Keywords: sans fil, réseau, mobilité, théorie de l'information, analyse complexe

Vitesse de propagation de l'information en fonction de la capacité de transport dans les réseaux mobiles ad hoc

. Nous montrons que la vitesse de propagation tend vers zero quand la capacité de transport tend vers l'infini. Nous comparons avec les performances d'un algorithme réaliste basé sur des angles.

Introduction

We consider a wireless network with a (potentially) infinite density of nodes on the plan. Each node has a wireless interface of nominal capacity C (expressed in bit per second). We assume that the access protocol generates consecutive sets of simultaneous transmitters such that any of those sets follows a uniform distribution of density equal to λ (expressed in inverse square meters). We consider asymptotic situations where λ → ∞. To fix the idea, the access protocol can be ALOHA-like with slotted time as described in [START_REF] Baccelli | An Aloha protocol for multihop mobile wireless networks[END_REF], but the result could be generalized to more sophisticated access schemes as did Gupta and Kumar in [START_REF] Gupta | The capacity of wireless networks[END_REF]. Indeed they show that in very general settings the efficient transmission radius r = r(λ) = α √ λ for some constant α > 0 depending on protocol, signal propagation, demodulation, etc.

If nodes were to transmit data only to their closest neighbors, then the useful capacity density of the network would be equal to βCλ (in bit per second per square meter) where β, depending on the access scheme, is the average number of retries before any succesful packet delivery. This would lead to an actual capacity density tending to infinity when λ → ∞. For the remaining of the paper we will assume β = 1, the case β < 1 does not actually impact our result.

But in general data are to be transmitted toward distant destinations; in this case the data must be relayed by other nodes. With shortest path routing, if L is the distance between source and destination, then the data would be relayed on a number of hops equal to L r(λ) ∼ L √ λ α when λ → ∞. If all datas were to be moved over an average distance L then each packet would be repeated L √ λ α times, and therefore the net capacity density of the network would be divided by this factor, hence Cα √ λ 1 L , which tends to zero when L → ∞. Due to the obvious intrication of wireless network with physical space, Gupta and Kumar in [START_REF] Gupta | The capacity of wireless networks[END_REF] introduced the concept of transport capacity. The transport capacity of a connection is equal to the product of the flow throughput with the euclidian distance to destination (expressed in bit meter per second). Similarly the transport capacity density is equal to the useful capacity density multiplied by the average distance to destination of the locally generated traffic. Gupta and Kumar result tells that such density does not depend on the average distances to destinations and is actually equal to Cα √ λ, as also shown in [START_REF] Baccelli | An Aloha protocol for multihop mobile wireless networks[END_REF]. This fundamental result shows that the transport capacity density still tends to infinity when λ → ∞.

From now we assume that the nodes are mobile and follow i.i.d motion processes. Grossglauser and Tse [START_REF] Grossglauser | Mobility increases the capacity of ad hoc wireless networks[END_REF] have shown that the mobility indeed increases capacity, stressing the fact that wireless networks also show a strong implication with the physical time. If the nodes are confined in a squared network map of size L × L and if the pairing between sources and destinations follows an uniform distribution, then the data delivery can be actually done with only two transmissions. A fundamental condition is that the motion process is ergodic on the whole network map. The trick is that the packet travels most of the distance in the memory buffer of a mobile relay instead of being moved like a "hot potatoe" as in [START_REF] Gupta | The capacity of wireless networks[END_REF]. We denote v the average node velocity. The consequence is that with Grossglauser Tse scheme, the useful capacity density is now C 2 λ and the transport density rises to C 2 λΘ(L). But Grossglauser Tse scheme shows two main drawbacks:

1. the transport capacity does not scale well, it would drop when destinations are distributed close to sources instead of being uniformly distributed over the square.

2. The average packet delivery delay which is of order L 2 vr(λ) = L 2 √ λ vα , does not significantly drop when the destinations are closer to the sources.

Notice that the average speed of information propagation, defined as the ratio of the average distance to destination over the average delivery time, is of order vr(λ) L with Grossglauser Tse algorithm and tends to zero when L or λ tends to infinity, which suggests that the information propagation speed tends to zero when the transport capacity tends to infinity.

Our aim is to show that, for a given targeted transport capacity density, there exists an upper bound v p (∆) of information propagation. Quantity v p (∆) depends of λ and on the average distance ∆ travelled by the packet toward the destination between two transmissions. In this case the transport capacity equals to Cλ∆. The speed also depends on the node mobility model. To simplify we assume that store and forward operations take zero time, which means that the propagation speed is infinite in Gupta and Kumar model. Assuming non zero time would only introduce minor complications.

When node velocity is bounded by v max , quantity ∆ ∆-r(λ) v max is a very trivial estimate of v p (∆). This evaluation is not satisfactory since it does not tend to zero when ∆ → ∞. We will exhibit a non trivial estimate of v p (∆) based on the roots of a multivariate complex function and such that v p (∆) → ∞ when ∆ → r(λ), and v p (∆) → 0 when ∆ → ∞.

Our result makes the link between [START_REF] Gupta | The capacity of wireless networks[END_REF] and [START_REF] Grossglauser | Mobility increases the capacity of ad hoc wireless networks[END_REF] and uses the space-time implication of wireless networks like in [START_REF] Jacquet | Information Propagation Speed in Mobile and Delay Tolerant Networks[END_REF]. The paper is organized as follow: we introduce the models and parameters in the next section, in the section "Methodology" we introduce an upper bound scheme and define the concept of packet journey as in [START_REF] Jacquet | Information Propagation Speed in Mobile and Delay Tolerant Networks[END_REF]. We develop our mathematical analysis in the section "Journey Analysis". In a last section we compare to an existing scheme based on angles.

Models and parameters

We assume that the nodes moves according to an i.i.d. random walk. They move on strait lines at constant speed v and change heading after an exponentially distributed time of parameter τ (expressed in inverse time). The average "free space" distance, i.e. the distance traveled in strait lines between two consecutive heading changes, is v τ . When the node change heading it takes a random direction whose angle is uniformly distributed on [0, 2π]. If we consider that nodes may have different speed with a maximum speed v max , we consider the optimal case where nodes move at constant speed v = v max .

We assume that the node motions are unpredictable. Therefore the packet delivery delay is a random variable. We denote T (L) the packet delivery delay of a packet to a destination at distance L. To simplify we will assume that the destination does not move, but extension to mobile destination will be given in the generalizations, and in this case L indicates the distance of the destination when it receives the packet.

INRIA

Definition 1. Quantity v p is a packet speed upper bound [START_REF] Jacquet | Information Propagation Speed in Mobile and Delay Tolerant Networks[END_REF] if for all c > v p :

lim L→∞ P T (L) < L c = 0 . (1)
We denote N (L) the number of times the packet is transmitted before being delivered to a destination at distance L. Let ∆ be a length. We call v p (∆) an upper bound constrained packet speed, defined as follows.

Definition 2. Quantity v p (∆), in length times inverse time unit, is a constrained packet speed upper bound if for all c > v p (∆):

lim L→∞ P T (L) < L c | N (L) < L ∆ = 0 . (2)
We notice that when N (L) < L ∆ then the transport capacity density is greater than Cλ∆. We should have v p (∆) → ∞ when ∆ → r(λ), since it corresponds to the classic store and forward scheme without carry phase.

Methodology

Transmission phases versus carry phases

Our aim is to find a packet speed upper bound for all possible schemes. A scheme is characterized by a sequence of transmit and carry phase. A transmit phase is when a relay transmits the packet to a next relay within the radius r = r(λ). A transmission phase can be made of several transmissions in series over a sequence of consecutive relays, each of them being within radius r from the previous one. We will suppose that the new relay has a random motion vector whose angle is uniformly distributed in [0, 2π]. This means that we assume that the transmitter is not aware of the motion vector of the next relay. We will later give a straightforward extension to the case where the transmitter selects with respect to the motion vector of the later.

A carry phase is a period when the relay carries the packet in its memory and proceeds in its random walk before retransmitting the packet to a new relay.

Upper bound schemes

We consider a source S and a destination D located at a distance L of S. In order to have a lower bound on the delay delivery, we assume that the packet is delivered when it reaches any point on the line containing D and orthogonal to the vector (S, D). We call this line, the "destination wall". we denote d the unitary vector oriented like (S, D). With respect to this model we consider the class of optimal schemes such that :

1. Packet is transmitted to the destination wall when a relay arrive at distance r from the destination wall;

2. otherwise in transmission phase, the packet is transmitted to the closest node to the destination wall, i.e. we assume it is always at distance r from the transmitter oriented according to d;

3. Transmission phases are made of an arbitrary number of emissions and occur only when the mobile relay changes heading.

It is relatively straightforward to see that the optimal transmission phase would occur on points on the mobile relay trajectory which correspond to local optima with respect to the distance to the destination wall.

Protocol versus journey analysis

We introduce the concept of journey as in [START_REF] Jacquet | Information Propagation Speed in Mobile and Delay Tolerant Networks[END_REF]. We consider a given destination wall and the unitary vector d. A journey J is a space-time trajectory made of a sequence of carry phases and transmission phases. We assume that the journey ends within transmission distance r of the destination wall. We denote s(J) the start point of the journey, called the source point. We denote t(J) the difference of time between the two end points, which is basically the packet delivery delay. We denote n(J) the number of packet transmissions made on the journey J. In figure 1 and 2 we display the spatial and time trajectory of a journey J with n(J) = 3. The fastest journey would take L r hops. Let a point S at distance L of the destination wall. Let T be a real number and N an integer. Our aim is to find if there exist a journey J such that s(J) = S, t(J) < T and n(J) ≤ N . We deduce that if there exist is a protocol which can deliver the packet from S to D in less than T time, with less than N transmissions. Therefore whatever the protocol, the probability to deliver the packet in less than T time with less than N transmissions is always smaller than the probability that there exists a journey J such t(J) < T , under the condition that s(J) = S and n(J) ≤ N . We call this probability P (L, T, N ). Since by [START_REF] Grossglauser | Mobility increases the capacity of ad hoc wireless networks[END_REF], we know that there is always a journey with n(J) ≤ N (provided that N ≥ 2), This is also the probability that there exists a journey J such t(J) < T and n(J) ≤ N , under the condition s(J) = S.

Our aim is to estimate an upper bound of the quantity P (L, T, N ). We denote by E(L, T, N ) the average number of journeys J that starts from a source S at distance L to the destination wall, such that t(J) < T and n(J) ≤ N . We have P (L, T, N ) ≤ E(L, T, N ). We will evaluate E(L, T, N ).

INRIA

Journey Analysis 4.1 Journey density function

We develop a tool comparable inspired from [START_REF] Jacquet | Information Propagation Speed in Mobile and Delay Tolerant Networks[END_REF], but adapted to dense networks. We are going to estimate the space-time density of journey starting from a point S at distance L to the destination wall. We call d(J) the end point of of a journey J. The are several journeys that can connect S to the destination wall. We want to know the average number of journeys that satisfies certain conditions on t(J) and n(J). We call f (x, t, n) the journey density function, i.e. the density function of the average number of journeys such that (S, d(J)). d = x, t(J) = t and n(J) = n. This is a density because the variables x and t stand on continuous space, contrary to the variable n which is discrete. We have the inequality

E(L, T, N ) ≤ N i=1 T 0 dt x>L-r f (x, t, i)dx , (3)
which translates the fact that at some time between 0 and T , the packet has been within r of the distance wall.

Journey space-time Laplace transform

We denote the Laplace transform

w(ρ, θ, u) = n ∞ 0 dt f (x, t, n) exp(-θt -ρx)u n dx .
Via inverse laplace transform we will get

E(L, T, N ) = 1 (2iπ) 3 du (1 -u)u N +1 (4)
× w(ρ, θ, u) e -T θ-(L-r)ρ dθdρ θρ

We will build function w(ρ, θ, u) via enumeration methods.

A transmission phase is made of an arbitrary number of emissions, all made within radius r from the last relay and oriented according to d. Therefore the space-time Laplace transform of a transmission phase is equal to

1 1-ue -rρ .
Notice that variable θ is not involved in the formula, since the transmission phase is supposed to take zero time.

A carry phase is made of an arbitrary, but non zero, number of free space moves. If the free space move is made at speed v heading with angle φ with d during time t then its space-time Laplace transform is exp(-vtρ cos(φ) -θt).

Integrating on an exponential time distribution of parameter τ for t, and for φ uniform on [0, 2π] we get the space-time transform of one free space move equal to τ √ (θ+τ ) 2 -ρ 2 v 2 . Since a journey J is an alternance of free space moves interleaved by transmission phase, it can be described in term of the formalism of grammar by J = X * (F X * ) * where the symbol X means an emission, and F a free space move. The Laplace transform is

1 1-ue -rρ 1 - 1 1-ue -rρ τ √ (θ+τ ) 2 -ρ 2 v 2 -1
. To get w(ρ, θ, u) one must multiply by the Laplace transform of the last interrupted free space move, namely τ

√ (θ+τ ) 2 -ρ 2 v 2 : w(ρ, θ, u) = 1 (1 -ue -ρr ) (θ + τ ) 2 -ρ 2 v 2 -τ . (5)
Let

D(ρ, θ, ω) = (1-e -ω-ρr ) (θ + τ ) 2 -ρ 2 v 2 -τ = 1 w(ρ,θ,,e -ω )
. We call Kernel K the set of tuples (ρ, θ, ω) such that D(ρ, θ, ω) = 0. Theorem 1. For all (ρ, θ, ω) ∈ K, for all ω > ω, there exist A such that E(L, T, N ) ≤ A exp(θT + ρL + ω N ) .

Proof. See appendix.

Theorem 2. For all (ρ, θ, ω) ∈ K the ratio -θ ρ+ ω ∆ is an information speed upper bound of v p (∆), as long as it is positive.

Proof. For a given inter-transmission distance ∆, the number of transmissions on a distance L should be smaller or equal to N = L ∆ . Let ω > ω > ω, according to Theorem 1 there exist A such that we have

P (L, T, N ) ≤ A exp(θT + ρL + ω L ∆ ). Let c(ρ, θ, ω) = -θ ρ+ ω ∆ , we have P (T < L c(ρ, θ, ω") | N < N ∆ ) ≤ E(L, L c(ρ, θ, ω") , L ∆ ) ≤ A exp(θ L c(ρ, θ, ω ) + ρL + ω L D ) = A exp((ω -ω ) L ∆ )
which tends to zero when L → ∞, and since c(ρ, θ, ω ) tends to c(ρ, θ, ω) when ω tends to ω, the quantity c(ρ, θ, ω) is a propagation speed upper bound of v p (∆).

INRIA

The Kernel is made of (ρ, θ, ω) such that θ ≥ 0, (θ + τ

) 2 > τ 2 + v 2 ρ 2 and ω = -rρ -log 1 - τ (θ + τ ) 2 -ρ 2 v 2 .
Thus the upper bound ratio has expression

R(ρ, θ) = -θ (1 -r ∆ )ρ -1 ∆ log 1 - τ √ (θ+τ ) 2 -ρ 2 v 2 . ( 6
)
We notice that the expression log

1 1- τ √ (θ+τ ) 2 -ρ 2 v 2
is equal to the logarithm of the Laplace transform of the space time vector between the position where a relay receives the packet and the position where it forwards it.

When ∆ → ∞ the ratio becomes -θ ρ and the minimum ratio comes when ρ and τ tend to zero. But in this case θ = O(ρ 2 ), therefore the minimum ratio is zero. When ∆ = r the ratio becomes

R(ρ, θ) = θr log 1 - τ √ (θ+τ ) 2 -ρ 2 v 2
, which is always negative, therefore there is no speed upper bound. This is confirmed by the fact that the speed is indeed infinite since the journey exclusively made of emission segments arrive to D in zero time.

Kernel Analysis in the Basic Case

We want to give an analytic expression for the minimum value of R(ρ, θ). We denote x = (θ + τ ) 2 -v 2 ρ 2 the Minkowski norm of vector (θ + τ, ρ). Let (x) = -log(1 -τ √ x ). We have

R(ρ, θ) = θ∆ (∆ -r)ρ + (x)
.

Therefore the optimal value of (ρ, θ) only depends on the difference ∆ -r. At the optimal point we must have ∂ ∂θ R(ρ, θ) = 0 and ∂ ∂ρ R(ρ, θ) = 0. This leads to the Lagrange system:

(∆ -r) ρ θ + 1 θ (x) -2(θ + τ ) (x) = 0 ∆ -r -2v 2 ρ (x) = 0 with x = (θ + τ ) 2 -ρ 2 v 2 .
The resultant of this system is:

2 (θ + τ )θ -v 2 ρ 2 (x) = (x)
which can be rewritten in x -(x) 2 (x) = (θ + τ )τ which gives a parametric description of the optimal tuples for x > τ 2 : Via straightforward application of analytic expansion when ∆ → ∞ we get

     θ = 1 τ (x -(x) 2 (x) ) -τ ρ = -1 v (θ + τ ) 2 -x 2 ∆ = r + 2v 2 ρ (x)
v p = 2v 2 τ ∆ 1 -log 2 + 2 log( ∆τ v ) + O(log log( ∆τ v )) ,
and when ∆ → r:

v p = v r ∆ -r (1 + O(∆ -r))

Speed aware next relaying

In this section we suppose that in a transmission phase the node transmits to the relay which optimizes its motion vector toward destination. For the INRIA upper bound speed we simply assume that the motion vector of the relay exactly heads to the destination. In this case the Laplace transform of the space-time displacement between the reception and the next packet transmission is

1 + 1 θ + τ + ρv 1 1 - τ √ (θ+τ ) 2 -ρ 2 v 2 and we have therefore R(ρ, θ) = θ∆ (∆-r)ρ+ (ρ,θ) with (ρ, θ) = log   1 + 1 θ + τ + ρv 1 1 - τ √ (θ+τ ) 2 -ρ 2 v 2   .
, v = 1, τ = 2 . Since ∂ ∂θ R(ρ, θ) = 0 and ∂ ∂ρ R(ρ, θ) = 0, we have -(∆ -r)ρ -+ θ ∂ ∂θ = 0 (∆ -r)ρ + ρ ∂ ∂ρ = 0 The Kernel equation is = θ ∂ ∂θ + ρ ∂ ∂ρ . (7)
But contrary to the previous case, the equation is not separable and we cannot apply Lagrange parametric reduction. Therefore we have to rely on a Euler finite element resolution.

Moving destination

The previous analysis was done under the hypothesis that the destination D of the packet is fixed. The analysis can be easily extended to the moving destination case. In this case the distance L must be understood as the distance between the original position of the source S at the packet generation and the final position of D, when the later eventually receives the packet. The modification consists into multiplying the journey space-time Laplace transform w(ρ, θ, u) by the space Laplace transform of the destination trajectory, i.e. In this case the new laplace transform has the same Kernel K, the roots of 1 w(ρ,θ,0) , (i.e. (θ + τ ) 2 -ρ 2 v 2 -τ ) being screened by the roots of D(ρ, θ, ω).

A sub-optimal angle protocol

This scheme has been introduced in [START_REF] Jacquet | Space and Time Curvature in Information Propagation in Massively Dense Ad Hoc Networks[END_REF]. The principle (for fixed destination) is the following, assuming θ c is a non negative protocol parameter strictly smaller than π:

• The relay keeps and carries the packet as long as the angle between its heading and the bearing to the destination is smaller than θ c .

• Otherwise it transmits the packet to the next hop to the node the closest to the destination.

• If the relay is in range of the destination it transmits the packet to the destination.

The angle protocol is shown to satisfy:

∆ = r + θ π sin(θ) π-θ v τ T = θ π-θ 1 τ ,
where T indicates here the average time between two consecutive transmissions. Notice that ∆ cannot exceed r + v τ . The average propagation speed is therefore v p = π-θ θ rτ + sin(θ) π v.

Conclusion

We have identified an upper bound propagation speed of information which is function of the transmission inter-distance ∆. This result makes the link between Gupta and Kumar result (where ∆ → r(λ)) and Grossglauser and Tse scheme where ∆ → ∞. This confirms the strong implication of space-time in the evaluation of transport capacity of a wireless network.

Figure 1 :

 1 Figure 1: The space trajectory of a journey from source S at distance L to a destination D. Transmission phases are in red.

Figure 2 :

 2 Figure 2: The time trajectory of a journey from source S at distance L to a destination D with average inter transmission ∆.

Figure 3 :

 3 Figure 3: Information upper bound speed versus ∆, compared to actual speed of angle protocol (dashed), for r = 0.1, v = 1, τ = 2 .

Figure 4 :

 4 Figure 4: Information slowness versus ∆, compared to actual slowness of angle protocol (dashed), for r = 0.1, v = 1, τ = 2 .

Figure 5 :

 5 Figure 5: Information speed versus ∆ in speed aware next relaying (red) compared to basic relaying (green), for r = 0.1, v = 1, τ = 2 .

1 √(

 1 θ+τ ) 2 -ρ 2 v 2 -τ, namely w(ρ, θ, 0). Therefore the new laplace transform w(ρ, θ) has expression w(ρ, θ, u) = w(ρ, θ, 0)w(ρ, θ, u) .

RR n°xxxx

Appendix

of theorem 1. we proceed via inverse Laplace 4 on complex tuples (u, θ, ρ):

where (u 0 , θ 0 , ρ 0 ) is a real tuple which belongs to the definition domain of w(ρ, θ, u). Let (ρ 0 , θ 0 , ω 0 ) ∈ K and let u 0 = e -ω with ω > ω 0 . Thus (ρ, θ, u) with |u| = e -ω , (θ) = θ 0 and (ρ) = ρ 0 is in the definition domaine of w(ρ, θ, u). The following inequality holds:

The function w(ρ,θ,u) θρ is absolutely integrable. Thus we don't need a sophisticated integration path as in [START_REF] Jacquet | Information Propagation Speed in Mobile and Delay Tolerant Networks[END_REF]. Indeed since we have |w(ρ, θ, u)| ≤ w( (ρ), (θ), |u|), a careful analysis gives

which confirms the absolute integrability of w(ρ,θ,u) θρ

. Thus

, w(ρ 0 , θ 0 , u 0 )} .


J Hermet

G Geneste

G Dezanneau

Molecular dynamics simulations of oxygen diffusion in GdBaCo 2 O 5.5

The mechanisms of oxygen diffusion in GdBaCo 2 O 5.5 compound are investigated by molecular dynamics simulations. The results confirm that diffusion is mainly bidimensional with oxygen moving in the ͑a,b͒ plane while diffusion along the c axis is much more difficult. Between 1000 and 1600 K, the activation energy for diffusion is about 0.6 eV, close to experimental values. Going deeper inside the oxygen diffusion mechanism, we see that this diffusion occurs mainly in the cobalt planes while most of the oxygen vacancies are kept in the Gd planes. Analysis of oxygen motions show that Gd planes can be seen as source-sink for the oxygen vacancies rather than as fast pathways. © 2010 American Institute of Physics. ͓doi:10.1063/1.3504250͔

The main issue in solid oxide fuel cells ͑SOFC͒ technology is to lower the operating temperature 1 in the range 500-750 °C. Due its high activation energy process, the cathode becomes a major source of losses at low temperature. 2 Recent experiments show the suitability of GdBaCo 2 O 5.5+x as a cathode material for intermediate temperature SOFCs. 3,4 To understand the particularities of this promising material, we study here the oxygen diffusion in GdBaCo 2 O 5.5 using molecular dynamics ͑MD͒ simulations.

GdBaCo 2 O 5.5+x is a double perovskite compound with Gd and Ba cations alternating on the A site along the c axis, leading to a tetragonal symmetry at high temperature. It generally contains oxygen vacancies and mixed valency on cobalt site. The diffusion of oxygen species is far from being understood in these compounds even if the presence of oxygen vacancies in the Gd plane, as found from neutron diffraction, 5 suggested preferential diffusion in this atomic plane. We thus performed here MD simulations to get deeper inside the oxygen diffusion mechanism. All simulations were run with the DLPOLY code 6-8 using a supercell of 4 ϫ 4 ϫ 2 unit cells ͑containing 304 atoms and 16 vacancies͒. This gives roughly the same size for the supercell box in all directions due to the doubling of the c parameter in the pristine structure. Interactions between ions were described by a long-range Coulombic term calculated by Ewald summation, and a short-range Buckingham pair potential. For Coulombic interactions, the formal charges +2, +3, +3, and Ϫ2 were used, respectively, for Ba, Gd, Co, and O ions. The use of formal ionic charge was shown to represent correctly the dynamic features of atoms in most oxide compounds. [9][10][11][12] The Buckingham potential is described by the following:

ij ͑r͒ = A ij expͩ- r ij ͪ-C ij r 6 , ͑1͒
where r is the distance between the atoms i and j, and A ij , ij , and C ij are potential parameters specific to each ion pair whose values are given in Table I. Two sets of potentials were used which were previously validated in MD simulations of cobaltite compounds. For the set no. 1, we used the potentials from Fisher et al. 9 for Ba-O, Co-O, O-O interactions, and we determined the parameters for gadoliniumoxygen pair by fitting structural data of Gd 2 O 3 . For the second set, we directly used the interatomic potentials from Rupasov et al., 10 Busker et al., 11 and Minervini et al. 12 The potentials were tested by performing energy minimizations with the GULP software 13 at a temperature of 0 K on a 4 ϫ 4 ϫ 6 supercell with vacancies located in the Gd plane. As results, the calculated cell a = b and c parameters expressed for a pseudotetragonal double perovskite cell, were 3.76͑1͒ Å and 7.58͑1͒ Å for the potentials set no. 1, and 3.87͑3͒ Å and 7.71͑6͒ Å for potentials set no. 2. These values are close to those extrapolated to 0 K from high temperature measurements ͑a=b=3.85337 Å and c = 7.5318 Å͒.

The system was first equilibrated at 1000, 1200, 1400, and 1600 K and at zero pressure for 10 000 time steps ͑with a time step of 2 fs͒ in the isothermal-isobaric ͑NPT͒ ensemble with Nosé-Hoover thermostat. Then 100 000 time steps allowed to find the equilibrium lattice parameters. While for the set of potentials no. 1, the tetragonal high temperature structure was maintained with cell parameters close to experimental values, for the set no. 2, a dramatic increase in the c cell parameter was observed leading to a collapse of the structure. In this case, we thus used a a͒ Author to whom correspondence should be addressed. Electronic mail: guilhem.dezanneau@ecp.fr. DL_POLY isotropic option during NPT simulations for which, the a = b and c parameters are allowed to vary in a non independent way. Then, 100 000 times steps were performed in the canonical ensemble ͑NVT͒ with Nosé-Hoover thermostat, keeping the lattice constants at their previously determined equilibrated values, and finally the diffusion properties were studied in the microcanonical ensemble ͑NVE͒, for 2 ϫ 10 6 time steps ͑up to 5 ϫ 10 6 time steps͒. The calculated thermal expansion was 15.2ϫ 10 -6 K -1 for set no. 1, while it was only 8.8ϫ 10 -6 K -1 for set no. 2. The first value is very close to the one experimentally found by Tarancón et al. 4 of 16.4ϫ 10 -6 K -1 for GdBaCo 2 O 5+x . This result would indicate that set no. 2 is probably not fully adapted for such simulations.

The oxygen diffusivity was determined from the evolution of the mean square displacement as a function of time at different temperatures using the Einstein's law. The values of D are represented in the Arrhenius plot of Fig. 1. We also plot the oxygen tracer diffusion coefficient as determined from Tarancon et al. 12 These two coefficients could be directly compared, neglecting the evolution of oxygen composition with temperature. We also present in Fig. 1 the chemical diffusion coefficients as determined from relaxation measurements to which was applied a correction. Indeed, the chemical diffusion coefficient was divided by the thermodynamic factor set to 75, corresponding to the value obtained from the oxygen content in GBCO in the range 950-1050 °C. 15 From our simulations, the oxygen transport can be described by an Arrhenius law with an activation energy of 0.67 eV and 0.56 eV, respectively, for sets 1 and 2. Theses values reasonably compare with the value of 0.60 eV found from isotope exchange. 14 They are also close to the values of oxygen chemical diffusion activation energies of 0.66 eV and 0.77 eV found, respectively, by Taskin et al. 16 and Choi et al. 17 from mass and conductivity relaxation measurements. Nevertheless, all the diffusion coefficients presented in Fig. 1 spread over several orders of magnitude for a same temperature, underlining the difficulty to determine experimentally the oxygen diffusion coefficient.

MD simulations also allow a deeper understanding of diffusion mechanism. Figure 2 compares the mean square displacement of oxygen along the c-axis and in the a-b plane at 1400 K. Oxygen atoms mostly diffuse in the a-b plane with a diffusivity of 1.1ϫ 10 -7 cm 2 s -1 . This diffusivity is almost ten times higher than the one found along the c-axis of 1.3ϫ 10 -8 cm 2 s -1 . This anisotropic diffusion can be explained by the absence of oxygen vacancies in the barium planes, making the efficient diffusion of oxygen vacancies across Ba planes difficult.

Barium planes logically do not contain oxygen vacancies. As foreseen from previous neutron diffraction studies, vacancies are mainly located in the gadolinium planes ͑ϳ95%͒, but significant amount of them is also in the cobalt planes ͑ϳ5%͒. When studying the individual displacements of oxygen atoms, we observe that they spend most of their time in the cobalt planes and occasionally hop into gadolinium planes ͑where they stay only for a few picoseconds while they can remain up to some nanoseconds in cobalt planes͒. Figure 3 presents the oxygen density ͑a͒ in ͑001͒ cobalt planes and ͑b͒ in ͑100͒ cobalt planes. Figure 3͑b͒ confirms that oxygen atoms in barium planes stay at the same position and do not diffuse in the neighboring cobalt planes. Besides, a count of the oxygen hops shows that oxygen atoms make roughly three times more jumps in the cobalt planes than going back and forth or cross a gadolinium plane. This would mean that diffusion mainly occurs in the Co plane and that vacancies in the Gd planes are mainly trapped. Their presence is nevertheless necessary to make one oxygen escaping from time to time from the Co plane leaving a highly diffusing oxygen vacancy in the Co plane. The first intuition that vacancies in Gd planes lead to an enhanced diffusion is then probably not fully correct. Actually, the diffusion in perovskite compounds involves nearestneighbor jumps between oxygen positions and can thus not be explained by the only consideration of vacancies in Gd planes. From these simulations, we indeed show that Gd planes serve as source-sink for oxygen vacancy rather than as real pathways.

In conclusion, MD simulations predict that oxygen transport in GdBaCo 2 O 5.5 is strongly anisotropic with an activation energy of around 0.6 eV in the temperature range of 1000-1600 K. Oxygen atoms mainly diffuse along the cobalt planes with short hops into gadolinium planes and very seldom cross a barium plane. Meanwhile, most of oxygen vacancies are kept into gadolinium planes while some of them are moving along the cobalt planes to allow oxygen diffusion.

This work was performed using HPC resources from GENCI-͓CCRT/CINES/IDRIS͔ ͑Project 2010x2010096468͒.

͑Received 3

 3 June 2010; accepted 27 September 2010; published online 25 October 2010͒

FIG. 1 .

 1 FIG. 1. Arrhenius plot of oxygen diffusivity in GdBaCo 2 O 5.5 : simulations ͑triangles͒ and comparison to diffusion coefficient as obtained from oxygen exchange ͑Ref. 12͒ and from oxygen chemical diffusion oxygen corrected by thermodynamic factor ͑Refs. 13 and 14͒.

TABLE I .

 I Sets 1 and 2 of Buckingham interatomic potential parameters for GdBaCo 2 O 5.5 .

			A		C
	Interaction		͑eV͒	͑Å͒	͑eV Å -6 ͒
	Ba 2+ . . .O 2-	Set 1 a	1214.4	0.3537	0.0
		Set 2 b	905.70	0.3976	0.0
	Gd 3+ . . .O 2-	Set 1 c	1458.38	0.3522	0.0
		Set 2 d	1885.75	0.3399	20.34
	Co 3+ . . .O 2-	Set 1 a	1329.82	0.3087	0.0
		Set 2 e	1226.32	0.3087	0.0
	O 2-. . .O 2-	Set 1 a	22764.3	0.149	43.0
		Set 2 e	9547.96	0.2192	32.0

Downloaded 10 Nov 2010 to 138.195.90.71. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions


Maryline Chetto
email: maryline.chetto@univ-nantes.fraudrey.queudet@univ-nantes.fr

Audrey Queudet

Scheduling and Power Management Facilities in Linux/RTAI for Real-time Systems powered by Ambient Energy Source

Keywords: RTAI, Linux-based operating systems, power aware scheduling, energy harvesting, component-based development

In this paper, we present a component-based development project dedicated to real-time embedded harvesting systems. Energy harvesting consists in converting the ambient energy into electricity to power electronic devices, making them self-sufficient. Instead of minimizing its energy consumption, such a system must operate in a so-called energy neutral mode by consuming only as much energy as harvested in the environment. Consequently, our objective is to develop and integrate power management and scheduling facilities specifically adapted to energy harvesting embedded systems with real-time constraints. For example, these system level facilities have to make on-line decisions such as : when should the system use energy (i.e. active state) and when should it recharge the energy storage (i.e. idle state) ? We are providing these facilities as open-source software components. Adding them to the existing open-source CLEOPATRE library based on Linux/RTAI will participate to the evolution of the open-source real-time technology.

Introduction

Scavenging energy from available sources offers the potential to power applications indefinitely without wires and extend the operating times of battery-powered systems. Ambient energy sources include light, heat differentials, mechanical vibration, transmitted RF signals, or any source that can produce an electrical charge through a transducer (photovoltaic cell , piezoelectric element...). There are many existing energy harvesting deployments all around us and most of them are real-time ones. In a real time system, performance is generally measured by the deadline miss rate. Lower is this rate, better is the Quality of Service (QoS) delivered by the system. The autonomous nature of operation makes it imperative that the system learns its own energy environment and adapts its power consumption accordingly. Indeed, an important issue in such Real-Time Harvesting (RTH) systems is the way they control their energy consumption, which directly affects their performance, as well as their possibility to provide a continuous service, as depicted in Figure 1. A good policy for power management and scheduling will have to make best effort decisions to adapt the quality of service according to the available energy profile. As the stored energy and the energy harvested from the environment are time-varying and thus very unstable, performance of a RTH system will heavily depend on its operating system and more precisely on all mechanisms for dynamically managing the processor activity and then maintain an acceptable quality of service even through a degraded mode. The paper is organized as follows : Section 2 describes the characteristics of real-time energy harvesting systems. In section 3, we present a brief state of the art about both real-time scheduling and RTOS under energy constraints. A new scheduling framework is introduced in section 4. We describe an open-source library, Cleopatre, which will integrate energy management software components. Section 6 concludes the paper.

FIG. 1 -A typical RTH system

Specificities of energy harvesting computers

Motivations

Most prior wireless monitoring systems in last decades have relied on continuous power supplied by batteries such as lithium-ion cells. Their disadvantage is that they become depleted, must be periodically replaced or recharged and consequently place hard restrictions on products' usability, lifetime, and cost of ownership. Moreover, while processing power roughly doubles ever two years, battery technology advances at a much more sluggish pace. Even if it is possible to increase their energy density by tenfold within a few years, we must still consider practical safety concerns. First, given improper use, batteries with extremely high energy densities can dangerous, explosive devices. Second, in many embedded applications, battery replacement is impractical or has high labor costs associated with maintenance. Harvesting energy in surrounding environment to power embedded systems for the lifespan appears to be the alternative to conventional batteries. In that context, super-capacitors or dynamically rechargeable batteries can be used to store the ambient energy. New applications of energy harvesting technology for embedded systems are beginning to drive economic development in many sectors, as seen in the newly created on line journal about energy harvesting products (http ://www.energyharvestingjournal.com). Given the limitations of battery power, there are commercial, scientific and military needs for systems which can operate effectively using alternative renewable power sources. It concerns as well the high technology sectors as the general public products in which wireless sensor networks are used in a variety of applications, such as environmental applications (forest fire and flood detection, monitoring of drinking water and level air pollution), military applications (battlefield surveillance, reconnaissance of enemy forces), health applications (tele-monitoring of human physiological data, tracking and monitoring of doctors and patients), home applications (intrusion detection), or commercial applications (monitoring of product quality, climate control in large buildings). While some of these applications are marginal today, they will become commonplace one day and will contribute to improve the quality of living while reducing the overall health care cost. But in order to make this technology affordable to the general population, the researchers have to develop efficient devices built with inexpensive off-the-shelf components for harvesting and storing environmental energy, for sensing, processing and communicating data. Low-cost, autonomous sensor networks will enrich our lives by providing valuable data about the status of our environment (including human body), and will do so with no reoccurring cost or impact on the environment.

Real-time scheduling and energy harvesting

Typical embedded real-time applications require periodic activities (in general sensing, actuating and communicating) that have to be cyclically executed at fixed rates and within specific deadlines. To cope with the timing constraints, the computing system must be provided with a scheduler in charge of dispatching and executing the activities (called tasks) at the good times. A lot of work has been done for the well known Rate Monotonic (RM) and Earliest Deadline First (EDF) scheduling strategies [START_REF] Liu | Scheduling algorithms for multiprogramming in a hard real-time environment[END_REF]. These ones are commonly implemented in commercialized and free real-time operating systems including Linux/RTAI, see [START_REF] Liu | Real-Time Systems[END_REF] for a survey on real time uniprocessor scheduling. However, under unpredictable conditions (i.e. workloads), their performance may be poor. As an example, EDF algorithm is one of the most widely used online scheduling policies and it is proven to be optimal in deterministic environments only. Hence, the same poor behavior can be expected under fluctuating energy approvisioning. In brief, despite the significant body of results, specificities of energy harvesting has not been supported by these real-time schedulers. Researchers have been motivated to design specific power management capabilities for RTH systems from less than 10 years only. The crucial issue is to find both power management and scheduling mechanisms that can adapt dynamically the activity of the embedded system according to the available energy. Up to now, when designing a real-time embedded system, the unique concern has been time, leaving energy efficiency as a hopeful consequence of empiric decisions. In the other hand, when designing an energy harvesting system, the unique concern has been energy efficiency in terms of harvesting and storing, abstracting for timing considerations. But the primary concern when designing a RTH system must be to consider both energy and time availabilities in order to avoid energy waste and energy lack and to guarantee timing requirements of real-time activities perpetually.

State of the Art

Real-time scheduling and energy management

Most of the research on energy-aware real-time scheduling focused on either minimizing the energy consumption or maximizing the system performance such as the lifetime achieved under the energy constraints [SCH 04]. In such works, rechargeability of the energy storage unit is always disregarded. These research activities use Dynamic Power Management techniques (DPM) and Dynamic Voltage and Frequency Selection (DVFS), [START_REF] Scmitz | System-Level Techniques for Energy Efficient Embedded Systems[END_REF]. Although both DPM and DVFS are able to reduce the power consumption of a device, they do not suffice for designing an efficient RTH system because there is a fundamental difference between power-aware and low-power technologies. In a RTH system, we have to make the best use of the available power and the goal of a scheduler is to assign real-time tasks to time slots such that all timing and power constraints are satisfied every time. We then say that the system operates in an energy neutral mode by consuming only as much energy as harvested. Little work has explored the problem of scheduling real-time tasks in a uni-processor rechargeable system. The work by Allavena et al. was certainly the first one to concentrate on a rechargeable system with hard real-time constraints [START_REF] Allavena | Scheduling of frame-based embedded systems with rechargeable batteries[END_REF]. They addressed the problem of finding the scheduling of frame-based embedded systems which is able to execute all the tasks within the deadline. It relies on the unrealistic assumption that the power scavenged by the energy source is constant and all tasks consume energy at a constant rate. But in a real application, instantaneous power consumed by tasks varies along time depending on circuitry and devices required by the tasks. A solution is presented that schedules tasks in such a way that the wasted recharging energy is minimized and the energy storage level is at all times within two limits. Later in 2006, Moser at al. focused on scheduling tasks with deadlines, periodic or not. They proposed LSA (Lazy Scheduling Algorithm) and prove it to be optimal [START_REF] Moser | Real-time scheduling for energy harvesting sensor nodes[END_REF]. In that work, the consumption power of the computing system is characterized by some maximum value which implies that the total energy consumption of every task is directly connected to its execution time through the constant power of the processing device. In a very recent work presented in 2009, we relaxed this restrictive hypothesis [START_REF] Chetto | Real-time Scheduling of periodic tasks in a monoprocessor system with rechargeable energy storage[END_REF].

RTOS and energy management

Several operating systems are available for wireless sensor networks : MANTIS from University of Colorado at Boulder, CONTIKI from the Swedish Institute of Computer Science, NANO-RK from Carnegie Mellon University and TinyOS from University of California at Berkeley. These open-source operating systems have been designed by the academic community for microcontrollers with small amounts of memory and have been used is a large variety of industrial projects. However, none of them provide software facilities for power management that jointly consider energy harvesting and timing constraints. It has been now twenty years that GNU tools are used for embedded software. The unmistakable advantage of this model lies in the earning in flexibility and the possible option to modify the code in the optics to answer specific needs. It lies as well in the training for several years, thanks to Internet notably, of a critical mass of developers. At the moment, companies already propose on the same CD-ROM, realtime kernels patches, network, development (editors, compilers, deboggers) and graphic tools allowing to realize applications and insure a technical support. Regrettably, neither the various versions of realtime Linux (RTAI [START_REF] Mantegazza | DIAPM RTAI for Linux : Why's, what's and how's[END_REF] , Xenomai, RTLinux, Litmus RT , Hard Hat Linux, Etlinux, Red Linux, KURT), nor the marketed owners executives do not answer suitably the very diversified requirements by the new applications powered through ambient energy source. Linux is now everywhere, including in places where there are no power outlets. An appropriate design and power management implementation is required though, to allow the system to operate with an ambient energy source and to consume as little power as possible to preserve our environment. By providing enhanced power management capabilities to RTAI, we will answer to the requirements of vendors of consumer electronics devices.

GreenPD : a new scheduling scheme

The intuition behind the GreenPD scheduler is to run tasks according to a Priority Driven (PD) rule but not necessarily as soon as possible. For example, GreenEDF will use Earliest Deadline First whereas GreenRM will use Rate Monotonic. However, before authorizing a task to execute, we must ensure that the energy level is sufficient to provide energy for all operations performed by occurring tasks (general processing, I/O operations, memory accesses,...) considering their worst case timing and energy requirements and the replenishment rate of the storage unit. When this condition is not verified, the processor has to be idle so that the storage unit recharges as much as possible and as long as the system will be able to meet all the deadlines i.e. the system will have available time to remain idle. We assume here that the system consumes no energy in the idle state. The way of scheduling tasks is based on the on-line computation of two quantities called slack time and slack energy, the latter one being a new concept dedicated to hard deadline tasks with energy constraints. The slack-based method for delaying tasks and making the processor inactive during recharging phases of the energy storage unit is the crucial point. On-line computing by how long the tasks should be delayed is possible thanks to a specific mechanism for computing the so-called slack time. GreenPD functional behavior is illustrated in figure 2.

FIG. 2 -GreenPD scheduling scheme

We have proved through simulation how to schedule the tasks of a given application so as to guarantee their timing constraints perpetually by suitably exploiting both the processor utilization and the available ambient energy. This work is conducted both under static and dynamic priorities. GreenPD algorithm, relative concepts such as slack energy and simulation results can be found in [START_REF] Ghor | A Real-Time Scheduling Framework for Embedded Systems with environmental energy harvesting[END_REF]. Furthermore, we are now considering quality of service facilities. Our objective is to provide them as open-source software components on an additional shelf dedicated to dynamic power management.

CLEOPATRE Library

General framework

From 2002 until 2006, a library of free software components was developed within the French National project CLEOPATRE (Software Open Components on the Shelf for Embedded Real-Time Applications) in order to provide more efficient and better service to real-time applications, [START_REF] Silly-Chetto | CLEOPATRE : Open-source Operating System Facilities for Real-Time Embedded Applications[END_REF]. Our purpose was to enrich the real-time facilities of real-time Linux versions. RTAI was the solution adopted for this project because we wanted the CLEOPATRE components to be distributed under the LGPL license which is also the one used in the RTAI project. The LGPL allows proprietary code to be linked to the GNU C library, glibc. When a program is linked with a library, whether statically or using a shared library, the combination of the two is legally speaking a combined work, a derivative of the original library. The ordinary General Public License (GPL) therefore permits such linking only if the entire combination fits its criteria of freedom. The LGPL license permits more lax criteria for linking other code with the library. Companies do not have to release the source to code which has been dynamically linked to an LGPLed library, which makes the use of such codes much more attractive. The CLEOPATRE library offers selectable COTS (Commercial-Off-The-Shelf) components dedicated to dynamic scheduling, aperiodic task service, resource control access, fault-tolerance and now, QoS scheduling. An additional task named TCL (Task Control Layer) interfaces all the CLEOPATRE components and is considered as the task of higher priority in the whole system. It has been added as a dynamic module in $RTAIDIR/modules/TCL.o, and interfaces with the legacy RTAI scheduler defined in $RTAIDIR/modules/rtai_sched.o, as depicted in figure 3. As the CLEOPATRE interface is totally independent from the RTAI core layer, it can be directly used with Xenomai which supports the RTAI API and easily adapted to any other Linux real-time extension. FIG. 3 -Cleopatre Library within Linux/RTAI CLEOPATRE applications are highly portable to any new CPU architecture thanks to this OS abstraction layer which makes the library of services, generic [START_REF] Chetto | Cléopatre : Real-Time facilities for Linux Based Control Applications[END_REF]. The CLEOPATRE Off-the-Shelf components are optional except the OS abstraction layer (TCL) and the scheduler. At most one component per shelf can be selected. Since all components of a given shelf have the same programming interface, they are interchangeable. Everything needed to use and develop CLEOPATRE can be downloaded from the web site of the project : http : //cleopatre.rts-software.org.

Power management COTS

Of particular interest are techniques applicable to running systems, adjusting power parameters on-thefly while ensuring real-time deadlines of running tasks are met. DPM software exploits recent advances in hardware to scale clocking information (such as CPU and core bus frequencies) and core voltages with low latency. This allows these parameters to be adjusted very frequently in order to realize power savings during brief idle periods or execution of tasks with lower performance and power demands. For GreenPD to be operational, the operating system has to automatically select desired power parameters based on system state, and more precisely two fundamental data : slack time and slack energy.

Conclusion

Ten years ago, the evolution of Linux passed by the transfer of the academic skills in real-time processing (scheduling, fault-tolerance, communication) and the contribution of answers to the needs of the new embedded applications (presence of overload conditions, quality of service guarantees, ...). Today, new generation applications are more energy consuming and rely on ambient energy sources in order to be "green". Our project is then to transfer new skills relative to open-source scheduling and power management facilities for such applications.


Mario R Silva-Junior

Stephan P A Sauer
email: sauer@kiku.dk

Marko Schreiber

Walter Thiel
email: thiel@mpi-muelheim.mpg.de

Basis set effects on coupled cluster benchmarks of electronically excited states: CC3, CCSDR(3) and CC2

Keywords: electronically excited states, benchmarks, coupled cluster methods, CC2, CCSDR electronically excited states, benchmarks, coupled cluster methods, CC2, CCSDR(3), CC3

Vertical electronic excitation energies and one-electron properties of 28 medium-size molecules from a previously proposed benchmark set are revisited using the augmented correlationconsistent triple-zeta aug-cc-pVTZ basis set in CC2, CCSDR(3), and CC3 calculations. The results are compared to those obtained previously with the smaller TZVP basis set. For each of the three coupled cluster methods, a correlation coefficient greater than 0.994 is found between the vertical excitation energies computed with the two basis sets. The deviations of the CC2 and CCSDR(3) results from the CC3 reference values are very similar for both basis sets, thus confirming previous conclusions on the intrinsic accuracy of CC2 and CCSDR(3). This similarity justifies the use of CC2-or CCSDR(3)-based corrections to account for basis set incompleteness in CC3 studies of vertical excitation energies. For oscillator strengths and excited-state dipole moments, CC2 calculations with the aug-cc-pVTZ and TZVP basis sets give correlation coefficients of 0.966 and 0.833, respectively, implying that basis set convergence is slower for these one-electron properties.

Basis set effects on coupled cluster benchmarks of electronically excited states: CC3, CCSDR(3) and CC2

Introduction

The importance of an accurate reference data set for excited-state energies and properties is unquestionable. Recently, we introduced a benchmark set [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF][START_REF] Silva-Junior | Benchmarks for electronically excited states: Time-dependent density functional theory and density functional theory based multireference configuration interaction[END_REF][START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF] for vertical electronic excitation energies of 28 medium-sized molecules which represent the most significant organic chromophores. Theoretical best estimates were derived [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF] from linear response CC3 and multistate CASPT2 calculations as well as high-level ab initio literature values. The main purpose of such a comprehensive reference data set is the validation of existing as well as newly developed methods and the parametrization of semiempirical or other adjustable methods. The proposed reference set has already been used in the validation of lower-level coupled cluster methods and approaches based on density functional theory [START_REF] Silva-Junior | Benchmarks for electronically excited states: Time-dependent density functional theory and density functional theory based multireference configuration interaction[END_REF][START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF][START_REF] Goerigk | Computation of accurate excitation energies for large organic molecules with double-hybrid density functionals[END_REF][START_REF] Jacquemin | Extensive TD-DFT Benchmark: Singlet-Excited States of Organic Molecules[END_REF].

In our previous work we employed a TZVP basis set [START_REF] Schäfer | Fully optimized contracted Gaussian basis sets for atoms Li to Kr[END_REF] which was considered sufficiently flexible for the low-lying valence excited states that were of prime interest in our benchmarking. This choice of a medium-size basis set also enabled us to treat the majority of the reference molecules at the CC3 level. Since the TZVP basis lacks diffuse functions, it is less appropriate for higher-lying valence states and for states which are spatially extended and have (partial) Rydberg character. Therefore, the proper identification of such states may become problematic when using the TZVP basis as in our previous studies [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF][START_REF] Silva-Junior | Benchmarks for electronically excited states: Time-dependent density functional theory and density functional theory based multireference configuration interaction[END_REF][START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF], because the character of these higher-lying states might change upon basis set extension. Rydberg states can often be described effectively by adding a single set of very diffuse functions at the center of the molecule, as has e.g. been done for benzene or other polycyclic aromatic hydrocarbons (see Refs. [START_REF] Bak | Atomic integral driven second order polarization propagator calculations of the excitation spectra of naphthalene and anthracene[END_REF][START_REF] Falden | Benchmarking second order methods for the calculation of vertical electronic excitation energies: Valence and Rydberg states in polycyclic aromatic hydrocarbons[END_REF] and references therein), but this is not a convenient option for molecules of arbitrary shape. Therefore, we previously studied [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF] the basis set dependence of the vertical excitation energy of the high-lying 1 1 B 1u singlet state in ethene, which is known to be rather diffuse and susceptible to valence-Rydberg mixing, using twelve different correlation consistent basis sets [START_REF] Dunning | Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen[END_REF][START_REF] Kendall | Electron affinities of first row atoms revisitedsystematic basis sets and wave functions[END_REF]. When comparing against the essentially converged excitation energies computed with the most extended aug-cc-pV5Z and d-aug-cc-pVQZ basis sets, we found differences of more than 0.4 eV for the results with the TZVP basis, but excellent agreement for those with the aug-cc-pVTZ basis (deviations of less than 0.1 eV).

In this work we use the aug-cc-pVTZ basis set [START_REF] Dunning | Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen[END_REF][START_REF] Kendall | Electron affinities of first row atoms revisitedsystematic basis sets and wave functions[END_REF] in a comprehensive study that includes all 28 original benchmark molecules in CC2 calculations and subsets of 21 and 8 molecules at the CCSDR(3) and CC3 levels, respectively. This basis set was chosen because it has been proven to be adequate in coupled cluster calculations of high-lying singlet states not only in the case of ethene [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF], but also for a number of other chromophores (see e.g. Refs. [START_REF] Christiansen | Response functions in the CC3 iterative triple excitation model[END_REF][START_REF] Christiansen | Large-scale calculations of excitation energies in coupled cluster theory: The singlet excited states of benzene[END_REF][START_REF] Hald | Triplet excitation energies in the coupled cluster singles and doubles model using an explicit triplet spin coupled excitation space[END_REF][START_REF] Öhrn | Electronic excitation energies of pyrimidine studied using coupled cluster response theory[END_REF][START_REF] Osted | Theoretical Study of the Electronic Gas-Phase Spectrum of Glycine, Alanine, and Related Amines and Carboxylic Acids[END_REF]). However, even the aug-cc-pVTZ basis set is not optimal for Rydberg states in general, and hence additional molecule-centered diffuse basis functions may be necessary for an accurate description of states with significant valence-Rydberg mixing.

The paper is organized as follows: Computational details are described in the next section. The results for the aug-cc-pVTZ basis are presented and compared to those for the TZVP basis in the following section which also provides various statistical evaluations. The final section offers a summary.

Computational Details

All calculations were performed at the MP2/6-31G* optimized ground-state geometries [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF] using the augmented aug-cc-pVTZ [START_REF] Dunning | Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen[END_REF][START_REF] Kendall | Electron affinities of first row atoms revisitedsystematic basis sets and wave functions[END_REF] basis. The CCSDR(3) [START_REF] Christiansen | Perturbative triple excitation corrections to coupled cluster singles and doubles excitation energies[END_REF][START_REF] Christiansen | Excitation Energies of H 2 O, N 2 and C 2 in Full Configuration Interaction and Coupled Cluster Theory[END_REF] and CC3 [START_REF] Christiansen | Response functions in the CC3 iterative triple excitation model[END_REF][START_REF] Koch | The CC3 model: An iterative coupled cluster approach including connected triples[END_REF][START_REF] Hald | CC3 triplet excitation energies using an explicit spin coupled excitation space[END_REF] computations were carried out with the Dalton 2.0 program package [START_REF]a molecular electronic structure program[END_REF]. Only singlet states were calculated, with the core electrons being frozen during the coupled cluster calculations which causes negligible deviations (less than 0.01 eV) compared with a fully correlated all-electron treatment [START_REF] Bak | Atomic integral driven second order polarization propagator calculations of the excitation spectra of naphthalene and anthracene[END_REF].

The CC2 [START_REF] Christiansen | The second-order approximate coupled cluster singles and doubles model CC2[END_REF][START_REF] Hald | Linear response CC2 triplet excitation energies[END_REF] calculations were done with the parallel version of program RICC2 [START_REF] Hättig | CC2 excitation energy calculations on large molecules using the resolution of the identity approximation[END_REF][START_REF] Hättig | Implementation of RI-CC2 triplet excitation energies with an application to trans-azobenzene[END_REF][START_REF] Hättig | First-order properties for triplet excited states in the approximated coupled cluster model CC2 using an explicitly spin coupled basis[END_REF][START_REF] Hättig | Transition moments and excited-state first-order properties in the coupledcluster model CC2 using the resolution-of-the-identity approximation[END_REF] from the TURBOMOLE package (version 5.10) [START_REF]a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH[END_REF], employing the resolution-of-identity approximation [START_REF] Vahtras | Integral approximations for LCAO-SCF calculations[END_REF][START_REF] Feyereisen | Use of approximate integrals in ab initio theory. An application in MP2 energy calculations[END_REF] and the appropriate auxiliary basis set provided by the TURBOMOLE library [START_REF] Weigend | Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations[END_REF][START_REF] Hättig | Optimization of auxiliary basis sets for RI-MP2 and RI-CC2 calculations: Core-valence and quintuple-ζ basis sets for H to Ar and QZVPP basis sets for Li to Kr[END_REF]. In the case of CC2, both singlet and triplet excitation energies as well as one-electron properties (dipole moments and oscillator strengths in dipole length representation) were computed. It should be noted that the presence of diffuse functions in the basis set makes the assignment of states more difficult. As in our previous work [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF][START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF], we used the expectation value r 2 as one of the criteria to determine the valence character of a given excited state and to identify valence-Rydberg mixings which can be pronounced especially in the high-lying states. TZVP aug-cc-pVTZ TZVP aug-cc-pVTZ TZVP aug-cc-pVTZ

A ′ (π → π * ) 4.80 4.69 1 1 A ′′ (n → π * ) 5.43 4.91 2 1 A ′′ (n → π * ) 5.01 5.33 3 1 A ′ (π → π * ) 5.71 5.56 4 1 A ′ (π → π * ) 6.65 6.29 5 1 A ′ (π → π * ) 6.94 6.62 Thymine 1 1 A ′′ (n → π * ) 4.94 4.83 2 1 A ′ (π → π * ) 5.
3 1 A ′ (π → π * ) 6.46 6.26 2 1 A ′′ (n → π * ) 6.33 6.16 4 1 A ′ (π → π * ) 6.80 6.50 4 1 A ′′ (n → π * ) 6.73 6.55 5 1 A ′′ (n → π * ) 7.18 7.00 5 1 A ′ (π → π * ) 7.71 7.47 Uracil 1 1 A ′′ (n → π * ) 4.91 4.81 2 1 A ′ (π → π * ) 5.52 5.33 3 1 A ′ (π → π * ) 6.43 6.24 2 1 A ′′ (n → π * ) 6.73 6.09 3 1 A ′′ (n → π * )
6.26 6.55 4 1 A ′ (π → π * ) 6.96 6.66 5 1 A ′′ (n → π * ) 7.12 6.95

5 1 A ′ (π → π * ) 7.66 7.48 Adenine 2 1 A ′ (π → π * ) 5.28 5.20 3 1 A ′ (π → π * ) 5.42 5.20 1 1 A ′′ (n → π * ) 5.27 5.13 2 1 A ′′ (n → π * ) 5.

Results

In Table 1 we present the results for singlet excited states obtained with the CC2, CCSDR(3) and CC3 methods and the aug-cc-pVTZ basis set. Those obtained previously with the TZVP basis set [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF][START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF] are also listed for comparison. Some higher-lying states of the original benchmark set (e.g. in formamide, formaldehyde, and naphthalene) are not included here due to problems in identifying these states when using the aug-cc-pVTZ basis set. Within an acceptable amount of compute time, we could afford to calculate a total of 22 states with the CC3 method and 99 states with the CCSDR(3) method. In both cases, the triples contributions scale formally as N 7 with the number of orbitals N . But while CC3 has an iterative triples contribution, CCSDR(3) is based on a pseudoperturbation theory expansion of the CC3 eigenvalue problem and thus includes essentially only a non-iterative triples correction which is added to the CCSD excitation energy. The efficient parallel implementation of the RI-CC2 method in the TURBOMOLE program allowed us to compute all the singlet excited states from the original benchmark set at the CC2 level.

In Figure 1 the correlation between the aug-cc-pVTZ and TZVP results is shown graphically for the three coupled cluster methods, while Figure 2 presents histograms for the deviations of the aug-cc-pVTZ from the TZVP results. The corresponding statistical data are given in Table 2.

Upon inspection of the numerical results and the histograms in Figure 2, one immediately sees a systematic lowering of the calculated excitation energies when going from the TZVP to the larger aug-cc-pVTZ basis, for each of the three coupled cluster methods. There are only three states at the CC2 level (4 1 A ′ in imidazole, 2 1 A ′′ in cytosine, and 3 1 A ′′ in uracil) and six at the CCSDR(3) level (2 1 A g in all-E-hexatriene, 2 1 A g in pyrazine, 1 1 A ′ 2 in s-triazine, 3 1 B 1g and 2 1 B 3g in s-tetrazine and 1 1 A u in p-benzoquinone) which have a higher energy with the aug-cc-pVTZ basis. These are mostly states with rather low single excitation weights, %R 1 , in the CC3/TZVP calculations [START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF]. The correlation (Figure 1) between the results with b Same subset of states as in CCSDR(3).

c Total number of considered states.

the two basis sets is almost perfect, with correlation coefficients of 0.9951, 0.9945, and 0.9967 for CC2, CCSDR(3), and CC3, respectively. Only for high-lying states with energies above 8 eV there are a few outliers. The largest one concerns the 3 1 B 2 state in pyridine at the CC2 (-1.34 eV) and CCSDR(3) (-1.05 eV) levels, which also has a low single excitation weight, %R 1 , in the CC3/TZVP calculations [START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF].

The second largest outliers are found for the 4 1 A 1 state in cyclopentadiene at the CC2 level (-0.67 eV) and for the 2 1 A 1 state in acetone at the CCSDR(3) and CC3 levels. For most other states there is close agreement between the aug-cc-pVTZ and TZVP results (see e.g. the 1 1 B 2u states in benzene and s-tetrazine with deviations of less than 0.05 eV for all three methods).

There is another particular outlier in pyrrole that requires some comment. A previous extensive coupled cluster study of pyrrole [START_REF] Christiansen | The electronic spectrum of pyrrole[END_REF] found significant valence-Rydberg mixing in the three lowest excited 1 B 2 states showing "similar diffuseness and oscillator strengths". We also see considerable mixing in our coupled cluster calculations: for example, both CC2/aug-cc-pVTZ and CCSDR(3)/aug-cc-pVTZ yield two strongly mixed 1 B 2 states at 6.25/6.85 eV and 6.23/6.81 eV, respectively. According to our standard assignment criteria, the more valence-like state occurs at higher energy for CC2 and at lower energy for CCSDR [START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF], by a small margin in each case, and the corresponding entries in Table 1 are thus 6.85 and 6.23 eV, respectively. At face value, this would seem to indicate a large discrepancy between the CC2 and CCSDR(3) results which are in reality quite similar (see above). We have refrained from imposing correlations between the assignments of separate calculations, to be as unbiased as possible, even though this will artificially deteriorate the statistical results to some extent. Our general conclusions are however not affected by this choice.

The mean and absolute mean deviations in Table 2 are of essentially the same magnitude (about 0.2 eV) for all three coupled cluster methods. This confirms that the basis set effects are very systematic. Looking at the finer details in Table 2, we note that the basis set dependence is slightly more pronounced for CC2 than for CCSDR(3) and CC3, both for the full set of CC2 results as well as for the subset of states that is shared between CC2 and CCSDR(3). Comparing CCSDR(3) and CC3, the statistical data in Table 2 as well as the individual numerical results indicate an essentially identical basis set dependence of the computed vertical excitation energies. This suggests an incremental scheme where expensive CC3 calculations that may only be feasible with a smaller basis set (such as TZVP) are augmented with a basis set correction obtained at the cheaper CC2 or CCSDR(3) level.

For the 22 states where CC3/aug-cc-pVTZ results are available, this correction scheme has been tested, with statistical results listed in Table 3. For the estimates obtained from the CC3/TZVP results plus the CCSDR(3) correction, the agreement with the CC3/aug-cc-pVTZ results is almost perfect, with negligible mean a Columns labeled with TZVP have CC3/TZVP results [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF] as reference data. Columns labeled with aug-cc-pVTZ have the present CC3/aug-cc-pVTZ results as reference data.

b Reference [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF].

c Subset of singlet states from Reference [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF], in order to compare directly with the present aug-cc-pVTZ results.

d Reference [START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF].

e Subset of singlet states from Reference [START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF], in order to compare directly with the present aug-cc-pVTZ results.

f Total number of considered states. Some data has been discarded whose assignment was considered unreliable.

and absolute mean deviations of 0.01 and 0.02 eV, respectively, and a maximum deviation of only 0.04 eV. The CC2 correction performs not quite so well: the mean and absolute mean deviations are still quite low (-0.02 and 0.08 eV, respectively), but there are outliers for states with a low CC3/TZVP single excitation weight (maximum deviation of -0.42 eV for 2 1 A g of butadiene and 1 1 E 2g of benzene, %R 1 below 70%).

In Table 4 we compare the performance of CC2 and CCSDR(3) relative to CC3, both for the TZVP [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF][START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF] and aug-cc-pVTZ basis sets. Table 5 provides an analogous comparison between CC2 and CCSDR(3), again for both basis sets. In each case, the given deviations refer to excitation energies calculated with a lower-level vs. a higher-level coupled cluster method (using the same basis). The mean and absolute deviations between the CCSDR(3) and CC3 results are essentially of the same magnitude (0.08±0.01 eV) for both basis sets. In the case of CC2, the mean absolute deviations from the CC3 and CCSDR(3) results are slightly smaller for the augcc-pVTZ basis (0.12 and 0.10 eV) than for the TZVP basis (0.16 and 0.11 eV) so that the higher-level results are reproduced slightly better with the larger basis.

Using the CC2 method we have also computed vertical excitation energies for triplet excited states as well as one-electron properties for singlet excited states (oscillator strengths and dipole moments). The correlation between the aug-cc- b CCSDR(3) results from reference [START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF] and CC2 results from reference [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF].

c Subset of singlet states of the CCSDR(3) results from reference [START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF] and CC2 results from reference [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF], in order to compare directly with the present aug-cc-pVTZ results.

d Total number of considered states. pVTZ and TZVP results is shown in Figures 3 and4. The corresponding statistical data for the differences between the aug-cc-pVTZ and TZVP results are given in Table 6. For the vertical triplet excitation energies, the CC2 results for the two basis sets exhibit very small mean and absolute mean deviations of -0.08 and 0.10 eV, respectively, as well as a very large correlation coefficient of 0.9977, which confirms the high quality of the previous CC2/TZVP results for the triplet states.

Obviously the vertical excitation energies for the triplets are even less basis set dependent than those for the singlet excited states.

The situation is less clear-cut for one-electron properties, i.e., for the oscillator strengths and dipole moments of the singlet excited states, which are apparently affected more strongly by the addition of further diffuse and polarization functions in the basis set. While extension of the basis set leads to a slight systematic lowering in the calculated vertical excitation energies, the one-electron properties change more randomly, although there is also a general overall trend towards lower absolute values (see Figure 4). Stronger variations may be caused by changes in the composition of states when further diffuse and/or polarization functions are included: Two states which are composed of +/-combinations of configuration state functions and thus give rise to bright/dark states could mix differently upon basis set extension and then evolve into two states with moderate intensity. Generally speaking, the oscillator strengths and excited-state dipole moments depend more than the energies on the proper description of the excited-state wavefunctions, and they should thus be more susceptible to basis set effects (especially in the case of diffuse high-lying states). The largest changes in the oscillator strengths indeed occur for the high-lying states of cytosine (4A ′ : -0.238) and uracil (5A ′ : -0.287) where the intensities of these bright states are lowered considerably. For the dipole moments, the largest changes are found for the high-lying states of cytosine (4A ′ : -2.38 D and 5A ′ : -3.51 D). Turning to the statistics again, the correlation coefficients are 0.9659 for the oscillator strengths and 0.8325 for excited-state dipole moments (see Figure 4), which are still reasonably large, but significantly smaller than those for the vertical excitation energies. In summary, the differences between the CC2/aug-cc-pVTZ and CC2/TZVP results for the one-electron properties are still appreciable, and convergence with regard to basis set extension thus appears to be slower for these properties than for the excitation energies.

Conclusions

We have revisited a benchmark set of electronically excited states. Vertical excitation energies to singlet excited states were calculated with the CC2, CCSDR(3) and CC3 coupled cluster linear response methods, using the same geometries as in previous studies, but with the augmented correlation-consistent aug-cc-pVTZ basis instead of the TZVP basis. A total of 143 states were covered at the CC2 level. Due to the high computational demands of CCSDR(3) and in particular CC3, a smaller number of states were investigated with these methods, i.e., 99 with CCSDR( and 22 with CC3. Inspection of the individual numerical results and statistical evaluations for the aug-cc-pVTZ and TZVP data show that the basis set effects on the calculated singlet excitation energies are rather uniform, for each of the three coupled cluster methods investigated. There is a systematic lowering of the computed energies upon basis set extension, of typically about 0.2 eV on average. This provides some justification for an incremental scheme where CC3 results obtained with a moderate basis are augmented by applying a basis set correction determined with a less costly coupled cluster method such as CC2 or CCSDR(3).

In the case of triplet excited states, the CC2 vertical excitation energies are even less basis set dependent than in the singlet case, since going from TZVP to aug-cc-pVTZ causes only a very minor lowering of typically 0.1 eV. By contrast, the changes in the CC2 oscillator strengths and dipole moments for singlet excited states are less systematic and sometimes rather random. Hence, further convergence studies are desirable for these one-electron properties.

To conclude, some of the calculated vertical excitation energies, in particular those to high-lying singlet states, are significantly lowered by the inclusion of additional diffuse and polarization functions in the aug-cc-pVTZ basis set (compared to TZVP), and further changes may be expected upon addition of molecule-centered diffuse functions for states with significant valence-Rydberg mixing. However, in an overall assessment of the present results, the effects of basis set extension on the vertical excitation energies of the chosen benchmark set are rather minor. Therefore, our previous conclusions on the performance of the coupled cluster methods CC2, CCSDR(3), and CC3 in excited-state studies remain valid [START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF][START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF].

(π → π * ) 0.062 0.086 3 1 A 1 (π → π * ) 0.446 0.410 1 1 B 2 (π → π * ) 0.023 0.029 2 1 B 2 (π → π * ) 0.476 0.515 1 1 B 1 (n → π * ) 0.006 0.005 Pyridazine 2 1 A 1 (π → π * ) 0.014 0.017 3 1 A 1 (π → π * ) 0.444 0.438 1 1 B 2 (π → π * ) 0.009 0.003 2 1 B 2 (π → π * ) 0.489 0.455 1 1 B 1 (n → π * ) 0.006 0.005 2 1 B 1 (n → π * ) 0.005 0.004 s-Triazine 1 1 E ′ (π → π * ) 0.441 0.410 1 1 A ′′ 2 (n → π * ) 0.017 0.015 s-Tetrazine 1 1 B 1u (π → π * ) 0.017 0.286 2 1 B 1u (π → π * ) 0.376 0.102 1 1 B 2u (π → π * ) 0.046 0.054 2 1 B 2u (π → π * ) 0.368 0.364 1 1 B 3u (n → π * ) 0.007 0.006 1 1 B 3u (n → π * ) 0.011 0.011 Formaldehyde 1 1 B 1 (σ → π * ) 0.079 0.001 Acetone 2 1 A 1 (π → π * ) 0.298 0.238 p-Benzoquinone 1 1 B 1u (π → π * ) 0.538 0.530 2 1 B 1u (π → π * ) 0.544 0.283 1 1 B 3u (n → π * ) 0.000 0.001 Formamide 1 1 A ′′ (n → π * ) 0.001 0.001 Acetamide 2 1 A ′ (π → π * ) 0.199 0.210 3 1 A ′ (π → π * ) 0.279 0.204 1 1 A ′′ (n → π * ) 0.001 0.000 Propanamide 2 1 A ′ (π → π * ) 0.138 0.196 3 1 A ′ (π → π * ) 0.189 0.099 Cytosine 2 1 A ′ (π → π * ) 0.049 0.050 3 1 A ′ (π → π * ) 0.165 0.144 4 1 A ′ (π → π * ) 0.632 0.395 5 1 A ′ (π → π * ) 0.168 0.228 1 1 A ′′ (n → π * ) 0.001 0.001 2 1 A ′′ (n → π * ) 0.002 0.001 Thymine 2 1 A ′ (π → π * ) 0.197 0.179 3 1 A ′ (π → π * ) 0.080 0.041 4 1 A ′ (π → π * ) 0.250 0.168 5 1 A ′ (π → π * ) 0.515 0.356 3 1 A ′′ (n → π * ) 0.
(π → π * ) 2.13 0.15 1 1 A 2 (n → π * ) 1.10 0.39 1 1 B 2 (π → π * ) 2.02 2.01 2 1 B 2 (π → π * ) 2.69 2.71 1 1 B 1 (n → π * ) 1.10 0.40 1 3 A 1 (π → π * ) 2.15 2.15 2 3 A 1 (π → π * ) 2.53 2.49 3 3 A 1 (π → π * ) 1.59 1.74 1 3 A 2 (n → π * ) 1.01 0.32 1 3 B 2 (π → π * ) 1.96 1.97 2 3 B 2 (π → π * ) 3.49 3.25 3 3 B 2 (π → π * ) 1.16 1.26 1 3 B 1 (n → π * ) 1.10 0.55 Pyrimidine Ground State 2.43 2.40 2 1 A 1 (π → π * ) 3.24 0.81 3 1 A 1 (π → π * ) 2.11 1.52 1 1 A 2 (n → π * ) 1.07 1.34 1 1 B 2 (π → π * ) 2.26 2.19 2 1 B 2 (π → π * ) 1.38 2.06 1 1 B 1 (n → π * ) 0.

Figure 1 .Figure 2 .

 12 Figure 1. Correlation plots for the vertical excitation energies (eV) of singlet excited states at the CC3, CCSDR(3), and CC2 levels: aug-cc-pVTZ vs. TZVP results.

Figure 3 .

 3 Figure 3. Correlation plots for the vertical excitation energies of triplet excited states: Basis set effects for CC2.

Figure 4 .

 4 Figure 4. Correlation plots for (left) oscillator strengths and (right) dipole moments of singlet excited states: Basis set effects for CC2.

  CC2 results for vertical excitation energies of triplet excited states.

a

  Results with the TZVP basis set from ref.[START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF] b Results with the aug-cc-pVTZ basis set from this work. Correlation plots for the vertical excitation energies (eV) of singlet excited states at the CC3, CCSDR(3), and CC2 levels: aug-cc-pVTZ vs. TZVP results. 255x67mm (600 x 600 DPI) Histograms (in %) of the deviations between aug-cc-pVTZ and TZVP vertical excitation energies (eV) of singlet excited states at the CC3, CCSDR(3), and CC2 levels 255x67mm (600 x 600 DPI) Correlation plot for the vertical excitation energies of triplet excited states: Basis set effects for CC2. 81x76mm (600 x 600 DPI) Correlation plots for (left) oscillator strengths and (right) dipole moments of singlet excited states: Basis set effects for CC2. 158x77mm (600 x 600 DPI)

Table 1 .

 1 : Vertical excitation energies ∆E (eV) of singlet excited states from coupled-cluster calculations with the TZVP a and aug-cc-pVTZ b basis sets.

	Molecule	State	CC2			CCSDR(3)	CC3
			TZVP aug-cc-pVTZ TZVP aug-cc-pVTZ TZVP aug-cc-pVTZ
	Ethene	1 1 B 1u (π → π * )	8.40	7.90	8.37	7.87	8.37	7.89
	E-Butadiene	1 1 B u (π → π * )	6.49	6.13	6.56	6.19
		2 1 A g (π → π * )	7.63	7.06	6.95	6.82	6.77	6.63
	all-E-Hexatriene	1 1 B u (π → π * )	5.41	5.18	5.56	5.30
		2 1 A g (π → π * )	6.67	6.43	6.04	6.09
	all-E-Octatetraene 2 1 A g (π → π * )	5.87	5.74
		1 1 B u (π → π * )	4.72	4.53
		2 1 B u (π → π * )	6.91	6.37
		3 1 A g (π → π * )	6.72	6.12
		4 1 A g (π → π * )	7.01	6.36
	Cyclopropene	1 1 B 1 (σ → π * ) 1 1 B 2 (π → π * )	6.96 7.17	6.73 6.72	6.89 7.10	6.67 6.68	6.90 7.10	6.67 6.68
	Cyclopentadiene	1 1 B 2 (π → π * ) 2 1 A 1 (π → π * ) 3 1 A 1 (π → π * )	5.69 7.05 8.86	5.47 6.80 8.19	5.72 6.76 8.72	5.48 6.64 8.17
	Norbornadiene	1 1 A 2 (π → π * ) 1 1 B 2 (π → π * )	5.57 6.37	5.30 6.09
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	Molecule Imidazole Pyridine Pyrazine	State 3 1 A 1 (π → π * ) 1 1 A ′′ (n → π * ) 2 1 A ′ (π → π * ) 3 1 A ′ (π → π * ) 2 1 A ′′ (n → π * ) 4 1 A ′ (π → π * ) 1 1 B 2 (π → π * ) 1 1 B 1 (n → π * ) 1 1 A 2 (n → π * ) 2 1 A 1 (π → π * ) 3 1 A 1 (π → π * ) 2 1 B 2 (π → π * ) 3 1 B 2 (π → π * ) 4 1 A 1 (π → π * ) 1 1 B 3u (n → π * ) 1 1 A u (n → π * ) 1 1 B 2u (π → π * ) 1 1 B 2g (n → π * ) 1 1 B 1g (n → π * ) 1 1 B 1u (π → π * ) 2 1 B 1u (π → π * ) 2 1 B 2u (π → π * )	Table 1 -Continued CC2 TZVP aug-cc-pVTZ TZVP aug-cc-pVTZ TZVP aug-cc-pVTZ CCSDR(3) CC3 8.44 7.81 8.20 7.63 6.86 6.69 6.73 6.40 7.28 6.91 8.00 7.64 8.62 8.68 5.32 5.26 5.20 5.17 5.12 4.97 5.12 5.02 5.39 5.27 5.55 5.46 6.88 6.64 6.88 6.63 7.72 7.43 7.72 7.41 7.61 7.32 7.61 7.35 9.37 8.03 9.09 8.04 9.00 8.59 9.00 8.65 4.26 4.12 4.31 4.20 4.95 4.86 5.11 5.05 5.13 5.07 5.07 5.02 5.92 5.79 5.86 5.77 6.70 6.61 6.86 6.80 7.10 6.86 7.10 6.86 8.13 7.92 8.09 7.89 8.07 7.82 8.08 7.84	Molecular Physics	13:43 Molecular Physics paper December 3, 2009
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	-Continued

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	Molecule s-tetrazine Formaldehyde Acetone	State 1 1 E ′ (π → π * ) 1 1 B 3u (n → π * ) 1 1 A u (π → π * ) 1 1 B 1g (n → π * ) 1 1 B 2u (π → π * ) 1 1 B 2g (n → π * ) 2 1 A u (n → π * ) 2 1 B 2g (n → π * ) 2 1 B 1g (n → π * ) 3 1 B 1g (n → π * ) 2 1 B 3u (n → π * ) 1 1 B 1u (π → π * ) 2 1 B 1u (π → π * ) 2 1 B 2u (π → π * ) 2 1 B 3g (π → π * ) 1 1 A 2 (n → π * ) 1 1 B 1 (σ → π * ) 1 1 A 2 (n → π * ) 1 1 B 1 (σ → π * ) 2 1 A 1 (π → π * )	Table 1 -Continued CC2 TZVP aug-cc-pVTZ TZVP aug-cc-pVTZ TZVP aug-cc-pVTZ CCSDR(3) CC3 8.06 7.88 8.07 7.87 2.47 2.39 2.61 2.56 2.53 2.46 3.67 3.61 3.88 3.86 5.10 5.00 5.15 5.08 4.97 4.87 5.20 5.16 5.20 5.18 5.12 5.08 5.53 5.45 5.51 5.47 5.34 5.28 5.50 5.40 5.56 5.49 6.32 6.23 6.43 6.40 6.23 6.16 6.91 6.82 6.98 6.94 6.87 6.80 7.64 7.42 7.60 7.64 6.70 6.61 6.77 6.72 6.67 6.60 7.60 7.51 7.54 7.27 7.45 7.18 7.75 7.65 7.83 7.65 7.79 7.59 8.65 8.45 8.58 8.40 8.97 8.85 8.86 8.90 4.09 4.00 3.94 3.89 3.95 3.88 9.35 9.19 9.19 9.06 9.18 9.05 4.52 4.45 4.39 4.39 4.40 4.38 9.29 9.11 9.17 9.04 9.74 9.31 9.66 8.92 9.65 8.90	Molecular Physics 8	December 3, 2009 13:43 Molecular Physics paper Molecular Physics paper December 3, 2009 13:43
	p-Benzoquinone	1 1 A u (n → π * )	2.92	2.88	3.01	3.02
	Continued on Next Page. . .

URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics F o r P e e r R e v i e w O n l y M. R. Silva-Junior et. al.

Table 1

 1

	-Continued

Table 1

 1

	-Continued

Table 2 .

 2 Deviations in the vertical excitation energies (eV) of singlet excited states: aug-cc-pVTZ vs. TZVP results a .

		CC2 CC2 b	CCSDR(3)	CC3
	Count c	143	99	99	22
	Mean	-0.22	-0.22	-0.18	-0.18
	Abs. Mean	0.23	0.22	0.18	0.18
	Std. Dev.	0.29	0.28	0.26	0.25
	Maximum (-)	1.34	1.34	1.05	0.75

a TZVP results from Refs.

[START_REF] Schreiber | Benchmarks for electronically excited states: CASPT2, CC2, CCSD and CC3[END_REF][START_REF] Sauer | Benchmarks for electronically excited states: A comparison of noniterative and iterative triples corrections in linear response coupled cluster methods: CCSDR(3) versus CC3[END_REF]

.

Table 3 .

 3 Deviations in the vertical excitation energies (eV) of singlet excited states: CC3/TZVP-based estimates with CC2 and CCSDR(3) basis set corrections vs. CC3/augcc-pVTZ results. See text.

		CC3 with CC2 correction a	CC3 with CCSDR(3) correction b
	Count c	22	22
	Mean	-0.02	0.01
	Abs. Mean	0.08	0.02
	Std. Dev.	0.15	0.02
	Maximum (-)	0.42	0.03
	Maximum (+)	0.32	0.04
	a Estimated CC3/aug-cc-pVTZ = CC3/TZVP + CC2/aug-cc-pVTZ -CC2/TZVP.
	b Estimated CC3/aug-cc-pVTZ = CC3/TZVP + CCSDR(3)/aug-cc-pVTZ -
	CCSDR(3)/TZVP.

c Total number of considered states.

Table 4 .

 4 Deviation of CC2 and CCSDR(3) excitation energies (eV) of singlet excited states with respect to CC3 results obtained with the same basis set a .

					Method
			CC2			CCSDR(3)
		TZVP b	TZVP c	aug-cc-pVTZ TZVP d	TZVP e	aug-cc-pVTZ
	Count f	118	22	22	118	22	22
	Mean	0.13	0.14	0.11	0.09	0.07	0.08
	Abs. Mean	0.16	0.15	0.12	0.09	0.07	0.08
	Std. Dev.	0.25	0.25	0.17	0.14	0.11	0.12
	Maximum (+)	0.95	0.86	0.43	0.52	0.28	0.25

  . Deviation of CC2 excitation energies (eV) of singlet excited states with respect to CCSDR(3) results obtained with the same basis set a .

		TZVP b	TZVP c	aug-cc-pVTZ
	Count d	118	99	99
	Mean	0.04	0.03	0.00
	Abs. Mean	0.11	0.11	0.10
	Std. Dev.	0.16	0.15	0.14
	Maximum (+)	0.68	0.68	0.62
	a Columns labeled with TZVP have CCSDR(3)/TZVP
	results [3] as reference data. Columns labeled with aug-
	cc-pVTZ have the present CCSDR(3)/aug-cc-pVTZ re-
	sults as reference data.

Table 6 .

 6 Deviations between the CC2/aug-cc-pVTZ and CC2/TZVP results for the vertical excitation energies (eV) of singlet and triplet excited states and for the oscillator strengths and dipole moments (D) of singlet excited states: Statistics for all 28 benchmark molecules.

			CC2/aug-cc-pVTZ
		Singlet states	Triplet states	Osc. strength Dipole moment
	Count a	143	71	95	127
	Mean	-0.22	-0.08	-0.032	-0.14
	Abs. Mean	0.23	0.10	0.047	0.57
	Std. Dev.	0.29	0.14	0.088	0.97
	Maximum (-)	1.34	0.54	0.287	3.51

a Total number of considered states. Some data has been discarded whose assignment was considered unreliable.

Table A1 .

 A1 : Vertical excitation energies (eV) of triplet excited states from CC2 calculations with the TZVP a and aug-cc-pVTZ b basis sets.

	December 3, 2009 13:43
	Molecule	State		CC2
			TZVP aug-cc-pVTZ
	Ethene	1 3 B 1u (π → π * )	4.52		4.56
	E-Butadiene all-E-Hexatriene all-E-Octatetraene 1 3 B u (π → π * ) 1 3 B u (π → π * ) 1 3 A g (π → π * ) 1 3 B u (π → π * ) 1 3 A g (π → π * ) F o r	3.40 5.25 2.78 4.40 2.40		3.43 5.24 2.80 4.42 2.42
	1 3 A g (π → π * ) 1 3 B 2 (π → π * ) 1 3 B 1 (σ → π * ) 1 3 B 2 (π → π * ) e P Cyclopropene Cyclopentadiene 1 3 A 1 (π → π * ) Norbornadiene 1 3 A 2 (π → π * ) 1 3 B 2 (π → π * ) e r	3.76 4.44 6.65 3.36 5.22 3.76 4.25		3.78 4.40 6.44 3.37 5.22 3.72 4.25
	Benzene Naphthalene	1 3 B 1u (π → π * ) 1 3 E 1u (π → π * ) R 1 3 B 2u (π → π * ) 1 3 E 2g (π → π * ) e 1 3 B 2u (π → π * ) 1 3 B 3u (π → π * ) 1 3 B 1g (π → π * ) 2 3 B 2u (π → π * ) v 4.31 5.14 6.08 7.99 3.27 4.38 4.64 4.88 i e 2 3 B 3u (π → π * ) 5.11 1 3 A g (π → π * ) 5.76 w	4.33 5.05 5.86 7.92 3.25 4.28 4.59 4.81 4.93 5.67
	Furan Pyrrole	2 3 B 1g (π → π * ) 2 3 A g (π → π * ) 3 3 A g (π → π * ) 3 3 B 1g (π → π * ) 1 3 B 2 (π → π * ) 1 3 A 1 (π → π * ) 1 3 A 1 (π → π * ) 1 3 B 2 (π → π * )	6.44 6.83 6.94 7.23 4.38 5.67 5.72 4.68	6.07 6.53 6.80 7.14 4.33 O n l 5.61 5.63 4.64 y
	Imidazole	1 3 A ′ (π → π * )	4.89		4.85
		2 3 A ′ (π → π * )	6.01		5.86
		1 3 A ′′ (n → π * )	6.44		6.32
		3 3 A ′ (π → π * )	6.74		6.57
	Continued on Next Page. . .

Table A1

 A1

	December 3, 2009 13:43 Molecular Physics paper
	-Continued TZVP aug-cc-pVTZ CC2 4 3 A ′ (π → π * ) State 7.68 7.51 2 3 A ′′ (n → π * ) 7.52 7.30 1 3 A 1 (π → π * ) 4.46 4.48 1 3 B 1 (n → π * ) 4.54 4.46 1 3 B 2 (π → π * ) 5.07 4.93 2 3 A 1 (π → π * ) 5.33 5.24 1 3 A 2 (n → π * ) 5.35 5.24 2 3 B 2 (π → π * ) 6.52 6.34 3 3 B 2 (π → π * ) 8.39 7.85 3 3 A 1 (π → π * ) 8.18 8.11 1 3 B 3u (n → π * ) 1.86 1.83 1 3 A u (n → π * ) 3.43 3.39 4.62 4.66 1 3 B 2u (π → π * ) 4.81 4.65 1 3 B 2g (n → π * ) 5.03 2 3 A u (n → π * ) 5.05 4.98 π * , π * ) 6.05 5.98 2 3 B 1g (n → π * ) 6.72 6.65 2 3 B 3u (n → π * ) 6.52 2 3 B 2u (π → π * ) 7.65 7.52 1 3 A 2 (π → π * ) 3.57 1 3 A 1 (π → π * ) 6.08 6.12 1 3 A 2 (n → π * ) 4.08 4.07 1 3 B 1g (n → π * ) 2.47 2.46 1 3 A u (n → π * ) 2.59 2.58 w w 1 3 A 1 (π → π * ) 6.27 6.31 v i e e i v 3.52 e e 6.43 R R 2 3 B 2g (n → π * ) 2 3 B 1u (π → π * ) 5.67 5.59 e r r e 1 3 B 3g (n, n → e p-Benzoquinone Molecule Pyridine s-Tetrazine Formaldehyde Acetone e 4.99 P P 1 3 B 1u (π → π * ) 1 3 B 1g (n → π * ) 4.30 4.27 F o r r o Table B1 -Continued Molecule State CC2 TZVP b aug-cc-pVTZ c 1 1 B 3u (n → π * ) 0.007 0.001 Molecular Physics F Pyrimidine 2 1 A 1
	Formamide Acetamide Propanamide	1 3 B 1u (π → π * ) 1 3 B 3g (π → π * ) 1 3 A ′′ (n → π * ) 1 3 A ′ (π → π * ) 1 3 A ′′ (n → π * ) 1 3 A ′ (π → π * ) 1 3 A ′′ (n → π * )	3.12 3.50 5.39 5.94 5.42 6.06 5.44	3.18 3.46 5.27 5.88 5.33 O n l O n l 5.88 5.35 y y
		1 3 A ′ (π → π * )	6.07	6.02
	a Results with the TZVP basis set from ref. [1]
	b Results with the aug-cc-pVTZ basis set from this work.

Taylor & Francis and I.T. Consultant

Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft (SFB 663, project C4). M.R.S.J. gladly acknowledges support by the Deutscher Akademischer Austauschdienst (DAAD). S.P.A.S. acknowledges support from the Danish Center for Scientific Computing (DCSC) and from the Danish Natural Science Research Council/The Danish Councils for Independent Research. We thank Prof. Christoph Hättig for helpful discussions.

Appendix B. CC2 results for oscillator strengths and dipole moments of singlet excited states.

Table B1.: Oscillator strengths (in dipole length representation) for optically allowed transitions from CC2 calculations a .

Molecule State CC2

TZVP b aug-cc-pVTZ c

Ethene