Review of the hyperfine structure theory of hydrogen molecular ions

Zhong Zhen-Xiang

doi:10.7498/aps.73.20241101

Abstract

The study of high-precision spectroscopy for hydrogen molecular ions enables the determination of fundamental constants, such as the proton-to-electron mass ratio, the deuteron-to-electron mass ratio, the Rydberg constant, and the charge radii of proton and deuteron. This can be accomplished through a combination of high precision experimental measurements and theoretical calculations. The spectroscopy of hydrogen molecular ions reveals abundant hyperfine splittings, necessitating not only an understanding of rovibrational transition frequencies but also a thorough grasp of hyperfine structure theory to extract meaningful physical information from the spectra. This article reviews the history of experiments and theories related to the spectroscopy of hydrogen molecular ions, with a particular focus on the theory of hyperfine structure. As far back as the second half of the last century, the hyperfine structure of hydrogen molecular ions was described by a comprehensive theory based on its leading-order term, known as the Breit-Pauli Hamiltonian. Thanks to the advancements in non-relativistic quantum electrodynamics (NRQED) at the beginning of this century, a systematic development of next-to-leading-order theory for hyperfine structure has been achieved and applied to $\text{H}_2^+$ and $\text{HD}^+$ in recent years, including the establishment of the $m\alpha^7\ln(\alpha)$ order correction. For the hyperfine structure of $\text{H}_2^+$, theoretical calculations show good agreement with experimental measurements after decades of work. However, for HD⁺, discrepancies have been observed between measurements and theoretical predictions that cannot be accounted for by the theoretical uncertainty in the non-logarithmic term of the $m\alpha^7$ order correction. To address this issue, additional experimental measurements are needed for mutual validation, as well as independent tests of the theory, particularly regarding the non-logarithmic term of the $m\alpha^7$ order correction.

Keywords:

hydrogen molecular ions /
hyperfine structure /
quantum electrodynamic (QED) corrections /
spin-orbit and spin-spin interactions

Authors and contacts

Zhong Zhen-Xiang^1,2

1.
Theoretical Physics Research Center, School of Physics and Optoelectronic Engineering, Hainan University, Haikou 570228, China
2.
Department of Theory and Interdisciplinary Research, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China

Corresponding author: Zhong Zhen-Xiang, zxzhong@hainanu.edu.cn

Funds: Project supported by the Key Program of the National Natural Science Foundation of China (Grant No. 12393821) and the National Key Research and Development Program of China (Grant No. 2021YFA1402103).

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Cited By

图 1 氢分子离子$ \text{H}_2^+ $振转态$ (v, L=1, 3) $的超精细劈裂示意图

Figure 1. Hyperfine structure of hydrogen molecular ion $ \text{H}_2^+ $ rovibrational states $ (v, L=1, 3) $.

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图 2 氢分子离子HD⁺振转态$ (v, L\geqslant2) $的超精细劈裂示意图

Figure 2. Hyperfine structure of hydrogen molecular ion HD⁺ rovibrational states $ (v, L\geqslant2) $.

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表 1 通过氢分子离子精密光谱测定质子-电子质量比

Table 1. Determination of the proton-electron mass ratio through precision spectroscopy of hydrogen molecular ions.

年份	作者(研究机构)	跃迁$ (v, L)\rightarrow (v', L') $	所测频率/MHz	理论频率/MHz	质量比
HD⁺
2007	Koelemeij et al. (HHU)^[16]	$ (0, 2)\rightarrow (4, 3) $	214978560.6(0.5)	214978560.88(7)^[62]	–
2016	Biesheuvel et al. (VU)^[18]	$ (0, 2)\rightarrow (8, 3) $	383407177.38(41)	383407177.150(15)^[20]	1836.1526695(53)
2018	Alighanbari et al. (HHU)^[63]	$ (0, 0)_{J=2}\rightarrow (0, 1)_{J'=3} $	1314935.8280(4)(3)^a	1314935.8273(10)^b	1836.1526739(24)
2020	Alighanbari et al. (HHU)^[23]	$ (0, 0)\rightarrow (0, 1) $	1314925.752910(17)	1314925.752896(18)(61)^b,c	1836.152673449(24) (25)(13)^d
2020	Patra et al. (VU)^[24]	$ (4, 2)\rightarrow (9, 3) $	415264925.5005(12)	415264925.4962(74)^b	1836.152673406(38)
2021	Kortunov et al. (HHU)^[64]	$ (0, 0)\rightarrow (1, 1) $	58605052.16424(16)(85)^e	58605052.1639(5)(13)^b,c	1836.152673384(11) (31)(55)(12)^f
2023	Alighanbari et al. (HHU)^[17]	$ (0, 0)\rightarrow (5, 1) $	259762971.0512(6) (0.00004)^e	259762971.05091^[41]	1836.152673463 (10)(35)(1)(6)^f
$ \text{H}_2^+ $
2024	Schenkel et al. (HHU)^[65]	$ (1, 0)\rightarrow (3, 2) $	124487032.7(1.5)	124487032.45(6) ^[40]	1836.152665(53)
CODATA推荐值
2014	CODATA group^[66]	CODATA 2014			1836.15267389(17)
2018	CODATA group^[22]	CODATA 2018			1836.15267343(11)
2022	CODATA group^g	CODATA 2022			1836.152673426(32)
注: ^a第一个误差为统计误差, 第二个为系统误差; ^b来自Korobov, 是该实验文章的共同作者; ^c第一个误差来自理论, 第二个来自CODATAk 2018基本物理常数; ^d第一个误差来自实验, 第二个来自理论, 第三个来自CODATAk 2018基本物理常数; ^e第一个误差来自实验, 第二个来自超精细结构理论; ^f第一个误差来自实验, 第二个来自QED理论, 第三个来自超精细结构理论, 第四个来自CODATA 2018基本物理常数; ^g来自NIST的基本物理常数表https://physics.nist.gov/cuu/Constants/index.html.

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表 2 基于HD⁺中不同的振转跃迁测量定出的质量比组合$ \mathcal{R}$之比较^[17]

Table 2. Comparison of the combined mass ratio $\mathcal{R}$ determined from the measurements of different rovibrational transitions in HD⁺^[17].

年份	振转跃迁	$ \mathcal{R} $	相对误差	引文
2020	$ (0, 0)\rightarrow (0, 1) $	1223.899228658(23)	$ 1.9\times10^{-11} $	HHU^[23]
2020	$ (0, 3)\rightarrow (9, 3) $	1223.899228735(28)	$ 2.3\times10^{-11} $	VU^[67]
2021	$ (0, 0)\rightarrow (0, 1) $	1223.899228711(22)	$ 1.8\times10^{-11} $	HHU^[64]
2023	$ (0, 0)\rightarrow (5, 1) $	1223.899228720(25)	$ 2.0\times10^{-11} $	HHU^[17]
—		1223.899228642(37)	$ 3.0\times10^{-11} $	潘宁阱^[25,26,68]
—		1223.899228723(56)	$ 4.8\times10^{-11} $	CODATA 2018^[22]

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表 3 $ \text{H}_2^+ $等效哈密顿量(9)中的超精细劈裂系数, 单位为kHz. 每个振转态的第一行是理论值, 随后是实验值. d₁和d₂需要乘以因子$ 3(2 L-1)(2 L+3) $才能与文献[51]中的值匹配

Table 3. The hyperfine splitting coefficients in the effective Hamiltonian of $ \text{H}_2^+ $, as appeared in Eq. (9), in units of kHz. The first line of each rovibrational state is for the theoretical values, followed by the experimental ones. d₁ and d₂ need to be multiplied by $ 3(2 L-1)(2 L+3) $ to match the values in Ref. [51].

L	v	b_F	c_e	c_I	d₁	d₂
1	0	922 930.1(9)^a	42 417.32(15)^b	–41.673^d	8566.174(17)^b	–19.837^d
		922 940(20)^[53]	42 348(29)^[53]	–3(15)^[53]	8550.6(1.7)^[53]
		923.16(21)^[54]
1	4	836 728.7(8)^a	32 655.32(11)^b	–35.826^c	6537.386(13)^b	–16.414^c
		836 729.2(8)^[52]	32 636^[52]	–34(1.5)^e	6535.6^[52]
1	5	819 226.7(8)^a	30 437.80(11)^b	–34.148^c	6080.400(12)^b	–15.531^c
		819 227.3(8)^[52]	30 421^[52]	–33(1.5)^e	6078.7^[52]
1	6	803 174.5(7)^a	28 280.95(10)^b	–32.385^c	5637.627(11)^b	–14.633^c
		803 175.1(8)^[52]	28 266^[52]	–31(1.5)^e	5636.0^[52]
1	7	788 507.5(7)^a
		788 507.9(8)^[52]	26 156^[52]	–29(1.5)^e	5204.9^[52]
1	8	775 171.2(7)^a
		775 172.0(8)^[52]	24 080^[52]	–27(1.5)^e	4782.2^[52]
注: ^a包含高阶修正的贡献, 来自文献[49]; ^b包含高阶修正的贡献, 来自文献[59]; ^c仅计算领头项$ m\alpha^4 $阶的Breit-Pauli哈密顿量的贡献, 误差为相应值乘以$ \alpha^2\approx5.3\times10^{-5} $, 来自文献[51]; ^d仅计算领头项$ m\alpha^4 $阶的Breit-Pauli哈密顿量的贡献, 误差为相应值乘以$ \alpha^2\approx5.3\times10^{-5} $, 来自文献[30]; ^e由Babb于1995年重新拟合实验数据获得, 来自文献[46].

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表 4 $ \text{H}_2^+ $在特定的转动量子数L下, 可能具有的不同总自旋量子数F和总角动量量子数J的值. n是相应的超精细劈裂态的数目, 见文献[93]表I

Table 4. Possible values of different total spin quantum number F and total angular momentum quantum number J that $ \text{H}_2^+ $ may have under specific rotational quantum number L. n is the number of corresponding hyperfine splitting states, see Table I in Ref. [93].

L	s_e	I	F	J	n
0	1/2	0	1/2	1/2	1
1	1/2	1	1/2	1/2, 3/2	5
			3/2	1/2, 3/2, 5/2
偶	1/2	0	1/2	$ L-1/2, L+1/2 $	2
奇	1/2	1	1/2	$ L-1/2, L+1/2 $	6
			3/2	$ L-3/2, L-1/2, L+1/2, L+3/2 $

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表 5 $ \text{H}_2^+ $在振转态$ (v=4—8, L=1) $下, 超精细劈裂态$ (F, J) $之间的跃迁频率理论和实验结果比较, 单位为MHz. 第一行是Korobov等^[58]计算的理论值; 第二行中的实验值来源于文献[52]. 对于$ v=4—6 $的跃迁$ (1/2, 3/2)—(1/2, 1/2) $, 理论值已于2022年得到更新^[59], 相应的实验值取自引文[94]

Table 5. Comparison of theoretical and experimental transition frequencies between hyperfine states of $ \text{H}_2^+ $ in rovibrational state $ (v=4—8, L=1) $, in MHz. The first row shows the theoretical values calculated by Korobov et al.^[58]; the experimental values in the second row are from the Ref. [52]. For transitions $ (F, J)=(1/2, 3/2)—(1/2, 1/2) $ for $ v=4—6 $, the theoretical values were updated in 2022^[59], and the corresponding experimental values are cited from Ref. [94].

v	$ \left(\dfrac{3}{2}, \dfrac{3}{2}\right)—\left(\dfrac{3}{2}, \dfrac{5}{2}\right) $	$ \left(\dfrac{3}{2}, \dfrac{3}{2}\right)—\left(\dfrac{3}{2}, \dfrac{1}{2}\right) $	$ \left(\dfrac{1}{2}, \dfrac{3}{2}\right)—\left(\dfrac{1}{2}, \dfrac{1}{2}\right) $	$ \left(\dfrac{3}{2}, \dfrac{5}{2}\right)—\left(\dfrac{1}{2}, \dfrac{3}{2}\right) $	$ \left(\dfrac{3}{2}, \dfrac{3}{2}\right)—\left(\dfrac{1}{2}, \dfrac{3}{2}\right) $
4	5.7202	74.0249	15.371316(56)^a	1270.5504	1276.2706
	5.721	74.027	15.371407(2)^b	1270.550	1276.271
5	5.2576	68.9314	14.381453(52)^a	1243.2508	1248.5084
	5.258	68.933	14.381513(2)^b	1243.251	1248.509
6	4.8168	63.9879	13.413397(48)^a	1218.1538	1222.9706
	4.817	63.989	13.413460(2)^b	1218.154	1222.971
7	4.3948	59.1626	12.4607	1195.1558	1199.5506
	4.395	59.164	12.461	1195.156	1199.551
8	3.9892	54.4238	11.5172	1174.1683	1178.1576
	3.989	54.425	11.517	1174.169	1178.159
注: ^a来自引文[59]的理论值; ^b来自引文[94]的实验值.

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表 6 实验涉及的HD⁺振转态超精细劈裂系数, 单位kHz. 振转态(0, 1)的系数E₁, E₆和E₇实验值 (第二个条目) 由Haidar等^[50]从实验数据^[23]中提取

Table 6. Hyperfine coefficients for rovibrational states of HD⁺, in kHz. Experimental values (the second entry) of coefficients E₁, E₆ and E₇ for rovibrational state (0, 1) were extracted by Haidar et al. in Ref. [50] from experimental data^[23].

(v, L)	(0, 0)	(0, 1)	(1, 1)	(6, 1)	(0, 3)	(9, 3)
E₁^[50]		31985.41(12)	30280.74(11)	22643.89(8)	31628.10(11)	18270.85(6)
		31984.9(1)
E₂^[47]		–31.345(8)	–30.463(8)	–25.356(7)	–30.832(8)	–21.304(6)
E₃^[47]		–4.809(1)	–4.664(1)	–3.850(1)^a	–4.733(1)	–3.225(1)
E₄^[49]	925394.2(9)	924567.7(9)	903366.5(8)	816716.1(8)	920480.0(9)	775706.1(7)
E₅^[49]	142287.56(8)	142160.67(8)	138910.27(8)	125655.51(7)	141533.07(8)	119431.93(7)
E₆^[50]		8611.299(18)	8136.859(17)	6027.925(13)	948.5421(20)	538.9991(12)
		8611.17(5)
E₇^[50]		1321.7960(28)	1248.9624(27)	925.2072(20)	145.5969(3)	82.7250(2)
		1321.72(4)
E₈^[47]		–3.057(1)	–2.945(1)	–2.369(1)^a	–0.335	–0.219
E₉^[50]		5.660(1)	5.653(1)	5.204(1)^a	0.612	0.501
注: ^a未见于文献中, 由本文作者计算.

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表 7 HD⁺振转态(v, L)可能具有的总自旋角动量F和总角动量J的值

Table 7. Possible total spin angular momentum F and total angular momentum J values for HD⁺ rotational state (v, L).

L	F	S	J
L	0	1	$ L-1, L, L+1 $
	1	0	L
		1	$ L-1, L, L+1 $
		2	$ L-2, L-1, L, L+1, L+2 $

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表 8 HD⁺超精细劈裂跃迁频率f_ij的理论值和实验值比较, 单位kHz. $ f_{ij}=f_j-f_i $, 这里f_i是振转跃迁$ (v, L)\rightarrow (v', L') $光谱的第i个超精细劈裂峰, 参考实验文献[23, 24, 64]. $ \varDelta_{ij}=f_{ij}^\text{exp}-f_{ij}^\text{theor} $是实验与理论之间的偏差, $ \sigma_{\mathrm{c}}=\{[u(f_{ij}^\text{exp})]^2+ $$ [u(f_{ij}^\text{theor})]^2\}^{1/2} $是实验与理论值之间的标准误差, 这里$ u(f) $表示f的相对误差

Table 8. Comparison between experimental and theoretical results for some hyperfine intervals, in kHz. $ f_{ij}=f_j-f_i $, where f_i is the i-th hyperfine component of rovibrational transition $ (v, L)\rightarrow (v', L') $, see Refs. [23, 24, 64]. $ \varDelta_{ij}=f_{ij}^\text{exp}- $$ f_{ij}^\text{theor} $ is the deviation between experimental and theoretical frequencies, and $ \sigma_{\mathrm{c}}=\{[u(f_{ij}^\text{exp})]^2+[u(f_{ij}^\text{theor})]^2\}^{1/2} $ is the standard deviation, with $ u(f) $ being the relative uncertainty in f.

i	$ FSJ\rightarrow F'S'J' $	j	$ FSJ\rightarrow F'S'J' $	$ f_{ij}^\text{exp} $	$ f_{ij}^\text{theor} $^[50]	$ \varDelta_{ij} $	$ \varDelta_{ij}/\sigma_c $
$ (v=0, L=0)\longrightarrow(v'=0, L'=1) $ $ f_{ij}^\text{exp} $来自引文[23]
12	122→121	14	100→101	2434.211(75)	2434.465(23)	–0.254	–3.2
12	122→121	16	011→012	31074.752(43)	31074.102(56)	–0.350	–4.9
12	122→121	19	122→123	43283.419(54)	43284.10(12)	–0.677	–5.0
12	122→121	20	122→122	44944.338(72)	44945.289(64)	–0.951	–9.8
12	122→121	21	111→112	44996.486(61)	44997.14(11)	–0.652	–5.3
14	100→101	16	011→012	6939.541(66)	6939.636(42)	–0.095	–1.2
14	100→101	19	122→123	1949.208(47)	1948.63(11)	–0.423	–3.2
14	100→101	20	122→122	20810.127(88)	20810.823(63)	–0.696	–6.5
14	100→101	21	111→112	20862.275(79)	20862.673(91)	–0.398	–3.3
16	011→012	19	122→123	12209.667(41)	12209.994(72)	–0.327	–4.0
16	011→012	20	122→122	13870.586(62)	13871.187(42)	–0.601	–7.9
16	011→012	21	111→112	13922.734(49)	13923.037(51)	–0.303	–4.3
19	122→123	20	122→122	1660.919(70)	1661.19(10)	–0.274	–2.2
19	122→123	21	111→112	1713.067(59)	1713.042(25)	0.025	0.4
20	122→122	21	111→112	52.148(6)	51.850(75)	0.298	2.8
$ (v=0, L=0)\longrightarrow(v'=1, L'=1) $ $ f_{ij}^\text{exp} $来自引文[64]
12	122→121	16	122→123	41294.06(32)	41293.66(12)	0.40	1.2
$ (v=0, L30)\longrightarrow(v'=9, L'=3) $ $ f_{ij}^\text{exp} $来自引文[24]
$ F=0 $	014→014	$ F=1 $	125→125	178254.4(9)	178245.89(28)	8.5	9.0

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