Abstract
All attosecond timeresolved measurements have so far relied on the use of intense nearinfrared laser pulses. In particular, attosecond streaking, laserinduced electron diffraction and highharmonic generation all make use of nonperturbative light–matter interactions. Remarkably, the effect of the strong laser field on the studied sample has often been neglected in previous studies. Here we use highharmonic spectroscopy to measure laserinduced modifications of the electronic structure of molecules. We study highharmonic spectra of spatially oriented CH_{3}F and CH_{3}Br as generic examples of polar polyatomic molecules. We accurately measure intensity ratios of even and oddharmonic orders, and of the emission from aligned and unaligned molecules. We show that these robust observables reveal a substantial modification of the molecular electronic structure by the external laser field. Our insights offer new challenges and opportunities for a range of emerging strongfield attosecond spectroscopies.
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Introduction
Attosecond and fewcycle laser pulses offer new opportunities for studying the electronic structure and dynamics of molecules (see, for example, refs 1, 2, 3, 4, 5, 6, 7). Highharmonic spectroscopy (HHS), on which we focus here, relies on the laserdriven recombination of an electron with its parent cation that characterizes both the electronic structure and dynamics of the molecule. The photorecombination step accesses the richness of electronicstructure information previously studied over decades by photoelectron and photoionization spectroscopy with the important additional benefit of temporal resolution.
An essential difference between highharmonic and photoionization spectroscopies is the presence of a strong laser field in the case of HHS. This field could thus be expected to dynamically modify the electronic structure of the target. However, no such effects have been reported to date and therefore HHS has been extensively used for obtaining fieldfree electronicstructure information^{8,9}. This working hypothesis was justified by solving the timedependent Schrödinger equation for atoms exposed to intense laser pulses and comparing the results with photorecombination crosssections of the parent ion^{8,10}. Examples of electronicstructure effects studied by HHS include Cooper minima^{9,11,12,13}, giant resonances^{14} and shape resonances^{15,16}. Basic quantummechanical considerations reveal that the similarity between photoionization and highharmonic spectroscopies cannot extend to large molecules because of their spatial extension and high polarizabilities.
In this article, we show how a strong laser field modifies the electronic structure of molecules and how to measure it. We chose methyl fluoride (CH_{3}F) and methyl bromide (CH_{3}Br) as examples because their large permanent dipole moments result in a strong, firstorder coupling to the laser electric field. The laser field is found to lift the degeneracy of the electronic ground state of the molecular cation (of Esymmetry in the C_{3v} point group) and to reshape the structure of its two wavefunction components. This laserinduced modification of the electronic structure of the molecules is verified experimentally by accurately measuring highharmonic spectra of aligned and oriented molecules. We determine intensity ratios between even and odd harmonics and between different alignment distributions. These ratios are found to be robust with respect to intensity and wavelength of the driving laser pulses. Our experimental observations are shown to strongly contrast with the prediction of the stateoftheart theory of HHS that relies on fieldfree models of photorecombination^{8,9,17}. We therefore develop a theory that consistently includes the laser field in all three steps of HHS: ionization, propagation and recombination. The model reproduces the experimental observations with quantitative accuracy. Our study reveals new opportunities for HHS to precisely characterize the electronic structure of molecules in intense laser fields. Finally, we also show that the observables of our experiment distinguish between different sets of linear combinations of a degenerate electronic state, which is impossible in the absence of an external field.
Results
Experimental results
We use impulsive orientation and/or alignment to characterize the electronic structure of the molecules. By varying the time delay between the nonresonant orientation and highharmonicgeneration pulses, we probe the orientationdependent highharmonic yield in the time domain. Since the optimal laser parameters for orientation and alignment differ slightly, we report two sets of measurements. In the first experiment, we impulsively align the molecules using temporally stretched (150 fs) onecolour (800 nm) laser pulses at nonionizing intensities. In the second experiment, we impulsively orient the molecules using shorter (80 fs) twocolour (800+400 nm) laser pulses. The molecules CH_{3}F or CH_{3}Br are diluted to 10% in a supersonic expansion in helium. Further details are given in the Methods section. In both experiments, we measure highorder harmonic spectra around the first full rotational revival of the molecules using laser pulses centred at 800, 1,275 or 1,330 nm and multiple intensities. The probelaser parameters are identical in the alignment and orientation experiments.
Figure 1 shows the results of the impulsive alignment experiment. The variations of the highharmonic signal generated by an 800nm probe pulse in the vicinity of the first full rotational revival are shown in Fig. 1a–d. Low and highharmonic orders display opposite modulations of the emitted intensity. The molecules CH_{3}F and CH_{3}Br also display opposite modulations with respect to each other. These results can be related to the orientation dependence of highharmonic emission by comparing them to the degree of axis alignment (〈cos^{2}θ〉), shown in Fig. 1e,f calculated for our experimental conditions^{18}. The degree of alignment first goes through a minimum when the C–X (X=F, Br) axis is aligned perpendicular to the laser field (antialignment), then through a maximum when the C–X axis lies parallel to the laser field (alignment). Hence, the loworder harmonics of CH_{3}F (for example, ninth harmonic (H9)) and the highorder harmonics of CH_{3}Br (for example, H17) are most intense when the C–X axis is close to perpendicular to the laser field. The highorder harmonics of CH_{3}F (for example, H17) and the loworder harmonics of CH_{3}Br (for example, H9) are most intense when the C–X axis is close to parallel to the laser field.
This harmonic signal modulation is further quantified by the intensity ratio of aligned to isotropically distributed molecules, the ‘alignedtoisotropic ratio’. This quantity, measured in multiple experiments using different driving laser intensities and central wavelengths, is displayed in Fig. 1g,h for CH_{3}F and CH_{3}Br, respectively. The ratios are found to cross unity between 16 and 18 eV in the case of CH_{3}F and between 18 and 21 eV in the case of CH_{3}Br. The reversal of the modulation is found to be independent of both the wavelength and the intensity of the probe pulse within the accuracy of the measurements. This excludes interference between the emission involving multiple electronic states of the cation as the origin of the observed revival inversion^{19,20}.
Figure 2a shows the highharmonic spectrum of oriented CH_{3}F molecules recorded at the delay of maximal orientation (19.5 ps) between the twocolour pump and the 800nm probe pulse. Besides the appearance of the commonly observed oddharmonic orders, even harmonics are also emitted, demonstrating the orientation of the molecular sample^{21}. Remarkably, the intensities of the evenharmonic orders follow a different envelope than those of the oddharmonic orders. This observation is further quantified by the intensity ratio of the evenharmonic orders (2 × I_{2n}) to the average of their two adjacent oddharmonic orders (I_{2n−1}+I_{2n+1}), which is shown in Fig. 2b (the ‘eventoodd ratio’). This ratio displays a sharp increase at low orders until H14 (22 eV) where a local maximum is observed. No evenharmonic emission was observed from CH_{3}Br under the same experimental conditions. Independent Coulombexplosion experiments showed that CH_{3}Br orients very weakly under the action of a femtosecond twocolour laser field (Ram et al., (manuscript in preparation)).
Theoretical results
In the following, we introduce a theory that accounts for the influence of the laser field on the electronic structure of the target molecule in all three steps of highharmonic generation. We then use this theory to demonstrate the substantial manifestations of laserinduced electronicstructure effects in highharmonic spectra. Our theoretical work builds on previous results that have hinted at the role of the laser field in strongfield spectroscopies. Laserinduced mixing of multiple electronic states was predicted to modify the angle dependence of strongfield ionization (SFI)^{22}. Different Stark shifts of the neutral and ionic species substantially modify the angledependent SFI rates of polar molecules^{23,24,25}. The same effect has also been predicted^{26} and recently observed^{16} to modify the phase of the emitted highharmonic signal. Finally, field distortions of the electronic states entering the photorecombination dipole moments were predicted to modify the emitted harmonic spectra^{27,28}. However, to the best of our knowledge, no direct experimental evidence of this effect has been reported yet.
The electronic ground state of CH_{3}F^{+} is well described by a onehole configuration with an electron removed from the highest occupied molecular orbital (HOMO) of Esymmetry. Consequently, we will use the term ‘orbitals’ to refer to the canonical Hartree–Fock orbitals of the neutral molecule instead of equivalently referring to the N−1 electron wave functions of the cation. Figure 3 shows the effect of an applied static laser field on the two components of the HOMO. The corresponding Dyson orbitals are virtually indistinguishable from the Hartree–Fock orbitals and are shown in the Supplementary Fig. 1.
The interaction of a molecule with an external electric field is given by the operator , where F is the electric field of amplitude F, β and γ are the Euler angles describing the rotation of the moleculefixed axis system relative to the laboratory frame^{29,30} as shown in Fig. 4b, and is a 2 × 2 matrix in the basis (φ^{a} and φ^{b}) of the two degenerate components of the fieldfree HOMO. The eigenvalues of this matrix are:
where U and V are coupling constants obtained from quantum chemical calculations with applied static electric fields (see Methods section). The corresponding eigenfunctions and are given by:
where S=sin(γ/2+π/4) and C=cos(γ/2+π/4). The Stark effect and the corresponding fieldinduced orbital mixing is illustrated in Fig. 4a, where the changes in the binding energies of and are displayed for β=π/2. Instead of the eigenfunctions that would look similar to Fig. 3b, we show isocontour representations of the eigenfunction amplitudes corresponding to particular values of γ. For each orientation of the molecules relative to the laser polarization, the laser field selects a particular γdependent linear combination of the fieldfree basis as new eigenfunctions and lifts the degeneracy (except for β=0 and π). Remarkably, the Stark shift of the binding energies depends only on β, while the mixing coefficients S and C depend only on γ. While this property is specific to the C_{3v} point group, the field effects described here are general for symmetrictop molecules and are easily extended to arbitrary asymmetrictop molecules.
We calculate the angular variation of the SFI rates of CH_{3}F and CH_{3}Br within the adiabatic theory^{31} because SFI of these molecules under the present experimental conditions is expected to be well described by the quasistatic approximation. More specifically, the weakfield asymptotic theory (WFAT) of tunnelling ionization introduced in ref. 29 and applied to linear^{30} and nonlinear^{32} molecules is generalised to degenerate orbitals of nonlinear molecules and to incorporate the linear Stark effect described in the preceding paragraph. The WFAT describes SFI of molecules as a multichannel eigenvalue problem in parabolic coordinates (ξ,η,φ). Within the WFAT the leadingorder term in the ionization rate, Γ=G_{00}(β,γ)^{2}W_{00}(F), factorizes into the structure factor G_{00}(β,γ) and the field factor W_{00}(F). The indices n_{ξ}=0 and m=0 of both quantities are the parabolic quantum numbers of the dominant ionization channel. The structure factor G_{00}(β,γ) depends on the molecular orbital, the associated Stark shift and the orientation of the molecule relative to the probe field. Its calculation is described in the following paragraph. We note that this factor also includes the sign of the asymptotic orbital wave function (colour coded in Fig. 4b), which accounts for the effect of the nodal structure of the orbitals on the orientationdependent phase of the continuum electron wave packet^{29,31}. This result of the WFAT for molecules was also found to be essential for modelling highharmonic spectra of polyatomic molecules^{33}. The field factor is given by
where , E_{0} is the fieldfree energy of the degenerate HOMOs.
An accurate calculation of the structure factors requires orbital wave functions with precise longrange tails that are very difficult to obtain in standard quantum chemistry methods. We therefore use specifically designed basis sets and variationally optimize the basisset exponents to minimize the Hartree–Fock energy of the HOMO^{34}. This procedure is applied to basis sets of five different quality levels ranging from unpolarized double to polarized quintuple zeta quality (pcn, n=0–4). Further details are given in the Methods section. The structure factor, , for each of the two components of the fieldmodified orbitals i=A, B is then calculated from the asymptotic value of the structure function,
Here is the zcomponent of the dipole of , where z_{i} is defined in Equation (1), and describes the rotation from the laboratory frame to the molecular frame. The extraction of the structure factor from the structure function proceeds as discussed in ref. 34 with additional details given in ref. 32: is determined at the maximal η where the results of pc(n−1) and pcn coincide. The squared structure factors of the fieldfree and fieldmodified orbitals are shown in the Supplementary Fig. 2.
Within our theory, SFI thus prepares the eigenstates of the instantaneous Hamiltonian while a possible coupling between the two onehole states occurs only in second order in the field and can be neglected. Consequently, the electron hole remains in the orbital in which it was created from the moment of ionization until the electron recombines. SFI is found to preferentially occur from the orbital with the largest dipole moment (), which corresponds to an electron density characterized by maxima in the F–C–H planes.
The contribution to the phase of the emitted radiation originating from the propagation step is evaluated from the quasiclassical action ^{35}. Since the laser field does not add a channelspecific contribution to the kinetic energy E_{kin}(t′′) of the continuum electron, we include the field as a firstorder correction to the I_{p}dependent action for each trajectory^{26}.
where the longpulse limit is used and F_{0} is the peak electricfield amplitude. t and t′ represent the harmonicorderspecific return and birth times, respectively. The calculation of the molecularframe photorecombination dipoles d_{rec}(Ω,β,γ) corresponding to the observed harmonic frequencies Ω is described in the Methods section. The effect of the laser field is taken into account by transforming the photorecombination dipole moments according to Equation (2).
This theory now allows us to calculate the induced dipole moment as product of the contributions from each step involved in the process of highharmonic generation. We coherently average the singlemolecule induced dipole moment, that is, the product of the ionization, propagation and recombination amplitudes, over the molecular axis distribution generated by the pump pulse and coherently sum over the emissions from and
Here a_{prop,i}(Ω,β) is the complex amplitude of the recombining continuum electron wave packet and A(β) is the calculated axis distribution of the molecules.
Discussion
We now apply this theory (‘full theory’ in what follows) to predict the experimental observables. Since we compare ratios, the spectral amplitude of the recombining photoelectron wave packet cancels and we use . Moreover, frequencydependent factors relating the induced dipole moment to the intensity^{36} play no role for obtaining ratios. We demonstrate the importance of laserinduced effects by contrasting the predictions of the full theory and the experiments to that of a ‘fieldfree’ theory that neglects the fieldinduced modification of the orbitals in both ionization and recombination and also neglects the Stark phase .
Figure 5a shows the measured intensity ratio of aligned to antialigned CH_{3}Br molecules, compared with the fieldfree and the full theory. Clearly, only the full theory agrees with the experimental observations, whereas the prediction of the fieldfree theory is qualitatively incorrect in the following way: the reversal of the alignmentinduced intensity modulation is observed at H11 (17 eV), whereas the fieldfree theory predicts the reversal at 24 eV. The fieldfree theory thus predicts an alignmentinduced modulation for H13 and H15 that is opposite to the experimental observation. The full theory, in contrast, correctly and quantitatively predicts the experimental observations.
The fieldfree theory further predicts an alignedtoantialigned ratio very close to one for H15–H21. Within the signaltonoise ratio of the experiment, this prediction would correspond to the absence of alignmentinduced intensity modulation in H15–H21. This prediction strongly contrasts with the experimental results as is illustrated in the inset of Fig. 5a. This inset further shows that the full theory is in quantitative agreement with the experiment, whereas the fieldfree theory is qualitatively incorrect because it essentially predicts the absence of alignmentinduced modulation.
Figure 5b shows the alignedtoantialigned ratio of CH_{3}F. In this case, the predictions of the full and fieldfree theories do not differ as much as in the case of CH_{3}Br. Importantly, the full theory is again in quantitative agreement with the experimental data. This validates our theory further by showing that it does not only quantitatively predict the observations relating to CH_{3}Br, but also those of CH_{3}F. The deviation of the fieldfree theory from the experimental data is clearest at H17–H21 (26.3–32.5 eV), where the line corresponding to the full theory lies on top of the experimental data points, whereas the fieldfree theory consistently underestimates the ratio. The agreement between fieldfree and full theories at H9–H13 can only be attributed to an accidental cancellation of laserinduced electronicstructure effects in these observables.
Figure 5c shows the eventoodd ratio of oriented CH_{3}F molecules compared with both theories. The fieldfree theory erroneously predicts an eventoodd ratio that is almost independent of the harmonic order and fails to reproduce the pronounced increase from H10 to H14. Moreover, the fieldfree theory strongly underestimates the absolute values of the eventoodd ratio. In this comparison, the degree of orientation was determined following the approach described in ref. 21 as a singleparameter fit of the full theory to the experimental data which yields 21.5%. Following adjustment of this single parameter, the full theory agrees very well with the measurements. In contrast, the fieldfree theory predicts an eventoodd ratio that is much smaller than the observed one. Bringing the fieldfree theory into remote agreement with the experiment requires a degree of orientation of 29%, much higher than in our calculations that predict 〈cos θ〉=0.15±0.01, that is, ∼20% orientation. The remaining deviation of the full theory from the experimental data at H10 and H12 is not entirely unexpected since they lie very close to the ionization threshold of CH_{3}F (13.04 eV), where recollisionbased theories do not necessarily yield accurate results.
Although Fig. 5 only shows results obtained with 800nm laser pulses, the theoretical results are independent of the intensity and wavelength of the pulses. This remarkable result follows from two properties: (i) the contribution of the linear Stark effect to the highharmonic phase is wavelength and intensity independent^{37} and (ii) the contributions from electronically excited states of the cation have been neglected. These two points are strongly supported by the experimentally observed agreement between intensity ratios measured using multiple central wavelengths and intensities (see Fig. 1g,h). Calculations including the first electronically excited state of the cation (that is, the next lowerlying orbital (HOMO1)) are shown in the Supplementary Fig. 3 and explained in the Supplementary Note 1. These calculations are found to only marginally deviate from the results presented in Fig. 5.
In conclusion, we have observed laserinduced electronicstructure effects in HHS. Generalizing fieldfree orientation to the nonlinear molecules CH_{3}F and CH_{3}Br, we have observed a reversal of the oddharmonic signal modulation as a function of the pump–probe delay and a local maximum in the eventoodd spectral ratio of CH_{3}F. All three characteristic features occur at photon energies that are independent of both the intensity and the wavelength of the driving laser field and are therefore not signatures of electronically excited states of the cation. The interpretation of these experimental results required the development of a theory, which explicitly included the modification of the bound electronic states of the molecule. The electric field was found to lift the degeneracy of the Esymmetry electronic ground state of the cation and to create a particular alignmentdependent linear combination of its spatial components. As a consequence of the modified binding energies and wave functions, different sets of linear combinations of a degenerate electronic state become experimentally distinguishable. The contribution of the Stark effect to the phase of highharmonic emission was also found to be essential in reproducing the experimental results. Although we demonstrated these effects on two symmetrictop molecules, the discussed phenomena are expected to be general in highharmonic and related strongfield spectroscopies, such as timeresolved studies of chemical dynamics^{38}. These effects are of fundamental importance to emerging techniques of attosecond science such as streaking, tunnelling and rescattering^{2,39,40,41,42,43}. They will moreover play a crucial role in extending these techniques to the general class of polar molecules.
Methods
Experimental setup
The experimental setup consists of an amplified femtosecond titanium:sapphire laser system and a vacuum chamber for generation and spectral characterization of highorder harmonic radiation. The output of the laser system (800 nm, 25 fs and 1 kHz) is either used directly to generate highharmonic radiation or to pump an opticalparametric amplifier (TOPASHE, Light Conversion) tuned to 1,275 or 1,330 nm that generates pulses of ∼40fs duration. The laser pulses are focused with a spherical (f=45 cm) mirror into a pulsed supersonic expansion of 10% CH_{3}F in He with 10–20 bar backing pressure. The molecules are either impulsively oriented by a twocolour laserpulse (800+400 nm, (3±1) × 10^{13} W cm^{−2} peak intensity, 80 fs) or impulsively aligned by a temporally stretched pulse (150 fs) with a lower, nonionizing intensity ((1±0.5) × 10^{13} W cm^{−2}). The alignment/orientation and probe beams are aligned parallel to each other on the spherical mirror with a vertical offset of ∼1 cm and intersect in the molecular beam. The high harmonics generated by the probe beam propagate into an XUV spectrometer consisting of a 120μmwide entrance slit, a concave aberrationcorrected grating (Shimadzu, 30002) and a microchannelplate detector backed with a phosphor screen. A chargecoupled device camera records the spectral images and transfers them to a computer for analysis. More detailed specifications of the setup and the conditions for generating twocolour laser pulses are given elsewhere^{21}.
Theoretical results
The theoretical results in Fig. 1e,f are obtained by solving the timedependent Schrödinger equation for a rigid symmetric top using the polarizabilities α_{}=17.92 a.u. and α_{}=16.29 a.u. (ref. 44) for CH_{3}F and α_{}=6.890 a.u. and α_{}=4.925 a.u. for CH_{3}Br from a B3LYP//augccpVQZ calculation. The molecularframe photorecombination matrix elements d_{rec,i}(Ω,β,γ) are calculated using the Schwinger variational method for the electronic continuum implemented in epolyscat^{45,46} with orbitals from a Hartree–Fock calculation using the QZVP basis set.
Calculation of molecular orbitals
The asymptotic behaviour of the structure function G_{00}^{i}(β,γ,η) associated with the HOMOs (i={A, B}; see Equation (4) of main text) at large η=r−z is the key quantity for the application of the WFAT^{29} for the tunnelling rate. We use a basisset quantumchemistry approach using the polarizationconsistent (pc) basis sets to obtain the HOMO. These basis sets have been designed and optimized for density functional theory, which have very similar basisset requirements as HF, and are available in five different quality levels from (unpolarized) double zeta to (polarized) pentuple zeta quality (pcn, n=0–4)^{47}. It has been shown that these basis sets are capable of reproducing gridbased numerical HF energies for diatomic systems for microHartree accuracy^{48}. All basis sets in the present work have been used in their uncontracted forms and variationally optimized with respect to all basisset exponents. Analytical gradients of the HF energy with respect to basis function exponents have been calculated with the DALTON programme^{49}. Basis exponent optimizations have been done using a pseudoNewton–Raphson method. Starting values were taken from the standard pcn basis sets^{47}. In ref. 34 it was shown that this procedure greatly improves the asymptotic behaviour of the wave function at the HF level of theory. All calculations used the experimental geometries of the neutral molecules.
Determination of Starkeffect coefficients
The Starkeffect coefficients U and V are obtained in HF//augccpVQZ calculations with applied static electric fields F=0–0.05 a.u. applied along the x, y or z axes defined in Fig. 3. A secondorder polynomial is fitted to the HOMO binding energies as a function of the applied field strengths. The U and V coefficients are taken as the average of the linear coefficients of these fits. This definition of U and V incorporates the effect of interelectron interaction, in agreement with the manyelectron theory^{50}. We obtain U=0.1606, a.u. and V=0.7568, a.u. in the case of CH_{3}F and U=0.0167, a.u. and V=−0.1301, a.u. in the case of CH_{3}Br.
Additional information
How to cite this article: Kraus, P.M. et al. Observation of laserinduced electronic structure in oriented polyatomic molecules. Nat. Commun. 6:7039 doi: 10.1038/ncomms8039 (2015).
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Acknowledgements
We gratefully acknowledge funding from the Swiss National Science Foundation (PP00P2_128274), ETH Zürich (ETH33 103), the Russian Foundation for Basic Research (Grant No. 140292110), the Ministry of Education and Science of Russia (State assignment No. 3.679.2014/K), the Danish Center for Scientific Computation, the VKR Centre of Excellence QUSCOPE and ERC Starting Grants (Project Nos. 277767TDMET and 307270ATTOSCOPE). We thank Paul Hockett for valuable discussions about the calculation of photorecombination matrix elements.
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P.M.K. and H.J.W. conceived and designed the experiments. O.I.T., T.M. and L.B.M. developed the theory and calculated the structure factors. P.M.K., D.B., A.R. and J.S. performed the experiments and analysed the data. P.M.K., A.R., J.S. and H.J.W. calculated the photorecombination matrix elements. F.J. calculated the molecular orbitals. C.Z.B. provided computer code for the calculations of rotational dynamics. P.M.K. and H.J.W. wrote the first draft of the manuscript that was finalised by all authors.
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Supplementary Figures 13, Supplementary Note 1 and Supplementary References (PDF 900 kb)
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Kraus, P., Tolstikhin, O., Baykusheva, D. et al. Observation of laserinduced electronic structure in oriented polyatomic molecules. Nat Commun 6, 7039 (2015). https://doi.org/10.1038/ncomms8039
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DOI: https://doi.org/10.1038/ncomms8039
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