linear and non-linear dielectric response of periodic systems from quantum monte carlo calculations....
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![Page 1: Linear and non-Linear Dielectric Response of Periodic Systems from Quantum Monte Carlo Calculations. Paolo Umari CNR CNR-INFM DEMOCRITOS Theory@Elettra](https://reader036.vdocument.in/reader036/viewer/2022062721/56649f1b5503460f94c31372/html5/thumbnails/1.jpg)
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
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In collaboration with:
•N. Marzari,
Massachusetts Institute of Technology
•G.Galli
University of California, Davis
•A.J. Williamson
Lawrence Livermore National Laboratory
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Outline
Motivations
Finite electric fields in QMC with PBCs
Results for periodic linear chains of H2
dimers: polarizability and second hyper-
polarizability
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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
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Motivations…
Periodic chains of conjugated polymers,DFT-GGA
overestimates:
Linear susceptibilities: >~2 times
Hyper susceptibilities: > orders of magnitude:
importance of electronic correlations
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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
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Diffusion - QMC
•Wavefunction as stochastic density of walker
•The sign of the wavefunction must be known
•We have errorbars
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….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
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…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
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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
?
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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
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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:
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•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
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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
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.... 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
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{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
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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):
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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
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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
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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)
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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
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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
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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
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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
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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)
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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)
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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)
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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
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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
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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)
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Perspectives…
•“Linear scaling”
•Testing critical cases
•understanding polarization effects in DFT
•....
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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
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•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.
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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
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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)
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…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)
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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