The Coupled Electron-Ion Monte Carlo study of hydrogen under extreme - PowerPoint PPT Presentation

The Coupled Electron-Ion Monte Carlo study of hydrogen under extreme conditions Carlo Pierleoni Maison de la Simulation, CEA-Saclay Paris, France and Dept. of Physical and Chemical Sciences University of L’Aquila, Italy

OUTLINE • Motivation • Hydrogen: the paradigmatic system • Electronic structure methods: DFT vs QMC • Ab-initio simulation methods: the CEIMC • Trial wave function for hydrogen • Liquid-liquid phase transition in hydrogen • Crystalline molecular hydrogen: CEIMC vs PIMD predictions • Conclusions & perspectives

Motivations • To contribute to develop First-Principles simulation methods to a predictive level : standard first principle simulation methods are mostly based on Density Functional Theory, in practice a mean-field theory , with uncontrolled approximations. Our aim is to develop simulation methods where all approximations can be controlled and improved, hence switching from non-predictive to predictive methods . • To study matter at extreme conditions beyond present experimental capabilities : those methods will be able to provide reliable predictions even in absence of experimental results. E xperiments at extreme conditions are difficult and extremely expensive, they often provide only partial information and different methods are often in disagreement. Predictive First Principle theories will greatly help our understanding and will reduce the cost of these activities. • Light elements like Hydrogen, Helium, Lithium are very fundamental: their study under extreme conditions requires considering explicitly the electronic correlation, a fully quantum treatment of nuclei

Hydrogen: the paradigmatic system • Hydrogen is the simplest element, i.e. the element with the simplest electronic structure. • Hydrogen is the most abundant element in the Universe: the Giant gas planets are comprised by 70-90% of hydrogen, plus helium and other heavier elements. Developing accurate planetary models requires accurate acknowledge of the equation of state of hydrogen, helium and their mixtures. • Hydrogen is relevant for energy applications: nuclear fusion, etc. • The hydrogen atom and and the hydrogen molecule have been the prototype models in developing Quantum Mechanics. Hydrogen is the ideal playground to develop new theoretical approaches and methods . • Being the simplest element, it is desirable to be able to predict its properties from first- principle (the Hamiltonian is known and simple) from a theoretical perspective. • Despite its simplicity Hydrogen under pressure presents a reach and difficult physics.

Hydrogen phase diagram from J. McMahon, M.A. Morales, C. Pierleoni and D.M. Ceperley, Rev Mod Phys (2012) -6 -4 -2 0 2 4 6 8 10 10 10 10 10 10 10 10 10 10 T F 0.1 T F classical TCP 5 5 degenerate TCP 10 10 Temperature (K) 4 4 fluid H 10 10 fluid H 3 3 10 10 I fluid H 2 solid H 2 2 10 10 III II solid H 2 -6 -4 -2 0 2 4 6 8 10 10 10 10 10 10 10 10 10 10 Pressure (GPa)

Hydrogen phase diagram 2000 static compr. dynamic compr. 1600 CEIMC Fluid H Temperature (K) Fluid H 2 1200 800 Solid H 2 I (LP) 400 IV IV’ - V II (BSP) III (H-A) VI- H 2 (PRE) MH 0 100 200 300 400 500 Pressure (GPa) Wigner-Huntington IMT Pierleoni, Morales, Rillo, Holzmann, Ceperley PNAS 113, 4953 (2016)

2000 static compr. dynamic compr. 1600 CEIMC Fluid H Temperature (K) Fluid H 2 1200 800 Solid H 2 I (LP) 400 IV IV’ - V II (BSP) III (H-A) VI- H 2 (PRE) MH 0 200 300 500 100 400 Pressure (GPa) Pierleoni, Morales, Rillo, Holzmann, Ceperley PNAS 113, 4953 (2016)

First-principles theoretical methods • first-principle methods based on Quantum Mechanics and Statistical Mechanics . • they treat nuclei and electrons explicitly and are unique methods to study systems in a large variety of chemical and physical states • they assume the non-relativistic Hamiltonian of the system of nuclei and electrons in a volume Ω at temperature T (in condensed phase) • Under rather general conditions, the energy scales for nuclei and electrons are widely separated: adiabatic approximation (Born-Oppenheimer) ˆ T n + ˆ ˆ H el = ˆ T n + ˆ T e + ˆ H = V , N n N e 2 2 ˆ � I ˆ ˆ ˆ � � T n = � I , T e = � � e i , kinetic energies ⇥ ⇥ I =1 i =1 z I z J 1 z I ˆ � � � Coulomb law V = + , r j | � | � R I � � r i � � | � r i � � R J | | � R I | I<J i<j i,I e , � e ¼ 1 = 2 , � I ¼ 1 = ð 2 M I Þ , mass and charge (in units of First principles McMahon, Morales, Pierleoni, Ceperley, Rev. Mod. Phys. 84, 1607 (2012)

Electrons : solve the electronic problem at given nuclear positions ground state energy H el = ˆ ˆ K el + ˆ V nuclear coordinates ( 3N n ) ˆ Schroedinger equation (SE) H el Φ 0 ( r | R ) = E 0 ( R ) Φ 0 ( r | R ) electronic coordinates ( 3N e ) ground state wave function Density Functional Theory : maps the interacting electrons problem onto a single electron problem in a self-consistent effective potential. Solve the single electron SE as an eigenvalue problem in 3 dimensions . Mean field solution , introduces uncontrolled approximations. Quantum Monte Carlo: assumes an explicit form of the many-electrons wave function based on physical insight and exploits the Variational Principle to control the accuracy T ( r | R ) ˆ R H el Ψ T ( r | R ) dr Ψ ∗ E 0 ( R ) ≤ E T ( R ) ≡ trial wave function dr | Ψ T ( r | R ) | 2 R i 2 h ˆ R T ( r | R ) H el − E T ( R ) Ψ T ( r | R ) dr Ψ ∗ 0 ≤ σ 2 T ( R ) ≡ dr | Ψ T ( r | R ) | 2 R

Ψ T ( r | R ) depends explicitly on some free parameters to be optimized using the variational principle: the lower the energy and the variance the better the quality of the solution. The variational principle provides an internal consistency check when comparing various trial functions. Imaginary time projection automatically optimizes but requires an approximation for fermions: the fixed node approximation but the method remain variational H | Ψ T i / | Φ 0 i t →∞ e − t ˆ lim BCC hydrogen: N p =54; 6x6x6 twist grid -0.367 -0.107 SJ -0.368 •Slater-Jastrow (SJ) -0.369 -0.1075 •three-body (3b) energy (h/at) SJ+3b -0.37 Energy -0.108 •backflow (BF) -0.371 •fixed-node (FN) FN-SJ -0.1085 -0.372 SJ+3b+BF -0.373 -0.109 0 0.05 0.1 -0.374 3D electron gas at r s =10 0 0.02 0.04 0.06 0.08 2 /at) variance (h Variance

DFT QMC Advantages: Advantages: • electron correlation is explicitly put in the • DFT is reasonably fast and accurate. wave function. • it is far more transferrable than the • the accuracy can be assessed by the effective potential approach. Variational Principle (internal check of the • electronic dynamical properties can also theory). be computed in the single-electron theory • efficient methods to compute properties (optical spectra, transport properties….) other than the energy are available (not fully justified). Limitations: Limitations: • projection introduces the fixed node • DFT misses an internal check on the approximation. But the method is still accuracy of the various approximations variational. • assessment of accuracy based on • it needs larger computer resources. comparison with experiments or more • electronic dynamics is more difficult but, accurate theories (like QMC). to some extend, can be dealt with by • dispersion interactions, band gaps and CFQMC ( see Li et al, PRL 2010 ) excited states are generally bad. • the development of community codes is • the approximation is an extra variable to much behind DFT, so the use of QMC choose. much less spread.

Nuclear sampling : use the electronic energy (or forces) to sample the nuclear configurational space at physical temperature (Boltzmann): classical nuclei are point particles ( P=1 ) quantum nuclei are paths in configuration space (P> 1 , closed for diagonal observables) Molecular Dynamics or ρ n ( R ) ∼ e − [ K n + E T ( R )] /k B T R ∈ R 3 N n P Monte Carlo + importance sampling BOMD (CPMD): uses DFT forces and MD to sample nuclear configuration space CEIMC : uses QMC energy and Metropolis MC for nuclear sampling QMCMD : uses QMC forces and LD for nuclear sampling

Coupled Electron-Ion Monte Carlo ( CEIMC) : an ab-initio simulation method with QMC accuracy CEIMC: Metropolis Monte Carlo for finite T ions. The BO energy in the Boltzmann distribution is obtained by a QMC calculation for ground state electrons. • Ground state electrons: • Variation Monte Carlo (VMC) & Reptation Quantum Monte Carlo (RQMC) • Twist Average Boundary Conditions (TABC) within CEIMC to reduce electronic (single particle) finite size effects. • Efficient energy difference method • Efficient RQMC algorithm: The bounce algorithm • Finite temperature ions: Noisy Monte Carlo The Penalty Method • Quantum Protons: Path Integral Monte Carlo (PIMC) within CEIMC • Moving the nuclei: two level sampling • The computational cost of CEIMC in the present implementation is quite higher than for BOMD (limited to small systems ~ 100 protons), but the scaling is the same ( ~N 3 ). • HPC Tier-0 systems are now available for this generation of calculations !