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

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