Atomic Structure of Nd 9 + for Highly Charged Ion Clocks

: The energy levels arising from the electronic orbital 5 p − 4 f crossing between the ground 5 p 2 4 f and excited 5 p 4 f 2 conﬁgurations in the Nd 9 + ion are investigated by using high-accuracy relativistic ab initio calculations. The accurate atomic data of the lifetime, g J factor, electric quadrupole moment, and hyperﬁne structure of the magnetic dipole are also presented. The long-lived states that are suitable for making narrow-linewidth (milli-Hz) clock lines are found. Dominant systematics caused by stray electromagnetic interactions in an experiment and the coefﬁcients of the relativistic sensitivityto variation of the ﬁne-structure constant α and of the Lorentz invariance violation are evaluated, thus validating that the Nd 9 + ion can be a new candidate for high-resolution spectroscopy and precision fundamental studies for probing new physics beyond the Standard Model.


Introduction
Highly charged ion (HCI) clocks have been suggested to be some of the most interesting candidates for future frequency metrology, with the aim of breaking precision limits lower than 10 −18 and aiding in the quest for powerful tools to be used research in physics beyond the Standard Model. The experimental advances in fabrication, cooling, and trapping of highly charged ions have enabled the measurement of the optical spectroscopy of several categories of highly charged ions, such as Nd-like ions [1], the Ho 14+ ion [2], the Pr 9+ ion [3], the Ni 11+ -Ni 14+ ions [4], etc. Very recently, the sympathetic cooling and the coherent laser spectroscopy of Ar 13+ were demonstrated [5]. All of this progress makes HCIs accessible for high-resolution spectroscopy and precision fundamental studies.
HCIs have high sensitivities to variations in fundamental constants as a consequence of their strong relativistic effects and higher ionization energies. Such high sensitivities can be further enhanced by the electronic-orbital-crossing phenomenon that occurs in inter-configurations. As an Sb-like lanthanide ion, the Nd 9+ ion has the core [1s 2 , ..., 4d 10 , 5s 2 ] and three valence electrons in the 5p and 4 f shells. The reordering of the 5p and 4 f electronic orbital binding energies along the Sb-like isoelectronic sequences generates rich optical transitions over inter-configurations. However, the energy levels due to such 5p − 4 f crossings may be very complicated. In particular, the 5s orbital has a binding energy that is close to those of the 5p and 4 f shells. This indicates that the two 5s electrons should be included in the valent field, considering the Nd 9+ ion as a five-valent system. For many-electron valent systems, the configuration interaction method (CI) is a widely adopted method, and, in principle, it has no limit for the number of valence electrons. However, calculations of many-valence-electron systems have many practical difficulties in terms of their computational techniques, especially when the number of valence electrons is beyond 4. The accurate prediction of the optical transition in HCIs with valence electrons in the 4 f shell is especially challenging. Berengut et al. studied the 4 f -5p level crossing in Nd 9+ [6]. Complete and consistent information about Nd 9+ ions remains scarce.
In this work, we adopted two different CI methods-the multi-reference configuration interaction (MRCI) method [7][8][9] and the configuration interaction plus many-body perturbation theory (CI+MBPT) theory [10,11]. The comparative computation based on the MRCI and CI+MBPT methods ensures the consistency and reliability of our predicted results with minimal uncertainty. Our results show that the 5p-4 f crossing generates wealthy laser-accessible states over the 5p 2 4 f and 5p4 f 2 configurations. The long-lived states that are suitable for making clocks are identified. Accurate data on the atomic properties, such as energies, lifetimes, the g J factor, the magnetic-dipole hyperfine structure (hfs) constants, and the electric quadrupole moment, are predicted. The clock-transition-related properties are calculated, thus validating the dominant systematics that are attainable to below 10 −19 . High coefficients of the relativistic sensitivity to variations in the fine-structure constant α and the violation of the local Lorentz invariance are found for the clock transitions of Nd 9+ , indicating its application to studies of a possible varying fine-structure constant α and the violation of the local Lorentz invariance.

MRCI
The calculations start with the Dirac-Hartree-Fock (DHF) calculation under the Dirac-Coulomb-Gaunt Hamiltonian that is given bŷ where ff and β are Dirac matrices, p is the kinetic momentum, m 0 c 2 is the resting mass energy of an electron with the speed of light c, V iA is the nuclear potential with the atomic mass number A, and r ij is the distance between the i-th and j-th electrons. The last term in Equation (1) is the Gaunt term, which is the leading term of the Breit interaction. The DHF calculation is combined with the relativistic all-electron correlation-consistent optimized basis sets developed by Dyall et al., which contain the {24s;19p;13d;8f;2g}, {30s;24p;16d;11f;3g;2h}, and {35s;30p;19d;13f;5g;3h;2i} functions in the 2ζ, 3ζ, and 4ζ basis sets, respectively [12]. Based on the single-electron wavefunctions obtained with the DHF calculation, a string-based Hamiltonian-direct configuration interaction [7][8][9] calculation was carried out. The electron excitation from the occupied to the virtual orbitals comprised configuration spaces with different sizes by tailoring the numbers of the active electrons and the correlated orbitals to balance the computational cost and accuracy. The maximum of 28 electrons that occupied the 3d, 4s, 4p, and 4d orbitals were set in order to allow for a single (S) excitation; five electrons in the 5s, 5p, and 4 f shells were assigned to be valence electrons for single and double (SD) excitations; virtual orbitals with energy less than 10 a.u. were included in the CI space, which contained 9sp8d f 6g (which indicates sand p-orbitals with n ≤ 9, dand f -orbitals with n ≤ 8, and g-orbitals with n ≤ 6) in the 2ξ basis set, 10sp9d f 6g7h in the 3ξ basis set, and 12sp10d f 7ghi in the 4ξ basis sets. The MRCI calculations were performed by using a relativistic ab initio electronic structure program, DIRAC [13].

CI+MBPT
The CI+MBPT calculation is based on the Dirac-Coulomb-Breit Hamiltonian, which is similar to Equation (1), with the one-electron Dirac-Fock operator and the Breit term where V DF is the potential of the N DF electrons included in the self-consistent DHF procedure. Herein, N DF is either all N electrons of the atom or some subset of them.
The choice of the potential may impact the calculation results. The QED interaction is included by adopting the radiative potential method, which was originally developed by Flambaum and Ginges [14]. The remaining valence and virtual orbitals (pseudostates) are constructed as a linear combination of B-spline basis functions. The configuration space is constructed by allowing SD excitation from the leading configurations, 5s 2 5p 2 4 f and 5s 2 5p4 f 2 , up to 8spd f , alongside the entirety of the SD excitations from the 4d shells for the large-side CSFs, while the small-side CSFs are restricted to 6spd f for the SD excitations and 8spd f for an additional S excitation. Correlations with the frozen core orbitals, i.e., those below 4d and the virtual orbitals beyond the valence basis set, are treated using the second-order MBPT. The B-spline basis set includes virtual orbitals up to the main quantum number n ≤ 30 and angular momentum l ≤ 4. We use the 'use-valence' flag to include valence-valence MBPT diagrams for orbitals above the valence basis set and below the MBPT basis set. The CI+MBPT calculation was carried out by using the AMBiT code [11].

Energy Levels
The excited energies (EEs) of the low-lying states in Nd 9+ obtained with the MRCI calculation are shown in Table 1. We first adopt an intermediate CI model space that correlates 18 core electrons (4s 2 4p 6 4d 10 ) at the S-excitation level and five valence electrons of the 5s, 5p, 4 f shells at the SD-excitation level; these are referred to as '(core18) 2ξ ', '(core18) 3ξ ', and '(core18) 4ξ ' for the different basis sets. The convergent energy values are obtained at the 4ξ basis set with a possible error due to the finite-size effect of the basis set, with ∆ basis being estimated by the difference between (core18) 3ξ and (core18) 4ξ . Two additional calculations are conducted for the 2ξ basis set: The first calculation includes the triple excitations of the 5s, 5p, 4 f valence electrons, referred to as '(core18) T '; this is used to estimate the possible correction due to the triple excitations, ∆ T , through the difference between '(core18)' and '(core18) T '; the second calculation extends the inner-core excitations up to the inclusion of 28 core electrons by adding those from the 3d shell, referred to as '(core28)'; then, the correction due to more core excitations, ∆ c , is estimated by the difference between '(core18)' and '(core28)'. The contribution of the QED interaction, ∆ QED , that is absent in the MRCI calculation is taken from the AMBiT calculation. The final results are recommended based on (core18) 4ξ plus ∆ QED , with the corresponding uncertainties assigned according to the rms of ∆ basis , ∆ T , and ∆ c . The EEs obtained by the CI+MBPT calculations are given in Table 2. The changes in the EEs under the V N−5 , V N−3 , and V N potentials are obvious, more than 2000-3000 cm −1 , bringing in certain uncertainty. The ∆ n considers the changes in the EEs when the valence basis set increases to 10spd f . The final results are given using the results under the V N potential plus ∆ QED and ∆ n .  The MRCI and CI+MBPT results for the EEs are consistent within a reasonable error range. The EE results prove that the Nd 9+ ion has a 5p 2 4 f ground configuration. The fine-structure splitting via (5p 2 4 f ) o 5/2 and (5p 2 4 f ) o 7/2 , labeled as G0 and G1, respectively, is around 6524(69) cm −1 (MRCI result), which is consistent with the earlier calculations [6,15]. Subsequently, the 5p − 4 f orbital crossing raises many of the 5p4 f 2 excited states for those below about 30,000 cm −1 , which are listed as E0 to E9 in Tables 1 and 2. Another fine-structure splitting state for the ground configuration, G2: 5p 2 4 f , is found in the CI+MBPT calculation. Table 3 compares the results obtained with MRCI and AMBiT for the lifetime τ, g j factor, electric quadrupole moment Θ, and magnetic-dipole hyperfine structure constant A for the energy levels in the Nd 9+ ion. The differences in level energies of the MRCI and AMBiT results change the transition wavelengths and then lead to differences in the τ values. The results for τ justify several long-lived excited states that are suitable for making clock transitions. One good example is the G0:(5p 2 4 f ) o 5/2 -E0:(5p4 f 2 ) o 9/2 transition, which occurs at 485 (80) nm (the MRCI result) and has a natural linewidth that is estimated to be about 2 milli-Hz in terms of the 67.21 s (the MRCI result) upper-state lifetime. In the case of the odd isotope, hyperfine mixing with faster decaying levels may need to be taken into account, as it can change the lifetime of E0 : (5p4 f 2 ) o 9/2 , or we can choose appropriate hyperfine sublevels F to adjust the hyperfine-mediate transitions. The results for the g J factor and the electric quadrupole moment Θ (a.u.), as they show excellent consistency between the MRCI and CI+MBPT calculations, are used for the evaluation of the systematic effect of the clock transition. The Nd element has rich, naturally stable isotopes- 142,143,144,145,146,148,150 Nd. The even isotopes have simple energy levels, which is advantageous for setting up the clock transition, whereas the hyperfine structures also have a wide range of usage for atomic clocks. For completeness, we provide the hyperfine structure constant A (MHz) of 143 Nd 9+ (nuclear spin I = 3/2 and nuclear magnetic moment µ I = −1.08µ N , where µ N is a nuclear magneton), which has the greatest natural abundance. Table 4 lists some clock-transition-related properties.  Table 4. The static scalar and tensor electric dipole polarizabilites, α E1 S and α E1 T (in a.u.), the magnetic dipole polarizabilites, α M1 (in a.u.), the values of the reduced matrix elements of J T (2) J (in a.u.), and the coefficient of the relativistic sensitivity K to variations in α.

Conclusions
In conclusion, we investigated the atomic structure of the Nd 9+ ion as a possible clock frequency standard. Many spectroscopic properties, such as energy levels, lifetimes, g J factors, electric quadrupole moments, electric dipole polarizabilities, and magnetic-dipole hyperfine structure constants of the low-lying atomic states, were provided by using comparative computations based on the MRCI and CI+MBPT methods. The dominant systematics in Nd 9+ are attainable below the level of 10 −19 . Moreover, a strong sensitivity to violation of the Lorentz invariance and variations in the fine-structure constant α was found in Nd 9+ . All of these findings strongly suggest the aptness of the Nd 9+ ion as a promising optical clock and cast its great potential for application for the probing of new physics beyond the Standard Model.