Non-Born–Oppenheimer study of the muonic molecule ion <sup>4</sup>He μ<sup>+</sup>

Cite this Article

Yang Hang, Wu Meng-Shan, Zhang Yi, Shi Ting-Yun, Varga Kalman, Zhang Jun-Yi. Non-Born–Oppenheimer study of the muonic molecule ion ⁴He μ⁺. Chinese Physics B, 2020, 29(4): 043102 Copy to clipboard

Permissions

Non-Born–Oppenheimer study of the muonic molecule ion ⁴He μ⁺

Yang Hang^{1, 2}, Wu Meng-Shan^{1, †}, Zhang Yi^{1, 2}, Shi Ting-Yun¹, Varga Kalman³, Zhang Jun-Yi¹

1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China

2University of Chinese Academy of Sciences, Beijing 100049, China

3Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA

† Corresponding author. E-mail: mswu@wipm.ac.cn

Project supported by the National Natural Science Foundation of China (Grant No. 11704399), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB21030300), and the National Key Research and Development Program of China (Grant No. 2017YFA0304402).

Abstract

Accurate non-Born–Oppenheimer variational calculations of all bound states of the positive muon molecular ion ⁴Heμ⁺ have been performed using explicitly correlated Gaussian functions in conjunction with the global vectors. All the energies obtained are accurate in the order of 10⁻⁶ Hartree (1 Hartree = 27.2114 eV). Compared with the binding energies obtained from calculations based on the Born–Oppenheimer potential with the mass-weighted adiabatic corrections (Chem. Phys. Lett. 110 487 (1984)), the largest relative deviation is up to 15%. By analyzing the average interparticle distances and possibility distributions of interparticle distances of this system, it is confirmed that the Born–Oppenheimer approximation is reasonable for this system and that ^4He μ⁺ can be regarded as a system of positive muon bound to a slightly distorted helium atom.

PACS: ;31.15.ac;;31.15.vn;;31.15.xt;

Keyword:;positive muon;;variational calculation;;explicitly correlated Gaussian;

Show Figures

1. Introduction

Positive muon μ⁺, the antiparticle of muon, has attracted considerable attention and has been extensively studied. As a lepton, it plays an important role for testing the theory of quantum electrodynamics.^[1–5] Moreover, it has also been used in searches for new physics beyond the Standard Model^[6–10] As required by these applications, μ⁺ should be trapped and cooled down to low temperature (around 10 eV) since the typical μ⁺ beams have relatively high energies and poor phase space qualities.^[11] Some high-precision and high-sensitivity μ⁺ experiments are in progress, such as the muonium (Mu) spectroscopy experiment using microwave project (MuSEUM) at Japan Proton Accelerator Research Complex (JPARC)^[5] and the Mu-MASS project at Paul Scherrer Institute (PSI).^[12,13] Among these experiments, the helium buff gas is usually used in the precooling process due to its efficiency.

⁴He μ⁺ was first observed by Fleming et al. in the thermalization process of μ⁺ in the low-pressure He gas in 1981.^[14] Interestingly, no muonium atom (μ⁺e) was observed in the same thermalization process. Later, using the model of charge exchange collisions, Senba^[15] well explained the experimental observation of Fleming et al. In addition, the slowing-down process of μ⁺ in gas mixtures He–Ar and He–H₂, the process during which μ⁺ is slowed from its initial MeV kinetic energy down to ∼ 10 eV by collisions with gas atoms, was also studied by Senba.^[16]

The calculation^[17] of ⁴He μ⁺ was performed by Fournier and Lassier-Govers in 1982 using the Born–Oppenheimer (BO) potential for He H⁺ reported by Kolos and Peek^[18] and combining with the appropriately mass-weighted adiabatic corrections calculated by Bishop and Cheung.^[19] In 1984, Fournier and Le Roy improved their calculations for all bound levels of ⁴He μ⁺ using a new interpolation procedure for the BO potential.^[20] Because the positive muon is much like proton chemically (with the same charge and about 1/9 of proton’s mass), it is acceptable to use the BO potential for He H⁺ in the calculations for ⁴He μ⁺. The BO potential for He H⁺ was further improved by Cencek et al.^[21] who expanded the wavefunctions at an internuclear distance of R = 1.46 a.u.\ (atomic unit) in terms of 600 explicitly correlated Gaussian (ECG) functions in 1995. Recently, using an expansion of 20000 generalized Heitler–London basis functions, Pachucki^[22] obtained a very accurate BO potential with the precision of about 2 × 10⁻⁷ cm⁻¹. Later, new accurate potential curves for ⁴He H⁺, ⁴He D⁺, ³He H⁺, ³He D⁺ were generated by Tung et al.^[23] using an expansion of 600 shifted centers ECG functions and including the adiabatic corrections. The accurate non-Born–Oppenheimer (non-BO) variational energies for bound states of ⁴He H⁺ with zero orbital angular momentum have been reported by Stanke et al.^[24,25] However, for ⁴He μ⁺, there is no ab initio calculation until now.

In this work, high-accuracy wavefunctions and energies for ⁴He μ⁺ have been obtained by ab-initio calculations using ECG basis functions. To avoid the complicated one-by-one coupling of the orbital angular momenta, the global vector representation^[26] is used. Convergence tests for the ground state energy and virial factor are given for examining the reliability of our calculations. The probability distributions and average values of interparticle distances are calculated in order to analyze the structure of ⁴He μ⁺.

2. Theory

The total non-relativistic Hamiltonian for ⁴He μ⁺ in the laboratory coordinate has the following form

where r_i, m_i, and q_i represent the position vector, the mass, and the charge of the i-th particle, respectively. Particle 1 and 2 refer to the He nucleus (α) and μ⁺ while the last two stand for electrons.

To describe the intrinsic excitations of this system, the center-of-mass motion is subtracted from the total Hamiltonian, and the internal Hamiltonian has the following form

where

x_i = r_{i + 1} − r₁ is the relative coordinate, and

is the momentum conjugate to x_i. U refers to the transformation matrix defined by

where x₄ is the center-of-mass coordinate.

The total wavefunction is expanded as

where P₃₄ means the permutation of two electrons and ϕ_k is the ECG basis function. ECG basis functions were first introduced to quantum-chemical calculations by Boys and Singer in 1960.^[27,28] Their matrix elements are analytically calculable and can be easily generalized for any N-body systems. Moreover, they are easily adaptable to the permutational symmetry of the interesting systems. Due to these major advantages, they are now widely used in the calculations of various few-body problems. In the present work, ϕ_k has the form

where x^T = (x₁,x₂,x₃) and A_k is the k-th parameter matrix, v = u^T x with u^T = (u₁,u₂,u₃) is the global vector.

is the spherical harmonics while χ is the spin function. The global vector representation^[26] simplifies the calculation of matrix elements by avoiding the complicated one-by-one coupling of the orbital angular momenta. Compared with the spherical ECG basis, the factor |v|^2K+L plays an important role in describing the increasing number of nodes for excited rovibrational states.

The energy and wavefunction for each state are obtained by minimizing the eigenvalue E through optimizing A_k, and by solving the general eigenvalue equation

The optimization was carried out by the stochastic variational method (SVM) which has been proved to be efficient and accurate by various applications,^[26–31] such as the calculation of the scattering length for the scattering between two ground state positronium atoms.^[32] For simplicity, the values of u are set manually instead of being optimized automatically.

In the present study, the mass used for α and μ⁺ are m_α = 7294.29954142 m_e and m_μ⁺ = 206.7682830 m_e, which are the CODATA 2018 recommended values. Atomic units (a.u.) are used throughout the paper unless stated otherwise.

3. Calculations, results, and discussion

3.1. Convergence test

The convergence of the ground state energy for ⁴He μ⁺ as a function of the basis-set size is shown in Table 1. As the reference for the binding energies of ⁴He μ⁺, the ground state energy of ⁴He obtained is −2.9033045577 a.u. by using 700 spherical ECG functions. In order to accurately describe the rovibration and orbital angular momentum coupling of ⁴He μ⁺, three sets of u^T are used, i.e., u^T = (1,0,0), (0.5,0.5,0), and (0.33,0.33,0.33). The energy obtained becomes lower as the basis-set size increases. Using 2800 ECGs, we get the ground state energy accurate in the order of 10⁻⁸ Hartree. Usually, the convergences of the wavefunctions are slower than those of energies. In order to check the quality of the ground state wavefunction, the virial factor η is also calculated and shown in Table 1. The virial factor is defined as

where ⟨T⟩ and ⟨V⟩ are the expectation values of the kinetic energy operator and potential energy operator, respectively. The closer to zero that η is, the better quality the ground state wavefunction has. From Table 1 we can see that both ⟨T⟩ and ⟨V⟩ have eight significant digits and the smallest η is in the order of 10⁻⁹.

Table 1.

The convergences of the total energy, the expectation values of kinetic energy operator and potential energy operator, and the virial factor for the ground state ⁴Heμ⁺.

3.2. Expectation values of energies

For each of the rovibrational excited states, the largest basis-set size and the global vectors used are the same as the corresponding for the ground state. Compared with the ground state, larger values of K in Eq. (5) are used for higher rovibrational states. For the ν-th vibrational state, 0 ≤ K ≤ ν is used. Consequently, it is more difficult and time-consuming for the calculations of high excited states. Due to the slower convergences, seven significant digits are obtained for the energies of some high excited states. Comparison of our results for binding energies (E^b) and those of Fournier and Le Roy^[20] () is shown in Table 2. The parameter represents the relative deviation. The rovibrational bound levels are denoted as (j,ν) with j and ν being the rotational and vibrational quantum numbers, respectively. All our non-BO values for the binding energies are larger than the previous BO results of Fournier and Le Roy as the relative deviations increase along with the increase of j and ν. The most weakly bound state is the (j = 4,ν = 2) state with the binding energy 14.576 cm⁻¹ for which the two approaches have the largest relative deviation, i.e., about 15%.

Table 2.

Comparisons of the binding energies we obtained (E^b, in units of cm⁻¹) with those of Fournier and Le Roy^[20] () for ⁴Heμ⁺. The parameter represents the relative deviation.

3.3. Expectation values of interparticle distances

The expectation values of interparticle distances and the virial factors for all the bound states of ⁴Heμ⁺ are shown in Table 3. In order to provide a direct-viewing feeling, the expectation values of interparticle distances are illustrated in Fig. 1. For simplicity, we number all bound states in sequence. From Fig. 1(a) we can see that the average distances between α and μ⁺ (⟨r_α–μ⁺⟩) increase quickly with the vibrational excitations and slowly with the rotational excitations. The (j = 1,ν = 3) state has the longest distance for ⟨r_α–μ⁺⟩. It can be seen from Fig. 1(b) that the distances between μ⁺ and the electron (⟨r_{μ⁺ – e} ⟩) have the same trend as ⟨r_α–μ⁺⟩. Moreover, the values of ⟨r_μ⁺–e⟩ are very close to those of ⟨r_α–μ⁺ ⟩, i.e., ⟨r_μ⁺–e⟩ ≈ ⟨r_α–μ⁺ ⟩ as shown in Table 3. Figure 1(c) shows that the distances between α and the electron (⟨r_α–e⟩) first slightly increase and then decrease as ν increase for 0 ≤ j ≤ 4. Generally, ⟨r_α – e⟩ changes slightly around 0.95 a.u. The distances between two electrons (⟨r_e–e⟩) show the same trend as ⟨r_α–e⟩, see Fig. 1(d). To make a comparison, ⟨r_α–e⟩ = 0.9296 a.u. and ⟨r_e–e ⟩ = 1.4222 a.u. are also calculated for the ground state of ⁴He atom. Consequently, the deviations for ⟨r_α–e ⟩ and ⟨r_e–e⟩ between ⁴He and ⁴He μ⁺ are small, i.e., ⟨r_α–e⟩_Heμ⁺ ≈ ⟨ r_α–e ⟩_He and ⟨r_e–e⟩_Heμ⁺ ≈ ⟨r_e–e⟩_He. Therefore, it is reasonable to conclude that ⁴Heμ⁺ can be treated as a μ⁺ particle weakly bound to a slightly distorted ⁴He.

	Figure Option View Download New Window
	Fig. 1. The average interparticle distances (in units of a.u.) for all bound states of ⁴He μ⁺.

Table 3.

The average interparticle distances (in units of a.u.) and the virial factors for all bound states of ⁴Heμ⁺.

No.	(j,ν)	⟨r_α–μ⁺⟩	⟨r_α–e⟩	⟨r_μ⁺–e⟩	⟨r_e–e⟩					η
1	(0,0)	1.6221	0.9416	1.6616	1.4184	2.7091	1.2043	3.1581	2.4654	9.55 × 10⁻⁹
2	(0,1)	2.0236	0.9547	2.0531	1.4495	4.3648	1.2623	4.8340	2.6050	5.12 × 10⁻⁹
3	(0,2)	2.7387	0.9529	2.7730	1.4545	8.1219	1.2714	8.7005	2.6465	1.54 × 10⁻⁷
4	(0,3)	5.0912	0.9354	5.1370	1.4308	28.3601	1.2151	29.2245	2.5545	2.05 × 10⁻⁷
5	(1,0)	1.6343	0.9427	1.6727	1.4205	2.7499	1.2078	3.1984	2.4734	3.47 × 10⁻⁹
6	(1,1)	2.0439	0.9553	2.0729	1.4508	4.4526	1.2646	4.9235	2.6107	1.96 × 10⁻⁸
7	(1,2)	2.7916	0.9525	2.8263	1.4543	8.4378	1.2707	9.0251	2.6462	2.50 × 10⁻⁷
8	(1,3)	5.6295	0.9341	5.6744	1.4289	35.0737	1.2103	35.9701	2.5462	2.55 × 10⁻⁷
9	(2,0)	1.6589	0.9449	1.6954	1.4247	2.8334	1.2148	3.2810	2.4891	1.41 × 10⁻⁹
10	(2,1)	2.0860	0.9564	2.1142	1.4533	4.6382	1.2691	5.1130	2.6215	3.99 × 10⁻⁸
11	(2,2)	2.9115	0.9516	2.9472	1.4535	9.1787	1.2684	9.7858	2.6440	2.97 × 10⁻⁷
12	(3,0)	1.6967	0.9480	1.7304	1.4308	2.9644	1.2251	3.4108	2.5122	4.36 × 10⁻¹⁰
13	(3,1)	2.1539	0.9579	2.1811	1.4568	4.9463	1.2751	5.4280	2.6366	2.66 × 10⁻⁸
14	(3,2)	3.1459	0.9496	3.1836	1.4512	10.7296	1.2625	11.3743	2.6361	9.01 × 10⁻⁹
15	(4,0)	1.7491	0.9518	1.7794	1.4385	3.1513	1.2383	3.5968	2.5422	7.10 × 10⁻⁹
16	(4,1)	2.2562	0.9592	2.2826	1.4605	5.4307	1.2817	5.9246	2.6540	5.58 × 10⁻⁸
17	(4,2)	3.7775	0.9443	3.8191	1.4442	15.7829	1.2458	16.5121	2.6094	2.11 × 10⁻⁷
18	(5,0)	1.8187	0.9563	1.8452	1.4476	3.4090	1.2541	3.8544	2.5782	1.06 × 10⁻⁸
19	(5,1)	2.4141	0.9599	2.4405	1.4636	6.2274	1.2872	6.7434	2.6704	1.48 × 10⁻⁸
20	(6,0)	1.9106	0.9609	1.9331	1.4575	3.7662	1.2716	4.2136	2.6189	3.54 × 10⁻⁹
21	(6,1)	2.7078	0.9584	2.7366	1.4637	7.8913	1.2865	8.4541	2.6770	4.99 × 10⁻⁸
22	(7,0)	2.0359	0.9653	2.0550	1.4675	4.2865	1.2898	4.7405	2.6624	1.00 × 10⁻⁸

Table 3.

The average interparticle distances (in units of a.u.) and the virial factors for all bound states of ⁴Heμ⁺.

Furthermore, the comparison of average interparticle distances between ⁴He μ⁺ and ⁴He H⁺ is shown in Table 4. Due to the lighter mass of μ⁺, for the case of the zero total angular momentum, ⁴He μ⁺ has only four bound states. However, there exist twelve bound states of ⁴He H⁺ for the same case. Though the average interparticle distances of the ground state ⁴He μ⁺ are close to those of ⁴He H⁺ the average interparticle distances for ⁴He μ⁺ are much larger for the excited states. In other words, ⁴He H⁺ are more tightly bounded than ⁴He μ⁺.

Table 4.

Comparison of the average interparticle distances (in units of a.u.) between ⁴He μ⁺ and ⁴He H⁺.^[33]

3.4. Probability density distribution of interparticle distances

Compared with the average interparticle distances, the probability density distribution of interparticle distances will give more information about the structure of ⁴Heμ⁺. The probability density of distance between particle i and j is given by

where Φ is the wavefunction of a bound state, r_i–j = r_i − r_j, ⟨···⟩ means the integration of relative coordinates (x₁,x₂,x₃) and ∫dΩ_{r_i–j} means the integration over the angle of r_i–j. ρ(r_i–j) can be obtained easily since the matrix elements for Dirac delta function δ can be analytically calculated.

Figures 2(a) and 2(b) plot the probability densities ρ(r_α–μ⁺) and ρ(r_α–e) for (j = 0,ν = 0,1,2) as functions of r_α–μ⁺ and r_α–e, respectively. ρ(r_α–μ⁺) of excited bound states have the larger spatial extension than that of the ground state while the number of nodes for ρ(r_α–μ⁺) is equal to the vibrational quantum number ν. As shown in Fig. 2(b), the electrons are centered around α and their probability distributions are almost the same for different states. In addition, ρ(r_α–e) of ⁴He is also shown in Fig. 2(b) to make a comparison. Generally, ρ(r_α–e) of ⁴He μ⁺ is only little different from ρ(r_α–e) of ⁴He. Interestingly, for Heμ⁺ with j = 0 and 0 ≤ ν ≤ 2, ρ(r_α–e) becomes larger along with the increase of ν for r_α–e < 1 a.u. while they are all smaller than ρ(r_α–e) of He in the same region of r_α–e. This can be explained with two reasons: Firstly, the electrons are attracted by μ⁺ and hence slightly deviate from α. Secondly, the attraction of μ⁺ becomes weaker for higher excited states as the distances between μ⁺ and electrons are larger.

	Figure Option View Download New Window
	Fig. 2. Probability density distributions of ρ(r_α–μ⁺) and ρ(r_α–e) for the (j = 0,ν = 0,1,2) states. In panel (b), ρ(r_α–e) for ⁴He is also shown.

4. Summary

For the first time, non-BO calculations are carried out for the structural properties of ⁴He μ⁺ using ECGs with the global vector representation together. Overall, the non-BO energies for the rovibrational bound states are accurate in the order of 10⁻⁶ Hartree. Compared with the BO values of binding energies obtained by Fournier and Le Roy, all the corresponding non-BO values are larger. The largest relative deviation between them is about 15% for the most weakly bound state with j = 4 and ν = 2. In addition, the expectations and probability density distributions of interparticle distances are discussed in detail to confirm that ⁴Heμ⁺ can be regarded as a system of μ⁺ bound to a slightly distorted ⁴He and hence that the BO approximation is reasonable for this system.

Reference

[1]	Rich A 1981 Rev. Mod. Phys. 53 127
[2]	Karshenboim S G 2005 Phys. Rep. 422 1
[3]	Jungmann K P 2006 Nucl. Phys. B-Proc. Suppl. 155 355
[4]	Khaw K S Antognini A Prokscha T Kirch K Liszkay L Salman Z Crivelli P 2016 Phys. Rev. 94 022716
[5]	Crivelli P 2018 Hyperfine Interact. 239 49
[6]	Willmann L Schmidt P V Wirtz H P et al. 1999 Phys. Rev. Lett. 82 49
[7]	Bennett G W Bousquet B Brown H N et al. (Muon (g-2) Collaboration) 2004 Phys. Rev. Lett. 92 161802
[8]	Karshenboim S G 2010 Phys. Rev. Lett. 104 220406
[9]	Karshenboim S G McKeen D Pospelov M 2014 Phys. Rev. 90 073004
[10]	Frugiuele C Pérez-Ríos J Peset C 2019 Phys. Rev. 100 015010
[11]	Taqqu D 2006 Phys. Rev. Lett. 97 194801
[12]	Bao Y Antognini A Bertl W Hildebrandt M Khaw K S Kirch K Papa A Petitjean C Piegsa F M Ritt S Sedlak K Stoykov A Taqqu D 2014 Phys. Rev. Lett. 112 224801
[13]	Belosevic I Antognini A Bao Y Eggenberger A Hildebrandt M Iwai R Kaplan D M Khaw K S Kirch K Knecht A Papa A Petitjean C Phillips T J Piegsa F M Ritjoho N Stoykov A Taqqu D Wichmann G 2019 Hyperfine Interact. 240 41
[14]	Fleming D G Mikula R J Garner D M 1981 Hyperfine Interact. 8 307
[15]	Senba M 1988 J. Phys. B: At. Mol. Opt. Phys. 21 3093
[16]	Senba M Arseneau D J Pan J J Fleming D G 2006 Phys. Rev. 74 042708
[17]	Fournier P G Lassier-Govers B 1982 J. Physique Lett. 43 483
[18]	Kołos W Peek J 1976 Chem. Phys. 12 381
[19]	Bishop D M Cheung L M 1979 J. Mol. Spectrosc. 75 462
[20]	Fournier P G Roy R J L 1984 Chem. Phys. Lett. 110 487
[21]	Cencek W Komasa J Rychlewski J 1995 Chem. Phys. Lett. 246 417
[22]	Pachucki K 2012 Phys. Rev. 85 042511
[23]	Tung W C Pavanello M Adamowicz L 2012 J. Chem. Phys. 137 164305
[24]	Stanke M Kedziera D Molski M Bubin S Barysz M Adamowicz L 2006 Phys. Rev. Lett. 96 233002
[25]	Stanke M Kedziera D Bubin S Adamowicz L 2008 Phys. Rev. 77 022506
[26]	Varga K Suzuki Y 1995 Phys. Rev. 52 2885
[27]	Mitroy J Bubin S Horiuchi W Suzuki Y Adamowicz L Cencek W Szalewicz K Komasa J Blume D Varga K 2013 Rev. Mod. Phys. 85 693
[28]	Bubin S Pavanello M Tung W C Sharkey K L Adamowicz L 2013 Chem. Rev. 113 36
[29]	Varga K Suzuki Y 1996 Phys. Rev. 53 1907
[30]	Usukura J Suzuki Y Varga K 1999 Phys. Rev. 59 5652
[31]	Mitroy J Zhang J Y Varga K 2008 Phys. Rev. Lett. 101 123201
[32]	Ivanov I A Mitroy J Varga K 2001 Phys. Rev. Lett. 87 063201
[33]	Bubin S 2006 Accurate Non-Born–Oppenheimer Variational Calculations Of Small Molecular Systems Ph.D. Dissertation Tucson The University Of Arizona