domain: phys chim

text:
Antoine Jay
email: ajay@laas.fr

Miha Gunde

Nicolas Salles

Matic Poberžnik

Layla Martin-Samos

Nicolas Richard

Stefano De Gironcoli

Normand Mousseau
email: normand.mousseau@umontreal.ca

Anne Hémeryck
email: anne.hemeryck@laas.fr

Activation-Relaxation Technique: an efficient way to find minima and saddle points of potential energy surfaces

Keywords: activation-relaxation technique, activated mechanisms, atomistic modelling, saddle-point search, catalysis reactions, atomistic diffusion

The Activation-Relaxation Technique (ARTn) is an efficient technique for finding the minima and saddle points of multidimensional functions such as the potential energy surface of atomic systems in chemistry. In this work we detail and illustrate significant improvements made to the algorithm, regarding both preprocessing and the activation process itself. As showcased, these advances significantly reduce ARTn computational costs, especially when applied with ab initio description. With these modifications, ARTn establishes itself as a very efficient method for exploring the energy landscape and chemical reactions associated with complex mechanisms.

Introduction

Understanding molecular reactions and the evolution of atomic structures is of crucial interest in modern chemistry and materials science. This objective requires a precise knowledge of the energy landscape associated with diffusion pathways, including the initial, final and transition states, as well as their relative free energies, i.e., barrier energy and entropy.

Mathematically, for purely activated events in which energy barriers are high compared to temperature, the initial and final states can be assigned to a local energy minimum on the potential energy surface (PES) of the system, whereas the transition states are assigned to a first-order saddle point, corresponding to the highest energy point on the minimum-energy path connecting these two minima. Once these states are known, the kinetics of the system can be described within the framework of the transition state theory [1,2]. The reader should refer to the Appendix A for a better understanding of some key words used in this work.

The challenge to identify these states is two-fold: first, finding local points of interest; second, sampling the landscape to ensure that all the relevant mechanisms are identified. For a chemical system composed of N at atoms, the PES has 3N at dimensions. While a local minimum of this PES can be easily reached by any algorithm that follows the slope in all these dimensions, saddle points are unstable along one particular dimension called the valley, and are therefore harder to identify. Indeed, reaching a saddle point implies a relaxation in the 3N at -1 other dimensions and a climb along this unknown dimension of negative curvature. The curvature is, by definition, positive around a local minimum and does not provide information concerning the nature and position of saddle points sitting on the ridge between two minima. Such information can become available only beyond the inflection line surrounding a minimum: above this line, the valleys leading to these saddle points start to form and to dissociate (see the shoulders area in Fig. A.7c of Appendix A). For all these reasons, identifying local transition states remains a formidable task.

The Activation Relaxation Technique nouveau (ARTn) [3,4] has demonstrated to be a very efficient and versatile approach for finding saddle points on a PES using only local information (energy and forces). Doing so, it addresses both challenges underlined in the previous paragraph: the identification of local saddle point as well as, through its unbiased search approach, the capacity to provide an extensive mapping of the PES through fully con-

⊥ Relax until F ⊥ < F Push X = X + αê Lanczos λ m , êm if λ min < 0 ê = êm α = ||F ||/λ m Input X ê = êrand α = Cste Saddle=X X = X ± êm relax to min A B C D E if ||F|| < if ||F|| > Figure 1:
The three main steps of ARTn (in pink) that find a saddle point from an initial input position X, a given random vector êrand , and a norm α for this random push.

The lowest eigenvalue λ m (m as minimum) of the Hessian matrix and its corresponding eigenvector êm are calculated with the Lanczos procedure. F ⊥ = (F•ê)ê is the component of the forces F that is parallel to the pushing direction ê and F ⊥ = F-F is the orthogonal component.

nected activated paths. Over the last 20 years, ARTn has been applied to a wide range of systems both to explore their energy landscape [5,6,7,8,9], and to generate long-time kinetic trajectories [10,11,12,13,14,15] when used as kinetic ART, an off-lattice kinetic Monte Carlo algorithm with on-the-fly cataloguing. When searching for a saddle point, a recent implementation of ARTn coupled with Density Functional Theory (DFT) (ARTn-DFT in the following) [16], was shown to result in more accurate saddle pointsi.e. with total force many orders of magnitude lower -and to be computationally less expensive than the climbing image-nudged elastic band (NEB) method [17]. Previous work has also shown that ARTn is more efficient than the dimer method [18] for generating a fully connected PES with multiple saddle points search [19]. The scope of this work is to present an overview of the ARTn method and the latest improvements on the core algorithm as well as pre-processing strategies when running multiple-saddle-point search. These improvements further reduce the number of energy and force evaluations while keeping whole numerical accuracy. The basic structural elements of ARTn are outlined in section 2, while latest improvements and pre-processing strategies are discussed in sections 3 and 4, respectively.

Standard Activation Relaxation Technique nouveau

The core of ARTn is based on three stages that are executed at every step of the search until convergence is achieved:

• Evaluation of the Hessian lowest eigenvalue λ m (lowest curvature), the one with the most negative value, or with the smallest value if they are all positive and its corresponding eigenvector êm (direction along which the curvature is the lowest);

• Uphill push, against the forces;

• Relaxation into the hyperplane perpendicular to the push.

This workflow is shown in Fig. 1 (pink blocks). The choice of the initial push (first grey block in Fig. 1) and the possibility for searching multiple-saddle points are, here, considered as pre-processing and post-processing operations, respectively. Fig. 2a illustrates ARTn search on a bi-dimensional (2D) potential energy surface model. The concept of the algorithm is simple. In practice, however, its performances (reduced number of energy and force evaluations, numerical stability, success rate and precision) rely on several numerical and mathematical tricks.

In this section we discuss a few central elements of ARTn.

Initial uphill push

Since a local energy minimum does not provide information regarding the position or nature of the saddle points connected to it, by default, ARTn generates a random push (ê rand ) from a local minimum to start an event search.

In the absence of strong mathematical justification, many approaches can be used for this initial deformation: the random displacement can be applied to a single atom, a local environment, or to the entire system dimensions. Since, irrespective of the number of atoms involved in the initial displacement, ARTn imposes no constraints on the atoms that can move, all saddle points can, in principle, be found, regardless of the initial push. In practice, however, "smart" choices of the initial push improve the sampling of the saddle point. For example, global pushes tend to oversample collective transition states often related to soft degree of freedom. On the contrary, very local pushes tend to sample transition states that only involve the breaking or formation of few bonds (hard degree of freedom). Since activated events away from phase transitions tend to be local in nature, typical initial random deformation are therefore applied on local environments counting a limited number of atoms.

While there is no formal proof that ARTn can generate an exhaustive list of activation barriers connected to a local minimum, the random search permits to limit the biases. In fact, it has been shown that a sufficient number of random explorations generates a larger catalogue of events than competing algorithms, while always retrieving previously known barriers [4,20,12].

Evaluation of the lowest curvature

During this stage, the lowest eigenvalue (λ m ) and corresponding eigenvector (ê m ) of the Hessian matrix are evaluated by means of Lanczos algorithm [21] (see Appendix B). This lowest eigenvalue determines if the system is below (λ m > 0), or above the inflection hyperplane (λ m < 0). Once above this hyperplane, the initial uphill push direction is switched to êm . To reduce the computational cost, in practice, if ARTn search starts in a minimum, the lowest curvature could be computed only after few uphill pushes as the curvature is positive near a minimum, by definition.

Uphill push

The uphill push is performed to (i) escape from the harmonic basin surrounding the minimum and (ii) to move towards a ridge. The push direction depends on the position of ARTn on the PES:

1. Below the inflection hyperplane, i.e., where are directions have positive curvature, the push direction is kept constant and equal to the initially chosen direction. 2. Above the inflection, i.e, where the lowest curvature λ m is negative, with absolute value larger than a given threshold λ thr , the push direction is updated to the corresponding eigenvector êm with an orientation set opposite to the force and the norm of the displacement set as in Ref. [22]:

dr = min size max , ||F || max(|λ min |, 0.5) , (1)
where F is the component of the force that is parallel to êmin and size max is a user defined threshold. This adaptive norm reduces the size of the displacement when the system is close to the saddle point and was shown to accelerate convergence.

Orthogonal relaxation

To reach convergence to a first-order saddle, the 3N at -1 directions orthogonal to êm must be in a local minimum. As for the uphill push, this orthogonal relaxation plays two different roles depending on whether the system is below or above the inflection plane.

1. Below the inflection, a weak orthogonal relaxation is applied to avoid too short interatomic distances that could result from the application of the uphill push. A small number of relaxation steps is then sufficient to assure stability, whereas a whole relaxation (numerical accuracy) is in general not recommended, because it might lead to the undesired exploration of soft valleys often associated with large elastic deformation. 2. Above the inflection, an orthogonal relaxation is essential to reach the actual valley in which F ⊥ , the component of the forces that is orthogonal to êm , is by definition null. To reduce its computational cost, this relaxation can be stopped when F ⊥ becomes smaller than the parallel component F by controlling its convergence threshold. However, if the system is not significantly above the inflection, it might happen that this relaxation brings the algorithm back below the inflection, meaning that the lowest mode becomes positive and is associated with elastic deformation, leading to an unsuccessful event. To mitigate the number of unsuccessful searches, ARTn follows a 3-fold strategy by using: (i) non-zero inflection threshold (below -0.5 eV Å2 ) to determine the point where the system is first above the inflection; (ii) a progressive number of relaxation steps: small close to the inflection and progressively larger; (iii) a smooth switch of the uphill push direction from the initial to the lowest-curvature eigenvector êm [16].

2.5. Reaching adjacent minimum and checking the existence of a connected path Once the saddle point -defined as a point where λ m < 0 and the total force is below a threshold near zero -has been reached, an adjacent minimum can be found through standard energy minimization, completing the event.

Since forces are zero at the saddle point, a push along the direction of the eigenvector êm is applied before starting a standard unconstrained minimization to reach the adjacent minimum. This push must be large enough to ensure that the forces are sufficiently large. Applying the push along the opposite direction allows to check the existence of a connected path between the two minima (blue arrows in Fig. 2b).

Improvement of the ARTn core: reducing the number of Lanczos iterations

The computation of the local curvature and associated eigenvectors represents the dominant computational cost for ARTn. This section presents recent developments focused on reducing this cost and improving ARTn's overall computational efficiency.

We refer the reader to a comparative benchmark of the efficiency of ARTn against other existing methods published recently by the authors to complement the benchmarks shown here (see Ref [16]).

A Lanczos chain starts with a guess basis vector L 1 (see Appendix B) for iteratively determining êm . This vector can be selected as random or based on informed knowledge. As the ARTn kernel calls the Lanczos algorithm after each push-relax, the eigenvector êi+1 m calculated at the ARTn step i + 1 is generally close to the one calculated at the previous step i. Therefore, if for the Lanczos of the initial ARTn step, the first basis vector L 1 is generated at random, the êi m obtained as a result of the Lanczos chain at ARTn step i can be used as the basis vector L 1 at ARTn step i+1 (L i+1 1 = êi m ). Indeed, in our experience, the dot product between two consecutive eigenvectors êi m •ê i+1 m is usually lower than 0.7 below the inflection line and higher than 0.95 above it, making the use of the previously converged eigenvector more efficient in this second case (see Fig. 4). Moreover, in this region, the lowest eigenvalue is negative above the inflection, opening up a gap in the spectrum between the first and second lowest eigenvalues. This decreases the mixing of eigenvectors with neighbouring eigenvalues and increases the ratio λ i /λ m (see Appendix B, eq. B.2). This means that, using an L 1 close to the eigenvector associated with the lowest curvature, in most cases, convergence can be obtained with a small tridiagonal matrix, as represented in Fig. 3. Since force calculations are much costlier than matrix diagonalisation, we implement an adaptive approach where, instead of keeping the Lanczos basis set fixed, as in previous versions of ARTn, the size of the basis is progressively increased (see Appendix B) until the lowest eigenvalue reaches convergence (currently set to λ m < 10 -2 eV/ Å2 ).

In most cases, this iterative approach leads to a significant reduction in the number of force calls associated with the Lanczos procedure with respect to the previous version of the code, despite its former use of the smart starting L 1 .

For the systems presented in the next section, for example, this number of force evaluations needed to converge the eigenvector is reduced from 16 (old version) to typically less than 10 below the inflection and less than 5 above the inflection (see black circles Fig. 4), since successive vectors are more parallel.

Pre-processing strategies: smart initial pushes

As mentioned previously, ARTn has been applied with success to many systems with a ranging level of complexity and nature. The reader is invited to look at these works for more details. In the following, we focus our attention on two simple examples on which pre-processing strategies can reduce even more the computational cost. In these examples, ARTn is coupled with the ab initio software Quantum Espresso v7.0 [23] for the evaluation of energies and forces. ARTn can be used with any other quantum mechanical code, including SIESTA [5], BigDFT [24,25] and VASP [9], as well any continuously derivable empirical potential.

Diffusion of Al adatom on Al(100) surface

In this first example, we look at the diffusion of an Al adatom on an Al (100) surface, a well studied system with empirical potentials [18]. This example shows how to adopt an event search strategy with minimum bias, in order to reduce the computational cost of the search without affecting the richness of results.

This system is known to be associated with non-intuitive diffusion mechanisms, so called exchange mechanisms, in addition to the simple hopping from adsorption site to adsorption site on the Al(100) surface [26,18]. These exchange mechanisms occur when the adatom takes the place of its first neighbour, which becomes the new adatom or pushes the second (3rd, or 4rth) to the surface. These two sets of mechanisms are respectively represented by red (hopping) and black (exchange) arrows in Fig. 5. For a full description of all the possible exchanges, see Fig. 8 of Ref. [18].

If the input vectors are completely random, then half of them shift the adatom in a direction that is opposite to the surface. In these cases, we can easily predict that all the ARTn searches would lead to the dissociation of the adatom from the surface, a physically important mechanism which is of little interest, however, if we are interested in diffusion. To avoid this event, it is possible to restrict the choices of the input vector. In this specific example, we reduce the displacement of the adatom so that it is oriented towards the surface, and restricted to only a quarter of the space since it is symmetrically equivalent to the others.

This constraint is obtained by adding a condition that accepts the random input vector only if the displacement of the adatom is in a cone for which the user gives the direction (11-1) and the half angle (45°), as presented Fig 5.

The push on this atom remains random, to preserve the exhaustiveness, but restricted, to save computational cost. This approach can be used on a single atom or generalized to a region, when appropriate.

The PES is calculated using a 300+1 atoms supercell of crystalline aluminum composed by 5×5 atoms along the surface, six bulk layers for which the two bottom layers are fixed at the interatomic distance (5.30 Å) and a 18 Å vacuum on top of the surface. The Γ centred sampling of the supercell Brillouin zone is 2 × 2 along the reciprocal surface directions, and an energy cutoff of 15 Ry limits the number of plane waves used to describe the wavefunctions. All explorations of the PES are run in parallel using four different directions simultaneously, with each exploration also taking advantage of the k-point and FFT parallelism capabilities of the DFT software. The total number of explorations is stopped when 50 saddle points are found, as the existence of only about ten low energy saddle points is an a posteriori knowledge. All the atoms in the first layer of the surface are taken into account in the initial random displacement.

In this work, ARTn is able to recover all previously identified diffusion mechanisms without their previous knowledge: events with saddle point energies lower than 1 eV are described in Table given in Fig. 5. Each generated events requires on average 170 force calls. Note that our DFT energy barriers are different than the ones that had been obtained with an empirical potential [18], but are similar as the ones previously obtained within DFT: 0.20 and 0.65 eV [26] respectively for the 1st neighbour exchange (Id 1) and the hopping (Id 2).

Chemical reaction in the gas phase

In this second example, a CH 3 Cl molecule is close enough to another Cl atom to form the well known Cl • ...CH 3 Cl complex [27] shown in Fig. 6a. From this configuration, the CH 3 can reverse to catch this Cl • while breaking its bond with the other Cl, thus forming the reversed ClCH 3 ...Cl • complex shown in Fig. 6c. To compute the energy barrier of this reaction, the system is placed in a large unit cell with 15 Å of vacuum in the three directions (to reduce image-image interactions, as Quantum ESPRESSO uses periodic boundary conditions). The position of the Cl atom initially bonded to CH 3 is kept fixed throughout the procedure to avoid translation of the system, while all the other atoms are allowed to move, fixing the coordinates of an atom not involved in the event has no impact on the result. The push size used to escape the area below the inflection is 0.2 Å, which is also the maximum push size above the inflection, the convergence condition on the forces used to define a saddle point is 0.05 eV/ Å.

We compare here four pre-processing procedures applied to ARTn in order to recover the saddle point structure detailed in Fig. 6b. Its energy calculated in this work is 0.74 eV which is thus the activation barrier of this chemical reaction. The basic ARTn-DFT approach corresponds to procedure #1. Procedures #2 to #4 implement additional constraints based on the knowledge of the system that save computational time by reducing the number of force calculations. The number of force calls for each of them is presented in the Table given in the Fig. 6.

Procedure #1: The only information used is that of the initial position. To generated the desired mechanism, the PES must therefore be explored following the standard unbiased ARTn algorithm shown in Fig. 1, generating different events resulting from each initial random displacement. Here, ARTn generates four different physically-relevant events for this system, for a total of 463 force calls. These events include the separation of one hydrogen atom from the carbon atom (not shown) and the binding transfer from one Cl to the other that we are looking for.

Procedure #2: It requires the knowledge of the general direction to the saddle point, which restricts the event search, in a similar spirit to the first example. Instead of launching searches in many random directions and waiting for the desired event to be generated, a specific displacement is directly inputted as the starting deformation: the carbon atom is moved in the exact direction of the isolated Cl. By reducing the number of events searches, this procedure drastically reduces the cost to 96 force calculations, as the algorithm only needs to be run once to find the correct saddle.

Procedure #3: It requires the knowledge of the final products of the reaction, which corresponds to a common case in chemistry. With this information, a guessed path can be constructed between the known reactant and the product. ARTn can then start with a structure that is an interpolation between the initial and final structures, which is usually already above the inflection line and close to the saddle point. Here, convergence to the saddle point is obtained in only 50 force evaluations, which is two times less than in procedure #2 and almost 10 times less than in procedure #1. This procedure is also known as r-ARTn [16] and was suggested as an alternative to the computationally expensive string methods.

Procedure #4: It requires the knowledge that the path is fully symmetrical. This is not true in general, but is sometimes the case for diffusion in crystalline system, catalysis and simple molecular reactions. Let C represent the point at the symmetric centre on the diffusion path. C is known and can be computed for example as C = (x i max + x f max )/2, where

x i(f )
max is the triplet of initial (final) coordinates (x,y,z) of the atom that have the maximal displacement, here the carbon atom. This symmetry implies that êm can be accurately estimated and provided to ARTn as a pre-processing input. Finally, as in procedure #3, the algorithm can start at an interpolated position, which is, by symmetry, exactly in the center between the two minima, X mid = (X i + X f )/2. This starting point is located on a ridge that is orthogonal to the valley due to the symmetry. Each atom relaxes on a straight line linking C with its starting middle position. The direction of this line is X mid C = X mid -(C, C, . . . , C), and êm is thus the component of the total displacement vector ∆X = X i -X f that is orthogonal to this direction:

êm = ∆X -(∆X • X mid C )X mid C (2)
Reaching the saddle point costs less than half the previous price with only 20 force calculations all called by one single perpendicular relaxation.

Conclusion

This paper presents a fundamental description of ARTn using an original point of view that clearly shows the link between the various steps of ARTn and the evolution of a system on its own potential energy surface. This knowledge was refined over the last few years, leading to various technical improvements that were presented recently [16], but never explicitly detailed and illustrated. The capacity of ARTn to explore the PES at the ab initio accuracy is demonstrated on two systems using a version of the code that includes all these latest improvements and that is embedded into Quantum Espresso [23]. These examples show how to make use of prior knowledge of PES to reduce the computational cost of an unbiased exploration and demonstrate the remarkable efficiency of ARTn local approach to characterize diffusion and activated mechanisms in complex environments.

Over the last two decades, ARTn has already been applied with great success to a wide range of complex systems ranging from proteins aggregation to amorphous surfaces reactions or diffusion in glassy materials. It can therefore be applied to any complex materials to look problems such as catalysis, chemical reaction, structural evolution and much more. With these latest improvements, the method now has unsurpassed accuracy and efficiency and is therefore one of the most powerful tools for the investigation and characterization of activated atomic mechanisms. These significant improvements to ARTn are now available with the Quantum Espresso electronic structure relaxation code. In order to help the development and dissemination of ART in the rest of the community, we are currently working on a more portable version of ART that can be plugged into any software capable of calculating energy and forces with DFT or simple empirical potentials. ARTn will then present itself as a valuable alternative tool to the more conventional methods widely used in the atomic scale community today both for single and double-ended problems. calculated by splitting the gradient in two components: ∇E = ∇E ⊥ + ∇E , where ∇E ⊥ = ∇E -(∇E • êm )ê m . Along the valley, the points of highest energy are the saddle points.

• The ridges, represented in white Fig . A.7c, are the set of points for which the gradient is orthogonal to the eigenvector êm , i.e. for which ∇E is null. The ridges cross the valleys at the saddle points and at the minima.

• A basin, represented in blue Fig. A.7b
, is a set of atomic positions for which an atomic relaxation moves the system toward a same minimum. This is a global definition contrary to all the others which are local. A basin is surrounded by steep energy rises that generally correspond to the ridges except around the extrema where forbidden crossings occur due to the use of eigenvectors in the ridges definition.

To obtain êm , the powers method is used. It states that any matrix elevated at a sufficiently high power k and multiplied by any vector L returns an approximation of the eigenvector associated to the highest eigenvalue (lowest for -k):

lim k→+∞ [H] -k L = êm (B.1)
It can be simply demonstrated by decomposing L on the eigen basis of [H], giving L = 3Nat i=1 c i êi where c 1 = c m (minimum), and by factorizing:

[H] -k L = λ -k m c m êm + 3Nat i=2 c i λ -k i êi (B.2) = λ -k m        c m êm + 3Nat i=2 c i λ i λ m -k êi ----→ k→+∞ 0       
Here the number of iterations k needed to converge depends on two conditions :

• The ratio λ i λm . If it is close to one, meaning that λ m is not separated from the other eigenvalues as it occurs below the inflection, then convergence is slow; on the contrary, if it is close to 0, meaning that λ m is separated from the other eigenvalues as it occurs above the inflection where one eigenvalue is negative, then convergence is faster.

• The amplitude of the coefficients c i =m . If they are small, this means that the initial vector L is already quite converged to êm , and less power iterations are needed to achieve the convergence.

With this approach, the algorithm finds an approximate eigenvector, that gets closer and closer to êm at each increase of k. To avoid handling large matrices, it is most cost efficient to use a recurrence scheme : L k+1 = [H]L k , where the initial L 1 can be random or, better, a reasonable guess for êm .

To reduce the number of iterations k needed to converge to êm , Lanczos added two tricks to the power method. First, [H] is reduced on its subspace generated by the basis of the L k vectors. This is done using a reduction matrix

[R] = (L 1 |L 2 | • • • |L k ), leading to a tridiagonal matrix [T ] = [R] T [H][R].
The size of [T ] is k × k and its lowest eigenvalue converges to λ min , because its basis contains more and more parts of êm . However the L k basis is not orthogonal. Hence, the second Lanczos idea is to chose each new vector L k such that it is orthogonal to the previous ones by removing their respective contribution:

L k+1 = [H]L k - k j=0 (L k+1 T L j )L j (B.3)
These vectors are called the Lanczos vectors and contain the part of êm coming from the power, but not the parts that are already taken into account in the previous basis vectors. Taking orthogonal vectors means that the next Lanczos vector focuses the power method on the components of êm that are still unknown, restricting the number of dimensions that still need to be converged. Thanks to recurrence, this form can be reduce to:

L k+1 = [H]L k -α k L k -β k-1 L k-1 (B.4)
where

α k = L k T [H]L k L k T L k and β k = L k-1 T [H]L k L k-1 T L k-1 (B.5)
Finally, the [T ] matrix can be simply written as: The cone is centred on the adatom, with the direction < 111 > and a 45°angle. An input vector êrand is accepted only if its displacement of the Al adatom is located into this cone. The two possible diffusion mechanisms are shown by the black (hopping -Event Id 2) and red (exchange -Event Id 1) arrows. Bottom table: Transition energies ∆E for diffusion events found with a barrier lower than 1 eV. The Id of each event is the same as the ones used in Fig. 8 of Ref. [18]: Id 1, 4 and 3 respectively for the exchange with the 1rst, 3rd and 2nd neighbour, and Id 2 for the hopping. Many other barriers lower than 1 eV have been found around 0.90 eV, but they do not correspond to a diffusion of the addatom, such as the creation of a second addatom or of a vacancy. Pos X, normalized L 1

T k =         α 1 β 2 0 • • • 0 β 2 α 2 β 3 . . . . . . 0 β 3 α 3 . . . 0 . . . . . . . . . . . . β k 0 • • • 0 β k α k         (B.
êi m L 1 =random êi m |L1 L 2 êi m |L2 L 3 êi m |L 3 L 4 êi m | L 4 êi T 2 êi T 3 êi T 4 ê i m | L 2 ê i m | L 3 ê i m | L 4 (a) Lanczos method at ARTn step i êi+1 m êi+1 T 1 =L 1 =ê i T last êi+1 m |L1 ∼1 L 2 ê i + 1 m | L 2 êi+1 T 2 (b)
α 1 = L 1 HL 1 L 2 =HL 1 -α 1 L 1 β k = ||L k-1 || L k = L k /||L k || α k = L k HL k L k =HL k-1 -α k-1 L k-1 -β k-1 L k-2 T =          α 1 β 2 0 • • • 0

6 )

 6 Upon diagonalization of the matrix [T ], the lowest eigenvalue λ T m and its corresponding eigenvector êTm are obtained. The eigenvalue λ T m approximates the lowest eigenvalue λ m of matrix [H], and the corresponding eigenvector êm can be constructed from the eigenvector êTm , by multiplying êm = [R]ê Tm . The size of the Lanczos basis (number of iterations k) is increased until reaching the convergence criterion, as represented Fig. B.8.

( a )

 a Figure2: A complete event generated with ARTn on a two-dimensional potential energy surface. (2a) An ARTn even is launched from a local energy minimum. Each black dot is an ARTn step. The black arrows are the pushing directions at each step. The first steps, below the inflection line defined by all the point for which the lowest eigenvalue λ m is zero (in blue), are pushes in a predetermined random direction followed, at each step, by a slight perpendicular relaxation to avoid collisions. Above the inflection line (in the red region), steps follow the direction associated with the negative curvature (associated with lowest eigenvalue of the hessian), while relaxing in the perpendicular direction until a first-order saddle point is found (large black dot in the red region). (2b) Once at the saddle point, the system is pushed along +ê m and -ê m to escape the zone where the forces are null (blue arrows) and relaxed toward the adjacent minima (black arrows). Top figures: 3D plot three dimensional landscape. Bottom figures: 2D plot of two-dimension projection of the path; the small black lines are the isovalue levels lines for which the energy is the same.

Figure 4 :

 4 Figure 4: Typical number of Lanczos iterations (number of force calls) at each new call of the Lanczos algorithm by ART in three different cases. First case (violet triangles): when the size of the Lanczos basis is fixed. Second case (red diamonds): when the size of the Lanczos basis is not predetermined and randomly initialized. Third case (black circles): same as second case but the Lanczos basis is initialized with the previously converged eigenvector (L i+1 1 = êi m ).

Figure 5 :

 5 Figure 5: Top figure: The two first layers of the Al (100) surface and the Al adatom.The cone is centred on the adatom, with the direction < 111 > and a 45°angle. An input vector êrand is accepted only if its displacement of the Al adatom is located into this cone. The two possible diffusion mechanisms are shown by the black (hopping -Event Id 2) and red (exchange -Event Id 1) arrows. Bottom table: Transition energies ∆E for diffusion events found with a barrier lower than 1 eV. The Id of each event is the same as the ones used in Fig.8of Ref.[18]: Id 1, 4 and 3 respectively for the exchange with the 1rst, 3rd and 2nd neighbour, and Id 2 for the hopping. Many other barriers lower than 1 eV have been found around 0.90 eV, but they do not correspond to a diffusion of the addatom, such as the creation of a second addatom or of a vacancy.

  path is symmetrical 20

Figure 6 :Figure A. 7 :

 67 Figure 6: Top: Atomic structures of the reactants, transition state and products of the chemical reaction of CH 3 Cl with a Cl atom.Below: Total number of force calls to find the saddle point for the above reaction using each of the four procedures described in the text. This number of force calls does not take into account the relaxation to the adjacent minimum, which is the same for all.

β 2 α 2 β 3 →Figure B. 8 :

 38 Figure B.8: Schematic representation of the Lanczos algorithm.

  Lanczos method at ARTn step i + 1 Figure Scheme of two consecutive Lanczos procedures: one at ARTn step i and the second at ARTn step i + 1. The starting Lanczos vector L 1 is random at ARTn step 0 and is the previously converged at ARTn step i + 1, which implies that fewer Lanczos vectors are needed to converge to êm at step i + 1. Each vector is the 2D projection of a 3N at D vector. The blue brackets êm |L i indicate the projection of the unknown êm on each of the Lanczos basis vectors L k . The previous algorithm was always using 16 Lanczos basis vectors.

				Silcon interstitial 65 atoms
	# of Lanczos iterations k (=force calls)	0 5 0 10 15 20 25 30	2 Case 2: {L k } size = variable, L 1 =e 4 6 ∧ Case 3: {L k } size = variable, L 1 = e 8 rand ∧i-1 m Case 1: {L k } size = Cste =16	Inflection	10	12
			# of Lanczos call (=ART iterations i)
				Case 1		Case 2	Case 3
		# of force calls Average Total Average Total Average Total
	before inflection	16	112	16-28	160	3-10	54
		after inflection	16	80	9-14	57	2-6	19
			Total		192		217	73

Acknowledgements

NM acknowledges partial support from a Discovery Grant by the Natural Sciences and Engineering Research Council of Canada (NSERC). The simulations were performed using HPC resources from CALMIP (Grant P1555 and P1418) and from GENCI (Grant A0090911942). Some preliminary work was made possible with the generous computer allocation of Calcul Québec and Compute Canada to NM. All the authors are active members of the Multiscale And Multi-Model Approach for MaterialS In Applied Science consortium (MAMMASMIAS consortium), and acknowledge the efforts of the consortium in fostering scientific collaborations.

Software availability

ARTn-DFT is available freely upon request to any of the authors, it includes a complete documentation and many examples. The present version is compatible with QE from QE-6.5 to the present QE-7.0, also available freely at https://www.quantum-espresso.org. We are currently developing a plugin version for an easier use, hands on, maintenance and portability for easy integration into a wide range of ab initio and empirical codes.

Appendix A. Few definitions regarding the potential energy surface

We provide here a few definitions for some of the concepts used in this article regarding the potential energy surface. There are largely based on Ref. [28]. For a summarized version, see Table A Its complementary part is the configuration space above the inflection, in which at least one Hessian eigenvalue is negative.

• The valleys, represented in black Fig . A.7c, are the set of points for which the gradient is parallel to the eigenvector êm , i.e for which the component of the gradient that is orthogonal to êm is null. This is

Properties of the main remarkable regions and points on the potential energy surface. λ i represents a Hessian eigenvalue.

Appendix B. Lanczos diagonalization

The Lanczos algorithm [21] is a particular case of the Arnoldi algorithm [29] that can be simplified when the matrix to be diagonalized is hermitian. This method can deliver the smallest eigenvalue λ m and its corresponding eigenvector êm of a potentially very large matrix through the knowledge of the matrix-vector products. In ARTn, this matrix is the Hessian

Knowledge of the full Hessian matrix is not needed, because the algorithm only needs its product with a given vector, [H]dL k , at each step k, where dL is a small displacement vector from a reference position. This product can be obtained in only one force calculation:

• Do a small distortion dr of the atomic positions along L (forming dL)

:

• Calculate the forces at this displaced position with the software (here Quantum Espresso): F(X L ) = -∇E(X L )

• Return the correct term:

[H]dL = ∆F = F(X L )-F(X) dr , where F(X) has already been calculated in the last minimization step, before entering the Lanczos loop.


PhD Alice Bellicha
email: alice.bellicha@u-pec.fr

MD Cécile Ciangura
email: cecile.ciangura@aphp.fr

PhD Célina Roda
email: celina.roda@parisdscartes.fr

MD Adriana Torcivia
email: adriana.torcivia@aphp.fr

PhD Pierre Portero
email: pierre.portero@u-pec.fr

MD , PhD Jean-Michel Oppert
email: jean-michel.oppert@aphp.fr

Changes in cardiorespiratory fitness after gastric bypass: relations with accelerometryassessed physical activity

kg/m 2 ) were included. Mean (SD) weight loss 6 months after RYGB was -27.5 (7.9) kg (P<0.001). Absolute V ̇O2peak decreased by 0.35 (0.50) L/min (P<0.001) and relative V ̇O2peak tended to increase by 1.7 (5.3) mL/kg/min (P=0.06), both with large interindividual variability. Based on objective accelerometry data, daily steps and moderate-tovigorous physical activity increased by 1,275 [START_REF] Ross | Importance of Assessing Cardiorespiratory Fitness in Clinical Practice: A Case for Fitness as a Clinical Vital Sign: A Scientific Statement From the American Heart Association[END_REF]164) steps/d and 7.6 (19.3) min/d, respectively (both P<0.05). Change in absolute V ̇O2peak was positively related to change in moderate-to-vigorous physical activity (r=0.35, P=0.03).

Conclusions:

The decrease in absolute CRF observed 6 months after RYGB might be prevented by increasing habitual physical activity.

Trial Registration: clinicaltrials.gov Identifier: NCT01113996

Key words

Physical fitness, V ̇O2max, physical activity patterns, accelerometry, gastric bypass, obesity

Introduction

Bariatric surgery induces marked weight loss, reduces mortality and improves obesity-related comorbidities as well as health-related quality of life [START_REF] Mechanick | Clinical practice guidelines for the perioperative nutritional, metabolic, and nonsurgical support of the bariatric surgery patient--2013 update: cosponsored by American Association of Clinical Endocrinologists, The Obesity Society, and American Society for Metabolic & Bariatric Surgery[END_REF][START_REF] Sjostrom | Bariatric surgery and long-term cardiovascular events[END_REF]. Current guidelines for clinical care in this setting include promotion of physical activity (PA) to maintain or improve overall health and physical fitness [START_REF] Mechanick | Clinical practice guidelines for the perioperative nutritional, metabolic, and nonsurgical support of the bariatric surgery patient--2013 update: cosponsored by American Association of Clinical Endocrinologists, The Obesity Society, and American Society for Metabolic & Bariatric Surgery[END_REF]. Physical fitness is a major health indicator [START_REF] Ross | Importance of Assessing Cardiorespiratory Fitness in Clinical Practice: A Case for Fitness as a Clinical Vital Sign: A Scientific Statement From the American Heart Association[END_REF]. The concept of physical fitness includes different dimensions such as cardiorespiratory fitness (CRF), muscular fitness, and flexibility [START_REF]Physical activity guidelines for Americans[END_REF]. The gold-standard method for measuring CRF is to perform a graded maximal exercise test with direct measurement of maximal oxygen consumption (V ̇O2max) [START_REF]American College of Sports Medicine's Guidelines for Exercise Testing and Prescription[END_REF], expressed either in absolute values (L/min) or relative to body weight (mL/kg/min). Estimates of CRF can also be calculated from heart rate response or exercise time to exhaustion during maximal or submaximal exercise tests [START_REF]American College of Sports Medicine's Guidelines for Exercise Testing and Prescription[END_REF]. Low CRF is recognized as a major risk factor for all-cause mortality and cardiovascular events, including in patients with obesity [START_REF] Kodama | Cardiorespiratory fitness as a quantitative predictor of all-cause mortality and cardiovascular events in healthy men and women: a meta-analysis[END_REF][START_REF] Fogelholm | Physical activity, fitness and fatness: relations to mortality, morbidity and disease risk factors. A systematic review[END_REF]. CRF is also a marker of functional capacity and ability to perform adequately the tasks of daily living [START_REF] Ross | Importance of Assessing Cardiorespiratory Fitness in Clinical Practice: A Case for Fitness as a Clinical Vital Sign: A Scientific Statement From the American Heart Association[END_REF]. In the bariatric surgery setting, low CRF has been associated with increased short-term post-operative complications [START_REF] Mccullough | Cardiorespiratory fitness and short-term complications after bariatric surgery[END_REF].

Patients seeking bariatric surgery are characterized by poor physical fitness as measured by low CRF, with mean V ̇O2max values below 20 mL/kg/min [START_REF] Seres | Increased exercise capacity after surgically induced weight loss in morbid obesity[END_REF]. Surgery-induced weight loss has been associated with a significant, although modest, increase in relative V ̇O2max [START_REF] Seres | Increased exercise capacity after surgically induced weight loss in morbid obesity[END_REF][START_REF] Kanoupakis | Left ventricular function and cardiopulmonary performance following surgical treatment of morbid obesity[END_REF][START_REF] De Souza | Effect of weight loss on aerobic capacity in patients with severe obesity before and after bariatric surgery[END_REF][START_REF] Wilms | Differential changes in exercise performance after massive weight loss induced by bariatric surgery[END_REF][START_REF] Lund | Increased post-operative cardiopulmonary fitness in gastric bypass patients is explained by weight loss[END_REF].

In contrast, absolute V ̇O2max (L/min) has been found either decreased [START_REF] Seres | Increased exercise capacity after surgically induced weight loss in morbid obesity[END_REF][START_REF] Kanoupakis | Left ventricular function and cardiopulmonary performance following surgical treatment of morbid obesity[END_REF][START_REF] Neunhaeuserer | Functional Evaluation in Obese Patients Before and After Sleeve Gastrectomy[END_REF] or unchanged [START_REF] Wilms | Differential changes in exercise performance after massive weight loss induced by bariatric surgery[END_REF] after bariatric surgery. However, published studies have not consistently reported changes in both absolute and relative V ̇O2max. Therefore, there is a need to better assess the magnitude of change in both absolute and relative CRF values after bariatric surgery.

PA is the main modifiable determinant of CRF but the response to PA varies greatly between individuals and also depends on genetic factors [START_REF] Bouchard | Individual differences in response to regular physical activity[END_REF]. We showed in a recent systematic review and meta-analysis that endurance-based exercise training programs performed after bariatric surgery help improving CRF [START_REF] Bellicha | Effectiveness of exercise training after bariatric surgery-a systematic literature review and meta-analysis[END_REF][START_REF] Da Silva | Exercise Training Does Improve Cardiorespiratory Fitness in Post-Bariatric Surgery Patients[END_REF]. Little is known about the role of habitual (i.e. spontaneous) PA in this setting. Change in self-reported PA has been found associated with change in calculated V ̇O2max estimates [START_REF] Tettero | Increase in Physical Activity After Bariatric Surgery Demonstrates Improvement in Weight Loss and Cardiorespiratory Fitness[END_REF]. However, association between objectivelymeasured PA and directly measured V ̇O2max has not been investigated.

Therefore, the aims of this analysis were to quantify changes in CRF (V ̇O2peak expressed as absolute value or relative to body mass) and in accelerometry-assessed PA patterns 6 months after RYGB and to examine whether change in CRF was related to change in habitual PA, with special attention to moderate-to-vigorous PA which forms the basis of current PA recommendations.

Methods

Study population

This analysis was conducted on a subgroup of participants from a randomized clinical trial examining the effect of exercise on physical fitness and body composition 6 months after RYGB (clinicaltrials.gov Identifier: NCT01113996) [START_REF] Oppert | Resistance training and protein supplementation increase strength after bariatric surgery: a randomized controlled trial[END_REF]. In that trial performed at a single academic center, patients undergoing RYGB and recruited from 2010 to 2014 were randomly assigned to usual care (controls, N=22), or usual care and additional oral protein intake (N=31), or usual care, additional protein intake and supervised resistance training (N=23). All patients were encouraged to accumulate 150 min/wk of moderate-intensity PA, as part of usual care [START_REF] Jensen | AHA/ACC/TOS guideline for the management of overweight and obesity in adults: a report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines and The Obesity Society[END_REF]. As reported elsewhere, analyses revealed no differences in changes in CRF and habitual PA between the three groups [START_REF] Oppert | Resistance training and protein supplementation increase strength after bariatric surgery: a randomized controlled trial[END_REF]. Therefore, data from patients included in the control group and protein group were pooled for analyses in the present study. The study protocol was approved by the ethics committee of our institution. All patients signed a written informed consent prior to initiating the study.

Measures

All measurements were performed before and 6 months after RYGB [START_REF] Oppert | Resistance training and protein supplementation increase strength after bariatric surgery: a randomized controlled trial[END_REF]. CRF was measured by indirect calorimetry (Cardiac Assessment System for Exercise, CASE, GE Medical Systems Information Technologies Inc.) during a graded maximal exercise test on a cycle ergometer (GE Healthcare), as previously described [START_REF] Oppert | Resistance training and protein supplementation increase strength after bariatric surgery: a randomized controlled trial[END_REF]. Maximally achieved oxygen consumption (V ̇O2peak) was determined as the highest attained V ̇O2 during the test and expressed in absolute values (L/min) and relative to body mass (mL/kg/min) and to lean body mass (mL/kg LBM/min).

Habitual PA and sedentary time were measured using the Actigraph GT3x accelerometer (Manufacturing Technology, Inc., FL, USA). Participants were asked to wear the Actigraph at the hip using the provided elastic belt for seven consecutive days during all waking hours except during water-based activities. Data were considered valid when the accelerometer was worn for at least 4 days during at least 8 h each day [START_REF] Bond | Pre-to postoperative physical activity changes in bariatric surgery patients: self report vs. objective measures[END_REF]. Analyses were restricted to patients who provided valid accelerometry data both before and 6 months after RYGB. Based on previous literature [START_REF] Freedson | Calibration of the Computer Science and Applications, Inc. accelerometer[END_REF], cut-points below 100 counts/min, between 100 and 1,951 counts/min, between 1,952 and 5,723 counts/min and over 5,724 counts/min were used to quantify sedentary behavior, light, moderate and vigorous intensity PA, respectively. Moderate-to-vigorous PA (MVPA) bouts were defined as 10 or more consecutive min above the MVPA activity counts threshold with allowance of 2 min below this threshold [START_REF] Troiano | Physical activity in the United States measured by accelerometer[END_REF]. Daily steps, corresponding to the total steps accumulated during wear time for a day, were calculated. As previously described, a conservative approach was used to determine step counts: only steps taken at an intensity ≥500 counts/min were registered [START_REF] Tudor-Locke | Accelerometer profiles of physical activity and inactivity in normal weight, overweight, and obese U.S. men and women[END_REF]. Patients were considered compliant to the 2008 US PA guidelines when MVPA accumulated in ≥ 10-min bouts was ≥ 150 min/wk [START_REF]Physical activity guidelines for Americans[END_REF] and compliant to the 2018 US PA guidelines when total MVPA was ≥ 150 min/wk [START_REF] Piercy | The Physical Activity Guidelines for Americans[END_REF].

Body composition (fat mass, lean body mass) was measured by DXA scanning (Hologic Discovery W, software v12.9; Hologic, Bedford,MA), as previously described [START_REF] Oppert | Resistance training and protein supplementation increase strength after bariatric surgery: a randomized controlled trial[END_REF]. Weight loss was expressed change in body weight and in body mass index (BMI). Diabetes was defined according to a fasting blood glucose >7 or >11 mM two hours after a 75-g oral glucose tolerance test, or the use of antidiabetic drugs, insulin or GLP-1 analogues [START_REF] Oppert | Resistance training and protein supplementation increase strength after bariatric surgery: a randomized controlled trial[END_REF].

Statistical analyses

Pre

Results

Participant characteristics

From the 53 participants included in the initial trial in the groups without PA intervention, 45 participants provided valid accelerometry data before and after surgery and 41 participants also performed a maximal exercise test at both time points. Participants who did not provide valid accelerometry and V ̇O2peak data (N=12) did not differ significantly from participants who provided valid data (data not shown). Before surgery, 11 (24.4%) patients had type 2 diabetes. Six months after RYGB, a significant weight loss (mean [SD]) -27.5 [7.5] kg), fat mass and lean body mass loss (all P<0.001) was observed (Table 1). The proportion of participants with type 2 diabetes and hypertension tended to decrease (P=0.07 and P=0.08, respectively).

Changes in CRF and in accelerometry-assessed physical activity

Absolute V ̇O2peak significantly decreased after RYGB by -0.35 (0.50) L/min (P<0.001) (Table 1), with values ranging from -1.30 to +0.53 L/min. V ̇O2peak relative to body weight tended to increase by 1.7 (5.3) mL/kg/min (P=0.06), with values ranging from -9.6 to +10.8 mL/kg/min. V ̇O2peak relative to lean body mass was not significantly different 6 months after RYGB.

Accelerometer wear time did not differ 6 months after RYGB. A significant increase in daily steps (all P<0.05) was observed. Mean duration of moderate-and vigorous-intensity PA significantly increased (P<0.05), which resulted in a +39.7 (84.5) % increase in total MVPA (P<0.05). Change in total MVPA ranged from -23 to +69 min/d. The duration of MVPA accumulated in ≥ 10-min bouts increased on average by more than two-fold (P<0.05). The proportion of MVPA accumulated in ≥ 10-min bouts also increased from 7% to 16% of total MVPA (P<0.05). The number (%) of patients who did not engage in any MVPA bout ≥ 10 min changed from 16 (36%) before RYGB to 8 (18%) after RYGB (P=0.08). However, the proportion of patients compliant to the 2008 and the 2018 US PA guidelines was not significantly different 6 months after RYGB (9 to 20%, P = 0.07 and 60 to 67%, P=0.60, respectively). The duration of sedentary time and light-intensity PA was not different after RYGB.

Relations between changes in CRF and MVPA

When adjusting for age and preoperative BMI, change in absolute V ̇O2peak after RYGB was positively related to change in total MVPA (r=0.35, P=0.03) (Figure 1a). Change in relative V ̇O2peak tended to be related to change in total MVPA (r=0.30, P=0.07) (Figure 1b).

Discussion

In this study, we assessed changes in CRF and in accelerometry-assessed habitual PA after bariatric surgery in women with obesity. Six months after RYGB, we observed a significant decrease in absolute V ̇O2peak and a significant increase in habitual PA, with large interindividual variability for both outcomes. We also found that change in CRF was positively related to change in MVPA.

Our findings strengthen the notion that absolute CRF decreases in the first months after bariatric surgery [START_REF] Seres | Increased exercise capacity after surgically induced weight loss in morbid obesity[END_REF][START_REF] Kanoupakis | Left ventricular function and cardiopulmonary performance following surgical treatment of morbid obesity[END_REF][START_REF] Neunhaeuserer | Functional Evaluation in Obese Patients Before and After Sleeve Gastrectomy[END_REF]. In contrast, because of massive weight loss, relative V ̇O2peak increases after bariatric surgery [START_REF] Seres | Increased exercise capacity after surgically induced weight loss in morbid obesity[END_REF][START_REF] Kanoupakis | Left ventricular function and cardiopulmonary performance following surgical treatment of morbid obesity[END_REF][START_REF] De Souza | Effect of weight loss on aerobic capacity in patients with severe obesity before and after bariatric surgery[END_REF][START_REF] Wilms | Differential changes in exercise performance after massive weight loss induced by bariatric surgery[END_REF][START_REF] Lund | Increased post-operative cardiopulmonary fitness in gastric bypass patients is explained by weight loss[END_REF]. Relative V ̇O2peak being a strong predictor of both mortality and functional impairment, this is an important benefit of bariatric surgery.

Previous studies had reported mean increases in V ̇O2peak 6 months after bariatric surgery ranging from +1.3 mL/kg/min [START_REF] De Souza | Effect of weight loss on aerobic capacity in patients with severe obesity before and after bariatric surgery[END_REF] to 5.2 mL/kg/min [START_REF] Kanoupakis | Left ventricular function and cardiopulmonary performance following surgical treatment of morbid obesity[END_REF]. The +1.7 mL/kg/min increase in V ̇O2peak observed in our study is therefore consistent with previous results. However, despite such increase, mean V ̇O2peak remains very low after surgery, with mean values around 20 to 25 mL/kg/min, corresponding to that usually observed in older women from the general population [START_REF] Kaminsky | Reference Standards for Cardiorespiratory Fitness Measured With Cardiopulmonary Exercise Testing Using Cycle Ergometry: Data From the Fitness Registry and the Importance of Exercise National Database (FRIEND) Registry[END_REF]. Further improving CRF in this context is therefore an important objective to optimize the benefits of bariatric surgery on health status and functional capacity.

Our data suggest that increasing habitual MVPA may be an important contributor to maintaining CRF in the first months after bariatric surgery. This in line with controlled trials that reported improvements in CRF after an exercise training program in patients undergoing bariatric surgery [START_REF] Bellicha | Effectiveness of exercise training after bariatric surgery-a systematic literature review and meta-analysis[END_REF]. In our study, MVPA increased on average by 8 min/d, with large between-subject variability in both the direction and magnitude of change. Six months after RYGB, two-third of patients were compliant with the latest US PA guidelines [START_REF] Piercy | The Physical Activity Guidelines for Americans[END_REF], and only 20% were compliant with previous recommendations that considered only MVPA accumulated in bouts of 10 minutes or longer [START_REF]Physical activity guidelines for Americans[END_REF]. Some previous studies have shown no significant change in accelerometry-assessed PA in the first months after bariatric surgery [START_REF] Bond | Pre-to postoperative physical activity changes in bariatric surgery patients: self report vs. objective measures[END_REF][START_REF] Giusti | Energy and macronutrient intake after gastric bypass for morbid obesity: a 3-y observational study focused on protein consumption[END_REF][START_REF] Berglind | Longitudinal assessment of physical activity in women undergoing Roux-en-Y gastric bypass[END_REF][START_REF] Afshar | Changes in physical activity after bariatric surgery: using objective and self-reported measures[END_REF][START_REF] Liu | Metabolic and neuroendocrine responses to Roux-en-Y gastric bypass. I: energy balance, metabolic changes, and fat loss[END_REF], in contrast with a number of studies that have consistently reported increases in self-reported PA [START_REF] Herring | Changes in physical activity behaviour and physical function after bariatric surgery: a systematic review and meta-analysis[END_REF]. Objectively-measured PA data suggest that weight loss alone might not be sufficient to substantially increase habitual PA. Behavioral interventions targeting daily PA might therefore be needed to help patients undergoing bariatric surgery reaching the recommended level of PA.

Another important finding of this study was that MVPA accumulated in bouts represented less than 10% of total MVPA before RYGB, and less than 20% six months after RYGB. Short bouts of MVPA were therefore largely predominant both before and after RYGB. Similar patterns of MVPA have been described in previous studies, with ≥10-min bouts MVPA representing 9% to 22% of total MVPA in patients after bariatric surgery [START_REF] Bond | Pre-to postoperative physical activity changes in bariatric surgery patients: self report vs. objective measures[END_REF][START_REF] Berglind | Longitudinal assessment of physical activity in women undergoing Roux-en-Y gastric bypass[END_REF][START_REF] Bond | Objective quantification of physical activity in bariatric surgery candidates and normalweight controls[END_REF][START_REF] King | Objective assessment of changes in physical activity and sedentary behavior: Pre-through 3 years post-bariatric surgery[END_REF]. Prolonged bouts of MVPA appear more prevalent in normal-weight or adults with overweight, in whom MVPA accumulated in ≥10-min bouts has been found to reach up to 40% of total MVPA [START_REF] Glazer | Sustained and shorter bouts of physical activity are related to cardiovascular health[END_REF]. Growing evidence suggests that MVPA is associated with health benefits, independently of bout duration: short bouts of MVPA have been associated with a reduction in mortality risk [START_REF] Saint-Maurice | Moderate-to-Vigorous Physical Activity and All-Cause Mortality: Do Bouts Matter?[END_REF], BMI or adiposity, blood pressure, blood lipids and with improved glycemic control [START_REF] Pagac | Physical Activity Guidelines Advisory Comittee Report[END_REF]. However, there is no consensus as to whether prolonged bouts of MVPA are more strongly associated with health outcomes than short bouts [START_REF] Pagac | Physical Activity Guidelines Advisory Comittee Report[END_REF]. Our data suggest that modest improvement in total MVPA might provide additional benefits on CRF after bariatric surgery, even though short bouts of MVPA remain predominant. In these subjects, maintaining long bouts of MVPA might be a particularly challenging task.

Therefore, promoting both incidental and prolonged PA may represent an important and reachable objective in patients with severe obesity undergoing bariatric surgery.

Strengths of the study include the objective measurement of CRF and PA before and after RYGB in a homogeneous group of women who received the same surgical procedure. The use of min-by-min accelerometry data provided new insights about accumulation of PA in these subjects, especially regarding bout analysis. Although the most widely used accelerometers in clinical and population studies, Actigraph accelerometers have not been specifically validated in subjects with severe obesity. Only women were included in the study, and our subjects underwent only one type of bariatric procedure, which prevents us from extrapolating results to all obese subjects and to those undergoing other types of bariatric surgery. In addition, patients were initially included in a clinical trial although we did not include subjects from the exercise training group in our analyses.

Conclusion

This study shows that CRF relative to body weight tends to increase in the early first months following bariatric surgery, but that absolute CRF decreases significantly. A significant, although modest, increase in habitual PA was also found. Changes in both CRF and PA were characterized by large inter-individual variability in both the direction and magnitude of change. The change in CRF was positively related to the change in PA, suggesting that habitual PA contributes to the improvement in physical fitness after bariatric surgery. These findings highlight the need to identify effective interventions for promoting PA after bariatric surgery that can be incorporated into routine clinical care.

Title:

  Changes in cardiorespiratory fitness after gastric bypass: relations with accelerometryassessed physical activity Abstract Background: Exercise training optimizes cardiorespiratory fitness (CRF) after bariatric surgery but the effect of spontaneous physical activity is not well known. This study aimed to objectively quantify changes in CRF and habitual physical activity 6 months after bariatric surgery and to examine whether change in CRF was related to change in physical activity. Methods: Secondary analyses were performed on data from women who underwent Rouxen-Y gastric bypass (RYGB) between 2010 and 2014. Measurements were performed before and 6 months after RYGB and included V ̇O2peak (graded maximal exercise test on a cycle ergometer), habitual physical activity (Actigraph GT3x accelerometer worn during 7 days), and body composition (DXA absorptiometry). Changes after RYGB were analyzed using Wilcoxon signed-rank test. Relations between change in CRF and change in physical activity were analyzed with Spearman correlations adjusted on age and preoperative BMI. Results: Forty-five women (median [P25-P75] age: 43.0 [38.0-51.0] y, BMI: 42.6 [40.0-45.5]

  -and post-operative values are presented as median values (25 th -75 th percentile) for continuous data and frequencies and percentages for categorical data. Pre-to post-operative changes are presented as mean (SD). Mean (min-max) values are presented to characterize interindividual variability in CRF and MVPA changes. Given the non-normality of most variables, non-parametric tests were used. Preoperative and post-operative values were compared with Wilcoxon signed-rank test for continuous variables and with McNemar test for categorical variables (proportion of participants with obesity comorbidities or compliant with PA recommendations). Relations between change in CRF and change in MVPA were analyzed with Spearman correlations adjusted for age and preoperative BMI. The null hypothesis was rejected at P<0.05. Analyses were performed using R version 3.4.3.

  Figure caption

Table 1 . Characteristics of study participants before and after RYGB (N=45) 442

 1

	Before RYGB	After RYGB	Change	P-value
			Mean (SD)

Acknowledgements

This work was supported by PHRC Programme Hospitalier de Recherche Clinique-Ile de France (AOR 09 059).

We recognize the contribution of the Clinical Research Unit (Unité de Recherche Clinique URC), Pitie-Salpetriere Hospital (AP-HP), Paris, France, in data acquisition and management.

We express our thanks to the patients for their participation in this study.

Non-blinded version of your COI: No conflicts of interest

Acknowledgements

This work was supported by PHRC Programme Hospitalier de Recherche Clinique-Ile de France (AOR 09 059).

We recognize the contribution of the Unité de Recherche Clinique/Clinical Unit Research, Pitie-Salpetriere Hospital, Paris, France, in data acquisition and management.

We express our thanks to the patients for their participation in this study Blinded conflict of interest disclosure statement

Data are median (25 th -75 th percentile) unless otherwise stated. P-value from Wilcoxon signed-rank test; bold values indicate significance with P<0.05. Censored steps were calculated by censoring steps taken at an intensity <500 counts/min [START_REF] Tudor-Locke | Accelerometer profiles of physical activity and inactivity in normal weight, overweight, and obese U.S. men and women[END_REF].

Abbreviations: BMI, body mass index; LBM, lean body mass; RYGB, Roux-en-Y gastric bypass; PA, physical activity; MVPA, moderate-to-vigorous physical activity.

Author 1, author 2, author 3, author 4, author 5 and author 6 declare no conflicts of interest.

Statements regarding ethics and consent

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.