# Fock-space perturbed relativistic coupled-cluster theory for electric dipole
polarizability of one-valence atomic systems: Application to Al and In

Ravi Kumar  Department of Physics, Indian Institute of Technology, Hauz Khas,
New Delhi 110016, India  D. Angom  Department of Physics, Manipur University,
Canchipur 795003, Manipur, India  Physical Research Laboratory, Ahmedabad -
380009, Gujarat, India  B. K. Mani<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India

###### Abstract

We have developed a Fock-space relativistic coupled-cluster theory based
method for the calculation of electric dipole polarizability of one-valence
atoms and ions. We employ this method to compute the ground-state and spin-
orbit coupled excited state electric dipole polarizability of Al and In. To
check the quality of many-electron wavefunctions, we also compute the
excitation energies of some low-lying states of Al and In. The effects of
Breit interaction and QED corrections from the Uehling potential and the self-
energy are included to improve the accuracy of $\alpha$ further. Our
recommended value of ground-state $\alpha$ for both atoms are in good
agreement with the previous theoretical results. From our computations, we
find that more than 65% of contributions come from the dipolar mixing of
$3p$($5p$) with $3d$($5d$) and $4s$($6s$)-electrons for Al(In). The largest
Breit and QED contributions are found to be 1.3% and 0.6%, respectively.

##  I Introduction

Group-13 elements are a promising candidates for accurate optical atomic
clocks as they are offer low fractional frequency errors Dehmelt (1982); Chou
_et al._ (2010); Safronova  _et al._ (2011); Zuhrianda  _et al._ (2012); Chen
_et al._ (2017). It is to be mentioned that, the recent experiments, Refs.
Brewer  _et al._ (2019a, b), on Al+ optical atomic clock have achieved a low
fractional frequency error of $9.4\times 10^{-19}$ Brewer  _et al._ (2019a, b)
which is, perhaps, the most accurate clock in existence today. The electric
dipole polarizability, $\alpha$, of an atom or ion is a key parameter in
estimating the accuracy of an atomic clock. It is used to estimate the
blackbody radiation (BBR) shift, one of the dominant environment induced
frequency shifts, in the transition frequencies of atoms and ions due to ac
Stark effect. Since the measurement of $\alpha$ for individual states is non-
trivial van Wijngaarden (1999), accurate values from precision theory
calculations play a crucial role in the development of new frequency and time
standards for atomic clocks. The other potential implications of $\alpha$
include, discrete symmetry violations in atoms and ions Khriplovich (1991);
Griffith  _et al._ (2009), condensates of dilute atomic gases Anderson  _et
al._ (1995); Bradley  _et al._ (1995); Davis  _et al._ (1995), high-harmonic
generation and ultrafast processes Lewenstein  _et al._ (1994, 1995); Blaga
_et al._ (2009); Lai  _et al._ (2018), and the search for the variation in the
fundamental constants Karshenboim and Peik (2010); Murphy  _et al._ (2007).

In this work, we have employed a Fock-space perturbed relativistic coupled-
cluster (FS-PRCC) theory to compute the properties of one-valence atomic
systems in the external perturbations. We employ this method to compute the
$\alpha$ for the ground state, ${}^{2}P_{1/2}$, and the spin-orbit
(SO)-splitted excited state, ${}^{2}P_{3/2}$, of Al and In. In the literature,
$\alpha$ for ${}^{2}P_{1/2}$ and ${}^{2}P_{3/2}$ states of Al and In have been
calculated using different methods Fleig (2005); Borschevsky  _et al._ (2012);
Buchachenko (2010); Lupinetti and Thakkar (2005); Fuentealba (2004); Chu and
Dalgarno (2004), including the coupled-cluster based methods like ours. One
common trend in the reported data is, however, a large variation in the
$\alpha$ values. For example, for ${}^{2}P_{1/2}$ state of Al, there is a
difference of $\approx$ 10% in the smallest Fleig (2005) and the largest Chu
and Dalgarno (2004) reported $\alpha$ values. The same trend is also observed
in the experimental values Milani  _et al._ (1990); Sarkisov  _et al._ (2006);
Guella  _et al._ (1984); Ma  _et al._ (2015). The recent experiment Sarkisov
_et al._ (2006) reports $\approx$ 20% larger $\alpha$ than Ref. Milani  _et
al._ (1990) for Al. It is to be emphasized that, unlike the closed-shell
atomic systems, calculation of $\alpha$ for an open-shell system is a
challenging task and requires the inclusion of core-core, core-valence and
valence-valence electrons correlations to the highest level of accuracy.
Moreover, the inclusion of correlation effects from Breit interaction and QED
corrections and the large basis sets are essential to tune the accuracy
further. The aim of the present work is to fill this gap. We aim to: develop a
Fock-space relativistic coupled-cluster (FS-RCC) theory based method to
accurately account for an external perturbation in the properties calculation
of one-valence atomic systems; compute the accurate value of $\alpha$ for Al
and In; and quantify the various electron correlation effects embedded in the
$\alpha$ of Al and In.

To test the accuracy of the wavefunctions, we have calculated to excitation
energies of few low-lying states of Al and In using RCC theory. RCC is one of
the most powerful many-body theories for atomic structure calculations. It
accounts for the electron correlation to all-orders of residual Coulomb
interaction, and has been used to calculate a plethora of properties in atomic
systems. The implementation of such a theory and a FORTRAN code for the
properties calculations of closed-shell and one-valence atomic systems without
external perturbation is reported in our previous work Mani  _et al._ (2017).
For the properties calculation in the presence of external perturbation, we
had reported a perturbed relativistic coupled-cluster (PRCC) theory for
closed-shell in the works Kumar  _et al._ (2020, 2021) and references therein.
One of the key merits of PRCC is that it does not employ the sum-over-state
Safronova  _et al._ (1999); Derevianko  _et al._ (1999) approach to
incorporate the effects of a perturbation. The summation over all the possible
intermediate states is subsumed in the perturbed cluster operators. Due to
important prospects associated with $\alpha$, it has been computed using a
variety of other many-body methods in the literature. The recent review
article by Mitroy et al. Mitroy  _et al._ (2010) provides a summary of
$\alpha$ for several atoms and ions computed using different methods. The
other reference which we found very useful is the Schwerdtfeger’s updated
table of $\alpha$ for neutral atoms Schwerdtfeger and Nagle (2019). The table
provides an exhaustive list of references on experimental and theoretical
values of $\alpha$ for several neutral atoms.

The remaining part of the paper is organized into five sections. In Sec. II,
we discussed the RCC and PRCC theories for one-valence atomic systems where we
derive the PRCC equations and also discuss in detail the contributing diagrams
to each terms. In Sec. III, we discuss the calculation of $\alpha$ using PRCC
theory. Here, we provide some dominant diagrams contributing to $\alpha$. The
basis set convergence and other calculational details are discussed in Sec. IV
of the paper. In Sec. V, we analyze and present our results of excitation
energy and dipole polarizability. Unless stated otherwise, all the results and
equations presented in this paper are in atomic units
($\hbar=m_{e}=e=1/{4\pi\epsilon_{0}}=1$).

##  II Methodology

###  II.1 One-valence RCC Theory

The many-electron ground state wavefunction of an one-valence atom or ion in
the RCC theory is expressed as

$|\Psi_{v}\rangle=e^{(T^{(0)}+S^{(0)})}\ |\Phi_{v}\rangle,$ (1)

where $|\Phi_{v}\rangle$ is the one-valence Dirac-Fock (DF) reference state,
and is obtained by adding an electron to the closed-shell reference state,
$|\Phi_{v}\rangle=a_{v}^{\dagger}|\Phi_{0}\rangle$. The operators $T^{(0)}$
and $S^{(0)}$ are the coupled cluster (CC) operators which act within the
Hilbert spaces of the closed-shell and open-shell systems, respectively. The
ground state $|\Psi_{v}\rangle$ is the solution of the eigenvalue equation

$\displaystyle H^{\text{DCB}}|\Psi_{v}\rangle=E_{v}|\Psi_{v}\rangle,$ (2)

where $H^{\text{DCB}}$ is the Dirac-Coulomb-Breit no-virtual-pair Hamiltonian
and $E_{v}$ is the exact energy of the one-valence system. For an atom with
$N$-electrons, $H^{\text{DCB}}$ is

$\displaystyle H^{\text{DCB}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\left[c\bm{\alpha}_{i}\cdot\mathbf{p}_{i}+(\beta_{i}-1)c^{2}-V_{N}(r_{i})\right]$
(3)
$\displaystyle+\sum_{i<j}\left[\frac{1}{r_{ij}}+g^{\text{B}}(r_{ij})\right],$

where $\bm{\alpha}$ and $\beta$ are the Dirac matrices, and $V_{N}(r_{i})$ is
the nuclear potential. And, the last two terms, $1/r_{ij}$ and $g^{\\\
text{B}}(r_{ij})$, are the Coulomb and Breit interactions, respectively. The
effects of the negative-energy continuum states are avoided by employing a
kinetically balanced finite Gaussian basis Mohanty  _et al._ (1991); Stanton
and Havriliak (1984).

In the RCC theory, the single and double excitations incorporate most of the
electron correlation effects and provide a good description of the properties.
Therefore, we can approximate $T^{(0)}=T^{(0)}_{1}+T^{(0)}_{2}$ and
$S^{(0)}=S^{(0)}_{1}+S^{(0)}_{2}$, which is referred to as the coupled-cluster
with singles and doubles (CCSD) approximation. These operators in the second
quantized notation are

$\displaystyle T^{(0)}_{1}$ $\displaystyle=$
$\displaystyle\sum_{ap}t_{a}^{p}a_{p}^{\dagger}a_{a},{\;\;\text{and}\;\;}T^{(0)}_{2}=\frac{1}{2!}\sum_{abpq}t_{ab}^{pq}a_{p}^{\dagger}a_{q}^{\dagger}a_{b}a_{a},$
(4a) $\displaystyle S^{(0)}_{1}$ $\displaystyle=$
$\displaystyle\sum_{p}s_{v}^{p}a_{p}^{\dagger}a_{v},{\;\;\text{and}\;\;}S^{(0)}_{2}=\sum_{apq}s_{va}^{pq}a_{p}^{\dagger}a_{q}^{\dagger}a_{a}a_{v}.$
(4b)

Here, the indices $ab\ldots$ and $pq\ldots$ represent the core and virtual
orbitals, respectively. And, $t_{\cdots}^{\cdots}$ and $s_{\cdots}^{\cdots}$
are the cluster amplitudes of the $T$ and $S$ operators, respectively. These
closed-shell and one-valence operators are obtained by solving a set of
coupled nonlinear equations and details are discussed in our previous works
Mani  _et al._ (2009); Mani and Angom (2010); Mani  _et al._ (2017). In the
Ref. Mani  _et al._ (2017), we have provided descriptions of the computational
implementation of RCC theory for the properties calculations of the closed-
shell and one-valence systems without an external perturbation.

###  II.2 One-valence PRCC theory

In the presence of an external perturbation, the wavefunction and the energy
of the system are modified. For the electric dipole polarizability, the
perturbation is due to the interaction between the external electric field
$\mathbf{E}_{\text{ext}}$ and the induced electric dipole moment of the system
$\mathbf{D}$. And, the interaction Hamiltonian is
$H_{1}=-\mathbf{D}\cdot\mathbf{E}_{\text{ext}}$. We refer the modified
eigenstate as the perturbed eigenstate, $|\tilde{\Psi}_{v}\rangle$, and the
modified energy as the perturbed energy, $\tilde{E}_{v}$. In the PRCC theory,
$|\tilde{\Psi}_{v}\rangle$ is expressed as

$\displaystyle|\tilde{\Psi}_{v}\rangle$ $\displaystyle=$ $\displaystyle
e^{T^{(0)}}\left[1+\lambda\mathbf{T}^{(1)}\cdot\mathbf{E}_{\text{ext}}\right]$
(5)
$\displaystyle\left[1+S^{(0)}+\lambda\mathbf{S}^{(1)}\cdot\mathbf{E}_{\text{ext}}\right]|\Phi_{v}\rangle,$

where, $\lambda$ is the perturbation parameter. The operators
$\mathbf{T}^{(1)}$ and $\mathbf{S}^{(1)}$ are referred to as the perturbed
closed-shell and one-valence cluster operators, respectively, and both are
rank one operators. The operator $\mathbf{T}^{(1)}$ is obtained by solving a
set of coupled perturbed equations within the Hilbert space of the occupied
electrons. The details on its tensor representation and the PRCC equations are
discussed in our previous works on the dipole polarizability of the closed-
shell atomic systems Chattopadhyay  _et al._ (2012); Kumar  _et al._ (2020).
So, here, we discuss only the tensor representation and PRCC equations of the
open shell cluster operator $\mathbf{S}^{(1)}$.

Figure 1: Diagrammatic representations of $\mathbf{T}_{1}^{(1)}$,
$\mathbf{T}_{2}^{(1)}$, $\mathbf{S}_{1}^{(1)}$ and $\mathbf{S}_{2}^{(1)}$
perturbed cluster operators.

Similar to the case of $T^{(0)}$ and $S^{(0)}$ operators, in the CCSD
approximation, we take $\mathbf{S}^{(1)}=S_{1}^{(1)}+S_{2}^{(1)}$. And, these
in the second quantized notations,

$\displaystyle\mathbf{S}_{1}^{(1)}$ $\displaystyle=$
$\displaystyle\sum_{p}\xi_{v}^{p}\mathbf{C}_{1}(\hat{r})a_{p}^{\dagger}a_{v},$
(6a) $\displaystyle\mathbf{S}_{2}^{(1)}$ $\displaystyle=$
$\displaystyle\sum_{apq}\sum_{lk}\xi_{va}^{pq}(l,k)\mathbf{C}_{l}(\hat{r}_{1})\mathbf{C}_{k}(\hat{r}_{2})a_{p}^{\dagger}a_{q}^{\dagger}a_{a}a_{v}.$
(6b)

Here, $\xi^{\cdots}_{\cdots}$ represents the cluster amplitude for the
operator $\mathbf{S}^{(1)}$. The one-body operator $\mathbf{S}_{1}^{(1)}$ is
an odd parity operator and expressed in terms of a rank-one
$\mathbf{C}$-tensor. It satisfies the orbital-parity and orbital-triangular
selection rules, $(-1)^{l_{v}+l_{p}}=-1$ and $|j_{v}-j_{p}|\leqslant
1\leqslant(j_{v}+j_{p})$, respectively. The tensor structure of the two-body
operator $\mathbf{S}_{2}^{(1)}$ involves two $\mathbf{C}$-tensors with ranks
$l$ and $k$ associated with its two-vertices. These two $\mathbf{C}$-tensors
are coupled to give a rank-one operator, $\mathbf{S}_{2}^{(1)}$. The allowed
orbital-parity and and orbital-triangular selections rules for
$\mathbf{S}_{2}^{(1)}$ are, $(-1)^{l_{v}+l_{p}}=-(-1)^{l_{a}+l_{q}}$ and
$|j_{v}-j_{p}|\leqslant l\leqslant(j_{v}+j_{p})$, $|j_{a}-j_{q}|\leqslant
k\leqslant(j_{a}+j_{q})$, respectively. The diagrammatic representations of
$\mathbf{T}^{(1)}$ and $\mathbf{S}^{(1)}$ are shown in the Fig.1.

In analogy with the Eq. (2), $|\tilde{\Psi}_{v}\rangle$ is the solution of the
eigenvalue equation

$\left(H^{\text{DCB}}+\lambda
H_{1}\right)|\tilde{\Psi}_{v}\rangle=\tilde{E}_{v}|\tilde{\Psi}_{v}\rangle,$
(7)

here, within the first-order time-independent perturbation theory, the
perturbed energy $\tilde{E}_{v}\equiv E_{v}$ as the first-order correction
vanishes due to the odd parity nature of $H_{1}$. Using Eq. (5) in the
eigenvalue equation (7) and by operating with $e^{-T^{(0)}}$ from left,
retaining the terms first order in $\lambda$, we get

$\displaystyle\left[e^{-T^{(0)}}H^{\text{DCB}}e^{T^{(0)}}\left(\mathbf{S}^{(1)}\cdot\mathbf{E}_{\text{ext}}\right)+e^{-T^{(0)}}H^{\text{DCB}}e^{T^{(0)}}\left(\mathbf{T}^{(1)}\cdot\mathbf{E}_{\text{ext}}\right)\left(1+S^{(0)}\right)+e^{-T^{(0)}}H_{1}e^{T^{(0)}}\left(1+S^{(0)}\right)\right]|\Phi_{v}\rangle$
(8)
$\displaystyle=\left[E_{v}\left(\mathbf{S}^{(1)}\cdot\mathbf{E}_{\text{ext}}\right)+E_{v}\left(\mathbf{T}^{(1)}\cdot\mathbf{E}_{\text{ext}}\right)\left(1+S^{(0)}\right)\right]|\Phi_{v}\rangle.$

Using the definition of the normal ordered Hamiltonian,
$H_{N}=H^{\text{DCB}}-\langle\Phi_{v}|H^{\text{DCB}}|\Phi_{v}\rangle$, and
dropping $E_{\text{ext}}$ for simplicity from both sides of the equation, we
can write

$\displaystyle\left[\bar{H}_{N}\mathbf{S}^{(1)}+\bar{H}_{N}\mathbf{T}^{(1)}\left(1+S^{(0)}\right)+\bar{H}_{1}(1+S^{(0)})\right]|\Phi_{v}\rangle=$
$\displaystyle\Delta
E_{v}\left[\mathbf{S}^{(1)}+\mathbf{T}^{(1)}\left(1+S^{(0)}\right)\right]|\Phi_{v}\rangle$
(9)

where, $\Delta E_{v},=E_{v}-\langle\Phi_{v}|H^{\text{DCB}}|\Phi_{v}\rangle$ is
the correlation energy of one-valence atom. And,
$\bar{H}_{N},=e^{-T^{(0)}}H_{N}e^{T^{(0)}}$, is a similarity transformed
Hamiltonian. Using the Wick’s theorem, it can be reduced to

$\displaystyle\bar{H}_{N}$ $\displaystyle=$ $\displaystyle
H_{N}+\\{\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
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3.41666pt}}H_{N}T^{(0)}T^{(0)}\\}$ (10)
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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=1.72218pt\vrule
width=9.37514pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.73938pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=26.89923pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=26.89923pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=18.99355pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=15.91125pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=41.28745pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=41.28745pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=29.16751pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=24.63869pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=6.88889pt\vrule
width=55.67566pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=6.88889pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=6.88889pt\vrule
width=55.67566pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=6.88889pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=6.88889pt\vrule
width=39.34148pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=6.88889pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=6.88889pt\vrule
width=33.36612pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=6.88889pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{N}T^{(0)}T^{(0)}T^{(0)}T^{(0)}\\}$

By projecting Eq. (9) with singly and doubly-excited determinants,
$\langle\Phi_{v}^{p}|$ and $\langle\Phi_{va}^{pq}|$, respectively, and using
the Wicks’s theorem to remove the disconnected terms, we obtain the PRCC
coupled equations for singles and doubles as

$\displaystyle\langle\Phi_{v}^{p}|\bar{H}_{1}+\\{\mathchoice{\vbox{\hbox
to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=10.71664pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=10.71664pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.54164pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=5.95831pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\bar{H}_{1}S^{(0)}\\}+\\{\mathchoice{\vbox{\hbox to0.0pt{\kern
5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.37514pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.73938pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=46.30373pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=46.30373pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=33.25444pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=26.89124pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}{\bar{H}}_{N}\mathbf{T}^{(1)}(1+S^{(0)})\\}+\\{\mathchoice{\vbox{\hbox
to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.02663pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.02663pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.19162pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.60829pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\bar{H}_{N}\mathbf{S}^{(1)}\\}\Phi_{v}\rangle$ $\displaystyle=$
$\displaystyle
E_{v}^{\text{att}}\langle\Phi_{v}^{p}|\mathbf{S}_{1}^{(1)}|\Phi_{v}\rangle,$
(11a)
$\displaystyle\langle\Phi_{va}^{pq}|\bar{H}_{1}+\\{\mathchoice{\vbox{\hbox
to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=10.71664pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=10.71664pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.54164pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=5.95831pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\bar{H}_{1}S^{(0)}\\}+\\{\mathchoice{\vbox{\hbox to0.0pt{\kern
5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.37514pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.73938pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=46.30373pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=46.30373pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=33.25444pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=26.89124pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}{\bar{H}}_{N}\mathbf{T}^{(1)}(1+S^{(0)})\\}+\\{\mathchoice{\vbox{\hbox
to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.02663pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.02663pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.19162pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.60829pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\bar{H}_{N}\mathbf{S}^{(1)}\\}\Phi_{v}\rangle$ $\displaystyle=$
$\displaystyle
E_{v}^{\text{att}}\langle\Phi_{va}^{pq}|\mathbf{S}_{2}^{(1)}|\Phi_{v}\rangle.$
(11b)

Here, $E_{v}^{\text{att}}$ is the attachment energy of the valence electron
and is expressed as $E_{v}^{\text{att}}=\epsilon_{v}+\Delta E_{v}$, where
$\epsilon_{v}$ is the single-particle energy. In deriving these equations we
have used the relations,
$\langle\Phi_{v}^{*}|\mathbf{T}^{(1)}|\Phi_{v}\rangle=0$ and
$\langle\Phi_{v}^{*}|\mathbf{T}^{(1)}S^{(0)}|\Phi_{v}\rangle=0$, as they do
not contribute, where $*$ represents the single and doubly excited
determinant. We solve these coupled nonlinear equations using the Jacobi
method. To remedy the slow convergence of this method we employ direct
inversion of the iterated subspace (DIIS) Pulay (1980).

###  II.3 Linearized PRCC

The Eqs. (11a) and (11b) contain all the CC terms associated with the PRCC
equations of the one-valence system. And, therefore, provides an accurate
description of the properties of the system. However, solving these equations
is computationally expensive due to the large number of many-body diagrams
arising from the contractions with multiple CC operators. One simple approach
to mitigate this is to retain terms which are linear in the CC operators. And,
this also provides reliable results as in most of the cases the contribution
from the nonlinear terms is small. So, retaining the terms linear in CC
operators, we can write Eqs. (11a) and (11b) as

$\displaystyle\langle\Phi_{v}^{p}|H_{1}+\\{\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=10.9788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=10.9788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.72516pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=6.0894pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{1}T^{(0)}\\}+\\{\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=10.71664pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=10.71664pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.54164pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=5.95831pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{1}S^{(0)}\\}+\\{\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
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width=13.28879pt,height=0.0pt,depth=0.50003pt\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=1.72218pt\vrule
width=9.37514pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.73938pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{N}\mathbf{T}^{(1)}\\}$ $\displaystyle+\\{\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.02663pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.02663pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.19162pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.60829pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{N}\mathbf{S}^{(1)}\\}\Phi_{v}\rangle=E_{v}^{\text{att}}\langle\Phi_{v}^{p}|\mathbf{S}_{1}^{(1)}|\Phi_{v}\rangle,$
(12a) $\displaystyle\langle\Phi_{va}^{pq}|H_{1}+\\{\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=10.9788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=10.9788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.72516pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=6.0894pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{1}T^{(0)}\\}+\\{\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=10.71664pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=10.71664pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.54164pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=5.95831pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{1}S^{(0)}\\}+\\{\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.37514pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.73938pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{N}\mathbf{T}^{(1)}\\}$ $\displaystyle+\\{\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.02663pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.02663pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.19162pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.60829pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{N}\mathbf{S}^{(1)}\\}\Phi_{v}\rangle=E_{v}^{\text{att}}\langle\Phi_{va}^{pq}|\mathbf{S}_{2}^{(1)}|\Phi_{v}\rangle.$
(12b)

We refer to these equations as the linearized perturbed coupled-cluster
(LPRCC) equations. The LPRCC incorporates all the important many-body effects
like random-phase approximation and provides a good description of the one-
valence atomic or ionic properties in the presence of perturbation.

###  II.4 PRCC diagrams

To solve the coupled-cluster amplitude Eqs.(11a) and (11b), we have to
evaluate all the possible terms arising from each of the matrix elements in
the equations. There are several terms and the book keeping is simplified with
the many-body Goldstone diagrammatic approach. In this section we describe the
Goldstone diagrams arising from the matrix elements and these are evaluated
manually. It is, however, possible to identify diagrams computationally as
well Fritzsche  _et al._ (2008). We have adopted the manual approach for the
convenience in evaluating the angular factors. For simplicity, from here
onwards, as it should be the case for $\alpha$, we use $\mathbf{D}$ in place
of $H_{1}$.

####  II.4.1 $\bar{\mathbf{D}}$

For the one-valence system, from the definition of the similarity transformed
Hamiltonian in Eq. (10), using the CSSD approximation we get

$\bar{\mathbf{D}}=\mathbf{D}+\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
3.81944pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.43578pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.81944pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.43578pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.67361pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=5.20503pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 1.90971pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=3.71788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathbf{D}T_{1}^{(0)}+\mathchoice{\vbox{\hbox to0.0pt{\kern
0.0pt\kern 3.81944pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.43578pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.81944pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.43578pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.67361pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=5.20503pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 1.90971pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=3.71788pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathbf{D}T_{2}^{(0)}.$ (13)

The terms with two or higher orders of $T^{(0)}$ do not contribute to the PRCC
equation for one-valence system. The first term, $\mathbf{D}$, is the bare
dipole operator and contributes to the equation of $\mathbf{S}_{1}^{(1)}$. The
two remaining terms represent the contraction with the unperturbed operator
$T^{(0)}$ and contribute to the PRCC equations of the $\mathbf{S}^{(1)}_{1}$
and $\mathbf{S}^{(1)}_{2}$, respectively. In total, there are 3 diagrams from
$\bar{\mathbf{D}}$, and these are shown in Fig. 2. Using the algebra of
evaluating the Goldstone diagrams Lindgren and Morrison (2nd Edition, 1986),
we can write the algebraic expressions of the diagrams as

$\displaystyle\langle\mathbf{D}\rangle_{v}^{p}+\langle\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.89479pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.89479pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=5.52635pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=3.94739pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathbf{D}T^{(0)}\rangle^{p}_{v}$ $\displaystyle=$
$\displaystyle\mathbf{r}_{pv}-\sum_{a}\mathbf{r}_{av}t_{a}^{p},$ (14a)
$\displaystyle\langle\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.27847pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.89479pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.89479pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=5.52635pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=3.94739pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathbf{D}T^{(0)}\rangle_{va}^{pq}$ $\displaystyle=$
$\displaystyle-\sum_{b}\mathbf{r}_{bv}t_{ab}^{qp},$ (14b)

respectively. Here, in atomic units $\mathbf{D}\equiv-\mathbf{r}$ and
$\mathbf{r}_{ij}=\langle i|\mathbf{r}|j\rangle$ is the electronic part of the
single-particle matrix element of the dipole operator. And,
$\langle\cdots\rangle_{v}^{p}$ and $\langle\cdots\rangle_{vb}^{pq}$ represent
the matrix elements $\langle\Phi_{v}^{p}|\cdots|\Phi_{v}\rangle$ and
$\langle\Phi_{va}^{pq}|\cdots|\Phi_{v}\rangle$, respectively.

Figure 2: Single and double PRCC diagrams contributing to the term
$\bar{\mathbf{D}}$ of Eqs. (11a) and (11b), respectively.

####  II.4.2  $\bar{\mathbf{D}}S^{(0)}$

Like the first term, consider the second term in Eqs.(11a) and (11b).
Expanding the similarity transformed operator $\bar{\mathbf{D}}$ in terms of
$T^{(0)}$, we can write

$\displaystyle\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.27847pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.63263pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.63263pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=5.34283pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=3.8163pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}{\bar{\mathbf{D}}}S^{(0)}$ $\displaystyle=$
$\displaystyle\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.27847pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.63263pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.63263pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=5.34283pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=3.8163pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathbf{D}S^{(0)}_{1}+\mathchoice{\vbox{\hbox to0.0pt{\kern
0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.63263pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.63263pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=5.34283pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=3.8163pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathbf{D}S^{(0)}_{2}+\mathchoice{\vbox{\hbox to0.0pt{\kern
0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.89479pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.89479pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=5.52635pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=3.94739pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.27847pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=22.02084pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=22.02084pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=15.5168pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=12.54373pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathbf{D}T^{(0)}_{1}S^{(0)}_{1}+\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.89479pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
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3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=22.02084pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
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width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathbf{D}T^{(0)}_{2}S^{(0)}_{1}$ (15)
$\displaystyle+\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.27847pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.27847pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=22.02084pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=15.5168pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=12.54373pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathbf{D}T^{(0)}_{1}S^{(0)}_{2}.$

The terms having higher than two orders of CC operators do not contribute. The
first three terms in the above equation contribute to the PRCC equation of
$\mathbf{S}^{(1)}_{1}$ and leads to 4 diagrams. The diagrams are shown in Fig.
3(a-d). Except for the first term, all the other terms contribute to the PRCC
equation of $\mathbf{S}^{(1)}_{2}$. In total there are 6 diagrams from these
terms and these are shown in Fig. 3(e-j). The corresponding algebraic
expressions are

$\displaystyle\langle\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathbf{D}S^{(0)}\rangle_{v}^{p}+\langle\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
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width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=22.02084pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
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3.41666pt}}\mathbf{D}T^{(0)}S^{(0)}\rangle_{v}^{p}=\sum_{q}\mathbf{r}_{pq}s_{v}^{q}+\sum_{aq}\mathbf{r}_{aq}(s_{va}^{pq}$
$\displaystyle\;\;\;-s_{va}^{qp}-t_{a}^{p}s_{v}^{q}),$ (16a)
$\displaystyle\langle\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
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3.41666pt}}\mathbf{D}S^{(0)}\rangle_{va}^{pq}+\langle\mathchoice{\vbox{\hbox
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3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.27847pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.99492pt\hbox{\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathbf{D}T^{(0)}S^{(0)}\rangle_{va}^{pq}=\sum_{r}(\mathbf{r}_{pr}s_{av}^{qr}+\mathbf{r}_{qr}s_{va}^{pr})$
$\displaystyle\;\;\;-\sum_{b}\mathbf{r}_{ba}s_{vb}^{pq}-\sum_{br}\mathbf{r}_{br}(t_{ab}^{qp}s_{v}^{r}+t_{a}^{r}s_{vb}^{pq}+t_{b}^{q}s_{va}^{pr}).$
(16b)

This term is an important one in PRCC theory as it subsumes the many-body
effects of the core-polarization.

Figure 3: Single and double PRCC diagrams contributing to the term
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.13922pt\hbox{\vrule
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3.41666pt}}\bar{\mathbf{D}}S^{(0)}$ of PRCC Eqs. (11a) and (11b),
respectively.

####  II.4.3 ${\bar{H}}_{N}\mathbf{T}^{(1)}$

Unlike the previous two terms where the dipole operator appears explicitly, in
this term, the effects of the perturbation is embedded in a rank one operator,
$\mathbf{T}^{(1)}$. We can expand this as

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3.41666pt}}{\bar{H}}_{N}\mathbf{T}^{(1)}=\mathchoice{\vbox{\hbox to0.0pt{\kern
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3.41666pt}}H_{N}T^{(0)}\mathbf{T}^{(1)}\\!+\frac{1}{2}\mathchoice{\vbox{\hbox
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3.41666pt}}H_{N}T^{(0)}T^{(0)}\mathbf{T}^{(1)}$ (17)

Consider first the PRCC equation of the $\mathbf{S}_{1}^{(1)}$. Only the first
two terms contribute. The PRCC diagrams from these terms are obtained by
invoking all the possible contractions between the $H_{N}$ and CC operators.
There are 8 diagrams and these are shown in Fig. 4. We do not consider the
diagrams arising from the one-body part of $H_{N}$. These do not contribute as
we use Dirac-Fock orbitals in our calculations. The algebraic expression of
these diagrams are given in Eq.(18). In the equation, $g_{ijkl}$ represents
the matrix element $\langle ij|1/r_{12}+g^{B}(r_{12})|kl\rangle$ and
$\tilde{g}_{ijkl},=(g_{ijkl}-g_{ijlk}$), is an antisymmetric matrix element.

Figure 4: Single PRCC diagrams contributing to the term
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3.41666pt}}{\bar{H}}_{N}\mathbf{T}^{(1)}$ of Eq. (11a).

In the PRCC equation of $\mathbf{S}_{2}^{(1)}$, all the terms contribute and
leads to 29 diagrams. These diagrams are shown in Fig. 5. Like in the case of
$\mathbf{S}^{(1)}$, we do not include the diagrams from the one-body part of
the $H_{N}$. The algebraic expression of these diagrams are given in the
Eq.(18b). In the equation we see the emergence of a trend. The number of terms
in this equation far exceed those in $\mathbf{S}_{1}^{(1)}$. This trend is
there in the remaining non-linear terms as well.

$\displaystyle\langle\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
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3.41666pt}}H_{N}\mathbf{T}^{(1)}\rangle_{v}^{p}+\langle\mathchoice{\vbox{\hbox
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
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3.41666pt}}H_{N}T^{(0)}\mathbf{T}^{(1)}\rangle_{v}^{p}=\sum_{aq}\tilde{g}_{apqv}\tau_{a}^{q}+\sum_{abq}\left[-\tilde{g}_{abqv}\tau_{ab}^{qp}+g_{abvq}\left(-t_{a}^{p}\tau_{b}^{q}+t_{a}^{q}\tau_{b}^{p}-t_{b}^{q}\tau_{a}^{p}+t_{b}^{p}\tau_{a}^{q}\right)\right]$
(18a) $\displaystyle\langle\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
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3.41666pt}}H_{N}\mathbf{T}^{(1)}\rangle_{va}^{pq}+\langle\mathchoice{\vbox{\hbox
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3.41666pt}}H_{N}T^{(0)}\mathbf{T}^{(1)}\rangle_{va}^{pq}+\langle\mathchoice{\vbox{\hbox
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3.41666pt}}H_{N}T^{(0)}T^{(0)}\mathbf{T}^{(1)}\rangle_{va}^{pq}=-\sum_{b}g_{bqva}\tau_{b}^{p}+\sum_{bc}g_{bcva}\tau_{bc}^{pq}-\sum_{br}\left[g_{bpvr}\tau_{ba}^{rq}+g_{bqvr}(\tau_{ba}^{pr}\right.$
$\displaystyle\;\;+\left.\tau_{b}^{p}t_{a}^{r}+t_{b}^{p}\tau_{a}^{r})\right]+\sum_{bc}g_{bcva}(\tau_{b}^{p}t_{c}^{q}+t_{b}^{q}\tau_{c}^{q})+\sum_{bcr}g_{bcvr}\left(-t_{ba}^{pq}\tau_{c}^{r}+t_{ab}^{qr}\tau_{c}^{p}+t_{ba}^{pr}\tau_{c}^{q}+t_{bc}^{pq}\tau_{a}^{r}+t_{ca}^{pq}\tau_{b}^{r}-t_{ca}^{rq}\tau_{b}^{p}+t_{ac}^{rq}\tau_{b}^{p}\right.$
$\displaystyle\;\;-\left.\tau_{ba}^{pq}t_{c}^{r}+\tau_{ab}^{qr}t_{c}^{p}+\tau_{ba}^{pr}t_{c}^{q}+\tau_{bc}^{pq}t_{a}^{r}+\tau_{ca}^{pq}t_{b}^{r}-\tau_{ca}^{rq}t_{b}^{p}+\tau_{ac}^{rq}t_{b}^{p}+\tau_{a}^{r}t_{b}^{p}t_{c}^{q}+\tau_{c}^{q}t_{b}^{p}t_{b}^{q}+\tau_{b}^{p}t_{a}^{r}\right)$
(18b)

Figure 5: Double PRCC diagrams contributing to the term
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.37514pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.73938pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}{\bar{H}}_{N}\mathbf{T}^{(1)}$ of Eq. (11b).

####  II.4.4  ${\bar{H}}_{N}\mathbf{T}^{(1)}S^{(0)}$

This term involves one each of the perturbed and unperturbed CC operators
$\mathbf{T}^{(1)}$ and $S^{(0)}$, respectively. And, the term contributes to
the nonlinear PRCC equation. By expanding $\bar{H}_{N}$, we obtain

$\displaystyle\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.37514pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.73938pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.41484pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.41484pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=19.36559pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=16.33572pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}{\bar{H}}_{N}\mathbf{T}^{(1)}S^{(0)}=\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.37514pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.73938pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.41484pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.41484pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=19.36559pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=16.33572pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{N}\mathbf{T}^{(1)}S^{(0)}+\mathchoice{\vbox{\hbox to0.0pt{\kern
0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.37514pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
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=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.677pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.677pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=19.5491pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=16.46681pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=41.80305pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=41.80305pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=29.53955pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=25.06316pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{N}T^{(0)}\mathbf{T}^{(1)}S^{(0)}.$ (19)

The higher order terms will not contribute. As to be expected, several
diagrams arise from this term due to various possible contractions between
$H_{\rm N}$ and CC operators. For the PRCC equation of $\mathbf{S}_{1}^{(1)}$,
there are 14 diagrams from this term. And, these are are shown in the left
panel of Fig. 6. The algebraic expressions of these diagrams are given in
Eq.(20).

$\displaystyle\langle\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
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width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.41484pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=19.36559pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
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width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}$ $\displaystyle H_{N}$
$\displaystyle\mathbf{T}^{(1)}S^{(0)}\rangle_{v}^{p}+\langle\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
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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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width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
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=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
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width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=41.80305pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=29.53955pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=25.06316pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{N}T^{(0)}\mathbf{T}^{(1)}S^{(0)}\rangle_{v}^{p}=\sum_{aqr}g_{apqr}(\tau_{a}^{q}s_{v}^{r}-\tau_{a}^{r}s_{v}^{q})+\sum_{aqbr}g_{abqr}\left[s_{v}^{r}(-\tau_{ab}^{qp}+\tau_{ab}^{pq})+\tau_{a}^{q}s_{bv}^{rp}+\tau_{b}^{q}s_{av}^{rp}-\tau_{a}^{q}s_{bv}^{pr}\right.$
(20) $\displaystyle+$
$\displaystyle\left.\tau_{a}^{p}s_{bv}^{qr}-\tau_{b}^{p}s_{av}^{qr}+\tau_{b}^{q}s_{av}^{pr}+(-\tau_{b}^{p}t_{a}^{q}-\tau_{a}^{q}t_{b}^{p}+\tau_{b}^{q}t_{a}^{p}+\tau_{a}^{p}t_{b}^{q})s_{v}^{r}\right]$
$\displaystyle\langle\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
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3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=16.33572pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}$ $\displaystyle H_{N}$
$\displaystyle\mathbf{T}^{(1)}S^{(0)}\rangle_{va}^{pq}+\langle\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.37514pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.73938pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.677pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.677pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=19.5491pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=16.46681pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=41.80305pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=41.80305pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=29.53955pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=25.06316pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{N}T^{(0)}\mathbf{T}^{(1)}S^{(0)}\rangle_{va}^{pq}=\sum_{rs}g_{pqrs}\tau_{a}^{s}s_{v}^{r}-\sum_{rb}(g_{pbra}\tau_{b}^{q}+g_{qbar}\tau_{b}^{p})s_{v}^{r}+\sum_{sbr}g_{pbsr}\left[s_{v}^{s}(\tau_{ba}^{rq}-\tau_{ab}^{rq})-s_{v}^{r}(\tau_{ab}^{qs}\right.$
(21) $\displaystyle+$
$\displaystyle\left.\tau_{ab}^{sp})\right]+\sum_{cbr}g_{cbar}\tau_{cb}^{qp}s_{v}^{r}+\sum_{brs}g_{bprs}(s_{va}^{sq}\tau_{b}^{r}-s_{va}^{rq}\tau_{b}^{s}-s_{bv}^{qs}\tau_{a}^{r}-s_{av}^{rs}\tau_{b}^{q}-s_{bv}^{pr}\tau_{a}^{s}-s_{bv}^{sp}\tau_{a}^{r}+s_{sp}^{av}\tau_{b}^{r}-s_{rs}^{va}\tau_{b}^{p}+s_{bv}^{rp}\tau_{a}^{s}$
$\displaystyle-$ $\displaystyle
s_{av}^{rp}\tau_{b}^{s})+\sum_{cbr}g_{cbar}[s_{bv}^{pr}\tau_{c}^{q}-s_{cv}^{qp}\tau_{b}^{r}+s_{cv}^{rp}\tau_{b}^{q}+s_{bv}^{qp}\tau_{c}^{r}-s_{bv}^{rp}\tau_{c}^{q}+s_{vc}^{rq}\tau_{b}^{p}]+\sum_{bcrs}g_{bcrs}\left[s_{cv}^{sp}\tau_{ab}^{qr}+s_{cv}^{rp}\tau_{ab}^{qs}-s_{cv}^{sp}\tau_{ab}^{rq}+s_{cv}^{qp}\tau_{ab}^{rs}\right.$
$\displaystyle+$
$\displaystyle\left.s_{av}^{sp}\tau_{bc}^{qr}-s_{cv}^{ps}\tau_{ab}^{qr}+s_{cv}^{rs}\tau_{ab}^{qp}+s_{vc}^{rp}\tau_{ab}^{qs}+s_{vc}^{sp}\tau_{ab}^{rq}-s_{vc}^{rs}\tau_{ab}^{qp}-s_{av}^{sp}\tau_{bc}^{rq}-s_{cv}^{qp}\tau_{ba}^{rs}\right]$
$\displaystyle\langle\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.37514pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.73938pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.677pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.677pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=19.5491pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=16.46681pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=41.80305pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=41.80305pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=29.53955pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=25.06316pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}$ $\displaystyle H_{N}$ $\displaystyle
T^{(0)}\mathbf{T}^{(1)}S^{(0)}\rangle_{va}^{pq}+\langle\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=13.28879pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=9.37514pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
width=7.73938pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.677pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=27.677pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=19.5491pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=16.46681pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=41.80305pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=41.80305pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=29.53955pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt\vrule
width=25.06316pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=5.16663pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}H_{N}T^{(0)}\mathbf{T}^{(1)}S^{(0)}\rangle_{va}^{pq}=\sum_{brs}g_{bprs}\left[(-t_{a}^{r}\tau_{b}^{q}-t_{b}^{q}\tau_{a}^{r})s_{v}^{s}-(t_{a}^{s}\tau_{b}^{p}+t_{b}^{p}\tau_{a}^{s})s_{v}^{r}\right]+\sum_{brc}g_{bcar}(t_{b}^{q}\tau_{c}^{p}$
(22) $\displaystyle+$ $\displaystyle
t_{c}^{p}\tau_{b}^{q})s_{v}^{r}+\sum_{brcs}g_{bcrs}\left[(-t_{ab}^{qr}\tau_{c}^{p}+t_{ab}^{qp}\tau_{c}^{r})s_{v}^{s}+(-t_{ab}^{qp}\tau_{c}^{s}+t_{ab}^{qs}\tau_{c}^{p})s_{v}^{r}+(t_{ab}^{rq}\tau_{c}^{p}+t_{ac}^{rp}\tau_{b}^{q}+t_{bc}^{qp}\tau_{a}^{r})s_{v}^{s}+(\tau_{ab}^{qs}t_{c}^{p}-\tau_{ab}^{qr}t_{c}^{p}\right.$
$\displaystyle+$
$\displaystyle\left.\tau_{ab}^{qp}t_{c}^{r}-\tau_{ac}^{qp}t_{b}^{r}+\tau_{ac}^{rp}t_{b}^{q}+\tau_{bc}^{qp}t_{a}^{r}+\tau_{ab}^{rq}t_{c}^{p})s_{v}^{s}+[-t_{b}^{r}\tau_{c}^{p}-t_{c}^{p}\tau_{b}^{r}+t_{b}^{p}\tau_{c}^{r}+\tau_{b}^{p}t_{c}^{r}]s_{va}^{sq}+(-t_{b}^{q}\tau_{a}^{r}-\tau_{b}^{q}t_{a}^{r})s_{cv}^{sp}+(t_{c}^{r}\tau_{b}^{q}\right.$
$\displaystyle+$
$\displaystyle\left.\tau_{c}^{r}t_{b}^{q})s_{av}^{sp}+(t_{b}^{s}\tau_{a}^{r}+\tau_{b}^{s}t_{a}^{r})s_{cv}^{qp}+(t_{a}^{r}\tau_{b}^{q}+\tau_{a}^{r}t_{b}^{q})s_{cv}^{ps}+(t_{a}^{r}\tau_{c}^{p}+\tau_{a}^{r}t_{c}^{p})s_{bv}^{qs}+(t_{b}^{q}\tau_{c}^{p}+\tau_{b}^{q}t_{c}^{p})s_{av}^{rs}\right]$

In the case of the PRCC equation for $\mathbf{S}_{2}^{(1)}$, there are 72
Goldstone diagrams arise from this term. Out of these, 36 diagrams each arise
from the first and second terms. These diagrams are shown in the left and
right panels, respectively, of the Fig. 7. The algebraic expression of the
diagrams in the left panel is given in Eq.(21). Similarly, the algebraic
expression of the diagrams in the right panel is given in Eq.(22).

Figure 6: Single PRCC diagrams contributing to the terms
$\mathchoice{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 2.28123pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}\mathchoice{\vbox{\hbox to0.0pt{\kern 0.0pt\kern
4.56248pt\hbox{\vrule width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
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width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt\vrule
width=19.36559pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=3.44444pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}{\bar{H}}_{N}\mathbf{T}^{(1)}S^{(0)}$ (panel (a)) and
$\mathchoice{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
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width=13.02663pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 4.56248pt\hbox{\vrule
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width=13.02663pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
6.83331pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 3.19374pt\hbox{\vrule
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width=9.19162pt,height=0.0pt,depth=0.50003pt\vrule
width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
4.78333pt}}{\vbox{\hbox to0.0pt{\kern 5.00002pt\kern 2.28123pt\hbox{\vrule
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width=0.50003pt,height=0.0pt,depth=1.72218pt}\hss}\vskip 2.15277pt\vskip
3.41666pt}}{\bar{H}}_{N}\mathbf{S}^{(1)}$ (panel (b)) of Eq. (11a). Figure 7:
Double PRCC diagrams contributing to the terms $\mathchoice{\vbox{\hbox
to0.0pt{\kern 0.0pt\kern 4.56248pt\hbox{\vrule
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6.83331pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 3.19374pt\hbox{\vrule
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4.78333pt}}{\vbox{\hbox to0.0pt{\kern 0.0pt\kern 2.28123pt\hbox{\vrule
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3.41666pt}}H_{N}T^{(0)}\mathbf{T}^{(1)}S^{(0)}$ (right panel) of Eq. (11b).

####  II.4.5  ${\bar{H}}_{N}\mathbf{S}^{(1)}$

This term contains a perturbed operator, $\mathbf{S}^{(1)}$, which subsumes
dominant effects of perturbation for one-valence atomic systems. Expanding
${\bar{H}}_{N}$

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