# Number conservation in odd-particle number random phase approximation and
extensions

Mitsuru Tohyama  Faculty of Medicine, Kyorin University, Mitaka, Tokyo
181-8611, Japan

###### Abstract

The number conservation law in the odd-particle number random-phase
approximation (oRPA) and its extension (EoRPA) is studied by applying them to
a pairing model and 16O. It is found in the application to 16O that the number
conservation law is not fulfilled in oRPA and EoRPA and that it is drastically
improved in EoRPA due to the inclusion of ground-state correlation effects.

###### pacs:

21.60.Jz

##  I Introduction

The standard approach to study the single-particle properties of atomic nuclei
may be the Green’s function method RS ; s21 . Various theoretical approaches
have been proposed to implement the coupling to low-lying collective modes
into the self-energy of the Green’s function: For example, the particle-
vibration coupling (PVC) models Giai ; bortignon ; litv ; colo , the Tamm-
Dancoff approximation (TDA) hrpa96  and the Faddeev random-phase approximation
(FRPA) dick02 . The odd-particle number random-phase approximation (oRPA) and
its extension (EoRPA) have also been used to study the single-particle
properties of 16O ts13 . oRPA and EoRPA provide us alternative approaches to
calculate the single-particle Green’s function: The single-particle Green’s
function is expressed by using the eigenvalues and the transition amplitudes
in oRPA and EoRPA. In the Green’s function method the occupation probabilities
of the single-particle states are obtained from the residues of its poles RS .
The conservation law of the total number of particles requires that the number
of particles depleted from the single-particle states below the Fermi level be
the same as the number of particles occupying the single-particle states above
the Fermi level. This is also known as the Luttinger theorem lutt2  which has
been attracting much theoretical and experimental interests in the fields of
condensed-matter physics heath  and atomic physics sogo ; urban ; pieri ;
durel . The proof of this theorem is given in Refs. lutt2 ; heath  for the
exact theory and a question is what happens to it when approximations are made
in realistic calculations where diagrams are partially summed up in any case.
This problem has been addressed in TDA hrpa96  and FRPA dick02  for finite
nuclei. It has also been discussed for nuclear matter cases ramos ; mahaux .
In this paper the number conservation law is studied for oRPA and EoRPA. EoRPA
is formulated based a correlated ground state given by the time-dependent
density-matrix theory (TDDM) toh20 . It is pointed out that the neglect of the
coupling to higher amplitudes in oRPA and EoRPA inevitably brings about the
violation of the Luttinger theorem and it is demonstrated that the effects of
ground-state correlations included in EoRPA drastically improves the number
conservation law. The paper is organized as follows: The formulation of EoRPA
is given in Sect. II, the properties of EoRPA and its relations to other
approaches are discussed in Sect. III and the applications of oRPA and EoRPA
to a pairing model rich  and 16O are presented in Sect.IV. Section V is
devoted to summary.

##  II Formulation

Let us consider a nucleus consisting of $N$ nucleons and assume that the total
Hamiltonian $H$ consists of the kinetic energy term and a two-body
interaction. Let us assume that $|0\rangle$ is the ground state of the $N$
nucleon system with even $N$ and energy $E_{0}$ and that $|\mu\rangle$ is an
exact eigenstate of the Hamiltonian for the $N-1$ system with an eigenvalue
$E_{\mu}$ ($H|\mu\rangle=E_{\mu}|\mu\rangle$).

###  II.1 Single-paticle Green’s function and transition amplitudes

First it is explained that the single-particle Green’s function is expressed
with the transition amplitudes $x^{\mu}_{\alpha}=\langle
0|a^{+}_{\alpha}|\mu\rangle$ and
$\chi^{\mu}_{\alpha}=\langle\mu|a_{\alpha}|0\rangle$, where
$a^{+}_{\alpha}~{}(a_{\alpha})$ is the creation (annihilation) operator of a
nucleon at a single-particle state $\alpha$. In the exact case
$x^{\mu}_{\alpha}=(\chi^{\mu}_{\alpha})^{*}$ holds. The Green’s function
$G_{\alpha\alpha^{\prime}}(t-t^{\prime})$ for time $t>t^{\prime}$ is given by

$\displaystyle G_{\alpha\alpha^{\prime}}(t-t^{\prime})=i\langle
0|a^{+}_{\alpha^{\prime}}(t)a_{\alpha}(t^{\prime})|0\rangle,$ (1)

where $a^{+}_{\alpha}(t)$ is $e^{iHt}a^{+}_{\alpha}e^{-iHt}$. Units such as
$\hbar=1$ are used hereafter. The Green’s function is written with
$x^{\mu}_{\alpha}$ and $\chi^{\mu}_{\alpha}$ as

$\displaystyle G_{\alpha\alpha^{\prime}}(t-t^{\prime})$ $\displaystyle=$
$\displaystyle i\langle
0|a^{+}_{\alpha^{\prime}}(t)a_{\alpha}(t^{\prime})|0\rangle$ (2)
$\displaystyle=$ $\displaystyle i\sum_{\mu}\langle
0|e^{iHt}a^{+}_{\alpha^{\prime}}e^{-iHt}|\mu\rangle\langle\mu|e^{iHt^{\prime}}a_{\alpha}e^{-iHt^{\prime}}|0\rangle$
$\displaystyle=$ $\displaystyle
i\sum_{\mu}x^{\mu}_{\alpha^{\prime}}\chi^{\mu}_{\alpha}e^{i\omega_{\mu}(t-t^{\prime})},$

where $\omega_{\mu}=E_{0}-E_{\mu}$. The occupation matrix
$n_{\alpha\alpha^{\prime}}$ is given by RS

$\displaystyle n_{\alpha\alpha^{\prime}}=-i\lim_{t\rightarrow
t^{\prime}}G_{\alpha\alpha^{\prime}}(t-t^{\prime})=\sum_{\mu,\alpha}x^{\mu}_{\alpha}\chi^{\mu}_{\alpha},$
(3)

and in the exact case $\sum_{\alpha}n_{\alpha\alpha}$ gives $N$.

###  II.2 Time-dependent density-matrix theory and number conservation

As mentioned above, EoRPA is formulated based on the TDDM ground state. In the
following it is shown that TDDM conserves the total number of particles. TDDM
consists of the coupled equations of motion for the occupation matrix
$n_{\alpha\alpha^{\prime}}$ and the two-body correlation matrix
$C_{\alpha\beta\alpha^{\prime}\beta^{\prime}}(C_{2})$ defined by

$\displaystyle n_{\alpha\alpha^{\prime}}$ $\displaystyle=$
$\displaystyle\langle 0|a^{+}_{\alpha^{\prime}}a_{\alpha}|0\rangle,$ (4)
$\displaystyle C_{\alpha\beta\alpha^{\prime}\beta^{\prime}}$ $\displaystyle=$
$\displaystyle\langle
0|:a^{+}_{\alpha^{\prime}}a^{+}_{\beta^{\prime}}a_{\beta}a_{\alpha}:|0\rangle,$
(5)

where $:~{}:$ implies that lower-order operators are subtracted such that

$\displaystyle:a^{+}_{\alpha^{\prime}}a^{+}_{\beta^{\prime}}a_{\beta}a_{\alpha}:$
$\displaystyle=$ $\displaystyle
a^{+}_{\alpha^{\prime}}a^{+}_{\beta^{\prime}}a_{\beta}a_{\alpha}-{\cal
A}(n_{\alpha\alpha^{\prime}}n_{\beta\beta^{\prime}})$ (6) $\displaystyle-$
$\displaystyle{\cal
AS}(n_{\alpha\alpha^{\prime}}a^{+}_{\beta^{\prime}}a_{\beta}).$

Here, ${\cal A}()$ (${\cal S}()$) antisymmetrizes (symmetrizes) the quantities
in parentheses under the exchange of the single-particle indices. The TDDM
equations have been used to calculate the correlated ground states of closed-
shell nuclei using an adiabatic method toh20 . The TDDM equations conserve the
total number of particles in the adiabatic method, where the residual
interaction $V$ gradually increases as a function of time $t$ such that
$H(t)=H_{0}+t/T\times V$. Here it is assumed that $T$ is large and that the
correlated ground state wavefunction is given by
$|\Psi_{0}(t)\rangle=\exp[-i\int_{0}^{t}Hdt]|\Psi_{0}\rangle$, where
$|\Psi_{0}\rangle$ is the ground state of $H_{0}$ (the Hartree-Fock (HF)
ground state is usually used for $|\Psi_{0}\rangle$). This adiabatic method is
based on the Gell-mann-Low theorem gell . The equation of motion for
$n_{\alpha\alpha^{\prime}}$ is obtained by replacing $|0\rangle$ in Eqs. (4)
and (5) with $|\Psi_{0}(t)\rangle$

$\displaystyle i\frac{dn_{\alpha\alpha^{\prime}}}{dt}$ $\displaystyle=$
$\displaystyle\langle\Psi_{0}(t)|[a^{+}_{\alpha^{\prime}}a_{\alpha},H]|\Psi_{0}(t)\rangle$
(7) $\displaystyle=$
$\displaystyle(\epsilon_{\alpha}(t)-\epsilon_{\alpha^{\prime}}(t))n_{\alpha\alpha^{\prime}}(t)$
$\displaystyle+$
$\displaystyle\sum_{\lambda_{1}\lambda_{2}\lambda_{3}}[\langle\alpha\lambda_{3}|v(t)|\lambda_{1}\lambda_{2}\rangle
C_{\lambda_{1}\lambda_{2}\alpha^{\prime}\lambda_{3}}(t)$ $\displaystyle-$
$\displaystyle
C_{\alpha\lambda_{3}\lambda_{1}\lambda_{2}}(t)\langle\lambda_{1}\lambda_{2}|v(t)|\alpha^{\prime}\lambda_{3}\rangle],$

where $v(t)=t/T\times V$ and the single-particle energy matrix

$\displaystyle\epsilon_{\alpha\alpha^{\prime}}=\epsilon^{0}_{\alpha}\delta_{\alpha\alpha^{\prime}}+\sum_{\lambda\lambda^{\prime}}\langle\alpha\lambda|v(t)|\alpha^{\prime}\lambda^{\prime}\rangle_{A}n_{\lambda^{\prime}\lambda}$
(8)

is assumed diagonal to simplify the equation. Here, $\epsilon^{0}_{\alpha}$ is
the HF single-particle energy. The equation of motion for $C_{2}$ couples to
the correlated part ($C_{3}$) of a three-body density matrix. To obtain a
closed set of the equations for $n_{\alpha\alpha^{\prime}}$ and $C_{2}$,
$C_{3}$ is usually approximated by the squares of $C_{2}$ toh20 . As easily
seen from Eq. (7), the total number of particles
$N=\sum_{\alpha}n_{\alpha\alpha}(t)$ is conserved, that is,
$dN/dt=\sum_{\mu}dn_{\alpha\alpha}/dt=0$ because the second term on the right-
hand side of Eq. (7) vanishes when the sum is taken over all single-particle
indices. Thus TDDM fulfills the number conservation low regardless of the
approximations for $C_{3}$.

###  II.3 Equations for transition amplitudes and total number conservation

In oRPA and EoRPA the single-fermion transition amplitudes are used to
calculated the occupation probabilities of the single-particle states. By
using the above time-dependent Hamiltonian $H(t)$ the conditions that the
transition amplitudes should satisfy to fulfill the number conservation law
are discussed below. For this purpose the time-dependent transition amplitudes
for $H(t)$ are considered

$\displaystyle x^{\mu}_{\alpha}(t)$ $\displaystyle=$
$\displaystyle\langle\Psi_{0}(t)|a^{+}_{\alpha}|\Psi_{\mu}(t)\rangle,$ (9)
$\displaystyle X^{\mu}_{\alpha\beta:\gamma}(t)$ $\displaystyle=$
$\displaystyle\langle\Psi_{0}(t)|:a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}:|\Psi_{\mu}(t)\rangle,$
(10)

and

$\displaystyle\chi^{\mu}_{\alpha}(t)$ $\displaystyle=$
$\displaystyle\langle\Psi_{\mu}(t)|a_{\alpha}|\Psi_{0}(t)\rangle,$ (11)
$\displaystyle{\cal X}^{\mu}_{\alpha\beta:\gamma}(t)$ $\displaystyle=$
$\displaystyle\langle\Psi_{\mu}(t)|:a^{+}_{\gamma}a_{\beta}a_{\alpha}:|\Psi_{0}(t)\rangle,$
(12)

where $|\Psi_{\mu}(t)\rangle=\exp[-i\int_{0}^{t}Hdt]|\Psi_{\mu}(0)\rangle$.
Here, $|\Psi_{\mu}(0)\rangle$ is the eigenstate of $H_{0}$ with the number
$N-1$. The time derivatives of $x^{\mu}_{\alpha}(t)$ and
$\chi^{\mu}_{\alpha}(t)$ give

$\displaystyle i\frac{dx^{\mu}_{\alpha}}{dt}$ $\displaystyle=$
$\displaystyle\langle\Psi_{0}(t)|[a^{+}_{\alpha},H]|\Psi_{\mu}(t)\rangle$ (13)
$\displaystyle=$ $\displaystyle-\epsilon_{\alpha}(t)x^{\mu}_{\alpha}(t)$
$\displaystyle+$
$\displaystyle\sum_{\lambda_{1}\lambda_{2}\lambda_{3}}\langle\lambda_{1}\lambda_{2}|v(t)|\alpha\lambda_{3}\rangle
X^{\mu}_{\lambda_{1}\lambda_{2}:\lambda_{3}}(t),$ $\displaystyle
i\frac{d\chi^{\mu}_{\alpha}}{dt}$ $\displaystyle=$
$\displaystyle\langle\Psi_{\mu}(t)|[a_{\alpha},H]|\Psi_{0}(t)\rangle$ (14)
$\displaystyle=$ $\displaystyle\epsilon_{\alpha}(t)\chi^{\mu}_{\alpha}(t)$
$\displaystyle-$
$\displaystyle\sum_{\lambda_{1}\lambda_{2}\lambda_{3}}\langle\alpha\lambda_{3}|v(t)|\lambda_{1}\lambda_{2}\rangle{\cal
X}^{\mu}_{\lambda_{1}\lambda_{2}:\lambda_{3}}(t).$

The occupation matrix is defined by

$\displaystyle n_{\alpha\alpha^{\prime}}(t)$ $\displaystyle=$
$\displaystyle\langle\Psi_{0}(t)|a^{+}_{\alpha^{\prime}}a_{\alpha}|\Psi_{0}(t)\rangle$
(15) $\displaystyle=$
$\displaystyle\sum_{\mu}\langle\Psi_{0}(t)|a^{+}_{\alpha^{\prime}}|\Psi_{\mu}(t)\rangle\langle\Psi_{\mu}(t)|a_{\alpha}|\Psi_{0}(t)\rangle$
$\displaystyle=$
$\displaystyle\sum_{\mu}x^{\mu}_{\alpha^{\prime}}(t)\chi^{\mu}_{\alpha}(t).$

The time derivative of the total number $N=\sum_{\alpha}n_{\alpha\alpha}$ is
given by

$\displaystyle i\frac{dN}{dt}$ $\displaystyle=i$
$\displaystyle\sum_{\alpha}\left(\frac{dx^{\mu}_{\alpha}}{dt}\chi^{\mu}_{\alpha}+x^{\mu}_{\alpha}\frac{d\chi^{\mu}_{\alpha}}{dt}\right)$
(16) $\displaystyle=$
$\displaystyle\sum_{\mu\alpha\lambda_{1}\lambda_{2}\lambda_{3}}[\langle\lambda_{1}\lambda_{2}|v|\alpha\lambda_{3}\rangle
X^{\mu}_{\lambda_{1}\lambda_{2}:\lambda_{3}}\chi^{\mu}_{\alpha}$
$\displaystyle-$
$\displaystyle\langle\alpha\lambda_{3}|v|\lambda_{1}\lambda_{2}\rangle
x^{\mu}_{\alpha}{\cal X}^{\mu}_{\lambda_{1}\lambda_{2}:\lambda_{3}}],$

where the notation $(t)$ is omitted for simplicity. From Eq. (16) it is found
that the necessary condition for number conservation is

$\displaystyle\sum_{\mu}X^{\mu}_{\alpha^{\prime}\beta^{\prime}:\beta}\chi^{\mu}_{\alpha}=\sum_{\mu}x^{\mu}_{\alpha^{\prime}}{\cal
X}^{\mu}_{\alpha\beta:\beta^{\prime}}.$ (17)

The quantities on the both sides of the above equation correspond to
$C_{\alpha\beta\alpha^{\prime}\beta^{\prime}}$ in Eq. (7). Equation (17)
requires that $C_{\alpha\beta\alpha^{\prime}\beta^{\prime}}$ consisting of the
one-fermion and three-fermion transition amplitudes should be independent of
how it is composed. When the conditions

$\displaystyle\chi_{\alpha}^{\mu}=(x_{\alpha}^{\mu})^{*}$ (18)

and

$\displaystyle{\cal
X}_{\alpha\beta:\gamma}^{\mu}=(X_{\alpha\beta:\gamma}^{\mu})^{*}$ (19)

hold, Eq. (17) is expressed with $x^{\mu}_{\alpha}$ and
$X^{\mu}_{\alpha^{\prime}\beta^{\prime}:\beta}$ as

$\displaystyle\sum_{\mu}X^{\mu}_{\alpha^{\prime}\beta^{\prime}:\beta}(x^{\mu}_{\alpha})^{*}=\sum_{\mu}x^{\mu}_{\alpha^{\prime}}({X}^{\mu}_{\alpha\beta:\beta^{\prime}})^{*}.$
(20)

This is the relation that is satisfied by exact solutions. As will be
discussed below, Eqs. (18) and (19) are satisfied both in oRPA and in EoRPA.
It will be also pointed out that the neglect of higher amplitudes in the
calculations of $X_{\alpha\beta:\gamma}^{\mu}$ causes the violation of Eq.
(20).

###  II.4 Equations for transition amplitudes I

Now the equations for the transition amplitudes $x^{\mu}_{\alpha}$ and
$X^{\mu}_{\alpha\beta:\gamma}$ that have been used in realistic calculations
ts13  are presented. Here the time-independent Hamiltonian $H$ consisting of
the HF part $H_{0}$ and the residual interaction $v$ is considered. The
transition amplitudes are defined by

$\displaystyle x^{\mu}_{\alpha}$ $\displaystyle=$ $\displaystyle\langle
0|a^{+}_{\alpha}|\mu\rangle,$ (21) $\displaystyle
X^{\mu}_{\alpha\beta:\gamma}$ $\displaystyle=$ $\displaystyle\langle
0|:a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}:|\mu\rangle,$ (22)

where $|0\rangle$ and $|\mu\rangle$ are the ground and excited states of $H$,
respectively. The occupation matrix $n_{\alpha\alpha^{\prime}}$ is given by

$\displaystyle n_{\alpha\alpha^{\prime}}=\langle
0|a^{+}_{\alpha^{\prime}}a_{\alpha}|0\rangle.$ (23)

From the equation-of-motion (EoM) relation ts13

$\displaystyle\langle 0|[H,a^{+}_{\alpha}]=\langle 0|a^{+}_{\alpha}(E_{0}-H),$
(24)

the equation for $x^{\mu}_{\alpha}$ is obtained as

$\displaystyle\langle 0|[H,a^{+}_{\alpha}]|\mu\rangle$ $\displaystyle=$
$\displaystyle\omega_{\mu}\langle
0|a^{+}_{\alpha}|\mu\rangle=\omega_{\mu}x^{\mu}_{\alpha},$ (25)

where $\omega_{\mu}=E_{0}-E_{\mu}$. Here, $E_{0}$ and $E_{\mu}$ are
respectively the energies of $|0\rangle$ and $\mu\rangle$. The commutator on
the left-hand side of the above equation includes terms with
$a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}$. Therefore, $x^{\mu}_{\alpha}$ couples
to $X^{\mu}_{\alpha\beta:\gamma}$. In a way analogous to that used in deriving
Eq. (25) the equation for $X^{\mu}_{\alpha\beta:\gamma}$ is given as

$\displaystyle\langle
0|[H,:a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}:]|\mu\rangle$ $\displaystyle=$
$\displaystyle\omega_{\mu}\langle
0|:a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}:|\mu\rangle$ (26) $\displaystyle=$
$\displaystyle\omega_{\mu}X^{\mu}_{\alpha\beta:\gamma}.$

When the coupling to higher amplitudes is neglected to close the equations
within $x^{\mu}_{\alpha}$ and $X^{\mu}_{\alpha\beta:\gamma}$, Eqs. (25) and
(26) are written in matrix form ts13 :

$\displaystyle\left(\begin{array}[]{cc}a&c\\\
b&d\end{array}\right)\left(\begin{array}[]{c}x^{\mu}\\\
X^{\mu}\end{array}\right)=\omega_{\mu}\left(\begin{array}[]{c}x^{\mu}\\\
X^{\mu}\end{array}\right),$ (33)

where the single-particle indices in $x^{\mu}_{\alpha}$ and
$X^{\mu}_{\alpha\beta:\gamma}$ are omitted for simplicity. The matrices $a$,
$b$, $c$ and $d$ are given in Appendix A. The effects of ground-state
correlations are included in $b$ and $d$ through $n_{\alpha}$ and $C_{2}$. For
later use the equation for $x^{\mu}_{\alpha}$ is explicitly given

$\displaystyle(\epsilon_{\alpha}-\omega_{\mu})x^{\mu}_{\alpha}$
$\displaystyle+$
$\displaystyle\sum_{\lambda_{1}\lambda_{2}\lambda_{3}}\langle\lambda_{1}\lambda_{2}|v|\alpha\lambda_{3}\rangle
X^{\mu}_{\lambda_{1}\lambda_{2}:\lambda_{3}}=0.$ (34)

In a similar way the equations for the transition amplitudes

$\displaystyle\chi^{\mu}_{\alpha}$ $\displaystyle=$
$\displaystyle\langle\mu|a_{\alpha}|0\rangle,$ (35) $\displaystyle{\cal
X}^{\mu}_{\alpha\beta:\gamma}$ $\displaystyle=$
$\displaystyle\langle\mu|:a^{+}_{\gamma}a_{\beta}a_{\alpha}:|0\rangle$ (36)

can be derived. Since the Hamiltonian matrix of Eq. (33) is not hermitian, the
eigenvalues of Eq. (33) can be complex. For these complex solutions the
relation $(\chi^{\mu},{\cal X}^{\mu})=((x^{\mu})^{*},(X^{\mu})^{*})$ does not
hold.

###  II.5 Equations for transition amplitudes II: Equation of motion approach
with an excitation operator

Equation (33) lacks some effects of higher configurations such as self-energy
contributions in the configuration $X^{\mu}_{\alpha\beta:\gamma}$, which
should be included when a correlated ground state is used. Another formulation
which is based on EoM Rowe ; s21  is presented to take account of such
effects. By introducing the excitation operator $q^{+}_{\mu}$

$\displaystyle
q^{+}_{\mu}=\sum_{\alpha}y_{\alpha}^{\mu}~{}a_{\alpha}+\sum_{\alpha\beta\gamma}Y_{\alpha\beta:\gamma}^{\mu}:a^{+}_{\gamma}a_{\beta}a_{\alpha}:$
(37)

and assuming, as usual, $q^{+}_{\mu}|0\rangle=|\mu\rangle$ and
$q_{\mu}|0\rangle=0$, the equations for $y^{\mu}$ and $Y^{\mu}$ are obtained
from Eqs. (25) and (26). They are written in matrix form as

$\displaystyle\left(\begin{array}[]{cc}A&C\\\
B&D\end{array}\right)\left(\begin{array}[]{c}y^{\mu}\\\
Y^{\mu}\end{array}\right)=\omega_{\mu}\left(\begin{array}[]{cc}N_{11}&N_{12}\\\
N_{21}&N_{22}\end{array}\right)\left(\begin{array}[]{c}y^{\mu}\\\
Y^{\mu}\end{array}\right),$ (46)

where the matrices are defined as

$\displaystyle A(\alpha:\alpha^{\prime})$ $\displaystyle=$
$\displaystyle\langle
0|\\{[H,a^{+}_{\alpha}],a_{\alpha^{\prime}}\\}|0\rangle,$ (47) $\displaystyle
B(\alpha\beta\gamma:\alpha^{\prime})$ $\displaystyle=$ $\displaystyle\langle
0|\\{[H,:a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}:],a_{\alpha^{\prime}}\\}|0\rangle,$
(48) $\displaystyle C(\alpha:\alpha^{\prime}\beta^{\prime}\gamma^{\prime})$
$\displaystyle=$ $\displaystyle\langle
0|\\{[H,a^{+}_{\alpha}:],:a^{+}_{\gamma^{\prime}}a_{\beta^{\prime}}a_{\alpha^{\prime}}\\}|0\rangle,$
(49) $\displaystyle
D(\alpha\beta\gamma:\alpha^{\prime}\beta^{\prime}\gamma^{\prime}$
$\displaystyle)$ $\displaystyle=\langle 0|\\{[H$ ,
$\displaystyle:a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}:],:a^{+}_{\gamma^{\prime}}a_{\beta^{\prime}}a_{\alpha^{\prime}}:\\}|0\rangle,$
(50) $\displaystyle N_{11}(\alpha:\alpha^{\prime})$ $\displaystyle=$
$\displaystyle\langle
0|\\{a^{+}_{\alpha},a_{\alpha^{\prime}}\\}|0\rangle=\delta_{\alpha\alpha^{\prime}},$
(51) $\displaystyle N_{12}(\alpha:\alpha^{\prime}\beta^{\prime}\gamma)$
$\displaystyle=$ $\displaystyle\langle
0|\\{a^{+}_{\alpha},:a^{+}_{\gamma^{\prime}}a_{\beta^{\prime}}a_{\alpha^{\prime}}:\\}|0\rangle=0,$
(52) $\displaystyle N_{21}(\alpha\beta\gamma:\alpha^{\prime})$
$\displaystyle=$ $\displaystyle\langle
0|\\{:a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}:,a_{\alpha^{\prime}}\\}|0\rangle=0,$
(53) $\displaystyle
N_{22}(\alpha\beta\gamma:\alpha^{\prime}\beta^{\prime}\gamma^{\prime}$
$\displaystyle)$ $\displaystyle=\langle
0|\\{:a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}:$ ,
$\displaystyle:a^{+}_{\gamma^{\prime}}a_{\beta^{\prime}}a_{\alpha^{\prime}}:\\}|0\rangle.$
(54)

Here, $\\{~{}\\}$ implies the anticommutator, $\\{A,B\\}=AB+BA$. The norm
matrix $N_{22}$ is given in Appendix B. The matrix elements in Eq. (46) can be
expressed by using those in Eq. (33) such as

$\displaystyle A=a\times N_{11}$ (55) $\displaystyle B=b\times N_{11},$ (56)
$\displaystyle C=c\times N_{22}.$ (57)

The matrix $D$ consists of the terms of two types, one expressed by
$D_{1}=d\times N_{22}$ and the other given by $D_{2}$, which originates from
the terms with five fermion operator
$:a^{+}_{\lambda_{5}}a^{+}_{\lambda_{4}}a^{+}_{\lambda_{3}}a_{\lambda_{2}}a_{\lambda_{1}}:$
in $[H,:a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}:]$:

$\displaystyle[$ $\displaystyle
H,:a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}:]=\sum_{\alpha^{\prime}}b(\alpha\beta\gamma:\alpha^{\prime})a^{+}_{\alpha^{\prime}}$
(58) $\displaystyle+$
$\displaystyle\sum_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}d(\alpha\beta\gamma:\alpha^{\prime}\beta^{\prime}\gamma^{\prime}):a^{+}_{\alpha^{\prime}}a^{+}_{\beta^{\prime}}a_{\gamma^{\prime}}:$
$\displaystyle+$
$\displaystyle\sum_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}\lambda_{5}}e(\alpha\beta\gamma:\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}\lambda_{5})$
$\displaystyle\times$
$\displaystyle:a^{+}_{\lambda_{1}}a^{+}_{\lambda_{2}}a^{+}_{\lambda_{3}}a_{\lambda_{5}}a_{\lambda_{4}}:.$

Using an additional norm matrix

$\displaystyle N_{32}$ $\displaystyle($
$\displaystyle\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}\lambda_{5}:\alpha\beta\gamma)$
(59) $\displaystyle=$ $\displaystyle\langle
0|\\{:a^{+}_{\lambda_{1}}a^{+}_{\lambda_{2}}a^{+}_{\lambda_{3}}a_{\lambda_{5}}a_{\lambda_{4}}:,:a^{+}_{\gamma}a_{\beta}a_{\alpha}:\\}|0\rangle,$

$D_{2}$ can be expressed as $e\times N_{32}$. Thus $D_{2}$ expresses the
coupling to five fermion configurations and includes the self-energy
contributions to the three-fermion configurations in
$Y_{\alpha\beta:\gamma}^{\mu}$ ts13 . In the numerical applications
$|0\rangle$ is approximated by the TDDM ground state and the three-body
correlation matrix $C_{3}$ contained in $N_{32}$ is expressed by the squares
of $C_{2}$ toh20 .

The transition amplitudes are given by

$\displaystyle\left(\begin{array}[]{c}x^{\mu}\\\
X^{\mu}\end{array}\right)=\left(\begin{array}[]{cc}N_{11}&N_{12}\\\
N_{21}&N_{22}\end{array}\right)\left(\begin{array}[]{c}y^{\mu}\\\
Y^{\mu}\end{array}\right).$ (66)

####  II.5.1 Symmetry properties

Here it is shown that the Hamiltonian matrix of Eq. (46) is hermitian. For
this purpose the operator identity

$\displaystyle\langle 0|\\{[H,\hat{A}],\hat{B}\\}|0\rangle$ $\displaystyle+$
$\displaystyle\langle 0|\\{[H,\hat{B}],\hat{A}\\}|0\rangle$ (67)
$\displaystyle=$ $\displaystyle\langle 0|[H,\\{\hat{A},\hat{B}\\}]|0\rangle$

is used. In the matrix $A$ of Eq. (46), the operators $\hat{A}$ and $\hat{B}$
are $a^{+}_{\alpha}$ and $a_{\alpha^{\prime}}$, respectively. Since
$\\{\hat{A},\hat{B}\\}$ is unity, the right-hand side of Eq. (67) vanishes,
which means $\langle 0|\\{[H,\hat{A}],\hat{B}\\}|0\rangle=-\langle
0|\\{[H,\hat{B}],\hat{A}\\}|0\rangle$ and

$\displaystyle A(\alpha:\alpha^{\prime})^{*}$ $\displaystyle=$
$\displaystyle-\langle
0|\\{[H,a_{\alpha}],a^{+}_{\alpha^{\prime}}\\}|0\rangle,$ (68)
$\displaystyle=$ $\displaystyle\langle
0|\\{[H,a^{+}_{\alpha^{\prime}}],a_{\alpha}\\}|0\rangle=A(\alpha^{\prime}:\alpha).$

In the case of the matrix $B$ in Eq. (46) $\hat{A}$ is
$:a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}:$ and $\hat{B}$ is
$a_{\alpha^{\prime}}$, and $\\{\hat{A},\hat{B}\\}$ is reduced to a one-body
operator. Due to the ground-state condition in TDDM

$\displaystyle i\dot{n}_{\alpha\alpha^{\prime}}$ $\displaystyle=$
$\displaystyle\langle 0|[a^{+}_{\alpha^{\prime}}a_{\alpha},H]|0\rangle=0,$
(69)

the right-hand side of Eq. (67) vanishes, which means $\langle
0|\\{[H,\hat{A}],\hat{B}\\}|0\rangle=-\langle
0|\\{[H,\hat{B}],\hat{A}\\}|0\rangle$ and

$\displaystyle B(\alpha\beta\gamma:\alpha^{\prime})^{*}$ $\displaystyle=$
$\displaystyle-\langle
0|\\{[H,:a^{+}_{\gamma}a_{\beta}a_{\alpha}:],a^{+}_{\alpha^{\prime}}\\}|0\rangle,$
(70) $\displaystyle=$ $\displaystyle\langle
0|\\{[H,a^{+}_{\alpha^{\prime}}],:a^{+}_{\gamma}a_{\beta}a_{\alpha}:\\}|0\rangle$
$\displaystyle=$ $\displaystyle C(\alpha^{\prime}:\alpha\beta\gamma).$

Similarly, for the matrix $D$ in Eq. (46) $\hat{A}$ is
$:a^{+}_{\alpha}a^{+}_{\beta}a_{\gamma}:$ and $\hat{B}$ is
$:a^{+}_{\gamma^{\prime}}a_{\beta^{\prime}}a_{\alpha^{\prime}}$, and
$\\{\hat{A},\hat{B}\\}$ is reduced to at most a two-body operator. Due to the
ground-state conditions Eq. (69) and

$\displaystyle i\dot{C}_{\alpha\beta\alpha^{\prime}\beta^{\prime}}$
$\displaystyle=$ $\displaystyle\langle
0|[:a^{+}_{\alpha^{\prime}}a^{+}_{\beta^{\prime}}a_{\beta}a_{\alpha}:,H]|0\rangle=0,$
(71)

the right-hand side of Eq. (67) vanishes, which implies

$\displaystyle
D(\alpha\beta\gamma:\alpha^{\prime}\beta^{\prime}\gamma^{\prime})^{*}=D(\alpha^{\prime}\beta^{\prime}\gamma^{\prime}:\alpha\beta\gamma).$
(72)

Therefore, the Hamiltonian matrix in Eq. (46) is hermitian. This means that
the left-hand eigenvectors $((\bar{y}^{\mu})^{*}~{}~{}(\bar{Y}^{\mu})^{*})$
are equivalent to $(({y}^{\mu})^{*}~{}~{}({Y}^{\mu})^{*})$ and that the right
and left transition amplitudes satisfy similar conjugate relations
$(\chi^{\mu}~{}~{}{\cal X}^{\mu})=((x^{\mu})^{*}~{}~{}(X^{\mu})^{*})$, leading
to

$\displaystyle\sum_{\mu}X^{\mu}_{\alpha^{\prime}\beta^{\prime}:\beta}\chi^{\mu}_{\alpha}$
$\displaystyle=$ $\displaystyle\sum_{\mu}(x^{\mu}_{\alpha}{\cal
X}^{\mu}_{\alpha^{\prime}\beta^{\prime}:\beta})^{*}.$ (73)

When the truncation of higher amplitudes is made in the calculations of
$X^{\mu}_{\alpha^{\prime}\beta^{\prime}:\beta}$ and ${\cal
X}^{\mu}_{\alpha\beta:\beta^{\prime}}$, however, the relation

$\displaystyle\sum_{\mu}(x^{\mu}_{\alpha}{\cal
X}^{\mu}_{\alpha^{\prime}\beta^{\prime}:\beta})^{*}=\sum_{\mu}x^{\mu}_{\alpha^{\prime}}{\cal
X}^{\mu}_{\alpha\beta:\beta^{\prime}}$ (74)

that implies symmetry under the exchange
$(\alpha,\beta)\leftrightarrow(\alpha^{\prime},\beta^{\prime})$ does not hold
exactly and, consequently, Eq. (17) is not completely fulfilled.

It has been pointed out in Ref. ts13  that the left-hand eigenvectors
$(({y}^{\mu})^{*}~{}~{}({Y}^{\mu})^{*})$ corresponds to the amplitudes for a
particle state with $N+1$. Thus Eq. (46) gives simultaneously the particle
states and the hole states. This is completely analogous to pp(hh)RPA (see
Ref. RS .)

####  II.5.2 Self energy

In the following it is shown that the self-energy of the Green’s function is
derived from the equation for the transition amplitude $x^{\mu}$. By using
$y^{\mu}=x^{\mu}$ and $Y^{\mu}=(N_{22})^{-1}X^{\mu}$ Eq. (46) for $Y^{\mu}$ is
expressed as

$\displaystyle
By^{\mu}+DY^{\mu}=bx^{\mu}+D(N_{22})^{-1}X^{\mu}=\omega_{\mu}X^{\mu}.$ (75)

This gives

$\displaystyle X^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\omega_{\mu}-D(N_{22})^{-1}}bx^{\mu}$ (76)
$\displaystyle=$
$\displaystyle\frac{1}{\omega_{\mu}-D(N_{22})^{-1}}N_{22}vx^{\mu},$

where $b=C^{*}=N_{22}v$ is used. By inserting this into Eq. (34) the equation
for $x^{\mu}$ is written as

$\displaystyle(\epsilon_{\alpha}-\omega_{\mu})x^{\mu}_{\alpha}+v\frac{1}{\omega_{\mu}-D(N_{22})^{-1}}N_{22}vx^{\mu}=0,$
(77)

where the sum over single-particle indices is implicit. The matrix

$\displaystyle
h(\omega)=\epsilon_{\alpha}+v\frac{1}{\omega-D(N_{22})^{-1}}N_{22}v,$ (78)

can be regarded as an effective Hamiltonian for $x^{\mu}$ and the second term
gives the self-energy of the Green’s function. $h(\omega)$ is hermitian in
EoRPA. An equation similar to Eq. (77) is used to obtain the eigenstates in
the Green’s function method based on the PVC approach litv .

####  II.5.3 Hartree-Fock approximation for the ground state

If the HF approximation is taken for the ground state $|0\rangle$, $N_{22}$ in
Eq. (54) becomes

$\displaystyle N_{22}(\alpha\beta\gamma$ $\displaystyle:$
$\displaystyle\alpha^{\prime}\beta^{\prime}\gamma^{\prime})=(\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}-\delta_{\alpha\beta^{\prime}}\delta_{\beta\alpha^{\prime}})\delta_{\gamma^{\prime}\gamma}$
(79) $\displaystyle\times$
$\displaystyle(n^{0}_{\gamma\gamma}+n^{0}_{\alpha\alpha}n^{0}_{\beta\beta}-n^{0}_{\gamma\gamma}n^{0}_{\alpha\alpha}-n^{0}_{\gamma\gamma}n^{0}_{\beta\beta}),$

where $n^{0}_{\alpha\alpha}$ is equal to 1 or 0. In HF, $N_{22}$ is non-
vanishing only for $Y_{\rm pp^{\prime}:h}^{\mu}$ and $Y_{\rm
hh^{\prime}:p}^{\mu}$. Here h and p refer to a hole state and a particle
state, respectively. These amplitudes $Y_{\rm pp^{\prime}:h}^{\mu}$ and
$Y_{\rm hh^{\prime}:p}^{\mu}$ correspond to the backward amplitudes of $y_{\rm
h}^{\mu}$ and $y_{\rm p}^{\mu}$, respectively. Hereafter this formulation
consisting of the four amplitudes, $y_{\rm h}^{\mu}$, $y_{\rm p}^{\mu}$,
$Y_{\rm pp^{\prime}:h}^{\mu}$ and $Y_{\rm hh^{\prime}:p}^{\mu}$ is referred to
as oRPA. For completeness the full expression of oRPA is given

$\displaystyle(\epsilon_{\alpha}$ $\displaystyle-$
$\displaystyle\omega_{\mu})y^{\mu}_{\alpha}$ (80) $\displaystyle+$
$\displaystyle\sum_{\rm pp^{\prime}h}\langle{\rm pp^{\prime}}|v|\alpha{\rm
h}\rangle{\tilde{Y}}^{\mu}_{\rm pp^{\prime}:h}$ $\displaystyle+$
$\displaystyle\sum_{\rm hh^{\prime}p}\langle{\rm hh^{\prime}}|v|\alpha{\rm
p}\rangle{\tilde{Y}}^{\mu}_{\rm hh^{\prime}:p}=0,$
$\displaystyle(\epsilon_{\rm p}+\epsilon_{\rm p^{\prime}}-\epsilon_{\rm h}$
$\displaystyle-$ $\displaystyle\omega_{\mu}){\tilde{Y}}^{\mu}_{\rm
pp^{\prime}:h}$ (81) $\displaystyle+$
$\displaystyle\sum_{\alpha}\langle\alpha{\rm h}|v|{\rm
pp^{\prime}}\rangle_{A}y^{\mu}_{\alpha}$ $\displaystyle+$
$\displaystyle\sum_{\rm p_{1}p_{2}}\langle{\rm p_{1}p_{2}}|v|{\rm
pp^{\prime}}\rangle{\tilde{Y}}^{\mu}_{\rm p_{1}p_{2}:h}$ $\displaystyle+$
$\displaystyle\sum_{\rm p_{1}h_{1}}[\langle{\rm hp_{1}}|v|{\rm
ph_{1}}\rangle_{A}{\tilde{Y}}^{\mu}_{\rm p_{1}p^{\prime}:h_{1}}$
$\displaystyle+$ $\displaystyle\langle{\rm hp_{1}}|v|{\rm
p^{\prime}h_{1}}\rangle_{A}{\tilde{Y}}^{\mu}_{\rm pp_{1}:h_{1}}]=0,$
$\displaystyle(\epsilon_{\rm h}+\epsilon_{\rm h^{\prime}}-\epsilon_{\rm p}$
$\displaystyle-$ $\displaystyle\omega_{\mu}){\tilde{Y}}^{\mu}_{\rm
hh^{\prime}:p}$ (82) $\displaystyle+$
$\displaystyle\sum_{\alpha}\langle\alpha{\rm p}|v|{\rm
hh^{\prime}}\rangle_{A}y^{\mu}_{\alpha}$ $\displaystyle-$
$\displaystyle\sum_{\rm h_{1}h_{2}}\langle{\rm h_{1}h_{2}}|v|{\rm
hh^{\prime}}\rangle{\tilde{Y}}^{\mu}_{\rm h_{1}h_{2}:p}$ $\displaystyle-$
$\displaystyle\sum_{\rm p_{1}h_{1}}[\langle{\rm ph_{1}}|v|{\rm
hp_{1}}\rangle_{A}{\tilde{Y}}^{\mu}_{\rm h_{1}h^{\prime}:p_{1}}$
$\displaystyle+$ $\displaystyle\langle{\rm ph_{1}}|v|{\rm
h^{\prime}p_{1}}\rangle_{A}{\tilde{Y}}^{\mu}_{\rm hh_{1}:p_{1}}]=0,$

where $y^{\mu}_{\alpha}$ denotes $y^{\mu}_{\rm h}$ or $y^{\mu}_{\rm p}$, and
${\tilde{Y}}^{\mu}_{\alpha\beta:\gamma}$ means
$Y^{\mu}_{\alpha\beta:\gamma}-Y^{\mu}_{\beta\alpha:\gamma}$. The Hamiltonian
matrix for Eqs. (80)–(82) is hermitian because Eqs. (69) and (71) are
satisfied by the HF condition and by
$C_{\alpha\beta\alpha^{\prime}\beta^{\prime}}=0$. In this approximation
${Y}^{\mu}_{\rm pp^{\prime}:h}$ and ${Y}^{\mu}_{\rm hh^{\prime}:p}$ do not
couple to other three-fermion amplitudes. Therefore, oRPA corresponds to the
Green’s function approach, TDA of Ref. hrpa96 .

Figure 1: (a) Equation for the vertex function $\Gamma$ (the squares with
$\Gamma$). The horizontal lines depict the Green’s functions, the dash line
means the residual interaction and $J$ the product of the two Green’s
functions. (b) The self-energy $\Sigma$. The thick lines depict the Green’s
functions. The vertical dot-dashed line in (b) which intersects three-fermion
lines represents $Y_{\rm pp^{\prime}:h}^{\mu}$ or $Y_{\rm
hh^{\prime}:p}^{\mu}$.

If correlations in ${Y}^{\mu}_{\rm pp^{\prime}:h}$ and ${Y}^{\mu}_{\rm
hh^{\prime}:p}$ are restricted to p–p and h–h pairs, oRPA corresponds to the
Green’s function approaches, the $G_{0}G_{0}t-$matrix pieri  and pp-RPA sogo ;
urban . In these Green’s function approaches the self-energy $\Sigma$ is
approximated by $G_{0}\Gamma$, where $G_{0}$ is the HF single-particle Green’s
function and $\Gamma$ the vertex function. The equation for $\Gamma$ and the
expression for $\Sigma$ are schematically shown in Fig. 1 (a) and (b),
respectively. In Fig. 1 the thick lines depict the Green’s functions, the dash
line means the residual interaction and $J$ the product of two Green’s
functions. In $G_{0}G_{0}t-$matrix pieri  and pp-RPA urban  all the Green’s
functions are given by $G_{0}$. The vertical dot-dashed line in Fig. 1 (b)
which intersects three-fermion lines corresponds to $Y_{\rm
pp^{\prime}:h}^{\mu}$ or $Y_{\rm hh^{\prime}:p}^{\mu}$. Since $\Sigma$ is
given by $G_{0}\Gamma$, the correlations between the two fermions, one in
$G_{0}$ and the other in $\Gamma$ are not considered in $G_{0}G_{0}t-$matrix
and pp-RPA. In oRPA these correlations are included as the p–h correlations in
$Y_{\rm pp^{\prime}:h}^{\mu}$ and $Y_{\rm hh^{\prime}:p}^{\mu}$. In the PVC
models the p–h correlations included in $Y_{\rm pp^{\prime}:h}^{\mu}$ and
$Y_{\rm hh^{\prime}:p}^{\mu}$ are expressed by the RPA phonons colo ; litv
but p–p and h–h correlations are not included.

As will be shown in Sect.III, the condition Eq. (17)

$\displaystyle\sum_{\mu}y_{\rm p}^{\mu}({Y}_{\rm
hh^{\prime}:p^{\prime}}^{\mu})^{*}=\sum_{\mu}(y_{\rm h}^{\mu})^{*}Y_{\rm
pp^{\prime}:h^{\prime}}^{\mu}$ (83)

is not satisfied in oRPA except for special cases such as the paring
Hamiltonian rich  where the hole space is symmetric to the particle space.
This may be understood in the following way. The two-body correlation matrix
$C_{\rm pp^{\prime}hh^{\prime}}$, for example, includes all p–p, h–h and p–h
correlations. On the other hand $\sum_{\mu}y_{\rm p}^{\mu}({Y}_{\rm
hh^{\prime}:p^{\prime}}^{\mu})^{*}$ includes full h–h and partial p–h
correlations but no p–p correlations are included as seen from Eq. (82) for
${Y}_{\rm hh^{\prime}:p^{\prime}}^{\mu}$. Similarly, $({y}_{\rm
h}^{\mu})^{*}Y_{\rm pp^{\prime}:h^{\prime}}^{\mu}$ includes full p–p and
partial p–h correlations but no h–h correlations are included (see Eq. (81)).
This asymmetry causes the violation of Eq. (83). In EoRPA $C_{2}$ included in
Eq. (46) plays a role in reducing the asymmetry between particle-space and
hole-space correlations. The coupling of $y^{\mu}_{\rm h}$ to $Y^{\mu}_{\rm
pp^{\prime}:h^{\prime}}$ included in $B$ (Eq. (46)) is shown in Fig. 2. The
vertical lines with arrows denote either a hole state or a particle state, the
dashed lines mean the residual interaction, the squares with $Y^{\mu}$ show
$Y^{\mu}_{\rm pp^{\prime}:h^{\prime}}$, and the horizontal bars with $C_{2}$
the two-body correlation matrix. The process (a) is included both in oRPA and
in EoRPA. The processes (b) and (c) are additionally included in EoRPA. These
processes are also called vertex corrections. As mentioned above, $C_{2}$ in
(b) and (c) includes full p–p, h–h and p–h correlations and thus can
supplement h–h correlations missing in the process (a). The two-body
correlation matrix $C_{2}$ included in $D$ and the three-body correlation
matrix $C_{3}$ in $D_{2}$ also represent additional correlations that are not
included in oRPA.

Figure 2: Coupling of $y^{\mu}_{\rm h}$ to $Y^{\mu}_{\rm
pp^{\prime}:h^{\prime}}$. The vertical lines with arrows denote either a hole
state or a particle state, the dashed lines mean the residual interaction, the
squares with $Y^{\mu}$ show $Y^{\mu}_{\rm pp^{\prime}:h^{\prime}}$, and the
horizontal bars with $C_{2}$ the correlation matrix. The process (a) is
included in oRPA, and the processes (b) and (c) are additionally included in
EoRPA.

When a correlated ground state is used, other three-fermion amplitudes such as
$Y_{\rm hh^{\prime}:h^{\prime\prime}}^{\mu}$ and $Y_{\rm ph:p^{\prime}}^{\mu}$
($Y_{\rm pp^{\prime}:p^{\prime\prime}}^{\mu}$ and $Y_{\rm
ph:h^{\prime}}^{\mu}$) which are the backward amplitudes of $Y_{\rm
pp^{\prime}:h}^{\mu}$ ($Y_{\rm hh^{\prime}:p}^{\mu}$) have non-vanishing
values of $N_{22}$, and such a formulation with the backward amplitudes may
correspond to FRPA of Ref. dick02 . However, the inclusion of such three-
fermion amplitudes with small values of $N_{22}$ sometimes causes unphysical
fragmentation of the single-particle strength hrpa96 ; ts13 . Therefore, the
same four amplitudes $y_{\rm h}^{\mu}$, $y_{\rm p}^{\mu}$, $Y_{\rm
pp^{\prime}:h}^{\mu}$ and $Y_{\rm hh^{\prime}:p}^{\mu}$ as those in oRPA are
used also in EoRPA.

####  II.5.4 Renormalization approximation

Here the approximation where the ground-state correlation effects are
partially included through $n_{\alpha\alpha^{\prime}}$ neglecting $C_{2}$ is
considered. Such a treatment of ground-state correlations may be called the
renormalization approximation Rowe . This approximation in the Green’s
function method is called r-RPA durel . When $C_{2}$ is neglected, $N_{22}$ in
Eq. (54) becomes

$\displaystyle
N_{22}(\alpha\beta\gamma:\alpha^{\prime}\beta^{\prime}\gamma^{\prime})$
$\displaystyle=$
$\displaystyle(\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}-\delta_{\alpha\beta^{\prime}}\delta_{\beta\alpha^{\prime}})\delta_{\gamma^{\prime}\gamma}$
(84) $\displaystyle\times$ $\displaystyle N(\alpha\beta\gamma)$

where
$N(\alpha\beta\gamma)=n_{\gamma}+n_{\alpha}n_{\beta}-n_{\gamma}n_{\alpha}-n_{\gamma}n_{\beta}$.
Here, $n_{\alpha\alpha^{\prime}}=\delta_{\alpha\alpha^{\prime}}n_{\alpha}$ is
assumed. The Hamiltonian matrix is given by

$\displaystyle A(\alpha:\alpha^{\prime})$ $\displaystyle=$
$\displaystyle\epsilon_{\alpha}\delta_{\alpha\alpha^{\prime}},$ (85)
$\displaystyle B(\alpha\beta\gamma:\alpha^{\prime})$ $\displaystyle=$
$\displaystyle C(\alpha^{\prime}:\alpha\beta\gamma)^{*}$ (86) $\displaystyle=$
$\displaystyle\langle\alpha^{\prime}\gamma|v|\alpha\beta\rangle_{A}N(\alpha\beta\gamma),$
$\displaystyle
D(\alpha\beta\gamma:\alpha^{\prime}\beta^{\prime}\gamma^{\prime})$
$\displaystyle=$
$\displaystyle\delta_{\gamma^{\prime}\gamma}[(\epsilon_{\alpha}+\epsilon_{\beta}-\epsilon_{\gamma})$
(87) $\displaystyle\times$
$\displaystyle(\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}-\delta_{\alpha\beta^{\prime}}\delta_{\beta\alpha^{\prime}})$
$\displaystyle+$
$\displaystyle\langle\alpha^{\prime}\beta^{\prime}|v|\alpha\beta\rangle_{A}(1-n_{\alpha}-n_{\beta})]$
$\displaystyle\times$ $\displaystyle
N(\alpha^{\prime}\beta^{\prime}\gamma^{\prime}),$

where the p–h correlations in $D$ are omitted similarly to r-RPA. $D$ is not
hermitian because $(1-n_{\alpha}-n_{\beta})$ differs from
$N(\alpha\beta\gamma)$. Removing the term $n_{\alpha}n_{\beta}$ from
$N(\alpha\beta\gamma)$ can make the Hamiltonian matrix hermitian. This
approximation corresponds to r-RPA durel . The condition Eq. (17) is not
fulfilled in the renormalization approximation because of the same reason as
in oRPA and EoRPA, and the number conservation law is violated durel .

##  III Applications

###  III.1 Pairing model

First the pairing Hamiltonian rich  is considered to show that EoRPA is a
reasonably good approximation. The pairing Hamiltonian is given as

$\displaystyle
H=\sum_{i=1}^{\Omega}\epsilon_{\alpha}(a^{+}_{i}a_{i}+a^{+}_{\bar{i}}a_{\bar{i}})-g\sum_{i\neq
j}^{\Omega}a^{+}_{i}a^{+}_{\bar{i}}a_{\bar{j}}a_{j}.$ (88)

Here $g$ is the strength of the pairing force acting in a space of $\Omega$
twofold degenerate equidistant orbitals with the single-particle energies
$\epsilon_{i}=(i-1)\Delta\epsilon$. This Hamiltonian has extensively been used
to investigate the validity of theoretical approaches s21 .

In Table 1 the occupation probabilities calculated in TDDM for $N=4$ and
$g/\Delta\epsilon=0.5$ are compared with the results in exact diagonalization
approach (EDA). In HF the $i=1$ and $2$ states are fully occupied and the
$i=3$ and $4$ states are empty. The TDDM results agree well with the EDA
results.

Table 1: Single-particle energies $\epsilon_{i}$ and occupation probabilities $n_{i}$ calculated in TDDM for the pairing model with $N=4$ and $g/\Delta\epsilon=0.5$. The results in EDA are given in parentheses. $i$ | $\epsilon_{i}/\Delta\epsilon$ | $n_{i}$
---|---|---
$1$ | 0 | 0.964 (0.965)
$2$ | 1 | 0.909 (0.901)
$3$ | 2 | 0.091 (0.099)
$4$ | 3 | 0.036 (0.035)

The transition strength
$S_{\alpha}^{\mu}=x^{\mu}_{\alpha}(x^{\mu}_{\alpha})^{*}$ of the $i=1$ state
in $N=4\rightarrow 5$ calculated in EoRPA (dotted lines) is shown in Fig. 3.
Since the $i=1$ state in the $N=4$ system is not fully occupied due to the
ground-state correlations, such a transition is allowed, and Eq. (46) can
handle both the single-particle removal and addition processes. Since the
interaction in the pairing Hamiltonian Eq. (88) is allowed for p–p or h–h
pairs, $y^{\mu}_{\rm h}$ can couple only to $Y^{\mu}_{\rm pp^{\prime}:h}$. The
results in oRPA and EDA are shown with the dot-dashed and solid lines,
respectively. In EDA the single-particle transition strength is calculated by
using the ground state for the $N=4$ system and the eigenstates for the $N=5$
system. The two peaks in EoRPA are shifted upward as compared with the oRPA
results and become closer to the results in EDA. This difference between the
EoRPA and oRPA is mainly attributed to the self-energy contributions included
in the $D$ matrix in EoRPA. The self-energy contributions play a role in
effectively increasing the energies of the three-fermion configurations in
$Y^{\mu}_{\rm pp^{\prime}:h}$.

Figure 3: Transition strength of the $i=1$ state in $N=4\rightarrow 5$
calculated in EoRPA with the TDDM ground state for the pairing model with
$g/\Delta\epsilon=0.5$ (the dotted lines). The results in oRPA with the HF
ground state are depicted with the dot-dashed lines. The EDA results are shown
with the solid lines.

The transition strength of the $i=2$ state in $N=4\rightarrow 5$ calculated in
EoRPA (dotted lines) is shown in Fig. 4. As in the case of the $i=1$ state the
two peaks in EoRPA are shifted upward as compared with the oRPA results and
become closer to the results in EDA. The transition strength distributions of
the $i=3$ and $4$ states in $N=4\rightarrow 3$ are the same as those of the
$i=2$ and $1$ states, respectively, and are not shown here. The occupation
probability of each single-particle state is shown in Table 2. The occupation
probabilities of the $i=1$ and $2$ are obtained from
$1-\sum_{\mu}S^{\mu}_{\alpha}$, where $S^{\mu}_{\alpha}$ is the transition
strength shown in Figs. 3 and 4. The results in EoRPA agree well with those in
TDDM. Due to the hole–particle symmetry of the pairing Hamiltonian, the
occupation probabilities of the $i=3$ and $4$ states are equal to $1-n_{2}$
and $1-n_{1}$, respectively, and the the number conservation law is fulfilled
both in oRPA and EoRPA as shown in the bottom line of Table 2, where
$N=\sum_{\alpha}n_{\alpha}$.

Figure 4: Same as Fig. 3 but for the $i=2$ state. Table 2: Occupation probabilities $n_{\alpha}$ calculated in TDDM, EoRPA and oRPA for the pairing Hamiltonian at $g/\Delta\epsilon=0.5$. i | $\epsilon_{i}/\Delta\epsilon$ | TDDM | EoRPA | oRPA
---|---|---|---|---
$1$ | 0 | 0.964 | 0.96606 | 0.97444
$2$ | 1 | 0.909 | 0.90685 | 0.91706
$3$ | 2 | 0.091 | 0.09315 | 0.08294
$4$ | 3 | 0.036 | 0.03394 | 0.02556
| $N$ | 4 | 4 | 4

###  III.2 16O

In this section EoRPA is applied to the single-particle states in 16O where
hole-particle symmetry is broken. It is shown that the number conservation law
is drastically improved by going from oRPA to EoRPA. The small single-particle
space consisting of the proton $1p_{3/2}$, $1p_{1/2}$ and $1d_{5/2}$ states is
used to facilitate the comparison with the results in EDA. In this single-
particle space $y^{\mu}_{\rm h}$ and $y^{\mu}_{\rm p}$ respectively couple
only to the backward amplitudes $Y^{\mu}_{\rm pp^{\prime}:h}$ ad $Y^{\mu}_{\rm
hh^{\prime}:p}$ due to parity selection rule. These couplings to the backward
amplitudes are essential to determine the occupation probabilities below and
above the Fermi level ts13 . The single-particle energies and wavefunctions
are obtained from the HF calculation with the Skyrme III force (SKIII). Only
the $t_{0}$ term of SKIII is used as the residual interaction. The value of
$t_{0}$ is reduced by 40 % to obtain ground-state correlations comparable to
the results of other theoretical calculations adachi ; utsuno . The TDDM
ground state of 16O is calculated by using the adiabatic method where $C_{3}$
is approximated by the square of $C_{2}$ toh20 . The occupation probabilities
are shown in Table 3 where the results in EDA are given in parentheses. In HF
the $1p_{3/2}$ and $1p_{1/2}$ states are full occupied and the $1d_{5/2}$
state is empty. TDDM well reproduce the results in EDA.

Table 3: Proton single-particle energies $\epsilon_{\alpha}$ and occupation probabilities $n_{\alpha}$ calculated in TDDM for 16O. The results in EDA are given in parentheses. Orbit |  $\epsilon_{\alpha}$ [MeV] | $n_{\alpha}$
---|---|---
$1p_{3/2}$ | -18.2 | 0.924 (0.922)
$1p_{1/2}$ | -12.0 | 0.836 (0.822)
$1d_{5/2}$ |  -3.8 | 0.106 (0.111)

The transition strength of the proton $1p_{3/2}$ state in 16O$\rightarrow$17F
calculated in EoRPA (dotted lines) is shown in Fig. 5. Since the proton
$1p_{3/2}$ state in 16O is not fully occupied due to the ground-state
correlations, such a single proton transfer is allowed, and Eq. (46) can
describe both the single-particle removal and addition processes. The results
in oRPA and EDA are shown with the dot-dashed and solid lines, respectively.
In EDA the single-particle transition strength is calculated by using the
ground state for the $N=6$ system and the eigenstates for the $N=7$ system.
The small peaks seen below 10 MeV in the oRPA result disappear in EoRPA and
the EoRPA strength is fragmented around 10 MeV, which is consistent with the
EDA result. This difference between the EoRPA and oRPA may be explained by the
self-energy contributions included in the $D$ matrix in EoRPA. However, EoRPA
and also oRPA cannot produce the strength distribution in high-energy region
seen in the EDA results. Apparently higher amplitudes such as $Y^{\mu}_{\rm
p_{1}p_{2}p_{3}p_{4}:h_{1}h_{2}h_{3}}$ whose unperturbed energies are from 20
to 40 MeV are needed to explain the strength in the high-energy region. The
five fermion amplitudes $Y^{\mu}_{\rm h_{1}h_{2}h_{3}:p_{1}p_{2}}$ can couple
to $Y^{\mu}_{\rm pp^{\prime}:h}$ but would have negligible effects because
their unperturbed energies are smaller than $-28$ MeV. The transition strength
of the proton $1p_{1/2}$ state in 16O$\rightarrow$17F calculated in EoRPA
(dotted lines) is shown in Fig. 6. As in the case of the $1p_{3/2}$ state, the
peaks in EoRPA are shifted upward from the oRPA positions and locate closer to
the EDA results but again EoRPA cannot produce the small strength distribution
in high-energy region seen in the EDA results.

The transition strength of the proton $1d_{5/2}$ state in 16O$\rightarrow$15N
calculated in EoRPA (dotted lines) is shown in Fig. 7. Since such transition
strength appears in the negative energy region ts13 , it is shown as a
function of $-\omega$. The partial occupation of the proton $1d_{5/2}$ state
in 16O due to the ground-state correlations enables such a transition. The
single-particle transition strength in EDA is calculated by using the ground
state for the $N=6$ system and the eigenstates for the $N=5$ system. As in the
case of the $1p_{3/2}$ state, the peaks in EoRPA are shifted upward and each
main peak in EoRPA is very closer to that in EDA.

Figure 5: Transition strength of the proton $1p_{3/2}$ state in 16O$\rightarrow$17F calculated in EoRPA with the TDDM ground state for 16O (dotted lines). The results in oRPA with the HF ground state for 16O are depicted with the dot-dashed lines. The EDA results are shown with the solid lines. Figure 6: Same as Fig. 5 but for the proton $1p_{1/2}$ state. Figure 7: Same as Fig. 5 but for the proton $1d_{5/2}$ state in 16O$\rightarrow$15N. Table 4: Occupation probabilities $n_{\alpha}$ calculated in TDDM, EoRPA and oRPA for 16O. Orbit |  $\epsilon_{\alpha}$ [MeV] | TDDM | EoRPA | oRPA
---|---|---|---|---
$1p_{3/2}$ | -18.2 | 0.92361 | 0.92524 | 0.95746
$1p_{1/2}$ | -12.0 | 0.83619 | 0.82733 | 0.89707
$1d_{5/2}$ |  -3.8 | 0.10553 | 0.10771 | 0.08298
| $N$ | 6 | 6.00188 | 6.12186

The number conservation law is discussed below. The occupation probabilities
of the single-particle states $1p_{3/2}$ and $1p_{1/2}$ are given by
$1-\sum_{\mu}S^{\mu}_{\alpha}$ where $S^{\mu}_{\alpha}$ is the strength shown
in Figs. 5 and 6. On the other hand the occupation probability of the
$1d_{5/2}$ state is equal to the sum of the strength in Fig. 7. The occupation
probabilities thus obtained are tabulated in Table 4, where $N$ in the bottom
line means $N=4\times n_{1p_{3/2}}+2\times n_{1p_{1/2}}+6\times n_{1d_{5/2}}$.
The occupation probabilities in EoRPA agree well with those in TDDM. In EDA
the occupation probabilities obtained from the transition strengths shown in
Figs. 5–7 are the same as those given in Table 3 and the total number is
conserved. The extent of the violation of the number conservation law is
measured by $\Delta N/N=(N-6)/6$. In oRPA $\Delta N/N$ is $2.03$ %. This is of
the same order of accuracy as TDA applied to 16O hrpa96  which gives $\Delta
N/N=6.56$ %. In EoRPA $\Delta N/N$ is drastically decreased to $0.031$ %.
Equation (33) that partially includes the effects of ground-state correlations
(Eq. (33) is equivalent to Eq. (46) without $D_{2}$) is also applied and
$\Delta N/N$ is found $0.88$ %. The comparison between the oRPA and EoRPA
results demonstrates that the ground-state correlations included in EoRPA
greatly alleviate the violation of the number conservation law. In the pp-RPA
like approximation that neglects the p–h correlations in oRPA $\Delta N/N$ is
$3.29$ %. The pp-RPA Green’s function formalism has been applied to boson-
fermion mixtures sogo  and strongly polarized fermion gases urban  and it was
shown that the Luttinger theorem is fulfilled. The only difference between
oRPA without p–h correlations and the pp-RPA formalism is that the latter
includes backward going p–p and h–h correlations. This corresponds to the
inclusion of the $Y^{\mu}_{\rm p_{1}p_{2}:p_{3}}$ and $Y^{\mu}_{\rm
h_{1}h_{2}:h_{3}}$ amplitudes in oRPA. However, these amplitudes have zero
values of $N_{22}$ in HF and cannot be included in oRPA. In the r-RPA like
treatment of EoRPA that uses the fractional occupation probabilities in Eqs.
(86) and (87) $\Delta N/N$ is decreased to $1.43$ % because the occupation
factors $N(\alpha\beta\gamma)$ and $(1-n_{\alpha}-n_{\beta})$ reduce p–p and
h–h correlations. The r-RPA Green’s function formalism durel  has also been
applied to strongly polarized fermion gases and it was found that the number
conservation law is violated to the same extent as in the r-RPA like treatment
of EoRPA. It was shown in Ref. pieri  that the fully self-consistent Green’s
function method called the $GGt$-matrix or the Luttinger-Ward method where all
$G_{0}$ including that closing the loop in the self-energy $\Sigma$ (see Fig.
1 (b)) are replaced by fully self-consistent $G$, the Luttinger theorem is
exactly fulfilled with the full Dyson equation. This might be due to the fact
that such a self-consistent approach includes diagrams which can only be
described by the higher amplitudes in EoRPA. According to Ref. mahaux  only
the approximation that includes the set of all the diagrams in the same order
of the interaction strength is number-conserving. The $GGt$-matrix does not
satisfy the condition of Ref. mahaux  because the correlations between the two
fermions, one outside and the other inside of the square box in Fig. 1 (b) are
not included in the self-consistent $GGt$-matrix. Exact solutions include such
correlations as a matter of course. The problem of the Luttinger theorem and
approximate Green’s function approaches seems to need further investigations.

The violation of number conservation is also related to the fact that the
relation

$\displaystyle C_{\alpha\beta\alpha^{\prime}\beta^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{\mu}({x}^{\mu}_{\alpha})^{*}{X}^{\mu}_{\alpha^{\prime}\beta^{\prime}:\beta}$
(89) $\displaystyle=$
$\displaystyle\sum_{\mu}{x}^{\mu}_{\alpha^{\prime}}({X}^{\mu}_{\alpha\beta:\beta^{\prime}})^{*}$
(90)

does not exactly hold. For example, for $\alpha=\beta=1p_{1/2}$ and
$\alpha^{\prime}=\beta^{\prime}=1d_{5/2}$ Eq. (89) gives $0.2200$ in EoRPA
while Eq. (90) gives $0.2198$. One may think that the violation of the number
conservation law be caused by the inaccuracy of the TDDM ground state and the
neglect of other three-fermion amplitudes in EoRPA. So a calculation is made
where $n_{\alpha}$, $C_{2}$ and $C_{3}$ calculated from the ground state in
EDA are inserted into the EoRPA equation including all three-fermion
amplitudes. The obtained value of $\Delta N/N$ is 0.054%, indicating that the
accuracy of the ground state and the neglect of other three-fermion amplitudes
in EoRPA is not the origin of the violation of the number conservation law. In
the single-particle space consisting of the $1p_{1/2}$, $1p_{3/2}$ and
$1d_{5/2}$ states $y^{\mu}_{\rm h}$ does not couple to the forward amplitude
$Y^{\mu}_{\rm hh^{\prime}:p}$ due to parity selection rule and also
$y^{\mu}_{\rm p}$ has no coupling to $Y^{\mu}_{\rm pp^{\prime}:h}$. The
couplings to the forward amplitudes $Y^{\mu}_{\rm hh^{\prime}:p}$ and
$Y^{\mu}_{\rm pp^{\prime}:h}$ cause fragmentation of a hole state and a
particle state, respectively. To estimate the influence of such forward
amplitudes on the number conservation law, additional oRPA and EoRPA
calculations are performed where the proton $1s_{1/2}$ state is added to the
single-particle space. The obtained occupation probabilities are shown in
Table 5. The results in EoRPA agree well with those in TDDM and EDA, and
$\Delta N/N$ is again drastically decreased by going from oRPA to EoRPA but it
is of the same order of magnitude as the EoRPA result in Table 4. Thus the
coupling to the forward amplitudes has little influence on the number
conservation law.

Table 5: Occupation probabilities $n_{\alpha}$ calculated in TDDM, EoRPA and oRPA for 16O including the proton $1s_{1/2}$ state. The results in EDA are given in parentheses. Orbit |  $\epsilon_{\alpha}$ [MeV] | TDDM | EoRPA | oRPA
---|---|---|---|---
$1s_{1/2}$ | -32.1 | 0.96924 (0.96873) | 0.97068 | 0.99348
$1p_{3/2}$ | -18.2 | 0.90814 (0.90553) | 0.91105 | 0.95550
$1p_{1/2}$ | -12.0 | 0.81183 (0.78752) | 0.79981 | 0.89707
$1d_{5/2}$ |  -3.8 | 0.13422 (0.14423) | 0.13653 | 0.10554
| $N$ | 8 | 8.00437 | 8.23635

Presumably higher-amplitudes such as $X^{\mu}_{\rm
p_{1}p_{2}p_{3}p_{4}:h_{1}h_{2}h_{3}}$ and $X^{\mu}_{\rm
h_{1}h_{2}h_{3}h_{4}:p_{1}p_{2}p_{3}}$ which are not considered in EoRPA are
needed to further improve the number conservation law because ${x}^{\mu}_{\rm
h_{1}}$ in $\sum_{\mu}({x}^{\mu}_{\rm h_{1}})^{*}{X}^{\mu}_{\rm
p_{1}p_{2}:h_{2}}$ and ${x}^{\mu}_{\rm p_{1}}$ in $\sum_{\mu}({x}^{\mu}_{\rm
p_{1}})^{*}{X}^{\mu}_{\rm h_{1}h_{2}:p_{2}}$ (see Eq. (89)) include the
coupling to ${X}^{\mu}_{\rm p_{3}p_{4}:h_{3}}$ and ${X}^{\mu}_{\rm
h_{3}h_{4}:p_{3}}$, respectively.

##  IV Summary

The number conservation law in the odd-particle number random-phase
approximation (oRPA) and its extension (EoRPA) was investigated by applying
them to the pairing model and 16O and comparing with the results in exact
diagonalization approach (EDA). It was found that the results in EoRPA
generally have good agreement with those in EDA. In the applications of oRPA
and EoRPA to 16O it was shown that the number conservation law is not
satisfied and that it is drastically improved by going from oRPA to EoRPA due
to the effects of ground-state correlations included in EoRPA. It was pointed
out that to fulfill the number conservation law the two-body correlation
matrix consisting of the one-fermion and three-fermion transition amplitudes
in EoRPA should be independent of how it is composed. Any extended oRPA
approaches that are based on the truncation of higher transition amplitudes
presumably cannot satisfy this condition, however.

##  Appendix A Matrix elements of Eq. (33)

$\displaystyle
a(\alpha:\alpha^{\prime})=\epsilon_{\alpha}\delta_{\alpha\alpha^{\prime}}$
(91) $\displaystyle b(\alpha\beta\gamma:\alpha^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{\lambda}\langle\alpha^{\prime}\lambda|v|\alpha\beta\rangle_{A}n_{\gamma\lambda}$
$\displaystyle-$
$\displaystyle\sum_{\lambda\lambda^{\prime}}[\langle\alpha^{\prime}\lambda^{\prime}|v|\alpha\lambda\rangle_{A}(n_{\lambda\beta}n_{\gamma\lambda^{\prime}}+C_{\gamma\lambda\lambda^{\prime}\beta})$
$\displaystyle+$
$\displaystyle\langle\alpha^{\prime}\lambda^{\prime}|v|\lambda\beta\rangle_{A}(n_{\lambda\alpha}n_{\gamma\lambda^{\prime}}+C_{\gamma\lambda\lambda^{\prime}\alpha})$
$\displaystyle-\langle\alpha^{\prime}\gamma|$ $\displaystyle v$
$\displaystyle|\lambda\lambda^{\prime}\rangle_{A}(n_{\lambda\alpha}n_{\lambda^{\prime}\beta}+\frac{1}{2}C_{\lambda\lambda^{\prime}\alpha\beta})],$
(92) $\displaystyle c(\alpha:\alpha^{\prime}\beta^{\prime}\gamma^{\prime})$
$\displaystyle=$
$\displaystyle\langle\alpha^{\prime}\beta^{\prime}|v|\alpha\gamma^{\prime}\rangle$
(93) $\displaystyle d$ $\displaystyle($
$\displaystyle\alpha\beta\gamma:\alpha^{\prime}\beta^{\prime}\gamma^{\prime})=(\epsilon_{\alpha}+\epsilon_{\beta}-\epsilon_{\gamma})\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}\delta_{\gamma\gamma^{\prime}}$
(94) $\displaystyle+$
$\displaystyle\frac{1}{2}\langle\alpha^{\prime}\beta^{\prime}|v|\alpha\beta\rangle_{A}\delta_{\gamma\gamma^{\prime}}$
$\displaystyle+$
$\displaystyle\sum_{\lambda}[\langle\lambda\alpha^{\prime}|v|\alpha\gamma^{\prime}\rangle_{A}n_{\gamma\lambda}\delta_{\beta\beta^{\prime}}-\langle\lambda\alpha^{\prime}|v|\beta\gamma^{\prime}\rangle_{A}n_{\gamma\lambda}\delta_{\alpha\beta^{\prime}}$
$\displaystyle+$
$\displaystyle\langle\gamma\beta^{\prime}|v|\lambda\gamma^{\prime}\rangle_{A}n_{\lambda\alpha}\delta_{\beta\alpha^{\prime}}-\langle\gamma\beta^{\prime}|v|\lambda\gamma^{\prime}\rangle_{A}n_{\lambda\beta}\delta_{\alpha\alpha^{\prime}}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\delta_{\gamma\gamma^{\prime}}(\langle\alpha^{\prime}\beta^{\prime}|v|\alpha\lambda\rangle_{A}n_{\lambda\beta}+\langle\alpha^{\prime}\beta^{\prime}|v|\lambda\beta\rangle_{A}n_{\lambda\alpha})].$

##  Appendix B Norm matrix $N_{22}$

The norm matrix $N_{22}$ is given as

$\displaystyle N_{22}(\alpha\beta\gamma$ $\displaystyle:$
$\displaystyle\alpha^{\prime}\beta^{\prime}\gamma^{\prime})=(\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}-\delta_{\alpha\beta^{\prime}}\delta_{\beta\alpha^{\prime}})n_{\gamma^{\prime}\gamma}$
(95) $\displaystyle+$
$\displaystyle\delta_{\gamma\gamma^{\prime}}(n_{\alpha\alpha^{\prime}}n_{\beta\beta^{\prime}}-n_{\alpha\beta^{\prime}}n_{\beta\alpha^{\prime}}+C_{\alpha\beta\alpha^{\prime}\beta^{\prime}})$
$\displaystyle-$
$\displaystyle\delta_{\alpha\alpha^{\prime}}(n_{\gamma^{\prime}\gamma}n_{\beta\beta^{\prime}}+C_{\gamma^{\prime}\beta\gamma\beta^{\prime}})$
$\displaystyle-$
$\displaystyle\delta_{\beta\beta^{\prime}}(n_{\gamma^{\prime}\gamma}n_{\alpha\alpha^{\prime}}+C_{\gamma^{\prime}\alpha\gamma\alpha^{\prime}})$
$\displaystyle+$
$\displaystyle\delta_{\alpha\beta^{\prime}}(n_{\gamma^{\prime}\gamma}n_{\beta\alpha^{\prime}}+C_{\gamma^{\prime}\beta\gamma\alpha^{\prime}})$
$\displaystyle+$
$\displaystyle\delta_{\beta\alpha^{\prime}}(n_{\gamma^{\prime}\gamma}n_{\alpha\beta^{\prime}}+C_{\gamma^{\prime}\alpha\gamma\beta^{\prime}}).$

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