Abstract
Expectation values of geometric structure parameters \(\delta (\mathbf{r} ),\;\delta (\mathbf{r} _{12}),\;\frac{1}{r},\;r,\;r^2,\;\frac{1}{r_{12}},\;r_{12},\;r_{12}^2 ,\;r_<, r_>,\;\cos {\theta _{12}},\;\theta _{12}\) for ground state, singlet and triplet singly excited \(1n2s\;^{1,3}S^e,1snp\;^{1,3}P^o\) states with n up to 5 of helium were calculated in detail with Hylleraas-B-spline basis. Our results of \(\delta (\mathbf{r} )\) and \(\delta (\mathbf{r} _{12})\) for helium ground state have 14 and 6 significant digits, respectively, and have at least 9 significant digits for other parameters. Expectation values of \(\delta (\mathbf{r} ),\;\frac{1}{r},\;\frac{1}{r_{12}}\) have at least 9 significant digits and expectation values of \(r_<,\;r_>\) reach 8 significant digits for singly excited S and P states. Expectation values of \(\cos {\theta _{12}},\;\theta _{12}\) are in coincidence with data in the previous work. Our results from independent calculation with Hylleraas-B-spline method are in good agreement with the previous works.
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This manuscript has no associated data or the data will not be deposited. [Authors’ comment: There is no separate data, all data is provided in the paper.]
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 12074295). The numerical calculations in this article have been done on the supercomputing system in the Supercomputing Center of Wuhan University.
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The first author Yu confirms contribution to the papers as follows: theoretical derivation, program writing, analysis and interpretation of results and paper writing. The second author Zhou confirms contribution to the papers as follows: program correction and modification, data collection, analysis and interpretation of results. The corresponding author Qiao confirms contribution to the papers as follows: ideal design and source of computing resources. The contribution of Yu is the most important part of the present work. All authors reviewed the results and approved the final version of the manuscript.
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Yu, Y., Zhou, C. & Qiao, H. Geometric structure parameters of ground and singly excited states of helium. Eur. Phys. J. D 76, 26 (2022). https://doi.org/10.1140/epjd/s10053-021-00317-y
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DOI: https://doi.org/10.1140/epjd/s10053-021-00317-y