† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 11874064), the Strategic Priority and the Research Program of the Chinese Academy of Sciences (Grant No. XDB21030300), the National Key Research and Development Program of China (Grant No. 2016YFA0302104). Yong Liu acknowledges the Project of Hebei Educational Department, China (Grant No. ZD2018015) and the Natural Science Foundation of Hebei Province, China (Grant No. A2019203507). Bing-Bing Suo acknowledges the financial support from the National Natural Science foundation of China (Grant Nos. 21673174 and 21873077).
Electric quadrupole moments of low-lying excited states of Yb+ are calculated by relativistic coupled-cluster theory with perturbations from external fields. The field-dependent energy differentiation provides accurate values of the electric quadrupole moments of 2P3/2, 2D3/2,5/2, and 2F5/2,7/2 states which agree well with experimental values. The important role of the electronic correlation to the electric quadrupole moments is investigated. Our calculations indicate the early dispute of the electric quadrupole moment of the Yb+(2F7/2) state for which the measured and theoretical values have a large discrepancy. These electric quadrupole moment values can help us to determine the electric quadrupole shifts in start-of-the-art experiments of the Yb+ ion.
The Yb+ ion has been investigated by many national standards laboratories due to its potential application in optical clock with fractional uncertainties below 10−17.[1–4] Two transitions of Yb+, e.g., E2: 2S1/2 ← 2D3/2 and E3: 2S1/2 ← 2F7/2, are accepted as secondary representations of the second transition, in which E2 is an electric quadrupole transition with the excitation wavelength of 436 nm, and E3 is an electric octupole transition with that of 467 nm. The E3 transition has an extremely long lifetime and has a natural linewidth on the order of nanohertz. The atomic clock based on the E3 transition has achieved a total systematical uncertainty of 3.2 × 10−18.[5] The 2S1/2 → 2D3/2 transition is suggested as a suitable system to study the parity non-conservation (PNC).[6–8] Moreover, a comparison of two clock transitions of Yb+ had set new constraints on the time variation of the proton-to-electron ratio and the fine structure constant.[4]
The 2D3/2 and 2F7/2 states have the electron configurations of [Xe]4f145d and [Xe]4f136s2, respectively. Both these two states have none-zero electric quadrupole moments (labeled as Θ). As one of the largest systematical energy shifts of the Yb+ clock, the values of Θ are used to estimate electric quadrupole shifts due to the interaction of electric quadrupole moment with external electric field gradient. Besides, precise data of electric quadrupole moments are also useful for analyzing nuclear structures. A precise measurement of the electric quadrupole moment of Yb+ has been implemented for two clock states, 2D3/2 and 2F7/2.[2,9] Comparisons of theoretical results with experimental measurements could serve as excellent tests to the relativistic atomic theories. However, calculation of the electric quadrupole moment of Yb+ is a challenging task because Yb+ is a heavy system which contains 69 electrons with 13–14 electrons occupied in the 4f orbital. Earlier theoretical studies on the Θ value of the Yb+ 2D3/2 state has given consistent results with measured values while theoretical values of the 2F7/2 state show large discrepancies from the measured results, implying some effects of electronic correlation that occurs for the 2F7/2 state other than the 2D3/2 state. Furthermore, the electric quadrupole moment is found very sensitive to the configuration interaction. The quadrupole moment is expected to be small due to large cancellations of one-electron contributions. This makes the result obtained even for the biggest CI space inconclusive.[8] Therefore, further investigations to validate the disagreement and to understand the role of the electron correlation on the electric quadrupole moment of low-lying excited states of Yb+ are strongly desired.
In this paper, we employ the relativistic coupled-cluster (RCC) method to treat the electron correlation in Yb+. We adopt the finite-field (FF) approach to obtain the electric quadrupole moment by numerical differentiation of the FF-dependent energies. We calculate the electric quadrupole moments of the 2P3/2, 2D3/2,5/2, and 2F5/2,7/2 states, and make comparisons with the experimented results and the early theoretical values. Our results for 2D3/2,5/2 state show an excellent agreement with the measured and early theoretical values, while for the 2F7/2 state, our result is in the line with the experimental measurement, but different with the early theoretical results obtained by the expectation-value method. Besides, we give a recommendation value of the electric quadrupole moment of the Yb+ 2P3/2 state which has not been reported before.
The atomic properties are evaluated as the numerical differentiation of energy with respect to externally applied fields or field gradients in the FF method. For an atom or ion in a purely quadrupole electric field, the corresponding energy shift is written as[10–12]
The field-dependent energies are calculated by using a two-step procedure. First, the four-component Dirac–Hartree–Fock (DHF) calculation is performed to generate the reference state and one-electron orbital for subsequent electron correlation calculations. Then, the Fock space coupled-cluster method (FSCC) is implemented by using single-electron orbitals from DHF.[13,14] For the 2P3/2 and 2D3/2,5/2 states, we carried out the DHF calculations for the closed-shell reference state of Yb2+. Then, the electron attachment calculation is implemented by FSCC with the model space composed of the 6s, 6p, and 5d orbitals, which determines the electron affinity (EA) energies at the corresponding valence states. The 2F7/2 state shows a multi-configurational manner and is troublesome. We first calculate the 1S ground state of the Yb atom via DHF, then the electron detachment calculation is carried out by FSCC, in which one electron is removed from the model space 4f146s2 and then ionic potentials (IP) are obtained. The FSCC calculations are augmented by the extrapolated intermediate Hamiltonian approach to avoid the convergence difficulties in the coupled-cluster iterations.[13] All calculations are carried out by using the electronic structure code DIRAC.[15] The electron correlations of the core shells are restricted for appropriate sizes of the core to save computational costs.
The Dyall’s uncontracted correlation consistent double-, triple-, quadruple-ζ basis sets are used in calculations, which are referred to as Xζ with X = 2, 3, and 4.[16] Arbitrary 4–8 field values are chosen in Fzz = [0,4.5 × 10−5] a.u. (a.u. is the atomic unit), where atomic unit of Fzz is 5.142250 × 1011 V/cm2. The fitting is checked to remove the dependence of the studied properties on sampling. The energy convergence criterion is chosen as 10−10 Hartree (1 Hartree = 4.3597 × 10−27 J) in calculations. We adopt a composite scheme to get the convergent property. The energies and properties from FSCC/4ζ are corrected by three terms, Δbasis, Δcore, and Δvirt to include contributions of the incomplete basis sets, truncations of the core, and virtual orbitals, respectively. We estimate Δbasis by the differential values of the results obtained at X = 3ζ and X = 4ζ. Moreover, Δcore means the electron correlation contribution from the increasing of the core electrons, and Δvirt estimates the contribution of the neglected virtual orbitals due to energy cutoff. The uncertainties in our final values are estimated by the dominant errors caused by the finite basis size and the neglected electronic correlations.
In Table
![]() | Table 1. Values of the electron affinity (EA) in a.u. at the Yb+ 2S1/2, 2P3/2, and 2D3/2,5/2 states obtained by using the RCC method. . |
Table
![]() | Table 2. Values of Θ(|γ J⟩), in units of a.u., of the 2P3/2 and 2D3/2,5/2 states of the Yb+ ion obtained by using the RCC method. . |
In Table
![]() | Table 3. Values of the ionic potentials (IP), in units of a.u., of the 2S1/2 and 2F5/2,7/2 electrons obtained by using the RCC method. . |
Table
![]() | Table 4. Values of Θ(|γ J⟩), in units of a.u., of the 2F5/2,7/2 states of the Yb+ ion obtained by using the RCC method. . |
In conclusion, we have performed the FF method on the FSCC level of theory to calculate the Θ(|γ J⟩) values of the low-lying 2P3/2, 2D3/2,5/2, 2F5/2,7/2 states of the Yb+ ion. The standard Dyall’s basis sets of varying size are adopted. In order to study the effects of the electron correlation on Θ(|γ J⟩), we have investigated the sensitivity of Θ(|γ J⟩) to various computational parameters, i.e., the size of the basis sets, the number of correlated electrons, and the chosen virtual energy cutoff, allowing us to estimate the uncertainty of our results. The Θ(|γ J⟩) values of the 2D3/2 and 2F7/2 states have been measured for their application in the E2 and E3 clock transitions, and there are also early theoretical studies by using the expectation-value method. For the 2D3/2 state, the expectation-value method gives the consistent results with the experimental measurements, while for the 2F7/2 states, there is large discrepancy between the measured and the expectation-value theoretical values. The insufficient electron correlation effect and lack of the response of wavefunction to the external field were supposed to be the reason for such a dispute. By combining FSCC with the FF approach, we obtained the values of the electric quadrupole moments of the D and F states that both show excellent agreement with the measured results. The theoretical Θ(|γ J⟩) values of the 2P3/2, 2D3/2,5/2, and 2F5/2,7/2 states can be used to estimate the related electric quadruple shift in the Yb+ experiments.
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