Transcript Van der Waals forces

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 (pressuredensity 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
dTemporary dipole resulting
from quantum fluctuation
d+
Gives net
attraction
d-
d+
Induced dipole, due to
presence of other dipole
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.
Hard-Core Repulsive Forces
Two neon atoms
Electron clouds overlap
when neon atoms are
brought together
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 / highdensity regime.
Aims of this Project
To compare the accuracy with which
competing electronic-structure and pairpotential 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 planewave 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 pairpotential 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 χ2 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
Neon dimer:
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.
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 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
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 HighPerformance Computing Facility.