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

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 H 2 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. 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

7. Diffusion - QMC Wavefunction as stochastic density of walker  The sign of the wavefunction must be known 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. In a periodic or extended system the linear electric potential is not compatible with periodic boundary conditions the Method: 1st challenge ?

11. 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)

12. 2nd challenge 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:

13. For every H(z i ) there is a corresponding z i+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

14. 3rd challenge Without stochastic error an iterative map can lead to the fixed point: In QMC, at every z i 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 {z i } provides the estimate for the fixed point The spread of the z i around the fixed point, depends upon the stochastic error:

16. {z i } 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. 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):

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

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

20. Results from quantum chemistry: dependence on correlations N7N7 =50.6 CCSD(T) N7N7 =53.6 MP4 N5N5 =47.6 CCD N5N5 =58.0 MP2 N 3,N =144.6 DFT-GGA Scaling costPolarizabiliy per H 2 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)

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) Basis setMP3MP4 (6)-31G 60135525683649 (6)-311G 64338376186813 (6)-31G(*)* 657295965776108 (6)-311G(*)*730024974683 54

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-H 2 periodic units cells 10 units20 units QC extrapolations  27.828.528.6  57.1 56.7

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.   = 27.0 +/- 0.5 a.u. From Q.C. extrapolations:  a.u.(*10 3 ) MP4  = 89.8 +/- 6.1 a.u. (*10 3 ) From Q.C. extrapolations:  =26.5 a.u. MP4  =25.7 a.u. CCSD(T)

26. 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. (*10 3 )

27. 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. (*10 3 )

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 money: DARPA-PROM For HF applications: S. de Gironcoli, Sissa, Trieste

33. For 10-H 2 : For 16-H 2 : Importance of nodal surface: from DFT For 22-H 2 :  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.

34. Electronic localization for H 2 periodic chain: Localization spread: For GGA-DFT: (Resta & Sorella, PRL ’99) 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 e iGX in diffusion QMC (Hammond, Lester & Reynolds ’94) e iGX does not commute with the Hamiltonian: we use forward walking Observable are samples after a projection time t