Finite element calculations for the helium atom∗ 1 Introduction

all states. The grid points are:
1s1s 1 S : 0.0, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 1.0, 1.2, 1.6,
2.0, 2.6, 3.2, 4.2, 6.0, 9.0, 15.0;
− 1.0, −0.6, −0.2, 0.2, 0.6, 1.0;
1s2s 1 S : 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0,
2.2, 2.4, 2.6, 3.0, 3.4, 3.8, 4.2, 4.8, 5.6, 8.0, 11.5, 15.0, 20.0;
− 1.0, −0.5, 0.5, 1.0;
1s2s 3 S : 0.0, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.4, 1.6,
1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0, 3.2, 3.4, 3.6, 4.0, 4.5, 5.5, 7.0, 10.0, 13.0,
16.0, 20.0, 25.0;
− 1.0, 0.0, 1.0;
1s2p 3 P : 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.2, 2.6, 3.0, 3.6, 5.0, 7.0,
10.0, 15.0, 20.0, 25.0;
− 1.0, 0.0, 1.0;
1s3d 3 D : 0.0, 0.2, 0.4, 0.6, 1.0, 1.6, 2.2, 3.0, 4.0, 6.0, 9.0, 13.0, 18.0, 24.0, 30.0;
− 1.0, 0.0, 1.0;
Table 1: Computational efforts.
state 1s1s 1 S 1s2s 1 S 1s2s 3 S 1s2p 3 P 1s3d 3 D
number of 57472 65320 68243 44954 36246
unknowns
Table 2: FEM results for the helium atom(a.u.).
state 1s1s 1 S 1s2s 1 S 1s2s 3 S 1s2p 3 P 1s3d 3 D
results in -2.90372437703411959 -2.1459740460544 -2.1752293782367 -2.133164190 -2.055636309453
references 83111594(4) [12] 188(21) [10] 913037(13) [10] 77927(1) [35] 261(4) [35]
this work -2.903724106 -2.1459740042 -2.1752293277 -2.1331633824 -2.05558078
Now, we show the behaviors of wave functions in a special case µ = 1(θ = 0 ◦ ). From
Figure 1-5, we can see that the domain where electrons appear frequently is considerably
small, so it is reasonable to solve the Schrödinger equation in bounded domains. Furthermore,
we can see that the figures of wave functions ψs, ψp constructed in (3.6) are
symmetric or antisymmetric according to singlets or triplets respectively. We didn’t add
this assumption a priori.
Acknowledgements: The authors are grateful to the anonymous referees for their
valuable comments and advices for our paper. The authors also thank professor Peizhu
Ding of Jilin University for discussing the problem with us and reading this paper.
References
[1] Ch.Froese-Fischer, T heHartree − F ock Method forAtoms, Wiley-Intersciece, New
York(1977).
[2] I.L.Hawk and D.L.Hardcastle, Comput. Phys. Commun. 16,159(1979).
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