Quantum Simulations of Materials Under Extreme Conditions

David M. CeperleyRichard M. Martin

Simone Chiesa Ed Bukhman

William D. Mattson*Xinlu Cheng

Department of PhysicsUniversity of Illinois at Urbana-Champaign

Not supported by the MURI grant *Thesis at University of Illinois, 2003 now at Army research Lab

Simulations of energetic materialsfrom the fundamental equations

• Simulation techniques are essential to “solve” many-body problems: e.g. classical simulations of atoms & molecules, reactions, thermal motion

• Combine Quantum Monte Carlo, DFT and Quantum Chemistry methods– Density Functional Theory (DFT)

• Most widely used approach for large scale simulations of nuclei and electrons

• In principle exact, but, in practice, limited by the approximate functionals

– Quantum Monte Carlo (QMC)• Most accurate method for large, many-electron systems • A wavefunction-based approach• Provides benchmark quality results for systems of 1000’s of valence

electrons • Can describe matter from plasmas to molecules to condensed matter• Provides improved functionals for DFT• DFT provides input for QMC trial functions

• Development of new methods --- Applications to energetic materials

Nitrogen under extreme conditions• DFT simulations as a function of pressure and temperature• SIESTA code – GGA functional• Dissociation and exotic behavior in shock waves

Squeezed&

Cooled

•Hot molecular liquid --- 58 Gpa 7600 K•Nitrogen molecules dissociate and reform

– Connected structures – non-molecular

– Two-fold (chain-like) and three-fold (cubic gauche-like) – Large energy barriers

– Glassy behavior and meta-stability at low temperature – Prediction of new structures at low temperature

Nitrogen under extreme conditions

Molecular N2 – N6

Hexagonal packedzig-zag chains

Volume/atom

Ene

rgy/

atom

Known phase

W. D. Mattson, D. Sanchez-Portal, S. Chiesa, R. M. Martin, Phys. Rev. Lett. (2004)

New low energy structures found in low temperature simulations

Previously predicted“Cubic Gauche”

Nitrogen: New structures predictedNew low energy crystal structures found from simulations at low temperature

GGA functional

Known phase

Hexagonal packedzig-zag chains

Molecular N2 – N6 structure

Top view

Side view

Fermi Surface of

Hexagonal packedzig-zag chains - Two types of bands

Oxygen: Prediction of energies of atomic phases at high pressure

• Calculations for simple metallic structures using same method as used for nitrogen – SIESTA with GGA

En

ergy

per

ato

m -

eV

-433

-432

-431

-430

-429

-428

-427

-426

-425

0 10 20 30 40 50

Simple cubic

BCC

Volume per atom - A3

Magnetic Transition

• Collaboration with Brenner to make improved potentials for O

Simple cubic is most stable

• Preliminary molecular calculations to study dissociation pathways

• Goal: full simulations in condensed phase at high temperature and pressure

Nitromethane: CH3-NO2

• Calculations using SIESTA with GGA• Related to work in recent papers

– Kabadi and Rice, J. Phys. Chem. A 108, 532 (2004)

– Manna, Reed, Fried, Galli, and Gygi, J. Chem. Phys. 120, 10146 ( 2004)

Quantum Monte Carlo (QMC) simulations of energetic materials

• Symbiosis between QMC & DFT-quantum chemistry approaches– QMC gives benchmark quality results for systems of 1000’s of valence

electrons – can describe condensed matter

• QMC denotes several stochastic methods:– Variational Monte Carlo ( T=0) – Projector Monte Carlo - diffusion MC – Path Integral Monte Carlo ( T>0) – Coupled electron-ion Monte Carlo (separating energy scales)

• What is “niche” for QMC in understanding energetic materials?– Systems with strong correlation such as

– Rearrangements of electrons during reactions – Nearly degenerate structures

– Disordered systems such as liquids – Significant electronic excitations or temperature effects

• New advances this year

New method for correcting size effects

• Able to treat anisotropic structures, metals, insulators,..

• Potential energy correction from low k-limit of charge-charge response function, S(k).

• Kinetic energy corrections from Brillouin zone integration within DFT.

Much smaller size dependence Hence, more accurate extrapolation to thermodynamic limit

Results for Nitrogen structures: QMC (with extrapolation) compared to DFT

•QMC supports our main result using PBE-GGA

•Energy of chain very close to cubic gauche; curves very similar

•QMC finds shifts in the total energy relative to the N2 molecule

Bond dissociation energiesof nitro and amino molecules

• QMC studies of energetic molecules in kcal/mol. • Reasonable numbers even for largest molecules. • Statistical error < 1 kcal/mol• More work needed on minimizing fixed-node error

molecule DMC Other theory Exp.Methylamine CH3-NH2 86.6 81.5 DFT 85.7

Nitroamine NH2-NO2 62.3 53.6 G2

Nitromethane CH3-NO2 64.2 54.6 DFT 60.8-63.7

DMN (CH3)2N-NO2 52.5 39.6-43.8

RDX C3H6N5O4-NO2 72.1 41.6 DFT

Long standing problem: forces in QMC

3

i j i j

ji ij

e e r rF

r

Hellman-Feynman forces have infinite variance.Our approach:

• inside core: fit p-wave electronic QMC density using a polynomial basis. • outside core: compute force directly with HF equation

• Exact if electronic density is exact. Need to use forward walking or reptation to get the density.• Method is local, very simple to program, and fast.• Is it accurate?

Accuracy of bond distances:comparison with other methods

• All other bond distances taken from the NIST website

• QMC predicts bond lengths to 0.4%

• As accurate as other approaches

• Slower convergence for large Z

• Goal: applications to structures of energetic materials

Chiesa, Ceperley, Zhang, Sept. 04, physics/0409087

Relative error wrt experiment

Coupled Ionic-Electronic Simulations

• Much progress in recent years with “ab initio” molecular dynamics simulations.

• However density functional theory is not always accurate enough.• Use power of current commodity processors to enhance accuracy

of simulations

– Empirical potentials (e.g. Lennard-Jones)– Local density functional theory or other mean field methods (Car-Parrinello or ab

initio MD)– Quantum Monte Carlo: CEIMC method

Method demonstrated on molecular and metallic hydrogen at extreme pressures and temperatures. Fast code!

CEIMC calculations on dense HTemperature dependence in CPMD-LDA is off by 100%.

e-p distribution function

At the same temperature LDA scaled by 2

Progress this year

• Calculation of energy of new solid nitrogen structures– New method for QMC finite size corrections – Comparison of QMC and DFT – Paper published in PRL

• Calculation of high pressure oxygen• Survey of nitro amines bond dissociation energies with QMC.

– Direct coupling of QMC with DFT calculations • New method for computing forces within QMC

– Combines simplicity with accuracy. – Paper submitted

• Major effort to produce next generation QMC codes.• CEIMC calculations of dense hydrogen showing major problems with

DFT temperature scale.

Plans for next year

• Develop new CEIMC/PIMC code able to treat systems beyond hydrogen.– Appropriate pseudopotentials– Appropriate trial functions– Able to use Teraflop resources effectively.– Apply to energetic materials

• DFT simulations of energetic materials at high temperature and pressure– Search for dissociation mechanisms and pathways – Molecules and condensed systems, e.g., nitromethane– Initiate studies of more complex systems, e.g., RDX

• Benchmark studies for chemical reactions using QMC molecular forces. • Feasibility study for full simulations of energetic liquids in detonation

conditions.