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Linear and non-Linear Dielectric
Response of Periodic Systems
from Quantum Monte Carlo
Calculations.
Paolo Umari
CNR
CNR-INFM DEMOCRITOS
Theory@Elettra Group
Basovizza, Trieste, Italy
In collaboration with:
•N. Marzari,
Massachusetts Institute of Technology
•G.Galli
University of California, Davis
•A.J. Williamson
Lawrence Livermore National Laboratory
Outline

Motivations

Finite electric fields in QMC with PBCs

Results for periodic linear chains of H2
dimers: polarizability and second hyperpolarizability
Motivations
DFT with GGA-LDA not always reliable for
dielectric properties:
0 2 4 6 8 10 12 14 16 18 20 22 24
Ge
Si
GaAs
GaP
AlAs


 10 m/V
AlN
2
12
GaP
Expt.
DFT-LDA
GaAs
AlP
C
Se
GaN
-100
0
100
200 300 400 500
Motivations…
Periodic chains of conjugated polymers,DFT-GGA
overestimates:
Linear susceptibilities: >~2 times
Hyper susceptibilities: > orders of magnitude:
importance of electronic correlations
Linear and non-linear optical
properties of extended systems
We want:
•Periodic boundary conditions: real extended
solids
•Accurate many-body description: conjugate
polymers
•Scalability: large systems
Quantum Monte Carlo
Diffusion - QMC
•Wavefunction as stochastic density of walker
•The sign of the wavefunction must be known
Y
•We have errorbars
….some diffusion-QMC basics
•We evolve a trial wave-function into imaginary
time:
t  it
ˆ

(H
E
)t
0
Y
(
t)
e
Y
(
0
)
•At large t, we find the exact ground state:
lim
Y
(t)cY
0
t

• Usually, importance sampling is used, we evolve f
in imaginary time:
f(t)
(t)
TY
…need for a new scheme

Static dielectric properties are defined as
derivative of the system energy with respect to a
static electric field

for describing extended systems periodic
boundary conditions are extremely useful

Perturbational approaches can not be (easily)
implemented within QMC methods
We need: finite electric fields AND
periodic boundary conditions
the Method: 1st challenge
In a periodic or extended system
the linear electric potential
V
ˆ
V(x)x
is not compatible with periodic
boundary conditions
x
L
The many-body electric enthalpy
•We don’t know how to define a linear potential
with PBCs, but the MTP provides a definition for the
polarization:
L
iGXˆ
P
Im ln Y e Y
G  2 / L
2
•With the N-body operator:
iGXˆ
e
Xˆ  xˆ1    xˆ N
•A legendre transform leads to the electric
enthalpy functional:
EY  E0 Y  PY 
PU & A.Pasquarello PRL 89, 157602 (02); I.Souza,J.Iniguez & D.Vanderbilt PRL 89, 117602(‘02)
R.Resta, PRL 80, 1800 (‘98); R.D. King-Smith & D. Vanderbilt PRB 47, 1651 (‘93)
2nd challenge
•We want to minimize the electric enthalpy functional
•We need an hermitian Hamiltonian
•We obtain a Hamiltonian which depends self-consistently
upon the wavefunctions:
ˆ
iG
X


L 
e


H
(
z
)

H

Im

0

 z
2




ˆ
iG
X

z Y
e Y


It’s a self-consistent many-body operator !
Iterative maps in the complex plane
•For every H(zi) there is a corresponding zi+1
•This define a complex-plane map: f(z)
•The solution to the self-consistent scheme and the
minimum of the electric enthalpy correspond to the
fixed point:
f z   z
•Gives access to the polarization in the presence of
the electric field : the solution of our problem
3rd challenge
•Without stochastic error an iterative map can lead to the
fixed point:
z1  z2  z3  z4  z5  
•In QMC, at every zi in the iterative sequence is
associated a stochastic error 
.... and solution
•We can assume that close to the fixed point, the
map can be assumed linear:
f (z)azb
•The average over a sequence of {zi}
provides the estimate for the fixed point
•The spread of the zi around the fixed point,
2

depends upon the stochastic error:
1 a
2
{zi} series in complex plane
•Electric field: 0.001 a.u., bond alternation 2.5 a.u.
•10 iterations of 40 000 time-steps 2560 walkers
Implementation: from DFT to QMC
First Step (DFT - HF):
Hilbert space single Slater determinants:
We implemented single-particle electric enthalpy in
the quantum-ESPRESSO distribution (publicly available at
www.quantum-espresso.org)
Second Step (QMC):
Wave functions are imported in the CASINO
variational and diffusion QMC code, where we
coded all the present development
(www.tcm.phy.cam.ac.uk/~mdt26/cqmc.html)
Validation: H atom
•We can compare our scheme with a simple sawtooth potential for an isolated system: polarizability
 of H atom
•Isolated H atom in a saw
4
.52

0
.05
a.u.
tooth potential:

•Same atom in P.B.C. via
our new formulation:
Exact:
4
.49

0
.03
a.u.
4.50a.u.
The true test: periodic H2 chains
2. a.u.
2.5 a.u.
P  E  6E
2. a.u..
2. a.u..
3. a.u.
4. a.u.
3
Results from quantum chemistry:
dependence on correlations
Polarizability for 2.5 a.u. bond alternation
Polarizabiliy per H2 unit
Scaling cost
DFT-GGA
=144.6
N3,N
MP2
=58.0
N5
CCD
=47.6
N5
MP4
=53.6
N7
CCSD(T)
=50.6
N7
Infinite chain limit; quantum chemistry results need to be extrapolated.
B. Champagne & al. PRA 52, 1039 (1995)
Results from quantum chemistry:
dependence on basis set
Second hyper-polarizability for 3. a.u. bond alternation at
MP3 and MP4 level
Basis set
MP3
MP4
(6)-31G
60135±52
56836±49
(6)-311G
64338±37
61868±13
(6)-31G(*)*
65729±59
65776±108
(6)-311G(*)*
73002±49
74683± 54
Infinite chain limit; quantum chemistry results need to be extrapolated.
B. Champagne & D.H. Mosley, JCP 105, 3592 (‘96)
QMC treatment
•2.5,3.,4. a.u. bond alternation
•Nodal surface and trial wavefunction from HF
•HF wfcs calculated in the presence of electric field
Convergence with respect to supercell
size
Results from HF, 3. a.u. bond alternation
10 units
20 units
QC
extrapolations

27.8
28.5
28.6
/1000.
57.1
57.1
56.7
We will consider 10-H2 periodic units cells
Test on linearity of f(z)
•
bond alternation 2.5 a.u., electric field 0.003 a.u.
• 2560 walkers 120 000 time steps / iteration
• 2560 walkers 40 000 time steps / iteration
Diffusion QMC results: 3. a.u. bond
alternation
•We apply electric fields of: 0.003 a.u. , 0.02 a.u.
= 27.0 +/- 0.5 a.u.
•=26.5 a.u. MP4
From Q.C. extrapolations: •=25.7 a.u. CCSD(T)
= 89.8 +/- 6.1 a.u. (*103)
From Q.C. extrapolations: •>74.7 a.u.(*103) MP4
Diffusion QMC results: 2.5 a.u. bond
alternation
•We apply electric fields of: 0.003 a.u. , 0.01 a.u.
= 50.6 +/- 0.3 a.u.
•=53.6 a.u. MP4
From Q.C. extrapolations: •=50.6 a.u. CCSD(T)
= 651.9 +/- 29.9 a.u. (*103)
Diffusion QMC results: 4. a.u. bond
alternation
•We apply electric fields of: 0.01 a.u. , 0.03 a.u.
= 16.0 +/- 0.1 a.u.
•=15.8 a.u. MP4
From Q.C. extrapolations: •=15.5 a.u. CCSD(T)
= 16.5 +/- 0.6 a.u. (*103)
Effects of correlation: polarizability
60
50
40
HF
30
DMC
20
10
0
2.5 a.u.
3.0 a.u.
4.0 a.u.
Exchange is the most important contribution
Effects of correlation: 2nd hyperpolarizability
700000
600000
500000
400000
HF
300000
DMC
200000
100000
0
2.5 a.u.
3.0 a.u.
4.0 a.u.
Correlations are important!!
Conclusions
•Novel approach for dielectric properties via QMC
•Implemented via diffusion QMC
•Validated in periodic hydrogen chains:very nice
agreement with the best quantum chemistry
results
•PRL 95, 207602 (‘05)
Perspectives…
•“Linear scaling”
•Testing critical cases
•understanding polarization effects in DFT
•....
Acknowledgments
•For the QMC CASINO software:
M.D. Towler and R.J. Needs, University of
Cambridge
•For HF applications:
S. de Gironcoli, Sissa, Trieste
•For money: DARPA-PROM
Importance of nodal surface:
from DFT
Bond alternation 2.5 a.u.
•For 10-H2:
DMC= 52.2 +/- 1.3 a.u.
GGA= 102.0 a.u.
•For 16-H2:
DMC= 55.4 +/- 1.2 a.u.
GGA= 123.4 a.u.
•For 22-H2:
DMC= 53.4 +/- 1.1 a.u.
GGA= 133.5 a.u.
From nodal surface HF: DMC= 50.6 +/- 0.3 a.u.
Electronic localization for H2
periodic chain:
•Localization spread:
2
L
2
2
 2ln
z
N
4

(Resta & Sorella, PRL ’99)
•For GGA-DFT:
 4.32a.u.
2
•For DMC-QMC:
2
.44

0
.01
a.u.
2
Finite electric fields in DFT
with finite field
Si (8atoms 4X4X10kpoints):
 14
P

12
.6
(Expt.
: 11
.4)
11
1
a.u.

5
.
1422
10
V/m
Solution for single particle Hamitonian:
Umari & Pasquarello PRL 89, 157602 (’02)
Souza, Iniguez & Vanderbilt PRL 89, 117602 (’02)
…DFT-Molecular Dynamics with
electric fields:
•Possible applications:
•Static Dielectric properties of liquids at finite
temperature, (Dubois, PU, Pasquarello, Chem. Phys. Lett. ’04)
•Dielectric properties of iterfaces
(Giustino, PU,Pasquarello,
PRL’04)
•Infrared spectra of large systems
•Non-resonant Raman and Hyper-Raman spectra of
large systems (Giacomazzi, PU, Pasquarello, PRL’05; PU, Pasquarello, PRL’05)
Sampling eiGX in diffusion QMC
•eiGX does not commute with the Hamiltonian:
we use forward walking
•Observable are samples after a projection time
t

e
e 
N
 
ˆ
iG
X
iGX
'j,


t
j
1
,N

(Hammond, Lester & Reynolds ’94)