Isotope effects in x-ray absorption spectra of liquid water
Chunyi Zhang, Linfeng Zhang, Jianhang Xu, Fujie Tang, Biswajit Santra, Xifan Wu
IIsotope effects in x-ray absorption spectra of liquid water
Chunyi Zhang, Linfeng Zhang, Jianhang Xu, Fujie Tang, Biswajit Santra, and Xifan Wu ∗ Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA Program in Applied and Computational Mathematics,Princeton University, Princeton, New Jersey 08544, USA
The isotope effects in x-ray absorption spectra of liquid water are studied by a many-body ap-proach within electron-hole excitation theory. The molecular structures of both light and heavywater are modeled by path-integral molecular dynamics based on the advanced deep-learning tech-nique. The neural network is trained on ab initio data obtained with SCAN density functionaltheory. The experimentally observed isotope effect in x-ray absorption spectra is reproduced semi-quantitatively in theory. Compared to the spectrum in normal water, the blueshifted and lesspronounced pre- and main-edge in heavy water reflect that the heavy water is more structured atshort- and intermediate-range of the hydrogen-bond network. In contrast, the isotope effect on thespectrum is negligible at post-edge, which is consistent with the identical long-range ordering inboth liquids as observed in the diffraction experiment.
I. INTRODUCTION
Water is one of the most important substances to makelife possible on earth . The unique hydrogen (H)-bondnetwork results in the distinctive properties of water andhas been at the center of scientific interest for decades .Normal water (H O) and heavy water (D O) only differslightly in the H-bond network ; however, the formeris essential for a living cell, while the latter can be harm-ful in many ways . Moreover, minute distortionsin the H-bond network can cause noticeable changes infunctionalities of numerous biological processes occurringin aqueous environments . Therefore, a precise pic-ture of the isotope effect of liquid water is crucial, whichalso serves as an important milestone to accurately un-derstand the intensely discussed microscopic structure ofwater .The last decade has witnessed a rapid emergence of thex-ray absorption spectroscopy (XAS) being applied toprobing the H-bond network of water . In the XASprocess, the time scale of the electron-hole interactionis much shorter than that of the molecular relaxation .Therefore, XAS carries out an instantaneous local finger-print of water structure, which is complementary to theaveraged structural information obtained in diffractionexperiments . Light and heavy water have beenextensively studied by various experimental techniques,however their differences in XAS only became availablevery recently by the increased spectral resolution in thetransmission-mode spectroscopy technique . It revealedthat they are similar but not identical. The discerniblespectral difference suggests H-bond networks in H O andD O are affected differently by nuclear quantum effects(NQEs) .With the significant advances in XAS experiment, thetheoretical exploration of XAS spectra is urgently neededto unambiguously associate spectral features to specificstructural motifs of water, which requires both an ac-curate molecular structure and a proper treatment ofelectron-hole interaction. Based on density functional theory (DFT) , Feynman path-integral ab initio molec-ular dynamics (PI-AIMD) provide an ideal plat-form to predict the liquid structure by including theNQEs. However, for decades, simulation of water hasbeen a formidable task. Extensive studies have identifiedthat the van der Waals interaction and exact exchangeare key ingredients to differentiate between water andice . To accommodate these fine effects, a non-local exchange-correlation functional should be adoptedin functional construction that requires higher rungs in the metaphorical Jacob’s ladder . In this regard,the modeling of water by SCAN functional has showngreat accuracy in comparison to experiment . In par-allel, the modeling of electron excitation in the opticalprocess stands as another major challenge that has beenunder active development for years . The excitedelectrons need to be treated as quasiparticles to solvethe Bethe-Salpeter equation (BSE) , whose Coulombinteractions are screened by the electron sea in water.The proper treatments of the electronic screening, suchas the Slater’s transition state theory or the more rig-orous Hedin’s GW approximation for the self-energyapproach, is found to be crucial to qualitatively repro-ducing experimental XAS spectra. However, due to thesignificantly increased computational burden in solvingBSE as well as in the PI-AIMD simulation, such theoret-ical studies so far remain elusive.To address the above issues, we compute the XAS spec-tra of both H O and D O at oxygen K edge based on theself-energy approximation to the BSE. In particular, theliquid structures are generated from path-integral deeppotential molecular dynamics (PI-DPMD) using a deepneural network-based potential energy model . ThePI-DPMD scheme preserves the accuracy of SCAN-DFTwith a computational cost comparable to that of empir-ical force fields. The resulting isotope effects in XASspectra are in good agreement with experiment , whichshows a stronger influence by NQEs in light water thanin heavy water. The pre-edge of the XAS spectra, a sig-nature of short-range ordering of H-bond network, showsa blueshift in the excitation energies and weaker spectral a r X i v : . [ c ond - m a t . d i s - nn ] S e p intensities in D O compared to H O, which originatesfrom the shorter covalent bond but a stronger H-bondingenvironment in D O. For intermediate-range ordering,the light water exhibits an enhanced degree of inhomo-geneity as revealed by local structure index analysis .Therefore, a more pronounced main-edge at lower en-ergy is identified in H O because a softer liquid struc-ture promotes the localization and stabilization of theexcitons. The post-edge of XAS as an indicator of long-range ordering, however, has a negligible isotope effect.This is consistent with the nearly identical structures be-yond second shell coordination as observed in the diffrac-tion experiment . This work simulates the isotopic dif-ferences of XAS spectra of liquid water. Our approach,combining accurate molecular dynamics simulations andthe electron-hole theory, provides an important theoret-ical lens to understand the fine structures and quantumfluctuations of water by XAS. II. METHOD
The PI-DPMD simulations were conducted in anisobaric-isothermal ensemble at 330 K and 1 bar for0.3 ns with a 128-molecule supercell. For both H Oand D O, one representative snapshot was selected fromPI-DPMD trajectories and was adopted for the cal-culation of XAS spectra using our recently developedenhanced static Coulomb-hole and screened exchangeapproximation . The XAS spectra of H O and D Owere aligned according to the position of the post-edgebecause the post-edges of H O and D O coincide inexperiment . More simulation details are described inthe Supplemental Material . III. RESULTS AND DISCUSSION
The theoretical XAS spectra of both liquids H O andD O are presented in Fig. 1(a) together with the exper-imental spectra . A good agreement can be seen be-tween the theory and experiment, not only on overallspectral shapes and all three features of pre-edge ( ∼ ∼
538 eV), and post-edge ( ∼
541 eV) inFig. 1(a), but also on the more delicate spectral differ-ences between the two isotopes in Fig. 1(b). (The smallpeak at ∼ ±
20 and 200 ±
20 meV , re-spectively. A close inspection of the spectral differencebetween H O and D O in Fig. 1(b) further reveals thatthe isotope effect in XAS is most significant in the pre-and main-edges, which decays rapidly and is negligible
Exp. (a)(b)(c)
Pre-edge I n t en s i t y ( a r b . un i t s ) D i ff e r en c e Main-edge Post-edge
Exp. H OExp. D OTheo. H OTheo.Theo. D O Energy (eV) D e r i v a t i v e ( a r b . un i t s ) Exp. H OExp. D OTheo. H OTheo. D O FIG. 1. (a) XAS spectra of H O (red) and D O (black)from theory (solid and dashed lines) and experiment (circlesand triangles). (b) D O-H O difference of the XAS spectrafrom theory (line) and experiment (circles). (c) First-orderderivatives of the XAS spectra with respect to the energy ofH O (red) and D O (black) from theory (solid and dashedlines) and experiment (circles and triangles). at the higher excitation energies in the post-edge. Apartfrom the XAS spectra and difference spectra, the first-order derivatives of the XAS spectra obtained in thiswork also agree well with experimental results as shownin Fig. 1(c). The successful prediction of the experimen-tal measurement indicates that the H-bond structuresand their signatures in electronic excitations of H O andD O are both accurately modeled.In experiment, heavy water is characterized as a morestructured liquid than light water . The subtle struc-tural differences are accurately predicted as displayed inFig. 2(a). The oxygen-oxygen pair distribution functions, g OO ( r ), show more prominent first and second coordina-tion shells in D O than those in H O, which agrees wellwith the diffraction measurement by Soper et al. . Since Exp. H OExp. D OTheo. H OTheo. D O -0.81602.52.01.0 3 4 5 6 71.5140120 V (Å) V (Å) (cid:31)(cid:30) (Å) g OO ( (cid:31) ) Θ ( ° ) P/P max (a) (b)H O D O (c) Θ d OH d O (cid:31)(cid:31)(cid:31) H V =d OH -d O (cid:31)(cid:31)(cid:31) H FIG. 2. (a) g OO ( r ) of liquids H O (red) and D O (black)from theory (solid and dashed lines) and experiment (circlesand triangles). The inset shows definitions of proton transfercoordinate ν and OH · · · O angle θ . Joint probability distribu-tions of θ as a function of ν for (b) H O and (c) D O. Thered dot located at ( − − θ and ν values withthe highest probability. both heavy and light water share the same electronic con-figuration, their structural difference arises entirely fromthe NQEs that restructure their H-bond networks by dif-ferent magnitudes. Under the influence of NQEs, theprotons are more delocalized and are able to probe theconfiguration space that is inaccessible to the classical nu-clei. The delocalized protons along the direction of thestretching mode promote the formation of H-bonds. Asshown by the proton transfer coordinate in Fig. 2, thetendency of a proton to approach the acceptor moleculeis increased under the larger NQEs in H O than in D O.On the contrary, the delocalized proton along the direc-tion of the libration mode facilitates the breaking of H-bonds by perturbing the OH · · ·
O angle, θ , further awayfrom 180 ◦ . The overall NQEs resulting from the twocompeting effects are dependent on the anharmonicity ofthe potential energy surface . In liquid water, NQEsactually soften the liquid structure with more broken H-bonds . Because of the heavier deuteron than proton,NQEs are suppressed in heavy water, which can be iden-tified by the narrower distribution of deuteron as func-tions of ν and θ comparing to that in light water as shownin Fig. 2(b) and (c). Not surprisingly, the H-bond net-work in D O is less destructed by the NQEs compared to H O. The slightly stronger H-bond of D O can be seenby the more parallelly aligned OH · · ·
O angle and a pro-ton transfer coordinate that is closer to zero than thoseof H O as shown by the red and black dots in Figs. 2 (b)and 2 (c). The less perturbed H-bond network of D Othan H O by NQEs is captured by distinct edge featuresin XAS spectra.The pre-edge is attributed to a bound exciton with a characteristic whose origin can be traced back to the firstelectronic excitation in water vapor . Once the wa-ter molecule is excited, a positive oxygen core-hole is leftbehind. The core-hole generates a strong potential thattraps excitonic states that are well localized within themolecule. Therefore, the pre-edge carries out a signatureof short-range ordering of the H-bond network . Theexcitation energies are sensitive to the relative positionbetween proton and oxygen as determined by the cova-lent bond length, which is under constant thermal andquantum fluctuations. When the bond length becomeslonger, the proton moves away from the excited oxygen.As a result, the enhanced electropositivity around theoxygen atom makes the core-hole Coulomb potential ef-fectively stronger, which stabilizes the exciton with lowerenergy. The opposite trend is true when the bond lengthbecomes shorter. The above effect gives rise to a nega-tive correlation between pre-edge excitation energies andcovalent bond lengths as shown by the red areas anddashed black lines in Fig. 3(a). Moreover, the above neg-ative correlation matches well with the anticorrelationbetween average covalent bond lengths and pre-edge en-ergies of H O and D O as shown by the dashed blue linein Fig. 3(a) which has a negative slope with tan α =0.006˚A/160 meV, where 160 meV is the blue-shift of the pre-edge and 0.006 ˚A is the bond length contraction whengoing from H O (1.005 ˚A) to D O (0.999 ˚A). The 0 . .
5% found in the neutron diffraction experiment of Zei-dler et al. (the 3% contraction found in the experimentof Soper et al. is likely to overestimate the bond con-traction as stated in Ref. ). At the same time, pre-edge of D O has a narrower spectral width than that ofH O as observed in Fig. 1, which is consistent with thesuppressed quantum delocalization in D O as shown inFig. 2.Besides the excitation energy, the spectral intensity isalso affected by the isotope substitution. According tothe selection rule, the transition matrix element is deter-mined by the p character in the excitation . Based onsymmetry analysis, the intensity of the pre-edge is weak,but not vanishing even for an intact H-bonding environ-ment in crystalline ice . Moreover, the pre-edge inten-sity is rather sensitive to local distortions of the H-bond.As shown in our analysis in Fig. 3(b), the spectral inten-sity in pre-edge is largely increased as the water structureis deviated from the ideal tetrahedron by more brokenH-bonds, which enhances the p character of quasiparti-cle exciton. As aforementioned, the NQEs weaken theH-bonding strength compared to classical nuclei. The A D A D I n t en s i t y ( a r b . un i t s ) D i s t r i bu t i on ( % ) P/P max (a) (b) (c) α O H OD O D O1.2 0204060534 535 536 B ond l eng t h ( Å ) Energy (eV) A D A D A D A D A D A D A D A D A D A D A D FIG. 3. (a) Joint probability distribution of O-H/O-D covalent bond lengths as a function of excitation energies of states with4 a character of liquids H O (red shadows) and D O (dashed black lines). (b) Averaged intensities of states with 4 a characteras a function of A i D j in H O. A i D j indicates the number of acceptor-type (A i ) and donor-type (D j ) H-bonds, respectively.Insets show representative H-bond environments and distributions of quasiparticle wavefunctions (QWs). The excited oxygenatom is shown in black. QWs with opposite signs are depicted in blue and yellow. (c) Distributions of A i D j in liquids H O(red) and D O (grey). heavy water is, therefore, less influenced by NQEs dueto the heavier nuclei mass as evidenced by the strongerH-bonding environment as displayed in Fig. 3(c). Con-sistently, a slightly weaker pre-edge intensity of D O ascompared to H O is seen in both experiment and theory.The main-edge has been assigned to exciton resonanceof b characteristic, which originates from the second ex-cited state in a water monomer . Because it is un-bound, its quasiparticle can no longer be confined withinexcited water molecules. Nevertheless, main-edge is rel-atively low in energy, and a certain degree of localiza-tion remains. As schematically plotted in the inset ofFig. 4(b), quasiparticles of main-edge are largely dis-tributed on the water molecules in first and second co-ordination shells, which gives rise to notable spectral in-tensities determining the main-edge feature. As a result,the main-edge serves as a probe of intermediate-rangeordering of water .In the intermediate-range, the softer liquid structurein H O is evidenced by both the less structured first andsecond coordination shells in the g OO ( r ) [Fig. 2(a)] andthe slightly larger density of 0.10623 (0.10007) atom/˚A of H O than 0.10581 (0.10000) atom/˚A of D O. In theabove, the number outside (within) parentheses denotesthe theoretical (experimental ) values. Therefore, morenonbonded water molecules in H O will flow into inter-stitial regions. Indeed, the light water has a slightly lessdeep first minimum in the g OO ( r ), which is observed inboth experiment and theory in Fig. 2(a). With its denserinterstitial regions, the light water is more disordered. Inorder to quantify the degree of inhomogeneity, we resortto the local structure index (LSI) analysis . The re-sulting distributions of LSI are shown in Fig. 4(a). Basedon the average LSI value (0.0326 ˚A ) of H O, we furtherdecompose the LSI distributions into low LSI (LSI L ) andhigh LSI (LSI H ) regions in Fig. 4(a), which are used toqualitatively describe the disordered and structured con- figurations, respectively. As expected, H O shows itselfas a more disordered liquid through the more prominentpeak in LSI L as compared to D O. This structural differ-ence is responsible for the observed isotope effect in XASspectra at the main-edge.Relative to the structured liquid in LSI H , the mag-nitude of disorder increases significantly in LSI L , whosequasiparticle wavefunctions become more localized in realspace simultaneously as shown in Fig. 4(c). The disorder-promoted excitation localization is a well-known effectin semiconductors due to the enhanced backscatteringprocesses . The similar mechanism applies in wa-ter; the surrounding water molecules, similar to defectsin semiconductors, serve as the scattering center. Be-sides, the enhanced localizations also stabilize the ex-citonic states and give rise to larger transition matrixelements. The above can be clearly seen by the system-atically red-shifted energies and much stronger spectralintensities at the main-edge of LSI L than those of LSI H in Fig. 4(b). The light water is composed of a slightlylarger fraction of LSI L , therefore, the resulting main-edgeof XAS in H O is located at lower energy with higher am-plitude relative to that of D O in Fig. 1(a).The electronic excitations at the post-edge are exci-ton resonant states as well, which share the same b in orbital characteristic . However, they are muchhigher in energy than those in the main-edge. Not sur-prisingly, the remaining localization in the main-edge iscompletely absent. At post-edge, the exciton resonancesbecome Bloch-like states that distribute over the entirespace of liquids . The above delocalized nature makesthe post-edge an indicator of the long-range ordering ofthe H-bond network of water. It can be seen in Fig. 1that the isotope effect at post-edge is negligible, which isconsistent with the almost identical g OO ( r ) beyond thesecond coordination shell [Fig. 2 (a)]. LSI H LSI L I n t en s i t y ( a r b . un i t s ) H O D O H LSI L P r obab ili t y ( % ) H OD OLSI L LSI H LSI L disordered structured LSI H I n t en s i t y ( a r b . un i t s ) Localization (Å)
FIG. 4. (a) Probability distributions of LSI of liquids H O (solid red line) and D O (dashed black line). The yellow and greenshadows divide H O molecules into two parts: LSI L and LSI H . Insets illustrate molecular configurations schematically, witheach dot represents a water molecule. The grey and red areas show the first and second coordination shells, respectively. (b)XAS spectra of liquid H O contributed by water molecules in part LSI L (solid orange line) and part LSI H (dashed green line).The XAS spectra of D O contributed by LSI L and LSI H molecules have the same trend and are presented in the SupplementalMaterial . Insets show representative QWs of excited water molecules in part LSI L (upper panel) and part LSI H (lower panel).(c) Intensities and excitation energies of excitons in the part LSI L (yellow circles) and part LSI H (green circles). Colors ofthe circles show the localization distance of QWs, which is defined as the radial distance from the excited oxygen atom thatincludes 80% of QWs. Excitons with excitation energy ∈ [537, 539] eV and intensity larger than 0.5, which contribute to theformation of the main-edge, are presented. IV. CONCLUSION
In conclusion, we have studied the isotope effect inXAS spectra of water by advanced theoretical methods.The electron-hole excitation was modeled by quasiparti-cle approach to solving the Bethe-Salpeter equation ap-proximately. Facilitated by machine learning techniques,the liquid structures were generated from PI-DPMD sim-ulations with the accuracy of the SCAN meta-generalizedgradient approximation functional. Our theoretical simu-lations have reproduced the isotope effect in XAS spectraof water semi-quantitatively, with the isotopic XAS spec-tral differences slightly larger than experimental results,which are expected to be improved in future studies withmore accurate methods in the description of molecularstructure and electron-hole interactions. The observedblueshifts of spectral energies with weaker intensities, onpre- and main-edge, indicate that the heavy water has aslightly more structured H-bond network in short- andintermediate-range than normal water. This is due tothe intricate competing effects from NQEs that affectthe heavy water slightly less than normal water. Thesuccessful theoretical modeling of the delicate isotope ef-fect on XAS spectra will provide an important means for further exploration of the delicate nature of the H-bondnetwork of water.
ACKNOWLEDGMENTS
This work was supported by National Science Founda-tion through Awards No. DMR-1552287. This researchused resources of the National Energy Research ScientificComputing Center, which is supported by the U.S. De-partment of Energy (DOE), Office of Science under Con-tract No. DE-AC02-05CH11231. The work of F. T. andL. Z. (Deep learning molecular dynamics) was supportedby the Computational Chemical Center: Chemistry inSolution and at Interfaces funded by the DOE underAward No. DE-SC0019394. This work of J. X. (SCAN-based PI-AIMD) was supported as part of the Center forthe Computational Design of Functional Layered Mate-rials, an Energy Frontier Research Center funded by theU.S. Department of Energy, Office of Science, Basic En-ergy Sciences, under Grant No. DE-SC0012575. 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