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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

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Linear and non-Linear Dielectric Response of Periodic Systems from Quantum Monte Carlo Calculations. Paolo Umari. CNR-INFM DEMOCRITOS Theory@Elettra Group Basovizza, Trieste, Italy. CNR. In collaboration with:. N. Marzari , Massachusetts Institute of Technology G.Galli - PowerPoint PPT Presentation

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Page 1: Paolo Umari

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

Page 2: Paolo Umari

In collaboration with:

•N. Marzari,

Massachusetts Institute of Technology

•G.Galli

University of California, Davis

•A.J. Williamson

Lawrence Livermore National Laboratory

Page 3: Paolo Umari

Outline

Motivations

Finite electric fields in QMC with PBCs

Results for periodic linear chains of H2

dimers: polarizability and second hyper-

polarizability

Page 4: Paolo Umari

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

Page 5: Paolo Umari

Motivations…

Periodic chains of conjugated polymers,DFT-GGA

overestimates:

Linear susceptibilities: >~2 times

Hyper susceptibilities: > orders of magnitude:

importance of electronic correlations

Page 6: Paolo Umari

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

Page 7: Paolo Umari

Diffusion - QMC

•Wavefunction as stochastic density of walker

•The sign of the wavefunction must be known

•We have errorbars

Page 8: Paolo Umari

….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

Page 9: Paolo Umari

…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

Page 10: Paolo Umari

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

?

Page 11: Paolo Umari

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

lnIm2 LG /2

eXiG ˆ

NxxX ˆˆˆ1

PEE 0

Page 12: Paolo Umari

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:

Page 13: Paolo Umari

•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

Page 14: Paolo Umari

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

Page 15: Paolo Umari

.... 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

Page 16: Paolo Umari

{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

Page 17: Paolo Umari

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):

Page 18: Paolo Umari

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

Page 19: Paolo Umari

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

Page 20: Paolo Umari

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)

Page 21: Paolo Umari

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

Page 22: Paolo Umari

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

Page 23: Paolo Umari

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

Page 24: Paolo Umari

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

Page 25: Paolo Umari

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)

Page 26: Paolo Umari

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)

Page 27: Paolo Umari

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)

Page 28: Paolo Umari

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

Page 29: Paolo Umari

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

Page 30: Paolo Umari

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)

Page 31: Paolo Umari

Perspectives…

•“Linear scaling”

•Testing critical cases

•understanding polarization effects in DFT

•....

Page 32: Paolo Umari

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

Page 33: Paolo Umari

•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.

Page 34: Paolo Umari

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

Page 35: Paolo Umari

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)

Page 36: Paolo Umari

…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)

Page 37: Paolo Umari

Sampling eiGX in diffusion QMC

(Hammond, Lester & Reynolds ’94)

NNj

iGX

XiG ee

tj

,1

,

•eiGX does not commute with the Hamiltonian:

we use forward walking

•Observable are samples after a projection time

t