# Relativistic coupled-cluster-theory analysis of the hyperfine interaction of
Ra+ isotopes

Fei-Chen Li,1,3 Yong-Bo Tang,2,∗ Hao-Xue Qiao,1,∗ and Ting-Yun Shi3
1Department of Physics, Wuhan University, Wuhan, 430072, China  2College of
Engineering Physics, Shenzhen Technology University, Shenzhen, 518118, China
3State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision
Measurement Science and Technology, Chinese Academy of Sciences, Wuhan,
430071, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Hyperfine-structure constants of odd Ra+ due to the interactions of nuclear
magnetic dipole, electric quadrupole, and magnetic octupole moments with the
electrons are investigated in the framework of relativistic coupled-cluster
method within single- and double-excitation approximation. The calculated
energies and magnetic dipole hyperfine-structure constants $A$ exhibit a good
agreement with available experimental values. Combining with the experimental
electric quadrupole hyperfine-structure constant, we also extracted the
electric quadrupole moments $Q$ of 209,211,221,223Ra. Our $Q$(221Ra) and
$Q$(223Ra) are consistent with the referenced values from a semi-empirical
analysis (Z. Phys. D: At., Mol. Clusters 11, 105 (1988)), but
$Q(^{211}$Ra)=$0.33(2)$ is smaller than the referenced value $0.48(4)$ by
about 30%. Furthermore, we also performed a procedure for assessing the
contributions of magnetic octupole moment to the hyperfine splitting. The
sensitivity of hyperfine-structure interval measurements in 223Ra+ that can
reveal the effect caused by the nuclear octupole moment are found to be on the
order of kHz.

###### pacs:

##  I INTRODUCTION

Hyperfine-structure(HFS) of the atomic energy level is caused by the
interaction between electrons and the electromagnetic multipole moments of the
nucleus. Theoretical studies of HFS are of great significance in precisely
determining different atomic properties and obtaining nuclear structure
information. For example, the interpretation of the atomic parity-
nonconservation (PNC) and electric dipole moment (EDM) experiments need to
theoretically provided the weak matrix elements of PNC amplitude and the
P,T-odd interaction constants of EDM which cannot be directly measured Nataraj
_et al._ (2008); Dzuba  _et al._ (2012); Porsev  _et al._ (2012). The accuracy
of these calculations can be reliably estimated by comparing the experimental
and theoretical values of the magnetic dipole HFS constant, because all of
them are sensitive to the wave function behavior near the nucleus, although
the three resulting mechanisms completely different. With the advancement of
spectroscopy technologies, the hyperfine splittings can be measured very
precisely. The comparison between the HFS spectrum and the corresponding high-
precision calculation can offer an alternative route to determine the
electromagnetic multipole moments of the nucleus in a nuclear model-
independent way. This is also currently one of the most accurate methods for
determining the nuclear moment, especially the nuclear quadrupole moments $Q$
and octupole moments $\Omega$ of heavy nuclei.

Ra+, as the heaviest alkaline earth metal ion in the periodic table, has been
proposed as a promising candidate for the measurement of atomic parity-
nonconservation (PNC) effect Geetha  _et al._ (1998); Ginges and Flambaum
(2004); Dzuba  _et al._ (2001); Versolato  _et al._ (2011); Dzuba and Flambaum
(2012); Nuñez Portela  _et al._ (2013); Gossel  _et al._ (2013); Roberts  _et
al._ (2014), for the high-precision atomic clock research Dzuba and Flambaum
(2000); Sahoo  _et al._ (2007, 2009); Versolato  _et al._ (2011); Holliman
_et al._ (2019) and quantum information processing Fan  _et al._ (2019). Up to
now, $35$ Ra isotopes have been discovered Dammalapati  _et al._ (2016).
Accurate knowledge of hyperfine-structure information of Ra+ is the first step
toward these high-precision researches.

Nuclear magnetic dipole moments $\mu_{I}$ of many Ra isotopes are known well.
They have been directly measured for 213Ra and 225Ra at ISOLDE by the
observation of Larmor precession of optically pumped atoms in a fast beam, and
the $\mu_{I}$ of other odd isotopes were deduced from that Arnold  _et al._
(1987). However, there is no direct measurement of the electric quadrupole
moments $Q$ of Ra isotopes to date. The widely referenced values for the
electric quadrupole moments $Q$ of Ra+ were based on a semi-empirical analysis
of HFS energy of the transition $7s_{1/2}\rightarrow 7p_{3/2}$ Neu  _et al._
(1988). Especially noteworthy is that the semi-empirical $Q$ value Neu  _et
al._ (1988) results in a about 40% difference between the experimental and
theoretical values of the electric quadrupole HFS constant of the $6d_{3/2}$
state of 211Ra+ Sahoo  _et al._ (2007); Versolato  _et al._ (2011). Although
the magnetic dipole and electric quadrupole HFS constants of the first low-
lying states of Ra+ were measured and calculated, the magnetic octupole HFS
constants and nuclear magnetic octupole moment $\Omega$ of Ra+ have not been
explored yet. With the advent of modern spectroscopic technologies, the
hyperfine splitting of 133Cs Gerginov  _et al._ (2003), 137Ba+ Lewty  _et al._
(2013), and 171Yb Singh  _et al._ (2013) were measured precisely, from which
the contributions from magnetic octupole interactions were extracted, and the
nuclear octupole moment $\Omega$ were determined by combining with the
theoretical HFS constant $C$. Thus, it is also very interesting to perform a
systematic investigation of magnetic octupole HFS constants of Ra+ isotopes.

In this work, we employed the relativistic coupled-cluster method within
single- and double-excitation approximation(CCSD), based on an even-tempered
Gaussian-type basis set, to calculate the HFS constants of Ra+ isotopes due to
the interactions of nuclear magnetic dipole, electric quadrupole, and magnetic
octupole moments and the electron. To understand the role of electron
correlation in the calculations, the intermediate results from lower-order
many-body perturbation-theory (MBPT) and the linear version of coupled-cluster
with single and double approximation (LCCSD) are also presented. The
electrical quadrupole moment $Q$ of 209,211,221,223Ra were extracted out by
combining our results with the measured HFS constants $B$ of the $7p_{3/2}$ or
$6d_{3/2}$ states. Using the HFS constants $C$ of Ra+ and the nuclear shell
model, the effect of magnetic octupole moment $\Omega$ on the hyperfine
structure has also been evaluated. The following section presents a brief
outline of the hyperfine structure theory and the coupled-cluster method for
one-electron attachment processes. Numerical results presented in section III
are compared with available theoretical and experimental data. Finally, a
summary is given in section IV.

##  II THEORETICAL METHOD

###  II.1 The hyperfine structure theory

The hyperfine interaction Hamiltonian for a relativistic electron takes the
form Schwartz (1955)

$H_{\mathrm{HFI}}=\sum_{k}T_{n}^{(k)}\cdot T_{e}^{(k)},$ (1)

where $T_{n}^{(k)}$ and $T_{e}^{(k)}$ are the spherical tensor operators with
rank $k$ ($k>0$) in the nuclear and electronic coordinates, respectively. The
hyperfine states $|(\gamma IJ)FM_{F}\rangle$ with total angular momentum F=I+J
are represented by coupling an atomic angular momentum eigenstate $|\gamma
J,M_{J}\rangle$ with a nuclear angular momentum eigenstate $|I,M_{I}\rangle$,
where $\gamma$ encapsulating the remaining electronic quantum numbers.
Restricted to $k\leq 3$, the first-order correction $W_{F,J}^{(1)}$ of
hyperfine interaction to the energy can be parameterized in terms of the HFS
constants $A$, $B$, $C$, and angular momentum coefficients

$\displaystyle W_{F,J}^{(1)}=$ $\displaystyle\sum_{k_{1}=1}^{3}\langle(\gamma
IJ)FM_{F}|T_{n}^{(k_{1})}\cdot T_{e}^{(k_{1})}|(\gamma IJ)FM_{F}\rangle$ (2)
$\displaystyle=$
$\displaystyle\underbrace{\frac{1}{2}KA}_{M1:\,{k_{1}=1}}+\underbrace{\frac{1}{2}\frac{3K(K+1)-4I(I+1)J(J+1)}{2I(2I-1)2J(2J-1)}B}_{E2:\,{k_{1}=2}}$
$\displaystyle+\frac{1}{[I(I-1)(2I-1)J(J-1)(2J-1)]}\times\left\\{(5/4){K}^{3}\right.$
$\displaystyle+5{K}^{2}+{K}\times[-3I(I+1)\times J(J+1)+I(I+1)$
$\displaystyle\underbrace{+J(J+1)+3]-5I(I+1)J(J+1)\\}C\qquad\qquad}_{M3:\,{k_{1}=3}}.$

Here the constants $A$, $B$, and $C$ are defined as

$\displaystyle A$ $\displaystyle=\mu_{N}\frac{\mu_{I}}{I}\frac{\langle{\gamma
J|}\big{|}{T_{e}^{(1)}}\big{|}{|\gamma J}\rangle}{\sqrt{J(J+1)(2J+1)}}$ (3)
$\displaystyle B$
$\displaystyle=2Q\left[\frac{2J(2J-1)}{(2J+1)(2J+2)(2J+3)}\right]^{1/2}{\langle{\gamma
J|}\big{|}{T_{e}^{(2)}}\big{|}{|\gamma J}\rangle}$ (4)
$\displaystyle=QR^{E2}(\gamma J),$

and

$\displaystyle C$
$\displaystyle=\Omega_{I}\left[\frac{J(2J-1)(J-1)}{(J+1)(J+2)(2J+1)(2J+3)}\right]^{1/2}{\langle{\gamma
J|}\big{|}{T_{e}^{(3)}}\big{|}{|\gamma J}\rangle}$ (5)
$\displaystyle=\Omega_{I}R^{M3}(\gamma J),$

where $K=F(F+1)-I(I+1)-J(J+1)$, and $\mu_{N}$ is the nuclear Bohr magneton. We
have defined two ratios, i.e.$R^{E2}(\gamma J)$=$B/Q$ and
$R^{M3}(J)$=$C/\Omega_{I}$. They can be determined directly from theoretical
calculations.

The contributions from the magnetic dipole$-$magnetic dipole ($M1-M1$) and
magnetic dipole$-$electric quadrupole ($M1-E2$) second-order hyperfine
interactions are generally on the same order as the magnetic octupole
contribution, thus we also take into account these two second-order
contribution terms. The second-order correction $W_{F,J}^{(2)}$ can be written
as Beloy  _et al._ (2008)

$\displaystyle W_{F,J}^{(2)}=$ $\displaystyle\sum_{\gamma
J^{\prime}}\sum_{k_{1},k_{2}}\frac{\langle(\gamma
IJ^{\prime})FM_{F}|T_{n}^{(k_{1})}\cdot T_{e}^{(k_{1})}|(\gamma
IJ)FM_{F}\rangle}{E_{\gamma J}-E_{\gamma^{\prime}J^{\prime}}}$ (6)
$\displaystyle\times\langle(\gamma IJ^{\prime})FM_{F}|T_{n}^{(k_{2})}\cdot
T_{e}^{(k_{2})}|(\gamma IJ)FM_{F}\rangle$ $\displaystyle\approx$
$\displaystyle\underbrace{\sum_{J^{\prime}}\left|\left\\{\begin{array}[]{lll}F&J&I\\\
k_{1}&I&J^{\prime}\end{array}\right\\}\right|^{2}\eta}_{M1-M1:\,k_{1}=k_{2}=1}+\underbrace{\sum_{J^{\prime}}\left\\{\begin{array}[]{lll}F&J&I\\\
k_{1}&I&J^{\prime}\end{array}\right\\}\left\\{\begin{array}[]{lll}F&J&I\\\
k_{2}&I&J^{\prime}\end{array}\right\\}\zeta}_{M1-E2:\,k_{1}=1,k_{2}=2},$

where

$\eta=\frac{(I+1)(2I+1)}{I}{\mu^{2}_{I}}\frac{|{\langle{\gamma
J^{\prime}|}\big{|}{T_{e}^{(1)}}\big{|}{|\gamma J}\rangle}|^{2}}{E_{\gamma
J}-E_{\gamma J^{\prime}}},$ (7) $\displaystyle\zeta=$
$\displaystyle\frac{(I+1)(2I+1)}{I}\sqrt{\frac{(2I+3)}{2I-1}}$ (8)
$\displaystyle\times{\mu_{I}}Q\frac{{\langle{\gamma
J^{\prime}|}\big{|}{T_{e}^{(1)}}\big{|}{|\gamma J}\rangle}{\langle{\gamma
J^{\prime}|}\big{|}{T_{e}^{(2)}}\big{|}{|J}\rangle}}{E_{\gamma J}-E_{\gamma
J^{\prime}}}.$

The single particle reduced matrix elements of the operators $T^{(1)}_{e}$,
$T^{(2)}_{e}$, and $T^{(3)}_{e}$ are given by

$\displaystyle\langle{\kappa_{i}}$ $\displaystyle\|T^{(1)}_{e}\|$
$\displaystyle\kappa_{j}\rangle=-\langle-\kappa_{i}\|C^{(1)}\|\kappa_{j}\rangle(\kappa_{i}+\kappa_{j})$
(9) $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}{dr\frac{P_{i}(r)Q_{j}(r)+P_{j}(r)Q_{i}(r)}{r^{2}}\times{F(r)}},$
$\displaystyle\langle{\kappa_{i}}\|T^{(2)}_{e}\|\kappa_{j}\rangle$
$\displaystyle=$ $\displaystyle-\langle\kappa_{i}\|C^{(2)}\|\kappa_{j}\rangle$
(10) $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}{dr\frac{P_{i}(r)P_{j}(r)+Q_{j}(r)Q_{i}(r)}{r^{3}}},$

and

$\displaystyle\langle{\kappa_{i}}\|T^{(3)}_{e}\|\kappa_{j}\rangle$
$\displaystyle=$
$\displaystyle-\frac{1}{3}\langle-\kappa_{i}\|C^{(3)}\|\kappa_{j}\rangle(\kappa_{i}+\kappa_{j})$
(11) $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}{dr\frac{P_{i}(r)Q_{j}(r)+P_{j}(r)Q_{i}(r)}{r^{4}}},$

where the relativistic angular-momentum quantum number
$\kappa=\ell(\ell+1)-j(j+1)-1/4$, $P$ and $Q$ are the large and small radial
components of Dirac wavefunction, respectively.

The $F(r)$ in Eq.(9) is a factor describing the correction of the finite
nuclear magnetization distribution called Bohr-Weisskopf effect (BW) Bohr and
Weisskopf (1950). When the point magnetization distribution (point) is used,
$F(r)=1$. It means that the BW effect is not considered in calculation. For
the magnetic dipole hyperfine structure constants of heavy atomic system, the
contribution of the BW correction cannot be ignored. The BW effect can usually
be evaluated by approximating the magnetization distribution of the finite
nucleus to a uniform sphere distribution model (sph) or a single-particle
model (SP) Roberts and Ginges (2020). For the uniformly magnetized sphere
(sph) model, $F(r)$ can be written as

$\displaystyle F_{sphere}(r)=\begin{cases}(\frac{r}{r_{N}})^{3}&r<r_{N}\\\
1&r\geq{r_{N}}\end{cases}.$ (12)

For the single-particle (SP) model, $F(r)$ is

$\displaystyle
F(r)=F_{sphere}(r)\times\begin{cases}1-\delta{F_{I}\ln(\frac{r}{r_{N}})}&r<r_{N}\\\
1&r\geq{r_{N}}\end{cases},$ (13)

with

$\displaystyle\delta{F_{I}}=\begin{cases}\frac{3(2I-1)}{8(I+1)}\frac{4(I+1)g_{L}-g_{S}}{g_{I}I}&I=L+1/2\\\
\frac{3(2I+3)}{8(I+1)}\frac{4Ig_{L}+g_{S}}{g_{I}I}&I=L-1/2\\\ \end{cases},$
(14)

where $r_{N}=\sqrt{5/3}r_{\rm rms}$ ($r_{\rm rms}$ is the root-mean-square
radius of the nucleus. ). $I$, $L$, and $S$ are the total, orbital, and spin
angular momentum of the unpaired nucleon. Taking $g_{L}=1$ for the proton and
$g_{L}=0$ for the neutron, the effective spin g factor, $g_{S}$, is determined
from the experimental value $g_{I}=\mu/(\mu_{N}I)$ using the formula

$g_{I}=\frac{1}{2}[g_{L}+g_{s}+(g_{L}-g_{s})\frac{L(L+1)-S(S+1)}{I(I+1)}].$
(15)

According to the Ref. Shabaev  _et al._ (1995, 1997), the SP model can make a
rough assessment of BW correction, but it can also be further improved by
including the spin-orbit interaction and founding the nucleon wavefunction in
a Woods-Saxon potential (SP$-$WS). Present work pays more attention to the
high-order hyperfine interaction, ie. $B/Q$ and $C/\Omega$, so here we only
use the sphere model and SP model to discuss the BW correction of the HFS
constant $A$. The required parameters of radium isotopes can be found in Ref.
Angeli (2004).

###  II.2 A brief description of the relativistic coupled-cluster approach

In the coupled-cluster theory, the atomic wave function
$|\Psi_{\upsilon}\rangle$ for a single valence($\upsilon$) open-shell system
is given by

$|\Psi_{\upsilon}\rangle={e^{S}}|\Phi_{\upsilon}\rangle.$ (16)

$|\Phi_{v}\rangle$ is the reference state which is set as the zero-order
Dirac-Fock wavefunction in present work, and $S$ represents the cluster
operator. Within the single- and double-excitation approximation, the cluster
operator $S$ can be partitioned into

$\displaystyle S$ $\displaystyle=$ $\displaystyle S^{(0,0)}+S^{(0,1)}$ (17)
$\displaystyle=$ $\displaystyle
S^{(0,0)}_{1}+S^{(0,0)}_{2}+S^{(0,1)}_{1}+S^{(0,1)}_{2},$

where $S^{(0,0)}$ and $S^{(0,1)}$ correspond to the excitation from core and
from valence electrons, respectively. The expected value of one-particle
operator $O$ for the state $|\Psi_{\upsilon}\rangle$ can be evaluated using
the following expression

${\bar{O}_{\upsilon}}=\frac{\langle{\Psi_{\upsilon}}|{O}|{\Psi_{\upsilon}}\rangle}{\langle{\Psi_{\upsilon}}|{\Psi_{\upsilon}}\rangle}=\frac{\langle{\Phi_{\upsilon}}|{e^{S^{\dagger}}Oe^{S}}|{\Phi_{\upsilon}}\rangle}{\langle{\Phi_{\upsilon}}|{e^{S^{\dagger}}e^{S}}|{\Phi_{\upsilon}}\rangle}.$
(18)

where the numerator and the denominator $e^{S^{\dagger}}Oe^{S}$ are not
terminating. In previous works Tang  _et al._ (2019); Lou  _et al._ (2019),
only linear terms of cluster operators were included in property calculations,
i.e.,

$\displaystyle e^{S{\dagger}}O{e^{S}}\approx$ $\displaystyle O$
$\displaystyle+\\{{O}{S^{(0,0)}_{1}}+{\rm c.c.}\\}+\\{{O}{S^{(0,1)}_{1}}+{\rm
c.c.}\\}$ $\displaystyle+$ $\displaystyle\\{{O}{S^{(0,1)}_{2}}+{\rm
c.c.}\\}+\\{S^{(0,0){\dagger}}_{1}O{S^{(0,1)}_{1}}+{\rm c.c.}\\}$
$\displaystyle+$ $\displaystyle
S^{(0,0){\dagger}}_{1}O{S^{(0,0)}_{1}}+\\{S^{(0,0){\dagger}}_{1}O{S^{(0,0)}_{2}}+{\rm
c.c.}\\}$ $\displaystyle+$
$\displaystyle\\{S^{(0,0){\dagger}}_{1}O{S^{(0,1)}_{2}}+{\rm
c.c.}\\}+S^{(0,0){\dagger}}_{2}O{S^{(0,0)}_{2}}$ $\displaystyle+$
$\displaystyle\\{S^{(0,0){\dagger}}_{2}O{S^{(0,1)}_{2}}+{\rm
c.c.}\\}+S^{(0,1){\dagger}}_{1}O{S^{(0,1)}_{1}}$ $\displaystyle+$
$\displaystyle\\{S^{(0,1){\dagger}}_{1}O{S^{(0,1)}_{2}}+{\rm
c.c.}\\}+S^{(0,1){\dagger}}_{2}O{S^{(0,1)}_{2}}$

and

$\displaystyle e^{S{\dagger}}{e^{S}}\approx$ $\displaystyle 1$
$\displaystyle+{S^{(0,0){\dagger}}_{1}}{S^{(0,0)}_{1}}+{S^{(0,1){\dagger}}_{1}}{S^{(0,1)}_{1}}$
(20) $\displaystyle+$
$\displaystyle{S^{(0,0){\dagger}}_{2}}{S^{(0,0)}_{2}}+{S^{(0,1){\dagger}}_{2}}{S^{(0,1)}_{2}}$

where ${\rm c.c.}$ stands for the complex conjugate part. For the LCCSD
method, Eq. II.2 and Eq. 20 are exact. But this approximation ignores the so-
called “dressing” from the nonlinear terms for the CCSD method. To account for
the contribution from the main non-linear terms of cluster operators, we use
an iteration strategy as proposed in Ref. Mani and Angom (2010). The basic
idea of this strategy is that the $e^{S^{(0,0)\dagger}}Oe^{S^{(0,0)}}$ was
expanded in terms of effective one-body, two-body, and three-body terms. These
terms are obtained by a self-consistently iteration calculation and stored as
an intermediate block. Then these intermediate blocks are constructed with
$S^{(0,1)\dagger}$ and $S^{(0,1)}$ to the final diagrams. It is worth noting
that one should avoid the repetition of any diagram in the iterative
procedure. In this work, we calculate the expected value using three coupled-
cluster approximations: $\rm LCCSD$, $\rm CCSD_{L}$, and $\rm CCSD$. $\rm
LCCSD$ and $\rm CCSD_{L}$ represent that the expected values are computed
using Eq. II.2 and Eq. 20 with the cluster amplitudes from LCCSD and CCSD
method respectively. $\rm CCSD$ corresponds to the self-consistently iteration
calculation.

###  II.3 Computational outline

The zero-order wave-function $|\Phi_{v}\rangle$ is obtained by the Dirac-Fock
calculation. The no-pair Dirac Hamiltonian is set as the starting point. Breit
interaction is considered at the same foot as coulomb interaction. Different
from the previous works Tang  _et al._ (2017, 2019), the lowest-order quantum
electrodynamics (QED) correction is also included in Dirac-Fock formalism
using a simple radiative potential proposed by Flambaum and Ginges Flambaum
and Ginges (2005). The radiative potential can be written as

$\Phi_{\rm
rad}(r)=\Phi_{U}(r)+\Phi_{g}(r)+\Phi_{f}(r)+\Phi_{l}(r)+\frac{2}{3}\Phi^{\rm
Simple}_{WC}(r)$ (21)

where $\Phi_{U}(r)$ is the Uehling potential, $\Phi_{g}(r)$ is the magnetic
form-factor contribution, $\Phi_{f}(r)$ is the high-frequency electric form-
factor contribution, $\Phi_{l}(l)$ is the low-frequency electric form-factor
contribution, and $\Phi^{\rm Simple}_{WC}(r)$ is the Wichmann-Kroll potential,
$\Phi_{U}(r)$ and $\Phi^{\rm Simple}_{WC}(r)$ are for the vacuum polarization,
and the other three parts correspond to self-energy corrections. The
expression of these effective operators can be found in Ref. Ginges and
Berengut (2016, 2016). To check our QED code, we reproduced almost all results
in Ref. Ginges and Berengut (2016, 2016).

In this work, we employed a finite basis set composed of even-tempered
Gaussian-type functions to expand the Dirac radial wave function, as in
Ref.Chaudhuri  _et al._ (1999). To avoid the spurious state and variational
collapse problem, the kinetic balance condition between the large and the
small components is used. The Gaussian-type function has the form

$G_{i,k}=\mathcal{N}_{i}r^{n_{k}}e^{-\eta_{i}r^{2}},$ (22)

where $\mathcal{N}_{i}$ is the normalization factor, $n_{k}=l-1$, and
$\eta_{i}=\eta_{0}\xi^{i-1}$. Then the large and small components of the
orbital can be expressed as

$\begin{array}[]{l}P(r)=\sum\limits_{i=1}^{N}C_{i}^{P_{\kappa}}G_{i,\kappa}^{r},\\\
Q(r)=\sum\limits_{i=1}^{N}C_{i}^{Q_{\kappa}}\mathcal{N}_{i}^{Q}(\frac{d}{dr}+\frac{\kappa}{r})G_{i,\kappa}^{r},\end{array}$
(23)

with

$\mathcal{N}_{i}^{Q}=\sqrt{\frac{\alpha_{i}}{2n_{\kappa}-1}[4(\kappa^{2}+\kappa-
n_{\kappa})-1]}.$ (24)

The Fermi nuclear distribution was employed to describe the Coulomb potential
between electrons and the nucleus. All the core orbitals and virtual orbitals
with energies smaller than 10000 a.u. are included in correlation
calculations. Table 1 lists the Gauss basis parameters. $N$ is the number of
basis set for each symmetry, and $N_{c}$ and $N_{v}$ represent the number of
core orbitals and virtual orbitals respectively.

Table 1: The parameters of the Gauss basis set. $N$ is the number of basis set for each symmetry. $N_{c}$ and $N_{v}$ represent the number of core orbitals and virtual orbitals, respectively.  | $s$ | $p$ | $d$ | $f$ | $g$ | $h$ | $i$ | $j$
---|---|---|---|---|---|---|---|---
$\eta_{0}\times 10^{3}$ | 0.42 | 0.42 | 0.42 | 0.57 | 7.6 | 86 | 95 | 96
$\xi$ | 1.71 | 1.69 | 1.71 | 1.74 | 2.0 | 2.0 | 2.0 | 2.0
$N$ | 55 | 45 | 42 | 35 | 25 | 20 | 15 | 10
$N_{c}$ | 6 | 5 | 3 | 1 | 0 | 0 | 0 | 0
$N_{v}$ | 29 | 28 | 27 | 25 | 17 | 14 | 13 | 10

##  III RESULTS AND DISCUSSION

###  III.1 Energies

Table 2:  Energy levels of Ra+ in ${\rm cm^{-1}}$. ${\rm E_{DF}}$ denotes the lowest-order Dirac-Fock energy. ${\rm E_{MBPT(2)}}$, ${\rm E_{LCCSD}}$, ${\rm E_{CCSD}}$, and ${\rm E_{CCSD+QED}}$ are the energies obtained using second-order MBPT, LCCSD, CCSD, and CCSD results with QED correction approximations, respectively. ${\rm\delta_{i}(\%)}$, where i=0, 1, 2, 3, 4 shows the differences between corresponding calculation results and experimental values.  Level | ${\rm E_{DF}}$ | ${\rm E_{MBPT(2)}}$ | ${\rm E_{LCCSD}}$ | ${\rm E_{CCSD}}$ | ${\rm E_{CCSD+QED}}$ |  ${\rm E_{Expt.}}$Kramida  _et al._ (2019) | $\delta_{\rm 0}(\%)$ | $\delta_{1}(\%)$ | $\delta_{\rm 2}(\%)$ | $\delta_{\rm 3}(\%)$ | $\delta_{\rm 4}(\%)$
---|---|---|---|---|---|---|---|---|---|---|---
$7s_{1/2}$ |  $-$75872.96 |  $-$83359.54 |  $-$82506.42 |  $-$82077.91 |  $-$81989.19 |  $-$81842.49 |  $-$7.29 | 1.85 | 0.81 | 0.29 | 0.18
$7p_{1/2}$ |  $-$56825.75 |  $-$60979.06 |  $-$60821.14 |  $-$60518.69 |  $-$60521.04 |  $-$60491.17 |  $-$6.06 | 0.81 | 0.55 | 0.05 | 0.05
$7p_{3/2}$ |  $-$52887.58 |  $-$56002.27 |  $-$55889.94 |  $-$55652.01 |  $-$55650.44 |  $-$55633.64 |  $-$4.94 | 0.66 | 0.46 | 0.03 | 0.03
$6d_{3/2}$ |  $-$62408.99 |  $-$71017.35 |  $-$70327.07 |  $-$69744.51 |  $-$69786.20 |  $-$69758.22 |  $-$10.54 | 1.80 | 0.82 |  $-$0.02 | 0.04
$6d_{5/2}$ |  $-$61662.25 |  $-$69101.11 |  $-$68593.87 |  $-$68071.05 |  $-$68103.28 |  $-$68099.50 |  $-$9.45 | 1.47 | 0.73 |  $-$0.04 | 0.00
Table 3:  Comparison of our calculated energies ${\rm E_{CCSD+QED}}$ with other available theoretical and experimental data in ${\rm cm^{-1}}$. Level | ${\rm E_{CCSD+QED}}$ |  ${\rm E_{CCSD}}$ Eliav  _et al._ (1996) |  ${\rm E_{SDpT}}$ Pal  _et al._ (2009); Safronova  _et al._ (2007) |  ${\rm E_{CPM}}$ Dzuba  _et al._ (2001) |  ${\rm E_{CPM+LD}}$ Dzuba (2013) |  ${\rm E_{Expt.}}$ Kramida  _et al._ (2019)
---|---|---|---|---|---|---
7s1/2 |  $-$81989.19 |  $-$82025 |  $-$81508 |  $-$81960 |  $-$81714 |  $-$81842.5
6d3/2 |  $-$69786.20 |  $-$69596 |  $-$69488 |  $-$70149 |  $-$69788 |  $-$69758.22
6d5/2 |  $-$68103.28 |  $-$67936 |  $-$67947 |  $-$68449 |  $-$68045 |  $-$68099.51
7p1/2 |  $-$60521.04 |  $-$60462 |  $-$60326 |  $-$60681 |  $-$60511 |  $-$60491.17
7p3/2 |  $-$55650.44 |  $-$55629 |  $-$55519 |  $-$55734 |  $-$55625 |  $-$55633.64

We have calculated the energies of some important low-lying states of Ra+
using different models including Dirac-Fock(DF), second-order many-body
perturbation theory (MBPT(2)), LCCSD, and CCSD calculations, and the predicted
energies are labeled by ${\rm E_{DF}}$, ${\rm E_{MBPT(2)}}$, ${\rm
E_{LCCSD}}$, and ${\rm E_{CCSD}}$, respectively. We also evaluated the
correction of the QED effect to the calculation of CCSD values labeled by
${\rm E_{CCSD+QED}}$. The results of the first five states are listed in table
2 and compared with experimental values from NIST Kramida  _et al._ (2019)
labeled as ${\rm E_{Expt.}}$. From table 2, one can easily find that: (i)
There are obvious differences between DF and CCSD energies indicating
significant contributions from electron correlation effects that are not
included in DF calculations. The largest contribution is approximately $10\%$
occurring at $6d$ states. (ii) The differences between CCSD results and the
experimental values are no more than $0.1\%$ except for the ground state,
showing a much better agreement than MBPT(2) and LCCSD results for all states.
This phenomenon indicates that the inclusion of nonlinear terms of the cluster
operators are of crucial importance in obtaining very accurate energy levels.
(iii) It is necessary to consider the QED effect for the energy improvement of
the low energy states, especially for the ground state, the energy improvement
is $0.1\%$.

There are also some other theoretical calculations about radium ion energies
Eliav  _et al._ (1996); Safronova  _et al._ (2007); Pal  _et al._ (2009);
Dzuba  _et al._ (2001); Dzuba (2013). Table 3 presents a comprehensive
comparison of our calculated energies with other $ab$ $initio$ results. Elaiv
et al. Eliav  _et al._ (1996) calculated the energies of the first low-lying
states of Ra+ and Ra using relativistic CCSD based on DCB Hamiltonian. Our
method is the same as theirs, but we use a larger basis set and partial waves.
Safronova et al. Safronova  _et al._ (2007) and Pal et al. Pal  _et al._
(2009) carried out their calculations using the all-order many-body
perturbation theory, in which the linear single- and double-excitation terms
are included and a part of triple-excitation are also considered using third-
order MBPT corrections (SDpT). The method adopted by Dzuba et al. Dzuba  _et
al._ (2001) is called the correlation potential method(CPM). In Ref. Dzuba
(2013), the ladder diagrams were added to the correlation potential method
(CPM+LD), which significantly improved the level of accuracy for the lowest
$D$ states. From Table 3, one can notice that our total CCSD results, about
0.2% better than the SDpT results Pal  _et al._ (2009) with respect to the
experimental values, are very close to the results by the CPM Dzuba  _et al._
(2001) and CPM+LD Dzuba (2013) which also include the QED corrections.

###  III.2 Hyperfine structure constants A of Ra+ isotopes.

Table 4: The HFS constants $A$ (MHz) of 223Ra+ at different correlation levels are given. The total results, ie. CCSD+QED+(BW$-$SP)CC, are our recommended values. The contributions of higher partial waves, QED correction, BW correction and high-order correlation beyond CCSD are estimated respectively and summarized in brackets as the uncertainties of the recommended values. Some other $ab$ $initio$ theoretical and experimental results are also listed for comparison.  Method | $7s_{1/2}$ | $6d_{3/2}$ | $6d_{5/2}$ | $7p_{1/2}$ | $7p_{3/2}$
---|---|---|---|---|---
DF | 2687 | 52.9 | 19.3 | 438 | 33.8
MBPT(3) | 3690 | 80.6 |  $-$19.3 | 665 | 53.5
LCCSD | 3663 | 82.7 |  $-$25.5 | 689 | 56.6
$\rm CCSD_{L}$ | 3621 | 79.7 |  $-$22.0 | 666 | 56.3
CCSD | 3608 | 82.0 |  $-$24.2 | 666 | 56.8
Estimated BW and QED
(BW$-$sph)DF |  $-$66.9 | 0.0 | 0.0 |  $-$1.62 | 0.0
(BW$-$sph)CC |  $-$90.1 |  $-$1.27 | 1.24 |  $-$2.24 |  $-$0.17
(BW$-$SP)DF |  $-$91.8 | 0.0 | 0.0 |  $-$2.26 | 0.0
(BW$-$SP)CC |  $-$123.5 |  $-$1.73 | 1.70 |  $-$3.12 |  $-$0.24
QED |  $-$59.7 |  $-$0.54 | 0.68 |  $-$2.72 |  $-$0.03
Estimated uncertainties
HO | 45 | 1.4 | 2.1 | 11.2 | 1.2
Basis | 1.8 | 0.2 | 0.1 | 0.4 | 0.1
$\rm BW_{c}$ | 62 |  $-$0.87 | 0.85 |  $-$1.56 |  $-$0.12
$\rm QED_{c}$ | 30 |  $-$0.27 | 0.34 |  $-$1.36 |  $-$0.02
Recommended values
Total | 3426(82) | 79.7(1.7) |  $-$21.8(2.3) | 660(11.4) | 56.5(1.2)
SDpT Pal  _et al._ (2009) | 3450 | 79.6 |  $-$24.1 | 671.5 | 54.4
CCSD(T) Sahoo  _et al._ (2007) | 3567.26 | 77.1 |  $-$23.9 |  |
Z-vector Sasmal (2017) | 3446.3 |  |  |  |
Expt.1987 Wendt  _et al._ (1987) |  |  |  | 667.1(2.1) | 56.5(8)
Expt.1988 Neu  _et al._ (1988) | 3404.0(1.9) |  |  |  |

Different isotopes of Ra have different hyperfine splittings resulted from
their different nuclear spin. 223Ra+ as an example, the HFS constant $A$ of
the first five states from various methods are list in table 4 in MHz. The
theoretical results in the first five rows are obtained within the point
magnetization distribution model. The following rows give the BW contributions
from the sphere model and SP model at DF and CCSD levels, together with the
QED correction from the radiation potential method. The following “Estimated
uncertainties” part provides the uncertainties due to the incomplete basis set
(“Basis”), high-order correlation effect (“HO”) beyond the CCSD method, BW
correction(“$\rm BW_{c}$”) and QED correction(“$\rm QED_{c}$”). Finally, the
“Total” row lists our recommended value obtained by CCSD+QED+(BW$-$SP)CC with
the uncertainty enclosed in parentheses. Some other $ab$ $initio$ theoretical
and experimental results are also listed for comparison.

In our calculation, the “Basis” uncertainty is estimated using the partial
wave extrapolation strategy based on calculating all quantities under the
conditions of $\ell_{max}=4,5,6,7$, respectively. The premise is that the
number of Gaussian bases we use is large enough to satisfy our calculations.
The contribution from high-order electron correlation effect beyond CCSD is
difficult to effective estimating, but it is usually not greater than the
contribution of the lower-order nonlinear terms (ie.$\rm CCSD-LCCSD$) which
makes up about 5% of the total electron correlation. Therefore, we take the 5%
of the total electron correlation contribution, ie.$\rm(CCSD-DF)\times 5\%$ as
the uncertainty. Ginges et al. Ginges  _et al._ (2017) investigated the BW
correction for the HFS A of the ground state of some alkali-metal-like ions
within sphere model, SP model, and SP$-$WS model respectively. The uncertainty
of BW correction does not exceed 30% of the maximum value of the two results
from SP$-$WS model and the sphere model. For 223Ra+, the BW correction
obtained by SP model of the five states in table are greater than those from
the sphere model, so we take 50% of the total BW effect within the SP model as
the conservatively estimated uncertainty. The radiation potential method here
is reliable for estimating QED correction to binding energy, but is not
rigorous for the short-distance operators such as hyperfine interaction
operators. Compared with the more rigorous evaluation in Ref. Ginges  _et al._
(2017), our calculation seems to overestimate the effect of QED and should be
compensated with a large uncertainty, ie. the QED$\times$50%.

From the table 4, we can find that the CCSD results show better agreement with
experimental values than the LCCSD and the MBPT(3) method, and the MBPT(3)
calculation yields the worst results. The inclusion of correlation effects by
our CCSD method improved the results of $A$ from the DF values by $35\%$ in
the $6d_{3/2}$ state and $180\%$ in the $6d_{5/2}$ state regardless of the
sign, which suggests that the correlation effect plays a significant role in
$d$ states. Moreover, the opposite sign of CCSD values and DF values of
$6d_{5/2}$ states reveals a strong cancellation from different correlation
effects. Compared with the experimental value of the ground state $7s_{1/2}$,
3404.0(1.9) MHz, our CCSD result, 3608 MHz, is overestimated by about 6%.
After further considering BW and QED corrections (both are negative values),
the result changes to 3426(82) MHz, then the difference is significantly
reduced to 0.6%, showing that these two corrections are extremely important to
further improve the accuracy of the A for the ground state, moreover, the
contribution of BW corrections seems in an even more dominant position than
the QED radiative correction. The BW effect also plays an important role in
$d$ states. Especially for $6d_{5/2}$, the total BW correction obtained by the
SP model accounts for 7% of the CCSD result, which is 2% larger than that
obtained from the sphere model. Consequently, the assessment of the BW is
necessary for the state with a relatively large correlation effect, and a more
accurate nuclear model is also required in this process. Some interesting
features about the BW correction of the states with high angular momentum can
also be found in the table. For $7p_{3/2}$, $6d_{3/2}$, and $6d_{5/2}$ states,
whether within the sphere distribution model or the SP model, the BW
corrections to DF are zero, and the total BW corrections only come from the
calculation of the correlation effect, which indicates that the accurate
calculation of the correlation effect is very important for the investigation
of the BW correction on these states. Overall, our calculations show that the
BW and QED corrections are important for the HFS constant A of Ra+. To further
obtain a more accurate wave function of different radium isotope ion and
reduce the current uncertainty, it is necessary and important to optimize the
magnetization distribution model and perform a rigorous calculation of the QED
effect.

The present CCSD, SDpT Pal  _et al._ (2009), CCSD(T) Sahoo  _et al._ (2007),
Z-vector Sasmal (2017) methods in Table 4 are all based on relativistic
coupled-cluster-theory, though they have some different treatments for
electronic correlation effects. In the SDpT method, other nuclear magnetic
distribution models are used to consider the BW correction, while there is no
description of the BW effect in the CCSD(T) and Z-Vector methods, and none of
these three methods evaluate the contribution of QED effect. Compared with
other results, our final value of the $7s_{1/2}$ and $7p_{3/2}$ states
including the BW and QED corrections are the closest to the experimental
value. The experimental value of the $7p_{1/2}$ state is between our total
value and the SDpT value, which indicates that there may be some offset among
the (BW + QED) corrections, non-linear terms and the higher-order excitations.
We find that all the theoretical HFS constant A of ground state are larger
than the experimental value.

###  III.3 Evaluation of the electric quadrupole moment Q for different Ra+
isotopes.

Due to the absence of direct measurements of electric quadrupole moments $Q$,
extracting the $Q$ value out by combining the measured HFS constant $B$ and
calculated electric field gradient or $R^{E2}(\gamma J)$ is usually considered
to be convenient and economical. From this point, a very precise measurement
of the hyperfine splitting and accurate theoretical determination of the ratio
value $R^{E2}(\gamma J)$ is needed to extract the $Q$ value accurately. The
validity of our CCSD calculation has been verified by the above analysis for
energies and HFS constants $A$. In the same theoretical framework, we
calculated the ratio $R^{E2}(\gamma J)$ (in MHz/b). The $R^{E2}(\gamma J)$ of
$7p_{3/2}$, $6d_{3/2}$ and $6d_{5/2}$ states from various methods are listed
in Table 5. For this quantity, the uncertainty is from the incomplete basis
set and the high-order correlation effect beyond CCSD method. The contributes
of QED correction is almost zero. Our recommended result of 309(5) MHz/b for
$R^{E2}$ of 6d3/2 state is very close to the value of 306.12 MHz/b in Ref.
Sahoo  _et al._ (2007). It implies that the 40% difference between the
experimental and theoretical values of the HFS constant $B$ of the $6d_{3/2}$
state of 211Ra+ Sahoo  _et al._ (2007); Versolato  _et al._ (2011) may be
caused by inaccurate estimation of $Q$ and prompted us to evaluate $Q$ in
combination with available experimental HFS constant $B$ and our CCSD
$R^{E2}(\gamma J)$.

Table 5:  $R^{E2}(\gamma J)$ (in MHz/b) of $7p_{3/2}$, $6d_{3/2}$ and $6d_{5/2}$ states at different levels of correlation. Our recommended value is listed in the $\rm{``Final"}$ column, where the uncertainty is in parentheses. Level | ${\rm DF}$ | ${\rm LCCSD}$ | ${\rm CCSD_{L}}$ | ${\rm CCSD}$ | Final
---|---|---|---|---|---
$7p_{3/2}$ | 398 | 714 | 692 | 693 | 693(15)
$6d_{3/2}$ | 213 | 309 | 311 | 309 | 309(5)
$6d_{5/2}$ | 227 | 386 | 386 | 384 | 384(8)
Table 6:  The electric quadrupole moment $Q$ (in b) of 209,211,221,223Ra. $Q_{\rm ccsd}$ are our values. ${Q_{\rm semi}}$ are from Ref. Neu  _et al._ (1988). Uncertainties are given in parentheses. The subscript $\rm{``E"}$ and $\rm{``T"}$ represent the uncertainty coming from the experiment value $B_{\rm Expt.}$ and the theoretical calculation $R^{E2}(\gamma J)$, respectively. | 6d 2D3/2 | 7p 2P${}^{o}_{3/2}$
---|---|---
|  209Ra+ |  211Ra+ |  221Ra+ |  223Ra+
${B_{\rm Expt.}}$ | 104(38) | 103(6) | 1364.2(5.1) | 864.8(1.9)
${Q_{\rm ccsd}}$ | 0.337$(123)_{\rm{E}}$$(6)_{\rm{T}}$ | 0.333$(19)_{\rm{E}}$$(6)_{\rm{T}}$ | 1.968$(7)_{\rm{E}}$$(34)_{\rm{T}}$ | 1.248$(3)_{\rm{E}}$$(22)_{\rm{T}}$
${Q_{\rm semi}}$ | 0.40$(2)_{\rm{E}}$$(4)_{\rm{T}}$ | 0.48$(2)_{\rm{E}}$$(4)_{\rm{T}}$ | 1.978$(7)_{\rm{E}}$$(106)_{\rm{T}}$ | 1.254$(3)_{\rm{E}}$$(66)_{\rm{T}}$

Since the HFS constants $B$ are only measured for $7p_{3/2}$ state of
221,223Ra+ and $6d_{3/2}$ state of 209,211Ra+, we evaluated the electric
quadrupole moment $Q$ for these states, labeled as $Q_{\rm ccsd}$, and
compared with a semi-empirical analysis, $Q_{\rm semi}$, from Ref. Neu  _et
al._ (1988) in table 6. The experimental values of electric quadrupole HFS
constants, $B_{\rm Expt.}$, are also listed in the table. The equation used to
evaluate the electric quadrupole moment $Q$ is

$Q=\frac{B_{\rm Expt.}}{R^{E2}}(\gamma J).$ (25)

Obviously, the accuracy of resulting $Q$ is dependent on the accuracy of
$B_{\rm Expt.}$ and $R^{E2}$. We use the results in Table 6. In Table 6,
uncertainties are given in parentheses. The subscript $\rm{``E"}$ and
$\rm{``T"}$ represent the uncertainties from $B_{\rm Expt.}$ and
$R^{E2}(\gamma J)$, respectively. Our $Q_{\rm ccsd}$ are consistent with
$Q_{\rm semi}$ for 221,223Ra with a difference of only $0.5\%$, but are
significantly smaller than $Q_{\rm semi}$ in the cases of 209,211Ra. One
should also note that the experimental HFS constants $B$ for the $6d_{3/2}$
state of 209,211Ra+ have larger uncertainties.

Furthermore, according to the ratio values provided by present work, if the
HFS constant $B$ of a state $|{\gamma^{\prime}J^{\prime}}\rangle$ of other Ra+
isotopes, denoted by $B_{1}(\gamma^{\prime}J^{\prime})$, are measured, we can
easily obtain their electric quadrupole moment $Q_{1}$ according to the
following relations:

$\frac{B_{1}(\gamma^{\prime}J^{\prime})}{B_{2}(\gamma
J)}=\frac{Q_{1}R^{E2}_{1}(\gamma^{\prime}J^{\prime})}{Q_{2}R^{E2}_{2}(\gamma
J)}$ (26)

where the $B_{2}(\gamma J)$ is the experimental HFS constant $B$ for a given
state $|\gamma J\rangle$ and $Q_{2}$ is the CCSD value for a certain Ra+
isotope.

###  III.4 Appraising nuclear octupole moment contributions to the hyperfine
structures.

In this section, we use the nuclear octupole moment $\Omega$ from the single-
particle shell model to roughly evaluate the HFS constant $C$ of some low-
lying states in Ra+. We expect to obtain the typical orders of magnitudes of
the hyperfine splitting caused by magnetic octupole hyperfine interaction.
This process is necessary to know the precisions required of these hyperfine
spectral measurements to extract the nuclear octupole moment $\Omega$ of Ra.
In the absence of reference values for $\Omega$ of Ra, we can estimate it
crudely by the single-particle model:

$\displaystyle\Omega^{\mathrm{sp}}=$ $\displaystyle\mu_{N}\left\langle
r^{2}\right\rangle\frac{3}{2}\frac{(2I-1)}{(2I+4)(2I+2)}$ (27)
$\displaystyle\times\left\\{\begin{array}[]{c}(I+2)[(I-\frac{3}{2})g_{L}+g_{S}],\text{
for }I=L+\frac{1}{2}\\\ (I-1)[(I+\frac{5}{2})g_{L}-g_{S}],\text{ for
}I=L-\frac{1}{2}\end{array}\right.$

For even-odd (even number of protons, odd number of neutrons) nuclei Ra,
$g_{L}$ =0, $g_{S}$ =$-$3.826. Here we set 223Ra as an example, possessing
nuclear ground states with a spin of $I=3/2$, and an orbital momentum $L=2$.
Then we may estimate the octupole moment to be

$\Omega^{\mathrm{sp}}=0.164\mu_{N}\left\langle r^{2}\right\rangle\approx
0.0525\mu_{N}\times\mathrm{b}.$ (28)

The root-mean-square radius $\left\langle r^{2}\right\rangle$1/2 of the
nucleus of 223Ra is 5.6602 $fm$. The ratio of magnetic octupole hyperfine
constants to magnetic octupole moments $R^{M3}(\gamma J)$ are listed in Table
7.

Table 7:  $R^{M3}(\gamma J)$ (in kHz/($\mu_{N}$$\times$b)) of $7p_{3/2}$, $6d_{3/2}$ and $6d_{5/2}$ states at different levels of correlation. Our recommended value is listed in the $\rm{``Final"}$ column, where the uncertainty is in parentheses. Level | ${\rm DF}$ | ${\rm LCCSD}$ | ${\rm CCSD_{L}}$ | ${\rm CCSD}$ | Final
---|---|---|---|---|---
$7p_{3/2}$ | 26.2 | 42.9 | 41.4 | 41.4 | 41.4(8)
$6d_{3/2}$ | 6.71 | 9.73 | 9.64 | 9.66 | 9.66(15)
$6d_{5/2}$ | 1.82 |  $-$6.17 |  $-$5.99 |  $-$6.23 |  $-$6.23(40)

It is also seen in the table that the trends in the correlation effects for
the calculations of $A$ and $C$ are almost the same ignoring the $d_{5/2}$
states which have a larger uncertainty. Comparing the magnitudes of the
calculated $C/\Omega$ values among the given states, the $C^{7p_{3/2}}$ is
large:

$C^{7p_{3/2}}=2172\ \mathrm{Hz}.$ (29)

For the $7p_{3/2}$ state, the hyperfine structure intervals $\delta
W_{F}=W_{F,7p_{3/2}}-W_{F-1,7p_{3/2}}$ can be expressed in terms of these
constants as

$\delta
W_{1}=A^{7p_{3/2}}-B^{7p_{3/2}}+56C^{7p_{3/2}}+\frac{\eta^{7p_{3/2}}}{36}-\frac{\sqrt{5}\zeta^{7p_{3/2}}}{60},$
(30) $\delta
W_{2}=2A^{7p_{3/2}}-B^{7p_{3/2}}-28C^{7p_{3/2}}+\frac{\eta^{7p_{3/2}}}{45}+\frac{2\sqrt{5}\zeta^{7p_{3/2}}}{75},$
(31) $\delta
W_{3}=3A^{7p_{3/2}}+B^{7p_{3/2}}+8C^{7p_{3/2}}-\frac{\eta^{7p_{3/2}}}{20}-\frac{\sqrt{5}\zeta^{7p_{3/2}}}{100}.$
(32)

Setting Eqs.(30)$\times$5-Eqs.(31)$\times$4+Eqs.(32), one can find

$C^{7p_{3/2}}=\frac{1}{400}\left(5\delta W_{1}-4\delta W_{2}+\delta
W_{3}+\frac{\sqrt{5}\zeta^{7p_{3/2}}}{5}\right).$ (33)

It can be deduced that the sensitivity of measurements on these hyperfine
splittings on the order of $\sigma_{\delta W}\approx 1\mathrm{kHz}$ would
result in an uncertainty in the HFS constant $C$ on the order of
$\sigma_{C}\approx 0.0162\sigma_{\delta W}\approx 20\mathrm{Hz}$, and the
effects of HFS constant $C$ of the predicted magnitude would be revealed
within this sensitively reliably.

It is also important to note that $C$ is related to $\zeta^{7p_{3/2}}$ factor,
which represents the second-order correction. If we want to evaluate
$\zeta^{7p_{3/2}}$, we have to know the off-diagonal elements which are also
one of the major systematics in the extractions of the $C$ values from the
measured hyperfine splittings. For this purpose, we have calculated these
matrix elements using the DF, LCCSD and CCSD methods and displayed them in
Table 8. Then we find $\eta^{7p_{3/2}}$=0.051$\mathrm{KHz}$,
$\zeta^{7p_{3/2}}$=7.09$\mathrm{KHz}$, and

$\Delta C^{7p_{3/2}}=\frac{\sqrt{5}\zeta^{7p_{3/2}}}{2000}=7.93\mathrm{Hz}.$
(34)

For the $\zeta^{7p_{3/2}}$ parameter, the contribution from off-diagonal
matrix elements between $7p_{3/2}$ and $7p_{1/2}$ states with the lowest
denominator of the energy difference is predominant, and the rest can be
ignored.

Table 8:  Important off-diagonal matrix elements (in MHz) among the fine-structure partners obtained using our DF, LCCSD and CCSD methods. Off-diagonal matrix | DF | LCCSD | $\rm CCSD_{L}$ | CCSD | Final
---|---|---|---|---|---
$\langle 7p_{1/2}||{O}^{(1)}||7p_{3/2}\rangle$ |  $-$229 |  $-$147 |  $-$123 |  $-$119 |  $-$119(5)
$\langle 6d_{3/2}||{O}^{(1)}||6d_{5/2}\rangle$ |  $-$125.5 |  $-$2536 |  $-$2350 |  $-$2449 |  $-$2449(116)
$\langle 7p_{1/2}||O^{(2)}||7p_{3/2}\rangle$ |  $-$1281 |  $-$2222 |  $-$2155 |  $-$2158 |  $-$2158(44)
$\langle 6d_{3/2}||O^{(2)}||6d_{5/2}\rangle$ |  $-$245 |  $-$334 |  $-$338 |  $-$334 |  $-$334(5)

##  IV Conclusion

In summary, we calculated energy levels and hyperfine-structure coupling
constants of Ra+ using Dirac-Fock (DF), lower-order many-body perturbation
theory (MBPT), coupled-cluster single- and double-excitation approximation
with (CCSD) and without non-linear terms (LCCSD). The calculated energies and
HFS constants $A$ show a good agreement with available experimental values. We
find that for high angular momentum states, such as $6d_{5/2}$ state, the BW
effect is very important, and this contribution mainly comes from the
calculation of the correlation effect. To more accurately calculate the
contribution of BW to the hyperfine structure constant of HFS constant $A$ in
the low energy state of radium ions, further optimization of the model is
needed, such as considering the wave function of the nucleus and the spin
interaction, etc. Combining with available experimental HFS constants $B$, we
evaluated the electric quadrupole moments $Q$ for 209,211,221,223Ra. For
221,223Ra isotopes, our $Q$ values are consistent with the values from Neu
_et al._ (1988) that are widely referenced today within $0.5\%$. We
recommended a value of $Q=0.33(2)~{}b$ for 211Ra nucleus, which is smaller
than the referenced value of 0.48(4) b about 30%. If HFS constants $B$ of
other isotopes are obtained, corresponding $Q$ can be also extracted by
combining our recommended results. In addition, considering the preliminary
value of $\Omega$ from the nuclear shell model, its contributions to the
hyperfine structures of some low-lying states in Ra+ are estimated. We
conclude that it would be capable of revealing the effects of the HFS constant
$C$ of the present magnitude of experimental measurement. Furthermore, the
expressions for the hyperfine splitting for some important low-lying states
derived in the present work are useful for extracting out the HFS constants
$A$, $B$, and $C$ when these hyperfine-structure levels in Ra+ is measured
with high-precision.

###### Acknowledgements.

We are grateful to Yong-jun Cheng of Shaanxi Normal University and Ji-Guang Li
of Institute of Applied Physics and Computational Mathematics for many useful
discussions. The work was supported by the National Natural Science Foundation
of China (No. 11504094, No. 11674253, No. 11774386), the Strategic Priority
Research Program of the Chinese Academy of Sciences Grant (No. XDB21030300),
the National Key Research and Development Program of China under Grant (No.
2017YFA0304402), the Post-doctoral research project of SZTU (No.
202028555301011), and the project of Educational Commission of Guangdong
Province of China (No.2020KTSCX124).

##

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# On the Importance of Karaka Framework in Multi-modal Grounding

Sai Kiran Gorthi
IIIT Hyderabad

Radhika Mamidi
IIIT Hyderabad
<EMAIL_ADDRESS>

###### Abstract

Computational Paninian Grammar (CPG) model helps in decoding a natural
language expression as a series of modiﬁer-modiﬁed relations and therefore
facilitates in identifying dependency relations closer to language/context
semantics compared to the usual stanford dependency relations. However, the
importance of this CPG dependency scheme has not been studied in the context
of multi-modal vision and language applications. At IIIT-H, we plan to perform
a novel study to explore the potential advantages and disadvantages of CPG
framework in a vision-language navigation task setting, a popular and
challenging multi-modal grounding task.

##  1 Introduction

The Computational Paninian Grammar (CPG) framework, inspired by inflectionally
rich language Sanskrit Bharati et al. (1996); Bharati and Sangal (1993),
provides the level of syntactico-semantic intepretation of a sentence and has
been successfully applied to several tasks Gupta et al. (2012); Akula et al.
(2013); Akula (2015); Palakurthi et al. (2015). Karaka based relations are
generated by this framework are known to facilitate relations between verbs
and other related syntactic constituents such as nouns in a language
expression/sentence. More concretely, CPG treats a language expression as a
series of modifier-modified relations Begum et al. (2008). This CPG parsing
framework has been shown to be very effective for morphologically richer
languages with relatively free word order such as Indian languages. In
addition, there is a prior work extending CPG framework to English resulting
in an elegant computational grammar Vaidya et al. (2009). However, the
benefits and limitations of this framework has not been recently studied in
the context of multi-modal grounding problems Antol et al. (2015); Akula
(2022); Akula et al. (2019c).

At IIIT-H, we critically examine the importance of CPG based karaka relations
in solving the Vision-and-Language Navigation (VLN) task, a fundamental and
challenging multi-modal problem Fried et al. (2018); Anderson et al. (2018).
VLN is an emerging research area aimed at building embodied agent that can
communicate with humans through natural language expressions in real 3D
environments. Specifically, the VLN agent navigates and interacts with the
surrounding objects in the scene to complete the task according to the given
input language instructions. Recent works showed that state-of-the-art models
fail at comprehending the underlying semantics of language instructions and
rather look at salient words such as nouns. We believe that this is one of the
primary reason for observing a very low navigation performance on these
datasets such as ALFRED Shridhar et al. (2020). We plan to conduct a study to
evaluate whether if the modifier-modified based syntactico-semantic karaka
relations in CPG could help in grounding the deeper semantics of language
expressions in VLN.

To achieve this, we pick the ET model Pashevich et al. (2021), a top-
performing VLN model on ALFRED benchmark and plug in karaka annotations during
pre-training the language tower. We compare the performance of this approach
with the parts-of-speech based pre-training, syntax tree based pre-training
and dependency based pre-training (see Section 2). We will draw several
interesting insights with this by conducting extensive ablations and analysis
on the generalization and interpretation capabilities Akula et al. (2019b) of
ET model with all these different language tower pre-training strategies.

Figure 1: ET model for ALFRED: Separate encoders for encoding representations
of language and visual information; and an additional multi-modal encoder to
attend over both language and visual representations Figure 2: Our Approach:
Enforcing language encoder to explicitly predict karaka relations to
facilitate in better grounding of spatial relations in the input language
instructions.

##  2 Importance of Karaka Relations in Multi-modal grounding

An embodied AI agent needs to strictly comprehend the language instructions to
be able to successfully solve the interaction and navigation tasks. As
discussed in prior works Zhang et al. (2020); Zhu et al. (2021), it is
sometimes possible for these agents to ignore language instructions and just
rely on visual images in navigating the environment. This is problematic as
this fails to test the language understanding abilities of these agents.
ALFRED Shridhar et al. (2020) benchmark introduces a complex object
interaction and navigation environment where it is relatively more difficult
to solve the task successfully without comprehending language instructions.
This is because, ALFRED consists of longer sequence of instructions with
diverse set of target objects. In this work, We focus on the ALFRED
environment and its defined tasks.

Transformer based architectures Vaswani et al. (2017), have been shown to be
successful in solving a wide range of multi-modal grounding problems including
VLN Akula et al. (2021b, a, 2020); Akula and Zhu (2022). In particular, they
efficiently map long horizon tasks to the target set of actions. For our
analysis, we choose the ET model Pashevich et al. (2021), a top-performing
Transformer based VLN model on ALFRED leaderboard. ET model relies on
attention-based multi-layer transformer encoders and observes the full history
of visual observations and previous actions (in predicting next next action).
More specifically, ET has separate encoders for encoding representations of
language and visual information; and an additional multi-modal encoder to
attend over both language and visual representations Akula and Zhu (2019);
Gupta et al. (2016); Akula et al. (2019a, 2018). The language encoder takes
the language instructions as input and outputs a sequence of contextualized
language embeddings. The visual encoder projects a sequence of visual
observations into its embedding. The multimodal encoder is a multi-layer
transformer encoder, outputs a new embedding by taking the concatenated
language and visual embeddings from the modality-specific encoders. Figure 1
demonstrates the high-level modules in ET.

Out goal is to improve the performance of ET model, which suffers greatly on
navigation sub-tasks Pashevich et al. (2021); Zhu et al. (2021), by explicitly
enforcing the spatial relations from the input instructions in the language
encoder pre-training. To do this, we plug in the CPG karaka annotations during
pre-training the language encoder in ET. We compare the performance of this
approach with the other syntax based enforcing methods such as parts-of-speech
based pre-training, syntax tree based pre-training and dependency based pre-
training. This allow us to observe several interesting insights by conducting
extensive ablations and analysis on the generalization capabilities of ET
model with all these different language tower pre-training strategies. We
demonstrate our approach in Figure 2.

##  3 Conclusion

In this paper, we proposed our plan to critically examine the importance of
CPG dependency scheme in the context of grounding semantics of language in
multi-modal environment. Through our novel study, we explored the potential
advantages and disadvantages of CPG framework in solving ALFRED tasks, a
popular and challenging VLN benchmark.

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