Theoretical study of polarizabilities and hyperpolarizabilities of Be<sup>+</sup> ions and Li atoms

Wang Ting; Jiang Li; Wang Xia; Dong Chen-Zhong; Wu Zhong-Wen; Jiang Jun

doi:10.7498/aps.70.20201386

Abstract

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 2s→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:

Authors and contacts

College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China

Corresponding author: Jiang Jun, phyjiang@yeah.net

Funds: Project supported by the National Key R&D Program of China (Grant No. 2017YFA0402300) and the National Natural Science Foundation of China (Grant Nos. 11774292, 11804280, 11864036)

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References

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Cited By

表 1 Be⁺离子和Li原子的截断参数${\rho _{l, j}}$(单位: a.u.)

Table 1. Cut-off parameters${\rho _{l, j}}$of Be⁺ ions and Li atoms (in a.u.).

State	j	ρ_{l, j}
State	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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表 2 Be⁺离子和Li原子基态和部分低激发态相对于原子实的能级, 实验值(Exp.)是来自于NIST的数据(单位: a.u.)

Table 2. Energy levels of the ground state and some low-lying states of Be⁺ ions and Li atoms relative to atomic core. Experimental values (Exp.) are from the NIST data (in a.u.).

State	j	Be⁺		Li
State	j	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

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表 3 Be⁺离子基态和部分低激发态之间跃迁的振子强度, “Diff.”表示用RCICP方法计算的结果与NIST结果之差的百分比

Table 3. Oscillator strengths of transitions between the ground state and some low-lying states of Be⁺ ions. “Diff.” represents the difference in percentage form calculated by RCICP method and NIST results.

Transitions	RCICP	NIST^[42]	Theor.^[25]	Diff./%
2s_1/2→2p_1/2	0.16624	0.16596	0.1661	0.17
2s_1/2→2p_3/2	0.33258	0.33198	0.3322	0.18
2s_1/2→3p_1/2	0.02760	0.02768	0.0277	0.29
2s_1/2→3p_3/2	0.05517	0.05540	0.0553	0.42
2p_1/2→3s_1/2	0.06434	0.06438	0.0644	0.06
2p_3/2→3s_1/2	0.06436	0.06438	0.0644	0.03
2p_1/2→4s_1/2	0.01022	0.01039	0.0102	1.64
2p_3/2→4s_1/2	0.01022	0.01039	0.0102	1.64
2p_1/2→3d_3/2	0.6320	0.6320	0.6319	0.00
2p_3/2→3d_3/2	0.0632	0.0632	0.0632	0.00
2p_3/2→3d_5/2	0.5689	0.5689	0.5688	0.00
3s_1/2→3p_1/2	0.2768	0.2767	0.2767	0.04
3s_1/2→3p_3/2	0.5538	0.5535	0.5535	0.05
3p_1/2→3d_3/2	0.08069	0.08113	0.0811	0.54
3p_3/2→3d_3/2	0.08059	0.08103	0.081	0.54
3p_3/2→3d_5/2	0.07256	0.07294	0.073	0.52
3p_1/2→4s_1/2	0.1346	0.1347	0.1346	0.07
3p_3/2→4s_1/2	0.1346	0.1347	0.1346	0.07

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表 4 Li原子基态和部分低激发态之间跃迁的振子强度, “Diff.”表示用RCICP方法计算的结果与NIST结果之间差别的百分比

Table 4. Oscillator strengths of transitions between the ground state and some low-lying states of Li atoms. “Diff.” represents the difference in percentage form calculated by RCICP method and NIST results.

Transitions	RCICP	NIST^[42]	Theor.^[29]	Diff./%
2s_1/2→2p_1/2	0.24915	0.24899	0.2490	0.06
2s_1/2→2p_3/2	0.49832	0.49797	0.4981	0.07
2s_1/2→3p_1/2	0.00157	0.00157	0.0016	0.00
2s_1/2→3p_3/2	0.00313	0.00314	0.0032	0.32
2p_1/2→3s_1/2	0.11058	0.11050	0.1106	0.07
2p_3/2→3s_1/2	0.11059	0.11050	0.1106	0.08
2p_1/2→4s_1/2	0.01285	0.01283	0.0128	0.16
2p_3/2→4s_1/2	0.01285	0.01283	0.0128	0.16
2p_1/2→3d_3/2	0.63876	0.63858	0.6386	0.03
2p_3/2→3d_3/2	0.06388	0.06386	0.0639	0.03
2p_3/2→3d_5/2	0.57489	0.57472	0.5747	0.03
3s_1/2→3p_1/2	0.40512	0.4051	0.405	0.00
3s_1/2→3p_3/2	0.81027	0.8100	0.810	0.03
3p_1/2→3d_3/2	0.07397	0.0733	0.0744	0.91
3p_3/2→3d_3/2	0.00740	0.00736	0.0074	0.54
3p_3/2→3d_5/2	0.06657	0.0663	0.0669	0.41
3p_1/2→4s_1/2	0.22325	0.2230	0.2232	0.11
3p_3/2→4s_1/2	0.22325	0.2230	0.2232	0.11
3d_3/2→4p_1/2	0.01453	0.01497	0.015	2.94
3d_3/2→4p_3/2	0.00290	0.00299	0.003	3.01
3d_5/2→4p_3/2	0.01743	0.01796	0.018	2.95

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表 5 Be⁺离子基态的电偶极极化率$\alpha _{\rm{0}}^{\rm{1}}$和超极化率${\gamma _{\rm{0}}}$, “Diff.”表示用RCICP方法计算的γ₀与其它理论数据之间差别的百分比, 括号内的值表示不确定度

Table 5. Electric-dipole polarizability and hyperpolarizability of the ground state of Be⁺ ions. “Diff.” represents the difference of γ₀ in percentage form calculated by RCICP and other theoretical method. The values in parentheses indicate the uncertainties.

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

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表 6 中间态对Be⁺离子基态超极化率的贡献, RCICP^C表示2s→2p_j, 2p_j→3d_j跃迁的约化矩阵元用NIST^[42]结果替换之后计算的结果, 括号内的值表示RCICP相对于RCICP^C的不确定度(单位: a.u.)

Table 6. Contributions to the hyperpolarizability of the ground state of Be⁺ ions. RCICP^C represents that the reduced matrix elements of the 2s→2p_j、2p_j→3d_j transitions are replaced by NIST^[42] results. The values in parentheses indicate the uncertainties of RCICP relative to RCICP^C (in a.u.).

Contr.	RCICP	RCICP^C	RMBT^[25]
$\tfrac{1}{18}$T (s, p_1/2, s, p_1/2)	34.34(2)	34.32	32.605(53)
$-\tfrac{1}{18}$T (s, p_1/2, s, p_3/2)	68.68(5)	68.63	68.886(92)
$-\tfrac{1}{18}$T (s, p_3/2, s, p_1/2)	68.68(5)	68.63	68.886(92)
$\tfrac{1}{18}$T (s, p_3/2, s, p_3/2)	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, p_1/2, d_3/2, p_1/2)	202.75(16)	202.59	202.031(121)
$\tfrac{1}{18\sqrt{10} }$T (s, p_1/2, d_3/2, p_3/2)	40.55(4)	40.51	40.403(18)
$\tfrac{1}{18\sqrt{10} }$T (s, p_3/2, d_3/2, p_1/2)	40.55(4)	40.51	40.403(18)
$\tfrac{1}{180}$T (s, p_3/2, d_3/2, p_3/2)	8.11(1)	8.10	8.080(3)
$\tfrac{1}{30}$ T (s, p_3/2, d_5/2, p_3/2)	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)

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表 7 Li原子基态的电偶极极化率$\alpha _{\rm{0}}^{\rm{1}}$和超极化率${\gamma _{\rm{0}}}$, 括号内的值表示不确定度(单位: a.u.)

Table 7. Electric-dipole polarizability and hyperpolarizability of the ground state of Li atoms. The values in parentheses indicate the uncertainties (in a.u.).

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)

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表 8 中间态对Li原子基态超极化率的贡献, RCICP^C表示2s→2p_j, 2p_j→3d_j跃迁的约化矩阵元用NIST^[42]结果替换之后计算的结果, “Diff.”表示RCICP与RCICP^C之间差别的百分比, 括号内的值表示RCICP相对于RCICP^C的不确定度

Table 8. Contributions to the hyperpolarizability of the ground state of Li atoms. RCICP^C represents that the reduced matrix elements of 2s→2p_j, 2p_j→3d_j transitions are replaced by NIST^[42] results. “Diff.” represents the difference in percentage form between RCICP method and RCICP^C. The values in parentheses indicate the uncertainties of RCICP relative to RCICP^C.

Contr.	RCICP/a.u.	RCICP^C/a.u.	Diff. /%
$ \frac{1}{18} $T (s, p_1/2, s, p_1/2)	8314(2)	8312	0.03
$ -\frac{1}{18} $T (s, p_1/2, s, p_3/2)	16629(5)	16624	0.03
$ -\frac{1}{18} $T (s, p_3/2, s, p_1/2)	16629(5)	16624	0.03
$ \frac{1}{18} $T (s, p_3/2, s, p_3/2)	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, p_1/2, d_3/2, p_1/2)	33812(13)	33799	0.04
$ \frac{1}{18\sqrt{10}} $T (s, p_1/2, d_3/2, p_3/2)	6762(3)	6759	0.04
$ \frac{1}{18\sqrt{10}} $T (s, p_3/2, d_3/2, p_1/2)	6762(3)	6759	0.04
$ \frac{1}{180} $T (s, p_3/2, d_3/2, p_3/2)	1352(0)	1352	0.00
$ \frac{1}{30} $ T (s, p_3/2, d_5/2, p_3/2)	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

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[1]	Kassimi E B, Thakkar A J 1994 Phys. Rev. A 50 2948 Google Scholar
[2]	Dür W, Briegel H J 2002 Phys. Rev. Lett. 90 067901 Google Scholar
[3]	Childress L, Taylor J M, Sørensen A S, Lukin M D 2005 Phys. Rev. A 72 052330 Google Scholar
[4]	Jiang L, Taylor J M, Sørensen A S, Lukin M D 2007 Phys. Rev. A 76 062323 Google Scholar
[5]	Gorshkov A V, Rey A M, Daley A J, Boyd M M, Ye J, Zoller P, Lukin M D 2009 Phys. Rev. Lett. 102 110503 Google Scholar
[6]	Wineland D J, Drullinger R E, Walls F L 1978 Phys. Rev. Lett. 40 1639 Google Scholar
[7]	Neuhauser W, Hohenstatt M, Toschek P E, Dehmelt H 1978 Phys. Rev. Lett. 41 233 Google Scholar
[8]	Flury J 2016 J. Phys. Conf. Ser. 723 012051 Google Scholar
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Theoretical study of polarizabilities and hyperpolarizabilities of Be⁺ ions and Li atoms

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Theoretical study of polarizabilities and hyperpolarizabilities of Be+ ions and Li atoms

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Theoretical study of polarizabilities and hyperpolarizabilities of Be⁺ ions and Li atoms