Physics Letters A 368 (2007) 476–479

www.elsevier.com/locate/pla

Bound states and dipole polarizability of hydrogen molecular ion H+2

in weakly coupled hot plasmas

Sabyasachi Kar ∗, Y.K. Ho

Institute of Atomic and Molecular Sciences, Academia Sinica, PO Box 23-166, Taipei 106, Taiwan, ROC

Received 27 December 2006; received in revised form 18 April 2007; accepted 18 April 2007

Available online 19 April 2007

Communicated by F. Porcelli

Abstract

The effects of hot-dense plasmas on the bound states and the dipole polarizability of the ground state of Coulomb three-body molecular ion H+2

have been investigated using highly correlated basis functions and by considering the Debye shielding approach of plasma modeling. The groundS state and the first excited P state energies along with the dipole polarizability for different shielding parameters are reported.© 2007 Elsevier B.V. All rights reserved.

PACS: 52.20.-j; 33.20.-t; 31.10.+z; 31.25.Nj

An atom/ion immersed in plasmas experiences various per-turbation from plasmas and recent theoretical investigations([1–8], references therein) on atomic processes in the weaklycoupled hot plasmas show a definitive distribution of the differ-ent atomic states compared to the free atom which providesuseful information for the determination of various macro-parameters (ion-temperature, electron density, etc.) for theprocesses in plasmas. In the present work we have investigatedthe effect of plasmas on the bound 1S and 3P states and thedipole polarizability of hydrogen molecular ion, H+

2 which isa fundamental there-body quantum system. Several theoreticalstudies ([9–22], references therein) and few experimental inves-tigations ([23,24], references therein) have been performed sofar for this system in the unscreened case. With the abundanceof H+

2 in interstellar matter and with recent experimental devel-opment in experiments of H+

2 using laser spectroscopy, it is ofgreat importance to study this simplest three-body system un-der the influence of plasma environments. Recently, Mukherjeeet al. [25] have studied the stability of H+

2 molecular ion un-der the influence of Debye plasmas. They have reported thebound ground 1S state energies of H+

2 in plasmas for differ-

* Corresponding author.E-mail address: skar@pub.iams.sinica.edu.tw (S. Kar).

0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.04.057

ent screening parameters. In this work, we have calculated thebound 1S and 3P states energies and the dipole polarizability ofthe ground states of H+

2 in plasmas for different Debye lengths.The 1S state energies for different screening parameters ob-tained from our calculation are lower than those reported byMukherjee et al. [25]. It is also important to mention here thatthe effect of Debye plasmas on the polarizability of two elec-tron atoms/ions has been reported in the literature [2,3,26].

The non-relativistic Hamiltonian describing the H+2 ion em-

bedded in Debye plasmas characterized by a parameter D,called the Debye length, is given by

(1)

H = − 1

2m

[∇21 + ∇2

2

] − 1

2∇2

3

−[

exp(−r31/D)

r31+ exp(−r32/D)

r32

]

+ exp(−r21/D)

r21,

where 1, 2, and 3 denote the two nuclei and the electron re-spectively, rij = |�ri − �rj | = rji and m = 1836.152701, m is thenuclear mass in the units of electron mass. The Debye lengthcan be represented as D = [kBT /4πn(Ze)2]1/2 [27,28], n de-notes plasma density and T its temperature. A set of plasmacondition can be simulated for different choice of D(n,T ). For

S. Kar, Y.K. Ho / Physics Letters A 368 (2007) 476–479 477

two-component plasmas near thermodynamic equilibrium, theDebye length D can be represented by [6,27,28]

(2)D = 1

μ=

[4π(1 + Z)e2ne

kBTe

]−1/2

,

kB is the Boltzmann constant, ne is the electron density in plas-mas and Te is the electron temperature of the plasma, and Z isthe nuclear charge and here its value is unity. For the presentproblem Debye plasma with electron density ne and with en-ergy of ED (in eV) can be written from Eq. (2) in the form

(3)ne = 9.8674 × 1021 ED cm−3.

D2

We have calculated the electron density due to 100 eV De-bye plasmas for different screening parameters for referenceand presented in Table 1. We have presented a form of Debyelength D in (2) on the view of the discussions in Section 4.4 ofRef. [28] on the ‘dressed particle’ consisting of a given parti-cle plus its correlated particle cloud in the treatment of many-particle systems such as plasmas. It has been pointed out thatthe ‘dress’ of an ion is a 50–50 mixture of electrons and otherions when their contributions to the Debye screening lengths arethe same. In other words, when the temperatures of the elec-trons and ions systems are the same for the Z = 1 case, theDebye wave numbers of the electrons and ions are the same. It

Table 1The bound 1Se and 3Po states energies and dipole polarizability (αd ) of hydrogen molecular ion H+

2 in plasmas for different Debye lengths and the electron density(ne) due to 100 eV Debye plasmas along with the H(1S) ground state energy. The notation x[+y] stands for x[+y] = x × 10y

D (a.u.) ne cm−3 1S state (a.u.) 3P state (a.u.) αd (a30) H(1S) [4]

∞ 0.0 −0.5971383 −0.5968724 3.174 −0.50000000−0.59713906a −0.5968737b 3.1687256c,3.171d,

−0.5949e 3.1680f

3.167969(15)g

100 9.87[+19] −0.587204 −0.586938 3.175 −0.490074519.9[+19]e −0.5850e −0.49008h

50 3.95[+20] −0.577399 −0.577133 3.178 −0.48029611

30 1.10[+21] −0.564523 −0.564257 3.184 −0.46748228

20 2.47[+21] −0.548736 −0.548471 3.195 −0.451816432.5[+21]e −0.5465e −0.45182h

15 4.39[+21] −0.533283 −0.533018 3.211 −0.43653060

10 9.87[+21] −0.503340 −0.503074 3.255 −0.407058039.9[+21]e −0.5011e −0.40705h

8 1.54[+22] −0.481691 −0.481427 3.297 −0.38587872

6 2.74[+22] −0.447080 −0.446816 3.387 −0.35225907

5 3.95[+22] −0.420654 −0.420391 3.477 −0.326808513.9[+22]e −0.4184e −0.32674h

4 6.17[+22] −0.383016 −0.382755 3.641 −0.290919596.2[+22]e −0.3808e −0.29076h

3 1.10[+23] −0.325303 −0.325044 4.007 −0.23683267

2.5 1.56[+23] −0.283355 −0.283100 4.401 −0.19837608

2 2.47[+23] −0.226965 −0.226717 5.223 −0.148117022.5[+23]e −0.2250e

1.8 3.05[+23] −0.198825 −0.198582 5.851 −0.12381303

1.6 3.85[+23] −0.166678 −0.166440 6.877 −0.09686159

1.5 4.34[+23] −0.148941 −0.148708 7.672 −0.08244734

1.4 5.04[+23] −0.130046 −0.129818 8.805 −0.06752960

1.3 5.84[+23] −0.110012 −0.109791 10.52 −0.05231507

1.2 6.85[+23] −0.088950 −0.088738 13.37 −0.03717849

1.1 8.16[+23] −0.067150 −0.066950 18.74 −0.02278129

1.0 9.87[+23] −0.045223 −0.045039 31.19 −0.010285799.9[+23]e −0.0443e 0.16598h

0.9 1.22[+24] −0.024406 −0.024256 −0.001757

0.8 1.54[+24] −0.007200 −0.007077

0.75 1.75[+24] −0.00147 −0.00138

0.74 1.80[+24] −0.00070 −0.00062

0.73 1.85[+24] −0.00009 −0.00003

a Refs. [17–19]. b Refs. [18,19]. c Ref. [13]. d Ref. [9]. e Ref. [25]. f Ref. [12]. g Ref. [24]. h Ref. [2].

478 S. Kar, Y.K. Ho / Physics Letters A 368 (2007) 476–479

is well demonstrated in the recent work of Zhang and Winkler[8] about the thermal equilibrium of electrons and ions. In theirimportant study entitled ‘Debye–Hückel screening and fluctua-tions’; they have defined the Debye length D with certain spe-cific conditions for singly ionized ion as D = [Teq/4πn0e

2]1/2,where Teq = TeTi/(Te + Ti), with Te and Ti being the tem-peratures of the electrons and ions, respectively. In this work,we have presented the electron density in Table 1 for differ-ent Debye lengths using relation (2) on view of the discussionsin Ref. [28] and following the demonstration in the work ofMukherjee et al. [25].

For the S and P states of the H+2 molecular ion, we have

considered the wave function

(4)

Ψ = (1 + SpnO21)

N∑i=1

CirL31PL(cos θ1)

× exp(−αir31 − βir32 − γir21),

where αi , βi , γi are the non-linear variation parameters, Ci

(i = 1, . . . ,N ) are the linear expansion coefficients, L = 0 forS state and L = 1 for P state, Spn = 1 denotes singlet states andSpn = −1 indicates triplet states, N is the number basis terms.The operator O21 is the permutation of the two-identical par-ticles. The non-linear variational parameters αi,βi and γi arechosen from a quasi-random process ([17], reference therein).The non-linear parameters αi,βi and γi are chosen from thethree positive interval [A1,A2], [B1,B2] and [C1,C2];

αi = ⟨⟨i(i + 1)

√7/2

⟩⟩(A2 − A1) + A1,

βi = ⟨⟨i(i + 1)

√11/2

⟩⟩(B2 − B1) + B1,

(5)γi = ⟨⟨i(i + 1)

√13/2

⟩⟩(C2 − C1) + C1,

where the symbol 〈〈· · ·〉〉 designates the fractional part of a realnumber.

We first carry out calculations to determine the bound stateof the hydrogen molecular ion in Debye plasma environments.To do so, one needs to obtain the solutions of the Schrödingerequation HΨ = EΨ , where E < 0 following the Rayleigh–Ritz variational principle. The lowest values of the bound S andP states energies are obtained with the optimized choice of thenon-linear variational parameters following the scheme (5). Thebound 1S and 3P state energies obtained from our calculationsare presented in Fig. 1, and in Table 1. From Fig. 1 and Ta-ble 1, it is clear that the bound states energies increase withincreasing plasma strengths. Also from Table 1, it seems thatour results differ by 7 × 10−7 for the S state and by 1.3 × 10−6

for P state as compared with the best results in the literature forthe unscreened case. For the 1S state calculations we have usedN = 600 terms and for the 3P states calculations, N = 1000terms. In Table 1 and Fig. 1, we have also shown the H(1S)threshold energies [2,4] for different screening parameters tocompare the plasma effect on the energy levels of the hydro-gen molecular ion with the H(1S) threshold energies. We havenot included the plot of the 3P state energies as it is similar tothe 1S state energies in Fig. 1. The rotation-vibration levels ofthe 1sσg electronic ground state of H+

2 have 1Se or 3Po sym-metries while the 2pσu excited states are either 3Se or 1Po. In

Fig. 1. The ground state energy of H+2 in plasmas for different Debye shielding

parameters along with the H(1S) ground state energy (dashed line).

the present work, we have calculated the lowest vibronic stateswith J = 0 and J = 1 (J being the total angular momentumof the system) of H+

2 below the first dissociation limit withinthe non-relativistic framework for different Debye length. Thedifferent vibronic states are partially important for the determi-nation of the fundamental mass ratio, estimation of the dipolepolarizability, etc. Details of such states for the unscreened caseare available the work of Hilico et al. [14].

Next we calculate the dipole polarizability of H+2 in plasmas

for different Debye lengths. The dipole polarizability obtainedfrom second order perturbation theory can be written as [12,18]

(6)αd = 2∑p =0

|〈Ψp| �d . ε|Ψs〉|Ep − Es

2

a30,

where Ψp is the one the intermediate states of P symmetry withassociated energy eigenvalues Ep , Ψs is the ground state wavefunction, ε is the direction of external electric field and �d isthe dipole operator defined by �d = �r1 + �r2 − �r3, the summationindicates the sum over all states including continuum. With theusual transformation, Eq. (6) can be written as [12]

(7)αd = 2

μ3m(1 + μm)2

∑p

|〈Ψp|z1 + z2|Ψs〉|Ep − Es

2

a30,

where the reduced mass μm = m/(m + 1), z1,2 = μm(�r1,2 −�r3) . ε.

Our results obtained for different Debye length using Eq. (7)are presented in Table 1. In the unscreened case the relativeuncertainty is within 0.2% as compared to the reported experi-mental result and the best theoretical results. We have used 600basis terms for S states and 900 basis terms for P states in ourpresent calculations. Both S and P states are singlet spin-stateswith respect to the protons. We should also mention that in ourpresent work, we have not found any bound singlet-spin P statesfor the unscreened case. In Fig. 2, we have plotted the dipole po-larizability of H+

2 for different Debye screening parameters. Itappears from Table 1 and Fig. 2 that the dipole polarizability

S. Kar, Y.K. Ho / Physics Letters A 368 (2007) 476–479 479

Fig. 2. The dipole polarizability of H+2 in plasmas for different Debye screening

parameters.

increases with the increase of plasma strength. The increasingtrend of dipole polarizability with increasing plasma strengthindicates that the system would become more polarizable whenthe plasma strength is increased.

In summary, we have investigated the effect of weakly cou-pled hot plasmas on the bound states and the dipole polarizabil-ity of the ground state of hydrogen molecular ion H+

2 . The firstexcited bound P state energies and the dipole polarizability ofH+

2 in plasmas for different shielding parameters are our newpredictions. Our ground S-state energies are lower than thosereported by Mukherjee et al. [25]. It is believed that our ener-gies are reasonably accurate as they are within the 6th placesafter the decimal as compared to the best reported values in theliterature for the unscreened case. For further improvement onthe accuracy, we need to include the complex variation para-meters in the basis function (4) as those used in [17]. Such anapproach is one of our future goals. The Debye shielding in thepresent investigation of plasma modeling is a simple and goodapproximation in weakly coupled hot plasmas and low-densitywarm plasmas. For determination of macro-parameters in plas-mas it is important to have accurate atomic data (e.g. energylevels) available in the literature. The Dipole polarizability isimportant in many branches of physics and chemistry such as

collision phenomena, interaction of matter and electromagneticfields. We hope our prediction will provide useful informationto research communities in astrophysics, plasma physics, andatomic and molecular physics.

Acknowledgement

The work is supported by the National Science Council ofTaiwan, ROC.

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