Ab-initio simulation of liquid water by quantum Monte Carlo

Transcription

1 Ab-initio simulation of liquid water by quantum Monte Carlo Sandro Sorella G. Mazzola & Y. Luo SISSA, IOM DEMOCRITOS, Trieste A. Zen, L. Guidoni U. of L Aquila, L Aquila 28 July 2014, Mike Towler Institute, Apuan Alps

2 Outline/Motivations à The correlated wave function for realistic systems: from Hubbard to Hydrogen and Water à Few examples on accuracy and achievements on small molecules H 2 (H 2 O) 2 à Large number of electrons/long simulations with QMC now possible. à MD for realistic liquid à example on Hydrogen at high pressures à Can we do also liquid water?

3 We assume that HTc superconductivity shows up in a correlated systems due to strong correlation. The paradigm wave function is the Gutzwiller partially projected BCS (Mean Field) wavefunction: ψ VMC = exp( g n R n R )P N MF R where MF is the ground state of the BCS hamiltonian H = k,σ Variational Gutzwiller ansatz k [ 2(cosk x + cosk y ) µ VMC ]c + k,σ c kσ + Δ(cosk x cosk y ) % P N BCS = AGP = ' f k c k & k + + c k c + + k c k There are only 3 variational parameters for f k. ( * ) N /2 0 + h.c.

4 Important to optimize Jastrow and BCS toghether Hubbard Model : BCS order parameter H + + = t ci σ c jσ U ni ni < ij>, σ i J BCS> BCS> Hubbard model U/t= Energy minimization Iterations In mean field (BCS) no way to have BCS>0 for U>0 Theorem Lieb 90 Qualitative new features appear if Jastrow and BCS optimized toghether: RVB insulator or supercond.

5 Generalization of the wave function to continuous ψ T = J MF MF> may be a standard Slater determinant J is the so called Jastrow correlation term: " J = exp g( r i, % $ r j )' # $ i< j &' g is a generic function of two el. cooredinates The peculiarity of our approach (TurboRVB) is to fully optimize MF> and J in a localized basis of simple atomic orbitals (e.g. Gaussians 1s,2p ).

6 Generalization to realityà the Hamiltonian is: H = i Δ i 2 Z j r i R + ij j i< j 1 r i r + Z iz j j R i R j {R i } are atomic classical coordinates within the Born-Oppenheimer approximation r 1, r 2, r 3 r " N J SD = exp g( r i, % $ r j )' # $ i< j &' Det "# ψ i( r j )% & Given that, one can apply Variational Monte Carlo and compute all correlation functions by a statistical method No further approximation required fully ab-initio (no U, no double counting) But how to parametrize the function g?

7 The Gutzwiller for realistic systems u lr (r) = 1 2 g( r, r!) = u lr ( r r! )+ λ a,b i, j ψ a i ( r)ψ b j (! a,b,,i, j e.g. ψ a k ( r) = exp" Z k r R # a 2 $ a,b %, i.e. localized atomic orbitals, #λ kl # atoms variational parameters (say~1000) determined by: min a λkl, SD r ) r 1+ Br "a(b)" labels atom positions R a (R b ) The non-homogeneous part a=b is local like Gutzw. and useful to decrease # parameters (no 4-body a b) SD JHJ SD SD J 2 SD

8 Quantum Monte Carlo vs DFT, is it worth? In H 2 clear With a very small basis (2 gaussians/atom) one gets the essentially exact dispersion for H 2

9 The main question we want to address: What happens when we apply large pressure to a hydrogen molecular liquid? When the average distance between molecules is comparable with their bond length (~1.4 a.u.) we have a transition to a system where the molecule is no longer defined (atomic). According to band theory, from an insulator 2el/unit (H 2 ) to an half-filled band 1el./unit (H)à Metal, Wigner and Heterington prediction 35.

10 Example: DFT failure for hydrogen

11 As we have learned by Car and Parrinello (1998) Phase Diagram of realistic systems à Ab-Initio Molecular dynamics with Born-Oppenheimer approx. Evaluation of Forces are required within QMC Algorithmic differentiation helped much and thanks also to Tapenade (automatic diff.): SS & Luca Capriotti, JCP 133, (2010)

12 Cpu time referenced to simple VMC (only energy) for computing all 3M force components in water. (CPU VMC+forces)/(CPU VMC) AD Finite difference # Water molecules x 4 Use of pseudopotentials straightforward

13 Just to clarify a bit what we mean by AD Just a black box programming discipline allowing to compute all derivatives of e L ( r 1, r 2,, r N, R 1, R 2, R M ) With respect to all 3N electron coordinates And the 3M ionic coordinates at the same ~cost of computing the local energy e L And the same for the wave function.

14 Dynamics: 1. Efficient QMC forces Check: Newtonian dynamics of a H2 molecule. Verlet integrator.

15 Dynamics: 2. Generalized Langevin We have efficient but still noisy forces. We use Langevin dynamics in order to sample the canonical ensamble for the ions. Fluctuation - dissipation Covariance matrix of the forces Now the noise does not prevent the possibility of doing MD. Only renormalize the friction!

16 Dynamics: 2. Generalized Langevin We have efficient but still noisy forces. We use Langevin dynamics in order to sample the canonical ensamble for the ions.

17 At large N the first order is evident from the g(r) 6 g (r) r s = 1.28 r s = 1.24 r s = 1.23 r s = 1.22 r s = 1.21 r s = Jump T= 600 K r r s = 1.19

18 Liquid-liquid transition T= 2300 K

19 Our quantum Monte Carlo phase diagram ( ) for the first time with N=256 Hydrogen, is now much different from DFT!!!! Weir (molecular) Experiment 1996 We find that the molecular fluid is unexpectedly stable and the transition towards a fully atomic liquid occurs at much higher pressures.

20 Comparison small basis/good basis 54 atoms rs=1.44 T=1000K increased accuracyà more stable molecular!!!

21 Why is so different? QMC vs DFT PES for a single molecule inside the bulk (54H)

22 How to distinguish a metal from insulator? One can compute the density matrix: D(r,r')= ψ i i (r)ψ i (r') ψ i (r) are the optimized molecular orbitals Metalà Fermi surface à D(R,r) ~ R-r -2 Insulatorà Gap à D(R,r) ~exp(- R-r /ξ) Thus DM = 1 # atoms dr 3 D(R a, r) R a Metal DM à Insulator DM à Finite

23 There is some interesting crossover at T=2400K But we do not have enough large size for the MIT

24 And now few slides on liquid water Why water liquid simulation? It is fundamental in biological life, e.g. life with No waterà Non sense Phase diagram of immense difficulties, low energy and competing (e.g. Hydrogen bond and vdw long range) scales and still many things to understand by computer simulations. DFT problems, g(r) overstructured, eq. density large (20% off), supercooled liquid (melting at ~400K)

25 Reduction of number of parameters In QMC optimization the number of parameters is proportional to the dimension of the basis. It is useful to reduce them by hybrid contraction ϕ a,l,m,n (r) = lmn c a ψ GTO (r) lmn lmn,a This is not useful instead in chemistry as the molecular HF basis is used instead. See A. Zen, Y. Luo, SS and L.Guidoni JCTC 2013

26 Water dimer test: NB in dynamics we are interested in relative forces, i.e. derivative of binding energy 2.0 Water dimer bond energy Bond energy [kcal/mol] VMC-JSD small basis VMC-JSD-converged basis LRDMC Relevant region Distance oxygens [Ang]

27 Choice of Great freedom properly damping fast modes large time step reducing the correlation time Our choice γ estimated by QMC contains info of Hessian α α = α I QMC = Noise is useful!!! 0 cov( γ + Δ f ) 1 = α( R) 2T 0 α QMC

28 Water dimer integration test: Δ 0 = 8a.u. 300K DFT has not the covariance of forces and is much less efficient (smaller time steps) than QMC (fs)

29 Following the BO 150K fitting 1000K fitting up to V 2 up to V 4 Y. Luo, A. Zen and S. Sorella submitted to JCP

30 E pur si muove (and yet it moves) (Galileo Galilei) Second order Langevin dynamics of 32 water molecules at 300K with Variational Monte Carlo G. Mazzola, A. Zen, Y. Luo, L. Guidoni, and S. Sorella in preparation (2014)

31 Some info on this simulation 2048 nodes on BG/Q, 30 days simulation: 24milion core hours Time step = 1.54fs, Total time ~ 10ps Each step MDà 10 Optimization steps ~12000 Variational parameters ~ Sampling measurements/step Each sample after 1024 Metropolis step A huge computation, impossible without HPC

32 A different phylosophy is to use DMC to correct DFT (Blyp-2) Alfe et al JCP 2013.

33 But the quantum effects should play a role from J. A. Morrone and R.Car PRL, 2008 i) Peak positions are not changed by quantum ii) The radial distribution is substantially broaden

34 QMC water radial distribution function More than 4000 iterations~ 7ps Remaining differences: Ionic quantum effects? More accuracy? Size effects? A. Zen, G. Mazzola,, Y. Luo, L. Guidoni, and S. Sorella in preparation (2014)

35 64 waters possible with nodes (> cores!!!) 5 first peak in g OO (r) convergence r MAX region of fit VMC 32-wat g OO (r) VMC 64-wat r [Ang] / # steps Neutron diffraction - Soper, Chem. Phys. 258,121 (2000) Neutron diffraction - Soper, ISRN Phys. Chem.,2013 (2013) X-ray diffraction - LB Skinner et al., JCP 138, (2013) 32 waters VMC-based NVT MD simulation 32 waters DFT/BLYP-based NVT MD simulation 64 waters DFT/BLYP-based NVT MD simulation

36 The pressure problem: At the right density the average pressure is 0.47Gpa>> 1atm(=10-5 Gpa!!!) Pressure (Gpa) VMC LRDMC DMC reduces much (and change sign) this value a 2

37 Conclusions Realistic simulation of liquids are now possible also within fully many-body wave function based approach. 256H (64 H 2 O) are not so far from what is currently done within DFT ~500H (128 H 2 O) à Peak positions of the rdf are finally reproduced: no other ab-initio first principle simulation is able à Accuracy of VMC probably not enough, and also quantum effects should play a role. à Several applications are now possible, allowing to falsify or improve DFT predictions. Liquid water is currently under investigation.

38 TurboRVB Quantum Monte Carlo package main developers: Sandro Sorella Michele Casula Claudio Attaccalite A. Zen Ye Luo G. Mazzola E. Coccia Funding: MIUR-CNR, Riken-Tokio, PRACE Computational support CINECA Bologna, SISSA