Paolo Umari

1. Linear and non-Linear Dielectric Response of Periodic Systems from Quantum Monte Carlo Calculations. Paolo Umari CNR-INFM DEMOCRITOS Theory@Elettra Group Basovizza, Trieste, Italy CNR

2. In collaboration with: • N. Marzari, Massachusetts Institute of Technology • G.Galli University of California, Davis • A.J. Williamson Lawrence Livermore National Laboratory

3. Outline Motivations Finite electric fields in QMC with PBCs Results for periodic linear chains of H2 dimers: polarizability and second hyper-polarizability

4. Motivations DFT with GGA-LDA not always reliable for dielectric properties:

5. Motivations… Periodic chains of conjugated polymers,DFT-GGA overestimates: Linear susceptibilities: >~2 times Hyper susceptibilities: > orders of magnitude: importance of electronic correlations

6. Linear and non-linear optical properties of extended systems We want: • Periodic boundary conditions: real extended solids • Accurate many-body description: conjugate polymers • Scalability: large systems Quantum Monte Carlo

7. Diffusion - QMC • Wavefunction as stochastic density of walker • The sign of the wavefunction must be known Y • We have errorbars

8. ….some diffusion-QMC basics • We evolve a trial wave-function into imaginary time: • At large t, we find the exact ground state: • Usually, importance sampling is used, we evolve f in imaginary time:

9. …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

10. the Method: 1st challenge ? In a periodic or extended system the linear electric potential is not compatible with periodic boundary conditions

11. The many-body electric enthalpy • We don’t know how to define a linear potential with PBCs, but the MTP provides a definition for the polarization: • With the N-body operator: • 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)

12. 2nd challenge • We want to minimize the electric enthalpy functional • We need an hermitian Hamiltonian • We obtain a Hamiltonian which depends self-consistently • upon the wavefunctions: It’s a self-consistent many-body operator !

13. Iterative maps in the complex plane • 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: • Gives access to the polarization in the presence of the electric field : the solution of our problem

14. 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 

15. .... and solution • We can assume that close to the fixed point, the map can be assumed linear: • The average over a sequence of {zi} provides the estimate for the fixed point • The spread of the ziaround the fixed point, depends upon the stochastic error:

16. {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

17. Implementation: from DFT to QMC First Step (DFT - HF): Hilbert space single Slater determinants: We implemented single-particle electric enthalpy in the quantum-ESPRESSO distribution (publicly available at www.quantum-espresso.org) Second Step (QMC): 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)

18. Validation: H atom • We can compare our scheme with a simple saw-tooth potential for an isolated system: polarizability  of H atom • Isolated H atom in a saw-tooth potential: • Same atom in P.B.C. via our new formulation: Exact:

19. The true test: periodic H2 chains 2.5 a.u. 2. a.u. 3. a.u. 2. a.u.. 4. a.u. 2. a.u..

20. Results from quantum chemistry: dependence on correlations Polarizabiliy per H2 unit Scaling cost DFT-GGA a=144.6 N3,N MP2 a=58.0 N5 CCD a=47.6 N5 MP4 a=53.6 N7 CCSD(T) a=50.6 N7 Polarizability for 2.5 a.u. bond alternation Infinite chain limit; quantum chemistry results need to be extrapolated. B. Champagne & al. PRA 52, 1039 (1995)

21. Results from quantum chemistry:dependence on basis set Second hyper-polarizability for 3. a.u. bond alternation at MP3 and MP4 level Infinite chain limit; quantum chemistry results need to be extrapolated. B. Champagne & D.H. Mosley, JCP 105, 3592 (‘96)

22. 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

23. Convergence with respect to supercell size Results from HF, 3. a.u. bond alternation We will consider 10-H2 periodic units cells

24. 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

25. Diffusion QMC results: 3. a.u. bond alternation • We apply electric fields of: 0.003 a.u. , 0.02 a.u. a = 27.0 +/- 0.5 a.u. • a=26.5 a.u. MP4 • a=25.7 a.u. CCSD(T) From Q.C. extrapolations: g = 89.8 +/- 6.1 a.u. (*103) From Q.C. extrapolations: • g >74.7 a.u.(*103) MP4

26. Diffusion QMC results: 2.5 a.u. bond alternation • We apply electric fields of: 0.003 a.u. , 0.01 a.u. a = 50.6 +/- 0.3 a.u. • a=53.6 a.u. MP4 • a=50.6 a.u. CCSD(T) From Q.C. extrapolations: g = 651.9 +/- 29.9 a.u. (*103)

27. Diffusion QMC results: 4. a.u. bond alternation • We apply electric fields of: 0.01 a.u. , 0.03 a.u. a = 16.0 +/- 0.1 a.u. • a=15.8 a.u. MP4 • a=15.5 a.u. CCSD(T) From Q.C. extrapolations: g = 16.5 +/- 0.6 a.u. (*103)

28. Effects of correlation: polarizability Exchange is the most important contribution

29. Effects of correlation: 2nd hyper-polarizability Correlations are important!!

30. 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)

31. Perspectives… • “Linear scaling” • Testing critical cases • understanding polarization effects in DFT • ....

32. Acknowledgments • For the QMC CASINO software: M.D. Towler and R.J. Needs, University of Cambridge • For HF applications: S. de Gironcoli, Sissa, Trieste • For money: DARPA-PROM

33. Importance of nodal surface: from DFT Bond alternation 2.5 a.u. • For 10-H2: aDMC= 52.2 +/- 1.3 a.u. aGGA= 102.0 a.u. • For 16-H2: aDMC= 55.4 +/- 1.2 a.u. aGGA= 123.4 a.u. • For 22-H2: aDMC= 53.4 +/- 1.1 a.u. aGGA= 133.5 a.u. From nodal surface HF: aDMC= 50.6 +/- 0.3 a.u.

34. Electronic localization for H2 periodic chain: • Localization spread: (Resta & Sorella, PRL ’99) • For GGA-DFT: • For DMC-QMC:

35. Finite electric fields in DFT Si (8atoms 4X4X10kpoints): with finite field Solution for single particle Hamitonian: Umari & Pasquarello PRL 89, 157602 (’02) Souza, Iniguez & Vanderbilt PRL 89, 117602 (’02)

36. …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)

37. Sampling eiGXin diffusion QMC • eiGX does not commute with the Hamiltonian: we use forward walking • Observable are samples after a projection time t (Hammond, Lester & Reynolds ’94)