Ultrafast inter-ionic charge transfer of transition-metal complexes mapped by femtosecond X-ray powder diffraction

Spin-crossover in a transition metal compound is a transition between electronic states with different numbers of unpaired electrons.¹ Most often, this is a transition between the low-spin (LS) singlet and high-spin (HS) quintet states of an octahedral metal complex with a d⁴–d⁷ electron count. In practice, six-coordinated Fe(II) complexes of nitrogen-donor ligands are most often used in spin-crossover research, since these exhibit the greatest structural differences between their HS and LS states. While spin transition from LS to HS states was first discovered to be induced by pressure and/or temperature, it was later found that it could also be induced under pulsed laser excitation of Fe(II) and Fe(III) complexes in solution.² This was followed by the observation that at cryogenic temperatures, the HS to LS relaxation slows down to such an extent that under visible light irradiation, Fe(II) complexes can be quantitatively trapped in the excited HS state, in what became known as the “Light-Induced Excited State Spin Trapping” (LIESST).³ This discovery led to an intense research activity,^4–6 in part due to the fact that being observed in crystalline molecular solids, LIESST opened a dynamic approach to the study of cooperative effects and bistability.

The dynamics of the light-induced spin transition as well as the lifetimes and relaxation pathways of the different electronic states have been addressed by ultrafast spectroscopy on transition metal complexes dissolved at low concentration in liquid solvents. Fe(II) complexes at room temperature are characterized by a manifold of singlet metal-to-ligand-charge-transfer (¹MLCT) states, which absorb in the visible range, lower lying and optically silent ³MLCT states, and so-called metal-centered (MC, also called ligand field) states, of which the high spin quintet (⁵T) state is the lowest excited state. In order to determine the electronic relaxation channels leading to the population of the ⁵T state, ultrafast experiments were implemented by McCusker and co-workers^7–10 and by Chergui and co-workers,^11–15 who combined ultrafast optical and X-ray absorption spectroscopy. Although both groups investigated somewhat different Fe(II) complexes, the overall dynamics seem quite identical. In particular, the study of the [Fe(bpy)₃]^{2 +} complex in solution^11–15 showed that upon excitation of the ¹MLCT state, an ultrafast intersystem crossing occurs to the ³MLCT state in 30 fs, which then decays directly into the ⁵T state within 150 fs, leaving it vibrationally excited. Thus, contrary to previous claims, the relaxation cascade avoids the ^{1, 3}MC states. Time-resolved X-ray absorption spectroscopy gave insight into structural changes in the excited ⁵T state and showed that the equilibrium length of the Fe–N bond increases by 20 pm with respect to the ground state, in good agreement with studies carried out using quasi-static X-ray absorption spectroscopy or X-ray diffraction on complexes having much longer lived quintet states.¹⁶ The ⁵T state of [Fe(bpy)₃]^{2 +} complexes in solution decays with a characteristic lifetime of the order of 650 ps, much longer than its formation time.

In such liquid phase studies, interactions between different transition metal complexes and/or between a complex and its counterion can safely be neglected because of the high dilution in the solvent. Most potential applications of spin crossover materials require, however, solid state materials^17,18 in which the complexes and their counterions are densely packed and, thus, strongly coupled via long-range Coulomb forces. Moreover, a potential cooperative response in spin-crossover solids deserves investigation, especially at ultrashort time scales. Collet and co-workers have studied the LS to HS photoswitching transformation process in a spin crossover solid using X-ray diffraction with a 100 ps time resolution.^19–23 They reported structural reorganizations, associated with the different steps of the spin crossover process, namely, the local molecular photoswitching with structural reorganization at constant volume, the volume relaxation with an inhomogeneous distribution of local temperatures, and at very long times (tens of ms) a homogenization of the crystal due to heating.

A real-time observation of the initial photoinduced structure changes necessitates a higher time resolution, as suggested by the solution phase studies. A key issue in this process is the spatial rearrangement of electronic charge within an electronically excited crystal and the role of the ionic environment consisting of unexcited complexes and counterions. This requires a measurement and analysis of transient electron density maps on ultrafast time scales. Recently, we have shown that femtosecond X-ray powder diffraction provides access to transient electron density maps with a sub-100 pm spatial and a 100 fs temporal resolution.^24–26 Here, we apply this method to determine charge relocations in crystalline [Fe(bpy)₃]^{2 +}(⁠

$PF 6−$⁠)₂. Our results provide a detailed picture of the transient electronic structure and reveal for the first time the delocalized character of the CT and a significant CT contribution of the counterions.

Iron(II)-tris(bipyridine)-bis(hexafluorophosphate) ([Fe(bpy)₃]^{2 +}(⁠

$PF 6−$⁠)₂) samples in powdered form were used in the experiment. The material has a centrosymmetric crystal structure and belongs to space group P

$\bar{3}$

$3¯$c1 (No. 165²⁷) with two formula units per unit cell (Figure 1). The hexagonal unit cell has dimensions a = 1.061 nm and c = 1.648 nm. The samples, about 100 μm thick, were held between two 20 μm-thin diamond windows. The linear absorption spectrum of the powder sample is shown in Figure 2 and agrees with the solution-phase spectrum.²⁸ 

The experiment was performed in an optical pump–X-ray probe scheme. An amplified Ti:sapphire laser system provided 40 fs pulses of 5 mJ energy at a 1 kHz repetition rate. A beamsplitter divided the beam to derive pump and probe pulses. On the probe arm, the major portion of the energy (4.75 mJ) was used to produce hard X-ray pulses (Cu Kα, E = 8.04 keV or λ = 0.154 nm, 100 fs pulse duration) from a laser-plasma source.²⁹ On the pump arm, the 800 nm optical pump was focused on the sample at an intensity of

$8×1011W/ cm 2$⁠. The ^1,3MLCT states and/or the underlying MC state³⁰ were populated via two-photon absorption (photon energy 1.55 eV) to ensure a sufficient penetration depth of the pump light into the powder sample.¹⁵ It has been shown for the similar [Ru(bpy)₃]^{2 +} complex that two-photon excitation populates the MLCT states.³¹ Because the latter are the doorway states to the ⁵T state in [Fe(bpy)₃]^{2 +}, it is fair to assume that the two photon excitation will lead to population of the HS state.

Photoinduced structural changes were mapped by diffracting the synchronized hard X-ray pulse from the powdered sample. The resulting ring-like diffraction patterns (Debye-Scherrer rings) were recorded with a large-area X-ray detector. To limit the photon counting noise and detect small pump-induced changes of diffracted X-ray intensity, integration times of the order of 8 h were used in individual measurements at particular pump-probe delays. The data set presented in the following was recorded with a total integration time of 230 h. The time-delay zero between pump and probe was measured by optical means at the beginning and the end of each measurement to allow for averaging data taken on different days.

The intensity of a particular Debye-Scherrer ring diffracted from a set of lattice planes hkl is given by I_hkl = M_hkl · LP_hkl · |F_hkl|², where M_hkl, LP_hkl, and F_hkl are the multiplicity, the Lorentz-polarization factor and the structure factor, respectively. M_hkl and LP_hkl can be obtained from textbooks.³² The change of the diffracted intensity ΔI_hkl(t) measured in the pump-probe experiment is connected to the corresponding structure factor F_hkl in the following way:²⁴ 

where

$Fhkl0$ and

$F_{hkl}^{ex}(t)$

$Fhklex(t)$ is the structure factor of the unit cell in its ground and excited state, respectively, and η is the fraction of modified unit cells. Since the crystal structure possesses inversion symmetry, the phases of all structure factors are ϕ_hkl = 0 or ϕ_hkl = π. In our experiments, only a small fraction of unit cells is modified and, thus, terms quadratic in η can be neglected, resulting in the following expression for the change of the structure factor:

Thus, Eq. (2) directly links the measured quantity ΔI_hkl(t) to the changes of the structure factors

$ΔFhklex(t)$⁠. The change of the structure factors for different sets of lattice planes (hkl) allows for determining the change

$\Delta \rho (\textbf {r},t)$

$Δρ(r,t)$ of electronic charge density via the Fourier series

Here a*, b*, and c* are the reciprocal lattice vectors of the lattice vectors defined in Figure 1. In this way, transient electron density maps are derived from the time-resolved diffraction data. In the experiment, Debye-Scherrer rings up to 2θ_max ≈ 50° are recorded. This results in a limited spatial resolution and, because of the abrupt end of the Fourier series, artifacts may occur such as the so-called Gibbs phenomenon (the appearance of ring-like structures around the atoms) in the charge-density distribution. Additionally, some structure factors cannot be measured individually since their corresponding rings overlap. These problems are addressed with the maximum entropy method. Details of this analysis are given in the supplementary material.⁴⁴ 

The inset of Figure 3(a) shows the stationary Debye-Scherrer pattern of the powder sample recorded with a total integration time of 8 h of the large-area detector. Integration over the individual diffraction rings gives the lineout of the reflections as a function of 2θ, covering an angular interval from approximately 10° to 50°. The different peaks in Figure 3(a) were assigned to their sets of lattice planes by comparing the powder pattern to literature data.³³ Upon photoexcitation, the intensities of the peaks change by up to 3 × 10⁻² while the angular positions are unchanged within the experimental accuracy of 0.1°. In Figure 3(b), we present an overview of the intensity changes measured on all reflections up to 2θ = 43°. In Figures 3(c)–3(f), the intensity changes of four selected reflections are shown as a function of pump-probe delay. A common feature in their temporal evolution is a step-like intensity change at zero time-delay.

From the diffraction patterns for each delay time we derived the three-dimensional differential electron density maps ηΔρ(r, t). In Figure 4 we show two-dimensional projections

of the 3D electron density around (a) the

$PF 6−$ anion and (b) the Fe^{2 +} and one bipyridine unit for a delay time of t = 250 fs. Figure 4(a) displays the projection of the electron density distribution of the

${\rm PF}_6^{\ -}$

$PF 6−$ ions on the xy-plane, i.e.,

$\hat{e}_u=\hat{e}_x$

$êu=êx$⁠,

$\hat{e}_v=\hat{e}_y$

$êv=êy$⁠, and

$\hat{e}_w=\hat{e}_z$

$êw=êz$ while Figure 4(b) shows a projection on a plane containing the Fe atom and one of the bipyridine units using an appropriate triple of unit vectors

$\hat{e}_u$

$êu$⁠,

$\hat{e}_v$

$êv$⁠, and

$\hat{e}_w$

$êw$ which is different from that used in Figure 4(a). The dashed lines in the two panels represent the contours of the static electron density convoluted with the spatial resolution of the experiment. A strong decrease of the electron density (blue areas) is observed on the

${\rm PF}_6^{\ -}$

$PF 6−$ anions and on the Fe atom whereas a significant increase of electron density occurs on the bipyridines [red part in Figure 4(b)]. Moreover, the arrangement of the charge changes (decrease on the inner and increase on the outer region of the map in the lower panel) on the bipyridines clearly points to an increase of the Fe-ligand bond length (see below).

For a quantitative estimate of the magnitude of the charge changes, the unit cell was divided in sub-volumes, each containing either the Fe atom, the bipyridines or the counterion,²⁵ cf. the supplementary material.⁴⁴ The total amount of charge contained in the boxes and the center of gravity of the charge were then calculated from electron density maps recorded at different delay times. The results displayed in Figure 5 reveal the strongest decrease of electron density on the

$PF 6−$ anion and a charge decrease on the Fe atom which is about a factor of three smaller. The charge is mainly transferred to the bipyridine units where an increase of electron density occurs. We point out that in half a unit cell there are one Fe atom, two

${\rm PF}_6^{\ -}$

$PF 6−$ anions, and three bipyridines, so that the sum of all negative and positive changes of electron density is equal to zero within the experimental accuracy. All transients in Figure 5 show a step-like time evolution with a rise around delay zero.

The time evolution of the Fe–N, Fe-pyridine ring and ring-ring distances within a bipyridine unit as derived from the center of gravity of the electron density changes are plotted in Figure 6. There are pronounced changes in the Fe-Py and Py-Py distances ηΔR and a small increase of ηΔR = 0.12 pm for the Fe–N bond length.

In addition, we performed all-optical pump-probe measurements on a 300-nm thick poly-crystalline thin film of [Fe(bpy)₃]^{2 +}(⁠

$PF 6−$⁠)₂. In Figs. 5 and 6 we show in each panel a scaled version of the transient of a (400-nm) pump–(530 nm) probe experiment (blue lines). The amplitude of the measured transient bleaching was ΔT/T₀ ≈ 0.01 obtained for a similar excitation density as in the x-ray diffraction experiment. The main purpose of this measurement was to demonstrate the very similar temporal shape of the all-optical measurements (blue lines) with that of the charge and position changes shown in Figs. 5 and 6. From this we conclude that the electronic character of the long-lived electronic state being characterized in our femtosecond x-ray diffraction experiment is identical for 1-photon vs. 2-photon excitation of poly-crystalline [Fe(bpy)₃]^{2 +}(⁠

${\rm PF}_6^{\ -}$

$PF 6−$⁠)₂.

The transient electron density maps derived from the diffraction data reveal both pronounced changes of electron density on the different molecular units and changes of the molecular geometries and distances. It is important to note that the

$PF 6−$ counterions play a key role in the charge relocation scenario. This behavior is markedly different from that of diluted liquid solutions of [Fe(bpy)₃]^{2 +}(⁠

${\rm PF}_6^{\ -}$

$PF 6−$⁠)₂ where the spatial separation of the separately solvated anions and cations is much larger.

The changes of electron density and geometry are initiated by two-photon excitation of a small fraction of transition-metal complexes, showing the same linear absorption spectrum as in solution (cf. Figure 2). Two-photon absorption populates the manifold of ^{1, 3}MLCT states in which charge is transferred to the bipyridine ligands. This initial process occurs within the time resolution of our experiment and, thus, a quasi-instantaneous step-like onset of the electron density and geometry changes is observed (cf. Figures 5 and 6). The character of the resulting charge relocation is visualized with the help of Figure 7 where the one-dimensional projection of the electron density ρ_LP and its change ηΔρ_LP are plotted along an axis linking the central Fe atom and one of the

$PF 6−$ counterions with one of the bipyridine units in-between. The electron density maps show a decrease of electron density on the Fe and

${\rm PF}_6^{\ -}$

$PF 6−$ and a corresponding increase on the bipyridine. In parallel, the distance between the Fe and the bipyridine units increases, leading to the dispersive shape of Δρ on the bipyridine which is evident from the electron density map in Figure 4(b). In the transient charge distribution, the differences in electron density and, thus, the local electric polarization between the different structural units are reduced compared to the electronic ground state. We recall that the total charge change in a [Fe(bpy)₃]^{2 +} and its 2

${\rm PF}_6^{\ -}$

$PF 6−$ counterions is zero within the experimental accuracy.

The predominant interaction between the ionic constituents of the crystal arises from long-range Coulomb forces. As a result, photoinduced changes of electron density affect not just the particular excited [Fe(bpy)₃]^{2 +} complex and its neighboring counterions but a larger range of the crystal structure. A schematic comparison of a local and a delocalized change of electron density is shown in Figures 8(a) and 8(b), respectively. To distinguish such two scenarios, we now analyze the observed changes of electron density and geometry in a more quantitative way. A key quantity in determining both the absolute values of Δρ_e(r, t) and distance changes ΔR is the fraction η of modified unit cells. For a delocalization of the structure changes over more than a single, i.e., the directly excited [Fe(bpy)₃]^{2 +} unit, η is expected to be substantially larger than the fraction ϕ of [Fe(bpy)₃]^{2 +} units that have been excited directly by the pump pulse. An upper limit of ϕ is estimated by assuming that the pump pulse is being completely two-photon absorbed in the powder sample and that the intensity of pump light scattered by the crystallites is negligibly small. From the present experimental parameters, one derives 0.004 < ϕ ⩽ 0.008.

We first consider the absolute values of Δρ_e(r, t) predicted under the assumption η = ϕ = 0.008 (cf. Figure 5). Here, one estimates a charge transfer Δρ_{e, max} = 0.26/0.008 = 32.5 e⁻ for the

$PF 6−$ counterion, a very high, clearly unphysical value. This demonstrates that the charge transfer behavior cannot be explained by simply taking into account a single [Fe(bpy)₃]^{2 +} unit. Instead, possible interactions between neighboring [Fe(bpy)₃]^{2 +} and between the latter and the counterions in the environment have to be taken into account.

A model of Coulomb interactions in the ionic crystal was developed to qualitatively reproduce the measured results. Fe atoms, bipyridines and

$PF 6−$ ions were modeled as capacitors³⁴ having energies E(q) which are nonlinear in q, the electronic charge. This technique is well-known in chemistry and is related to the concept of electronegativity and chemical hardness (the chemical hardness is the reciprocal of the capacitance).^35–38 The values for E(q) were derived from literature values, the calculated or measured ionization energies and electron affinities. Here, the values for bipyridine and

${\rm PF}_6^{\ -}$

$PF 6−$ display a substantial uncertainty. The model was completed by adding the Madelung energy to describe the electrostatic interactions between the different ions in the crystal. The result of simulations without and with the Madelung energy for a supercell containing many unit cells is shown in Figures 8(c) and 8(d), respectively. The contour plots show the energy per [Fe(bpy)₃]^{2 +}(⁠

${\rm PF}_6^{\ -}$

$PF 6−$⁠)₂ unit as a function of both the charge of the

${\rm PF}_6^{\ -}$

$PF 6−$ ions (x-axis) and that of the bipyridine units (y-axis). Due to neutrality the charge on the Fe atoms is

$Q_{\rm Fe} = -3 Q_{\rm bpy}- 2 Q_{\rm PF_6}$

$Q Fe =−3Q bpy −2Q PF 6$⁠. In general, the many-body system tends to minimize the total energy by exchanging charge between the different groups. The calculation without the Madelung energy [Figure 8(c)] corresponds to the situation in which the cations and anions are well separated in real space leading to ionic charges, i.e., Fe^{2 +} and

${\rm PF}_6^{\ -}$

$PF 6−$⁠. The situation changes drastically when we go to the crystalline environment as shown in Figure 8(d). Here, the energy per [Fe(bpy)₃]^{2 +}(⁠

${\rm PF}_6^{\ -}$

$PF 6−$⁠)₂ unit shows a double minimum in the two-dimensional space of bipyridine and

${\rm PF}_6^{\ -}$

$PF 6−$ charges. Electrostatic equilibrium is found at the somewhat deeper minimum at

$Q_{\rm PF_6}\approx -1.5 e_0$

$Q PF 6≈−1.5e0$ and Q_bpy ≈ +0.4e₀. This situation is, however, extremely sensitive to small perturbations. Our calculations show that lowering the chemical hardness of just one Fe^{2 +} ion within the entire super cell consisting of many unit cells shifts the electrostatic equilibrium towards the second minimum at

$Q_{\rm PF_6}\approx -0.5 e_0$

$Q PF 6≈−0.5e0$ and Q_bpy ≈ −0.4e₀. Please note that the calculated ratio of the charge changes

$\Delta Q_{\rm PF_6}/\Delta Q_{\rm bpy}=1.25$

$ΔQ PF 6/ΔQ bpy =1.25$ is almost identical with that observed in our experiments shown in Figures 5 and 7(b). For instance, photo-excitation of an Fe^{2 +} ion reduces the ionization energy of its outermost valence electron which in turn weakens its bond to the ion core which is nothing else than an increase of the capacitance of that Fe atom. As a result, the photo-excited electron abandons the Fe atom and the system compensates for the missing charge on the Fe^{3 +} ion by relocating the charge on the neighboring units. The charge rearrangement takes place not only on the [Fe(bpy)₃]^{2 +} containing the modified Fe atom but in a much larger surrounding volume. This finding points to a highly delocalized character of charge transfer, in line with the cartoon in Figure 8(b).

The concept of electronic cloud deformation or electronic polaron (self-trapped charge-transfer exciton) has been a subject of research for a long time.^26,39–42 The electronic polaron is a quasiparticle composed of an electron together with the associated polarization induced in the electronic clouds of the surrounding units or atoms. In our case, the strong electric field produced by the original CT polarizes the neighboring units inducing a CT also on them. Additional evidence for the cooperative character of CT comes from the following estimate of η. If we allow that only one electron is involved in the

$PF 6−$ CT, then we obtain η = 0.25 to be compared with the estimated fraction of directly excited [Fe(bpy)₃]^{2 +} of ϕ = 0.008. The large difference between these two numbers is due to to the Coulomb-mediated many-body effects, involving on the average a number of η/ϕ ≈ 30 units. Considering that

$3\sqrt {30} \approx 3$

$330≈3$⁠, the radius of the electronic polaron is about 3 [Fe(bpy)₃]^{2 +}(⁠

${\rm PF}_6^{\ -}$

$PF 6−$⁠)₂ units.

The many-body character of CT implies that the transient changes of distance between the different groups (cf. Figure 6) represent averages over all modified unit cells. One may assume that distance changes are most pronounced in the directly excited [Fe(bpy)₃]^{2 +} complex and become smaller with increasing spatial separation from it. An upper limit for the geometry change in the directly excited [Fe(bpy)₃]^{2 +} can be estimated by assuming η = ϕ corresponding to a CT fully localized on this unit. Such estimate gives a maximum increase of the F–N bond length [Figure 6(a)] of ΔR = 0.12/0.008 = 15 pm, in the same order of magnitude as the value of 20 pm reported for the ⁵T state of [Fe(bpy)₃]^{2 +} in solution.¹⁵ Under the same assumption, the data in Figure 6(b) suggest a seven times larger maximum change in distance between Fe and bipyridine which again is unphysical and shows that the picture of a local geometry change is inappropriate in the crystalline phase. Thus, the analysis of both the electron density and the geometry changes underlines the many-body character of CT in the crystals studied here. While the precise character of the electronic polaron state cannot be determined with the present data at hand, this state bears some similarity with the ⁵T state of the [Fe(bpy)₃]^{2 +} in solution: the charge transfer between Fe and bipyridine has the same direction and the Fe-bpy distance increases. However, a basic new aspect in the crystal consists in the pronounced contribution of the

$PF 6−$ counterions to the overall CT.

Finally, our results may explain the issue of the high efficiency of the ^{1, 3}MLCT-⁵T spin conversion process observed in the solution phase studies.^7–15 The crossing of the ⁵T₂ state and the MLCT states in the Franck-Condon region is just as favourable as the crossing of the intermediate metal-centred ³T₂ and ¹T₁, which in addition have more favourable spins. Furthermore, de Graaf and Souza³⁰ estimated the spin-orbit coupling matrix elements between the ³MLCT and these states to be comparable as that with the ⁵T state. Thus, it is remarkable that the MLCT states should decay exclusively into the ⁵T₂ state. De Graaf and Souza³⁰ have suggested that the latter has more ionic character than the triplet and singlet MC states. The formal electronic configuration of the MLCT states is Fe 3d⁵L⁺¹(where L is the bipyridine ligand), whereas the more ionic MC states are Fe 3d⁶L. On the other hand, the triplet and singlet ligand-field states have important contributions from the Fe 3d⁷L⁻¹ electronic configurations. This could lead to a more effective decay of the ³MLCT into the more similar ⁵T₂ state (single electron transfer) than through the triplet or singlet MC states, which require a certain degree of double electron transfer. The observed partial charge transfer from the Fe to the bipyridine reported here (Figure 4) would be the first experimental confirmation of this hypothesis, which calls for detailed quantum-chemical calculations about the nature of the ^{1, 3, 5}MC states.

In conclusion, we recorded transient electron density maps of transition-metal complexes in the densely packed crystalline phase by means of time-resolved powder X-ray diffraction with femtosecond time and atomic spatial resolution. Upon femtosecond two-photon excitation, [Fe(bpy)₃]^{2 +}(⁠

$PF 6−$⁠)₂ complexes display a transfer of electron density from both the Fe atoms and the

${\rm PF}_6^{\ -}$

$PF 6−$ counterions to the bipyridine units of the structure, accompanied by changes of the spatial separation between the different units. An analysis of the transient electron density maps demonstrates the many-body character of charge transfer, i.e., a delocalization of the changes in electron density over approximately 30 [Fe(bpy)₃]^{2 +} complexes around the directly excited one. This many-body behavior arises from the strong Coulomb interaction within the crystal and can be visualized by the concept of an electronic polaron. While cooperative effects in other transition-metal complexes were attributed to entropic origins, to internal pressure, elastic interactions⁵ or to strong intermolecular interaction,⁴³ our results underline the important role played by weakly shielded, long-range Coulombic forces. Furthermore, the weak charge transfer character of the HS state reported here may explain the high efficiency of the MLCT-⁵T transition in the liquid phase.^13–15 

We thank A. Hauser (Geneva) for providing us with the powdered samples, K. Reimann for the Madelung energy calculations, and R. M. van der Veen for the help in the experiments. This research has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 247051 (T.E.) and from the Deutsche Forschungsgemeinschaft (DFG) (Grant No. WO 558/13-1) (M.W.). M.C. is grateful to the Alexander von Humboldt Foundation for support.