Electronic properties of water Giulia Galli University of California, Davis

1. Electronic properties of water Giulia Galli University of California, Davis http://angstrom.ucdavis.edu/

2. 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

3. Hydrogen Bonds ~ 3.6 bonds /molecule, consistent with several expt. Standard model is challenged by recent XAS experiments Tetrahedral network The first coordination shell contains~ 4.2 molecules Comparison between XAS spectrameasured 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 Ph.Wernet et al. Science 2004 (A. Nilsson’s group, Stanford) Ab-initio simulations of water at interfaces are carried out at 350/400 K instead of 300 K Basic physical picture as provided by standard, quasi-tetrahedral model, is reproduced by DFT/GGA

4. Flat valence “bands”; highly dispersive low-lying conduction states with delocalized character (poorly described by MD cells with less than 256 molecules) Isolated LUMO The electronic properties of water are qualitatively similar to those of ice—important details are different D.Prendergast and G.G, JCP 2005. 64 molec.; G pt. “Isolated” excited state (LUMO) found in “small” cell/G 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.

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

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

7. 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)

8. 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)

9. 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 core conduction Calculations of XAS spectra within Density Functional Theory/GGA D.Prendergast and G.G, PRL 2006 Very good agreement between theory and experiment for ice (cubic and hexagonal); good, qualitative agreement for water (salient features reproduced)

10. Both disorder of oxygen lattice and broken hydrogen bonds determine differences between ice and water XAS • 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) Broken Hydrogen Bonds Disorder

11. Possible asymmetry in HB of liquid water (?) 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.

12. 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

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

14. A Spectral representation of Green functions GWa= generalization of the HF approximation, with a dynamically screened Coulomb interaction Plasmon-pole approx. (Hybertsen and Louie, 1986) F. Aryasetiawan and O.Gunnarson, review on “The GW method”, Rep. Phys. 1998

15. Bethe Salpeter to describe electron-hole interaction Quasi particle corrections to LDA energies Scaling N4 (N, number of electrons) 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.

16. Excited state properties of water using the GW approximation • Geometry of 16 equilibrated TIP4P water molecules generated from classical simulations • Unit cell size: 14.80 a.u.3 • DFT - GGA (PBE) • Norm-conserving PSP (TM) • Kinetic energy cutoff: 30 Ha • K-point sampling: 4x4x4 uniform grid • Code: ABINIT + parallelization

17. shift 1.22 eV Eg=4.52 eV EgGW=8.66 eV shift -2.92 eV GW correction on water band gap GGA band structure GW correction

18. GW band gap at  point [1]. V. Garbuio, et al. , Phys. Rev. Lett. 97:137402, 2006. [2]. A. Bernas, et al., Chem. Phys., 222:151, 1997. Deyu Lu et al. 2007

19. Dielectric matrix and eigen modes Alder-Wiser formalism The size of the matrix scales as npw2nqn Decompose the static dielectric matrix into eigenmodes:

20. 32 48 local dielectric response construct MLWFs 16 Locality of the dielectric modes dielectric band structure • 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 65-96 involve nearest neighbors. +

21. How many dielectric eigenmodes are needed to determine the quasi-particle band gap? GW implementation starting from DFT/GGA orbitals 1 denotes (x1, y1, z1, t1) Convergence is slow!

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

23. + + + + Decomposition of the dielectric modes

24. 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.

25. 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 IBM@Watson), NERSC