# Many-body theory calculations of positronic-bonded molecular dianions

J. P. Cassidy<EMAIL_ADDRESS>Centre for Light-Matter Interactions,
School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7
1NN, Northern Ireland, United Kingdom  J. Hofierka  Centre for Light-Matter
Interactions, School of Mathematics and Physics, Queen’s University Belfast,
Belfast BT7 1NN, Northern Ireland, United Kingdom  B. Cunningham  Centre for
Light-Matter Interactions, School of Mathematics and Physics, Queen’s
University Belfast, Belfast BT7 1NN, Northern Ireland, United Kingdom  D. G.
Green<EMAIL_ADDRESS>Centre for Light-Matter Interactions, School of
Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, Northern
Ireland, United Kingdom

###### Abstract

The energetic stability of positron di-anion systems [A${}^{-};e^{+};$A-] is
studied via many-body theory, where $A^{-}$ includes H-, F-, Cl- and the
molecular anions (CN)- and (NCO)-. Specifically, the energy of the system as a
function of ionic separation is determined by solving the Dyson equation for
the positron in the field of the two anions, using a positron-anion self
energy as constructed in [J. Hofierka, B. Cunningham, C. M. Rawlins, C. H.
Patterson and D. G. Green, _Nature_ 606 688 (2022)] that accounts for
correlations including polarization, screening, and virtual-positronium
formation. Calculations are performed for a positron interacting with
H${}_{2}^{2-}$, F${}_{2}^{2-}$, and Cl${}_{2}^{2-}$, and are found to be in
good agreement with previous theory. In particular, we confirm the presence of
two minima in the potential energy of the [H${}^{-};e^{+}$;H-] system with
respect to ionic separation: one a positronically-bonded [H${}^{-};e^{+}$;H-]
local minimum at ionic separations $r\sim 3.4$ Å, and a global minimum at
smaller ionic separations $r\lesssim 1.6$ Å that gives overall instability of
the system with respect to dissociation into a H2 molecule and a positronium
negative ion, Ps-. The first predictions are made for positronic bonding in
dianions consisting of molecular anionic fragments, specifically for
(CN)${}_{2}^{2-}$, and (NCO)${}_{2}^{2-}$. In all cases we find that the
molecules formed by the creation of a positronic bond are stable relative to
dissociation into A- and $e^{+}$A- (positron bound to a single anion), with
bond energies on the order of 1 eV and bond lengths on the order of several
ångstroms.

##  I Introduction

In their seminal 2018 paper, Charry, Varella and Reyes predicted that
introducing a positron to two otherwise repelling hydride anions H- would lead
to stabilization of the molecular complex via a _positronic covalent bond_
Charry, do N Varella, and Reyes (2018). Their analysis of the potential energy
curves (PEC) calculated via the configuration interaction (CI) method for the
[H-;$e^{+}$;H-] system revealed a minimum in the total system energy below the
threshold for dissociation into a free hydride anion H- and a positron bound
to the other anion, $e^{+}$H-, demonstrating atomic matter fragments bound by
antimatter.

The potential stability of the [H-;$e^{+}$;H-] system was subsequently also
investigated in so-called “multi-component Quantum Theory of Atoms in
Molecules” calculations by Goli and ShahbazianGoli and Shahbazian (2019), and
Monte Carlo calculations by Ito _et al._ Ito  _et al._ (2020), and by
BressaniniBressanini (2021a), with Bressanini showing that a local energy
minimum occurred for anionic separations greater than 1.7 Å, but that the
system was ultimately unstable against dissociation into H2 and the
positronium negative ion Ps-.

Since the original publication in 2018, there have been numerous studies of
anionic bonding via antimatter. Moncada _et al._ Moncada  _et al._ (2019)
showed that positronic bonds could be formed in homo- and heteronuclear
dihalide systems (they considered fluoride F-, chloride Cl- and bromide Br-
anions), and Ito _et al._ Ito  _et al._ (2023) showed that a positron could
stabilize two Li- anions. Bressanini went on to show that two positrons could
form a chemical bond in (PsH)2Bressanini (2021b) in a similar way to how the
electrons from two hydrogen atoms can form a bond to create the H2 molecule,
provided that the positrons involved have opposite spins (the conventional
notation PsA is used to denote the positron-anion bound system Schrader
(1998)). However, analysis of the ${}^{1}\Sigma_{g}$ PEC for the (PsH)2
molecule formed revealed that for internuclear distances less than 3.6 a.u.
the molecule would dissociate into H2 and Ps-. Charry _et al._ Charry  _et
al._ (2022) showed that a distinct non-electronic three-centre, two positron
bond could be formed in a system containing two positrons and three hydride
ions.

Recently, a number of us developed the many-body theory of positron binding to
polyatomic molecules implemented via a Gaussian-basis representation in our
EXCITON+ code Hofierka  _et al._ (2022). The approach enables the accurate
account of the positron-molecule correlations including polarization,
screening of the electron-positron Coulomb interaction, and the important
process of virtual-positronium formation (where an electron temporarily
tunnels to the positron). After successful benchmarking against explicitly
correlated Gaussian and quantum Monte Carlo calculations for small molecules
(LiH and formaldehyde respectively), it gave the first _ab initio_
calculations in agreement with over two decades of pioneering experiments,
delineated the role of distinct correlations, and provided fundamental
insight, e.g., the contribution of individual molecular orbitals and the role
of the positron-molecule potenial anisotropy, and made predictions for systems
including DNA bases. More recently it has provided _ab initio_ description of
trends within molecular familiesCassidy  _et al._ (2023), quantified the
effects of high-order diagrams to positron-atom scattering and annihilation
properties Hofierka  _et al._ (2023a), and has been successfully extended to
positron scattering and annihilation on non-binding molecules Rawlins  _et
al._ (2023).

Here, we apply our many-body approach to study the formation of positronic
bonds between selected anions. We first consider the H${}_{2}^{2-}$,
F${}_{2}^{2-}$, Cl${}_{2}^{2-}$ anionic systems, comparing with existing
results. We then go beyond previous studies by making predictions of
positronic bonding in dianion systems containing molecular anion fragments,
specifically (CN)${}_{2}^{2-}$, and (NCO)${}_{2}^{2-}$. We find that all the
systems are capable of forming positronic bonds that are stable relative to
dissociation into A- and $e^{+}$A- (positron bound to a single anion), with
corresponding bond energies on the order of 1 eV and bond lengths on the order
of a few ångstroms. In particular, we find that the positronically bonded
[H-;$e^{+}$;H-] system is unstable relative to dissociation into H2 and Ps-
for small ionic separations $r\lesssim 1.6$ Å, agreeing with the previous work
of BressaniniBressanini (2021a).

The paper is organized as follows. In Sec. II we present the theoretical
details of the present study, including a brief overview of the many-body
theory approach and numerical details for the present calculations. Section
III presents the results, including bond energy curves and the positron
density at the bond length separation of each anion system, comparing to other
theoretical studies where possible. We conclude in Sec IV.

##  II Theory

Throughout this work we use atomic units unless otherwise stated, and employ
the Born-Oppenheimer approximation for all systems under consideration.

In this work we consider the stability of the [A${}^{-};e^{+}$;A-] systems
relative to the dissociation [A${}^{-};e^{+}$;A${}^{-}]\to$
A${}^{-}+\text{PsA}$, i.e. into one free anion and a positron bound to the
other anion. As we discuss in Sec III, at small ionic separations $r\lesssim
1.7$ Å the [H${}^{-};e^{+}$;H-] system’s lowest energy dissociation channel is
into H${}_{2}+$ Ps-Bressanini (2021a). At these small separations, the
H${}_{2}+$ Ps- PEC is below that of the [H${}^{-};e^{+}$;H-] PEC, resulting in
the system not being absolutely stable. However, for $r>1.7$ Å there is a
local energy minimum of the [H${}^{-};e^{+}$;H-] system which is below the
PECs of the other dissociation channels. We are able to compute this local
minimum in this work using many-body theory, and we discuss it in order to
benchmark against the results in the existing literature.

Assuming the lowest dissociation channel is to A${}^{-}+$PsA, the _bond
energy_ of the positron and molecular dianion system is

$\displaystyle B(r)$ $\displaystyle=$ $\displaystyle
E_{\text{A}^{-};e^{+};\text{A}^{-}}(r)-E_{\text{PsA+A}^{-}},$ (1)

where $E_{\text{PsA+A}^{-}}$ is the total energy of a free anion A- and PsA,
and $E_{\text{A}^{-};e^{+};\text{A}^{-}}(r)$ is that of the positron-dianion
system. Thus, negative positronic bond energies indicate a stable complex
relative to dissociation into the free anion A- and PsA, and
$\lim_{r\to\infty}B(r)=0$. The positronic bond length $R$ is the value of $r$
at which $B(r)$ has a minimum. In practice we determine the total energy of
the positron-dianion system as (see Fig. 1)111In this approximation we account
for positron-anion correlations, but assume the electron orbitals are
unaffected by the presence of the positron. The good agreement we obtain with
other methods suggests an _a posteorori_ justification for this.

$\displaystyle
E_{\text{A}^{-};e^{+};\text{A}^{-}}(r)=E_{\text{A}^{-}\text{A}^{-}}(r)-|\varepsilon_{b}^{(2)}(r)|,$
(2)

and the positron-anion plus anion threshold energy as

$\displaystyle
E_{\text{PsA+A}^{-}}=2E_{\text{A}^{-}}-|\varepsilon_{b}^{(1)}|.$ (3)

Here we calculate the all-electronic energy terms
$E_{\text{A}^{-}\text{A}^{-}}(r)$ and $E_{\text{A}^{-}}$ in the Hartree-Fock
approximation. The positron binding energies to a single anion
$\varepsilon_{b}^{(1)}$ and to two anions (as a function of the interionic
distance) $\varepsilon_{b}^{(2)}(r)$ are obtained using many-body theory Dzuba
_et al._ (1996); Müller and Cederbaum (1990); Gribakin and Ludlow (2004);
Green, Ludlow, and Gribakin (2014); Green and Gribakin (2013, 2015); Hofierka
_et al._ (2022) accounting for important correlation effects, via solution of
the Dyson equation for the positron in the field of the anion (or atom or
molecule) Fetter and Walecka (2003); Dickhoff and Neck (2008); Green, Ludlow,
and Gribakin (2014); Hofierka  _et al._ (2022)

Figure 1: Schematic of the potential energy curves of positronic-bonded
dianions, as in Eq. (1). $E_{\text{A}^{-}\text{A}^{-}}(r)$ (black line) is the
PEC of two anions separated by a distance $r$ in the absence of the positron,
which in the limit of infinite $r$ has the threshold energy of twice the
energy of a single anion $E_{\text{A}^{-}}$. The blue curve is for the
positron-dianion system, which in the limit of infinite $r$ has the threshold
energy of a free anion A- and PsA. $\varepsilon_{b}^{(2)}(r)$ is the positron
binding energy to the dianionic molecule at separation $r$, while
$\varepsilon_{b}^{(1)}$ is the positron binding energy to a single anion. $R$
is the bond length, i.e., the point at which $E_{\text{A}^{-}\text{A}^{-}}(r)$
is minimum.

$(\hat{H}_{0}+\hat{\Sigma}_{\varepsilon})\psi_{\varepsilon}(\mathbf{r})=\varepsilon\psi_{\varepsilon}(\mathbf{r}).$
(4)

Figure 2: Diagrams the calculation of the positron self-energy. All lines
directed to the right (left) represent Green’s functions of particles (holes)
propagating on the $N$ electron ground state of the anion(s). Single wavy
lines represent Coulomb interactions. (a) The ‘$GW$’ diagram, which is a
product of the positron Green’s function $G_{\nu}$ and the dressed Coulomb
interaction $W$, accounts for polarization of the target by the positron,
electron-hole interactions and screening of the Coulomb interaction; (b) the
virtual-positronium formation diagram, which includes the infinite series of
electron-positron interactions (the $\Gamma$-block); (c) the positron-hole
interaction diagram (the $\Lambda$-block). See [Hofierka  _et al._ (2022)] for
full details.

Here, $\hat{H}_{0}$ is the zeroth order Hamiltonian of the positron in the
static field of the anion(s), $\hat{\Sigma}_{\varepsilon}$ is the non-local,
energy-dependent correlation potential that is equivalent to the irreducible
self-energy of the positron in the field of the ion,
$\psi_{\varepsilon}(\mathbf{r})$ is the positron (Dyson) wave function and
$\varepsilon$ is the positron energy. For negative positron energy eigenvalues
we define the positron binding energy as $\varepsilon_{b}=-\varepsilon$. We
have calculated the positron binding energies $\varepsilon_{b}^{(1)}$ to H-,
F- and Cl- in (Hofierka  _et al._ , 2023b) and use those results here. Full
details on the construction of $\Sigma$ and the solution of the Dyson
equation, including numerical implementation, can be found in the recent work
by some of the authorsHofierka  _et al._ (2022). Briefly, we expand the self
energy $\Sigma$ diagrammatically in the residual electron-electron and
electron-positron interactions. We calculate three infinite classes of
diagrams, shown in Fig. 2. The first is the $GW$ diagram, which we calculate
in the Bethe-Salpeter Equation approximation, accounting for polarization of
the electron cloud by the positron, screening of the Coulomb interaction
(random phase approximation) and screened electron-hole interactions. The
second is the virtual positronium (vPs) formation diagram that includes the
infinite ladder series of electron-positron Coulomb interactions, the
‘$\Gamma$-block’. The third is the positron-hole repulsion diagram, which is
of similar structure to the vPs diagram, but instead includes an infinite
series of positron-hole Coulomb interactions.

We use the atomic aug-cc-pVQZ Gaussian basis sets of Dunning Kendall, Dunning
Jr, and Harrison (1992) to expand the electron and positron molecular orbital
wave functions. These basis set centres coincide with the anionic centres. In
order to ensure the convergence of the vPs contribution to the self energy, we
employ additional hydrogen-type aug-cc-pVQZ bases on “ghost atom” centres to
provide effective higher angular-momenta Swann and Gribakin (2018). For the
[F-;$e^{+}$;F-], [Cl-;$e^{+}$;Cl-], [(CN)-;$e^{+}$;(CN)-], and
[(NCO)-;$e^{+}$;(NCO)-], systems we used a single ghost centred between the
anions along the bond axis. For the [H-;$e^{+}$;H-] system we used 5 ghosts:
one centred between the two anions on the bond axis, and 4 more placed 1Å away
from the bond axis forming a square plane normal to the bond axis. We used the
additional ghosts in the [H-;$e^{+}$;H-] system in order to compensate for the
lack of $g$-type angular momentum functions in the H aug-cc-pVQZ basis which
are present in the corresponding F, Cl, C, N, and O basis sets.

##  III Results and discussion

The total energy curve for the [H-;$e^{+}$;H-] system, along with the positron
wave function densities at the total energy minima are shown in Fig. 3. The
bond energy curves and the positron wave function densities at the bond energy
minima for the [F-;$e^{+}$;F-] and [Cl-;$e^{+}$;Cl-] systems are presented
together in Fig. 4. Those for [(CN)-;$e^{+}$;(CN)-], and
[(NCO)-;$e^{+}$;(NCO)-] are shown in Fig. 5, along with data for positron
binding to the single anions (CN)- and (NCO)-. We now discuss each system in
turn.

###  III.1 H${}_{2}^{2-}$

The positron plus hydrogen molecular dianion system has been studied in
previous works by Charry et al.Charry, do N Varella, and Reyes (2018), Ito et
al.Ito  _et al._ (2020) and BressaniniBressanini (2021a). All confirmed the
stability of the positronic-bonded [H-; $e^{+}$; H-] system with respect to
dissociation into H- and PsH.

Figure 3a shows the total energy of the [H-;$e^{+}$;H-] system for ionic
separations ranging from $r=0.4$ Å to $r=4.0$ Å. Two minima are evident, one
at 0.7 Å and one at 3.4 Å, matching the so-called “M1” and “M2” minima
described by Ito et al.Ito  _et al._ (2020)

As shown in Fig. 3a and Table 1, at separations $r\gtrsim 2$ Å we find that
the system is stable with respect to dissociation into H- and PsH, with a bond
energy of magnitude 0.70 eV and equilibrium bond length $R=3.4$ Å. The
magnitude of the bond energy is in good agreement with the results of the
aforementioned studies, e.g., the 0.68 eV full configuration interaction (FCI)
and configuration interaction up to quadruple excitation (CISDTQ) results of
Charry et al., the 0.61 eV result of the multicomponent diffusion Monte Carlo
(MC DMC) calculation by Ito et al., and the 0.64 eV diffusion Monte Carlo
result of Bressanini. Our calculated bond length $R=3.4$ Å is in very close
agreement with the best results of Charry et al.Charry, do N Varella, and
Reyes (2018) and Ito et al.Ito  _et al._ (2020), who both reported bond
lengths of $R=3.26$ Å, and in excellent agreement with Bressanini’s range of
$R=3.39\pm 0.21$ ÅBressanini (2021a). The positronic density for the system at
the $r=3.4$ Å separation is shown in Fig. 3c, and shows a clear preference for
the positron to be found in the intermediate region between the two H- anions.
This is in perfect agreement with the original findings of Charry et al., that
first presented such a positronic bond.

Table 1:  Calculated positronic bond lengths $R$ (Å) and the magnitudes of the bond energies $B(R)$ (eV) for positronic molecular dianions as in Eq. (1). The bond energies are relative to dissociation into an infinitely separated free anion and a positron bound to the remaining anion [A-;$e^{+}$;A-]$\to$A${}^{-}+\text{PsA}$. System | Method | $R$ | $B(R)$
---|---|---|---
H${}_{2}^{2-}$ | FCI Charry, do N Varella, and Reyes (2018) | 3.26 | 0.68
| CISDTQ Charry, do N Varella, and Reyes (2018) | 3.26 | 0.68
| CBS Charry, do N Varella, and Reyes (2018) | 3.26 | 0.81
| MC VMC Ito  _et al._ (2020) | 3.26 | 0.43
| MC DMC Ito  _et al._ (2020) | 3.26 | 0.61
| VMC Bressanini (2021a) | 3.39$\pm 0.21$ | 0.62
| DMC Bressanini (2021a) | 3.39$\pm 0.21$ | 0.64
| MBT (present work) | 3.4 | 0.70
F${}_{2}^{2-}$ | APMO/REN-PP3Moncada  _et al._ (2019) | 3.09 | 1.13
| MBT (present work) | 3.2 | 1.42
Cl${}_{2}^{2-}$ | APMO/REN-PP3 Moncada  _et al._ (2019) | 3.87 | 0.86
| MBT (present work) | 4.2 | 0.94
(CN)${}_{2}^{2-}$ | MBT (present work) | 3.9 | 0.90
(NCO)${}_{2}^{2-}$ | MBT (present work) | 3.5 | 1.28
Figure 3: (a) The total energy of the [H-;$e^{+}$;H-] system as a function of
the ionic separation distance $r$. The blue dashed curve is the PEC for the
bare H2 molecule, calculated by Kolos and WolniewiczKolos and Wolniewicz
(1968), shifted vertically so that the minima coincide. The black dashed line
is the PsH+H- energy threshold; (b) the positron Dyson wave function density
(in a.u.) for H${}_{2}^{2-}$, $|\psi_{\varepsilon}(\mathbf{r})|^{2}$ in the
$xy$-plane, at the separation $r=0.7$ Å. Contours are drawn at 10%, 25%, 50%
and 80% of the maximum density value; (c) the positron Dyson wave function
density (in a.u.) for H${}_{2}^{2-}$, $|\psi_{\varepsilon}(\mathbf{r})|^{2}$
in the $xy$-plane, at the bond length separation $R=3.4$ Å. Contours are drawn
at 10%, 25%, 50% and 80% of the maximum density value.

For separations $r<1.6$ Å we found that positronium formation was no longer
virtual, meaning that dissociation channels involving real Ps and Ps- were
available. This corroborates the work of Bressanini Bressanini (2021a), who
found that below 1.7 Å, the lowest energy dissociation threshold involved
formation of a Ps- anion, i.e., the positron was able to escape the system
bound to electrons that originated from the bare anions. The sharp peak in the
curve at $r\sim 1.6$ Å corresponds to the crossing of the H2+Ps- and H-+PsH
PECs. Along with the total energy curve of [H-;$e^{+}$;H-], we also show the
PEC of the H2 molecule (originally calculated by Kolos and WolniewiczKolos and
Wolniewicz (1968)), shifted vertically so that the minimum of the H2 PEC
coincides with the [H-;$e^{+}$;H-] energy minimum. Given that the shapes of
the curves are in very good agreement, this suggests that the [H-;$e^{+}$;H-]
system has dissociated into bare H2, with the two surplus electrons forming
Ps- with the positron. We originally attempted to compare our [H-;$e^{+}$;H-]
PEC with that of H2 shifted down by the absolute value of the ground state
energy of Ps- (0.262 a.u.), and found that the resultant H2+Ps- PEC had a
minimum 0.137 a.u. below our [H-;$e^{+}$;H-] minimum. This discrepancy can be
interpreted as the expectation energy of the kinetic energy of the Ps- being
described (there will also be some discrepancy due to the fact that we only
consider the all-electron energy terms in the Hartree-Fock approximation).
Figure 3b shows the positron density at the $r=0.7$ Å separation. In order to
ensure an accurate description of the positronic density around the entire
molecule, we used an extra two ghosts, each ghost 1 Å away from the H- anions
along the bond axis. In our description of the system at this separation, the
positron is delocalized around the entirety of the molecule (but it is still
bound to the system). The spatial distribution of the positron density around
the entire molecule, coupled with the shape of the total [H-;$e^{+}$;H-]
energy curve matching that of bare H2, provides further evidence that a Ps--
like object has been formed. We do note however that our description of such a
Ps--like object is imperfect. Our $\Gamma$-block diagram contribution to the
self-energy is only rigorous in cases where Ps formation is virtual, and
strictly speaking it does not specifically allow for two electrons to tunnel
near the positron. Furthermore, the use of Gaussian basis functions that decay
quickly from the basis function centres mean we cannot describe the positron
(and thus Ps-) over all space far from the anions. However, a useful
consequence of the decay of the Gaussian basis functions is the artificial
confinement of the positron to regions of space near the anions, and this is
why the many-body theory in its current implementation is able to at least
_partially_ describe the Ps- system.

With regards to the experimental realization of the positronically-bonded
[H-;$e^{+}$;H-] system, our results corroborate the suggestion by
BressaniniBressanini (2021a) that the formation of the [H-;$e^{+}$;H-] complex
could be evidenced by resonant scattering in collisions of Ps- with
vibrationally excited H2.

Figure 4: Positronic bond energy curves for the [F-;$e^{+}$;F-] and
[Cl-;$e^{+}$;Cl-] systems, with negative bond energies indicating a stable
complex with respect to dissociation into a free anion A- and PsA. The bond
energy is defined in Eq. (1). The position of the bond energy minimum defines
the positronic bond length, $R$. Bottom panels show the positron Dyson wave
function density (in a.u.), $|\psi_{\varepsilon}(\mathbf{r})|^{2}$ in the
$xy$-plane, at the bond length separation $R$. Contours are drawn at 10%, 25%,
50% and 80% of the maximum density values for each system.

###  III.2 F${}_{2}^{2-}$

We next consider halide anions. For these systems we are only aware of one
other study by Moncada et al.Moncada  _et al._ (2019) Owing to the more
complicated many-electron structure of the halide anions and the resultant
high computational cost compared to H${}_{2}^{2-}$, their study used
“thermodynamic cycles” to determine the positronic bonding energies in terms
of positron binding energies to single anions and molecular anions, electron
binding energies and the dissociation energies of purely electronic molecular
dianions. Their positron binding energies were calculated using the APMO/REN-
PP3 propagator method, which is a renormalized third-order approximation to
the diagonal elements of the self energy. In contrast, our present many-body
theory approach considers the full self energy of the positron, i.e.,
including off-diagonal elements in the matrix elements of the diagrams in Fig.
2.

Our bond energy curve for the F${}_{2}^{2-}$ system is shown in Fig. 4. Our
results confirm that the system is stable with respect to dissociation into F-
and PsF, with a bond energy of magnitude 1.42 eV and a bond length of 3.2 Å.
The bond energy is in reasonable agreement with the result of Moncada et al.,
1.13 eV, with our reported bond length in very good agreement with their
reported 3.09 Å. Figure 4 also shows the positronic density for the system,
with the positron most likely to be found in the region between the F- anions,
indicating a positronic bond.

Figure 5: (a) The positron density along the molecular axis of a single CN-
(black curve) and NCO- (red curve) molecular anion. Also shown are the
positron binding energies for these molecular anions, $\varepsilon_{b}^{(1)}$,
and the corresponding Dyson wave functions at 90% of the wave function
maximum. The locations of zero positron density coincide with the positions of
the nuclei in each molecular anion (in the order as shown in the ball-and-
stick models); (b) the positronic bond energy curve for the (CN)${}_{2}^{2-}$
system along with the positron Dyson wave function at 80% of the maximum wave
function value. $r$ is taken to be the distance between the directly opposing
C atoms; (c) the positronic bond energy curve for the (NCO)${}_{2}^{2-}$
system along with the positron Dyson wave function at 80% of the maximum wave
function value. $r$ is taken to be the distance between the directly opposing
N atoms; (d) the positron Dyson wave function density (in a.u.) for
(CN)${}_{2}^{2-}$ , $|\psi_{\varepsilon}(\mathbf{r})|^{2}$ in the $xy$-plane,
at the bond length separation $R=3.9$ Å. Contours are drawn at 10%, 25%, 50%
and 80% of the maximum density value; (e) the positron Dyson wave function
density (in a.u.) for (NCO)${}_{2}^{2-}$ ,
$|\psi_{\varepsilon}(\mathbf{r})|^{2}$ in the $xy$-plane, at the bond length
separation $R=3.5$ Å. Contours are drawn at 10%, 25%, 50% and 80% of the
maximum density value.

###  III.3 Cl${}_{2}^{2-}$

Our final comparison with a previously studied system is Cl${}_{2}^{2-}$,
which also featured in the study of Moncada et al.Moncada  _et al._ (2019) We
again confirm the stability of the system with respect to dissociation into
Cl- and PsCl, and report a positronic bond energy of 0.94 eV, in very close
agreement with the 0.86 eV reported by Moncada et al. There is also good
agreement in the reported bond lengths, with the present MBT bond length of
4.2 Å less than $10$% larger than the 3.87 Å reported by Moncada et al. It is
important to note that both studies agree in that the positronic
F${}_{2}^{2-}$ system is more energetically stable than the positronic
Cl${}_{2}^{2-}$ system, demonstrating consistency in the predictions made by
both methods. As shown in Fig. 4, the positron is again most likely to be
found in the region between the chloride anions, confirming the formation of a
positronic bond.

###  III.4 (CN)${}_{2}^{2-}$ and (NCO)${}_{2}^{2-}$

We now turn our attention to the systems comprised of CN- and NCO- anions. To
our knowledge, our calculations for these systems are not only the first for
these systems in the context of positronic bonding, but they are also the
first calculations of positronic bonding where the bonded anionic fragments
are themselves molecular in nature. We found (via many-body calculations using
the full self energy of Fig. 2) that a single CN- anion binds the positron
with a binding energy of $\varepsilon_{b}^{(1)}=5.30$ eV, with the maximum
positron density localised near carbon. For a single NCO- anion we calculated
the positron binding energy to be $\varepsilon_{b}^{(1)}=4.82$ eV, with the
maximum positron density localised near nitrogen. The positronic density along
the molecular axis for both anions is shown in Fig. 5a.

Considering now the molecular dianions, we orientate the anionic fragments
such that the molecular axes are collinear, with the carbon atoms of CN-
facing each other for [(CN-);$e^{+}$;(CN)-], and the nitrogen atoms of NCO-
facing each other for [(NCO-);$e^{+}$;(NCO)-] (see Figs. 5b-c). For these
systems the separation distance $r$ in the figures is the distance between the
directly facing C or N atoms. We confirm the stability of the systems with
respect to dissociation as shown in Figs. 5b-c. For (CN)${}_{2}^{2-}$ we
report a bond energy of 0.90 eV and corresponding bond length of 3.9 Å. For
(NCO)${}_{2}^{2-}$ we report a higher bond energy of 1.28 eV and a
corresponding bond length of 3.5 Å. The positron density in the $xy$ plane for
(CN)${}_{2}^{2-}$ is shown in Fig 5d, and for (NCO)${}_{2}^{2-}$ in Fig. 5e.
Both plots illustrate the positronic bond, consisting of positron density
between the anionic fragments. In the cases of (CN)${}_{2}^{2-}$, and
(NCO)${}_{2}^{2-}$, there still remains the question of whether the
positronically-bonded systems are stable with respect to dissociation to Ps
and Ps-, which would be of interest for future works.

##  IV summary and conclusions

Many-body theory was applied to study the phenomenon of positronic bonding in
molecular dianions. For the [H-;$e^{+}$;H-], [F-;$e^{+}$;F-], and
[Cl-;$e^{+}$;Cl-] positron-dianion systems the calculated bond energies and
bond lengths were found to be in good agreement ($\lesssim$10% error) with
previous theoretical studies. For [H-;$e^{+}$;H-] we also observed the system
to be unstable to dissociation into H2 and Ps-, corroborating prevoius studies
. We presented predictions for systems of positronic bonding of _molecular_
fragments [(CN)-;$e^{+}$;(CN)-] and [(NCO)-;$e^{+}$;(NCO)-], finding positron-
bonded complexes stable relative to the anion A- and PsA dissociation
threshold. For these calculations we arranged the molecular fragments such
that they were collinear and the positron-attracting ends of the single anions
were directly facing each other. This orientation was chosen based on the fact
that in a single CN- anion the positron was localized mainly on the C atom
while in a single NCO- anion the positron was localized mainly on the N atom.
It is possible that this orientation is not the most energetically preferable
for the dimer system, either with or without the positron. Therefore, it may
be of interest to investigate the stability of both dianion systems as a
function of, for example, two orientation parameters– one being similar to the
separation parameter, $r$, as discussed in this paper, and an angular
parameter to describe rotations of one of the fragments away from the linear
orientation used here.

The many-body theory calculations used Hartree-Fock dianion potential energy
curves. Improved calculations could be performed by accounting for eletron-
electron correlations, e.g., in a $GW$@BSE calculation of the total energies,
or via a self-consistent calculation in which the electron orbitals are
allowed to adapt to the presence of the positron. Further studies of interest
(beyond the orientation dependence of the stability of the dimer systems as
described above) could include considering larger dianion molecular complexes,
the rovibrational dependence and electronic-vibrational coupling in the
dianion molecular fragment systems, and time-dependent studies of ionization,
electron and positron scattering, and nuclear dynamics on such positronic-
bonded complexes. Beyond fundamental interest, understanding these
interactions could elucidate positron physics and chemistry in e.g., the
interstellar medium Prantzos  _et al._ (2011), where anions are known to be
present Millar, Walsh, and Field (2017).

###### Acknowledgements.

We thank Gleb Gribakin for useful discussions. JPC is funded by a Department
for the Economy Northern Ireland postgraduate research studentship. This work
was funded by the European Research Council grant 804383 ’ANTI-ATOM’ (DGG).

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# RPG: Learning Recursive Point Cloud Generation

Wei-Jan Ko
<EMAIL_ADDRESS>Hui-Yu Huang
<EMAIL_ADDRESS>Yu-Liang Kuo
<EMAIL_ADDRESS>Chen-Yi Chiu
<EMAIL_ADDRESS>Li-Heng Wang
<EMAIL_ADDRESS>Wei-Chen Chiu
<EMAIL_ADDRESS>National Chiao Tung University

###### Abstract

In this paper we propose a novel point cloud generator that is able to
reconstruct and generate 3D point clouds composed of semantic parts. Given a
latent representation of the target 3D model, the generation starts from a
single point and gets expanded recursively to produce the high-resolution
point cloud via a sequence of point expansion stages. During the recursive
procedure of generation, we not only obtain the coarse-to-fine point clouds
for the target 3D model from every expansion stage, but also unsupervisedly
discover the semantic segmentation of the target model according to the
hierarchical/parent-child relation between the points across expansion stages.
Moreover, the expansion modules and other elements used in our recursive
generator are mostly sharing weights thus making the overall framework light
and efficient. Extensive experiments are conducted to demonstrate that our
proposed point cloud generator has comparable or even superior performance on
both generation and reconstruction tasks in comparison to various baselines,
as well as provides the consistent co-segmentation among 3D instances of the
same object class.

##  1 Introduction

As the increasing usage of 3D sensors nowadays, understanding the 3D data and
the rich geometric information of it becomes a crucial problem in many
applications such as robotics and autonomous vehicles. Among various formats
of 3D data (e.g. meshes, voxels, and point clouds), the point cloud is one of
the most intuitive and popular representations, not only because of being the
default output of several 3D sensors (e.g. LiDARs) but also its ability of
well capturing the geometric details. In this paper, we focus on better
understanding the underlying geometry of 3D data via learning the generative
procedure of point clouds.

Without loss of generality, understanding the geometry of 3D point clouds can
be categorized into three levels of granularity from fine to coarse, as
sequentially described in the following: (1) _Modelling structure details_ ,
where many works [22, 8, 3, 21, 10, 1] proposed for generating or
reconstructing point clouds belongs to such level, in which various modelling
techniques are adopted to describe the geometric details. For instance, [22,
8] utilize the transformation of multiple 2D grids/patches to reconstruct the
3D object surface, while [3] builds the probabilistic model based on point
densities for generating sophisticated surfaces through sampling; (2)
_Extracting semantic parts_ , where a point cloud is represented as the
(hierarchical) combination of semantic parts. For instance, both [9] and [18]
propose tree-structured generative models of point clouds, where the former
explicitly constructs hierarchical part-based representations while the latter
produces part segmentation of the generated point cloud via tracing the tree
built upon graph convolutions; and (3) _Discovering shape correspondences_ ,
where the co-analysis of multiple point clouds of the same class is applied to
discover the correspondences across shapes. For example, [6] learns the co-
segmentation while [5] finds the semantic-aware structural points shared
across object instances.

Figure 1: With the novel design of having recursive point expansion stages,
our proposed model is able to not only produce coarse-to-fine generation of
point clouds, but also unsupervisedly discover their semantic parts, which
provides more fine-grained understanding on the geometry of 3D shapes.

While most of the research works of understanding the geometry of point clouds
only focus on one of the aforementioned levels (with few of them taking care
of two levels), in this paper we aim to model the generative procedure of
point cloud data with including all the granularities in a unified framework.
We propose a recursive point cloud generator (RPG) which takes a latent vector
(standing for the representation of the whole target 3D shape) as input, and
generates the point clouds in a coarse-to-fine manner via a sequence of point
expansion stages, as shown in Figure 1. Basically, the generation starts from
initializing the simplest/coarsest point cloud composed of only a single 3D
point at origin, the expansion module at each expansion stage recursively
duplicates every point obtained from the previous stage certain times to form
a finer point cloud (i.e. having more points to detailedly represent the
target 3D shape). The point clouds produced from all the stages naturally form
a tree-structure, where we can partition the points at an arbitrary stage
according to their ancestor assignments on a previous stage thus resulting the
hierarchical part segmentation in an unsupervised way. Moreover, as the RPG is
learnt on multiple 3D instances at the same time, the shape correspondences
across point clouds can be automatically discovered. With having the expansion
modules sharing weights across expansion stages by the recursive nature of our
RPG, our proposed framework results to have small model size thus leading to
more efficient model training. We experimentally show that our RPG framework
is able to achieve comparable generation and reconstruction performance with
respect to the state-of-the-art baselines, and produce compact and
semantically-meaningful part segmentation/co-segmentation on point clouds.

##  2 Related Work

In the following we describe the related works for understanding geometry of
3D point cloud data from three granularities, as aforementioned in the
introduction.

Reconstruction and Generation. Though the popularity of using point cloud to
represent 3D data, the tasks of reconstructing or generating the point cloud
data are still considered to be challenging due to its irregularity (e.g.
permutation-invariance). Various research works have been proposed these
years, for instance: [22] and [8] represent a 3D shape as the deformation of
2D grid points or the 2D parametric surface elements, which however cannot
well reconstruct the local details of point cloud data; [21] models the point
clouds as a two-level hierarchy of distributions via adopting the frameworks
based on continuous normalizing flow; [10] develops a progressive
deconvolution network to generate the coarse-to-fine point clouds, which is
trained via adversarial learning; and [3] proposes an energy-based
probabilistic framework to model the distribution of 3D points via learning
the gradient field of the point densities. While these prior works improve the
fidelity of generated or reconstructed point clouds over the recent years, the
information or the segmentation of semantic parts are not taken into
consideration for their model designs.

Hierarchical and Part-based Representation. Treating the 3D shape as a
hierarchical decomposition of parts has facilitated humans’ understanding on
the shape structures, and thus have attracted plenty research attentions. For
instance, [14] and [15] represent 3D objects as hierarchical $n$-ary trees and
train the graph neural network [12] and the conditional variational
autoencoder respectively to learn the hierarchical structures, where they
require the dataset with groundtruth part annotations (e.g. PartNet
[mo2019partnet]) for the model learning. In contrast, Hertz _et al_. [9]
learns the hierarchical part-based representation of point clouds in an
unsupervised manner, where they adopt the tree-structured Gaussian mixture
distribution to probabilistically approximate the point cloud surface.
However, their model struggles to produce small sharp details in the generated
point cloud, as indicated by themselves in [9]. Similarly, TreeGAN proposed by
[18] learns a point cloud generator built upon tree-structured graph
convolution network, but only outputs the point cloud with part segmentation
at the final layer and lacks of the intermediate point clouds during the
generation, i.e. having no hierarchical representation.

Cross-Object Shape Correspondence. Discovering the semantic correspondences
across 3D object instances helps to investigate and study the variances among
3D shapes thus leading to more high-level understanding. [7, 16, 19] learn the
decomposition of shapes with adopting the primitive-based representation,
where the parts are however limited to the primitives such as cuboids,
superquadrics, or convex hulls. [5] instead proposes an unsupervised method to
discover the 3D structural points which are shared among the shape instances
with similar structures and show the semantic consistency. And [6] leverages a
branched autoencoder to achieve co-segmentation between similar shapes, where
each branch represents a recurring part in the implicit field. Although
producing semantic correspondences across shapes, [5, 6] do not support the
point cloud generation.

In contrast to all the aforementioned works, our proposed recursive point
cloud generator (RPG) is able to perform reconstruction and generation of
point cloud data with high fidelity, produce hierarchical part-based
segmentation without requiring any supervision on part annotations, and
discover the semantic correspondence across instances of the same category,
with all achieved by a lightweight model.

Figure 2: Illustration for the basic concept behind our proposed Recursive
Point Cloud Generator (RPG). Assuming now we are aiming to reconstruct the
point cloud of a 3D chair model, with being given its latent/structural
representation $z$. Our generation starts from having
$\mathbb{S}^{(0)}=\\{\mathbf{0}\\}$ and $\mathbb{H}^{(0)}=\\{z\\}$. The
expansion module $\mathcal{E}$ in the Stage 0 firstly expands from the origin
point $\mathbf{0}$ into $k(0)=5$ points
$\mathbb{S}^{(1)}=\\{s^{(1)}_{i}\\}_{i=1}^{k(0)}$, which are roughly
distributed to the locations of the chair back and chair legs, as well as
evolves $z$ into $k(0)$ structural representations
$\mathbb{H}^{(1)}=\\{h^{(1)}_{i}\\}_{i=1}^{k(0)}$ indicating that now the
chair model is assembled by $k(0)$ parts. The expansion module $\mathcal{E}$
in the Stage 1 can further expand every point in $\mathbb{S}^{(1)}$ and
representation in $\mathbb{H}^{(1)}$ by $k(1)$ times in order to derive
$\mathbb{S}^{(2)}$ and $\mathbb{H}^{(2)}$, where now the point cloud for the
chair contains more points thus becoming more fine-grained. With having the
expansion module recursively applied $D$ times, we obtain $\mathbb{S}^{(D)}$
as the resultant point cloud for the 3D chair model, in which there are
$\prod_{d=0}^{D-1}k(d)$ points in the point cloud. The overall procedure of
our RPG shows the coarse-to-fine generation of the 3D point cloud, and we can
easily obtain the part segmentation of the point cloud according to the
corresponding ancestors (e.g. in $\mathbb{S}^{(1)}$) for each point in
$\mathbb{S}^{(D)}$, as shown in the right-most figure.

##  3 Recursive Point Cloud Generator (RPG)

The goal of our proposed method is to unsupervisedly learn a point cloud
generator, named as Recursive Point Cloud Generator (RPG), which is able to
generate 3D point clouds in a recursive coarse-to-fine manner. The learning of
RPG is based on the autoencoder framework: Given a point cloud data
$\mathcal{P}$, an encoder $E$ (which is built upon the PointNet [17] followed
by a fully-connected layer in our implementation) first maps it into a latent
representation $z\in\mathbb{R}^{U}$, our proposed RPG then takes $z$ as input
and attempts to reconstruct $\hat{\mathcal{P}}$ which ideally should be
identical to $\mathcal{P}$. The reconstruction error between
$\hat{\mathcal{P}}$ and $\mathcal{P}$ is thus the objective to drive the
training of encoder $E$ and our RPG.

Now we detail the generative process of our proposed RPG for generating the
point cloud data from the latent representation $z$. Basically, we decompose
the generative process into $D$ sequential stages of point expansion, as
formulated in Eq. 1. At each stage $d$, the 3D points $\mathbb{S}^{(d)}$
together with their corresponding structural representations
$\mathbb{H}^{(d)}$ are expanded by $k(d)$ times into $\mathbb{S}^{(d+1)}$ and
$\mathbb{H}^{(d+1)}$ respectively via the expansion module $\mathcal{E}$,
where $d=0,1,\cdots,D-1$. In other words, the number of 3D points in
$\mathbb{S}^{(d+1)}$, denoted as $\left|\mathbb{S}^{(d+1)}\right|$, is $k(d)$
times more than the one in $\mathbb{S}^{(d)}$. The 3D points obtained after
the last stage of expansion, i.e. $\mathbb{S}^{(D)}$, become the output
$\hat{\mathcal{P}}$ of our RPG, in which $\left|\mathbb{S}^{(D)}\right|$ can
be also calculated easily by $\prod_{d=0}^{D-1}k(d)$.

Stage 0:
$\displaystyle\quad\mathbb{S}^{(1)},\mathbb{H}^{(1)}=\mathcal{E}(\mathbb{S}^{(0)},\mathbb{H}^{(0)})$
(1) Stage 1:
$\displaystyle\quad\mathbb{S}^{(2)},\mathbb{H}^{(2)}=\mathcal{E}(\mathbb{S}^{(1)},\mathbb{H}^{(1)})$
$\displaystyle\qquad\small{\vdots}$ Stage $D-1$:
$\displaystyle\quad\mathbb{S}^{(D)},\mathbb{H}^{(D)}=\mathcal{E}(\mathbb{S}^{(D-1)},\mathbb{H}^{(D-1)}).$

The basic concept behind these sequential stages can be described in an
intuitive way, as illustrated in Figure 2: Assuming that we are now asked to
build the 3D point cloud of a specific chair, but at the beginning we can only
use $5$ points to represent the structure of that chair. It then become a
reasonable choice to distribute these $5$ points to the chair back and $4$
chair legs respectively. Such scenario is analogous to our stage 0 of point
expansion with taking $\mathbb{H}^{(0)}=\\{z\\}$ and
$\mathbb{S}^{(0)}=\\{\mathbf{0}\\}$, where the $k(0)=5$ points (denoted as
$\mathbb{S}^{(1)}=\\{s^{(1)}_{i}\\}^{k(0)}_{i=1}$) are expanded from the
origin $\mathbf{0}$ according to the latent structural representation $z$ of
the whole chair, and $z$ is evolved into
$\mathbb{H}^{(1)}=\\{h^{(1)}_{i}\\}^{k(0)}_{i=1}$ representations indicating
that now the chair is assembled by $k(0)=5$ parts. When stepping further with
being allowed to use more points to represent each part of this chair, these
points would ideally be distributed to better cover the volume of their
corresponding parts. Similarly, our stage 1 is analogous to this scenario, in
which for every point $s^{(1)}_{i}\in\mathbb{S}^{(1)}$ we can expand it into
$k(1)$ more points as well as evolve its structural representation
$h^{(1)}_{i}\in\mathbb{H}^{(1)}$ into $k(1)$ corresponding structural
representations, indicating that now the part related to $s^{(1)}_{i}$ is
represented by $k(1)$ more sub-parts. All the points and structural
representations expanded from $\mathbb{S}^{(1)}$ and $\mathbb{H}^{(1)}$ are
denoted as $\mathbb{S}^{(2)}$ and $\mathbb{H}^{(2)}$ respectively, in which
$\mathbb{S}^{(2)}$ shows more finer-grained 3D point cloud of the chair in
comparison to $\mathbb{S}^{(1)}$. Following the similar principle as previous
stages, more stages of point expansion can be recursively applied to build the
3D point cloud in a coarse-to-fine manner.

###  3.1 Expansion Module $\mathcal{E}$

The architecture of the expansion module $\mathcal{E}$ used in RPG is
illustrated in Figure 3. Given a 3D point $s_{i}^{(d)}\in\mathbb{S}^{(d)}$
with its corresponding structural representation $h_{i}^{(d)}$ and scaling
factor $\alpha_{i}^{(d)}$, the expansion module $\mathcal{E}$ expands
$s_{i}^{(d)}$, $h_{i}^{(d)}$, and $\alpha_{i}^{(d)}$ respectively by $k(d)$
times:

$\displaystyle\left\\{s^{(d,i)\rightarrow(d+1)}_{m},h^{(d,i)\rightarrow(d+1)}_{m},\alpha^{(d,i)\rightarrow(d+1)}_{m}\right\\}^{k(d)}_{m=1}$
(2)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad=\mathcal{E}(s^{(d)}_{i},h^{(d)}_{i},\alpha^{(d)}_{i})$

where
$h^{(d,i)\rightarrow(d+1)}_{1}=h^{(d,i)\rightarrow(d+1)}_{2}=\cdots=h^{(d,i)\rightarrow(d+1)}_{k(d)}$.
Please note that in Eq. 1 we skip the scaling factors $\alpha$ for simplicity.
With denoting all the 3D points and structural representations expanded from
$s_{i}^{(d)}$ and $h_{i}^{(d)}$ respectively as
$\mathbb{S}^{(d,i)\rightarrow(d+1)}$ and $\mathbb{H}^{(d,i)\rightarrow(d+1)}$,
the overall points and their corresponding structural representations obtained
at the stage $d$ are then accumulated respectively by:

$\displaystyle\mathbb{S}^{(d+1)}$
$\displaystyle=\bigcup\limits_{i=1}^{\left|\mathbb{S}^{(d)}\right|}\mathbb{S}^{(d,i)\rightarrow(d+1)}$
(3) $\displaystyle\mathbb{H}^{(d+1)}$
$\displaystyle=\bigcup\limits_{i=1}^{\left|\mathbb{H}^{(d)}\right|}\mathbb{H}^{(d,i)\rightarrow(d+1)}$

Figure 3: Architecture of our expansion module $\mathcal{E}$. Please refer to
Section 3.1 for more detailed description.

We now provide more detailed explanation for how such expansion module works
as well as the physical meanings behind each variable. We say, the parent
point of $s^{(d)}_{i}$ (denoted as $P(s^{(d)}_{i})\in\mathbb{S}^{(d-1)}$,
which $s^{(d)}_{i}$ is expanded from) is in charge of representing a certain
portion $\mathcal{V}$ of the target 3D model (e.g. a 3D chair). According to
Eq. 2, all the sibling points of $s^{(d)}_{i}$, which are also expanded from
$s^{(d)}_{i}$, do share the same structural representation $h_{i}^{(d)}$ as
$s^{(d)}_{i}$. Therefore, $h_{i}^{(d)}$ actually carries the structural
information of $\mathcal{V}$. Now, when we would like to further expand
$s^{(d)}_{i}$ into $k(d)$ points, a function $\mathcal{F}$ is adopted to
extract from $h_{i}^{(d)}$ the structural information ${h}^{\prime}$ for the
sub-portion $\mathcal{V}^{\prime}$ of $\mathcal{V}$ that $s^{(d)}_{i}$ is
taking care of.

${h}^{\prime}=\mathcal{F}(s^{(d)}_{i},h^{(d)}_{i})=\text{tanh}(\mathcal{M}_{s}\ast
s^{(d)}_{i}+\mathcal{M}_{h}\ast h^{(d)}_{i})$ (4)

where $s^{(d)}_{i}\in\mathbb{R}^{3\times 1}$,
$\mathcal{M}_{s}\in\mathbb{R}^{U\times 3}$, $h^{(d)}_{i}\in\mathbb{R}^{U\times
1}$, and $\mathcal{M}_{h}\in\mathbb{R}^{U\times U}$ ($U$ is set to $512$ in
all our experiments). Then, as all the child points expanded from
$s^{(d)}_{i}$ would share the same ${h}^{\prime}$, we duplicate it $k(d)$
times, and denote them as
$\mathbb{H}^{(d,i)\rightarrow(d+1)}=\\{h^{(d,i)\rightarrow(d+1)}_{m}\\}_{m=1}^{k(d)}$.
Regarding the scaling factor $\alpha^{(d)}_{i}$ of the point $s^{(d)}_{i}$, it
is to constraint the range of movement for $s^{(d)}_{i}$ when $s^{(d)}_{i}$ is
expanded from its parent $P(s^{(d)}_{i})$ as $\mathcal{V}$ is just a portion
of the whole target 3D model, i.e. $s^{(d)}_{i}$ should not exceed the spatial
range of $\mathcal{V}$.

Regarding the way of distributing $k(d)$ points in the sub-portion
$\mathcal{V}^{\prime}$ that $s^{(d)}_{i}$ takes care of, we particularly
introduce the embeddings $\\{e^{(d)}_{m}\\}_{m=1}^{k(d)}$ which serve as the
high-level representations for an initial spatial-arrangement, the
concatenation of each $e^{(d)}_{m}$ with ${h}^{\prime}$ is then taken as the
input for a multiple-layer perceptron (MLP) to produce the offset
$o^{(d,i)\rightarrow(d+1)}_{m}\in\mathbb{R}^{3\times 1}$ and the scaling
factor $\alpha^{(d,i)\rightarrow(d+1)}_{m}$ of the $m$-th point expanded from
$s^{(d)}_{i}$. Please note that now the scaling factor
$\alpha^{(d,i)\rightarrow(d+1)}_{m}$ has been multiplied with
$\alpha^{(d)}_{i}$ after MLP. The resultant 3D point coordinate
$s^{(d,i)\rightarrow(d+1)}_{m}$ of $m$-th point is computed by:

$s^{(d,i)\rightarrow(d+1)}_{m}=s^{(d)}_{i}+\frac{o^{(d,i)\rightarrow(d+1)}_{m}}{o_{\text{max}}}\ast\alpha^{(d,i)\rightarrow(d+1)}_{m}$
(5)

where
$o_{\text{max}}=\max(\\{\left\|o^{(d,i)\rightarrow(d+1)}_{m}\right\|\\}_{m=1}^{k(d)})$.

Figure 4: Illustration of the tree-structure built upon the point clouds
obtained from the expansion stages of our RPG.

Insights.  As the points in $\mathbb{S}^{(d)}$ are expanded the ones in
$\mathbb{S}^{(d-1)}$, with leveraging the hierarchical/parent-child relation
between them, we can straightforwardly construct a tree-structure upon all the
point clouds (i.e. $\\{\mathbb{S}^{(d)}\\}_{d=0}^{D-1}$) obtained during the
process of our recursive point generator, as illustrated in Figure 4. Note
that, such tree-structure will form a perfect $k$-ary tree if all $k(d)$ are
equivalent $\forall d$. Moreover, when we consider the path of tracing each
point in $\mathbb{S}^{(D)}$ (i.e. leave node) back to the origin point in
$\mathbb{S}^{(0)}$ (i.e. root node), the function $\mathcal{F}$ with respect
to all the points and their structural representations in such path actually
form a simplest recurrent neural network (RNN), in which the points $s$ and
structural representations $h$ in the path are the inputs and hidden states
respectively, while $\mathcal{M}_{s}$ and $\mathcal{M}_{t}$ are the parameters
of such RNN.

Lastly, when we consider the cluster assignments for all the points in
$\mathbb{S}^{(d)}$ according to their corresponding ancestor points in
$\mathbb{S}^{({d}^{\prime})}$, where ${d}^{\prime}<d$, we can obtain the
segmentation result of $\mathbb{S}^{(d)}$ with $|\mathbb{S}^{({d}^{\prime})}|$
clusters. Examples of using $\mathbb{S}^{(1)}$ as the reference of the
ancestor points to perform segmentation on various $\mathbb{S}^{(d)}$ with
$d>1$ are shown in Figure 1.

###  3.2 Training Objective

The main objective for our RPG training simply adopts the Chamfer Distance
(CD) [2] for evaluating the reconstruction error between $\mathcal{P}$ and our
RPG output $\hat{\mathcal{P}}=\mathbb{S}^{(D)}$.

$\displaystyle\mathcal{L}_{\text{CD}}=\text{CD}(\mathcal{P},\hat{\mathcal{P}})$
$\displaystyle=\frac{1}{|\mathcal{P}|}\sum_{{s}\in{\mathcal{P}}}\min_{{s}^{\prime}\in\hat{\mathcal{P}}}||s-{s}^{\prime}||_{2}^{2}$
(6)
$\displaystyle+\frac{1}{|\hat{\mathcal{P}}|}\sum_{{s}^{\prime}\in{\hat{\mathcal{P}}}}\min_{s\in\mathcal{P}}||s-{s}^{\prime}||_{2}^{2}$

Moreover, we introduce a regularization term to generally encourage smaller
scaling factors, in order to reduce the potential overlapping between
different portions (i.e. $\mathbb{S}^{(d,i)\rightarrow(d+1)}$ and
$\mathbb{S}^{(d,j)\rightarrow(d+1)}$, $i\neq j$) on each stage.

$\mathcal{L}_{\text{reg}}=\frac{1}{D}\sum_{d=0}^{D-1}\left(\frac{1}{|\mathbb{S}^{(d)}|}\sum_{i=1}^{|\mathbb{S}^{(d)}|}\mathbb{A}^{(d)}\right)$
(7)

where $\mathbb{A}^{(d)}$ denotes all the scaling factors obtained at stage
$d$. The overall objective then becomes
$\mathcal{L}_{\text{CD}}+\lambda\mathcal{L}_{\text{reg}}$, where $\lambda$ is
set to $5\mathrm{e}{-5}$ in our experiments.

Please especially note that, the parameters of $\mathcal{F}$ (i.e.
$\mathcal{M}_{s}$ and $\mathcal{M}_{t}$) and the ones of MLP are shared
globally across all the stages of point expansion, while the embeddings
$\\{e^{(d)}_{m}\\}_{m=1}^{k(d)}$ in stage $d$ are shared among all expansion
operations of the points in $\mathbb{S}^{(d)}$ (where
$e^{(d)}_{m}\in\mathbb{R}^{64\times 1}$ in our experiments). The initial value
for the scaling factor $\alpha^{0}$ at the stage $0$ is set to $1$. The batch
size is set to $64$, and we use the AdamW [13] optimizer with learning rate
$1\mathrm{e}{-3}$. All our source code and models will be publicly available
upon paper acceptance.

##  4 Experiments

We first introduce the dataset and the evaluation metrics used in our
experiments. We then provide the experimental results as well as the
comparisons with respect to baseline(s), on tasks of point cloud
reconstruction, generation, part segmentation, and co-segmentation.

#### Datasets.

The experiments and evaluations are conducted on ShapeNet [4], which is a
large scale dataset composed of 51,127 pre-aligned 3D shapes from 55
categories. We follow the experimental setting as [3, 21] to have 35,708/5,158
shapes for the train/test split of ShapeNet. Since the shapes in ShapeNet are
originally in the format of meshes, we follow the common practice as [17] to
sample random points on the triangles from the meshes in order to build up the
point cloud data of each shape for our experiments, where the number of points
in each point cloud data is 2048 unless stated otherwise. All the point clouds
are zero-centered (i.e. aligning the point cloud center to the origin) and
normalized to be within the unit sphere (i.e. coordinates re-scaled by
dividing with the distance between the farthest point and the point cloud
center).

#### Evaluation metrics.

We follow the prior works [3, 21] to adopt the symmetric Chamfer Distance (CD)
for evaluating the performance of point cloud reconstruction. Regarding the
point cloud generation, we also follow [3, 21] to use the Minimum Matching
Distance (MMD), Coverage (COV), and 1-Nearest Neighbor Accuracy (1-NNA) as
metrics to access the quality of generated point clouds. While taking the test
split of ShapeNet as the reference set, MMD measures the averaged Chamfer
distances from each reference point cloud to its nearest neighbor in the set
of generated point clouds, COV measures the fraction of reference point clouds
which are matched to at least one sample in the set of generated point clouds,
while 1-NNA mainly considers the similarity between the reference set and the
generated point clouds in terms of their shape distributions. Note that MMD
and 1-NNA are the smaller the better, while COV is instead the larger the
better.

#### Our RPG variants.

Two variants are adopted for our RPG training. The first one, denoted as
RPG3125, is trained on the point cloud data with having 3125 points per point
cloud, where there are 5 expansion stages with setting all $k(d)$ equally to
5. Another one, denoted as RPG2048, is instead based on the points clouds of
2048 points, where there are also 5 expansion stages but setting $k(1)$ to 8
and $k(2)=k(3)=k(4)=k(5)$ to 4. We mainly adopt RPG2048 for evaluating the
tasks of reconstruction and generation in order to have fair comparison to the
baselines, while using RPG3125 to illustrate the part segmentation for better
visualization (where the segmentation into $k(1)=5$ parts is easier for clear
colorization of points).

###  4.1 Reconstruction

We compare the reconstruction ability of auto-encoding the point cloud data
among our proposed RPG framework and various baselines, i.e. AtlasNet [8] with
patches and with sphere, PointFlow [21] (abbreviated as PF), and the current
state-of-the-art ShapeGF [3], based on the whole test set of ShapeNet as well
as the three categories of it (i.e. Airplane, Car, and Chair). Please note
that, we follow the same experimental setting as [21, 3] where the number of
points in each point cloud used for evaluation is 2048. The reconstruction
errors in terms of Chamfer distance as well as the number of parameters for
each method are provided in Table 1. Note here the performance of our RPG
framework is based on RPG2048, where the performance obtained by RPG3125 is
almost the same. It is shown that our proposed RPG framework has the
comparable performance of point cloud reconstruction with respect to all the
baselines, while having only 1.8M model parameters (with 0.7M from the
PointNet encoder $E$), being almost half of the current state-of-the-art
ShapeGF [3]. In particular, although the PointFlow [21] baseline has smaller
number of model parameters than ours, but our reconstruction performance is
superior to it on all the datasets. Qualitative examples of reconstruction
produced by our RPG are provided in Figure 5.

Table 1: Experimental results of the reconstruction task conducted on the
ShapeNet dataset and three categories of it. The reconstruction errors between
the input and reconstructed output are given in Chamfer distance multiplied by
$10^{4}$.

| AtlasNet [8] | PF [21] | ShapeGF [3] | Ours
---|---|---|---|---
Dataset | Sphere | Patches
Airplane | 1.002 | 0.969 | 1.208 | 0.96 | 0.924
Chair | 6.564 | 6.693 | 10.120 | 5.599 | 7.493
Car | 5.392 | 5.441 | 6.531 | 5.328 | 5.711
ShapeNet | 5.301 | 5.121 | 7.551 | 5.154 | 5.607
Params | 12M | 12M | 1.5M | 3.5M | 1.8M

Figure 5: Examples of reconstruction by our RPG. For each pair, the shape on
the left is the input while the output is on the right.

###  4.2 Generation

We follow the experimental setting as [3] to conduct the evaluation in terms
of generation ability on two categories (i.e. Airplane and Chair) in ShapeNet.
Several baselines are adopted for making comparison, including r-GAN [1], GCN
[20], TreeGAN [18], PF [21], and ShapeGF [3]. In order to equip our RPG
framework with the ability of generation, our model training is extended from
the autoencoder to the variation autoencoder (VAE [11]), where the
distribution of latent representations of 3D shapes are modelled as a normal
distribution, thus our RPG can take noise vectors sampled from such normal
distribution as $\mathbb{H}^{(0)}$ and output the generated point clouds. The
quantitative results of generation quality for various approaches as well as
their model sizes are provided in Table 2. Please note here we report the
performance of our RPG2048, which has almost the same performance as RPG3125.
We can see that, our proposed RPG framework achieves superior performance with
respect to r-GAN, GAN, TreeGAN, and PF on almost all the metrics for both
categories. While our RPG has comparable performance with ShapeGF, the amount
of parameters of our RPG model is smaller than all the baselines thus
contributing to less training time and lower hardware requirements. We also
provide the qualitative examples of point clouds generated by our proposed RPG
in Figure 7 to visualize their quality as well as the diversity. Moreover, in
Figure 8(b) we provide the 5 example sets of interpolation between two 3D
shapes of the same category (i.e. shapes generated from the interpolation
between two noise vectors which can lead to the point clouds of the same
category), where we can see the smooth and semantically-consistent transition
between generated point clouds during the interpolation. In Figure 8(a) we
visualize the point clouds generated on all the expansion stages for
interpolated shapes. While in Figure 8(c) we provide even the examples of
interpolation between different object categories.

Table 2: Experimental results of the generation task conducted on the Airplane
and Chair categories of the Shape dataset. Three metrics (i.e. MMD, COV, and
1-NNA) are used to access the quality of generated point clouds, where both
MMD and 1-NNA are the lower the better, while COV is the higher the better.
The MMD numbers are multiplied with $10^{4}$ here.

| Model | Params | MMD($\downarrow$) |  | COV ($\uparrow$) |  | l-NNA($\downarrow$)
---|---|---|---|---|---|---|---
Airplane | r-GAN [1] | 6.9M | 1.657 |  | 38.52 |  | 95.80
GCN [20] | 7.0M | 2.623 |  | 9.38 |  | 95.16
TreeGAN [18] | 40M | 1.466 |  | 44.69 |  | 95.06
PF [21] | 1.5M | 1.408 |  | 39.51 |  | 83.21
ShapeGF [3] | 3.5M | 1.285 |  | 47.65 |  | 85.06
Ours | 1.1M | 1.288 |  | 42.47 |  | 84.57
Chair | r-GAN [1] | 6.9M | 18.187 |  | 19.49 |  | 84.82
GCN [20] | 7.0M | 23.098 |  | 6.95 |  | 86.52
TreeGAN [18] | 40M | 16.147 |  | 40.33 |  | 74.55
PF [21] | 1.5M | 15.027 |  | 40.94 |  | 67.60
ShapeGF [3] | 3.5M | 14.818 |  | 46.37 |  | 66.16
Ours | 1.1M | 15.686 |  | 45.62 |  | 66.39

###  4.3 Part Segmentation and Co-Segmentation

As what can be seen from both Figure 7 and 8, the points in the generated
point clouds are colorized according to their ancestor assignments in
$\mathbb{S}^{(1)}$, which particularly highlight the most important feature,
i.e. part segmentation, that distinguishing our proposed RPG from the
baselines such as PointFlow [21] and ShapeGF [3] that can only generate point
clouds without having any part segmentation. In particular, the part
segmentation is naturally obtained during the inference of our proposed RPG
framework, according to the tree-structure built upon the point clouds
produced in the sequential stages of expansion (cf. Figure 4) without any
supervision nor manual annotation on the object parts.

Moreover, as the generative procedures of different object instances are
learnt by the same recursive process of expansions in our RPG framework,
regardless of the tasks of generation or reconstruction, we are therefore able
to discover the common patterns of part segmentation shared across object
instances thus straightforwardly achieving the co-segmentation, where the
common parts (e.g. the left and right wings, tail, and nose of airplanes) are
segmented out among objects as shown in Figure 9. The co-segmentation helps us
to understand the intra-class variance and similarity from a more fine-grained
perspective according to the common parts (e.g. different forms of wings among
airplanes).

Furthermore, as TreeGAN proposed in [18] shares several similarities in high-
level concepts with our RPG, here we particularly highlight the differences.
Basically, TreeGAN adopts the adversarial learning framework to train the
generator built upon tree-structured graph convolution network, where multiple
layers of branching operations and the graph convolutions are applied to
generate the final point cloud. Due to its tree-structured design, the part
segmentation of the generated point cloud is also available. Our work
distinguishes from TreeGAN by the following aspects: (1) As the graph
convolutions used in different layers of TreeGAN are not shared, its model
size is relatively large; while our RPG framework introduces the novel
recursive expansion module which is repeatedly applied on all the expansion
stages (i.e. weights sharing), the model size of our RPG is much smaller than
TreeGAN (cf. Table 2) thus leading to more efficient training; (2) Our RPG
framework not only provides better performance in generation (cf. Table 2) but
also shows more compact and semantically-meaningful part segmentation, as
shown in Figure 6. We can see that TreeGAN’s part segmentation would mix
between the chair back and chair surface (or between chair legs and chair
surfaces) or even produce some subtle parts, while ours better decomposes
chair legs, surface, and back into different parts thus getting closer to the
human interpretation. Moreover, while leveraging the part annotation provided
in ShapeNetPart [23] dataset to evaluate the purity of part segmentation on
the point clouds generated by TreeGAN and our RPG, our RPG achieves superior
performance than TreeGAN (i.e. 88% vs. 74% in purity, please refer to
supplementary materials for more details); (3) TreeGAN can only output point
cloud on the final layer of generator, while our RPG is able to obtain all the
intermediate point clouds thus providing better understanding on the
hierarchical segmentation and generative procedure of point cloud data.

Figure 6: Comparison between part segmentations obtained from TreeGAN (left)
and our proposed RPG (right), where the center image of the left or right part
is the generated 3D shape, surrounded by its 8 segmented parts. It shows that
our RPG is able to provide more compact and semantically-meaningful parts.
Figure 7: Qualitative examples of the point clouds generated by our proposed
recursive point cloud generator (RPG). Figure 8: Examples for our
interpolation between different shapes: (a) Rows sequentially show the point
clouds generated on all the expansion stages while interpolating between the
chairs on the bottom-left and bottom-right corners; (b) Each row shows
interpolation between two 3D shapes of the same object category; (c) Each row
shows interpolation between two shapes from different categories. Figure 9:
Visualization of co-segmentation results among object instances from Car,
Chair and Airplane categories in ShapeNet. For each object category, the rows
sequentially highlight different common parts with green color shared across
the instances.

##  5 Conclusion

In this paper we propose the recursive point cloud generator (RPG) which is
capable of generating or reconstructing point clouds in a coarse-to-fine
manner. Based on the generative procedure composed of multiple expansion
stages, our RPG produces point clouds with semantic parts without requiring
any supervision of part annotations. Our proposed RPG achieves comparable
performance with respect to state-of-the-art baselines in terms of point cloud
generation and reconstruction, with having relatively small model size thanks
to the design of sharing weights of the expansion module across all the
stages. Moreover, the part segmentation and co-segmentation of point clouds
naturally obtained from our RPG provide the potential of understanding point
clouds from a more fine-grained perspective.

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