Electronic properties of water Giulia Galli University of California, Davis

Outline Electronic properties of water as obtained using DFT/GGA Interpretation of X-Ray-Absorption (XAS) spectra Electronic structure properties beyond GGA: GW results for water and approximate dielectric matrices

Hydrogen Bonds ~ 3.6 bonds /molecule, consistent with several expt. Basic physical picture as provided by standard, quasi-tetrahedral model, is reproduced by DFT/GGA Tetrahedral network The first coordination shell contains~ 4.2 molecules Ab-initio simulations of water at interfaces are carried out at 350/400 K instead of 300 K Standard model is challenged by recent XAS experiments Ph.Wernet et al. Science 2004 (A. Nilsson’s group, Stanford) Comparison between XAS spectra measured for ice, ice surface and water with those obtained using structures from simulations and electronic structure from DFT, was used to suggest liquid water has only ~ 2 instead of ~4 HB/molecule

The electronic properties of water are qualitatively similar to those of ice—important details are different “ Isolated” excited state (LUMO) found in “small” cell/  point calculations of liquid water is unphysical [origin is numerical accuracy, e.g. k-point/BZ folding effect] and unrelated to LUMO of water dimer. Isolated LUMO 64 molec.; G pt. Flat valence “bands”; highly dispersive low-lying conduction states with delocalized character (poorly described by MD cells with less than 256 molecules) D.Prendergast and G.G, JCP 2005.

Occupied and empty single particle electronic states in ice Band structure No “lone” state in ice

Water and ice band structures Lone state found in “small” cell/  point calculations is a k-point (BZ folding) effect Representative config. of liquid water Ice

Convergence of unoccupied e-subspace of water requires several k-points in 32 (64) molecule cells or simulations with at least 256 molecules Electronic structure calculations on long classical trajectories (TIP4P )

Convergence of unoccupied e-subspace of water requires several k-points in 32 (64) molecule cells or simulations with at least 256 molecules Electronic structure calculations on long classical trajectories (TIP4P )

Convergence of unoccupied e-subspace of water requires several k-points in 32 (64) molecule cells or simulations with at least 256 molecules Electronic structure calculations on long classical trajectories (TIP4P )

Calculations of XAS spectra within Density Functional Theory/GGA Electronic excitations described by Fermi golden rule; excited electron in conduction band treated explicitly Pseudopotential approximation TIP4P MD (1 ns) for cells with 32 water molecules 10 uncorrelated snapshots; average over 320 computed XAS spectra Up to 27 k-pts to sample BZ conduction core Very good agreement between theory and experiment for ice (cubic and hexagonal); good, qualitative agreement for water (salient features reproduced) D.Prendergast and G.G, PRL 2006

Both disorder of oxygen lattice and broken hydrogen bonds determine differences between ice and water XAS Broken Hydrogen Bonds Disorder All current theoretical approaches (FCH, HCH, XCH) are consistent with available measurements and with quasi-tetrahedral model Experimental results only partially understood Improvement in the theory (description including SIC and possibly beyond DFT) needed to fully understand experimental data. L.Pettersson’s group (Sweden) E.Artachos’s group (Cambridge, UK) R.Car’s group (Princeton) R.Saykally’s group (UCB)

Standard model of water withstand x-ray probe

No evidence justifying the dismissal of quasi- tetrahedral model, based on current interpretations of XAS experiments Measured XAS spectra are only partially understood. Open question: how to get to a thorough, complete account of measured XAS using a sound electronic structure theory. This is first and foremost an electronic structure problem, not (or at least not yet) a structural determination problem. Once we have solved in a robust and convincing fashion the electronic structure problem, if issues in the interpretation of measured XAS remain, we may go back and ask questions about current structural models. Possible asymmetry in HB of liquid water (?)

Electronic properties of inorganic molecules in water: DFT and QMC study Removing water does not alter trend in absorption gap Solvated Unsolvated The impact of H 2 O screening on optical properties of Si clusters is negligible; thermal strain provides dominant impact D.Prendergast, J.Grossman et al. JACS 2004

Excited state properties of water beyond DFT/GGA QMC may work for optical gaps and other specific energy differences (e.g. Stoke shifts), but it is difficult to generalize to spectra calculations Need for affordable and accurate calculations of excited state properties beyond DFT is widespread (e.g. realistic environment –solvation model for excited states; nanostructures for a variety of applications; systems under pressure; molecular electronics….): —GW results —Approximate dielectric matrices

Hamiltonian of the system Kohn-Sham equations Quasi-particles Green Functions and Perturbation Theory Dyson Equation Quick reminder on GW approx.

Spectral representation of Green functions GWa= generalization of the HF approximation, with a dynamically screened Coulomb interaction F. Aryasetiawan and O.Gunnarson, review on “The GW method”, Rep. Phys Plasmon-pole approx. (Hybertsen and Louie, 1986) A

Bethe Salpeter to describe electron-hole interaction Quasi particle corrections to LDA energies M.Plummo et al. review on “The Bethe Salpether equation: a first principles approach for calculating surface optical spectra”, J.Phys Cond Matt. 2004; and Rev.Mod.Phys. Reining et al. Scaling N 4 (N, number of electrons)

 Geometry of 16 equilibrated TIP4P water molecules generated from classical simulations  Unit cell size: a.u. 3  DFT - GGA (PBE)  Norm-conserving PSP (TM)  Kinetic energy cutoff: 30 Ha  K-point sampling: 4x4x4 uniform grid  Code: ABINIT + parallelization Excited state properties of water using the GW approximation

GGA band structure GW correction Eg=4.52 eV shift 1.22 eV shift eV Eg GW =8.66 eV GW correction on water band gap

GW band gap at  point # of H 2 O # of k points GGA gap (eV) ∆GW HOMO ∆GW LUMO ∆GW gap (eV) GW gap (eV) this work Ref. [1] config config config EXP [2] 8.7±0.5 [1]. V. Garbuio, et al., Phys. Rev. Lett. 97:137402, [2]. A. Bernas, et al., Chem. Phys., 222:151, Deyu Lu et al. 2007

Dielectric matrix and eigen modes Alder-Wiser formalism Decompose the static dielectric matrix into eigenmodes: The size of the matrix scales as npw 2  nq  n 

Locality of the dielectric modes the first 64 eigen modes belong to intra-molecular screening (O,O-H,lone pair). the higher modes correspond to inter-molecular screening. in particular, the screening of modes involve nearest neighbors. local dielectric response construct MLWFs + 32 dielectric band structure

How many dielectric eigenmodes are needed to determine the quasi-particle band gap? Convergence is slow! GW implementation starting from DFT/GGA orbitals 1 denotes (x 1, y 1, z 1, t 1 )

dielectric eigenmodes of the system construct V i, (q) according to the orthogonality condition, and  i (q) from the Penn model model dielectric response + the Penn model D. Penn, Phys. Rev., 128: 2093,1962. Decomposition of the dielectric modes

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GW calculations and approximate dielectric matrices  The locality of the static dielectric matrix of liquid water has been characterized by the MLDMs.  The effect of the dielectric response can be separated into localized (intra-molecular screening and inter-molecular screening within nearest neighbors) modes and delocalized modes.  The contribution of the delocalized modes can be replaced by model dielectric response.  Hybrid dielectric matrices including only a small number of true dielectric eigenmodes yield good accuracy in quasiparticle energy calculations.

Many thanks to my collaborators David Prendergast (UCB) Deyu Lu (UCD) Francois Gygi (UCD) Thank you! Support from DOE/BES, DOE/SciDAC and LLNL/LDRD Computer time: LLNL, INCITE AWARD (ANL and NERSC