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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.
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QMC and DFT Studies of Solid Neon Neil Drummond and Richard Needs TCM Group, Cavendish Laboratory, Cambridge, UK ESDG meeting, 9th 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 Two electrically neutral, closed-shell atoms • 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. d- d+ d- d+ Induced dipole, due to presence of other dipole Temporary dipole resulting from quantum fluctuation Gives net attraction
Hard-Core Repulsive Forces Electron clouds overlap when neon atoms are brought together Two neon atoms • 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.
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 2nd 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 χ2values 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. 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. r Neon dimer:
Using Pair Potentials • To get the static-lattice energy per atom: • Add up pair potential U(r) between red atom and each white atom inside the sphere. • Integrate 4πr2U(r) from radius of sphere to infinity & multiply by density of atoms to get contribution from atoms outside sphere. • 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 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.