X-ray absorption of liquid water by advanced ab initio methods

Oxygen K-edge x-ray absorption spectra of liquid water are computed based on configurations from advanced ab initio molecular dynamics simulations, as well as an electron excitation theory from the GW method. One the one hand, the molecular
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  PHYSICAL REVIEW B  96 , 104202 (2017) X-ray absorption of liquid water by advanced  ab initio  methods Zhaoru Sun, 1 Mohan Chen, 1 Lixin Zheng, 1 Jianping Wang, 1 Biswajit Santra, 2 Huaze Shen, 3 Limei Xu, 3 Wei Kang, 4 Michael L. Klein, 1,5,6 and Xifan Wu 1,6,* 1  Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA 2  Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA 3  International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China 4 College of Engineering, Peking University, Beijing 100871, China 5  Department of Chemistry, Temple University, Philadelphia, Pennsylvania 19122, USA 6  Institute for Computational Molecular Science, Temple University, Philadelphia, Pennsylvania 19122, USA (Received 17 April 2017; revised manuscript received 6 July 2017; published 11 September 2017)Oxygen  K -edge x-ray absorption spectra of liquid water are computed based on configurations from advanced ab initio  molecular dynamics simulations, as well as an electron excitation theory from the GW method.One the one hand, the molecular structures of liquid water are accurately predicted by including both vander Waals interactions and a hybrid functional (PBE0). On the other hand, the dynamic screening effects onelectron excitation are approximately described by the recently developed enhanced static Coulomb-hole andscreened-exchange approximation of W. Kang and M. S. Hybertsen [Phys. Rev. B  82 , 195108 (2010)]. The resulting spectra of liquid water are in better quantitative agreement with the experimental spectra due to thesoftened hydrogen bonds and the slightly broadened spectra srcinating from the better screening model.DOI: 10.1103/PhysRevB.96.104202 I. INTRODUCTION Water is arguably one of the most important materials onEarth and needs to be thoroughly understood [1]. However, the understanding of liquid water is itself a challenge in manyaspects. Unlike other liquids, water shows a lot of anomaliessuch as the density maximum at 4 ◦ C and the isobaric heat-capacity minimum at 35 ◦ C, among many other things [2,3]. Understanding the microscopic structures and dynamics of liquid water, in particular its hydrogen-bond (HB) network,is the key to understanding these anomalies [4–7]. Recently, high-resolution oxygen  K -edge core-level spectroscopy, suchasx-rayabsorptionspectra(XAS)andx-rayRamanscattering,has emerged as a powerful experimental technique to probethe electronic structure and infer the molecular structure of water and ice, as well as aqueous solutions [6,8–22]. Excited from the oxygen 1 s  core level, the electron excitation probesthe unoccupied electronic states, which are antibonding alongthe covalent OH bonds and particularly sensitive to the HBs.Therefore, the XAS technique serves as a local probe for theHB structures of liquid water and ice.The experimental XAS of water have three distinct featuresas a function of increasing excitation energies: a preedge startsfrom the absorption threshold at 533 to 536 eV with a peakcentered at 535 eV, a main edge spans from 537 to 539 eV, anda postedge exists from 539 eV and beyond [11,18,20–22]. Ex- perimentally,thepreedgefeatureispresentinbothliquidwaterandicebutismoreintenseintheformer.Therelativeintensitiesbetween the main edge and postedge of liquid water andcrystalline ice Ih are substantially different [6,8–11,20–24]. As one of the most qualitative differences, the intensity of themain edge is higher than that of the postedge in the XAS of liquid water, while the opposite trend is true for the spectra inthe ice. * Corresponding author: The unambiguous assignments of the XAS features to theunderlying HB structures are prerequisites for the physical in-terpretationoftheexperimentalspectra,whichcanbeachievedby first-principles methods including both the modeling of the molecular structure of liquid water and the electron-holeexcitation process. With snapshots of the represented molec-ular configurations from an equilibrated molecular dynamicstrajectory, the XAS can be computed with the knowledge of electronic structures of the excited core hole. In this regard,variousapproximations[17,25,26]forexcitedelectronicstates havebeenproposedwithintheframeworkofdensityfunctionaltheory (DFT) [27,28]. In the seminal work of Prendergast and Galli [17], the proposed excited electron and core-hole approximation yielded XAS in close agreement with exper-imental measurements. More rigorously, the XAS of liquidwatercanbecomputedbysolvingtheBethe-Salpeterequation(BSE)describingtheelectron-hole interaction[15,29,30].The BSEapproachinvolvescalculationsoftheself-energyoperatorand quasiparticles, which are, in general, computationallyexpensive for liquid water. An approximate way of solvingthe BSE with fewer computational resources was introducedby Chen  et al.  [9] based on a model electron-screeningfunction in the static Coulomb-hole and screened-exchange(COHSEX) approximation [31], which was evaluated usingthe maximally localized Wannier functions as basis [32], reducing the computational cost significantly. In the abovework, the molecular srcins of the spectra features as well asthe intensity difference between the main edge and postedgein water and ice have been validated [9]. Despite these recent developments, two uncertainties stillremaininthetheoreticalXASofliquidwaterobtainedwiththeapproximate solutions of the BSE. The first uncertainty comesfrom the drawback in modeling the liquid-water structure byemploying the generalized gradient approximation (GGA) inthe framework of DFT [33–43]. It is known that GGA predicts overstructured liquid water [40–43]. Thus, a significantly elevated temperature is often adopted to generate a softer HB 2469-9950/2017/96(10)/104202(9) 104202-1 ©2017 American Physical Society  ZHAORU SUN  et al.  PHYSICAL REVIEW B  96 , 104202 (2017) structureclosertotheexperimentalmeasurement[41–43].The lack of physics in accurately describing the water structure isdue to the neglected van der Waals (vdW) interactions and thespurious self-interaction error [44] in the GGA functional. Specifically, by including the vdW interactions, the waterpopulation in the interstitial region between the first andsecond coordination shells of water molecules is increasedto better match experiment [43]. Furthermore, by mitigatingthe self-interaction error through the hybrid functional, thedirectional HB strength between water molecules is weakenedso it is closer to the experiment; as a result, the protonsare less easily donated to neighboring water molecules [43]. However, the effects of this improved water structure on thetheoretical XAS have not been elucidated. Second, in theseries of computational works adopting the static COHSEXapproximation for the electron-hole excitation of liquid water,a homogeneous electronic screening model was first adopted[9] and then extended by using the Hybertsen-Louie ansatz [45] to account for the inhomogeneous screening effects from themolecularenvironment[10].However,somediscrepanciesstill exist, and it is not yet clear to what extent the dynamicscreening effect will affect the quasiparticle wave functions(QWs) and the computed XAS. For example, it was observedthat the width of the theoretical XAS by static COHSEX isslightly narrower than the experimental data [10]. In an effort to address the above issues, we adopted asystematic way to study the XAS of liquid water at ambientconditions.Specifically,weusedmoreadvanced abinitio mod-eling of molecular structures and electronic excitations. Wegeneratedliquid-watertrajectoriesfromthe abinitio moleculardynamics (AIMD) [46] simulations by employing a hierarchyof exchange-correlation (XC) functionals of Perdew, Burke,and Ernzerhof (PBE) [47], PBE with the vdW interactions in the form of Tkatchenko and Scheffler (PBE + vdW) [48], and the hybrid functional PBE0 [49,50] with vdW interactions (PBE0 + vdW). By utilizing the enhanced static COHSEXmethod to treat the excitations [51], we find that the XAScomputed from the snapshot generated by the PBE0 + vdWfunctional agree well with the experiment among the three XCfunctionals studied. The vdW interactions soften the waterstructures by increasing the population of water molecules inthe interstitial region, while the hybrid functional mitigatesthe self-interaction error and weakens the HB strength so itis closer to experiment [43]. Both structural corrections and improved excitation theory are crucial in giving rise to anoverall improvement of the three edges of XAS. In particular,the postedge feature in the high-energy region of XAS is inslightly better agreement with experiment because of betterscreening modeling using the enhanced static COHSEX. Inaddition, we also compare a set of XAS computed fromdifferent excitation theories to the experiment in order toshow the importance of self-consistently diagonalized QWsin capturing the qualitative features of XAS. II. METHODS We performed AIMD simulations to generate liquid-water trajectories using a modified version of the  QUANTUMESPRESSO  package [52]. We simulated 128 water molecules ina periodic cubic cell with a cell length of 15.68 ˚A using theCar-Parrinello molecular dynamics (CPMD) [46] within the canonical ( NVT  ) ensemble. We employed norm-conservingpseudopotentials in the form of Troullier and Martins [53] and set the kinetic-energy cutoff of the electronic wave functionsas 71 Ry. We used a hierarchy of XC functionals, includingPBE, PBE + vdW, and PBE0 + vdW, as mentioned. The hybridfunctional PBE0 with a mixing of 25% exact exchange wasevaluated in a linear-scaling manner by taking advantageof maximally localized Wannier functions [32]. The ionictemperatures were controlled through Nosé-Hoover chainthermostats with a chain length of 4 for each ion [54–56]. All AIMD simulations were performed at 330 K, where anincrease of 30 K has been found to mimic the nuclear quantumeffectinstructuralquantitiessuchastheoxygen-oxygenradialdistribution function in DFT-based simulations of liquid water[57]. A time step of 4.0 a.u. and a fictitious electron mass of  300 a.u. were chosen.We calculated the x-ray absorption cross section usingFermi’s golden rule: σ  ( ω )  =  4 π 2 α 0 ¯ hω  f  | M  if  | 2 δ ( ω if   −  ω ) ,  (1)where  α 0  is the fine-structure constant and ¯ hω  is theabsorbed photon energy matching the energy difference¯ hω if   =  E f   −  E i .  E f   and  E i  are the eigenvalues of the finaland initial states, respectively.  M  if   are the transition matrixelements between the initial state  | φ i   and final state  | φ f   ,which can be evaluated within the electric-dipole approxima-tion as  M  if   ∼  φ i | x  | φ f   , averaging over the three Cartesiandirections.Wetakethe1 s  atomiccorewavefunctionfromDFTcalculations as the initial state  | φ i  . For the final state  | φ f   , weapply a self-consistent diagonalization procedure within theenhanced static COHSEX approach [51], and the details areas follows.The final state  | φ f    is obtained by utilizing the enhancedCOHSEX approach, which has been implemented withinthe framework of the CPMD [46] scheme. Specifically, the COHSEXmethodhasbeenimplementedintheCPMDmodulewithin the  QUANTUM ESPRESSO  package [52]. For each set of  input wave functions  | φ f   , we fix the ion positions and dampthe wave functions of the system in a self-consistent way,as explained below. Note that the formulas we describe hereare suitable only for the excitation theory we adopt and areindependent of the other CPMD simulations we performedwith vdW and PBE0 functionals.First, the Lagrangian from the Car-Parrinello approach is L  = µ 2  i   ˙ ψ i |  ˙ ψ i  +  12  I  M  I   ˙ R 2 I   −  E tot ( R , { ψ } ) + λ ij  (  ψ i | ψ j   −  δ ij  ) ,  (2)where  ν  is a fictitious mass of electrons and  ψ i  is the orbitalof state  i .  M  I   is the mass for atom  I   that is located at  R I  . E tot ( R , { ψ } ) is the total energy calculated from first-principlesmethods. The last part is the orthogonality constraint imposedon orbitals by the Lagrangian multiplier to be  λ ij  . Note thatthe initial ion positions are fixed, and only wave functions aregenerated. Therefore, we need to damp the wave functionstowards the ground state, which is realized via the equations 104202-2  X-RAY ABSORPTION OF LIQUID WATER BY ADVANCED . .. PHYSICAL REVIEW B  96 , 104202 (2017) of motion of electrons in plane-wave basis set, µ  ¨ ψ i  = −  H  ( R , { ψ } ) ψ i  +  j  λ ij  ψ j  .  (3)Here,  H  ( R , { ψ } ) is the Hamiltonian matrix of the system.Furthermore, the Hamiltonian part can be evaluated in realspace as  H  ψ ( r )  =  − 12 ∇  2 ( r )  +  V  ext ( r , R )  +  V  H  ( r )  ψ ( r ) (4) +    d  r ′  ( r , r ′ ,E ) ψ ( r ′ ) ,  (5)where  V  ext ( r , R ) is the external potential and  V  H  ( r ) is theHartree potential. In particular,   ( r , r ′ ,E ) is the self-energyoperator that is nonlocal in real space and depends on theself-energy  E . In the static COHSEX approximation, theself-energy operator can be approximated as  staticCOHSEX ( r , r ′ )  =   staticCOH ( r , r ′ )  +   staticSEX  ( r , r ′ ) .  (6)The first part is  staticCOH ( r , r ′ )  =  12 δ ( r  − r ′ ) W  p ( r , r ′ ; E  =  0) (7) =  12 δ ( r  − r ′ )( W   −  v ) ,  (8)where  W  p  is the Coulomb hole,  W   is the screened Coulombinteraction, and  v  is the bare Coulomb interaction. TheHybertsen-Louie ansatz [45] proposed that  W   generallyfollows the local charge density and has the form W  ( r , r ′ ; E  =  0)  =  12 { W  [ r −  r ′ ; ρ ( r ′ )]  +  W  [ r ′  −  r ; ρ ( r )] } , (9)where  W   can be written as W  [ r ′  − r ; ρ ( r )]  = 12 π 3    ǫ − 1 [ q ; ρ ( r )] v ( q ) e i q · ( r − r ′ ) d  q .  (10)Here, we use the Bechstedt model [58] for the dielectric function, 0 ǫ [ q ,ρ ( r )]  =  1  +  ( ǫ 0  −  1) − 1 +  α ( q/q TF ) 2 +  q 4  43 k 2 F  q 2TF  − 1 ,  (11)where  q TF  is the Thomas-Fermi wave vector,  k F  is the Fermiwave vector,  ǫ 0  is taken from experiment, and  α  is fixed bymatching the Bechstedt model to the  q 2 dependence of thePenn model [59]. Next, we can transform  W   to W  [ r ′  − r ; ρ ( r )]  = v ( r  − r ′ ) ǫ 0 − 1 a ( x 1  −  x 2 ) | r ′  − r |×  e i √  x 1 | r ′ − r | x 1 − e i √  x 2 | r ′ − r | x 2  .  (12)Here,  x 1 , 2  =  ( − b  ±√  b 2 −  4 ac ) / 2 a ,  a  =  ( 43 k 2 F  q 2TF ) − 1 ,  b  = α/q 2TF , and  c  =  ǫ 0 / ( ǫ 0  −  1). The second part is  staticSEX  ( r ′ , r )  = − occ  i ψ i ( r ) ψ ∗ i  ( r ′ ) W  ( r , r ′ ; E  =  0) .  (13)Numerically, it has been shown that most of the error fromusing the static COHSEX approximation, when compared tothe GW approximation, comes from the short-wavelength partof the assumed adiabatic accumulation of the Coulomb hole W  p , namely, the COH part, while the SEX term in the staticCOHSEXapproximationyieldsvaluesrelativelyclosetothoseof the GW calculations [51]. Therefore, the enhanced staticCOHSEX was proposed to introduce a universal function f   to approximately include the dynamics screening in thesrcinal static model in the COHSEX formula. Specifically,the enhanced static COH term can be evaluated as  newCOH ( r , r ′ )  = δ ( r −  r ′ )2    W  p ( q ; E  =  0) f  ( q/k f  ) e − i q · r d  q , (14)where q  isaplanewaveand k f   istheFermivector.Thescalingfunction is f  ( x )  = 1  +  a 1 x  +  a 2 x 2 +  a 3 x 3 +  a 4 x 4 +  a 5 x 5 +  a 6 x 6 1  +  b 1 x  +  b 2 x 2 +  b 3 x 3 +  b 4 x 4 +  b 5 x 5 +  b 6 x 6 , (15)where  a 1  =  1 . 9085,  a 2  = − 0 . 542572,  a 3  = − 2 . 45811,  a 4  = 3 . 08067,  a 5  = − 1 . 806,  a 6  =  0 . 410031,  b 1  =  2 . 01317,  b 2  =− 1 . 55088,  b 3  =  1 . 58466,  b 4  =  0 . 368325,  b 5  = − 1 . 68927,and  b 6  =  0 . 599225.Above,wedescribedtheprocedurestocomputeXASbasedon an excited water molecule in a snapshot. We then excitedevery water molecule in the 128-molecule supercell in orderto sample the different local environments of the disorderedliquid-water structure. In our case, we found converged XASafter randomly exciting 64 water molecules in the snapshot.Wenotethatinapreviousstudy,10individualanduncorrelatedsnapshots of 32 water molecules were chosen, and onlysmalldifferenceswereobservedbetweenthesesnapshots[17]. Therefore, we consider one snapshot of a large cell to besufficient to yield meaningful results and take a representativesnapshotfromeachAIMDtrajectoryreflectingtheequilibratedstructure of liquid water to compute XAS.Due to the different local environments in liquid water foreach excited oxygen atom, we adopted the core-hole energyshift of each excitation by following Ref. [60], which is astandard approach to compute the core-level shifts. We usedGaussian broadening of 0.4 eV for all spectra, which wasused in previously calculated XAS of liquid water [9,10]. The computed XAS were aligned to the onset of the preedge(535eV)andthennormalizedtothesameareaofexperimentaldata ranging from 533 to 546 eV.To analyze the real-space locations of QWs in terms of the three edges in XAS of liquid water, we also define theone-dimensional density of the QW as ρ i ( r )  =    | ψ i ( r,θ,φ ) 2 | dθdφ  (16)along the radial direction, and the origin is taken to bethe position of the excited oxygen with a core hole. Inthe equation,  i  represents the index of the conduction bandwith excitation energy  ε i . In order to present the differentlocalizations of the QWs, the index  i  is chosen so that ε i  ∈  [534 . 5 eV , 535 . 5 eV] for QWs representing the preedge 104202-3  ZHAORU SUN  et al.  PHYSICAL REVIEW B  96 , 104202 (2017) pre-edge0123g OO (r) main-edge0123g OO (r) post-edge0 1 2 3 4 5 6 7 Distance (Å) 0123g OO (r) (a)(b)(c) FIG. 1. DensitydistributionsoftheQW(greendashedline)of(a)the preedge, (b) main edge, and (c) postedge as a function of oxygen-oxygen distance computed from the snapshot of the PBE0 + vdWtrajectory using the enhanced static COHSEX method. The  g OO ( r )(red line) from the PBE0 + vdW trajectory is shown for comparison.The insets show the representative QWs of the three edges around theexcited water molecule. Water molecules residing within the secondcoordination shell of the excited oxygen are shown. Red, white, andyellow spheres represent oxygen, hydrogen, and oxygen atoms witha core hole, respectively. QWs with opposite signs are depicted inblue and green. of XAS,  ε i  ∈  [537 eV , 539 eV] for QWs representing themainedge of XAS, and  ε i  ∈  [540 eV , 542 eV] for QWsrepresenting the postedge of XAS. III. RESULTS AND DISCUSSIONA. HB structure probed by the preedge, main edge,and postedge of XAS The QWs can be strongly perturbed by the local liquid-water structures. The preedge, main edge, and postedge of XAS are found to have distinguishable molecular signaturesthat relate to different spatial regions of the HB network[8,9,18]; the preedge has 4 a 1  character, while the main edgeand postedge have  b 2  character, both of which srcinate fromthe molecular excitations in the gas phase [8]. In order to quantitatively study the spatial regions in terms of differentXAS edges, we present the density distributions of QWs as afunction of oxygen-oxygen distance in Fig. 1. The oxygen- oxygen radial distribution function  g OO ( r ) and an excitedoxygen with QWs distributed within the HB network (insets)are also shown for comparison. Overall, Figs. 1(a), 1(b), and 1(c) show that the density distributions of QWs becomemore delocalized from the preedge to the main edge to thepostedge.ThedensitydistributionofQWsofthepreedgeillustratedinFig. 1(a) has the highest peak at 1.7 ˚A and is mostly localizedwithin 2.75 ˚A, the latter of which is the first peak positionof   g OO ( r ). The inset in Fig. 1(a) shows that the QW of thepreedge resembles the first excited state of a water molecule inthe gas phase with 4 a 1  symmetry. In this regard, our result isconsistent with a previous assignment [9] of the preedge to abound exciton state,wherethe electronorbitalwasfound tobemostly localized within the first coordination shell. Therefore,the preedge features can be largely affected by the short-rangestructures, such as broken HBs and covalent bond strength,aroundtheexcitedoxygens. WiththeweakenedHBsdescribedby the hybrid DFT functional (PBE0) and vdW interactions,that is, the short-range HB network, therefore, the computedpreedge of XAS is expected to be improved in both energiesand intensities.As shown in Fig. 1(b), the density distribution of the QWof the main edge is more delocalized than that of the preedge.The inset shows that the QW of the main edge can be foundnot only on the excited molecule itself but also on its first-and second-shell neighbors. In addition, a clear  b 2  charactercan be identified for a typical QW of the main edge. Theabove is consistent with the fact that the main edge wasfound to srcinate from the second excited state of a watermolecule in the gas phase. By comparing the localization of the main-edge density of the QW to that of the preedge, it canbe seen that the former is more localized between the first andsecondcoordinationshellsoftheliquid-waterstructure.Inthisregard, it is expected that the main-edge feature of XAS willbe sensitive to the intermediate-range order ofthe liquid-waterstructure, i.e., water molecules in the interstitial region.In contrast to the density distributions of QWs of thepreedge and main edge, the QW of the postedge shown inFig. 1(c) is the most delocalized one but still preserves  b 2 character. The strong delocalization can be clearly seen by theincreased density of the QW as a function of the distanceaway from the excited water molecule. Hence, the watermolecules in long-range order are critical in determining thepostedge features. Using a small simulation cell containing 32water molecules fails to yield a postedge intensity of XAS incloseagreementwithexperimentaldata[9].Thedelocalizationfeatureisconsistentwiththefactthatthepostedgeisaresonantexciton state. B. XAS calculated from PBE, PBE + vdW,and PBE0 + vdW AIMD trajectories Among the three levels of XC functionals investigated,the XAS computed from the snapshot obtained with PBEshow the least agreement with the experimental spectra inFig. 2(a). Four major discrepancies can be identified. First,the intensity of the computed preedge is underestimatedcompared to experiment. Second, the theoretical main edgeis centered around 538.5 eV, showing a large blueshift (around1 eV) compared to the experimental value at 537.5 eV.Third, a significantly overestimated postedge intensity leadsto the fact that the main edge and post edge have almostthe same intensities, which contradicts the experimental fact 104202-4  X-RAY ABSORPTION OF LIQUID WATER BY ADVANCED . .. PHYSICAL REVIEW B  96 , 104202 (2017) (e) (f) 2 3 4 5 6 r (Å) 00.511.522.53   g    O  -   O    (  r   ) Exp.PBE2 3 4 5 6 r (Å) 00.511.522.53   g    O  -   O    (  r   ) Exp.PBE+vdW534 536 538 540 542 544 546 Energy (eV) 02468101214    I  n   t  e  n  s   i   t  y   (   A  r   b .   U  n   i   t  s   ) Exp.PBE534 536 538 540 542 544 546 Energy (eV) 02468101214    I  n   t  e  n  s   i   t  y   (   A  r   b .   U  n   i   t  s   ) Exp.PBE+vdW534 536 538 540 542 544 546 Energy (eV) 02468101214    I  n   t  e  n  s   i   t  y   (   A  r   b .   U  n   i   t  s   ) Exp.PBE0+vdW (a) (b) (c)(d) 2 3 4 5 6 r (Å) 00.511.522.53   g    O  -   O    (  r   ) Exp.PBE0+vdW FIG. 2. Computed XAS and  g OO ( r ) from three levels of exchange-correlation functionals used in the AIMD simulations, namely, PBE,PBE + vdW, and PBE0 + vdW. A representative snapshot consisting of 128 water molecules from each equilibrated AIMD trajectory was usedfor spectra calculation. The enhanced static COHSEX method is adopted as the excitation theory. The experimental (Exp.) data of XAS [21]and  g OO ( r ) [61] are also shown for comparison. that the main edge is more prominent than the postedgein liquid water. Fourth, the width of the computed XAS isslightly narrower than the experimental XAS, especially in thehigh-energy region. Experimentally, the XAS of crystallineice Ih with an intact HB network show a more prominentpeak of the postedge than that of the main edge. Therefore,the discrepancies imply that the HB network of liquid waterfrom the PBE AIMD trajectory is overstructured. Indeed, g OO ( r ) computed from the PBE AIMD trajectory significantlydeviates from experimental measurement [61], as shown in Fig. 2(d). Specifically, the first and second peaks of   g OO ( r )from simulations are significantly overestimated, and the firstminimum is largely underestimated. Consistently, the averagenumber of HBs per water molecule is found to be 3.76 inthe PBE trajectory according to the popular HB definitionproposed by Luzar and Chandler [62], which is the highestamongallthefunctionalsstudiedherein.Alltheaboveindicatethat the HB network of liquid water is overstructured from thePBE AIMD trajectory. Not surprisingly, the theoretical XASpredicted by an overstructured HB network from the PBEAIMD trajectory yield icelike spectra with a relatively moreprominent postedge feature.As shown in Fig. 2(b), the XAS computed from the PBE + vdW trajectory are largely improved with regard to theexperimentalspectracomparedtothoseobtainedfromthePBEtrajectory. The improvement can be seen by a higher preedgeintensity, a shift of the main edge towards lower energy, anda lower postedge intensity. The better agreement is attributedto the improved description of the HB network by includingthe vdW interactions in the AIMD simulation. An explicitaccount of vdW forces strengthens the attractive interactionsamong water molecules, which significantly increase thepopulation of water molecules in the interstitial region. Theincreased population of water molecules brings  g OO ( r ) incloser agreement with experiment within the first and secondcoordination shells. Moreover, the increased population of water molecules in the interstitial region weakens the HBsamong the water molecules in the first coordination shell,resulting in a reduced first peak in  g OO ( r ). The averagenumber of HBs per molecule is found to be 3.56, whichis about 5% smaller than that of PBE. Hence, an excitedoxygen atom experiences a more disordered environment bythe surrounding water molecules. First of all, the preedgeintensity is increased due to the breaking of more HBs. Inorder to verify this, we selected two excited water molecules,one with broken HBs and the other one with four intact HBs,and plotted their density distributions of QWs in Fig. 3(a).The QW of the preedge from the excited molecule with thebroken HBs is more localized with enhanced  p  character dueto the more disordered short-range molecular environment.Therefore, larger amplitudes of transition matrices  M  ij   areobtainedaccordingtoFermi’sgoldenrule,asshowninEq.(1). Asaresult,thepreedgeintensityisincreasedwithmorebrokenHBs. Second, the main-edge intensity is also enhanced byan increased population of water molecules in the interstitialregionduetovdWinteractions.Inordertofurtherexplainhowthe water molecules in the interstitial region affect the main-edge features, we also selected two representative excitedwater molecules and calculated their density distributions of QWs; one of the excited water molecules has four intact HBs,and the other one has extra neighboring water moleculeswithin the interstitial region in addition to the four intactHBs. Figure 3(b) suggests that the density of the main-edgeQW of the excited molecule with extra neighboring water 104202-5
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