Guillaume Carlier
email: carlier@ceremade.dauphine.fr

Lina Mallozzi
email: mallozzi@unina.it

Softening bilevel problems via two-scale Gibbs measures

Keywords: bilevel optimization, Stackelberg games, Gibbs measures, Γ-convergence, Laplace method. MS Classification: 49K35, 90C26, 60B05

We introduce a new, and elementary, approximation method for bilevel optimization problems motivated by Stackelberg leader-follower games. Our technique is based on the notion of two-scale Gibbs measures. The first scale corresponds to the cost function of the follower and the second scale to that of the leader. We explain how to choose the weights corresponding to these two scales under very general assumptions and establish rigorous Γ-convergence results. An advantage of our method is that it is applicable both to optimistic and to pessimistic bilevel problems.

Introduction

Bilevel optimization is defined as mathematical programming where an optimization problem contains another one as a constraint. It consists of decision making problems with hierarchical leader-follower structure and has a natural interpretation in game theory. Bilevel problems have a long history that dates back to von Stackelberg [START_REF] Stackelberg | Marktform und Gleichgewicht[END_REF] and have been intensively studied from a theoretical point of view as well as in applications to various domains including traffic planning, security, supply chain management, principal-agent models, production planning, market deregulation, optimal taxation, parameter estimation, see the recent surveys [START_REF] Colson | An overview of bilevel optimization[END_REF][START_REF] Dempe | Bilevel optimization: theory, algorithms and applications[END_REF] and the references therein. In the context of Cournot duopolies, von Stackelberg investigated the leaderfollower model where the leader firm maximizes profit under the constraint that the follower firm reacts with an optimal choice of the quantity that is supposed to be unique. Later on, Leitmann [START_REF] Leitmann | On generalized Stackelberg strategies[END_REF] discussed the case where the optimal solutions for the follower's problem form a set that the leader has to take into account in order to solve her own optimization problem.

In case the follower's program has several solutions, we see that there is some ambiguity even in the definition of the leader's program. In the literature, several concepts have been considered. The optimistic (or strong) Stackelberg solution assumes a cooperative like behavior between the agents: the leader expects the follower to choose solutions leading to the best outcome for her. On the contrary, the pessimistic (or weak) Stackelberg solution assumes that the follower always breaks ties by choosing the worst actions for the leader which corresponds to a security strategy for her, see [START_REF] Başar | Dynamic noncooperative game theory[END_REF][START_REF] Breton | Sequential Stackelberg equilibria in two-person games[END_REF]. Some intermediate cases can also be considered. In [START_REF] Aboussoror | Strong-weak Stackelberg problems in finite-dimensional spaces[END_REF], a cooperation degree is assumed leading to the optimization of a convex combination of the best and the worst payoff value for the leader, while, in [START_REF] Mallozzi | Oligopolistic markets with leadership and demand functions possibly discontinuous[END_REF], a probabilistic information about the follower's behavior is assumed resulting in the optimization of an average payoff.

Both optimistic and pessimistic bilevel programs are challenging and often difficult to solve in practice. In the present paper, we present a new and quite simple (unconstrained) approximation scheme for such problems based on the notion of two-scale Gibbs measures. Our method is directly inspired by the classical Laplace method: Gibbs probability measures which have a density proportional to e -λu with respect to a reference measure with full support concentrate on the set where u is minimal as λ → ∞. We refer to Hwang [START_REF] Hwang | Laplace's method revisited: weak convergence of probability measures[END_REF] for a fine study of the method and precise statements in smooth finite-dimensional situations. In the context of bilevel optimization, we have to take into account the objective of both the leader and the follower and a single parameter λ is not enough capture the nested structure of the leaderfollower problem. This is why we introduce two-scale Gibbs measures where the first scale (with weight λ) takes into account the follower's objective and the secondary one (with a smaller weight to be chosen properly) takes into account the leader's objective. We investigate in details convergence issues (both in the pointwise and Γ-convergence sense) and the choice of the secondary scale only in terms of the reference measure and the modulus of continuity of the leader and follower objective functions. Our method is flexible enough to cope with both the optimistic and the pessimistic case.

The paper is organized as follows: the setting of our analysis is introduced in section 2. In section 3, we recall some basic facts on Gibbs measures and analyze the convergence of two-scale Gibbs measures in a simple case. In section 4, we give a construction for the weights which guarantees convergence of the corresponding two-scale Gibbs measures under general assumptions. We then establish Γ-convergence results, first for the optimistic case in section 5 and then in the pessimistic case in section 6. Finally, section 7 concludes with some remarks and examples.

Setting

Throughout this paper, X (strategy set for the leader) and Y (strategy set for the follower) will be two compact metric spaces. We will also assume that the cost functions of both the leader and the follower are continuous, (2.1)

ϕ ∈ C(X × Y ), ψ ∈ C(X × Y ) will
Under our assumptions, it is obvious that (2.1) admits at least one solution but finding such solutions in practice is a challenging task due to the constraint y ∈ argmin ψ(x, .). Of course, one can rewrite (2.1) as a minimization problem with respect to x only:

inf x∈X ϕ * (x) (2.2)
where ϕ * (x) := min ϕ(x, y) : y ∈ argmin ψ(x, .) .

(2.3) Problem (2.1) is usually refered to as the optimistic problem since it assumes that in case the follower has several optimal strategies she will break ties by chosing one which is optimal for the leader. The pessimistic problem consists, on the contrary, in assuming that the follower actually breaks ties by chosing strategies which are the worst for the leader. The corresponding bilevel pessimistic program therefore consists in

inf x∈X ϕ * (x) (2.4)
where ϕ * (x) := max ϕ(x, y) : y ∈ argmin ψ(x, .) .

(2.5)

In general, the pessimistic value ϕ * is not lower-semicontinuous (lsc) so (2.4) does not necessarily admit solutions which makes the pessimistic problem more involved than the optimistic one and requires some suitable relaxation of ϕ * . Our goal is to approximate the bilevel problem (2.1) by a family of unconstrained ones (we will also address the approximation of the pessimistic bilevel problems (2.4) in section 6). We shall indeed prove that the somehow rough function ϕ * can be approximated by a family of more regular ones defined by an integral depending on a parameter. By approximated we mean both in the pointwise sense and in the sense of Γ-convergence we recall below (see [START_REF] Braides | Γ-convergence for beginners[END_REF] or [START_REF] Dal | An introduction to Γ-convergence, volume 8 of Progress in Nonlinear Differential Equations and their Applications[END_REF] for an overview of Γ-convergence and its applications): Definition 2.1. Let F : X → R and let for every λ > 0, F λ : X → R, then F λ is said to Γ-converge to F as λ → +∞ if the following two conditions are satisfied:

• for every x ∈ X and every family (x λ ) λ>0 converging to x as λ → +∞, one has the Γ-liminf inequality:

F (x) ≤ lim inf λ→+∞ F λ (x λ ),
• for every x ∈ X, there exists a (so-called recovery) family (x λ ) λ>0 converging to x as λ → +∞ such that the following Γ-limsup inequality holds

F (x) ≥ lim sup λ→+∞ F λ (x λ ).
Our approximation is a variant of the celebrated Laplace method, which as far as we know, has not been investigated in the bilevel framework. First of all, we give ourselves a Borel probability measure ν on Y (which we will denote ν ∈ P(Y )) and assume that it has full support spt(ν) = Y.

(2.6)

For r > 0 and y ∈ Y , we denote by B r (y) the closed ball of radius r centered at y and set for every r ≥ 0,

α ν (r) := inf y∈Y ν(B r (y)). (2.7)
Note that the full support assumption (2.6) and the compactness of Y ensure that α ν (r) > 0 for every r > 0. Given x ∈ X, λ > 0 and δ > 0, we consider the probability measure on Y µ λ,δ (dy|x) := Z λ,δ (x)e -λ(ψ(x,y)+δϕ(x,y)) ν(dy) (2.8) where Z λ,δ (x) is the normalizing constant which makes µ λ,δ (.|x) a probability measure i.e. Z λ,δ (x) := 1

Y e -λ(ψ(x,y)+δϕ(x,y)) ν(dy)

.

Our main result is that one can choose the secondary scale δ = δ λ with lim λ→+∞ δ λ = 0, lim

λ→+∞ λδ λ = +∞ (2.9)
in such a way that the family

x → ϕ λ (x) := Y ϕ(x, y)µ λ,δ λ (dy|x)
Γ-converges and converges pointwise to ϕ * as λ → ∞. Our construction of δ λ only depends on the function α ν defined in (2.8) and a modulus of continuity of ϕ and ψ, it will be detailed in section 4.

3 On Gibbs measures

On standard Gibbs measures

In this section, we temporarily leave the approximation of Stackelberg problems and focus on the asymptotic behavior of Gibbs measures. Given ν ∈ P(Y ) with full support as in (2.6), λ > 0 and w ∈ C(Y ), we define the Gibbs measure ν λ,w := Z λ,w e -λw ν, Z λ,w := 1 Y e -λw(y) ν(dy) .

Of course, ν λ,w is unchanged if one adds a constant to w so there is no loss of generality in normalizing w in some way, and the most natural way is to assume that its minimum is 0. The following elementary result will be used intensively in the sequel: Lemma 3.1. Let w ∈ C(Y ) be such that min Y w = 0 and let ν λ,w be defined by (3.1) then for every ε > 0 there holds

ν λ,w ({w ≥ ε}) ≤ e -λε 2 ν({w ≤ ε 2 }) . (3.2)
Proof. Let us write ν λ,w ({w ≥ ε}) = {w≥ε} e -λw(y) ν(dy) {w≥ε} e -λw(y) ν(dy) + {w<ε} e -λw(y) ν(dy)

so that 1 ν λ,w ({w ≥ ε}) = 1 + {w<ε} e -λw(y) ν(dy)
{w≥ε} e -λw(y) ν(dy) then observe that {w≥ε} e -λw(y) ν(dy) ≤ e -λε and {w<ε} e -λw(y) ν(dy) ≥

{w≤ ε 2 } e -λw(y) ν(dy) ≥ e -λε 2 ν({w ≤ ε 2 })
so putting eveything together yields (3.2).

Since, for every λ, ν λ,w is a probabilty measure and Y is compact, there is a sequence λ n → ∞ and ν ∈ P(Y ) such that, ν λn,w weakly1 star converges to ν as n → ∞. Since the set {w > ε} is open, it follows from Portmanteau's theorem (see [START_REF] Billingsley | Convergence of probability measures[END_REF]) that

ν({w > ε}) ≤ lim inf n ν λn,w ({w > ε}).
Hence, for every ε > 0 thanks to (3.2) and the fact that ν({w ≤ ε 2 }) > 0 (because ν has full support and the minimum of w is 0), we get ν({w > ε}) = 0 letting ε → 0 + and using the fact that w is continuous, we conclude that ν concentrates on the set where w = 0. We thus recover the well-known fact that Gibbs measures concentrate on the set where the potential is minimal: Corollary 3.2. Let w ∈ C(Y ) and ν λ,w be defined by (3.1), then any weak star cluster point of ν λ,w as λ → ∞ has its support in argmin Y w.

Convergence of two-scale Gibbs measures in a simple case

To understand how to approximate bilevel problems with Gibbs measures, we first have to understand the following question. Given two functions u and v in C(Y ), we want to find a weight δ λ such that δ λ → 0, λδ λ → +∞ as λ → +∞ in such a way that the two-scale Gibbs measure ν λ,u+δ λ v concentrates when λ → ∞ on the double argmin set:

argmin argmin u v := {y ∈ argmin u : v(y) = min{v(z) : z ∈ argmin u}}. (3.
3) We will give a general constructive answer in paragraph 4 (depending on the modulus of continuity of u and v and the function α ν in (2.7)). Yet, for now, we prefer to focus on a rather simple case where the explicit choice δ λ := 1 √ λ works (as well as many other simple ones, see Remark 3.4 below). This simple case corresponds to the extra assumptions that both u and v are Hölder continuous and the function α ν is bounded from below by a power function. Denoting by dist the distance on Y and diam(Y ) its diameter, these assumptions mean that there exist C > 0, α ∈ (0, 1] and

β ∈ R * + such that max(|u(y) -u(z)|, |v(y) -v(z)|) ≤ C dist(y, z) α , ∀(y, z) ∈ Y 2 , (3.4)
and

α ν (r) ≥ r β C , ∀r ∈ [0, diam(Y )]. (3.5)
To shorten notations, let us set

ν λ := Z λ e -λu- √ λv ν, Z λ := 1 Y e -λu- √ λv ν (3.6)
which corresponds to the the two-scale Gibbs measure ν λ,u+ 1

ε > 0, for λ large enough {u > ε} ⊂ {w λ > ε 2 } ⊂ {w λ > min Y w λ + ε 2 } and {u ≤ ε 8 } ⊂ {w λ ≤ min Y w λ + ε 4 ). It then follows from the inequality (3.2) of Lemma 3.1 that ν λ ({u > ε}) ≤ ν λ ({w λ > min Y w λ + ε 2 }) ≤ e -λε 4 ν({w λ ≤ min Y w λ + ε 4 }) ≤ e -λε 4 ν({u ≤ ε 8 })
so that, again thanks to Portmanteau's Theorem, ν({u > ε}) = 0 and ν is supported by argmin u = {u = 0}. In particular, with (3.7), v ≥ 0 on spt(ν).

To conclude, we thus have to show that for every ε > 0, ν({v > ε}) = 0. Since u ≥ 0 and min Y w λ ≤ 0, we have {v > ε} ⊂ {w λ > min Y w λ + ε √ λ } so using Lemma 3.1 again we get

ν λ ({v > ε}) ≤ e - √ λε 2 ν({w λ ≤ min Y w λ + ε 2 √ λ }) . (3.8)
Let y λ be a point where w λ achieves its minimum, then it follows from (3.4) that for λ ≥ 1,

w λ ≤ min Y w λ + ε 2 √
λ in the ball of center y λ and radius ( ε

4C √ λ ) 1 α , hence ν({w λ ≤ min Y w λ + ε 2 √ λ }) ≥ α ν ε 4C √ λ 1 α ≥ 1 C ε 4C √ λ β α
where the last inequality follows from (3.5). With (3.8), this yields

ν λ ({v > ε}) ≤ ε -β α 4 β α C 1+ β α √ λ β α e - √ λε 2
since for every ε > 0 the right hand side tends to 0 as λ → ∞, Portmanteau's Theorem again allows us to conclude that ν({v > ε}) = 0.

Remark 3.4. Our choice δ λ = 1 √ λ above is just for illustrative purpose and by no means the only possible one or optimal in any sense. It is indeed straightforward to check, with the same proof as above, that under the assumptions of Proposition 3.3, any choice of δ λ such that

lim λ→+∞ δ λ = 0, lim λ→+∞ λδ λ = +∞, lim λ→+∞ log(δ λ ) λδ λ = 0 (3.9)
guarantees that the corresponding two-scale measures ν λ,u+δ λ v tend to concentrate on the double argmin set (3.3) as λ → +∞. In particular any power choice for δ λ i.e. δ λ = λ -γ with γ ∈ (0, 1) (or much larger weights such as

δ λ = 1 log(λ) , δ λ = 1 log log(λ)
) ensures convergence to the double argmin set. Note that smaller weights such as δ λ = log(λ) λ violate condition (3.9). Assumptions (3.4) and (3.5) are essential if one wishes to use power like weights. In section 7, we will consider examples where α ν is much smaller than a power function. In such cases, the choice δ λ = 1

Chosing the weights under general assumptions

Now we consider the general case where u and v are continuous and ν has full support. We wish to find secondary weights δ λ satisfying (2.9) in such a way that defining the two-scale Gibbs measures,

ν λ := Z λ e -λ(u+δ λ v) ν, Z λ := 1 Y e -λ(u+δ λ v) ν (4.1)
every weak star cluster point of ν λ as λ → ∞ is supported by the double argmin set (3.3). First of all, we fix a common modulus of continuity ω for both u and v in the sense that:

|u(y) -u(z)| + |v(y) -v(z)| ≤ ω(dist(y, z)), ∀(y, z) ∈ Y 2 (4.2)
without loss of generality, we may assume that ω is strictly increasing and concave (hence continuous) on the whole of R + and of course ω(t) → 0 + as t → 0. We the denote by Γ, R * + → R * + the inverse of ω, Γ := ω -1 , by construction we thus have

z ∈ B Γ(ε) (y) ⇒ |u(y) -u(z)| + |v(y) -v(z)| ≤ ε. (4.3)
Recalling that α ν is defined by (2.7), we define for every t > 0

θ(t) := lim δ→0 + log(α ν (Γ(t -δ))). (4.4)
Note then that θ is nondecreasing, lsc (this is why we define it as a left limit) and θ(t) → -∞ as t → 0 + . Lemma 4.1. Let θ be given by (4.4), for every λ > 0 define

t λ := inf{t > 0 : θ(t) + λt 2 ≥ 0} (4.5)
and

δ λ := 2 √ t λ (4.6)
then λ → δ λ is decreasing, satisfies (2.9) and

lim λ→∞ θ δ 2 λ 2 + λδ 2 λ 2 = +∞. (4.7)
Proof. First of all, it is clear that t λ is well defined since t → θ(t) + λt 2 is an increasing function which tends to -∞ as t → 0 + and +∞ as t → +∞. Moreover since we chose θ lsc we have

θ(t λ ) + λt λ 2 ≤ 0 (4.8)
and since 2t λ > t λ we also have

θ(2t λ ) + λt λ ≥ 0. (4.9)
It is obvious by monotonicity that λ → t λ is nonincreasing and dividing (4.8) by λ and letting λ → +∞ one easily sees that t λ converges to 0 as λ → +∞, this gives θ(2t λ ) → -∞ as λ → +∞. Hence, thanks to (4.9), we have

lim λ→+∞ λt λ = +∞. ( 4.10)
The fact that δ λ decreases to 0 and λδ λ tends to +∞ immediately follow. Finally, by definition of δ λ and using (4.9), we have

θ δ 2 λ 2 + λδ 2 λ 2 = θ(2t λ ) + 2λt λ ≥ λt λ
so (4.7) directly follows from (4.10).

Proposition 4.2. Let δ λ be defined as in Lemma 4.1 and ν λ be defined by (4.1). Then any weak star cluster point of ν λ as λ → ∞ has its support included in the double argmin set (3.3).

Proof. Again we can normalize u and v so that (3.7) holds and set w λ := u + δ λ v, since δ λ → 0, one can proceed as in the proof of Proposition 3.3 to show that for every ε > 0, ν λ ({u > ε}) tends to 0 as λ → +∞ and thus deduce that any weak star cluster point of ν λ as λ → ∞ has its support included in {u = 0} = argmin u. To conclude that such weak star cluster points are in fact supported by the double argmin set (3.3), it is enough to show that for every ε > 0, ν λ ({v > ε}) tends to 0 as λ → +∞. To prove this, we observe that since u ≥ 0, {v > ε} ⊂ {w λ ≥ δ λ ε} and since min Y w λ ≤ 0 (because of (3.7)), we have {v > ε} ⊂ {w λ ≥ min Y w λ + δ λ ε}.

Since ν λ = ν λ,w λ , our basic inequality (3.2) in Lemma 3.1 gives:

ν λ ({v > ε}) ≤ ν λ ({w λ ≥ min Y w λ + δ λ ε}) ≤ e -λδ λ ε 2 ν({w λ ≤ min Y w λ + δ λ ε 2 })
.

(4.11) Choose now λ large enough so that δ λ ≤ 1, doing so ω is a modulus of continuity of w λ . Hence if y λ is a minimum point of w λ , the ball

B Γ( δ λ ε 2 ) (y λ ) is contained in {w λ ≤ min Y w λ + δ λ ε
2 } hence by definition of Γ, α ν and θ we get

ν({w λ ≥ min Y w λ + δ λ ε 2 }) ≥ α ν Γ( δ λ ε 2 ) ≥ e θ( δ λ ε 2 )
replacing in (4.11) gives

ν λ ({v > ε}) ≤ e -λδ λ ε 2 -θ( δ λ ε 2 ) (4.12)
for λ large enough, δ λ ≤ ε so that, using the monotonicity of θ:

λδ λ ε 2 + θ( δ λ ε 2 ) ≥ λδ 2 λ 2 + θ δ 2 λ 2
and since the right hand side tends to +∞ as λ → +∞ thanks to (4.7), we obtain that for every ε > 0

lim λ→+∞ ν λ ({v > ε}) = lim λ→+∞ e -λδ λ ε 2 -θ( δ λ ε 2 ) = 0,
which gives the desired conclusion.

Remark 4.3. In fact the proof of Proposition 4.2 gives a more precise information than the fact that weak cluster points of ν λ concentrate on the double argmin set (3.3). Thanks to (4.12), we in fact have

ν λ ({v > min argmin u v + ε}) ≤ e -λδ λ ε 2 -θ( δ λ ε 2 ) (4.13)
and by construction for every ε > 0:

λδ λ ε 2 + θ( δ λ ε 2 ) → +∞ as λ → ∞. (4.14)
Now observe that the quantity in (4.14) only depends on u and v through the function θ i.e. through the modulus ω. So the speed at which ν λ ({v > min argmin u v +ε}) converges to 0 is uniform with respect to v and u admitting ω as modulus of continuity. This uniform behavior will be useful in section 6 devoted to the pessimistic case.

Remark 4.4. Let us also emphasize that what really matters for convergence is the fact that the left-hand side of (4.13) tends to 0 as λ → +∞ for every ε > 0 and not really the precise construction of δ λ . Our construction based on Lemma 4.1 is just a cooking recipe which guarantees this property, and we are not claiming that it is optimal in any sense. In fact, the choice δ λ = 2 √ t λ in Lemma 4.1 is a bit arbitrary, taking δ λ = t γ λ with γ ∈ (0, 1) or δ λ = -t λ log(t λ ) would have worked just as well.

Γ-convergence to the bilevel problem

We are now ready to go back to the bilevel problem (2.2) of paragraph 2. Since both ϕ and ψ are continuous on X × Y which is compact, we can find ω, a (strictly increasing and concave) modulus of continuity of ϕ(x, .), ψ(x, .) which is uniform with respect to x ∈ X i.e.:

|ϕ(x, y) -ϕ(x, z)| + |ψ(x, y) -ψ(x, z)| ≤ ω(dist(y, z)), ∀(x, y, z) ∈ X × Y 2
(5.1) we then define Γ := ω -1 as in the previous paragraph, θ as in (4.4) and then the weights δ λ by (4.5)-(4.6) as in Lemma 4.1. Recalling (2.8), let us set for every x ∈ X, µ λ (dy|x) := µ λ,δ λ (dy|x)

i.e. µ λ (dy|x) := Z λ (x)e -λ(ψ(x,y)+δ λ ϕ(x,y)) ν(dy)

with Z λ (x) := Z λ,δ λ (x) = 1
Y e -λ(ψ(x,y)+δ λ ϕ(x,y)) ν(dy)

.

Finally, let us define

ϕ λ (x) := Y ϕ(x, y) µ λ (dy|x), ∀x ∈ X. (5.2)
Our main result is then the following Theorem 5.1. Let ϕ * be defined by (2.3) and ϕ λ be as above. Then ϕ λ Γ-converges and converges pointwise to ϕ * as λ → +∞.

Proof. The pointwise convergence follows directly from Proposition 4.2. The Γ-limsup inequality straighforwardly follows (taking the constant x λ = x as recovery sequence). It remains to show the Γ-liminf inequality. Let then x ∈ X, x λ → x as λ → +∞, since ϕ(x λ , .) converges uniformly to ϕ(x, .), we first have lim inf Let us set f n := ψ(x λn , .) + δ λn ϕ(x λn , .) to shorten notations. Thanks to (3.2) again, we have

λ→∞ ϕ λ (x λ ) = lim inf λ→∞ Y ϕ(x, y) µ λ (dy|x λ ) (5.
µ n { ψ(x, .) > min Y ψ(x, .) + ε} ≤ e -λnε 4 ν({f n ≤ min Y f n + ε 4 })
and since f n converges uniformly to ψ(x, .), for large enough n, {f n ≤ min Y f n + ε 4 } contains a neighbourhood of a minimizer of ψ(x, .), we obtain

lim n µ n { ψ(x, .) > min Y ψ(x, .) + ε} = 0.
Portmanteau's theorem thus implies that µ { ψ(x, .) > min Y ψ(x, .) + ε} = 0 which ends the proof.

As a consequence of the previous Γ-convergence, we immediately have: and if x λ is a minimizer of ϕ λ and x is a cluster point of x λ as λ → +∞ then x is a minimizer of ϕ * .

The pessimistic case

Let us now address the pessimistic case and the approximation via Gibbsmeasures of the pessimistic value function ϕ * defined in (2.5). Since ϕ * is not necessarily lsc, we have to relax it and consider its lsc envelope (that is the largest lsc function lying below ϕ * on X) which we denote by ϕ * and is given by ϕ * (x) := lim inf

x →x ϕ * (x ) = lim r→0 + inf Br(x) ϕ * . (6.1)
We construct δ λ exactly as in paragraph 5, and for every x ∈ X, we consider the Gibbs measure:

µ + λ (dy|x) := Z + λ (
x)e -λ(ψ(x,y)-δ λ ϕ(x,y)) ν(dy) with Z + λ (x) := 1 Y e -λ(ψ(x,y)-δ λ ϕ(x,y)) ν(dy)

.

We consider then the approximations

ϕ + λ (x) := Y ϕ(x, y) µ + λ (dy|x), ∀x ∈ X. (6.2)
Our convergence result concerning the pessimistic value is the following: Theorem 6.1. Let ϕ * be defined by (2.5) and ϕ + λ be as above. Then, as λ → +∞, ϕ + λ converges pointwise to ϕ * and Γ-converges to its lsc envelope, ϕ * , defined in (6.1).

Proof. The pointwise convergence of ϕ + λ to ϕ * follows directly from Proposition 4.2.

Let us then introduce the Γ-liminf and Γ-limsup:

Γ -lim inf λ ϕ + λ (x) := inf lim inf λ ϕ + λ (x λ ) : x λ → x and Γ -lim sup λ ϕ + λ (x) := inf lim sup λ ϕ + λ (x λ ) : x λ → x .
Since ϕ + λ converges pointwise to ϕ * we obviously have Γ -lim sup

λ ϕ + λ ≤ ϕ * but since Γ -lim sup λ ϕ + λ is lsc (see Proposition 1.
28 in [START_REF] Braides | Γ-convergence for beginners[END_REF]) this also implies Γ -lim sup

λ ϕ + λ ≤ ϕ * . (6.3)
We now aim to show that

Γ -lim inf λ ϕ + λ ≥ ϕ * . (6.4)
Let us argue by contradiction assuming that (6.4) is false, then there exists x ∈ X, r > 0, ε > 0, x λ converging to x as λ → +∞ such that lim inf

λ ϕ + λ (x λ ) ≤ inf Br(x) ϕ * -4ε,
To ease notations, choose a sequence x n = x λn with λ n → +∞ in such a way that lim inf

λ ϕ + λ (x λ ) = lim n Y ϕ(x n , y) µ + λn (dy|x n ).
and set ϕ n := ϕ(x n , .), ψ n := ψ(x n , .) and µ + n := µ + λn (.|x n ). Since for n large enough x n ∈ B r (x), we have for large enough n

Y ϕ n (y)µ + n (dy) ≤ ϕ * (x n ) -3ε = max argmin ψn ϕ n -3ε. (6.5)
But it follows from (4.13)-(4.14) in remark 4.3 (actually applied to

u = ψ n and v = -ϕ n ) that lim n µ + n ({ϕ n < ϕ * (x n ) -ε}) = 0 and since Y ϕ n (y)µ + n (dy) ≥ -ϕ ∞ µ + n ({ϕ n < ϕ * (x n ) -ε}) +(ϕ * (x n ) -ε)µ + n ({ϕ n ≥ ϕ * (x n ) -ε}) for large enough n, we have Y ϕ n (y)µ + n (dy) ≥ ϕ * (x n ) -2ε
which contradicts (6.5) and therefore establishes (6.4). Finally, it is wellknown (see Theorem 1.17 in [START_REF] Braides | Γ-convergence for beginners[END_REF]) that (6.3)-(6.4) imply that ϕ λ Γ-converges to ϕ * as λ → +∞.

Examples and remarks

If one wishes to implement our approximation on concrete problems (which, at the moment, we leave for future works), a certain number of issues have to be seriously addressed. Among them one can think of the choice of the reference measure ν as well as of the numerical method to efficiently compute the integrals which appear in ϕ λ . But the first question that naturally comes to mind is the choice of the weight δ λ for the secondary scale. Since δ λ captures the tradeoff between the upper and lower level, a small δ λ will result in a good accuracy for the follower's optimizing behavior but might too slowly take into account the leader's objective. We already saw that δ λ cannot be chosen too small for the convergence to be guaranteed but choosing it too large may affect the speed of convergence to the value function of the leader. A universallly good choice for δ λ is certainly impossible and the aim of this final section is precisely to illustrate on some particular examples the behavior of our approximations.

7.1 A case where δ λ cannot be too small We first consider an example where the assumption (3.5) of a power-like lower bound on α ν is relaxed. This example shows that one cannot hope for a universal choice of δ λ , in particular δ λ = λ -1 2 does not guarantee convergence to the double argmin set if α ν happens to be too small near 0. • v vanishes at 0 only.

We thus have argmin u = K∪{0} and the double argmin set argmin argmin u v is reduced to {0}. Now choose ν absolutely continuous with respect to the 1-dimensional Lebesgue measure dν = f dy with f bounded, f > 0 Lebesgue almost everywhere so that ν has full support but f extremely small in a neighbourhood of 0, more precisely:

f (y) = 3 y 4 e -1
|y| 3 , for |y| > 0 small and , f (0) = 0. so that for small δ > 0 ν([-δ, δ]) = 2e -1 δ 3 .

Consider now the two-scale measure with a square root weight for the secondary function v:

ν λ := Z λ e -λu- √ λv ν with Z λ := 1 Y e -λu- √ λv ν . First observe that Y e -λu- √ λv ν ≥ K f e - √ λv ≥ K f e -max K v √ λ =: Ce -M √ λ
then for δ > 0 small we have

ν λ ([-δ, δ]) ≤ 2e M √ λ
ν λ ([-δ, δ]) ≤ 2C -1 e √ λM (e -λ 3/4 δ 3 + e -λ 3/4 δ f ∞ )
since the right hand-side goes to 0 as λ → +∞, we deduce that there exists a neighbourhood of 0 which has zero measure for any weak cluster point of ν λ , in particular none of these cluster points can concentrate on {0}. In an example like this one, one typically has θ(t) ∼ -1 t 3 , so applying the cooking recipe of Lemma 4.1 one finds t λ ∼ λ -1 4 and weights δ λ which guarantee convergence are δ λ = λ -γ with γ ∈ (0, 1 4 ) or δ λ = λ -1 4 log(λ).

7.2 A case where δ λ cannot be power-like

We now consider a variant of the previous example where any power choice for δ λ fails to guarantee convergence of the two-scale measure to the double argmin set.

Example 7.2. Consider exactly the same u and v as in example 7.1, but with an even worse reference measure whose density f is of the form

f (y) = 1 y 2 exp 1 |y| -exp( 1 |y| )
in a neighbourhood of 0 (and bounded and bounded away from zero elsewhere, say) so that for small δ > 0, ν([-δ, δ]) = 2 exp(-exp(δ -1 )) and α ν (r) ∼ exp(-exp( 1 r )) for small r. Let us try to take a power function λ γ (i.e. δ λ = λ γ-1 ) for some γ ∈ (0, 1) as a weight on the function v i.e. consider

ν λ := Z λ e -λu-λ γ v ν with Z λ := 1 Y e -λu-λ γ v ν .
As in example 7.1, we have

δ δ σ e -λy f ≤ exp(-exp( σ δ )) + f ∞ e -λδ σ choosing σ = λ 1-γ 2
and replacing in (7.1), we arrive at

ν λ ([-δ, δ]) ≤ 2 C e M λ γ exp(-exp(δ -1 λ 1-γ 2 )) + f ∞ exp(-δλ 1+γ
2 ) and since γ < 1+γ 2 , we reach the conclusion that ν λ ([-δ, δ]) tends to 0 as λ → ∞, ruling out the convergence of ν λ to the Dirac mass at 0 (recall that {0} is the double argmin set in this example). In other words, no power like secondary weight gives convergence. But using Lemma 4.1 and Remark 4.4 the (very slowly decaying!) weight

δ λ = log(log(log(λ))) log(log(λ))
ensures the desired convergence. Of course, one may think that it is crazy to use such a pathological reference measure, but a rough behavior of u, v or the boundary of the set Y in higher dimensions may generate similar pathologies as well.

7.3 A case where the pessimistic value is not lsc

The following explicit example illustrates the convergence of the approximations in a case where the pessimistic value is not lsc. Since ϕ and ψ are Lipschitz and ν is the Lebesgue measure any choice of δ λ of the form δ λ = λ -γ with γ ∈ (0, 1) ensures the validity of our convergence result. In the optimistic case, the leader minimizes the optimistic value ϕ * (x) = min y∈R(x)

x + y = x and the solution is 0. Now, the approximation scheme proposed in the paper can be explicitly computed. For a given λ > 0, consider

δ λ = 1 √ λ , µ λ = Z λ (x)e -λxy- √ λ(x+y) dy, where Z λ (x) = 1 1 0 e -λxy- √ λ(x+y) dy = e √ λx λx + √ λ [1 -e -(λx+ √ λ) ] then ϕ λ (x) = Zλ (x) 1 0 (x + y)e -λxy- √ λ(x+y) dy = x + e -λx- √ λ e -λx- √ λ -1 + 1 λx + √ λ converges to ϕ * (x) = x as λ → +∞.
In the pessimistic case, the leader minimizes the pessimistic value

ϕ * (x) = max y∈R(x) x + y = 1 if x = 0 x if x > 0
which is not lsc at 0 and the infimum of ϕ * which is 0 is not achieved. Note that the lsc envelope of ϕ * coincides with the optimistic value ϕ * . In this case, for a given λ > 0, consider

δ λ = 1 √ λ , µ + λ = Z + λ (x)e -λxy+ √ λ(x+y) dy where Z + λ (x) = 1 1 0 e -λxy+ √ λ(x+y) dy = e √ λx [e (-λx+ √ λ) -1] -λx+ √ λ if x = 1 √ λ 1 e if x = 1 √ λ then ϕ + λ (x) = Zλ (x) 1 0 (x+y)e -λxy+ √ λ(x+y) dy = x+ e -λx+ √ λ e -λx+ √ λ -1 -1 -λx+ √ λ if x = 1 √ λ 1 √ λ + 1 2 if x = 1 √ λ .
We know that ϕ + λ pointwise converges to ϕ * and Γ-converges to ϕ * as λ → +∞. The pointwise convergence is of course slower near 0 and we have tested various exponents for δ λ (the square root as above but also γ = 1 4 , for which the convergence is even slower and γ = 3 4 which seems to give more accurate approximations).

The choice of δ λ is critical in practice

Even in the case where argmin ψ(x, .) is a singleton for any x ∈ X, so that optimistic and pessimistic solutions coincide, the optimistic and pessimistic λapproximations converge, with a different convergence speeed, to the common value ϕ * = ϕ * , as shown in the next example in which X is two-dimensional and for which the choice of δ λ seems to be crucial for practical convergence. The Stackelberg solution is x = (0, 0) and y = 0. Since ϕ and ψ are Lipschitz we can chose any power function for δ λ , δ λ = λ -γ . Our approximation is given by: ϕ λ (x) = Y ϕ(x, y)e -λψ(x,y)-λ 1-γ ϕ(x,y) dy Y e -λψ(x,y)-λ 1-γ ϕ(x,y) dy (and the pessimistic approximation ϕ + λ is given by a smilar formula, just by changing the sign of the term involving ϕ) which converges as λ → +∞ to ϕ * which achieves its minimum at (0, 0). We illustrate the convergence with various exponents and values of λ. The convergence turns out to be very bad for γ = 1 2 but very good for γ close to 1 as in the case γ =

  denote the cost function of the leader (who chooses x ∈ X) and the follower (choosing y ∈ Y ) respectively. The Stackelberg problem is the program of the leader which reads inf (x,y)∈X×Y ϕ(x, y) : y ∈ argmin ψ(x, .) .

Proposition 3 . 3 .

 33 Assume that u and v satisfy (3.4), that ν satisfies (3.5) and define ν λ by (3.6) then any weak star cluster point of ν λ as λ → ∞ has its support in the double argmin set argmin argmin u v. Proof. To ease notations, let us normalize u and v in such a way that min Y u = 0, min{v(y) : y ∈ Y, u(y) = 0} = 0. (3.7) Also define w λ := u + v √ λ and observe that (3.7) implies that min Y w λ ≤ 0. Let then λ n → ∞, ν ∈ P(Y ) such that ν λn weakly star converges to ν. Let

√λ

  may rule out the desired convergence property (see Example 7.1). Even worse, it may be the case that no power-like weight converges to the double argmin set (see Example 7.2). An immediate consequence of Proposition 3.3 is Corollary 3.5. Under the same assumptions as in Proposition 3.3, we have lim λ→∞ Y v(y) ν λ (dy) = min{v(y) : y ∈ argmin u}.

3 )

 3 we may then find a sequence λ n → +∞ in such a way that µ n := µ λn (.|x λn ) weakly star converges to some µ ∈ P(Y ) andlim inf λ→∞ Y ϕ(x, y) µ λ (dy|x λ ) = lim n Y ϕ(x, y) µ λn (dy|x λn ) = Y ϕ(x, y)µ(dy) so if we show that ψ(x, .) = min Y ψ(x, .) µ-a.e.(5.4)this will show that Y ϕ(x, y)µ(dy) ≥ ϕ * (x) by definition of ϕ * yielding the desired Γ-liminf inequality. To prove (5.4), we take ε > 0 and observe that for large enough n, the set {y ∈ Y : ψ(x, y) > min Y ψ(x, .) + ε} is included in {y ∈ Y : ψ(x λn , y) + δ λn ϕ(x λn , y) > min Y {ψ(x λn , .) + δ λn ϕ(x λn , .)} + ε 2 }.

Corollary 5 . 2 .

 52 Let ϕ * be defined by (2.3) and ϕ λ be as above, then lim λ→+∞ min X ϕ λ = min x∈X ϕ * = min{ϕ(x, y), (x, y) ∈ X × Y, y ∈ argmin ψ(x, .)}

Example 7 . 1 .

 71 Consider the case where Y = [-1, 1], and let u and v Lipschitz, nonnegative with • u = 0 exactly on K := [-3/4, -1/2] and at 0, u(y) = |y| in a neighbourhood of 0 (for instance u(y) = min(|y|, d K (y)) where d K is the distance to K),

4 e 3 +

 43 -λy f ≤ e -λ 3/4 δ e -λ 3/4 δ f ∞ we finally get

Ye 2 C e M λ γ δ 0 e. 1 )For any σ > 1 we have δ 0 e -λy f = δ σ 0 e

 20100 -λu-λ γ v ν ≥ Ce -M λ γ hence ν λ ([-δ, δ]) ≤ -λy f (y)dy.(7-λy f +

Example 7 . 3 .

 73 Let us consider the case X = Y = [0, 1], take ν to be the Lebesgue measure on Y and ϕ(x, y) = x + y, ψ(x, y) = xy, then R(x) = argmin y∈Y xy = 0 if x ∈ (0, 1] [0, 1] if x = 0.

Figure 1 :

 1 Figure 1: Graph of ϕ λ for λ = 36, λ = 144 (circle), λ = 225 (square) with δ λ = 1 √ λ (Example 7.3).

Figure 2 :

 2 Figure 2: Graph of ϕ + λ for λ = 36, λ = 144 (circle), λ = 225 (square) with δ λ = 1 √ λ (Example 7.3).

Figure 3 :

 3 Figure 3: Graph of ϕ + λ for λ = 36, λ = 144 (circle), λ = 225 (square) with δ λ = 1

Figure 4 : 1 + 4x 2 y -x 1 x 2 ψx 2 4 + x 1 -x 2 .x 2 4 + x 1 -x 2 whose

 412412412 Figure 4: Graph of ϕ + λ for λ = 36, λ = 144 (circle), λ = 225 (square) with δ λ = 1 4 √ λ (Example 7.3).

Figure 5 :

 5 Figure 5: Optimistic case for δ λ = 1 √ λ (Example 7.4).

Figure 6 :Figure 7 : 10 √ λ 9 (Figure 8 : 10 √ λ 9 (

 671098109 Figure 6: Pessimistic case for δ λ = 1 √ λ (Example 7.4).

Figure 9 : 4 √ λ 3 (Figure 10 : 4 √ λ 3 (

 9431043 Figure 9: Optimistic case for δ λ = 1 4 √ λ 3 (Example 7.4).

Recall that (ν n ) n ∈ P(Y ) N weakly star converges to ν if Y u(y)ν n (dy) → Y u(y)ν(dy) for every u ∈ C(Y ).

√ λ 3 (Example 7.3).


Kin Long

Kelvin Lee

Sven Thorwirth

Marie-Aline Martin-Drumel

Michael C Mccarthy

Generation and structural characterization of Ge carbides GeC n (n = 4, 5, 6) by laser ablation, broadband rotational spectroscopy, and quantum chemistry †

come

Introduction

Small carbon clusters have been extensively studied by experiment and theory, and found almost exclusively to possess either linear, or highly symmetrical ring structures [1][2][3][4] . Substitution of one or more of these carbon atoms with the isovalent group 14 element silicon or germanium has profound consequences to molecular structure and polarity, which in turn often imparts these species with strong rotational spectra, a property that has enabled a number of small silicon carbides to be detected in circumstellar shells around certain evolved carbon stars 5 bon and tetracarbon, for example, are only known experimentally to possess linear structures 1 , but SiC 2 , 6 Si 2 C, 7 , and Si 3 8 have bent or T-shaped geometries. For SiC 3 , both linear and rhombic-shaped isomers are known, [9][10][11][12] . For still larger SiC n clusters (n = 4 -8), linear chains have been detected at high spectral resolution. 12 In contrast, much less is known about the structure and bonding properties of the analogous Ge-bearing species. The interest in these species is to understand the underlying electronic structure of group 14 elements, and the nature of their chemical bonding. Until recently, only two Ge-species have been observed at high spectral resolution: germanium carbide (GeC) by low-temperature electronic emission 13 , and one symmetric chain, GeC 3 Ge, using infrared free-jet spectroscopy and a laser ablation source 14 . At lower spectral resolution, Graham and coworkers used Fourier transform infrared matrix isolation to measure the vibrational spectra of a number of Ge n C m clusters, including GeC 3 Ge, 15 GeC 5 Ge, 16 and three odd-numbered chains

1-10 | 1
Using chirped-pulsed (CP) and cavity Fourier-transform (FT) microwave spectroscopies 19 , the rotational spectrum of GeC 2 , the isovalent analog to C 3 and SiC 2 , was reported by several of us recently. The species is produced so abundantly by laser ablation of a Ge:C rod that rotational lines of its 13 C isotopic species were observed in natural abundance in the chirped spectrum, despite the low fractional abundance of this isotope (1.1% natural abundance). This work, the first pure rotational investigation of a Gecarbide cluster, enabled an unambiguous structure determination of the compound. Indeed, in contrast to prior quantum-chemical calculations [20][21][22][23][24] predicting an L-shaped structure as the global minimum on the potential energy surface, this carbide is found to be T-shaped instead. The structural controversy that surrounded GeC 2 , much like early work on SiC 2 25 , illustrates the importance of developing accurate theoretical descriptions of bonding that involve group 14 elements; the relevance of Si-and Gedoped materials in semiconductor technology and nano/microelectronics [26][27][28] , and the plethora of possible structures that are predicted even for small clusters, is another motivation to pursue accurate descriptions of their electronic structure. [29][30][31][32][33][34] For these reasons, there is great practical utility in studies of other small Ge carbides to benchmark theoretical methods so that chemical models with greater predictive power can be developed.

In this paper, we report the detection and characterization of three larger Ge-carbides, linear forms of GeC 4 , GeC 5 , and GeC 6 (Fig. 1). Guided by new coupled-cluster calculations of their structure, evidence was found for GeC 4 and GeC 5 chains in the broadband CP spectrum that initially enabled GeC 2 detection. 19 Subsequent measurements using cavity FT spectroscopy enabled both to extend the observations to higher frequency, and, based on high-level quantum-chemical calculations and empirical scaling factors, to specifically target and detect GeC 6 . Precise rotational constants were determined for the most abundant Ge isotopic species without hyperfine interactions ( 74 Ge , 72 Ge, [START_REF] Lefebvre-Brion | Perturbations in the Spectra of Diatomic Molecules[END_REF] Ge, 76 Ge), from which the Ge-C bond length has been derived to high accuracy in all three chains. In agreement with theoretical predictions and analogy to the pure carbon and silicon-carbon chains, clusters with an odd number of atoms (GeC 4 and GeC 6 ) are described by a closed-shell electronic configuration, while odd-numbered GeC 5 is an open-shell (triplet multiplicity) species, although its spectrum is otherwise indistinguishable from that of a 1 Σ molecule owing to a very large spin-spin constant λ . Finally, we draw parallels in the growth and relative abundance of germanium-carbon chains by comparison with analogous SiC n experiments.

Experimental Methods

Chirped-pulse Fourier-transform microwave (CP-FTMW) spectroscopy 35 followed by its cavity variant 36 were used to de- tect and analyze products formed in the laser ablation of a Ge:C rod. The combined approach of CP-and cavity FT spectroscopy has recently been used with good success to detect entirely new molecules, 37 such as HC 4 C(O)H, 38 GeC 2 , 19 and new vibrationally excited states of highly abundant species, such as C 2 S and C 3 S. 39 A real strength of this approach is that it exploits the advantages of both spectroscopies, in which the CP spectrum is used as a survey tool to detect "bright" resolution elements over a very wide frequency range, and cavity spectroscopy can then be efficiently used to rapidly analyze and characterize these smaller number of spectral features, owing to its higher instantaneous sensitivity and spectral resolution.

The laser ablation and microwave setups used for this experiment have been described in the previous publication on GeC 2 19 and are essential identical in design to that used previously to detect rotational lines of TiO 2 . 40 In the present study, a 0.25 in diameter rod was formed by pressing a mixture of Ge and graphite powders (in the molar ratio 1:2) and a small amount of epoxy binder in a metal mold, and by applying two tons of pressure for 15 min. After curing for approximately 24 h, the rod was ablated using the second harmonic output of an Nd:YAG laser, corresponding to an unfocused pulse energy of 20 mJ/pulse. The 532 nm light was then focused onto the GeC rod with a 75 cm focal length lens; the energy density of the focused beam was found to optimize the production of SiC 2 using a rod of similar stoichiometry in preliminary experiments. The laser ablation pulse is synchronized with a pulsed nozzle so that the plume of ablated matter is entrained in an inert gas (Ne) and then expands adiabatically into a large vacuum chamber where both the CP and cavity spectrometers are co-located, but aligned along perpendicular axes. As is almost universally done in laser ablation experiments of this kind, the rod was continuously rotated and translated so that each laser pulse ablates a new spot on the rod. The repetition rate of the laser and pulsed valve were 5 Hz.

|

1-10

Chirped-Pulse Measurements

The broadband spectrum of the laser ablation products from a Ge:C rod was recorded between 7 and 18 GHz by mixing a 4 µs chirp between 1 and 11.5 GHz with a phase-stable 19 GHz local oscillator. After amplifying the lower sideband (18-7 GHz), the chirp was sent to a 200 W traveling wave tube (TWT) for further amplification before it was broadcast between two matched horns, through which the gas pulse passes. The free induction decay (FID) was detected with a low-noise amplifier, but both a PIN diode and high isolation microwave switch are inserted just after the receive horn to protect this amplifier during the excitation pulse.

To improve the signal-to-noise ratio (SNR), each gas pulse was probed by 10 microwave chirps, each separated by 20 µs, where the FIDs associated with each are co-averaged on a fast digital oscilloscope (20 GHz) prior to transfer to a Linux workstation for real-time processing and analysis. At a sampling rate of 50 GSa/s, each FID consists of 750,000 points, yielding an instrumental resolution of 100 kHz. Because the phase stability is generally quite good, deep averages of at least several 100,000 gas pulses are feasible, although experimental factors such as changes in the nozzle tension and laser alignment often limit the effectiveness of very long integrations (i.e., >24 h). In total, the spectrum was accumulated for 17 h, corresponding to ∼300,000 gas pulses at a repetition rate of 5 Hz. Subsequently, the intensity and frequency of all features in a specified bandwidth and with a specified SNR can be generated and monitored in real-time by using an automated peak finding algorithm. Upon the completion of the chirped experiment, instrumental artifacts and common contaminants are removed, and the resulting line list can be trivially converted to a batch file that can been be automatically executed with the cavity spectrometer software. 37 Known species are also systematically removed: for example, transitions of GeC 2 which were already assigned by Zingsheim et al. 19

Cavity Measurements

Upon completion of the CP experiment, follow-up cavity measurements were immediately performed. This changeover is rapid (i.e., <3 min) because both spectrometers share much of the same equipment, including nozzle source, gas-flow system, timing circuitry, etc., and the only physical change required is to manually pullback a highly absorbent curtain that is placed in front of one of the cavity mirrors to spoil the cavity Q during CP operation. The cavity instrument used in the present investigation has been described in detail elsewhere, 41,42 but it operates between 5 and 26 GHz, so in addition to analysis and characterization, it can be used to extend the range of measurement, often to higher frequency.

The operation of the cavity spectrometer is similar to the CP instrument in that a short (1 µs) pulse of microwave radiation coherently excites molecules as they pass through the center of a high-Q (10 4 -10 5 ) Fabry-Perot cavity. If a molecule possesses a rotational transition within the narrow frequency range of the pulse (∼1 MHz), the FID is detected with a sensitive microwave receiver and the corresponding Fourier transform (performed on the computer) contains the frequency-domain spectrum. Owing to the high finesse of the cavity, its sensitivity per unit MHz is roughly 40 times greater than that of the CP spectrometer. 35 In addition, the linewidths of cavity-recorded transitions, normally determined by the time-of-flight through the active volume, 43 are typically at least 10 times sharper (2-4 kHz FWHM) than those in a CP spectrum, a feature which often allows closely-spaced hyperfine structure to be resolved.

Computational Methods

Quantum-chemical calculations of GeC 4 , GeC 5 , and GeC 6 were performed using coupled-cluster with single, double, and perturbative triple excitations [CCSD(T)] 44 using the CFOUR suite of programs [START_REF] Cfour | Watts and the integral packages MOLECULE[END_REF][START_REF] Harding | [END_REF] in combination with Dunning's hierarchies of correlation-consistent polarized valence 47,48 and polarized corevalence basis sets. [49][50][51][52] Structure calculations were performed using the cc-pVXZ (X =D, T, Q) basis sets in the frozen core (fc) approximation, and with basis sets as large as cc-pwCVQZ when considering all-electrons (ae) in the CCSD(T) treatment. Harmonic, cubic, and semi-quartic force fields were calculated using the cc-pVXZ (X =D, T) bases in the fc approximation and analytic second-derivatives procedures 53 together with numerical differentiation for the third and fourth derivatives. 54,55 Due to the large computational expense associated with these calculations, the cc-pVTZ basis was only used for the smallest (GeC 4 ) species as a point of comparison. For the chains with an even number of carbon atoms, a closed-shell Hartree-Fock (HF) reference function was used, whereas an unrestricted HF (UHF) reference was used for GeC 5 , which has a triplet ground state. For the latter, using the cc-pwCVQZ basis set, appreciable spin contamination is observed at the UHF level (<S 2 >=2.40 translating into an approximate spin multiplicity of some 3.25) whereas at the unrestricted CCSD (UCCSD) level this is almost entirely resolved (projected <S 2 >=2.11 and approximate spin multiplicity of 3.07).

Results

As illustrated in Fig 2, the most prominent features in the 7-18 GHz spectrum are the fundamental a-type rotational lines of T-shaped GeC 2 near 15.3 GHz, owing to the large number of germanium isotopes with significant fractional abundance (21% [START_REF] Lefebvre-Brion | Perturbations in the Spectra of Diatomic Molecules[END_REF] Ge, 28% 72 Ge, 8% 73 Ge, 36% 74 Ge, and 8% 76 Ge). Beyond these features, nearly 100 other spectral features were identified with a SNR of three or greater. Nearly one half of these were assigned to known molecules including GeS, three cyanopolyyne chains HC 3 N, HC 5 N, and HC 7 N, and two methyl acetylene chains CH 3 CCH and CH 3 C 4 H using SPECDATA, a recently developed inhouse database query system that rapidly assigns transitions of known species in an experimental spectrum. [START_REF] Oliveira | SPECDATA: Automated Analysis Software for Broadband Spectra[END_REF] The remaining features were subject to further scrutiny by cavity spectroscopy, and of these, roughly ten were either not observed within a few minutes of integration, not laser dependent, or exhibit narrow line (hyperfine) structure that is not consistent with the elemental composition of the pressed powder rod, suggesting that they likely arise from the binder, one of its by-products, or some other contaminant. However, approximately 20 lines, most of which lying between 7 and 10 GHz, were observed only when the laser was present and were structureless, suggesting the carrier or carriers must be Ge-bearing molecules, C-bearing, or both. Tables of observed frequencies and molecular parameters can be found in the Electronic Supplementary Information.

GeC 4

Guided by the quantum-chemical calculations in Sec. 3 (Table 1), a total of six lines of either 70 GeC 4 , 72 GeC 4 or 74 GeC 4 were assigned among the unidentified features, and closer inspection of the CP spectrum (Fig. 2) reveals additional features, albeit at lower SNR, at the expected frequencies of the less abundant 73 GeC 4 species, or at either higher or lower predicted frequencies for the two most abundant species. Ultimately, at least five rotational transitions between 6 and 22 GHz were measured with the cavity spectrometer for all but 73 GeC 4 (Table S1); owing to the larger rotational partition function due to hyperfine structure from the I = 9/2 nuclear spin of 73 Ge, no evidence was found for 73 GeC 4 at the present SNR.

The observed lines were trivially analyzed using a standard Hamiltonian for a linear molecule. The best-fit rotational constants are summarized in Table 1 along with those predicted at the CCSD(T)/cc-pwCVQZ level of theory. The agreement between the sets of rotational constants is exceptional; the calculated values reproduce the measurements within ∼0.1% even with the lower quality force field (fc-CCSD(T)/cc-pVDZ).

GeC 5

Following discovery of GeC 4 in the CP spectrum in Fig. 2, attempts were made to identify lines of linear structures of GeC 3 , GeC 5 , and GeC 6 . As illustrated in the plots at the bottom of this Figure, several harmonic series of lines near in frequency to those predicted for isotopic GeC 5 were readily identified; six of these were among the 20 or so lines that were previously found to be laser-dependent unidentified features in the cavity categorization tests. Because the lines are separated in frequency by ratios of integers, additional features were soon found with the cavity spectrometer. Table S3 provides a summary of the measurements, which range from N = 6 to 17, for [START_REF] Lefebvre-Brion | Perturbations in the Spectra of Diatomic Molecules[END_REF] GeC 5 , 72 GeC 5 , and 74 GeC 5 . 400.88084(1) 400.904 a Uncertainties in parentheses are 1σ in the units of the last significant digit. The complete set of spectroscopic constants, derived from the measurements in Tables S1-3, is given in Table S7. † b Calculated from the CCSD(T)/cc-pwCVQZ structure and fc-CCSD(T)/cc-pVDZ zero-point vibrational corrections, see supplementary material, except when otherwise noted. c Calculated from the CCSD(T)/cc-pwCVQZ structure and fc-CCSD(T)/cc-pVTZ zero-point vibrational corrections, see supplementary material. d The percentage difference, calculated as

B 0 exp -B 0 calc B 0 exp .
Because the spectrum of GeC 5 is otherwise indistinguishable from that of 1 Σ molecule, magnetic tests were performed to establish that its lines are indeed sensitive to the presence of a weak magnetic field, as required for a radical with a 3 Σ ground state. As shown in Fig. 3, the intensity of the 7 6 → 6 5 line (and others) are significantly diminished when a permanent magnet is brought near the supersonic expansion. Although weak, the resulting line profiles are similar to those of radicals that have partially-resolved Zeeman structure (with ∆M J = ±1 transitions).

Like isovalent SiC 5 , GeC 5 is predicted to possess a 3 Σ ground state. Thus, by analogy to other species with triplet multiplicity such as C 6 S and C 8 S, [START_REF] Gordon | [END_REF] lines from the F 2 (J = N) and F 3 (J = N + 1) spin components are expected to lie significantly higher in energy relative to those of the F 1 (J = N -1) component owing to the large spin-spin constant λ . As a consequence, the rotational spectrum at low rotational temperature approaches that of 1 Σ molecule when λ B 0 , in which only levels of the F 1 component are thermally populated, and the pattern-forming quantum number is J; Hund's case (a) limit. For both SiC 5 and odd-carbon chains up to C 8 S, however, small but significant deviations from harmonicity -beyond those expected from centrifugal distortion alone -are manifest in the spectrum, and allow λ to be determined in a least-squares fit. For GeC 5 , however, no such deviations are apparent: line frequencies only differ from ratios of integers by that expected from D 0 , implying a very large value of λ . The rotational lines of 74 GeC 5 , for example, can be reproduced to a fraction of the 2 kHz measurement un-Fig. 2 The broadband rotational spectrum between 7 and 18 GHz in which a Ge:C rod (1:2 by molar ratio) was ablated with the 532 nm frequency-doubled output of a Nd:YAG laser, and the products, entrained in Ne buffer gas, adiabatically expanded into a large vacuum chamber. The top trace displays the entire 11 GHz spectrum with contribution from electronic artifacts and known molecules removed. The 1 0,1 → 0 0,0 transition of 74 GeC 2 and its less abundant Ge isotopic species are conspicuous near 15.3 GHz in the top spectrum and thus the intensity axis has been normalized to the strongest transition of this species (consequently, most transitions of GeC 2 are been truncated in the displayed spectrum). Three series of harmonically-related lines are visible in lower traces (portions of the broadband spectrum in which the vertical scale has been expanded by a factor 5); the strongest features were ultimately assigned to 74 GeC 5 and 72 GeC 5 , and the weaker one to 76 GeC 5 .

certainty with only two free parameters, B 0 =605.75309(6) MHz and D 0 = 8.3(5) × 10 -6 MHz. By comparison, closed-shell SiC 6 has very similar rotational and centrifugal distortion constants [B 0 =611.25102(6) MHz and D 0 = 5.5(2) × 10 -6 MHz] 12 . From simulations using a 3 Σ Hamiltonian and different values for λ , this parameter must be at least 770 GHz, and, as a consequence, was simply constrained in the final fits. The best-fit rotational constants for GeC 5 using a 1 Σ Hamiltonian are summarized in Table 1.

GeC 6

While the longest GeC n chain considered in this work was not directly observed in the broadband spectrometer, the search for 74 GeC 6 was carried out based on quantum chemical predic-tions. Confidence based on the GeC 4 experimental and theoretical work, we attempted to exploit the systematic deviation between the electronic structure prediction and the experimental rotational constant: by calculating B e at the CCSD(T)/cc-pwCVTZ level of theory for both GeC 4 and GeC 6 , the search began by scaling the ab initio B e for GeC 6 by the ratio B 0 /B e (1.0018) derived from the GeC 4 results. The basis for this scaling is to account for the vibrational corrections to B e , with the assumption that the electronic structure is described equally well and indifferent between the two species. Somewhat later in the study, the CCSD(T)/cc-pwCVQZ equilibrium rotational constant and CCSD(T)/cc-pVDZ vibrational correction also became available. The B 0 rotational constants derived using these two different methods are 403.745 and 404.727 MHz, respectively. Relative Int.

Frequency Offset (kHz)

Fig. 3 The 7 6 → 6 5 rotational line of 74 GeC 5 recorded in the absence (blue) and presence (red) of a weak magnetic field. The spectrum is displayed as a frequency offset from the rest frequency of 7269.030 MHz, and its line shape is instrumental in origin; it consists of two well-resolved Doppler components because the supersonic jet expands along the axis of the Fabry-Peŕot cavity. Both spectra are co-averages of approximately 1750 gas pulses, requiring about 6 min of integration at the 5 Hz repetition rate of the nozzle.

A subsequent search using the cavity spectrometer revealed six lines with relatively high SNR with approximately harmonic separation. By scaling the predicted rotational constants for other isotopologues, additional transitions belonging to 70 GeC 6 , 72 GeC 6 , and 76 GeC 6 were quickly found. Analogous to the GeC 4 work earlier in this paper, the observed transitions were analyzed and fit to a linear molecule Hamiltonian comprising only of B 0 and D 0 terms. The determined B values of GeC 6 are shown in Table 1. We note here that, while the empirical scaling factor was sufficiently accurate for a deep cavity search, the results highlight the importance of accurate vibrational corrections from first-principles, which provided a calculated B 0 substantially closer to the final experimental value.

Determination of Ge-C bond lengths

While the experimental data sets collected here do not permit derivations of complete empirical molecular (semi-experimental, r emp e ) structures [START_REF] Vázquez | Equilibrium Molecular Structures -From Spectroscopy to Quantum Chemistry[END_REF] , from the rotational constants of isotopic GeC 4 , GeC 5 and GeC 6 it is possible to determine highly accurate Ge-C bond lengths for all three carbides. To do this, the experimental rotational constants were corrected for zero-point vibrational effects at the fc-CCSD(T)/cc-pVTZ (GeC 4 ) and CCSD(T)/cc-pVDZ (all three chains) levels of theory, the C--C bond lengths were constrained to values derived at the ae-CCSD(T)/cc-pwCVQZ level, and the Ge-C bond length was then least-squares-optimized [START_REF] Kisiel | [END_REF] to reproduce the semi-experimental equilibrium moments of inertia of the isotopic species (see supplementary material). This procedure yields a Ge-C bond length of 1.7757/1.7742 Å for GeC 4 (for the cc-pVTZ/cc-pVDZ zero-point vibrational corrections, respectively), 1.8175 Å for GeC 5 , and 1.7820 Å for GeC 6 , with 1σ statistical uncertainties well below 10 -4 Å.

Since the C--C bond lengths were constrained in the optimization and the experimentally derived rotational constants of GeC 5 are model dependent (i.e., on the assumed value of λ ), more conservative uncertainties are on the order of a few mÅ for the singlet species (see, e.g., Ref. 60 and references therein) and presumably of order 10 mÅ (1 pm) for GeC 5 . Nevertheless, all Ge-C bond lengths derived here from experiment compare very favorably to those predicted at the ae-(U)CCSD(T)/cc-pwCVQZ level of theory, and more generally indicative of a double bond between these two atoms. For comparison, methyl germane, CH 3 GeH 3 , may serve as a prototypical system featuring a Ge-C single bond. From microwave experiments 61 and at the CCSD(T)/cc-pwCVQZ level of theory (calculated in this work) bond lengths of 1.9453(5) Å and 1.9454 Å, respectively, have been obtained, corresponding to an elongation of more than 0.1 Å compared to the three chains studied here. By way of further comparison, the Ge-C r emp e semi-experimental bond length is 1.770(1) Å in GeC 3 Ge, 14 and 1.77579 (21) Å in GeCH 62 suggesting beyond diatomic GeC (1.805 Å) 13 that the Ge-C double bond length in cumulenes is fairly insensitive to the length of the carbon chain and therefore implying that electronic density between Ge-C remains relatively localized from the rest of the conjugated chain. The species with triplet multiplicity (GeC/GeC 5 ) possess slightly longer heavy atom-carbon bonds (∼1.81 Å) compared to the singlet chains (GeC 4 /GeC 6 ∼1.78 Å) -a finding very similar to the situation in isovalent SiC n species 63,64 -and is consistent with repulsion due to the unpaired electronic configuration.

Comparison of SiC n and GeC n production

Because identical cavity experiments were performed with Si:C and Ge:C rods, a systematic comparison of the production of the two group 14 carbides by laser ablation is possible. Since Si 3 8 and other Si carbides such as Si 2 C 7 and rhomboidal SiC 3 10 have been previously studied by cavity spectroscopy and have strong, low-J transitions in the 7-18 GHz frequency range, it is also possible to quantify their production under the same experimental conditions.

For each species, relative intensities were first corrected for the natural abundance of isotopic species, and then converted to relative abundances by factoring the respective rotational partition functions at T rot =5 K and dipole moments. The temperature was chosen to reflect the estimated rotational temperature of our molecular beam, determined by measurements with laser ablation experiments on TiO 2 . For the SiC n species, literature

6 | 1-10
dipole moments were used while for the GeC n species the dipole moments were calculated at the CCSD(T)/cc-pwCVTZ level; the values used are organized into Table 2. The resulting abundances (relative to XC 2 , X = Ge,Si) are shown in Figure 4. Two important results arise from Figure 4: (I) the abundances of each carbon chain are quite similar between Ge and Si chains, and (II) the linear fits show two different slopes/rates in the abundance. Between one and four units of carbon, the abundance of both Si and Ge species span approximately three orders of magnitude while from four to eight carbon atom chains, the abundance decreases only by an order of magnitude. While the dynamics of formation are not entirely clear, a main contributing factor in their detection would be the dipole moment, which appears to grow steadily with the length of carbon chain (Table 2). Thus, while the production of longer species is expected to diminish, Figure 4 shows that the abundance of long Si and Ge chains, with up to eight atoms for Si-C, is sufficiently high that they remain detectable following a feasible (∼30 minutes) integration time in a cavity experiment, providing accurate predictions of the rotational transitions are available.

Discussion

Based on a combination of CP and cavity FTMW spectroscopies, we were able to characterize three new Ge-C chains, two of which were readily abundant and detectable by CP FTMW spectroscopy (GeC 4 and GeC 5 ), and the remaining GeC 6 was subsequently found in the cavity. As a general comment, it appears that the electronic structure of Ge-C chains are similar to Si-C and other carbon-chains, displaying the same odd-even triplet/singlet electronic ground state alternation, although as we shall discuss in the following passages, there are a few subtleties regarding the electronic structure of Ge-C species that require further elabora- GeC n SiC n Fig. 4 The abundance of SiC n (blue squares) and GeC n (red circles) species, relative to their respective dicarbides as observed in the microwave cavity experiments. Error bars represent 2σ uncertainties. The SiC 3 point is given as the cumulative abundance of the linear and rhombic forms. The estimated detection limit is based on the noise floor of a 1 MHz window following 30 min of integration in the cavity instrument.

tion.

It is somewhat surprising that the ground state of GeC 5 radical is so close to the Hund's case (a) limit that its spin-spin coupling constant can not be determined even from high-resolution rotational data with sub-ppm fractional accuracy. As a point of comparison, this constant is 31.5 GHz for SiC 5 , while a lower limit of ∼770 GHz is derived here. This constant includes second-order spin-orbit interactions between the triplet ground state and a lowlying singlet electronic state, [START_REF] Gordy | Microwave Molecular Spectra[END_REF] so such a large value likely indicates that the unpaired electrons have significant density at the heteroatom, owing to the ∼10 fold increase in the spin-orbit constant of Ge or Ge + (904 and 1178 cm -1 , respectively) [START_REF] Lefebvre-Brion | Perturbations in the Spectra of Diatomic Molecules[END_REF] relative to either Si or Si + (149 and 191 cm -1 , respectively). Although the energy gap between the two states may also be smaller in GeC 5 , no theoretical calculations or experimental data on electronically excited states are available for Ge carbides beyond GeC.

The present work suggests that despite the heavier mass of Ge and increased importance of relativistic factors for third-row elements, CCSD(T)/cc-pwCVXZ (X =T, Q) can still be used with considerable confidence to predict the structures of other small Gebearing species -that is to say that the electronic structure is adequately described by coupled-cluster theory with modestly-sized basis sets. For the three Ge-C n chains reported here, the theoretical rotational constants reproduce the experimentally determined values to within ∼0.2%. Further evidence that this agreement is not fortuitous is provided by the close agreement between the 1-10 | 7 calculated and best-fit Ge-C bond lengths.

Where the rotational constants have been used to bridge the theory and experiment, additional comparisons could be made on the dipole moment of these species; as a property related to the electronic density, and therefore the quality of the electronic wavefunction, measurement of the dipole moment would allow a direct comparison between theory and experiment. In this light, an experimental determination of the electric dipole of GeC n may prove illuminating -in addition to the usual, wellknown difficulties in retrieving electron correlation in metal carbide chains 63 , the inclusion of a third-row element provides an opportunity to benchmark relativistic contributions to molecular electronic wavefunctions. Owing to the low fractional ionization of laser ablation, the dipole moment of SiC 2 , produced in a similar fashion than in the present work, was measured long ago using FT cavity spectroscopy and a Stark cage. 65 Such a measurement for GeC 2 would appear to be possible: its fundamental line is very intense and a Stark cage [START_REF] Emilsson | [END_REF] was made available for use in our CP chamber. Since GeC 4 and GeC 5 were observable in the CP spectrum, conceivably the dipole moments of GeC 2 , GeC 4 , and GeC 5 could be measured simultaneously in one experiment.

In addition to testing ab initio theories, the experimental work here furthers the detection of long chain metal carbide species. As seen in Figure 4, the production of Si-and Ge-carbides is remarkably similar in our experiments: both dicarbides are produced in very high abundance, while the longer four-, five-, and six-carbon carbides, while readily detectible, are roughly present at a few percent of this level. The absence of rotational lines of linear SiC 3 , however, and the apparent absence of lines that can be attributed to analogous GeC 3 species is slightly puzzling. Previous studies of the SiC 3 isomers suggested that kinetic factors favored formation of the linear chain compared to the ringlike isomers at short gas durations and lower pressure pressures, presumably owing to entropic factors. At more typical expansion conditions, however, the thermodynamically more stable product, the oblate isomer of rhombic SiC 3 . Drawing a parallel between the earlier SiC n work and the new frontiers of GeC n , the apparent absence of GeC 3 , along with GeC 7 and GeC 8 appears technical rather than fundamental; both species should be present, albeit at somewhat lower abundance compared to their longer carbides counterparts. Subsequent searches can be carried out, providing ab initio predictions of their structures are sufficiently accurate to constrain the search range to feasible integration time. Additionally, based on the observed decrease in abundance decrement, the detection of progressively longer chains XC n , where X =Ge,Si and n > 8, would similarly be achievable. Efforts in this direction will likely contribute to the understanding of how metal-bearing carbon chains grow, from a thermodynamic and kinetic perspective.

Laser ablation has been used successfully to generate and detect diatomic and relatively small polyatomic species in the ra-dio band; there are by comparison relatively few examples were larger metal-bearing species have been detected by similar means. It is well known from mass spectrometric investigations, however, that much larger clusters are readily formed under similar conditions. Presumably the limitation in detecting their pure rotational transitions is a combination of less efficient production with increasing molecule size and the sensitivity of the method. As the present work demonstrates, with improvements in one or both areas, modestly larger clusters can now be detected, suggesting that the same basic approach might be used with good success to characterize other large metal-bearing species.

Conclusions

Combining CP-and cavity FTMW spectroscopy with laser ablation, we present the discovery and spectroscopic characterization of three new germanium-bearing carbon chain species. Upon discovery of GeC 2 in the CP spectrum 19 , evidence for the GeC 4 and GeC 5 chains were found soon afterwards in the same spectrum based on high-level quantum-chemical calculations. An additional third species, GeC 6 , was discovered with cavity measurements, once again relying on high-level calculations. Drawing parallels to the closely related SiC n chains, the spectroscopy of the three species was reproduced readily with simple linear 1 Σ Hamiltonians, including the curious case of GeC 5 , which should require a 3 Σ model. Despite the fact that the value of λ for this species is thought to be large, the experimental frequencies are wellreproduced using an effective 1 Σ Hamiltonian, which may motivate first principles determination of this parameter. Finally, the relative abundances of Ge-C and Si-C chains in the ablation experiment were compared using cavity FTMW measurements. The results show a remarkable similarity between the two metal bearing carbon chains, suggesting that the analogous GeC 3 , GeC 7 , and GeC 8 species should, in theory, be present and detectable. The decrement in abundance towards longer XC n (where X = Ge, Si and n > 4) carbon chains appears to be fairly flat, contrary to the rapid decrease in relative abundance of the shorter (n < 4) chains. The fortuitous sustained detectability will hopefully inspire not only subsequent searches for the remaining GeC n species, but also longer chains of SiC n as well as other metal-bearing carbon-rich molecules.

Fig. 1

 1 Fig. 1 Optimized structures of the GeC n linear chains reported in this work. Geometries are adapted from CCSD(T)/cc-pwCVQZ calculations (italics). Empirical Ge-C bond lengths are given in bold face. See the Discussion section for details on the experimental structural determination.

2

 2 Estimated detection limit at 30 min. integration

  . Tricar-

	a Center for Astrophysics | Harvard-Smithsonian, 60 Garden Street, Cambridge,
	Massachusetts 02138, USA; E-mail: kinlee@cfa.harvard.edu
	b I. Physikalisches Institut, Universität zu Köln, Zülpicher Straße 77, 50937 Köln,
	Germany

c Institut des Sciences Moléculaires d'Orsay (ISMO), CNRS, Univ. Paris-Sud, Université Paris-Saclay, F-91405 Orsay, France † Electronic Supplementary Information (ESI) available: [Includes tables of measured rotational lines and best-fit spectroscopic constants].

Table 1

 1 Measured and ab initio rotational constants of GeC 4 , GeC 5 , and GeC 6 (in MHz).

	Species	B 0 exp	a	B 0 calc	b	Q d
	70 GeC 4 72 GeC 4 74 GeC 4 76 GeC 4	1036.4053(2) 1027.0435(2) 1018.1423(1) 1009.6690(2)	1035.718/1036.406 c 1026.365/1027.047 c 1017.473/1018.148 c 1009.009/1009.678 c	0.07/-6×10 -5
	70 GeC 5 72 GeC 5 74 GeC 5 76 GeC 5	617.48402(3) 611.47529(1) 605.75309(6) 600.29762(2)	619.630 613.596 607.850 602.372	-0.3
	70 GeC 6 72 GeC 6 74 GeC 6 76 GeC 6	412.90688(3) 408.70900(1) 404.70546(1)	412.926 408.730 404.727	-0.005

Table 2

 2 Dipole moments, in Debye, used to calculate the relative abundances of SiC n and GeC n species. With the exception of SiC 2 (which has been experimentally determined), the values tabulated here are calculated using various ab initio methods. Dipole moments for the GeC n chains, which are unique to this work, are calculated at the CCSD(T)/cc-pwCVTZ level. Corresponds to the linear, cyclic oblate, and cyclic prolate forms of SiC 3 , respectively.

	n a	SiC n	Ref.	GeC n	Ref.
	1	0.9		-
	2	2.4	65	3.1	19
	3	4.8, 2.1, 4.2 b	66-68	-
	4	6.4	63	7.0 this work
	5	6.6	66	6.6 this work
	6	8.3	63	8.8 this work
	7	7.3		-
	8	10.0		-

a Number of carbon atoms b

| 1-10

Acknowledgments

We thank Michael E. Harding for helpful discussions, and Oliver Zingsheim for help in the original CP experiments. M. C. M. and K. L. K. L. acknowledge NSF grant AST-1615847 for financial support. Research of carbon clusters in Cologne is carried out within the Collaborative Research Centre 956, sub-project B3, funded by the Deutsche Forschungsgemeinschaft (DFG) -project ID 184018867. S. T. is grateful to the DFG for additional support through grants TH 1301/3-2 and SCHL 341/15-1 (Cologne Cen-8 | 1-10 ter for Terahertz Spectroscopy). We also thank the Regional Computing Center of the Universität zu Köln (RRZK) for providing computing time on the DFG-funded High Performance Computing (HPC) system CHEOPS. M.-A. M.-D. is thankful to the Programme National "Physique et Chimie du Milieu Interstellaire" (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES for support.