bound states of helium atom in dense plasmas

Bound States of Helium Atom in DensePlasmas

SABYASACHI KAR, Y. K. HOInstitute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei, Taiwan 106,Republic of China

Received 10 June 2005; accepted 22 July 2005Published online 4 November 2005 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.20822

ABSTRACT: We have obtained the bound 1s2 1S, 1s2s 1,3S, and 1s2p 1,3P statesenergies of helium atom in dense plasma environments in accurate variationcalculations. A screened Coulomb potential to represent the Debye model is used forthe interaction between the charged particles. A correlated wave function consisting of ageneralized exponential expansion has been used to take care of the correlation effect.The 1s2 1S, 1s2s 1,3S, and 1s2p 1,3P states energies along with the ionization potential, theenergy splitting between the 1s2s 3S, and 1s2s 1S states, transition energies between theground state and low-excited states of He estimated for various Debye lengths, arereported. The results show high degree of accuracy even under strong plasmaconditions. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem 106: 814–822, 2006

Key words: dense plasma; Debye model; bound excited states; ionization potentials;variational calculations

Introduction

T he study of atomic systems in the dense (non-ideal) plasmas consisting of atomic ions and

electrons has received considerable attention in the-oretical investigations. The plasma has a screeningeffect on the Coulomb potential. The temperatureand electron density of the charged particles deter-mine the screening parameter. Because of the longrange of the Coulomb potential, an atom is always

under the simultaneous influence of many plasmaelectrons and ions, which partially shield or screenone another. The study of two-electron systems inthe model plasma environments is important, as thecorrelation effects between the charged particlescan be identified and calculated. Several investiga-tions have been performed to estimate energy levelsof one-electron atoms with screened Coulomb po-tentials. Two relevant studies for such systems arethe calculations of the bound eigenstates of thestatic screened Coulomb potential by Rogers et al.[1], and the variation solutions of Schrodinger’sequation for the static screened Coulomb potentialby Roussel and O’Connell [2]. Few investigationshave also been performed so far for the bound

Correspondence to: S. Kar; e-mail: [email protected] grant sponsor: National Science Council of Taiwan,

Republic of China.

International Journal of Quantum Chemistry, Vol 106, 814–822 (2006)© 2005 Wiley Periodicals, Inc.

states and resonance states of two electron atoms.The influence of plasmas on the 11S bound states oftwo electron systems was studied by Lam andVarshini [3], Hashino et al. [4], Winkler [5], Saha etal. [6], and Kar and Ho [7, 8]. Hashino et al. [4] havemade a calculation on the energy eigenvalues forthe bound excited 21S and 23S states of He-likeatoms in dense plasmas. The bound states of H�

and He� have been studied by Mercero et al. [9].Dai et al. [10] have performed calculations of prop-erties for the screened He-like systems using corre-lated wave functions for bound excited P and Dstates. They have reported the numerical results forthree values of the screening parameters. In numer-ical pair function calculations, Wang and Winkler[11] reported the stability of hydrogen negative ionand the energy splitting of several 1sns and 1snpstates of helium atom, using a many-body pertur-bation theory. Zhang and Winkler [12] have studiedsome aspects of the influence of a strong Debyeplasma environment on the negative hydrogen ionand the neutral helium atom, but no numericalresults for helium have been presented. The groundstate of positronium negative ion (Ps�) for variousDebye lengths has been investigated by Saha et al.[13] and Kar and Ho [14]. Very recently, Kar andHo [8, 15] investigated the resonance states of H�

[8, 15], of He [7], and of Ps� [14] in dense plasmas.The effect of Debye plasmas on the doubly excitedstates of highly stripped ions has been studied bySil and Mukherjee [16]. Screened Coulomb poten-tials are widely used in simple problems to approx-imate complicated many-body interactions, for in-stance, in the scattering of electrons from atoms, inplasmas, liquid metals, and electrolyte solutions[12].

In the present work, we have calculated the en-ergy eigenvalues of the bound 11S, 21S, 23S, 21P, 23Pstates of helium embedded in Debye plasmas envi-ronments. The energy eigenvalues of helium havebeen obtained in an accurate variational calculationin the framework of the Rayleigh–Ritz variationprinciple. A screened Coulomb potential obtainedfrom the Debye model is used to represent theinteraction in the Hamiltonian. Correlated wavefunctions expanded in terms of exponential typeinvolving inter-particle coordinates are used to rep-resent the correlation effects between the threecharged particles. Bound states of helium have beeninvestigated in several studies [3, 4, 6, 7, 9–11], butour present calculations are lower than those re-ported in the literature. In the present study, wealso present the ionization potential of helium atom

for various shielding parameters, as well as theenergy splitting between the bound 1s2s 1S and 1s2s3S states. Transition energies between the groundstate and some low-lying excited states have alsobeen estimated. The convergence of the calculationsis examined with increasing number of terms in thebasis expansion. The atomic unit (a.u.) has beenused throughout the present work, and all calcula-tions are performed in quadruple precision (32 sig-nificant figures) on DEC-ALPHA workstations inthe UNIX environments.

Theoretical Details

The nonrelativistic Hamiltonian describing thehelium atom embedded in Debye plasmas charac-terized by a parameter D is given by

H � �12 �1

2 �12 �2

2 � 2�exp��r1/D�

r1�

exp��r2/D�

r2�

�exp��r12/D�

r12, (1)

where r1 and r2 are the radial coordinates of the twoelectrons, and r12 is their relative distance. A par-ticular value of the screening parameter D corre-sponds to the range of plasma conditions, as theDebye parameter is a function of electron densityand electron temperature. For two-component plas-mas near thermodynamic equilibrium, the Debyelength D can be written as [4, 6, 16, 17]

D � �4��Z* � 1�e2ne/kBTe��1/ 2, (2)

where Z* is the effective charge, ne is the numberdensity of the electron, kB is the Boltzmann constantand Te is the electron temperature. The smallervalues of D are associated with stronger screening.A parameterized screening potential approximatesthe effects of the plasma charges on the interactionbetween the bound electron and the atomic nuclei.

For the S and P states of He atom, we haveconsidered the wave function

� � �1 � SpnO12� �i�1

N

Cir1LPL�cos�1�

exp��ir1 � �ir2 � �ir12��, (3)

BOUND STATES OF HELIUM ATOM IN DENSE PLASMAS

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 815

where �i, �i, �i are the nonlinear variation pa-rameters, Ci (i � 1, . . . , N) are the linear expan-sion coefficients, L � 0 for S states and L � 1 forP states, N is the number of basis terms, Spn � 1indicates singlet states, and Spn � �1 assignstriplet states, and O12 is the permutation operatordefined by O12f(r1, r2, r12, �1) � f(r2, r1, r12, �2).The wave functions of Eq. (3) have been widelyused in several bound-state and resonance statecalculations of two electron systems in modelplasma environments [5– 8, 10, 13–15, and refer-ences therein]. In this regard, it is worth mention-ing that Thakkar and Smith [18, 19] used thiswave function in their calculations of the bound11S, 21S, 23S, 21P, and 23P states in the un-screened case of helium-like ions from H�

through Mg10. In the present work, we use aquasi-random process suggested by Frolov [20] tochoose the nonlinear variation parameters �i, �i,and �i. According to the multi-box strategy forconstructing highly accurate bound-state wavefunctions for three-body systems [20], the param-eters �i, �i, and �i are chosen from the threepositive interval [A1

(k), A2(k)], [B1

(k), B2(k)], and [C1

(k),C2

(k)], where k � mod(i, 3) 1, 1 i N.

�i � 1�k��1

2 i(i � 1)�2�� [A2�k� � A1

�k�] � A1�k��

�i � 2�k��1

2 i(i � 1)�3�� [B2�k� � B1

�k�] � B1�k�� (4)

�i � 3�k��1

2 i(i � 1)�5�� [C2�k� � C1

�k�] � C1�k�� ,

where the symbol ��. . .�� designates the fractionalpart of a real number. The positive scaling factors1

(k), 2(k), and 3

(k) are equal to 1 in the first stage, andin the second stage they will be varied. But for thepresent investigation, it has been observed that thebetter optimization can be obtained by selecting A1

(k)

� 0, A2(k) � a; B1

(k) � 0, B2(k) � b; C1

(k) � 0, C2(k) � c; and

1(k) � 1, 2

(k) � 1, 3(k) � �. Ultimately, four variation

parameters, a, b, c, and �, are used in all the calcu-lations. The necessary integrals involving screenedCoulomb potentials for calculations of Hamiltonianand overlap matrix elements using basis functions(3) have been discussed in the literature (Ref. [10]and references therein).

Results and Discussion

The energy levels for S and P states of helium arecalculated by diagonalizing the Hamiltonian (1)with the basis functions (3). Following the quasi-random process in Eq. (4), we have varied the non-linear parameters a, b, and c to obtain the minimumeigenenergies for the bound S and P states. It isinteresting to note that, in the unscreened case, wehave first reproduced the reported results ofThakkar and Smith [18, 19], using the same wavefunctions as those used in their calculations. In thevacuum case, our final results for the S and P statesare fairly comparable to the best results in the lit-erature [21], with a difference of 10�10 a.u. Theconvergence of our calculations with the increasingbasis terms for the bound 1s2 1S and 1s2s 1,3S and1s2p 1,3P state eigenenergies are presented in TableI for various Debye lengths. We have optimized the1s2 1

S and 1s2s 1S states, using 700 basis functions ofEq. (3), and 500-term and 600-term basis functions,respectively, for the 1s2s 3S, 1s2s 1,3P states of Hewith various Debye lengths. For a given state, thesame set of parameters is used to examine the con-vergence for lower number of terms. Table I showsthat our results converge very well for various val-ues of D and for different bound S and P states. Weestimate the uncertainty of our final results, for themost part, is no more than a few parts in the lastquoted digit. The uncertainty for the results withsmall D values (D 3) is estimated at a few parts inthe second of the last quoted digit.

The ground 1s2 1S state eigenenergies of He(EHe(1s2 1S) for various Debye lengths are presentedin Table II, along with the He(1S) thresholdenergies (EHe(1S)) and the ionization potential[EHe(1S) � EHe(1s2 1S)] of the He atom. We alsopresent the results of the ground state energies ofHe for various Debye length D in Figure 1(a) and forvarious Debye parameters 1/D in Figure 1(b) alongwith the He(1S) threshold energies. It is evidentfrom Table II and Figure 1 that the ground-stateenergy increases with increasing plasma strength1/D. The ionization potential of He atom in Table IIis rapidly decreased with the increase of plasmastrength. The results presented in Table II and Fig-ure 1 are lower than those reported in the literature[4, 6, 7]. To estimate the ground 1s2 1S state eigenen-ergy of He for each Debye length, we have used thesame set of the nonlinear parameters with a � 2.84,b � 2.53, c � 4.2, and � � 1. We have also calculatedthe He(1S) threshold energies by diagonalizing the

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816 VOL. 106, NO. 4

atomic Hamiltonian with the standard Slater-typeorbital, and they are nicely comparable with theresults of Rogers et al. [1] and Roussel andO’Connell [2].

Next, we present the bound 1s2s 1S [EHe(1s2s1S)],1s2s 3S [EHe(1s2s3S)], 1s2p 1P [EHe(1s2p1P)], and 1s2p 3P(EHe(1s2p3P)) states energies in Table III. It is seen thatall the energies increase with increasing plasmastrength. The 1s2s 1S [EHe(1s2s1S)] and 1s2s 3S[EHe(1s2s3S)] states energies are presented for variousDebye length D in Figure 2(a) and for various De-bye parameters 1/D in Figure 2(b). Figure 3 pre-sents the 1s2p 1P [EHe(1s2p1P)] and 1s2p 3P [EHe(1s2p3P)]

state energies for various Debye length D [in Fig.3(a)] and for different values of 1/D (in Figure 3(b)).Both in Figures 2 and 3 we also show the He (1S)threshold energies. The behavior of the bound ex-cited S- and P-state energies of He are very similarto the 1s2 1S state He. The bound excited 1s2s 1,3Sand 1s2p 1,3P energies are very close to He (1S)threshold energies when D is 1 and 5, respec-tively. The optimum value of the nonlinear param-eters a for the 1s2s 1,3S states is 2.84 for all Debyelengths. The value of b is chosen from the interval[0.01, 2.53] for the 1s2s 1,3S states, whereas the valueof c is taken from the interval [0.01, 2.2] for the 1s2s

TABLE I ______________________________________________________________________________________________Convergence of the bound 1s2 1S, 1s2s 1,3S, and 1s2p 1,3P-state energies with the increase of basis terms fora fixed set of parameters for each Debye length.

States DBound-state

energies

N

500 600 700

1S � � EHe(1s1s1S) 2.90372437696 2.90372437699 2.90372437700� EHe(1s2s1S) 2.14597404468 2.14597404602 2.14597404604

20 � EHe(1s1s1S) 2.75654881108 2.75654881111 2.75654881112� EHe(1s2s1S) 2.00368072484 2.00368072807 2.00368072809

3 � EHe(1s1s1S) 2.01993692201 2.01993692203 2.01993692204� EHe(1s2s1S) 1.4096866 1.4096889 1.4096890

2 � EHe(1s1s1S) 1.65540131507 1.65540131509 1.65540131511� EHe(1s2s1S) 1.163694 1.163743 1.163745

1 � EHe(1s1s1S) 0.81821418276 0.81821418279 0.81821418280� EHe(1s2s1S) 0.59247 0.59253 0.59255

0.5 � EHe(1s1s1S) 0.05158 0.05163 0.05169

N

300 400 500

3S � � EHe(1s2s3S) 2.175229378231 2.175229378236 2.17522937823720 � EHe(1s2s3S) 2.032006503712 2.032006503717 2.0320065037183 � EHe(1s2s3S) 1.41575534482 1.41575534492 1.415755344942 � EHe(1s2s3S) 1.16368 1.16369 1.163751 � EHe(1s2s3S) 0.592471 0.592471 0.592475

N

400 500 600

1P � � EHe(1s2p1P) 2.12384308637 2.12384308645 2.1238430864720 � EHe(1s2p1P) 1.98143717096 1.98143717105 1.9814371710610 � EHe(1s2p1P) 1.85270354221 1.85270354254 1.852703542585 � EHe(1s2p1P) 1.631695939 1.631695961 1.631695964

3P � � EHe(1s2p3P) 2.13316419069 2.13316419074 2.1331641907620 � EHe(1s2p3P) 1.99021175592 1.99021175596 1.9902117559810 � EHe(1s2p3P) 1.86009784173 1.86009784182 1.860097841845 � EHe(1s2p3P) 1.6347490133 1.6347490150 1.6347490154



1S state and [0.01, 0.2] for the 1s2s,3S state. Thevalues of b and c are the same for Debye length upto D � 20 for the 1s2s 1S state (values 2.53 and 2.2,respectively) and D � 4 for the 1s2s,3S state (valuesare 2.53 and 0.20). For the 1s2s 1S state, the opti-mized values of b are 2.48, 2.42, 2.32, 2.32, 2.32, 2.32,2.02, 0.01, and 0.01 for D � 15, 10, 8, 6, 5, 4, 3, 2, and1, respectively. The optimized values of c are 2.1,1.9, 1.8, 1.7, 1.2, 1.0, 0.2, 0.01, and 0.01 for D � 15, 10,8, 6, 5, 4, 3, 2, and 1, respectively. The optimizedvalues of � is equal to 1 for the values of D up to 2and is equal to 0.5 for D � 1. For the 1s2s 3S state,the optimized values of b are 2.2, 0.01, and 0.01 for

D � 3, 2, and 1, respectively, and the optimizedvalues of c are 0.1, 0.01, and 0.01 for D � 3, 2, and1, respectively. The optimized values of � is equalto 1 for the values of D up to 3 and is equal to 0.5 forD � 2 and 1. The final results for the 1s2s 3S statewith D � 2 and D � 1, shown in Table III, arecalculated using 700-term basis function of Eq. (3)with � � 0.4, and the other parameters are the sameas those used in the 500-term basis functions. Theoptimum values of the nonlinear parameters a and� for the 1s2p 1,3P states are 2.07 and 1.0, respec-tively, for all Debye lengths. For the 1s2p 1,3P states,we use b � 2.81 and c � 0.61 for the values of D

TABLE II ______________________________________________________________________________________________Ground-state energy of He [EHe(1s2 1S)] for various Debye lengths along with the He� (1S) threshold energies[EHe�(1S)] and ionization potential [EHe�(1S) � EHe(1s2 1S)] of He.

D � EHe � (1S) � EHe(1s2 1S) EHe � (1S) � EHe(1s2 1S)

� 2.00000000000 2.90372437700 0.903724377002.90372437703a —

100 1.98007475170 2.87383879453 0.893764042832.8738306b —

70 1.97158091085 2.86110041616 0.8895195053150 1.96029802699 2.84418057552 0.8838825485340 1.95046490933 2.82943605013 0.8789711408030 1.93415761310 2.80498623696 0.8708286238620 1.90184477572 2.75654881112 0.85470403540

2.75655c —15 1.86992912020 2.70871996994 0.8387908497410 1.80726571410 2.61485294693 0.80758723283

2.61485c —8 1.76126804403 2.54598690844 0.784718864417 1.72892615283 2.49758558778 0.768659434956 1.68645744772 2.43405530526 0.747597857545 1.62823212245 2.34700618425 0.71877406180

2.34700c —4 1.54351488113 2.22046867051 0.676953789383 1.40903628503 2.01993692204 0.610900637012.5 1.30723404548 1.86845054569 0.56121650021

1.86845c —2 1.16367835009 1.65540131511 0.49172296502

1.6552679b —1 0.59246808756 0.81821418280 0.22574609524

0.8159995b —0.81821c —

0.5 0.04114315996 0.05169 0.01054684004�0.012209b —

0.05034c —0.45 0.00703048 0.0081 0.00106952

a Ref. [21].b Ref. [6].c Ref. [4].

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818 VOL. 106, NO. 4

FIGURE 1. Ground-state energy of helium in plasmas for various Debye length D in (a) and Debye parameters 1/Din (b). Dashed line denotes the He � (1S) threshold energy.

TABLE III _____________________________________________________________________________________________Bound 1s2s 1,3S and 1s2p 1,3P-state eigenenergies of helium for various Debye lengths along with the energysplitting between the 1s2s 3S and 1s2s 1S states.

D � EHe � (1S)

� EHe(1s2s1S)

(N � 700)� EHe(1s2s3S)

(N � 500)EHe(1s2s3S) �

EHe(1s2s1S)

� EHe(1s2p1P)

(N � 600)� EHe(1s2p3P)

(N � 600)

� 2.00000000000 2.14597404604 2.175229378237 �0.029255332197 2.12384308647 2.133164190762.14597404605a 2.175229378237a — 2.123843086498a 2.13316419078a

100 1.98007475170 2.11630001491 2.145513403147 �0.029213388237 2.09416329441 2.1034598077470 1.97158091085 2.10377784771 2.132948780257 �0.029170932547 2.08163531181 2.0909069136750 1.96029802699 2.08725861976 2.116351529880 �0.029092910120 2.06510552556 2.0743313156740 1.95046490933 2.07296680575 2.101972274301 �0.029005468551 2.05080212860 2.0599765436530 1.93415761310 2.04947574728 2.078297887213 �0.028822139933 2.02728777130 2.0363544022320 1.90184477572 2.00368072809 2.032006503718 �0.028325775628 1.98143717106 1.99021175598

— 2.0037b 2.0320b — 1.981216c 1.990202c

15 1.86992912020 1.95939361362 1.987067984541 �0.027674370921 1.9370952865 1.9454866052510 1.80726571410 1.87503634005 1.901012327825 �0.025975987775 1.8527035426 1.86009784184

— 1.8750b 1.9010b — — —8 1.76126804403 1.81520303624 1.839614107486 �0.024411071246 1.7929961705 1.799475928556 1.68645744772 1.72148066891 1.742841178045 �0.021360509135 1.7000667624 1.704769331815 1.62823212245 1.65149042284 1.670095755304 �0.018605332464 1.631695964 1.634749015

— 1.6509b 1.6703b — — —4 1.54351488113 1.55407966134 1.568177953014 �0.014098291674 — —3 1.40903628503 1.4096890123 1.41575534494 �0.00606633264 — —2 1.16367835009 1.163745 1.16384d �0.000095 1.157886c 1.158562c

1 0.59246808756 0.59255 0.59258d �0.00003 — —— 0.5649b 0.5747b — — —

a Best results [21].b Hashino et al. [4].c Dai et al. [10].d Using 700-term basis functions of Eq. (3).



ranging from infinity to 8. For D � 6 and 5, the setsof (b, c) values are (2.51, 0.36) and (2.46, 0.18), re-spectively. It is also interesting to note that a smallvariation on the nonlinear parameters in the neigh-borhood of the optimized values does not affectsignificantly on the best energies as shown in Ta-bles II and III.

Table III also presents the energy splitting be-tween the 1s2s 3S and 1s2s 1S states of He atom. Theenergies of the 1s2s 3S and 1s2s 1S states increasewith increasing plasma strength. The energy split-ting between these two electronic states are de-creasing with the decreasing values of D. The en-

ergy values of the 1s2s 1,3S states and the energysplitting between these two states are comparableto the reported results of Lopez et al. [22]. However,the results of Lopez et al. [22] for these two elec-tronic states have been reported without consider-ing the screening on electron–electron correlationterms. We have compared the available results ofHashino et al. [4] and Dai et al. [10] in Table III. Allour calculated energies presented in Table III arelower than those available in the literature, except atthe 1s2s 3S state energies for D � 5. In this case,Hashino et al. [4] obtained the bound 1s2s 3S stateenergy as �1.6703 a.u., whereas from our calcula-

FIGURE 2. Bound 1s2s 3S (solid line) and 1s2s 1S (dashed line) states energies of helium in plasmas for variousDebye length D in (a) and Debye parameters 1/D in (b). Dash-dotted line denotes the He � (1S) threshold energy.

FIGURE 3. Bound 1s2p 3P (solid line) and 1s2p 1P (dashed line) states energies of helium in plasmas for variousDebye length D in (a) and Debye parameters 1/D in (b). Dash-dotted line denotes the He � (1S) threshold energy.

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820 VOL. 106, NO. 4

tions, we have found this energy as �1.670095755304a.u. In Table III, it is seen that the estimated energyvalues by Hashino et al. [4] for the bound 1s2s 1,3Sstates with D � 1, and by Dai et al. [10] for the bound1s2p 1,3P states with D � 2, are higher than the He

(1S) threshold energies. However, for the D valuesmentioned above, we have not found in our presentwork any bound states with energies lying below therespective He (1S) threshold. We have also presentedthe He (1S) threshold energies in Table III for com-parison of the results. A “state” with which the en-ergy value [4, 10] is located above the He (1S) thresh-old may manifest itself as a shape resonance in e�–He scattering. For a resonance state, the energybecomes complex, with the real part is related to theresonance position and the imaginary part to thewidth. More work is needed to shed light on thesubject of a bound state becoming a shape resonancefor changing Debye lengths. Such investigations,however, are outside the scope of our present work.

Table IV presents the transition energies betweenthe 1s2p 1,3P, 1s2s 1,3S, and the ground state of Heatom under various screening conditions. Transi-tion energies between the 1s2p 1,3P state and theground state, and between the 1s2p 3P and 1s2s 3Sstates, are in decreasing trend with the increase ofplasma strength, whereas the transition energiesbetween the 1s2p 1P and 1s2s 1S states increase up tothe value of D around 10 and then decrease for D 10. The transition between the 1s2p 3P and 1s2 1Sstates is dipole forbidden. Such a transition mayoccur in the plasma environments and in fact it is avaluable tool for plasma modeling and plasma di-agnostics [11].

Summary and Conclusions

In the present work, we have calculated thebound 1s2 1S, 1s2s 1,3S, and 1s2p 1,3P state energiesof helium in plasmas for various Debye lengths,using a quasi-random process in the frameworkof Rayleigh–Ritz variation principle. A correlatedwave function consisting of a generalized exponen-tial expansion has been used. The ionization poten-tial of the helium atom, the energy splitting be-tween the 1s2s 3S and 1s2s 1S states, and transitionenergies between the 1s2s 1,3S and 1s2p 1,3P statesare also presented for various Debye lengths. Ourcalculated bound-state energies are lower thanthose available in the literature. We believe that ourfindings will provide useful information to the re-search communities of plasma physics, astrophys-ics, and atomic physics.

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TABLE IV _____________________________________________________________________________________________Transition energy between the ground states and some low-excited states of He.

DEHe(1s2p1P) �

EHe(1s1s1S)

EHe(1s2p3P) �EHe(1s1s1S)



� 0.77988129053 0.77056018624 0.02213095957 0.042065187477100 0.77967550012 0.77037898679 0.02213672050 0.04205359540770 0.77946510435 0.77019350249 0.02214253590 0.04204186658750 0.77907504996 0.76984925985 0.02215309420 0.04202021421040 0.77863392153 0.76945950648 0.02216467715 0.04199573065130 0.77769846566 0.76863183473 0.02218797598 0.04194348498320 0.77511164006 0.76633705514 0.02224355703 0.04179474773815 0.77162468344 0.76323336469 0.02229832712 0.04158137929110 0.76214940433 0.75475510509 0.02233279745 0.0409144859858 0.75299073794 0.74651097989 0.02220686574 0.0401381789366 0.73398854286 0.72928597345 0.02141390651 0.0380718462355 0.71531022025 0.71225716925 0.01979445884 0.035346740304



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KAR AND HO

822 VOL. 106, NO. 4

bound states of helium atom in dense plasmas

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