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The wave functions, energy levels, and oscillator strengths of Be +ions and Li atoms are calculated by using a relativistic potential model, which is named the relativistic configuration interaction plus core polarization method (RCICP). The calculated energy levels in this work are in good agreement with experimental levels tabulated in NIST Atomic Spectra Database, and the difference appears in the sixth digit after the decimal point. The present oscillator strengths are in good agreement with the existing theoretical and experimental results. By means of these energy levels and oscillator strengths, the electric-dipole static polarizabilities and hyperpolarizabilities of the ground states are determined. The contributions of different intermediate states to the hyperpolarizabilities of the ground state are further discussed. For Be +ions, the present electric-dipole polarizability and hyperpolarizability are in good agreement with the results calculated by Hartree-Fock plus core polarization method, the finite field method and relativistic many-body method. The largest contribution to the hyperpolarizability is the term of
$\alpha _{\text{0}}^{\text{1}}{\beta _0}$ . For Li atoms, the present electric-dipole polarizability is in good agreement with the available theoretical and experimental results. However, the present hyperpolarizability is different from the other theoretical results significantly. Moreover, the hyperpolarizabilities calculated by different theoretical methods are quite different. The biggest difference is more than one order of magnitude. In order to explain the reason for these differences, we analyze the contributions of different intermediate states to the hyperpolarizability in detail. It is found that the sum of the contributions of the 2s→ np j$\left( {n \geqslant 3} \right)$ and np j→ nd j$\left( {n \geqslant 3} \right)$ to hyperpolarizability is approximately equal to that term of$\alpha _{\text{0}}^{\text{1}}{\beta _0}$ . The total hyperpolarizability, which is the difference between the sum of the contributions of the 2 s→ np j$\left( {n \geqslant 3} \right)$ and np j→ nd j$\left( {n \geqslant 3} \right)$ to hyperpolarizability and$\alpha _{\text{0}}^{\text{1}}{\beta _0}$ , is relatively small. Consequently, this difference magnifies the calculated error. If the uncertainties of the transition matrix elements are less than 0.1%, the uncertainty of hyperpolarizability is more than 100%. Therefore, the differences of hyperpolarizabilities for the ground state of Li atoms, calculated by various theoretical methods, are more than 100% or one order of magnitude.-
Keywords:
- electric-dipole polarizability/
- hyperpolarizability/
- Be+ions/
- Li atoms/
- the relativistic model potential method
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State j ρl,j Be+ Li 2s 1/2 0.9552 1.40880 2p 1/2 0.8789 1.28466 3/2 0.8775 1.28396 3d 3/2 0.1287 2.324 5/2 0.1284 2.330 
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State j Be+ Li RCICP Expt.[42] RCICP Expt.[42] 2s 1/2 –0.66924767 –0.66924755 –0.1981419 –0.1981419 2p 1/2 –0.52376962 –0.52376949 –0.1302358 –0.1302358 3/2 –0.52373967 –0.52373953 –0.1302343 –0.1302343 3s 1/2 –0.26719384 –0.26723337 –0.0741684 –0.0741817 3p 1/2 –0.22954214 –0.22958234 –0.0572264 –0.0572354 3/2 –0.22953331 –0.22957356 –0.0572260 –0.0572354 3d 3/2 –0.22247809 –0.22247805 –0.0556055 –0.0556057 5/2 –0.22247565 –0.22247565 –0.0556051 –0.0556055 4s 1/2 –0.14313397 –0.14315285 –0.0386096 –0.0386151 4p 1/2 –0.12811380 –0.12813485 –0.0319693 –0.0319744 3/2 –0.12811009 –0.12813115 –0.0319691 –0.0319744 4d 3/2 –0.12512357 –0.12512455 –0.0308153 –0.0312735 5/2 –0.12512257 –0.12512345 –0.0308152 –0.0312734 5s 1/2 –0.08905659 –0.08906605 –0.0236202 –0.0236365 5p 1/2 –0.08159826 –0.08160960 –0.0203583 –0.0203739 3/2 –0.08159637 –0.08160765 –0.0203583 –0.0203739 5d 3/2 –0.08006698 –0.08006725 –0.0124153 –0.0200122 5/2 –0.08006648 –0.08006670 –0.0124152 –0.0200122 下载:
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Transitions RCICP NIST[42] Theor.[25] Diff./% 2s1/2→2p1/2 0.16624 0.16596 0.1661 0.17 2s1/2→2p3/2 0.33258 0.33198 0.3322 0.18 2s1/2→3p1/2 0.02760 0.02768 0.0277 0.29 2s1/2→3p3/2 0.05517 0.05540 0.0553 0.42 2p1/2→3s1/2 0.06434 0.06438 0.0644 0.06 2p3/2→3s1/2 0.06436 0.06438 0.0644 0.03 2p1/2→4s1/2 0.01022 0.01039 0.0102 1.64 2p3/2→4s1/2 0.01022 0.01039 0.0102 1.64 2p1/2→3d3/2 0.6320 0.6320 0.6319 0.00 2p3/2→3d3/2 0.0632 0.0632 0.0632 0.00 2p3/2→3d5/2 0.5689 0.5689 0.5688 0.00 3s1/2→3p1/2 0.2768 0.2767 0.2767 0.04 3s1/2→3p3/2 0.5538 0.5535 0.5535 0.05 3p1/2→3d3/2 0.08069 0.08113 0.0811 0.54 3p3/2→3d3/2 0.08059 0.08103 0.081 0.54 3p3/2→3d5/2 0.07256 0.07294 0.073 0.52 3p1/2→4s1/2 0.1346 0.1347 0.1346 0.07 3p3/2→4s1/2 0.1346 0.1347 0.1346 0.07 下载:
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Transitions RCICP NIST[42] Theor.[29] Diff./% 2s1/2→2p1/2 0.24915 0.24899 0.2490 0.06 2s1/2→2p3/2 0.49832 0.49797 0.4981 0.07 2s1/2→3p1/2 0.00157 0.00157 0.0016 0.00 2s1/2→3p3/2 0.00313 0.00314 0.0032 0.32 2p1/2→3s1/2 0.11058 0.11050 0.1106 0.07 2p3/2→3s1/2 0.11059 0.11050 0.1106 0.08 2p1/2→4s1/2 0.01285 0.01283 0.0128 0.16 2p3/2→4s1/2 0.01285 0.01283 0.0128 0.16 2p1/2→3d3/2 0.63876 0.63858 0.6386 0.03 2p3/2→3d3/2 0.06388 0.06386 0.0639 0.03 2p3/2→3d5/2 0.57489 0.57472 0.5747 0.03 3s1/2→3p1/2 0.40512 0.4051 0.405 0.00 3s1/2→3p3/2 0.81027 0.8100 0.810 0.03 3p1/2→3d3/2 0.07397 0.0733 0.0744 0.91 3p3/2→3d3/2 0.00740 0.00736 0.0074 0.54 3p3/2→3d5/2 0.06657 0.0663 0.0669 0.41 3p1/2→4s1/2 0.22325 0.2230 0.2232 0.11 3p3/2→4s1/2 0.22325 0.2230 0.2232 0.11 3d3/2→4p1/2 0.01453 0.01497 0.015 2.94 3d3/2→4p3/2 0.00290 0.00299 0.003 3.01 3d5/2→4p3/2 0.01743 0.01796 0.018 2.95 下载:
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Method $\alpha _{\rm{0}}^{\rm{1}}$/a.u. γ0/a.u. Diff./% RCICP 24.504(32) –11529.971(84) Coulomb approximation[43] 24.77 Variation-perturbation Hylleraas CI[44] 24.5 Hylleraas[24] 24.489 Asymptotic correct wave function[45] 24.91 Variation-perturbation FCCI[46,47] 24.495 Hartree-Fock plus core polarization[22] 24.493 –11511 0.16 Hylleraas[22] 24.4966(1) –11521.30(3) 0.08 Relativistic many-body calculation[25] 24.483(4) –11496(6) 0.29 The finite field method[30] 24.5661 –11702.31 1.49 下载:
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Contr. RCICP RCICPC RMBT[25] $\tfrac{1}{18}$T(s, p1/2, s, p1/2) 34.34(2) 34.32 32.605(53) $-\tfrac{1}{18}$T(s, p1/2, s, p3/2) 68.68(5) 68.63 68.886(92) $-\tfrac{1}{18}$T(s, p3/2, s, p1/2) 68.68(5) 68.63 68.886(92) $\tfrac{1}{18}$T(s, p3/2, s, p3/2) 137.35(10) 137.25 137.669(109) $T({\rm{s, }}{{\rm{p}}_{j'}}, {\rm{ s}}, {\rm{ }}{{\rm{p}}_{j''}})$ 308.04(12) 308.83 308.046(178) $\tfrac{1}{18}$T(s, p1/2, d3/2, p1/2) 202.75(16) 202.59 202.031(121) $\tfrac{1}{18\sqrt{10} }$T(s, p1/2, d3/2, p3/2) 40.55(4) 40.51 40.403(18) $\tfrac{1}{18\sqrt{10} }$T(s, p3/2, d3/2, p1/2) 40.55(4) 40.51 40.403(18) $\tfrac{1}{180}$T(s, p3/2, d3/2, p3/2) 8.11(1) 8.10 8.080(3) $\tfrac{1}{30}$T(s, p3/2, d5/2, p3/2) 437.85(40) 437.45 438.434(148) $T({\rm{s}}, {{\rm{p}}_{j'}}, {{\rm{d}}_j}, {{\rm{p}}_{j''}})$ 729.79(43) 729.17 729.351(192) $\alpha _{\rm{0}}^{\rm{1}}{\beta _0}$ 1999.67(6.95) 1992.72 1995.743(382) γ0(2 s) –11529(84) –11456 –11496(6) 下载:
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Method $\alpha _{\rm{0}}^{\rm{1}}$ γ0 RCICP 164.05(8) 1920(3264) The coupled cluster (all single, double and triple substitution)[1] 164.19 2880 Finite-field quadratic configuration interaction[1] 164.32 1020 Hylleraas[31] 164.112(1) 3060(40) The relativistic coupled-cluster method[48] 164.23 Relativistic variation perturbation[49] 164.084 Relativistic all-order methods[29] 164.16(5) Variation perturbation[33] 164.10 3000 Semiempirical pseudopotentials[26] 164.08 65000 Frozen core Hamiltonian with a semiempirical polarization potential[50] 164.21 Finite-field fourth-order many-body perturbation theory[34] 164.5 4300 Configuration interaction[35] 164.9 37000 Relativistic ab initio methods[51] 164.0(1) The restricted Hartree-Fock[32] 170.1 –55000 The Rydberg-Klein-Rees inversion method with the quantum defect theory[52] 164.14 3390 Exp.[53] 164(3) Exp.[54] 164.2(11) 下载:
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Contr. RCICP/a.u. RCICPC/a.u. Diff. /% $ \frac{1}{18} $T(s, p1/2, s, p1/2) 8314(2) 8312 0.03 $ -\frac{1}{18} $T(s, p1/2, s, p3/2) 16629(5) 16624 0.03 $ -\frac{1}{18} $T(s, p3/2, s, p1/2) 16629(5) 16624 0.03 $ \frac{1}{18} $T(s, p3/2, s, p3/2) 33259(11) 33248 0.03 $T({\rm{s}}, {{\rm{p}}_{j'}}, {\rm{s}}, {{\rm{p}}_{j''}})$ 74833(13) 74809 0.02 $ \frac{1}{18} $T(s, p1/2, d3/2, p1/2) 33812(13) 33799 0.04 $ \frac{1}{18\sqrt{10}} $T(s, p1/2, d3/2, p3/2) 6762(3) 6759 0.04 $ \frac{1}{18\sqrt{10}} $T(s, p3/2, d3/2, p1/2) 6762(3) 6759 0.04 $ \frac{1}{180} $T(s, p3/2, d3/2, p3/2) 1352(0) 1352 0.00 $ \frac{1}{30} $T(s, p3/2, d5/2, p3/2) 73033(40) 72993 0.05 $T({\rm{s}}, {{\rm{p}}_{j'}}, {{\rm{d}}_j}, {{\rm{p}}_{j''}})$ 121723(42) 121661 0.03 $\alpha _{\rm{0}}^{\rm{1}}{\beta _0}$ 196396(268) 196128 0.14 γ0(2 s) 1920(3264) 4109 170 下载:
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