Linear and non-Linear Dielectric
Response of Periodic Systems
from Quantum Monte Carlo
Calculations.
Paolo Umari
CNRCNR-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 hyper-
polarizability
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
AlP
C
GaN
-100 0 100 200 300 400 500
Se
GaAs
GaP
AlN
Expt.DFT-LDA
m/V10 122
Motivations…
Periodic chains of conjugated polymers,DFT-GGA
overestimates:
Linear susceptibilities: >~2 times
Hyper susceptibilities: > orders of magnitude:
importance of electronic correlations
We want:
•Periodic boundary conditions: real extended
solids
•Accurate many-body description: conjugate
polymers
•Scalability: large systems
Linear and non-linear optical properties of
extended systems
Quantum Monte Carlo
Diffusion - QMC
•Wavefunction as stochastic density of walker
•The sign of the wavefunction must be known
•We have errorbars
….some diffusion-QMC basics
•We evolve a trial wave-function into imaginary
time:
)0()( )ˆ( 0 tEHet
•At large t, we find the exact ground state:
0)(lim
ctt
• Usually, importance sampling is used, we evolve f
in imaginary time:)()( T ttf
itt
…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
xxV ˆ)(
L
V
x
In a periodic or extended system
the linear electric potential
is not compatible with periodic
boundary conditions
the Method: 1st challenge
?
The many-body electric enthalpy
•With the N-body operator:
•We don’t know how to define a linear potential
with PBCs, but the MTP provides a definition for the
polarization:
•A legendre transform leads to the electric
enthalpy functional:
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)
eXiGL
Pˆ
lnIm2 LG /2
eXiG ˆ
NxxX ˆˆˆ1
PEE 0
2nd challenge
XiG
XiG
ez
z
eLHzH
ˆ
ˆ
0 Im2
)(
It’s a self-consistent many-body operator !
•We want to minimize the electric enthalpy functional
•We need an hermitian Hamiltonian
•We obtain a Hamiltonian which depends self-consistently
upon the wavefunctions:
•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:
Iterative maps in the complex plane
•Gives access to the polarization in the presence of
the electric field : the solution of our problem
zzf
3rd challenge
•Without stochastic error an iterative map can lead to the
fixed point:
•In QMC, at every zi in the iterative sequence is
associated a stochastic error
54321 zzzzz
.... and solution
•We can assume that close to the fixed point, the
map can be assumed linear:
bazzf )(
•The average over a sequence of {zi}
provides the estimate for the fixed point
•The spread of the zi around the fixed point,
depends upon the stochastic error:2
2
1 a
{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
Hilbert space single Slater determinants:
We implemented single-particle electric enthalpy in
the quantum-ESPRESSO distribution (publicly available at
www.quantum-espresso.org)
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)
Second Step (QMC):
Implementation: from DFT to QMC
First Step (DFT - HF):
Validation: H atom
•Isolated H atom in a saw-
tooth potential: a.u. 05.052.4
•Same atom in P.B.C. via
our new formulation:
a.u. 03.049.4 Exact:
a.u. 50.4
•We can compare our scheme with a simple saw-
tooth potential for an isolated system: polarizability
of H atom
The true test: periodic H2
chains
2. a.u.2.5 a.u.
4. a.u.
3. a.u.2. a.u..
2. a.u..
36 EEP
Results from quantum chemistry: dependence on
correlations
N7=50.6CCSD(T)
N7=53.6MP4
N5=47.6CCD
N5=58.0MP2
N3,N=144.6DFT-GGA
Scaling costPolarizabiliy per H2 unit
Infinite chain limit; quantum chemistry results need to be extrapolated.
Polarizability for 2.5 a.u. bond alternation
B. Champagne & al. PRA 52, 1039 (1995)
Results from quantum chemistry:
dependence on basis setSecond hyper-polarizability for 3. a.u. bond alternation atMP3 and MP4 level
Infinite chain limit; quantum chemistry results need to be extrapolated.
B. Champagne & D.H. Mosley, JCP 105, 3592 (‘96)
Basis set MP3 MP4
(6)-31G 6013552 5683649
(6)-311G 6433837 6186813
(6)-31G(*)* 6572959 65776108
(6)-311G(*)* 7300249 74683 54
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
We will consider 10-H2 periodic units cells
10 units 20 units QC extrapolations
27.8 28.5 28.6
57.1 57.1 56.7
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.
From Q.C. extrapolations:
• a.u.(*103) MP4
= 89.8 +/- 6.1 a.u. (*103)
From Q.C. extrapolations:
•=26.5 a.u. MP4•=25.7 a.u. CCSD(T)
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.
From Q.C. extrapolations: •=53.6 a.u. MP4•=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.
From Q.C. extrapolations: •=15.8 a.u. MP4•=15.5 a.u. CCSD(T)
= 16.5 +/- 0.6 a.u. (*103)
Effects of correlation: polarizability
Exchange is the most important contribution
0
10
20
30
40
50
60
2.5 a.u. 3.0 a.u. 4.0 a.u.
HF
DMC
Effects of correlation: 2nd hyper-polarizability
Correlations are important!!
0
100000
200000
300000
400000
500000
600000
700000
2.5 a.u. 3.0 a.u. 4.0 a.u.
HF
DMC
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 money: DARPA-PROM
•For HF applications:
S. de Gironcoli, Sissa, Trieste
•For 10-H2:
•For 16-H2:
Importance of nodal surface: from DFT
•For 22-H2:
DMC
= 52.2 +/- 1.3 a.u. GGA
= 102.0 a.u.
DMC
= 55.4 +/- 1.2 a.u. GGA
= 123.4 a.u.
DMC
= 53.4 +/- 1.1 a.u. GGA
= 133.5 a.u.
Bond alternation 2.5 a.u.
From nodal surface HF: DMC
= 50.6 +/- 0.3 a.u.
Electronic localization for H2 periodic chain:
•Localization spread:
2
2
22 ln
4z
N
L
•For GGA-DFT:
a.u. 32.42
(Resta & Sorella, PRL ’99)
•For DMC-QMC:
a.u. 01.044.22
Finite electric fields in DFT
)4.11:Expt.(
6.1241
P
Si (8atoms 4X4X10kpoints):with finite field
V/m101422.5a.u. 1 11
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
(Hammond, Lester & Reynolds ’94)
NNj
iGX
XiG ee
tj
,1
'ˆ
,
•eiGX does not commute with the Hamiltonian:
we use forward walking
•Observable are samples after a projection time
t