QMC and DFT Studies QMC and DFT Studies of Solid Neonof Solid Neon

Neil Drummond and Richard NeedsNeil Drummond and Richard Needs

TCM Group, Cavendish Laboratory, TCM Group, Cavendish Laboratory, Cambridge, UKCambridge, UK

ESDG meeting, 9th of November, 2005

Solid NeonSolid NeonNeonNeon is a is a noble gasnoble gas. It has no partially filled shells . It has no partially filled shells of electrons.of electrons.The chemistry of neon is therefore particularly The chemistry of neon is therefore particularly simple: a competition between simple: a competition between van der Waalsvan der Waals attraction and attraction and hard-corehard-core repulsion. repulsion.At very low temperatures or high pressures, neon At very low temperatures or high pressures, neon forms a crystalline solid with the FCC structure.forms a crystalline solid with the FCC structure.Because of its simplicity, and the fact that highly Because of its simplicity, and the fact that highly accurate experimental data are available, neon is an accurate experimental data are available, neon is an excellent test system for theoretical methods.excellent test system for theoretical methods.In particular, a lot of effort has been put into In particular, a lot of effort has been put into constructing accurate interatomic pair potentials for constructing accurate interatomic pair potentials for neon, which can be used to calculate a wide range neon, which can be used to calculate a wide range of properties (e.g., phase diagram, diff. const., …)of properties (e.g., phase diagram, diff. const., …)

Neon in Diamond Anvil CellsNeon in Diamond Anvil CellsNeon is widely used as a Neon is widely used as a pressure-conducting pressure-conducting mediummedium in in diamond-anvil-celldiamond-anvil-cell experiments. experiments.

Its zero-temperature equation of state (pressure-Its zero-temperature equation of state (pressure-density relationship) is therefore of some practical density relationship) is therefore of some practical importance.importance.

Diamond anvil

Metal gasket

Pressure-conducting medium, e.g. neon

Sample

Van der Waals ForcesVan der Waals Forces

Although Van der Waals forces are weak, they are Although Van der Waals forces are weak, they are often the often the onlyonly attractive force between molecules. attractive force between molecules.

VdW forces are not described by Hartree-Fock VdW forces are not described by Hartree-Fock theory, because they are due to theory, because they are due to correlationcorrelation effects. effects.

The dependence on the charge density is The dependence on the charge density is nonlocalnonlocal, , so the usual approximations in DFT are poor.so the usual approximations in DFT are poor.

Two electrically neutral, closed-shell atoms

Gives net attraction

Temporary dipole resulting from quantum fluctuation

Induced dipole, due to presence of other dipole

Hard-Core Repulsive ForcesHard-Core Repulsive Forces

Exchange Exchange effects give rise to strong repulsive forces effects give rise to strong repulsive forces when noble-gas atoms are brought sufficiently close when noble-gas atoms are brought sufficiently close together that their electron clouds overlap.together that their electron clouds overlap.

There is no reason why this There is no reason why this hard-corehard-core repulsion repulsion should not be well-described by DFT or HF theory.should not be well-described by DFT or HF theory.

HCR may be poorly described by semiempirical pair HCR may be poorly described by semiempirical pair potentials, however, because there are limited potentials, however, because there are limited experimental data in the small-separation / high-experimental data in the small-separation / high-density regime.density regime.

Electron clouds overlap when neon atoms are brought together

Two neon atoms

Aims of this ProjectAims of this ProjectTo compare the accuracy with which To compare the accuracy with which competing electronic-structure and pair-competing electronic-structure and pair-potential methods describe van der Waals potential methods describe van der Waals forces and hard-core repulsion.forces and hard-core repulsion.To calculate an accurate equation of state for To calculate an accurate equation of state for solid neon using the solid neon using the diffusion Monte Carlodiffusion Monte Carlo method.method.To use the DMC method to generate a new To use the DMC method to generate a new pair potential for neon, and to assess the pair potential for neon, and to assess the performance of this pair potential.performance of this pair potential.

DFT CalculationsDFT Calculations

Plane-wave basis set (CASTEP).Plane-wave basis set (CASTEP).

LDA and PBE-GGA XC functionals.LDA and PBE-GGA XC functionals.

Ultrasoft neon pseudopotentials.Ultrasoft neon pseudopotentials.

Ensured convergence with respect to plane-Ensured convergence with respect to plane-wave cutoff energy and wave cutoff energy and kk-point sampling.-point sampling.

Ensured convergence of Hellmann-Feynman Ensured convergence of Hellmann-Feynman force constants with respect to atomic force constants with respect to atomic displacements and supercell size in phonon displacements and supercell size in phonon calculations.calculations.

QMC Calculations IQMC Calculations IUsed DFT-LDA orbitals in a Used DFT-LDA orbitals in a Slater-Jastrow trial wave Slater-Jastrow trial wave function (CASINO).function (CASINO).Used HF neon pseudopot.Used HF neon pseudopot.Appreciable time-step bias in Appreciable time-step bias in DMC energies in EoS DMC energies in EoS calculations. (Not in pair-calculations. (Not in pair-potential calcs, where a much potential calcs, where a much smaller time step was used.)smaller time step was used.)Used same time step in all Used same time step in all DMC EoS calculations; bias in DMC EoS calculations; bias in energy nearly same at each energy nearly same at each density; density; hence there is very hence there is very little bias in the pressurelittle bias in the pressure..Verified this by calculating EoS Verified this by calculating EoS at two different time steps: at two different time steps: clear that EoS has converged.clear that EoS has converged.

QMC Calculations IIQMC Calculations IIThe QMC energy per atom in an The QMC energy per atom in an infiniteinfinite crystal differs crystal differs from the energy per atom in a from the energy per atom in a finitefinite crystal subject to crystal subject to periodic boundary conditions.periodic boundary conditions.Difference is due to Difference is due to single-particle finite-size effectssingle-particle finite-size effects (i.e. (i.e. kk-point sampling) and -point sampling) and Coulomb finite-size effectsCoulomb finite-size effects (interaction of electrons with their periodic images).(interaction of electrons with their periodic images).The former are negligible in our QMC results.The former are negligible in our QMC results.Latter bias is assumed to go as 1/Latter bias is assumed to go as 1/NN, where , where NN is the is the number of electrons.number of electrons.Vinet EoSs are fitted to QMC results in simulation Vinet EoSs are fitted to QMC results in simulation cells of 3x3x3 and 4x4x4 primitive unit cells, and the cells of 3x3x3 and 4x4x4 primitive unit cells, and the assumed form of the finite-size bias is used to assumed form of the finite-size bias is used to extrapolate the EoS to infinite system size.extrapolate the EoS to infinite system size.

Lattice DynamicsLattice DynamicsThe The zero-point energyzero-point energy of the lattice-vibration of the lattice-vibration modes is significant in solid neon.modes is significant in solid neon.The phonon frequencies and lattice thermal free The phonon frequencies and lattice thermal free energy were calculated within the energy were calculated within the quasiharmonic quasiharmonic approx.approx. using the using the method of finite displacementsmethod of finite displacements..(Displace one atom in a periodic supercell, and (Displace one atom in a periodic supercell, and evaluate the forces on the other atoms; write down evaluate the forces on the other atoms; write down Newton’s 2Newton’s 2ndnd law for the atoms and look for a law for the atoms and look for a normal-mode solution with wave vector normal-mode solution with wave vector kk; obtain an ; obtain an eigenvalue problem for the squared phonon eigenvalue problem for the squared phonon frequencies; each frequency corresponds to an frequencies; each frequency corresponds to an independent harmonic oscillator: use statistical independent harmonic oscillator: use statistical mechanics to calculate the free energy of each mechanics to calculate the free energy of each harmonic oscillator; integrate over harmonic oscillator; integrate over kk.).)DFT Hellmann-Feynman forces or forces from the DFT Hellmann-Feynman forces or forces from the pair potential were used in our phonon calculations.pair potential were used in our phonon calculations.

MiscellaneaMiscellanea

VinetVinet EoS models give lower EoS models give lower χχ22 values than values than Birch-MurnaghanBirch-Murnaghan models when fitted to models when fitted to DFT or QMC DFT or QMC EE((VV) data for solid neon.) data for solid neon.

We compared the energies of We compared the energies of FCCFCC and and hexagonalhexagonal phases of solid neon within DFT, phases of solid neon within DFT, but were unable to resolve any phase but were unable to resolve any phase transition.transition.

Experimentally, the FCC phase is observed Experimentally, the FCC phase is observed up to at least 100 GPa, and so we have used up to at least 100 GPa, and so we have used this structure in all of our calculations.this structure in all of our calculations.

Pair PotentialsPair PotentialsHFD-BHFD-B: “Best” semiempirical pair potential in : “Best” semiempirical pair potential in the literature, due to Aziz and Aziz & Slaman. the literature, due to Aziz and Aziz & Slaman. Fitted to a wide range of experimental data.Fitted to a wide range of experimental data.CCSD(T)CCSD(T): a fit of the form of potential : a fit of the form of potential proposed by Korona to the results of CCSD(T) proposed by Korona to the results of CCSD(T) quantum-chemistry calculations for a neon quantum-chemistry calculations for a neon dimer performed by Cybulski and Toczydimer performed by Cybulski and Toczyłłowski.owski.DMCDMC: a fit of the form of potential proposed by : a fit of the form of potential proposed by Korona to our DMC results.Korona to our DMC results.

r Calculate total fixed-nucleus energy E(r) using DMC. Gives pair potential within Born-Oppenheimer approx., up to a constant. Constant is a fitting parameter.

Neon dimer:

Using Pair PotentialsUsing Pair PotentialsTo get the static-lattice energy per atom:

1. Add up pair potential U(r) between red atom and each white atom inside the sphere.

2. Integrate 4πr2U(r) from radius of sphere to infinity & multiply by density of atoms to get contribution from atoms outside sphere.

3. Divide by two, to undo double counting.

The radius of the sphere is increased until the The radius of the sphere is increased until the static-lattice energy per atom has converged.static-lattice energy per atom has converged.The ZPE is calculated using a periodic The ZPE is calculated using a periodic supercell, finite displacements of atoms and supercell, finite displacements of atoms and the quasiharmonic approximation.the quasiharmonic approximation.

Phonon Dispersion Curves: High Phonon Dispersion Curves: High DensityDensity

At high densities the phonon dispersion At high densities the phonon dispersion curves obtained using DFT and the pair curves obtained using DFT and the pair potentials are in good agreementpotentials are in good agreement

Phonon Dispersion Curves: Low Phonon Dispersion Curves: Low DensityDensity

The phonon dispersion curves calculated using the The phonon dispersion curves calculated using the different methods don’t agree so well at low densities.different methods don’t agree so well at low densities.

We assume the HFD-B curve to be the most accurate.We assume the HFD-B curve to be the most accurate.

Pressure due to Zero-Point EnergyPressure due to Zero-Point Energy

Pressure due to ZPE is significant.Pressure due to ZPE is significant.

All methods are in good agreement.All methods are in good agreement.

Einstein approximation gives good results.Einstein approximation gives good results.

Band Gap of Solid NeonBand Gap of Solid Neon

Solid neon has one of the highest metallisation densities of Solid neon has one of the highest metallisation densities of any material.any material.Hawke Hawke et al.et al. used a magnetic flux compression device to used a magnetic flux compression device to demonstrate that neon is still an insulator at 500 GPa.demonstrate that neon is still an insulator at 500 GPa.Our DFT calculations predict the metallisation pressure of Our DFT calculations predict the metallisation pressure of neon to be around 366 TPa.neon to be around 366 TPa.

The Equation of State IThe Equation of State I

The Equation of State IIThe Equation of State II

The Equation of State IIIThe Equation of State III

The DFT-LDA and DFT-PBE EoSs are very The DFT-LDA and DFT-PBE EoSs are very different from one another at low to different from one another at low to intermediate densities, indicating that DFT intermediate densities, indicating that DFT gives a poor description of vdW forces.gives a poor description of vdW forces.

The DMC EoS is highly accurate at all The DMC EoS is highly accurate at all densities, although the HFD-B pair potential is densities, although the HFD-B pair potential is also accurate at low densities.also accurate at low densities.

The CCSD(T) and DMC pair potentials do not The CCSD(T) and DMC pair potentials do not give very accurate EoSs, unlike the HFD-B give very accurate EoSs, unlike the HFD-B pair potential (at low pressure at least). pair potential (at low pressure at least).

ConclusionsConclusionsDMC gives an DMC gives an accurateaccurate description of both description of both van der Waals forces and hard-core repulsion van der Waals forces and hard-core repulsion in solid neon, whereas DFT gives a in solid neon, whereas DFT gives a poorpoor description of van der Waals attraction.description of van der Waals attraction.

It is reasonable to expect that these It is reasonable to expect that these conclusions will hold in other systems where conclusions will hold in other systems where van der Waals forces are important.van der Waals forces are important.

DMC and CCSD(T) pair potentials do not give DMC and CCSD(T) pair potentials do not give especially good EoS results for neon: especially good EoS results for neon: therefore therefore non-pairwise non-pairwise effects must be effects must be significant in solid neon.significant in solid neon.

AcknowledgmentsAcknowledgments

We thank John Trail for providing the We thank John Trail for providing the relativistic Hartree-Fock neon relativistic Hartree-Fock neon pseudopotentials used in this work.pseudopotentials used in this work.We have received financial support from We have received financial support from the Engineering & Physical Sciences the Engineering & Physical Sciences Research Council (EPSRC), UK.Research Council (EPSRC), UK.Computing resources have been provided Computing resources have been provided by the Cambridge-Cranfield High-by the Cambridge-Cranfield High-Performance Computing Facility.Performance Computing Facility.