# Fock-space relativistic coupled-cluster calculations of clock transition
properties in Pb2+

Palki Gakkhar  Department of Physics, Indian Institute of Technology, Hauz
Khas, New Delhi 110016, India  Ravi Kumar  Department of Chemistry, University
of Zurich, Switzerland  D. Angom  Department of Physics, Manipur University,
Canchipur 795003, Manipur, India  B. K. Mani<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi
110016, India

###### Abstract

We have employed an all-particle Fock-space relativistic coupled-cluster
theory to probe $6s^{2}{\;{}^{1}}S_{0}-6s6p{\;{}^{3}P^{o}_{0}}$ clock
transition in an even isotope of Pb2+. We have computed the excitation
energies for several low lying states, E1 and M1 transition amplitudes, and
the lifetime of the clock state. Moreover, we have also calculated the
electric dipole polarizability of the ground state using perturbed
relativistic coupled-cluster theory. To improve the accuracy of results, we
incorporated the corrections from the relativistic and QED effects in all our
calculations. Contributions from the triple excitations are accounted
perturbatively. Our computed excitation energies are in excellent agreement
with the experimental values for all states. Our computed lifetime,
$10.45\times 10^{6}$ s, of clock state is $\approx$ 16% larger than the
previously reported value using CI+MBPT [Phys. Rev. Lett. 127, 013201 (2021)].
Based on our analysis, we find that the contributions from the valence-valence
electron correlations from higher energy configurations and the corrections
from the perturbative triples and QED effects are essential to get accurate
clock properties in Pb2+. Our computed value of dipole polarizability is in
good agreement with the available theory and experimental data.

###### pacs:

31.15.bw, 11.30.Er, 31.15.am

##  I Introduction

Optical atomic clocks are one of the most accurate time measurement
instruments in existence today Ludlow  _et al._ (2015); De and Sharma (2023).
Due to their unprecedented accuracies as frequency and time standards, they
can be used as important probes to several fundamental and technological
applications. Some examples where atomic clocks are of vital importance
include, the variation in the fundamental constants Safronova (2019); Prestage
_et al._ (1995); Karshenboim and Peik (2010), probing physics beyond the
standard model of particle physics Dzuba  _et al._ (2018); Berengut  _et al._
(2018), navigation systems Grewal  _et al._ (2013); Major (2013), quantum
computer Weiss and Saffman (2017); Wineland  _et al._ (2002), the basis for
redefining the second Karshenboim and Peik (2010); Riehle  _et al._ (2018),
and more Ludlow  _et al._ (2015); De and Sharma (2023). For single ion optical
clocks, the hyperfine induced $3s^{2}{\;{}^{1}}S_{0}-3s3p{\;{}^{3}}P^{o}_{0}$
(267.4 nm) transition based 27Al+ is demonstrated to be one of the best
clocks, with a fractional frequency uncertainty of $9.4\times 10^{-19}$ Brewer
_et al._ (2019). The high accuracy in 27Al+ could be attributed to the low
sensitivity to electromagnetic fields, narrow natural linewidth and small room
temperature black-body radiation (BBR) shift in the clock transition frequency
Kállay  _et al._ (2011); Safronova  _et al._ (2011); Kumar  _et al._ (2021a).
Among the neutral atoms, a lattice clock based on degenerate fermionic 87Sr
atoms with a hyperfine-induced $5s^{2}{\;{}^{1}}S_{0}-5s5p{\;{}^{3}}P^{o}_{0}$
(698 nm) transition is reported to be one of the best neutral atoms’ clocks.
The smallest fractional frequency error achieved is $\approx$ 2.0 $\times$
10-18 Nicholson  _et al._ (2015); Bothwell  _et al._ (2019).

In the search for achieving a new and improved frequency standard, an optical
clock based on 6s${}^{2}{\;{}^{1}S_{0}}-6s6p{\;{}^{3}}P^{o}_{0}$ transition,
mediated through a two photon E1+M1 channel, in doubly ionized even isotope of
lead could be a promising candidate. Like the case of 27Al+, the clock
transition is an electric dipole forbidden transition between two $J=0$
states, providing a strong resistance to the environmental perturbations. In
addition, unlike 27Al+, the nuclear spin quantum number $I$ is zero. This is
crucial, as it prevents clock transition from the nonscalar perturbations
which may arise though the coupling of electrons’ and nuclear multipole
moments. Despite this important prospect with lead for an accurate optical
atomic clock, the clock transition properties are not well explored. For
instance, in terms of theory calculations, to the best of our knowledge, there
is only one study on the lifetime of the clock state Beloy (2021). The work
Beloy (2021), employing a combined method of configuration interaction (CI)
and many-body perturbation theory (MBPT), has computed the lifetime of clock
state, ${\;{}^{3}}P^{o}_{0}$, as $9.0\times 10^{6}$ s. Considering there are
no experimental inputs, more theory calculations, specially using the accurate
methods like relativistic coupled-cluster (RCC), would be crucial to get more
accurate insights into the clock properties. Moreover, the inclusion of
relativistic and QED corrections to the properties calculations are must to
get reliable results. It can thus be concluded that there is a clear research
gap in terms of the scarcity of accurate properties results for
${}^{1}S_{0}-{{}^{3}}P^{o}_{0}$ clock transition in doubly ionized lead.

In this work, we have employed an all-particle multireference Fock-space
relativistic coupled-cluster (FSRCC) theory to accurately examine the clock
transition properties in Pb2+. It is to be noted that, RCC theory is one of
the most reliable many-body theories for atomic structure calculations. It
accounts for electron correlation effects to all-orders of residual Coulomb
interaction and has been employed to obtain accurate properties results in
several closed-shell and one-valence atoms and ions Pal  _et al._ (2007); Mani
_et al._ (2009); Nataraj  _et al._ (2011); Kumar  _et al._ (2020). The
application of RCC for two-valence atomic systems, as the present case of Pb2+
clock transition, is, however, limited to few studies Eliav  _et al._ (1995a,
b); Mani and Angom (2011). And, the reason for this, perhaps, is the
complications associated with the implementation of FSRCC theory for multi-
reference systems Eliav  _et al._ (1995a, b); Mani and Angom (2011); Kumar
_et al._ (2021a). To address the clock transition properties in a
comprehensive way, using FSRCC theory Mani and Angom (2011); Kumar  _et al._
(2021a), we carried out precise calculations of the excitation energies and E1
and M1 transition amplitudes associated with ${}^{1}S_{0}-{{}^{3}}P^{o}_{0}$
transition in Pb2+. Using these results, we have then calculated the lifetime
of the ${}^{3}P^{o}_{0}$ clock state. In addition, as electric dipole
polarizability is a crucial parameter for estimating BBR shift in clock
frequency, we have also calculated the ground state polarizability of Pb2+
using perturbed relativistic coupled-cluster (PRCC) theory Kumar  _et al._
(2020, 2021b). Moreover, in all these properties calculations, we have
incorporated and analyzed the contributions from the Breit interaction, QED
corrections and perturbative triples.

The remainder of the paper is organized into five sections. In Sec. II, we
provide a brief description of the FSRCC theory for two-valence atomic
systems. We have given the coupled-cluster working equation for two-valence
systems. In Sec. III, we provide and discuss the expression for E1M1 decay
rate. The results obtained from our calculations are presented and analyzed in
Sec. IV. Theoretical uncertainty in our computed results is discussed in Sec.
V of the paper. Unless stated otherwise, all results and equations presented
in this paper are in atomic units ( $\hbar=m_{e}=e=1/4\pi\epsilon_{0}=1$).

##  II Two-valence FSRCC Theory

As the clock transition involves atomic state functions (ASFs) of two-valence
nature, we need an accurate multireference theory to calculate these
wavefunctions and corresponding many-body energies. For this, in the present
work, we have employed a FSRCC theory for two-valence, developed and reported
in our previous works Mani and Angom (2011); Kumar  _et al._ (2021a). In Refs.
Mani and Angom (2011); Mani  _et al._ (2017); Kumar  _et al._ (2021a), we have
discussed in detail the implementation of FSRCC theory in the form of
sophisticated parallel codes and have also given the working equations and
Goldstone diagrams contributing to the theory. So, here, for completeness, we
provide a very brief description of FSRCC theory for two-valence atoms and
properties calculations using it in the context of Pb2+.

The atomic state function for a two-valence atom or ion is obtained by solving
the many-body Schrodinger equation

$H^{\rm DCB}|\Psi_{vw}\rangle=E_{vw}|\Psi_{vw}\rangle,$ (1)

where $|\Psi_{vw}\rangle$ is the exact many-body wavefunction and $E_{vw}$ is
the corresponding exact energy. The indices $v,w,\cdots$ represent the valence
orbitals. $H^{\rm DCB}$ is the Dirac-Coulomb-Breit no-virtual-pair Hamiltonian
used in all calculations, and expressed as

$\displaystyle H^{\rm DCB}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\left[c\bm{\alpha}_{i}\cdot\mathbf{p}_{i}+(\beta_{i}-1)c^{2}-V_{N}(r_{i})\right]$
(2) $\displaystyle+\sum_{i<j}\left[\frac{1}{r_{ij}}+g^{\rm B}(r_{ij})\right],$

where $\bm{\alpha}$ and $\beta$ are the Dirac matrices, and $1/r_{ij}$ and
$g^{\rm B}(r_{ij})$ are the Coulomb and Breit interactions, respectively. In
FSRCC theory, $|\Psi_{vw}\rangle$ is written as

$|\Psi_{vw}\rangle=e^{T}\left[1+S_{1}+S_{2}+\frac{1}{2}\left({S_{1}}^{2}+{S_{2}}^{2}\right)+R_{2}\right]|\Phi_{vw}\rangle,$
(3)

where $|\Phi_{vw}\rangle,=a^{\dagger}_{w}a^{\dagger}_{v}|\Phi_{0}\rangle,$ is
the Dirac-Fock reference state for a two-valence system. Operators $T$, $S$
and $R$ are the electron excitation operators, referred to as the coupled-
cluster (CC) operators, for closed-shell, one-valence and two-valence sectors,
respectively. The subscripts $1$ and $2$ with these operators represent the
single and double excitations, referred to as the coupled-cluster with singles
and doubles (CCSD) approximation. The FSRCC theory with CCSD approximation
subsumes most of the electron correlation effects in atomic structure
calculations and provides an accurate description of the atomic properties. In
the second quantized representation, the CC operators can be expressed as

$\displaystyle T_{1}=\sum_{ap}t_{a}^{p}a_{p}^{\dagger}a_{a}{\;\;\rm
and\;\;}T_{2}=\frac{1}{2!}\sum_{abpq}t_{ab}^{pq}a_{p}^{\dagger}a_{q}^{\dagger}a_{b}a_{a},$
(4a) $\displaystyle S_{1}=\sum_{p}s_{v}^{p}a_{p}^{\dagger}a_{v}{\;\;\rm
and\;\;}S_{2}=\sum_{apq}s_{va}^{pq}a_{p}^{\dagger}a_{q}^{\dagger}a_{a}a_{v},$
(4b) $\displaystyle
R_{2}=\sum_{pq}r_{vw}^{pq}a_{p}^{\dagger}a_{q}^{\dagger}a_{w}a_{v}.$ (4c)

Here, the indices $a,b,\ldots$ and $p,q,\ldots$ represent the core and virtual
orbitals, respectively. And, $t_{\ldots}^{\ldots}$, $s_{\ldots}^{\ldots}$ and
$r_{\ldots}^{\ldots}$ are the cluster amplitudes corresponding to $T$, $S$ and
$R$ CC operators, respectively. The diagrammatic representation of these
operators is shown in Fig. 1. It is to be mentioned that the dominant
contributions from triple excitations are, however, also included in the
present work using perturbative triples approach Kumar  _et al._ (2021a).

Figure 1: The diagrammatic representation of closed-shell, one-valence and
two-valence single and double CC operators.

The operators for closed-shell and one-valence sectors are obtained by solving
the set of coupled nonlinear equations discussed in Refs. Mani  _et al._
(2009) and Mani and Angom (2010), respectively. The two-valence CC operator,
$R_{2}$, is obtained by solving the CC equation Mani and Angom (2011); Kumar
_et al._ (2021a)

$\displaystyle\langle\Phi^{pq}_{vw}|\bar{H}_{\rm N}+\\{\mathchoice{\vbox{\hbox
to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\bar{H}_{\rm N}S^{{}^{\prime}}\\}+\\{\mathchoice{\vbox{\hbox
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 2.28123pt\hbox{\vrule
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3.41666pt}}\bar{H}_{\rm N}R_{2}\\}|\Phi_{vw}\rangle=$ $\displaystyle E^{\rm
att}_{vw}\langle\Phi^{pq}_{vw}|\Bigl{[}S^{{}^{\prime}}+R_{2}\Bigr{]}|\Phi_{vw}\rangle.$
(5)

Here, for compact notation we have used
$S^{\prime}=S_{1}+S_{2}+\frac{1}{2}(S^{2}_{1}+S^{2}_{2})$. $E^{\rm att}_{vw}$
is the two-electron attachment energy and it is expressed as the difference
between the correlated energy of $(n-2)-$electron (closed-shell) and
$n-$electron (two-valence) sectors, $E_{vw}-E_{0}$.

Figure 2: The CC diagrams contributing to the linearized FSRCC theory for two-
valence atomic systems. Diagrams (l) - (n) are referred to as the folded
diagrams and arise from the renormalization terms in CC equations for
multireference systems.

In Fig. 2, we have given the Goldstone diagrams contributing to linearized
FSRCC theory for two-valence systems. These are obtained by considering the
terms with only one order of CC operators in Eq. (5) and then contracting
residual Coulomb interaction with these CC operators using Wick’s theorem. The
CC diagrams $(l)-(n)$ are referred to as the folded diagrams and they arise
due to renormalization terms on the right hand side of Eq. (5). The presence
of folded diagrams in open-shell systems constitute one of the main
differences from the CC theory of closed-shell systems. The rectangular
portion represents the effective energy diagrams arising from the one-valence
(diagram (l)) and two-valence (diagrams (m) and (n)) sectors. The Goldstone
diagrams in Fig. 2 correspond to the algebraic expression

$\displaystyle\langle$ $\displaystyle H_{\rm N}$
$\displaystyle\rangle^{pq}_{vw}+\langle\mathchoice{\vbox{\hbox to0.0pt{\kern
0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{\rm N}T\rangle^{pq}_{vw}+\langle\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=8.54164pt,height=0.0pt,depth=0.50003pt\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=6.95831pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{\rm
N}S^{{}^{\prime}}\rangle^{pq}_{vw}+\langle\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
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3.41666pt}}H_{\rm N}R_{2}\rangle^{pq}_{vw}-$ (6)
$\displaystyle\langle\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
3.97916pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.97916pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.7854pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=10.68193pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 1.98956pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
width=9.09026pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}E^{\rm
att}_{vw}S^{{}^{\prime}}\rangle^{pq}_{vw}-\langle\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 3.97916pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.7854pt\hbox{\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 1.98956pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=4.30554pt\vrule
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width=0.50003pt,height=0.0pt,depth=4.30554pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}E^{\rm att}_{vw}R_{2}\rangle^{pq}_{vw}=g_{pqvw}+g_{pqrw}s^{r}_{v}$
$\displaystyle-
g_{aqvw}t^{p}_{a}+\tilde{g}_{pavr}s^{rq}_{aw}+g_{pqvw}\epsilon_{p}+g_{pqrs}r^{rs}_{vw}$
$\displaystyle-
g_{aqvr}s^{pr}_{aw}+g_{abvw}t^{pq}_{ab}-g_{aqrw}s^{rp}_{va}-E^{\rm
att}_{w}r^{pq}_{vw}$ $\displaystyle-E^{\rm att}_{vw}s^{p}_{v}-E^{\rm
att}_{vw}r^{pq}_{vw},$

where $\tilde{g}_{ijkl}=g_{ijkl}-g_{ijlk}$. Since the Dirac-Fock orbitals are
used in our calculations, diagram (f) does not contribute and therefore not
included in the expression.

##  III E1M1 Decay Rate Using FSRCC

Since $I=J=F=0$ for clock transition, ${}^{1}S_{0}-{{}^{3}}P^{o}_{0}$, in
Pb2+, the transition is allowed through a two-photon  E1+M1 channel. As shown
in the schematic diagram, Fig. 3, in the first route, the initial state
$|\Psi_{i}\rangle$ can couple to a same parity state through a magnetic dipole
operator (photon with energy $\omega_{1}$) and then connect to the final state
$|\Psi_{f}\rangle$ through an electric dipole operator (photon with energy
$\omega_{2}$). Alternatively, in the second route, the initial state
$|\Psi_{i}\rangle$ can couple to an opposite parity state via an electric
dipole operator first and then connect to the ground state through a magnetic
dipole operator. Mathematically, the E1+M1 decay rate between
$|\Psi_{i}\rangle$ and $|\Psi_{f}\rangle$ ASFs can be expressed in terms of
the reduced matrix elements of electric and magnetic dipole operators, as
Craig and Thirunamachandran (1998); Santra  _et al._ (2004)

$\displaystyle\Gamma_{E1M1}=\frac{8}{27\pi}\alpha^{6}$
$\displaystyle\int_{0}^{\infty}d\omega_{1}\omega_{1}^{3}\int_{0}^{\infty}d\omega_{2}\omega_{2}^{3}$
(7)
$\displaystyle\times\bigg{|}\frac{\langle\Psi_{f}||{\mathbf{D}}||\Psi_{n}\rangle\langle\Psi_{n}||M_{1}||\Psi_{i}\rangle}{E_{n}+\omega_{1}-E_{i}}$
$\displaystyle+\frac{\langle\Psi_{f}||M_{1}||\bar{\Psi}_{n}\rangle\langle\bar{\Psi}_{n}||{\mathbf{D}}||\Psi_{i}\rangle}{E_{\bar{n}}+\omega_{2}-E_{i}}\bigg{|}^{2}$
$\displaystyle\times\delta(E_{i}+\omega_{1}+\omega_{2}-E_{f}).$

Here for Pb2+: $|\Psi_{i}\rangle=6s6p{\;{}^{3}}P^{o}_{0}$,
$|\Psi_{f}\rangle=6s^{2}{\;{}^{1}}S_{0}$,
$|\Psi_{n}\rangle=6s6p{\;{}^{3}}P^{o}_{1},6s6p{\;{}^{1}}P^{o}_{1}$ and
$|\bar{\Psi}_{n}\rangle=6s7s{\;{}^{3}}S_{1},6s6d{\;{}^{3}}D_{1}$. Since the
transition is allowed through two photons, the energy difference of final and
initial states satisfies the relation $E_{f}-E_{i}=\omega_{1}+\omega_{2}$.

The reduced matrix elements in Eq. (7) are calculated using the FSRCC theory.
Properties calculation using FSRCC theory is explained in detail in our work
Kumar  _et al._ (2021a). However, to illustrate it briefly in the present
work, we consider the example of dipole matrix element. Using the RCC wave
function from Eq. (3), the expression for dipole matrix elements is

$\displaystyle\langle\Psi_{f}|{\mathbf{D}}|\Psi_{n}\rangle=\sum_{kl}{c^{f}_{k}}^{*}c^{n}_{l}\left[\langle\Phi_{k}|\tilde{\mathbf{D}}+\tilde{\mathbf{D}}\left(S^{{}^{\prime}}\right.\right.$
$\displaystyle\left.\left.+R_{2}\right)+\left(S^{{}^{\prime}}+R_{2}\right)^{\dagger}\tilde{\mathbf{D}}+\left(S^{{}^{\prime}}+R_{2}\right)^{\dagger}\right.$
$\displaystyle\left.\;\tilde{\mathbf{D}}\left(S^{{}^{\prime}}+R_{2}\right)|\Phi_{l}\rangle\right],$
(8)

where the coefficients $c^{f}_{k}$ represent the mixing coefficients in the
expansion of a multireference configuration state function $|\Phi_{f}\rangle$,
and obtained by diagonalizing the $H^{\rm DCB}$ matrix within the chosen model
space. The dressed operator
$\tilde{\mathbf{D}},=e{{}^{T}}^{\dagger}{\mathbf{D}}e^{T},$ is a non
terminating series in closed-shell CC operator $T$. Including all orders of
$T$ in the dressed operator is practically challenging. In Ref. Mani and Angom
(2010), we developed an algorithm to include a class of dominant diagrams to
all order in $T$, iteratively, in the dressed Hamiltonian. Based on this
study, we concluded that the terms higher than quadratic in $T$ contribute
less than $0.1\%$ to the properties. So, in the present work we truncate
$\tilde{\mathbf{D}}$ after the second-order in $T$ and include the terms
$\widetilde{\mathbf{D}}\approx{\mathbf{D}}+{\mathbf{D}}T+T^{\dagger}{\mathbf{D}}+T^{\dagger}{\mathbf{D}}T$
in the properties calculations.

Figure 3: Schematic energy level diagram for $6s^{2}\;{}^{1}S_{0}\rightarrow
6s6p\;^{3}P^{o}_{0}$ clock transition in Pb2+ via a two photon E1+M1
transition.

##  IV Results and discussions

###  IV.1 Single-particle basis and convergence of properties

An accurate description of single-electron wave functions and corresponding
energies are crucial to obtain the reliable results using FSRCC theory. In the
present work, we have used the Gaussian-type orbitals (GTOs) Mohanty  _et al._
(1991) as the single-electron basis for FSRCC calculations. The GTOs are used
as the finite basis sets in which the single-electron wave functions are
expressed as a linear combination of the Gaussian-type functions (GTFs). More
precisely, the GTFs of the large component of the wavefunction are expressed
as

$g^{L}_{\kappa p}(r)=C^{L}_{\kappa i}r^{n_{\kappa}}e^{-\alpha_{p}r^{2}},$ (9)

where $p=0$, $1$, $2$, $\ldots$, $N$ is the GTO index and $N$ is the total
number of GTFs. The exponent $\alpha_{p}$ is further expressed as
$\alpha_{0}\beta^{p-1}$, where $\alpha_{0}$ and $\beta$ are the two
independent parameters. The parameters $\alpha_{0}$ and $\beta$ are optimized
separately for each orbital symmetry so that the single-electron wavefunctions
and energies match well with the numerical values obtained from the GRASP2K
Jönsson  _et al._ (2013). The small components of wavefunctions are derived
from the large components using the kinetic balance condition Stanton and
Havriliak (1984).

Table 1: The single-particle and SCF energies (in a.u.) from GTO compared with GRASP2K and B-Spline results. The optimized $\alpha_{0}$ and $\beta$ parameter for the even tempered basis used in our calculations are also provided.  Orbitals | GTO | GRASP2K | B-Spline
---|---|---|---
$1s_{1/2}$ | -3257.41150 | -3257.40298 | -3257.41571
$2s_{1/2}$ | -589.39795 | -589.39605 | -589.39835
$2p_{1/2}$ | -565.00395 | -565.00364 | -565.00295
$2p_{3/2}$ | -484.41530 | -484.41512 | -484.41548
$3s_{1/2}$ | -145.26466 | -145.26400 | -145.26472
$3p_{1/2}$ | -134.31668 | -134.31639 | -134.31638
$3p_{3/2}$ | -116.10663 | -116.10639 | -116.10665
$3d_{3/2}$ | -98.32473 | -98.32446 | -98.32473
$3d_{5/2}$ | -94.48445 | -94.48420 | -94.48445
$4s_{1/2}$ | -35.49304 | -35.49280 | -35.49305
$4p_{1/2}$ | -30.72284 | -30.72269 | -30.72276
$4p_{3/2}$ | -26.20738 | -26.20725 | -26.20737
$4d_{3/2}$ | -18.43141 | -18.43130 | -18.43141
$4d_{5/2}$ | -17.57859 | -17.57861 | -17.57859
$4f_{5/2}$ | -7.44501 | -7.44494 | -7.44502
$4f_{7/2}$ | -7.25330 | -7.25331 | -7.25331
$5s_{1/2}$ | -7.63856 | -7.63855 | -7.63857
$5p_{1/2}$ | -5.94459 | -5.94458 | -5.94458
$5p_{3/2}$ | -5.09665 | -5.09664 | -5.09666
$5d_{3/2}$ | -2.62373 | -2.62373 | -2.62373
$5d_{5/2}$ | -2.51852 | -2.51852 | -2.51852
$E_{\rm SCF}$ | -20910.40152 | -20910.37469 | -20910.40151
| $\alpha_{0}$ | $\beta$ | GTOs
$s$ | 0.00450 | 1.805 | 40
$p$ | 0.00478 | 1.792 | 38
$d$ | 0.00605 | 1.855 | 34
$f$ | 0.00355 | 1.845 | 28

In Table 1, we have provided the optimized values of $\alpha_{0}$ and $\beta$
parameters for Pb2+ and have compared the values of single-electron and self-
consistent field (SCF) energies with GRASP2K Jönsson  _et al._ (2013) and
B-spline Zatsarinny and Fischer (2016) results. It is to be mentioned that,
the single-electron basis used in the properties calculations also
incorporates the effects of Breit interaction, vacuum polarization and the
self-energy corrections. As evident from the table, the single-particle and
SCF energies are in excellent agreement with GRASP2K and B-spline results. The
largest difference at the level of SCF and single-particle energies are
0.0001% and 0.0003%, respectively.

Since GTOs are a mathematically incomplete basis, convergence of the
properties results with basis size must be checked to get reliable results
using FSRCC. To show the convergence of properties results, in Table 8 of
Appendix, we have listed the values of electric dipole polarizability and E1
and M1 transition reduced matrix elements for increasing basis size. To
achieve a converged basis, we start with a moderate basis size and add
orbitals systematically in each symmetry until the change in the properties is
less than or equal to $10^{-3}$ in respective units. For example, as evident
from the table, the change in E1 amplitude of
$\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$ transition is of the order of
$10^{-3}$ a.u. when basis is augmented from 158 ($24s21p18d13f8g7h$) to 169
($25s22p19d14f9g8h$) orbitals. So, to minimize the compute time, we consider
the basis set with 158 orbitals as optimal, and use it for further FSRCC
calculations where the corrections from the Breit interaction, vacuum
polarization and the self-energy are incorporated.

Figure 4: Convergence of transition amplitudes with basis size, panel (a).
Dominant percentage contributions from Breit, perturbative triples and QED
corrections to the reduced matrix elements, panel (b), and the lifetime of the
clock state, panel (c). Dominant contributions from core orbitals to the
dipole polarizability, panel (d).

###  IV.2 Excitation Energy

The eigen energies obtained from the solution of many-electron Schrodinger
equation (1) using FSRCC are used to calculate the excitation energies. The
excitation energy of a general state
$nln^{{}^{\prime}}l^{{}^{\prime}}{\;{}^{(2S+1)}}L_{J}$ is defined as

$\Delta
E_{nln^{{}^{\prime}}l^{{}^{\prime}}{\;{}^{(2S+1)}}L_{J}}=E_{nln^{{}^{\prime}}l^{{}^{\prime}}{\;{}^{(2S+1)}}L_{J}}-E_{ns^{2}{\;{}^{1}}S_{0}},$
(10)

where $E_{ns^{2}{\;{}^{1}}S_{0}}$ and
$E_{nln^{{}^{\prime}}l^{{}^{\prime}}{\;{}^{(2S+1)}}L_{J}}$ are the exact
energies of the ground and excited states, respectively. In Table 2, we have
provided the excitation energies from our calculations along with other theory
and experimental data for comparison. To account for valence-valence
correlations more accurately, we have also included $6p^{2}$, $6s6d$ and
$6s7s$ high energy configurations in the model space. For a quantitative
assessment of electron correlations, we have listed the contributions from
Breit and QED corrections separately.

Table 2: The ground state energy of ${}^{1}S_{0}$ (cm-1) and excitation
energies of some low lying excited states of Pb2+. For quantitative analysis
of electron correlations, contributions from Breit and QED corrections are
given separately.

States | DC-CCSD | Breit | Self-energy | Vac-pol | Total | Other cal. | NISTNIS (2013) | % Error
---|---|---|---|---|---|---|---|---
$6s^{2}$ ${}^{1}S_{0}$ | 599355.44 | 12.00 | -0.67 | 188.97 | 599556 | 600984Safronova  _et al._ (2012) | 598942 | 0.1
$6s6p$ ${}^{3}P^{o}_{0}$ | 59624.80 | 121.57 | -0.58 | 81.41 | 59827 | 61283Safronova  _et al._ (2012), 60653Curtis  _et al._ (2001) | 60397 | 0.94
$6s6p$ ${}^{3}P^{o}_{1}$ | 64146.99 | 116.92 | -0.53 | 82.84 | 64346 | 65089Safronova  _et al._ (2012), 65683Migdalek and Baylis (1985) | 64391 | 0.06
|  |  |  |  |  | 60387Migdalek and Bojara (1988),58905Chou and Huang (1992) |  |
|  |  |  |  |  | 64609Curtis  _et al._ (2001) |  |
$6s6p$ ${}^{3}P^{o}_{2}$ | 79478.69 | 83.67 | -0.33 | 90.27 | 79652 | 80029Safronova  _et al._ (2012), 79024Curtis  _et al._ (2001) | 78985 | -0.8
$6s6p$ ${}^{1}P^{o}_{1}$ | 95045.89 | 83.07 | -0.63 | 84.30 | 95213 | 95847Safronova  _et al._ (2012), 97970Migdalek and Baylis (1985) | 95340 | 0.13
|  |  |  |  |  | 91983Migdalek and Bojara (1988), 95537Chou and Huang (1992) |  |
|  |  |  |  |  | 95535Curtis  _et al._ (2001) |  |
$6p^{2}$ ${}^{3}P_{0}$ | 142922.33 | 236.01 | -1.41 | 169.97 | 143327 | 143571Safronova  _et al._ (2012) | 142551 | -0.54
$6s7s$ ${}^{3}S_{1}$ | 149898.62 | 4.90 | -0.32 | 64.10 | 149967 | 151183Safronova  _et al._ (2012) | 150084 | 0.07
$6s6d$ ${}^{1}D_{2}$ | 152651.06 | 108.27 | -0.70 | 135.00 | 152894 | 153614Safronova  _et al._ (2012) | 151885 | -0.6
$6s7s$ ${}^{1}S_{0}$ | 153901.92 | 3.47 | -0.28 | 62.88 | 153968 | 155054Safronova  _et al._ (2012) | 153783 | -0.12
$6p^{2}$ ${}^{3}P_{1}$ | 155401.22 | 189.27 | -1.16 | 170.87 | 155760 | 156610Safronova  _et al._ (2012) | 155431 | -0.2
$6s6d$ ${}^{3}D_{1}$ | 157523.12 | 17.37 | -0.55 | 92.43 | 157632 | 158439Safronova  _et al._ (2012) | 157444 | -0.1
$6s6d$ ${}^{3}D_{2}$ | 157902.47 | 5.93 | -0.48 | 87.81 | 157996 | 159134Safronova  _et al._ (2012) | 157925 | -0.04
$6s6d$ ${}^{3}D_{3}$ | 159147.60 | 0.04 | -0.38 | 86.03 | 159233 | 160530Safronova  _et al._ (2012) | 158957 | -0.17
$6p^{2}$ ${}^{3}P_{2}$ | 164987.49 | 117.07 | -0.99 | 143.34 | 165247 | 165898Safronova  _et al._ (2012) | 164818 | -0.26
$6p^{2}$ ${}^{1}D_{2}$ | 179412.09 | 139.08 | -1.17 | 166.70 | 179717 | 179646Safronova  _et al._ (2012) | 178432 | -0.7
$6p^{2}$ ${}^{1}S_{0}$ | 189714.06 | 161.46 | -1.17 | 179.67 | 190054 | 190061Safronova  _et al._ (2012) | 188615 | -0.7

a Ref. Safronova  _et al._ (2012) \- CI + all-order , b Ref. Curtis  _et al._
(2001) \- MCDHF, c Ref. Migdalek and Baylis (1985) \- CIRHF + CP, d Ref.
Migdalek and Bojara (1988) \- CIRHF + CP, e Ref. Chou and Huang (1992) \-
MCRRPA

As evident from the table, our computed energies are in excellent agreement
with the experimental values. The largest relative error in our calculation is
$\approx$ 0.9%, in the case of ${{}^{3}}P^{o}_{0}$ state. However, for other
ASFs, specially for those which contribute to the lifetime of the clock state,
the errors are much smaller. The ASFs ${{}^{3}}P^{o}_{1}$,
${{}^{1}}P^{o}_{1}$, ${{}^{3}}S_{1}$ and ${{}^{3}}D_{1}$, which couple either
via E1 or M1 operator in clock transition, have the relative errors of 0.06%,
0.13%, 0.07% and -0.10%, respectively. This is crucial, as these energies
contribute to the lifetime of the clock state. Among all the previous theory
results listed in the table, Ref. Safronova  _et al._ (2012) is close to ours
in terms of the many-body methods used, however, with an important difference.
Ref. Safronova  _et al._ (2012) uses a linearized CCSD method, whereas the
present work employs a nonlinear CCSD, which accounts for electron correlation
effects more accurately in the calculation. The relative errors in the
reported excitation energies for ${{}^{3}}P^{o}_{1}$, ${{}^{1}}P^{o}_{1}$,
${{}^{3}}S_{1}$ and ${{}^{3}}D_{1}$ states in Ref. Safronova  _et al._ (2012)
are 1.08%, 0.53%, 0.73% and 0.63%, respectively. Remaining results are mostly
based on the multi-configuration Hartree-Fock and its variations and, in
general, not very consistent in terms of treating electron correlations.

Examining the contributions from high energy configurations, we observed an
improvement in the excitation energies of ${{}^{3}}P^{o}_{1}$ and
${{}^{1}}P^{o}_{1}$ states due to accounting of valence-valence correlation
more accurately. We find that the relative error has reduced from 0.7% (0.6%)
to 0.4% (0.3%) for ${{}^{3}}P^{o}_{1}$(${{}^{1}}P^{o}_{1}$) state. Among the
contributions from Breit, vacuum polarization and self-energy corrections, the
former two are observed to contribute more. The largest cumulative
contribution of about $0.3$% from Breit and vacuum polarization is observed in
the case of ${}^{3}P^{o}_{0}$. Self-energy contributions are of opposite phase
and negligibly small.

###  IV.3 E1 Reduced Matrix Elements

In Table 3, we have provided the values of E1 reduced matrix elements from our
calculations for all dominant transitions which contribute to the lifetime of
the clock state. Since there are more data on oscillator strengths from
literature, we converted E1 reduced matrix elements to oscillator strength and
tabulated in the table for comparison with experiments and other theory
calculations. The contributions from Breit+QED and triples are provided
separately in the table. As evident from the table, as to be expected, DC-CCSD
gives the dominant contribution to all the matrix elements. The contributions
from Breit, QED and perturbative triples are small but important to get
accurate results of the properties. A quantitative analysis on these is
provided later in the section.

Table 3: The E1 and M1 reduced matrix elements (a.u.) and oscillator strengths
for some allowed transitions in Pb2+. For comparison, data from experiments
and other theory calculations are also provided.

States | DC-CCSD | Breit + QED | P-triples | Total | Other cal. | Expt.
---|---|---|---|---|---|---
E1 Reduced Matrix Elements |
$\langle^{1}S_{0}||D||^{3}P^{o}_{1}\rangle$ | 0.5359 | -0.0007 | -0.0130 | 0.5222 | 0.706Beloy (2021), 0.644Safronova  _et al._ (2012) |
$\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$ | 1.9854 | -0.0006 | -0.0131 | 1.9717 | 2.350Beloy (2021), 2.384Safronova  _et al._ (2012) |
$\langle^{3}S_{1}||D||^{3}P^{o}_{0}\rangle$ | 0.5363 | -0.0005 | -0.0232 | 0.5126 | 0.963Safronova  _et al._ (2012) |
$\langle^{3}D_{1}||D||^{3}P^{o}_{0}\rangle$ | -1.4985 | 0.0155 | -0.0676 | -1.5506 | -1.516Safronova  _et al._ (2012) |
Oscillator Strengths |
$\langle^{1}S_{0}||D||^{3}P^{o}_{1}\rangle$ | 5.5716[-2] | 0.0025[-2] | -0.2327[-2] | 5.3414[-2] | 8.11[-2]Safronova  _et al._ (2012), 7.40[-2]Alonso-Medina  _et al._ (2009), 5.44[-2]Migdalek and Baylis (1985), | (7.3 $\pm$ 0.5)[-2]Pinnington  _et al._ (1988), Ansbacher  _et al._ (1988)
|  |  |  |  | 7.55[-2]Glowacki and J.Migdalek (2003), 6.15[-2]Chou and Huang (1992), 6.15[-2]Chou and Huang (1997) |
|  |  |  |  | 5.52[-2]Migdalek and Bojara (1988) |
$\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$ | 1.1317 | 0.0013 | -0.0092 | 1.1238 | 1.65Safronova  _et al._ (2012), 1.24Alonso-Medina  _et al._ (2009), 1.64Migdalek and Baylis (1985), 1.51Glowacki and J.Migdalek (2003), | (1.01 $\pm$ 0.20)Andersen  _et al._ (1972)
|  |  |  |  | 2.45Chou and Huang (1992), 1.42Chou and Huang (1997), 1.43Migdalek and Bojara (1988) |
$\langle^{3}S_{1}||D||^{3}P^{o}_{0}\rangle$ | 0.0788 | 0 | -0.0069 | 0.0719 | 0.229Alonso-Medina  _et al._ (2009) |
$\langle^{3}D_{1}||D||^{3}P^{o}_{0}\rangle$ | 0.6664 | -0.0130 | 0.0589 | 0.7123 | 0.93Alonso-Medina  _et al._ (2009) |
M1 Reduced Matrix Elements |
$\langle^{3}P^{o}_{1}||M1||^{3}P^{o}_{0}\rangle$ | -1.3117 | -0.0002 | 0.0001 | -1.3118 | -0.674Beloy (2021) |
$\langle^{1}P^{o}_{1}||M1||^{3}P^{o}_{0}\rangle$ | 0.4959 | -0.0003 | 0.0001 | 0.4957 | 0.205Beloy (2021) |
$\langle^{1}S_{0}||M1||^{3}S_{1}\rangle$ | 0.0046 | -0.0004 | -0.0002 | 0.0040 |  |
$\langle^{1}S_{0}||M1||^{3}D_{1}\rangle$ | -0.0116 | -0.0019 | 0.0006 | -0.0129 |  |

a Ref. Beloy (2021) \- CI + MBPT , b Ref. Alonso-Medina  _et al._ (2009) \- IC
+ RHF +CP, c Ref. Glowacki and J.Migdalek (2003) \- CIDF - MP, d Ref. Chou and
Huang (1997) \- MCRRPA, e Ref. Safronova  _et al._ (2012) \- CI + all-order, f
Ref. Colón and Alonso-Medina (2000) \- IC + RHF, g Ref. Migdalek and Bojara
(1988) \- CIRHF + CP, h Ref. Chou and Huang (1992) \- MCRRPA, i Ref. Migdalek
and Baylis (1985) \- CIRHF + CP,

From previous theory calculations, to the best of our knowledge, we found two
works, Refs. Beloy (2021)-CI+MBPT and Safronova  _et al._ (2012)-CI+all-order,
for comparison of E1 reduced matrix elements. The values of our E1 reduced
matrix elements are slightly on the smaller sides of Refs. Beloy (2021) and
Safronova  _et al._ (2012) for all the listed transitions. The reason for this
could be attributed to the different treatment of electron correlations in
these methods. In Ref. Beloy (2021), MBPT is used to treat core-core and core-
valence correlations, whereas valence-valence correlation is treated in the
framework of configuration interaction (CI). In Ref. Safronova  _et al._
(2012), however, the core-core and core-valence correlations are accounted
using a linearized CCSD method. The present work, however, employs a nonlinear
CCSD theory to account for the core-core and core-valence correlations and,
therefore, is more accurate. The valence-valence correlation is however
treated in the same way as in Refs. Beloy (2021); Safronova  _et al._ (2012).
The other two important inclusions in the present work are, the use of high
energy configurations ($6p^{2}$, $6s7s$ and $6s6d$) in the model space and the
corrections from the Breit, QED and perturbative triples.

For oscillator strengths, from other theory calculations, there are several
data in the literature for $\langle^{1}S_{0}||D||^{3}P^{o}_{1}\rangle$ and
$\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$ transitions for comparison. As
evident from the table, there is, however, a large variation in the reported
values. For example, for $\langle^{1}S_{0}||D||^{3}P^{o}_{1}\rangle$
transition, the smallest value $5.44\times 10^{-2}$ from Ref. Migdalek and
Baylis (1985) differs by $\approx$ 33% with the largest value $8.11\times
10^{-2}$ reported in the Ref Safronova  _et al._ (2012). The similar trend is
also observed for $\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$ transition. The
smallest value, 1.24 Alonso-Medina  _et al._ (2009), is almost half the
largest value, 2.45 Chou and Huang (1992). The reason for this large variation
could be attributed to the different many-body methods employed in these
calculations. It is to be noted that, none of the previous calculations use
FSRCC theory, like the case of the present work. Except Ref. Safronova  _et
al._ (2012), which uses CI + all-order, the other calculations are mostly
based on the MCDF and its variations. The large difference among these results
clearly indicates the inherent inconsistencies with MCDF methods in terms of
incorporating electron correlation effects in many-body calculation. For
$\langle^{1}S_{0}||D||^{3}P^{o}_{1}\rangle$ transition, our result,
$5.34\times 10^{-2}$, lies in the range of the previous values. Whereas, for
transition $\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$, our value, 1.12, is
smallest among all results listed in the table.

From experiments, to the best of our knowledge, there is one result each for
oscillator strength for $\langle^{1}S_{0}||D||^{3}P^{o}_{1}\rangle$ Pinnington
_et al._ (1988) and $\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$ Andersen  _et
al._ (1972) transitions. Both of these experiments use the beam-foil technique
to study atomic spectra. For $\langle^{1}S_{0}||D||^{3}P^{o}_{1}\rangle$
transition, our calculated result, $5.34\times 10^{-2}$, is of the same order
of magnitude as the experimental value, $(7.3\pm 0.5)\times 10^{-2}$, but
smaller by about 27%. Among the previous theory calculations, the MCDF
calculations Refs. Alonso-Medina  _et al._ (2009) and Glowacki and J.Migdalek
(2003) are more closer to the experiment. For
$\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$ transition, however, among all the
theory results listed in table, our result, 1.12, shows the best match to the
experimental result, $1.01\pm 0.20$ Andersen  _et al._ (1972).

Table 4: Termwise contributions to E1 and M1 reduced matrix elements (a.u.) from different terms in FSRCC theory. The operator $\hat{O}$ is either electric or magnetic dipole operator. Terms + h. c. | $\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$ | $\langle^{3}P^{o}_{1}||M1||^{3}P^{o}_{0}\rangle$
---|---|---
DF | $2.1553$ | $1.3586$
1v diagrams | $-0.4040$ | $0.0122$
$\hat{O}R_{2}$ | $0.1049$ | $-0.0199$
${R_{2}}\hat{O}{R_{2}}$ | $0.0711$ | $-0.0423$
${S_{1}}\hat{O}{R_{2}}+{S_{2}}\hat{O}{R_{2}}+{S_{1}^{2}}\hat{O}{R_{2}}$ | $0.0074$ | $0.0081$
$\hat{O}S_{2}$ | $0.0063$ | $-0.0003$
${S_{2}}\hat{O}{S_{2}}+{S_{2}}\hat{O}{S_{1}}+{S_{2}}\hat{O}{S_{1}^{2}}$ | $0.0310$ | $-0.0046$
${T_{1}}\hat{O}{R_{2}}$ | $-0.0001$ | $0$
${S_{2}}\hat{O}{T_{2}}+{S_{2}}\hat{O}{T_{1}}$ | $0.0007$ | $0$
${T_{2}}\hat{O}{T_{2}}+{T_{1}}\hat{O}{T_{2}}$ | $0.0003$ | $0$
Total | $1.9729$ | $1.3118$

###  IV.4 M1 Reduced Matrix Elements

Next, we present and analyze the M1 reduced matrix elements for transitions
which contribute dominantly to the lifetime of the clock state. These are
given in the Table 3. As to be expected, like the case of E1 matrix elements,
the most dominant contribution is from DC-CCSD for all the transitions. The
cumulative contribution from Breit, QED and perturbative triples are small but
important to get a reliable transition properties results.

Unlike the E1 matrix elements, for M1 there are not much data available from
literature for comparison. To the best of our knowledge, there is only one
theory calculation using CI+MBPT Beloy (2021), which reports the values of M1
reduced matrix elements for $\langle^{3}P^{o}_{1}||M1||^{3}P^{o}_{0}\rangle$
and $\langle^{1}P^{o}_{1}||M1||^{3}P^{o}_{0}\rangle$ transitions.
Interestingly, unlike the E1 reduced matrix elements where the two works are
comparable, our results for M1 reduced matrix elements differ by a factor of
two or more to Ref. Beloy (2021). This is expected to lead to a large
difference between the lifetimes calculated using the two data. Since there
are no other theory or experimental results to compare for Pb2+, to cross
check and validate our M1 implementation within FSRCC, we calculated and
compared results for other atomic systems. Ref. Dzuba  _et al._ (2018), using
a combined method of configuration interaction (CI) and perturbation theory
(PT), reports the value of M1 transition rate for
$\langle^{3}P^{o}_{2}||M1||^{3}P^{o}_{1}\rangle$ transition in neutral Yb as
$6.7\times 10^{-2}$ s-1. And our computed value is $5.3\times 10^{-2}$ s-1. It
should be noted that the difference, $\approx$ 21%, is much smaller than that
of our M1 reduced matrix elements with Ref. Beloy (2021) in the case of Pb2+.
Reason for this could be attributed to the more accurate treatment of electron
correlations in FSRCC theory. In an another work, Ref. Derevianko (2001) using
CI+RPA has computed and reported the value of transition rate for
$\langle^{3}P^{o}_{2}||M1||^{3}P^{o}_{1}\rangle$ transition in neutral Sr. The
reported value, $8.26\times 10^{-4}$, compares well with our computed value,
$8.88\times 10^{-4}$, except with a small difference of about 7% which could
again be due to a better accounting of electron correlations in FSRCC. In yet
another seminal work, Ref. Safronova  _et al._ (1999a) carried out a second-
order MBPT calculation of M1 reduced matrix elements for
$\langle^{3}P^{o}_{0}||M1||^{3}P^{o}_{1}\rangle$ and
$\langle^{3}P^{o}_{0}||M1||^{1}P^{o}_{1}\rangle$ transitions in Fe22+. The
reported values, 1.40 and 0.22, respectively are in good agreement with our
computed values, 1.37 and 0.28.

###  IV.5 Lifetime of Clock State

Next, using the E1 and M1 reduced matrix elements from Table 3 and excitation
energies from Table 2 in Eq. (7), we calculated the E1M1 decay rate ($\Gamma$)
for $6s^{2}{\;{}^{1}}S_{0}\rightarrow 6s6p{\;{}^{3}}P^{o}_{0}$ clock
transition, inverse of which gives the lifetime ($\tau$) of the clock state.
The $\Gamma$ and $\tau$ obtained from our calculations are given in the Table
5. To assess the effect of valence-valence correlation, we have separately
given the contributions from $6s6p$, $6s7s$ and $6s6d$ configurations. Our
computed lifetime, $10.45\times 10^{6}$ s, is about 16% larger than the only
calculation Ref. Beloy (2021). The reason for this could be attributed to the
more accurate treatment of electron correlations in our calculations. It
should be noted that, our calculations also incorporate the contributions from
E1 and M1 matrix elements from higher energy ASFs, $6s6d{\;{}^{3}}D_{1}$ and
$6s7s{\;{}^{3}}S_{1}$. This has a significant cumulative contribution of
$\approx$ 2.8% to the total lifetime. The other key difference from Ref. Beloy
(2021) is, the inclusion of the corrections from Breit, QED and perturbative
triples in our calculations. It is to be noted that the contribution from
perturbative triples is significant, it contributes approximately about 10% to
the lifetime. This is consistent with our previous work on Al+ clock where we
observed a large contribution of about 6% to the lifetime of the clock state
Kumar  _et al._ (2021a). Contributions from Breit and QED corrections are
small but important to get accurate value of lifetime. The combined
contribution from Breit+QED is observed to be $\approx$ 0.5%. As discernible
from the Fig. 4(c), DF alone contributes about 30% of the total value. The
most significant contribution comes from the electron correlations arising
from the residual Coulomb interaction through FSRCC within CCSD framework. The
cumulative contribution from DF and CCSD is about 90% of the total lifetime.

Table 5: E1M1 decay rate and lifetime of the clock state ${}^{3}P^{o}_{0}$. Tabulated results also incorporate the contributions from the Breit, QED and perturbative triples corrections. Configurations |  $\Gamma$ (sec-1) |  $\tau$ (sec)
---|---|---
6s2 \+ 6s6p | $9.931\times 10^{-8}$ | $10.195\times 10^{6}$
6s2 \+ 6s6p + 6s7s | $9.902\times 10^{-8}$ | $10.224\times 10^{6}$
6s2 \+ 6s6p + 6s7s + 6s6d | $9.661\times 10^{-8}$ | $10.448\times 10^{6}$
Recommended | $9.661\times 10^{-8}$ | $(10.45\pm 0.50)\times 10^{6}$
Others |  |  $9\times 10^{6}$Beloy (2021)

###  IV.6 Dipole Polarizability

The electric dipole polarizability, $\alpha$, is a measure of the interaction
between electrons’ and nuclear moments, and provides an important probe to
several fundamental properties Khriplovich (1991); Griffith  _et al._ (2009);
Karshenboim and Peik (2010). In the context of the present work, $\alpha$ is
needed for calculating the BBR shift in the clock transition frequency. In
Table 6, we present our result on $\alpha$ for the ground state of Pb2+ and
compare with data in the literature. To calculate this, we have used the PRCC
theory developed and presented in our previous works Kumar  _et al._ (2020,
2021b). Table 6 also shows the contributions from various correlation terms
embedded in the PRCC theory. The term estimated identifies the contribution
from the orbitals from $i$, $j$ and $k$-symmetries. As to be expected, the
dominant contribution is from the DF term. It contributes about 116% of the
total value. The PRCC value is $\approx$ 13% smaller than the DF value. The
reason for this is the cancellation due to opposite contributions from
electron correlation.

Table 6: The value of $\alpha$ (a.u.) for ground state, $6s^{2}\;{}^{1}S_{0}$,
of Pb2+ from PRCC calculation. The available data from experiment and other
theory calculations are also provided for comparison.

State | Present work |
---|---|---
| Method | $\alpha$ |
$6s^{2}\;{}^{1}S_{0}$ | DF | $16.246$ |
| PRCC | $14.173$ |
| PRCC(T) | $14.166$ |
| PRCC(T)+Breit | $14.173$ |
| PRCC(T)+Breit+QED | $14.064$ |
| Estimated | $14.016$ |
| Recommended | $14.02\pm 0.21$ |
| Other cal. | $13.3\pm 0.4^{\rm a}$ |
| Expt. | $13.62\pm 0.08^{\rm b}$ |

aRef.Safronova  _et al._ (2012)-CI + all-order, bRef.Hanni  _et al._
(2010)-Expt.
---

Table 7: Termwise contributions to $\alpha$ (a.u.) from different terms in PRCC theory. Terms + h. c. | $\alpha$
---|---
${{\mathbf{T}}_{1}^{(1)\dagger}{\mathbf{D}}}$ | $16.6849$
${\mathbf{T}_{1}}{{}^{(1)\dagger}}{\mathbf{D}}T_{2}^{(0)}$ | $-1.0835$
${\mathbf{T}_{1}}{{}^{(1)\dagger}}{\mathbf{D}}T_{1}^{(0)}$ | $-0.2746$
${\mathbf{T}_{2}}{{}^{(1)\dagger}}{\mathbf{D}}T_{1}^{(0)}$ | $-0.0052$
${\mathbf{T}_{2}}{{}^{(1)\dagger}}{\mathbf{D}}T_{2}^{(0)}$ | $0.4145$
Normalization | $-1.5633$
Total | $14.1733$

Comparing our result with data in literature, to the best of our knowledge, we
found one result each from the experiment and other theory calculation. Our
recommended value, 14.02, is in good agreement with the experimental value,
13.62$\pm$0.08, reported in Ref. Hanni  _et al._ (2010). The difference from
the experiment is about 3%. From theory calculation, Safronova and
collaborators Safronova  _et al._ (2012) have reported the value of 13.3 using
the CI+all-order method. Our recommended result is about 6% larger than Ref.
Safronova  _et al._ (2012). As mentioned earlier, reason for this difference
could be attributed to the more accurate treatment of electron correlations in
the present calculation. The other key merit in the present calculation is
that, it does not employ the sum-over-state Safronova  _et al._ (1999b);
Derevianko  _et al._ (1999) approach to incorporate the effects of
perturbation. The summation over all the possible intermediate states is
accounted through the perturbed cluster operators Kumar  _et al._ (2020,
2021a). In addition, the present work also incorporates the effects of Breit,
QED and perturbative triples corrections in the calculation of $\alpha$.

###  IV.7 Electron Correlations in FSRCC and PRCC Theories and Corrections
from Breit, QED and Perturbative Triples

Next, we analyze and present the trend of contributions from various
correlation terms in FSRCC and PRCC theories, and from the Breit and QED
corrections. As mentioned earlier, FSRCC method is used to calculate the
lifetime of the metastable clock state, whereas PRCC theory is employed to
calculate $\alpha$ for the ground state of Pb2+.

In Table 4, we have provided the term-wise contributions from FSRCC for one
each from E1 and M1 matrix elements. As expected, the leading order (LO)
contribution to both the matrix elements is from the Dirac-Fock (DF). It
contributes $\approx$ 109 and 104% of the total value for
$\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$ and
$\langle^{3}P^{o}_{1}||M1||^{3}P^{o}_{0}\rangle$, respectively. As the next
leading order (NLO) contribution, the two matrix elements show a different
trend. For $\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$, the NLO contribution
of opposite phase of $\approx$ -20% is observed from the one-valence sector.
Whereas, for $\langle^{3}P^{o}_{1}||M1||^{3}P^{o}_{0}\rangle$, the NLO
contribution is observed from the two-valence sector, where an opposite
contribution of $\approx$ -3% is observed from the ${R_{2}}\hat{O}{R_{2}}$
term. As the next dominant contribution, the term $\hat{O}{R_{2}}+{\rm h.c.}$
contributes $\approx$ 5 and 1.5%, respectively, to
$\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$ and
$\langle^{3}P^{o}_{1}||M1||^{3}P^{o}_{0}\rangle$ matrix elements.

For the contributions from the Breit and QED corrections to matrix elements,
the largest combined contribution is observed to be $\approx$ 1.0%, in the
case of $\langle^{3}D_{1}||D||^{3}P^{o}_{0}\rangle$ transition. Whereas, the
largest contribution from the perturbative triples is observed to be $\approx$
4.4%, in the case of $\langle^{3}D_{1}||D||^{3}P^{o}_{0}\rangle$. Combining
these two, the largest cumulative contribution from Breit+QED+perturbative
triples is observed to $\approx$ 5.5%. Considering the high accuracies
associated with atomic clocks, this is a significant contribution and
important to be included in the calculations to get accurate clock properties
results.

To understand the nature of electron correlations embedded in $\alpha$, in
Table 7 we have listed the termwise contributions from PRCC theory. As evident
from the table, the LO term, ${{\mathbf{T}}_{1}^{(1)\dagger}{\mathbf{D}}}$ \+
h. c., contributes $\approx$ 118% of the total value. This is expected, as it
includes the DF and dominant contribution from core-polarization. For a better
illustration, in Fig. 4(d), we have shown the five dominant contributions from
core orbitals. As discernible from the figure, about 87% of the LO
contribution comes from $6s$ valence-electrons through the dipolar mixing with
$6p$ states. This is expected due to the larger radial extent of $6s$ orbital.
In remaining LO contribution, a contribution of about 8.5% is observed from
the $5d$ core-electrons, through the dipolar mixing with $6p$ and
$4f$-electrons. The NLO term is
${\mathbf{T}_{1}}{{}^{(1)\dagger}}{\mathbf{D}}T_{2}^{(0)}$, contributes
$\approx$ 8%. It is to be noted that, it accounts for the dominant pair-
correlation effects through the $T_{2}^{(0)}$ operator. The next dominant
contribution of about 3%, which also include some part of pair-correlation, is
from the term ${\mathbf{T}_{2}}{{}^{(1)\dagger}}{\mathbf{D}}T_{2}^{(0)}$.

The perturbative triples and Breit each contributes about 0.04% to $\alpha$.
The contribution from QED is, however, significant, about 0.7%. So, the
cumulative contribution from Breit+QED+perturbative triples is $\approx$ 0.8%.
The contribution from the higher symmetry orbitals is estimated to be about
0.34% of the total value.

##  V Theoretical Uncertainty

The uncertainty in the calculated lifetime of clock state will depend on the
uncertainties in the E1 and M1 matrix elements and the energy denominators, as
they contribute in Eq. (7). As the experimental results are not available for
all the E1 and M1 reduced matrix elements, we have identified four different
sources which can contribute to the uncertainty in E1 and M1 matrix elements.
The first source of uncertainty is due to the truncation of the basis set in
our calculation. As discussed in the basis convergence section, our calculated
values of E1 and M1 reduced matrix elements converge to the order of $10^{-3}$
or smaller with basis. Since this is a very small change, we can neglect this
uncertainty. The second source of uncertainty arises due to the truncation of
the dressed Hamiltonian $\tilde{H}^{\rm e}_{\rm hfs}$ to the second order in
$T^{(0)}$ in the properties calculation. In our earlier work Mani and Angom
(2010), using an iterative scheme, we found that the terms with third and
higher orders in $T^{(0)}$ contribute less than 0.1%. So, we consider 0.1% as
an upper bound for this source. The third source is due to the partial
inclusion of triple excitations in the properties calculation. Since
perturbative triples account for the leading order terms of triple excitation,
the contribution from remaining terms will be small. Based on the analysis
from our previous works Kumar  _et al._ (2020); Chattopadhyay  _et al._ (2015)
we estimate the upper bound from this source as 0.72%. The fourth source of
uncertainty could be associated with the frequency-dependent Breit interaction
which is not included in the present calculations. However, in our previous
work Chattopadhyay  _et al._ (2014), using a series of computations using
GRASP2K we estimated an upper bound on this uncertainty as $0.13$% in Ra. So,
for the present work, we take $0.13$% as an upper bound from this source.
There could be other sources of theoretical uncertainty, such as the higher
order coupled perturbation of vacuum polarization and self-energy terms,
quadruply excited cluster operators, etc. However, in general, these all have
much lower contributions to the properties and their cumulative theoretical
uncertainty could be below 0.1%. Uncertainty in the energy denominator is
estimated using the relative errors in the energy difference of
${}^{3}P^{o}_{1}$, ${}^{1}P^{o}_{1}$, ${}^{3}S_{1}$ and ${}^{3}D_{1}$
intermediate states with respect to ${}^{3}P^{o}_{0}$. Among all the
intermediate states, ${}^{1}P^{o}_{1}$ and ${}^{3}D_{1}$ states contribute
dominantly, through routes 1 and 2, respectively, to the lifetime. The
relative errors in the energy difference of these states with
${}^{3}P^{o}_{0}$ are 1.27% and 0.78%, respectively. Since they correspond to
the dominant contributions, we have taken them as the uncertainty in the
energy denominator. By combining the upper bounds of all the uncertainties,
the theoretical uncertainty associated with the lifetime of the clock state is
$\approx$ 4.8%. It should, however, be noted that the uncertainty in the value
of $\alpha$ is much smaller, about 1.5% Kumar  _et al._ (2022)

##  VI Conclusions

We have employed an all-particle multireference Fock-space relativistic
coupled-cluster theory to examine the clock transition properties in Pb2+. We
have computed the excitation energies for several low lying states and E1 and
M1 transition amplitudes for all the allowed transitions in the considered
model space. These were then used to calculate the lifetime of the clock
state. Moreover, using PRCC theory, we also calculated the electric dipole
polarizability for the ground state of Pb2+. In all these calculations, to
obtain accurate properties results, we incorporated the corrections from the
relativistic and QED effects. The dominant contribution from triple
excitations was incorporated though perturbative triples and a fairly large
basis sets were used to achieve the convergence of the properties.

Our computed excitation energies are in excellent agreement with experimental
values for all the states. Our calculated value of lifetime is about 16%
larger than the previously reported CI+MBPT value in Ref. Beloy (2021). The
reason for the higher lifetime in the present calculation could partially be
attributed to the more accurate treatment of core-core and core-valence
electron correlations in FSRCC theory. In addition, to account for the
valence-valence correlation more accurately, we also incorporated the
contributions from the higher energy configurations $6s6d$ and $6s7s$ in our
calculation. Based on our analysis, we find that this contributes about 2.8%
of the total lifetime. Also, from our study, we find that the contributions
from the perturbative triples and Breit+QED corrections are crucial to get
reliable lifetime of the clock state. They are observed to contribute
$\approx$ 10% and 0.5%, respectively. Our recommended value of dipole
polarizability is in good agreement with the available experimental value,
with a small difference of about 3%. Based on our analysis on theoretical
uncertainty, the upper bound on uncertainty for calculated lifetime is about
4.8%, whereas for polarizability it is about 1.5%.

###### Acknowledgements.

The authors wish to thank Suraj Pandey for the useful discussion. Palki
acknowledges the fellowship support from UGC (BININ04154142), Govt. of India.
B. K. M acknowledges the funding support from SERB, DST (CRG/2022/003845).
Results presented in the paper are based on the computations using the High
Performance Computing clusters Padum at IIT Delhi and PARAM HIMALAYA facility
at IIT Mandi under the National Supercomputing Mission of Government of India.

##  Appendix A Convergence of $\alpha$ and E1 and M1 matrix elements

In Table 8, we provide the trend of the convergence of dipole polarizability
and E1 and M1 matrix elements as a function of the basis size. As it is
evident from the table, all the properties converge to the order of $10^{-3}$
or less in the respective units of the properties.

Table 8: Convergence trend of $\alpha$, and E1 and M1 matrix elements as a function of basis size. Basis | $\alpha$ | $\langle^{1}S_{0}||D||^{1}P^{o}_{1}\rangle$ | $\langle^{3}P^{o}_{1}||M1||^{3}P^{o}_{0}\rangle$
---|---|---|---
$96:20s15p12d7f4g0h$ | $15.919$ | 2.0512 | 1.3158
$103:19s16p13d8f3g2h$ | $15.660$ | 2.0268 | 1.3153
$114:20s17p14d9f4g3h$ | $14.979$ | 1.9966 | 1.3198
$125:21s18p15d10f5g4h$ | $14.467$ | 1.9873 | 1.3134
$136:22s19p16d11f6g5h$ | $14.274$ | 1.9888 | 1.3126
$147:23s20p17d12f7g6h$ | $14.205$ | 1.9883 | 1.3121
$158:24s21p18d13f8g7h$ | $14.175$ | 1.9875 | 1.3118
$169:25s22p19d14f9g8h$ | $14.173$ | 1.9854 | 1.3117

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