# Long-range additive and nonadditive potentials in a hybrid system: Ground
state atom, excited state atom, and ion

Pei-Gen Yan  Department of Physics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3, Canada  Li-Yan Tang<EMAIL_ADDRESS>State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China  Zong-Chao Yan  Department of Physics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3, Canada  State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China  James F. Babb<EMAIL_ADDRESS>ITAMP, Center for Astrophysics | Harvard & Smithsonian, MS 14, 60 Garden St., Cambridge, MA 02138, USA

###### Abstract

We report a theoretical study on the long-range additive and nonadditive
potentials for a three-body hybrid atom-atom-ion system composed of one ground
$S$ state Li atom, one excited $P$ state Li atom and one ground $S$ state Li+
ion, Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li${}^{+}(1\,^{1}S$). The interaction
coefficients are evaluated with highly accurate wave functions calculated
variationally in Hylleraas coordinates. For this hybrid system the three-body
nonadditive interactions (appearing in second-order) induced by the energy
degeneracy and enhanced by the induction effect of the Li+ ion through the
internal electric field can be strong and even stronger than the two-body
additive interactions at the same order. We find that for particular
geometries the two-body additive interactions of the system sum to zero
leaving only three-body nonadditive interactions thus making the present
system potentially a platform to explore quantum three-body effects. We also
extract by first-principles the leading coefficients of the long-range
electrostatic, induction, and dispersion energies of $\text{Li}_{2}^{+}$
electronic states correlating to
$\text{Li}^{+}(1\,{}^{1}S)$–$\text{Li}(2\,{}^{2}P)$, which until now were not
available in the literature. The results should be especially valuable for the
exploration of schemes to create trimers with ultracold atoms and ions in
optical lattices.

Long-range interactions; Van der Waals interaction; First-principles
calculations

##  I Introduction

This paper provides detailed results for the long-range interactions between
three atomic systems, specifically, a ground state atom, a (low-lying) excited
state atom, and a ground state ion, for the particular case of lithium,
specifically, Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li${}^{+}(1\,^{1}S$). Most studies
of hybrid systems consisting of ground or low-lying state atoms have been
concerned with pair-wise cases, i.e., an atom and an ion (reviewed in Ref.
[1]), or a diatomic molecule and an ion [2, 3, 4]. Recently, properties of
low-lying states of triatomic cations were systematically studied [5]. And,
some studies, while considering excitation of atoms, do not consider ions. For
three atoms, with at least one atom in a Rydberg state, there are a number of
studies, such as Refs. [6, 7]. Other studies have considered three-body
interactions of diatomic molecules in a trapping potential [8, 9].

We explore another possibility—three atomic systems that are in the long-range
domain (sufficiently separated such that electron exchange is small) with one
constituent charged and one constituent electronically excited. There are two
main results: First, we give expressions for the long-range potentials as
expansions in inverse-powers of separation distances and corresponding
precisely evaluated coefficients for two-body (dipole-dipole and van der
Waals) and three-body (van der Waals) long-range additive and nonadditive
interactions, in a manner similar to, but extending our previous work on three
atoms [10, 11] and on two atoms and a ground state ion [12]. While in the
present work the derived formulas are generally applicable to the hybrid
$A(n_{0}S)$-$A(n_{0}^{\prime}L)$-$A^{Q+}(n_{0}^{\prime\prime}S)$ systems, even
involving Rydberg states, we choose the particular states of lithium because
we can evaluate the coefficients precisely using accurate wave functions. We
discuss applications for quantum chemical studies of $\text{Li}_{3}^{+}$ and,
as a consequence of our formulation, for long-range potential energies of
$\text{Li}_{2}^{+}$ electronic states correlating to
$\text{Li}^{+}(1\,{}^{1}S)$–$\text{Li}(2\,{}^{2}P)$. Second, different from
previous studies on the weak nonadditive interactions for three-body systems
composed of atoms [13, 14, 10, 11, 15, 16, 17, 18, 19] or of two atoms and an
ion [12], here we find theoretical evidence of a new pure quantum three-body
effect that might have influence on constructing accurate potential surfaces.
Specifically, for the Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li${}^{+}(1\,^{1}S$)
system, we find that at particular geometries the two-body additive
interactions disappear leaving only three-body nonadditive interactions. These
net effects of two- and three-body interactions are quite similar to those for
two- and three-body interactions in the case of polar molecules confined in
lattice traps [8, 9] or three Rydberg atoms under the influence of an external
electric field [7], where the same goal—removal of two-body interactions—was
pursued. To provide necessary context we begin with some general contextual
background from molecular (chemical) physics and from ultra-cold science.

###  I.1 General aspects of triatomic systems

The intrinsic complexity of triatomic molecules111 A widely-known aphorism
(from 1981), attributed to Schawlow, warns atomic physicists that, “a diatomic
molecule is one atom too many” [20], but it may be predated by an earlier
observation (from 1971) attributed to Herschbach: “The trouble with triatomic
molecules is, they have one atom too many!” [21]. Recently, Gao [22]
emphasized the emergence of chemical complexity beginning with three atoms.
produces interesting phenomena such as conical intersections and geometric
phases [23, 24] and the Renner-Teller effect [25], while consideration of
three atoms at ultra-low energies leads to Efimov [26, 27], Borromean [9], and
Pfaffian [28, 8] states and makes the description of collisional processes,
such as atom-diatom collisions [29, 30] and three-body recombination loss [31,
32, 33] challenging. The demands for understanding the spectroscopy and
collisional processes of specific important triatomic molecules at thermal
collisional energies also continue to drive progress. For example, ozone
($\textrm{O}_{3}$) is a vital atmospheric constituent of the planet, with
quantum-mechanical collisional cross sections recently reported (see Ref. [34]
and references therein), tricarbon ($\textrm{C}_{3}$) is prominent in comets
and other astrophysical [35] and laboratory realms [36], and $\textrm{H}_{3}$,
$\textrm{H}_{3}^{+}$, and their isotopologues, serve as long-standing
theoretical benchmark systems [23, 37] and are important in astrophysical
applications such as, for example, to the cooling of hydrogen gas in molecular
clouds [38] and in the evolution of the early Universe [39].

Detailed procedures for calculating and constructing potential energy surfaces
(and other properties) of triatomic systems have been developed, exemplified
(for the representative molecules discussed above) by recent works such as for
tricarbon [40], for ozone [41, 42, 43], and for $\mbox{H}_{3}$ [44]. A
successful strategy to construct three-atom potential energy surfaces using
semi-empirical methods requires input calculations of atom-dimer and three-
atom long-range potentials [45, 46, 47]. To understand the dynamics of low-
energy (ultra-cold) collisions, consideration of the long-range potentials is
paramount, see, for example, for atom-molecule systems [48, 2, 3, 49, 50] and
for atom-molecular-ion systems [51, 52, 53]. Next, we provide an overview of
the lithium dimer and trimer cations.

###  I.2 Homonuclear lithum dimer and trimer cations: Excited electronic
states

We provide a brief overview of relevant work on the lithium homonuclear
systems $\text{Li}_{2}^{+}$ and $\text{Li}_{3}^{+}$ in order to demonstrate
that the present work provides data previously not available in the
literature.

For the diatomic lithium cation $\text{Li}_{2}^{+}$, four electronic states
(ignoring fine-structure) correlate to the separated pair
$\text{Li}^{+}(1\,{}^{1}S)$–$\text{Li}(2\,{}^{2}P)$, namely,
$2\,{}^{2}\Sigma_{g}$, $2\,{}^{2}\Sigma_{u}$, $1\,{}^{2}\Pi_{g}$, and
$1\,{}^{2}\Pi_{u}$. Model potential method calculations were given by Magnier
et al. [54] and by Rabli and McCarroll [55]; a CASSCF/MRCI calculation was
reported in Ref. [56] (and references therein for earlier work). Magnier et
al. [54] calculated long-range potential curves as functions of internuclear
distance $R$, including the exchange energies and electrostatic, induction,
and dispersion terms up to $\mathcal{O}(R^{-8})$, but did not give the long-
range potential coefficients. The emphasis of the present paper is on the
three-body system, but because the two-body interactions are available from
our calculations, as will be shown in Secs. II.7–II.9, we will extract the
values of the long-range potential coefficients of the four states of
$\text{Li}_{2}^{+}$.

For the triatomic lithium cation $\text{Li}_{3}^{+}$, because we have found no
previous quantum chemical studies of the excited electronic states
corresponding to those reported here, we present a summary of calculations on
the ground electronic state of $\text{Li}_{3}^{+}$. In a series of works,
Searles, Dunne, and von Nagy-Felsobuki [57, 58, 59, 60] calculated the ground
state potential energy and dipole moment surfaces, which were utilized to
calculate ro-vibrational spectra [61, 62]. Surprisingly, we have found few
subsequent studies on the $\text{Li}_{3}^{+}$ ground electronic state [63];
although, very recently as part of a systematic study exploring alkali-metal
and alkali-earth-metal hybrid ion-atom diatomic and triatomic systems,
Śmiałkowski and Tomza [5] calculated equilibrium properties of the ground
${}^{1}A_{1}$ and lowest triplet ${}^{3}B_{2}$ states of $\text{Li}_{3}^{+}$.
Our previous paper [12] supplies the long-range interactions for the ground
and lowest triplet states of $\text{Li}_{3}^{+}$. In advance of awaited ab
initio quantum chemical calculations, in the present work, we calculate the
long-range interaction potentials of $\text{Li}_{3}^{+}$ when one Li atom is
$\text{Li}(2\,^{2}P)$.

###  I.3 Similarity to lattice studies

Büchler et al. derived a Hubbard model for cold polar molecules trapped in an
optical lattice [64], with the intent of realizing a system that could be used
to model Hamiltonians that exhibit exotic ground state properties [8, 9]. In
terms of the intermolecular interactions within the lattice, they write

$U_{ij}=U_{0}a^{3}|\mathbf{R}_{i}-\mathbf{R}_{j}|^{-3}+U_{1}a^{6}|\mathbf{R}_{i}-\mathbf{R}_{j}|^{-6}$
(1)

and

$W_{ijk}=W_{0}a^{6}|\mathbf{R}_{i}-\mathbf{R}_{j}|^{-3}|\mathbf{R}_{i}-\mathbf{R}_{k}|^{-3},$
(2)

where $U_{0}$, $U_{1}$, and $W_{0}$ are certain energy scales, $a$ is a length
scale, ($i$, $j$, $k$) label the particles, the indices $i,j,k$ are cyclically
permuted, and $\mathbf{R}_{i}$ are certain position vectors of the lattice
site (see Ref. [64] for the complete definitions). By appropriate “dressing”
of the cold molecules by an external static electric field and a microwave
field, they show that the two-body interactions may be tuned “from repulsive
to attractive, and even switched off, while the three-body terms remain
repulsive and strong.” We will derive two equations for the present system,
Eqs. (25) and (42), respectively, that are of the same form—but with
additional terms—as Eqs. (1) and (2). We will show that at specific geometries
we recover exactly Eqs. (1) and (2). The anisotropies of the interactions in
the present system due to the ion charge and the excited $\text{Li}(2\,^{2}P)$
atom are similar to the anisotropies due to the intermolecular dipole-dipole
interactions in the optical-lattice-trapped cold polar molecular system [65,
66, 67, 68]. Further discussion will be given in Sec. III.5.

##  II Theoretical Formulation

The geometry of the three-body system is shown in Fig. 1, in which the three
particles define a plane with the two neutral atoms labeled as 1 and 2 and the
ion labeled as 3. It is important to note that due to the degeneracy of atoms
1 and 2 we can’t specify which one is in the ground state or excited state.
The interior angles of the configuration are $\alpha$, $\beta$ and $\gamma$.

Figure 1:  Configuration of the Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li+($1\,^{1}S$)
system. The three particles define the $x\text{--}y$ plane with the two
neutral atoms labeled as 1 and 2 and the ion labeled as 3, $R_{IJ}$ are the
internuclear distances, and $\alpha$, $\beta$ and $\gamma$ are the interior
angles.

###  II.1  Coulomb potential expansion

In the present work, we take the electrostatic interaction $V_{123}$ between
pairs of particles for the
Li($n_{0}\,S$)-Li($n_{0}\,L$)-Li+($n_{0}^{\prime}\,S$) system as a
perturbation,

$H^{\prime}=V_{123}=V_{12}+V_{23}+V_{31}\,,$ (3)

where $V_{12}$, $V_{23}$ and $V_{31}$ are the two-body mutual electrostatic
interactions between atoms 1 and 2 and ion 3. For three well-separated atoms
or ions, the mutual interaction energy $V_{IJ}$ can be expanded with the same
method as used in Refs. [10, 11, 12], thus,

$V_{IJ}=\sum_{l_{I}l_{J}}\sum_{m_{I}m_{J}}T_{\text{{\hbox{l_I-
m_I}}}}(\boldsymbol{\sigma})T_{\text{{\hbox{l_Jm_J}}}}(\boldsymbol{\rho})W_{l_{I}l_{J}}^{m_{I}-m_{J}}(IJ)\,,$
(4)

where the geometry factor is

$\displaystyle W_{l_{I}l_{J}}^{m_{I}-m_{J}}(IJ)$ $\displaystyle=$
$\displaystyle\frac{4\pi(-1)^{l_{J}}}{R_{IJ}^{l_{I}+l_{J}+1}}\frac{(l_{I}+l_{J}-m_{I}+m_{J})!(l_{I},l_{J})^{-1/2}}{[(l_{I}+m_{I})!(l_{I}-m_{I})!(l_{J}+m_{J})!(l_{J}-m_{J})!]^{1/2}}P_{l_{I}+l_{J}}^{m_{I}-m_{J}}(\cos\theta_{IJ})$
(5) $\displaystyle\times$ $\displaystyle\exp[{i(m_{I}-m_{J})\Phi_{IJ}}]\,,$

where ${\bf R}_{IJ}={\bf R}_{J}-{\bf R}_{I}$ is the relative position vector
from particle $I$ to particle $J$, the notation
$(l_{I},l_{J},\ldots)=(2l_{I}+1)(2l_{J}+1)\ldots$, and
$P_{l_{I}+l_{J}}^{m_{I}-m_{J}}(\cos\theta_{IJ})$ is the associated Legendre
function with $\theta_{IJ}$ representing the angle between ${\bf R}_{IJ}$ and
the $z$-axis. The 2ℓ-pole transition operator of an atom consisting of $n+1$
charged particles, in the laboratory frame is defined as in Ref. [69],

$\displaystyle T_{{\ell m}}=\sum_{i=0}^{n}q_{i}{\rho}_{i}^{\ell}Y_{\ell
m}(\hat{\boldsymbol{\rho}}_{i})\,,$ (6)

where $q_{i}$ is the charge of the $i$-th sub-particle of the atom. In the
center of mass frame [69], $\boldsymbol{\rho}_{i}$ becomes

$\displaystyle\boldsymbol{\rho}_{i}=\sum_{j=1}^{n}\epsilon_{ij}\textbf{r}_{j}\,,$
(7)

where $\textbf{r}_{i}=\boldsymbol{\rho}_{i}-\boldsymbol{\rho}_{0}$,
$\epsilon_{ij}=\delta_{ij}-m_{j}/M_{T}$, $i=0,1,2,...,n$, $j=1,2,...,n$, and
$M_{T}$ represents the total mass of the system. Using the formula

$\displaystyle{Y}_{{\ell
m}}(\hat{\textbf{r}})=\sqrt{\frac{3}{4\pi}}\prod_{i=1}^{\ell-1}\bigg{(}\sqrt{\frac{2i+3}{i+1}}\bigg{)}(\underbrace{\hat{\textbf{r}}\otimes\hat{\textbf{r}}\otimes\cdots\hat{\textbf{r}}}_{\ell})_{m}^{(\ell)}\,,$
(8)

where $\otimes$ denotes the coupling between two irreducible tensor operators,
the 2ℓ-pole transition operator can be simplified as

$\displaystyle{T}_{{\ell}}=\sqrt{\frac{3}{4\pi}}\prod_{m=1}^{\ell-1}\bigg{(}\sqrt{\frac{2m+3}{m+1}}\bigg{)}\sum_{j_{1},\cdots,j_{\ell}}\bigg{(}\sum_{i=0}^{n}q_{i}\epsilon_{ij_{1}}\epsilon_{ij_{2}}\cdots\epsilon_{ij_{\ell}}\bigg{)}(\underbrace{\hat{\textbf{r}}_{j_{1}}\otimes\hat{\textbf{r}}_{j_{2}}\otimes\cdots\hat{\textbf{r}}_{j_{\ell}}}_{0})_{m}^{(\ell)}\,.$
(9)

For a four-body system, the explicit forms of transition operators $T_{\ell}$
with $\ell$ up to 3 can be found in Ref. [70].

###  II.2 The Hylleraas basis set

The nonrelativistic Hamiltonian of the Li atom in the centre of mass frame
[71] can be written as

$\displaystyle
H=-\frac{1}{2\mu}\sum_{i=1}^{3}\nabla_{i}^{2}-\frac{1}{m_{0}}\sum_{i>j\geqslant
1}^{3}\nabla_{i}\cdot\nabla_{j}+q_{0}\sum_{i=1}^{3}\frac{q_{i}}{r_{i}}+\sum_{i>j\geqslant
1}^{3}\frac{q_{i}q_{j}}{r_{ij}}\,,$ (10)

where $\mu=m_{e}m_{0}/(m_{e}+m_{0})$ is the reduced mass between an electron
$m_{e}$ and the nucleus $m_{0}$. The basis set is constructed in Hylleraas
coordinates,

$\displaystyle\phi({\textbf{r}_{1},\textbf{r}_{2},\textbf{r}_{3}})={r}_{1}^{j_{1}}{r}_{2}^{j_{2}}{r}_{3}^{j_{3}}{r}_{12}^{j_{12}}{r}_{23}^{j_{23}}{r}_{31}^{j_{31}}e^{-\alpha
r_{1}-\beta r_{2}-\gamma
r_{3}}\mathcal{Y}_{(\ell_{1}\ell_{2})\ell_{12},\ell_{3}}^{(LM)}(\hat{\textbf{r}}_{1},\hat{\textbf{r}}_{2},\hat{\textbf{r}}_{3})\mathcal{X}(1,2,3)\,,$
(11)

where

$\displaystyle\mathcal{Y}_{(\ell_{1}\ell_{2})\ell_{12},\ell_{3}}^{(LM)}(\hat{\textbf{r}}_{1},\hat{\textbf{r}}_{2},\hat{\textbf{r}}_{3})=\sum_{m_{i}}$
$\displaystyle\langle\ell_{1}m_{1};\ell_{2}m_{2}|\ell_{1}\ell_{2};\ell_{12}m_{12}\rangle\langle\ell_{12}m_{12};\ell_{3}m_{3}|\ell_{12}\ell_{3};LM_{L}\rangle$
(12) $\displaystyle\times
Y_{\ell_{1}m_{1}}(\hat{\textbf{r}}_{1})Y_{\ell_{2}m_{2}}(\hat{\textbf{r}}_{2})Y_{\ell_{3}m_{3}}(\hat{\textbf{r}}_{3})\,$

is a vector-coupled product of spherical harmonics to form an eigenstate of
the total angular momentum $L$ and component $M_{L}$, and $\mathcal{X}(1,2,3)$
is the three-electron spin-1/2 function. The variational wave function of the
Li atom is a linear combination of basis functions $\phi$ antisymmetrized.
With some truncations to avoid the numerical linear dependence, all terms in
Eq. (11) are included such that

$\displaystyle j_{1}+j_{2}+j_{3}+j_{12}+j_{23}+j_{31}\leqslant\Omega\,,$ (13)

where $\Omega$ is an integer, and the convergence for the energy eigenvalue is
studied by increasing $\Omega$ progressively. The reduced matrix elements for
various transition operators can be evaluated with the following basic
integral

$\displaystyle\int$ $\displaystyle
d{\textbf{r}}_{1}d{\textbf{r}}_{2}d{\textbf{r}}_{3}{r}_{1}^{j_{1}}{r}_{2}^{j_{2}}{r}_{3}^{j_{3}}{r}_{12}^{j_{12}}{r}_{23}^{j_{23}}{r}_{31}^{j_{31}}e^{-\alpha
r_{1}-\beta r_{2}-\gamma r_{3}}$ $\displaystyle\times
Y^{*}_{\ell^{\prime}_{1}m^{\prime}_{1}}({\textbf{r}}_{1})Y^{*}_{\ell^{\prime}_{2}m^{\prime}_{2}}({\textbf{r}}_{2})Y^{*}_{\ell^{\prime}_{3}m^{\prime}_{3}}({\textbf{r}}_{3})Y_{\ell_{1}m_{1}}({\textbf{r}}_{1})Y_{\ell_{2}m_{2}}({\textbf{r}}_{2})Y_{\ell_{3}m_{3}}({\textbf{r}}_{3})\,.$

The details of computational method for this integral are developed in Refs.
[71, 72]. Similarly, for the Li+ ion, we also use the Hylleraas variational
method to obtain the energies, wavefunctions and transition matrix elements.
The detailed Hylleraas method for a two-electron atom is given in Ref. [73].

###  II.3 The zeroth-order wave function

For the degenerate Li($n_{0}\,S$)-Li($n_{0}\,L$)-Li+($n_{0}^{\prime}\,S$)
system with energy
$E_{n_{0}n_{0}n_{0}^{\prime}}^{(0)}=E_{n_{0}S}^{(0)}+E_{n_{0}L}^{(0)}+E_{n_{0}^{\prime}S}^{(0)}$,
the zeroth-order wave function can be written as

$\displaystyle\left|\Psi^{(0)}\right\rangle=a\left|{n_{0}}L;{n_{0}}0;{n_{0}^{\prime}}0\right\rangle+b\left|{n_{0}}0;{n_{0}}L;{n_{0}^{\prime}}0\right\rangle\,,$
(15)

where $a$ and $b$ are the expansion coefficients of the zeroth-order wave
function in the basis set
{$\left|{n_{0}}L;{n_{0}}0;{n_{0}^{\prime}}0\right\rangle$,
$\left|{n_{0}}0;{n_{0}}L;{n_{0}^{\prime}}0\right\rangle$} with
$\left|{n_{0}}0\right\rangle$, $\left|{n_{0}}L\right\rangle$, and
$\left|{n^{\prime}_{0}}0\right\rangle$, respectively, being the initial states
for Li($n_{0}\,S$), Li($n_{0}\,L$) and Li+($n_{0}^{\prime}\,S$). The
corresponding zeroth-order wave functions (or the values of $a$ and $b$)
depend on the geometrical configuration formed by the three particles and are
determined by diagonalizing the perturbation in this basis set. Then using the
degenerate perturbation theory, we can obtain the long-range part of the
interaction potential for the Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li+($1\,^{1}S$)
system, which can be written as

$\displaystyle\Delta E=\Delta E^{(1)}_{\text{add}}+\Delta
E^{(2)}_{\text{add}}+\Delta E^{(2)}_{\text{non}}\,,$ (16)

where $\Delta E^{(1)}_{\text{add}}$ and $\Delta E^{(2)}_{\text{add}}$ are,
respectively, the first-order and second-order additive interactions and
$\Delta E^{(2)}_{\text{non}}$ is the second-order nonadditive interaction.

###  II.4 The first-order additive interactions

The first-order additive interaction $\Delta E^{(1)}_{\text{add}}$ is given by

$\displaystyle\Delta
E^{(1)}_{\text{add}}=-\frac{C_{3}^{(12)}(1,M)}{R_{12}^{3}}-\frac{C_{3}^{(23)}(1,M)}{R_{23}^{3}}-\frac{C_{3}^{(31)}(1,M)}{R_{31}^{3}}\,,$
(17)

where $C_{3}^{(12)}(1,M)$ describes the dipole-dipole interaction between two
neutral atoms. $C_{3}^{(23)}(1,M)$ and $C_{3}^{(31)}(1,M)$, respectively,
describe the electrostatic interaction between the charge of the ion labeled
as 3 and the quadrupole moments of atom 2 and atom 1; the quadrupole moment
comes from the excited Li($2\,^{2}P$) atom, which can be atom 1 or atom 2 due
to the degeneracy of the three-body system. These leading long-range
interaction coefficients are given by

$\displaystyle C_{3}^{(12)}(1,M)$ $\displaystyle=$
$\displaystyle(a^{*}b+b^{*}a)\frac{4\pi(-1)^{1+M}}{9(1-M)!(1+M)!}|\langle{n_{0}}0\|T_{1}\|{n_{0}}1\rangle|^{2}\,,$
(18) $\displaystyle C_{3}^{(23)}(1,M)$ $\displaystyle=$
$\displaystyle|b|^{2}Q(-1)^{1+M}\sqrt{\frac{\pi}{5}}\left(\begin{array}[]{ccc}1&2&1\\\
-M&0&M\\\ \end{array}\right)\langle{n_{0}}1\|T_{2}\|{n_{0}}1\rangle\,,$ (21)
$\displaystyle C_{3}^{(31)}(1,M)$ $\displaystyle=$
$\displaystyle|a|^{2}Q(-1)^{1+M}\sqrt{\frac{\pi}{5}}\left(\begin{array}[]{ccc}1&2&1\\\
-M&0&M\\\ \end{array}\right)\langle{n_{0}}1\|T_{2}\|{n_{0}}1\rangle\,,$ (24)

where $Q$ is the charge of the ion, $M$ represents the magnetic quantum number
of the excited Li($2\,^{2}P$) atom, and $T_{\ell}$ is the 2ℓ-pole transition
operator, which is defined in Sec. II.1.

###  II.5 The second-order additive interactions

The second-order additive interaction $\Delta E^{(2)}_{\text{add}}$ is given
by

$\displaystyle\Delta E^{(2)}_{\text{add}}=$ $\displaystyle-$
$\displaystyle\frac{C_{4}^{(23)}(1,M)}{R_{23}^{4}}-\frac{C_{4}^{(31)}(1,M)}{R_{31}^{4}}-\frac{C_{6}^{(12)}(1,M)}{R_{12}^{6}}$
(25) $\displaystyle-$
$\displaystyle\frac{C_{6}^{(23)}(1,M)}{R_{23}^{6}}-\frac{C_{6}^{(31)}(1,M)}{R_{31}^{6}}-\cdots\,,$

where $C_{4}^{(23)}(1,M)$ and $C_{4}^{(31)}(1,M)$, respectively, describe the
induction interactions between the ion 3 and the neutral atoms 2 and 1. The
dispersion interaction coefficient between the neutral atoms 1 and 2 is given
by $C_{6}^{(12)}(1,M)$, while $C_{6}^{(23)}(1,M)$ and $C_{6}^{(31)}(1,M)$,
respectively, describe the interactions between the ion 3 and the two neutral
atoms 2 and 1, including both the induction and the dispersion interaction
coefficients. The corresponding expressions for the additive coefficients are

$\displaystyle C_{4}^{(23)}(1,M)$ $\displaystyle=$
$\displaystyle|a|^{2}\mathbb{T}_{1}+|b|^{2}\mathbb{T}_{3}(M)\,,$ (26)
$\displaystyle C_{4}^{(31)}(1,M)$ $\displaystyle=$
$\displaystyle|a|^{2}\mathbb{T}_{3}(M)+|b|^{2}\mathbb{T}_{1}\,,$ (27)
$\displaystyle C_{6}^{(12)}(1,M)$ $\displaystyle=$
$\displaystyle|a|^{2}\mathbb{T}_{4}(M)+|b|^{2}\mathbb{T}_{4}(M)\,,$ (28)
$\displaystyle
C_{6}^{(23)}(1,M)=|a|^{2}\mathbb{T}_{2}+|b|^{2}\mathbb{T}_{5}(M)\,,$ (29)
$\displaystyle
C_{6}^{(31)}(1,M)=|a|^{2}\mathbb{T}_{5}(M)+|b|^{2}\mathbb{T}_{2}\,,$ (30)

where

$\displaystyle\mathbb{T}_{1}=\frac{4\pi
Q^{2}}{9}\sum_{\begin{subarray}{c}n_{t}\end{subarray}}^{\prime}\frac{|\langle{n_{0}}0\|T_{1}\|{n_{t}}1\rangle|^{2}}{E_{{n_{t}1}}-E_{n_{0}0}^{(0)}}\,,$
(31) $\displaystyle\mathbb{T}_{2}$ $\displaystyle=$ $\displaystyle\frac{4\pi
Q^{2}}{25}\sum_{\begin{subarray}{c}n_{t}\end{subarray}}^{\prime}\frac{|\langle{n_{0}}0\|T_{2}\|{n_{t}}2\rangle|^{2}}{E_{{n_{t}2}}-E_{n_{0}0}^{(0)}}+\frac{32\pi^{2}}{27}\sum_{\begin{subarray}{c}n_{t}n_{u}\end{subarray}}^{\prime}\frac{|\langle{n_{0}}0\|T_{1}\|{n_{t}}1\rangle|^{2}|\langle{n_{0}^{\prime}}0\|T_{1}\|{n_{u}}1\rangle|^{2}}{(E_{n_{t}1}-E_{n_{0}0}^{(0)})+(E_{n_{u}1}-E_{n_{0}^{\prime}0}^{(0)})}\,,$
(32)
$\displaystyle\mathbb{T}_{3}(M)=\frac{Q^{2}}{4\pi}\sum_{n_{t}L_{t}}\frac{G_{1}(L_{t},0;1,1;1,M)|\langle{n_{0}}1\|T_{1}\|{n_{t}}L_{t}\rangle|^{2}}{E_{n_{t}L_{t}}-E_{n_{0}1}^{(0)}}\,,$
(33)
$\displaystyle\mathbb{T}_{4}(M)=\sum_{n_{t}n_{u}L_{t}}\frac{G_{1}(L_{t},1;1,1;1,M)|\langle{n_{0}}1\|T_{1}\|{n_{t}}L_{t}\rangle|^{2}\langle{n_{0}^{\prime}0}\|T_{1}\|{n_{u}}1\rangle|^{2}}{(E_{n_{t}L_{t}}-E_{n_{0}1}^{(0)})+(E_{n_{u}1}-E_{n_{0}^{\prime}0}^{(0)})}\,,$
(34) $\displaystyle\mathbb{T}_{5}(M)$ $\displaystyle=$
$\displaystyle\sum_{n_{t}n_{u}L_{t}}\frac{G_{1}(L_{t},1;1,1;1,M)|\langle{n_{0}}1\|T_{1}\|{n_{t}}L_{t}\rangle|^{2}\langle{n_{0}^{\prime}0}\|T_{1}\|{n_{u}}1\rangle|^{2}}{(E_{n_{t}L_{t}}-E_{n_{0}1}^{(0)})+(E_{n_{u}1}-E_{n_{0}^{\prime}0}^{(0)})}$
(35) $\displaystyle+$
$\displaystyle\frac{Q^{2}}{4\pi}\sum_{n_{t}L_{t}}\bigg{\\{}\frac{G_{1}(L_{t},0;2,2;1,M)|\langle{n_{0}}1\|T_{2}\|{n_{t}}L_{t}\rangle|^{2}}{E_{n_{t}L_{t}}-E_{n_{0}1}^{(0)}}$
$\displaystyle+$
$\displaystyle\frac{G_{1}(L_{t},0;1,3;1,M)\langle{n_{0}}1\|T_{1}\|{n_{t}}L_{t}\rangle^{*}\langle{n_{0}}1\|T_{3}\|{n_{t}}L_{t}\rangle}{E_{n_{t}L_{t}}-E_{n_{0}1}^{(0)}}$
$\displaystyle+$
$\displaystyle\frac{G_{1}(L_{t},0;3,1;1,M)\langle{n_{0}}1\|T_{3}\|{n_{t}}L_{t}\rangle^{*}\langle{n_{0}}1\|T_{1}\|{n_{t}}L_{t}\rangle}{E_{n_{t}L_{t}}-E_{n_{0}1}^{(0)}}\bigg{\\}}\,,$

where $G_{1}$-function is defined by

$\displaystyle G_{1}(L_{i},L_{j},\ell_{k},\ell_{k}^{\prime};L,M)$
$\displaystyle=$
$\displaystyle\frac{16\pi^{2}(\ell_{k},\ell_{k}^{\prime})^{-1/2}}{(2L_{j}+1)^{2}}\sum_{M_{i}M_{j}m_{k}}\left(\begin{array}[]{ccc}L&\ell_{k}&L_{i}\\\
-M&m_{k}&M_{i}\\\
\end{array}\right)\left(\begin{array}[]{ccc}L&\ell_{k}^{\prime}&L_{i}\\\
-M&m_{k}&M_{i}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\frac{(L_{j}+\ell_{k}-M_{j}+m_{k})!(L_{j}+\ell_{k}^{\prime}-M_{j}+m_{k})!P_{L_{j}+\ell_{k}}^{M_{j}-m_{k}}(0)P_{L_{j}+\ell_{k}^{\prime}}^{M_{j}-m_{k}}(0)}{(L_{j}+M_{j})!(L_{j}-M_{j})![(\ell_{k}+m_{k})!(\ell_{k}-m_{k})!(\ell_{k}^{\prime}+m_{k})!(\ell_{k}^{\prime}-m_{k})!]^{1/2}}\,.$

The detailed derivations are given in the Supplemental Material [74]. We note
that these formulas can also be used to calculate long-range interaction
coefficients for other two-body or three-body systems such as the two-body
Li($2\,^{2}S$)-Li($2\,^{2}P$) system, the two-body
Li($2\,^{2}S$)-Li+($1\,^{1}S$) system, the two-body
Li($2\,^{2}P$)-Li+($1\,^{1}S$) system, and the three-body
Li($2\,^{2}S$)-Li($2\,^{2}S$)-Li+($1\,^{1}S$) system. We will provide specific
examples below in Secs. II.7–II.9.

###  II.6 The second-order nonadditive potentials

Due to the degeneracy of the three-body system, the nonadditive potential
$\Delta E^{(2)}_{\text{non}}$ starts at the second-order, and is given by

$\displaystyle\Delta E_{\text{non}}^{(2)}=$ $\displaystyle-$
$\displaystyle\frac{C_{3,3}^{(12,23)}(1,M)}{R_{12}^{3}R_{23}^{3}}-\frac{C_{3,3}^{(23,31)}(1,M)}{R_{23}^{3}R_{31}^{3}}-\frac{C_{3,3}^{(31,12)}(1,M)}{R_{31}^{3}R_{12}^{3}}$
(42) $\displaystyle-$
$\displaystyle\frac{C_{4,2}^{(12,23)}(1,M)}{R_{12}^{4}R_{23}^{2}}-\frac{C_{2,4}^{(31,12)}(1,M)}{R_{31}^{2}R_{12}^{4}}-\cdots\,,$

where $C_{3,3}^{(23,31)}(1,M)$ represents the dispersion nonadditive
interaction coefficient. The remaining terms are the nonadditive induction
interactions. The detailed expressions are given by

$\displaystyle C_{3,3}^{(23,31)}(1,M)$ $\displaystyle=$
$\displaystyle\sum_{M_{u}}(-1)^{M_{u}+M}G_{4}(1,M_{u};1,M)\bigg{\\{}(a^{*}b)\exp[i(M_{u}-M)\gamma]+(b^{*}a)\exp[-i(M_{u}-M)\gamma]\bigg{\\}}$
(43)
$\displaystyle\times\sum_{n_{u}}\bigg{[}\frac{|\langle{n_{0}}1\|T_{1}\|{n_{0}}0\rangle|^{2}|\langle{n_{0}^{\prime\prime}}0\|T_{1}\|{n_{u}}1\rangle|^{2}}{(E_{n_{u}1}-E_{n_{0}^{\prime\prime}0})+(E_{n_{0}0}-E_{n_{0}1})}+\frac{|\langle{n_{0}}0\|T_{1}\|{n_{0}}1\rangle|^{2}|\langle{n_{0}^{\prime\prime}}0\|T_{1}\|{n_{u}}1\rangle|^{2}}{(E_{n_{u}1}-E_{n_{0}^{\prime\prime}0})+(E_{n_{0}1}-E_{n_{0}0})}\bigg{]}\,,$
$\displaystyle C_{4,2}^{(12,23)}(1,M)$ $\displaystyle=$
$\displaystyle|a|^{2}\sum_{M_{t}}(-1)^{M_{t}+M}G_{5}(1,M_{t};2;1,M;Q)\cos(M_{t}\beta)$
(44)
$\displaystyle\times\sum_{n_{t}}\frac{\langle{n_{0}}1\|T_{2}\|{n_{0}}1\rangle|\langle{n_{0}}0\|T_{1}\|{n_{t}}1\rangle|^{2}}{E_{n_{s}1}-E_{n_{0}0}}$
$\displaystyle+$
$\displaystyle\sum_{M_{t}m_{2}^{\prime}}G_{6}(2,M_{t};1,m_{2}^{\prime};1,M;Q)\bigg{\\{}(a^{*}b)\exp[{-i(m_{2}^{\prime})\beta}]+(b^{*}a)\exp[{i(m_{2}^{\prime})\beta}]\bigg{\\}}$
$\displaystyle\times\sum_{n_{t}}\frac{\langle{n_{0}}0\|T_{1}\|{n_{0}}1\rangle\langle{n_{0}}0\|T_{2}\|{n_{t}}2\rangle^{*}\langle{n_{0}}1\|T_{1}\|{n_{t}}2\rangle}{E_{n_{t}2}-E_{n_{0}1}}$
$\displaystyle-$
$\displaystyle\sum_{M_{t}}G_{7}(1,M_{t};2;1,M;Q)\bigg{\\{}(a^{*}b)\exp[{i(M_{t})\beta}]+(b^{*}a)\exp[{-i(M_{t})\beta}]\bigg{\\}}$
$\displaystyle\times\sum_{n_{t}}\frac{\langle{n_{0}}0\|T_{1}\|{n_{0}}1\rangle^{*}\langle{n_{0}}1\|T_{2}\|{n_{t}}1\rangle^{*}\langle{n_{0}}0\|T_{1}\|{n_{t}}1\rangle}{E_{n_{t}1}-E_{n_{0}0}}\,,$
$\displaystyle C_{2,4}^{(31,12)}(1,M)$ $\displaystyle=$
$\displaystyle|b|^{2}\sum_{M_{s}}(-1)^{M_{s}+M}G_{5}(1,M_{s};2;1,M;Q)\cos(M_{s}\alpha)$
(45)
$\displaystyle\times\sum_{n_{s}}\frac{\langle{n_{0}}1\|T_{2}\|{n_{0}}1\rangle|\langle{n_{0}}0\|T_{1}\|{n_{s}}1\rangle|^{2}}{E_{n_{s}1}-E_{n_{0}0}}$
$\displaystyle+$
$\displaystyle\sum_{M_{s}m_{1}^{\prime}}G_{6}(2,M_{s};1,m_{1}^{\prime};1,M;Q)\bigg{\\{}(a^{*}b)\exp[{i(m_{1}^{\prime})\alpha}]+(b^{*}a)\exp[{-i(m_{1}^{\prime})\alpha}]\bigg{\\}}$
$\displaystyle\times\sum_{n_{s}}\frac{\langle{n_{0}}0\|T_{1}\|{n_{0}}1\rangle\langle{n_{0}}0\|T_{2}\|{n_{s}}2\rangle^{*}\langle{n_{0}}1\|T_{1}\|{n_{s}}2\rangle}{E_{n_{s}2}-E_{n_{0}1}}$
$\displaystyle-$
$\displaystyle\sum_{M_{s}}G_{7}(1,M_{s};2;1,M;Q)\bigg{\\{}(a^{*}b)\exp[{i(M_{s})\alpha}]+(b^{*}a)\exp[{-i(M_{s})\alpha}]\bigg{\\}}$
$\displaystyle\times\sum_{n_{s}}\frac{\langle{n_{0}}0\|T_{1}\|{n_{0}}1\rangle^{*}\langle{n_{0}}1\|T_{2}\|{n_{s}}1\rangle^{*}\langle{n_{0}}0\|T_{1}\|{n_{s}}1\rangle}{E_{n_{s}1}-E_{n_{0}0}}\,,$
$\displaystyle C_{3,3}^{(12,23)}(1,M)$ $\displaystyle=$
$\displaystyle\sum_{M_{t}m_{2}^{\prime}}G_{6}(1,M_{t};2,m_{2}^{\prime};1,M;Q)\bigg{\\{}(a^{*}b)\exp[{-i(m_{2}^{\prime})\beta}]+(b^{*}a)\exp[{i(m_{2}^{\prime})\beta}]\bigg{\\}}$
(46)
$\displaystyle\times\sum_{n_{t}}\frac{\langle{n_{0}}0\|T_{1}\|{n_{0}}1\rangle\langle{n_{0}}0\|T_{1}\|{n_{t}}1\rangle^{*}\langle{n_{0}}1\|T_{2}\|{n_{t}}1\rangle}{E_{n_{t}1}-E_{n_{0}1}}$
$\displaystyle-$
$\displaystyle\sum_{M_{t}}G_{7}(2,M_{t};1;1,M;Q)\bigg{\\{}(a^{*}b)\exp[{i(M_{t})\beta}]+(b^{*}a)\exp[{-i(M_{t})\beta}]\bigg{\\}}$
$\displaystyle\times\sum_{n_{t}}\frac{\langle{n_{0}}0\|T_{1}\|{n_{0}}1\rangle^{*}\langle{n_{0}}1\|T_{1}\|{n_{t}}2\rangle^{*}\langle{n_{0}}0\|T_{2}\|{n_{t}}2\rangle}{E_{n_{t}2}-E_{n_{0}0}}\,,$

and

$\displaystyle C_{3,3}^{(31,12)}(1,M)$ $\displaystyle=$
$\displaystyle\sum_{M_{s}m_{1}^{\prime}}G_{6}(1,M_{s};2,m_{1}^{\prime};1,M;Q)\bigg{\\{}(a^{*}b)\exp[{i(m_{1}^{\prime})\alpha}]+(b^{*}a)\exp[{-i(m_{1}^{\prime})\alpha}]\bigg{\\}}$
(47)
$\displaystyle\times\sum_{n_{s}}\frac{\langle{n_{0}}0\|T_{1}\|{n_{0}}1\rangle\langle{n_{0}}0\|T_{1}\|{n_{s}}1\rangle^{*}\langle{n_{0}}1\|T_{2}\|{n_{s}}1\rangle}{E_{n_{s}1}-E_{n_{0}1}}$
$\displaystyle-$
$\displaystyle\sum_{M_{s}}G_{7}(2,M_{s};1;1,M;Q)\bigg{\\{}a^{*}b\exp[{i(M_{s})\alpha}]+b^{*}a\exp[{-i(M_{s})\alpha}]\bigg{\\}}$
$\displaystyle\times\sum_{n_{s}}\frac{\langle{n_{0}}0\|T_{1}\|{n_{0}}1\rangle^{*}\langle{n_{0}}1\|T_{1}\|{n_{s}}2\rangle^{*}\langle{n_{0}}0\|T_{2}\|{n_{s}}2\rangle}{E_{n_{s}2}-E_{n_{0}0}}\,,$

where the functions $G_{4}$, $G_{5}$, $G_{6}$, and $G_{7}$ are defined by

$\displaystyle G_{4}(L_{i},M_{i};L,M)$ $\displaystyle=$ $\displaystyle
16\pi^{2}\frac{[P_{L_{i}+L}^{M_{i}-M}(0)(L_{i}+L-M_{i}+M)!(L_{i},L)^{-1}]^{2}}{(L_{i}+M_{i})!(L_{i}-M_{i})!(L+M)!(L-M)!}\,,$
(48) $\displaystyle G_{5}(L_{i},M_{i};\ell_{k};L,M;Q)$ $\displaystyle=$
$\displaystyle\frac{8\sqrt{\pi^{3}}QP_{\ell_{k}+L_{i}}^{M_{i}}(0)P_{L_{i}}^{M_{i}}(0)(\ell_{k}+L_{i}-M_{i})!}{(2L_{i}+1)^{2}\sqrt{2\ell_{k}+1}(l_{1})!(L_{i}+M_{i})!}\left(\begin{array}[]{ccc}L&\ell_{k}&L\\\
-M&0&M\\\ \end{array}\right)\,,$ (51) $\displaystyle
G_{6}(L_{i},M_{i};\ell_{k},m_{k};L,M;Q)$ $\displaystyle=$
$\displaystyle\frac{8\sqrt{\pi^{3}}Q(\ell_{k})^{-1/2}}{(2L+1)(2L_{i}+1)}\left(\begin{array}[]{ccc}L&\ell_{k}&L_{i}\\\
-M&-m_{k}&M_{i}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\frac{P_{L+L_{i}}^{-M+M_{i}}(0)P_{\ell_{k}}^{m_{k}}(0)(L+L_{i}+M-M_{i})!(\ell_{k}-m_{k})!}{[(L+M)!(L-M)!(L_{i}+M_{i})!(L_{i}-M_{i})!(\ell_{k}+m_{k})!(\ell_{k}-m_{k})!]^{1/2}}\,,$
$\displaystyle G_{7}(L_{i},M_{i};\ell_{k};L,M;Q)$ $\displaystyle=$
$\displaystyle\frac{8\sqrt{\pi^{3}}Q(\ell_{k})^{-1/2}}{(2L+1)(2L_{i}+1)}\sum_{m_{k}}\left(\begin{array}[]{ccc}L&\ell_{k}&L_{i}\\\
-M&m_{k}&M_{i}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\frac{P_{L+\ell_{k}}^{M-m_{k}}(0)P_{L_{i}}^{M_{i}}(0)(L+\ell_{k}-M+m_{k})!(L_{i}-M_{i})!}{[(L+M)!(L-M)!(L_{i}+M_{i})!(L_{i}-M_{i})!(\ell_{k}+m_{k})!(\ell_{k}-m_{k})!]^{1/2}}\,.$

The detailed derivations are given in the Supplemental Material [74]. From
Eqs. (18)–(47), we see that all of these coefficients depend on the atomic
states of the three-body system because they include $a$ and $b$. In other
words, these additive and nonadditive coefficients show a dependence on the
configurations of the three-body system. This is clearly a kind of quantum
three-body collective effect. In the following subsection, we show that these
three-body nonadditive interactions significantly influence the total
interaction potentials. Because of the enhancement through the induction
effect, the nonadditive interactions are large enough to be comparable to (or
even stronger than) the additive interactions at the same order.

In the present paper, we only consider long-range interaction for the
Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li+($1\,^{1}S$) system up to
$\mathcal{O}(R^{-6})$, since the next terms are $C_{7}/R^{7}$, which come from
the third-order perturbation theory.

###  II.7 Specific results extracted from the general expressions

With the zeroth-order wave functions as shown in Eq. (15), the present work
can be easily related to the calculations of long-range interactions for other
two-body or three-body systems. For example, if we set $a={1}$, $b=0$ and
remove the terms involving the Li+($1\,^{1}S$) ion, the formulae can be used
to describe the long-range interactions for the two-body
Li($2\,^{2}S$)-Li($2\,^{2}S$) system; if we set $a=\frac{1}{\sqrt{2}}$,
$b=\pm\frac{1}{\sqrt{2}}$ and remove the terms involving the Li+($1\,^{1}S$)
ion, the formulae can be used to describe the long-range interactions for the
two-body Li($2\,^{2}S$)-Li($2\,^{2}P$) system; if we set $a={1}$, $b=0$ and
remove the terms involving the Li($2\,^{2}P$) atom, the formulae can be used
to describe the long-range interactions for the two-body
Li($2\,^{2}S$)-Li+($1\,^{1}S$) system; if we set $a={1}$, $b=0$ and remove the
terms involving the Li($2\,^{2}S$) atom, the formulae can be used to describe
the long-range interactions for the two-body Li($2\,^{2}P$)-Li+($1\,^{1}S$)
system; and if we set $a={1}$, $b=0$ and $L=0$, the formulae can be used to
describe the long-range interactions for the three-body
Li($2\,^{2}S$)-Li($2\,^{2}S$)-Li+($1\,^{1}S$) system. For these long-range
additive interaction coefficients, we have arranged the following formulae to
show these connections:

$\displaystyle C_{3}^{(12)}(1,M)$ $\displaystyle=$ $\displaystyle
C_{3,\text{dip}}^{(P-S)}\,,$ (60) $\displaystyle C_{3}^{(23)}(1,M)$
$\displaystyle=$ $\displaystyle|b|^{2}C_{3,\text{elst}}^{(P-S^{+})}\,,$ (61)
$\displaystyle C_{3}^{(31)}(1,M)$ $\displaystyle=$
$\displaystyle|a|^{2}C_{3,\text{elst}}^{(P-S^{+})}\,,$ (62) $\displaystyle
C_{4}^{(23)}(1,M)=|a|^{2}C_{4,\text{ind}}^{(S-S^{+})}+|b|^{2}C_{4,\text{ind}}^{(P-S^{+})}(M)\,,$
(63) $\displaystyle
C_{4}^{(31)}(1,M)=|a|^{2}C_{4,\text{ind}}^{(P-S^{+})}(M)+|b|^{2}C_{4,\text{ind}}^{(S-S^{+})}\,,$
(64) $\displaystyle C_{6}^{(12)}(1,M)=C_{6,\text{disp}}^{(P-S)}(M)\,,$ (65)
$\displaystyle
C_{6}^{(23)}(1,M)=|a|^{2}\bigg{\\{}C_{6,\text{ind}}^{(S-S^{+})}+C_{6,\text{disp}}^{(S-S^{+})}\bigg{\\}}+|b|^{2}\bigg{\\{}C_{6,ind}^{(P-S^{+})}(M)+C_{6,\text{disp}}^{(P-S^{+})}(M)\bigg{\\}}\,,$
(66)

and

$\displaystyle
C_{6}^{(31)}(1,M)=|a|^{2}\bigg{\\{}C_{6,\text{ind}}^{(P-S^{+})}(M)+C_{6,\text{disp}}^{(P-S^{+})}(M)\bigg{\\}}+|b|^{2}\bigg{\\{}C_{6,\text{ind}}^{(S-S^{+})}+C_{6,\text{disp}}^{(S-S^{+})}\bigg{\\}}\,,$
(67)

where $C_{3,\text{dip}}^{(P-S)}$ and $C_{6,\text{disp}}^{(P-S)}$,
respectively, represent the dipolar and dispersion interaction coefficients
for the two-body Li($2\,^{2}S$)-Li($2\,^{2}P$) system, which have been given
in the Ref. [10] (also see Eqs. (51) and (52) in the Supplemental Material
[74]). $C_{2n,\text{ind}}^{(S-S^{+})}$ and $C_{2n,\text{disp}}^{(S-S^{+})}$
represent the long-range induction and dispersion coefficients for the
Li($2\,^{2}S$)-Li+($1\,^{1}S$) system, which have been given in the Ref. [12]
(also see Eqs. (48)–(50) in the Supplemental Material [74]). For the two-body
Li($2\,^{2}P$)-Li+($1\,^{1}S$) system, we provide more detail below in Sec.
II.9 as no previous numerical results for the coefficients were found in the
literature. In short, $C_{3,\text{elst}}^{(P-S^{+})}$ represents the
electrostatic interaction between the charge of the ion and the quadrupole
moment of the neutral atom; $C_{2n,\text{ind}}^{(P-S^{+})}$ and
$C_{2n,\text{disp}}^{(P-S^{+})}$ represent the long-range induction and
dispersion coefficients for the Li($2\,^{2}P$)-Li+($1\,^{1}S$) system, where
the formulae of these coefficients are given by Eqs. (220)–(223) in subsection
II.9. Clearly, with these formulae, we can easily relate the long-range
additive interactions of the three-body
Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li+($1\,^{1}S$) system to those of other two-
body or three-body systems. On the other hand, the nonadditive interactions of
the three-body Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li+($1\,^{1}S$) system are
induced by the degeneracy and cannot be decomposed in terms of diatomic
subsystems. This is in contrast to the nondegenerate
Li($2\,^{2}S$)-Li($2\,^{2}S$)-Li+($1\,^{1}S$) system [12], where the
nonadditive interactions start from the third-order energy correction and may
still be used to predict contributions to the long-range potentials between
the $\text{Li}(2\,^{2}S)$ atom and the excited state dimer
$\text{Li}_{2}^{+}(2\,{}^{2}\Sigma_{g,u}^{+},1\,{}^{2}\Sigma_{g,u})$ or
between the $\text{Li}(2\,^{2}P)$ atom and the ground state dimer
$\text{Li}_{2}^{+}(1\,{}^{2}\Sigma_{g/u}^{+})$. We note that
$C_{4,2}^{(12,23)}(1,M)$ [see Eq. (44)] and $C_{2,4}^{(31,12)}(1,M)$ [see Eq.
(45)] may be very important in the study of the interactions between the
cation $\text{Li}^{+}$($1\,^{1}S$) and the excited dimer
$\text{Li}_{2}^{+}(2\,{}^{2}\Sigma_{g,u}^{+},1\,{}^{2}\Sigma_{g,u})$.

###  II.8 Orientation-dependence considerations

In this subsection, we describe the orientation-dependence of the long-range
interactions due to the anisotropic charge distribution of the excited Li
atom. To illustrate the orientation-dependence, we use the two-body
Li($2\,^{2}S$)-Li($2\,^{2}P$) system as an example. The rotation of the two-
body system is illustrated in Fig. 2, where the two-body system is rotated
from $z$-axis to $x$-axis (or one of the two atoms is rotated from “p1” to
“p3”). In this process, we find that the zeroth-order wave functions of the
two-body system $\Psi_{S-P}^{(0)}(M)$ and the corresponding long-range
interaction coefficients change. Thus we can easily get the following
inequality relation

$\displaystyle C_{n,p_{1}}^{(S-P)}(M)\neq C_{n,p_{2}}^{(S-P)}(M)\neq
C_{n,p_{3}}^{(S-P)}(M)\,.$ (68)

Figure 2:  Simple illustration of the rotation of the two-body system from $z$-axis to $x$-axis.  Table 1: The long-range interaction coefficients (in atomic units) of the ∞Li($2\,^{2}S$)-∞Li($2\,^{2}P$) system for the two atoms lying on the $z$-axis and the $x$-axis, respectively, (“p1”) and (“p3”) as shown in Fig. 2. The numbers in parentheses represent the computational uncertainties.  “p1” | $C_{3,z}^{(S-P)}(M=0)$ | $C_{3,z}^{(S-P)}(M=\pm 1)$ | $C_{6,z}^{(S-P)}(M=0)$ | $C_{6,z}^{(S-P)}(M=\pm 1)$
---|---|---|---|---
$\Psi_{S-P,z}^{(0)}$($\beta$=1) | 11.000221(2) |  $-$5.500111(1) | 2075.40(3) | 1406.68(3)
$\Psi_{S-P,z}^{(0)}$($\beta$=-1) |  $-$11.000221(2) | 5.500111(1) | 2075.40(3) | 1406.68(3)
“p3” | $C_{3,x}^{(S-P)}(M=0)$ | $C_{3,x}^{(S-P)}(M=\pm 1)$ | $C_{6,x}^{(S-P)}(M=0)$ | $C_{6,x}^{(S-P)}(M=\pm 1)$
$\Psi_{S-P,x}^{(0)}$($\beta$=1) |  $-$5.500111(1) | 2.750054(1) | 1406.68(3) | 1741.06(5)
$\Psi_{S-P,x}^{(0)}$($\beta$=-1) | 5.500111(1) |  $-$2.750054(1) | 1406.68(3) | 1741.06(5)

In our previous work, we have given the numerical values of the long-range
interaction coefficients for the two atoms lying on the $z$-axis (see Table IX
in Ref. [70]), which would correspond to “p3” in Fig. 2. In the present work,
we use the coordinates of Fig. 1, represented as “p1” in Fig. 2, which
corresponds to the two atoms lying on the $x$-axis. The comparison of these
long-range interaction coefficients is given in Table 1, where the present
values were obtained using highly accurate variational wave functions for the
Li atom in Hylleraas coordinates with finite nuclear mass effects [70]. For
these two specific situations (“p1” and “p3”), we also find the following
relations

$\displaystyle C_{n,p_{3}}^{(S-P)}(M=0)=C_{n,p_{1}}^{(S-P)}(M=\pm 1)\,,$ (69)

which are obeyed by our numerical values of these coefficients shown in Table
1.

In general, the long-range interaction coefficients are given by

$\displaystyle
C_{n,p_{i}}^{(S-P)}(M)=C_{n,p_{1}}^{(S-P)}(M,\cos\theta_{i})\,,$ (70)

where $p_{i}$ is the position i of the atom as shown in Fig. 2 and
$\theta_{i}$ is the corresponding polar angle. For the leading coefficients
$C_{3}$, the formulas are simplified as

$\displaystyle
C_{3,p_{i}}^{(S-P)}(M)=C_{3,p_{1}}^{(S-P)}(M)P_{2}(\cos\theta_{i})\,,$ (71)

where $P_{2}(\cos\theta_{i})$ is the Legendre polynomial. For the other
coefficients $C_{n}$ with $(n>3)$, the parts containing the polar angle
$\theta_{i}$ would be coupled with the virtual states, which cannot be
separated. But we can still utilize the present formulas (calculated at the
“p1” orientation) to give the general formulas by changing the Legendre
polynomial from $P_{l_{I}}^{m_{I}}(0)$ to $P_{l_{I}}^{m_{I}}(\cos\theta_{i})$.
For example, we can use the formula of Eq. (52) in the Supplemental Material
[74] to get the general leading dispersion coefficient
$C_{6,p_{i}}^{(S-P)}(M)$ at orientation pi. Similarly, for other excited Li
dimer and trimers, the long-range interactions also contain such orientation
dependencies. In the next subsection, we will apply these ideas to derive the
long-range potentials for the
$\text{Li}(2\,{}^{2}P)$–$\text{Li}^{+}(1\,{}^{1}S)$ system in the “p3”
orientation. In Sec. III, we will consider the three-body system in detail.

###  II.9 The long-range potentials for the
$\text{Li}(2\,{}^{2}P)$–$\text{Li}^{+}(1\,{}^{1}S)$ system

In this subsection, we use our results to calculate the long-range potentials
of the four states of $\text{Li}_{2}^{+}$, $1\,{}^{2}\Pi_{u}$,
$1\,{}^{2}\Pi_{g}$, $2\,{}^{2}\Sigma_{g}^{+}$, and $2\,{}^{2}\Sigma_{u}^{+}$,
correlating to $\text{Li}(2\,{}^{2}P)$–$\text{Li}^{+}(1\,{}^{1}S)$ system. We
begin by writing down the long-range potential as calculated in our
coordinates, see Fig. 1, with $V^{(P-S^{+})}(R;M)$ corresponding to the two-
body $\text{Li}(2\,{}^{2}P)$–$\text{Li}^{+}(1\,{}^{1}S)$ system in the
orientation “p3” of Fig. 2, which can be written as

$\displaystyle
V^{(P-S^{+})}(R;M)=-\frac{C_{3}(M)}{R^{3}}-\frac{C_{4}(M)}{R^{4}}-\frac{C_{6}(M)}{R^{6}}-\cdots\,,$
(72)

where $C_{3}(M)$ represents the electrostatic interaction between the charge
of the $\text{Li}^{+}(1\,{}^{1}S)$ ion and the quadrupole moment of the
excited $\text{Li}(2\,{}^{2}P)$ atom, $C_{4}(M)$ represents the leading long-
range induction coefficient, which is related to the dipole polarizability of
the $\text{Li}(2\,{}^{2}P)$ atom, and the $C_{6}(M)$ is the sum of long-range
induction coefficients $C_{6,\text{ind}}$ and dispersion coefficients
$C_{6,\text{disp}}$. The formulae of these coefficients are given by

$\displaystyle
C_{3}(M)=C_{3,\text{elst}}^{(P-S^{+})}(M)=Q(-1)^{1+M}\sqrt{\frac{\pi}{5}}\left(\begin{array}[]{ccc}1&2&1\\\
-M&0&M\\\ \end{array}\right)\langle{n_{0}}1\|T_{2}\|{n_{0}}1\rangle\ \,,$ (75)
$\displaystyle C_{4}(M)=C_{4,\text{ind}}^{(P-S^{+})}(M)$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{4\pi}\sum_{n_{t}L_{t}}\frac{G_{1}(L_{t},0;1,1;1,M)|\langle{n_{0}}1\|T_{1}\|{n_{t}}L_{t}\rangle|^{2}}{E_{n_{t}L_{t}}-E_{n_{0}1}^{(0)}}\,,$
(76) $\displaystyle
C_{6}(M)=C_{6,\text{disp}}^{(P-S^{+})}(M)+C_{6,\text{ind}}^{(P-S^{+})}(M)\,,$
(77)

where

$\displaystyle C_{6,\text{disp}}^{(P-S^{+})}(M)$ $\displaystyle=$
$\displaystyle\sum_{n_{t}n_{u}L_{t}}\frac{G_{1}(L_{t},1;1,1;1,M)|\langle{n_{0}}1\|T_{1}\|{n_{t}}L_{t}\rangle|^{2}\langle{n_{0}^{\prime}0}\|T_{1}\|{n_{u}}1\rangle|^{2}}{(E_{n_{t}L_{t}}-E_{n_{0}1}^{(0)})+(E_{n_{u}1}-E_{n_{0}^{\prime}0}^{(0)})}\,,$
(78)

and

$\displaystyle C_{6,\text{ind}}^{(P-S^{+})}(M)$ $\displaystyle=$
$\displaystyle\frac{Q^{2}}{4\pi}\sum_{n_{t}n_{u}L_{t}}\bigg{\\{}\frac{G_{1}(L_{t},0;2,2;1,M)|\langle{n_{0}}1\|T_{2}\|{n_{t}}L_{t}\rangle|^{2}}{E_{n_{t}L_{t}}-E_{n_{0}1}^{(0)}}$
(79) $\displaystyle+$
$\displaystyle\frac{G_{1}(L_{t},0;1,3;1,M)\langle{n_{0}}1\|T_{1}\|{n_{t}}L_{t}\rangle^{*}\langle{n_{0}}1\|T_{3}\|{n_{t}}L_{t}\rangle}{E_{n_{t}L_{t}}-E_{n_{0}1}^{(0)}}$
$\displaystyle+$
$\displaystyle\frac{G_{1}(L_{t},0;3,1;1,M)\langle{n_{0}}1\|T_{3}\|{n_{t}}L_{t}\rangle^{*}\langle{n_{0}}1\|T_{1}\|{n_{t}}L_{t}\rangle}{E_{n_{t}L_{t}}-E_{n_{0}1}^{(0)}}\bigg{\\}}\,,$

and the $G_{1}$ function is defined in Eq. (II.5).

Table 2: The long-range interaction coefficients (in atomic units) of the Li($2\,^{2}P$)-Li+($1\,^{2}S$) system for the two particles lying on $z$-axis and $x$-axis, respectively, “p1” and “p3” as shown in Fig. 2. Position “p1” corresponds to standard molecular $\Sigma$, $\Pi$ nomenclature where the $z$-axis joins the atom and the ion. The numbers in parentheses represent the computational uncertainties. “p1” | $C_{3,z}^{(P-S^{+})}(M=0)$ | $C_{3,z}^{(P-S^{+})}(M=\pm 1)$ | $C_{4,z}^{(P-S^{+})}(M=0)$ | $C_{4,z}^{(P-S^{+})}(M=\pm 1)$ | $C_{6,z}^{(P-S^{+})}(M=0)$ | $C_{6,z}^{(P-S^{+})}(M=\pm 1)$
---|---|---|---|---|---|---
∞Li | 10.8199392(2) |  $-$5.4099696(1) | 61.8515(2) | 64.2838(2) | 9811.485(6) |  $-$1820.6261(3)
7Li | 10.8192592(2) |  $-$5.4096296(1) | 61.8385(2) | 64.2911(2) | 9811.274(6) |  $-$1820.7205(2)
6Li | 10.8191462(2) |  $-$5.4095731(1) | 61.8364(2) | 64.2925(2) | 9811.239(6) |  $-$1820.7362(3)
“p3” | $C_{3,x}^{(P-S^{+})}(M=0)$ | $C_{3,x}^{(P-S^{+})}(M=\pm 1)$ | $C_{4,x}^{(P-S^{+})}(M=0)$ | $C_{4,x}^{(P-S^{+})}(M=\pm 1)$ | $C_{6,x}^{(P-S^{+})}(M=0)$ | $C_{6,x}^{(P-S^{+})}(M=\pm 1)$
∞Li |  $-$5.4099696(1) | 2.7049847(1) | 64.2838(2) | 63.0676(3) |  $-$1820.6261(3) | 3995.429(3)
7Li |  $-$5.4096296(1) | 2.7048148(1) | 64.2911(2) | 63.0648(2) |  $-$1820.7205(2) | 3995.276(3)
6Li |  $-$5.4095731(1) | 2.7047866(1) | 64.2925(2) | 63.0643(3) |  $-$1820.7362(3) | 3995.251(3)
Figure 3:  Long-range potentials (in atomic units) of the Li($2\,^{2}P$)-Li+($2\,^{1}S$) system calculated in the coordinate system of Fig. 1, which corresponds to “p3” of Fig. 2.  Table 3: Convergence of the long-range interaction coefficients $C_{3,x}^{(P-S^{+})}(M)$ for the ∞Li($2\,^{2}P$)-∞Li+($1\,^{2}S$) system, where two particles lie on the $x$-axis (“p3”) as shown in Fig. 2. $N_{P}$ denotes the size of the basis for the $P$ state of the ∞Li($2\,^{2}P$) atom. $N_{P}$ | $C_{3,x}^{(P-S^{+})}(M=0)$ | $C_{3,x}^{(P-S^{+})}(M=\pm 1)$
---|---|---
1174 |  $-$5.409965844 | 2.704982922
2091 |  $-$5.409968720 | 2.704984360
3543 |  $-$5.409969427 | 2.704984713
5761 |  $-$5.409969563 | 2.704984781
Extrapolated |  $-$5.4099696(1) | 2.7049847(1)
Table 4: Convergence of the long-range interaction coefficients $C_{4,x}^{(P-S^{+})}(M)$ for the ∞Li($2\,^{2}P$)-∞Li+($1\,^{2}S$) system, where two particles lie on $x$-axis (“p3”) as shown in Fig. 2. $N_{P}$ denotes the size of basis for the $P$ state of the ∞Li atom. $N_{S}$, $N_{(pp)P}$, and $N_{D}$, respectively, denote the sizes of the bases for the corresponding intermediate states of symmetries $S$, $P$, and $D$, and $(pp)P$ stands for the main configuration of two $p$ electrons coupled to form a total angular momentum of $P$ (since the contribution from the $(pp)P$ configuration is well converged at $N_{(pp)P}=3413$, we did not increase $N_{(pp)P}$ any further [70]). $(N_{P},N_{S},N_{(pp)P},N_{D})$ | $C_{4,x}^{(P-S^{+})}(M=0)$ | $C_{4,x}^{(P-S^{+})}(M=\pm 1)$
---|---|---
(1174,1589,1106,1174) | 64.282596 | 63.061604
(2091,2625,2002,2091) | 64.283174 | 63.066184
(3543,4172,3413,3543) | 64.283498 | 63.067033
(5761,6412,3413,5761) | 64.283634 | 63.067338
Extrapolated | 64.2838(2) | 63.0676(3)
Table 5: Convergence of the long-range interaction coefficients $C_{6,x}^{(P-S^{+})}(M)$ for the ∞Li($2\,^{2}P$)-∞Li+($1\,^{2}S$) system, where two particles lie on the $x$-axis (“p3”) as shown in Fig. 2. $N_{P}$ denotes the sizes of bases for the $P$ state of the ∞Li atom. $N_{S}$, $N_{(pp)P}$, and $N_{D}$, respectively, are the sizes of basis for the corresponding intermediate states of symmetries $S$, $P$, and $D$, and $(pp)P$ stands for the main configuration of two $p$ electrons coupled to form a total angular momentum of $P$ (since the contribution from the $(pp)P$ configuration is well converged at $N_{(pp)P}=3413$, we did not increase $N_{(pp)P}$ any further [70]). $N_{S}^{+}$ and $N_{P}^{+}$, respectively, denote the sizes of the bases for the ground state and the intermediate states of symmetry $P$ of ∞Li+. $(N_{S}^{+},N_{P}^{+};N_{P},N_{S},N_{(pp)P},N_{D})$ | $C_{6,x}^{(P-S^{+})}(M=0)$ | $C_{6,x}^{(P-S^{+})}(M=\pm 1)$
---|---|---
(504,728,1174,1589,1106,1174) |  $-$1820.631774 | 3995.37910
(744,1120,2091,2625,2002,2091) |  $-$1820.627179 | 3995.40911
(1050,1632,3543,4172,3413,3543) |  $-$1820.626701 | 3995.42357
(1430,2280,5761,6412,3413,5761) |  $-$1820.626499 | 3995.42669
Extrapolated |  $-$1820.6261(3) | 3995.429(3)

The molecular states for $\text{Li}_{2}^{+}$ correlating to
$\text{Li}(2\,{}^{2}P)$–$\text{Li}^{+}(1\,{}^{1}S)$ system are the
$2\,{}^{2}\Sigma_{g}^{+}$, $2\,{}^{2}\Sigma_{u}^{+}$, $1\,{}^{2}\Pi_{g}$ and
$1\,{}^{2}\Pi_{u}$ states (we do not consider fine structure). We calculated
the long-range interaction coefficients using Eqs. (220)–(223), which include
electrostatic, induction, and dispersion energies up to $\mathcal{O}(R^{-6})$;
the corresponding numerical values are given in Table 2 and plotted in Fig. 3.
These correspond to position “p3” of the $\text{Li}(2\,^{2}P)$ atom as
indicated in Fig. 2. Convergence studies of these long-range interaction
coefficients $C_{3,x}^{(P-S^{+})}(M)$, $C_{4,x}^{(P-S^{+})}(M)$, and
$C_{6,x}^{(P-S^{+})}(M)$ are given in, respectively, Tables 3, 4, and 5. In
these tables, $N_{P}$ denotes the size of the basis for the $P$ state of the
∞Li atom, and $N_{L}$ denotes the size of the basis for the corresponding
intermediate states of symmetry $L$. Similarly, $N_{S}^{+}$ and $N_{P}^{+}$
denote respectively the sizes of the bases for the ground state and the
intermediate states of symmetry $P$ of the ∞Li+ ion.

Table 6: Comparison of the polarizability components ($\alpha^{p_{1}}_{zz}$, $\alpha^{p_{1}}_{xx}$) and ($\alpha^{p_{3}}_{zz}$, $\alpha^{p_{3}}_{xx}$) (in atomic units) for the excited state $2\,^{2}P$ of ${}^{\infty}\text{Li}$. For the two coordinate systems, we have $\alpha^{p_{1}}_{xx}$= $\alpha_{1}$\+ $\alpha_{1}^{T}$ or $\alpha^{p_{1}}_{zz}$= $\alpha_{1}-2\alpha_{1}^{T}$(see Refs. [75, 76, 77, 78]) and $\alpha^{p_{3}}_{zz}$= $\alpha_{1}$\+ $\alpha_{1}^{T}$ or $\alpha^{p_{3}}_{xx}$= $\alpha_{1}-\frac{1}{2}\alpha_{1}^{T}$, respectively, where the electric field lies in $z$ or $x$ direction, expressed in terms of the principal polarizabilities $\alpha_{1}$ (scalar) and $\alpha{{}_{1}^{T}}$ (tensor). Reference | $\alpha^{p_{1}}_{zz}=2C_{4,z}^{(P-S^{+})}{(M=0)}$ | $\alpha^{p_{1}}_{xx}=2C_{4,z}^{(P-S^{+})}{(M=\pm 1)}$
---|---|---
Pipin & Bishop [79](1993) | 123.634 | 128.449
Rérat et al. [77] (1997) | 131 | 129
Cohen & Themelis [80] (2005) | 122.94 | 128.13
Johnson et al. [81] (2008) | 123.81 | 128.580
This work | 123.703(4) | 128.5676(4)
| $\alpha^{p_{3}}_{zz}=2C_{4,x}^{(P-S^{+})}{(M=0)}$ | $\alpha^{p_{3}}_{xx}=2C_{4,x}^{(P-S^{+})}{(M=\pm 1)}$
Pipin & Bishop [79](1993) | 128.449 | 126.0415
Rérat et al. [77] (1997) | 129 | 131
Cohen & Themelis [80] (2005) | 128.13 | 125.535
Johnson et al. [81] (2008) | 128.580 | 126.195
This work | 128.5676(4) | 126.1352(6)

In order to apply the results to the $\Sigma$ and $\Pi$ molecular states of
standard molecular nomenclature, where the $z$-axis joins the atom and ion, we
must first apply the considerations of the previous Sec. II.8 to express our
results in terms of position “p1” of Fig. 2. The analysis yields the
coefficients, with the numerical values given in Table 2. The corresponding
general coefficients are defined by

$\displaystyle
C_{3,p_{i}}^{(P-S^{+})}(M)=C_{3,p_{1}}^{(P-S^{+})}(M)P_{2}(\cos\theta_{i})\,,$
(80) $\displaystyle
C_{4,p_{i}}^{(P-S^{+})}(M=0)=\frac{1}{2}\alpha_{zz}^{p_{i}}=\frac{1}{2}[\alpha_{1}-2\alpha^{T}_{1,p_{1}}P_{2}(\cos\theta_{i})]\,,$
(81)

and

$\displaystyle C_{4,p_{i}}^{(P-S^{+})}(M=\pm
1)=\frac{1}{2}\alpha_{xx}^{p_{i}}=\frac{1}{2}[\alpha_{1}+\alpha^{T}_{1,p_{1}}P_{2}(\cos\theta_{i})]\,,$
(82)

where ($\alpha^{p_{i}}_{zz}$, $\alpha^{p_{i}}_{xx}$) are, respectively, the
polarizability components, along the $z$ and $x$ directions of an applied
electric field [75, 76, 77, 78], $\alpha_{1}$ and $\alpha_{1}^{T}$ are the
principal scalar and tensor polarizibilities of Li($2\,^{2}P$) atom [70], and
$P_{2}(\cos\theta_{i})$ is the Legendre polynomial. The comparison of the
polarizability components ($\alpha^{p_{1}}_{zz}$, $\alpha^{p_{1}}_{xx}$) and
($\alpha^{p_{3}}_{zz}$, $\alpha^{p_{3}}_{xx}$) is given in Table. 6. In the
present configuration (as shown in Fig. 1), these components can be related to
the leading induction coefficients $C_{4,x}(M)$ (as shown in Table 2) by
$\alpha^{p_{3}}_{zz}=2C_{4,x}(M=0)$ and $\alpha^{p_{3}}_{xx}=2C_{4,x}(M=\pm
1)$. According to the symmetry of the degenerate system [75, 76, 77, 78], we
can connect the polarizability components ($\alpha^{p_{3}}_{zz}$,
$\alpha^{p_{3}}_{xx}$) with the principal polarizabilities, scalar
($\alpha_{1}$) and tensor ($\alpha_{1}^{T}$), of the Li($2\,^{2}P$) atom by
$\alpha_{1}=\frac{1}{3}(\alpha^{p_{3}}_{zz}+2\alpha^{p_{3}}_{xx})$ and
$\alpha_{1}^{T}=\frac{2}{3}(\alpha^{p_{3}}_{zz}-\alpha^{p_{3}}_{xx})$. For
example, using the present data from Table 6, we find $\alpha_{1}=126.9460$
and $\alpha_{1}^{T}=1.6216$ in agreement with Ref. [70, Table VIII]. Further
details concerning the polarizability components ($\alpha^{p_{1}}_{zz}$,
$\alpha^{p_{1}}_{xx}$) for the two particles lying on the $z$-axis are given
in, for example, Refs. [75, 76, 77, 78]. For the coefficient $C_{6}$, we can
use the formula of Eq. (77) by changing the Legendre polynomial from
$P_{l_{I}}^{m_{I}}(0)$ to $P_{l_{I}}^{m_{I}}(\cos\theta_{i})$ to get the
general $C_{6,p_{i}}^{(P-S^{+})}(M)$ at orientation pi.

The long-range potential energy functions expressed relative to “p1” follow
from Table 2. For example, for the $2\,{}^{2}\Sigma_{g,u}$ states of the
${}^{\infty}\text{Li}_{2}$, we have

$\displaystyle
V^{(P-S^{+})}(R;\Sigma)=-\frac{10.8199392}{R^{3}}-\frac{61.8515}{R^{4}}-\frac{9811.485}{R^{6}}-\cdots\,,$
(83)

and for the $1\,{}^{1}\Pi_{g,u}$ states, we have

$\displaystyle
V^{(P-S^{+})}(R;\Pi)=\frac{5.4099696}{R^{3}}-\frac{64.2838}{R^{4}}+\frac{1820.6261}{R^{6}}-\cdots\,.$
(84)

Similar expressions may be written for ${}^{6}\text{Li}_{2}^{+}$ and
${}^{7}\text{Li}_{2}^{+}$ using Table 2. To the best of our knowledge, the
expressions in Eqs. (83) and (84) are unavailable in the literature.

Magnier et al. [54] calculated the long-range potentials with the inclusion of
exchange, electrostatic, induction, and dispersion interactions up to
$\mathcal{O}(R^{-8})$; the results were presented graphically. While a direct
comparison of long-range coefficients is not possible, we can calculate the
exchange energies using the expressions given by Magnier et al. and add that
to our long-range potentials to compare with their total potentials for each
of the four molecular states. When the exchange energy and the long-range
potential energy have opposite signs, a long-range well or barrier results;
these singular features provide good quantitative checks between Magnier et
al. and the present work. From Magnier et al. [54], Figs. (8)–(10), it is
evident that the exchange energies are positive for the
$2\,{}^{2}\Sigma_{u}^{+}$ and $1\,{}^{2}\Pi_{g}$ states and negative for the
$2\,{}^{2}\Sigma_{g}^{+}$ and $1\,{}^{2}\Pi_{u}$ states, while the exchange
splitting for the pair of $2\,{}^{2}\Sigma^{+}$ states is larger by a factor
of $R/2$ compared to the pair of $1\,{}^{2}\Pi$ states, where $R$ is the
internuclear distance. Calculations show that the two $\Sigma$ states and the
$1\,{}^{2}\Pi_{u}$ state form potential wells, while the $1\,{}^{2}\Pi_{g}$
state is purely repulsive [55]. It is evident from the data in Eq. (84) that
the net positive long-range potential and positive exchange energy completely
account for the repulsive $1\,{}^{2}\Pi_{g}$ state. Of the three states with
potential wells, the $2\,{}^{2}\Sigma_{u}^{+}$ state well exists at the
greatest internuclear distance, about $25\,a_{0}$, with a depth of only
$127\,\text{cm}^{-1}$ according to a recent model potential calculation [55].
With our long-range expansion of Table 2 evaluated for ${}^{\infty}\text{Li}$
as in Eq. (83) and using Eq. (8) of Ref. [54], to estimate the contribution of
the exchange energy222 By close comparison of Eqs. (2.13) and (3.7) of Ref.
[82], we believe that the factor “2” in the denominator of $D$ given in Eq.
(8) of Ref. [54] should be replaced by “$m!$”. We evaluated the exchange
energy splittings with this evident correction included. , we find a well of
depth $119\,\text{cm}^{-1}$ at $R=25.8\,a_{0}$ to be compared to the depth
$124\,\text{cm}^{-1}$ at $25.7\,a_{0}$ obtained by Magnier et al. using a
long-range expansion and the exchange energy. We also obtain for the the
$1\,{}^{2}\Pi_{u}$ state using Eq. (84) a potential barrier of
$36\,\text{cm}^{-1}$ at $R=23.9\,a_{0}$, compared to $40\,\text{cm}^{-1}$ at
$23.4\,a_{0}$ found by Magnier et al. The agreement of the well and the
barrier positions and energies calculated using Eqs. (83) and (84), with the
similar calculations of Magnier et al. is satisfactory. The
$2\,{}^{2}\Sigma_{u}^{+}$ state is an example of a long-range molecular state
[83]. Moreover, we do not attempt to reproduce the wells of the
$2\,{}^{2}\Sigma_{u}^{+}$ or $1\,{}^{2}\Pi_{u}$ states, because it is evident
from Ref. [54, Figs. 8–9] that these potential wells are fully realized with
the inclusion of charge overlap (i.e. in quantum-chemical calculations [54,
55]).

Having thus demonstrated that two-body long-range interaction potentials can
be extracted from our results, and providing some coefficients that were
previously unavailable in the literature [viz. Table 2 and Eqs. (83) and
(84)], we turn back to the three-body system.

##  III Results and Discussions

As in Secs. II.8–II.9, we use highly accurate variational wave functions for
lithium atoms and ions in Hylleraas coordinates with finite nuclear mass
effects to evaluate the numerical values [70]. We note that in general the
zeroth-order wave functions are obtained by using degenerate perturbation
theory through Eq. (10) in the Supplemental Material [74] and there are
intrinsic geometrical dependencies that complicate the analysis. In
particular, the zeroth-order wave functions change with the geometry (interior
angles and interatomic separations) of the three-body system. However, when
$R_{23}=R_{31}=R$, we have the matrix elements $\Delta_{12}$=$\Delta_{21}$,
see Eq. (10) in the Supplemental Material [74], and the geometrical
dependencies don’t appear in the zeroth-order wave functions, simplifying the
analysis of the three-body system. Therefore, in this section, we consider the
Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li${}^{+}(1\,^{1}S$) system for the
configurations where $R_{23}=R_{31}=R$.

In Sec. III.1 we introduce the zeroth-order wave functions and in Sec. III.2
provide the numerical values of these additive coefficients. In Sec. III.3, we
focus on the two specific arrangements of the three particles, collinear and
in an equilateral triangle, providing the nonadditive coefficients.

###  III.1 Additive coefficients: Wave functions

With respect to the “p1” orientation as shown in Fig. 2, we calculate the
long-range additive potentials for the three-body system lying collinearly on
the $z$-axis. According to degenerate perturbation theory, the corresponding
zeroth-order wave functions are

$\displaystyle\Psi_{1,z}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\bigg{[}\left|{n_{0}}1_{z};{n_{0}}0;{n_{0}^{\prime}}0\right\rangle+\left|{n_{0}}0;{n_{0}}1_{z};{n_{0}^{\prime}}0\right\rangle\bigg{]}\,,$
(85) $\displaystyle\Psi_{2,z}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\bigg{[}\left|{n_{0}}1_{z};{n_{0}}0;{n_{0}^{\prime}}0\right\rangle-\left|{n_{0}}0;{n_{0}}1_{z};{n_{0}^{\prime}}0\right\rangle\bigg{]}\,,$
(86)

where the symbol $z$ indicates the three-particles lying on the $z$-axis for
the configurations of $R_{23}=R_{31}=R$.

For three particles lying in the $x$-$y$ plane as shown in Fig. 1, the
corresponding zeroth-order wave functions are

$\displaystyle\Psi_{1,\bot}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\bigg{[}\left|{n_{0}}1;{n_{0}}0;{n_{0}^{\prime}}0\right\rangle+\left|{n_{0}}0;{n_{0}}1;{n_{0}^{\prime}}0\right\rangle\bigg{]}\,,$
(87) $\displaystyle\Psi_{2,\bot}^{(0)}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\bigg{[}\left|{n_{0}}1;{n_{0}}0;{n_{0}^{\prime}}0\right\rangle-\left|{n_{0}}0;{n_{0}}1;{n_{0}^{\prime}}0\right\rangle\bigg{]}\,,$
(88)

where the symbol $\bot$ indicates specificity to the the $x$-$y$ planar
configuration with $R_{23}=R_{31}=R$. Note that Eqs. (87) and (88) include the
special case of the three particles lying collinearly on the $x$-axis, i.e.
the orientation “p3”.

###  III.2 Additive coefficients: Evaluation

Table 7: The long-range additive interaction coefficients (in atomic units) of the Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li+($2\,^{1}S$) system for two different types of the zeroth-order wave functions, where the three particles lie collinearly on the $z$-axis (similar to the “p1” orientation of the two-body system shown in Fig. 2). The numbers in parentheses represent the computational uncertainties. Coefficients | $\Psi_{1,z}^{(0)}$ | $\Psi_{2,z}^{(0)}$ | $\Psi_{1,z}^{(0)}$ | $\Psi_{2,z}^{(0)}$ | $\Psi_{1,z}^{(0)}$ | $\Psi_{2,z}^{(0)}$
---|---|---|---|---|---|---
|  ∞Li |  7Li |  6Li
$C_{3,z}^{(12)}(1,M=0)$ | 11.000221(2) |  $-$11.000221(2) | 11.001853(2) |  $-$11.001853(2) | 11.002125(2) |  $-$11.002125(2)
$C_{3,z}^{(23)}(1,M=0)$ | 5.4099696(1) | 5.4099696(1) | 5.4096296(1) | 5.4096296(1) | 5.4095731(1) | 5.4095731(1)
$C_{3,z}^{(31)}(1,M=0)$ | 5.4099696(1) | 5.4099696(1) | 5.4096296(1) | 5.4096296(1) | 5.4095731(1) | 5.4095731(1)
$C_{4,z}^{(23)}(1,M=0)$ | 71.9539(4) | 71.9539(4) | 71.9594(4) | 71.9594(4) | 71.9604(4) | 71.9604(4)
$C_{4,z}^{(31)}(1,M=0)$ | 71.9539(4) | 71.9539(4) | 71.9594(4) | 71.9594(4) | 71.9604(4) | 71.9604(4)
$C_{6,z}^{(12)}(1,M=0)$ | 2075.40(3) | 2075.40(3) | 2076.08(7) | 2076.08(7) | 2076.19(7) | 2076.19(7)
$C_{6,z}^{(23)}(1,M=0)$ | 5263.218(3) | 5263.218(3) | 5263.151(3) | 5263.151(3) | 5263.140(3) | 5263.140(3)
$C_{6,z}^{(31)}(1,M=0)$ | 5263.218(3) | 5263.218(3) | 5263.151(3) | 5263.151(3) | 5263.140(3) | 5263.140(3)
$C_{3,z}^{(12)}(1,M=\pm 1)$ |  $-$5.500111(1) | 5.500111(1) |  $-$5.500926(1) | 5.500926(1) |  $-$5.501062(1) | 5.501062(1)
$C_{3,z}^{(23)}(1,M=\pm 1)$ |  $-$2.7049847(1) |  $-$2.7049847(1) |  $-$2.7048148(1) |  $-$2.7048148(1) |  $-$2.7047866(1) |  $-$2.7047866(1)
$C_{3,z}^{(31)}(1,M=\pm 1)$ |  $-$2.7049847(1) |  $-$2.7049847(1) |  $-$2.7048148(1) |  $-$2.7048148(1) |  $-$2.7047866(1) |  $-$2.7047866(1)
$C_{4,z}^{(23)}(1,M=\pm 1)$ | 73.1701(4) | 73.1701(4) | 73.1859(4) | 73.1859(4) | 73.1885(4) | 73.1885(4)
$C_{4,z}^{(31)}(1,M=\pm 1)$ | 73.1701(4) | 73.1701(4) | 73.1859(4) | 73.1859(4) | 73.1885(4) | 73.1885(4)
$C_{6}^{(12)}(1,M=\pm 1)$ | 1406.68(3) | 1406.68(3) | 1407.15(5) | 1407.15(5) | 1407.20(2) | 1407.20(2)
$C_{6,z}^{(23)}(1,M=\pm 1)$ |  $-$552.8371(7) |  $-$552.8371(7) |  $-$552.8460(5) |  $-$552.8460(5) |  $-$552.8472(7) |  $-$552.8472(7)
$C_{6,z}^{(31)}(1,M=\pm 1)$ |  $-$552.8371(7) |  $-$552.8371(7) |  $-$552.8460(5) |  $-$552.8460(5) |  $-$552.8472(7) |  $-$552.8472(7)
Table 8: The long-range additive interaction coefficients (in atomic units) of the Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li+($2\,^{1}S$) system for two different types of the zeroth-order wave functions, where the three particles lie in the $x$-$y$ plane with $R_{23}=R_{31}=R$ as shown in Fig. 1. Note this includes the special case of the three particles collinear on the $x$-axis. The numbers in parentheses represent the computational uncertainties.  Coefficients | $\Psi_{1,\bot}^{(0)}$ | $\Psi_{2,\bot}^{(0)}$ | $\Psi_{1,\bot}^{(0)}$ | $\Psi_{2,\bot}^{(0)}$ | $\Psi_{1,\bot}^{(0)}$ | $\Psi_{2,\bot}^{(0)}$
---|---|---|---|---|---|---
|  ∞Li |  7Li |  6Li
$C_{3}^{(12)}(1,M=0)$ |  $-$5.500111(1) | 5.500111(1) |  $-$5.500926(1) | 5.500926(1) |  $-$5.501062(1) | 5.501062(1)
$C_{3}^{(23)}(1,M=0)$ |  $-$2.7049847(1) |  $-$2.7049847(1) |  $-$2.7048148(1) |  $-$2.7048148(1) |  $-$2.7047866(1) |  $-$2.7047866(1)
$C_{3}^{(31)}(1,M=0)$ |  $-$2.7049847(1) |  $-$2.7049847(1) |  $-$2.7048148(1) |  $-$2.7048148(1) |  $-$2.7047866(1) |  $-$2.7047866(1)
$C_{4}^{(23)}(1,M=0)$ | 73.1701(4) | 73.1701(4) | 73.1859(4) | 73.1859(4) | 73.1885(4) | 73.1885(4)
$C_{4}^{(31)}(1,M=0)$ | 73.1701(4) | 73.1701(4) | 73.1859(4) | 73.1859(4) | 73.1885(4) | 73.1885(4)
$C_{6}^{(12)}(1,M=0)$ | 1406.68(3) | 1406.68(3) | 1407.15(5) | 1407.15(5) | 1407.20(2) | 1407.20(2)
$C_{6}^{(23)}(1,M=0)$ |  $-$552.8371(7) |  $-$552.8371(7) |  $-$552.8460(5) |  $-$552.8460(5) |  $-$552.8472(7) |  $-$552.8472(7)
$C_{6}^{(31)}(1,M=0)$ |  $-$552.8371(7) |  $-$552.8371(7) |  $-$552.8460(5) |  $-$552.8460(5) |  $-$552.8472(7) |  $-$552.8472(7)
$C_{3}^{(12)}(1,M=\pm 1)$ | 2.750054(1) |  $-$2.750054(1) | 2.750462(1) |  $-$2.750462(1) | 2.750530(1) |  $-$2.750530(1)
$C_{3}^{(23)}(1,M=\pm 1)$ | 1.3524924(1) | 1.3524924(1) | 1.3524074(1) | 1.3524074(1) | 1.3523932(1) | 1.3523932(1)
$C_{3}^{(31)}(1,M=\pm 1)$ | 1.3524924(1) | 1.3524924(1) | 1.3524074(1) | 1.3524074(1) | 1.3523932(1) | 1.3523932(1)
$C_{4}^{(23)}(1,M=\pm 1)$ | 72.5620(5) | 72.5620(5) | 72.5727(5) | 72.5727(5) | 72.5745(5) | 72.5745(5)
$C_{4}^{(31)}(1,M=\pm 1)$ | 72.5620(5) | 72.5620(5) | 72.5727(5) | 72.5727(5) | 72.5745(5) | 72.5745(5)
$C_{6}^{(12)}(1,M=\pm 1)$ | 1741.06(5) | 1741.06(5) | 1741.59(4) | 1741.59(4) | 1741.68(4) | 1741.68(4)
$C_{6}^{(23)}(1,M=\pm 1)$ | 2355.190(2) | 2355.190(2) | 2355.152(2) | 2355.152(2) | 2355.146(2) | 2355.146(2)
$C_{6}^{(31)}(1,M=\pm 1)$ | 2355.190(2) | 2355.190(2) | 2355.152(2) | 2355.152(2) | 2355.146(2) | 2355.146(2)

Using the degenerate perturbation theory, we find that different from the
ground state Li${}_{3}^{+}$ trimer (where there is no analogous quantum three-
body effect for these long-range additive coefficients [12]), the atomic
states ($a$, $b$) and the corresponding additive coefficients are changing
with different geometries of the three-body system for the excited
Li${}_{3}^{+}$ trimer. This phenomenon is absolutely a kind of quantum three-
body effect, which is caused by the degeneracy of the excited Li${}_{3}^{+}$
trimer. While for the specific geometries with $R_{23}=R_{31}=R$, we find that
the atomic states and the corresponding additive interaction coefficients
would remain unchanged due to the symmetry of the three-body system. This
feature makes these coefficients be useful in the quantum chemistry studies.
In the present paper, we give the calculations of the long-range coefficients
for these specific configurations. The additive coefficients $C_{n}^{(IJ)}$
[to be in Eqs. (17) and (25)] are calculated for the specific three-body
Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li${}^{+}(1\,^{1}S$) system lying on the
$z$-axis or in the $x$-$y$ plane. The numerical values are shown in Tables 7
and 8, where we can also find the orientation dependence (that are
demonstrated in Sec. II.8 for two-body system) of the long-range interaction
coefficients for the excited Li${}_{3}^{+}$ trimer.

For the numerical values of these additive coefficients, we note that the
leading long-range interaction coefficient between two neutral atoms
$C_{3}^{(12)}(1,M)$ can be positive (attractive) or negative (repulsive)
corresponding to the different atomic states related to the symmetry of the
system. The leading terms $C_{3,z}^{(23)}(1,M=\pm 1)$, $C_{3,z}^{(31)}(1,M=\pm
1)$ (see Table 7) and $C_{3}^{(23)}(1,M=0)$, $C_{3}^{(31)}(1,M=0)$ (see Table
8) are always negative, which represents the repulsive interactions between
the charge of the Li${}^{+}(1\,^{1}S$) ion and the permanent electric
quadruple moments of the Li($2\,^{2}P)$ atom generated by its anisotropic
charge distribution along the $z$-axis for $M=\pm 1$ and $M=0$ states,
respectively. The leading terms $C_{3,z}^{(23)}(1,M=0)$,
$C_{3,z}^{(31)}(1,M=0)$ (see Table 7) and $C_{3}^{(23)}(1,M=\pm 1)$,
$C_{3}^{(31)}(1,M=\pm 1)$ (see Table 8) are always positive (attractive),
which is caused by induction effect of the Li+($1\,^{1}S)$ atom. Similarly,
the inductive terms $C_{4,z}^{(23)}(1,M)$, $C_{4,z}^{(31)}(1,M)$ (see Table 7)
and $C_{4}^{(23)}(1,M)$, $C_{4}^{(31)}(1,M)$ (see Table 8) are also always
positive (attractive). Their numerical values are the linear combinations of
the inductive interactions of the Li($2\,^{2}S$)-Li${}^{+}(1\,^{1}S$) system
and the Li($2\,^{2}P$)-Li${}^{+}(1\,^{1}S$) system [Eqs. (63) and (64)]. The
$C_{6,z}^{(23)}(1,M)$, $C_{6,z}^{(31)}(1,M)$ (see Table 7) and
$C_{6}^{(23)}(1,M)$, $C_{6}^{(31)}(1,M)$ (see Table 8) are also the linear
combinations of inductive and dispersion interactions of the
Li($2\,^{2}S$)-Li${}^{+}(1\,^{1}S$) system and the
Li($2\,^{2}P$)-Li${}^{+}(1\,^{1}S$) system [Eqs. (66) and (67)].

###  III.3 Nonadditive coefficients: Collinear and equilateral triangle

Table 9: The long-range nonadditive interaction coefficients (in atomic units) of the Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li+($2\,^{1}S$) system for two different types of the zeroth-order wave functions, where the three particles form an equally-spaced collinear configuration with the ion in the middle lying on the $x$-axis. The numbers in parentheses represent the computational uncertainties.  Coefficients | $\Psi_{1,\bot}^{(0)}$ | $\Psi_{2,\bot}^{(0)}$ | $\Psi_{1,\bot}^{(0)}$ | $\Psi_{2,\bot}^{(0)}$ | $\Psi_{1,\bot}^{(0)}$ | $\Psi_{2,\bot}^{(0)}$
---|---|---|---|---|---|---
|  ∞Li |  7Li |  6Li
$C_{4,2}^{(12,23)}(1,M=0)$ |  $-$1873.904(5) | 3205.671(5) |  $-$1874.274(5) | 3206.351(5) |  $-$1874.334(6) | 3206.464(5)
$C_{2,4}^{(31,12)}(1,M=0)$ |  $-$1873.904(5) | 3205.671(5) |  $-$1874.274(5) | 3206.351(5) |  $-$1874.334(6) | 3206.464(5)
$C_{3,3}^{(12,23)}(1,M=0)$ | 244.58680(3) |  $-$244.58680(3) | 244.65297(5) |  $-$244.65297(5) | 244.66399(5) |  $-$244.66399(5)
$C_{3,3}^{(23,31)}(1,M=0)$ | 1.0592047(2) |  $-$1.0592047(2) | 1.0597875(3) |  $-$1.0597875(3) | 1.0598847(2) |  $-$1.0598847(2)
$C_{3,3}^{(31,12)}(1,M=0)$ | 244.58680(3) |  $-$244.58680(3) | 244.65297(5) |  $-$244.65297(5) | 244.66399(5) |  $-$244.66399(5)
$C_{4,2}^{(12,23)}(1,M=\pm 1)$ | 936.951(3) |  $-$1602.836(3) | 937.136(3) |  $-$1603.176(3) | 937.167(3) |  $-$1603.231(2)
$C_{2,4}^{(31,12)}(1,M=\pm 1)$ | 936.951(3) |  $-$1602.836(3) | 937.136(3) |  $-$1603.176(3) | 937.167(3) |  $-$1603.231(2)
$C_{3,3}^{(12,23)}(1,M=\pm 1)$ | 611.46703(6) |  $-$611.46703(6) | 611.6324(1) |  $-$611.6324(1) | 611.6600(1) |  $-$611.6600(1)
$C_{3,3}^{(23,31)}(1,M=\pm 1)$ | 2.6480119(5) |  $-$2.6480119(5) | 2.649468(1) |  $-$2.649468(1) | 2.6497127(6) |  $-$2.6497127(6)
$C_{3,3}^{(31,12)}(1,M=\pm 1)$ | 611.46703(6) |  $-$611.46703(6) | 611.6324(1) |  $-$611.6324(1) | 611.6600(1) |  $-$611.6600(1)
Table 10: The long-range nonadditive interaction coefficients (in atomic units) of the Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li+($2\,^{1}S$) system for two different types of the zeroth-order wave functions, where the three particles form an equilateral triangle. The numbers in parentheses represent the computational uncertainties.  Coefficients | $\Psi_{1,\bot}^{(0)}$ | $\Psi_{2,\bot}^{(0)}$ | $\Psi_{1,\bot}^{(0)}$ | $\Psi_{2,\bot}^{(0)}$ | $\Psi_{1,\bot}^{(0)}$ | $\Psi_{2,\bot}^{(0)}$
---|---|---|---|---|---|---
|  ∞Li |  7Li |  6Li
$C_{4,2}^{(12,23)}(1,M=0)$ |  $-$936.951(3) | 1602.836(3) |  $-$937.136(3) | 1603.176(3) |  $-$937.167(3) | 1603.231(2)
$C_{2,4}^{(31,12)}(1,M=0)$ |  $-$936.951(3) | 1602.836(3) |  $-$937.136(3) | 1603.176(3) |  $-$937.167(3) | 1603.231(2)
$C_{3,3}^{(12,23)}(1,M=0)$ | 244.58680(3) |  $-$244.58680(3) | 244.65297(5) |  $-$244.65297(5) | 244.66399(5) |  $-$244.66399(5)
$C_{3,3}^{(23,31)}(1,M=0)$ | 1.0592047(2) |  $-$1.0592047(2) | 1.0597875(3) |  $-$1.0597875(3) | 1.0598847(2) |  $-$1.0598847(2)
$C_{3,3}^{(31,12)}(1,M=0)$ | 244.58680(3) |  $-$244.58680(3) | 244.65297(5) |  $-$244.65297(5) | 244.66399(5) |  $-$244.66399(5)
$C_{4,2}^{(12,23)}(1,M=\pm 1)$ | 468.476(1) |  $-$801.417(1) | 468.567(2) |  $-$801.587(1) | 468.584(1) |  $-$801.616(2)
$C_{2,4}^{(31,12)}(1,M=\pm 1)$ | 468.476(1) |  $-$801.417(1) | 468.567(2) |  $-$801.587(1) | 468.584(1) |  $-$801.616(2)
$C_{3,3}^{(12,23)}(1,M=\pm 1)$ |  $-$214.01346(2) | 214.01346(2) |  $-$214.07137(3) | 214.07137(3) |  $-$214.08101(3) | 214.08101(3)
$C_{3,3}^{(23,31)}(1,M=\pm 1)$ |  $-$0.9268041(2) | 0.9268041(2) |  $-$0.9273142(2) | 0.9273142(2) |  $-$0.9273991(2) | 0.9273991(2)
$C_{3,3}^{(31,12)}(1,M=\pm 1)$ |  $-$214.01346(2) | 214.01346(2) |  $-$214.07137(3) | 214.07137(3) |  $-$214.08101(3) | 214.08101(3)
Figure 4:  Long-range additive interaction potentials (in atomic units) of the
∞Li($2\,^{2}S$)-∞Li($2\,^{2}P$)-∞Li+($1\,^{1}S$) system for two types of the
zeroth-order wave functions, where three particles lie collinearly on the
$z$-axis. For each curve labeled by a wave function, the plotted curve is the
sum of $\Delta E^{(1)}$ and $\Delta E^{(2)}$. Figure 5: Long-range interaction
potentials (in atomic units) of the
∞Li($2\,^{2}S$)-∞Li($2\,^{2}P$)-∞Li+($1\,^{1}S$) system for two types of the
zeroth-order wave functions, where three particles lie collinearly on the
$x$-axis. The plotted potentials include all electrostatic, dispersion, and
induction type interactions (additive and nonadditive) up to
$\mathcal{O}(R^{-6})$. For each curve labeled by a wave function, the plotted
curve is the sum of $\Delta E^{(1)}$ and $\Delta E^{(2)}$. Figure 6: Long-
range interaction potentials (in atomic units) of the
∞Li($2\,^{2}S$)-∞Li($2\,^{2}P$)-∞Li+($1\,^{1}S$) system for two types of the
zeroth-order wave functions, where three particles form an equliteral triangle
on the $x$-$y$ plane. The plotted potentials include all electrostatic,
dispersion, and induction type interactions (additive and nonadditive) up to
$\mathcal{O}(R^{-6})$. For each curve labeled by a wave function, the plotted
curve is the sum of $\Delta E^{(1)}$ and $\Delta E^{(2)}$.

The nonadditive interaction coefficients of Eq. (42) show a dependence on the
interior angles of the three-body system. It is not practical to calculate the
nonadditive coefficients for arbitrary cases when $R_{23}=R_{31}=R$. However,
for the collinear and the equilateral triangle configuration, which
fortunately are probably the most interesting configurations, we can evaluate
specific values. In this subsection, these coefficients are given for two
geometries: an equally-spaced collinear configuration with the ion in the
center ($R_{23}=R_{31}=R$), see Table 9, and an equilateral triangle
configuration ($R_{23}=R_{31}=R_{12}=R$), see Table 10. Different from the
ground state Li${}_{3}^{+}$ trimer demonstrated in Ref. [12], the long-range
nonadditive interactions of the current excited Li${}_{3}^{+}$ trimer appear
in the second-order correction, not in the third-order correction. This
phenomenon is caused by the degeneracy of the three-body system introduced by
the presence of the excited Li($2\,^{2}P$) atom. From Tables 9 and 10, we can
find that most of the nonadditive coefficients are indeed different from each
other for these two geometries, even with the same atomic states as shown in
Eqs. (87) and (88). This kind of three-body effect is caused by the different
interior angles of the two geometries accociated with the magnetic quantum
number $M$ of the Li($2\,^{2}P$) atom, which can also be easily figured out
from Eqs. (43)–(47). And for the different interior angles of the geometries
and for the different magnetic quantum number $M$, these nonadditive terms can
be attractive or repulsive.

Due to the induction effect of the Li${}^{+}(1\,^{1}S$) cation, some of these
nonadditive coefficients are enhanced. For example, from Table 9, we find that
the inductive nonadditive coefficients
$|C_{4,2}^{(12,23)}(1,M=0)|=|C_{2,4}^{(31,12)}(1,M=0)|=1873.904(5)$ a.u. are
much larger than the dispersion nonadditive one
($|C_{3,3}^{(23,31)}(1,M=0)|=1.0592047(2)$ a.u. from Table 9) and are even
larger than some of the additive dispersion
[$|C_{6}^{(12)}(1,M=0)|=1406.68(3)$ a.u. from Table 8], and additive inductive
ones [$|C_{6}^{(23)}(1,M=0)|=552.8371(7)$ a.u. from Table 8] at the same
order. The competition between the additive attractive and nonadditive
repulsive terms of $C_{6}$ for particular geometries will also be discussed in
the following section. These large nonadditive inductive interactions would be
indispensable in constructing potential surfaces and be very useful in studies
of quantum three-body effect for the excited Li${}_{3}^{+}$ trimers.

###  III.4 Long-range potentials: Results

Evaluating the additive and nonadditive long-range potentials using the
coefficients given in Tables 7–10, the potential functions are displayed for
two geometries: An equally-spaced collinear configuration with
$R_{23}=R_{31}=R$, see Figs. 4 and 5, and an equilateral triangle with sides
of length $R$, see Fig. 6. We should indicate that the nonadditive
interactions of the present paper are all evaluated for the geometries lying
on the $x$-$y$ plane as shown in Fig. 1. Thus for the three particles lying on
the $z$-axis, only the additive potentials are shown in Fig. 4 with respect to
the two-body p1 situation (see Fig. 2). For the collinear configuration lying
on the $x$-$y$ plane, the total additive and nonadditive potentials are
displayed in Fig. 5. The separations between the $M=0$ and $M=\pm 1$ states
are mainly caused by the leading repulsive or attractive electrostatic
interaction involving $C_{3}$ between the ion and the excited atom. In Fig. 6,
we display the total long-range potentials (additive and nonadditive) for the
geometry of equilateral triangle lying on the $x$-$y$ plane, where a barrier
of about $115\,\text{cm}^{-1}$ at internuclear distance of $17\,a_{0}$ is
found for the $\Psi_{1,\bot}^{(0)}(1,M=0)$ state. The barrier results from the
interplay of the repulsive leading terms involving $C_{3}$ and the attractive
induction interaction involving $C_{4}$. For the other states, the long-range
potentials are attractive at all internuclear distances. Note that the data
presented in this subsection does not include exchange energies, which may
contribute at these internuclear distances. We will discuss their
contributions in the following subsection.

###  III.5 The strong nonadditive potentials and “switch-off” of the additive
potentials

Figure 7:  Comparison of long-range additive potentials [left side, (A) and
(C)] and nonadditive (collective) potentials [right side, (B) and (D)] (in
atomic units) of the
${}^{\infty}\text{Li}(2\,^{2}S)$-${}^{\infty}\text{Li}(2\,^{2}P)$-${}^{\infty}\text{Li}^{+}(1\,^{1}S)$
system for two types of the zeroth-order wave functions with
$R_{23}=R_{31}=R$: equilateral triangle (A) and (B), equally-spaced collinear
configurations (C) and (D). At the labeled points, the two-body additive
potentials sum to zero leaving only the net nonadditive collective potentials.

As we discussed before, the nonadditive collective effect of the three-body
system is caused by its degeneracy, which is introduced by the presence of the
excited Li($2\,^{2}P$) atom. Meanwhile, the presence of the Li+($1\,^{1}S$)
ion introduces the induction effect, which strongly enhances the nonadditive
(collective) interaction, as demonstrated in Section III.3. In this
subsection, we present a graphical comparison of the additive and nonadditive
potentials for the equilateral triangle and collinear ($R_{23}=R_{31}=R$)
configurations, see Fig. 7.

The figure illustrates that the nonadditive potentials are significant and can
even be stronger than the net contribution from the additive potentials. For
example, for the equilateral triangle configuration, the magnitude of the
additive contribution [Fig. 7(A)] becomes less than the nonadditive
contribution [Fig. 7(C)] around $R\sim 14\,a_{0}$. Indeed, we find that there
are specific internuclear distances at which the additive contributions sum to
zero leaving only nonadditive contributions. Denoting these special distances
by $\bar{R}$, for the equilateral triangle configuration, the additive
cancellation occurs at $\bar{R}=13.58\,a_{0}$ for the $\Psi_{1,\Delta}(1,M=0)$
state with a net energy of $48.42\,\text{cm}^{-1}$. For the collinear
configuration the additive cancellation occurs at $\bar{R}=23.68\,a_{0}$ for
the $\Psi_{1,\Delta}(1,M=0)$ state with a net energy of $0.25\,\text{cm}^{-1}$
and at $\bar{R}=30.69\,a_{0}$ for the $\Psi_{2,\Delta}(1,M=0)$ state with a
net energy of $-0.09\,\text{cm}^{-1}$.

We now, as promised in the Introduction, Sec. I.3, draw a comparison with
trapped cold polar molecules. When the additive (two-body) contributions sum
to zero at a distance $\bar{R}$, Eq. (16) reduces to

$\Delta E(\bar{R})=\Delta E_{\text{non}}^{(2)}(\bar{R}).$ (89)

Comparing Eq. (42) for $\Delta E_{\text{non}}^{(2)}$ with Eq. (2) for the
three-body lattice interaction, we observe that they are precisely the same
form. Since our results are specific to three-particles, the collinear case is
most similar to the case of trapped polar molecules in a linear configuration,
such as shown in Fig. 1(A) of Ref. [64]. Our intriguing result deserves
further study. In retrospect, we can understand the appearance of a
cancellation analogous to that found for trapped polar molecules: The
anisotropy of the present system due to the $\text{Li}(2\,^{2}P)$ atom in the
presence of the $\text{Li}^{+}(1\,^{1}S)$ ion charge is physically similar to
the dipole-dipole interaction in the presence of an external electric field in
the optical lattice case. To gauge precisely the physical potential energies
at the special distances $\bar{R}$, treatment of the exchange energy
contributions, or equivalently quantum-chemical calculations, would be
desirable. However, by analogy with the
$\text{Li}^{+}(1\,^{1}S)$–$\text{Li}(2\,^{2}P)$ results that we presented in
Sec. II.9, we observe that the values of $\bar{R}$ are probably sufficiently
large so that it is likely that only exchange energies will contribute.
Nevertheless, the present result suggests an intriguing similarity between the
$\text{Li}(2\,^{2}S)$-$\text{Li}(2\,^{2}P)$-$\text{Li}^{+}(1\,^{1}S)$ system
and the trapped cold polar molecule scenario.

##  IV Conclusion

The long-range additive and nonadditive interaction potentials for the
Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li+($1\,^{1}S$) system were calculated by using
degenerate perturbation theory. We found that all the first-order and second-
order additive and nonadditive interaction coefficients show a dependence on
the geometrical configurations of the system. The nonadditive interactions
depend on both the atomic states and the interior angles of the
configurations. The degeneracy of the system caused by the presence of the
$\text{Li}(2\,^{2}P)$ atom leads to the three-body collective effect. The
presence of the $\text{Li}^{+}(1\,^{1}S)$ ion was found to enhance this
collective effect, which makes the three-body nonadditive collective
interactions of the system even stronger than the two-body additive
interactions for some specific configurations of the three-body system. For
the two particular configurations with $R_{23}=R_{31}=R$, the equilateral
triangle configuration and the equally-spaced collinear configuration, the
interaction coefficients were evaluated with highly accurate wave functions
calculated variationally in Hylleraas coordinates. In addition, for the
Li($2\,^{2}S$)-Li($2\,^{2}P$)-Li+($1\,^{1}S$) system, the two-body additive
interaction can be “switched off” leaving only three-body nonadditive
interactions for particular geometries, which makes this three-body system a
prospective platform to study the quantum collective effect. We demonstrated
how two-body interaction potentials can be extracted from our results and gave
explicit expressions for the long-range potentials of the
$\text{Li}^{+}(1\,^{1}S)$–$\text{Li}(2\,^{2}P)$ system. The present high-
precision results can serve as benchmarks for future quantum-chemical
calculations and may be of interest for constructing precise potential energy
surfaces. The general formulae for
$A(n_{0}S)$-$A(n_{0}^{\prime}L)$-$A^{Q+}(n_{0}^{\prime\prime}S)$ are listed in
the Supplemental Material.

###### Acknowledgements.

This work was supported by NSERC of Canada, and by the Strategic Priority
Research Program of the Chinese Academy of Sciences under Grant No.
XDB21030300. J.F.B. was supported in part by the U.S. NSF through grant
PHY-1521560 for the Institute of Theoretical Atomic, Molecular, and Optical
Physics at Harvard University and Smithsonian Astrophysical Observatory.

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Supplemental Material: Long-range additive and nonadditive potentials in a
hybrid system: Ground state atom, excited state atom, and ion
Pei-Gen Yan,1 Li-Yan Tang,2 Zong-Chao Yan,1,2 and James F. Babb3

1Department of Physics, University of New Brunswick,
Fredericton, New Brunswick, E3B 5A3, Canada
2State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Wuhan Institute of Physics and Mathematics,
Innovation Academy for Precision Measurement Science and Technology,
Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China
3ITAMP, Center for Astrophysics | Harvard & Smithsonian,
MS 14, 60 Garden St., Cambridge, MA 02138, USA
(Dated: )

This Supplemental Material provides additional details on the calculations. We
introduce elements of the perturbative approach in Secs. IV.1–IV.3 and give
specific expressions for $\text{Li}^{+}(1\,^{1}S)$–$\text{Li}(2\,^{2}S)$
system in Sec. IV.4, for $\text{Li}(2\,^{2}S)$–$\text{Li}(2\,^{2}P)$ system in
Sec. IV.5, for $\text{Li}(2\,^{2}P)$–$\text{Li}^{+}(1\,^{1}S)$ system in Sec.
IV.6 and for
$\text{Li}(2\,^{2}S)$–$\text{Li}(2\,^{2}P)$–$\text{Li}^{+}(1\,^{1}S)$ system
in Sec. IV.7. In the present work, we take the electrostatic interaction
$V_{123}$ between pairs of particles for the $A(n_{0}S)$
-$A(n_{0}^{\prime}L)$-$A^{Q+}(n_{0}^{\prime\prime}S)$ system as a perturbation

$H^{\prime}=V_{123}=V_{12}+V_{23}+V_{31}\,,$ (90)

where $V_{12}$, $V_{23}$ and $V_{31}$ are the two-body mutual electrostatic
interactions between atoms 1 and 2 and ion 3. For three well-separated
systems, the mutual interaction energy $V_{IJ}$ can be expanded with the same
method as used in Refs. [10, 11, 12],

$V_{IJ}=\sum_{l_{I}l_{J}}\sum_{m_{I}m_{J}}T_{\text{{\hbox{l_I-
m_I}}}}(\boldsymbol{\sigma})T_{\text{{\hbox{l_Jm_J}}}}(\boldsymbol{\rho})W_{l_{I}l_{J}}^{m_{I}-m_{J}}(IJ)\,,$
(91)

where the multipole transition operators are

$\displaystyle T_{l_{I}-m_{I}}(\boldsymbol{\sigma})$ $\displaystyle=$
$\displaystyle\sum_{i}Q_{i}\sigma^{l_{I}}_{i}Y_{l_{I}-m_{I}}(\hat{\boldsymbol{\sigma}_{i}})\,,$
(92) $\displaystyle T_{l_{J}m_{J}}(\boldsymbol{\rho})$ $\displaystyle=$
$\displaystyle\sum_{j}q_{j}\rho^{l_{J}}_{j}Y_{l_{J}m_{J}}(\hat{\boldsymbol{\rho}_{j}})\,,$
(93)

where $Q_{i}$ and $q_{j}$, respectively, are the charges of the $i$-th and
$j$-th sub-particles of the atoms $I$ and $J$. The geometry factor is

$\displaystyle W_{l_{I}l_{J}}^{m_{I}-m_{J}}(IJ)$ $\displaystyle=$
$\displaystyle\frac{4\pi(-1)^{l_{J}}}{R_{IJ}^{l_{I}+l_{J}+1}}\frac{(l_{I}+l_{J}-m_{I}+m_{J})!(l_{I},l_{J})^{-1/2}}{[(l_{I}+m_{I})!(l_{I}-m_{I})!(l_{J}+m_{J})!(l_{J}-m_{J})!]^{1/2}}P_{l_{I}+l_{J}}^{m_{I}-m_{J}}(\cos\theta_{IJ})$
(94) $\displaystyle\times$ $\displaystyle\exp[{i(m_{I}-m_{J})\Phi_{IJ}}]\,,$

where ${\bf R}_{IJ}={\bf R}_{J}-{\bf R}_{I}$ is the relative position vector
from particle $I$ to particle $J$, the notation
$(l_{I},l_{J},\ldots)=(2l_{I}+1)(2l_{J}+1)\ldots$, and
$P_{l_{I}+l_{J}}^{m_{I}-m_{J}}(\cos\theta_{IJ})$ is the associated Legendre
function with $\theta_{IJ}$ representing the angle between ${\bf R}_{IJ}$ and
the $z$-axis. If we now choose the $z$ axis to be normal to the plane of the
three particles, i. e., $\theta_{12}=\theta_{23}=\theta_{31}=\pi/2$, the
associated Legendre functions can be simplified as

$\displaystyle P_{l}^{m}(0)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{l+1}}[1+(-1)^{l+m}](-1)^{\frac{l+m}{2}}(l+m)!\bigg{[}\bigg{(}\frac{l+m}{2}\bigg{)}!\bigg{]}^{-1}\bigg{[}\bigg{(}\frac{l-m}{2}\bigg{)}!\bigg{]}^{-1}\,.$
(95)

$\Phi_{IJ}$ denotes the angle between ${\bf R}_{IJ}$ and the $x$-axis. It
shows the dependence of the mutual dipole-dipole interaction between two atoms
on the orientation of the interacting dipoles relative to the line connecting
them [14]. The expressions for $V_{JK}$ and $V_{KI}$ are similar to $V_{IJ}$.
For simplicity, in this work, we transform all $\Phi_{IJ}$ into interior
angles ($\alpha$, $\beta$, $\gamma$) of the triangle formed by the three
lithium nuclei with the same method as used in Ref. [10].

###  IV.1 The zeroth-order wave function

According to degenerate perturbation theory, the zeroth-order wave function of
the unperturbed system $A(n_{0}S)$
-$A(n_{0}^{\prime}L)$-$A^{Q+}(n_{0}^{\prime\prime}S)$ can be written as

$\displaystyle\left|\Psi^{(0)}\right\rangle=a\left|\phi_{1}\right\rangle+b\left|\phi_{2}\right\rangle\,,$
(96)

where the two orthonormalized degenerate eigenvectors of the unperturbed
Hamiltonian with the energy eigenvalue
$E_{n_{0}n_{0}^{\prime}n_{0}^{\prime\prime}}^{(0)}=E_{n_{0}S}^{(0)}+E_{n_{0}^{\prime}L}^{(0)}+E_{n_{0}^{\prime\prime}S}^{(0)}$
can be writen as,

$\displaystyle\left|\phi_{1}\right\rangle$ $\displaystyle=$
$\displaystyle\left|{n_{0}^{\prime}}L;{n_{0}}0;{n_{0}^{\prime\prime}}0\right\rangle\,,$
(97) $\displaystyle\left|\phi_{2}\right\rangle$ $\displaystyle=$
$\displaystyle\left|{n_{0}}0;{n_{0}^{\prime}}L;{n_{0}^{\prime\prime}}0\right\rangle\,.$
(98)

The expansion coefficients $a$, $b$ are determined by diagonalizing the
perturbation in the basis set {$\phi_{1}$, $\phi_{2}$}, which depend on the
geometrical configuration formed by the three particles. In the following, we
show that all the long-range interaction coefficients would contain $a$, $b$
or one of them. This leads to the dependence of these coefficients on the
configurations of the three particles. Thus the zeroth-order energy correction
is obtained by the perturbation matrix with respect to
$\\{\phi_{1},\phi_{2}\\}$

${H}^{\prime}=\left(\begin{array}[]{ccc}\Delta_{11}&\Delta_{12}\\\
\Delta_{12}^{*}&\Delta_{22}\\\ \end{array}\right)\,,$ (99)

where

$\displaystyle\Delta_{11}$ $\displaystyle=$
$\displaystyle\langle\phi_{1}|V_{123}|\phi_{1}\rangle=\langle\phi_{1}|V_{12}|\phi_{1}\rangle+\langle\phi_{1}|V_{23}|\phi_{1}\rangle+\langle\phi_{1}|V_{31}|\phi_{1}\rangle$
(102) $\displaystyle=$
$\displaystyle\sum_{l_{1}}\frac{Q(-1)^{L-M}}{R_{31}^{l_{1}+1}}\sqrt{\frac{{4\pi}}{2l_{1}+1}}P_{l_{1}}(0)\left(\begin{array}[]{ccc}L&l_{1}&L\\\
-M&0&M\\\
\end{array}\right)\langle{n^{\prime}_{0}}L\|T_{l_{1}}\|{n^{\prime}_{0}}L\rangle\,,$
$\displaystyle\Delta_{12}$ $\displaystyle=$
$\displaystyle\langle\phi_{1}|V_{123}|\phi_{2}\rangle=\langle\phi_{1}|V_{12}|\phi_{2}\rangle+\langle\phi_{1}|V_{23}|\phi_{2}\rangle+\langle\phi_{1}|V_{31}|\phi_{2}\rangle$
(103) $\displaystyle=$
$\displaystyle\frac{4\pi}{R_{12}^{2L+1}}\frac{(-1)^{L+M}(2L)!P_{2L}{(0)}}{(2L+1)^{2}(L-M)!(L+M)!}|\langle{n_{0}}0\|T_{L}\|{n_{0}^{\prime}}L\rangle|^{2}\,,$
$\displaystyle\Delta_{22}$ $\displaystyle=$
$\displaystyle\langle\phi_{2}|V_{123}|\phi_{2}\rangle=\langle\phi_{2}|V_{12}|\phi_{2}\rangle+\langle\phi_{2}|V_{23}|\phi_{2}\rangle+\langle\phi_{2}|V_{31}|\phi_{2}\rangle$
(106) $\displaystyle=$
$\displaystyle\sum_{l_{2}}\frac{Q(-1)^{L-M}}{R_{23}^{l_{2}+1}}\sqrt{\frac{{4\pi}}{2l_{2}+1}}P_{l_{2}}(0)\left(\begin{array}[]{ccc}L&l_{2}&L\\\
-M&0&M\\\
\end{array}\right)\langle{n^{\prime}_{0}}L\|T_{l_{2}}\|{n^{\prime}_{0}}L\rangle\,.$

We solve this eigenvalue problem to get the eigenvalues and corresponding
eigenfunctions.

###  IV.2 The first-order energy correction

According to perturbation theory, the first-order energy correction is

$\displaystyle\Delta E^{(1)}=\langle\Psi^{(0)}|V_{123}|\Psi^{(0)}\rangle$
$\displaystyle=|a|^{2}\langle\phi_{1}|V_{123}|\phi_{1}\rangle+|b|^{2}\langle\phi_{2}|V_{123}|\phi_{2}\rangle+(a^{*}b+b^{*}a)\langle\phi_{1}|V_{123}|\phi_{2}\rangle$
$\displaystyle=|a|^{2}\sum_{l_{1}}\frac{Q(-1)^{L-M}}{R_{31}^{l_{1}+1}}\sqrt{\frac{{4\pi}}{2l_{1}+1}}P_{l_{1}}({0})\left(\begin{array}[]{ccc}L&l_{1}&L\\\
-M&0&M\\\
\end{array}\right)\langle{n^{\prime}_{0}}L\|T_{l_{1}}\|{n^{\prime}_{0}}L\rangle$
(109)
$\displaystyle+|b|^{2}\sum_{l_{2}}\frac{Q(-1)^{L-M}}{R_{23}^{l_{2}+1}}\sqrt{\frac{{4\pi}}{2l_{2}+1}}P_{l_{2}}(0)\left(\begin{array}[]{ccc}L&l_{2}&L\\\
-M&0&M\\\
\end{array}\right)\langle{n^{\prime}_{0}}L\|T_{l_{2}}\|{n^{\prime}_{0}}L\rangle$
(112)
$\displaystyle+(a^{*}b+b^{*}a)\frac{4\pi}{R_{12}^{2L+1}}\frac{(-1)^{L+M}(2L)!P_{2L}{(0)}}{(2L+1)^{2}(L-M)!(L+M)!}|\langle{n_{0}}0\|T_{L}\|{n_{0}^{\prime}}L\rangle|^{2}\,.$
(113)

###  IV.3 The second-order energy correction

The second-order energy correction is given by

$\displaystyle\Delta E^{(2)}$ $\displaystyle=$
$\displaystyle-\sum_{n_{s}n_{t}n_{u}}\sum_{L_{s}L_{t}L_{u}}\sum_{M_{s}M_{t}M_{u}}\frac{|\langle\Psi^{(0)}|V_{123}|{n_{s}}L_{s};{n_{t}}L_{t};{n_{u}}L_{u}\rangle|^{2}}{E_{n_{s}L_{s};n_{t}L_{t};n_{u}L_{u}}-E_{n_{0}S;n_{0}^{\prime}L;n_{0}^{\prime\prime}S}^{(0)}}$
(114) $\displaystyle=$ $\displaystyle
V_{12}^{(2)}+V_{23}^{(2)}+V_{31}^{(2)}+V_{12,23}^{(2)}+V_{23,31}^{(2)}+V_{31,12}^{(2)}\,,$

where $|{n_{s}}L_{s};{n_{t}}L_{t};{n_{u}}L_{u}\rangle$ is an intermediate
state of the system with the energy eigenvalue
$E_{n_{s}L_{s};n_{t}L_{t};n_{u}L_{u}}=E_{n_{s}L_{s}}+E_{n_{t}L_{t}}+E_{n_{u}L_{u}}$.
It is noted that the above summations should exclude terms with
$E_{n_{s}L_{s};n_{t}L_{t};n_{u}L_{u}}=E_{n_{0}S;n_{0}^{\prime}L;n_{0}^{\prime\prime}S}^{(0)}$.
The three additive interaction terms, denoted by $V_{12}^{(2)}$,
$V_{23}^{(2)}$, $V_{31}^{(2)}$ become, respectively,

$\displaystyle V_{12}^{(2)}$ $\displaystyle=$
$\displaystyle-|a|^{2}\sum_{n_{s}n_{t}}\sum_{L_{s}L_{t}l_{1}l_{1}^{\prime}}\sum_{M_{s}M_{t}m_{1}}\frac{16\pi^{2}}{R_{12}^{2L_{t}+l_{1}+l_{1}^{\prime}+2}}\left(\begin{array}[]{ccc}L&l_{1}&L_{s}\\\
-M&m_{1}&M_{s}\\\
\end{array}\right)\left(\begin{array}[]{ccc}L&l_{1}^{\prime}&L_{s}\\\
-M&m_{1}&M_{s}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\frac{P_{L_{t}+l_{1}}^{M_{t}-m_{1}}(0)P_{L_{t}+l_{1}^{\prime}}^{M_{t}-m_{1}}(0)(L_{t}+l_{1}-M_{t}+m_{1})!(L_{t}+l_{1}^{\prime}-M_{t}+m_{1})!(L_{t},L_{t})^{-1}(l_{1},l_{1}^{\prime})^{-1/2}}{(L_{t}+M_{t})!(L_{t}-M_{t})![(l_{1}+m_{1})!(l_{1}-m_{1})!(l_{1}^{\prime}+m_{1})!(l_{1}^{\prime}-m_{1})!]^{1/2}}$
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}^{\prime}}L\|T_{l_{1}}\|{n_{s}}L_{s}\rangle^{*}\langle{n_{0}^{\prime}}L\|T_{l_{1}^{\prime}}\|{n_{s}}L_{s}\rangle|\langle{n_{0}}0\|T_{L_{t}}\|{n_{t}}L_{t}\rangle|^{2}}{E_{n_{s}L_{s}}+E_{n_{t}L_{t}}-E_{n_{0}S}^{(0)}-E_{n_{0}^{\prime}L}^{(0)}}$
$\displaystyle-$
$\displaystyle|b|^{2}\sum_{n_{s}n_{t}}\sum_{L_{s}L_{t}l_{2}l_{2}^{\prime}}\sum_{M_{s}M_{t}m_{2}}\frac{16\pi^{2}}{R_{12}^{2L_{s}+l_{2}+l_{2}^{\prime}+2}}\left(\begin{array}[]{ccc}L&l_{2}&L_{t}\\\
-M&m_{2}&M_{t}\\\
\end{array}\right)\left(\begin{array}[]{ccc}L&l_{2}^{\prime}&L_{t}\\\
-M&m_{2}&M_{t}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\frac{P_{L_{s}+l_{2}}^{M_{s}-m_{2}}(0)P_{L_{s}+l_{2}^{\prime}}^{M_{s}-m_{2}}(0)(L_{s}+l_{2}-M_{s}+m_{2})!(L_{s}+l_{2}^{\prime}-M_{s}+m_{2})!(L_{s},L_{s})^{-1}(l_{2},l_{2}^{\prime})^{-1/2}}{(L_{s}+M_{s})!(L_{s}-M_{s})![(l_{2}+m_{2})!(l_{2}-m_{2})!(l_{2}^{\prime}+m_{2})!(l_{2}^{\prime}-m_{2})!]^{1/2}}$
$\displaystyle\times$
$\displaystyle\frac{|\langle{n_{0}}0\|T_{L_{s}}\|{n_{s}}L_{s}\rangle|^{2}\langle{n_{0}^{\prime}}L\|T_{l_{2}}\|{n_{t}}L_{t}\rangle^{*}\langle{n_{0}^{\prime}}L\|T_{l_{2}^{\prime}}\|{n_{t}}L_{t}\rangle}{E_{n_{s}L_{s}}+E_{n_{t}L_{t}}-E_{n_{0}S}^{(0)}-E_{n_{0}^{\prime}L}^{(0)}}$
$\displaystyle-$ $\displaystyle
a^{*}b\sum_{n_{s}n_{t}}\sum_{L_{s}L_{t}l_{1}l_{2}^{\prime}}\sum_{M_{s}M_{t}m_{1}m_{2}^{\prime}}\frac{16\pi^{2}(-1)^{L_{s}+l_{2}^{\prime}-M_{s}-M_{t}}}{R_{12}^{l_{1}+L_{s}+L_{t}+l_{2}^{\prime}+2}}\left(\begin{array}[]{ccc}L&l_{1}&L_{s}\\\
-M&-m_{1}&M_{s}\\\
\end{array}\right)\left(\begin{array}[]{ccc}L&l_{2}^{\prime}&L_{t}\\\
-M&m_{2}^{\prime}&M_{t}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\frac{P_{L_{t}+l_{1}}^{M_{t}+m_{1}}(0)P_{L_{s}+l_{2}^{\prime}}^{M_{s}-m_{2}^{\prime}}(0)(L_{t}+l_{1}-M_{t}-m_{1})!(L_{s}+l_{2}^{\prime}-M_{s}+m_{2}^{\prime})!{(L_{s},L_{t})^{-1}(l_{1},l_{2}^{\prime})^{-1/2}}}{[(L_{s}+M_{s})!(L_{s}-M_{s})!(L_{t}+M_{t})!(L_{t}-M_{t})!(l_{1}+m_{1})!(l_{1}-m_{1})!(l_{2}^{\prime}+m_{2}^{\prime})!(l_{2}^{\prime}-m_{2}^{\prime})!]^{1/2}}$
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}^{\prime}}L\|T_{l_{1}}\|{n_{s}}L_{s}\rangle^{*}\langle{n_{0}}0\|T_{L_{t}}\|{n_{t}}L_{t}\rangle^{*}\langle{n_{0}}0\|T_{L_{s}}\|{n_{s}}L_{s}\rangle\langle{n_{0}^{\prime}}L\|T_{l_{2}^{\prime}}\|{n_{t}}L_{t}\rangle}{E_{n_{s}L_{s}}+E_{n_{t}L_{t}}-E_{n_{0}S}^{(0)}-E_{n_{0}^{\prime}L}^{(0)}}$
$\displaystyle-$ $\displaystyle
b^{*}a\sum_{n_{s}n_{t}}\sum_{L_{s}L_{t}l_{1}^{\prime}l_{2}}\sum_{M_{s}M_{t}m_{1}^{\prime}m_{2}}\frac{16\pi^{2}(-1)^{L_{s}+l_{2}-M_{s}-M_{t}}}{R_{12}^{l_{1}^{\prime}+L_{t}+L_{s}+l_{2}+2}}\left(\begin{array}[]{ccc}L&l_{1}^{\prime}&L_{s}\\\
-M&-m_{1}^{\prime}&M_{s}\\\
\end{array}\right)\left(\begin{array}[]{ccc}L&l_{2}&L_{t}\\\ -M&m_{2}&M_{t}\\\
\end{array}\right)$ $\displaystyle\times$
$\displaystyle\frac{P_{L_{s}+l_{2}}^{M_{s}-m_{2}}(0)P_{L_{t}+l_{1}^{\prime}}^{M_{t}+m_{1}^{\prime}}(0)(L_{t}+l_{1}^{\prime}-M_{t}-m_{1}^{\prime})!(L_{s}+l_{2}-M_{s}+m_{2})!{(L_{s},L_{t})^{-1}(l_{1}^{\prime},l_{2})^{-1/2}}}{[(L_{s}+M_{s})!(L_{s}-M_{s})!(L_{t}+M_{t})!(L_{t}-M_{t})!(l_{1}^{\prime}+m_{1}^{\prime})!(l_{1}^{\prime}-m_{1}^{\prime})!(l_{2}+m_{2})!(l_{2}-m_{2})!]^{1/2}}$
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}}0\|T_{L_{s}}\|{n_{s}}L_{s}\rangle^{*}\langle{n_{0}^{\prime}}L\|T_{l_{2}}\|{n_{t}}L_{t}\rangle^{*}\langle{n_{0}^{\prime}}L\|T_{l_{1}^{\prime}}\|{n_{s}}L_{s}\rangle\langle{n_{0}}0\|T_{L_{t}}\|{n_{t}}L_{t}\rangle}{E_{n_{s}L_{s}}+E_{n_{t}L_{t}}-E_{n_{0}S}^{(0)}-E_{n_{0}^{\prime}L}^{(0)}}$
$\displaystyle=$
$\displaystyle-\bigg{\\{}|a|^{2}\sum_{n_{s}n_{t}}\sum_{L_{s}L_{t}l_{1}l_{1}^{\prime}}\frac{F_{1}(n_{s},n_{t},L_{s},L_{t};l_{1},l_{1}^{\prime};L,M)}{R_{12}^{2L_{t}+l_{1}+l_{1}^{\prime}+2}}+|b|^{2}\sum_{n_{s}n_{t}}\sum_{L_{s}L_{t}l_{2}l_{2}^{\prime}}\frac{F_{1}(n_{t},n_{s},L_{t},L_{s};l_{2},l_{2}^{\prime};L,M)}{R_{12}^{2L_{s}+l_{2}+l_{2}^{\prime}+2}}$
$\displaystyle+$ $\displaystyle
a^{*}b\sum_{n_{s}n_{t}}\sum_{L_{s}L_{t}l_{1}l_{2}^{\prime}}\frac{F_{3}(n_{s},n_{t},L_{s},L_{t};l_{1},l_{2}^{\prime};L,M)}{R_{12}^{L_{s}+L_{t}+l_{1}+l_{2}^{\prime}+2}}+b^{*}a\sum_{n_{s}n_{t}}\sum_{L_{s}L_{t}l_{1}^{\prime}l_{2}}\frac{F_{3}^{*}(n_{s},n_{t},L_{s},L_{t};l_{1}^{\prime},l_{2};L,M)}{R_{12}^{L_{s}+L_{t}+l_{1}^{\prime}+l_{2}+2}}\bigg{\\}}\,,$

$\displaystyle V_{23}^{(2)}$ $\displaystyle=$
$\displaystyle-|a|^{2}\sum_{n_{t}n_{u}}\sum_{L_{t}L_{u}}\sum_{M_{t}M_{u}}\frac{16\pi^{2}}{R_{23}^{2L_{t}+2L_{u}+2}}\frac{[P_{L_{t}+L_{u}}^{M_{t}+M_{u}}(0)(L_{t}+L_{u}-M_{t}-M_{u})!]^{2}(L_{t},L_{u})^{-2}}{(L_{t}+M_{t})!(L_{t}-M_{t})!(L_{u}+M_{u})!(L_{u}-M_{u})!}$
$\displaystyle\times$
$\displaystyle\frac{|\langle{n_{0}}0\|T_{L_{t}}\|{n_{t}}L_{t}\rangle|^{2}|\langle{n_{0}^{\prime\prime}}0\|T_{L_{u}}\|{n_{u}}L_{u}\rangle|^{2}}{E_{n_{t}L_{t}}+E_{n_{u}L_{u}}-E_{n_{0}S}^{(0)}-E_{n_{0}^{\prime\prime}S}^{(0)}}$
$\displaystyle-$
$\displaystyle|b|^{2}\sum_{n_{t}n_{u}}\sum_{L_{t}L_{u}l_{2}l_{2}^{\prime}}\sum_{M_{t}M_{u}m_{2}}\frac{16\pi^{2}}{R_{23}^{2L_{u}+l_{2}+l_{2}^{\prime}+2}}\left(\begin{array}[]{ccc}L&l_{2}&L_{t}\\\
-M&m_{2}&M_{t}\\\
\end{array}\right)\left(\begin{array}[]{ccc}L&l_{2}^{\prime}&L_{t}\\\
-M&m_{2}&M_{t}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\frac{P_{L_{u}+l_{2}}^{M_{u}-m_{2}}(0)P_{{L_{u}+l_{2}^{\prime}}}^{M_{u}-m_{2}}(0)(L_{u}+l_{2}-M_{u}+m_{2})!(L_{u}+l_{2}^{\prime}-M_{u}+m_{2})!(L_{u},L_{u})^{-1}(l_{2},l_{2}^{\prime})^{-1/2}}{(L_{u}+M_{u})!(L_{u}-M_{u})![(l_{2}+m_{2})!(l_{2}-m_{2})!(l_{2}^{\prime}+m_{2})!(l_{2}^{\prime}-m_{2})!]^{1/2}}$
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}^{\prime}}L\|T_{l_{2}}\|{n_{t}}L_{t}\rangle^{*}\langle{n_{0}^{\prime}}L\|T_{l_{2}^{\prime}}\|{n_{t}}L_{t}\rangle|\langle{n_{0}^{\prime\prime}}0\|T_{L_{u}}\|{n_{u}}L_{u}\rangle|^{2}}{E_{n_{t}L_{t}}+E_{n_{u}L_{u}}-E_{n_{0}^{\prime\prime}S}^{(0)}-E_{n_{0}^{\prime}L}^{(0)}}$
$\displaystyle=$
$\displaystyle-\bigg{\\{}|a|^{2}\sum_{n_{t}n_{u}}\sum_{L_{t}L_{u}}\frac{F_{2}(n_{t},n_{u},L_{t},L_{u})}{R_{23}^{2L_{t}+2L_{u}+2}}+|b|^{2}\sum_{n_{t}n_{u}}\sum_{L_{t}L_{u}l_{2}l_{2}^{\prime}}\frac{F_{1}(n_{t},n_{u},L_{t},L_{u};l_{2},l_{2}^{\prime};L,M)}{R_{23}^{2L_{u}+l_{2}+l_{2}^{\prime}+2}}\bigg{\\}}\,,$

$\displaystyle V_{31}^{(2)}$ $\displaystyle=$
$\displaystyle-|a|^{2}\sum_{n_{s}n_{u}}\sum_{L_{s}L_{u}l_{1}l_{1}^{\prime}}\sum_{M_{s}M_{u}m_{1}}\frac{16\pi^{2}}{R_{31}^{2L_{u}+l_{1}+l_{1}^{\prime}+2}}\left(\begin{array}[]{ccc}L&l_{1}&L_{s}\\\
-M&m_{1}&M_{s}\\\
\end{array}\right)\left(\begin{array}[]{ccc}L&l_{1}^{\prime}&L_{s}\\\
-M&m_{1}&M_{s}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\frac{P_{L_{u}+l_{1}}^{M_{u}-m_{1}}(0)P_{L_{u}+l_{1}^{\prime}}^{M_{u}-m_{1}}(0)(L_{u}+l_{1}-M_{u}+m_{1})!(L_{u}+l_{1}^{\prime}-M_{u}+m_{1})!(L_{u},L_{u})^{-1}(l_{1},l_{1}^{\prime})^{-1/2}}{(L_{u}+M_{u})!(L_{u}-M_{u})![(l_{1}+m_{1})!(l_{1}-m_{1})!(l_{1}^{\prime}+m_{1})!(l_{1}^{\prime}-m_{1})!]^{1/2}}$
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}^{\prime}}L\|T_{l_{1}}\|{n_{s}}L_{s}\rangle^{*}\langle{n_{0}^{\prime}}L\|T_{l_{1}^{\prime}}\|{n_{s}}L_{s}\rangle|\langle{n_{0}^{\prime\prime}}0\|T_{L_{u}}\|{n_{u}}L_{u}\rangle|^{2}}{E_{n_{s}L_{s}}+E_{n_{u}L_{u}}-E_{n_{0}^{\prime\prime}S}-E_{n_{0}^{\prime}L}}$
$\displaystyle-$
$\displaystyle|b|^{2}\sum_{n_{s}n_{u}}\sum_{L_{s}L_{u}}\sum_{M_{s}M_{u}}\frac{16\pi^{2}}{R_{31}^{2L_{s}+2L_{u}+2}}\frac{[P_{L_{u}+L_{s}}^{M_{u}+M_{s}}(0)(L_{u}+L_{s}-M_{u}-M_{s})!]^{2}(L_{u},L_{s})^{-2}}{(L_{u}+M_{u})!(L_{u}-M_{u})!(L_{s}+M_{s})!(L_{s}-M_{s})!}$
$\displaystyle\times$
$\displaystyle\frac{|\langle{n_{0}}0\|T_{L_{s}}\|{n_{s}}L_{s}\rangle|^{2}|\langle{n_{0}^{\prime\prime}}0\|T_{L_{u}}\|{n_{u}}L_{u}\rangle|^{2}}{E_{n_{s}L_{s}}+E_{n_{u}L_{u}}-E_{n_{0}S}-E_{n_{0}^{\prime\prime}S}}$
$\displaystyle=$
$\displaystyle-\bigg{\\{}|a|^{2}\sum_{n_{s}n_{u}}\sum_{L_{s}L_{u}l_{1}l_{1}^{\prime}}\frac{F_{1}(n_{s},n_{u},L_{s},L_{u};l_{1},l_{1}^{\prime};L,M)}{R_{31}^{2L_{u}+l_{1}+l_{1}^{\prime}+2}}+|b|^{2}\sum_{n_{s}n_{u}}\sum_{L_{s}L_{u}}\frac{F_{2}(n_{s},n_{u},L_{s},L_{u})}{R_{31}^{2L_{s}+2L_{u}+2}}\bigg{\\}}\,.$

The three nonadditive interaction terms, denoted by $V_{12,23}^{(2)}$,
$V_{23,31}^{(2)}$, $V_{31,12}^{(2)}$ become, respectively,

$\displaystyle V_{12,23}^{(2)}$ $\displaystyle=$
$\displaystyle-|a|^{2}\sum_{n_{t}L_{t}M_{t}}\sum_{l_{1}}\frac{8\sqrt{\pi^{3}}Q(-1)^{L_{t}-M_{t}+L-M}}{R_{12}^{l_{1}+L_{t}+1}R_{23}^{L_{t}+1}}\left(\begin{array}[]{ccc}L&l_{1}&L\\\
-M&0&M\\\ \end{array}\right)cos(M_{t}\beta)$ $\displaystyle\times$
$\displaystyle\frac{P_{l_{1}+L_{t}}^{M_{t}}(0)P_{L_{t}}^{M_{t}}(0)(l_{1}+L_{t}-M_{t})!(L_{t})^{-2}(l_{1})^{-1/2}}{(l_{1})!(L_{t}+M_{t})!}$
$\displaystyle\times$
$\displaystyle\frac{\langle{n^{\prime}_{0}}L\|T_{l_{1}}\|{n^{\prime}_{0}}L\rangle|\langle{n_{0}}0\|T_{L_{t}}\|{n_{t}}L_{t}\rangle|^{2}}{E_{n_{t}L_{t}}-E_{n_{0}S}}$
$\displaystyle-$
$\displaystyle\sum_{n_{t}L_{t}M_{t}}\sum_{l_{2}^{\prime}m_{2}^{\prime}}\frac{8\sqrt{\pi^{3}}Q}{R_{12}^{L+L_{t}+1}R_{23}^{l_{2}^{\prime}+1}}\left(\begin{array}[]{ccc}L&l_{2}^{\prime}&L_{t}\\\
-M&-m_{2}^{\prime}&M_{t}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\\{a^{*}b\exp[{-i(m_{2}^{\prime})\beta}]+b^{*}a\exp[{i(m_{2}^{\prime})\beta}]\\}$
$\displaystyle\times$
$\displaystyle\frac{P_{L+L_{t}}^{-M+M_{t}}(0)P_{l_{2}^{\prime}}^{m_{2}^{\prime}}(0)(L+L_{t}+M-M_{t})!(l_{2}^{\prime}-m_{2}^{\prime})!(l_{2}^{\prime})^{-1/2}(L,L_{t})^{-1}}{[(L+M)!(L-M)!(L_{t}+M_{t})!(L_{t}-M_{t})!(l_{2}^{\prime}+m_{2}^{\prime})!(l_{2}^{\prime}-m_{2}^{\prime})!]^{1/2}}$
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}}0\|T_{L}\|{n_{0}^{\prime}}L\rangle\langle{n_{0}}0\|T_{L_{t}}\|{n_{t}}L_{t}\rangle^{*}\langle{n_{0}^{\prime}}L\|T_{l_{2}^{\prime}}\|{n_{t}}L_{t}\rangle}{E_{n_{t}L_{t}}-E_{n_{0}^{\prime}L}}$
$\displaystyle-$
$\displaystyle\sum_{n_{t}L_{t}M_{t}}\sum_{l_{2}m_{2}}\frac{8\sqrt{\pi^{3}}Q(-1)^{L}}{R_{12}^{L+l_{2}+1}R_{23}^{L_{t}+1}}\left(\begin{array}[]{ccc}L&l_{2}&L_{t}\\\
-M&m_{2}&M_{t}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\\{a^{*}b\exp[{i(M_{t})\beta}]+b^{*}a\exp[{-i(M_{t})\beta}]\\}$
$\displaystyle\times$
$\displaystyle\frac{P_{L+l_{2}}^{M-m_{2}}(0)P_{L_{t}}^{M_{t}}(0)(L+l_{2}-M+m_{2})!(L_{t}-M_{t})!(l_{2})^{-1/2}(L_{t},L)^{-1}}{[(L+M)!(L-M)!(l_{2}+m_{2})!(l_{2}-m_{2})!(L_{t}+M_{t})!(L_{t}-M_{t})!]^{1/2}}$
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}}0\|T_{L}\|{n_{0}^{\prime}}L\rangle^{*}\langle{n_{0}^{\prime}}L\|T_{l_{2}}\|{n_{t}}L_{t}\rangle^{*}\langle{n_{0}}0\|T_{L_{t}}\|{n_{t}}L_{t}\rangle}{E_{n_{t}L_{t}}-E_{n_{0}S}}$
$\displaystyle=$
$\displaystyle-\bigg{\\{}|a|^{2}\sum_{n_{t}L_{t}M_{t}}\sum_{l_{1}}\frac{F_{5}(n_{t},L_{t},M_{t};l_{1};L,M;Q)cos(M_{t}\beta)}{R_{12}^{l_{1}+L_{t}+1}R_{23}^{L_{t}+1}}$
$\displaystyle+$
$\displaystyle\sum_{n_{t}L_{t}M_{t}}\sum_{l_{2}^{\prime}m_{2}^{\prime}}\frac{\\{a^{*}b\exp[{-i(m_{2}^{\prime})\beta}]+b^{*}a\exp[{i(m_{2}^{\prime})\beta}]\\}F_{6}(n_{t},L_{t},M_{t};l_{2}^{\prime},m_{2}^{\prime};L,M;Q)}{R_{12}^{L+L_{t}+1}R_{23}^{l_{2}^{\prime}+1}}$
$\displaystyle+$
$\displaystyle\sum_{n_{t}L_{t}M_{t}}\sum_{l_{2}}\frac{\\{a^{*}b\exp[{i(M_{t})\beta}]+b^{*}a\exp[{-i(M_{t})\beta}]\\}F_{7}(n_{t},L_{t},M_{t};l_{2};L,M;Q)}{R_{12}^{L+l_{2}+1}R_{23}^{L_{t}+1}}\bigg{\\}}\,,$

$\displaystyle V_{23,31}^{(2)}$ $\displaystyle=$
$\displaystyle-\sum_{n_{u}L_{u}M_{u}}\frac{16\pi^{2}(-1)^{L_{u}+L+M_{u}-M}}{R_{23}^{L_{u}+L+1}R_{31}^{L_{u}+L+1}}\frac{[P_{L_{u}+L}^{M_{u}-M}(0)(L_{u}+L-M_{u}+M)!(L_{u},L)^{-1}]^{2}}{(L_{u}+M_{u})!(L_{u}-M_{u})!(L+M)!(L-M)!}$
$\displaystyle\times$
$\displaystyle\\{(a^{*}b)\exp[i(M_{u}-M)\gamma]+(b^{*}a)\exp[-i(M_{u}-M)\gamma]\\}$
$\displaystyle\times$
$\displaystyle\frac{|\langle{n_{0}^{\prime}}L\|T_{L}\|{n_{0}}0\rangle|^{2}|\langle{n_{0}^{\prime\prime}}0\|T_{L_{u}}\|{n_{u}}L_{u}\rangle|^{2}}{E_{n_{u}L_{u}}-E_{n_{0}^{\prime\prime}S}+E_{n_{0}S}-E_{n_{0}^{\prime}L}}$
$\displaystyle-$
$\displaystyle\sum_{n_{u}L_{u}M_{u}}\frac{16\pi^{2}(-1)^{L_{u}+L+M_{u}+M}}{R_{23}^{L_{u}+L+1}R_{31}^{L_{u}+L+1}}\frac{[P_{L_{u}+L}^{M_{u}+M}(0)(L_{u}+L-M_{u}-M)!(L_{u},L)^{-1}]^{2}}{(L_{u}+M_{u})!(L_{u}-M_{u})!(L+M)!(L-M)!}$
$\displaystyle\times$
$\displaystyle\\{(a^{*}b)\exp[-i(M_{u}+M)\gamma]+(b^{*}a)\exp[i(M_{u}+M)\gamma]\\}$
$\displaystyle\times$
$\displaystyle\frac{|\langle{n_{0}}0\|T_{L}\|{n_{0}^{\prime}}L\rangle|^{2}|\langle{n_{0}^{\prime\prime}}0\|T_{L_{u}}\|{n_{u}}L_{u}\rangle|^{2}}{E_{n_{u}L_{u}}+E_{n_{0}^{\prime}L}-E_{n_{0}S}-E_{n_{0}^{\prime\prime}S}}$
$\displaystyle=$
$\displaystyle-\sum_{n_{u}L_{u}M_{u}}\bigg{\\{}\\{(a^{*}b)\exp[i(M_{u}-M)\gamma]\\}+\\{(b^{*}a)\exp[-i(M_{u}-M)\gamma]\\}\bigg{\\}}\frac{F_{4}(n_{u},L_{u},M_{u};L,M)}{R_{23}^{L_{u}+L+1}R_{31}^{L_{u}+L+1}}\,,$

$\displaystyle V_{31,12}^{(2)}$ $\displaystyle=$
$\displaystyle-|b|^{2}\sum_{n_{s}L_{s}M_{s}}\sum_{l_{2}}\frac{8\sqrt{\pi^{3}}Q(-1)^{L-M+L_{s}-M_{s}}}{R_{31}^{L_{s}+1}R_{12}^{L_{s}+l_{2}+1}}\left(\begin{array}[]{ccc}L&l_{2}&L\\\
-M&0&M\\\ \end{array}\right)cos({M_{s}\alpha})$ $\displaystyle\times$
$\displaystyle\frac{P_{L_{s}+l_{2}}^{M_{s}}(0)P_{L_{s}}^{M_{s}}(0)(L_{s}+l_{2}-M_{s})!(L_{s})^{-2}(l_{2})^{-1/2}}{(L_{s}+M_{s})!(l_{2})!}$
$\displaystyle\times$
$\displaystyle\frac{\langle{n^{\prime}_{0}}L\|T_{l_{2}}\|{n^{\prime}_{0}}L\rangle^{*}|\langle{n_{0}}0\|T_{L_{s}}\|{n_{s}}L_{s}\rangle|^{2}}{E_{n_{s}L_{s}}-E_{n_{0}S}}$
$\displaystyle-$
$\displaystyle\sum_{n_{s}L_{s}M_{s}}\sum_{l_{1}m_{1}}\frac{8\sqrt{\pi^{3}}Q(-1)^{L}}{R_{12}^{l_{1}+L+1}R_{31}^{L_{s}+1}}\left(\begin{array}[]{ccc}L&l_{1}&L_{s}\\\
-M&-m_{1}&M_{s}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\\{a^{*}b\exp[{i(M_{s})\alpha}]+b^{*}a\exp[{-i(M_{s})\alpha}]\\}$
$\displaystyle\times$
$\displaystyle\frac{P_{l_{1}+L}^{m_{1}+M}(0)P_{L_{s}}^{M_{s}}(0)(l_{1}+L-m_{1}-M)!(L_{s}-M_{s})!(L_{s},L)^{-1}(l_{1})^{-1/2}}{[(l_{1}+m_{1})!(l_{1}-m_{1})!(L+M)!(L-M)!(L_{s}+M_{s})!(L_{s}-M_{s})!]^{1/2}}$
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}}0\|T_{L}\|{n_{0}^{\prime}}L\rangle^{*}\langle{n_{0}^{\prime}}L\|T_{l_{1}}\|{n_{s}}L_{s}\rangle^{*}\langle{n_{0}}0\|T_{L_{s}}\|{n_{s}}L_{s}\rangle}{E_{n_{s}L_{s}}-E_{n_{0}S}}$
$\displaystyle-$
$\displaystyle\sum_{n_{s}L_{s}M_{s}}\sum_{l_{1}^{\prime}m_{1}^{\prime}}\frac{8\sqrt{\pi^{3}}Q}{R_{12}^{L_{s}+L+1}R_{31}^{l_{1}^{\prime}+1}}\left(\begin{array}[]{ccc}L&l_{1}^{\prime}&L_{s}\\\
-M&-m_{1}^{\prime}&M_{s}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\\{a^{*}b\exp[{i(m_{1}^{\prime})\alpha}]+b^{*}a\exp[{-i(m_{1}^{\prime})\alpha}]\\}$
$\displaystyle\times$
$\displaystyle\frac{P_{L_{s}+L}^{m_{s}-M}(0)P_{l_{1}^{\prime}}^{m_{1}^{\prime}}(0)(L_{s}+L-M_{s}+M)!(l_{1}^{\prime}-m_{1}^{\prime})!(l_{1}^{\prime})^{-1/2}(L_{s},L)^{-1}}{[(L_{s}+M_{s})!(L_{s}-M_{s})!(L+M)!(L-M)!(l_{1}^{\prime}+m_{1}^{\prime})!(l_{1}^{\prime}-m_{1}^{\prime})!]^{1/2}}$
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}}0\|T_{L_{s}}\|{n_{s}}L_{s}\rangle^{*}\langle{n_{0}}0\|T_{L}\|{n_{0}^{\prime}}L\rangle\langle{n_{0}^{\prime}}L\|T_{l_{1}^{\prime}}\|{n_{s}}L_{s}\rangle}{E_{n_{s}L_{s}}-E_{n_{0}^{\prime}L}}$
$\displaystyle=$
$\displaystyle-\bigg{\\{}|b|^{2}\sum_{n_{s}L_{s}M_{s}}\sum_{l_{2}}\frac{F_{5}(n_{s},L_{s},M_{s};l_{2};L,M;Q)cos({M_{s}\alpha})}{R_{31}^{L_{s}+1}R_{12}^{L_{s}+l_{2}+1}}$
$\displaystyle+$
$\displaystyle\sum_{n_{s}L_{s}M_{s}}\sum_{l_{1}}\frac{\\{a^{*}b\exp[{i(M_{s})\alpha}]+b^{*}a\exp[{-i(M_{s})\alpha}]\\}F_{7}(n_{s},L_{s},M_{s};l_{1};L,M;Q)}{R_{12}^{l_{1}+L+1}R_{31}^{L_{s}+1}}$
$\displaystyle+$
$\displaystyle\sum_{n_{s}L_{s}M_{s}}\sum_{l_{1}^{\prime}m_{1}^{\prime}}\frac{\\{a^{*}b\exp[{i(m_{1}^{\prime})\alpha}]+b^{*}a\exp[{-i(m_{1}^{\prime})\alpha}]\\}F_{6}(n_{s},L_{s},M_{s};l_{1}^{\prime},m_{1}^{\prime};L,M;Q)}{R_{12}^{L_{s}+L+1}R_{31}^{l_{1}^{\prime}+1}}\bigg{\\}}\,.$

In the above Eqs. (IV.3)-(IV.3), the $F_{1}$, $F_{2}$, and $F_{3}$ functions
and the corresponding $G_{1}(L_{i},L_{j},\ell_{k},\ell_{k}^{\prime};L,M)$,
$G_{2}(L_{i},L_{j},\ell_{k_{1}},\ell_{k_{2}}^{\prime};L,M)$, and
$G_{3}(L_{i},L_{j})$ are defined in Ref. [10]. The $F_{4}$, $F_{5}$, $F_{6}$,
and $F_{7}$ functions and the corresponding $G_{4}(L_{i},M_{i};L,M)$,
$G_{5}(L_{i},M_{i};\ell_{k_{1}};L,M;Q)$,
$G_{6}(L_{i},M_{i};\ell_{k_{1}},m_{k_{1}};L,M;Q)$ and
$G_{7}(L_{i},M_{i};\ell_{k_{1}};L,M;Q)$ are are defined by

$\displaystyle F_{1}(n_{s},n_{t},L_{s},L_{t};l_{1},l_{1}^{\prime};L,M)$
$\displaystyle=$ $\displaystyle
G_{1}(L_{s},L_{t},l_{1},l_{1}^{\prime};L,M)\langle{n_{0}^{\prime}}L\|T_{l_{1}}\|{n_{s}}L_{s}\rangle^{*}$
(169) $\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}^{\prime}}L\|T_{l_{1}^{\prime}}\|{n_{s}}L_{s}\rangle|\langle{n_{0}(n_{0}^{\prime\prime})0}\|T_{L_{t}}\|{n_{t}}L_{t}\rangle|^{2}}{E_{n_{s}L_{s}}+E_{n_{t}L_{t}}-E_{n_{0}(n_{0}^{\prime\prime})S}^{(0)}-E_{n_{0}^{\prime}L}^{(0)}}\,,$
$\displaystyle F_{2}(n_{s},n_{t},L_{s},L_{t};l_{1},l_{2}^{\prime};L,M)$
$\displaystyle=$
$\displaystyle(-1)^{L_{s}+l_{2}^{\prime}}G_{2}(L_{s},L_{t},l_{1},l_{2}^{\prime};L,M)$
(170) $\displaystyle\times$
$\displaystyle\langle{n_{0}(n_{0}^{\prime\prime})}0\|T_{L_{s}}\|{n_{s}}L_{s}\rangle\langle{n_{0}^{\prime}}L\|T_{l_{2}^{\prime}}\|{n_{t}}L_{t}\rangle$
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}^{\prime}}L\|T_{l_{1}}\|{n_{s}}L_{s}\rangle^{*}\langle{n_{0}(n_{0}^{\prime\prime})}0\|T_{L_{t}}\|{n_{t}}L_{t}\rangle^{*}}{E_{n_{s}L_{s}}+E_{n_{t}L_{t}}-E_{n_{0}(n_{0}^{\prime\prime})S}^{(0)}-E_{n_{0}^{\prime}L}^{(0)}}\,,$
$\displaystyle F_{3}(n_{s},n_{t},L_{s},L_{t})$ $\displaystyle=$ $\displaystyle
G_{3}(L_{s},L_{t})\frac{|\langle{n_{0}}0\|T_{L_{t}}\|{n_{t}}L_{t}\rangle|^{2}|\langle{n_{0}^{\prime\prime}}0\|T_{L_{u}}\|{n_{u}}L_{u}\rangle|^{2}}{E_{n_{t}L_{t}}+E_{n_{u}L_{u}}-E_{n_{0}S}^{(0)}-E_{n_{0}^{\prime\prime}S}^{(0)}}\,,$
(171) $\displaystyle F_{4}(n_{u},L_{u},M_{u};L,M)$ $\displaystyle=$
$\displaystyle(-1)^{L_{u}+L+M_{u}+M}G_{4}(L_{u},M_{u};L,M)$ (172)
$\displaystyle\times$
$\displaystyle\bigg{[}\frac{|\langle{n_{0}^{\prime}}L\|T_{L}\|{n_{0}}0\rangle|^{2}|\langle{n_{0}^{\prime\prime}}0\|T_{L_{u}}\|{n_{u}}L_{u}\rangle|^{2}}{E_{n_{u}L_{u}}-E_{n_{0}^{\prime\prime}S}+E_{n_{0}S}-E_{n_{0}^{\prime}L}}$
$\displaystyle+$
$\displaystyle\frac{|\langle{n_{0}}0\|T_{L}\|{n_{0}^{\prime}}L\rangle|^{2}|\langle{n_{0}^{\prime\prime}}0\|T_{L_{u}}\|{n_{u}}L_{u}\rangle|^{2}}{E_{n_{u}L_{u}}+E_{n_{0}^{\prime}L}-E_{n_{0}S}-E_{n_{0}^{\prime\prime}S}}\bigg{]}\,,$
$\displaystyle F_{5}(n_{t},L_{t},M_{t};l_{1};L,M;Q)$ $\displaystyle=$
$\displaystyle(-1)^{L_{t}+L+M_{t}+M}G_{5}(L_{t},M_{t};l_{1};L,M;Q)$ (173)
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}^{\prime}}L\|T_{l_{1}}\|{n_{0}^{\prime}}L\rangle|\langle{n_{0}}0\|T_{L_{t}}\|{n_{t}}L_{t}\rangle|^{2}}{E_{n_{t}L_{t}}-E_{n_{0}S}}\,,$
$\displaystyle F_{6}(n_{t},L_{t},M_{t};l_{2}^{\prime},m_{2}^{\prime};L,M;Q)$
$\displaystyle=$ $\displaystyle
G_{6}(L_{t},M_{t};l_{2}^{\prime},m_{2}^{\prime};L,M;Q)$ (174)
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}}0\|T_{L}\|{n_{0}^{\prime}}L\rangle\langle{n_{0}}0\|T_{L_{t}}\|{n_{t}}L_{t}\rangle^{*}\langle{n_{0}^{\prime}}L\|T_{l_{2}^{\prime}}\|{n_{t}}L_{t}\rangle}{E_{n_{t}L_{t}}-E_{n_{0}^{\prime}L}}\,,$
$\displaystyle F_{7}(n_{t},L_{t},M_{t};l_{2};L,M;Q)$ $\displaystyle=$
$\displaystyle(-1)^{L}G_{7}(L_{t},M_{t};l_{2};L,M;Q)$ (175)
$\displaystyle\times$
$\displaystyle\frac{\langle{n_{0}}0\|T_{L}\|{n_{0}^{\prime}}L\rangle^{*}\langle{n_{0}^{\prime}}L\|T_{l_{2}}\|{n_{t}}L_{t}\rangle^{*}\langle{n_{0}}0\|T_{L_{t}}\|{n_{t}}L_{t}\rangle}{E_{n_{t}L_{t}}-E_{n_{0}S}}\,,$
$\displaystyle G_{1}(L_{i},L_{j},\ell_{k},\ell_{k}^{\prime};L,M)$
$\displaystyle=$
$\displaystyle\frac{16\pi^{2}(\ell_{k},\ell_{k}^{\prime})^{-1/2}}{(2L_{j}+1)^{2}}\sum_{M_{i}M_{j}m_{k}}\left(\begin{array}[]{ccc}L&\ell_{k}&L_{i}\\\
-M&m_{k}&M_{i}\\\
\end{array}\right)\left(\begin{array}[]{ccc}L&\ell_{k}^{\prime}&L_{i}\\\
-M&m_{k}&M_{i}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\frac{(L_{j}+\ell_{k}-M_{j}+m_{k})!(L_{j}+\ell_{k}^{\prime}-M_{j}+m_{k})!P_{L_{j}+\ell_{k}}^{M_{j}-m_{k}}(0)P_{L_{j}+\ell_{k}^{\prime}}^{M_{j}-m_{k}}(0)}{(L_{j}+M_{j})!(L_{j}-M_{j})![(\ell_{k}+m_{k})!(\ell_{k}-m_{k})!(\ell_{k}^{\prime}+m_{k})!(\ell_{k}^{\prime}-m_{k})!]^{1/2}}\,,$
$\displaystyle G_{2}(L_{i},L_{j},\ell_{k},\ell_{k}^{\prime};L,M)$
$\displaystyle=$
$\displaystyle\frac{16\pi^{2}(\ell_{k},\ell_{k}^{\prime})^{-1/2}}{(2L_{i}+1)(2L_{j}+1)}\sum_{M_{i}M_{j}m_{k}m_{k}^{\prime}}\left(\begin{array}[]{ccc}L&\ell_{k}&L_{i}\\\
-M&-m_{k}&M_{i}\\\
\end{array}\right)\left(\begin{array}[]{ccc}L&\ell_{k}^{\prime}&L_{j}\\\
-M&m_{k}^{\prime}&M_{j}\\\ \end{array}\right)$ $\displaystyle\times$
$\displaystyle\frac{(-1)^{M_{i}+M_{j}}P_{L_{j}+\ell_{k}}^{M_{j}+m_{k}}(0)P_{L_{i}+\ell_{k}^{\prime}}^{M_{i}-m_{k}^{\prime}}(0)}{[(L_{i}+M_{i})!(L_{i}-M_{i})!(L_{j}+M_{j})!(L_{j}-M_{j})!]^{1/2}}$
$\displaystyle\times$
$\displaystyle\frac{(L_{j}+\ell_{k}-M_{j}-m_{k})!(L_{i}+\ell_{k}^{\prime}-M_{i}+m_{k}^{\prime})!}{[(\ell_{k}+m_{k})!(\ell_{k}-m_{k})!(\ell_{k}^{\prime}+m_{k}^{\prime})!(\ell_{k}^{\prime}-m_{k}^{\prime})!]^{1/2}}\,,$