Vittorio Giammarino
email: vittoriogiammarino@gmail.com

Simone Baldi
email: s.baldi@tudelft.nl

Paolo Frasca
email: paolo.frasca@gipsa-lab.fr

Maria Laura

Delle Monache
email: ml.dellemonache@inria.fr

† S Baldi

Traffic Flow on a Ring with a Single Autonomous Vehicle: an Interconnected Stability Perspective

Keywords: String stability, control of traffic flow, stability of traffic, autonomous vehicle, ring roadway

In recent years, field experiments have been performed on ring roadways with human-driven vehicles or with a mix of human-driven and autonomous vehicles. While these experiments demonstrate the potential for controlling traffic flows by a small number of autonomous vehicles, the theoretical framework about such a possibility is to a large extent incomplete. Indeed, most work on mixed traffic focused on classical asymptotical stability notions, neglecting that human drivers are prone to the interconnected instability known in the literature as string instability. This work aims to enhance the existing theories to meet the questions raised by the field experiments. It starts from the observation that the standard notion of string stability on a ring roadway is too demanding for a mixed traffic scenario: therefore, a new interconnected stability definition, named weak ring stability, is proposed. This new interconnected stability notion, in combination with classical stability, is able to explain phenomena observed in field experiments and to highlight possibilities and limitations of traffic control via sparse autonomous vehicle. Furthermore, it allows designing AV controllers with improved string stability specifications, at the price of reducing the sparsity of the autonomous vehicles.

Introduction

Several experimental settings have reproduced spontaneous emergence of stop-and-go waves in traffic flow. The first experiment of this kind is due to Sugiyama et al. in [START_REF] Sugiyama | Traffic jams without bottlenecks experimental evidence for the physical mechanism of the formation of a jam[END_REF], who conducted an experiment with twenty-two Human-driven Vehicles (HVs) on a single-lane ring roadway as an approximation for an infinite open road. This experiment demonstrates how human driving behavior can be responsible of triggering stop-and-go waves. Recent advances in automation have brought forward the idea of exploiting Autonomous Vehicles (AVs) to dissipate stop and go waves and control traffic [START_REF] Firooznia | Co-design of controller and communication topology for vehicular platooning[END_REF][START_REF] Ge | Dynamics of connected vehicle systems with delayed acceleration feedback[END_REF][START_REF] Ge | Optimal control of connected vehicle systems with communication delay and driver reaction time[END_REF][START_REF] Gong | Cooperative platoon control for a mixed traffic flow including human drive vehicles and connected and autonomous vehicles[END_REF][START_REF] Knorn | Passivitybased control for multi-vehicle systems subject to string constraints[END_REF][START_REF] Swaroop | String stability of interconnected systems[END_REF]. In 2018, Stern et al. [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF] demonstrated experimentally that a single autonomous vehicle is able to dampen stop-and-go waves in the ring setup of [START_REF] Sugiyama | Traffic jams without bottlenecks experimental evidence for the physical mechanism of the formation of a jam[END_REF]. In this ring setup, which approximates an open road in which AVs are sparsely introduced, controlling the AV dynamics is sufficient to prevent and even dissipate stop-and-go-waves.

Despite this practical evidence of the possibility of controlling traffic flows via a limited number of autonomous vehicles, the theoretical analysis around these experiments is still not complete. In fact, research on the effects of AVs on traffic has mostly focused on classical asymptotical stability notions: that is, perturbations around the equilibrium flow should be disappear as time goes by [START_REF] Khalil | Nonlinear systems[END_REF]. These studies, in fact, neglect another type of instability known in literature as interconnected or string instability, which is often observed in multi-vehicle platoons [START_REF] Besselink | String stability and a delay-based spacing policy for vehicle platoons subject to disturbances[END_REF][START_REF] Feng | String stability for vehicular platoon control: Definitions and analysis methods[END_REF][START_REF] Middleton | String instability in classes of linear time invariant formation control with limited communication range[END_REF][START_REF] Swaroop | String stability of interconnected systems[END_REF]: a flow is said string stable if the effect (e.g. energy or magnitude) of a disturbance acting on one vehicle is not amplified throughout the platoon interconnection. Even when the flow is stable in the classical asymptotic sense, a string unstable flow can amplify disturbances over transients as per effect of vehicle interaction. To the best of our knowledge, the only paper looking at a single-lane lane string stability for heterogeneous platoons is [START_REF] Wu | Stabilizing Traffic with Autonomous Vehicles[END_REF], whose authors study the linearized model presented in [START_REF] Treiber | Congested traffic states in empirical observations and microscopic simulations[END_REF] and derive a numerical bound for the penetration rate of the AVs to stabilize traffic. In this work, we extend the existing theories and provide a theoretical framework for the aforementioned experiments by studying both classical and interconnected stability on the ring roadway. Our framework is based on Linear Time Invariant (LTI) systems theory: this choice is consistent with most analytic studies on the topic of mixed traffic [START_REF] Delle Monache | Feedback control algorithms for the dissipation of traffic waves with autonomous vehicles[END_REF][START_REF] Ge | Connected Cruise Control Design in Mixed Traffic Flow Consisting of Human-Driven and Automated Vehicles[END_REF][START_REF] Wang | Controllability analysis and optimal controller synthesis of mixed traffic systems[END_REF][START_REF] Wu | Stabilizing Traffic with Autonomous Vehicles[END_REF][START_REF] Zheng | Smoothing traffic flow via control of autonomous vehicles[END_REF], relying on linearization of the nonlinear vehicle dynamics around the equilibrium flow. Furthermore, LTI systems allow us to work in the frequency domain, in which string stability lends itself to a natural characterization based on the ratio of two transfer functions. Nonlinear analysis methods have been considered in literature [START_REF] Orosz | Traffic jams: dynamics and control[END_REF], which however do not give insight on the design of stabilizing AVs.

Our analysis starts from the observation that the standard notion of string stability on a ring roadway is too demanding for a mixed traffic scenario, since human drivers are prone to string instability (and this drawback also affects current ACC systems [START_REF] Gunter | Are commercially implemented adaptive cruise control systems string stable?[END_REF], [START_REF] Makridis | Response time and time headway of an adaptive cruise control. an empirical characterization and potential impacts on road capacity[END_REF]). A new definition, named weak ring stability, is therefore proposed (Section 2). Classical asymptotic stability and weak ring stability are analyzed, first for the homogeneous HVs case and then for the mixed HVs-AV scenario (Sections 3 and 4, respectively). For both scenarios, it is shown that recently proposed sufficient conditions for classical stability [START_REF] Cui | Stabilizing traffic flow via a single autonomous vehicle: Possibilities and limitations[END_REF][START_REF] Zheng | Smoothing traffic flow via control of autonomous vehicles[END_REF] can be very conservative, while a necessary and sufficient condition for classical stability for a ring of homogeneous HVs is formulated. It is shown that the proposed weak ring stability notion, in combination with classical stability, can explain the oscillatory phenomena occurring in the field experiments of [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF][START_REF] Sugiyama | Traffic jams without bottlenecks experimental evidence for the physical mechanism of the formation of a jam[END_REF]. The weak ring stability notion also completes the currently-known range of possibilities and limitations of traffic control via sparse autonomous vehicle [START_REF] Cui | Stabilizing traffic flow via a single autonomous vehicle: Possibilities and limitations[END_REF]: most notably, it indicates how to design mixed traffic scenarios with improved string stability specifications (dampening any oscillation). Key to this development is studying the AV controller originally proposed in [START_REF] Delle Monache | Feedback control algorithms for the dissipation of traffic waves with autonomous vehicles[END_REF], which employs a Proportional Integral (PI) control with saturation: once its limitations are understood, we can propose a novel design that partly overcomes them. Indeed, string stability can be ensured at the price of reducing the sparsity of the autonomous vehicles (Section 5).

Summarizing, the contributions of this work are: (i) a necessary and sufficient condition for classical stability for a ring of homogeneous HVs (Theorem 1); (ii) a theoretical framework for classical and string stability of HVs and mixed HVs-AV traffic on ring roadways (Definition 4, Theorem 2); (iii) a novel design procedure for AVs with improved performance (Theorem 3); and (iv) a discussion of the experimental findings in the light of our theoretical developments (Section 6).

Notation: The following notation is used throughout the paper: C, R, N are the sets of complex, real and natural numbers, || • || ∞ is the H ∞ -norm in systems theory, O n×m is the n × m zero matrix, and I n is the n × n identity matrix. The product of elements in a set {x j : j = 1, . . . , n} is written as n j=1 x j , where by convention the product is 1 if the set is empty, i.e.

j∈∅ x j = 1. i i+1 v i v i+1
x i+1 -x i u r y i+1 y i

String stability definitions

In this section we provide the string stability definitions that are relevant to our work. We begin our presentation by recalling the relevant notions for vehicles on a line, before moving to our case of interest of the ring.

String Stability on the Line

Fig. 1 provides a graphical representation of the interaction between two adjacent vehicles in a predecessor-follower topology (also known as one-vehicle look-ahead interconnection). Let us recall some standard interconnected stability definitions with such a topology, noting that different topologies might require modified versions of the definitions stated in this section [START_REF] Feng | String stability for vehicular platoon control: Definitions and analysis methods[END_REF]. Let us assume for each vehicle i the linear stable vehicle model

y i (s) = P i (s)u r (s), s ∈ C (1)
where y i (s) and u r (s) are respectively the Laplace transforms of the vehicle output y i (t) and of the exogenous input u r (t) in Fig. 1. Typical outputs and inputs are velocities and acceleration commands, respectively, although different choices are also possible [START_REF] Ploeg | l p string stability of cascaded systems: Application to vehicle platooning[END_REF]: for example, u r (s) can also be seen as an exogenous disturbance acting on the vehicles. From (1), it is possible to define

y i (s) = Γ i (s)y i+1 (s), P i (s) = Γ i (s)P i+1 (s), (2)
being Γ i (s) the transfer function between two adjacent vehicles outputs. A platoon is said to be "homogeneous" if all vehicles have identical dynamics in (2) (Γ i = Γ ∀i), and "heterogeneous" otherwise. The interconnection of N possibly heterogeneous vehicles is given in Fig. 2 and leads to the transfer function:

y i (s) = P N (s) N -1 j=i Γ j (s)u r (s). (3)
Based on (1)-( 3), two notions of string stability on the line arise, named for brevity as line stability: Definition 1 (Strong Line Stability (SLS) [START_REF] Ploeg | l p string stability of cascaded systems: Application to vehicle platooning[END_REF]). Consider the linear line interconnected system whose input-output relation is described by ( 1) and (2). Then, the system (3) is said to be strong line stable if

||P N (jω)|| ∞ is finite; (4)
||Γ i (jω)|| ∞ ≤ 1 ∀i ∈ {1, . . . , N -1}, ∀N. (5)
Definition 1 prevents amplification of the disturbance u r between a vehicle i and its predecessor i

+ 1 (||P i || ∞ = ||Γ i P i+1 || ∞ ≤ ||P i+1 || ∞ ).
A weaker definition requires decreasing H ∞ -norm on a fixed-length line, which does not exclude ||Γ i || ∞ > 1:

Definition 2 (Weak Line Stability (WLS)). Consider the linear line interconnected system whose input-output relation is described by (1) and (2). Then, the system (3) is said to be weak line stable if there exists a number N of vehicles such that

||P N (jω)|| ∞ is finite; (6)
||P i (jω)|| ∞ ≤ ||P i+1 (jω)|| ∞ ∀i ∈ {1, . . . , N -1}. ( 7
)
From a physical point of view, the H ∞ -norm represents the L 2 induced gain, i.e. the ratio between the energy of output and input signals [START_REF] Feng | String stability for vehicular platoon control: Definitions and analysis methods[END_REF], [START_REF] Ploeg | l p string stability of cascaded systems: Application to vehicle platooning[END_REF]. Both strong and weak string stability require decreasing the induced gain as per [START_REF] Feng | String stability for vehicular platoon control: Definitions and analysis methods[END_REF]. However, Definition 1 asks [START_REF] Feng | String stability for vehicular platoon control: Definitions and analysis methods[END_REF] to hold for any arbitrary N (uniformity with respect to N ), whereas Definition 2 asks [START_REF] Feng | String stability for vehicular platoon control: Definitions and analysis methods[END_REF] to hold for a fixed string length N : in fact, [START_REF] Feng | String stability for vehicular platoon control: Definitions and analysis methods[END_REF] can fail for too large N when ||Γ i || ∞ > 1 (lack of uniformity with respect to N ). Different relaxations to Definition 1 have been studied in literature, such as the head-to-tail string stability [START_REF] Ge | Dynamics of connected vehicle systems with delayed acceleration feedback[END_REF]. In this work we consider a relaxation in the sense of Definition 2 motivated by analogous definitions for platoons on a ring, which are the object of the next subsection. These two notions motivate two analogous definitions for platoons on a ring, which are the object of the next subsection.

String Stability on the Ring

The concept of string stability on a ring road topology is now considered, named for brevity as ring stability. The block diagram representation for N possibly heterogeneous vehicles on a ring is given in Fig. 3, and leads to the following transfer function:

y i (s) = F (N ) i (s)u r (s), F (N ) i (s) = Γ i (s)F (N ) i+1 (s), (8)
F (N ) i (s) = N -1 j=i Γ j (s) 1 -N j=1 Γ j (s) P N (s) (9)
Even with the same number of vehicles N , ring and line present different structures, as the distribution of poles in the complex plane differs between (3) and [START_REF] Gazis | Nonlinear follow-theleader models of traffic flow[END_REF]. In ( 3), all poles are certainly in the left half-plane; the same does not hold in general for [START_REF] Gazis | Nonlinear follow-theleader models of traffic flow[END_REF], as unstable poles might arise regardless of having stable poles in P i (s). Consequently, one cannot directly apply definitions given for a line: the notion of ring stability is less studied in literature and the most common one involves homogeneous platoons with Γ i = Γ, ∀i and

F (N ) i (s) = Γ N -i (s) 1 -Γ N (s) P N (s). ( 10
)
Definition 3 (Strong Ring Stability (SRS) [START_REF] Peters | Cyclic interconnection for formation control of 1-d vehicle strings[END_REF]). Consider the linear homogeneous ring interconnection whose input-output relation is described by (8) (with Γ i = Γ, ∀i). Further, assume ||Γ|| ∞ ≤ 1. Then, the system ( 10) is said to be strong ring stable if there exists c > 0 such that

||F (N ) i (jω)|| ∞ ≤ c i ∈ {1, . . . , N } ∀N. (11)
It is shown in [START_REF] Peters | Cyclic interconnection for formation control of 1-d vehicle strings[END_REF] that SRS is equivalent to SLS for homogeneous platoons. Definition 3 is too restrictive in the applications, for several important reasons:

(a) It assumes ||Γ|| ∞ ≤ 1 for all vehicles. Unfortunately, all human-driven vehicle models require ||Γ|| ∞ > 1 [START_REF] Ge | Connected Cruise Control Design in Mixed Traffic Flow Consisting of Human-Driven and Automated Vehicles[END_REF].

(b) It is based on homogeneity, i.e. it does not allow to consider a mixed traffic situation in which automated vehicles appear together with human-driven vehicles.

(c) Even though ||Γ|| ∞ ≤ 1 is sufficient to ensure stability of (10), it is not necessary, since despite ||Γ|| ∞ > 1, F (N ) i
(s) might have all its poles in the open-left-half plane. These cases are excluded by Definition 3.

These drawbacks make the definition unsuitable to be applied to Sugiyama's or Stern's experiments [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF][START_REF] Sugiyama | Traffic jams without bottlenecks experimental evidence for the physical mechanism of the formation of a jam[END_REF]. To overcome these drawbacks, we propose a weaker notion of ring stability: Definition 4 (Weak Ring Stability (WRS)). Consider the linear possibly heterogeneous ring interconnection whose input-output relation is described by [START_REF] Firooznia | Co-design of controller and communication topology for vehicular platooning[END_REF]. Then, for fixed N , system (9) is said to be weakly ring stable if

||F N i (jω)|| ∞ is finite; (12)
||F (N ) i (jω)|| ∞ ≤ ||F (N ) i+1 (jω)|| ∞ i ∈ {1, . . . , N }. ( 13
)
Remark 1 (Benefits of WRS ). Definition 4 implies that the effect of the worst-case disturbance is not amplified. Notably, as compared to Definition 3, we have that Definition 4:

(a) can address platoons of heterogeneous vehicles;

(b) does not require ||Γ i || ∞ ≤ 1;
(c) covers the cases where (9) is stable although

||Γ i || ∞ > 1.

Traffic with Human Drivers

This section examines human-driven vehicles on a ring, by providing first their model and then classical stability and ring stability properties.

Human-driven Vehicle (HV) Model

The Optimal Velocity-Follow The Leader (OV-FTL) is a popular model for human driving behavior, taking the form

ẋi = v i i ∈ {1, . . . , N }, vi = v i+1 -v i (x i+1 -x i ) 2 a + [V (x i+1 -x i ) -v i ]b (14)
where x i is the position of vehicle i and v i its velocity. Define h i = x i+1 -x i as the headway, and note that N + 1 = 1, being the platoon on a ring. The nonlinear function V (h i ) is

V (h i ) = tanh(h i -l v -d s ) + tanh(l v + d s ) 1 + tanh(l v + d s ) v max (15)
being l v the vehicle length and d s > 0 a safety distance between vehicles; (15) determines the desired speed since it tends to zero for small h i and approaches v max for large h i . The parameters a > 0 and b > 0 in [START_REF] Gong | Cooperative platoon control for a mixed traffic flow including human drive vehicles and connected and autonomous vehicles[END_REF] represent weights between the OV and the FTL models [START_REF] Bando | Analysis of optimal velocity model with explicit delay[END_REF][START_REF] Bando | Dynamical model of traffic congestion and numerical simulation[END_REF][START_REF] Gazis | Nonlinear follow-theleader models of traffic flow[END_REF]. Dynamics [START_REF] Gong | Cooperative platoon control for a mixed traffic flow including human drive vehicles and connected and autonomous vehicles[END_REF] has equilibrium

x i+1 -x i = h * = l r (N ) N v i = v i+1 = v * = V (h * ) (16)
where l r (N ) is the circumference of the ring. Note that, in order to keep the same h * when varying N , we vary l r proportionally, that is why we write l r (N ). Around this equilibrium the dynamics can be linearized as

ẋi = v i i ∈ {1, . . . , N } (i = N + 1 = 1) vi = (v i+1 -v i )ā + [(x i+1 -x i ) k -c -v i ] b (17)
The linearization coefficients in [START_REF] Knorn | Passivitybased control for multi-vehicle systems subject to string constraints[END_REF] are calculated as:

k = ∂V (h * ) ∂h i = 1 -tanh 2 (h * -l v -d s ) 1 + tanh(l v + d s ) v max , ā = a h 2 * , b = b, c = -kh * + V (h * ). ( 18
)
The linearized model ( 17) broadly appears in traffic flow literature (cf. [START_REF] Ge | Connected Cruise Control Design in Mixed Traffic Flow Consisting of Human-Driven and Automated Vehicles[END_REF] and reference therein). In this literature it is common to consider a homogeneous human driving behavior: under such simplification, the HV dynamics in [START_REF] Knorn | Passivitybased control for multi-vehicle systems subject to string constraints[END_REF] result in the transfer function

Γ(s) = ās + bk s 2 + (ā + b)s + bk ( 19
)
derived by taking v i as the output y i in (2). It can be verified that the same transfer function is obtained using When put in a ring topology as in Fig. 4, the linearized OV-FTL model gives rise to a linear system which is the equivalent state-space formulation of ( 19):

y i = v i -v i-
      χ1 . . . χN-1 χN       =       A 1 A 2 O 2×2 • • • O 2×2 . . . . . . O 2×2 • • • O 2×2 A 1 A 2 A 2 O 2×2 • • • O 2×2 A 1       A       χ 1 . . . χ N -1 χ N       +       0 . . . 0 B r       B u r (20)
where

A 1 = 0 -1 bk -ā -b A 2 = 0 1 0 ā B r = 0 1 , (21)
χ i = x i+1 -x i -h * , v i -v * T
and where u r in ( 20) represents an external disturbance, which is taken for convention acting on vehicle N .

Classical stability criteria

Before proceeding to the analysis of stability of [START_REF] Olson | Circulant matrices and their application to vibration analysis[END_REF] in the classical (asymptotic) sense, one observation is in order. As already observed in [START_REF] Zheng | Smoothing traffic flow via control of autonomous vehicles[END_REF], the matrix A in [START_REF] Olson | Circulant matrices and their application to vibration analysis[END_REF] necessarily has one eigenvalue equal to 0 and its corresponding nontrivial eigenspace is spanned by the eigenvector u = [1, k, . . . , 1, k] T . This fact means that the dynamics of χ and the dynamics of χ + αu coincide for any scalar α: note that replacing χ with χ + αu would mean increasing all intervehicle distances and vehicle speeds by quantities whose ratio is 1/ k. However, we are assuming the ring length to be fixed and therefore by definition

N i=1 χ i = [0, N i=1 v i -N v * ]:
this constraint implies that the dynamics actually cannot evolve along the eigenvector u. For this reason, in our stability analysis we shall disregard the presence of this zero eigenvalue, similarly to what was done in [START_REF] Rogge | Vehicle platoons through ring coupling[END_REF][START_REF] Stüdli | Vehicular platoons in cyclic interconnections[END_REF]: this choice is equivalent to removing the redundant states that arise from the ring structure by a suitable change of variable. To remind the reader that classical stability shall be established up to this structurally unavoidable zero eigenvalue, we shall say that matrix A is structurally stable when all its eigenvalues are negative, except the zero eigenvalue associated to u: also, whenever unambiguous, we will refer to this classical stability notion simply as stability. Note that literature conveniently assesses structural stability in the time domain via eigenvalue analysis of matrix A [START_REF] Cui | Stabilizing traffic flow via a single autonomous vehicle: Possibilities and limitations[END_REF][START_REF] Wang | Controllability analysis and optimal controller synthesis of mixed traffic systems[END_REF][START_REF] Zheng | Smoothing traffic flow via control of autonomous vehicles[END_REF]; on the other hand, the different notions of string stability were conveniently studied in the frequency domain via transfer function analysis (cf. [START_REF] Ploeg | l p string stability of cascaded systems: Application to vehicle platooning[END_REF][START_REF] Rödönyi | Heterogeneous string stability of unidirectionally interconnected mimo lti systems[END_REF] and [START_REF] Peters | Cyclic interconnection for formation control of 1-d vehicle strings[END_REF][START_REF] Rogge | Vehicle platoons through ring coupling[END_REF][START_REF] Stüdli | Vehicular platoons in cyclic interconnections[END_REF] for line and ring configurations).

A sufficient condition for structural stability of ( 20)-( 21) is known in literature.

Lemma 1 (Sufficient condition for stability [START_REF] Zheng | Smoothing traffic flow via control of autonomous vehicles[END_REF]). Consider the linear statespace system (20) arising from the ring interconnection of Fig. 4. Then, its equilibrium [START_REF] Khalil | Nonlinear systems[END_REF] Although [START_REF] Peters | Cyclic interconnection for formation control of 1-d vehicle strings[END_REF] provides a simple analytical result to check stability of ( 20)-( 21), it can be conservative, especially for relatively small N (conservativeness of ( 22) will be illustrated in Section 3.3). Therefore, we propose a necessary and sufficient condition which removes this conservativeness.

is structurally stable if 2ā + b ≥ 2 k (22)
Theorem 1 (Necessary and sufficient condition for stability). Consider the linear state-space system (20) arising from the ring interconnection of Fig. 4. Then, its equilibrium ( 16) is structurally stable if and only if

- 1 2 γ i ± 1 2 r 2 i + (η i + 2γ i φ i ) 2 + r i 2 1/2 ≤ 0, (23)
with

γ i = ā + b -ā cos 2π(i -1) N , (24)
φ i = -ā sin 2π(i -1) N , η i = 4 bk sin 2π(i -1) N , (25)
r i = γ 2 i -φ 2 i -4 bk 1 -cos 2π(i -1) N , (26)
for all i ∈ {1, . . . , N } and the inequality is strict except for the structural zero eigenvalue that can be found by putting i = 1 and "+" in [START_REF] Ploeg | l p string stability of cascaded systems: Application to vehicle platooning[END_REF].

Proof. See Appendix.

Numerical analysis of stability

Let us investigate numerically the stability and WRS of ( 20)-( 21) for different pairs (a, b) and by varying the number of vehicles N on the ring. The parameters in [START_REF] Gunter | Are commercially implemented adaptive cruise control systems string stable?[END_REF] are as in Tab. 1. In line with the experiment of Sugiyama, the headway equilibrium is taken as h * = 11.81m, resulting in k = 1.2163.

We begin by studying stability of ( 20)-( 21) via Theorem 1 for different vehicle numbers N . In the (a,b)-plane, Fig. 5 illustrates that the stability region decreases by increasing N . The color-bar on the right-hand side associates colors with different numbers of vehicles N . For instance, the dark-blue area on the top-left is the instability region related to N = 3. Increasing to N = 5, an additional area (blue) turns unstable. Eventually, for N → ∞ the instability region converges to the condition in [START_REF] Peters | Cyclic interconnection for formation control of 1-d vehicle strings[END_REF] that is represented by a white line. The conservativeness of ( 22) arises from the fact that it is equivalent to ||Γ|| ∞ ≤ 1 and therefore does not take into account the cases in which, despite ||Γ|| ∞ > 1, the equilibrium of ( 20) is in fact stable. Next, we proceed to fix some specific values of (a, b) and let N vary. Namely, we select a = 20 b = 0.5 (27) which have been proposed in literature for human drivers from field data [START_REF] Delle Monache | Feedback control algorithms for the dissipation of traffic waves with autonomous vehicles[END_REF][START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF] and consider the two exemplary cases of N = 3 and N = 22. For N = 3, we can see that we have stability (cf. the × sign in Fig. 5). According to Definition 4, Fig. 6a shows that the peaks of F (N ) i (s) decrease while moving from vehicle 3, where the disturbance acts, to vehicle 1. Fig. 6b verifies the numerical analysis by simulating the nonlinear OV-FTL [START_REF] Gong | Cooperative platoon control for a mixed traffic flow including human drive vehicles and connected and autonomous vehicles[END_REF] with an impulse disturbance acting on vehicle 3 at t = 60s. The disturbance is not amplified between vehicles and its effect is rejected in about 40s, indicating both stability and weak ring stability. Instead, for N = 22 as

Singular Values

Frequency (rad/s) in Sugiyama's experiment, system ( 20)-( 21) resulting from ( 27) is unstable. These evidences suggest that the wide oscillations seen in Sugiyama's experiment should be an outcome of the instability of the equilibrium, which consequently produces the onset of nonlinear oscillatory phenomena (Fig. 7 shows Sugiyama's experiment reproduced in simulations).

Traffic with Single Autonomous Vehicle

This section studies a dynamics that describes the experiments in [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF], where an AV is placed in the ring. We adopt the following convention: the AV is taken as the N -th vehicle, i.e. the AV follows vehicle 1 and precedes vehicle N -1. Consequently, we indicate with Γ AV the transfer function from v 1 to v N . Consider the following state-space formulation for the linearized platoon with N -1 HVs and a single AV:

      χ1 . . . χN-1 χN       =       A 1 A 2 O 2×2 . . . . . . A 1 A 2 -A AV . . . O A AV             χ 1 . . . χ N -1 χ N       +       0 . . . 0 B r       u r , (28)
with A 1 , A 2 and B r as in [START_REF] Orosz | Traffic jams: dynamics and control[END_REF] and

A AV = [0 1; 0 0].
The literature provides us with the following stabilizability result.

Lemma 2 (Stabilizability of mixed platoons [START_REF] Zheng | Smoothing traffic flow via control of autonomous vehicles[END_REF]). The linear state-space system in (28) is structurally stabilizable for all N ∈ N.

Lemma 2 states that it is possible to stabilize any platoon of N vehicles by means of a single AV, even though the stabilization of a large number of vehicles would incur performance limitations underlined in [START_REF] Cui | Stabilizing traffic flow via a single autonomous vehicle: Possibilities and limitations[END_REF]. Since Lemma 2 is not a constructive result, it is important to have a criterion to verify whether a specific form of AV dynamics stabilizes the platoon. A sufficient condition for stability of a mixed HV-AV platoon is conjectured in [START_REF] Cui | Stabilizing traffic flow via a single autonomous vehicle: Possibilities and limitations[END_REF], as an extension of the sufficient condition in Lemma 1. Such a conjecture is now proven: Theorem 2 (Sufficient condition for stability of mixed platoons [START_REF] Cui | Stabilizing traffic flow via a single autonomous vehicle: Possibilities and limitations[END_REF]). Consider Γ(s) as in [START_REF] Middleton | String instability in classes of linear time invariant formation control with limited communication range[END_REF] and Γ AV (s) the transfer function of the AV dynamics. The equilibrium of (28) is structurally stable if

|Γ(jω)| 1-1 N • |Γ AV (jω)| 1 N ≤ 1 ω ∈ R ( 29
)
where 1 N represents the ratio between HVs and AVs.

Proof. See Appendix.

Theorem 2 poses two natural questions.

• Since it was shown that the condition in Lemma 1 is conservative for HVs (cf. Fig. 5), it is interesting to see to what extent ( 29) is conservative for mixed platoons.

• Some AV designs have been proposed in the literature [START_REF] Delle Monache | Feedback control algorithms for the dissipation of traffic waves with autonomous vehicles[END_REF], and even used for the experiments in [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF], but it is unclear if these designs meet condition [START_REF] Sugiyama | Traffic jams without bottlenecks experimental evidence for the physical mechanism of the formation of a jam[END_REF].

Let us now address these two questions. We first focus on the AV controller, recalling AV designs in literature and proposing a novel design. Next, we investigate how conservative ( 29) is, showing that the condition is tight for large platoons.

Analysis of state-of-the-art AV control: PI with saturation

The starting point is the PI with saturation AV introduced in [START_REF] Delle Monache | Feedback control algorithms for the dissipation of traffic waves with autonomous vehicles[END_REF], whose nonlinear dynamics are

ẋN = v N vN = K veh (αv target + (1 -α)v 1 -v N ) (30)
where v target and α are defined via

v target = V d + min max x 1 -x N -7 δ , 0 , 1 V d = v N + v 1 2 ∆x s = max 2(v 1 -v N ), 4 α = min max x 1 -x N -∆x s γ , 0 , 1 . (31)
As commonly considered in [START_REF] Zhu | Analysis of mixed traffic flow with humandriving and autonomous cars based on car-following model[END_REF], δ > h * -7, which leads to the following linearization

ẋN = v N vN = K veh (1 -α)v 1 + α v N + v 1 2 + α x 1 -x N -7 δ -v N = K veh α δ (x 1 -x N ) -K veh 1 - α 2 v N + K veh 1 - α 2 v 1 -7 K veh α δ .
The corresponding transfer function is therefore

Γ AV (s) = K veh [(1 -α 2 )s + α δ ] s 2 + K veh (1 -α 2 )s + K veh α δ . ( 32
)
Using a condition similar to [START_REF] Peters | Cyclic interconnection for formation control of 1-d vehicle strings[END_REF], it is possible to verify that ||Γ AV || ∞ > 1 for any choice of the parameters. Since in practice it happens that ||Γ|| ∞ > 1 for the HVs, we can conclude that, according to the linear criteria proposed in this work, the stabilizing AV in the experiments of [START_REF] Delle Monache | Feedback control algorithms for the dissipation of traffic waves with autonomous vehicles[END_REF] does meet weak ring stability criteria (we will further elaborate on our interpretation of the these experiments in Section V). In order to overcome this apparent lack of string stability, clearly it is preferable for the AV to have ||Γ AV || ∞ ≤ 1, so as to damp the frequency peak of the HVs according to [START_REF] Giammarino | On a weaker notion of ring stability for mixed traffic with human-driven and autonomous vehicles[END_REF]. Note from (32) that the AV is merely a second-order system with little degrees of freedom with respect to frequency shaping. In order to improve the overall platoon performance, it is therefore crucial to build on the previous analysis to improve the design in [START_REF] Delle Monache | Feedback control algorithms for the dissipation of traffic waves with autonomous vehicles[END_REF] and accomplish some form of ring stability.

Proposed AV control design

In view of the poor properties of AV dynamics [START_REF] Swaroop | String stability of interconnected systems[END_REF], we propose the following improved design

ẋN = v N (33) vN = K veh (αv target + (1 -α)v 1 -v N ) + c(v AV * -v N )
with variables as in [START_REF] Treiber | Congested traffic states in empirical observations and microscopic simulations[END_REF], being v AV * the desired velocity of the AV at equilibrium and c > 0 to be designed. Similarly to the previous design, δ > h * -7 leads to the linearization

ẋN = v N vN = K veh α δ (x 1 -x N ) -K veh 1 - α 2 + c v N + K veh 1 - α 2 v 1 - 7K veh α δ -cv AV * (34)
and the following transfer function:

Γ AV (s) = K veh [(1 -α 2 )s + α δ ] s 2 + K veh (1 -α 2 ) + c s + K veh α δ . (35)
It is not difficult to verify that ||Γ AV || ∞ ≤ 1, provided that

K veh 1 - α 2 + c 2 -K 2 veh 1 - α 2 2 -2K veh α δ ≥ 0, which is satisfied if c is large enough: c > -K veh 1 - α 2 + K 2 veh 1 - α 2 2 + 2K veh α δ (36)
For this controller, the following stability result holds.

Theorem 3 (AV design for stability). Let us consider HVs as in [START_REF] Middleton | String instability in classes of linear time invariant formation control with limited communication range[END_REF] with (a,b) such that ||Γ|| ∞ > 1 and Γ AV as in [START_REF] Zheng | Smoothing traffic flow via control of autonomous vehicles[END_REF]. Then, condition (29) holds if (36) is satisfied and

K veh < 1 -α 2 cω 2 Γ -α δ ω2 Γ α 2 δ 2 + 1 -α 2 2 ω2 Γ (||Γ|| 2(N -1) ∞ -1) + ωΓ -1 - α 2 2 ω4 Γ - α δ 2c 1 - α 2 ω2 Γ + α δ c + (ω 2 Γ + c 2 ) α 2 δ 2 + 1 - α 2 2 ω2 Γ ||Γ|| 2(N -1) ∞ α 2 δ 2 + 1 -α 2 2 ω2 Γ (||Γ|| 2(N -1) ∞ -1) (37)
where ωΓ = arg max ω |Γ(jω)|.

Proof. See Appendix.

It can be shown that the righthand side of (37) is strictly positive thanks to ||Γ|| ∞ > 1.

Conservativeness of the sufficient condition of stability

Here we investigate numerically the conservativeness of Theorem 2, resulting in condition (37). The numerical values used for the analysis are the same as in Tab. 1, with (a,b) in [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF]. Condition (37) is compared with the indirect Lyapunov method applied on the system

      χ1 . . . χN-1 χN       =       A 1 A 2 O 2×2 . . . . . . A 1 A 2 A AV 2 . . . 0 A AV 1             χ 1 . . . χ N -1 χ N       +       0 . . . 0 B r       u r , (38)
where A 1 , A 2 and B r are as in [START_REF] Orosz | Traffic jams: dynamics and control[END_REF], and

A AV 1 =   0 -1 αK veh δ -K veh (1 -α 2 ) + c   (39)
A AV 2 = 0 1 0 K veh (1 -α 2 )
.

System (38) represents the HVs with the linearized AV dynamics as N -th row, with design parameters δ = 23, α = 0.9, c = 0.5 for the AV. The numerical analysis in Fig. 8 shows that condition (37) is conservative for very small N , but works well for N ≥ 5, which is the scope of interest of [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF][START_REF] Sugiyama | Traffic jams without bottlenecks experimental evidence for the physical mechanism of the formation of a jam[END_REF]. For N ≤ 4 the system is stable for any K veh > 0.

Simulations on mixed HV-AV platoons

This section investigates the role of WRS (Definition 4) for mixed platoons with a single AV. Assuming an exogenous disturbance acting on the AV, the transfer function F

(N ) i

(s) for a vehicle i from disturbance to velocity is

F AV (N ) i (s) = Γ N -i (s) 1 -Γ AV (s)Γ N -1 (s) P N (s) (41)
whose stability can be tested via Theorem 3. For N = 22, we find that a stabilizing gain is K veh = 0.0029. We simulate the ring with the stabilizing gain and the remaining parameters as in the previous section. Fig. 9a illustrates how the peak increases moving throughout the platoon denoting lack of weak ring stability according to Definition 4.

The lack of weak ring stability is confirmed by the nonlinear simulations. Fig. 9b shows that equilibrium is reached from zero initial conditions, indicating stability. However, a small 1s impulse disturbance on the AV ac- Figure 10: For the mixed traffic scenario of Fig. 9, the effect of a white noise on a single HV (Fig. 10a) is persistently amplified due to lack of weak ring stability. Similar effects occur when noises affect all HVs (Fig. 10b). This situation can be compared with experimental results of [START_REF] Firooznia | Co-design of controller and communication topology for vehicular platooning[END_REF] (Fig. 10c). Vehicle velocities of [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF] have been reproduced using the open source codes and data provided by the authors in [START_REF] Stern | Dissipation of stopand-go waves via control of autonomous vehicles: Experimental results: Data[END_REF]): our simulations qualitatively replicate the dynamics in the Autonomy phase after 150s. creates oscillations that persistently appear and disappear. Indeed, Fig. 10b qualitatively replicates the behavior observed in the experiment [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF] (cf. Fig. 3 in [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF], reproduced in this work as Fig. 10c by using the open source code and data provided in [START_REF] Stern | Dissipation of stopand-go waves via control of autonomous vehicles: Experimental results: Data[END_REF]). Despite the different equilibrium velocities of the two flows, which are due to the different AV settings used in simulations and in field experiments, Fig. 10b is similar to the Autonomy phase of Fig. 10c, where oscillations experience transient amplification. In conclusion, our simulations qualitatively replicate the dynamics in the Autonomy phase in [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF], and show that AVs (34) cannot achieve both stability and ring stability when too sparse (1 AV out of 22 vehicles).

Ring stability becomes possible, instead, if the sparsity of AVs is reduced. For N = 4, the indirect Lyapunov method shows that (38) is stable for any K veh > 0 (Fig. 8). We find that K veh = 15 results in a decrease in the peaks of (41) according to Definition 4, which denotes weak ring stability (cf. Fig. 11a). Fig. 11b shows that a stable system that is also weak ring stable has a remarkably good response even with additive noise. In other words, the proposed WRS analysis explains how to design mixed traffic scenarios with improved string stability specifications (Fig. 10b vs Fig. 11b): the price to be paid is reducing the sparsity of the autonomous vehicles. Figure 11: WRS analysis and nonlinear simulation with N = 4 for a mixed traffic with modified AV [START_REF] Wu | Stabilizing Traffic with Autonomous Vehicles[END_REF]. The peaks of F (N ) i (s) are not amplified denoting weak ring stability. Differently from Fig. 10b, when white noise is added to all the HVs, its effect is rejected throughout the platoon.

Stability in Sugiyama's and Stern's experiments

This concluding section aims to summarize what our system-theoretic framework tells us about the experiments in [START_REF] Sugiyama | Traffic jams without bottlenecks experimental evidence for the physical mechanism of the formation of a jam[END_REF] and [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF] and more generally about stabilizing traffic by AVs.

Sugiyama's experiment [START_REF] Sugiyama | Traffic jams without bottlenecks experimental evidence for the physical mechanism of the formation of a jam[END_REF] has reproduced the spontaneous emergence of stop-and-go waves. Our study of the corresponding linearized dynamics shows that the dynamics is unstable, triggering the onset of large periodic oscillations in the nonlinear dynamics. These theoretical findings and simulations, illustrated in Section 3, are consistent with the experimental evidence of stop-and-go waves. In Stern's experiment, the presence of an AV prevents and dampen stop-and-go waves. In Section 4, our study of the corresponding linearized dynamics has shown that the dynamics is (asymptotically) stable, but not string stable. Therefore, noise induces oscillations that are amplified only transiently, so that stop-and-go behavior is not triggered. Our theoretical findings and simulations are again consistent with the experiments that show a lack of stop-and-go waves but the presence of smaller, irregular, oscillations.

In order to guarantee better platooning performance, we have designed an improved AV controller that is able to ensure both stability and string stability for platoons with N ≤ 4 vehicles. This ability has been theoretically verified and validated by simulations. Our analysis suggests that string stability can be achieved in platoons with HVs and AV, provided the ratio HV/AV is large enough. In fact, in contrast with the experiment in [START_REF] Stern | Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments[END_REF] where only 1/22 = 4.5% of the vehicles is controlled (but string stability is not ensured), our result requires at least 1/4 = 25% of the vehicles to be controlled to ensure string stability (compare Fig. 10b with Fig. 11b).

It remains an open question whether better controllers can deliver string stability with fewer autonomous vehicles. Another open question is how to use communication among vehicles to improve string stability without increasing the number of AVs [START_REF] Firooznia | Co-design of controller and communication topology for vehicular platooning[END_REF][START_REF] Zhang | Motif-based design for connected vehicle systems in presence of heterogeneous connectivity structures and time delays[END_REF].

A Proof of Theorem 1

For the sake of brevity we define α 1 = bk , α 2 = ā + b, α 3 = ā. Being [START_REF] Olson | Circulant matrices and their application to vibration analysis[END_REF] a block circulant matrix [START_REF] Olson | Circulant matrices and their application to vibration analysis[END_REF], an expression for its eigenvalues can be found as the eigenvalues of

Λ i = A 1 + w i-1 A 2 = 0 w (i-1) -1 α 1 -α 2 + w (i-1) α 3 , Λ i -λI 2 = -λ 1
α 1 (w (i-1) -1) -α 2 + w (i-1) α 3 -λ , λ 2 +(α 2 -α 3 • w (i-1) )λ -α 1 (w (i-1) -1) = 0, with w = exp (2πj/N ) and i ∈ {1, . . . , N }. The eigenvalues of Λ i are λ i,± = -α 2 + α 3 w (i-1) 2 ± 1 2 (α 2 -α 3 w (i-1) ) 2 + 4α 1 (w (i-1) -1)

1/2 . ( 42
)
To ensure asymptotical stability we require that:

Re -α 2 + α 3 w (i-1) 2 ± Re 1 2 (α 2 -α 3 w (i-1) ) 2 + 4α 1 (w (i-1) -1)

1/2
≤ 0, which is equivalent to [START_REF] Ploeg | l p string stability of cascaded systems: Application to vehicle platooning[END_REF].

B Proof of Theorem 2

The poles of a mixed platoon with a ring interconnection are given by the roots (λ) of 1 -Γ AV (λ)Γ N -1 (λ) = 0 (43)

where λ = µ + jξ. For all N , it can be shown that all the roots of (43) lie in the following subset of the complex plane:

C = {λ ∈ C : |Γ(λ)| 1-γ |Γ AV (λ)| γ = 1}, (44)
where we defined as γ = 1/N the ratio between the single AV and all vehicles. Function |Γ(λ)| 1-γ |Γ AV (λ)| γ is a meromorphic function, which means that, given an open subset D of the complex plane, the function is holomorphic on all D except for the poles of the functions [START_REF] Burckel | An introduction to classical complex analysis[END_REF]. By definition, both Γ and Γ AV have the poles in the open left half plane, indicating that |Γ(λ)| 1-γ |Γ AV (λ)| γ is holomorphic in the closed right half plane. That is: the function is complex differentiable at every point of the considered subset and its maximum value lies along the edge of its domain. Being the domain the closed right half plane, its edge is the imaginary axis [START_REF] Burckel | An introduction to classical complex analysis[END_REF]. Consequently, ensuring |Γ(jξ)| 1-γ |Γ AV (jξ)| γ ≤ 1 for all ξ ∈ R we can ensure that does not exist any λ = µ + jξ with µ > 0 such that |Γ(jξ)| 1-γ |Γ AV (jξ)| γ = 1. In other words, we ensure that there does not exist any λ ∈ C which is at the same time root of (43) and has strictly positive real part.

Figure 1 :

 1 Figure 1: Scheme and notation for interconnected vehicles.

Figure 2 :

 2 Figure 2: Block-diagram for interconnected vehicles on a line.

Figure 3 :

 3 Figure 3: Block-diagram for interconnected vehicles on a ring.

  1 .[START_REF] Ploeg | l p string stability of cascaded systems: Application to vehicle platooning[END_REF][START_REF] Rogge | Vehicle platoons through ring coupling[END_REF].

Figure 4 :

 4 Figure 4: Schematic model of the ring road topology.

Figure 5 :

 5 Figure 5: Stability of (20)-(21) in the (a,b)-plane, when changing N (c.f. color bar on the right). The stability region decreases when increasing N . For N → ∞ the region tends to the sufficient condition (22), represented by a white line (the region above this line corresponds to ||Γ|| ∞ > 1).

Figure 6 :

 6 Figure 6: WRS analysis and nonlinear simulation of the ring with N = 3, a = 20 and b = 0.5. An impulse disturbance at t = 60s is not amplified denoting weak ring stability.

  Evolution of velocities over time. (b) Evolution of positions over time.

Figure 7 :

 7 Figure 7: Nonlinear simulations with N = 22, a = 20 and b = 0.5, showing backward-travelling stop-and-go waves in line with Sugiyama's experiment.

Figure 8 :

 8 Figure 8: Comparison between the largest (structurally) stabilizing K veh obtained from (37) and those obtained using the indirect Lyapunov method. For N ≤ 4 the system is stable for any K veh > 0.

  i = 19, 20, 21, 22. (b) Response to impulse disturbance at t = 60s.

Figure 9 :

 9 Figure 9: WRS analysis and nonlinear simulations with N = 22 for a mixed traffic with modified AV (33). The peaks of F (N ) i (s) are amplified denoting ring instability. As a result, an impulse disturbance acting on the AV is amplified throughout the platoon.

  (a) Response to white noise on only the (N -1)-th HV with 15dB signal-noise-ratio (b) Response to white noise on all the HVs with 20dB signalnoise-ratio. (c) Vehicles velocities measured during the experiment [27].

  Response to white noise with 20dB signal-noise-ratio.

Table 1 :

 1 Standard values used for the numerical analysis.

	Variable name	Symbol	Value
	Vehicle length	l v	4.5m
	Maximum vehicle velocity	v max	9.75m/s
	Safety distance	d s	6m

Acknowledgements

This work was initiated by a Descartes Fellowship by the French Embassy in the Netherlands under project AMAS. S. Baldi was supported by the Fundamental Research Funds for the Central Universities grant no.4007019109 (RECON-STRUCT), and by the special guiding funds for "double firstclass" grant no.4007019201. M.L. Delle Monache was supported by the IDEX-IRS 2018 project "MAVIT -Modeling autonomous vehicles in traffic flow". P. Frasca was partly supported by ANR via project HANDY, number ANR-18-CE40-0010. The authors are grateful to mr. Maolong Lv for some preliminary HVs simulations that were included in an early and incomplete account of this research [13].

C Proof of Theorem 3

The most critical point for condition [START_REF] Sugiyama | Traffic jams without bottlenecks experimental evidence for the physical mechanism of the formation of a jam[END_REF] is given by ||Γ|| ∞ , the peak of [START_REF] Middleton | String instability in classes of linear time invariant formation control with limited communication range[END_REF]. This peak occurs at frequency ωΓ . Thus, condition (29) leads to the following inequality

By developing (45), the following inequality is obtained

which yields (37).


Franck Gam

C W Liu

Samia Kahlal

Jean-Yves Saillard

Electron counting and bonding patterns in assemblies of three and more silver-rich superatoms

DFT calculations were carried out on a series of cluster cores, the framework of which being made of the condensation of several Pt@Ag12 centered icosahedra. Icosahedra condensations through vertex-sharing, face-sharing and interpenetration were considered and their favored electron counts were determined from their stable closed-shell configurations. A large number of the computed assemblies of n icosahedral superatomic units can be considered as isolobal analogs of stable closed-shell n-atom molecules, most of them obeying the octet rule. The larger the degree of fusion between icosahedra, the stronger the interaction between them. It was for example possible to design 3-icosahedral supermolecular cores analogous to CO2, SF2 or [I3] -, but also to the not yet isolated cyclic O3. Supermolecules equivalent to non-stable molecules can also be designed. Indeed, differences exist between atoms and superatoms, and original icosahedra assemblies with no "molecular" analog are also likely to exist, especially with compact structures and/or systems made of a large number of fused superatoms.

Introduction

When in low oxidation states, gold, silver and other late transition-metals tend to close-pack in a spherical way to give rise to stable species when "passivated" by a ligand protecting shell, which, in some cases, incorporates also metal centers in a higher oxidation state. The metal-metal bonding in such clusters was first rationalized by Mingos and coworkers [1][2][3][4][START_REF] Mingos | The Chemical Bond I[END_REF][START_REF] Wales | [END_REF] who pointed on the fact that it is dominated by the 6s gold orbitals and that their electron counts can be interpreted within the framework of the spherical jellium model. 7 This non-LCAO approach approximates the electron/nuclei interaction to a radial potential leading to the description of the electronic structure of such clusters in terms of jellium orbitals. The jellium orbitals somehow resemble atomic orbitals and extend substantially over the whole cluster sphere.

Table 1. Mingos' analogy between diatomic molecules and gold clusters made of two fused centered icosahedra (adapted from Ref. 35). Pt atoms are in green. Encapsulated atoms are in green and shared atoms are in red.

(a) The energy ordering of the occupied levels is arbitrary. Of course, many late transition-metal clusters, especially high nuclearity species, are not spherical and consequently their electronic structure cannot be rationalized within the spherical jellium model. At the end of the last century, Teo and coworkers proposed the concept of clusters of clusters 20,21 for describing the non-spherical metallic cores of group 11 clusters made of vertex-sharing centered icosahedra. A question that arises then is: can we look at assemblies of superatoms in the same way as one looks at assemblies of atoms, i.e. molecules. In other words, in the same way as, for example, the 14-electron F 2 molecule is made of two atoms of valence configuration 2s 2 2p

Number of electrons

5

, is there a stable closed-shell isolobal 22 equivalent cluster, made of two fused superatoms, each of them of jellium configuration 1S 2 1P

5

? It turns out that the answer to this question lies already in the literature. Several authors have described the electronic structure of real or hypothetical clusters made of two fused superatoms as analogous to that of diatomic molecules. [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] As regards gold-rich clusters made of two fused centered icosahedra, an illuminating rationalization of their electron counts was proposed by Mingos in 2015. 35 Its major conceptual developments are summarized in Table 1. One can see that an important parameter affecting the cluster electron count is its degree of fusion (vertex-sharing, face-sharing or interpenetrating), in a similar way as the bond order does for main group molecules.

A c c e p t e d M a n u s c r i p t To our knowledge, apart from a pioneering limited case study, 39 no systematic investigation of electron counting and electronic structure of late transition-metal clusters made of the condensation of more than two superatoms has been made so far. Recently, we published the synthesis, characterization and electronic structure of the trisicosahedral cluster Pt 3 Ag 44 (dtp) 22 (Figure 1). 38 Its core structure is a linear assembly of three Pt@Ag 12 icosahedra sharing two Ag vertices. Our density functional theory (DFT) calculations indicated that this 22-electron species is an isolobal analog of the linear [I 3 ] -ion. In this paper we extend our investigation to the possibility for various fused tris-icosahedral structures adopting favored closed-shell electron counts that would render them analogous to regular stable triatomic molecules (Table 2). Tetra-icosahedral and some larger assemblies are subsequently considered.

Computational Details

All calculations were carried out at the DFT level with the Amsterdam Density Functional (ADF) program. [START_REF] Baerends | ADF2018[END_REF] The gradient-corrected Becke-Perdew (BP86) exchange correlation functional [START_REF] Becke | [END_REF]44 was used, together with the triple-ζ, plus two polarization functions (STO-TZ2P) Slater type basis set. The zeroth-order regular approximation (ZORA) was employed in the calculations to account for scalar relativistic effects. 45 The frozen core approximation was applied to the [1s ] for Ag. Geometry optimizations were performed assuming a maximum cartesian step smaller than 10 -3 Å and an energy convergence criterion of 10 -5 Ha. Owing to the fact that the computed cluster cores are highly charged cations, solvent effects through the conductor-like screening model (COSMO) 46,47 was considered, to mimic the effects of counterions. Such a process has been proved to be efficient to treat metal cations. 48 The considered solvent was dichloromethane.

A c c e p t e d M a n u s c r i p t

Results and Discussion

Assembling three icosahedra

We have shown previously that computing the only superatomic core of a cluster in its actual oxidation state, i.e.

without its passivating shell, allows rationalizing the cluster electronic structure in terms of a superatom or assembly of superatoms. 15,38 Below we report our search for cluster cores made of three fused Pt@Ag 12 icosahedral units that would be closed-shell energy minima secured by significant HOMO-LUMO gaps (≥ 0.5 eV, unless specified in the text). The Pt@Ag 12 unit is that encountered in our first characterized tris-icosahedral cluster. Test calculations on isoelectronic structures made of Ag@Ag 12 units provided similar results. On the other hand, the presence of Pt at the icosahedra centers helps reducing the cationic charge of the computed models. Thus, two parameters were varied. One is the degree of icosahedra condensation: vertex-sharing, face-sharing and interpenetration. Edge-sharing was discarded owing to the fact that, to our knowledge, no experimental report exists. The other parameter is the electron count, i.e. the cluster charge.

Vertex-sharing.

There are two different open triangular configurations possible, depending which ones are the two vertex type that the central icosahedron shares with its two congeners: second or third neighbors (sharing two first neighbors would lead to a cyclic structure; see below). Assuming ideal assemblies of ideal icosahedra, they would correspond to Pt-Pt-Pt angles of 117° and 180°, respectively. The low degree of icosahedra condensation lets to expect weak interaction between them, thus large electron counts. Indeed, the only favored electron counts were found to be 22 and 24. The 24-electron count corresponds to the trivial case of three non-interacting saturated 8-electron superatoms, thus isolobal to a system made of three van der Waals bonded Ne atoms (Table 2). The lowest energy configuration (Figure 2) is bent, i.e. the two shared vertices of the central icosahedron are second neighbors. Interestingly, the Pt-Pt-Pt angle (137°) is much larger than the ideal value of 117°. This is because the (Ag-Pt-Ag-Pt-Ag-Pt-Ag) backbone is also bent at the shared Ag atoms (160° in Figure 2). The absence of backbone bending at the Ag atoms would have resulted in all-equal inter-icosahedral Ag…Ag contacts of ~ 3.4 Å.

The linear (Pt-Pt-Pt = 180°) 24-electron isomer is found to lie only 4.5 kcal/mol above the bent one (Figure 2). It is thus likely that for this electron count the bent vs. linear arrangement is going to be largely dependent on the structural requirements of the passivating shell. In fact, to our knowledge only one 24 . Its superatomic core adopts the linear arrangement.

The linear configuration was found to be the most stable in the case of the 22-electron count (Figure 2). We previously showed that such a species is isolobal to the linear [I 3 ] -anion. 38 12+ is the superatomic core of the fully characterized Pt 3 Ag 44 (dtp) 22 (Figure 1). 38 To our knowledge, no other 22-electron species has been mentioned so far. -, as illustrated in Figure 3). For this electron count, the bent configurations are significantly disfavored. The 20-electron count is found to be favored for the bent configuration corresponding to an ideal Pt-Pt-Pt angle of 138°, the optimized value being 140° (Figure 2). This cluster core is isolobal to SF 2 , as illustrated in Figure 4.

Interpenetration of icosahedra and the relationship with vertex-sharing.

Two interpenetrated centered icosahedra (Figure 7, left) share a five-vertex pentagonal ring, as well as an icosahedron center which is a vertex for the other one, and conversely. There are two ways of adding a third interpenetrating centered icosahedron to get a Pt 3 Ag 22 non-cyclic framework, as illustrated on the right side of Figure 7. In fact, as already pointed out a long time ago by Teo, 33 the linear D 5d 25-atom framework of Figure 7 is nothing else than the 25-atom vertex-sharing bisicosahedral arrangement of Table 1. Removing the connection lines between the facing red pentagons of the former structure would produce that of the latter. Assuming ideally staggered ideal icosahedra,

A c c e p t e d M a n u s c r i p t

the bisicosahedral structure of Table 1 would correspond to inter-icosahedra distances equal to only 1.18 times the icosahedra edge. Since the structure has not so much spatial flexibility to adjust these distances, both views (2 vertexsharing centered icosahedra and 3 interpenetrated centered icosahedra) can be considered as two different descriptions of the same geometrical object. As a matter of fact, the favored electron count of the D 5d Pt 3 Ag 22 model of Figure 7 is the same as that of the vertex-sharing bisicosahedral structure of Table 1, namely 16 electrons (Figure 1). Thus, [Pt 3 Ag 22 ] 6+ can be viewed as either isolobal to the Ne 2 van der Waals dimer (two vertex-sharing superatoms) or to CO 2 (three interpenetrated superatoms). The two approaches are not excluding each other since both Ne 2 and CO 2 have similar valence electron configurations in the way that their 16 electron occupy the same number of σ g (two), σ u (two), π g (one)

and π u (one) levels. The compactness of the central icosahedron is a reasonable geometrical indicator for choosing the most appropriate description. In the case of our D 5d [Pt 3 Ag 22 ] 6+ model (Figure 2), the contacts between the two outer icosahedra (3.165 Å) are less than 1.1 time the outer icosahedra edges, indicative significant compactness of the central icosahedron and therefore favoring the isolobal analogy with CO 2 that is illustrated in (Figure 8). It is possible to build an equilateral assembly of three interpenetrating icosahedra by adding one supplementary icosahedron to a framework made of two interpenetrating icosahedra as illustrated at the bottom of Figure 7. The resulting Pt 3 Ag 20 architecture has D 3h symmetry. This 23-atom structure has been shown to constitute the core of the Pd 37 (CO) 28 {P(p-Tolyl) 3 } 12 nanocluster. 58 For this arrangement, the unique favored closed-shell electron count is 20, i.e. [Pt 3 Ag 20 ]. There is no isoelectronic triatomic molecular equivalent. On the other hand, 20 is a "magic" closed-shell superatom electron number, corresponding to the 1S 2 1P 6 1D 10 2S 2 configuration. It turns out that this is a quite compact structure, although its shape is

Assembling more than three icosahedra.

The number of configurational possibilities to generate assemblies of four (or more) icosahedra is very large. This is why we have limited our investigation to the linear and star-shaped (one central icosahedron connected to three outer congeners) assemblies of four Pt@Ag 12 units, in order to verify that the analogy with simple main-group molecules holds also for larger systems. Thus, six structures were generated, depending on whether the four icosahedra are vertex-sharing, face-sharing or interpenetrating. They are depicted in Figure 10, together with their favored electron counts and their Kohn-Sham orbital diagrams are provided in Figure S1. As expected, the 32 = 4 x 8 count is compatible with the two vertexsharing structures (linear and star-shaped), since it is consistent with weakly interacting closed-shell superatoms.

Interestingly, the linear configuration is also favorable to the lower with a comparable substantial HOMO-LUMO gap of 0.72 eV (Figure S2).

The two investigated face-sharing models were found having the same unique 28-electron favored count [Pt 4 Ag 39 ] 11+ (Figure 10). The star-shaped framework is isolobal to hypervalent tetrahalogenides such as ClF 3 . The linear form has a more intriguing electronic structure which could be tentatively described as an analogue to an hypothetical linear I 4 hypervalent species, in which the two bonding electron pairs are delocalized over three bonding contacts.

The favored electron count for the linear [Pt 4 Ag 27 ] 5+ model with interpenetrating icosahedra is 22. Examination of its Kohn-Sham orbitals indicates it is the analog of a hypothetical linear [N 4 ] 2-ion 60 (isoeletronic to C 2 F 2 ). The star-shaped framework has a favored count of 20 electrons and an electronic structure related to that of the trimethylene methane dication, but, unlike the latter, with high-lying vacant π-type orbitals.

To our knowledge, none of the cluster cores shown in Figure 10 have been experimentally evidenced, so far. A few larger superatom assemblies have been however characterized by Zhu and collaborators. + (Dppp = diphenylphosphinepropane) 62 appears much more intriguing. This 30-electron cluster possesses an Au 8 Ag 41 inner core of ideal D 2d symmetry that can be described as a cyclic assembly of eight interpenetrating Au@Ag 12 icosahedra (left side of Figure 11). It is a 49-atom metal core because the silver atom at the center is shared by the eight icosaheda (49 = 13

x 8 -8 x 7 + 1).

We performed calculations on the , see above) is not helpful since 40 is not a "magic" number within the spherical jellium model. This is not surprising, owing to the substantially flattened (oblate) shape. On the other hand, 30 is a "magic" number within the approximation of the planar (2D) harmonic quantum dot model. 63 This would correspond to the 1S Clearly, one is in an intermediate situation between the spherical (3D) and planar (2D) jellium approximation. As regard electronic structure, some relationship can also be drawn with planar boron clusters. 64,65 Although this structure is likely to be unique, one can anticipate that other architectures exhibiting important oblate or prolate distortion could be stabilized with specific jellium electron counts.

Conclusions

Our DFT calculations on various assemblies of n icosahedral superatomic units (n > 2) confirm that it is possible to conceive stable closed-shell supermolecules that would be analogs to stable n-atomic molecules, most of them obeying the octet rule. The larger the degree of fusion between icosahedra, the stronger the interaction between them. It is also possible to design stable superatomic assemblies that can be analogous to unknown molecules, as exemplified by [Pt 3 Ag 28 ] 10+ (C 3v ) which is the analog of the not yet isolated cyclic ozone isomer. 56,57 Supermolecules equivalent to nonstable molecules could also be designed, such as [Pt 4 Ag 27 ] 5+ which is the analog of linear [N 4 ] 2-ion, whose relaxed structure is bent. In any case, the atom/superatom equivalence should not be taken too far. There are differences and these differences add up when the number of considered atoms/superatoms increases. This can be also the case for ring structures. In general, cyclic molecules have no appreciable density at their center. Because of their size and depending on their degree of fusion, cyclic assemblies of icosahedra can extend their atomic framework at (or near) the center of the ring, thus forcing delocalization across the ring. In such a case, the molecule/supermolecule equivalence does not hold anymore.

The whole cluster framework is better described as a superatom per se. This is the case of our D 3h

Figure 1 .

 1 Figure 1. Left: The structure of [Pt 3 Ag 44 {S 2 P(OPr) 2 } 22 ] (OPr groups omitted for clarity); right: Its 22-electron [Pt 3 Ag 34 ] 12+

Figure 2 .

 2 Figure 2. Stable closed-shell assemblies of three fused icosahedra (ΔE H-L = HOMO-LUMO gap). Pt atoms are in green. Silver atoms in red and black are shared between 2 and 3 icosahedra, respectively.

A c c e p t e d M a n u s c r i p t

  Figure 3 illustrates the correspondence between the [I 3 ] -MOs and the [Pt 3 Ag 34 ] 12+ "supermolecular" orbitals. The former are linear combinations of atomic orbitals and the latter are combinations of superatomic orbitals. Consistently with the rather large 22-electron count, the bent configuration lies only 3 kcal/mol above the linear one. As mentioned previously, the linear [Pt 3 Ag 34 ]

Figure 3 .

 3 Figure 3. Kohn-Sham orbital diagrams of the 22-electron [Pt 3 Ag 34 ] 12+ (D 5d , left), [Pt 3 Ag 30 ] 8+ (D 3d , right), and [I 3 ] -(D ∞h , middle). The HOMOs of [I 3 ] -are the π* orbitals and the LUMO is the unique vacant σ* orbital.

A c c e

  p t e d M a n u s c r i p t

Figure 4 .

 4 Figure 4. Kohn-Sham orbital diagrams of the C 2v 20-electron [Pt 3 Ag 30 ] 10+ and SF 2 . The HOMO and HOMO-1 of SF 2 are the πtype and σ-type S "lone pairs". The LUMO and LUMO+1 are the σ*(S-F) orbitals.

. 3 . 1 . 2 .

 312 Face-sharing. With face-sharing icosahedra there are three non-cyclic configurations that are possible for a Pt 3 Ag 30 (30 = 3 x 12 -6) core, depending which faces of the central icosahedron are shared and without leading to steric overcrowding. Assuming ideal assemblies of ideal icosahedra, they would correspond to Pt-Pt-Pt angles of 109°, 138° and 180°, respectively. The linear configuration was found to be favored for the unique 22-electron count (Figure2), thus making this cluster core isolobal to [I 3 ]

Figure 5 .

 5 Figure 5. The making of a C 3v cyclic Pt 3 Ag 28 framework by the fusion of a Pt@Ag 12 icosahedron to a Pt 2 Ag 21 unit made of 2 edge-sharing icosahedra. Pt atoms are in green. Silver atoms in red and black are shared between 2 and 3 icosahedra, respectively.

Figure 6 .

 6 Figure 6. Kohn-Sham orbital diagrams of the 18-electron [Pt 3 Ag 28 ] 10+ (C 3v ) and hypothetical cyclic O 3 isomer (D 3h ). The HOMOs of O 3 are the π* combinations and the three LUMOs are the vacant σ* orbital.

Figure 7 .

 7 Figure 7. The two ways of making of a non-cyclic (top and middle) and cyclic (bottom) Pt 3 Ag 22 framework resulting from the interpenetration of a supplementary icosahedron to a bisicosahedral Pt 2 Ag 17 unit (2 interpenetrating icosahedra). Pt atoms are in green. Silver atoms in red and black are shared between 2 and 3 icosahedra, respectively.

Figure 8 .

 8 Figure 8. Kohn-Sham orbital diagrams of the 16-electron [Pt 3 Ag 22 ] 6+ core and CO2. It is of note that in the isoelectronic and isostructural D 5d [Pt 2 Ag 23 ] 7+ model, generated by simply replacing the central Pt atom of [Pt 3 Ag 22 ] 6+ by Ag + , the contacts between the two outer icosahedra (3.318 Å) are 1.14 times the outer icosahedra edges. This is indicative of a weaker compactness, thus somehow favoring the analogy with Ne 2 . This model actually constitutes the core of the structurally characterized 16-electron cluster Pt 2 Ag 33 (dtp) 17 .38The above analogy between a staggered vertex-sharing bisicosahedral unit and a structure made of three interpenetrating icosahedra can be extended to any linear n-vertex sharing framework, which can be described as made of 2n -1 interpenetrating icosahedra. The 24-electron linear tris-icosahedral structure [Pt 3 Ag 34 ] 10+ described above can then be viewed as either the isolobal analog to a Ne3 van der Waals molecule or to carbon suboxide O=C=C=C=O. Interestingly, the linear 22-electron [Pt 3 Ag 34 ] 12+ (Figure2) has no simple 5-atom main-group stable analog. The analogy of linear nicosahedral vertex sharing structures with linear 2n -1 main-group atom linear molecules works preferentially for 8nelectron species.

A c c e

  p t e d M a n u s c r i p t somehow non-spherical (oblate-distorted). It exhibits an MO diagram (Figure9) consistent with this view, with a 1D HOMO and a LUMO of 1F lineage. In this particular case, the 2S orbital lies below the 1D block.

Figure 9 .

 9 Figure 9. Kohn-Sham orbital diagram of the D 3h 20-electron [Pt 3 Ag 20 ].

Figure 10 .

 10 Figure 10. Summary of different tetra-icosahedra linear and propeller assemblies. Pt atoms are in green. Silver atoms in red and black are shared between 2 and 3 icosahedra, respectively.

Table 2 .

 2 Stable non-cyclic triatomic main group molecules.

	Number of electrons 16 = 3 x 8 -8	18 = 3 x 8 -6	20 = 3 x 8 -4	22 = 3 x 8 -2	24 = 3 x 8
	Triatomic systems	CO 2 , [N 3 ]	-	SO 2 , [NO 2 ]	-	SF 2		XeF 2, [I 3 ]	-	3I -, 3Ne
	Bond order	2		1.5		2		0.5	0 (van der Waals)
	Bond angle	linear (sp)	bent (sp 2 )		bent (sp 3 )	linear	-
	Their shell ordering spans as 1S < 1P < 1D < 2S < 1F < 2P... As for atomic systems, closed-shell (noble gas-like)
	stability is achieved for specific (so-called "magic") electron counts, spanning 2, 8, 18, 20, 34, 40… Such an atom-like
	description of clusters was conceptualized as the superatom model 8-11 and the jellium orbitals are often called superatomic
	orbitals. One of the most frequent close-packed spherical motif encountered in group 11 cluster chemistry is the centered
	icosahedron,	12-17 which is generally associated with the 8-electron "magic" number. Typical group 11 examples are
	[Au 25 (SR) 18 ] -13,14,18,19 and [Ag 21 (dtp) 12 ] + (dtp = dithiophosphate = S 2 P(OR) 2 ).	15 Both species can be viewed as made of an
	[M 13 ]

5+

(M = Au, Ag) centered icosahedral superatomic core (8 s-type valence electrons) stabilized by a "passivating shell" made of 18 [SR]

-ligands and 12 Au + atoms in the former case, and 12 [S 2 P(OR) 2 ] -ligands and 8 Ag + atoms in the latter case.

In both systems, the outer M + atoms interact weakly with the [M 13 ] 5+ core, but are strongly coordinated to sulphur lone pairs, making locally stable linear 14-electron or trigonal planar 16-electron d 10 metal centers.

  ). It results that the outer and central icosahedra have short Ag…Ag contacts (2.90 Å) on one side (top side of bent [Pt 3 Ag 34 ]10+ in Figure2) and much longer ones (3.94 Å) on the other side (bottom side of bent [Pt 3Ag 34 ]

	10+

  non-interacting 8-electron superatoms. It turns out that this vertex-sharing triangular architecture has been reported for five related 24-electron clusters, namely [(p-Tol 3 P) 12 Au 18 Ag 20 Cl 14 ], [(Ph 3 P) 14 Au 18 Ag 20 Cl 12 ] 2+ , [Cu 3 Au 34 (PPh 3 ) 13 (tBuPhCH 2 S) 6 S 2 ] 3+ , [Pt 3 Ag 33 (PPh 3 ) 12 Cl 8 ] + and [Pt 3 Ag 21 Au 12 (PPh 3 ) 12 Cl 8 ]

+ 20,21,29,51,52

  Ag 44 {S 2 P(OPr) 2 } 22 ] (Figure1) with one more repeat unit, provided the same electronic structure as its [Pt 4Ag 45 ]

	30-electron count ([Pt 4 Ag 45 ]	15+ ). Examination of its
	electronic structure indicates that it is isolobal to the linear 30-electron tetraiodide anion [I 4 ] 2-59 . This is in line with the
	results obtained for the one-icosahedron shorter linear [Pt 3 Ag 34 ] 12+ model (Figure 2), which is isolobal to [I 3 ] -. The 30-
	electron ligated model [Pt 4 Ag 55 (S 2 PH 2 ) 27 ] 2-, which contains the [Pt 4 Ag 45 ] 15+ core, and built as an extrapolation of
	[Pt 3

15+ core;

  61,62 One of them is [Au 60 Se 2 (Ph 3 P) 10 (SePh) 15 ] + . 61 the metal framework of which consists of the cyclic assembly of five vertex-sharing icosahedra. This [Au 60 ] 20+ ring of icosahedra possesses 40 = 8 x 5 electrons, i.e. exhibits weak interactions between the superatoms, as previously demonstrated by Lin and coworkers. 39 It is the so far largest element of the family of 8n-electron clusters made of n vertex-sharing icosahedra. Another recently published cluster from this group, namely [Au 8 Ag 57 (Dppp) 4 (C 8 H 11 S) 33 Cl 2 ]

	M a n u s c r i p t
	A c c e p t e d

  30-electron [Au 8 Ag 41 ] 19+ core and on a simplified model of [Au 8 Ag 57 (Dppp) 4 (C 8 H 11 S) 33 Cl 2 ] + in which the phenyl and cyclohexyl groups were replaced by methyl substituents. Their computed HOMO-LUMO gaps are similar (0.60 and 0.47 eV, respectively). Both systems have related electronic structures, confirming the qualitative description of [Au 8 Ag 57 (Dppp) 4 (C 8 H 11 S) 33 Cl 2 ] + as made of a [Au 8 Ag 41 ] 19+ core "passivated" by an inner shell made of 4 Dppp, 33 (C 8 H 11 S)

-, 2 Cl -and 16 Ag + . The question that arises then is how to rationalize the 30-electron count of the metal core. No simple 8-member cyclic molecule has this electron count.

  Au 8 Ag 41 ] 19+ (Figure11, right side). Rather, the π-type 1P z and 1D xz , yz components appear among them, conferring the cluster with 6-electron π-type aromaticity. The jellium configuration can be written as 1S

		2 1P σ+π	6 1D σ+π	8 2S 2 2P σ	4 1F σ	4 1G σ
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	A c c e p t e d
					2 1P	4 1D	4 2S	2
	2P	4 1F 4 1G 4 2D 4 3S

2

. It should be noted that the planar approximation, no π-type P, D, F or G orbital is considered (shells closed with 4 electrons). It turns out that the expected 2D and 3S orbitals are missing within the occupied orbitals of [4 . Figure 11. left side: The [Au 8 Ag 41 ] 19+ core of [Au 8 Ag 57 (Dppp)4(C 8 H 11 S) 33 Cl 2 ] + ) (from ref. 62), made of eight interpenetrating icosahedra (ideal D 2d symmetry). Au atoms are in green and occupy the icosahedra centers. Silver atoms in red, black and magenta are shared between 2, 3 and 8 icosahedra, respectively. Right side: Its Kohn-Sham orbital diagram.

  20-electron [Pt 3 Ag 20 ] model, for example. Such compact assemblies of fused icosahedra can importantly deviate from spherical shape, as for example the [Au 8 Ag 41 ] 19+ inner core of the experimentally characterized [Au 8 Ag 57 (Dppp) 4 (C 8 H 11 S) 33 Cl 2 ] + nanocluster. With its specific degree of oblaticity, this 30-electron species has a specific closed-shell electron count. We hope that this work will encourage efforts to grow ligand-protected poly-icosahedral clusters and open new opportunities for practical experimentations.

	M a n u s c r i p t
	A c c e p t e d

Acknowledgements

This work was supported by the France-Taiwan ANR-MOST 2018 program (projectNanoalloys) and Ministry of Science and Technology in Taiwan (MOST 108-2923-M-259-001). The GENCI is acknowledged for HPC resources (Project A0050807367).

Notes and references

Conflicts of interest

There are no conflicts to declare.