Lawrence Livermore National Laboratory

Physical Sciences

Quantum Monte Carlo studies of metals and

materials with properties determined by weak

dispersive interactions

Randolph Q. Hood

Performance Measures x.x, x.x, and x.x

2Physical Sciences

Weak dispersive interaction

+ + - -+ + - -++ - - -

+

++ -+ + - - + - + - -

+ -

6

1~)(R

RV

Quantum mechanically induced dipoles type of van der Waals interaction

Important in life processes such as genetic replication and proteins, and for several types of proposed H2 storage

R

3Physical Sciences

Weak dispersive interaction neglected in mean field DFT

DFT typically predicts accurate structures, but• van der Waals not included in mean field DFT• LDA & GGA qualitatively disagree on binding• Need “beyond DFT” approaches

])[,( nvxc rLDA ))(( rnvxc

GGA ))(),(( rr nnvxc

Quantum Monte Carlo gives correct description of van der Waals interactions

4Physical Sciences

Overview

Describe quantum Monte Carlo – DMC

Argon – dimers, trimers, and FCC solid phase

Applications for H2 storage. H2 on carbon absorbents - benzene, coronene, and graphene

Applications in metals, FCC aluminum

5Physical Sciences

Electron correlations treated directly, non-perturbative approach

QMC solves…

Ground state of full many-body Schrödinger equation

),...,,(),...,,( 2121 NN EH rrrrrr

||||||

1

2

1

,

2

rrrrrr

ZZZH

i iji jiii

6Physical Sciences

Variational Monte Carlo (VMC)

Single particle orbitals from DFT and parameters in are determined using variance minimization

}){,( RT

JNT eDD ),...,,( 21 rrr

kji

kjiji

jii

i fffJ ),,( ),()( 321 rrrrrr

)(...)()(

:::

)(...)()(

21

12111,

NNNN

N

D

rrr

rrr

Slater-Jastrow trial wavefunction

7Physical Sciences

DFT inputs for “production” runs

• PWSCF

• LDA & GGA exchange-correlation functionals

• Plane wave basis sets (150 - 400 Rydberg cutoff)

• Norm-conserving, Troullier-Martins pseudopotentials (Casula scheme to maintain variational principle)

• Experimental structures (no optimization)

8Physical Sciences

Fe with 1024 electrons, timings using blips in seconds

Old (ver. 2.1) New (ver. 2.2.*)

Improvements to CASINO ver. 2.2.* for large systems

blips )1(

wavesplane )( )( orbital of timeEvaluation

O

NOr

VMC

(total time)

WFDET

New 130.7 10.4

Old 232.8 86.1

DMC

(total time)

WFDET

New 673.4 54.9

Old 1196.5 487.5

WFDET in new version is 8-9 times faster

WFDET only 8% of total computing time (Jastrow 39%)

9Physical Sciences

Distributing storage of blips in CASINO ver. 2.2.*

Blips in large systems can require large amounts of memory

Share blip orbitals among a set of CPUs

CPU 1 WFDET CPU 2r r

r

r

Time

(r,r) (r,r)share

evaluate orbitals{φ1(r), φ2(r), φ1(r), φ2(r)} {φ3(r), φ4(r), φ3(r), φ4(r)}

swap orbitals

{φ1(r), φ2(r)}

{φ3(r), φ4(r)}

φ1(r)

φ2(r)

φ3(r)

φ4(r)

φ1(r)

φ2(r)

φ3(r)

φ4(r)

Swaps (using MPI) can be done at different points in code

10Physical Sciences

Overhead of sharing blips in CASINO ver. 2.2.*

Fe with 1024 electrons, timings using blips in seconds on 64 CPUs

Number of CPUs in group

VMC VMC swap overhead

DMC DMC swap overhead

1 170.8 541.7

2 220.1 28.9 % 607.8 12.3 %

4 249.2 46.0 % 582.8 7.6 %

8 312.6 83.1 % 639.5 18.1 %

16 408.4 139.1 % 790.8 46.0 %

11Physical Sciences

FCC argon bound by weak dispersion interactions

• Argon (closed electronic shell) very inert

• Noble atom solid, argon melts at 84 K

• Well characterized experimentally

FCC argon

12Physical Sciences

Argon dimer- compare DMC and CCSD(T)

• Simple system to study the weak dispersive interaction

• DMC and highly converged CCSD(T) agree at all separations

Ar Ar

d

13Physical Sciences

Argon dimer- compare DMC and CCSD(T)

DMC fixed-node error

independent of separation d

Two-body potential from K. Patkowski, et. al., Mol. Phys., 103, 2031 (2005)

14Physical Sciences

Argon dimer- compare DMC and CCSD(T)

Lennard-Jones potential

612

4)(dd

dV

For Å Lennard-Jones potential agrees with DMC5d

15Physical Sciences

Including only two-body contributions to FCC argon

Aziz† V2 DMC V2 Exp

(Without ZPE)

A0 5.21 5.22 5.25 Å

Ecoh 94.3 95.2 88.9 meV

B0 37.5 37.9 31.9 kbar

†R.A. Aziz, J. Chem. Phys. 99, 4518 (1993)

16Physical Sciences

Argon trimer – probing 3-body term

Ar

Ar

Ard0

x

3-body term- 8% of cohesive energy in FCC argon

17Physical Sciences

FCC argon- high precision DMC

Our statistical error bars are 5 times smaller and time-step 4 times smaller

FCC Ne : N.D. Drummond and R.J. Needs, Phys. Rev. B 73, 024107 (2006)

Probed volumes 10 times larger

N

VbVEVE SL

NSL )(

)()(

Eliminate finite-size bias

18Physical Sciences

FCC argon - DMC and DFT

LDA severely overbinds while GGA is significantly underbound

DMC results not sensitive to nodes

CV

VB

V

VB

B

VBVE

3/1

00

3/1

002

0

00 1)1(2

3exp1)1(

2

31

)1(

4)(

Vinet EOS gave best fit to DMC

19Physical Sciences

FCC argon – comparison with experiment

LDA GGA DMC Exp(Without ZPE)

A0 5.0 6.0 5.28(2) 5.25 Å

Ecoh 140 22 79(2) 88.9 meV/atom

B0 61 3.7 31(1) 31.9 kbar

error of 10 meV/atom = 0.2 kcal/mole sub-chemical accuracy

meV )1(4)( N

EEDMCE solidatomcoh error 2.0 kcal/mole

Variational principle – get better cancellation of fixed-node error by computing EOS

)(DMCEcoh

20Physical Sciences

Fixed-node error in DMC

55 molecules (G1 basis set) DMC error =130 meV/atom = 2.9 kcal/moleJ.C. Grossman, J. Chem. Phys. 117, 1434 (2002)

Si Ge C

LDA 5.28 4.59 8.61

DMC 4.63(2) 3.85(2) 7.46(1)

Exp. 4.62(2) 3.85 7.37

Binding energies in semiconductors (eV/atom)

Binding energies in molecules

Computing binding energies using EOS approach would likely give sub-chemical accuracy

21Physical Sciences

Many-body terms in FCC argon

Argon many-body effects reduce the binding energy and the bulk modulus of FCC argon

22Physical Sciences

Hydrogen economy requires effective hydrogen storage

• Ideal storage is at room temperature

• High density requires non-hydrogen elements (1liter gasoline has 64% more H than 1liter of liquid H)

• Range of H2 binding energies suitable:

0.1 - 0.5 eV/(H2 molecule)

BMW Hydrogen 7

23Physical Sciences

Understanding physisorption of H2 on carbon substrates

Focus :: H2 adsorbed on

Benzene

CoroneneGraphene

LDA and GGA unable to correctly describe H2 binding in these systems

24Physical Sciences

H2 on benzene

Single H2 binding energy is ~52 ± 8 meV

25Physical Sciences

H2 on coronene

Single H2 binding energy is ~200 ± 12 meV

26Physical Sciences

H2 on planar Graphene (1/3 filling )

128 atom super cell

cuticonst

cutini

vdW

rv

rr

cv

|| if

|| if |)(|0

Rr

RrRr

Methods to treat van der Waals interactions accurately within DFT is an active area of research

27Physical Sciences

(a) C2H2 dimer, (b & c) C2H2-H2, (d) C02 dimer, (e) C6H6-H2, (f) C6H6-H20, (g & h) C6H6 dimer

vdW CCSD(T) LDA GGA

vdW potentials are transferable

140 structures of DNA base pairs vdW errors of 0.5 kcal/mole

28Physical Sciences

In progress / future directions

• Carbon based materials offer many possibilities for tuning binding energetics of H2

• curvature, damage, doping, decorating, charging

• Metal-organic frameworks (MOFs) have shown promise for H2 storage

29Physical Sciences

Applying DMC to metals

First important application of DMC to electronic systems was

homogeneous electron gas at LLNL (D.M Ceperley and B.J. Alder, Phys.

Rev. Lett. 45, 566 (1980)) Third most cited Physical Review Letters Results form basis of LDA and GGA approaches

There have been few calculations of the EOS of inhomogeneous metals

Li†,Al* – VMC †(G. Yao, et. al., Phys. Rev. B 54, 8393 (1996)),

*(R. Gaudoin, et. al., J. Phys.: Condens. Matter 14, 8787 (2002)) Mg – DMC (M. Pozzo and D. Alfé, Phys. Rev. B 77, 104103 (2008))

30Physical Sciences

Challenges for DMC - inhomogeneous metals

Numerous semiconductors and insulators have been studied using QMC over the past 20 years

Inhomogeneous metals have a Fermi surface requiring larger supercells containing more electrons

Partial occupation of orbitals at Fermi level cannot be directly translated into a real used in DMC. Have an “open shell” which breaks symmetries

JT eDD )(R

31Physical Sciences

DMC of FCC Al

FCC Al with 256 atoms, 768 electrons

Statistical error bars 20 times smaller than previous VMC

32Physical Sciences

DMC of FCC Al using single determinant

Discontinuity in EOS caused by band crossing which changes symmetry of nodes at a=3.97 Å when using a single determinant trial wavefunction J

T eDD

T

33Physical Sciences

DMC of FCC Al using multiple determinants

Jii

iiT eDDa

6

1

T

optimized using variance minization

Obtain smooth EOS but not the lowest energy at all “a” despite having greater variational freedom

34Physical Sciences

DMC of FCC Al using mulitiple determinants

Jii

iiT eDDa

6

1

T

optimized using energy minimization*

Obtain lowest energy smooth EOS

*M.P. Nightingale and V. Melik-Alaverdian, Phys. Rev. Lett. 87, 043401 (2001)

C.J. Umrigar, et. al., Phys. Rev. Lett. 98, 110201 (2007) J. Toulouse and C.J. Umrigar, J. Chem. Phys. 126, 084102 (2007)

35Physical Sciences

DMC EOS of FCC Al

LDA DMC Exp(Without ZPE)

A0 3.96 3.94(1) 4.022 Å

Ecoh 4.21 3.55(1) 3.43 eV/atom

B0 0.802 1.0(2) 0.813 Mbar20

2

0 V

EVB

• B0 depended sensitively on the fit

• Size of error in Ecoh consistent with fixed-node error

• Our value for A0 is close to previous VMC calculation

• Understanding errors in A0 is a WIP

36Physical Sciences

Conclusions

• DMC is only feasible approach capable of directly treating the weak dispersive interaction for systems with more than a few atoms

• DMC calculated EOS of FCC argon agrees closely with experiment, while DFT fails

• Van der Waals interactions play a key role in H2 absorption in planer hydrocarbon absorbents

• Computed EOS of FCC aluminum

37Physical Sciences

Acknowledgments

Jonathan Dubois (LLNL)Norm Tubman (Northwestern)Sebastien Hamel (LLNL)Eric Schwegler (LLNL)

Shengbai Zhang (RPI)Yiyang Sun (RPI)Yong Hyun Kim (NREL)

38Physical Sciences

Comparison of first-principles methods

Method Ecorr

Ebind

% errors Scaling Time for C10

HF 0 50 % N3 14

LDA N/A 15-25 % N3 1

VMC 85 % 2-10 % N3 16

DMC 95 % 1-4 % N3 300

CCSD(T)* 75 % 10-15 % N7 1500

*With 6-311G* basis

W.M.C. Foulkes, et. al., Rev. Mod. Phys. 73, 33 (2001)