Supplemental information-v3
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Supplemental information: Benchmark oxygen-oxygen pair-distribution function of ambient water from x-ray diffraction measurements with a wide Q-range Lawrie B. Skinner, Congcong Huang, Daniel Schlesinger, Lars G.M. Pettersson, Anders Nilsson , Chris J. Benmore Contents: 1. Sensitivity analysis of gOO(r). 2. Numerical data description. 1. Sensitivity analysis of gOO(r). Below we demonstrate numerically the stability of πππ (π) for π > 2 Å with respect to assumptions on the self-scattering, intramolecular scattering and Compton scattering (CS) subtractions. For the purpose of the stability investigation we can rearrange equation 13 in the manuscript to give [πππ (π) − 1] ≅ [[πΌπ (π) + πΌππΆπ (π)] ππππ πΆπ πππ‘ππ (π) − πΌπ‘βππ (π) − πΆ(π) − 4ππ (π)ππ» (π)[πππ» − 1]] /ππ2 (π) where πΆ(π) is the molecular form factor, which includes self-scattering and intramolecular scattering ππππ contributions, [πΌπ (π) + πΌππΆπ (π)] is the measured x-ray differential cross-section including Compton πΆπ scattering, ππΌ (π) is the x-ray form factor of element α, and πΌπ‘βππ (π) is the theoretical x-ray Compton scattering curve, which is subtracted to account for the contribution present in the measurement πΌππΆπ (π). πππ‘ππ (π) πππ» is the intermolecular O-H structure factor, where the stability with respect to using different πππ‘ππ (π) πππ» curves is already shown in figures 7 and 8 of the manuscript. πΆπ In figures s1-s3 below the effects of varying ππΌ (π), πΆ(π) and πΌπ‘βππ (π) are demonstrated. Figure s1 is the extreme case where the intramolecular scattering has not been subtracted. Figure s2 includes in addition the effect of not modifying the ππΌ (π) form factors (i.e. using the IAA instead of MAFF form factors). Finally figure s3 shows the effects shown in figures s1, s2 plus the effect of using the (erroneous) πΆπ (π) approximation πΌπ‘βππ is a constant. In spite of these rather extreme approximations, as shown in figures s1-s3, the resulting errors in πππ (π) are largely confined to π < 2 Å, since the main effect is additive errors, which are slowly varying in Q. An exception is the effect of neglecting the MAFF modification of the ππ (π). Errors in ππ (π) result in a multiplicative error to πππ (π). However even the effect of completely neglecting to modifiy the IAA form factors, is shown to have only a 1-3% effect on πππ (π) at intermolecular distances. Further the errors caused by inaccurate subtractions are very obvious in the πππ (π), ππ (π) and πΌπ (π) functions. Hence the accuracy of the subtractions and normalization can be checked by confirming these behave correctly. Checks include that (1) the measured differential cross-section tends to the correct Compton scattering level at high Q, and (2) to the correct compressibility limit at low Q, whilst (3) the resulting πππ (π) oscillates around 1. In addition, the effect of the modified Lorch transform, and r < 1.1 Å oscillations to the SOO(Q) can be seen by comparing the raw data, to the back-transform of the modified Lorch transformed data, where r < 1.1 Å has been set to zero. This comparison is shown in figure s4. Figure s1. πΊπΆπΆ (πΈ), and ππΆπΆ (π) generated with a direct, unmodified Fourier transform, using the Wang C(Q) (black), and the MAFF generated C(Q) whilst completely neglecting the intramolecular terms in the latter C(Q) (red dashed). Figure s2. πΊπΆπΆ (πΈ), and ππΆπΆ (π) generated with a direct, unmodified Fourier transform, using the Wang C(Q) (black), and the IAA generated C(Q) whilst completely neglecting intramolecular terms in the latter C(Q) (red dashed), i.e. like figure 1, but no MAFF modification has been made to the fO, fH. Figure s3. πΊπΆπΆ (πΈ), and ππΆπΆ (π) generated with a direct, unmodified Fourier transform using the Wang C(Q) (black), and the IAA generated C(Q) whilst completely neglecting the intramolecular terms in the latter C(Q), and in addition assuming Compton scattering is a Q-independent constant (red dashed), i.e. like figure 2, but with a worst case Compton scattering subtraction. Figure s4. Left: π[πππ (π) − 1] functions form the raw data (red), and after back-transformation of the modified πππ (π) (black line). Right: π[πππ (π) − 1] functions generated with an unmodified Fourier transform (red) and with the variable modification fucntion, after having had the low-r set to zero (black). i.e. The pairs of black and red lines are related by direct unmodified Fourier transform, and modified backtransformed SOO(Q) (black) has effectively been Fourier-filtered to remove unphysical high and low Q-freqencies. 2. Numerical data description Included in the “Ambient_water_xray_data.txt” file is the numerical APS πΌπ (π), πππ (π) and πππ (π) data. The πππ (π) has been generated using the Zeidler et al. πππ» (π) measurement which was obtained using oxygen isotopes and neutron diffraction (see references 33,34 of the main manuscript). The C(Q) πΆπ molecular form factor and πΌπ‘βππ (π), Compton scattering were taken from Wang et al. (ref. 15). The neutral independent atom (IAA) ππ (π), ππ» (π) form factors were taken from the relativistic Dirac-Fock calculations of Su & Coppens (ref. 22), and modified for charge redistribution according to equation 4 in the main manuscript with δ =2.0 Å-2, and ππ , ππ» = −1, +0.5. The πππ (π) function given was generated using a direct Fourier transform with the modification function described in the main manuscript.
