QMC and DFT Studies of Solid Neon Neil Drummond and Richard Needs TCM Group, Cavendish Laboratory, Cambridge, UK ESDG meeting, 9 th of November, 2005

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

Neon in Diamond Anvil Cells Neon is widely used as a pressure-conducting medium in diamond-anvil-cell experiments. Its zero-temperature equation of state (pressure- density relationship) is therefore of some practical importance. Diamond anvil Metal gasket Pressure-conducting medium, e.g. neon Sample

Van der Waals Forces Although Van der Waals forces are weak, they are often the only attractive force between molecules. VdW forces are not described by Hartree-Fock theory, because they are due to correlation effects. The dependence on the charge density is nonlocal, 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 Forces Exchange effects give rise to strong repulsive forces when noble-gas atoms are brought sufficiently close together that their electron clouds overlap. There is no reason why this hard-core repulsion should not be well-described by DFT or HF theory. HCR may be poorly described by semiempirical pair potentials, however, because there are limited experimental data in the small-separation / high- density regime. Electron clouds overlap when neon atoms are brought together Two neon atoms

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

DFT Calculations Plane-wave basis set (CASTEP). LDA and PBE-GGA XC functionals. Ultrasoft neon pseudopotentials. Ensured convergence with respect to plane- wave cutoff energy and k-point sampling. Ensured convergence of Hellmann-Feynman force constants with respect to atomic displacements and supercell size in phonon calculations.

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

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

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

Miscellanea Vinet EoS models give lower χ 2 values than Birch-Murnaghan models when fitted to DFT or QMC E(V) data for solid neon. We compared the energies of FCC and hexagonal phases of solid neon within DFT, but were unable to resolve any phase transition. Experimentally, the FCC phase is observed up to at least 100 GPa, and so we have used this structure in all of our calculations.

Pair Potentials HFD-B: “Best” semiempirical pair potential in the literature, due to Aziz and Aziz & Slaman. Fitted to a wide range of experimental data. CCSD(T): a fit of the form of potential proposed by Korona to the results of CCSD(T) quantum-chemistry calculations for a neon dimer performed by Cybulski and Toczyłowski. DMC: a fit of the form of potential proposed by 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 Potentials To 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 4πr 2 U(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 static-lattice energy per atom has converged. The ZPE is calculated using a periodic supercell, finite displacements of atoms and the quasiharmonic approximation.

Phonon Dispersion Curves: High Density At high densities the phonon dispersion curves obtained using DFT and the pair potentials are in good agreement

Phonon Dispersion Curves: Low Density The phonon dispersion curves calculated using the different methods don’t agree so well at low densities. We assume the HFD-B curve to be the most accurate.

Pressure due to Zero-Point Energy Pressure due to ZPE is significant. All methods are in good agreement. Einstein approximation gives good results.

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

The Equation of State I

The Equation of State II

The Equation of State III The DFT-LDA and DFT-PBE EoSs are very different from one another at low to intermediate densities, indicating that DFT gives a poor description of vdW forces. The DMC EoS is highly accurate at all densities, although the HFD-B pair potential is also accurate at low densities. The CCSD(T) and DMC pair potentials do not give very accurate EoSs, unlike the HFD-B pair potential (at low pressure at least).

Conclusions DMC gives an accurate description of both van der Waals forces and hard-core repulsion in solid neon, whereas DFT gives a poor description of van der Waals attraction. It is reasonable to expect that these conclusions will hold in other systems where van der Waals forces are important. DMC and CCSD(T) pair potentials do not give especially good EoS results for neon: therefore non-pairwise effects must be significant in solid neon.

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