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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)