1. 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

2. 2 Physical Sciences Weak dispersive interaction + + - - ++ - - - + ++ - + + - - + - + - - + - 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 H 2 storage

3. 3 Physical 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 Quantum Monte Carlo gives correct description of van der Waals interactions

4. 4 Physical Sciences Overview  Describe quantum Monte Carlo – DMC  Argon – dimers, trimers, and FCC solid phase  Applications for H 2 storage. H 2 on carbon absorbents - benzene, coronene, and graphene  Applications in metals, FCC aluminum

5. 5 Physical Sciences Electron correlations treated directly, non-perturbative approach QMC solves … Ground state of full many-body Schrödinger equation

6. 6 Physical Sciences Variational Monte Carlo (VMC) Single particle orbitals from DFT and parameters in are determined using variance minimization Slater-Jastrow trial wavefunction

7. 7 Physical 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)

8. 8 Physical 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 VMC (total time) WFDET New130.7 10.4 Old232.8 86.1 DMC (total time) WFDET New 673.4 54.9 Old1196.5487.5 WFDET in new version is 8-9 times faster WFDET only 8% of total computing time (Jastrow 39%)

9. 9 Physical 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 2 r r r r Time (r,r)(r,r) (r,r)(r,r) share evaluate orbitals {φ 1 (r), φ 2 (r), φ 1 (r), φ 2 (r)} {φ 3 (r), φ 4 (r), φ 3 (r), φ 4 (r)} swap orbitals {φ 1 (r), φ 2 (r)} {φ 3 (r), φ 4 (r)} φ1(r)φ1(r) φ2(r)φ2(r) φ3(r)φ3(r) φ4(r)φ4(r) φ1(r)φ1(r) φ2(r)φ2(r) φ3(r)φ3(r) φ4(r)φ4(r) Swaps (using MPI) can be done at different points in code

10. 10 Physical 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 VMCVMC swap overhead DMCDMC swap overhead 1170.8541.7 2220.1 28.9 %607.8 12.3 % 4249.2 46.0 %582.8 7.6 % 8312.6 83.1 %639.5 18.1 % 16408.4139.1 %790.8 46.0 %

11. 11 Physical 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

12. 12 Physical 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 d

13. 13 Physical 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)

14. 14 Physical Sciences Argon dimer- compare DMC and CCSD(T) Lennard-Jones potential For Å Lennard-Jones potential agrees with DMC

15. 15 Physical Sciences Including only two-body contributions to FCC argon Aziz † V 2 DMC V 2 Exp (Without ZPE) A0A0 5.215.225.25Å E coh 94.395.288.9meV B0B0 37.537.931.9kbar † R.A. Aziz, J. Chem. Phys. 99, 4518 (1993)

16. 16 Physical Sciences Argon trimer – probing 3-body term Ar d0d0 x 3-body term- 8% of cohesive energy in FCC argon

17. 17 Physical 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 Eliminate finite-size bias

18. 18 Physical Sciences FCC argon - DMC and DFT LDA severely overbinds while GGA is significantly underbound DMC results not sensitive to nodes Vinet EOS gave best fit to DMC

19. 19 Physical Sciences FCC argon – comparison with experiment LDAGGADMCExp (Without ZPE) A0A0 5.06.05.28(2)5.25Å E coh 1402279(2)88.9meV/atom B0B0 613.731(1)31.9kbar error of 10 meV/atom = 0.2 kcal/mole sub-chemical accuracy error 2.0 kcal/mole Variational principle – get better cancellation of fixed-node error by computing EOS

20. 20 Physical Sciences Fixed-node error in DMC 55 molecules (G1 basis set) DMC error =130 meV/atom = 2.9 kcal/mole J.C. Grossman, J. Chem. Phys. 117, 1434 (2002) SiGeC LDA5.284.598.61 DMC4.63(2)3.85(2)7.46(1) Exp.4.62(2)3.857.37 Binding energies in semiconductors (eV/atom) Binding energies in molecules Computing binding energies using EOS approach would likely give sub-chemical accuracy

21. 21 Physical Sciences Many-body terms in FCC argon Argon many-body effects reduce the binding energy and the bulk modulus of FCC argon

22. 22 Physical 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 H 2 binding energies suitable: 0.1 - 0.5 eV/(H 2 molecule) BMW Hydrogen 7

23. 23 Physical Sciences Understanding physisorption of H 2 on carbon substrates Focus :: H 2 adsorbed on Benzene Coronene Graphene LDA and GGA unable to correctly describe H 2 binding in these systems

24. 24 Physical Sciences H 2 on benzene Single H 2 binding energy is ~52 ± 8 meV

25. 25 Physical Sciences H 2 on coronene Single H 2 binding energy is ~200 ± 12 meV

26. 26 Physical Sciences H 2 on planar Graphene (1/3 filling ) 128 atom super cell Methods to treat van der Waals interactions accurately within DFT is an active area of research

27. 27 Physical Sciences (a) C 2 H 2 dimer, (b & c) C 2 H 2 -H 2, (d) C0 2 dimer, (e) C 6 H 6 -H 2, (f) C 6 H 6 -H 2 0, (g & h) C 6 H 6 dimer vdW CCSD(T) LDA GGA vdW potentials are transferable 140 structures of DNA base pairs vdW errors of 0.5 kcal/mole

28. 28 Physical Sciences In progress / future directions Carbon based materials offer many possibilities for tuning binding energetics of H 2 curvature, damage, doping, decorating, charging Metal-organic frameworks (MOFs) have shown promise for H 2 storage

29. 29 Physical 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) )

30. 30 Physical 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

31. 31 Physical Sciences DMC of FCC Al FCC Al with 256 atoms, 768 electrons Statistical error bars 20 times smaller than previous VMC

32. 32 Physical 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

33. 33 Physical Sciences DMC of FCC Al using multiple determinants optimized using variance minization Obtain smooth EOS but not the lowest energy at all “a” despite having greater variational freedom

34. 34 Physical Sciences DMC of FCC Al using mulitiple determinants 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)

35. 35 Physical Sciences DMC EOS of FCC Al LDADMCExp (Without ZPE) A0A0 3.963.94(1)4.022Å E coh 4.213.55(1)3.43eV/atom B0B0 0.8021.0(2)0.813Mbar B 0 depended sensitively on the fit Size of error in E coh consistent with fixed-node error Our value for A 0 is close to previous VMC calculation Understanding errors in A 0 is a WIP

36. 36 Physical 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 H 2 absorption in planer hydrocarbon absorbents Computed EOS of FCC aluminum

37. 37 Physical Sciences Acknowledgments Jonathan Dubois(LLNL) Norm Tubman (Northwestern) Sebastien Hamel(LLNL) Eric Schwegler(LLNL) Shengbai Zhang (RPI) Yiyang Sun (RPI) Yong Hyun Kim (NREL)

38. 38 Physical Sciences Comparison of first-principles methods MethodE corr E bind % errorsScalingTime for C 10 HF 0 50 % N 3 14 LDA N/A15-25 % N 3 1 VMC 85 % 2-10 % N 3 16 DMC 95 % 1-4 % N 3 300 CCSD(T)* 75 %10-15 % N 7 1500 *With 6-311G* basis W.M.C. Foulkes, et. al., Rev. Mod. Phys. 73, 33 (2001)