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
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
Hydrogen Bonds
~ 3.6 bonds
/molecule,
consistent
with several
expt.
Tetrahedral network
The first
coordination
shell
contains~ 4.2
molecules
Standard model is challenged by recent XAS experiments
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
Ph.Wernet et al. Science 2004 (A. Nilsson’s group, Stanford)
The electronic properties of water are qualitatively
similar to those of ice—important details are different
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.
64 molec.; G pt.
Isolated LUMO
“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.
Occupied and empty single particle
electronic states in ice
Band structure
No “lone” state in ice
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
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
D.Prendergast and G.G, PRL 2006
core
Very good agreement between theory and
experiment for ice (cubic and hexagonal); good,
qualitative agreement for water (salient features
reproduced)
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)
Broken Hydrogen Bonds
Disorder
E.Artachos’s group (Cambridge, UK)
R.Car’s group (Princeton)
R.Saykally’s group (UCB)
No evidence justifying the dismissal of quasitetrahedral 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
Quick reminder on GW approx.
Hamiltonian of the system
Kohn-Sham equations
Quasi-particles
Green Functions and Perturbation Theory
Dyson Equation
Spectral representation of Green functions
A
GWa= generalization of the HF approximation, with a dynamically
screened Coulomb interaction
  iG(1,2)W (1 ,2)
W (1,2) 
1
d(3)

(1,3)v(3,2)


Plasmon-pole approx.
(Hybertsen and Louie, 1986)
F. Aryasetiawan and O.Gunnarson, review on “The GW method”, Rep. Phys. 1998
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.
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
GW correction on water band gap
GGA band structure
GW correction
shift 1.22 eV
Eg=4.52 eV
EgGW=8.66 eV
shift -2.92 eV
GW band gap at G point
# of
H2O
# of k
points
GGA gap
(eV)
∆GW
HOMO
16
64
4.52
-2.92
1.22
4.14
8.66
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
this work
Ref.[1]
∆GW ∆GW gap GW gap
LUMO
(eV)
(eV)
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



M  1/1
0,0 (0,0)
Dielectric matrix and eigen modes
4
0
G,G' (q; )  G,G' 
G,G'
(q; )

q  G q  G'
M
this work
EXP[1]
1.72
~ 1.8
[1]: see B. Bagchi, Chem. Rev., 105:3197, 2005.
*

(G)

k,q
k,q (G')
0
G,G' (q; )  2  f n (1 f m )
  n (k)  m (k  q)
k n,m
k,q (G)  k,n ei(q G)r k  q,m
Alder-Wiser formalism
The size of the matrix scales as npw2nqn

Decompose the static
dielectric matrix into
eigenmodes:
1
G,G'
(q)Ui,G' (q)  1
i Ui,G (q)
1
*
G,G'
(q)  G,G'  Ui,G (q)1
1
U
 i,G' (q)
i
i

Locality of the dielectric modes
dielectric band structure
Ui,G (G)
local dielectric
response
32
construct MLWFs
•
•
•
the first 64 eigen modes
48
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.
16
+
How many dielectric eigenmodes are needed
to determine the quasi-particle band gap?

GW implementation starting from   iG(1,2)W (1 ,2)
DFT/GGA orbitals
W (1,2)   d(3) 1(1,3)v(3,2)
1 denotes (x1, y1, z1, t1)
n
1
*
 G,G'
(q)  G,G'  U i,G (q)1
1
U

i
i,G' (q)
i1
N
*
 U i,G (q)1
1
U

i
i,G' (q)
i n 1

Convergence is slow!

Decomposition of the dielectric modes
n
ˆ (q)   U i (q) 
1
i 1
1
i
N
U i (q)
+
1
V
(
q
)

 i
i Vi ( q )
i  n 1
model dielectric response
dielectric eigenmodes of
construct Vi,(q) according to the
the system
orthogonality condition, and i(q)
from the Penn model
the Penn model
 Ep 


 Eg 
 penn (q  G)  1  
2
  E p  q  G 2

1/ 2



 F 
F 1 


E
k
  g  F 

q  G i  Vi (q) (q  G)Vi (q)
F 1 14 (E g / E f )
D. Penn, Phys. Rev., 128: 2093,1962.
1/ 2
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)
Thank you!
Francois Gygi (UCD)
Support from DOE/BES, DOE/SciDAC and LLNL/LDRD
Computer time: LLNL, INCITE AWARD (ANL and IBM@Watson), NERSC