Geometric structure parameters of ground and singly excited states of helium

Yuewu Yu; Chengdong Zhou; Haoxue Qiao

doi:10.1140/epjd/s10053-021-00317-y

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2024 Impact factor 1.5

Atomic, Molecular, Optical and Plasma Physics

Eur. Phys. J. D (2022) 76: 26
https://doi.org/10.1140/epjd/s10053-021-00317-y

Regular Article – Atomic Physics

Geometric structure parameters of ground and singly excited states of helium

Yuewu Yu, Chengdong Zhou and Haoxue Qiao^c

School of physics and technology, Wuhan University, 430072, Wuhan, China

^c qhx@whu.edu.cn

Received: 25 August 2021
Accepted: 30 November 2021
Published online: 11 February 2022

Abstract

Expectation values of geometric structure parameters $δ (r), δ (r_{12}), \frac{1}{r}, r, r^{2}, \frac{1}{r_{12}}, r_{12}, r_{12}^{2}, r_{<}, r_{>}, cos θ_{12}, θ_{12}$ $\delta (\mathbf{r} ),\;\delta (\mathbf{r} _{12}),\;\frac{1}{r},\;r,\;r^2,\;\frac{1}{r_{12}},\;r_{12},\;r_{12}^2 ,\;r_<, r_>,\;\cos {\theta _{12}},\;\theta _{12}$ for ground state, singlet and triplet singly excited $1 n 2 s^{1, 3} S^{e}, 1 s n p^{1, 3} P^{o}$ $1n2s\;^{1,3}S^e,1snp\;^{1,3}P^o$ states with n up to 5 of helium were calculated in detail with Hylleraas-B-spline basis. Our results of $δ (r)$ $\delta (\mathbf{r} )$ and $δ (r_{12})$ $\delta (\mathbf{r} _{12})$ for helium ground state have 14 and 6 significant digits, respectively, and have at least 9 significant digits for other parameters. Expectation values of $δ (r), \frac{1}{r}, \frac{1}{r_{12}}$ $\delta (\mathbf{r} ),\;\frac{1}{r},\;\frac{1}{r_{12}}$ have at least 9 significant digits and expectation values of $r_{<}, r_{>}$ $r_<,\;r_>$ reach 8 significant digits for singly excited S and P states. Expectation values of $cos θ_{12}, θ_{12}$ $\cos {\theta _{12}},\;\theta _{12}$ 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.