Electronic properties of water

Giulia Galli

University of California, Davis

http://angstrom.ucdavis.edu/

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.; 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)

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)

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 (?)

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 systemKohn-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. 1998

Plasmon-pole approx.

(Hybertsen and Louie, 1986)

A

iG(1,2)W (1,2)

W (1,2) d(3) 1(1,3)v(3,2)

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 N4

(N, number of

electrons)

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

Excited state properties of water using the GW approximation

GGA band structure GW correction

Eg=4.52 eV

shift 1.22 eV

shift -2.92 eV

EgGW=8.66 eV

GW correction on water band gap

GW band gap at point

# of H2O

# of k points

GGA gap (eV)

∆GW HOMO

∆GW LUMO

∆GW gap (eV)

GW gap (eV)

this work 16 64 4.52 -2.92 1.22 4.14 8.66

Ref.[1]

config. 1 17 8 5.09 -1.67 1.61 3.28 8.37

config. 2 17 8 4.71 -1.64 1.60 3.24 7.95

config. 3 17 8 5.29 -1.70 1.60 3.30 8.59

EXP[2] 8.7±0.5

[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

Dielectric matrix and eigen modes

Alder-Wiser formalism

G,G '(q;) G,G' 4

q G q G'G,G'

0 (q;)

G,G '0 (q;) 2 fn

n,m

k

(1 fm )k,q (G)k,q

* (G')

n (k) m (k q)

k,q (G) k,n e i(q G)r k q,m

G,G ' 1 (q)U i,G '(q) i

1U i,G (q)

G,G ' 1 (q) G,G ' Ui,G (q) i

1 1 U i,G'* (q)

i

Decompose the static dielectric matrix into eigenmodes:

The size of the matrix scales as npw2nqn

Dielectric Matrix and Eigenmodes

G,G'(q;) G,G' 4

q G q G'G,G'

0 (q;)

G,G'0 (q;) 2 fn

n,m

k

(1 fm)k,q (G)k,q

* (G')

n(k) m (k q)

k,q (G) k,n e i(q G)r k q,m

)()()(ˆ 1

1 qUqUq iii

i

iii qUqUq )()()(ˆ 11

decompose the static dielectric matrix into eigenmodes

the size of the matrix scales as npw2nqn

M 1/0,0 1 (0,0)

[1]: see B. Bagchi, Chem. Rev., 105:3197, 2005.

M ~ 1.81.72

EXP[1]this work

Alder-Wiser formalism

Dielectric Matrix and Eigenmodes

G,G'(q;) G,G' 4

q G q G'G,G'

0 (q;)

G,G'0 (q;) 2 fn

n,m

k

(1 fm)k,q (G)k,q

* (G')

n(k) m (k q)

k,q (G) k,n e i(q G)r k q,m

)()()(ˆ 1

1 qUqUq iii

i

iii qUqUq )()()(ˆ 11

decompose the static dielectric matrix into eigenmodes

the size of the matrix scales as npw2nqn

M 1/0,0 1 (0,0)

[1]: see B. Bagchi, Chem. Rev., 105:3197, 2005.

M ~ 1.81.72

EXP[1]this work

Alder-Wiser formalism

Locality of the dielectric modes

16

48• 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.

U i,G () local dielectric response

construct MLWFs

+

32

dielectric band structure

How many dielectric eigenmodes are needed to determine the quasi-particle band gap?

G,G ' 1 (q) G,G ' Ui,G (q) i

1 1 Ui,G'* (q)

i1

n

Ui,G (q) i 1 1 Ui,G '

* (q)in 1

N

Convergence is slow!

iG(1,2)W (1,2)

W (1,2) d(3) 1(1,3)v(3,2)GW implementation starting from DFT/GGA orbitals

1 denotes (x1, y1, z1, t1)

dielectric eigenmodes of the system construct Vi,(q) according to the

orthogonality condition, and i(q) from the Penn model

model dielectric response

+

n

iiii qUqUq

1

11 )()()(ˆ

N

niiii qVqV

1

1 )()(

the Penn model

F 1 14 (Eg / E f )

q Gi Vi(q) (q G)Vi(q)

2/1

2/1

22

11)(

F

k

Gq

E

EF

E

EGq

Fg

p

g

ppenn

D. Penn, Phys. Rev., 128: 2093,1962.

Decomposition of the dielectric modes

++ + +

Decomposition of the dielectric modes

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 IBM@Watson), NERSC