Alexandre Rege

Propagation of velocity moments and uniqueness for the magnetized Vlasov-Poisson system

In this paper we present two results regarding the three-dimensional Vlasov-Poisson system in the full space with a general bounded magnetic field. First, we study the propagation of velocity moments for solutions to the system. We rely on Pallard's optimal result regarding the unmagnetized Vlasov-Poisson system and we combine it with an induction procedure depending on the cyclotron frequency

. This induction procedure, similar to the one used by the author in the case of a constant magnetic field, is necessary because we can only get satisfactory estimates on a small time scale compared to the cyclotron frequency. Second, we manage to extend a result by Miot regarding uniqueness for Vlasov-Poisson to the magnetized case. This result relied heavily on the second-order structure of the Cauchy problem for the characteristics. The main difficulty in the magnetized case is that we lose this second-order structure.

Introduction

We study the Cauchy problem for the three-dimensional Vlasov-Poisson system with a general bounded magnetic field (which we will call magnetized Vlasov-Poisson system for the rest of the paper), given by the following set of equations:

∂ t f + v • ∇ x f + (E + v ∧ B) • ∇ v f = 0, f (0, x, v) = f in (x, v) ≥ 0. (1.1)
where f in is a positive measurable function and f := f (t, x, v) is the distribution function of particles at time t ∈ R + , position x ∈ R 3 and velocity v ∈ R 3 . The self-consistent electric field E := E(t, x) is given by:

E = -∇ x G 3 * ρ, (1.2)
The first main result of this paper, which is the continuation of [START_REF] Rege | The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity[END_REF], is to prove the propagation of velocity moments for (1.1) with a general magnetic field B. We manage to obtain this result by combining Pallard's method [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF] with an induction argument using the cyclotron period similar to the one in [START_REF] Rege | The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity[END_REF]. However, in this paper we don't obtain explicit singularities like in [START_REF] Rege | The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity[END_REF] because we use the "Lagrangian" point of view (study of the characteristics) from [START_REF] Pfaffelmoser | Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data[END_REF] instead of using the "Eulerian" point of view (study of the distribution function) from [START_REF] Lions | Propagation of moments and regularity for the 3dimensional Vlasov-Poisson system[END_REF][START_REF] Rege | The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity[END_REF]. This means we need to work on a small time scale compared to the cyclotron period T c = 1 B ∞ to obtain estimates analogous to those in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF]. This is due to the fact that on time scales comparable to T c or greater than T c , the variations of the characteristics are large and so the method in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF] fails. The second difficulty is to show that the estimates from [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF] are also valid in our framework up to a certain time depending on T c , and that these estimates depend only on quantities that don't prevent us from using the induction argument. Finally, propagation of velocity moments also implies propagation of the regularity of the initial data which means we have existence of classical solutions to (1.1). This result is detailed in [START_REF] Rege | The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity[END_REF] (theorem 2.5) for a constant magnetic field under additional conditions on f in (f in decays faster in velocity) but can be easily extended to the case of a general B.

Now we turn to results regarding uniqueness. First let's mention the major contribution by Loeper [START_REF] Loeper | Uniqueness of the solution to the Vlasov-Poisson system with bounded density[END_REF] who proved, for the unmagnetized Vlasov-Poisson system, that the set of solutions with bounded microscopic density was a uniqueness class. This result was also extended to (1.1) for a constant B in [START_REF] Rege | The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity[END_REF] and we discuss how to prove a similar result for a general B below. Then in [START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF], Miot improved the result by showing uniqueness under the condition that the L p norms of the macroscopic density grow at most linearly with respect to p. This allows for solutions with unbounded macroscopic density, more precisely with logarithmic blow-up. This paper's second main result is to prove that this uniqueness condition is also valid for (1.1), but only with added assumptions on the velocity moments of the initial data. In [START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF], a key point was exploiting the second-order structure of the characteristics of the Vlasov-Poisson system. This explains why the uniqueness condition from [START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF] doesn't apply to the two-dimensional Euler model for incompressible fluids, which presents many similarities with Vlasov-Poisson, whereas the condition from [START_REF] Loeper | Uniqueness of the solution to the Vlasov-Poisson system with bounded density[END_REF] works for both models. It is simply due to the fact that the characteristics of the Euler model only verify a first-order ODE. In our case, the main difficulty is that the added magnetic field breaks the second-order structure of the Cauchy problem for the characteristics. We manage to get around this by proving that the characteristics in the magnetized case can be controlled by assuming more regularity on B and with the additional assumptions on the moments of the initial data mentioned above. With these additional assumptions, we deduce a new uniqueness condition which is actually the same as the sufficient condition imposed on the initial data to verify the uniqueness criterion in [14, theorem 1.2] Outline of the paper: This paper will be organized as follows. We will finish this section by giving some notations and the classical a priori estimates satisfied by (1.1). In section 2 the main results of the paper will be presented. Then section 3 will be devoted to the proof of propagation of velocity moments (1.1). More precisely, we will explain how we find estimates that are equivalent to those in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF] up to a time T B depending on the cyclotron frequency and show how we can then use the same induction argument as in [START_REF] Rege | The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity[END_REF] to conclude. We finish with section 4 where we detail our proof of uniqueness for solutions to (1.1), highlighting how additional assumptions on the moments of the initial data allow us to control the added terms due to the added magnetic field in the analysis.

Preliminaries

Let's first detail the two main a priori bounds that we can deduce from system (1.1). The first bound is a direct consequence of the Vlasov equation where the coefficients are divergence-free.

f (t) p = f in p (1.5)
for all time t and exponents p ∈ [0 , +∞].

The second bound is the conservation of the energy E(t) of the system, with

E(t) := 1 2 R 3 ×R 3 |v| 2 f (t, x, v)dxdv + 1 2 R 3 |E(t, x)| 2 dx = E(0) < +∞. (1.6)
Furthermore, thanks to the conservation of the energy E(t), we have the following bounds.

LEMMA 1.1. For all t ≥ 0, we have M 2 (t) ≤ C 1 and ρ(t) 5

3

≤ C 2 with the constants

C 1 , C 2 depending only on E(0), f in 1 , f in ∞ .
We also present the standard notation for velocity moments: for any k > 2 and t ≥ 0 we define:

M k (t) = sup 0≤s≤t |v| k f (s, x, v)dvdx. (1.7)
As said before we will use the Lagrangian formulation detailed in [START_REF] Pfaffelmoser | Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data[END_REF], so we define the characteristics (X, V ) of (1.1) which are solutions to the following Cauchy problem:

     d ds X(s; t, x, v) = V (s; t, x, v), d ds V (s; t, x, v) = E(s, X(s; t, x, v)) + V (s; t, x, v) ∧ B(s, X(s; t, x, v)), (1.8)
with (X(t; t, x, v), V (t; t, x, v)) = (x, v).

(1.9)

Then like in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF], we define for any t > 0 and δ ∈ ]0 , t[.

Q(t, δ) := sup t t-δ |E(s, X(s; 0, x, v))| ds, (x, v) ∈ R 3 × R 3 . (1.10)
For the unmagnetized Vlasov-Poisson system, Q(t, δ) quantifies the evolution of the characteristics on the interval [t -δ , t]. However, in our context with the added magnetic field, Q(t, δ) will only quantify a part of the characteristic evolution.

Results

We now give the first main result of this section, which is the propagation of velocity moments of order k > 2, extending theorem 1 in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF] to the magnetized Vlasov-Poisson system.

THEOREM 2.1 (Propagation of moments). Let k 0 > 2, T > 0, f in = f in (x, v) ≥ 0 a.e. with f in ∈ L 1 ∩ L ∞ (R 3 × R 3
) and assume that

R 3 ×R 3 |v| k 0 f in dxdv < ∞. (2.11)
Then there exists a weak solution

f ∈ C(R + ; L p (R 3 × R 3 )) ∩ L ∞ (R + ; L p (R 3 × R 3 )) (2.12)
(1 ≤ p < +∞) to the Cauchy problem for the Vlasov-Poisson system with magnetic field

(1.1) in R 3 × R 3 such that sup 0≤t≤T R 3 ×R 3 |v| k 0 f (t, x, v)dvdx ≤ C (2.13)
with C that depends only on

T, k 0 , B ∞ , E(0), f in 1 , f in ∞ , R 3 ×R 3 |v| k 0 f in dxdv. (2.14)
REMARK 2.2. If f in satisfies the assumptions of the previous theorem, then all the moments of order k such that 0 ≤ k < k 0 are also propagated for the solution f , simply because of the following Hölder inequality

|v| k f (t, x, v)dvdx ≤ f k 0 -k k 0 1 |v| k 0 f (t, x, v)dvdx k k 0 (2.15)
where we use the decomposition

|v| k f = f k 0 -k k 0 |v| k f k k 0 and the exponents p = k 0 k 0 -k , q = k 0 k .
Like in [START_REF] Lions | Propagation of moments and regularity for the 3dimensional Vlasov-Poisson system[END_REF][START_REF] Rege | The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity[END_REF], we assume we have smooth solutions to conduct the proof in section 3, and since the a priori estimates depend only on (2.14) we can pass to the limit in the approximate Vlasov-Poisson system first introduced in [START_REF] Arsenev | Global existence of a weak solution of Vlasov's system of equations[END_REF]. In fact, theorem 2.1 will be a consequence of the main estimate in this paper which will only hold for these smooth solutions because it is an estimate on Q (given in (1.10)), which isn't necessarily well-defined for functions in Lebesgue spaces. We now give this estimate on Q.

Main estimate on Q:

For all T > 0 we have

N (T ) := sup 0≤t≤T Q(t, t) ≤ C, (2.16)
with C that depends on the constants in (2.14). In the following remark, we explain how theorem 2.1 is a consequence of (3.31).

REMARK 2.3. The estimate on propagation of velocity moments (2.13) in theorem 2.1 follows from the estimate on N (T ) (2.16) because we have:

R 3 ×R 3 |v| k f (t, x, v)dvdx = R 3 ×R 3 |V (t; 0, x, v)| k f in (x, v)dvdx ≤ R 3 ×R 3 (|v| + N (T )) k exp(kt B ∞ )f in (x, v)dvdx ≤ 2 k-1 exp(kt B ∞ ) R 3 ×R 3 |v| k f in (x, v)dvdx + N (T ) k f in 1
The first inequality above is obtained through a Grönwall inequality on |V (t; 0, x, v)|, indeed thanks to (1.8) we can write which implies

V (t; 0, x, v) = v + t 0 E(s, X(s; 0, x, v))ds + t 0 V (s; 0, x, v) ∧ B(s, X(s; 0, x, v))ds (2.
|V (t; 0, x, v)| ≤ |v| + Q(t, t) + B ∞ t 0 |V (s; 0, x, v)| ds ≤ |v| + N (T ) + B ∞ t 0 |V (s; 0, x, v)| ds
This is the classical Grönwall inequality which allows us to conclude that

|V (t; 0, x, v)| ≤ (|v| + N (T )) exp(t B ∞ ). (2.18)
The second inequality is just due to the fact that

2 k-1 (1 + x k ) ≥ (1 + x) k for x ≥ 0.
We finish this section by mentioning that, contrary to what is done in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF], we don't state any results on to the periodic case. This is simply explained by the fact that there are no results related to the existence of weak solutions to (1.1) in the periodic case. One possibility would be to adapt [START_REF] Batt | A Rigorous Stability Result for the Vlasov-Poisson System in Three Dimensions[END_REF] to the magnetized case, and then combine the results from [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF] for the periodic case and the proof of theorem 2.1 to show propagation of moments in the periodic case.

Uniqueness

For all the uniqueness results and so also in section 4, we assume the same regularity on B (1.3).

We first mention a very important result by Loeper [START_REF] Loeper | Uniqueness of the solution to the Vlasov-Poisson system with bounded density[END_REF] where it was shown that the boundedness of the macroscopic density ρ was a sufficient condition for uniqueness in the Vlasov-Poisson system. This result was extended to the Vlasov-Poisson system with constant magnetic field in [START_REF] Rege | The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity[END_REF] and the proof can be adapted to the case of a general magnetic field verifying (1.3). However, in the magnetized case, we require extra conditions on the velocity moments and space moments of the initial data f in , as shown in the following theorem:

THEOREM 2.4. Let B verify (1.3), let f in = f in (x, v) ≥ 0 a.e. with f in ∈ L 1 ∩L ∞ (R 3 ×R 3 )
and assume that

R 3 ×R 3 |v| 6 f in dxdv < ∞ and R 3 ×R 3 |x| 4 f in dxdv < ∞.
(2. [START_REF] Rege | The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity[END_REF] Assume further that the weak solution f to (1.1) provided by theorem 2.1 satisfies

ρ ∈ L ∞ ([0 , T ] × R 3 x ) (2.20)
for all T > 0.

Then the weak solution f is unique.

However, to exploit this result, one needs to build solutions to (1.1) with bounded macroscopic density. Hence, in the next proposition, we give an explicit condition that guarantees the existence of solutions to (1.1) with a bounded ρ. PROPOSITION 2.5. Let B verify (1.3) and let f in satisfy the assumptions of theorem 2.1 with k 0 > 6. We also assume that f in is such that for all R > 0 and T > 0

g R (t, x, v) ∈ L ∞ ([0 , T ] × R 3 x , L 1 (R 3 v )), (2.21)
where g R (t, x, v) = sup (y,w)∈S t,x,v,R f in (y + vt, w) (2.22) with S t,x,v,R = (y, w) : |y -x| ≤ (R + B ∞ |v|)t 2 e B ∞ t , |w -v| ≤ (R + B ∞ |v|)te B ∞ t .
(2.23) Then the weak solution f of (1.1) provided by theorem 2.1 verifies

ρ ∈ L ∞ ([0 , T ] × R 3 x )
for all T > 0.

This proposition was shown in [19, proposition 2.7] in the case of a constant magnetic field and remains unchanged when we take a general B. Now we present a theorem which is the second main result of this paper, where we show that the uniqueness criterion proved in [START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF] (theorems 1.1 and 1.2) also applies to (1.1) with B verifying (1.3), improving theorem 2.4 because it allows for solutions with unbounded macroscopic density. THEOREM 2.6. Let T > 0 and B verify (1.3). Furthermore, let f in ≥ 0 a.e. with

f in ∈ L 1 ∩ L ∞ (R 3 × R 3 ) and such that R 3 ×R 3 |v| m f in (x, v)dxdv < +∞, (2.24)
for some m > 6.

Now let f ∈ L ∞ ([0 , T ], L 1 ∩ L ∞ (R 3 × R 3
)) be a weak solution provided by theorem 2.1 with initial data

f in . If f in satisfies ∀k ≥ 1, R 3 ×R 3 |v| k f in (x, v)dxdv ≤ (C 0 k) k 3 , (2.25)
for some constant C 0 independent of k, then f the solution to the Cauchy problem for the magnetized Vlasov-Poisson system is unique and verifies

sup [0 ,T ] sup p≥1 ρ(t) p p < +∞. (2.26) REMARK 2.7.
In our framework, an important difference with [START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF] is that the uniqueness criterion isn't given by the inequality on the macroscopic density (2.26) but rather the stronger assumption on the moments of the solution (2.25).

As mentioned above, the assumptions of theorem 2.6 are less restrictive than the condition (2.20) and thus allow us to consider initial data with unbounded macroscopic density. This result is illustrated by the following theorem ([14, theorem 1.3]): THEOREM 2.8 (Miot, [START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF]). There exists f in ≥ 0 a.e. such that

f in ∈ L 1 ∩ L ∞ (R 3 × R 3 )
satisfying the assumptions of theorem 2.6 and such that

ρ 0 (x) = 4π 3 ln -(|x|), ∀x ∈ R 3 , (2.27)
where ln -= max(-ln(x), 0) is the negative part of the function ln.

In section 4, we will detail the proof of theorem 2.6 first because it is the main result of this section. Then we will present the proof of theorem 2.4 where ingredients from the proof of theorem 2.6 are used, notably the boundedness of the velocity characteristic. However in theorem 2.4 we require a condition on the space moment of the initial data (2.19) which isn't the case in theorem 2.6.

Finally, with regards to uniqueness for Vlasov-Poisson, a major open problem is finding a uniqueness condition in the periodic case. Indeed, it would be very interesting to see if the conditions found in [START_REF] Loeper | Uniqueness of the solution to the Vlasov-Poisson system with bounded density[END_REF][START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF] could be adapted to the periodic case.

Proofs regarding propagation of velocity moments

In this section, we shall denote by C a constant that can change from one line to another but that only depends on

E(0), f in 1 , f in ∞ . (3.28)
As mentioned above, the whole proof is conducted using smooth solutions. We consider k > 2 and ε > 0 small enough, say ε ∈ ]0 , ε 0 [ with ε 0 ≤ (k-2) 2k . As said in the introduction, the main difference with the analysis in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF] is that we're going to show propagation of moments for all time by using an induction argument using the cyclotron period

T c = 1 B ∞
. We begin with the initialization, so we're first going consider T > 0 with T ≤ T B , where T B is the unique real number such that

T B ∈ R * + and B ∞ T B exp(T B B ∞ ) = a, (3.29)
with a > 0. In our method, since we can only obtain estimates on Q for T B T c , we just need a small enough so we set a = 2 -10 .

Thus, we show propagation of velocity moments on [0 , T B ] using the following result.

PROPOSITION 3.1. For all T > 0 such that T ≤ T B , (2.16) is verified. More precisely we have the following estimate on Q(t, t)

Q(t, t) ≤ C exp(T B ∞ ) 2 5 (T 1 2 + T 7 5 ) (3.30)
with C that only depends on

k, E(0), f in 1 , f in ∞ , M k (0). REMARK 3.2.
This estimate is the analogous of the estimate (13) in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF]. In our magnetized framework, we only manage to generalize this result up to the time T B .

The following section will be devoted to the proof of this proposition, and just like in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF] the proof is done in three steps which correspond to proposition 3.3, proposition 3.6 and proposition 3.7.

The case T ≤ T

B PROPOSITION 3.3. For any 0 ≤ δ ≤ t ≤ T ≤ T B we have: Q(t, δ) ≤ C(δQ(t, δ) 4 3 + δ 1 2 (1 + M 2+ε (T )) 1 2 ) (3.31) Proof. Let (t, x * , v * ) ∈ [0 , T ] × R 3 × R 3 and set (X * , V * )(s) = (X, V )(s; t, x * , v * ). For any δ ∈ [0 , t] we have by definition of E t t-δ |E(s, X(s; t, x * , v * ))| ds ≤ t t-δ ρ(s, x)dx 4π |x -X * (s)| 2 ds
Our objective in the rest of this section will be to estimate the integral:

I * (t, δ) := t t-δ ρ(s, x)dx |x -X * (s)| 2 ds = t t-δ f (s, x, v)dvdx |x -X * (s)| 2 ds (3.32)
Now we will use a procedure that is inspired from [START_REF] Schäffer | Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions[END_REF] which consists in splitting [t -δ , t] × R 3 × R 3 into three parts. Here the partition is slightly different because, following [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF], we introduce ε > 0.

G = {(s, x, v) : min(|v| , |v -V * (s)|) < P } , B = {(s, x, v) : |x -X * (s)| ≤ Λ ε (s, v)} \G, U = [t -δ , t] × R 3 × R 3 \(G ∪ B), with P = 2 10 Q(t, δ) exp(δ B ∞ ) and Λ ε (s, v) = L(1 + |v| 2+ε ) -1 |v -V * (s)| -1 (3.33)
and L > 0 to be fixed later. The main difference here with [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF] is the definition of P , because the added magnetic modifies the evolution of the characteristic in velocity V (s). Furthermore, we take the same numerical constant 2 10 in the definition of P as in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF] is (in truth this constant just needs to be large enough). Using obvious notations, we write

I * = I G * + I B * + I U * .
The first two integrals will be more straightforward to estimate than I U * , which involves the set U considered as the "ugly set" according to [START_REF] Glassey | The Cauchy Problem in Kinetic Theory[END_REF].

The first two contributions I G * , I B * are treated the same in both magnetized and unmagnetized cases, simply because the modifications made to the sets G, B to take into account the added magnetic field don't change the computations required to estimate I G * and I B * . We succinctly present how to control both integrals following the calculations from [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF]. The first bound is obtained by using a standard functional inequality.

For

κ ∈ L ∞ (R 3 ) ∩ L 5 3 (R 3 ) we have κ * |•| -2 ∞ ≤ c κ 5 9 5 3 κ 4 9 ∞ , (3.34)
with c a numerical constant. We apply (3.34) to the quantity:

ρ G (s, x) = B(0,P )∪B(V * (s),P ) f (s, x, v)dv ≤ ρ(s, x), (3.35)
which implies the following control on I G * (t, δ):

I G * (t, δ) ≤ C(δP 4
3 ).

(3.36)

To estimate the contribution on B, we first integrate in the space variable using a spherical change of variable.

I B * (t, δ) ≤ t t-δ v |x-X * (s)|≤Λε(s,v) f (s, x, v)dx |x -X * (s)| 2 dvds, ≤ t t-δ v 4π Λε(s,v) 0 f in ∞ r 2 r 2 dr dvds, ≤ C t t-δ v L (1 + |v| 2+ε ) |v -V * (s)| dvds, ≤ C t t-δ |v|≤|v-V * (s)| L (1 + |v| 2+ε ) |v| dv + |v|>|v-V * (s)| L (1 + |v -V * (s)| 2+ε ) |v -V * (s)| dv ds, ≤ C t t-δ v L (1 + |v| 2+ε ) |v| dvds, ≤ C t t-δ R 3 Lr (1 + r 2+ε ) drds, ≤ CδL.
The last contribution I U * (t, δ) can be written

I U * (t, δ) = t t-δ f (s, x, v)1 U (s, x, v) |x -X * (s)| 2 dvdxds = t t-δ 1 U (s, X(s), V (s)) |X(s) -X * (s)| 2 dsf (t, x, v)dvdx (3.
37) where we have the obvious notation (X, V )(s) = (X, V )(s; t, x, v). Estimating this quantity is difficult and will occupy us for the rest of the proof of proposition 3.3.

The following lemma is very important in our proof because it highlights why we need to use the induction procedure mentioned above. In the unmagnetized case , we estimate I U * by noticing that because of the definition of U , the characteristic V (s) stays close to v on [t -δ , t] because v is large compared to P and P is much larger Q(t, δ) which quantifies the total variation of V (s) on [t -δ , t]. However in the magnetized case, this stays true only under the condition (3.29) because if the magnetic field is large than the variations of V (s) on [t -δ , t] can also be very large compared to P .

LEMMA 3.4. Let s 1 ∈ [t -δ , t] such that (s 1 , X(s 1 ), V (s 1 )) ∈ U , then for all s ∈ [t -δ , t] we have 2 -1 |v| ≤ |V (s)| ≤ 2 |v| , (3.38) and 2 -1 |v -v * | ≤ |V (s) -V * (s)| ≤ 2 |v -v * | . ( 3
[t -δ , t] V (s) -V (s 1 ) = s s 1 E(τ, X(τ ))dτ + s s 1 V (τ ) ∧ B(τ, X(τ ))dτ
Furthermore, one of the properties of the characteristics is that we have for all (x, v) ∈ R 3 ×R 3 and τ, t ∈ R + X(τ ; t, x, v) = X(τ ; 0, X(0; t, x, v), V (0; t, x, v))

and also that the function

(x, v) → X(τ ; t, x, v) is a C 1 -diffeomorphism which means that sup t t-δ |E(s, X(s; t, x, v))| ds, (x, v) ∈ R 3 × R 3 = sup t t-δ |E(s, X(s; 0, x, v))| ds, (x, v) ∈ R 3 × R 3 = Q(t, δ).
Thus we can write

|V (s) -V (s 1 )| ≤ Q(t, δ) + B ∞ δ V (s 1 ) + s s 1 |V (τ ) -V (s 1 )| dτ
which is a Grönwall inequality, so we finally have for all s ∈ [t -δ , t]

||V (s)| -|V (s 1 )|| ≤ |V (s) -V (s 1 )| ≤ (Q(t, δ) + B ∞ δ |V (s 1 )|) exp(δ B ∞ ). (3.41)
This last inequality highlights the main difference with the unmagnetized case, indeed when we use V (s 1 ) as a reference point to quantity the variation of V (s), we see that the added term

B ∞ δ |V (s 1 )| exp(δ B ∞ )
, which is just the added variation of the velocity characteristic resulting from the magnetic field, is potentially unbounded. This is due to the fact that even if 0 ≤ δ ≤ T , B ∞ is potentially large. This is the reason we introduce the time T B which depends on the cyclotron frequency B ∞ . Now using this last inequality, thanks to the relation between P and Q(t, δ) given in (3.33), to (3.40), and to (3.29) (because t ≤ T B ) we have

|V (s)| ≤ |V (s 1 )| (1 + B ∞ δ exp(δ B ∞ )) + 2 -10 P ≤ |V (s 1 )| 1 + 2 -10 + 2 -10 = |V (s 1 )| 1 + 2 -9
and using the same relations but this time for -|V (s 1 )| we can write

|V (s 1 )| 1 -2 -10 -2 -10 ≤ |V (s 1 )| (1 -B ∞ δ exp(δ B ∞ )) -2 -10 P ≤ |V (s 1 )| (1 -B ∞ δ exp(δ B ∞ )) -Q(t, δ) exp(δ B ∞ ) ≤ |V (s)|
These inequalities are valid for all s ∈ [t -δ , t] and so in particular for s = t. And so we can write which is equivalent to (3.38). Now let's look at the inequality (3.39). Like for (3.38), we try to write a Grönwall inequality but this time on

Z(s) = |(V (s) -V * (s)) -(V (s 1 ) -V * (s 1 ))|. (V (s) -V * (s)) -(V (s 1 ) -V * (s 1 )) = s s 1 E(τ, X(τ ))dτ + s s 1 V (τ ) ∧ B(τ, X(τ ))dτ - s s 1 E(τ, X * (τ ))dτ - s s 1 V * (τ ) ∧ B(τ, X * (τ ))dτ.
This allows us to write

Z(s) ≤ 2Q(t, δ) + s s 1 (V (τ ) ∧ B(τ, X(τ )) -V * (τ ) ∧ B(τ, X * (τ ))) dτ ≤ 2Q(t, δ) + s s 1 |V (τ ) ∧ (B(τ, X(τ )) -B(τ, X * (τ )))| + |(V * (τ ) -V (τ )) ∧ B(τ, X * (τ ))| dτ ≤ 2Q(t, δ) + 2δ B ∞ 2 |V (s 1 )| + B ∞ δ |(V * (s 1 ) -V (s 1 ))| + s s 1 |(V (τ ) -V * (τ )) -(V (s 1 ) -V * (s 1 ))| dτ
where in the second term in the last inequality we used the bound

|V (s)| ≤ |V (s 1 )| (1+2 -9 ) ≤ 2 |V (s 1 )| that we established just before.
Thus we have our Grönwall inequality on Z(s) which gives us

Z(s) ≤ (2Q(t, δ) + 4δ B ∞ |V (s 1 )| + B ∞ δ |(V * (s 1 ) -V (s 1 ))|) exp(δ B ∞ ). ( 3
Z(s) ≤ (2Q(t, δ) + 5 B ∞ δ |(V * (s 1 ) -V (s 1 ))|) exp(δ B ∞ ), (3.44)
and so by following exactly the same method as above to obtain (3.38), namely using (3.33), (3.40), and (3.29), from the last inequality we can write

|V (s) -V * (s)| ≤ |V (s 1 ) -V * (s 1 )| (1 + 5 B ∞ δ exp(δ B ∞ )) + 2 -9 P ≤ |V (s 1 ) -V * (s 1 )| 1 + 5 • 2 -10 + 2 -9 ≤ |V (s 1 ) -V * (s 1 )| 1 + 2 -6 and |V (s 1 ) -V * (s 1 )| 1 -2 -7 -2 -7 ≤ |V (s 1 ) -V * (s 1 )| 1 -5 • 2 -10 -2 -9 ≤ |V (s 1 ) -V * (s 1 )| (1 -5 B ∞ δ exp(δ B ∞ )) -2 -9 P ≤ |V (s 1 ) -V * (s 1 )| (1 -5 B ∞ δ exp(δ B ∞ )) -2Q(t, δ) exp(δ B ∞ ) ≤ |V (s) -V * (s)|
Now we look at the other case.

• Second case |V (s 1 ) -V * (s 1 )| < |V (s 1 )|:
From (3.43), we deduce

Z(s) ≤ (2Q(t, δ) + 5 B ∞ δ |V (s 1 )|) exp(δ B ∞ ). ( 3
|V (s) -V * (s)| ≤ |V (s 1 ) -V * (s 1 )| + |V (s 1 )| 5 B ∞ δ exp(δ B ∞ ) + 2 -9 P ≤ |V (s 1 )| 1 + 5 • 2 -10 + 2 -9 ≤ |V (s 1 )| 1 + 2 -6 .
We use exactly the same inequalities to get a lower bound on |V (s) -V * (s)|, but this time the hypothesis is used in the form -|V (s 1 ) -

V * (s 1 )| > -|V (s 1 )| |V (s 1 ) -V * (s 1 )| 1 -2 -7 -2 -7 ≤ |V (s 1 ) -V * (s 1 )| 1 -5 • 2 -10 -2 -9 ≤ |V (s 1 ) -V * (s 1 )| -|V (s 1 )| 5 • 2 -10 + 2 -9 ≤ |V (s 1 ) -V * (s 1 )| (1 -5 B ∞ δ exp(δ B ∞ )) -2 -9 P ≤ |V (s 1 ) -V * (s 1 )| (1 -5 B ∞ δ exp(δ B ∞ )) -2Q(t, δ) exp(δ B ∞ ) ≤ |V (s) -V * (s)| LEMMA 3.5. For any (x, v) ∈ R 6 we have t t-δ 1 U (s, X(s), V (s)) |X(s) -X * (s)| 2 ds ≤ C 1 + |v| 2+ε L . (3.46)
Proof. If (s, X(s), V (s)) / ∈ U for all s ∈ [t -δ , t] then the estimate (3.46) is verified. Now we assume that there exists s 1 ∈ [t -δ , t] such that (s 1 , X(s 1 ), V (s 1 )) ∈ U , then thanks to lemma 3.4 we can write

Λ ε (s, V (s)) ≥ L(1 + (2 |v|) 2+ε ) -1 (2 |v -v * |) -1 ≥ 2 -3-ε Λ ε (t, v) (3.47)
and hence

1 U (s, X(s), V (s)) |X(s) -X * (s)| 2 ≤ 1 R 3 \B(X * (s),2 -3-ε Λε(t,v)) (X(s)) |X(s) -X * (s)| 2 ≤ h(|Y (s)|), (3.48)
where Y (s) = X(s)-X * (s) and h(u

) = min(|u| -2 , 4 3+ε Λ ε (t, v) -2
|Y (s)| ≥ Y (s 0 ) + (s -s 0 )Y (s 0 ) - s s 0 (s -u)Y (u)du ≥ Y (s 0 ) + (s -s 0 )Y (s 0 ) -2 |s -s 0 | (Q(t, δ) + δ B ∞ |v -v * |). Now we consider s = s 0 that minimizes |Y (s)| 2 when s ∈ [t -δ , t], then this implies (s - s 0 )Y (s 0 ) • Y (s 0 ) ≥ 0 and so Y (s 0 ) + (s -s 0 )Y (s 0 ) 2 ≥ Y (s 0 ) 2 |s -s 0 | 2 (3.
Y (s 0 ) -2(Q(t, δ) + δ B ∞ |v -v * |) ≥ |v -v * | (2 -1 -2 -8 |v -v * | exp(-δ B ∞ ) -2δ B ∞ ) ≥ |v -v * | (2 -1 -2 -8 -2δ B ∞ ).
We need the quantity (2 -1 -2 -8 -2δ B ∞ ) to be strictly positive and so once again we need the condition (3.29) for δ B ∞ to be small and this inequality to be verified. Now we have

|Y (s)| ≥ α |v -v * | |s -s 0 | with α > 0.
Just as in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF], we bring this inequality into (3.48), integrate with respect to the time variable and estimate the integral as follows to obtain (3.46):

t t-δ h(|Y (s)|)ds ≤ t t-δ h(α |v -v * | |s -s 0 |)ds, ≤ +∞ 0 h(α |v -v * | r)dr, = (α |v -v * |) -1 +∞ 0 h(r)dr, = (α |v -v * |) -1 2 -3-ε Λε(t,v) 0 4 3+ε Λ ε (t, v) -2 dr + +∞ 2 -3-ε Λε(t,v) 1 r 2 dr , = (α |v -v * |) -1 (2 3+ε Λ ε (t, v) -1 + 2 3+ε Λ ε (t, v) -1 ), ≤ C 1 + |v| 2+ε L .
Now integrating in x, v and using the mass conservation, we finally obtain

I U * (t, δ) ≤ CL -1 (1 + M 2+ε (T )). (3.50)
We gather all the above estimates to conclude

I * (t, δ) ≤ C(δ (Q(t, δ) exp(δ B ∞ )) 4 3 + δL + L -1 (1 + M 2+ε (T )))
≤ C(δQ(t, δ)

4 3 + δL + L -1 (1 + M 2+ε (T )))
where the last inequality is justified by the fact that thanks to (3.29) we have exp(δ B ∞ ) ≤ exp(g(2 -10 )) where g is the inverse of the function x → x exp(x) on R + . We conclude in the same way as in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF], firstly by optimizing the parameter L and then by noticing that the pair (x * , v * ) is arbitrary so that we have

sup I * (t, δ), (x * , v * ) ∈ R 3 × R 3 ≥ Q(t, δ). (3.51)
Finally, we obtain (3.31).

The next two propositions allow us to conclude. The proof of proposition 3.6 is identical to the one in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF] because it doesn't rely on the characteristics of the system, but rather on real analysis arguments. In an effort of clarity, and also because some arguments of the proof are more detailed in this paper than in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF], we place the proof of proposition 3.6 in the appendix.

PROPOSITION 3.6. For any t ∈ [0 , T ] with T ≤ T B , we have

Q(t, t) ≤ C(t 1 2 + t)(1 + M 2+ε (T )) 4 7 .
(3.52)

Now we state the last result necessary in the proof of proposition 3.1.

PROPOSITION 3.7. There exists τ (ε, k) > 0 such that for any t ∈ [0 , T ] we have 5 ).

Q(t, t) ≤ C(1 + M k (0)) τ (ε,k) exp(T B ∞ ) 2 5 (T 1 2 + T 7
(3.53) REMARK 3.8. We notice that we obtain the same estimate (3.53) as in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF] when the magnetic field B is zero.

Proof. Using the same argument as in (2.3), we can write

M k (t) ≤ 2 k-1 exp(kt B ∞ ) M k (0) + N (T ) k f in 1 (3.54)
with N (T ) defined in theorem 2.1. Now to obtain the desired estimate we have to bound M 2+ε (t), which we manage with the Hölder inequality. Thus for any t ∈ [0 , T ] we have

|v| 2+ε f (t, x, v)dxdv ≤ |v| 2 f (t, x, v)dxdv 2+ε-k 2-k |v| k f (t, x, v)dxdv ε k-2 .
(3.55) With the conservation of the energy E(t), this Hölder inequality implies that

M 2+ε (T ) ≤ CM k (T ) ε k-2 (3.56)
and bringing this inequality into (3.54) it yields

M 2+ε (T ) ≤ C exp( kε k -2 T B ∞ ) M k (0) + N (T ) k f in 1 ε k-2 . (3.57)
Now thanks to (3.52) we can deduce

M 2+ε (T ) ≤ C exp( kε k -2 T B ∞ ) M k (0) + N (T ) k f in 1 ε k-2 ≤ C exp( kε k -2 T B ∞ ) M k (0) + (T 1 2 + T ) k (1 + M 2+ε (T )) 4k 7 f in 1 ε k-2 ≤ C exp( kε k -2 T B ∞ )(1 + M k (0)) ε k-2 (T 1 2 + T ) kε k-2 (1 + M 2+ε (T )) 4kε 7(k-2)
Like in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF], we write σ( , k) = 4kε 7(k-2) and notice that if we take ε small enough we have

σ( , k) ∈ ]0 , 1[. More precisely, if ε < ε 0 then σ( , k) ≤ 2 7 , kε k-2 ≤ 1 2 and ε k-2 ≤ 1 2k
and we find

M 2+ε (T ) ≤ C(1 + M k (0)) 1 2k exp(T B ∞ ) 1 2 (1 + T ) 1 2 (1 + M 2+ε (T )) 2 7 , (3.58)
where we used T 1 2 + T ≤ 2(1 + T ). Now since the right term in the last inequality is larger than 1 up to a constant C, we deduce that

(1 + M 2+ε (T )) 5 7 ≤ C(1 + M k (0)) 1 2k exp(T B ∞ ) 1 2 (1 + T ) 1 2 (3.59) which finally yields 1 + M 2+ε (T ) ≤ C(1 + M k (0)) 7 10k exp(T B ∞ ) 7 10 (1 + T ) 7 10 .
(3.60)

Then, using (3.52) again we deduce 5 ).

Q(t, t) ≤ C(T 1 2 + T )(1 + M k (0)) 2 5k exp(T B ∞ ) 2 5 (1 + T ) 2 5 ≤ C(1 + M k (0)) 2 5k exp(T B ∞ ) 2 5 (T 1 2 + T 7
This concludes the proof of proposition 3.7 and proposition 3.1.

The case T ≥ T B

We conclude the proof of (

Q(t, t) ≤ n-1 p=0 Q((p + 1)T B , T B ) + Q(t, t r ) ≤ C n p=0 (1 + M k (pT B )) 2 5k exp(T B B ∞ ) 2 5 (T 1 2 B + T 7 5 B ).
Furthermore, we can show by an immediate induction that for all p ∈ N with p ≤ n, M k (pT B ) is bounded such that (2.14). This concludes the proof of proposition 3.9 and theorem 2.1.

M k (pT B ) ≤ C p (k, B ∞ , E(0), f in 1 , f in ∞ , M k (0)). (3.61) This is just because M k (pT B ) ≤ C 1 ⇒ Q((p + 1)T B , T B ) ≤ C 2 ⇒ M k ((p + 1)T B ) ≤ C 3 with C 1 , C 2 , C 3 depending on

Proofs regarding uniqueness

The subsections 4.1, 4.2 and 4.3 will be devoted to the proof of theorem 2.6 and subsection 4.4 will be devoted to the proof of theorem 2.4. In this section, we shall denote by C a constant that can change from one line to another but that only depends on

E(0), f in 1 , f in ∞ , T, |v| m f in . (4.62)
4.1 Proof of the estimate on the L p norms of ρ (2.26)

We consider f in that satisfies the assumptions of theorem 2.6 and let f be the solution given by theorem 2.1 with initial data f in . By construction, we have propagation of moments:

sup t∈[0 ,T ] R 3 ×R 3 |v| m f (t, x, v)dxdv < +∞. (4.63)
Now thanks to a classical velocity moment inequality, we show how to control the L p norms of the macroscopic density with velocity moments, this inequality is given by

ρ(t) k+3 3 ≤ C f (t) k k+3 ∞ M k (t) 3 k+3 . (4.64)
with C independent of k. Since we want ρ to verify (2.26), this means that we need to prove

∀k ≥ 1, sup t∈[0 ,T ] ( f (t) k k+3 ∞ M k (t) 3 k+3 ) ≤ Ck. (4.65) Since the solution f ∈ L ∞ ([0 , T ], L ∞ (R 3 × R 3 )), we finally need to show ∀k ≥ 1, sup t∈[0 ,T ] M k (t) 3 k+3 ≤ Ck. (4.66)
First, we recall that thanks to (4.63) where m > 6 we can infer that ρ ∈ L ∞ ([0 , T ], L p (R 3 )) with p = m+3 3 > 3 and following (4.86

) we have E ∈ L ∞ ([0 , T ], L ∞ (R 3 )). Then we write d dt |V (t, x, v)| k = k |V (t, x, v)| k-1 V (t, x, v) • V (t, x, v) |V (t, x, v)| ,
and thanks to the bound on E and the definition of the characteristics (1.8) we can infer that for all k > m

|V (t, x, v)| k ≤ |v| k + k t 0 |V (s, x, v)| k-1 (E(s, X(s, x, v)) + V (s, x, v) ∧ B(s, X(s, x, v))) • V (s, x, v) |V (s, x, v)| ds ≤ |v| k + k E ∞ t 0 |V (s, x, v)| k-1 ds.
Since the contribution of magnetic field B vanishes, the following computations are the same as in the unmagnetized case [START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF]. In an effort to be clear, we explicit these computations nonetheless.

Integrating this last inequality with respect to f in (x, v)dxdv we get

M k (t) ≤ M k (0) + k E ∞ t 0 M k-1 (s)ds. (4.67)
Thus by induction we deduce that sup t∈[0 ,T ] M k (t) is finite for all k > m. furthermore, by another classical velocity moment inequality we obtain that

M k-1 (s) ≤ f (s) 1 k 1 M k (s) k-1 k . (4.68) Since f (t) 1 is conserved, we get M k (t) ≤ M k (0) + Ck t 0 M k (s) k-1 k ds, . (4.69)
Differentiating this inequality allows us to write

M k (t) ≤ CkM k (t) k-1 k ⇔ d dt (M k (t) 1 k ) ≤ C ⇒ sup t∈[0 ,T ] M k (t) 1 k ≤ M k (0) 1 k + C.
By assumption on M k (0) we find for all t ∈ [0 , T ]

M k (t) 1 k ≤ (C 0 k) 1 3 + C ≤ (Ck) 1 3 ≤ (Ck) 1 3 + 1 k , (4.70)
which finally implies that sup

t∈[0 ,T ] M k (t) 3 k+3 ≤ Ck. (4.71)

Estimate on the characteristics

We consider two solutions

f 1 , f 2 ∈ L ∞ ([0 , T ], L 1 ∩ L ∞ (R 3 × R 3 )) such that ρ 1 , ρ 2 verify ρ 1 , ρ 2 ∈ L ∞ ([0 , T ], L p (R 3 )) (4.72)
for some p > 3. This regularity on ρ 1,2 is guaranteed by the condition (2.24) thanks to the estimate (4.64). Then we write Y 1 = (X 1 , V 1 ) and Y 2 = (X 2 , V 2 ) for the corresponding characteristics, which are both solutions to (1.8) with t = 0. This means we can simplify the notation and will write Y i (t; 0, x, v) = Y i (t, x, v), i = 1, 2. Regarding the existence of such characteristics, the condition (4.72) yields sufficient regularity on the electric field E i , i = 1, 2, so that with the added regularity assumption on the magnetic field (1.3) we can define weak characteristics thanks to theorem III.2 (section III.2) in [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]. Now we introduce the distance

D(t) = R 3 ×R 3 |X 1 (t, x, v) -X 2 (t, x, v)| f in (x, v)dxdv. (4.73)
From (1.8) we can infer that

X 1 (t, x, v) -X 2 (t, x, v) = t 0 s 0 E 1 (τ, X 1 (τ, x, v)) -E 2 (τ, X 2 (τ, x, v)) + V 1 (τ, x, v) ∧ B(τ, X 1 (τ, x, v)) -V 2 (τ, x, v) ∧ B(τ, X 2 (τ, x, v))dτ ds (4.74)
which yields that

D(t) ≤ t 0 s 0 R 6 |E 1 (τ, X 1 (τ, x, v)) -E 2 (τ, X 2 (τ, x, v))| + |V 1 (τ, x, v) ∧ B(τ, X 1 (τ, x, v)) -V 2 (τ, x, v) ∧ B(τ, X 2 (τ, x, v))| f in (x, v)dxdvdτ ds ≤ t 0 s 0 R 6 |E 1 (τ, X 1 (τ, x, v)) -E 2 (τ, X 2 (τ, x, v))| f in (x, v)dxdvdτ ds + B ∞ t 0 s 0 R 6 |V 1 (τ, x, v) -V 2 (τ, x, v)| f in (x, v)dxdvdτ ds + t 0 s 0 R 6 |V 2 (τ, x, v)| |B(τ, X 1 (τ, x, v)) -B(τ, X 2 (τ, x, v))| f in (x, v)dxdvdτ ds = I(t) + J(t) + K(t)
The term I(t) is the quantity estimated thanks to the method in [START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF]. As for the other two terms J(t) and K(t), since we want to use the same method as in [START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF] which is to exploit the fact that the characteristics of the Vlasov equation verify an ODE of order 2, we need them to be controlled by

t 0 s 0 D(τ ) 1-3
p dτ ds. This is true and the estimates are given in the following proposition. PROPOSITION 4.1. For all t ∈ [0 , T ] and for all p > 3, we have the following estimates: (4.77) with

I(t) ≤ CpC ρ 1 ,ρ 2 t 0 s 0 D(τ )
C ρ 1 ,ρ 2 = max 1 + ρ 1 L ∞ ([0 ,T ],L p ) , 1 + ρ 2 L ∞ ([0 ,T ],L p ) , K B,p = 2 B W 1,∞ E 2 ∞ exp(T B ∞ ), K B = 2 B W 1,∞ exp(T B ∞ ),
and where C denotes a constant that depends only on T, f in ∞ , f in 1 . Proof of proposition 4.1. As said above, the term I(t) is the quantity estimated thanks to the method in [START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF], so we treat it identically to find the estimate (4.75).

Let's first look at the term K(t). Since B ∈ W 1,∞ (R + × R 3 ) then for all t ∈ [0 , T ] and

α ∈ ]0 , 1] B(t) ∈ C 0,α (R 3 ) (4.78) with Hölder coefficient C B(t) verifying C B(t) ≤ max(2 B ∞ , ∇B ∞ ) ≤ 2 B W 1,∞ .
Then we simply have for all p > 3

K(t) ≤ 2 B W 1,∞ t 0 s 0 R 6 |V 2 (τ, x, v)| |X 1 (τ, x, v) -X 2 (τ, x, v)| 1-3 p f in (x, v)dxdvdτ ds (4.
79) Now we need to estimate the velocity characteristic V 2 , and using (1.8) we can write once again

|V (t, x, v)| ≤ |v| + t 0 |E(s, X(s, x, v))| ds + B ∞ t 0 |V (s, x, v)| ds ≤ |v| + T E ∞ + B ∞ t 0 |V (s, x, v)| ds.
This classical Grönwall inequality yields for all t ∈ [0 , T ]

|V (t, x, v)| ≤ (|v| + T E ∞ ) exp(t B ∞ ).
(4.80)

So that we can write Then we estimate K 1 (t) by writing

K(t) ≤ 2 B W 1,∞ exp(T B ∞ ), t 0 s 0 R 6 (|v| + T E 2 ∞ ) |X 1 (τ, x, v) -X 2 (τ, x, v)| 1-3 p f in (x, v)dxdvdτ ds, = K 1 (t) + K 2 (t).
f in = (f in ) 1 p (f in ) 1 p
where 1 p + 1 p = 1, so that with the Hölder inequality applied to |v| (f in )

1 p and |X 1 (τ, x, v) -X 2 (τ, x, v)| 1-3 p (f in )
1 p with the exponents p and p we have This concludes the proof of (4.77).

K 1 (t) ≤ K B R 6 |v| p f in (x, v)dxdv 1 p t 0 s 0 R 6 |X 1 (τ, x, v) -X 2 (τ, x, v)| (1-3 p )p f in (x, v)dxdv
To estimate the last term J(t), we also use (1.8) to obtain a Grönwall inequality on |V 1 (t) -V 2 (t)|, and since the computations are complicated we write V 1,2 (s), X 1,2 (s) for the characteristics. First we write

|V 1 (t) -V 2 (t)| ≤ t 0 |E 1 (s, X 1 (s)) -E 2 (s, X 2 (s))| ds + B ∞ t 0 |V 1 (s) -V 2 (s)| ds + t 0 |V 2 (s)| |B(s, X 1 (s)) -B(s, X 2 (s))| ds.
Now using (4.78) and (4.80) we deduce

|V 1 (t) -V 2 (t)| ≤ t 0 |E 1 (s, X 1 (s)) -E 2 (s, X 2 (s))| ds + B ∞ t 0 |V 1 (s) -V 2 (s)| ds + (K B |v| + K B,p ) t 0 |X 1 (s) -X 2 (s)| 1-3 p ds
which is just the Grönwall inequality on |V 1 (t) -V 2 (t)| we were looking for and which yields

|V 1 (t) -V 2 (t)| ≤ t 0 |E 1 (s, X 1 (s)) -E 2 (s, X 2 (s))| + (K B |v| + K B,p ) |X 1 (s) -X 2 (s)| 1-3 p ds + t 0 s 0 |E 1 (τ, X 1 (τ )) -E 2 (τ, X 2 (τ ))| + (K B |v| + K B,p ) |X 1 (τ ) -X 2 (τ )| 1-3 p dτ × B ∞ exp((t -s) B ∞ )ds.
Now we insert this inequality in the definition of J(t) to obtain

J(t) ≤ B ∞ t 0 s 0 τ 0 R 6 |E 1 (u, X 1 (u)) -E 2 (u, X 2 (u))| + (K B |v| + K B,p ) |X 1 (u) -X 2 (u)| 1-3 p × f in (x, v)dxdvdudτ ds + B 2 ∞ exp(T B ∞ )× t 0 s 0 τ 0 u 0 R 6 |E 1 (w, X 1 (w)) -E 2 (w, X 2 (w))| + (K B |v| + K B,p ) |X 1 (w) -X 2 (w)| 1-3 p × f in (x, v)dxdvdwdudτ ds. ( 4
(t) to bound K B |v| |X 1 -X 2 | 1-3 p .
This gives the desired estimate (4.76) on J(t):

J(t) ≤ B ∞ t 0 s 0 τ 0 (CpC ρ 1 ,ρ 2 + (CpK B + K B,p )) D(u) 1-3 p dudτ ds + B 2 ∞ exp(T B ∞ ) t 0 s 0 τ 0 u 0 (CpC ρ 1 ,ρ 2 + (CpK B + K B,p )) D(w) 1-3 p dwdudτ ds.

A second order inequality on D(t)

We begin by looking at the dependence of K B,p with respect to p. The only term in K B,p which depends on p is E 2 ∞ , and since ρ 2 ∈ L ∞ ([0 , T ], L p (R 3 )) with p > 3, then we can deduce the desired L ∞ bound on E 2 because for all t ∈ [0 , T ]

E 2 (t) ∞ ≤ 1 |x|≥1 ∇G 3 ∞ ρ 2 (t) 1 + 1 |x|<1 ∇G 3 q ρ 2 (t) p (4.85) with 1 p + 1 q = 1.
From this last inequality we can finally deduce

E 2 ∞ ≤ C(1 + ρ 2 L ∞ ([0 ,T ],L p ) ), (4.86)
where C depends only on f in 1 . Now we consider that the solutions f 1 , f 2 verify the assumptions of theorem 2.6. This means that max( ρ 1 L ∞ ([0 ,T ],L p ) , ρ 2 L ∞ ([0 ,T ],L p ) ) ≤ Cp for all p ≥ 1, and so thanks to (4.86) and proposition 4.1 we have for all p > 3

D(t) ≤ C 1 p 2 t 0 s 0 D(τ ) 1-3 p dτ ds + C 2 (p 2 + p) t 0 s 0 τ 0 D(u) 1-3 p dudτ ds + C 3 (p 2 + p) t 0 s 0 τ 0 u 0 D(w) 1-3 p dwdudτ ds, (4.87) where C 1 , C 2 , C 3 are constants that depend on T, f in ∞ , f in 1 , B ∞ , ∇B ∞ . Let F(t) = t 0 s 0 D(τ ) 1-3
p dτ ds. Since F is increasing by construction and since p > 3 we can finally conclude that

D(t) ≤ Cp 2 t 0 s 0 D(τ ) 1-3 p dτ ds, (4.88)
with C that depends on T, f in ∞ , f in 1 , B ∞ , ∇B ∞ . Finally, we obtain the same second order differential inequality as in [START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF], for all t ∈ [0 , T ] we have:

F (t) ≤ Cp 2 F(t). ( 4

.89)

From this inequality, we use the same method as in [START_REF] Miot | A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system[END_REF] to conclude that for all t ∈ [0 , T ] we have f 1 (t) = f 2 (t) a.e. on R 3 × R 3 . This concludes the proof of theorem 2.6.

Proof of theorem 2.4

We finish this section with the proof of theorem 2.4, which is the extension of Loeper's result [START_REF] Loeper | Uniqueness of the solution to the Vlasov-Poisson system with bounded density[END_REF] to the magnetized Vlasov-Poisson system. As said above, this proof was already done for B constant in [START_REF] Rege | The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity[END_REF]. In fact in [START_REF] Rege | The Vlasov-Poisson system with a uniform magnetic field: propagation of moments and regularity[END_REF], it was already proved that under the assumptions (2.21), (2.22), and (2.23), the macroscopic density ρ is bounded (2.20), and this proof doesn't change in the case of a general magnetic field.

Like in the theorem 2.6, we require additional assumptions on the moments of f in to obtain uniqueness (compared to the unmagnetized case). However, these assumptions on the moments aren't as strong as in theorem 2.6 because the boundedness of ρ is already a strong assumption.

To prove our theorem, we only need to adapt subsection 3.2 from [START_REF] Loeper | Uniqueness of the solution to the Vlasov-Poisson system with bounded density[END_REF]. Thus we consider two solutions of (1.1) f 1 , f 2 with initial datum f in that verifies the assumptions of theorem 2.4. Like in the previous proof, we write the corresponding densities, electric fields, and characteristics ρ 1 , ρ 2 , E 1 , E 2 , and Y 1 (t, x, v), Y 2 (t, x, v) = (X 1 (t, x, v), V 1 (t, x, v)), (X 2 (t, x, v), V 2 (t, x, v)).

To simplify the presentation, we will write Y i (t) for the characteristics. We define the following quantity Q:

Q(t) = 1 2 R 6 f in (x, v) |Y 1 (t, x, v) -Y 2 (t, x, v)| 2 dxdv. (4.90)
Now we differentiate Q (which we couldn't do with the distance D (4.73)) splitting the magnetic part of the Lorentz force V ∧ B like in the previous section:

Q(t) = R 6 f in (x, v)(Y 1 (t) -Y 2 (t)) • ∂ t (Y 1 (t) -Y 2 (t))dxdv, = R 6 f in (x, v)(X 1 (t) -X 2 (t)) • (V 1 (t) -V 2 (t))dxdv + R 6 f in (x, v)(V 1 (t) -V 2 (t)) • (E 1 (t, X 1 (t)) -E 2 (t, X 2 (t)))dxdv + R 6 f in (x, v)(V 1 (t) -V 2 (t)) • [V 2 (t) ∧ (B 1 (t, X 1 (t)) -B 2 (t, X 2 (t))] dxdv + R 6 f in (x, v)(V 1 (t) -V 2 (t)) • [(V 1 (t) -V 2 (t)) ∧ B(t, X 1 (t))] dxdv.
First, we notice that the last term is null, which means we only need to control the second to last term (due to the added magnetic field) which we denote P (t). The first term is bounded by Q(t) and the second term can be estimated using the analysis from [START_REF] Loeper | Uniqueness of the solution to the Vlasov-Poisson system with bounded density[END_REF] and is bounded by Q(t) ln( 1 Q(t) ). To control P (t) we first use the bound on the velocity characteristic (4.80).

P (t) ≤ B W 1,∞ R 6 f in (x, v) |V 1 (t) -V 2 (t)| |V 2 (t)| |X 1 (t) -X 2 (t)| dxdv, ≤ B W 1,∞ R 6 f in (x, v) |V 1 (t) -V 2 (t)| (|v| + T E 2 ∞ )e T B ∞ |X 1 (t) -X 2 (t)| dxdv, = R(t) + S(t).
We recall that since ρ 1,2 ∞ ≤ +∞ we can bound E 1,2 ∞ thanks to (4.86) and the interpolation inequality:

E i ∞ ≤ C( ρ 1 , ρ ∞ ) := C ρ , (4.91)
with i = 1, 2. This means we can simply estimate S(t) with the Cauchy-Schwarz inequality applied on the functions (f in )

S(t) ≤ T C ρ C B,T R 6 f in (x, v) |V 1 (t) -V 2 (t)| 2 1 2 R 6 f in (x, v) |X 1 (t) -X 2 (t)| 2 1 2 ≤ T C ρ C B,T Q(t), with C B,T = B W 1,∞ e T B ∞ .
To control R(t) we first use the Cauchy-Schwarz inequality and then the bound on the velocity characteristic (4.80), which also gives us a bound on the position characteristic. With this inequality we can show, using standard Grönwall type arguments, that Q(0) = 0 ⇒ Q(t) = 0 for all t ≥ 0, which concludes the proof of theorem 2.4.

R(t) ≤ C B,T R 6 f in (x, v) |v| |Y 1 (t) -Y 2 (t)| 2 dxdv ≤ C B,T R 6 f in (x, v) |v| |Y 1 (t) -Y 2 (t)| |V 1 | 2 + |V 2 | 2 + |X 1 | 2 + |X 2 | 2 1 2 dxdv ≤ C B,T Q(t) R 6 f in (x, v) |v| 2 |V 1 | 2 + |V 2 | 2 + |X 1 | 2 + |X 2 | 2 dxdv ≤ C B,T Q(t) R 6 f in (x, v) |v| 2 2 (|v| + T C ρ ) 2 e 2T B ∞ + |x| + T (|v| + T C ρ )e T B ∞

Appendix

As said above, we present a slightly more detailed version of the proof of proposition 3.6 compared to the one found in [START_REF] Pallard | Moment propagation for weak solutions to the Vlasov-Poisson system[END_REF].

17

 17

  )

  .39) Proof. First, because of the definition of U , we can write min(|V (s 1 )| , |V (s 1 ) -V * (s 1 )|) ≥ P. (3.40) Let's start by proving the first bound (3.38), thanks to (1.8) we have for all s ∈

  .43) We notice the term 4δ B ∞ |V (s 1 )| in the inequality that will make the analysis a bit more complicated compared to what was done to obtain (3.38), because we will have to compare |V (s) -V * (s)| to both |V (s 1 ) -V * (s 1 )| and |V (s 1 )|. To do this we distinguish between the two cases |V (s 1 ) -V * (s 1 )| ≥ |V (s 1 )| and |V (s 1 ) -V * (s 1 )| < |V (s 1 )|. • First case |V (s 1 ) -V * (s 1 )| ≥ |V (s 1 )|: From (3.43), we deduce

  49) and thanks to (3.39) we get |Y (s 0 )| ≥ 2 -1 |v -v * | and when we evaluate (3.39) in s 1 this also yields Q(t, δ) ≤ 2 -9 |v -v * | exp(-δ B ∞ ) so we have

0 ( 0 ( 3 p

 003 ) ≤ (CpK B + K B,p ) CpC ρ 1 ,ρ 2 + (CpK B + K B,p )) D(u) CpC ρ 1 ,ρ 2 + (CpK B + K B,p )) D(w)1dwdudτ ds.

( 4 .

 4 81)By applying Jensen's inequality for concave functions to x → x

R 6 1 p ≤ (C 0 p) 1 3≤

 611 |v| p f in (x, v)dxdv Cp. Furthermore, we can once again use the Jensen inequality because (1 -3 p )p = p

  .84) Like previously, we can use the Jensen inequality to bound the terms K B,p |X 1 -X 2 | 1-3 p , the relation (4.75) to bound the terms |E 1 -E 2 | and the Hölder inequality used to estimate K 1

1 2

 1 |V 1 (t) -V 2 (t)| and (f in

  )

1 2

 1 |X 1 (t) -X 2 (t)|.

.

  Thanks to the assumption (2.19) of theorem 2.4, I is bounded because we haveI ≤ C f in |v| 6 , f in |x| 4 . (4.92)From these estimates, we conclude thatd dt Q(t) ≤ CQ(t) 1 + ln 1 Q(t) (4.93) with C := C T, B W 1,∞ , ρ 1 , ρ ∞ , f in |v| 6 , f in |x| 4 .

  ). Since h is a non-increasing function, we look for a lower bound on |Y (s)|. For any s 0 ∈ [t -δ , t] we have, thanks to(3.39)

  2.16) by showing that Q(t, t) is bounded for all time. (0), we can reiterate the procedure on any time interval I p = [pT B , (p + 1)T B ]. Indeed, T, k and B ∞ are constants f (t) 1 and f (t) ∞ are conserved in time and the energy E(t) is bounded. This means we can write

PROPOSITION 3.9. The inequality (2.16) is valid for all T ≥ T B .

Proof. For all t ∈ [0 , T ], we write t = nT B +t r with n ∈ N and t r ∈ [0 , T B [. Since the constant C in proposition 3.1 depends only on T, k, B ∞ , f in 1 , f in ∞ , E(0) and M k

-1 |V (s)| ≤ |V (s)| 1 -2 -9 1 + 2 -9 ≤ |v| ≤ |V (s)| 1 + 2 -9 1 -2 -9 ≤ 2 |V (s)| (3.42)

Proof of proposition 3.6. Let t ∈ [0 , T ]. We note here H = 1 + M 2+ε (T ) and for any δ ∈ [0 , t] we define N 1 (t, δ) = δQ(t, δ) 4 3 and N 2 (t, δ) = (δH) 1 2 as in the left hand side of inequality (3.31). We set:

(4.98)

Then for any such t, we can write t = nc * H -1 7 + r with n ∈ N * and r < c * H -1 7 and thanks to the last inequality we obtain

c * (rH)

So that finally for all t ∈ [c * H -


James C Fettinger

Heinz-Bernard Kraatz

Elsje Rinaldo Poli
email: poli@u-bourgogne.fr

Alessandra Quadrelli

Raymund C Torralba

B Elsje

Prof Rinaldo Poli

Electrophilic Addition vs Electron Transfer for the Interaction of Ag + with Molybdenum

come

Introduction

Transition metal hydrides pose an interesting question in terms of their reaction with electrophiles. The electrophilic reagent can either attack the electron density at the hydride ligand, or the metal directly (when there are filled metal-based orbital), in both cases affording electrophilic addition products. In addition, an electron transfer process may occur (Scheme 1).

Scheme 1

The protonation reaction, an extensively studied process, affords either classical M(H) 2 or non classical M(H 2 ) dihydrides by pathways a and b, respectively (see Scheme 1). [1][2][3][4][5][6][7][8][9][10][11][12][13] The 3c-2e nonclassical product may be unstable and collapse to the dihydrido complex or reductively eliminate dihydrogen (see Scheme 1, paths b.i and b.ii, respectively). The hydride abstraction reaction by the trityl cation (E + = Ph 3 C + ) [14][15][16][17] can also be viewed as an exemple of a reaction following path b.ii.

Gold-based electrophiles, especially AuPPh 3 + , have also been extensively investigated. 18,19 These typically afford 3c-2e M(µ-H)Au moieties that can be seen as the isolobal analogues of the nonclassical dihydrides. 20 The substitution of H + for AuPh 3 + typically affords more stable products. This has been the case for the particular complex CpMoH(CO) 2 (PMe 3 ), one of the rare mononuclear hydride complexes for which the comparative reactivity with both the proton and the AuPPh 3 + reagent has been investigated. 21 The silver ion has also been extensively investigated as an electrophilic reagent toward transition metal hydride complexes. The typical pathway is the attack at the M-H electron density, to afford complexes analogous to those obtained with the gold reagents (i.e. path b in Scheme 1). [22][23][24][25][26][27][28][29][30][31][32][33] However, silver also has oxidizing properties and a possible alternative reaction pathway is electron transfer to afford metallic silver and the 17-electron hydride complex. This transformation can occur in principle with or without prior coordination of the silver ion to the nucleophilic M-H bond (paths b.iii and c in Scheme 1, corresponding to outerand inner-sphere electron transfer pathways, respectively). An example of an inner sphere electron transfer via formation of a M(µ-H)Ag moiety has been shown recently. 34 The CpMoH(CO) n (PMe 3 ) 3-n series of complexes have attracted our attention because there is a ca. 2V difference between the oxidation potentials of the two extreme members of the series (n = 3 and 0), thus potentially influencing the reactivity of the Mo-H functionality toward the Ag + reagent. This paper presents the results of our investigations of the reactivity of complex CpMoH(PMe 3 ) 3 , 1, with the silver ion. The corresponding reactivity of complex CpMoH(CO) 2 (PMe 3 ) and a comparison between the two systems is presented in Part II. 35 The studies reported here also provided new findings of relevance to the decomposition of 17electron hydride complexes by the competitive deprotonation and disproportionation mechanisms.

Experimental

General. All operations were carried out under an atmosphere of dinitrogen with standard Schlenk-line techniques. Solvents were purified by conventional methods and distilled under argon prior to use (THF and Et 2 O from Na/benzophenone, heptane from Na, acetone from CaSO 4 , CH 2 Cl 2 from P 4 O 10 and MeCN form CaH 2 ). Deuterated solvents (C 6 D 6 , CD 3 CN, THF-d 8 and CD 3 COCD 3 ) were degassed by three freeze-pump-thaw cycles and then stored over 4Å molecular sieves under dinitrogen. NMR spectra were obtained with Bruker WP200 and AF400 spectrometers. The peak positions are reported with positive shifts downfield of TMS as calculated from the residual solvent peaks ( Cp 2 Fe + /Cp 2 Fe couple, which was introduced in the cell at the end of each measurement. The EPR spectroelectrochemistry studies were performed with a home-built cell quite similar to one described in the literature, 36 the difference being that the aqueous reference electrode was replaced with a silver wire pseudo-reference. Stopcocks and O-ring connections were added to allow all operations to be carried out under an inert atmosphere. Measurements were carried out on the Brucker ER200 spectrometer mentioned above and the EPR species was generated from 0.1-0.2 M solutions by application of a constant current of 1 mA with an EG&EG 363 galvanostat. A control experiment of the nitrobenzene radical anion, generated at room temperature, yielded an EPR spectrum with greater resolution relative to that obtained with the analogous cell described in the literature. 36 Elemental (C, H, N) analyses were performed by MHW laboratories (Phoenix, AZ) and the analytical service at the LSEO at the Université de Bourgogne. Compound 1 was prepared as previously published. 37 Since this is very soluble in saturated hydrocarbons and cannot be easily crystallized, it was always prepared in situ in THF from equimolar amounts of CpMoCl(PMe 3 ) 3 and LiBHEt 3 , followed by evaporation to dryness, extraction into n-heptane, filtration, and evaporation to dryness. In each case, the product purity was verified by 1 H NMR spectroscopy. AgBF 4 , FcPF 6 , and Cp 2 Co were purchased from Aldrich and used as received.

Formation of

+ [BF4] - , 4
. This procedure is a modification of that previously reported. 38 To a solution of 1 (0.367 mmol) in toluene ( 10mL) cooled to -60°C, HBF 4 (54% in diethyl ether) (51 µL, 0.367 mmol) was added via microsyringe causing the formation of an off-white precipitate. Stirring was continued for 10 minutes. Addition of heptane (5 mL) caused the precipitation of the product, which was filtered, washed with heptane (3 x 5 mL) and dried under vacuum to yield 134 mg (76%) of

[CpMoH 2 (PMe 3 ) 3 ][BF 4 ]
. The 1 H-NMR at room temperature corresponds to that previously reported. 38 Decoalescence of the hydride resonances is observed upon cooling to 238 K.

Results and Discussion

(a) Electrochemical Oxidation of CpMoH(PMe 3 ) 3 .

Compound 1 undergoes a reversible oxidation process at E 1/2 = -1.46 V vs the ferrocene standard in MeCN, contrary to the corresponding irreversible oxidation processes of complexes CpMoH(CO) 3 and CpMoH(CO) 2 L (L = PMe 3 , PPh 3 ). 16,40 As expected, the oxidation potential of 1 is shifted negatively with respect to the potentials required to oxidize CpMoH(CO) 2 (PMe 3 ) (E p,a = +0.26 V) 40 and CpMoH(CO) 3 (E p,a = +0.80 V). 16 This corresponds to a shift of ca. -0.54 V from the tricarbonyl to the dicarbonyl derivative, and to an average of ca. -0.86 V per CO/PMe 3 substitution from the dicarbonyl to the tris(phosphine) derivative. A shift of -0.56 V was correspondingly observed on going from CpWH(CO) 3 to CpWH(CO) 2 (PMe 3 ). 16 Similar trends upon CO/PR 3 substitution have been observed before for other systems, for instance CpRuHLL' (L, L' = CO, tertiary phosphine) [41][42][43][44] and Cp* analogues, 45 and a variety of octahedral, d 6 dinitrogen and  2dihydrogen complexes. 46 The identity of the oxidation product of 1 as the corresponding 17-electron cation [ derivatives are well-known to adopt a "four-legged piano stool" geometry (see for instance the X-ray structures of [CpMoCl(PMe 3 ) 3 ] + ). 37,47 If this is also the geometry adopted by [1] + , the phosphine ligands should give rise to a doublet of triplets rather than a quartet. However, hydride ligands are often highly mobile in the coordination sphere of transition metal complexes and isomerization processes are generally even faster in radical species, [START_REF]Organometallic radical processes[END_REF] for this class of half-sandwich Mo(III) complexes. 50 Therefore, we consider it more likely that fluxionality is the reason for the EPR equivalence of the three phosphorus nuclei. The EPR spectrum of [1] + becomes broader upon warming and only a broad line with no distinguishable hyperfine splittings is observed at room temperature. The compound slowly decomposes at room temperature in THF as indicated by the decrease of EPR intensity (t 1/2 ≈ 30 min), while it appears to be indefinitely stable at -70°C. The 1:1 interaction of 1 with either Ag + or Cp 2 Fe + in THF results in electron transfer, with formation of elemental silver or ferrocene, respectively, and the unstable radical complex

[1] + , see eq. 1, as shown by EPR spectroscopy. The interaction is instantaneous, without the formation of an observable hydride-bridged silver adduct.

CpMoH(PMe 3 ) 3 + X + X + [CpMoH(PMe 3 ) 3 ] + (1)

1 [1] + (X = Ag, Cp 2 Fe)
When reaction 1 is run in MeCN, the formation of [1] + is not observed by EPR, presumably because of its fast decomposition in this solvent. The final products of the reaction are Ag or ferrocene, and a 1:1 mixture of complexes [CpMo(PMe 3 ) 3 (MeCN)] + , and [CpMo(PMe 3 ) 3 H 2 ] + , see eq. 2.

2 CpMoH(PMe 3 ) 3 + 2 X + + MeCN [CpMo(PMe 3 ) 3 (MeCN)] + + 1 This result is consistent with the known reduction potentials of the trityl cation (+ 0.08 V vs.

+ [CpMo(PMe 3 ) 3 H 2 ] + + 2 X ( 2
Fc/Fc + , 51 ) and 1 (-1.46 V vs. Fc/Fc + , vide supra). This electron transfer reaction also leads to a number of unidentified by-products, which may result from side-reactions such as attack of the trityl radical at the cyclopentadienyl ligand. 43 The other expected end product is the trityl radical, in equilibrium with its head-to-tail dimer, the latter being observed in the The formation of the products of reaction 2 is in agreement with the general reactivity observed for a series of closely related systems. 16,44 It is worth noticing that the solvent adduct is the only product of the oxidation of the very similar CpMoH(CO) 2 (PMe 3 ), the reaction otherwise proceeding with the same stoichiometry (1 equiv of oxidizing agent per Mo center) under anhydrous conditions. 40 This is so because the hypothetical product of proton transfer, [CpMoH 2 (CO) 2 (PMe 3 )] + , is unstable toward loss of H 2 . The substitution of the two carbonyl ligands with as many phosphine ligands renders the dihydride product sufficiently electron-rich to prevent the expulsion of molecular hydrogen.

When the oxidation is carried out in the presence of two equivalents of oxidant (either Fc + or Ag + ) in either THF or MeCN, dicationic hydride complexes [CpMoH(PMe 3 ) 3 (S)] 2+ (S = THF, MeCN) are obtained (see eq. 3). These complexes have been isolated as PF 6 -salt for S = THF (compound 2) and as BF 4 -salt for S = MeCN (compound 3). Complex

[CpMoH(PMe 3 ) 3 (MeCN)] 2+ has also been observed as a kinetic product during the oxidation with only one equivalent of oxidant. When the reaction mixture was set for crystallization in the early stages of the oxidation, compound 3 was recovered as a the major product in a 38% yield.

CpMoH(PMe 3 ) 3 + 2 X + + S 2 X + [CpMoH(PMe 3 ) 3 (S)] 2+

(3) 1 (X = Ag, Cp 2 Fe; S = THF, MeCN) Compound 2 is rather unstable in solution, preventing a spectroscopic characterization and leading to an analytically impure solid. Its isolation is permitted by its sparing solubility under the synthetic conditions. However, recovery of the precursor complex 1 by cobaltocene reduction of isolated 2 comes in strong support of our proposed formulation. The scant solubility of the dication and the variety of decomposition products have discouraged us to investigate this decomposition process in more detail. One of the various final products of decomposition has been recognized as the oxo complex [CpMoO(PMe 3 ) 2 ] + , indicating that adventitious water favorably competes with THF for coordination to the Mo(IV) center of the dicationic hydrido complex. The role of water as a ligand during the oxidation of compound 1 under a variety of conditions will be the subject of a separate contribution. 52 A THF adduct deriving from the oxidation of CpRe(PPh 3 ) 2 H 2 in THF, [CpRe(PPh 3 ) 2 (THF)H] + , has been described. 53 Compound 3 is much more stable than compound 2 and could be recrystallized from MeCN/Et 2 O, yielding analytically pure crystals that were investigated by NMR and X-ray crystallography. The inequivalent phosphine ligands give rise to two resonances in a 2:1 ratio in both the 1 H and the 31 P NMR spectrum, in agreement with the crystallographically determined structure (vide infra). While no resonance was observable in the 1 H NMR for the hydridic proton, its presence was confirmed by selective decoupling 31 P NMR experiments.

The hydride resonance could not be hidden under the Cp* or PMe 3 resonances, because the latter ones were saturated in the selective decoupling experiment which reveals the P-hydride coupling. The coupling constants between the hydride and the two inequivalent sets of phosphorus nuclei are the same and relatively small (10 Hz), consistent with the observed cis arrangement. The geometry of the dication in the structurally characterized 3 . MeCN (see Figure 2) may be described as a pseudo-octahedron when considering the Cp ligand as occupying a single coordination position. This geometry is analogous to that established for a number of other Cp-containing 18-electron complexes of Mo(IV) and W(IV). [54][55][56][57][58][59][60][61] As may be expected from trans-influence arguments, the Mo-P(2) bond is significantly longer than the Mo-P(1) bond. No noticeable trans-effect, on the other hand, is exerted by the hydride ligand on the Cp ring, since the Mo-CNT distance of 1.97(1) or 1.98(1) Å for the two orientation of the ring in 3 are essentially indentical to the distance of 1.99(1) Å reported for complex [CpMoFCl(dppe)(MeCN)] + , where the ring is located trans relative to the weakly transdirecting fluoride ligand.

(c) Disproportionation mechanism

Upon one-electron oxidation of a metal hydride complex [M]-H, the formation of a 1:1 mixture of the protonation product {[M]H 2 } + and the solvato complex {[M](S)} + , as in equation 2, is frequently observed. There have been two established mechanisms for this process, both involving the decomposition of the primary oxidation product, the 17-electron {[M]-H} +. complex. These are the deprotonation mechanism 16 and a less commonly observed disproportionation 44,62 mechanism (paths a and b in Scheme 2, respectively). This disproportionation pathway has previously been proposed for the decomposition of complexes 44 The deprotonation mechanism involves rate-determining proton transfer to the unoxidized parent hydride complex, while the disproportionation mechanism involves rate-determining electron transfer between two molecules of the primary oxidation product. The disproportionation mechanism also involves a proton transfer, but only after the electron transfer generates a supposedly more acidic {[M](S)H} 2+ species. For the deprotonation mechanism to occur, the proton transfer from the 17-electron {[M]-H} +. complex to the neutral 18-electron precursor must be very rapid and compete with the oxidation reaction itself. The results of the reactions 1-3 show that the 17-electron complex [1] + is not rapidly deprotonated by 1, allowing the one-electron oxidation to proceed quantitatively. Then complex [1] + may only decompose by disproportionation since there is no residual 1 left in solution. This proposition is in agreement with kinetic studies, which are presented in the next section. The present system has allowed, for the first time, the isolation and characterization of the dicationic hydride intermediate of the disproportionation process. Previously, the existence of this intermediate had only been inferred from electrochemical studies. 44

[MH(dppe) 2 ] + (M = Rh, Ir), 62 [IrH(CO)(PPh 3 ) 3 ] + , 62 [CpReH 2 {P(p-XC 6 H 4 ) 3 } 2 ] + (X = H, Me, F, OMe), 53 and [CpRuHL 2 ] + (L = PPh 3 or L 2 = Ph 2 PCH 2 CH 2 CH 2 PPh 2 ).

Scheme 2

The dicationic hydride product is selectively obtained if the oxidation is carried out in the presence of two rather than one equivalent of oxidizing agent. Coordination of the solvent to the 17-electron primary oxidation product, to afford a more easily oxidizable 19-electron species probably occurs as shown in Scheme 2. 44 This is indicated by the longer lifetime of the 17-electron species [1] + in the weaker coordinating THF solvent (t 1/2 ≈ 30 min) relative to MeCN (complete in less than 7 min). A similar solvent effect has been noticed 44 for the decomposition of [CpRuH(PMe 3 ) 2 ] + by disproportionation.

The observation of the dication (and indeed its isolation as 3), even when only one equivalent of oxidant is used, shows that the final proton transfer step of the disproportionation mechanism is slow. This has been independently verified by mixing equivalent amounts of the [M ]-H

{[M ]H 2 } + [M ] .

+S -S [M ](S) .

[ox]

{[M ](S)} + {[M ](S)H } 2+ [M ]-H {[M ]-H} +. {[M ](S)} + {[M ]H 2 } + + slow slow rate = k obs [{[M ]-H} +. ] k obs = k dep rot [[M ]-H] rate = k disp rop [{[M ]-H} +. ] 2
isolated complexes 1 and 3 (eq. 4). The steric bulk of the reagents probably hinders the proton transfer. Deprotonation of the dication also occurs slowly upon addition of triethylamine. This reaction affords a spectroscopically pure solution of a single product, corresponding to the complex that forms together with the dihydride cation [CpMoH 2 (PMe 3 ) 3 ] + during the decomposition reaction, and confirms the assignment of this product to the MeCN adduct [CpMo(PMe 3 ) 3 (MeCN)] + (eq. 2). Under the reasonable assumption that this reaction follows a second order rate law, the measured half-life under pseudo-first order conditions (see Experimental section) yields an approximate rate constant of 3 . 10 -3 s -1 M -1 for the proton transfer to NEt 3 . A qualitative comparison indicates that the proton transfer to 1 is slower than to NEt 3 .

[CpMoH(PMe A kinetic handle on the decomposition of [1] + was gained through the time evolution of the EPR spectrum for the reaction carried out in THF (Figure 3, curve (a)). The data fit well a second order decay equation (correlation coefficient R 2 = 0.983), while the first order (exponential) decay gives a much poorer fit (R 2 = 0.929). This is as expected for the disproportionation mechanism (rate = k disprop [1 + ] 2 ; k disprop = 7.5 (8) . 10 -3 s -1 M -1 ). The isolation of 2 when using two equivalents of oxidant is consistent with this finding and leads us to believe that the kinetic results obtained in THF (rate law and mechanism) may be extrapolated to MeCN, where the decomposition is too fast to be followed by EPR spectrocopy.

Complex [CpMo(PMe 3 ) 3 (S)] + is unfortunately stable only when S = MeCN, whereas the THF compound leads to further complex decomposition. The decay of the EPR signal was also studied in the presence of residual neutral starting material 1, which may act as a base according to Scheme 2 (path a). A much faster decay was observed under these conditions (see Figure 3, curve (b)), suggesting that a faster proton transfer decomposition mechanism is indeed likely to take place under these conditions. As shown in Scheme 2, the rate law should show a first order dependence on the radical species and a first order dependence on the neutral complex for the deprotonation mechanism. In addition, we know that the deprotonated radical complex, CpMo(PMe 3 ) 3 , does not accumulate in solution since the EPR monitoring only shows the decay of the signal of [1] + . Thus, it is likely that CpMo(PMe 3 ) 3 is further oxidized (possibly after coordination of a solvent molecule) by the residual [1] + according to equation 5.

CpMo(PMe 3 ) 3 + [1] + + S [CpMo(PMe 3 ) 3 (S)] + + 1 (5)
The comparison of path a in Scheme 2 and equation 5 illustrates that compound 1 is consumed by the proton transfer step but is later regenerated by the electron transfer step. Thus, compound 1 merely acts as a catalyst of the decomposition of 1 + by the deprotonation mechanism. In other words, the concentration of 1 remains constant during the decomposition and a clean first order decay is expected. On the other hand, the data give a poor fit for both the first (R 2 = 0.955) and the second (R 2 = 0.953) order decay equations. We then reasoned that the decomposition of [1] + , under these conditions, would be taking place by a combination of the two independent mechanisms. The first order decay should become prevalent toward the end of the decomposition, when the concentration of 1 + is low, while the decay at the very beginning should follow the disproportionation pathway to a greater extent. [START_REF] Wilkins | Kinetics and Mechanism of Reactions of Transition Metal Complexes[END_REF] The decay should therefore follow the biphasic rate law in eq. 6, which provides the integrated rate law shown eq. 7 in a straightforward manner. In this equation, [1 + ] 0 is the starting concentration of the radical complex, k 2 = k disp , and k 1 = k deprot [1].

-

d[1 + ]/dt = k 1 [1 + ] + k 2 [1 + ] 2 (6)
Fitting of the data by equation 7 was carried out by imposing k 2 to be identical with the rate constant obtained from the fit of curve (a). This yields curve (b) in Figure 3 for an improved correlation coefficient R 2 = 0.968. The value obtained for k 1 is 1.00(7) s -1 , which in turns gives

k deprot = k 1 /[1] = 4.4(3) . 10 -2 s -1 M -1 .
The ratio k disp /k deprot = 5.8 (7) means that, when the concentration of 1 is the same as that of 1 + , the deprotonation pathway is ca. 5.8 times faster (in the THF solvent) than the disproportionation pathway or, in other words, that 1 + is ca. 5.8

times more likely to transfer a proton to 1 than to oxidize another molecule of 1 + (in equilibrium with [CpMoH(PMe 3 ) 3 (THF)] + ) in the THF solvent. In MeCN, the disproportionation pathway should be accelerated as predicted by the better coordinating power of the solvent. 44 The rate of proton transfer from 1 + to 1, on the other hand, should be essentially solvent independent.

If anything, the deprotonation should become slower in MeCN in view of the more favorable equilibrium of formation of the 19-electron solvent adduct, which should in turn have a lower acidity because of the higher metal electron density. In conclusion, the decomposition of 1 + takes place by disproportionation in MeCN, whereas there is a competition between deprotonation and disproportionation in THF, the former being prevalent in the presence of 1.

This appears to be the first hydride system for which a competition between the two decomposition mechanisms has been quantitatively assessed.

The effect of the PMe 3 substitution on the stability of the 17-electron hydride radical species, [CpMoH(CO) n (PMe 3 ) 3-n ] + , toward deprotonation is remarkable. This increased

ln k 1 +k 2 [1 + ] [1 + ] -ln k 1 +k 2 [1 + ] 0 [1 + ] 0 = k 1 t (7)
stabilization should probably be attributed to the steric crowding by the bulkier PMe 3 ligand and not to the decreased acidity of the hydride complex. In fact, while the acidity of [CpMoH(CO) n (PMe 3 ) 3-n ] + decreases as n decreases, the basicity of the proton acceptor CpMoH(CO) n (PMe 3 ) 3-n correspondingly increases, so that the thermodynamic driving force for the proton transfer should not be greatly affected by the CO/PMe 3 substitution. A similar steric argument has previously been advanced to explain the lack of proton transfer reactivity for the 17-electron [Cp*FeH(dppe)] + complex. [START_REF] Hamon | [END_REF] It is also interesting to compare the rates of proton transfer of the 17-electron complex [1] + and the 18-electron complex [CpMoH(PMe 3 ) 3 (MeCN)] 2+ to 1. The deprotonation of [1] + (k = 5.5 . 10 -2 s -1 M -1 ) is faster than that of the dicationic hydride (k < 3 . 10 -3 s -1 M -1 ), at odds with an expected greater thermodynamic acidity for the dicationic product. We can tentatively rationalize this penomenon, once again, on steric grounds. The increased steric crowding resulting from the MeCN coordination may render the proton less accessible and the deprotonation kinetically slower.

(e) Protonation of 1

While the addition of H + to compound CpMoH(CO) 2 (PMe 3 ) affords an unstable [CpMo(H 2 )(CO) 2 (PMe 3 )] + complex which immediately expels H 2 , 40 the protonation of the electron richer complex 1 affords the stable dihydride complex [CpMoH 2 (PMe 3 ) 3 ] + . 38 This protonation reaction was previously investigated in Et 2 O/H 2 O. 38 We find the same result in MeCN, confirming that the observed complex [CpMo(PMe 3 ) 3 H 2 ] + may be obtained in this solvent by proton transfer following the oxidation of 1. The BF 4 salt (compound 4) was isolated in high yields by protonation with HBF 4 in toluene. The NMR characterization that we have obtained for this complex completes that previously reported 38 and shows that the compound is a classical dihydride with inequivalent hydride ligands (decoalesced at 238 K in the 1 H NMR). A pseudo-octahedral structure similar to that described for compound 3, with one pseudo-axial and one pseudo-equatorial hydride ligand is suggested by the NMR double resonance experiments (see supplemental figure). The proposed arrangement is equivalent to that observed by X-ray crystallography for the isoelectronic complex CpMoH 3 (PMe 2 Ph) 2 . 61 A similar geometry has also been found through heteronuclear double-resonance experiments and X-ray diffraction study for the isoelectronic complex (C 6 H 5 Me)WH 2 (PMe 3 ) 3 2+ . 65 It is rather remarkable that the dihydride cation does not have any tendency to evolve dihydrogen. No change in the NMR spectrum was observed upon standing in CD 3 CN for several months or upon heating. The difference in H 2 evolution reactivity between [CpMoH(CO) n (PMe 3 ) 3-n ] + (n = 1 and 3) is in line with theoretical studies 66 on the equilibrium between classical and nonclassical polyhydrides for cyclopentadienyl complexes.

Conclusions

We have shown in this contribution that complex 1 undergoes a one-electron oxidation when reacted with Ag + or Fc + , to afford a relatively stable 17-electron hydride product, [1] + .

Hydride complexes with a 17-electron configuration are usually rather unstable. Complex [1] + is, to the best of our knowledge, the first such complex for Mo(III) that is sufficiently stable to be characterized spectroscopically. A kinetic study of its decomposition reaction by EPR spectroscopy shows a solvent assisted disproportionation pathway (faster in MeCN relative to THF). The key intermediate of this pathway, a dicationic hydrido Mo(IV) complex, has been isolated and structurally characterized. The presence of unoxidized 1, however, catalytically triggers a faster decomposition (in THF) by a deprotonation pathway. It was not possible to establish whether the silver oxidation is direct (outer-sphere) or whether it takes place via the formation of an intermediate Mo-(µ-H)-Ag adduct (inner-sphere mechanism), because no intermediate of this type could be observed. In the next contribution, 35 we shall compare the current findings with those obtained for the oxidation of the related CpMoH(CO) 2 (PMe 3 ) complex.

  Electrochemical oxidation of 1 in THF: An aliquot of a THF solution (5 mL) of compound 1 (0.6 mmol) was introduced in the spectroelectrochemical cell containing Bu 4 NPF 6 and the probe was cooled to -70C, followed by anodic oxidation at a costant current of 1 mA for 5 seconds. EPR: g = 2.018 (dq with Mo satellites, a d = 13.3 G; a q = 29.3 G; a Mo = 27 G). The signal intensity remains constant at low temperature, whereas the line broadens and then slowly disappears upon warming to room temperature. (b) Reaction of 1 with Ag + in THF. Solid AgBF 4 (68 mg; 0.35 mmol) was added to a THF solution of 1 (0.35 mmol) at -78°C, resulting in the immediate formation of a dark brown precipitate. An aliquot of this reaction mixture was transferred and tested for EPR activity at -78 o C. The resulting EPR is identical to that obtained in the spectrochemical cell (see previous section) and assigned to [1] + . Upon warming to room temperature, the signal broadens and slowly disappears. A number of diamagnetic products appeared in the NMR spectrum after the solution was allowed to warm up to room temperature. The nature of these decomposition products was not further investigated. (c) Reaction of 1 with Cp2Fe + in THF. Preparation of [CpMoH(PMe3)3-(THF)](PF6)

A) 2 , 3 .

 23 complete discoloration ensued. After 20 min, the NMR spectra revealed the nearly exclusive formation of [CpMoH(PMe 3 ) 3 (CD 3 CN)] 2+ , and ferrocene. 1 H NMR of [CpMoH(PMe 3 ) 3 (CD 3 CN)] 2+ (): 5.31 (m, J = 1 Hz, 5H, Cp), 1.61 (d, J P = 9Hz, 18H, PMe 3 ), 1.57 (d, J P = 9Hz, 9H, PMe 3 ). 31 P{ 1 H} NMR (CD 3 CN, ): 3.6 (t, J = 26 Hz, 1P); 0.5 (d, J = 26 Hz, 2P). 31 P{selec-1 H} NMR (CD 3 CN, ): 3.6 (td, J P = 26 Hz, J H = 10 Hz, 1P); 0.5 (dd, J P = 26 Hz, J H = 10 Hz, 2P).Reaction of 1 with one equivalent of Ag+ in MeCN: synthesis of [CpMo(PMe3)Compound 1 (0.12 mmol) and AgBF 4 (24 mg, 0.12 mmol) were mixed in CD 3 CN (1.2 mL). The immediate reaction produced black solid and bleached the ferrocenium color. The NMR investigation of the orange supernatant showed the formation of [CpMo(PMe 3 ) 3 (CD 3 CN)H] 2+ as the major product (≈ 85 % relative to Fc + ). The black precipitate was removed from the mother solution through filtration, and diethylether was then added to the solution. Precipitation of an orange solid ensued, which was recovered by filtration and vacuum drying. The solid was washed with dichloromethane (2 x 2 mL), the red filtrates were removed and the yellow microcrystalline solid was dried under vacuum (24 mg, 38 % yield as 3 . Et 2 O). An aliquot of the solid exhibited the resonances of [CpMoH(PMe 3 ) 3 (CD 3 CN)] 2+ and Et 2 O in relative intensities as expected for a 1:1 molar ratio. Elem. Anal.: Calcd. for C 20 H 43 D 3 B 2 F 8 NMoOP 3 C 35.2; H 6.35; D 0.88; N 2.05. Found : C 35.6; H 6.92; N 1.57. Crystals of 3 . MeCN suitable for an X-ray crystallographic study were obtained from acetonitrile/ether solutions stored at -20C for two days. Reaction of 1 with two equivalents of Ag + in CD3CN: Addition of AgBF 4 (35 mg, 181 µmol) to a solution of 1 (90 µmol) in CD 3 CN (0.5 mL) generated of [CpMoH(PMe 3) 3 -(CD 3 CN)] 2+ as the main product. The 1 H-NMR spectra were monitored over time (> 2 days). No transformation to [CpMo(PMe 3 ) 3 (CD 3 CN)] + and [CpMoH 2 (PMe 3 ) 3 ] + was observed. Reaction of 3 with NEt3 Crystals of compound 3 (ca. 2.0 mg, 3.1 . 10 -3 mmol) were dissolved in CD 3 CN (0.45 mL) and their purity was checked through 1 H-NMR. NEt 3 (2.0 µL, 3.2 . 10 -2 mmol) was added to the yellow solution. The resonances of [CpMoH(PMe 3 ) 3 -(CD 3 CN)] 2+ slowly disappeared (t 1/2 ≈ 2 hrs), while those of [CpMo(PMe 3 ) 3 (CD 3 CN)] + increased. 1 H NMR (): 4.79 (s, 5H, Cp), 1.5 (vb, PMe 3 ). 31 P NMR (): 1 (vb, w 1/2 = 100 Hz, 1P), -1 (vb, w 1/2 = 160 Hz, 2P). No resonances attributable to other secondary products were detected by NMR. Reaction of 3 with 1. A yellow solution of 1 (50 µmol) in CD 3 CN (0.8 mL) was split in two halves. To one of them AgBF 4 (10 mg; 51 µmol) was added, forming [CpMoH(PMe 3 ) 3 (CD 3 CN)] 2+ and Ag. Afer the precipitate settled, the orange supernatant was added to the yellow original solution and the reaction was monitored through 1 H-NMR. The formation of [CpMo(PMe 3 ) 3 (CD 3 CN)] + and [CpMoH 2 (PMe 3 ) 3 ] + was discernible within the first minutes of the addition, but it was not complete until after more than 24 hrs. Protonation of compound 1. (a) In toluene. Synthesis of mer-[CpMoH 2 (PMe 3 ) 3 ]

  1] + has been verified by EPR spectroscopy. Constant-current bulk oxidation in THF at -70°C in an EPR spectroelectrochemical cell rapidly afforded the EPR spectrum shown in Figure 1. The spectrum consists of a main doublet of quartets at g = 2.014 because of coupling to the hydride nucleus (a H = 12.9 G) and the three phosphorus nuclei (a P = 28.6 G). The main signal is flanked by satellites due to coupling to the I = 5/2 95 Mo and 97 Mo nuclei (a Mo = 27 G) in the corresponding isotopomers (total abundance ca. 25%). Half-sandwich, 17-electron Mo(III)

  therefore the three phosphine ligands may be rendered equivalent on the EPR time scale by a rapid fluxional process. Alternatives accounting for the equivalence of the phosphorus couplings are an accidental degeneracy of the two hyperfine splittings, or a static structure with equivalent phosphine ligands, namely a pseudo-tbp structure with Cp and H in the axial positions. However, all related tris-phosphine Mo(III) complexes have significantly different phosphorus hyperfine splittings (for [CpMoCl(PMe 3 ) 3 ] + : a P(d) = 26.0 G, a P(t) = 19.5 G; 37 for [CpMoCl-(triphos)] + : a P(d) = 28.0 G, a P(t) = 14.5 G 49 ), and a pseudo-tbp structure has never been observed

Figure 1 .

 1 Figure 1. EPR spectrum of the [CpMoH(PMe 3 ) 3 ] + species in THF (T = 193 K).

  ) (X = Ag, Cp 2 Fe) Complex [CpMo(PMe 3 ) 3 H 2 ] + has been previously reported from the protonation reaction of CpMoH(PMe 3 ) 3 and has been further investigated by us (see section e). Complex [CpMo(PMe 3 ) 3 (MeCN)] + , on the other hand, has not been previously described. It shows the two expected 31 P resonances in a 1:2 ratio and the correct integration for the Cp and PMe 3 resonances in the 1 H NMR spectrum. The 1 H NMR resonance of the MeCN ligand could not be observed, most likely because of a rapid exchange with the CD 3 CN solvent. In an effort to independently synthesize this complex, we attempted to abstract the hydride ligand from 1 with the trityl cation. A similar reaction on the related CpMoH(CO) 2 (PR 3 ) (R = Me, Ph) cleanly afforded the expected [CpMo(CO) 2 (PR 3 )(MeCN)] + . 15,40 Conversely, this reaction yields a mixture of [CpMo(PMe 3 ) 3 H 2 ] + , [CpMo(MeCN)(PMe 3 ) 3 ] + , and [CpMo(PMe 3 ) 3 H(MeCN)] 2+ , even when carried out at lower than ambient temperatures.

1 H

 1 NMR spectrum. A spectroscopically pure solution of [CpMo(PMe 3 ) 3 (MeCN)] + was eventually obtained by deprotonation of isolated [CpMo(PMe 3 ) 3 H(MeCN)] 2+ (vide infra).

60

 60

Figure 2 .

 2 Figure 2. A view of the cation for compound [CpMoH(PMe 3 ) 3 (MeCN)] 2+ with the

3 ) 3 ( 4 )(B = 1 ,

 3341 MeCN)] 2+ + B [CpMo(PMe 3 ) 3 (MeCN)] + + BH + (NEt 3 ) (d) Kinetic studies on the decomposition of complex [1] + in THF.

Figure 3 .

 3 Figure 3. Plot versus time of the EPR signal double integrals for: (a) [1] + (0.02 M); (b) [1] +

EPR monitoring of the decomposition of [1] + in THF. 80

	spectra were recorded over time. The relative concentration of [1] + was monitored by double
	integration of each spectrum (see Results).
	2, 2. A blue suspension of Cp 2 FePF 6 (52 mg, 0.16 mmol) was prepared in THF
	(3 mL) and cooled to -80C in a cryostat. A yellow THF solution (5 mL) of 1 (0.17 mmol) was
	cooled to -80C and added dropwise to the ferrocenium suspension. The green mixture slowly
	bleached the blue color of the ferrocenium and induced the precipitation of a clear solid. After
	the suspension was let stir overnight in the cryostat, the light orange supernatant was filtered
	off. An aliquot showed weak EPR activity in the form of overlapping unassigned multiplets.
	Compound 2 was recovered through filtration as a light pink solid, washed with Et 2 O (3 x 3
	mL), and dried under vacuum (27 mg, 45% yield relative to ferrocenium). Anal. Calcd for
	C 18 H 41 F 12 MoOP 5 : C 28.74; H 5.49. Found C 28.07; H 5.19. The solid is only slightly soluble
	in THF and acetonitrile and is not soluble in benzene and toluene.
	Reduction of 2 to 1. The solid obtained by the procedure described in the previous section
	(15 mg; 20 µmol) was added to a solution of Cp 2 Co (ca. 10 mg; 50 µmol) in C 6 D 6 (0.4 mL),
	inducing the solubilization of the solid. The resulting solution was EPR silent and showed the
	resonances of THF and compound 1 in the correct relative intensities for a 1:1 molar ratio in
	the 1 H-NMR spectrum.
	µmol of 1 were suspended in
	THF (4 mL). FcPF 6 (26 mg, 79 µmol) was added to the solution and immediate bleaching of
	the oxidant ensued. An aliquot of the orange solution was transferred to an EPR tube, and EPR

EPR monitoring of the decomposition of [1] + in the presence of 1 in THF 162

	µmol
	of 1 were suspended in THF (4 mL). FcPF 6 (24 mg; 72 µmol) was added to the solution and
	immediate bleaching of the oxidant ensued. An aliquot of the orange solution was transferred
	in an EPR tube, and EPR spectra were recorded over time and double-integrated (see Results).

Attempt at the EPR monitoring of the decomposition of [1] + in MeCN. Compound 1

	(0.12 mmol) was dissolved in MeCN (5 mL). FcPF 6 (39 mg; 0.12 mmol) was added and
	discoloration of the oxidant blue color ensued. An aliquot of the solution was transferred in an
	EPR tube. The first EPR spectrum, taken 7 minutes after the addition of the oxidant, did not
	show any activity. The short lifetime of [1] + in MeCN prevented the kinetic analysis of its
	decomposition by EPR.

Reaction of 1 with one equivalent of Fc + in MeCN: formation of [CpMo(PMe3)3(MeCN)] + and [CpMo(PMe3)3(H)2] + . A stock solution of 1 was prepared in

	PMe 3 ). 31 P NMR (): 1 (vb, w 1/2 = 100 Hz, 1P), -1 (vb, w 1/2 = 160 Hz, 2P)] and
	[CpMo(PMe 3 ) 3 (H) 2 ] + [ 1 H NMR (): 4.81 (m, J = 1.2 Hz, 5H, Cp), 1.47 (d, J P = 8.4 Hz, 27 H,
	PMe 3 ), -4.0 (vb, 1H, Mo-H). 31 P NMR (): 8 (vb, w 1/2 = 200 Hz)]. Both complexes were
	independently synthesized (vide infra).
	CD 3 CN (3 mL; 0.131 M), and its title was checked through integration against an interal
	standard of ferrocene. A 605 µL aliquot (79 µmol) was added to FcPF 6 (26 mg; 79 µmol).
	Immediate dissolution of the oxidant ensued to give an orange solution. The first acceptable
	NMR spectrum was obtained after 10 minutes from the addition. Earlier spectra were affected
	by poor shimming due to the presence of paramagnetic species in solution. The solution
	contained, in addition to ferrocene ( 1 H NMR resonance at  4.14), a 1:1 mixture of two
	molybdenum species: [CpMo(PMe 3 ) 3 (MeCN)] + [ 1 H NMR (): 4.79 (s, 5H, Cp), 1.5 (vb,

Reaction of 1 with two equivalents of Fc + in MeCN: formation of [CpMo(PMe3)3(CD3CN)H]

	2+ . FcPF 6 (52 mg; 157 µmol) was introduced in a NMR tube. A
	602 µL aliquot of the CD 3 CN stock solution of 1 (see previous section, 79 µmol) was added.

Table 1 .

 1 Selective decoupling experiments ( 1 H{selec-1 H} and 1 H{selec-31 P}) allowed all the individual coupling constants to be determined. A simulation of the experimental spectrum proved satisfactory. Complex 4 is stable in acetonitrile: no H 2 loss was observed over several months at room temperature or for 18 hours at 80°C. To a solution of 1 (75 µmol) in CD 3 CN, HBF 4 (54% in diethyl ether) (10 µL,72 mmol) was added. Quantitative formation of 4 ensued. No transformation to [CpMo(PMe 3 ) 3 (CD 3 CN)] + was observed over several days. 5U (parent) for the Me H atoms, initially determined with a rotational difference-Fourier map about the central carbon atom]. The hydride was located directly from a difference-Fourier map and its parameters (xyzU) were all allowed to refine freely. During the final stages of the fullmatrix least-squares refinement, all of the non-H atoms were refined anisotropically. Several restraints were applied to force the thermal parameters of the two sets of atoms comprising the Cp ring to remain positive definite. The function minimized was w(Fo Crystal data and structure refinement for compound 3

	(b) In wet MeCN. To a solution of 1 (80 µmol) in CD 3 CN (0.60 mL) was added H 2 O (2
	µmol, 0.1 mmol). The 1 H-NMR resonaces of 1 were not affetcted. Addition of HBF 4 (54% in
	diethyl ether) (10 µL,72 mmol) resulted in its transformation to 4 and did not shift the water
	peak resonance.

1

H-NMR (, CD 3 CN, 238 K): 4.81 (s, 5H, Cp), 1.46 (d, J HP = 8.4 Hz, 27H, PMe 3 ), -2.48 (virtual qd, J HP = 52.5 Hz, J HH = 8.4 Hz, 1H, Mo-H eq ) , -5.65 (virtual tt, J HP = 46.4 Hz, J HH = 8.4 Hz, 1H, Mo-H ax ).

31

P-NMR: (, CD 3 CN, 233 K): 10 (b, w 1/2 = 50 Hz, 2P), 4 (b, w 1/2 = 50 Hz, 1P). (b) In MeCN.

X-ray crystallography for compound 3 . MeCN. A yellow needle with approximate dimensions 0.50 x 0.10 x 0.10 mm was placed on the Enraf-Nonius CAD-4 diffractometer. The cell parameters and crystal orientation matrix were determined from 25 reflections in the range 10.0 <  < 16.2° and further confirmed with axial photographs. Crystal parameters and selected collection and refinement parameters are collected in Table

1

. Intensity standards showed no significant variation with time and no decay correction was applied. The data were corrected for Lorentz and polarization factors, and for absorption on the basis of the -scan method.

39

Systematic absences indicated the centrosymmetric space group Pnma (no. 62) or the noncentrosymmetric space group Pna2 1 (no.33), intensity statistics clearly favoring the centric case. The structure was determined by direct methods in SHELXTL, with the successful location of the four heavy atoms (Mo, 3P) and several carbon and nitrogen atoms. The remaining non-hydrogen atoms were found from several additional difference-Fourier maps. occupied, while the second one and the interstitial acetonitrile molecule were disordered in symmetry-related positions. Hydrogen atoms were allowed to ride on the parent carbon atom [d(C-H)= 0.950 Å; U H = 1.2U (parent) for the Cp H atoms; d(C-H)= 0.980 Å, U H = 1.

correction for extinction was also attempted but found to produce a negative value and was not applied. Selected bond distances and angles are collected in Table

2

. . MeCN.

Table 2 .

 2 Selected bond distances (Å) and angles (°) for compound 3 . MeCN.

	Mo(1)-CNT	1.980(11)	Mo(1)-P(2)	2.5108(11)
	Mo(1)-CNT2	1.966(13)	Mo(1)-N(7)	2.144(5)
	Mo(1)-P(1)	2.476(2)	Mo(1)-H(1A)	1.61(6)
	P(1)-Mo(1)-P(2)	90.28(3)	CNT-Mo(1)-P(2)	110.27(9)
	P(1)-Mo(1)-N(7)	143.36(13)	CNT-Mo(1)-N(7)	110.1(8)
	P(1)-Mo(1)-H(1A)	66(2)	CNT-Mo(1)-H(1A)	173(2)
	P(2)-Mo(1)-N(7)	77.26(5)	CNT2-Mo(1)-P(1)	106.9(8)
	P(2)-Mo(1)-H(1A)	70.7(3)	CNT2-Mo(1)-P(2)	110.23(10)
	N(7)-Mo(1)-H(1A)	77(2)	CNT2-Mo(1)-N(7)	109.7(8)
	CNT-Mo(1)-P(1)	106.6(8)	CNT2-Mo(1)-H(1A)	173(2)

Acknowledgments. We are grateful to the DOE (grant DEFG059ER14230) for support of this work.

Supporting Information