Calibrating TDDFT Calculations of the X-ray Emission Spectrum of Liquid Water: The Effects of Hartree–Fock Exchange

The structure and dynamics of liquid water continue to be debated, with insight provided by, among others, X-ray emission spectroscopy (XES), which shows a split in the high-energy 1b1 feature. This split is yet to be reproduced by theory, and it remains unclear if these difficulties are related to inaccuracies in dynamics simulations, spectrum calculations, or both. We investigate the performance of different methods for calculating XES of liquid water, focusing on the ability of time-dependent density functional theory (TDDFT) to reproduce reference spectra obtained by high-level coupled cluster and algebraic-diagrammatic construction scheme calculations. A metric for evaluating the agreement between theoretical spectra termed the integrated absolute difference (IAD), which considers the integral of shifted difference spectra, is introduced and used to investigate the performance of different exchange-correlation functionals. We find that computed spectra of symmetric and asymmetric model water structures are strongly and differently influenced by the amount of Hartree–Fock exchange, with best agreement to reference spectra for ∼40–50%. Lower percentages tend to yield high density of contributing states, resulting in too broad features. The method introduced here is useful also for other spectrum calculations, in particular where the performance for ensembles of structures are evaluated.


TDDFT spectra of selected functionals
Fig. S4 shows the HDL and LDL spectra for six different xc-functionals, in particular from regions where the fraction of HF exchange yields large changes in IAD and 1b1 split.Comparing results using BxLYP with 20 and 25% HF exchange, the split increases as a result of a shift in which LDL peak in the 1b 1 is most intense.
Increasing the exchange to 30% changes the features to be somewhat more defined (for the LDL clusters), with a 1b1 split larger than for 25%.Further increasing the fraction of HF exchange makes the features more welldefined, and the 1b1 split varies smoothly.Instead increasing the amount of long-range HF exchange (i.e. using CAM-B3LYP) does not lead to more defined features, but the relative intensities of the states change and the 1b1 split is seen to decrease from 0.62 eV (B3LYP) to 0.30 eV (CAM-B3LYP).

Spectra from GS-DFT
The X-ray emission spectra obtained using ADC(3) and TDDFT and GS-DFT are shown in Fig. S5, utilizing tailored BxLYP functionals with 40% HF exchange.Results are shown for the gas phase and for 5 HDL and 5 LDL clusters.For the gas phase the TDDFT and ADC(3) spectra are very similar, while the GS-DFT features are a bit more compressed (in particular the relative position of 1b2).The experimental energy difference between 1b1 and 1b2 is ~6.1 eV, and we obtain shifts of 5.8, 6.6, and 6.7 eV for GS-DFT, TDDFT, and ADC(3), respectively.
For the hexamer clusters, the differences in the spectra are more pronounced, with the TDDFT and ADC(3) results being relatively similar (at least in the region of the 1b1 peak), but the GS-DFT results in a significantly broadened spectrum, in particular for the LDL clusters.The broader GS-DFT features result from relatively delocalized molecular orbitals, which leads to intensity being distributed over a wider energy region.By comparison, using a two-step approach where a core-hole reference state is formed, MOs are more localized to the individual water molecule, and thus take a more "molecule"-like behavior.For calculations on individual molecules or singly-hydrogen-bonded systems this delocalization is less of an issue, and GS-DFT has thus been successfully applied for such systems.However, when considering solutions the GS-DFT approach may not be suitable, and explicit core-hole calculations with accompanying MO localizations are recommended.

Impact of the size of the clusters
The ADC(3) and CCSD spectra when using clusters with 4, 6, and 8 molecules are shown in Fig. S6.The results obtained using 4 and 8 molecules are compared to that of 6 molecules, and only relatively small differences are visible, primarily present for the smallest structure.Fig. S7 illustrates the impact of cluster size when using TDDFT, considering the SRC2-R1 and CAM-QTP00 xcfunctional.Here, we note small differences between the structures with 4, 6, and 8 molecules (in particular between 6 and 8), but a more significant blue-shift when including 32 molecules.This shift is different for the HDL and LDL structures, and we show the results when considering ten different structures with 6 or 32 molecules.A consistent blue-shift is noted, featuring ~0.08 eV for HDL structures, and ~0.18 eV for LDL structures.This indicates an increase in the 1b1 split of ~0.1 eV for larger clusters, at least for TDDFT and when considering highly asymmetric HDL and tetrahedral LDL structures.As such, we suggest that an increase in the 1b1 split of 0.1 eV may be an upper limit when moving from small to large clusters.

IAD when shifting to different 1b 1 peaks
In Fig. S8 we illustrate the integrated absolute difference (IAD) trends when shifting spectra to different features before calculating the IAD.Left panel shows results when shifting to the HDL 1b1 (as used throughout this work), and the middle panel when shifted to the LDL 1b1.Finally, in the right panel we report the results when shifting HDL and LDL spectra individually.We see that the trends are very similar, with the main differences being in the relative IADs for high fractions of HF exchange, as well as some of the fluctuations for low levels of HF exchange.The IAD minima are relatively unaffected by the precise shift in use, and we consider the present choice of an HDL reference to be appropriate.Including effective core potentials on selected oxygens Q-Chem does not allow different descriptions (all-electron or ECP) for different atoms of the same element.
Thus, we performed the present calculations by assigning the surrounding oxygens as Neon atoms, but represented with an Oxygen basis set and an Oxygen effective core-potential (ECP) called Neon, but with four electrons removed to give the effective nuclear charge of an Oxygen atom, rather than removing two as would be done for Neon.This is fully consistent with an ECP description of Oxygen and eliminated the 1s core levels of all oxygens except for the selected central water molecule, thus making its 1s state unique.

Geometries
Geometries of the isolated water molecule and of 10 asymmetric (HDL) and 10 tetrahedral hexamers are given below, as expressed in Ångström.The first oxygen is the central, probed, oxygen.

Fig. S3 :
Fig. S3: X-ray emission spectra of the isolated water molecule and a six-molecule cluster, as calculated with ADC(2), ADC(2)-x, and ADC(3).Position of each state is marked with a black cross, and bar spectra are included where the bars are colored by the single-excitation amplitude character (|v1| 2 ) of each state.

Fig. S4 :
Fig. S4: Summed X-ray emission spectra of 10 HDL and 10 LDL water clusters, as obtained using TDDFT with six different exchange-correlation functionals.Including the peak separation between the 1b1 features, i.e. the 1b1 split.

Fig. S5 :
Fig. S5: X-ray emission spectra of an isolated water molecule, and summed spectra of 5 HDL and 5 LDL clusters, as computed using ADC(3) and TDDFT/GS-DFT with a BxLYP functional with 40% HF exchange.Results are aligned to 1b 1 of ADC(3).

Fig. S6 :
Fig. S6: X-ray emission spectra of one asymmetric (HDL) and one tetrahedral (LDL) cluster, using ADC(3) and CCSD and considering different cluster sizes.Black lines showing spectra of the current cluster size, and dashed blue line shows the results using 6 molecules.

Fig. S7 :
Fig.S7: X-ray emission spectra of 1 or 10 HDL and LDL structures, obtained using TDDFT with two different xcfunctionals.For comparisons using one structure, spectra using different cluster sizes (black line) are compared to results obtained when using six water molecules (dashed blue line).For the ensembles of 10 structures we report the shift in 1b1 energies.

Fig. S8 :
Fig. S8: Integrated absolute differences (IADs) of X-ray emission spectra calculated using TDDFT with BxLYP, and calculated when comparing to ADC(3) results.Showing trends when shifting BxLYP spectra to align with ADC(3) for the HDL 1b1 (left), LDL 1b1 (middle), and with the HDL and LDL features shifted individually (right).