Coupled Electron-Ion Monte Carlo Study of Hydrogen Fluid Kris Delaney, David Ceperley University of Illinois at Urbana-Champaign Carlo Pierleoni Universita del l’Aquila, l’Aquila, Italy Supported by NSF DMR 04-04853 and NSF DMR 03-25939
Dec 21, 2015
Coupled Electron-Ion Monte Carlo Study of Hydrogen Fluid
Kris Delaney, David CeperleyUniversity of Illinois at Urbana-Champaign
Carlo PierleoniUniversita del l’Aquila, l’Aquila, Italy
Supported by NSF DMR 04-04853 and NSF DMR 03-25939 ITR
Overview
• Features of hydrogen fluid
• Simulation Method:– The CEIMC approach– Choosing and sampling the ensemble– Energy differences with noise– The trial wavefunction
• Simulating phase transition in finite systems
• Results– Plasma phase transition– Finite-size error assessment
• Conclusions
Why Hydrogen?• Important in planetary science and high-pressure
physics• Most abundant element• Theoretically clean:
– 1 electron and 1 proton per atom– No pseudopotential required
• A rich variety of properties, including:– Metal-insulator transition in fluid– Possible liquid-liquid phase transition– Possibility of superconducting and superfluid phases
• Equation of state not yet fully described
Features of Hydrogen Fluid• Current phase diagram:
• Open problems:– Liquid-liquid phase transition
– Shape of melt curve
– Possibility of quantum fluid
The CEIMC Approach• Assume Born-Oppenheimer (BO) valid:
– Separates electronic and ionic degrees of freedom– Electronic subsystem adiabatically remains in ground-state
• May split system into two coupled Monte Carlo simulations:– Classical or quantum (PI) simulation for ions at finite T– T=0K QMC for electrons
The CEIMC Approach• Assume Born-Oppenheimer (BO) valid:
– Separates electronic and ionic degrees of freedom– Electronic subsystem adiabatically remains in ground-state
• May split system into two coupled Monte Carlo simulations:– Classical or quantum (PI) simulation for ions at finite T– T=0K QMC for electrons
The CEIMC Approach• Assume Born-Oppenheimer (BO) valid:
– Separates electronic and ionic degrees of freedom– Electronic subsystem adiabatically remains in ground-state
• May split system into two coupled Monte Carlo simulations:– Classical or quantum (PI) simulation for ions at finite T– T=0K QMC for electrons
• BO good for T<<TF (~150,000K)
Choice of Ensemble
• For thermodynamic observables, first generate ionic configurations to a phase-space probability distribution of some ensemble.
• All ensemble averages equivalent in thermodynamic limit– Far from simulation regime– Careful choice required
• Use constant T: energy fluctuates in exchange with environmental heat bath.
• Use constant N: theoretical simplicity, especially in wavefunction generation.
• Choices: NPT or NVT.
The NVT Ensemble• We choose NVT for these studies
• Pros:– Algorithmically simple– Fewer computations per MC step than NPT
• Cons:– May bias transitions involving crystalline phases if simulation cell
doesn’t conformOK for fluid studies
– Doesn’t capture density fluctuations beyond length-scale of simulation cell: NPT or VT (GCE) would.
Must monitor error due to finite cell size
NVT Partition Function• The NVT partition function for classical ions is:
• Monte Carlo:– The momentum part is analytically soluble:
– Sample only configuration space, not phase-space.– Generate samples according to:
– Metropolis:• Propose uniform move in 3D box.
• Accept according to:
SSUA ,exp,1min
)(exp SU
N
i
i
VD
iNVT m
ppdSUdS
NZ
1
2
2expexp
!
1
2
3
2N
mP
BO energy difference of ion configurations
S,S’
Spatial configuration
of ions
The CEIMC Approach• Assume Born-Oppenheimer (BO) valid:
– Separates electronic and ionic degrees of freedom– Electronic subsystem adiabatically remains in ground-state
• May split system into two coupled Monte Carlo simulations:– Classical or quantum (PI) simulation for ions at finite T– T=0K QMC for electrons
• BO good for T<<TF (~150,000K)
QMC Energy Differences• Compute U(S,S’) with either VMC or RQMC
• RQMC: projector-based method with no mixed-estimator– More accurate than VMC– Unbiased for energy differences.
• Finite-size errors on energy difference are reduced by using twist-averaged boundary conditions– Improves electron kinetic energy part of U
– Does not improve potential energy error
• Remaining issues:– Estimate of U will have a statistical uncertainty– A fast and accurate trial function is required
Dealing with Noise• 3 Approaches:
– Run QMC for longer. U noise lower– Reduce noise through correlated sampling methods– Tolerate noise with modified acceptance criterion
• Correlated Sampling:– Direct energy difference; noise often large compared with U
– Importance sampling; QMC walker (R) samples combination of S1,S2
2
221
2121,
U
SUSUSSU
222211
21 ERERQERERRp LL
22
21 Q
Penalty Method• Metropolis acceptance ratio is biased if U has noise
• Adjust using “Penalty Method”:– Satisfy detailed balance on average– If U is Gaussian distributed (CLT) acceptance ratio becomes
– Tolerates noisy estimates of energy differences• Extra rejections through noise.
– In practice, estimated → further terms– J. Chem. Phys. 110, 9812 (1999)
• Correlated sampling methods still important noise lowered fewer noise rejections
2',exp,1min
2 SSUA
Trial Wavefunction• BO energy is defined as:
• Approximation for . Choose Slater-Jastrow:
• The {i}s obey a single-particle equation
• Veff can be:
– Some parameterized potential. Parameters optimized with EVMC.
Fast, but free parameters
– Full self-consistent Kohn-Sham potential with approximate Vxc (eg, LDA, GGA)
No free parameters, but slow
RSJSrRS ji ,;det,
SVSh
SrSrSh
eff
iii
2
2
)(ˆ
;;ˆ
RSSHRSSU BO ,ˆ,
where
Choices:•RPA•Optimizable:
•1-body•2-body
Choices:•Analytic backflow•Gaussian orbitals (molecular configurations)•DFT-LDA•Optimizable OEP bands
Trial Wavefunction: PW basis
• We employ a plane-wave basis to represent the single-body Hamiltonian:
• Fourier transform of potential is often possible analytically. For example, OEP potential in PBCs:
cell
effGGri
GGk SrVedGkGShG ;1
2
1'ˆ )'.(
'
2
i
iOEPeff RrVSrV ;
i
GGRiOEPeff
ieGGVSGGV '.';',
Cell structure factor.The only S-dependence
Trial Wavefunction: Cusp
• Solve for {i} in plane-wave basis (PWs)
– Many basis functions, few {i} band-by-band iterative diagonalization
– Sampling eln-ion cusp difficult with PWs: coefficients decay slowly with wavevector magnitude
• Remove eln-ion cusp from {i} using
RPA on the incomplete FFT grid.
• Cusp is returned analytically in
Jastrow term. No basis error.
• High-K less important. Truncate
basis-set after removal. Each
Slater determinant much faster.
CEIMC Summary• Propose move S→S’
• Compute new (S’)– Expand h in plane-wave basis– Diagonalize h– Iterate for new h if DFT-LDA– Cusp removal + basis truncation
• Compute U(S,S’)– Correlated sampling– VMC or RQMC (fixed node)– TABC – typically 216 twists, 15,000 total electron moves
• Accept/reject– Penalty method
• Repeat to sample Boltzmann distribution– Typically 8,000--20,000 moves needed
Optimizing OEP• Choose an electron-nuclear potential including effective
screening with free parameters:– Yukawa– Gaussian– Yukawa + Gaussian– …?
• Variationally optimize free potential parameters (eg. Yukawa screening)– Either on each CEIMC nuclear move (slow!)– A variety of static nuclear configurations, including:
• Crystals
• Disordered
• Molecular + non-molecular states
• Range of densities
• Try to discover parameter trends
Potential Optimization
• Parameters optimized for molecular & non-molecular crystals, and disordered systems at densities ranging from rs=1-4
• Bare Coulomb interaction was always optimal or near-optimal
Wavefunction Test• Test wavefunctions on static crystals:
– Veff = Bare Coulomb or DFT-LDA
– Pure Gaussian orbitals– Backflow
• Variational principle: lowest energy best wavefunction
Simulating Phase Transitions
• 1st-order phase-transitions hard to locate precisely:– Finite-size cell: periodic correlations suppress interface between pure
phases close to transition; forces system into a metastable state.– Finite simulation time: probability of changing from one phase to
another is related to F (Helmholtz F for NVT) and height of barrier.– Result: hysteresis; signature of first-order transition.
• Higher order transitions: usually no hysteresis .• Close to a critical point:
– System behaviour is singular– Fluctuations in density become
unbounded.– Finite-cell simulations more
unreliable, especially NVT.
The Plasma Phase Transition
• Study nature of transition from molecular to non-molecular fluid using CEIMC
• Simulations at T=2000K with P=50-200GPa
Results (U from VMC)• Simulation details:
– 32 atoms, NVT ensemble– T = 2000K
– P = 50 – 200 GPa
– VMC for U
– 216 twist angles
• Circles: simulations started from molecular fluid
• Crosses: from non-molecular fluid
Clear hysteresis
Results (U from RQMC)• Simulation details:
– 32 atoms, NVT ensemble– T = 2000K– P = 50 – 200 GPa– RQMC for U– 216 twist angles
• Circles: simulations started from molecular fluid
• Crosses: from non-molecular fluid
No hysteresis
Molecular Order Parameter
• Molecular order parameter, , defined as:
• VMC: Hysteresis; probably 1st order.• RQMC: No hysteresis; continuous transition.
rgrgrg nonmolmol 1)(
Finite-Size Error Assessment
•Pronounced finite-size error•Consistent with first order behaviour
•No significant finite-size error•Consistent with continuous transition
Circles: 32-atom cellCrosses: 54-atom cell
VMC RQMC
Conclusions
• More accurate solution of Hamiltonian changes the nature of the transition from molecular to non-molecular fluid
• VMC results indicate first-order at T=2000K– Compatible with CPMD but at different pressure
• RQMC results indicate continuous transition at T=2000K
• FSE not large for PPT, even when 1st order. No long-range nucleation.
• Future work:– Investigate metallization of fluid– Effects of quantum nuclei