Pascal Raymond
email: pascal.raymond@imag.fr

⋆ Verimag

Recognizing Regular Expressions by means of Dataflow Networks

This paper addresses the problem of building a Boolean dataflow network (sequential circuit) recognizing the language described by a regular expression. The main result is that both the construction time and the size of the resulting network are linear with respect to the size of the regular expression.

Introduction

"Grep" machine: Let Σ be a vocabulary, L be a regular language on Σ. A "grep" machine is a machine receiving a sequence s 0 , s 1 , . . . , s n , . . . of symbols (s i ∈ Σ) and computing a sequence b 0 , b 1 , . . . , b n , . . . of Booleans, such that b n is true if and only if the word s 0 s 1 . . . s n belongs to L 1 . This paper addresses the problem of building a "grep" machine for languages described by regular expressions. This problem is rather classical [START_REF] Brzozowski | Derivatives of regular expressions[END_REF][START_REF] Thompson | Regular expression search algorithm[END_REF][START_REF] Mcnaughton | Regular expressions and state graphs for automata[END_REF][START_REF] Berry | From regular expressions to deterministic automata[END_REF][START_REF] Aho | Algorithms for finding patterns in strings[END_REF][START_REF] Antimirov | Partial derivatives of regular expressions and finite automata constructions[END_REF]. We propose a solution which, to our knowledge, is new: Informally, it consists of building, from a regular expression E, a "circuit" (or Boolean data-flow network) exploring all the branches of a non-deterministic automaton recognizing L(E). The relations between regular languages and sequential circuits are also studied [START_REF] Brzozowski | On equations for regular languages, finite automata, and sequential networks[END_REF][START_REF] Floyd | The compilation of regular expressions into integrated circuits[END_REF]. But the important point here, is that the automaton is never explicit (it is represented implicitly by a system of Boolean equations) nor deterministic. As a consequence, both the construction time and the size of the circuit are linear with respect to the size of the expression E.

Our method is presented in a general framework based on linear systems of equations. We first review the classical methods (deterministic finite automaton, non-deterministic finite automaton without or with empty-labelled transitions), by expressing them in this framework. We show that this classification coincides with the classes of complexity (exponential, quadratic and linear).

1 Regular expressions

Regular languages

Let Σ be a finite set of symbols; Σ * denotes the set of all finite strings of symbols in Σ. The empty string (string of length 0) in Σ * is denoted by ε. A language over Σ is any subset of Σ * . The empty language is denoted by ∅. The set of regular languages over Σ is the least class of languages containing all the finite languages over Σ, and closed by union, concatenation (denoted by •) and Kleene star operator (denoted by * ). Any regular language can be denoted by a regular expression:

E ::= ∅ | a | E • E | E + E | E *
The semantics of regular expressions is given by a function L which associates to each expression E the language described by E2 :

L(∅) = ∅ L(a) = {a} L(E + F ) = L(E) ∪ L(F ) L(E • F ) = L(E) • L(F ) L(E * ) = µℓ.{ε} ∪ (ℓ • L(E))
We note ≃ the semantic equivalence over regular expressions, i.e.:

E ≃ F ⇔ L(E) = L(F )

More "useful" regular expressions

We are interested in "non trivial" regular languages, i.e. any language L which is neither empty (L = ∅) nor reduced to the empty string (L = {ε}). We will then use a more useful syntax, which forbids the representation of trivial languages:

E ::= a | E • E | E + E | E * | E ε
The semantic of the power-ε operator is given by: L

(E ε ) = ε ∪ L(E).
We note R⌉ §√ the set of regular expressions, and we also define a function

Λ : R⌉ §√ → IB = {true, false}, such that Λ(E) is true if and only if the empty string belongs to L(E). Λ(a) = false Λ(E ε ) = true Λ(E * ) = true Λ(E + F ) = Λ(E) ∨ Λ(F ) Λ(E • F ) = Λ(E) ∧ Λ(F )
In this section we show that the problem of building a grep machine from a regular expression can be expressed as follows: build a system of language equations of the form "X = t 1 + ... + t n ", where each t i is a language term of a "special" form. The various known methods for building grep machines can be related to the form of the terms. This classification is largely inspired by [START_REF] Brzozowski | On equations for regular languages, finite automata, and sequential networks[END_REF][START_REF] Chang | From regular expressions to DFA's using compressed NFA's[END_REF].

In this paper, we use the following notations: let θ be a system of equations, we note V(θ) the set of language identifiers defined in θ, A(θ) ∈ V(θ) the main language (the axiom), and for each X ∈ V(θ), θ(X) the set of terms t i defining X in θ. We note L(θ) the solution (if it exists) of θ, i.e. the language denoted by the axiom A(θ).

Suffix-linear systems

In such systems, a term is either ε, or of the form a • Y , where a is a symbol and Y is a language identifier. More precisely, each equation is of the form:

X = a 1 • Y 1 + ... + a n • Y n [+ ε]
Such a system θ is equivalent to a finite automaton (Q, I, F, T Σ ), where:

-Q = V(θ) is the set of states, -I = A(θ) is the (unique) initial state, -F = {X ∈ V(θ) | ε ∈ θ(X)} is the set of final states, -T Σ ∈ Q × Σ × Q is the set of symbol-labelled transitions, with: (X, a, Y ) ∈ T Σ ⇔ (a • Y ) ∈ θ(X).
Deterministic automaton (DFA): Moreover, if in each equation

X = a 1 •Y 1 +...+ a n • Y n [+ ε],
all the symbols a i are different, the automaton is deterministic, and each Y i is said to be the a i -derivative of X. Such a deterministic system can be built by computing derivatives of regular expressions. Brzozowski has formally defined this construction [START_REF] Brzozowski | Derivatives of regular expressions[END_REF]. The size of the resulting deterministic automaton (and thus the cost of the construction) is, in the worst case, exponential with respect to the size of the regular expression (number of operators in the regular expression).

Non-deterministic automaton (NFA):

If in each equation X = a 1 • Y 1 + ... + a n • Y n [+ ε]
, some a i are equal, the resulting automaton is non-deterministic.

McNaughton and Yamada defined the construction of such an automaton [START_REF] Mcnaughton | Regular expressions and state graphs for automata[END_REF].

Berry and Sethi gave an efficient algorithm to build such an automaton from a regular expression [START_REF] Berry | From regular expressions to deterministic automata[END_REF]. Antimirov defined a notion of partial derivatives, which generalizes the notion of derivative for non-deterministic automata [START_REF] Antimirov | Partial derivatives of regular expressions and finite automata constructions[END_REF]. For those algorithms, the size of the resulting non-deterministic automaton (and thus the cost of the construction) is, in the worst case, quadratic with respect to the size of the regular expression. More precisely, the number of states is linear, but the number of transitions can be quadratic.

Non-deterministic automaton with ε-transitions:

The previous definition can be extended with terms of the form Y . Such systems are equivalent to a nondeterministic automaton with ε-transitions: (Q, I, F, T Σ , T ε ), where:

-T ε ∈ Q × Q = {(X, Y ) | Y ∈ θ(X)}
Thompson has defined the construction of such machines from regular expressions [START_REF] Thompson | Regular expression search algorithm[END_REF]. The main result is that both the size of the automaton and the cost of its construction are linear with respect to the size of the regular expression.

Our goal is to define a linear algorithm that builds a grep machine from a regular expression, so we choose the third kind of linear systems (i.e. nondeterministic automaton with ε-transitions).

Prefix-linear systems

A completely dual definition can be given by using terms of the form Y •a instead of a•Y . Let us see the example of a non-deterministic prefix-linear system with εtransitions. Such a system θ is equivalent to a finite automaton (Q, I, F, T Σ , T ε ), where:

-Q = V(θ) is the set of states, -I = {X ∈ V(θ) | ε ∈ θ(X)} is the set of initial states, -F = A(θ) is the (unique) final state, -T Σ ∈ Q × Σ × Q is the set of symbol-labelled transitions, with: (Y, a, X) ∈ T Σ ⇔ (Y • a) ∈ θ(X), -T ε ∈ Q × Q = {(X, Y ) | Y ∈ θ(X)} is the set of ε-labelled transitions.
In this paper, we chose to build prefix-linear systems. Indeed, the problems of building prefix-or suffix-linear systems are completely equivalent, but, as we will show in a following section, prefix-linear systems are "equivalent" to sequential circuits, and then, in some sense, directly "executable".

3 From regular expressions to prefix-linear systems

Abstract syntax for left-linear systems

We propose here a "functionnal-like" syntax which allows a system of equations to be represented by a single equation:

system ::= X = terms terms ::= ε | X | X • a | terms + terms | let X = terms in terms | rec X = terms Note that rec X = terms is a macro-notation for let X = terms in X.
The semantic function L is naturally extended to this new syntax: L(σ) is the language denoted by the system σ. We also extend the semantic function for terms, which are, in some sense, "partially built" systems: L(τ ) is (if it exists) the language denoted by the system X = τ , where X is any identifier not appearing in τ .

The basic algorithm

The idea is to define a function Θ which transforms a regular expression into a prefix-linear system, i.e. which verifies:

L(E) = L(Θ(E))
Moreover, we want the cost of the translation to be linear with respect to the size of the regular expression; the translation should also introduce few variables.

For that purpose, we introduce a recursive function Γ (X, E) whose parameters are a regular expression (E) and a language identifier (X) representing the "prefixes" of E. The result of Γ is a partially built system of equations (i.e. a "terms" according to the previously defined abstract syntax). This will allow us to introduce variables only when needed.

The meaning of Γ is quite simple: intuitively, Γ (X, E) "denotes" the same language as "X • E". More formally, let L X be the language denoted by X:

L(Γ (X, E)) = L X • L(E)
The definition of Γ is given by induction on the structure of regular expressions:

Γ (X, a) = X • a (1) Γ (X, E ε ) = X + Γ (X, E) (2) Γ (X, E + F ) = Γ (X, E) + Γ (X, F ) (3)
Let Y be a new identifier:

Γ (X, E • F ) = let Y = Γ (X, E) in Γ (Y, F ) (4) Γ (X, E * ) = rec Y = X + Γ (Y, E) (5)
The system corresponding to the regular expression E is obtained by computing Γ (Z, E) with Z = ε, and then, by naming the result; let X be a new identifier:

Θ(E) = X = (let Z = ε in Γ (Z, E))

Example

Let us consider the expression (a * + b) * • a:

Γ (Z, (a * + b) * • a) = let Y = Γ (Z, (a * + b) * ) in Γ (Y, a) = let Y = Γ (Z, (a * + b) * ) in Y • a Γ (Z, (a * + b) * ) = rec T = Z + Γ (T, a * + b) Γ (T, a * + b) = Γ (T, a * ) + Γ (T, b) = (rec W = T + Γ (W, a)) + T • b = (rec W = T + W • a) + T • b
A classical system of equations is obtained by extracting the sub-equations and naming the axiom with X:

X = Y • a Y = T T = Z + W + T • b W = T + W • a Z = ε

Automaton size and comparison

The size of Θ(E) (or of the equivalent NFA (Q, I, F, T Σ , T ε )) can be easily related to the size of E: let o be the number of symbol occurrences, d be the number of dots, e be the number of power-εoperators, and k the number of Kleene stars, we have:

|Q| = d + k + 2 |T Σ | = o |T ε | = k + e
The two additional states are the initial state introduced by the starting rule ("let Z = ε in..."), and a final state corresponding to the axiom.

It is interesting to compare this solution where the states correspond to the dot and star operators, to the McNaughton and Yamada automata (or normalized NFA's) where the states correspond to the symbol occurences (in a normalized NFA, we have |Q| = o + 1) [START_REF] Mcnaughton | Regular expressions and state graphs for automata[END_REF][START_REF] Berry | From regular expressions to deterministic automata[END_REF].

The same notion of states (attached to dots and stars) exists in the Antimirov's work: the algorithm can then be viewed as a simple way to compute Antimirov's partial derivatives [START_REF] Antimirov | Partial derivatives of regular expressions and finite automata constructions[END_REF]. However, Antimirov's algorithm does not produce ε-transitions, so the resulting automaton is, in the worst case, quadratic with respect of the size of the regular expression.

The use of ε-transitions makes this algorithm close to the Thompson's one. But the basic Thompson NFA is, in general, bigger: between r and 2r states, and between r and 4r transitions, where r is the length of the regular expression. Our algorithm can be viewed as a way to directly produce an optimized Thompson NFA.

Boolean networks

We present here a simple model for sequential circuits. It consists of a Boolean operator network, together with a simple data-flow semantics. The networks are built with classical Boolean operators (true, false, not, and, or) and a delay binary operator fby, whose first argument is the initial value, and the second the Boolean value to be delayed.

Syntax

We define a syntax that allows the representation of a network (a cyclic graph in general) by a simple syntactic tree. This is done by using a "functional-like" syntax (X stands for any identifier):

net ::= X | true | false | not net | net and net | net or net | net fby net | let X = net in net | rec X = net
Just like for the linear systems of equations ( § 3.1), a network can also be described by a set of equations. For instance, the system of Boolean equations: is equivalent to the network shown (in both syntactic and graphic form) in Fig. 1. We only consider networks without combinational loop: recursive definitions can only appear under a fby operator. The set of correct networks is denoted by N .

Synchronous semantics

The set of free variables (inputs) of a network n is denoted by Free(n). A valuation of Free(n) is a function which associates a Boolean value to each variable of Free(n): ρ : Free(n) → IB. A trace over Free(n) is a finite sequence of valuation: ν = ρ 1 , ..., ρ k .

The synchronous semantics models the behavior of idealized sequential circuits: given an input trace ν of length k, a network n produces a sequence of k Boolean values. More precisely, each sub-network (each node in the network) produces a sequence of k Boolean values. For instance, the constant true produces a sequence of k "true". Let ν = ρ 1 , ..., ρ k be the input trace, the input x produces the sequence ρ 1 (x), ..., ρ k (x). The classical operators operate pointwise on sequences; for instance, if n produces the sequence b 1 , ..., b k and n ′ the se-

quence b ′ 1 , ..., b ′ k , then n or n ′ produces b 1 ∨ b ′ 1 , ..., b k ∨ b ′ k .
At last, the fby operator delays its second operand, while its first operand defines the initial value:

n fby n ′ produces b 1 , b ′ 1 , ..., b ′ k-1 .
The absence of combinational loops in the network n is a sufficient property for this semantics to be operational. In other terms, if n has no combinational loop, then for any input trace ρ, the corresponding output sequence is completely determined. We then consider that any network n without combinational loop is a function from traces to Boolean sequences, and we note n(ν) the output sequence corresponding to the trace ν.

Language recognized by a Boolean network

A trace is said to be recognized by the network n if the corresponding output Boolean sequence ends with "true". The set of traces recognized by n is called the behavior recognized by n, denoted by B(n):

B(n) = {ν | n(ν) = b 1 , ..., true}
In terms of languages, the set of valuation can be viewed as an alphabet Σ, a trace as a word in Σ * , and a behavior as a language over Σ. Since a network is clearly a finite state machine (with at most 2 m states, where m is the number of fby operators in the network), the behavior recognized by a network is a regular language over Σ.

Prefix-linear systems and networks

In this section, we will map a prefix-linear system of equations θ to an Boolean network n = Ω(θ).

Encoding

A prefix-linear system describes a language over some alphabet Σ, while the networks describe behaviors over a set of Boolean variables. So the initial alphabet has to be encoded using Boolean variables. We consider here a "trivial" one-to-one encoding (i.e. we identify the alphabet to a set of Boolean variables). Indeed, the method can easily be adapted for more powerful encoding such as logarithmic encoding.

Empty trace

As they are defined, the networks cannot recognize the empty trace, so the resulting network cannot be "completely equivalent" to the source system. Let θ be a prefix-linear system, the resulting network Ω(θ) simply verifies:

B(Ω(θ)) = L(θ) \ {ε}

Initial states

Intuitively, each term in a system of equations will be replaced by a network recognizing its non-empty traces. In order to treat terms of the form X • a, we need to know if the empty string belongs to X, since, if ε ∈ L(X):

L(X • a) \ {ε} = (L(X) \ {ε}) • a + a
The set of variables X such that ε ∈ L(X) is called the set of initial states (referring to the terminology of NFA's). This set is denoted by I(θ) and is recursively defined by:

X ∈ I(θ) ⇔ (ε ∈ θ(X)) ∨ (∃Y ∈ I(θ) | Y ∈ θ(X))
In fact, this set can be computed during the construction of the system, using the function Λ ( § 1.2) and this computation has a linear cost (with respect to the size of the regular expression).

From prefix-linear systems to networks

The definition of the mapping function Ω is the following:

Ω(ε) = false Ω(X) = X Ω(τ + τ ′ ) = Ω(τ ) or Ω(τ ′ ) Ω(X • a) = (true fby X) and a if X ∈ I(θ) Ω(X • a) = (false fby X) and a if X / ∈ I(θ) Ω(let X = τ in τ ′ ) = (let X = Ω(τ ) in Ω(τ ′ ))
Let us come back to the example of § 3.

Combinational loops and normalization

We have defined a method for building a sequential circuit form a regular expression: Ω(Θ(E)).

As it is computed, the intermediate system Θ(E) (and then the resulting network too) may contain combinational loops. For instance, in the example of § 3.3 (θ = Θ((a * + b) * • a)), we have W ∈ θ(T ) and T ∈ θ(W ). Such combinational loops appear in Θ(E) if and only if there exists in E a sub-expression of the form (F * ) such that Λ(F ). In order to avoid combinational loops, we define a function Norm on regular expressions, which recursively replaces each sub-expression of the form (F * ) by an equivalent expression (G * ) such that ¬Λ(G).

Norm(a) = a Norm(E + F ) = Norm(E) + Norm(F ) Norm(E • F ) = Norm(E) • Norm(F ) Norm(E ε ) = Norm(E) ε Norm(E * ) = NormBis(E) *
where:

NormBis(E) = Norm(E) if ¬Λ(E) NormBis(E + F ) = NormBis(E) + NormBis(F ) if Λ(E + F ) NormBis(E • F ) = NormBis(E) + NormBis(F ) if Λ(E • F ) NormBis(E ε ) = NormBis(E) NormBis(E * ) = NormBis(E)
The function Norm is a simple parsing which applies the "interesting" function NormBis to the Kleene star operators.

Notice that this normalization does not replace the expression E * by an expression F * such that L(F ) = L(E) \ ε. Such a transformation would have a quadratic cost since, for all regular languages ℓ and ℓ ′ containing the empty string: (ℓ

• ℓ ′ ) \ ε = (ℓ \ ε) • ℓ ′ + ℓ • (ℓ ′ \ ε).
The normalization simply replaces the expression E * by an expression F * such that L(F * ) = L(E * ) and ¬Λ(F ). The correctness of the algorithm is based on the following lemma:

(ε ∈ ℓ ∧ ε ∈ ℓ ′ ) ⇒ (ℓ • ℓ ′ ) * = (ℓ + ℓ ′ ) * .
Here are some examples of normalization:

Norm((a * + b) * ) = (a + b) * Norm(((a * ) * ) * ) = a * Norm((a ε • b * + c * ) * ) = (a + b + c) * 6
From regular expressions to Boolean networks Finally, let E be a regular expression over an alphabet Σ interpreted as a set of Boolean variables, we have defined a method which computes a correct "equivalent" network n, i.e. a network without combinational loop recognizing all the non-empty traces of the language L(E):

B(Ω(Θ(Norm(E)))) = L(E) \ ε
This method involves the computation of the function Λ on each sub-expression of E, the normalization of E, and also the computation of the initial states in the system Θ(Norm(E)). But the main result is that the cost of all those treatments is linear with respect to the size of the source regular expression.

In order to outline this linear complexity, we define in this section a function Φ which directly produces a network from a regular expression. Like the function Θ, the function Φ is defined using a recursive function, Υ , which takes a special parameter representing the prefixes. This function takes two parameters to describe the prefixes:

Υ : N × IB × R⌉ §√ → N (X, b, E) → n
The first parameter (X) is a network which is supposed to recognize the nonempty traces of the prefixes, while the second one (b) is a Boolean value indicating whether the empty trace belongs to the prefixes. Intuitively, this new parameter b allows us to directly compute the set of "initial states" ( § 5.3). The normalization of the regular expression ( § 5.5) is also performed during the construction. For that purpose, a special recursive function Υ * is applied to the Kleene star operands.

Φ(E) = let Z = false in Υ (Z, true, E) Υ (X, b, a) = (b fby X) and a Υ (X, b, E + F ) = Υ (X, b, E) or Υ (X, b, F ) Υ (X, b, E ε ) = X or Υ (X, b, E)
Let Y be a new identifier: The origin of this work takes place in the domain of synchronous programming [START_REF] Halbwachs | Synchronous programming of reactive systems[END_REF]. Synchronous programming offers an idealized framework for programming reactive systems. For instance, idealized sequential circuits are synchronous programs. This idealized framework also permits to perform formal verifications on programs. Critical properties are described by means of Boolean synchronous programs called observers [START_REF] Halbwachs | Synchronous observers and the verification of reactive systems[END_REF]. Such observers can be expressed with any existing synchronous language (Lustre, Argos, Esterel), but it appears that the classical regular constructs (sequence, iteration) are also very useful to describe safety properties.

The idea was to translate, with a minimal cost, a safety property expressed with regular constructs into a suitable synchronous program, such as a Boolean network.

A tool called reglo has been designed for this purpose. This tool translates a set of regular expressions, describing traces over a set of Boolean variables (i.e. the input alphabet is supposed to be already Boolean encoded), into an equivalent Boolean dataflow network, expressed in the language Lustre.

sFig. 1 .

 1 Fig. 1. A Boolean network and the corresponding expression

  3 (θ = Θ((a * + b) * • a)). We first compute I(θ) = {Z, T, Y, W }, then we obtain the network (given as a set of equations): X = (false fby Y ) and a Y = T T = Z or W or (true fby T ) and b W = T or (true fby W ) and a Z = false

ΥFigure 2

 2 Figure2shows the result of Υ (false, true, (a * + b) * • a). One can see that the resulting network has the same structure as the regular expression.

Fig. 2 .

 2 Fig. 2. The network of (a * + b) * • a

µℓ.f (ℓ) denotes the least fixpoint of the equation ℓ = f (ℓ)


David A Bonhommeau
email: david.bonhommeau@univ-reims.fr

Zero-point averaged dynamics of Ar + n He 1000 : Dopant-size influence on potential-energy surfaces, mass spectra and fragmentation patterns

Keywords: mixed quantum-classical dynamics, helium nanodroplets, argon clusters, zero-point energy, ion ejection, ion trapping, recombination

The fragmentation of Ar + n He 1000 (n = 3, 5) is investigated by zero-point averaged dynamics (ZPAD) and compared with recent results on Ar + 4 He 1000 to investigate the influence of the dopant size. At the core of ZPAD, effective He-He potentials converged for He 1000 and Ar n He 1000 are found alike which suggests that ZPAD simulations on argon-doped helium nanodroplets (HNDs) could restrict to effective potentials obtained for pure HNDs. The fragmentation dynamics of Ar + n He 1000 clusters share a number of similarities, like the absence of Ar + -containing fragments, the occurrence of ion ejections and trappings, Ar + n electronic relaxation typically extending on a picosecond time scale, or primary fragmentation events which are not necessarily representative of the final distribution of fragments. However, Ar + 3 He 1000 clusters also exhibit unexpected behaviors like an inefficient electronic relaxation when the dynamics starts on the uppermost Ar + 3 electronic state which makes the dopant trapped within the droplet.

Introduction

Although the first observation of helium nanodroplets (HNDs) by Becker and colleagues dates back to the 1960's [START_REF] Becker | On the History of Cluster Beams[END_REF], the doping of HNDs with clusters or molecules only emerged thirty years later. Among the first dopants to be considered were rare-gas clusters, ideal systems to investigate the capture and coagulation of species within HNDs [START_REF] Scheidemann | Capture of Neon Atoms by 4 He Clusters[END_REF][START_REF] Lewerenz | Successive capture and coagulation of atoms and molecules to small clusters in large liquid helium clusters[END_REF][START_REF] Callicoatt | Capture and ionization of argon within liquid helium droplets[END_REF], the physical processes behind the dopant ionization after bombarding the surface of the doped HND with electrons [START_REF] Ruchti | Charge transfer and fragmentation of liquid helium clusters that contain one or more neon atoms[END_REF][START_REF] Ruchti | Charge transfer and fragmentation of liquid helium droplets doped with xenon[END_REF] or high-energy photons [START_REF] Fröchtenicht | The photoionization of large pure and doped helium droplets[END_REF][START_REF] Kim | Photoionization of helium nanodroplets doped with rare gas atoms[END_REF], and the subsequent fragment distributions, signature of the fragmentation mechanisms at play. Beside the common observation of progressions of peaks steadily spaced out by 4 u, massspectrometry experiments also demonstrated the existence of "magic numbers", namely sizes of particularly stable fragments, for all common rare-gas clusters although these anomalies were less abundant and intense for xenon than for argon or krypton [START_REF] Bartl | On the stability of cationic complexes of neon with helium -solving an experimental discrepancy[END_REF][START_REF] Bartl | On the Size and Structure of Helium Snowballs Formed around Charged Atoms and Clusters of Noble Gases[END_REF] On the theoretical side, the modeling of doped HNDs confronts with the need for describing large systems (N = 10 3 -10 6 in typical experiments) of quantum nature. Takayanagi and Shiga tackled the photodissociation of Cl 2 in helium clusters composed of 200 atoms by combining a wavepacket dynamics for Cl 2 with a path-integral molecular dynamics of helium nuclei. Chlorine dissociated in most of trajectories, a result later confirmed by Vilá et al. who carried out TDDFT simulations on Cl 2 He 500 [START_REF] Vilà | Photodissociation Dynamics of Homonuclear Diatomic Molecules in Helium Nanodroplets. The Case of Cl 2 @( 4 He) N[END_REF]. In an attempt to free from the nanodroplet size, Bonhommeau et al. adopted a phenomenological description of helium atoms represented as a friction force that applied to the dopant, a neon cluster, when its speed exceeded the Landau critical velocity [START_REF] Bonhommeau | Fragmentation dynamics of ionized neon clusters (Ne n , n = 3 to 14) embedded in helium nanodroplets[END_REF]. Despite some qualitative agreement with experiments regarding the size of neon fragments, this method was not suited to discuss helium evaporation, magic numbers, or fragmentation mechanisms, and a more explicit method remained to be devised.

In this context, Bonhommeau et al. proposed another approach called zeropoint averaged dynamics (ZPAD) to investigate the fragmentation of Ne + n He 100 clusters (n = 4 -6) [START_REF] Bonhommeau | Modeling the fragmentation dynamics of ionic clusters inside helium nanodroplets: The case of He 100 Ne + 4[END_REF][START_REF] Bonhommeau | Fragmentation of ionized doped helium nanodroplets: Theoretical evidence for a dopant ejection mechanism[END_REF]. The cornerstone of this method is the design of an effective He-He interaction potential that takes into account zeropoint effects in an average manner. By combining this effective potential with the dopant potential-energy surface and dopant-He interactions, the mixed quantum-classical dynamics of the doped HND can be achieved at reasonable computational cost. Simulations on neon-doped HNDs proved the existence of fast ion ejections [START_REF] Bonhommeau | Fragmentation of ionized doped helium nanodroplets: Theoretical evidence for a dopant ejection mechanism[END_REF], a fragmentation process which also resulted in the ejection of helium atoms with kinetic energies of ∼ 28 -45 cm -1 , well above the typical binding energy of a helium atom to the HND (∼ 5 cm -1 ) [START_REF] Lewerenz | Successive capture and coagulation of atoms and molecules to small clusters in large liquid helium clusters[END_REF]. The smallness of the cluster prevented any observation of trappings or recombinations since abundances of ionic fragments mostly converged after a few tens of picoseconds due to ion ejections. In contrast, recent ZPAD simulations on Ar + 4 He 1000 demonstrated the occurrence of a multitude of fragmentation patterns, from fast ion ejections, favored by high electronic excitations of the dopant, to long-term ion trappings, favored by low electronic excitations of the dopant [START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF]. Geminate recombinations were also identified as a source of Ar + 3 and Ar + 4 fragments although the electronic relaxation of the initially excited Ar + 4 dopant mainly produced Ar + 2 . No Ar + -containing fragments were found whereas such fragments are abundant in experimental mass spectra [START_REF] Bartl | On the Size and Structure of Helium Snowballs Formed around Charged Atoms and Clusters of Noble Gases[END_REF], which seemed to indicate that such fragments might come from HNDs doped with smaller dopants (eg, Ar + 3 ), more prone to dissociate into Ar + in the gas phase [START_REF] Bonhommeau | Fragmentation dynamics of argon clusters (Ar n , n = 2 to 11) following electron impact ionization: Modeling and comparison with experiment[END_REF], secondary ionizations of large Ar p He q fragments [START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF], or multiple ionization of the initial HND.

In the present paper, ZPAD simulations are carried out on Ar + n He 1000 (n = 3, 5). The essential features of the ZPAD approach applied to argon-doped HNDs are first recalled in Section 2. In the same section, effective He-He interaction potentials converged for He 1000 , Ar 3 He 1000 , and Ar 5 He 1000 are compared with each other and with the effective He-He potential recently obtained for Ar 4 He 1000 to evaluate the influence of the dopant size on these potentials. Mass spectra and fragmentation patterns of Ar + n He 1000 (n = 3 -5) are then discussed in terms of dopant size and initial electronic excitation in Section 3. Concluding remarks about the influence of superfluidity on the results and possible improvements of ZPAD to model the dynamics of argon-doped, and more generally rare-gas-doped, HNDs are collected in Section 4.

Method

Origin of ZPAD

In the mid-1990's Sterling et al. proposed to model the dynamics of solid parahydrogen, pure or doped with Li, by describing H 2 molecules as gaussian particles [START_REF] Sterling | Simulations of quantum crystals by classical dynamics[END_REF]. For this purpose, they constructed an average interaction potential for solid H 2 by convolving a reference potential with a gaussian function which represented the H 2 probability density [START_REF] Sterling | Simulations of quantum crystals by classical dynamics[END_REF]. Slavíček et al. later generalized this approach to investigate the photodissociation dynamics of hydrogen halides in floppy neon clusters by iteratively calculating the neon wave function rather than assuming it of gaussian type [START_REF] Slavíček | Pickup and Photodissociation of Hydrogen Halides in Floppy Neon Clusters[END_REF]. Ne-Ne effective potentials were then fully converged after as few as four iterations. ZPAD is a direct extension of the method introduced by Slavíček et al. to HNDs doped with rare-gas clusters. It encompasses two methods: an iterative procedure devoted to the construction of an effective He-He interaction potential representative of the neutral doped HND (potential part of ZPAD), and the subsequent mixed quantum-classical dynamics of the neutral and ionized doped HNDs based on this effective potential (dynamical part of ZPAD). A comprehensive presentation of ZPAD is provided in our previous publications on neon-and argon-doped HNDs [START_REF] Bonhommeau | Modeling the fragmentation dynamics of ionic clusters inside helium nanodroplets: The case of He 100 Ne + 4[END_REF][START_REF] Bonhommeau | Fragmentation of ionized doped helium nanodroplets: Theoretical evidence for a dopant ejection mechanism[END_REF][START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF] and we mainly recall here essential features of the method.

Potential part of ZPAD

In the case of argon-doped HNDs, the reference system used to build the effective He-He interaction potential is Ar n He 1000 (n ≤ 5). The corresponding potential-energy surface is

V neut (R) = n i=1 n j=1 j>i V Ar-Ar (R ij ) + n i=1 1000 a=1 V He-Ar (R ia ) + 1000 a=1 1000 b=1 b>a V He-He (R ab ) (1)
where V Ar-Ar (R) is the HFDID1 Ar-Ar potential [START_REF] Aziz | A highly accurate interatomic potential for argon[END_REF], V He-Ar (R) is the HFD-B He-Ar potential proposed by Keil et al. [START_REF] Keil | On obtaining interatomic potentials from multiproperty fits to experimental data[END_REF][START_REF] Douketis | Intermolecular forces via hybrid Hartree-Fock-SCF plus damped dispersion (HFD) energy calculations. An improved spherical model[END_REF], and V He-He (R) is the SAPT2 He-He potential built by Janzen and Aziz [START_REF] Janzen | An accurate potential energy curve for helium based on ab initio calculations[END_REF].

The first step of the iterative potential part of ZPAD is a molecular dynamics (MD) equilibration at T = 0.38 K [START_REF] Grebenev | Superfluidity Within a Small Helium-4 Cluster: The Microscopic Andronikashvili Experiment[END_REF] for 3 ns (3 × 10 5 steps of 10 fs carried out with a velocity verlet algorithm), the last nanosecond being devoted to the calculation of the classical He-He radial distribution function (rdf). In every iteration except iteration 1, this classical rdf is convolved with the squared helium wavefunction from the previous iteration, then yielding a less structured semiclassical rdf that includes quantum effects. The convolution of the SAPT2 He-He potential with the rdf (classical for iteration 1 and semiclassical otherwise) then enables the formation of the average radial potential felt by one helium atom in the mean field due to the others. This radial potential serves as input in a radial Schrödinger equation that can be variationally solved by using a basis set composed of sine functions or Laguerre polynomials. The squared eigenfunction is finally convolved with the SAPT2 He-He potential to produce an effective He-He potential. This procedure, from equilibration to effective potential generation, is then repeated until convergence is achieved [START_REF] Slavíček | Pickup and Photodissociation of Hydrogen Halides in Floppy Neon Clusters[END_REF].

Effective He-He potentials obtained for He 1000 , Ar 3 He 1000 , and Ar 5 He 1000 are depicted in Figure 1. A truncation of helium wavefunctions at 1.6 Å was required during the iterative procedure because of the high zero-point energy of helium clusters that made convergence hard to reach. About thirty iterations were needed to get an energy deviation lower than 10 -3 cm -1 between the well depths of two successive effective He-He potentials. Data points for these potentials were fitted on the following analytical form:

V eff He-He = V rep (R) + T (R) [V well (R) -V rep (R)] , (2)
where

V rep (R) = A exp -α 1 R/R e -α 2 (R/R e ) 2 , (3a)
V well (R) = 6 i=3 c 2i (R/R e ) -2i , (3b)
T (R) = 0.5(1 + tanh[a(R -b)]). (3c)
V well (R) reproduces accurately the well and long-range part of the potential while V rep (R) is added to better model the repulsive wall and T (R) is a switching function. The fitting parameters of the effective potentials obtained for He 1000 , Ar 3 He 1000 , and Ar 5 He 1000 are reported in Table 1. Standard deviations (SD) between data points and the fitted analytical function remain below 0.003 cm -1 in the region of negative energies (SD -) and do not exceed 0.72 cm -1 in the region of positive energies (SD + ), which corresponds to the repulsive part of the potential. Moreover, according to Figure 1a, all the effective potentials have very similar shapes, and their well depths and equilibrium distances differ by less than 0.05 cm -1 and 0.04 Å, respectively. Therefore, Ar n dopants seem to have little influence on effective He-He potentials, a result that should hold true for larger HNDs. Indeed, the larger the droplet, the smaller the influence of a given dopant on the average He-He radial potential. This observation suggests that ZPAD could be fed by effective He-He potentials representative of pure HNDs, all the more since effective potentials obtained for Ar n He 1000 are probably not more relevant to model the fragmentation dynamics of the ionic system than those converged for pure HNDs.

However, the potential part of ZPAD suffers from several limitations. Firstly, it is only suitable for large helium clusters (N 100) for which defining an average rdf and radial potential is relevant. Secondly, as illustrated in Figure 1, the He-He well depth is about 1.3 cm -1 which means that very small clusters, and especially ions detected in mass spectra, could be more stable than in experiments. This latter drawback might not have a significant influence on the results since the cohesion of small ions is governed by dopant-He rather than He-He interactions, then making the smallest ions solid-like. In particular, Bartl et al found magic numbers for Ar + He q (q = 12, 32 and 44) interpreted as icosahedral layers about Ar + [START_REF] Bartl | On the Size and Structure of Helium Snowballs Formed around Charged Atoms and Clusters of Noble Gases[END_REF]. Similarly, Döppner et al found magic numbers for Ag + He q (q = 10, 12, 32 and 44) when investigating silver-doped droplets, although they had not discussed the structure of the corresponding fragments [START_REF] Döppner | Ion induced snowballs as a diagnostic tool to investigate the caging of metal clusters in large helium droplets[END_REF].

Dynamical part of ZPAD

Ar n He 1000 is first equilibrated for at least 10 ps on the potential-energy surface presented in Eq.( 1) where the SAPT2 He-He potential is replaced by the newly converged effective He-He potential. The ionization by electron bombardment is then modeled as a white-light vertical ionization that excites similarly the 3n electronic states of the Ar + n dopant, these states being calculated by diagonalizing a diatomics-in-molecules (DIM) matrix [START_REF] Wadt | The geometry and spectral properties of Ne + 3 , Ar + 3 , Kr + 3 , and Xe + 3[END_REF][START_REF] Kuntz | A DIM model for homogeneous noble gas ionic clusters[END_REF]. The mixed quantum-classical dynamics of Ar + n He 1000 is achieved by coupling a surfacehopping technique for Ar + n , the molecular dynamics with quantum transitions (MDQT) method [START_REF] Tully | Molecular dynamics with electronic transitions[END_REF][START_REF] Hammes-Schiffer | Proton transfer in solution: Molecular dynamics with quantum transitions[END_REF], with a classical dynamics of helium atoms. The dynamics of the ion starts on a randomly selected potential-energy surface j

V j ion (R) = V j + V j He-Ar + n + 1000 a=1 1000 b=1 b>a V eff He-He (R ab ), (4)
where

V j He-Ar + n = n i=1 1000 a=1 q (j) i V He-Ar + (R ia ) + 1 -q (j) i V He-Ar (R ia ) . (5)
In Eq.( 4), V j is the jth DIM adiabatic energy of Ar + n and V He-Ar + is a weighted sum of the analytical functions proposed by Carrington et al. for the X 2 Σ + and A 2 Π electronic states of He-Ar + [START_REF] Carrington | Microwave spectroscopy and interaction potential for the longrange He• • • Ar + ion[END_REF]. The multisurface nature of the system is taken into account by allowing for hops between Ar + n adiabatic surfaces. These hops influence the delocalization of the charge carried by Ar + n and, subsequently, the couplings between the dopant and helium atoms in Eq. [START_REF] Ruchti | Charge transfer and fragmentation of liquid helium clusters that contain one or more neon atoms[END_REF]. Velocity adjustment at a hopping event is achieved along the direction of

g He = ∇(V k He-Ar + n -V j He-Ar + n
) where j and k are the indices of the two surfaces involved in the hop. This nonadiabatic dynamics is performed for 100 ps to enable electronic relaxation of the system and it is followed by a 400-ps adiabatic dynamics.

Results

Fragment abundances

Theoretical mass spectra obtained for Ar + 3 He 1000 and Ar + 5 He 1000 at t = 500 ps are plotted in Figure 2. They are typically composed of a narrow set of peaks at low masses (m 200 u), a broad distribution of peaks at masses 200 u m 3500 u, and another narrow set of peaks at masses close to the parent ion mass (m 4000 u). The two first peak distributions were recently discussed within the context of the fragmentation dynamics of Ar + 4 He 1000 [START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF]: The lower-mass peaks were attributed to fast ion ejections while higher masses mainly corresponded to ion trappings in droplets composed of hundreds of helium atoms. However, the analysis of Ar + 4 He 1000 trajectories for several cutoff radii and velocity adjustment techniques had not revealed significant amounts of ions with masses close to the parent ion mass. Such trajectories are also rare in Ar + 5 He 1000 calculations but they represent roughly 14% of Ar + 3 He 1000 trajectories. Among these trajectories, ∼ 80% started on the most excited Ar + 3 electronic state (ie, state 9 in our state numbering). This point is worth noting because the highest-energy excited state of Ar + 3 is expected to contribute significantly to the production of Ar + fragments in the gas phase [START_REF] Bonhommeau | Fragmentation dynamics of argon clusters (Ar n , n = 2 to 11) following electron impact ionization: Modeling and comparison with experiment[END_REF] and the trapping of excited Ar + 3 consequently harms the formation of Ar + He q fragments. To ensure that discrepancies between Ar + 3 He 1000 and Ar + 5 He 1000 mass spectra are not artifacts induced by our choice of velocity adjustment vector at hopping events, additional 160 trajectories are performed for these two systems along the direction of g kj = ∇(V k -V j ), namely the gradient of the energy difference between the two Ar + n adiabatic states involved in the hop [START_REF] Blais | Trajectory-surface-hopping study of Na(3p 2 P ) + H 2 → Na(3s 2 S) + H 2 (v ′ , j ′ , θ)[END_REF][START_REF] Blais | The effect of a conical intersection on cross sections for collision-induced dissociation[END_REF][START_REF] Hack | Quantum Mechanical and Quasiclassical Trajectory Surface Hopping Studies of the Electronically Nonadiabatic Predissociation of the à State of NaH 2[END_REF]. As depicted in Figure S1, the same trends arise on the new mass spectra, then confirming the results obtained for g He .

While assigning an experimental peak to the corresponding species requires high-resolution mass spectrometers [START_REF] Bartl | On the Size and Structure of Helium Snowballs Formed around Charged Atoms and Clusters of Noble Gases[END_REF], this information can be easily extracted from ZPAD simulations. The contributions of the many ionic fragments to the total mass spectra of Ar + n He 1000 (n = 3 -5) are depicted with color bars in Figure 3, where distinct colors are assigned to ionic fragments with respect to the size of their Ar + p ionic core (p ≤ n). Note that data obtained for Ar + 4 He 1000 are also reported because such an analysis had not been completely performed in our former publication on this system [START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF]. For a given value of p, Ar + p He q (p ≤ n) fragments are sorted into four groups: bare ions (q = 0), small ions (0 < q ≤ 50), medium-sized ions (50 < q ≤ 500) and large ions (q > 500). This sorting yields 16 families of Ar + p He q fragments whose average abundances are plotted as insets in Figure 3.

The fragmentation of Ar + 3 He 1000 produces Ar + 2 He q in 71% of the trajectories, half of these fragments corresponding to small ions (q ≤ 50). The remaining 29% of the trajectories yield Ar + 3 He q , three fourths of these fragments corresponding to large ions (q > 500). As expected from gas-phase data [START_REF] Bonhommeau | Fragmentation dynamics of argon clusters (Ar n , n = 2 to 11) following electron impact ionization: Modeling and comparison with experiment[END_REF][START_REF] Bonhommeau | Fragmentation of rare-gas clusters ionized by electron impact: New theoretical developments and comparison to experiments[END_REF] where larger argon clusters tend to form larger ionic fragments, the proportion of Ar + 3 He q coming from Ar + 4 He 1000 or Ar + 5 He 1000 fragmentation (∼ 29% and 37%, respectively) increases at the expense of the abundance of Ar + 2 He q (∼ 65% and 40%, respectively) and fragments with larger ionic cores (ie, Ar + 4 He q and Ar + 5 He q ) emerge (∼ 5% and 23%, respectively).

Moreover, for all Ar + n He 1000 parent ions investigated here, Ar + p ionic cores (p ≥ 3) of large helium clusters (q > 500) are rather slightly bound complexes connecting an Ar + 2 ion with Ar atoms (eg, Ar ), in agreement with the equilibrium distance of the gas phase dimer in its ground electronic state [START_REF] Ha | The lowest electronic states of Ne + 2 , Ar + 2 and Kr + 2 : comparison of theory and experiment[END_REF]. As pointed out elsewhere [START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF], the formation of such slightly bound complexes can be attributed to limitations of the effective He-He potentials, that remains too deep to reproduce perfectly liquid helium, to helium snowballs about argon atoms, and to the neglect of superfluidity in ZPAD simulations.

No fragments containing an Ar + ionic core are formed during the fragmentation process of Ar + n He 1000 . This result would intuitively suggest that Ar + He q (q < 1000) produced by single ionization of a HND should only come from smaller dopants (ie, Ar or Ar 2 ) unless the inclusion of spin-orbit couplings drastically modifies theoretical fragment abundances, a point that will be tackled in future works. However, Figure S2 shows that large ArHe q and Ar 2 He q clusters can form during the fragmentation of Ar + n He 1000 and an additional ionization of these species, which is not modeled here, would increase the abundance of small Ar + He q fragments with no requirement for initial HNDs doped with Ar or Ar 2 . Moreover, HNDs are typically composed of 10 5 -10 6 helium atoms in recent experiments using rare gases as dopants [START_REF] Bartl | On the stability of cationic complexes of neon with helium -solving an experimental discrepancy[END_REF][START_REF] Bartl | On the Size and Structure of Helium Snowballs Formed around Charged Atoms and Clusters of Noble Gases[END_REF] and ions produced upon ionization of ArHe N or Ar 2 He N (N = 10 5 -10 6 ) might be too large to appear on experimental mass spectra whose mass range has an upper limit. Thence, the origin of small Ar + He q fragments in experiments is not clear and would need further investigation.

The lack of Ar + He q fragments and the large abundance of bare ions (Ar + 2 and Ar +

3 ) in our simulations together with the limited number of trajectories (160 per system) make impossible any quantitative comparison with experimental magic numbers. However, we can evaluate whether some Ar + p He q fragments (2 ≤ p ≤ 5, 1 ≤ q < 1000) produced frequently (in at least 5 trajectories) correspond to identified magic numbers. For Ar [START_REF] Bartl | On the Size and Structure of Helium Snowballs Formed around Charged Atoms and Clusters of Noble Gases[END_REF]. This disagreement confirms the need for making the statistics on a much broader range of HND and dopant sizes, a larger number of trajectories, and longer times all the more since we cannot exclude that some of the aforementioned Ar + p He q fragments would have only increased the abundance of Ar + 2 and Ar + 3 if ZPAD simulations had been extended beyond the 500-ps time limit.

Fragmentation patterns

The sudden ionization of the embedded neutral argon cluster, whose structure is far from the global minimum structure of the ionic species, makes the dopant dissociate. The average time needed for electronic relaxation of the dopant is depicted in Figure 4a as a function of the initial state number j (see Eq.( 4)). The average DIM energies of initial states j are also displayed for the sake of completeness. For Ar + 5 He 1000 , the electronic relaxation is completed within at most ∼ 2 ps when ZPAD starts on one of the thirteen lower-energy states while it can last up to ∼ 40 ps when the highest-energy state is excited. A similar behavior is observed for Ar + 4 He 1000 , the highest-energy state leading to a relaxation time of ∼ 30 ps. However, this behavior contrasts with that observed for Ar + 3 He 1000 . Although the relaxation time increases with energy up to states 5 -6, it decreases sharply for states 7 and 8 and no data is available for state 9 since all the trajectories starting on this state remained blocked in an excited state after 100 ps, which also explains why these trajectories correspond to trapped ions with masses close to the parent ion mass (see Section 3.1). A qualitative explanation for the faster electronic relaxation from states 7 and 8 can be inferred from Figures 4b andS3 where the nonadiabatic dynamics of Ar + 3 and Ar + 5 excited electronic states are illustrated. For Ar + 5 the decay to the ground state is slower for the higher-energy states, then following the energy ordering of excited states. For Ar + 3 , this behavior is roughly maintained up to state 6. In fact, it appears that the coupling of states 7 and 8 with states 5 and 6 is efficient while the coupling from these states to lower states (states 1 to 4) is weaker, as suggested by the plateaus observed for states 5 and 6 below t = 2 ps in Figure 4b. However, the decay of states 7 and 8 to the lowest-energy states seems to benefit from the kinetic energy gained during the preliminary decay to states 5 and 6, reminding that the probability for surface hops depends on the scalar product between nonadiabatic coupling vectors and atomic velocities. Such a behavior is not observed for larger dopants and should rely on the equilateral geometry of the Ar 3 precursor.

Upon completion of the dopant electronic relaxation, an ion can be ejected from the HND or trapped within the droplet. In this context, fragmentation processes can be sorted into three main families [START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF]: (i) fast ejections of small ions (Ar + p He q with p < n and q ≤ 50) within the first 100 ps of the dynamics, (ii) long-term trappings where a large ion (Ar + p He q with p ≤ n and q > 500) slowly loses helium atoms by evaporation on long time scales (t > 500 ps), and (iii) intermediate processes encompassing all other kinds of processes like delayed ejections (t > 100 ps), trappings in smaller droplets (q ≤ 500), or ejections mediated by cycles of trappings and recombinations. A statistics of trajectories subject to these three types of fragmentation processes is reported in Figure 5 for Ar + n He 1000 (n = 3 and 5). It is clear that trappings are always favored by the low-energy excitations whereas fast ejections are favored by high energies, which confirms results recently obtained for Ar + 4 He 1000 [START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF]. A slight deviation occurs for intermediate processes whose distribution of trajectories is rather uniform in the case of Ar + 4 He 1000 while a peak arises at mid-range energies in calculations devoted to Ar + 3 He 1000 and Ar + 5 He 1000 . However, this feature would require confirmation by increasing the number of trajectories for the three above-mentioned systems.

Point-cloud diagrams in Figures 5c and5d confirm that fast ejections rather produce small ions while large ions especially come from long-term trappings, intermediate processes filling the mass range in between. Nevertheless, a few cases deserve more attention, namely large Ar + 2 He q and small Ar + 4 He q produced by fast ejections (see red circles at m ≈ 400 u and m ≈ 1850 u in Figure 5c and red triangles in Figure 5d). Trajectory inspection reveals that the ionic fragment with mass m ≈ 400 u comes from the fast ejection of a rotating Ar + 2 He 2 fragment with little translational energy that is captured by the burst HND in its path to form a smaller compact droplet. Ar + 2 recombines with the trapped argon atom to form Ar + 3 and eventually produces an Ar + 2 He 80 fragment because of the energy released by the geminate recombination that makes helium atoms evaporate. Masses at m ≈ 1850 u follow a simpler pattern where and Ar + 2 ion, possibly surrounded by a few helium atoms, is ejected and captured by the burst HND to form and Ar + 2 He q cluster with m ≈ 440 u. Finally, small Ar + 4 He q fragments typically come from the swift ejection of an argon atom leaving an Ar + 4 or Ar + 3 • • • Ar behind. Then, the trapped ion can be ejected as an Ar + 4 He q fragment (q 10) due to the recoil energy of the primarily ejected argon atom or the geminate recombination of the slightly bound Ar + 3 and Ar subsystems. Similar fragmentation patterns were also observed for Ar + 4 He 1000 [START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF]. Such examples illustrate the great variety of fragmentation processes at play and the fact that a comprehensive understanding of the primary fragmentation events induced by the dopant ionization is not sufficient to account for final fragment abundances.

Conclusion

Zero-point averaged dynamics (ZPAD) of doped HNDs are based on the construction of an effective He-He interaction potential representative of the neutral droplet. Similarities between effective potentials converged for He 1000 and Ar n He 1000 made us conclude that focusing on the pure HND should be sufficient to get reliable results. This observation is essential because convergence of a single potential may need one month with the procedure followed for Ar n He 1000 , each and every iteration being checked before running the next iteration. Therefore, it opens new avenues toward simulations of large doped HNDs provided that the dopant potential-energy surface is known.

In the case of Ar + n He 1000 (n = 3-5) a number of similarities emerge, from the absence of Ar + He q fragments in mass spectra to the occurrence of fast ion ejections and long-term ion trappings, although some discrepancies may arise when focusing on details, like the uppermost Ar + 3 electronic state that does enable electronic relaxation within the first 100 ps of the dynamics. The comparison of these results with experimental data remains limited by the number of trajectories (∼10-20 by Ar + n electronic state), the short time scale (t max = 0.5 ns) that precludes any completion of helium evaporation, and the fact that experimental spectra are averaged over an unknown distribution of HND and dopant sizes.

In recent mass-spectrometry experiments the largest fragments reported contain up to eight argon atoms [START_REF] Bartl | On the Size and Structure of Helium Snowballs Formed around Charged Atoms and Clusters of Noble Gases[END_REF] which defines a lower limit for the maximum size of the Ar n dopant. From our results and gas-phase data [START_REF] Bonhommeau | Fragmentation dynamics of argon clusters (Ar n , n = 2 to 11) following electron impact ionization: Modeling and comparison with experiment[END_REF][START_REF] Bonhommeau | Fragmentation of rare-gas clusters ionized by electron impact: New theoretical developments and comparison to experiments[END_REF], we can reasonably infer that applying the current ZPAD approach to Ar + n He 1000 (6 ≤ n 10) should not yield different trends: fragmentation patterns should be alike, Ar + He q fragments should not form, and the abundance of fragments with large Ar + p ionic cores (p ≥ 3) should increase at the expense of Ar + 2 He q fragments. Comparisons with experiments may however be harmed by the neglect of superfluidity in ZPAD. Indeed, this property is expected to be relevant at two stages of the experiments: (i) during the capture and coagulation of argon atoms by the pure HND, a physical process which is not modeled here, and (ii) at long times ( 1ns) when cooled doped HNDs may subsist and undergo recombinations of embedded argon atoms. For instance, a molecular beam with speed 500 m s -1 crossing an ionization chamber of 10 cm long is exposed to high-energy electrons for 0.2 ms. Putting aside the possibility for multiple ionization of the HND, this time is much longer than the 0.5 ns of current ZPAD simulations. In other words, even if superfluidity is not taken into account, argon atoms embedded in small HNDs (100 N < 1000) modeled with ZPAD might have enough time to recombine and release small ions in the gas phase if simulations are extended to experimental time scales. However, this does not involve that the subsequent distribution of ions would be exactly the same as that obtained in experiments using the same droplet sizes.

Beside the question of superfluidity, ZPAD can be refined by constructing more fluid-like effective He-He potentials for pure HNDs closer to experimental sizes (N = 10 3 -10 6 ), by performing simulations beyond the nanosecond time scale to model the slow helium evaporation, and by including spin-orbit couplings in the DIM matrix representative of Ar + n dopants. The latter improvement would make the matrix complex-valued [START_REF] Cohen | Ground and excited states of Ne 2 and Ne + 2 . I. Potential curves with and without spin-orbit coupling[END_REF][START_REF] Amarouche | Structure and stability of Xe + n clusters[END_REF], double the number of electronic states, and require the rewriting of MDQT equations [START_REF] Bonhommeau | Dynamique de fragmentation d'agrégats de gaz rares monochargés en phase gazeuse et à l'intérieur de nanogouttes d'hélium[END_REF] since nonadiabatic coupling vectors will be complex-valued. We could also construct effective He-Ar potentials in complement to He-He ones, an improvement which has not be done here for the sake of comparison with previous calculations performed on Ar + 4 He 1000 . All these refinements are currently considered to improve ZPAD and make it more versatile.

Supplementary material

Theoretical mass spectra of Ar + 3 He 1000 and Ar + 5 He 1000 at t = 500 ps averaged over 160 trajectories when using g kj as velocity adjustment vector (Figure S1). Average number of neutral fragments per trajectory at t = 500 ps for Ar + n He 1000 (n = 3 -5) parent ions (Figure S2). Nonadiabatic dynamics of Ar + 5 excited electronic states (Figure S3).

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request. p Heq fragments into families which depend on the dopant size p (p = 2: black ; p = 3: red ; p = 4: blue ; p = 5: green) and on the number of surrounding helium atoms q (bare ions: q = 0, label B ; small ions: 0 < q ≤ 50, label S ; medium-sized ions: 50 < q ≤ 500, label M ; large ions: 500 < q ≤ 1000, label L).

3 (

 3 + n (n = 3, 5) electronic state number (a and b) and corresponding point-cloud diagrams showing the mass of ionic fragments which contain Ar + 2 (circles), Ar + squares), Ar + 4 (triangles), and Ar + 5 (diamonds) (c and d). In plot b, fractions of trajectories are computed by gathering states 1 to 14 by pairs (ie, (1,2), • • • ,[START_REF] Bonhommeau | Modeling the fragmentation dynamics of ionic clusters inside helium nanodroplets: The case of He 100 Ne + 4[END_REF][START_REF] Bonhommeau | Fragmentation of ionized doped helium nanodroplets: Theoretical evidence for a dopant ejection mechanism[END_REF]) because of noise on the results related to the limited number of trajectories by electronic state. In point-cloud diagrams, the color of symbols refers to the dynamical process involved to produce the fragments (red: fast ejection; blue: long-term trapping; and black: intermediate process) and trajectories where the Ar + n ground electronic state is reached before t = 100 ps are only considered. . . . . . . . . . . .[START_REF] Janzen | An accurate potential energy curve for helium based on ab initio calculations[END_REF]

Figure 1 :

 1 Figure 1: (color online) Analytical curves of effective He-He interaction potentials obtained for He 1000 , Ar 3 He 1000 , Ar 4 He 1000 [15], and Ar 5 He 1000 with a wave function cutoff radius of 1.6 Å(a). Potential energies (black circles) derived from the iterative potential part of ZPAD and corresponding fitted analytical functions (red lines) are plotted for He 1000 (b), Ar 3 He 1000 (c), and Ar 5 He 1000 (d). Standard deviations (SD) are also indicated for the regions of positive energies (R 4 Å, SD denoted SD + ) and negative energies (R 4 Å, SD denoted SD -).

Figure 3 :

 3 Figure 3: (color online) Contributions of Ar + p Heq (2 ≤ p ≤ 5, 0 ≤ q ≤ 1000) fragments to theoretical mass spectra of Ar + 3 He 1000 (a), Ar + 4 He 1000 (b) and Ar + 5 He 1000 (c) at t = 500 ps. Histograms plotted in insets are built by sorting Ar +p Heq fragments into families which depend on the dopant size p (p = 2: black ; p = 3: red ; p = 4: blue ; p = 5: green) and on the number of surrounding helium atoms q (bare ions: q = 0, label B ; small ions: 0 < q ≤ 50, label S ; medium-sized ions: 50 < q ≤ 500, label M ; large ions: 500 < q ≤ 1000, label L). Figurebis partly reproduced from Ref.[START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF], with the permission of AIP Publishing.

  Figure 3: (color online) Contributions of Ar + p Heq (2 ≤ p ≤ 5, 0 ≤ q ≤ 1000) fragments to theoretical mass spectra of Ar + 3 He 1000 (a), Ar + 4 He 1000 (b) and Ar + 5 He 1000 (c) at t = 500 ps. Histograms plotted in insets are built by sorting Ar +p Heq fragments into families which depend on the dopant size p (p = 2: black ; p = 3: red ; p = 4: blue ; p = 5: green) and on the number of surrounding helium atoms q (bare ions: q = 0, label B ; small ions: 0 < q ≤ 50, label S ; medium-sized ions: 50 < q ≤ 500, label M ; large ions: 500 < q ≤ 1000, label L). Figurebis partly reproduced from Ref.[START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF], with the permission of AIP Publishing.

20

 20

Figure 4 :

 4 Figure 4: (color online) Average electronic relaxation time t r,j of Ar + 3 (black solid line), Ar + 4 (black solid line) [15], Ar + 5 (red solid line) as a function of the initial state number (a) and nonadiabatic dynamics of Ar + 3 excited electronic states (b). In Figure a, the average initial DIM energies V j corresponding to the 3n states of the Ar + n dopants are also reported (black squares: Ar + 3 ; blue squares: Ar + 4 ; red squares: Ar + 5 ). In Figure b, state numbers are averaged over all the trajectories. Figure a is partly reproduced from Ref. [15], with the permission of AIP Publishing.

  He 6 and Ar + 3 He 4 when n = 5. The closest experimental magic numbers are Ar + 2 He 7 , Ar + 2 He 12 , and Ar + 3 He 8

	+ n He 1000 parent ions, common 2 He 10 when n = 3, Ar + 2 He and Ar + fragments are Ar + 2 He and Ar + 2 He 2 when n = 4, Ar + 2 He, Ar + 2

Table 1 :

 1 Parameters of the effective He-He interaction potentials converged for He 1000 , Ar 3 He 1000 , and Ar 5 He 1000 with a wave function cutoff radius Rcut = 1.6 Å. He 1000 Ar 5 He 1000 Fraction of trajectories subject to fast ejection of small ionic fragments (red lines), long-term trapping of Ar + p (2 ≤ p ≤ 5, blue lines), or intermediate processes (black lines) as a function of the initial Ar

	Parameters He 1000 Ar 3 R e ( Å) 4.40136 4.40034	4.39113
	A (cm -1 )	42864.1	1944.72	1942.29
	α 1	-0.922778 -7.98855	-8.01035
	α 2	13.021	17.2366	17.2557
	c 6 (cm -1 )	-2.41494	-2.34751	-2.35672
	c 8 (cm -1 )	7.72659	7.33127	7.32458
	c 10 (cm -1 ) -15.5743	-14.8025	-14.8356
	c 12 (cm -1 ) 8.99999 a ( Å-1 ) 41.7872	8.53158 18.446	8.56887 18.425
	b ( Å)	3.56143	3.6181	3.6136

Acknowledgments

Prof. Marius Lewerenz is gratefully acknowledged for providing me with the set of routines devoted to the convergence of the effective He-He potential-energy curves. Dr Nadine Halberstadt is warmly acknowledged for fruitful discussions and strong interest in this work.

List of Figures

1

(color online) Analytical curves of effective He-He interaction potentials obtained for He 1000 , Ar 3 He 1000 , Ar 4 He 1000 [START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF], and Ar 5 He 1000 with a wave function cutoff radius of 1.6 Å(a). Potential energies (black circles) derived from the iterative potential part of ZPAD and corresponding fitted analytical functions (red lines) are plotted for He 1000 (b), Ar (color online) Contributions of Ar + p He q (2 ≤ p ≤ 5, 0 ≤ q ≤ 1000) fragments to theoretical mass spectra of Ar + 3 He 1000 (a), Ar + 4 He 1000 (b) and Ar + 5 He 1000 (c) at t = 500 ps. Histograms plotted in insets are built by sorting Ar + p He q fragments into families which depend on the dopant size p (p = 2: black ; p = 3: red ; p = 4: blue ; p = 5: green) and on the number of surrounding helium atoms q (bare ions: q = 0, label B ; small ions: 0 < q ≤ 50, label S ; medium-sized ions: 50 < q ≤ 500, label M ; large ions: 500 < q ≤ 1000, label L). Figure b is partly reproduced from Ref. [START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF], with the permission of AIP Publishing. . . . . . . . . 20 4 (color online) Average electronic relaxation time t r,j of Ar + 3 (black solid line), Ar + 4 (black solid line) [START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF], Ar + 5 (red solid line) as a function of the initial state number (a) and nonadiabatic dynamics of Ar + 3 excited electronic states (b). In Figure a, the average initial DIM energies V j corresponding to the 3n states of the Ar + n dopants are also reported (black squares: Ar + 3 ; blue squares: Ar + 4 ; red squares: Ar + 5 ). In Figure b, state numbers are averaged over all the trajectories. Figure a is partly reproduced from Ref. [START_REF] Halberstadt | Fragmentation dynamics of Ar 4 He 1000 upon electron impact ionization: Competition between ion ejection and trapping[END_REF], with the permission of AIP Publishing. . . . . . 21