Above Saddle-Point Regions of Order in a Sea of Chaos in the

Article Cite This: J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Above Saddle-Point Regions of Order in a Sea of Chaos in the Vibrational Dynamics of KCN Published as part of The Journal of Physical Chemistry virtual special issue “Manuel Yáñez and Otilia Mó Festschrift”. H. Párraga,*,† F. J. Arranz,*,† R. M. Benito,*,† and F. Borondo*,‡,¶ †

Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain ‡ Instituto de Ciencias Matemáticas, Cantoblanco, 28049 Madrid, Spain ¶ Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain ABSTRACT: The dynamical characteristics of a region of regular vibrational motion in the sea of chaos above the saddle point corresponding to the linear C−N−K configuration is examined in detail. To explain the origin of this regularity, the associated phase space structures were characterized using suitably defined Poincaré surfaces of section, identifying the different resonances between the stretching and bending modes, as a function of excitation energy. The corresponding topology is elucidated by means of periodic orbit analysis.

1. INTRODUCTION A wealth of dynamical information concerning molecules and chemical processes coming both from experiments and theory can be found nowadays in the literature. Its existence has fostered the interest in the theoretical study of (nonlinear) dynamics of realistic molecular models. The dynamics of floppy molecules, where the motion is often chaotic, clearly fall into this category.1 These dynamics are relevant for different interesting chemical processes, such as isomerization,2−4 photodissociation,5 intramolecular energy relaxation,6−8 or reactions.9 The associated processes can be studied experimentally thanks to the developments made in laser technology10 and other related experimental techniques.11 From a mathematical perspective, the vibration of molecules can be viewed as arising from the motion of a collection of coupled anharmonic oscillators. In this view, molecules are see as Hamiltonian systems whose dynamics can be adequately followed in the classical phase space, which is organized according to the celebrated Kolmogorov−Arnold−Moser (KAM) and Poincaré−Birkhoff theorems.12 According to KAM, the dynamic of generic systems consists of regular motions, taking place in invariant tori, which are embedded in a surrounding chaotic sea formed by the remnants of the tori that were destroyed by the perturbation induced by the excitation energy. As the vibrational energy (that can be viewed as a perturbation) grows, more and more regular tori turn into chaos, eventually joining the different stochastic regions originated in this form. This gives rise to widespread chaos in which the trajectories explore ample regions of phase space, thus allowing the acting of the couplings among the different vibrational modes and the occurrence of the associated intramolecular energy flow. In the chaotic regions there are also © XXXX American Chemical Society

cantori, which are partially destroyed tori with fractal dimensionality, which act as bottlenecks for the flux of trajectories across.13 However, the Poincaré−Birkhoff theorem accounts for the dynamics in the regions of phase space in which specific resonant interactions take place. They consist of chains of islands, its number being given by the order of the resonances, surrounded by a band of chaos organized by the corresponding homoclinic intersections, as described by Poincaré at the turn of the 19th century. In these regions dominated by resonances, the coupling between specific modes is very efficient, and then the resonance fosters the corresponding energy transfer. In the center of the islands of stability one finds the corresponding stable periodic orbit (PO), and between those centers the corresponding unstable one generates the mentioned homoclinic tangle. These structures and processes can be studied by using composite surfaces of sections in two-dimensional systems and in general by using frequency map analysis14 or other chaotic indicators maps.15,16 Similar studies can be performed in a full quantum setup,17,18 by using, for example, suitable quasiprobability density distributions in phase space, such as the Wigner19 or the Husimi functions,20 which correspond to different (averaged) transforms of the usual quantum wave function giving joint quasiprobability distributions in phase space, which accurately include all quantum effects, or even plain quantum wavepacket propagations.21,22 Received: January 4, 2018 Revised: March 11, 2018 Published: March 13, 2018 A

DOI: 10.1021/acs.jpca.8b00113 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

molecular motions. This fact brings mathematical simplicity into the problem without losing the essence of chaos. 2.1. Hamiltonian. The rotationless, that is, J = 0, 2D Hamiltonian model describing the vibrations of KCN can be written in Jacobi coordinates (R, r, θ) as

Concerning this type of classical-quantum correspondence studies in realistic models for the vibrations of molecular systems,23 we presented in a previous paper24 an accurate global analytic potential energy surface (PES) for the KCN isomerizing system, based on accurate ab initio quantum chemistry calculations at the QCISD(T)/6-311+G(2d) level of calculation, which has greatly updated the previous studies existing in the literature, which were performed either at a very low level of calculation, incorrectly predicting the existence of only one minimum, or restricted to the calculation only of the different equilibrium points in the PES, not being then suitable for dynamical studies like ours.24 In that paper we also presented a first exploration of the structure of the KCN classical phase space. A very early onset of widespread chaos was found, and the route to it was described. The correspondence between these classical structures and the nodal pattern of the corresponding quantum wave functions was considered as well. Another relevant result in our study was the surprising existence at high energies of regular classical motion above the saddle point of the PES corresponding to the linear C−N−K configuration, with even a conspicuous chain of islands structure around it. These above saddle-point regions of order embedded in a chaotic sea have been described previously in other similar dynamical Hamiltonian systems,25−32 including the restricted three-body problem of Celestial Mechanics, H atom in a circularly polarized microwave field, Ar clusters, and chemical reactions. This is certainly a topic of much interest, especially in view of the results reported recently.33,34 Accordingly, the aim of the present paper is to investigate this interesting dynamical behavior further. The organization of the paper is as follows. In the next Section we briefly describe the model chosen for our study, giving the essential dynamical details relevant to this work, such as the potential function used24 and the way in which Poincaré surfaces of section12 are defined to study the KCN phase space. In Section 3 we present and discuss our results concerning the region of regularity described in the abstract. In particular, we examine the evolution of the phase space structure as a function of the excitation energy. Finally, we conclude the paper in Section 4 by summarizing our main results and presenting some ideas for related future work.

H=

PR 2 1⎛ 1 1 ⎞ 2 ⎟⎟Pθ + V (R , θ ) + ⎜⎜ + 2μ1 2 ⎝ μ1R2 μ2 re 2 ⎠

(1)

where μ1 = mK(mC + mN)/(mK + mC + mN) and μ2 = mCmN/ (mC + mN) are reduced masses, mX are the corresponding atomic masses, and PR and Pθ are the conjugate momenta associated with R and θ, respectively. The PES is V(R, θ), which in our case is given by the analytical function 5

V (R , θ ) =

18

∑ ∑ Cn,m[1 − e−α(R− R )]n cos(mθ) 0

n=0 m=0

(2)

where the parameters Cn,m, α, and R0, which were obtained by fitting to accurate high-level ab initio calculations, can be found in ref 24. This KCN PES is shown in the top part of Figure 1, as

2. MODEL In this Section we briefly discuss the model used in our study. Jacobi coordinates will be used to describe the molecular vibrational modes. In particular, the stretching K−CN vibration is described by the R coordinate, giving the distance between the K atom and the center of mass of the C−N fragment. The C−N stretching is described by the coordinate r, giving the distance between the C and the N atoms. Finally, the bending motion is described by the angular coordinate θ representing the angle between the two previous vectors, being θ = 0 at the K−C−N linear configuration. The vibrational dynamics of KCN are a very interesting problem in nonlinear research for several reasons. It can be described realistically, and it is very highly nonlinear. Moreover, as described in ref 24 the vibrations of KCN can be adequately described in two dimensions (2D) by freezing the C−N degree of freedom to its equilibrium value, r = re. This is due to the high value of the vibrational frequency of this mode, which then effectively decouples from the rest of

Figure 1. (top) Potential energy surface for KCN. The minimum energy path (red solid line), has been plotted superimposed. (bottom) Potential profile along the minimum energy path. The linear C−N−K saddle energy, relevant for our study, is 940 cm−1.

a contour plot. As can be seen it presents two local minima, one at θ = 0 corresponding to the slightly stable linear isomer K− C−N, another at θ ≃ π/2 rad, corresponding to a deeper stable triangular configuration, and a saddle point at the C−N−K linear configuration. In the plot we also include in red line the minimum energy path (MEP) connecting the two linear configurations. The potential profile along this MEP is shown in the bottom part of the figure. Full details on the topology of the KCN PES function were given in ref 24. B


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Figure 2. Composite Poincaré surface of section for KCN at different values of the vibrational excitation energy using the coordinates defined in eq 4. See Subsection 2.2 for more details.

2.2. Molecular Vibrations Dynamics. To study the dynamics corresponding to Hamiltonian (1) classical trajectories are obtained by numerical integration of the associated Hamilton equations of motion. To get a suitable graphical representation of the corresponding phase space, Poincaré surfaces of section (PSSs) are computed at different values of the energy, taking the MEP as the sectioning plane. For this purpose, we use the analytical expression for the MEP given by

3. RESULTS 3.1. The Global KCN Classical Dynamics in Phase Space. In Figure 2 we present a representative selection of composite PSSs at different values of the vibrational excitation energy, ranging from E = 45 to 2400 cm−1, to get an idea of the global structure and evolution of the KCN phase space, when the perturbation induced by the excitation vibrational energy grows, and also about their dynamics. The panels in the top row illustrate the onset of chaos as the perturbation induced by the growing of the vibrational energy increases. To facilitate the visual distinction of order from chaos, we used a dark blue color for the former and red for latter. The obtained results can be rationalized with the aid of the Poincaré−Birkhoff and KAM theorems.12 Thus, for E = 45 cm−1 the behavior of the system is seen to be completely regular, and it corresponds to molecular vibrations around the stable triangular KCN configuration at the global minimum (or well) of the PES. All visible structures are either invariant tori or correspond to chains of islands. At E = 65 cm−1, the principal (central) elliptic point undergoes a pitchfork bifurcation, at which the original fixed point becomes hyperbolic (unstable), and two new elliptic (stable) points are born. As energy increases, for example, at E = 250 cm−1 the separatrix of this new central hyperbolic point breaks down, giving rise to a noticeable stochastic band, marking the existence of widespread chaos (Chirikov’s mechanism35). Notice that in this system the phase space is already mixed at very low values of the energy, with large regions of chaos existing between the regular tori. As discussed in ref 24, this low-energy threshold is relevant for the (very irregular) nodal structure of the corresponding quantum eigenstates. The panel

9

R eq(θ ) =

∑ an cos(nθ) n=0

(3)

with the values for the coefficients an reported in ref 24, and make the PSSs an area-preserving map by means of the following canonical transformation: ρ = R − R eq(θ ), ϑ = θ , Pρ = PR , Pϑ = Pθ + PR [dR eq(θ )/dθ ]

(4)

In this way, at a given energy E, the PSS consists of the series of (ϑ, Pϑ) pairs along the trajectories recorded every time that ρ = 0, being at the same time Pρ in a predetermined branch that arises from the Hamiltonian conservation H(ρ, ϑ, Pρ, Pϑ) = E (in the present calculations the negative one of the secondorder equation for Pρ). Finally, all the PSS points are folded into the interval ϑ ∈ (0, π) to take into account the symmetry of the molecular system. By combining in the same plot the results obtained from propagating a bunch of trajectories all with the same energy but different initial conditions sweeping the available phase space the so-called composite PSS are obtained. C


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Figure 3. Enlarged view of the composite Poincaré surface of sections presented in Figure 2 in the vicinity of the C−N−K linear configuration for E = 1200−1500 cm−1. Let us remark that here we have not folded the angular coordinate into the fundamental interval ϑ ∈ (0, π). Notice also the small interval of the coordinates ϑ and Pϑ that is being considered.

in the left bottom part of the figure shows how at the (very modest) value of E = 1300 cm−1 all phase space is filled with irregular motion. As the energy is increased further, to E = 1500 cm−1 (bottom central panel), the regions corresponding to the triangular and linear minima configurations are connected, and then the K atom can rotate around the C−N fragment. Also, it is seen how an area of regularity appears around the linear K−N−C isomer (ϑ = 0), while the dynamics around the triangular minimum remain chaotic. This is obviously not surprising, since the PES there presents a local minimum. But more remarkably, a small regular structure (tori) emerges in the linear configuration at the potential saddle (ϑ, Pϑ) = (π, 0) in a region that was fully chaotic at lower energies. Even more remarkable is the fact that this region persists, and its area grows when the energy increases, as can be seen in the last panel corresponding to E = 2400 cm−1. Although it is known that on top of saddle points there may be a quantum regularity due to recurrences,18,36 a phenomenon described for the first time by Heller37 and known as scarring,38,39 to the best of our knowledge the existence of this kind of classical regular areas over saddle regions is not common, if they exist at all, in molecular systems. Accordingly, we decided to do some research about this issue, and the results are presented in the next three subsections. 3.2. Region of Order around the C−N−K Linear Configuration. To investigate the phenomenon of the creation of an island of regularity on top of the C−N−K linear configuration we computed and plotted composite PSSs (CPPSs) for different values of the vibrational excitation energy in the range of 1200−5500 cm−1. The results are shown in Figures 3−5. In the first one of them, Figure 3, we observe the birth of the regular region under study at ϑ ≃ π rad. As can be seen, at E = 1200 and 1300 cm−1 all motion in the plotted phase space range is irregular, and the hyperbolic structure around the central saddle point (ϑ = π rad) is readily apparent. Notice the small range of the variables ϑ and Pϑ that is considered in the plot. At E = 1400 cm−1 the central fixed point has become stable, having developed an area of regularity around it, which is filled with invariant tori with the shape in this composite PSS of horizontally elongated ellipses. This regular region is seen to grow in the last panel of the figure, corresponding to E = 1500 cm−1. Let us remark that this growth in the regular region over the saddle point of the PES is not derived from the increase of energy that we are considering, which takes place at larger

values of the coordinates, but rather is a genuine dynamical effect arising from another reason, which is of dynamical origin. In Figures 4 and 5 we show a similar series of composite PSSs in the energy range of 1600−5500 cm−1 to analyze the complete evolution of the region over the saddle point of the PES. We see in the first panels (E = 1600−1800 cm−1) how the regular area in this region continues growing, while still being filled only with invariant tori. However, at E = 1900 cm−1 we start to observe very close to the boundary between the inner regular and outer chaotic regions the appearance of the first chain of islands corresponding to resonances between the stretching and bending coordinates. Actually, it corresponds to the nR/nθ = 1:6 resonance. The corresponding chain of islands is seen to continue developing as the energy grows (see panels for E = 2000−2700 cm−1) and the associated structures migrates toward the center. At the same time, other higherorder resonances appear close to the phase space boundary, for example, the one corresponding to 1:14 can be seen in the E = 2300 cm−1 panel. Furthermore, in the range of E = 2500−2700 cm−1 the regular motion around the 1:6 resonance is seen to be progressively destroyed by the perturbation induced by the growth of vibrational excitation. For example, at E = 2500 cm−1 an extensive homoclinic chaotic structure around the islands is apparent, and at E = 2700 cm−1 the last remnants of (1:6) order can be seen as swallowed by the surrounding chaotic sea. Also, at E = 2600 cm−1 the chain of islands corresponding to a 1:10 resonance is incipient, which is seen to be well-developed by E = 2700 cm−1. More interesting is that above E = 2800 cm−1 the whole structure of the inner regular islands dramatically changes due to the appearance of a very conspicuous 1:4 resonant shape around the central fixed point. Also, at this energy a 1:8 chain of small islands is visible next to the border of the chaotic sea. The area of the four-island structure around the central 1:1 fixed point grows until E = 3000 cm−1, filling all the regular portion of the phase space. Beyond this value of the energy, the 1:4 stable island starts to disintegrate, giving rise to chaos, and finally disappearing after E = 3400 cm−1. Meanwhile, the region of regularity around the central 1:1 resonance remains approximately maintaining its area, which is filled with ellipses elongated along a diagonal axis. The final fate of this central regularity island is illustrated in Figure 5. In the first panel corresponding to E = 3600 cm−1 two resonances are seen to develop. The outermost one, which was incipient at E = 3500 cm−1, corresponds to 1:10, and the innermost one corresponds to 1:6. The former quickly disappears, being barely visible when E reaches 3800 cm−1, D


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Figure 4. Same as Figure 3 for higher values of the vibrational excitation energy. Notice that here the range of the coordinates ϑ and Pϑ considered is larger than in Figure 3.

our plots the 1:8 and 1:14 resonances are visible. Obviously the 1:10 and 1:12 should be between them, although they have not been picked up by our initial condition procedure. After that, we see the 1:8 resonance being developed and destroyed, in the range of E = 4200−4400 cm−1, and the 1:12 afterward at E > 4500 cm−1.

into a broad band of heteroclinic chaos, which is clearly seen at E = 3700 and 3800 cm−1. For E = 3900−4100 cm−1 the 1:6 resonance is at the border of the regular region, and in this energy range we see its destruction process into a new band of homoclinic chaos. This process is finished by E = 4200 cm−1, and during it other inner chains of islands have been formed. In E


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Figure 5. Continuation of Figure 4 for higher values of the vibrational excitation energy.

But more interesting is that at E = 4300 cm−1 we see that the central fixed point has turned unstable in a pitchfork bifurcation, at which two new stable orbits are born. This fact dramatically changes again the global structure of the regular region under study; a band of homoclinic chaos is also born around the two islands of regularity that exists around the newly created POs with the period doubled. The band does not

start to be visible until the panel for E = 4600 cm−1. In the meanwhile, other resonant chains of islands are visible both around the central (now unstable) PO, that is, 1:10 at E = 4300−4400 cm−1 and 1:12 at E = 4500 cm−1, and the newly born pair, that is, 1:6 at E = 4600−4700 cm−1. Finally, above E = 4800 cm−1 chaos begins to be widespread, as the two newly F


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corresponding to the area of this region for different values of the excitation vibrational energy. Notice that it has obviously the units of an action, angular in this case, since it represents the area of a surface of section in phase space. The corresponding values for the area have been crudely estimated by numerically integrating the area inside the manually digitalized boundaries of the blue regions in the composite PSS plots of Figures 4 and 5. As can be seen, the obtained results are very interesting, since the whole curve is seen to consist of five different portions, which approximately coincide with the changes in the phase space structure that was discussed in Subsection 3.1. The separations between these portions are indicated by vertical dashed gray lines in the plot. Indeed, the area first grows in an approximately linear fashion from E = 1300 to 2100 cm−1. The area values then stabilize in a second piece of the curve consisting of a plateau going up to E = 2900 cm−1. The curve then suddenly drops to a second plateau with a small peak in the middle. The energy range spanned by this third part of the curve is from E = 3000 to 4200 cm−1, where the curve initiates a fourth part consisting of a steep linear decrease that finishes at E = 4800 cm−1. At this value of energy the curve stabilizes or very slowly decreases up to E = 5400 cm−1, where we last see two tiny islands of regularity, forming the fifth and last piece of the curve. When comparing the energy values (thresholds) separating the five different pieces in the curve, namely, Eth = 2200, 3000, 4200, and 4800 cm−1, with the conclusions of

born stable region are destroyed progressively, until they cease to be visible for E > 5500 cm−1. 3.3. Area of the Region of Order around the C−N−K Linear Configuration. To continue our study of the region of regular motion appearing in the phase space at high values of the vibrational energy, we present in Figure 6 the results

Figure 6. Evolution of the area of the region of order on the linear C− N−K isomer with the vibrational energy.

Figure 7. Some representative POs of the C−N−K linear isomer corresponding to chosen composite Poincaré surfaces of sections in Figures 4 and 5. The PO notation nR:nθ s/u indicating the corresponding resonance order and its stable/unstable character is given in each panel. G


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present certainly deserve further investigation, and this will be the subject of a future publication.

Subsection 3.1, we see that they coincide with the points at which the phase space structure: (i) starts to create chaos by destroying the outermost resonances around the central fixed point corresponding to C−N−K linear motion, (ii) starts to create chaos by destruction of the conspicuous four islands structure existing at these energies, (iii) starts to create chaos inside the regular region by a pitchfork bifurcation, and (iv) the islands of regularity born in that bifurcation get isolated. The values of the regular area (or angular action) Jθ in Figure 6 are seen to be larger than ℏ2 in most of the energy range considered. This indicates that several eigenstates with a regular nodal structure should be located in the region of regularity studied classically in this paper. This interesting issue was studied in detail in our previous work, 24 where the corresponding eigenstates localized over the C−N−K were described. 3.4. Relevant Periodic Orbits around the C−N−K Linear Configuration. To conclude our study of the regular motion on the phase space region corresponding to the K−C− N linear isomer, we present in Figure 7 some characteristic POs. For simplicity, only POs that are symmetrical with respect to the θ = π plane are considered. From them, some are retracing with two ending turning points, and others correspond to (more or less complicated) closed loops. Notice that the latter are really two different POs, depending if they are traced in the clockwise or anticlockwise sense. Although they appear as different in the composite PSSs, they are however identical in the configuration plot of the trajectory. We indicated that fact in the figure by using a “+” notation in the corresponding panels. In Figure 7 we show (from top left to bottom right): (i) the stable and unstable 1:8 POs at E = 2100 cm−1, which are at the border of the regular region at this energy, (ii) the stable and unstable 1:6 POs at E = 2200 cm−1, which are inside that region at this energy, (iii) the stable and unstable 1:4 PO at E = 2900 cm−1, whose chain of islands structure really shapes the regular region at this energy, (iv) the stable and unstable 1:3 closed loop and open POs at E = 3800 cm−1, which are in the border of the regular region at this energy, (v) the stable and unstable 1:8 POs at E = 4200 cm−1 also at the border of the regular region, and finally (vi) the stable 1:2 and unstable 0:1 POs born in the pitchfork bifurcation. With this figure we have a complementary, yet more complete, view of the structure and extension of the region of regularity on the C−N−K linear configuration that has been described in this Section of the paper. Moreover, it provides some useful information regarding the possible origin of the regular molecular vibrational motion taking place on top of a saddle point of the PES, which is described in the present paper. For one thing the amplitude of this regular motion is seen to occur in an extremely narrow range of angular values, 0.9π < θ < 1.1π rad. In this range the potential energy function can be assumed constant, and the equipotentials limiting the motion in R can be assumed as straight and parallel lines. As a result, the dynamics should be similar to that of a particle in a rectangular stadium, and more specifically to the so-called bouncing-ball trajectories, which are known to be marginally stable.12 However, this is only part of the explanation, since as we see in the PO trajectories of Figure 7 a strong dynamical coupling is present, making the orbits to strongly bend backward/forward in the angular coordinate θ when they bounce, respectively, in the outer/inner equipotentials curves. This, and other, dynamical effects that are not clear to us at

4. SUMMARY AND CONCLUSIONS The vibrational dynamics of KCN are a very interesting research topic. First, the C−N frequency is quite high, and then easily decouples from the rest of vibrational modes in the molecule. Accordingly, a 2D model, freezing the C−N stretching at its equilibrium distance, is quite adequate for its study. This enables the use of an ample battery of nonlinear dynamics tools12 to perform the corresponding research, both at classical and quantum level.40,41 In this sense, KCN is very similar to other floppy molecules, such as LiCN, HCN, or HO2. Second, the KCN dynamics are also interesting, since classical chaos sets in at very low values of the excitation energy (even below the vibrational ground state). This early onset of chaos makes this system ideally suited for quantum chaos and classical-quantum correspondence studies. Third, we recently reported24 an accurate analytical potential energy surface, based on high-level quantum chemistry calculations, for this system, which allows a realistic description of this system. In this paper, we have studied in detail a very interesting feature of the classical vibrational dynamics of KCN, namely, the existence of a fairly big region of regular motion in the vicinity of the K−N−C linear configuration in the energy range of E = 1300−5500 cm−1. The existence of this region is quite surprising, since it is on the top of a saddle point of the KCN PES. This should give rise to exponential separation of trajectories, and then to chaos, as it happens above and below the previously mentioned range of regularity. By finely sweeping the different values of the excitation energy, we have followed the appearance of order and destruction of regular tori in the regularity energy range, and we have also studied the evolution of the area spanned in the PSS of the regular region. In particular, we have described the structure of the corresponding phase space, identifying the different resonances between the stretching R and angular θ coordinates, which are more prominent at the different values of E. We have also detected and calculated the trajectories of the corresponding stable and unstable POs. By examination of the topology of these trajectories we have been able to sketch a first approximation to the origin of the regularity studied in this paper. This explanation still needs to be put in more firm and formal grounds, by using, for example, the techniques used in refs 33 and 34 in which a similar dynamical situation was considered.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: *E-mail: *E-mail: *E-mail:

[email protected]. (H.P.) [email protected]. (F.J.A.) [email protected]. (R.M.B.) [email protected]. (F.B.)

ORCID

F. Borondo: 0000-0003-3094-8911 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research leading these results has received funding from the Ministerio de Economiá y Competitividad (MINECO) under Contract No. MTM2015-63914-P, from ICMAT Severo H


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Ochoa under Contract No. SEV-2015-0554, and from the European Union’s Horizon 2020 research and innovation programme under Grant No. 734557.

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