X-ray Absorption Spectrum Simulations of Hexagonal Ice

Transcription

1 X-ray Absorption Spectrum Simulations of Hexagonal Ice Iurii Zhovtobriukh a, Patrick Norman b, Lars G. M. Pettersson a* a FYSIKUM, Stockholm University, AlbaNova University Center, SE Stockholm, Sweden. b Department of Theoretical Chemistry and Biology, School of Engineering Sciences in Chemistry, Biotechnology and Health, KTH Royal Institute of Technology, SE Stockholm, Sweden. * lgm@fysik.su.se; Tel: Abstract: We calibrate basis sets and performance of two theoretical approaches to compute X-ray absorption spectra (XAS) of condensed water by comparison to experiment on hexagonal ice Ih. We apply both the transition-potential half-core-hole approach and the complex polarization propagator using four different models of the crystal with increasing oxygen and proton disorder, but find poor agreement with experiment. We note that there are large variations in experimental spectra depending on detection mode and how the ice samples were prepared, which leads us to critically investigate what structures were actually prepared and measured in each case. This is done by using a Monte Carlo-based fitting technique which fits the spectra based on a library of precomputed spectra and assigns weights to contributions from different model structures. These are then used to generate O-O and O-H radial distribution functions and tetrahedrality parameter associated with each of the measured spectra. We find that all spectra are associated with sharp peaks at the oxygen positions in the perfect lattice, but with significant disorder around these positions. We suggest that presently available XAS spectra of hexagonal ice are not fully 1

2 representative of the perfect crystalline lattice, but contain varying amounts of defects and possibly contributions from low-density amorphous ice. 2

3 I. INTRODUCTION Water is the most common liquid on our planet and essential to life as we know it. It exhibits anomalous behavior in comparison with typical liquids in terms of, e.g., increasing density upon melting, density maximum at 4 C, high surface tension as well as anomalous compressibility, heat capacity, and thermal expansivity 1. The explanation of such complex behaviors is to be found in the microscopic water structure and the peculiarities of the local H-bond network 2-3. The traditional textbook picture of liquid water is with predominantly tetrahedral coordination with local distortions due to the thermal motion. According to this picture, each water molecule should have on average two donated and two accepted hydrogen-bonds (H-bonds) with distortions and occasional interstitial molecules enhancing the density. In recent years, X-ray spectroscopy and scattering measurements have contributed significant information on the local structure of the liquid 4-5, but many questions around the interpretation of the data remain 6. Based on X-ray absorption spectroscopy (XAS) and X-ray Raman scattering (XRS) combined with theoretical density-functional theory (DFT) spectrum calculations, Wernet et al. 7 proposed an alternative picture of ambient water where the majority of water molecules have asymmetric H-bond coordination with one strong donated and one strong accepted H-bond. Later, based on X-ray absorption and emission spectroscopy data coupled with results of X-ray scattering measurements 2-3,8, the picture of ambient liquid water was connected to theories of water in the deeply supercooled region where two structurally different species have been proposed to exist 9-10 : low density liquid (LDL) with tetrahedral local coordination and high-density liquid (HDL) with asymmetric local H-bond coordination and distorted first hydration shell with broken or weakened H-bonds. The anomalous water properties can then be explained based on fluctuations between local HDL and LDL structural species 2, but currently the true liquid water structure is under strong debate. The XAS water spectrum is usually divided into three main parts - sharp but weak pre-edge at 535 ev, strong main-edge centered at 537 ev and strong and extended post-edge located around 3

4 541 ev. The pre-edge is interpreted in connection with broken or weakened H-bonds 7,11-18, while the post-edge is connected to water molecules with fully tetrahedral local H-bond coordination 7,11, The main-edge spectral feature is sensitive to H-bond distortions and linked to a collapse of the second hydration shell 2,19. A large number of theoretical approaches have been applied to simulate the XAS liquid water spectrum, such as transition-potential density functional theory (TP- DFT) 20-23, Bethe-Salpeter equation 24-25, time-dependent density functional theory (TDDFT) 26, and the GW approach with COHSEX approximation Although all these approaches generate a correct interpretation of pre-edge, main-edge and post-edge spectral features, none of them can reproduce the absorption spectrum correctly over the whole spectrum range and, furthermore, there is a lack of agreement on the degree of distortions from tetrahedrality that is required to generate the spectral features characteristic of liquid water 7,15,18-19, This disagreement may be connected with the theoretical methods to compute spectra or with the underlying molecular dynamics (MD) model structures used in the spectrum simulations. For a full understanding of the structural properties of water, it is important to identify the cause of deficiencies in spectrum simulations. Compared to liquid water, it is much more convenient to calibrate XAS calculations against hexagonal ice for which the crystalline structure is well known 37. Oxygen atoms in ice Ih form a hexagonal spatial lattice where each oxygen is bound by H-bonding to its four closest oxygens lying at the corners of an almost perfect tetrahedron. The oxygen-oxygen distance in the lattice is about 2.75 Å and the intra-molecular bond angles are about Unlike the oxygen positions there is no long-range order in the orientation of the hydrogen bonds. The hydrogens can occupy different sites according to the Bernal-Fowler ice rules which state that in ice each oxygen is covalently bonded with two hydrogens and between two neighboring oxygens only one hydrogen is allowed to be. Consequently, there is a large number of possible ways of hydrogen ordering that have approximately the same energy and resulting in zero-point entropy 39. This proton disorder should be taken into account when the ice absorption spectrum is computed. 4

5 The experimental XAS spectrum of hexagonal ice has been reported by many 7,14,40-46, obtained either for the bulk phase or for ice grown on different substrates. Although all measurements give qualitatively the same absorption spectrum profile, the different ice spectra differ in the values of pre-edge, main-edge and post-edge intensities and their relative ratios. The reason for such variations may be connected with different procedures used to prepare the ice samples or with experimental techniques used to record the spectra 45. The absorption spectrum can be measured using the direct transmission scheme, secondary-yield techniques, or X-ray Raman scattering (XRS) 13. The direct transmission mode requires samples with small uniform thickness and homogeneous structure as can be obtained using scanning transmission X-ray microscopy (STXM) 13. The secondary-yield modes of the XAS spectrum use an indirect method via secondary processes due to core-hole decay via radiative or non-radiative processes. The detection scheme based on the radiative decay process is denoted fluorescence yield (FY) while the non-radiative decay mode corresponds to Auger electron yield (AEY) or total electron yield (TEY). This secondary-yield mode requires the absorption within the detected sample to be small, allowing a Taylor expansion of the exponential in Beer s law. Violating this leads to saturation effects in the most intense spectrum features 47. The XRS technique is governed by the same transition matrix element as the XAS process when the magnitude of the momentum transfer of the inelastically scattered photon is small. But, even in case of small momentum transfer, the non-dipole contribution cannot be fully excluded, mainly resulting in enhancement of the pre-edge features 13. Recently we applied the complex polarization propagator (CPP) approach in conjunction with Kohn-Sham density functional electronic structure theory (KS-DFT) to calculate the XAS spectrum for differently H-bonded water structures It was shown that this CPP-DFT approach provides qualitatively correct absorption spectra of differently H-bonded water molecules, in close agreement with previous assignments. The obtained absorption spectrum profile for water molecules with asymmetric H-bond environment demonstrates a sharp and well resolved pre-edge, 5

6 intense main-edge and less intense post-edge, while the computed spectrum for tetrahedrally coordinated molecules is dominated by post-edge intensity and a very weak pre-edge. Although a correct assignment for the model water structure sets were obtained, the overall shape of the computed spectrum for liquid water showed distinct disagreements with experiment. We therefore here benchmark the theoretical method against experimental results for crystalline ice where the uncertainty of structural parameters is much reduced compared to the case of liquid water and presumably given only by proton disorder. We choose to study hexagonal ice as it is frequently used to benchmark H-bonded systems in various respects, including the assessment of theoretical methods for XAS spectrum calculations. It is important to note, however, that experimental XAS spectra of ice vary notably depending on the adopted technique of recording and sample preparation. This raises the question whether the measurements are performed on the assumed perfect crystalline ice or there could be structural defects in the samples. For this reason we will use our large sample of herein computed spectra from various ice models to fit the experimental XAS spectra in order to investigate for which ice structures the different XAS spectra are actually measured and thus shed some light on the ice structures that underlie the experiments. II. METHODOLOGY AND COMPUTATIONAL DETAILS The XAS spectra have been computed using two approaches, namely transition potential 20,50 and complex polarization propagator based density functional theory (TP-DFT and CPP-DFT). A. TP-DFT TP-DFT is an approximation to the Slater transition-state method that was originally devised to calculate electronic excitation energies. The Slater transition energy is evaluated as the orbital energy difference obtained for the transition state prepared by removing one-half electron from the initial orbital and adding it in various excited states. This energy is shown to include relaxation effects up to second order in the orbital response, but the approach requires state-by-state 6

7 calculations that are unfeasible in XAS spectrum simulations requiring hundreds of states to cover the valence, Rydberg, and continuum excitations. In the transition potential approach 50 the onehalf excited electron is simply neglected and the whole spectrum is obtained in one variational calculation for the state with one-half electron in the core orbital. Excitation energies are then computed as orbital energy differences, while transition probabilities are proportional to the dipole matrix element with respect to the core and unoccupied orbitals. When a large number of individual spectrum contributions are summed together in order to predict the bulk ice spectrum a common absolute energy scale should be defined. In case of the TP-DFT approach the absolute energy scale is introduced via a Δ-KS procedure by computing the energy difference between ground and fully variationally relaxed first core-excited state 56. The lowest energy in the TP spectrum is replaced by the Δ-KS energy while the rest of the TP spectrum is shifted accordingly. This procedure gives a reliable energy scale for individual spectra under the assumption that relaxation effects for the rest of the spectrum are similar to the case of the first excited state. Other energy discrepancies associated with relativistic effects and the approximate nature of the exchange-correlation functional can be taken into account via the difference between calculated and experimental core-electron binding energy (CEBE) for the gas phase molecule which is added as overall shift to the previously shifted TP spectrum 20,57. In the present work all TP-DFT spectrum calculations have been performed using the demon2k code 58. Each ice XAS spectrum has been computed using clusters containing 32 water molecules extracted from the different structural models of ice. The Perdew, Burke, Ernzerhof (PBE) gradient-corrected functional 59 has been chosen as exchange-correlation functional for the DFT calculations. The oxygen in the central cluster water molecule has been described with the IGLO-III basis set 60, while the remaining oxygen atoms have been described via the MWB effective core potential 61 and triple-ζ [3s3p1d] basis sets. Using the effective core potential ensures that only core excitations from the central water molecule are taken into account and reduces the computational cost as well. The triple-ζ [3s1p] IGLO-II basis set has been utilized for all hydrogens 7

8 in the ice cluster. The final TP-DFT XAS spectrum has been obtained using the double-basis-set technique where, after convergence of the half-occupied core-state, the orbital basis set has been augmented by a large diffuse [19s19p19d] basis set and one additional diagonalization of the Kohn-Sham matrix has been done. This approach ensures a better description of the spatially extended Rydberg and continuum states. Each transition in the transition-potential XAS spectrum has been convoluted by a Lorentzian with half-width at half-maximum (HWHM) equal to ev. B. CPP-DFT Within the framework of response theory, the complex molecular polarizability can be written ] where 0 and denote the ground and excited states, is the transition energy and γ is a relaxation (or damping) parameter that ensures a physically sound polarization propagator also when the external-field frequency comes close to one of the excitation frequencies. The CPP expression is here expressed in the basis of exact states, but in practice different approximate electronic structure methods are adopted and lead to corresponding forms of matrix expressions 51,53,66 that are solved without directly resolving the excited states. The linear absorption cross section,, is evaluated according to the expression 67 All CPP-DFT calculations have been performed using the Dalton code 68 and the CAM-B3LYP functional 69 with modified parameters [α = 0.19, β= 0.81, μ= 0.33] A common lifetime broadening γ=0.124 ev has been utilized in all CPP calculations. As in the TP-DFT calculations, the calculations were performed using clusters containing 32 water molecules. The oxygen of the central molecule was described using the IGLO-III 60 [7s6p2d] basis set or different Dunning 8

9 correlation consistent basis sets 70 (which basis set is employed will be indicated in each separate case) while for the hydrogens the IGLO-II [3s1p] basis was used. The remaining oxygen atoms were described via the MWB effective core potential 61. The six nearest water molecules were described using the triple-ζ IGLO-II basis set for hydrogens, while the oxygens were described with a triple-ζ basis set obtained from the double-ζ MWB basis set 49. The remaining water molecules in the ice cluster were described by a double-ζ basis set (2s) 71 for hydrogens and the MWB double-ζ basis set 61 employed for oxygens with the last p-function removed 48. In order to improve the description of the excited states, the basis set was augmented with sets of additional diffuse functions (19s19p19d, 19s19p, and 7s7p) placed at the central oxygen atom. The 19s19p19d and 19s19p sets were modified based on the basis set used for the central oxygen in such a way that exponents from the augmentation basis which overlapped the original basis set were removed. Such changed augmentation will be marked accordingly and explanation will be given at the first occurrence. Core-excitation energies computed with the TDDFT approach are usually too low in comparison with experiment due to self-interaction errors 72, and for comparison with experiment, all XAS spectra were shifted by ~15 ev in order to fit the experimental energy region. These shifts are shown in each case. C. Ice Structures The ice geometries used for the spectrum simulations have been obtained from four different sources. The first ice structure model is obtained from a path-integral molecular dynamics (PIMD) simulation of water ice Ih at 100 K with 1,536 water molecules in the simulation box 20. The second ice structure model was generated from an ice unit cell containing 768 water molecules. Oxygen positions in the cell were generated from the basic orthorhombic unit cell with eight water molecules by replicating it by translation nx, ny, nz times along the x, y and z axes. Hydrogen positions were generated based on the Bernal-Fowler ice rules with the additional constraint that the cell dipole moment be equal to zero. The detailed description of the cell-generating algorithm can be found elsewhere 73. The geometrical parameters were ROH= Å, ROO=2.76 Å and 9

10 θ= The third ice structure set has been obtained using the GenIce code, which has been designed to generate various hydrogen-disordered ice structures 74. The GenIce cell with 2,000 water molecules was obtained by replicating the Ih unit cell 5 times along each Cartesian axis and using the TIP3P water model for the dipole moments. As in the previous case the total dipole moment has been minimized. The last ice structure set was the hexagonal ice cell with 1,944 water molecules of Ref. 75 where it was obtained by empirical potential structure refinement (EPSR) to neutron scattering data recorded for ice at 258 K. In all cases, except for the EPSR derived structure, the covalent O-H distances were adjusted such that the O-H distance probability distribution for each model corresponded to the quantum distribution of a harmonic oscillator with frequency 3300 cm -1, which is close to the OH-stretch peak maximum in hexagonal ice. This was done in order to have a more realistic, quantum mechanical sampling of the internal OH-bond distances in the structures, in particular for the forcefield models; the EPSR structure already obeyed the OH quantum distribution as it was derived from neutron diffraction data. The internal OH-distances were furthermore readjusted to 0.98 Å before performing the statistical broadening around this separation. The resulting distance distributions and how they were sampled are shown in Fig. S6 in the Supplementary Information (SI). We note that the OH-stretch vibration in hexagonal ice is sharply peaked, but quite broad (~400 cm -1 ). However, using different frequencies in the range cm -1 has negligible effects on the associated OH-distance probability distribution (see Fig. S7 in the SI). Since only the lowest vibrational level ( =0) is populated, the harmonic approximation is adequate. D. SpecSwap-RMC Simulations In order to find which particular ice local structures contribute to different experimental XAS ice spectra, SpecSwap reverse Monte-Carlo (SpecSwap-RMC) simulations have been performed. The details of the SpecSwap-RMC approach can be found elsewhere and the program can be downloaded from Ref. 78. In brief, the SpecSwap-RMC performs a fit to experimental data using 10

11 a library of structures with associated precomputed (here) XAS spectra. A subset of the library is selected and the spectrum contributions summed and compared with experiment. Random replacements of the structures in the subset are then made and accepted or rejected according to a Metropolis Monte Carlo criterion on the 2 deviation from the experimental reference. The subset is sampled regularly to obtain statistics on the importance of each contribution, i.e. how often a particular structure is found in the selected subset. The resulting weights are then used to rescale the library to determine which structural aspects are important for obtaining agreement with experiment. In the present case, the SpecSwap library included 614 different local ice structures collected from all available ice models with XAS spectrum contributions computed using the CPP-DFT approach and the IGLO-III+19s19p19d basis set for the central oxygen. Since the exchange of structures between subset and library is done as replacements, each structure in the library was replicated eight times to allow multiple contributions from any given structure. The resulting library size was thus 4,912 out of which 140 structures were included in the trial subset in order to get smooth theoretical spectra. SpecSwap-RMC simulations targeting each available measured XAS ice spectrum were then performed separately using the same library. A total number of 1.6*10 9 trial attempts was made in each case with around 4.4*10 8 accepted moves. The obtained weights were then used to reweigh the library O-O and O-H radial distribution functions (RDF), as well as the q-parameter 79 measuring tetrahedrality, in order to determine the local ice configuration pertaining to each measured spectrum. The q-parameter is given by 79 1, where is the angle between the central oxygen position and two of its nearest four oxygen neighbors. For a perfect tetrahedron the q-value becomes one, while for a completely disordered system the value is zero. III. RESULTS AND DISCUSSION A. Basis set investigation 11

12 We begin by investigating the effects of basis set on the CPP-DFT computed spectra as shown in Fig. 1. Only the basis set description of the central, core-excited oxygen will be varied which should capture the main effects on the spectra considering that an overlap with the 1s-orbital is required for a significant transition probability. Extending the basis set also on the surrounding centers has been shown to improve the sampling particularly of higher continuum states 20 while not dramatically affecting lower-lying excitations. The experimental reference used here is the ice spectrum from Ref. 45 measured using STXM on a thin ice sample contained between Si3N4 windows while the structures for the spectrum calculations were obtained from PIMD simulations of ice Ih, as described above, using the same extracted structures for all calculations. Since the final computed spectra will be a sum of contributions from a sampling of various local H-bond arrangements, we will also for the basis set comparison use sums of 47 computed spectra in each case. All spectra, both computed and experimental, are normalized by area between 534 and ev. In Fig. 1A we compare theoretical spectra obtained with the aug-cc-pvdz and the somewhat more extended d-aug-cc-pvdz basis set with the experimental spectrum measured using STXM 45. The spectral features at the pre-edge (535 ev) and main-edge (~537 ev) are much too sharp and intense compared with experiment using the smaller aug-cc-pvdz basis set and, in addition, we obtain a sharp and intense peak at ~539 ev between the main-edge and post-edge (~ ev). Going to the d-aug-cc-pvdz basis shifts main-edge intensity upwards, resulting in a broad peak intermediate between main-edge and post-edge. The post-edge appears at too high energy with a significant dip in the position of the post-edge in experiment, but the width of the spectrum is improved with enhanced intensity beyond the post-edge. Extending the basis set to triple- quality as in the d-aug-cc-pvtz basis (Fig. 1B) shifts some intensity towards the main-edge and gives a sharp and intense post-edge in approximately the right position, but with significant lack of intensity between the two and somewhat surprisingly less intensity at higher energy compared to the d-aug-cc-pvdz basis (Fig. 1A). Increasing the number 12

13 of added diffuse functions to three of each kind (s, p, d, f) as in t-aug-cc-pvtz shifts intensity from the main-edge, but enhances the dip in intensity between main- and post-edge. FIG. 1 XAS spectra computed for the same ice structures, but with different basis sets used for description of the core-excited oxygen compared to (black line) the experimental spectrum recorded at T=232 K for a thin ice film between Si 3N 4 windows in STXM mode 45. Panel A - summed ice spectra computed for aug-cc-pvdz and d-aug-ccpvdz correlation-consistent Dunning basis sets assigned to the core-excited oxygen. Panel B - summed ice spectra computed for d-aug-cc-pvtz and t-aug-cc-pvtz correlation-consistent Dunning basis sets. Panel C - summed ice spectra with d-aug-cc-pvtz basis set. The f-functions do not give significant contribution to the final summed spectra so we treat the d-aug-cc-pvtz basis without f functions in combination with different augmentation bases (7s7p, 15s15p is obtained from 19s19p via removing close exponents, 15s15p15d this basis set is obtained from the 19s19p19d basis by removing close exponents see SI for details). Panel D - summed ice spectra with IGLO-III basis set in combination with different augmentation basis sets 16s16p and 19s19p19d. Adding additional diffuse functions to the d-aug-cc-pvtz (Fig. 1C) leads to improvements in 13

14 particular around the main-edge which can be understood as the diffuse functions contributing to the description of excited, unbound electrons with low kinetic energy which have a long de Broglie wave length that can be well-described by slowly varying functions. The 7s7p optimized Rydberg set (6s6p from ref. 80 with one higher-exponent s and p function added to cover the region between the main- and post-edge, see Supplementary Information (SI)) and even-tempered 15s15p basis sets are in very good agreement until the post-edge while beyond the post-edge the somewhat higher initial exponents in the larger basis give enhanced intensity at higher kinetic energies. Both basis sets, however, result in too sharp features which were not ameliorated by adding additional structures to the summation. In Fig. 1D we show the results using the IGLO-III basis in combination with a 16s16p eventempered diffuse basis obtained from the larger 19s19p19d basis by removing the d-functions and removing exponents that are close to exponents in the IGLO-III basis. The latter is otherwise done automatically in the program when checking for basis set linear dependencies, but was deemed prudent for the purpose of calibration. Comparing the results with and without d-functions in the added diffuse set shows improved agreement in the post-edge region from including also the large set of d-functions. Since 1s transitions to d-states are dipole-forbidden, this is due to secondary effects on the unoccupied orbitals from the larger orbital space. Since with this basis we largely remove the discrete sharp, spurious peaks in the post-edge region it was decided to proceed with the IGLO-III+19s19p19d basis set when computing XAS for the different theoretical models; this is in agreement with earlier experience 62-63,81. B. Theoretical Structure Models In Fig. 2 we show computed XAS using both CPP-DFT and TP-DFT for the four different structural models described in the methodology section above. Fig. 2A uses the perfect hexagonal ice Ih lattice, 2B the lattice obtained using GenIce which introduces some additional spread in the oxygen positions, 2C samples structures from a PIMD simulation and 2D samples the EPSRderived structure based on neutron scattering. In Fig. 3 the corresponding O-O RDFs are exhibited 14

15 with blue denoting the RDF of the full cell while red shows the distribution obtained from the sampled structures. The number of computed structures was 23 (Fig. 2A), 110 (Fig. 2B), 129 (Fig. 2C) and 62 (Fig. 2D). FIG. 2 Summed ice XAS theoretical spectra computed with CPP-DFT (red) and TP-DFT (blue) approaches for different ice models compared to the experimental XAS ice spectrum (black) taken from reference 45 (recorded in STXM mode for an ice film between Si 3N 4 windows at T=232 K ). In all cases in the CPP-DFT calculations the IGLO- III+19s19p19d basis set has been used for the central oxygen in the ice cluster models. Panel A - spectra computed for ice structures generated from the perfect ice cell 73, panel B - spectra computed for ice structures obtained using the GenIce program 74, panel C spectra computed for ice structures derived from PIMD SPC/E trajectories, panel D spectra computed for structures generated from EPSR simulation data 75. Theoretical CPP-DFT XAS spectra were aligned with respect to the experimental main-edge position and the corresponding shift is shown in each panel, while in case of TP-DFT spectra the absolute energy scale is accessible. Starting with the perfect ice lattice in Fig. 2A and focusing on the CPP-DFT results we find a very sharp and intense post-edge peak, consistent with there being minimal spread in O-O 15

16 distances in the perfect lattice (Fig. 3A). This is accompanied by a very sharp main-edge and we find very poor agreement with the experimental spectrum which here, as in Fig. 1, was obtained in transmission mode using STXM 45. Using structures obtained through the GenIce procedure, where there is some additional spread in O-O distances (Fig. 3B), we find a very similar resulting computed spectrum. The PIMD simulated structure introduces additional disorder, in particular around 4-6 Å (Fig. 3C), which brings down the post-edge somewhat and gives some additional intensity between main- and post-edge, but still in unsatisfactory agreement with experiment. Finally, the EPSR-derived structure model results in a spectrum with significant intensity in and below the pre-edge region and a quite weak post-edge. This is consistent with the non-negligible probability of finding O-O distances in the interstitial region 3-4 Å which is characteristic of more close-packed, HDL-like structures with weakened or broken H-bonds which contribute to the preedge region 13,20. However, none of the theoretical structure models for which spectra are computed using the CPP-DFT approach succeeds in reproducing the measured spectrum. For the post-edge region and the very disordered EPSR-derived structure, the TP-DFT spectra are in very good agreement with those obtained with the more stringent and costly CPP-DFT approach. However, the TP-DFT approach lacks intensity in the main-edge region for the more structured models where instead CPP-DFT exhibits a clear feature, albeit sharper than in experiment. This lack of agreement in the main-edge region for the TP-DFT approach has been noted earlier

17 FIG. 3 Oxygen-Oxygen radial distribution functions calculated for the sampled structures (red) from different ice models (blue). Panel A - RDF for ice structures derived from the perfect hexagonal ice cell plotted with RDF obtained for all oxygens in the cell, panel B calculated RDF for ice structures generated by the GenIce program together with full ice cell RDF, panel C computed RDF for ice structures selected from PIMD SPC/E simulation compared with RDF calculated for full PIMD SPC/E snapshots, panel D RDF calculated for selected ice structures from EPSR ice cell plotted together with RDF computed for the full EPSR ice cell. It is clear that neither CPP-DFT nor TP-DFT is able to reproduce the experimentally measured spectra based on either of the four models investigated here. This could be due to deficiencies in the computational setup, but in particular CPP-DFT has been very successful in reproducing spectra for molecular systems where the structure has been known 72. However, the size of the used cluster models (32 molecules) could be an issue; although earlier investigations have shown only minor effects of extending the cluster size 20, we have also noted some effects of a dielectric embedding when computing a single structure 48 which we here have assumed will average out 17

18 when summing spectra from many different structures. We investigate this by recomputing all structures used in the summed CPP-DFT spectrum for the PIMD SPC/E structures shown in Fig. 2c using a QM/MM approach. Each QM 32-molecule cluster was thus extended to a spherical cluster with 20 Å radius and fixed number (1050) of molecules using the Ahlström polarizable force-field 82. The obtained CPP-DFT spectrum is presented in Fig. S5 in the SI showing minor changes compared to Fig. 2c. We find that introduction of the MM environment smears out the post-edge region somewhat and shifts the main-edge to lower energy (by 0.4 ev). As obtained, the overall shape of the QM/MM XAS spectrum does not change significantly and we need to look elsewhere for the origin of the discrepancy between theory and experiment. An alternative possibility is that none of the model structures corresponds directly to the structure of the samples that have been measured. Indeed, perfect hexagonal ice is notoriously difficult to prepare for X-ray measurements and measured spectra do show variations due to preparation and detection modes

19 FIG. 4 Comparison of experimental XAS ice spectra using different detection modes and sample preparation processes. Black line spectrum detected using secondary electron yield (SEY) mode for an ice sample prepared on a BaF 2(111) surface 45 at T=144 K, red line spectrum of a thin ice film recorded in scanning transmission X-ray microscopy (STXM) detection mode 45 at T=232 K, blue line ice absorption spectrum registered in SEY mode for an ice film grown on a Pt(111) surface 42 at T=130 K, orange line ice absorption spectrum detected in SEY mode for bulk ice 7 prepared at T=125 K. All experimental spectra were area normalized over the energy interval between 534 and 545 ev. C. Experimental Spectra and SpecSwap-RMC Modeling The different experimental ice spectra are shown in Fig. 4. All spectra have qualitatively similar shape, but differ significantly in spectrum width and ratio between pre-, main- and post-edge intensities, which may be attributed to different detection modes or sample structures, although all measurements are nominally on hexagonal ice Ih. The ice spectrum registered for a thin ice film between Si3N4 windows using STXM at T=232 K (red line in Fig. 4) is characterized by small pre- 19

20 edge, large main-edge intensity and dominating post-edge intensity 45. This ice film is suggested to have inhomogeneous structure with significant contributions from defects and grain boundaries. This conclusion has been obtained based on comparison with the ice spectrum registered for a crystalline ice film on Pt(111) prepared by isothermal heating of amorphous ice at 150 K and which is supposed to consist of three-dimensional crystallites 45. The ice spectrum recorded in SEY mode for an ice film grown on the BaF2 surface at T=144 K (black line in Fig. 4) demonstrates very weak pre-edge, reduced main-edge intensity in comparison with the STXM spectrum and intense and extended post-edge with significant shoulder towards higher energies 45. The ice film prepared on a hydrophobic surface, such as BaF2, is characterized by fewer numbers of defects and grain boundaries in comparison with other preparation approaches which is manifested through corresponding spectrum changes (reduced pre-edge and enhanced post-edge intensity). On the other hand, the spectrum recorded in SEY mode for the ice film grown on Pt(111) at T=130 K (blue line in Fig. 4) 42 shows slightly higher pre-edge intensity and more intense post-edge in comparison with the previous cases, which is furthermore shifted to lower energy with maximum at ev (the STXM spectrum has post-edge maximum at ev and the SEY spectrum at the BaF2 surface has the maximum at 541 ev). The absorption spectrum registered for bulk ice at T=125 K using SEY mode (orange line in Fig. 4) shows intermediate pre-edge intensity, reduced main-edge intensity and very intense and sharp post-edge located at ev. Such large spectrum variations imply that in each case the investigated sample has different local spatial structure that may differ in the fraction of defects, grain boundaries or admixture of amorphous ice. Another possibility can be connected with possible changes in the local ice structure through, e.g., heating by the X-ray photon beam. The assumption about deviations from the ideal crystal lattice in the experimental ice samples is supported by the clear differences in spectra and the results of section B (theoretical structure models) where spectra of calculated model structures demonstrate strong and sharp post edge for the perfect ice lattice while model PIMD structures with some variations of oxygen-oxygen distances give spectra with smeared out and decreasing post-edge. 20

21 Based on these observations, it is interesting to try to investigate which ice structures are actually measured by XAS in each case by using the spectrum contributions from each of the molecules in the generated models to fit the spectra and thus deduce which structures are likely to be present in the experimental samples. One possible way to do this is to use the SpecSwap-RMC approach 76 discussed in the methodology section. The key idea is to fit each experimental XAS spectrum using a subset of the total number of computed spectra and assign weights to each based on how often they are selected in the Monte Carlo fit to the data. These weights are then used to build O-O and O-H radial distribution functions (RDFs) where the contribution from each individual structure is weighted by its associated weight before summation; we thus assume that for each local structure we can reliably predict the spectrum contribution and, since each contribution is dependent on the underlying structure, thus derive the likely structure that was actually obtained in each experiment. The SpecSwap-RMC fits and weighted O-O RDFs are shown in Figs. 5 and 6. In Fig. 5 the results of the SpecSwap-RMC fit are presented for the STXM and SEY (on BaF2) ice spectra. In both cases (see the top panels) the weighted spectra reproduce the experimental spectra quantitatively while the unweighted sum over all the computed structures in the four structure models shows significant discrepancies although all four structure models were based on hexagonal ice, either as the perfect crystal or with different degrees of disorder around the lattice positions. Based on the weights obtained from the fit to the experimental spectra, the weighted O- O RDFs are shown in the bottom panels. The first peak in both weighted RDFs is quite broad and smeared out compared to the ideal lattice (see Fig. 3A). The maximum of the first peak in the weighted RDF based on the fit to the STXM spectrum is at 2.74 Å while in case of the SEY spectrum on BaF2 surface this peak is shifted to shorter distance with maximum at 2.67 Å. 21

22 FIG. 5 Data derived from SpecSwap simulations. Column a presents the SpecSwap data obtained from fitting to the experimental ice spectrum measured in STXM mode at T=232 K for a thin ice film between Si 3N 4 windows. Top panel reference experimental STXM spectrum (black line), weighted library spectrum with weights derived from SpecSwap (red line), unweighted spectrum calculated as sum of all library contributions (blue line). Bottom panel weighted OO library RDF with weights derived from XAS SpecSwap fitting (red line), unweighted OO library RDF (blue line). Column b SpecSwap data obtained from fitting to the SEY experimental spectrum measured for an ice film grown on a BaF 2(111) surface at T=144 K. Top panel reference SEY experimental spectrum on the BaF 2 surface (black line), weighted library XAS spectrum (red line), unweighted library XAS spectrum (blue line). Bottom panel weighted OO RDF with weights derived from XAS fitting (red line), unweighted OO RDF (blue line). Fig. 6 contains the results of the SpecSwap fitting of the SEY ice spectrum obtained for the ice film on Pt(111) at T=130 K (column a) and for the bulk ice spectrum prepared on Pt(111) at T=125 K (column b). The weighted XAS spectrum reproduces the reference SEY spectrum on Pt(111) quite well, while for the bulk ice spectrum the weighted spectrum cannot fit completely 22

23 the main and post-edge intensities although a significant redistribution of intensity compared to the unweighted library is observed. The bottom panels contain the corresponding weighted oxygen-oxygen RDF with weights obtained from the fits to the absorption spectra. The first peak of the weighted RDF is similar in shape in both cases and has the maximum at 2.74 Å. It is clear that to reproduce the experimental spectra based on the computed spectrum contributions we need to include significant disorder around the lattice positions of the perfect hexagonal ice crystal structure and that this is significantly larger than what can be expected of thermal motion at the experimental temperatures. FIG. 6 Data derived from SpecSwap simulations. Column a shows the SpecSwap data obtained from fitting to the experimental ice spectrum measured in SEY mode for a thin ice film grown on Pt(111) at T=130 K. Top panel reference SEY ice spectrum (black line), weighted XAS library spectrum (red line), unweighted spectrum calculated as sum of all library contributions (blue line). Bottom panel weighted O-O library RDF with weights derived from XAS SpecSwap fitting (red line), unweighted O-O library RDF (blue line). Column b SpecSwap data obtained from fitting the SEY experimental spectrum measured for bulk ice at T=125 K. Top panel reference SEY experimental 23

24 spectrum (black line), weighted library XAS spectrum (red line), unweighted library XAS spectrum (blue line). Bottom panel weighted O-O RDF with weights derived from XAS fitting (red line), unweighted O-O RDF (blue line). In addition to the weighted O-O RDF, we also plotted the weighted and unweighted distribution of the tetrahedrality q-parameter 79 presented in Fig. 7. The q-distribution derived from the STXM spectrum has maximum at 0.97, while for the other experimental spectra the resulting q-distributions have maximum around A better measure is the weighted average of the q- distributions which is the lowest (0.9499) for the ice measured on BaF2 and highest (0.9675) for the SEY spectrum of bulk ice 7. Although the spread between them is quite small, the weighted q-distributions are consistent with the weighted O-O RDFs and experimental reference spectra. Spectra which are characterized by a very intense post-edge (e.g., the SEY bulk ice spectrum and SEY ice film on Pt(111)) also have the highest maximum of the q-distribution located at q=0.98. The SEY bulk ice spectrum with the most intense post-edge intensity gives a sharp first peak in the weighted O-O RDF and at the same time the most sharp q-distribution and highest q-average, which indicates that the corresponding spatial structure around the first hydration shell should be close to tetrahedral. However, none of the q-distributions shows a single peak around q=1.0 as would be the case of perfect hexagonal ice. Also the radial O-O RDFs demonstrate smeared-out peaks for all experimental spectra. This suggests that the measured experimental spectra do not correspond to perfect hexagonal ice samples, but rather correspond to samples with a number of defects, such as grain boundaries and some admixing of amorphous ice. 24

25 FIG.7 q-parameter distribution calculated with weights derived from XAS SpecSwap simulations. Panel a weighted (red line) and unweighted (blue line) q-distribution calculated with weights obtained for the SEY spectrum recorded for an ice film on BaF 2, panel b weighted and unweighted q-distribution obtained with weights derived from the SEY ice spectrum recorded for an ice film grown on Pt(111), panel c weighted and unweighted q-distribution obtained with weights for the STXM ice spectrum, panel d weighted and unweighted q-distribution obtained with weights derived from the SEY spectrum for bulk ice. Q average in each panel gives the average q-value derived from the weighted q-distribution. In Fig. 8 we show the weighted O-H RDFs which again exhibit clear differences between the distributions that are required to reproduce the respective spectrum. We emphasize that, although there will always be uncertainties in the calculations related to various aspects of the computational approach, the differences between the resulting structures should be reliable. We observe that for 25

26 the two SEY spectra measured on Pt(111) (blue 42 and yellow 7 ) the distribution of internal O-H distances is the most narrow which is consistent with the post-edge intensity being the highest and sharpest in these two spectra. It has earlier been shown (see Fig. 3 in Ref. 20) that the intensity in this region is very sensitive to the internal O-H distribution with the intensity going down as the O-H distribution becomes broader. In all of the model structures the internal O-H distances were adjusted so as to reproduce the zero-point O-H distribution of a harmonic O-H oscillator at a frequency of 3300 cm -1 and this distribution is largely retained by these two SEY spectra (see SI for O-H SpecSwap results) while for the STXM spectrum 45 and in particular the spectrum of ice on BaF2 45 the reweighted distributions are significantly broadened in particular towards shorter O- H distances indicating a less well-defined H-bonding and an effective O-H stretch frequency higher than the assumed 3300 cm -1. In agreement with this, the intermolecular H-bond distance, peaking around 1.75 Å, is shifted towards shorter distances for the two SEY (blue and yellow) spectra consistent with more well-defined intermolecular H-bonding. 26

27 FIG. 8 Weighted O-H RDFs calculated using weights derived from XAS SpecSwap-RMC simulations in order to fit the different experimental spectra. Black line fit to SEY experimental spectrum of a thin ice film grown on the BaF 2 surface 45 at T=144 K, red line fit to the STXM experimental spectrum for a thin ice film between Si 3N 4 windows 45 measured at T=232 K, blue line fit to the SEY ice spectrum measured for a thin ice film grown on a Pt(111) surface 42 at T=130 K, orange line fit to the SEY spectrum measured for bulk ice 7 at T=125 K. The weights used to plot the O-H RDF are the same as those which have been used to plot the O-O RDF in Figures 5 and 6 and the q-values in Figure 7. IV. CONCLUSIONS In the present work, we attempt to reproduce the experimental XAS spectrum of hexagonal ice by means of theoretical spectrum calculations performed on model ice structures with account made of proton disordering. We use two different DFT-based approaches, namely the transitionpotential (TP) 20,50 and complex polarization propagator (CPP) 48,51-52,72 methods and find better agreement with experiment for the latter. Water and ice provide specific challenges in terms of their X-ray spectra due to the simultaneous presence of both localized excitonic excitations at the onset of the spectra (pre-edge) and more delocalized excitations into the H-bond network 11,13,83. We investigate the basis set dependence based on traditional sequences of molecular basis sets through the d-aug-cc-pvxz (x=d, T) basis sets with different extensions to describe Rydberg and continuum states. We find that the underlying molecular basis is of less importance than the extension to cover diffuse and continuum states. We settle for the IGLO-III 60 basis to describe core relaxation and the 19s19p19d even-tempered basis 50,62 for the excited states, although it might be possible to obtain similar accuracy with a more limited basis set. It should be noted here that adding diffuse functions mainly improve the continuum region in a limited energy range near the edge since diffuse Gaussian functions can only be combined to give long wave-length oscillations corresponding to outgoing electrons with low kinetic energy 20. We compute spectra from four different models of proton-disordered hexagonal ice including the perfect crystallographic structure, a computer-generated crystallographic structure with some 27

28 disorder in the oxygen positions, an MD simulation of ice, and an EPSR-derived structure based on neutron diffraction data. We do not observe a satisfactory agreement between theoretical and experimental spectra and these discrepancies are arguably larger than one would expect at the adopted level of theory, see Ref. 72 for a recent account of the performance of the complex polarization propagator method based on DFT for XAS spectrum calculations. A possible origin of errors might be the size of the models used, which is limited to 32 molecules. However, in earlier work we have found only minor differences between this size cluster and larger models 20 and discrepancies at higher energies for individual structures due to extensions or embedding 48 are expected to average out when a large number of contributions are summed. We thus focus on the significant differences in experimentally measured XAS spectra of hexagonal ice prepared using different recipes and use our computed spectra from all the different structures to reverse-engineer what structures could potentially be present in the experimental samples in order to produce the measured spectra. We apply the SpecSwap-RMC 76,78,84 approach to a large set of computed spectra and their associated model structures to statistically determine the relative contributions of the different structures that are necessary in order to fit the experimental spectra. We find for each of the measured spectra a sharp peak in the predicted O-O RDFs at the expected positions for the perfect hexagonal ice Ih crystal structure, but with significant disorder around the peaks which also increases at longer distance. This could be indicative of some contribution of low-density amorphous (LDA) ice in the samples 85 or the presence of other defects, potentially induced by the X-ray beam in the absorption measurement. Such defects would give rise to more diffuse scattering in an X-ray diffraction measurement, which might not be detectable if the sensitivity of the detector has been limited so as to avoid damage at the intense Bragg spots. Crucially, in order to reliably calibrate techniques to accurately compute X-ray spectra, it is indispensable to have a reliable structure determination of the samples used in experiments, both before and after measurement of the spectra. 28