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Chemical Physics Letters 449 (2007) 246–248

Effect of Debye plasmas on the dispersion coefficients C6

for interactions among H and He atoms

Sabyasachi Kar *, Y.K. Ho

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

Received 21 June 2007; in final form 16 October 2007Available online 22 October 2007

Abstract

The effect of Debye plasmas on the dispersion coefficient C6 for interactions among H and He atoms has been investigated for the firsttime using highly correlated exponential basis functions. The C6 coefficient for the interactions among H and He are reported for variousDebye lengths.� 2007 Elsevier B.V. All rights reserved.

The investigation on the Van der Waals two-bodydispersion coefficients in the multipole expansion of secondorder long-range interaction between a pair of atoms isimportant for quantitative interpretation of the equilibriumproperties of gasses and crystals, of transport phenomenain gasses, and of phenomena occurring in slow atomicbeams [1]. The leading term of the interaction betweentwo atoms at large separation R is dipole–dipole interac-tion decreasing as R�6. This term has a coefficient com-monly called dispersion coefficient C6. Several studieshave been performed so far to calculate the dispersion coef-ficient C6 for the interactions among H and He [2–6], andbetween two He atoms [7–14]. In the present work, we haveinvestigated the effect of screened Coulomb (Yukawa)potentials on the dispersion coefficient C6 for interactionsamong the hydrogen and helium atoms in the frameworkof Debye shielding approach of plasma modeling.Recently, several studies have been performed on thebound states and other structural properties for H andHe embedded in Debye plasma environments ([15–24],references therein). We have reported the effect of Debyeplasmas on the resonance states of helium atom [21,25,26]. Detail applications of Debye screening on atomic

0009-2614/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2007.10.052

* Corresponding author. Fax: +886 2 2362 0200.E-mail address: skar@pub.iams.sinica.edu.tw (S. Kar).

and molecular systems are available from the earlier works[15–34].

The long-range part of interaction between two atoms a

and b in their ground states can be written in the form of aseries of inverse powers of the separation R as

V ab ¼ �C6

R6� C8

R8� C10

R10� � � � ; ð1Þ

with

C6 ¼3

2

Xnm

f ðlaÞn0 f ðlbÞ

m0

Ean0Eb

m0ðEan0 þ Eb

m0Þ; ð2Þ

where Ein0 ¼ Ei

n � Ei0 is the excitation energy for atom i and

is positive for the atoms in the ground state, and the 2l-poleoscillator strength f ðlÞn0 being defined by

f ðlÞn0 ¼8p

2lþ 1ðEn � E0Þ W0

Xi

rli P lðcos#iÞ

����������Wn

* +����������2

; ð3Þ

where i = 1 for hydrogen atom and i = 2 for helium atom.For hydrogen atom, we consider the standard Slater-

type basis set

W ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2lþ 1p

4p

XN

i¼l

Diriþle�krP lðcos h1Þ; ð4Þ

and for helium atom, we consider the wave functions

Table 1Dispersion coefficient C6 for different Debye lengths

kD H–H (20 terms) He–He (600terms)

H–He (20 terms–600terms)

1 6.4990267054058 1.460977836 2.8213439136.4990267054058a 1.4609778376a 2.821343915a

100 6.5035294875818 1.461349496 2.82259668350 6.5167846196348 1.462450764 2.82629205830 6.5475298244956 1.465022106 2.83487855520 6.6062097337940 1.469957179 2.85128126015 6.6867639767567 1.476750033 2.87378377610 6.9126586654976 1.495743943 2.9366111578 7.1397293994878 1.514625267 2.9992187886 7.6356432658368 1.554864621 3.1338002565 8.1502774867681 1.595062930 3.2702877884 9.1579954887588 1.669296716 3.5286163723 11.697469308730 1.833983497 4.1342647882.5 14.92800054106 2.008887454 4.8309880382 23.58193378 2.36332308 6.428127301.5 69.686464 3.32322241 12.2219391.0 8.831501

a Ref. [2].

0.00 0.08 0.16 0.24 0.32 0.406

7

8

9

10

11

12

13

14

15

C6

μ

H-H

Fig. 1. The dispersion coefficient C6 in terms of the screening parameter lfor the interaction between two H atoms in their ground states.

S. Kar, Y.K. Ho / Chemical Physics Letters 449 (2007) 246–248 247

W ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2lþ 1p

4pð1þ O12Þ

XN

i¼1

Dirl1P lðcos h1Þ

� expð�air1 � bir2 � cir12Þ; ð5Þ

where k, ai, bi, ci are the non-linear variation parameters,l = 0, 1 for S, P states, respectively, Di (i = 1, . . ., N) arethe linear expansion coefficients, and O12 is the permutationoperator on the subscripts 1 and 2 representing the twoelectrons. Here r1 and r2 are the radial coordinates of thetwo electrons and r12 is their relative distance. We haveused a quasi-random process ([12,20–26,29,31,33,34],references therein) to optimized the non-linear variationalparameters ai, bi and ci. The parameters ai, bi and ci arechosen from the three positive intervals [a1, a2], [b1, b2]and [d1, d2];

ai ¼1

2iðiþ 1Þ

ffiffiffi2p� �� �

ða2 � a1Þ þ a1;

bi ¼1

2iðiþ 1Þ

ffiffiffi3p� �� �

ðb2 � b1Þ þ b1;

ci ¼1

2iðiþ 1Þ

ffiffiffi5p� �� �

ðd2 � d1Þ þ d1;

ð6Þ

where the symbol ÆÆ� � �ææ designates the fractional part of areal number.

To investigate the effect on the dispersion coefficient C6

under Debye screening, we assume that the leading term inthe Van der Waals interaction between two atoms a and b

in their ground states still has a form of R�6, as

V ab ¼ �C6ðlÞ

R6: ð7Þ

Here the plasma effect on Vab will be reflected on the valueof C6, which now depends on the screening parameter l,and is denoted by C6(l). The parameter l(=1/kD) is calledthe Debye shielding parameter, and kD is called the Debyelength. To calculate C6 coefficient for interactions amonghydrogen and helium atoms in their ground states, oneneeds to obtain the energy levels and wave functions foreach of hydrogen and helium atoms separately in their Sand P states with the optimum choices of non-linearparameters. To obtain the energy levels for hydrogenatom with different Debye lengths, we diagonalize theHamiltonian

H ¼ � 1

2r2 � expð�r=kDÞ

r; ð8Þ

with the wave functions (4). For the helium atom we diag-onalize the Hamiltonian

H ¼ � 1

2r2

1 �1

2r2

2 � 2expð�r1=kDÞ

r1

þ expð�r2=kDÞr2

� �

þ expð�r12=kDÞr12

; ð9Þ

with wave functions (5). After calculating the energy levelsand eigen-functions, we use Eqs. (2) and (3) to calculate thecoefficient C6 for the H–H, He–He and H–He interactions.

We present our calculated results for different screeningparameters in Table 1 and in Figs. 1–3. For the unscreenedcase, our results compare well with other values availablein the literature [1–14], especially with the best variationalresults [2]. For the screened cases, our results show thatthe C6 dispersion coefficients for the interactions betweenH and H, He and He, and H and He in their ground statesincrease with increasing plasma strength. Our findings indi-cate that when the plasma screening effect increases, the di-pole polarizability (DP) for the individual atom increasesbecause the ground state wave functions become more dif-fused. Now when two such diffused atoms come together,the dispersion coefficient C6 will be increased, in analogto the free-atom case that the C6 for the H–H case (withDP = 4.5 for H) is larger than that for the He–He case(DP = 1.383192174 for He).

0.0 0.1 0.2 0.3 0.4 0.51.4

1.6

1.8

2.0

2.2

2.4

C6

μ

He-He

Fig. 2. The dispersion coefficient C6 in terms of the screening parameter lfor the interaction between two He atoms in their ground states.

0.0 0.1 0.2 0.3 0.4 0.52.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

C6

μ

H-He

Fig. 3. The dispersion coefficient C6 as a function l for the interactionbetween H and He atoms in their ground states.

248 S. Kar, Y.K. Ho / Chemical Physics Letters 449 (2007) 246–248

In summary, we have obtained accurate dispersion C6

coefficients for the interactions among H and He atomsin their ground states for the unscreened case as well asfor the screened cases. In the screened cases, our resultsare new to the best of our knowledge. There have not beendetailed investigations in the literature for the dispersioncoefficients for the plasma-embedded atoms until now.The Van der Waals force constants, particularly the leadingterm C6 arising from the induced dipoles, are of great

theoretical and experimental interest in atomic, molecularphysics [1–14]. We hope our findings will provide newinsight and useful information to the communities inatomic, molecular and chemical physics.

Acknowledgement

This work was supported by the National ResearchCouncil of Taiwan, ROC.

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