#  Universal and Non-Universal Correction Terms of Bose Gases in Dilute
Region: a Quantum Monte Carlo Study

Akiko Masaki-Kato1,2 Yuichi Motoyama3 and Naoki Kawashima3,4
<EMAIL_ADDRESS>Condensed Matter Physics
Laboratory1Computational Condensed Matter Physics Laboratory RIKEN RIKEN Wako
Wako Saitama 351-0198 Saitama 351-0198 Japan
2Computational Materials Science Research Team Japan
2Computational Materials Science Research Team RIKEN Center for Computational
Science (R-CCS) RIKEN Center for Computational Science (R-CCS) Kobe Kobe Hyogo
650-0047 Hyogo 650-0047 Japan
3The Institute for Solid State Physics Japan
3The Institute for Solid State Physics The University of Tokyo The University
of Tokyo Chiba 277-8581 Chiba 277-8581 Japan
4Trans-scale Quantum Science Institute Japan
4Trans-scale Quantum Science Institute The University of Tokyo The University
of Tokyo Bunkyo Bunkyo Tokyo 113-0033 Japan Tokyo 113-0033 Japan

###### Abstract

We study dilute gases of interacting bosons at zero-temperature in the region
where the system is characterized only by the $s$-wave scattering length. We
carry out quantum Monte Carlo simulation of the Bose Hubbard model and a
continuous-space hard-core model. Fitting the extended Lee-Huang-Yang formula
to the Monte Carlo results establishes the detailed mapping from the lattice
model to the continuous field theory characterized only by the $s$-wave
scattering length. Our estimate of the intrinsic $s$-wave scattering length
$a_{s}$ of the Bose-Hubbard model is $a_{s}/a_{l}=0.316(2)$ where $a_{l}$ is
the lattice constant. It turned out that inclusion of the universal second
correction term of $O(n\log n)$ makes the fitting worse while it is restored
by inclusion of the non-universal third correction term, of which the
existence was predicted analytically.

##  1 Introduction

Ground-state properties of homogeneous dilute bosons have been well studied
since Bogoliubov theory was established[1]. It is widely believed that
interacting bosons show universal behavior in the dilute limit, i.e., their
thermodynamic properties do not depend on the details of the interaction but
are characterized only by the number density $n$ and the s-wave scattering
length $a_{s}$. Specifically, for the ground-state energy as a function of the
gas parameter $\sqrt{n{a_{s}}^{3}}$, Lee, Huang and Yang obtained the first-
order correction to the Bogoliubov mean-field approximation (LHY correction)
using the perturbation theory[2, 3], which was followed by higher-order terms
obtained by Wu[4],

$\displaystyle\frac{E}{N}$ $\displaystyle=$
$\displaystyle\frac{2\pi\hbar^{2}na_{s}}{m}\left[1+\left(\frac{128}{15\sqrt{\pi}}\right)\sqrt{na_{s}^{3}}\right.$
(1) $\displaystyle+$
$\displaystyle\left.\frac{8(4\pi-3\sqrt{3})}{3}na_{s}^{3}\log(na_{s}^{3})+O(na_{s}^{3})\right],$

where $a_{s}$, $n$ and $m$ are the $s$-wave scattering length, the number
density and the mass of a boson, respectively. The condensate density $n_{0}$
was obtained by Bogoliubov[1] as

$\displaystyle n_{0}=1-\frac{8}{3\sqrt{\pi}}\sqrt{na_{s}^{3}}.$ (2)

It was shown [5, 6, 7, 8] that the first three terms (up to the one with the
logarithmic factor) do not depend on the character of the two-body
interaction, whereas the higher terms do. Below we call the first three terms
in (1) “universal” and the low-density region in which the energy is well
approximated by the first three terms the universal region. While we are
mainly interested in the universal behaviors, in what follows, we study a
broader range that seems to require non-universal terms for a full
explanation.

Recent developments in low-temperature atomic physics shed light on the
fundamental properties of bosonic systems. The static properties of bosonic
systems have been actively investigated by experiments and theories on ultra-
cold atoms over the past two decades. Numerical simulations are effective for
accurately treating inter-atomic interactions with various types of potentials
and with a wide range of strength to corroborate the analytical solutions. For
the continuous-space systems, finite-temperature quantum Monte Carlo studies
have provided estimates of the critical temperature as a function of
$na_{s}^{3}$ for the 3D hard-sphere gas in the homogeneous space[9] and with
the external parabolic potential[10]. Using the diffusion Monte Carlo (DMC)
method at $T=0$ Giorgini $et$ $al$ [11] presented a detailed comparison of the
numerical results with the LHY corrections to the Bogoliubov mean-field. They
obtained results in the universal region while they also found that the
logarithmic correction by Wu[4] is insufficient for predicting experimental
results for cold bosons and liquid 4He with realistic densities[11]. This may
be because the particle density in the experiments was too high for the
perturbative expansion Eq.(1) to be a quantitatively good approximation.

As mentioned above, Bose gas with the delta-function potential in continuous
space and the Bose-Hubbard model on the cubic lattice are the two most
fundamental models for bosons, and each of them has been studied extensively
for many years. While it is obvious from the theoretical point of view that
the two models are essentially equivalent in the diluted limit, where the
short-range details of the system are irrelevant, the quantitative
correspondence between them has never been studied in detail. Once the
quantitative correspondence has been established, one will be able to benefit
equally well from the study of either one of the models. In the universal
region, we expect that the behaviors of the two models are characterized only
by the s-wave scattering length. In the present short paper, by studying the
dilute region of Bose gases using the quantum Monte Carlo (QMC) methods for
lattice and continuous spaces, we establish the quantitative correspondence
between the two, and verify the validity of the analytical low-density
expansion by Lee, Huan and Yang and other groups. In particular, we obtain a
concrete estimate of the s-wave scattering length $a_{s}$, in the unit of the
lattice constant, $a_{l}$.

##  2 Lattice and Continuous Models

For the continuous system, we consider the Hamiltonian of interacting bosons
expressed as

${\cal
H}_{c}=-\frac{\hbar^{2}}{2m}\sum_{i}\frac{\mathrm{d}^{2}}{\mathrm{d}{\bf
r}_{i}^{2}}+\sum_{i<j}V({\bf r}_{i},{\bf r}_{j}),$ (3)

where ${\bf r}_{i}$ is the spatial coordinate of an $i$-th particle and $V$ is
a two-body interaction. For the lattice system, we employ the hard-core Bose-
Hubbard model in the cubic lattice given by

$\displaystyle{\cal H}_{l}=-t\sum_{\langle i,j\rangle}b^{\dagger}_{i}b_{j},$
(4)

where $t$ is the hopping constant, and $b_{i}(b_{i}^{\dagger})$ the
annihilation (creation) operator. We restrict the Hilbert space to ensure the
hard-core condition $n_{i}=0,1$, where $n_{i}$ is the number operator at $i$th
site. In our calculation of the lattice systems, we introduce the chemical
potential and simulate the grand-canonical ensemble. However, to make the
comparison with previous fixed-number[11] calculations more accurate, we
extract the canonical-ensemble averages from the output of our grand-canonical
simulation. Therefore, the numerical results presented below are the
canonical-ensemble averages.

Figure 1:  (Color) The dimensionless superfluid density $\rho_{s}a_{s}^{3}$ of
the continuous and the lattice models plotted against the density. The grand-
canonical ensemble averages are plotted. Purple diamonds represent $\rho_{s}$
of the lattice system with $L/a_{l}=32,\beta t=64$ and $\gamma=0.3151$. (See
Table I) Other points are for the continuous-space system with $L/a_{s}=20,30$
and $\beta\epsilon_{0}=80-180$ respectively. The dashed line corresponds to
the “perfect” superfluid ($\rho_{s}/n=1$).

At first, we show the comparison between the superfluid densities of the
continuous-space system and the lattice system. The superfluid density is
defined as the free-energy response to the infinitesimal flux. As is well
known, the superfluid density can be expressed with the winding number of
worldlines with the periodic boundary condition[12] as,
$\rho_{s}=\frac{m}{\hbar^{2}}\frac{\langle|{\bf W}|^{2}\rangle}{d\beta
L^{d-2}}$. To study the connection between the different systems, we examine
the dimensionless superfluid density

$\rho_{s}a_{s}^{3}=\frac{\langle|{\bf
W}|^{2}\rangle}{2d\beta\epsilon_{0}\tilde{L}},$ (5)

where $d$ is the space dimension, $\tilde{L}\equiv L/a_{s}$ is the
dimensionless linear size of the system. For the unit of the energy, we adopt
$\epsilon_{0}\equiv\frac{\hbar^{2}}{2ma_{s}^{2}}$. Here we introduce $\gamma$
as the $s$-wave scattering length of the lattice system measured in the unit
of the lattice constant, i.e., $\gamma\equiv a_{s}/a_{l}$. The s-wave
scattering length is the key-parameter because it is the only parameter that
characterizes the system in the universal low-density region. While its value
is obvious in the continuous-space delta-function model, it is not obvious for
the Bose Hubbard model. In fact, to our knowledge, its value has never been
estimated before. In order to establish the quantitative correspondence
between the continuous-space and the discrete-space systems, it is the key
quantity. The hopping constant $t$ for the hyper-cubic lattice is related to
$m$ by $t=\frac{\hbar^{2}}{2ma_{l}^{2}}$. As an order parameter of the
superfluid state, $\rho_{s}$ is more convenient in the simulation methods we
use for the present work than $N_{0}$, which is an off-diagonal quantity. For
continuous-space model (Eq. (3)), we adopt the hard-core interaction

$V({\bf r},{\bf r}^{\prime})=\begin{cases}\infty&\text{if}\quad|{\bf r}-{\bf
r}^{\prime}|\leq r_{\text{c}}\\\ 0&\text{otherwise}\end{cases},$ (6)

with the hard-core radius $r_{\text{c}}$, which is the same as the $s$-wave
scattering length $a_{s}$. the world-line Monte Carlo (WLMC) method based on
the continuous-time directed-loop algorithm for the lattice system[13, 14,
15], and WLMC method with worm update for the continuous-space model [16, 17].
For the discretization step of the imaginary time in simulations of the
continuous-space model, we have used $\Delta\tau=0.05\epsilon_{0}^{-1}$, which
we find small enough to ensure the convergence of the results. Points for the
continuous system in Fig. 1 represent particle density dependence of the
superfluid density for several combinations of system size and inverse
temperature, i.e., $L/a_{s}=20,30$ and $\beta\epsilon_{0}=4L/a_{s},6L/a_{s}$.
In the scale shown in Fig. 1, the size and the inverse-temperature dependence
is hardly recognizable. By using $\gamma=0.3151$ (See Table I), obtained by
the fitting based on the Bayesian regression as mentioned below, the lattice
results and the continuous-space results agree with each other in the low-
density region $na_{s}^{3}\lesssim 10^{-4}$. In this region, the system is a
nearly-perfect condensate as is indicated by the dashed line, i.e., almost all
particles are in the superfluid component.

##  3 LHY Formula Fitting to Lattice Model

Next, we consider the connection between the $s$-wave scattering length and
the lattice constant. Let us consider the ground-state energy per particle in
the continuous space and in the discrete space. To compare the two systems on
equal footing, we define the dimensionless energy per particle
$\tilde{\epsilon}\equiv N^{-1}\langle{\cal H}\rangle/\epsilon_{0}$. We can
express this quantity by the universal dimensionless function:

$\tilde{\epsilon}=f(na_{s}^{3}),$ (7)

where $n=N/\Omega$ is the particle density. By comparing the energy densities
of the two systems we can estimate the unknown parameter $\gamma$. In this
study we assume a form that includes the analytical prediction as a special
case. Specifically, we consider the following form for $f$ in (7).

$\displaystyle f(x)$ $\displaystyle=$ $\displaystyle 4\pi
x\times\left(1+c_{1}\frac{128}{15\sqrt{\pi}}\sqrt{x}\vphantom{+c_{2}\frac{8(4\pi-3\sqrt{3})}{3}x\ln
x}\right.$ (8) $\displaystyle\left.\qquad+c_{2}\frac{8(4\pi-3\sqrt{3})}{3}x\ln
x+c_{3}x\right),$

In (8), the analytical predictions are $c_{1}=1$[2] and $c_{2}=1$ [4]. It was
argued that $c_{1}$ and $c_{2}$ are universal, independent of the shape of the
model’s potential, whereas $c_{3}$ is not[4, 7].

Because our calculation simulates the grand-canonical ensemble, we can obtain
canonical-ensemble averages for many different values of $n$ by a single
Monte-Carlo run. For this conversion, we accumulate measured energy (and other
quantities as well) separately for each total particle number during our
grand-canonical Monte Carlo simulation. The separate accumulation makes it
possible to compute the canonical averages of the energy as a function of the
density in a narrow but finite range centered around the grand canonical
average of the density.

In our simulation, the range of the system size is from $L=16$ up to $64$, and
the inverse temperature $\beta=32$–$256$. Although the data for the largest
system sizes are shown in Fig. 2 together with the data for smaller systems,
we could not collect a sufficiently large number of independent samples that
could contribute to the minute fitting discussed below, and hence they are not
used in the fittings that require precision. Only the system of the size
$(L,\beta)=(32,64),(32,96)$ and $(48,96)$ are used in the other figures and
Table I. For each value of the chemical potential, we have confirmed that the
numerical estimates have reached the saturation as a function of $L$ and
$\beta$ within the statistical error, i.e., the results shown in all the
figures are for sufficiently large $L$ and $\beta$ so that they can be
regarded as those of the thermodynamic limit at zero temperature in the scale
adopted for each figure.

Together with the estimated correction to the Bogoliubov mean-field values, we
also show the fitting function Eq. (8). In particular, we have explored the
dilute region where the systems’ behaviors do not depend on the specific form
of the inter-particle interaction nor the discreteness of the space, apart
from non-universal corrections such as the $c_{3}$ term mentioned above. In
the WLMC simulation for the lattice system, the total energy can be measured
through the number of kinks $N_{k}$ in world lines:

$\langle{\cal H}_{l}\rangle=-N_{k}/\beta.$ (9)

Figure 2:  (Color online) The canonical ensemble average of the dimensionless
energy per particle of the Bose-Hubbard model in three dimensions as a
function of the parameter $na_{s}^{3}$. The errors are shown but too small to
be visible. The fitting curves are obtained from (8) by setting the
coefficients in Table I, except for $\gamma$. In all cases, the s-wave
scattering length was set $\gamma\equiv a_{s}/a_{l}=0.32$, which is not
necessarily the optimal value for each case in Table I, although the deviation
from the optimal fitting is negligible in the shown scale.

Figure 2 shows $\tilde{\epsilon}$ as a function of $na_{s}^{3}$. We can see
that our WLMC results roughly agree with the previous DMC results of the
continuous-space model up to the density around $na_{s}^{3}\sim 0.003$, while
the deviation becomes detectable at larger density $na_{s}^{3}\sim 0.001$,
which is consistent with the previous observation[11].

Figure 3:  (Color online) The interpolated canonical ensemble average of the correction to the Bogoliubov mean-field value of the scaled energy, $\tilde{\epsilon}-4\pi na_{s}^{3}$, as a function of the dimensionless parameter $na_{s}^{3}$. We use $f(x)-4\pi x$ with $f$ in Eq (8) for the fitting function with $\gamma$ and $c_{3}$ as the fitting parameters while $c_{1}=c_{2}$ are fixed to be 1. The estimates of these parameters are $\gamma=0.3151$ and $c_{3}=119$ optimized for $f^{\rm({IV})}$.  fit. func. |  $\gamma$ | $c_{1}$ | $c_{2}$ | $c_{3}$
---|---|---|---|---
I | 0.33709(6) | 0 | 0 | 0
II | 0.31178(5) | 1 | 0 | 0
III | 0.32574(7) | 1 | 1 | 0
IV | 0.3151(2) | 1 | 1 | 119(3)
Table 1:  Values of the parameter $\gamma$ and the coefficients used in
drawing curves shown in the figures. They are estimated using the canonical
ensemble data from the system sizes $(L,\beta)=(32,64),(32,96)$ and $(48,96)$.

In order to see the differences among various fittings more clearly, we show
in Fig. 3 the result of $\tilde{\epsilon}$ with the leading term (the
Bogoliubov mean-field value) subtracted. We note here that, for this finer
comparison, the original canonical ensemble averages are too noisy, because,
as we stated above, we have split a grand-canonical ensemble simulation into
many smaller pieces of canonical ensemble simulation, which makes the number
of statistically independent data contributing each canonical average small.
Therefore, for finer comparison such as Figures 3 and 4, we apply the linear
fitting to the resulting canonical averages (coming from the same simulation
at the fixed chemical potential) as a function of the density. Based on the
interpolation using this fitting, we obtained for each grand-canonical
ensemble simulation the interpolated canonical average of the energy at the
central value of the density. However, because the system sizes and the
inverse temperatures are sufficiently large, the difference between the
canonical ensemble and the grand-canonical ensemble is not significant in any
of the figures.

There are three curves in Fig. 3 corresponding to different places of
truncation of Eq. (8). While the parameters used for each curve and their
numerical estimates are obtained by optimizing the fitting to the (un-
interpolated) canonical ensemble data, which have a one-to-one correspondence
with the interpolated canonical data shown in Fig. 3. The actual figures thus
obtained are summarized in Table 1. In the case I, II and III, the relative
$s$-wave scattering length $\gamma$ is the only fitting parameter whereas, for
IV, $\gamma$ and $c_{3}$ are optimized. The optimal values and the associated
statistical errors of the parameters were obtained by the grid search. The
functional form II with only the first universal correction seems to explain
the numerical data rather well, with a slight but statistically significant
deviation as we see in Fig.3. The curve III shows that the inclusion of the
universal $c_{2}$ term makes the whole fitting worse. The reasonable fitting
in a broad range of the density is recovered by including the $c_{3}$ term
(IV). This term was predicted analytically as a non-universal correction term
that may depend on the short-range physics[4, 18, 7]. However, the numerical
estimate of its amplitude has been missing. The fitting IV has the broadest
range of good fitting as we see in Fig.3. A more detailed comparison between
the different functional forms can be found in Appendix.

While IV is the most consistent with the analytical predictions, the estimates
of Table I are based on the data from different $L$ and $\beta$, assuming that
the size dependence can be negligible relative to the statistical error of the
Monte Carlo simulation. Although we have observed no apparent inconsistency in
this assumption, to obtain a more reliable estimate, we fit the functional
form IV to the data from different system sizes separately. As a result we
have obtained $(\gamma,c_{3})=(0.3151(3),116(4)),(0.315(1),130(40))$ and
$(0.316(2),125(5))$, for $(L,\beta)=(32,64),(32,96)$ and $(48,96)$,
respectively. We did not use data from the larger systems because they are too
noisy and showed no significant difference from the estimate quoted above
beyond the statistical error. From these results, we quote the following
estimate that covers all the three mentioned above, as our conservative but
reliable estimate of the intrinsic $s$-wave scattering length and the
coefficient $c_{3}$ of the Bose Hubbard model:

$a_{s}/a_{l}=0.316(2)\quad\mbox{and}\quad c_{3}=130(40).$ (10)

##  4 Summary and Remarks

In summary, we carried out quantum Monte Carlo simulations of the Bose-Hubbard
model on the square lattice and the continuous-space hard-core Bose gas model.
By comparing the numerical results with the analytical prediction of the LHY
formula with additional non-universal correction, we determine two unknown
parameters, i.e., the $s$-wave scattering length and the amplitude of the non-
universal correction, thereby establishing the quantitative mapping from the
lattice model to the field-theoretical model characterized only by the
$s$-save scattering length. Our estimate for the intrinsic $s$-wave scattering
length is $0.316(2)$ in the unit of the lattice constant.

We note here that there is an analytical prediction that the amplitude of the
universal second correction is different[19, 20] from the value used in this
paper. The prediction was made based on the formalism discussed
previously[21]. However, this formalism was criticized recently[22] for its
multiple counting of the diagram of the single-particle Green’s function.
Since the independent and precise numerical estimation of the coefficients
$c_{1}$ and $c_{2}$ in (8) requires better accuracy and computation of much
larger systems, we have to leave it for a subject of the future work. Until
then, our estimates of the parameters should be regarded as the ones obtained
with the assumption that the analytical form (8) is correct (with
$c_{1}=c_{2}=1$).

The present method for obtaining model parameters by comparison between the
lattice model and the continuous model can be used in the estimation of the
strength of the tight-binding hopping not only in dilute cases but also in
more general cases. For example, simulations of liquid 4He absorbed on the
graphite[23] attract attention, especially in the second layer where the
existence of novel quantum phase has been discussed[23]. However, it is not
easy to obtain complete convergence with respect to system size, inverse
temperature and equilibration, in simulations of continuous systems[24]. While
it is not trivial to construct a lattice model that explains the realistic
Helium system with high accuracy, the present work may be useful in doing so.
Since parallelization technique has been developed for lattice systems[25],
this is a promising direction for future works.

This work was supported in part by Grants-in-Aid for Scientific Research
No.17K14361 and 18H01183. We are thankful for providing computational
resources of the K computer in RIKEN R-CCS through the HPCI System Research
Project (Project Nos.: hp160152, hp170213, hp180098, hp180129 and hp180225).
The computations were performed also on supercomputers at Advanced Center for
Computing and Communications (ISC), RIKEN, and at Supercomputer Center, ISSP,
the University of Tokyo. NK was supported by ImPACT Program of Council for
Science, Technology and Innovation (Cabinet Office, Government of Japan).

## References

  * [1] N. N. Bogoliubov, J. Phys. 11, 23 (1947).
  * [2] T. D. Lee, K. Huang and C. N. Y. Yang Phys. Rev. 106, 1135 (1957).
  * [3] A. L. Fetter and J. D. Walecka, “Quantum Theory Of Many-Particle Systems”, Dover Publication inc..
  * [4] T. T. Wu, Phys. Rev. 115, 1390 (1959).
  * [5] K. A. Brueckner and K. Sawada, Phys. Rev. 106, 1117 (1957); 106, 1128 (1957).
  * [6] S. T. Beliaev, Zh. Eksp. Teor. Fiz. 34, 433 (1958) (Sov. Phys. JETP 7, 299 (1958))
  * [7] N. M. Hugenholtz and D. Pines, Phys. Rev. 116, 489 (1959).
  * [8] E. H. Lieb, Phys. Rev. 130, 2518 (1963).
  * [9] P. Gürter and D. Ceperley, Phys. Rev. Lett. 79, 3549 (1997).
  * [10] S. Pearson, T. Pang and C. Chen, Phys. Rev. A 58, 4796 (1998).
  * [11] S. Giorgini, J. Boronat and J. Caulleras, Phys. Rev. A 60, 5129 (1999).
  * [12] E. L. Pollock and D. M. Ceperley, Phys. Rev. B 36, 8343 (1987).
  * [13] N. Kawashima and K. Harada, J. Phys. Soc. Jpn. 73, 1379 (2004).
  * [14] Y. Kato, T. Suzuki, and N. Kawashima, Phys. Rev. E 75, 066703 (2007).
  * [15] Y. Motoyama, K. Yoshimi, A. Masaki-Kato, T. Kato and N. Kawashima, Comput. Phys. Comm. 264 107944 (2021). (https://github.com/issp-center-dev/dsqss)
  * [16] D. M. Ceperley, Rev. Mod. Phys 67, 279 (1995).
  * [17] M. Boninsegni, N. V. Prokof’ev and B. V. Svisunov, Phys. Rev. E 74, 036701 (2006).
  * [18] K. Sawada, Phys. Rev. 116, 1344 (1959).
  * [19] K. Tsutsui and T. Kita, J. Phys. Soc. Jpn 82, 063001 (2013).
  * [20] K. Tsutsui, Y. Kato, and T. Kita: J. Phys. Soc. Jpn. 85, 124004 (2016).
  * [21] T. Kita, Phys. Rev. B 80, 214502 (2009); T. Kita, Phys. Rev. B 81, 214513 (2010).
  * [22] S. Watabe, New J. Phys. 22, 103010 (2020).
  * [23] S. Nakamura, K. Matsui, T. Matsui, and H. Fukuyama, Phys. Rev. B, 94, 180501(R) (2016).
  * [24] P. Corboz, M. Boninsegni, L. Pollet and M. Troyer, Phys. Rev. B 78, 245414 (2008).
  * [25] A. Masaki-Kato, T. Suzuki, K. Harada, S. Todo and N. Kawashima, Phys. Rev. Lett. 112, 140603 (2013).

##  Appendix A More detailed comparison

Figure 4:  (Color online) More detailed comparison of the numerical and the
analytical result for the energy. The interpolated canonical ensemble averages
are shown. Panels (II), (III) and (IV) corresponds to the fitting functions
II, III, and IV, respectively in Table I. The data for the continuous space
calculation are taken from the work by Giorgini et al.[11].

In order to examine the effect of each correction term more closely, we show
fittings corresponding to the three sets of parameters listed as II, III and
IV in Table 1. The three panels in Figure 4 show the grand canonical data from
the Monte Carlo simulation with the first term subtracted and normalized so
that the second term is represented by a constant, i.e.,

$g(x)\equiv\frac{f(x)/(4\pi x)-1}{\sqrt{x}}$ (11)

Note that we fit only the data of the lattice system, while the behavior
agrees well with the continuous-space system in the dilute region. The LHY
analytical prediction for the coefficient is used in (II), as represented by
the horizontal straight line. Even though the results of the lattice system
and the continuous system are close to each other in the region
$na_{s}^{3}<10^{-3}$ with the best estimate of $\gamma$, the functional form
in II does not represent the numerical data well. In the case of III, which
includes up to the logarithmic correction term, the lattice system and
continuous system agree within the statistical errors in the region
$na_{s}^{3}<10^{-4}$. However, the analytical results (fitting curve) rapidly
deviate above $na_{s}^{3}\sim 10^{-5}$. In the case of IV, the fitting
function includes the potential-specific non-universal coefficient $c_{3}$.
Using the value of $\gamma$ obtained from the fitting with this non-universal
correction term, the lattice system can be fitted by the analytical curve up
to $na_{s}^{3}\sim 10^{-3}$. In this region, both the lattice and the
continuous systems agree with each other by 10% difference, though this may be
outside the universal region.

These observations suggest that while the universal analytic expansion,
$f^{\rm({III})}$, explains the energy in the extreme dilute region
$na^{3}_{s}<10^{-5}$, the non-universal correction is required for explaining
the behavior of the lattice system up to $na^{3}_{s}\sim 10^{-3}$.


# Ultrastable optical, XUV and soft-x-ray clock transitions in
open-shell highly charged ions

Chunhai Lyu<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Christoph H. Keitel  Max-Planck-
Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Zoltán
Harman  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germany

(August 28, 2024)

###### Abstract

Highly charged ions (HCIs) are insensitive to external perturbations and are
attractive for the development of ultrastable clocks. However, only a few HCI
candidates are known to provide optical clock transitions. In this Letter, we
show that, as a result of strong relativistic effects, there are more than 100
suitable optical HCI clock candidates in more than 70 elements. Their
transitions are embedded in the fine-structure splitting of the $nd^{4}$,
$nd^{5}$ and $nd^{6}$ ground-state configurations with $n=3,4,5$ being the
principal quantum numbers. The corresponding high multipolarity transitions in
these ions have lifetimes and quality factors many orders of magnitude longer
and larger, respectively, than those in state-of-the-art clocks. Their
polarizabilities are also orders of magnitude smaller, rendering them more
stable against external electromagnetic fields. Furthermore, within the same
electronic configurations, the clock transitions in heavy ions scale up to the
XUV and soft-x-ray region, thus enable the development of clocks based on
shorter wavelengths. The existence of multiple clock transitions in different
charge states of a single element, as well as in a whole isoelectronic
sequence, would significantly enrich the detection of fine-structure constant
variations, the search for new physics and the test of nuclear theories via
high-precision spectroscopy.

High-precision optical clocks have wide applications in fields such as the new
definition of time units [1, 2, 3], the test of variations of the fine-
structure constant [4, 5, 6], and the constrain of a conjectured fifth force
[7, 8, 9] and other new physics phenomena [10, 11]. The elements of these
clocks are either neutral atoms or singly charged ions [12] based on
hyperfine-induced $ns^{2}\leavevmode\nobreak\
{}^{1}S_{0}-nsnp\leavevmode\nobreak\ {}^{3}P_{0}$ [13, 14, 15, 16, 17, 18] and
electric quadrupole $(n+1)s\leavevmode\nobreak\
{}^{1}S_{1/2}-nd\leavevmode\nobreak\ {}^{2}D_{5/2}$ transitions [19, 20, 21,
22, 23] with a quality factor $Q=\nu/\Delta\nu$ in the range of $10^{14}\sim
10^{17}$. Here, $n$ is the principal quantum number, $\nu$ the transition
frequency and $\Delta\nu$ the corresponding natural linewidth. For specific
cases such as Lu+ and Yb+, electric quadrupole and octupole transitions
between $6s^{2}-6s5d$ [24, 25] and between $4f^{13}6s^{2}-4f^{14}6s$ [26, 27]
with $Q$ factor in the range of $10^{21}$ and $10^{23}$, respectively, are
also employed.

Nevertheless, these clocks suffer from large systematic shifts and
uncertainties induced by black body radiation [28, 29] and external trapping
fields [30, 12]. To overcome these difficulties, clocks based on a nuclear
transition in the 227Th isomer [31, 32, 33] and electronic transitions in
highly charged ions (HCIs) [34, 35, 36, 37] are put forward as they bear
smaller polarizabilities and are sensitive to $\alpha$ variations. While the
radioactive 227Th nuclear clock, with $Q=2\times 10^{20}$, is still to be
implemented for a direct laser excitation [38], a HCI clock based on Ar13+
with a modest $Q=4\times 10^{13}$ has already been demonstrated via quantum
logic spectroscopy [35].

One category of HCI candidates are based on hyperfine transitions of the $1s$
electron in hydrogenlike ions [39, 40]. This transition has an energy of 6
$\mu$eV for the hydrogen atom, but scales up to 1.4 eV (886 nm) for ions as
heavy as 207Pb81+. The second variety of HCI clock transitions are from fine-
structure splittings of $np$ valence electron systems [41, 42, 43, 44]. Though
these ions have polarizabilities many orders of magnitude smaller than neutral
atoms or singly charged ions, their clock transitions are of magnetic dipole
transitions with $Q\sim 10^{13}$ [40, 35]. Thus, one needs a long
interrogation time to achieve high accuracy. Another type of HCI candidates
comes from level crossings between the $4f-5s$ [45, 46, 47], $4f-5p$ [48, 49,
50], and $5f-6p$ [51, 52] orbitals. However, only a small number of ions close
to the crossing points would provide optical clock transitions, not to mention
that the close lying orbitals usually induce large polarizabilities [53, 41].

Figure 1:  Low-lying levels of (a) U50+, (b) U52+ and (c) U51+ ions. Clock
transitions are depicted by blue lines. Similar level structures exist for
[Ar]$3d^{m}$ and [Xe]$4f^{14}5d^{m}$ ($m=4,5,6$) ions.

In this Letter, we put forward a category of ultrastable HCI clocks that
possess $Q$ factors beyond $10^{23}$ and electric polarizabilities in the
range of $10^{-42}$ J m2/V2. As shown in Fig. 1, the clock transitions come
from fine-structure splittings in ions with one of the open-shell $nd^{4}$,
$nd^{5}$ and $nd^{6}$ ground-state configurations. These configurations have
tens of terms with total angular momenta ranging from $J=0$ and 1/2 to $J=6$
and 13/2. For light ions, the energy levels order linearly according to the
values of the total angular momentum, and thus the lifetimes of the excited
states are dominated by $M1$ transitions to $J\pm 1$ states. These forbidden
transitions have found numerious applications in the diagnostics of
astrophysical and laboratory plasma properties [54, 55, 56]. However, for
heavy ions which are also highly charged, the strong relativistic influence
leads to a reordering of the terms and renders it possible to obtain
ultrastable clocks based on high-multipolarity $E2/M3$ forbidden transitions.

For the case of $nd^{6}$, optical clock transitions exist throughout the whole
isoelectronic sequences up to U68+, U50+ and U18+ for $n=3,4,5$, respectively,
providing nearly a hundred of HCI clocks. Some HCIs, such as U68+, might be
challenging to produce, but in principle can be obtained within an electron
beam ion trap [57] or a HITRAP [58]. High-precision spectroscopy of these
transitions would enable an accurate determination of nuclear parameters for
more than 70 elements and their isotopes ranging from _22^46Ti to _92^238U and
beyond, and test state-of-the-art nuclear theories [59, 60, 61, 62, 63]. For
$nd^{4}$ and $nd^{5}$, tens of optical clock transitions also exist in light
elements, but the transition energies scale up to the XUV and soft-x-ray
region for heavy elements. Considering that the stability of optical lasers is
approaching the fundamental limit set by thermal fluctuations of reference
cavities [64, 65, 66, 67], clocks based on lasers with shorter wavelengths
[68, 69] would be one of the approaches to further improve the clock
performance. Furthermore, most of these ions also have more than two clock
transitions with distinct sensitivities to the variation of $\alpha$,
rendering them good candidates for the test of new physics beyond the standard
model [10].

Figure 2:  Excitation energies of the $nd^{6}\leavevmode\nobreak\
{}^{5}D_{3,2,1,0}$ state in (a) Cr- and (b) Mo-like ions as functions of
atomic number $Z$. With a ground state of ${}^{5}D_{4}$, term reordering leads
to two clock states with $J=2$ and $J=0$, respectively. Insets are the
enlarged depiction around the crossing points.

MCDHF-RCI calculations — To accurately predict the transition energies and
lifetimes of these clock states, we employ ab initio fully relativistic
multiconfiguration Dirac–Hartree–Fock (MCDHF) and configuration interaction
(RCI) methods implemented in the GRASP2018 codes [70, 71]. Within this scheme,
the many-electron atomic state functions (ASFs) are expanded as a linear
combination of configuration state functions (CSFs) which are $jj$-coupled
Slater determinants of one-electron orbitals. The radial wavefunction for each
orbital is obtained via solving the self-consistent MCDHF equations under the
Dirac–Coulomb Hamiltonian. Then, the RCI method is applied to account for
corrections arising from mass shift, quantum electrodynamic and Breit
interactions.

Table 1: Predicted transition energy $\hbar\omega$, lifetime $\tau$, sensitivity to $\alpha$ variations $K$, and differential polarizability $\Delta\alpha_{eg}$ ($10^{-40}$ J m2/V2) of clock states in selected [Kr]$4d^{6}$ ions.  ions | $|e\rangle$ |  $\hbar\omega$ (eV) |  $\tau$ (s) | $K$ | $\Delta\alpha_{eg}$
---|---|---|---|---|---
Ce16+ | ${}^{3}D_{2}$ | 1.70(1) | 1156 | $-0.55$ | $-0.22$
${}^{3}P_{0}$ | 3.23(1) | 602 | $-1.50$ | $-0.45$
Dy24+ | ${}^{3}D_{2}$ | 2.20(1) | 545 | 0.06 | $-0.10$
${}^{3}P_{0}$ | 5.98(2) | 11.7 | $-1.02$ | $-0.20$
W32+ | ${}^{3}D_{2}$ | 2.67(1) | 332 | 0.01 | $-0.05$
${}^{3}P_{0}$ | 8.69(3) | 1.63 | $-0.61$ | $-0.10$
Pb40+ | ${}^{3}D_{2}$ | 3.23(1) | 208 | 0.05 | $-0.031$
${}^{3}P_{0}$ | 11.25(4) | 0.58 | $-0.38$ | $-0.062$
U50+ | ${}^{3}D_{2}$ | 3.99(1) | 125 | 0.07 | $-0.017$
${}^{3}P_{0}$ | 14.31(4) | 0.27 | $-0.23$ | $-0.035$

For the low-lying states in $nd^{6}$, $nd^{4}$ and $nd^{5}$, the scaling of
their excitation energies with respect to the atomic number $Z$ are plotted in
Figs. 2-4, respectively, where a clear reordering of the levels is observed.
These values are obtained via simple calculations that only account for
correlation effects within the corresponding ground-state configurations.
However, for specific ions where accurate values are needed, such as the ions
around the crossing points discussed below and the ions shown in Table 1,
large-scale calculations with millions of CSFs are performed. This is obtained
by allowing single and double excitations from the $ns$, $np$, and $nd$
orbitals up to the $8k$ orbitals. As a consequence, the calculated energies
and lifetimes bear a relative accuracy of 0.5% and 20%, respectively [72].
Increasing the size of the CSF basis set systematically shifts the values of
the excitation energies depicted in Figs. 2-4 downward by up to $10\%$, but
has little effects on the crossing points of the levels.

$nd^{6}$ ions — For ions in this configuration, there are 34 atomic states
with total angular momentum ranging from $J=0$ to 6, with a ${}^{5}D_{4}$ term
being the ground state. In light ions, the corresponding low-lying excited
states are ${}^{5}D_{3,2,1,0}$, respectively, and decay via $M1$ transitions.
Usually, the energies of the fine-structure multiplet scale with $Z^{2}$ [34],
which is the case for $J=1,3$ shown in Fig. 2. However, for heavy elements,
strong relativistic effects leads to large splitting between the $nd_{3/2}$
and $nd_{5/2}$ relativistic orbitals (the subscripts are the corresponding
single-electron total angular momenta). As a consequence, the ASFs for states
$J=0,2$ become dominated by the $nd_{3/2}^{4}nd_{5/2}^{2}$ configuration [73],
and their energies only increase linearly with respect to $Z$, and become the
first two excited states. These two levels decay via slow $E2$ transitions and
thus represent the two clock states illustrated in Fig. 1(a). We notice that a
similar unusual scaling of the fine-structure splittings also allows clock
transitions in low charged, so-called group-16-like ions [74, 75, 76].

For Mo-like ions ([Kr]$4d^{6}$) shown in Fig. 2(b), the level reordering of
the $J=2$ state happens at Ba14+. The transition energy between ${}^{5}D_{4}$
and ${}^{3}D_{2}$ is around 1.70 eV for Ce16+, and scales up to 3.99 eV for
U50+. With a lifetime of 625 and 63 s for the two ions (see Table 1),
respectively, these transitions have a $Q$ factor of $1.6\times 10^{18}$ and
$3.8\times 10^{17}$, respectively, which are orders of magnitude higher than
those in current optical clocks [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].
Particularly, they are more than 4 orders of magnitude higher than the first
HCI clock based on Ar13+ [40, 43]. Thus, this isoelectronic system would
contain more than 36 ion candidates suitable for optical HCI clocks operating
at various wavelengths.

For Cr-like ions ([Ar]$3d^{6}$) as presented in Fig. 2(a), the $J=2$
metastable state starts to emerge from Pd22+. As the electrons are more
tightly bound, the transition energies are relatively larger than those in Mo-
like ions but are still accessible for optical lasers. Specifically, the
corresponding transition energy and lifetime are 3.78 eV and 437 s,
respectively, for Cd24+, and scale to 11.4 eV and 18.3 s, respectively, for
U68+. This corresponds to $Q$ factors around $10^{18}$ for more than 46
optical HCI clocks in this isoelectronic sequence.

Figure 3:  Excitation energies of the $nd^{4}\leavevmode\nobreak\
{}^{5}D_{1,2,3,4}$ states in (a) Ti- and (b) Zr-like ions as functions of
atomic number $Z$. Term reordering leads to long-lived $J=4$ state. Insets are
the enlarged depiction around the crossing points.

Furthermore, the second metastable state (labelled by $J=0$) lies 3.23-14.3 eV
and 6.75-38.6 eV above the ground state for Mo- and Cr-like ions,
respectively. While it decays to the ground state via a highly forbidden $E4$
transition, its lifetime is dominated by an $E2$ transition to the $J=2$
excited state and ranges from 64 ms to 800 s. As shown in Table 1, this $J=0$
state is much more sensitive to the variation of the fine-structure constant
than that of the $J=2$ state. Taking the Mo-like Dy24+ as an example, with the
sensitivity defined by $K=(\Delta\omega/\omega)(\alpha/\Delta\alpha)$, one has
a $K=0.07$ for the ${}^{5}D_{4}-{}^{3}D_{2}$ transition, but a $K=-1.02$ for
the ${}^{5}D_{4}-{}^{3}P_{0}$ transition. Though the absolute sensitivities
are smaller in comparison to some clock transitions near level crossings [45,
46, 47, 48, 49, 50, 51, 52], the distinct $\alpha$ disproportionality of two
clock transitions in a single ion would largely reduce systematic
uncertainties in detecting $\alpha$ variations. Similar dual-clock transitions
also exist in Yb+ [5, 6], which provides the current most stringent constraint
of $\alpha$ variations. Nevertheless, one of the clock states in Yb+ has a
lifetime of only 7.3 ms ($Q=2.2\times 10^{14}$) that limits further
improvements.

Another advantage of these clock ions is that they are immune to external
fields. This property is mainly characterized by the small static dipole
polarizability defined by $\alpha_{a}=\sum_{k\neq
a}-3\hbar^{2}f_{ka}/2m(\hbar\omega_{ka})^{2}$ [77] Here, $|a\rangle$ is either
the ground state $|g\rangle$ or the excited clock state $|e\rangle$ , $\hbar$
the reduced Planck constant, $m$ the electron rest mass, $f_{ka}$ the electric
dipole oscillator strength of transition from $|a\rangle$ to an intermediate
state $|k\rangle$, and $\hbar\omega_{ja}$ the corresponding transition energy
in units of eV. The typical polarizability values for optical clocks are
larger than $10^{-40}$ J m2/V2 [78, 79]. However, for ions like U50+, the
lowest $|k\rangle$ state is from the $4d^{5}4f$ configuration which lies 270
eV above the two clock states. Making use of the Thomas–Reiche–Kuhn sum rule
$\sum_{k\neq a}f_{ka}=N$ ($N$ being the number of electrons in the ion) [80],
one can derive $\alpha_{e,g}<10^{-41}$ J m2/V2 which are orders of magnitudes
smaller in comparison to those in neutral atoms and singly charged ions.
Accordingly, the differential polarizabilities
$\Delta\alpha_{eg}=\alpha_{e}-\alpha_{g}$, as shown in Table 1, are also much
smaller.

Figure 4:  Excitation energies of the $nd^{5}\leavevmode\nobreak\
{}^{4}G_{5/2,7/2,9/2,11/2}$ states in (a) V- and (b) Nb-like ions as functions
of atomic number $Z$. Term reordering leads to long-lived $J=11/2$ state.
Insets are level structures of given ions.

$nd^{4}$ ions — As a counterpart of the above mentioned $nd^{6}$ isoelectronic
systems, the $nd^{4}$ ions have similar spectroscopic terms but with a
reversed level ordering. As shown in Fig. 3, these ions start with
${}^{5}D_{0}$ being the ground state and ${}^{5}D_{1,2,3,4}$ being the
subsequent excited states for light elements. Though the energies of all the
states scale dominantly with $Z^{2}$, the ${}^{5}D_{4}$ state gradually
becomes lower than the ${}^{5}D_{3}$ and ${}^{5}D_{2}$ states and becomes a
long-lived clock state shown in Fig. 1(b). Since their energies span from 9.0
to 200 eV, they can be probed with XUV frequency combs [68, 69] and are
suitable for future XUV and soft-x-ray ionic clocks. The higher transition
energy would further increase the $Q$ factor to the range of $10^{22}$ and
beyond, thus improve the short-term stability of a clock.

In the case of Zr-like ions ([Kr]$4d^{4}$), the clock transition energy varies
from 9.0 eV in Gd24+ to 60 eV in U52+. Through Gd24+ to Ta33+, the
${}^{3}H_{4}$ state is still above the ${}^{5}D_{2}$ state. Therefore, its
lifetime is mainly determined by the $E2$ transition and are of 5 hours to 20
days. With an energy of 9 to 20 eV, these clock transitions have a quality
factor of $Q=10^{20}\sim 10^{22}$. For heavier ions starting from W34+, the
${}^{3}H_{4}$ state becomes lower than the ${}^{5}D_{2}$ state. Thus, the
lifetime of this metastable state can be as long as tens to hundreds of years
and the quality factor of the transitions would go far beyond $10^{22}$. Such
a long lifetime makes excitation by lasers challenging, but the position of
these states can be determined with a sub-eV accuracy through mass
spectroscopy [81, 72]. Furthermore, the application of twisted lasers could
also enhance the transition rates of such high-multipolarity transitions [82].

Similarly, for Ti-like ions with a configuration of [Ar]$3d^{4}$, the
${}^{3}H_{4}$ state lies between the ${}^{5}D_{2}$ and ${}^{5}D_{3}$ states
for ions throughout Xe32+ to Pm39+, with a quality factor of $Q=10^{20}\sim
10^{23}$ for transition energies between 17.9 and 33.0 eV. When the
${}^{3}H_{4}$ state becomes lower than the ${}^{5}D_{2}$ state in Sm40+, its
lifetime again becomes as long as hundreds of years. However, for the 202-eV
metastable state in 92U70+, its lifetime reduces to 48 days via an $E4$
spontaneous emission to the ground state.

$nd^{5}$ ions — For V- and Nb-like systems, the lowest few states in neutral V
and Nb atoms are formed by the $nd^{4}(n+1)s$ and $nd^{3}(n+1)s^{2}$
configurations and have $M1$ forbidden transitions. However, for the singly
charged Cr+ and Mo+ ions, the $nd^{5}\leavevmode\nobreak\ {}^{6}S_{5/2}$ term
becomes the ground state. Nevertheless, the first excited state is still a
long-lived ${}^{6}D_{1/2}$ state from the $nd^{4}(n+1)s$ configuration and
represents a narrow near-infrared clock transition around 1.5 eV. Starting
from V-like Mn2+ and Nb-like Tc2+, the low-lying excited states become
${}^{4}G_{11/2-5/2}$ from the $nd^{5}$ configuration.

As shown in Fig. 4, for light elements, the separations between these excited
states are smaller than 10 meV that even the $M1$ decay channel are extremely
slow. As a consequence, all of these states are ultrastable clock states.
Especially, through V-like Mn2+ to Br12+ and Nb-like Tc2+ to Ag6+, our MCDHF-
RCI calculations show that all the $nd^{5}\leavevmode\nobreak\ {}^{4}G$ terms
have lifetimes longer than 1 s, with energies ranging from 3.0 to 9.4 eV. From
Kr13+ to Y16+ and Cd7+ to Sb10+, even though the lifetime of the
${}^{4}G_{5/2}$ state becomes shorter than 1 s, the other three states still
provide highly forbidden clock transitions for each ion in the range of 4.7 to
11.5 eV. These multiple optical clock transitions can be employed to infer the
$\alpha$ variations and may also find applications in searching for a fifth
force with a multidimensional King-plot [83].

For elements heavier than V-like Mo19+ and Nb-like Xe13+, the energies of the
excited states scale quadratically with the atomic number $Z$ (except for
slightly abnormal behaviour of the $J=5/2$ state). The splittings between
these states are far beyond 0.1 eV, thus fast $M1$ decay emerges. As a result,
only the state with $J={11/2}$ that decays via a slow $E2/M3$ transition
remains metastable. As shown in Fig. 1(c) for Nb-like U51+, this clock state
lies 57.2 eV above the ground state. With a lifetime of 8.3 days, it
represents an XUV clock with $Q=6.3\times 10^{22}$. In the case of V-like
U69+, the transition energy scales to 197 eV. With a lifetime of 12 hours,
this soft-x-ray clock transition bears a quality factor of $Q=1.3\times
10^{22}$.

We note that, such a metastable state in Nb-like Pb41+ [72] has been detected
in a recent high-precision mass spectroscopy experiment. The measured energy
agrees with our predicted 31.68(13)-eV value obtained via large-scale MCDHF-
RCI calculations. With a calculated lifetime of 26 days, it represents an XUV
clock transition with $Q=1.1\times 10^{23}$, and can be effectively probed
with state-of-the-art XUV frequency combs [36, 68, 69, 84, 85, 86, 87].

$5d^{4,5,6}$ ions — The above discussions also apply to Hf-, Ta-, and W-like
ions through Hg6+ to U20+. Though most of them are radioactive, the energy and
lifetime of their clock state varies from 1.06 to 11.0 eV and from 2 s to 500
days, respectively, with $Q=10^{16}-10^{23}$. Particularly, the dual clock
states in W-like ions, with energies of 1.06 and 1.92 eV in Hg6+, and 1.70 and
5.11 eV in U18+, respectively, are also disproportional to $\alpha$
variations, with the sensitivity of the $J=0$ states being around $-0.61$ to
$-1.23$.

Conclusion — We have shown that there exist more than 100 ultrastable optical
HCI clock candidates. They arise from fine-structure splittings in the open-
shell $nd^{4}$, $nd^{5}$ and $nd^{6}$ ($n=3,4,5$) ions, for more than 70
elements and their isotopes, ranging from _22^46Ti to _92^238U and beyond. The
corresponding transitions have $Q$ factors and polarizabilities many orders of
magnitude larger and smaller, respectively, than for current optical clocks,
and thus would allow for more accurate time keeping. Most of the ions have
multiple clock states with opposite dependence on fine-structure constant
variations. High-precision spectroscopy of these clock transitions would
enrich the search for new physics beyond the standard model, as well as the
test of state-of-the-art nuclear theories. Moreover, for heavy HCIs, these
clock transitions also scale up to the XUV and soft-x-ray range, and could
enable the construction of short-wavelength clocks. Similar clock transitions
are also found in open-shell but more complicated $4f^{m}$ ions [88].

## References

  * Collaboration [2021] BACONB Collaboration, Frequency ratio measurements at 18-digit accuracy using an optical clock network, Nature 591, 564 (2021).
  * Zheng  _et al._ [2022] X. Zheng, J. Dolde, V. Lochab, B. N. Merriman, H. Li, and S. Kolkowitz, Differential clock comparisons with a multiplexed optical lattice clock, Nature 602, 425 (2022).
  * Bothwell  _et al._ [2022] T. Bothwell, C. J. Kennedy, A. Aeppli, D. Kedar, J. M. Robinson, E. Oelker, A. Staron, and J. Ye, Resolving the gravitational redshift across a millimetre-scale atomic sample, Nature 602, 420 (2022).
  * Peik  _et al._ [2004] E. Peik, B. Lipphardt, H. Schnatz, T. Schneider, C. Tamm, and S. G. Karshenboim, Limit on the present temporal variation of the fine structure constant, Phys. Rev. Lett. 93, 170801 (2004).
  * Fischer  _et al._ [2004] M. Fischer, N. Kolachevsky, M. Zimmermann, R. Holzwarth, T. Udem, T. W. Hänsch, M. Abgrall, J. Grünert, I. Maksimovic, S. Bize, H. Marion, F. P. D. Santos, P. Lemonde, G. Santarelli, P. Laurent, A. Clairon, C. Salomon, M. Haas, U. D. Jentschura, and C. H. Keitel, New limits on the drift of fundamental constants from laboratory measurements, Phys. Rev. Lett. 92, 230802 (2004).
  * Godun  _et al._ [2014] R. M. Godun, P. B. R. Nisbet-Jones, J. M. Jones, S. A. King, L. A. M. Johnson, H. S. Margolis, K. Szymaniec, S. N. Lea, K. Bongs, and P. Gill, Frequency ratio of two optical clock transitions in ${}^{171}{\mathrm{Yb}}^{+}$ and constraints on the time variation of fundamental constants, Phys. Rev. Lett. 113, 210801 (2014).
  * Berengut  _et al._ [2018] J. C. Berengut, D. Budker, C. Delaunay, V. V. Flambaum, C. Frugiuele, E. Fuchs, C. Grojean, R. Harnik, R. Ozeri, G. Perez, and Y. Soreq, Probing new long-range interactions by isotope shift spectroscopy, Phys. Rev. Lett. 120, 091801 (2018).
  * Counts  _et al._ [2020] I. Counts, J. Hur, D. P. L. Aude Craik, H. Jeon, C. Leung, J. C. Berengut, A. Geddes, A. Kawasaki, W. Jhe, and V. Vuletić, Evidence for nonlinear isotope shift in ${\mathrm{yb}}^{+}$ search for new boson, Phys. Rev. Lett. 125, 123002 (2020).
  * Solaro  _et al._ [2020] C. Solaro, S. Meyer, K. Fisher, J. C. Berengut, E. Fuchs, and M. Drewsen, Improved isotope-shift-based bounds on bosons beyond the standard model through measurements of the ${{}^{2}\mathrm{D}}_{3/2}-{{}^{2}\mathrm{D}}_{5/2}$ interval in ${\mathrm{Ca}}^{+}$, Phys. Rev. Lett. 125, 123003 (2020).
  * Safronova  _et al._ [2018] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, Search for new physics with atoms and molecules, Rev. Mod. Phys. 90, 025008 (2018).
  * Sanner  _et al._ [2019] C. Sanner, N. Huntemann, R. Lange, C. Tamm, E. Peik, M. S. Safronova, and S. G. Porsev, Optical clock comparison for Lorentz symmetry testing, Nature 567, 204 (2019).
  * Ludlow  _et al._ [2015] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Optical atomic clocks, Rev. Mod. Phys. 87, 637 (2015).
  * Brewer  _et al._ [2019] S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt, ${}^{27}{\mathrm{Al}}^{+}$ quantum-logic clock with a systematic uncertainty below ${10}^{-18}$, Phys. Rev. Lett. 123, 033201 (2019).
  * Takamoto  _et al._ [2005] M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, An optical lattice clock, Nature 435, 321 (2005).
  * Yamaguchi  _et al._ [2019] A. Yamaguchi, M. S. Safronova, K. Gibble, and H. Katori, Narrow-line cooling and determination of the magic wavelength of cd, Phys. Rev. Lett. 123, 113201 (2019).
  * Ohtsubo  _et al._ [2019] N. Ohtsubo, Y. Li, K. Matsubara, N. Nemitz, H. Hachisu, T. Ido, and K. Hayasaka, Optical clock based on a sympathetically-cooled indium ion, Hyperfine Interact. 240, 1 (2019).
  * Lemke  _et al._ [2009] N. D. Lemke, A. D. Ludlow, Z. W. Barber, T. M. Fortier, S. A. Diddams, Y. Jiang, S. R. Jefferts, T. P. Heavner, T. E. Parker, and C. W. Oates, Spin-$1/2$ optical lattice clock, Phys. Rev. Lett. 103, 063001 (2009).
  * McFerran  _et al._ [2012] J. J. McFerran, L. Yi, S. Mejri, S. Di Manno, W. Zhang, J. Guéna, Y. Le Coq, and S. Bize, Neutral atom frequency reference in the deep ultraviolet with $\mathrm{\text{fractional uncertainty}}=5.7\times{}{10}^{-15}$, Phys. Rev. Lett. 108, 183004 (2012).
  * Huang  _et al._ [2016] Y. Huang, H. Guan, P. Liu, W. Bian, L. Ma, K. Liang, T. Li, and K. Gao, Frequency comparison of two ${}^{40}{\mathrm{Ca}}^{+}$ optical clocks with an uncertainty at the ${10}^{-17}$ level, Phys. Rev. Lett. 116, 013001 (2016).
  * Margolis  _et al._ [2004] H. Margolis, G. Barwood, G. Huang, H. Klein, S. Lea, K. Szymaniec, and P. Gill, Hertz-level measurement of the optical clock frequency in a single ${}^{88}{\mathrm{Sr}}^{+}$ ion, Science 306, 1355 (2004).
  * Diddams  _et al._ [2001] S. A. Diddams, T. Udem, J. Bergquist, E. Curtis, R. Drullinger, L. Hollberg, W. M. Itano, W. Lee, C. Oates, K. Vogel, _et al._ , An optical clock based on a single trapped ${}^{199}{\mathrm{Hg}}^{+}$ ion, Science 293, 825 (2001).
  * Dzuba  _et al._ [2021] V. A. Dzuba, S. O. Allehabi, V. V. Flambaum, J. Li, and S. Schiller, Time keeping and searching for new physics using metastable states of Cu, Ag, and Au, Phys. Rev. A 103, 022822 (2021).
  * Holliman  _et al._ [2022] C. A. Holliman, M. Fan, A. Contractor, S. M. Brewer, and A. M. Jayich, Radium ion optical clock, Phys. Rev. Lett. 128, 033202 (2022).
  * Arnold  _et al._ [2016] K. J. Arnold, R. Kaewuam, A. Roy, E. Paez, S. Wang, and M. D. Barrett, Observation of the ${}^{1}{S}_{0}$ to ${}^{3}{D}_{1}$ clock transition in ${}^{175}{\mathrm{Lu}}^{+}$, Phys. Rev. A 94, 052512 (2016).
  * Kaewuam  _et al._ [2020] R. Kaewuam, T. R. Tan, K. J. Arnold, S. R. Chanu, Z. Zhang, and M. D. Barrett, Hyperfine averaging by dynamic decoupling in a multi-ion lutetium clock, Phys. Rev. Lett. 124, 083202 (2020).
  * Huntemann  _et al._ [2012] N. Huntemann, M. Okhapkin, B. Lipphardt, S. Weyers, C. Tamm, and E. Peik, High-accuracy optical clock based on the octupole transition in ${}^{171}\mathrm{Yb}^{+}$, Phys. Rev. Lett. 108, 090801 (2012).
  * Fürst  _et al._ [2020] H. A. Fürst, C.-H. Yeh, D. Kalincev, A. P. Kulosa, L. S. Dreissen, R. Lange, E. Benkler, N. Huntemann, E. Peik, and T. E. Mehlstäubler, Coherent excitation of the highly forbidden electric octupole transition in ${}^{172}{\mathrm{Yb}}^{+}$, Phys. Rev. Lett. 125, 163001 (2020).
  * Arnold  _et al._ [2018] K. J. Arnold, R. Kaewuam, A. Roy, T. R. Tan, and M. D. Barrett, Blackbody radiation shift assessment for a lutetium ion clock, Nat. Commun. 9, 1 (2018).
  * Golovizin  _et al._ [2019] A. Golovizin, E. Fedorova, D. Tregubov, D. Sukachev, K. Khabarova, V. Sorokin, and N. Kolachevsky, Inner-shell clock transition in atomic thulium with a small blackbody radiation shift, Nat. Commun. 10, 1 (2019).
  * Huntemann  _et al._ [2016a] N. Huntemann, C. Sanner, B. Lipphardt, C. Tamm, and E. Peik, Single-ion atomic clock with $3\times{}{10}^{-18}$ systematic uncertainty, Phys. Rev. Lett. 116, 063001 (2016a).
  * Campbell  _et al._ [2012] C. J. Campbell, A. G. Radnaev, A. Kuzmich, V. A. Dzuba, V. V. Flambaum, and A. Derevianko, Single-ion nuclear clock for metrology at the 19th decimal place, Phys. Rev. Lett. 108, 120802 (2012).
  * Seiferle  _et al._ [2019] B. Seiferle, L. von der Wense, P. V. Bilous, I. Amersdorffer, C. Lemell, F. Libisch, S. Stellmer, T. Schumm, C. E. Düllmann, A. Pálffy, _et al._ , Energy of the ${}^{229}{\mathrm{Th}}$ nuclear clock transition, Nature 573, 243 (2019).
  * Beeks  _et al._ [2021] K. Beeks, T. Sikorsky, T. Schumm, J. Thielking, M. V. Okhapkin, and E. Peik, The thorium-229 low-energy isomer and the nuclear clock, Nat. Rev. Phys. 3, 238 (2021).
  * Kozlov  _et al._ [2018] M. Kozlov, M. Safronova, J. R. Crespo López-Urrutia, and P. Schmidt, Highly charged ions: Optical clocks and applications in fundamental physics, Rev. Mod. Phys. 90, 045005 (2018).
  * King  _et al._ [2022] S. A. King, L. J. Spieß, P. Micke, A. Wilzewski, T. Leopold, E. Benkler, R. Lange, N. Huntemann, A. Surzhykov, V. A. Yerokhin, _et al._ , An optical atomic clock based on a highly charged ion, Nature 611, 43 (2022).
  * Lyu  _et al._ [2020] C. Lyu, S. M. Cavaletto, C. H. Keitel, and Z. Harman, Interrogating the temporal coherence of EUV frequency combs with highly charged ions, Phys. Rev. Lett. 125, 093201 (2020).
  * [37] B.-B. Suo, Y.-M. Yu, B. K. Sahoo, Highly charged ion (HCI) clocks: Frontier candidates for testing variation of fine-structure constant, arXiv:2211.12162 doi.org/10.48550/arXiv.2211.12162.
  * Kraemer  _et al._ [2022] S. Kraemer, J. Moens, M. Athanasakis-Kaklamanakis, S. Bara, K. Beeks, P. Chhetri, K. Chrysalidis, A. Claessens, T. E. Cocolios, J. M. Correia, _et al._ , Observation of the radiative decay of the 229Th nuclear clock isomer, arXiv preprint arXiv:2209.10276 (2022).
  * Schiller [2007] S. Schiller, Hydrogenlike highly charged ions for tests of the time independence of fundamental constants, Phys. Rev. Lett. 98, 180801 (2007).
  * Yudin  _et al._ [2014] V. I. Yudin, A. V. Taichenachev, and A. Derevianko, Magnetic-dipole transitions in highly charged ions as a basis of ultraprecise optical clocks, Phys. Rev. Lett. 113, 233003 (2014).
  * Yu and Sahoo [2016] Y.-M. Yu and B. K. Sahoo, Scrutinizing al-like ${}^{51}\mathrm{V}^{10+},\phantom{\rule{1.60004pt}{0.0pt}}^{53}\mathrm{Cr}^{11+},\phantom{\rule{1.60004pt}{0.0pt}}^{55}\mathrm{Mn}^{12+},\phantom{\rule{1.60004pt}{0.0pt}}^{57}\mathrm{Fe}^{13+},\phantom{\rule{1.60004pt}{0.0pt}}^{59}\mathrm{Co}^{14+},\phantom{\rule{1.60004pt}{0.0pt}}^{61}\mathrm{Ni}^{15+}$, and ${}^{63}\mathrm{Cu}^{16+}$ ions for atomic clocks with uncertainties below the ${10}^{-19}$ level, Phys. Rev. A 94, 062502 (2016).
  * Windberger  _et al._ [2016] A. Windberger, F. Torretti, A. Borschevsky, A. Ryabtsev, S. Dobrodey, H. Bekker, E. Eliav, U. Kaldor, W. Ubachs, R. Hoekstra, J. R. Crespo López-Urrutia, and O. O. Versolato, Analysis of the fine structure of ${\mathrm{Sn}}^{11+}-{\mathrm{Sn}}^{14+}$ ions by optical spectroscopy in an electron-beam ion trap, Phys. Rev. A 94, 012506 (2016).
  * Micke  _et al._ [2020] P. Micke, T. Leopold, S. King, E. Benkler, L. Spieß, L. Schmoeger, M. Schwarz, J. R. Crespo López-Urrutia, and P. Schmidt, Coherent laser spectroscopy of highly charged ions using quantum logic, Nature 578, 60 (2020).
  * Rehbehn  _et al._ [2021] N.-H. Rehbehn, M. K. Rosner, H. Bekker, J. C. Berengut, P. O. Schmidt, S. A. King, P. Micke, M. F. Gu, R. Müller, A. Surzhykov, and J. R. Crespo López-Urrutia, Sensitivity to new physics of isotope-shift studies using the coronal lines of highly charged calcium ions, Phys. Rev. A 103, L040801 (2021).
  * Berengut  _et al._ [2010] J. C. Berengut, V. A. Dzuba, and V. V. Flambaum, Enhanced laboratory sensitivity to variation of the fine-structure constant using highly charged ions, Phys. Rev. Lett. 105, 120801 (2010).
  * Berengut  _et al._ [2011] J. C. Berengut, V. A. Dzuba, V. V. Flambaum, and A. Ong, Electron-hole transitions in multiply charged ions for precision laser spectroscopy and searching for variations in $\alpha$, Phys. Rev. Lett. 106, 210802 (2011).
  * Safronova  _et al._ [2015] U. I. Safronova, V. V. Flambaum, and M. S. Safronova, Transitions between the $4f\text{-core-excited}$ states in ${\mathrm{Ir}}^{16+},{\mathrm{Ir}}^{17+}$, and ${\mathrm{Ir}}^{18+}$ ions for clock applications, Phys. Rev. A 92, 022501 (2015).
  * Safronova  _et al._ [2014] M. S. Safronova, V. A. Dzuba, V. V. Flambaum, U. I. Safronova, S. G. Porsev, and M. G. Kozlov, Highly charged ions for atomic clocks, quantum information, and search for $\alpha$ variation, Phys. Rev. Lett. 113, 030801 (2014).
  * Bekker  _et al._ [2019] H. Bekker, A. Borschevsky, Z. Harman, C. Keitel, T. Pfeifer, P. Schmidt, J. R. Crespo López-Urrutia, and J. Berengut, Detection of the $5p-4f$ orbital crossing and its optical clock transition in Pr9+, Nat. Commun. 10, 1 (2019).
  * Yu  _et al._ [2021] Y.-M. Yu, D. Pan, S. Chen, H. Guan, K. Gao, and J. Chen, Highly charged ${\mathrm{Nd}}^{9+}$ ion: A potential candidate of $\mu$Hz linewidth optical clocks for probing fundamental physics, arXiv preprint arXiv:2107.10520 (2021).
  * Berengut  _et al._ [2012] J. C. Berengut, V. A. Dzuba, V. V. Flambaum, and A. Ong, Optical transitions in highly charged californium ions with high sensitivity to variation of the fine-structure constant, Phys. Rev. Lett. 109, 070802 (2012).
  * Dzuba and Flambaum [2017] V. Dzuba and V. Flambaum, Highly charged ions for atomic clocks and search for variation of the fine structure constant, in  _TCP 2014_ (Springer, 2017) pp. 79–86.
  * Nandy and Sahoo [2016] D. K. Nandy and B. K. Sahoo, Highly charged $\mathrm{W}^{13+},\mathrm{Ir}^{16+}$, and $\mathrm{Pt}^{17+}$ ions as promising optical clock candidates for probing variations of the fine-structure constant, Phys. Rev. A 94, 032504 (2016).
  * Feldman  _et al._ [1991] U. Feldman, P. Indelicato, and J. Sugar, Magnetic dipole line from U LXXI ground-term levels predicted at 3200 Å, J. Opt. Soc. Am. B 8, 3 (1991).
  * Morgan  _et al._ [1995] C. A. Morgan, F. G. Serpa, E. Takács, E. S. Meyer, J. D. Gillaspy, J. Sugar, J. R. Roberts, C. M. Brown, and U. Feldman, Observation of visible and uv magnetic dipole transitions in highly charged xenon and barium, Phys. Rev. Lett. 74, 1716 (1995).
  * López-Urrutia [2008] J. R. Crespo López-Urrutia, The visible spectrum of highly charged ions: A window to fundamental physics, Can. J. Phys. 86, 111 (2008).
  * Micke  _et al._ [2018] P. Micke, S. Kühn, L. Buchauer, J. Harries, T. M. Bücking, K. Blaum, A. Cieluch, A. Egl, D. Hollain, S. Kraemer, _et al._ , The heidelberg compact electron beam ion traps, Rev. Sci. Instrum. 89, 063109 (2018).
  * Herfurth  _et al._ [2015] F. Herfurth, Z. Andelkovic, W. Barth, W. Chen, L. Dahl, S. Fedotova, P. Gerhard, M. Kaiser, O. Kester, H. Kluge, _et al._ , The HITRAP facility for slow highly charged ions, Physica Scripta 2015, 014065 (2015).
  * Yang  _et al._ [2022] X. Yang, S. Wang, S. Wilkins, and R. G. Ruiz, Laser spectroscopy for the study of exotic nuclei, Prog. Part. Nucl. Phys. 129, 104005 (2023).
  * Malbrunot-Ettenauer  _et al._ [2022] S. Malbrunot-Ettenauer, S. Kaufmann, S. Bacca, C. Barbieri, J. Billowes, M. L. Bissell, K. Blaum, B. Cheal, T. Duguet, R. F. G. Ruiz, W. Gins, C. Gorges, G. Hagen, H. Heylen, J. D. Holt, G. R. Jansen, A. Kanellakopoulos, M. Kortelainen, T. Miyagi, P. Navrátil, W. Nazarewicz, R. Neugart, G. Neyens, W. Nörtershäuser, S. J. Novario, T. Papenbrock, T. Ratajczyk, P.-G. Reinhard, L. V. Rodríguez, R. Sánchez, S. Sailer, A. Schwenk, J. Simonis, V. Somà, S. R. Stroberg, L. Wehner, C. Wraith, L. Xie, Z. Y. Xu, X. F. Yang, and D. T. Yordanov, Nuclear charge radii of the nickel isotopes ${}^{58-68,70}\mathrm{Ni}$, Phys. Rev. Lett. 128, 022502 (2022).
  * Sommer  _et al._ [2022] F. Sommer, K. König, D. M. Rossi, N. Everett, D. Garand, R. P. de Groote, J. D. Holt, P. Imgram, A. Incorvati, C. Kalman, A. Klose, J. Lantis, Y. Liu, A. J. Miller, K. Minamisono, T. Miyagi, W. Nazarewicz, W. Nörtershäuser, S. V. Pineda, R. Powel, P.-G. Reinhard, L. Renth, E. Romero-Romero, R. Roth, A. Schwenk, C. Sumithrarachchi, and A. Teigelhöfer, Charge radii of ${}^{55,56}\mathrm{Ni}$ reveal a surprisingly similar behavior at $n=28$ in ca and ni isotopes, Phys. Rev. Lett. 129, 132501 (2022).
  * Rodríguez  _et al._ [2020] L. V. Rodríguez, D. L. Balabanski, M. L. Bissell, K. Blaum, B. Cheal, G. De Gregorio, J. Ekman, R. F. Garcia Ruiz, A. Gargano, G. Georgiev, W. Gins, C. Gorges, H. Heylen, A. Kanellakopoulos, S. Kaufmann, V. Lagaki, S. Lechner, B. Maaß, S. Malbrunot-Ettenauer, R. Neugart, G. Neyens, W. Nörtershäuser, S. Sailer, R. Sánchez, S. Schmidt, L. Wehner, C. Wraith, L. Xie, Z. Y. Xu, X. F. Yang, and D. T. Yordanov, Doubly-magic character of ${}^{132}\mathrm{Sn}$ studied via electromagnetic moments of ${}^{133}\mathrm{Sn}$, Phys. Rev. C 102, 051301 (2020).
  * Yordanov  _et al._ [2020] D. T. Yordanov, L. V. Rodríguez, D. L. Balabanski, J. Bieroń, M. L. Bissell, K. Blaum, B. Cheal, J. Ekman, G. Gaigalas, R. F. Garcia Ruiz, _et al._ , Structural trends in atomic nuclei from laser spectroscopy of tin, Commun. Phys. 3, 107 (2020).
  * Numata  _et al._ [2004] K. Numata, A. Kemery, and J. Camp, Thermal-noise limit in the frequency stabilization of lasers with rigid cavities, Phys. Rev. Lett. 93, 250602 (2004).
  * Jiang  _et al._ [2011] Y. Jiang, A. Ludlow, N. D. Lemke, R. W. Fox, J. A. Sherman, L.-S. Ma, and C. W. Oates, Making optical atomic clocks more stable with $10^{-16}$-level laser stabilization, Nat. Photon. 5, 158 (2011).
  * Matei  _et al._ [2017] D. G. Matei, T. Legero, S. Häfner, C. Grebing, R. Weyrich, W. Zhang, L. Sonderhouse, J. M. Robinson, J. Ye, F. Riehle, and U. Sterr, $1.5\text{ }\text{ }\mu\mathrm{m}$ lasers with sub-10 mHz linewidth, Phys. Rev. Lett. 118, 263202 (2017).
  * Kedar  _et al._ [2022] D. Kedar, J. Yu, E. Oelker, A. Staron, W. R. Milner, J. M. Robinson, T. Legero, F. Riehle, U. Sterr, and J. Ye, Frequency stability of cryogenic silicon cavities with semiconductor crystalline coatings, arXiv preprint arXiv:2210.14881 (2022).
  * Pupeza  _et al._ [2013] I. Pupeza, S. Holzberger, T. Eidam, H. Carstens, D. Esser, J. Weitenberg, P. Rußbüldt, J. Rauschenberger, J. Limpert, T. Udem, _et al._ , Compact high-repetition-rate source of coherent 100 ev radiation, Nat. Photon. 7, 608 (2013).
  * Benko  _et al._ [2014] C. Benko, T. K. Allison, A. Cingöz, L. Hua, F. Labaye, D. C. Yost, and J. Ye, Extreme ultraviolet radiation with coherence time greater than 1 s, Nat. Photon. 8, 530 (2014).
  * Grant [2007] I. P. Grant,  _Relativistic quantum theory of atoms and molecules: theory and computation_ (Springer, 2007).
  * Fischer  _et al._ [2019] C. F. Fischer, G. Gaigalas, P. Jönsson, and J. Bieroń, GRASP2018–A Fortran 95 version of the general relativistic atomic structure package, Comput. Phys. Commun. 237, 184 (2019).
  * [72] K. Kromer, C. Lyu, M. Door, _et al._ , Observation of a low-lying metastable electronic state in highly charged lead by Penning-trap mass spectrometry, In submission .
  * [73] Supplemental Materials .
  * Beloy  _et al._ [2020] K. Beloy, V. A. Dzuba, and S. M. Brewer, Quadruply ionized barium as a candidate for a high-accuracy optical clock, Phys. Rev. Lett. 125, 173002 (2020).
  * Allehabi  _et al._ [2022] S. O. Allehabi, S. M. Brewer, V. A. Dzuba, V. V. Flambaum, and K. Beloy, High-accuracy optical clocks based on group-16-like highly charged ions, Phys. Rev. A 106, 043101 (2022).
  * Chen  _et al._ [2022] S. Chen, Z. Zhou, J. Li, T. Zhang, C. Li, T. Shi, Y. Huang, H. Guan, and K. Gao, Precision Measurement of M1 Optical Clock Transition in Ni12+, arXiv: , 2303.04552 (2023).
  * Sussman [2011] B. J. Sussman, Five ways to the nonresonant dynamic stark effect, Am. J. Phys. 79, 477 (2011).
  * Safronova  _et al._ [2011] M. S. Safronova, M. G. Kozlov, and C. W. Clark, Precision calculation of blackbody radiation shifts for optical frequency metrology, Phys. Rev. Lett. 107, 143006 (2011).
  * Huntemann  _et al._ [2016b] N. Huntemann, C. Sanner, B. Lipphardt, C. Tamm, and E. Peik, Single-ion atomic clock with $3\times{}{10}^{-18}$ systematic uncertainty, Phys. Rev. Lett. 116, 063001 (2016b).
  * Johnson [2007] W. R. Johnson,  _Atomic structure theory_ (Springer, 2007).
  * Schüssler  _et al._ [2020] R. X. Schüssler, H. Bekker, M. Braß, H. Cakir, J. R. Crespo López-Urrutia, M. Door, P. Filianin, Z. Harman, M. W. Haverkort, W. J. Huang, _et al._ , Detection of metastable electronic states by penning trap mass spectrometry, Nature 581, 42 (2020).
  * Lange  _et al._ [2022] R. Lange, N. Huntemann, A. A. Peshkov, A. Surzhykov, and E. Peik, Excitation of an electric octupole transition by twisted light, Phys. Rev. Lett. 129, 253901 (2022).
  * Berengut  _et al._ [2020] J. C. Berengut, C. Delaunay, A. Geddes, and Y. Soreq, Generalized king linearity and new physics searches with isotope shifts, Phys. Rev. Res. 2, 043444 (2020).
  * Pupeza  _et al._ [2021] I. Pupeza, C. Zhang, M. Högner, and J. Ye, Extreme-ultraviolet frequency combs for precision metrology and attosecond science, Nat. Photon. 15, 175 (2021).
  * Nauta  _et al._ [2021] J. Nauta, J.-H. Oelmann, A. Borodin, A. Ackermann, P. Knauer, I. Muhammad, R. Pappenberger, T. Pfeifer, and J. C. López-Urrutia, XUV frequency comb production with an astigmatism-compensated enhancement cavity, Opt. Exp. 29, 2624 (2021).
  * Zhang  _et al._ [2020] J. Zhang, L.-Q. Hua, Z. Chen, M.-F. Zhu, C. Gong, and X.-J. Liu, Extreme ultraviolet frequency comb with more than 100 $\mu$W average power below 100nm, Chin. Phys. Lett. 37, 124203 (2020).
  * Dreissen  _et al._ [2019] L. S. Dreissen, C. Roth, E. L. Gründeman, J. J. Krauth, M. Favier, and K. S. E. Eikema, High-precision ramsey-comb spectroscopy based on high-harmonic generation, Phys. Rev. Lett. 123, 143001 (2019).
  * Lyu  _et al._ [2023] C. Lyu, C. H. Keitel, and Z. Harman, In preparation, .

##  I Supplemental Materials

This material summizes the mixing coefficients for low-lying states in
selected $4d^{6}$, $4d^{5}$, and $4d^{4}$ ions. For heavy elements, the strong
relativistic effect leads to large splittings between the $4d_{3/2}$ and
$4d_{5/2}$ relativistic orbitals, with the subscripts being the corresponding
single-electron total angular momenta. As the atomic number $Z$ increases, the
energies of atomic states dominated by the $4d_{3/2}$ orbital scale much
slower than those of the states involving high fraction of the $4d_{5/2}$
orbital. This impropotional scaling effect leads to the reordering of fine-
structure terms, thus the emergence of metastable states. The notation
$4d_{3/2}^{2}(0)$ means there are 2 electrons in the $4d_{3/2}$ orbital, and
the total angular momentum of these two electrons is 0.

###  I.1 $4d^{6}$ ions

For Mo and Mo-like Sn8+, the ASFs are dominated by the
$4d^{3}_{3/2}(3/2)4d^{3}_{5/2}(3/2)$ configuration where $d_{3/2}$ and
$d_{5/2}$ orbitals are equally occupied. However, for U50+, the energy of the
$4d_{3/2}$ orbital is 55.8 eV lower than that of the $4d_{5/2}$ orbital. Thus,
the dominate CSFs, for the low-lying $J=0,2,4$ states, are given by
$4d^{4}_{3/2}(0)4d^{2}_{5/2}(0)$, $4d^{4}_{3/2}(0)4d^{2}_{5/2}(4)$, and
$4d^{4}_{3/2}(0)4d^{2}(4)_{5/2}$, respectively. However, the
$4d^{4}_{3/2}4d^{2}_{5/2}$ confituration does not allow angular coupling with
$J=1,3$. Therefore, the terms with $J=1,3$ have more electrons occupying the
$4d_{5/2}$ orbital, and thus will have much higher energy than the other three
terms. Similar analysis also applies to the $4d^{5}$ and $4d^{4}$ ions shown
in Sec. I.2 and I.3, respectively.

Table 2: Absolute binding energies of the $4d_{3/2}$ and $4d_{5/2}$ relativistic orbitals in $4d^{6}$ ions.  ion |  $E_{4d_{3/2}}$ (a.u.) |  $E_{4d_{5/2}}$ (a.u.) |  $\Delta E$ (eV)
---|---|---|---
42Mo0+ | 0.214 | 0.207 | 0.190
50Sn8+ | 5.916 | 5.866 | 1.361
92U50+ | 107.599 | 105.548 | 55.811
Table 3: CSF components of the low-lying $J=0$ state.  ion | ASF | $c_{k}$
---|---|---
42Mo0+ | 0 + | 0.76080 | 0.37843 | -0.35227 | 0.28425 | 0.27035
|  | 3 | 1 | 2 | 5 | 4
50Sn8+ | 0 + | 0.74400 | 0.37938 | 0.36036 | -0.32716 | 0.25620
|  | 3 | 5 | 4 | 2 | 1
92U50+ | 0 + | 0.98227 | 0.16656 | -0.07640 | 0.03946 | 0.00383
|  | 5 | 4 | 2 | 3 | 1
CSF |  | 1 | 2 | 3 | 4 | 5
|  | $4d^{6}(0)_{5/2}$ | $4d_{3/2}^{2}(0)4d^{4}_{5/2}(0)$ | $4d_{3/2}^{2}(2)4d^{4}_{5/2}(2)$ | $4d_{3/2}^{3}(3/2)4d^{3}_{5/2}(3/2)$ | $4d_{3/2}^{4}(0)4d^{2}_{5/2}(0)$
Table 4: CSF components of the state with $J=1$.  ion | ASF | $c_{k}$ |
---|---|---|---
42Mo0+ | 1 + | 0.64164 | -0.51824 | -0.41117 | -0.38816 |
|  | 1 | 3 | 4 | 2 |
50Sn8+ | 1 + | 0.57585 | 0.50747 | -0.50443 | 0.39550 |
|  | 3 | 4 | 1 | 2 |
92U50+ | 1 + | 0.79804 | 0.59318 | 0.09266 | -0.05178 |
|  | 4 | 3 | 2 | 1 |
CSF |  | 1 | 2 | 3 | 4 |
|  | $4d_{3/2}(3/2)4d^{5}_{5/2}(5/2)$ | $4d_{3/2}^{2}(2)4d^{4}_{5/2}(2)$ | $4d_{3/2}^{3}(3/2)4d^{3}_{5/2}(5/2)$ | $4d_{3/2}^{3}(3/2)4d^{3}_{5/2}(3/2)$ |
Table 5: CSF components of the state with $J=2$.  ion | ASF | $c_{k}$
---|---|---
42Mo0+ | 2 + | 0.72324 | 0.37426 | -0.32648 | -0.30319 | -0.27253
|  | 5 | 1 | 8 | 6 | 4
50Sn8+ | 2 + | 0.62622 | -0.49708 | -0.35745 | 0.29182 | -0.26834
|  | 5 | 8 | 6 | 1 | 4
92U50+ | 2 + | 0.99339 | 0.09839 | -0.03938 | 0.03373 | -0.02196
|  | 8 | 6 | 2 | 7 | 5
CSF |  | 1 | 2 | 3 | 4 |
|  | $4d_{3/2}(3/2)4d^{5}_{5/2}(5/2)$ | $4d_{3/2}^{2}(0)4d^{4}_{5/2}(2)$ | $4d_{3/2}^{2}(2)4d^{4}_{5/2}(0)$ | $4d_{3/2}^{2}(2)4d^{4}_{5/2}(2)$ |
|  | 5 | 6 | 7 | 8 |
|  | $4d_{3/2}^{2}(2)4d^{4}_{5/2}(4)$ | $4d_{3/2}^{3}(3/2)4d^{3}_{5/2}(5/2)$ | $4d_{3/2}^{3}(3/2)4d^{3}_{5/2}(3/2)$ | $4d_{3/2}^{4}(0)4d^{2}_{5/2}(2)$ |
Table 6: CSF components of the state with $J=3$.  ion | ASF | $c_{k}$
---|---|---
42Mo0+ | 3 + | 0.74515 | -0.56412 | 0.22897 | -0.20396 | -0.14349
|  | 6 | 3 | 4 | 1 | 5
50Sn8+ | 3 + | 0.74515 | -0.56412 | 0.22897 | -0.20396 | -0.14349
|  | 6 | 3 | 4 | 1 | 5
92U50+ | 3 + | 0.89099 | 0.32023 | -0.30164 | -0.10551 | 0.02863
|  | 6 | 4 | 5 | 3 | 2
CSF |  | 1 | 2 | 3 | 4 |
|  | $4d_{3/2}(3/2)4d^{5}_{5/2}(5/2)$ | $4d_{3/2}^{2}(2)4d^{4}_{5/2}(2)$ | $4d_{3/2}^{2}(2)4d^{4}_{5/2}(4)$ | $4d_{3/2}^{3}(3/2)4d^{3}_{5/2}(5/2)$ |
|  | 5 | 6 |  |  |
|  | $4d_{3/2}^{3}(3/2)4d^{3}_{5/2}(3/2)$ | $4d_{3/2}^{3}(3/2)4d^{3}_{5/2}(9/2)$ |  |  |
Table 7: CSF components of the state with $J=4$.  ion | ASF | $c_{k}$
---|---|---
42Mo0+ | 4 + | 0.67191 | 0.47875 | -0.40263 | -0.29717 | -0.18941
|  | 6 | 7 | 4 | 2 | 1
50Sn8+ | 4 + | 0.65905 | 0.59695 | -0.31364 | -0.26429 | 0.13655
|  | 6 | 7 | 4 | 2 | 5
92U50+ | 4 + | 0.99493 | 0.08798 | -0.04455 | -0.01577 | -0.01165
|  | 7 | 6 | 2 | 4 | 3
CSF |  | 1 | 2 | 3 | 4 |
|  | $4d_{3/2}(3/2)4d^{5}(5/2)$ | $4d_{3/2}^{2}(0)4d_{5/2}^{4}(4)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{4}(2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{4}(4)$ |
|  | 5 | 6 | 7 |  |
|  | $4d_{3/2}^{3}(3/2)4d_{5/2}^{3}(5/2)$ | $4d_{3/2}^{3}(3/2)4d_{5/2}^{3}(9/2)$ | $4d_{3/2}^{4}(0)4d_{5/2}^{2}(4)$ |  |

###  I.2 $4d^{5}$ ions

Table 8: Absolute binding energies of the $4d_{3/2}$ and $4d_{5/2}$ relativistic orbitals in $4d_{5/2}^{5}$ ions.  ion |  $E_{4d_{3/2}}$ (a.u.) |  $E_{4d_{5/2}}$ (a.u.) |  $\Delta E$ (eV)
---|---|---|---
42Mo0+ | 0.554 | 0.547 | 0.190
50Sn8+ | 6.758 | 6.706 | 1.145
92U50+ | 110.328 | 108.162 | 58.940
Table 9: CSF components of the ground state with $J=5/2$.  ion | ASF | $c_{k}$
---|---|---
42Mo+ | 5/2 + | 0.61759 | 0.50663 | -0.45259 | 0.21928 | -0.19577
|  | 7 | 9 | 3 | 10 | 4
50Sn9+ | 5/2 + | 0.58952 | 0.57196 | -0.37560 | 0.29939 | -0.19366
|  | 7 | 9 | 3 | 10 | 4
92U51+ | 5/2 + | 0.99305 | 0.09767 | -0.05647 | 0.02543 | 0.01799
|  | 10 | 9 | 4 | 7 | 6
CSF |  | 1 | 2 | 3 | 4 | 5
|  | $4d_{5/2}^{5}(5/2)$ | $4d_{3/2}(3/2)4d_{5/2}^{4}(2)$ | $4d_{3/2}(3/2)4d_{5/2}^{4}(4)$ | $4d_{3/2}^{2}(0)4d_{5/2}^{3}(5/2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{3}(5/2)$
|  | 6 | 7 | 8 | 9 | 10
|  | $4d_{3/2}^{2}(2)4d_{5/2}^{3}(3/2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{3}(9/2)$ | $4d_{3/2}^{3}(3/2)4d_{5/2}^{2}(2)$ | $4d_{3/2}^{3}(3/2)4d_{5/2}^{2}(4)$ | $4d_{3/2}^{4}(0)4d_{5/2}(5/2)$
Table 10: CSF components of the excited state with $J=5/2$.  ion | ASF | $c_{k}$
---|---|---
42Mo+ | 5/2 + | 0.49676 | -0.43361 | 0.35679 | -0.31041 | 0.29820
|  | 6 | 8 | 2 | 9 | 7
50Sn9+ | 5/2 + | 0.68269 | 0.39504 | -0.36717 | 0.27866 | -0.24829
|  | 10 | 6 | 8 | 3 | 9
92U51+ | 5/2 + | 0.87733 | 0.45249 | 0.11797 | -0.08731 | -0.05064
|  | 9 | 8 | 7 | 10 | 3
CSF |  | 1 | 2 | 3 | 4 | 5
|  | $4d_{5/2}^{5}(5/2)$ | $4d_{3/2}(3/2)4d_{5/2}^{4}(2)$ | $4d_{3/2}(3/2)4d_{5/2}^{4}(4)$ | $4d_{3/2}^{2}(0)4d_{5/2}^{3}(5/2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{3}(5/2)$
|  | 6 | 7 | 8 | 9 | 10
|  | $4d_{3/2}^{2}(2)4d_{5/2}^{3}(3/2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{3}(9/2)$ | $4d_{3/2}^{3}(3/2)4d_{5/2}^{2}(2)$ | $4d_{3/2}^{3}(3/2)4d_{5/2}^{2}(4)$ | $4d_{3/2}^{4}(0)4d_{5/2}(5/2)$
Table 11: CSF components of the excited state with $J=7/2$.  ion | ASF | $c_{k}$
---|---|---
42Mo+ | 7/2 + | 0.59145 | -0.47959 | 0.39208 | -0.32789 | 0.28363
|  | 5 | 7 | 2 | 6 | 4
50Sn9+ | 7/2 + | 0.65387 | -0.46013 | 0.38918 | -0.31272 | -0.24884
|  | 7 | 5 | 6 | 2 | 4
92U51+ | 7/2 + | 0.92695 | 0.37150 | -0.03240 | -0.02816 | -0.02348
|  | 7 | 6 | 2 | 4 | 1
CSF |  | 1 | 2 | 3 | 4 | 5
|  | $4d_{3/2}(3/2)4d_{5/2}^{4}(2)$ | $4d_{3/2}(3/2)4d_{5/2}^{4}(4)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{3}(5/2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{3}(3/2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{3}(9/2)$
|  | 6 | 7 |  |  |
|  | $4d_{3/2}^{3}(3/2)4d_{5/2}^{2}(2)$ | $4d_{3/2}^{3}(3/2)4d_{5/2}^{2}(4)$ |  |  |
Table 12: CSF components of the state with $J=9/2$.  ion | ASF | $c_{k}$
---|---|---
42Mo+ | 9/2 + | 0.81369 | -0.44351 | 0.37558 | 0.01020 | 0.00565
|  | 4 | 5 | 1 | 2 | 3
50Sn9+ | 9/2 + | 0.77477 | -0.55675 | 0.29631 | 0.03996 | 0.01902
|  | 4 | 5 | 1 | 2 | 3
92U51+ | 9/2 + | 0.99487 | -0.08321 | -0.04440 | -0.02742 | -0.02423
|  | 5 | 4 | 2 | 3 | 1
CSF |  | 1 | 2 | 3 | 4 | 5
|  | $4d_{3/2}(3/2)4d_{5/2}^{4}(4)$ | $4d_{3/2}^{2}(0)4d_{5/2}^{3}(9/2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{3}(5/2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{3}(9/2)$ | $4d_{3/2}^{3}(3/2)4d_{5/2}^{2}(4)$
Table 13: CSF components of the state with $J=11/2$.  ion | ASF | $c_{k}$
---|---|---
42Mo+ | 11/2 + | 0.71539 | -0.53453 | 0.44999
|  | 2 | 3 | 1
50Sn9+ | 11/2 + | 0.67722 | -0.64170 | -0.35999
|  | 3 | 2 | 1
92U51+ | 11/2 + | 0.99809 | -0.04816 | -0.03862
|  | 3 | 2 | 1
CSF |  | 1 | 2 | 3
|  | $4d_{3/2}(3/2)4d_{5/2}^{4}(4)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{3}(9/2)$ | $4d_{3/2}^{3}(3/2)4d_{5/2}^{2}(4)$

###  I.3 $4d^{4}$ ions

Table 14: Absolute binding energies of the $4d_{3/2}$ and $4d_{5/2}$ relativistic orbitals in $4d_{5/2}^{4}$ ions.  ion |  $E_{4d_{3/2}}$ (a.u.) |  $E_{4d_{5/2}}$ (a.u.) |  $\Delta E$ (eV)
---|---|---|---
42Mo0+ | 0.968 | 0.960 | 0.218
50Sn8+ | 7.628 | 7.574 | 1.469
92U50+ | 113.079 | 110.888 | 59.620
Table 15: CSF components of the state with $J=0$.  ion | ASF | $c_{k}$
---|---|---
42Mo2+ | 0 + | 0.72165 | 0.51071 | -0.36020 | 0.22064 | 0.19994
|  | 4 | 5 | 3 | 1 | 2
50Sn10+ | 0 + | 0.71207 | 0.58598 | -0.34109 | 0.14457 | 0.11108
|  | 5 | 4 | 3 | 1 | 2
92U52+ | 0 + | 0.99720 | -0.06621 | 0.03466 | 0.00303 | -0.00088
|  | 5 | 3 | 4 | 1 | 2
CSF |  | 1 | 2 | 3 | 4 | 5
|  | $4d_{5/2}^{4}(0)$ | $4d_{3/2}(3/2)4d_{5/2}^{3}(3/2)$ | $4d_{3/2}^{2}(0)4d_{5/2}^{2}(0)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{2}(2)$ | $4d_{3/2}^{4}(0)$
Table 16: CSF components of the state with $J=1$.  ion | ASF | $c_{k}$ |
---|---|---|---
42Mo2+ | 1 + | 0.74455 | -0.45667 | 0.35526 | 0.33301 |
|  | 4 | 1 | 3 | 2 |
50Sn10+ | 1 + | 0.84828 | -0.36950 | 0.29390 | 0.23983 |
|  | 4 | 1 | 3 | 2 |
92U52+ | 1 + | 0.99761 | -0.05523 | 0.03652 | 0.01960 |
|  | 4 | 1 | 3 | 2 |
CSF |  | 1 | 2 | 3 | 4 |
|  | $4d_{3/2}(3/2)4d_{5/2}^{3}(5/2)$ | $4d_{3/2}(3/2)4d_{5/2}^{3}(3/2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{2}(2)$ | $4d_{3/2}^{3}(3/2)4d_{5/2}(5/2)$ |
Table 17: CSF components of the state with $J=2$.  ion | ASF | $c_{k}$
---|---|---
42Mo2+ | 2 + | 0.75176 | -0.42854 | 0.26385 | -0.26042 | 0.24193
|  | 7 | 8 | 2 | 6 | 1
50Sn10+ | 2 + | 0.75312 | -0.50378 | -0.23758 | 0.22462 | -0.18064
|  | 7 | 8 | 6 | 2 | 4
92U52+ | 2 + | 0.98830 | -0.14298 | -0.04338 | 0.02404 | 0.01494
|  | 8 | 7 | 2 | 5 | 3
CSF |  | 1 | 2 | 3 | 4 |
|  | $4d_{5/2}^{4}(2)$ | $4d_{3/2}(3/2)4d_{5/2}^{(}5/2)$ | $4d_{3/2}(3/2)4d_{5/2}^{(}3/2)$ | $4d_{3/2}^{2}(0)4d_{5/2}^{2}(2)$ |
|  | 5 | 6 | 7 | 8 |
|  | $4d_{3/2}^{2}(2)4d_{5/2}^{2}(0)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{2}(2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{2}(4)$ | $4d_{3/2}^{3}(3/2)4d_{5/2}(5/2)$ |
Table 18: CSF components of the state with $J=3$.  ion | ASF | $c_{k}$
---|---|---
42Mo2+ | 3 + | 0.67715 | -0.62242 | -0.31231 | 0.19152 | -0.10257
|  | 5 | 3 | 6 | 1 | 4
50Sn10+ | 3 + | 0.71626 | -0.53100 | -0.40474 | 0.17041 | -0.08183
|  | 5 | 3 | 6 | 1 | 4
92U52+ | 3 + | 0.99140 | -0.11531 | -0.04676 | -0.03413 | 0.01854
|  | 6 | 5 | 4 | 1 | 3
CSF |  | 1 | 2 | 3 | 4 |
|  | $4d_{3/2}(3/2)4d_{5/2}^{3}(5/2)$ | $4d_{3/2}(3/2)4d_{5/2}^{3}(3/2)$ | $4d_{3/2}(3/2)4d_{5/2}^{3}(9/2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{2}(2)$ |
|  | 5 | 6 |  |  |
|  | $4d_{3/2}^{2}(2)4d_{5/2}^{2}(4)$ | $4d_{3/2}^{3}(3/2)4d_{5/2}(5/2)$ |  |  |
Table 19: CSF components of the state with $J=4$.  ion | ASF | $c_{k}$
---|---|---
42Mo2+ | 4 + | 0.64516 | -0.46912 | -0.39396 | 0.32122 | 0.25876
|  | 3 | 6 | 1 | 4 | 7
50Sn10+ | 4 + | 0.55959 | -0.52906 | 0.40707 | 0.33947 | -0.28611
|  | 3 | 6 | 7 | 4 | 1
92U52+ | 4 + | 0.99728 | -0.04564 | 0.04427 | -0.02746 | 0.01816
|  | 7 | 2 | 4 | 6 | 5
CSF |  | 1 | 2 | 3 | 4 |
|  | $4d_{5/2}^{4}(4)$ | $4d_{3/2}(3/2)4d_{5/2}^{3}(5/2)$ | $4d_{3/2}(3/2)4d_{5/2}^{3}(9/2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{2}(4)$ |
|  | 5 | 6 | 7 |  |
|  | $4d_{3/2}^{2}(2)4d_{5/2}^{2}(2)$ | $4d_{3/2}^{2}(2)4d_{5/2}^{2}(4)$ | $4d_{3/2}^{3}(3/2)4d_{5/2}(5/2)$ |  |