτ,(11) The computations of Nielsen et al. [20] predict the kinetic energy 4 3 a marginally stable (ca. 10 mK) He2 He trimer 2 − 2 − 2 species with an average of 28 b for the 4–4 sepa- = r12 r23 r31 T φ(r12)φ(r23)φ(r31) ration and 38 b for each 4–3 separation. While the 2r23r31 choice of the parameter λ44 is quite straightforward, × φ(r12)φ (r23)φ (r31) dτ, (12) given the dimer results, no corresponding assign- and the deltafunction integral ment of λ43 canbemadeonthebasisofourmodel. 2 Based on the reduced mass ratio, we might approx- D = φ(r )φ(r )φ(r ) δ(r − r ) dτ. (13) 12 23 31 12 0 imate λ43 ≈−0.9207. This successfully accounts for the instability of the 4He 3He dimer, but this value is The integration element in r r r -space is given by 12 23 31 much too large when used in trimer computations. 2 dτ = 8π r12r23r31 dr12 dr23 dr31 (14) (We found, with some embarassment, that this parameter choice implied that the 443 trimer was stable with the limits r ={|r − r |, r + r }, r = 12 31 23 31 23 23 by 64 mK and that even the 433 trimer was weakly {0, ∞}, r ={0, ∞}. It has been found to be com- 31 bound.) putationally convenient to take all three integrals Our strategy is instead to fit the approximate over {0, ∞} while enforcing the triangle inequali- wave function ties with a product of Heaviside functions θ(−r12 + r23 + r31)θ(r12 − r23 + r31)θ(r12 + r23 − r31). (r12, r23, r31) = φ44(r12)φ43(r23)φ43(r31) (16)

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 421 LOHR AND BLINDER to the average 4–4 and 4–3 distances. With the pa- Aziz and Slaman [6a], they obtain mean particle rameters κ44 = 0.0092 and κ43 = 0.0056, we obtain radii from the center of mass of 49.0 and 10.2 b = = = 4 4 r12 28.0 and r23 r31 38.1. The parameter for He2 and He3, respectively. These values cor- λ43 canthenbe“tuned”to−0.77 to give a binding respond to particle separations of 98.0 and 17.7 b, energy of 11.4 mK, in approximate agreement with respectively, compared with our mean values of 98.1 4 3 accurate computations. Clearly neither the 433 nor and 24.3 b, shown in Table I. For the He2 He mixed the 333 trimers should be stable. trimer, constructing a triangle from our rij values, the mean radius from the center of mass is 19.9 b, virtually identical to the value of 20 b calculated by Summary Nielsen et al., although their averaging procedure differs from ours. Thus our method yields trimers We have revised the pairwise deltafunction po- very similar in size to those obtained by the Faddeev tential we previously employed [25] to obtain bet- methods. ter estimates of the binding energy of the helium trimer. We now calculate binding energies of 96.1 4 and 11.4 mK for the ground states of He3 and 4 3 4 3 References He2 He, respectively. The isotopomers He He2 3 and He3 are found to be unbound, in agreement 1. Luo, F.; McBane, G. C.; Kim, G.; Giese, C. F.; Gentry, W. R. 4 with other studies. The species He3 is characterized J Chem Phys 1993, 98, 3564. 4 3 by rij values of 24.3 b, while He2 He is character- 2. Luo, F.; Giese, C. F.; Gentry, W. R. J Chem Phys 1996, 104, ized by values of 28.0 b for its 4He–4He separation 1151. and 38.1 b for each of its 4He–3He separations. The 3. Schöllkopf, W.; Toennies, J. P. Science (Washington, DC) 4 binding energy of the species He3 increases slightly 1994, 266, 1345; J Chem Phys 1996, 104, 1155. to 99.8 mK when our simple product wave func- 4.Grisenti,R.E.;Schöllkopf,W.;Toennies,J.P.;Hegerfeldt, tion is augmented by a Hylleraas-type correlation G. C.; Köhler, T.; Stoll, M. Phys Rev Lett 2000, 85, 2284. function, Eq. (14). The latter variational method also 5. Elrod, M. J.; Saykalley, R. J. Chem Rev 1994, 94, 1975; Suhm, yields an excited state for 4He with a binding en- M. A.; Nesbitt, D. J. Chem Soc Rev 1995, 24, 45; Liu, K.; 3 Cruzan, J. D.; Sakally, R. J. Science (Washington, DC) 1996, ergy of 2.4 mK, comparable to values found by the 271, 929. Faddeev methods. 6. (a) Aziz, R. A.; Slaman, M. J. J Chem Phys 1991, 94, 8047. (b) Table I summarizes the energies and geometric Metrologia 1990, 27, 211. parameters for all predicted stable helium dimers 7. Liu, B.; McLean, A. D. J Chem Phys 1989, 91, 2348; Vos, R. J.; and trimers. The ± values represent standard devi- van Lenthe, J. H.; van Duijneveldt, F. B. J Chem Phys 1990, = 2 − 2 1/2 93, 643. ations σ ( rij rij ) . Nielsen et al. [20] calculated expectation values 8. Gdanitz, R. J Chem Phys 2000, 113, 5145. of various operators related to geometrical struc- 9.Anderson,J.B.;Traynor,C.A.;Boghosian,B.M.JChem Phys 1993, 99, 345. ture from their numerical solutions of the Faddeev 10. Lim, T. K.; Damert, W. C.; Duffy, K. Chem Phys Lett 1977, equations. Based on the LM2M2 dimer potential of 45, 377. 11. Lim, T. K.; Duffy, K.; Damert, W. C. Phys Rev Lett 1977, 38, 341. TABLE I 12. Lim, T. K.; Duffy, K. J Chem Phys 1978, 68, 655. Helium dimers and trimers. 13. Huber, H. S.; Lim, T. K. J Chem Phys 1978, 68, 1006. 14. Duffy, K.; Lim, T. K. J Chem Phys 1979, 70, 4778. Molecule −E (mK) rij±σ (b) 15. Lim, T. K.; Duffy, K.; Nakaichi, S.; Akaishi, Y.; Tanaka, H. J Chem Phys 1979, 70, 4782. 4 He2 1.310 98.1 ± 90.6 16. Cornelius, Th.; Glöckle, W. J Chem Phys 1986, 85, 3906. 4 He3 99.8 24.3 ± 17.2 17. Jee, Y.-J.; Larsen, S. Y.; Intemann, R. L.; Fabre de la Ripelle, ∗ M. Phys Rev A 1994, 49, 1912. 4He 2.4 103.6 ± 53.6 3 18. Esry, B. D.; Lin, C. D.; Greene, C. H. Phys Rev A 1996, 54, 4He 3He 11.4 28.0 ± 21.6a 2 394. ± b 38.1 34.9 19. Motovilov, A. K.; Soﬁanos, S. A.; Kolganova, E. A. Chem Phys Lett 1997, 275, 168. a r44 . 20. Nielsen, E.; Federov, D. V.; Jensen, A. S. J Phys B At Mol Opt b r43 . Phys 1998, 31, 4085.

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21. Roudnev, V.; Yakovlev, S. Chem Phys Lett 2000, 328, 97. 25. Lohr, L. L.; Blinder, S. M. Int J Quantum Chem 1995, 53, 413. 22. Blume, D.; Greene, C. H.; Esry, B. D. J Chem Phys 2000, 113, 26.Luo,F.;Kim,G.;McBane,G.C.;Giese,C.F.;Gentry,W.R. 2145. J Chem Phys 1993, 98, 9687. 23. Faddeev, L. D. Sov Phys JETP 1961, 12, 1014. 27. Lohr, L. L.; Blinder, S. M. Mol Phys 2000, 98, 397. 24. Faddeev, L. D. Sov Phys Dokl 1961, 6, 384; 1963, 7, 600. 28. Blinder, S. M.; Lohr, L. L. Mol Phys 2001, 99, 53.

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