Chemical Physics Letters 506 (2011) 282–285

Contents lists available at ScienceDirect

Chemical Physics Letters

journal homepage: www.elsevier .com/locate /cplet t

Borromean binding in Hþ2 with screened Coulomb potentials

Sabyasachi Kar a,⇑, Y.K. Ho b

a Center for Theoretical Atomic and Molecular Physics, The Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, Harbin 150080,People’s Republic of Chinab Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei, Taiwan 106, Republic of China

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 January 2011In final form 6 March 2011Available online 9 March 2011

0009-2614/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.cplett.2011.03.007

⇑ Corresponding author. Fax: +86 451 8640 2588.E-mail address: skar@hit.edu.cn (S. Kar).

The stability of the bound P and D states of the Hþ2 system, where the nuclei and the electron interactthrough a screened Coulomb (Yukawa-type) potential exp(�lr)/r, has been studied for different valuesof the screening parameters l using correlated exponential wave functions in the framework of Ritz var-iational principle. The critical values of the screening parameters for the bound S, P and D states arereported for which Hþ2 system is stable, while all the possible fragments are unbound, that is, it showsBorromean binding.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

A three-body system is called Borromean if it is bound while allthe two-body subsystems are unbound [1,2]. The word ‘Borro-mean’ has been proposed in nuclear physics to identify boundstates whose all the possible subsystems are unbound [3]. It comesfrom the Borromean rings, which are interlaced in a subtle topolog-ical way such that if any one of them is removed, the two other be-come unlocked. Richard [1] has proposed the followinggeneralization to the N-body system: a bound state is Borromeanif there is no path to build it via a series of stable state by addingthe constituents one by one. The Borromean binding is intimatelyrelated to two other fascinating phenomena, viz. Efimov effects [4]and Thomas collapse [5]. The existence of Borromean binding inthe ground states of different atomic and molecular systems hasbeen discussed in a number of articles [6–15]. Borromean systemshave also appeared in other areas such as nuclear physics [16],molecular physics [17], chemical physics [18] and DNA [19]. How-ever the stability of the bound P and D states of the hydrogenmolecular ion interacting with screened Coulomb (Yukawa) poten-tials has not been studied yet except our earlier studies [9] on the3Po states of the molecular Hþ2 ion with screened Coulomb poten-tials. Several theoretical studies ([20–29], references therein) andfew experimental investigations [30,31] have been performed sofar for this system to investigate various properties of Hþ2 molecu-lar ion in the pure Coulomb case. With abundances of the Hþ2 ionsin interstellar matter and with the recent experimental advance-ments in the experiments of Hþ2 using laser spectroscopy, it is ofimportant to study various properties of this simplest three-bodysystem under the influence of screened Coulomb potentials. Detail

ll rights reserved.

applications of the screened Coulomb potentials have been high-lighted in the several previous works ([32–38], references therein).

In this work we have investigated the stability of the bound Pand D states of the molecular Hþ2 ion, where the nuclei and theelectron interact with a screened Coulomb (Yukawa-type) poten-tial exp(�lr)/r, where l is the screening parameter. The calcula-tions have been carried out in the framework of Rayleigh–Ritzvariational principle employing correlated exponential wave func-tions with quasi-random choice of nonlinear parameters. The exis-tence of Borromean binding is established for the lowest two Pstates and lowest two D-states of the molecular ion Hþ2 which isa fundamental three-body quantum system. The stability of theD-states of Hþ2 molecular ion interacting via Yukawa potential is re-ported for the first time. Convergence of the calculations has beenexamined with the increasing number of terms in the wave func-tions. The atomic unit has been used throughout the work.

2. Hamiltonian and wave functions

The non-relativistic screened Hamiltonian for Hþ2 system can bewritten in the form

H ¼ � 12mr2

1 þr22

h i� 1

2r2

3 �expð�lr31Þ

r31þ expð�lr32Þ

r32

� �

þ expð�lr21Þr21

ð1Þ

where 1, 2, and 3 denote the two nuclei and the electron respec-tively, rij = |ri � rj| = rji, m = 1836.152701 and m is the nuclear massin the units of electron mass. When the helium atom is placed invacuum, we have l = 0. In plasma physics the parameter l (=1/kD,kD is called Debye length) is known as the Debye screeningparameter.

S. Kar, Y.K. Ho / Chemical Physics Letters 506 (2011) 282–285 283

For triplet P and singlet D states of the hydrogen molecular ion,we employ the wave functions [37–39]

W ¼ 1þ SpnP12

� �XNb

i¼1

XL

l1¼0

AiYl1 ;l2LM ðr31; r32Þ expð�air31 � bir32

� cir21Þ ð2Þ

with Yl1 ;l2LM ðr31; r32Þ ¼ rl1

31rl232

Xm1 ;m2

CLMl1m1 ;l2m2

Yl1m1 ðr31ÞYl1m2 ðr32Þ

ð3Þ

where the functions Yl1 ;l2LM ðr31; r32Þ are the bipolar harmonics or Sch-

wartz harmonics, ri ¼ ri=ri (i = 31, 32, 21), YlimiðriÞ denotes the usual

spherical harmonics, CLMl1m1 ;l2m2

are the Clebsch–Gordan coefficients,ai;bi; ci are the nonlinear variational parameters, Aiði ¼ 1; ::::;NÞare the linear expansion coefficients, l1 þ l2 ¼ L, Nb is number of ba-sis terms, and the operator P12 is the permutation of the two iden-tical particles 1 and 2. Spn denotes the spin functions. The nonlinearvariational parameters ai;bi and ci are chosen from a quasirandomprocess [9,33–35,37–39]. The parameters ai; bi and ci are chosenfrom the three positive interval [a1, a2], [b1, b2] and [d1, d2];

ai ¼ hh12 iðiþ 1Þffiffiffiffiffiffi19piiða2 � a1Þ þ a1

bi ¼ hh12 iðiþ 1Þffiffiffiffiffiffi23piiðb2 � b1Þ þ b1

ci ¼ hh12 iðiþ 1Þffiffiffiffiffiffi29piiðd2 � d1Þ þ d1

ð4Þ

where the symbol hh:::ii designates the fractional part of a realnumber.

Table 1Bound P and D states energies of the hydrogen molecular ion for different screening para

1/l 3Po (m = 0, J = 1) 3Po (m = 1, J = 1)

1 �0.596873 �0.586887�0.5968737a �0.586904a

100 �0.586939 �0.57695350 �0.577134 �0.56715030 �0.564258 �0.55427720 �0.548471 �0.53849615 �0.533018 �0.52305010 �0.503075 �0.4931308 �0.481427 �0.4715056 �0.446817 �0.4369435 �0.420392 �0.4105694 �0.382755 �0.3730283 �0.325045 �0.3155322.5 �0.283100 �0.2738082 �0.226717 �0.2178401.8 �0.198582 �0.1899791.6 �0.166441 �0.1582241.5 �0.148708 �0.1407451.4 �0.129819 �0.1221671.3 �0.109791 �0.1025251.2 �0.088739 �0.0819611.1 �0.066950 �0.0608001.0 �0.045039 �0.0397180.9 �0.024256 �0.0200550.88 �0.020406 �0.0164990.86 �0.016733 �0.0131320.85 �0.014970 �0.0115380.84 �0.013260 �0.0099970.82 �0.010025 �0.0071210.8 �0.007077 �0.004570.78 �0.004477 �0.002420.76 �0.002293 �0.000760.75 �0.00139 �0.000150.748 �0.00123 �0.000060.74 �0.000630.73 �0.00005

a Ref. [28].

3. Results

We first carry out calculations to determine bound P and Dstates energies of the hydrogen molecular ion for different choicesof the screening parameters. To do so, one needs to obtain the solu-tions of the Schrödinger equations HW ¼ EW, where E < 0 follow-ing the Rayleigh–Ritz variational principle. The lowest values ofthe bound P and D states energies are obtained with the optimizedchoice of the nonlinear variational parameters following thescheme (4). To designate molecular states it is convenient to usethe rotational (J) and vibrational (m) quantum numbers. Thethree-body states can also be classified as gerade or ungeradedepending on the combined effect of the spin (S) and parity (p),p(�1)S, with the resulting plus sign denoting gerade and minussign denoting ungerade. The atomic levels for the J = 0, 1 and 2 ger-ade states are 1Se, 3Po, and 1De, respectively. In this work, we deter-mine the bound 3Po (m = 0, J = 1), 3Po (m = 1, J = 1), 1De (m = 0, J = 2),1De (m = 1, J = 2) states energies of the hydrogen molecular ion fordifferent values of the screening parameters. We have reportedthe bound 1Se (m = 0, J = 0), 3Po (m = 0, J = 1) of the hydrogen molec-ular ion in our earlier work [9]. The 3Po (m = 0, J = 1) state energiesobtained in this work have been improved over the last quoted dig-its. The bound states energies obtained from the present work arepresented in Table 1 and Figures 1 and 2. From Table 1 and Figures.1 and 2, it is clear that the Hþ2 molecular ion is bound but thehydrogen atom is not bound for the stronger screening. This showsthe existence of the Borromean binding for the Hþ2 molecular ion.Next, to estimate the window of the Borromean binding for eachof S, P and D states, we have fitted the energy levels with a 9th

meters.

1De (m = 0, J = 2) 1De (m = 1, J = 2) H(1S)

�0.596344 �0.58637 �0.50000000�0.5963452a �0.58640a

�0.586409 �0.57644 �0.49007451�0.576605 �0.56663 �0.48029611�0.563729 �0.55376 �0.46748228�0.547942 �0.53798 �0.45181643�0.532490 �0.52253 �0.43653060�0.502547 �0.49261 �0.40705803�0.480900 �0.47099 �0.38587872�0.446291 �0.43643 �0.35225907�0.419868 �0.41006 �0.32680851�0.382234 �0.37252 �0.29091959�0.324530 �0.31503 �0.23683267�0.282593 �0.27332 �0.19837608�0.226223 �0.21737 �0.14811702�0.198097 �0.18952 �0.12381303�0.165968 �0.15778 �0.09686159�0.148244 �0.14031 �0.08244734�0.129365 �0.12174 �0.06752960�0.109351 �0.10211 �0.05231507�0.088316 �0.08157 �0.03717849�0.066551 �0.06043 �0.02278129�0.044671 �0.03937 �0.01028579�0.023920 �0.01972 �0.001757�0.020096 �0.01621 �0.000815�0.016436 �0.01285 �0.000213�0.014679 �0.01126 �0.000055�0.012976 �0.00972�0.009756 �0.00686�0.006824 �0.0043�0.00422 �0.0020�0.00206 �0.0003�0.00117�0.00098

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

E (

a.u.

)

μ (a0

−1)

3Po (ν=0, J=1)3Po (ν=1, J=1)

H (1s 2S)

Figure 1. The bound 3Po (m = 0, J = 1) and 3Po (m = 1, J = 1) states of the molecular Hþ2ion in terms of the screening parameters l along with H(1s 2S) threshold energies.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

E (

a.u.

)

μ (a0

−1)

1De (ν=0, J=2)1De (ν=1, J=2)

H(1s 2S)

Figure 2. The bound 1De (m = 0, J = 2) and 1De (m = 1, J = 2) states of the molecular Hþ2ion as functions of the screening parameters l along with H(1s 2S) thresholdenergies.

284 S. Kar, Y.K. Ho / Chemical Physics Letters 506 (2011) 282–285

degree polynomial to obtain critical values of l. The critical valuelc of l obtained from this work for different systems are presentedin Table 2 along with the other reported results. The rangelH

c � l � lHþ2c is commonly known as the Borromean window.

Table 2 also shows the windows, lHc � l � lHþ2

c for Borromeanbinding for the bound S, P, D states of Hþ2 . From tables and figures,it is also clear that the energies of Hþ2 is lower than the energies of

Table 2Windows for Borromean binding, lH

c � l � lHþ2c for S, P, D states of Hþ2 .

System lHc lHþ2

c

Hþ2 (1Se) (m = 0, J = 0) 1.1878 1.37341.1899a 1.373 (4)a

1.19b 1.35b

1.1900c 1.37c

Hþ2 (3Po) (m = 0, J = 1) 1.1878 1.3715

Hþ2 (3Po) (m = 1, J = 1) 1.1878 1.3389

Hþ2 (1De) (m = 0, J = 2) 1.1878 1.3626

Hþ2 (1De) (m = 1, J = 2) 1.1878 1.3223

a Bertini et al. [7].b Bressanini et al. [6].c Ghoshal and Ho [15].

the corresponding H atom, and the critical parameter for Hþ2 isgreater than that for H. For l < lH

c the hydrogen-like atom is stable,and H plus H+ would give the relevant threshold. This implies thatHþ2 is stable against dissociation: Hþ2 ? H + H+. For l > lH

c thehydrogen-like atom becomes unbound. But forlH

c � l � lHþ2c , the

system is stable against the dissociation: Hþ2 ? H+ + H+ + e�, or intoany other two-body subsystems, as in this range the H-like atom isnot bound. Since Hþ2 is stable even H is not stable, Hþ2 can be con-sidered as a prototype for the Borromean systems. Our results areconverged up to the quoted digits. In the unscreened case our cal-culated bound P and D states energies are accurate up to some partof 10�6 compared to the best reported results [20–29]. The H (1s2S) energies are taken from our earlier work [9]. We have used1200-term and 1400-term wave functions for the P-state and theD-state calculations, respectively.

4. Summary and conclusions

In this work, we have investigated the stability of the bound Pand D states of the molecular Hþ2 ion interacting with screenedCoulomb (Yukawa-type) potentials in the framework of Ritz varia-tional principle. We have determined the reasonably accurate val-ues of the bound 3Po (m = 0, J = 1), 3Po (m = 1, J = 1), 1De (m = 0, J = 2),and 1De (m = 1, J = 2) states energies for different values of thescreening parameters using highly correlated exponential wavefunctions. The 3Po (m = 1, J = 1), 1De (m = 0, J = 2), 1De (m = 1, J = 2)state energies for different screening parameters are reported forthe first time in the literature. The 3Po (m = 0, J = 1) state energyare improved slightly over our previous work [9]. The existenceof Borromean states are found in the S, P and D states of Hþ2 molec-ular ion. Windows for the Borromean bindings are also reported foreach S, P and D states.

Acknowledgments

SK wishes to thank Natural Scientific Research Innovation Foun-dation in Harbin Institute of Technology for financial support(Grant No. HIT.NSRIF.2008.01). YKH is supported by NSC of Taiwan.

References

[1] J.M. Richard, Phys. Rev. A 67 (2003) 034702.[2] J.M. Richard, Few-Body Syst. 38 (2006) 79.[3] M.V. Zhukov, B.V. Danilin, D.V. Fedorov, J.M. Bang, I.S. Thompson, J.S. Vaagen,

Phys. Rep. 231 (1993) 151.[4] V. Efimov, Phys. Lett. 33B (1970) 563.[5] L.H. Thomas, Phys. Rev. 47 (1935) 903.[6] D. Bressanini, G. Morosi, L. Bertini, M. Mella, Few-Body Syst. 31 (2002) 199.[7] L. Bertini, M. Mella, D. Bressanini, G. Morosi, Phys. Rev. A 69 (2004) 042504.[8] P.K. Mukherjee, S. Fritzsche, B. Fricke, Phys. Lett. A 360 (2006) 287.[9] S. Kar, Y.K. Ho, Phys. Lett. A 368 (2007) 476.

[10] S. Kar, Y.K. Ho, Phys. Rev. A 75 (2007) 062509.[11] S. Kar, Y.K. Ho, Euro. Phys. J. D 48 (2008) 157.[12] Y.K. Ho, S. Kar, Few-Body Syst. 44 (2008) 253.[13] M. Pawlak, M. Byliciki, J.K. Saha, P.K. Mukherjee, J. Phys. B 42 (2009) 215701.[14] A. Ghoshal, Y.K. Ho, J. Phys. B. 43 (2010) 115007.[15] A. Ghoshal, Y. K. Ho, Int. J. Quantum Chem. (2010), published online.[16] A. Lemasson, A. Navin, N. Keeley, et al., Phys. Rev C 82 (2010) 044617.[17] Y. Li, W.J. Zhang, Q.D. Gou, H.W. Song, T.Y. Shi, Phys. Rev. A 82 (2010) 022515.[18] C.D. Meyer, R.S. Forgan, K.S. Chichak, et al., Chem.-A Euro. J. 16 (2010) 12570.[19] D.R. Han, S. Pal, Y. Liu, et al., Nature Nano. 5 (2010) 712.[20] A.K. Bhatia, Phys. Rev. A 58 (1998) 2787.[21] A.K. Bhatia, R.J. Drachman, Phys. Rev. A 59 (1999) 205.[22] J.M. Taylor, A. Dalgarno, J.F. Babb, Phys. Rev. A 60 (1999) R2630.[23] L. Hilico, N. Bily, B. Gremaud, D. Delande, Euro. Phys. J. D 12 (2000) 449.[24] R.E. Moss, Phys. Rev. A 61 (2000) 040501.[25] A.M. Frolov, J. Phys. B 35 (2001) L331.[26] Z.-C. Yan, J.-Y. Zhang, Y. Li, Phys. Rev. A 67 (2003) 062504.[27] M.M. Cassar, G.W.F. Drake, J. Phys. B 37 (2004) 2485.[28] J.Ph. Karr, L. Hilico, J. Phys. B 30 (2006) 2095.[29] V.I. Korobov, Phys. Rev. A 74 (2006) 052506.[30] W.G. Sturrus, E.A. Hessels, P.W. Arcuni, S.R. Lundeen, Phys. Rev. A 44 (1991)

3032.

S. Kar, Y.K. Ho / Chemical Physics Letters 506 (2011) 282–285 285

[31] P.L. Jacobson, R.A. Komara, W.G. Sturrus, S.R. Lundeen, Phys. Rev. A 62 (2000)012509.

[32] B. Saha, T.K. Mukherjee, P.K. Mukherjee, Chem. Phys. Lett. 373 (2003) 218.[33] S. Kar, Y.K. Ho, Chem. Phys. Lett. 402 (2005) 544.[34] S. Kar, Y.K. Ho, Chem. Phys. Lett. 424 (2006) 403.

[35] S. Kar, Y.K. Ho, Chem. Phys. Lett. 449 (2007) 246.[36] A.N. Sil, S. Canuto, P.K. Mukherjee, Adv. Quantum Chem. 58 (2009) 115.[37] S. Kar, Y.K. Ho, J. Phys. B 44 (2011) 015001.[38] S. Kar, Y. K. Ho, Phys. Rev. A 83, in press.[39] A.M. Frolov, V.H. Smith Jr, Phys. Rev. A 53 (1996) 3853.