Geometric quantities of lower doubly excited bound states of helium

Chengdong Zhou(周成栋)^{1}, Yuewu Yu(余岳武)^{1}, Sanjiang Yang(杨三江)^{2,†}, and Haoxue Qiao(乔豪学)^{1,‡}

1 School of Physics and Technology, Wuhan University, Wuhan 430072, China; 2 College of Physics and Electronic Science, Hubei Normal University, Huangshi 435002, China

Abstract Expectation values of single electron and interelectronic geometric quantities such as $\langle r\rangle$, $\langle r_{12}\rangle$, $\langle r_<\rangle$, $\langle r_>\rangle$, $\langle \cos\theta_{12}\rangle$ and $\langle \theta_{12}\rangle$ are calculated for doubly excited $2{\rm p}n{\rm p}\,{}^1P^{\,\rm e}\,(3\leq n\leq5),\, 2{\rm p}n{\rm p}\,{}^3\!P^{\,\rm e}\,(2\leq n\leq5)$ and $2{\rm p}n{\rm d}\,{}^{1,3}D^{\,\rm o}\,(3\leq n\leq5)$ states of helium using Hylleraas-$B$-spline basis set. The energy levels converge to at least 10 significant digits in our calculations. The extrapolated values of geometric quantities except for $\langle \theta_{12}\rangle$ reach 10 significant digits as well; $\langle \theta_{12}\rangle$ reaches at least 7 significant digits using a multipole expansion approach. Our results provide a precise reference for future research.

(Calculations and mathematical techniques in atomic and molecular physics)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 12074295). The authors thank Yongbo Tang for the meaningful discussion. The numerical calculations in this article were performed on the supercomputing system in the Supercomputing Center of Wuhan University.

Chengdong Zhou(周成栋), Yuewu Yu(余岳武), Sanjiang Yang(杨三江), and Haoxue Qiao(乔豪学) Geometric quantities of lower doubly excited bound states of helium 2022 Chin. Phys. B 31 030301

[1] Compton K T and Boyce J C 1928 J. Franklin Inst.205 497 [2] Kruger P G 1930 Phys. Rev.36 855 [3] Wu T Y 1944 Phys. Rev.66 291 [4] Hilger R, Merckens H P and Kleindienst H 1996 Chem. Phys. Lett.262 400 [5] Ho Y K and Bhatia A K 1993 Phys. Rev. A47 2628 [6] Ho Y K and Bhatia A K 1997 J. Phys. B:At. Mol. Opt. Phys.30 3597 [7] Saha J K and Mukherjee T K 2009 Phys. Rev. A80 022513 [8] Kar S and Ho Y K 2010 Phys. Rev. A82 036501 [9] Eiglsperger J, Piraux B and Madroñero J 2010 Phys. Rev. A81 042527 [10] Eiglsperger J, Piraux B and Madroñero J 2010 Phys. Rev. A81 042528 [11] Bylicki M and Nicolaides C A 2000 Phys. Rev. A61 052508 [12] Bylicki M and Nicolaides C A 2000 Phys. Rev. A61 052509 [13] Bürgers A and Rost J M 1996 J. Phys. B:At. Mol. Opt. Phys.29 3825 [14] Ordóñez-Lasso A F, Cardona J C and Sanz-Vicario J L 2013 Phys. Rev. A88 012702 [15] Koga T 2004 J. Chem. Phys.121 3939 [16] Koga T 2002 Chem. Phys. Lett.363 598 [17] Koga T, Matsuyama H and Thakkar A J 2011 Chem. Phys. Lett.512 287 [18] Matsuyama H and Koga T 2007 Theor. Chem. Acc.118 643 [19] Jiao L G, Zan L R, Zhu L and Ho Y K 2018 Comput. Theor. Chem.1135 1 [20] Jiao L G, Zan L R, Zhu L, Zhang Y Z and Ho Y K 2019 Phys. Rev. A100 022509 [21] Yang S J, Mei X S, Shi T Y and Qiao H X 2017 Phys. Rev. A95 062505 [22] Wu F F, Yang S J, Zhang Y H, Zhang J Y, Qiao H X, Shi T Y and Tang L Y 2018 Phys. Rev. A98 040501 [23] Yang S J, Tang Y B, Zhao Y H, Shi T Y and Qiao H X 2019 Phys. Rev. A100 042509 [24] Gradshteyn I S and Ryzhik I M 2000 Table of Integrals, Series, and Products, 6th edn. (New York:Elsevier) p. 801 [25] Kar S and Ho Y K 2009 Phys. Rev. A79 062508 [26] Bhattacharyya S, Mukherjee T K, Saha J K and Mukherjee P K 2008 Phys. Rev. A78 032505

Altmetric calculates a score based on the online attention an article receives. Each coloured thread in the circle represents a different type of online attention. The number in the centre is the Altmetric score. Social media and mainstream news media are the main sources that calculate the score. Reference managers such as Mendeley are also tracked but do not contribute to the score. Older articles often score higher because they have had more time to get noticed. To account for this, Altmetric has included the context data for other articles of a similar age.