(a) Three molecular form factor calculations, C(Q) (labeled C). The blue dotted line is the C(Q) generated using the independent atom approximation (IAA), the red dashed line is the C(Q) from the modified atomic form factors (MAFF), and the black solid line is the C(Q) generated from the quantum calculations of Wang et al. 15 The gray line is the measured I X (Q) from the APS measurement, and the black dashed line is the Compton scattering contribution per molecule (labeled CS), which is subtracted from the scattering pattern to obtain the I X (Q) measurement. (b) Inset: The three C(Q)’s divided by the 9B(Q) weighting factor. This highlights the differences between the different C(Q) curves. (c) The resulting structure factors from using the MAFF (red dashed) and Wang (black) C(Q) functions. (d) The PDF functions corresponding to the structure factors plotted in (c), transformed without a modification function, using a maximum Q of 19.6 Å−1.
Four recent synchrotron x-ray diffraction measurements of liquid H2O at ambient temperatures. They are: APS dataset (22 °C) measured using 114.76 keV photons and an amorphous silicon area detector (APS: light blue line), 17,25 SPring-8 dataset (26 °C) measured at beam line 04B2 using 61.62 keV photons and a Ge detector (SPring-8: black dashed line), 26 and SSRL1 (25 °C) measured using 17 keV photons, a Ge analyzer and a photo multiplier tube detector (SSRL1: black line), 27 and SSRL2 data (23 °C) measured using 19.6 keV photons, a curved graphite analyzer and position-sensitive detector (SSRL2: red dotted line 3 ). (a) Scattering intensity I X (Q) in electron units, normalized to the molecular form factor. (b) The x-ray weighted structure factors, . This is plotted again in (c) as to emphasize the structure at large Q-values (the data sets in (c) are shifted vertically by 0.5 from each other for clarity). Also note that the datasets differ in their Q-step. The SSRL2 data have a Q-step of 0.2 Å−1, the SSRL1 data have a Q-step of 0.1 Å−1 while both the SPring8 and APS data are plotted with a 0.025Å−1 Q-step.
Comparison of the total molecular PDF with different modification functions in the Fourier transform of : (1) without a modification function (black solid line), (2) a Lorch modification function (light blue dotted line), and (3) a variable Lorch function (red dashed line). The variable Lorch function, shown in the inset, is tuned so that it does not broaden the around r = 2.8 Å. Using the APS dataset as a demonstration, the variable Lorch function reduces the Fourier transform oscillations, which have a frequency of about π/Q max in , whilst leaving the first neighbor O-O peak intact.
Demonstration of the effect of a limited maximum Q-value (Q max ) in the Fourier transform of S OO (Q), using the APS dataset. (a) The S OO (Q) data are truncated at S OO (Q) = 0 nodes indicated by the red squares. The first-neighbor region of the corresponding g OO (r) functions is shown in (b). The arrow indicates the trend with increasing Q max . The position of the first O-O peak maximum (r 1) and the peak height (g 1) are plotted as a function of Q max in (c) and (d), respectively. Both peak position, r 1, and height, g 1, are found to be sensitive to the value of Q max for values smaller than 20 Å−1.
Comparison of the g OO (r) functions obtained from the Fourier transform of S OO (Q), using the APS dataset, without using a modification function. The data were Fourier transformed using various Q max values. For clarity, the g OO (r) functions are shifted vertically by 0.2 from each other. The Q max values from bottom to top are 11.3, 12.4, 13.4, 15.9, 17.1, 18.1, 19.6, and 21.0 Å−1.
Comparison of x-ray data based upon the per-molecule scheme (red solid lines) and the per-atom scheme (black dashed lines): (a) Q[S X (Q) − 1] and (b) g X (r), where the APS dataset is tested as an example. A Q max value of 23 Å−1 was used in the Fourier transform. As expected the intensity of the intramolecular O-H peak disappears as the per-molecule scheme is applied. The structure at r-values greater than 1.5 Å is essentially identical between these two schemes.
Comparison of the different g OH (r) (a) and the corresponding S OH (Q) (b) from the literature. The light blue dashed line is the result from RMC fitting in the present work to both the APS dataset and neutron diffraction data from Ref. 12 , the red dotted line is the result from the TIP4P/2005 MD simulation, 36 and the black solid line is the result from recent neutron diffraction data with oxygen isotope substitution. 33,34 Note that the intramolecular O-H contribution is not included in this figure. A Q max = 24 Å−1 was used in the Fourier transforms to derive g OH (r) from the neutron diffraction data. 33 As a consequence, the O-H peak around 2 Å is artificially broadened in the neutron g OH (r) measurement, whereas the RMC and TIP4P/2005 MD g OH (r) functions are not obtained by the Fourier transform and hence are not artificially broadened. (c) Inset: The Q-dependent partial weighting factors of the O-O correlations (black solid line), the O-H correlations (blue dashed line), and the H-H correlations (red dotted line).
Subtraction of the O-H contribution from the APS x-ray dataset. In both (a) and (b), the red dotted line is the S OO (Q) or g OO (r) result calculated using the RMC generated , the black line is the S OO (Q) or g OO (r) result calculated using the TIP4P/2005 MD 36 generated , and the light blue dashed line is the S OO (Q) or g OO (r) result calculated using the oxygen isotope neutron diffraction data. (a) Compares the (gray line), and the three S OO (Q) functions obtained using the three different functions from Figure 7 . (b) Comparison of the (gray line) and the g OO (r) functions derived from Fourier transform of the S OO (Q) functions in (a). A Q max = 23 Å−1 was used in the Fourier transform.
Uncertainty in g OO (r) using the APS dataset and Eq. (14) to estimate the statistical error. (a) g OO (r) and (b)rD OO (r) = 4πρr 2[g OO (r) − 1] (ρ = 0.03338 Å−3), which emphasizes the oscillations at large r. A Q max = 23 Å−1 was used in the Fourier transform.
Comparison of the g OO (r) derived from the four x-ray datasets shown in Figure 2 . The APS data (light blue line), the SPring-8 data (black dashed line), SSRL1 data (black line), and SSRL2 data (red dotted line) were analyzed in the same way, and their resulting g OO (r) are very similar. The 1st O-O peak is broader and blunter in the SSRL2 dataset, compared to the others. This is expected to be due to the limited useful Q max ∼ 13 Å−1 of the SSRL2 measurement. This plot was produced using Q max values of 23, 18, 15.5, and 13 Å−1 in the Fourier transforms of the APS, Spring-8, SSRL1, and SSRL2 datasets, respectively. For clarity, the g OO (r) functions are shifted vertically by 0.5 from each other.
g OO (r) or rD OO (r) functions with uncertainty derived from the SSRL1, SPring-8, and APS x-ray experiments (black lines and blue shaded areas).
Positions and intensities of the 1st maximum (r 1, g 1), 1st minimum (r 2, g 2), and 2nd maximum (r 3, g 3) and average coordination (n OO ), in g OO (r). The SSRL2 data were not included in this table as its smaller useable Q-range and larger Q-step make it less precise. The errors were estimated using Eq. (14) and the expected normalization and calibration uncertainties. The APS data agree, within errors, with the SPring-8 and SSRL1 data when transformed using the lower Q max value of these data sets. For example the increase in r 1 and decrease in g 1 with lower Q max are shown in Figure 4 . The average coordination number was calculated from the integral of 4πρr 2 g OO (r) between 2.4 and 3.30 Å, where the r 2 g OO (r) function is minimum.
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