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