The Quantum Monte The Quantum Monte The Quantum Monte The Quantum Monte The Quantum Monte The Quantum Monte The Quantum Monte The Quantum Monte

Carlo MethodCarlo MethodCarlo MethodCarlo MethodCarlo MethodCarlo MethodCarlo MethodCarlo Method

Neil DrummondNeil Drummond

HPCx Fourth Annual Seminar

e-Science Institute, Edinburgh

Wednesday 4th October, 2005

ElectronicElectronicElectronicElectronicElectronicElectronicElectronicElectronic--------Structure CalculationStructure CalculationStructure CalculationStructure CalculationStructure CalculationStructure CalculationStructure CalculationStructure Calculation

Goal: solve the many-electron Schrödinger equation to obtain the ground-state energy and distribution of electrons for a given arrangement of nuclei in a molecule or solid.

Much of chemistry, materials science and condensed-matter

physics can in principle be obtained from such data.

R 0 R

E

O atom O atom

Equilibrium bond length

Curvature gives

vibration frequencyExample: calculate total energyE of an oxygen molecule as afunction of nuclear separation R.

Bond

energy

ElectronicElectronicElectronicElectronicElectronicElectronicElectronicElectronic--------Structure MethodsStructure MethodsStructure MethodsStructure MethodsStructure MethodsStructure MethodsStructure MethodsStructure Methods

1 10 100 10,000 1,000,000

0

0.0001 eV

Qualitative

Topological

0.1 eV

Tight binding

Empirical potentials

DFT

QMC

Accuracy

Number of atoms in simulation

QMC is at the high-accuracy, high-cost end of the spectrum of available methods for studying material properties.

Monte Carlo IntegrationMonte Carlo IntegrationMonte Carlo IntegrationMonte Carlo IntegrationMonte Carlo IntegrationMonte Carlo IntegrationMonte Carlo IntegrationMonte Carlo Integration

�� QMC methods use QMC methods use random samplingrandom sampling..

�� Random sampling is the most efficient method Random sampling is the most efficient method

for evaluating multidimensional integrals.for evaluating multidimensional integrals.

Example of Monte Carlo integration: area of a shape in 2D

Random points, uniformly distributed

Area of this shape is approximately given

by the area of the rectangle multiplied by

the fraction of random points in the shape.

Estimate of the area becomes exact in the limit that

the number of points goes to infinity. For finite

numbers of sampling points, one can evaluate the

statistical error in the estimate of the area.

Variational Monte CarloVariational Monte CarloVariational Monte CarloVariational Monte CarloVariational Monte CarloVariational Monte CarloVariational Monte CarloVariational Monte Carlo

ΨΨ

ΨHΨE

ˆ

≤0

( )

∫∫∫∫

−

∗

=

=

dRΨ

dRΨHΨΨ

dRΨ

dRΨHΨ

ΨΨ

ΨHΨ

2

12

2

ˆ

ˆˆ

Variational principleVariational principle: ground: ground--state energy is less than or equal state energy is less than or equal to the expectation value of the to the expectation value of the Hamiltonian with respect to a Hamiltonian with respect to a trial trial wave functionwave function; equality holds if the ; equality holds if the wave function is exact.wave function is exact.

VMC: VMC: use use Metropolis Metropolis

algorithmalgorithm to evaluate to evaluate

the expectation value.the expectation value.

We We optimiseoptimise wave functions, wave functions,

e.g. by minimising the energy e.g. by minimising the energy

expectation value.expectation value.

Diffusion Monte CarloDiffusion Monte CarloDiffusion Monte CarloDiffusion Monte CarloDiffusion Monte CarloDiffusion Monte CarloDiffusion Monte CarloDiffusion Monte Carlo

Configurations in regions of low potential energy tend to multiply, while those in regions of high potential tend to die.

x

x

x

t

V(x)

Ψ (x)0

Ψ (x)0

τ

�� SchrSchröödinger equation in dinger equation in imaginary timeimaginary time describes a describes a combination of combination of diffusiondiffusion and and branchingbranching processes.processes.

�� Simulate these to project out groundSimulate these to project out ground--state wave function.state wave function.

t

ΨΨH

∂

∂−=ˆ

QMC on Parallel ComputersQMC on Parallel ComputersQMC on Parallel ComputersQMC on Parallel ComputersQMC on Parallel ComputersQMC on Parallel ComputersQMC on Parallel ComputersQMC on Parallel Computers�� VMC is perfectly parallelVMC is perfectly parallel: just need to average the : just need to average the

results obtained in independent random walks.results obtained in independent random walks.

�� DMC is highly parallelisableDMC is highly parallelisable: each processor has its : each processor has its own population of configurations that undergo the own population of configurations that undergo the diffusion and branching processes.diffusion and branching processes.

�� Branching probability depends on energy data from Branching probability depends on energy data from the entire configuration population; hence processors the entire configuration population; hence processors must communicate after each time step.must communicate after each time step.

�� Every now and again we transfer configurations Every now and again we transfer configurations between processors to even up the load.between processors to even up the load.

The CASINO ProgramThe CASINO ProgramThe CASINO ProgramThe CASINO ProgramThe CASINO ProgramThe CASINO ProgramThe CASINO ProgramThe CASINO Program

�� The VMC and DMC methods are implemented in the The VMC and DMC methods are implemented in the

Cambridge QMC code Cambridge QMC code CASINOCASINO..

�� CASINO is used by research groups in Australia, CASINO is used by research groups in Australia,

Japan, Spain, Sweden, Taiwan and the USA, as well Japan, Spain, Sweden, Taiwan and the USA, as well

as the UK.as the UK.

�� It is available to collaborators on request.It is available to collaborators on request.

�� See See www.tcm.phy.cam.ac.uk/~mdt26/casino2.htmlwww.tcm.phy.cam.ac.uk/~mdt26/casino2.html..

Homogeneous Electron GasHomogeneous Electron GasHomogeneous Electron GasHomogeneous Electron GasHomogeneous Electron GasHomogeneous Electron GasHomogeneous Electron GasHomogeneous Electron Gas

�� The HEG is of fundamental The HEG is of fundamental importance in solidimportance in solid--state physics: state physics: simplest fully interacting quantum simplest fully interacting quantum manymany--body system and basic model body system and basic model of free electrons in metals and of free electrons in metals and semiconductors.semiconductors.

�� The HEG consists of a set of The HEG consists of a set of electrons moving in a uniform, electrons moving in a uniform, neutralising background.neutralising background.

�� At high densities the HEG exists in At high densities the HEG exists in the wellthe well--known known Fermi fluidFermi fluid phase.phase.

�� At low densities the HEG undergoes At low densities the HEG undergoes a transition to a a transition to a Wigner crystalWigner crystalphase to minimise the electronphase to minimise the electron--electron repulsion.electron repulsion.

Charge density

of fluid phase of

HEG: uniform!

2D HEG in Fujitsu

recessed ohmic power

FET structure

Wigner CrystalsWigner CrystalsWigner CrystalsWigner CrystalsWigner CrystalsWigner CrystalsWigner CrystalsWigner Crystals

�� Wigner crystals are a Wigner crystals are a

brokenbroken--symmetry state in symmetry state in

which the charge density which the charge density

is inhomogeneous.is inhomogeneous.

�� 2D Wigner crystals have 2D Wigner crystals have

been observed on been observed on

droplets of liquid helium droplets of liquid helium

and in MOSFET devices.and in MOSFET devices.

�� It has been suggested It has been suggested

that Wigner crystals could that Wigner crystals could

be used in quantum be used in quantum

computing devices.computing devices.

Charge density of a 2D Wigner crystal with a defect.

Crystallisation Density of HEGCrystallisation Density of HEGCrystallisation Density of HEGCrystallisation Density of HEGCrystallisation Density of HEGCrystallisation Density of HEGCrystallisation Density of HEGCrystallisation Density of HEG

We have performed the first precise calculation of the crystallisation density of the 3D HEG.

The graph shows the total energy of the fluid phase as a function of density (black line) and the total energy of the crystal phase as a function of density (red line). The crystallisation density is the point at which the curves cross.

We are currently studying the phase behaviour of the 2D electron gas.

Stability of FullerenesStability of FullerenesStability of FullerenesStability of FullerenesStability of FullerenesStability of FullerenesStability of FullerenesStability of FullerenesFullerenes are cage-like carbon molecules, which exhibit a rich variety

of physical and chemical properties.

An interesting and important question

is, “what is the smallest fullerene that is more stable than the competing

ring and sheet structures?”

QMC studies show that C26 cages have lower energies than rings or sheets and are probably the smallest stable fullerenes. Less accurate electronic-structure methods are not capable of making consistent predictions about the stability of fullerenes.

C26

C24

Nanoscience with QMC INanoscience with QMC INanoscience with QMC INanoscience with QMC INanoscience with QMC INanoscience with QMC INanoscience with QMC INanoscience with QMC IThe properties of nanometre-sized particles often differ from the properties of the corresponding bulk material. In recent years

there has been much interest in exploiting these properties in

nanotechnological applications.

Very recently it has become possible to study these new

materials using QMC. This is useful because nanoparticles are

often difficult or expensive to manufacture in the laboratory.

We have studied the optical and chemical properties of carbon nanoparticles called diamondoids, which potentially have useful

applications in sensors and display devices.

NanoscienceNanoscienceNanoscienceNanoscienceNanoscienceNanoscienceNanoscienceNanoscience with QMC IIwith QMC IIwith QMC IIwith QMC IIwith QMC IIwith QMC IIwith QMC IIwith QMC II

�� Optical gapOptical gap: difference : difference between groundbetween ground--state and state and firstfirst--excitedexcited--state energy state energy levels.levels.

�� Diamond has a band gap in Diamond has a band gap in the UV range.the UV range.

�� Quantum confinementQuantum confinement model model suggests the optical gap of a suggests the optical gap of a nanoparticlenanoparticle should be higher should be higher than the band gap of the than the band gap of the corresponding bulk material.corresponding bulk material.

�� New range of UV sensors?New range of UV sensors?

GS

1st ES

OG

E

Silicon substrate

Metal film Nanoparticle film

SiO2

NanoscienceNanoscienceNanoscienceNanoscienceNanoscienceNanoscienceNanoscienceNanoscience with QMC IIIwith QMC IIIwith QMC IIIwith QMC IIIwith QMC IIIwith QMC IIIwith QMC IIIwith QMC III�� Some surfaces of diamond have Some surfaces of diamond have negativenegative electron electron

affinitiesaffinities; hope that ; hope that diamondoidsdiamondoids have this property.have this property.

�� Negative electron affinity Negative electron affinity →→ energetically energetically unfavourable to form a negative ion.unfavourable to form a negative ion.

�� Conduction electrons should Conduction electrons should ““fall outfall out”” of a surface of a surface with a negative electron affinity: useful in electronwith a negative electron affinity: useful in electron--emission devices, e.g. for flatemission devices, e.g. for flat--screen displays.screen displays.

Electron emitter

Row electrode

Phosphor screen

Electron beam

Gate electrode

Nanoscience with QMC IVNanoscience with QMC IVNanoscience with QMC IVNanoscience with QMC IVNanoscience with QMC IVNanoscience with QMC IVNanoscience with QMC IVNanoscience with QMC IVC29H36 molecule

Highest occupied

molecular orbital

Lowest unoccupied

molecular orbital

Our QMC calculations show that:

1. Diamondoids have negative electron affinities, making them ideal for

use in electron-emission devices.

2. Diamondoid optical gaps fall off rapidly with particle size, so

diamondoids will not be as useful as had been hoped in UV sensors.

QMC has played an important role in establishing the properties of

these new materials.

NeonNeonNeonNeonNeonNeonNeonNeon�� The chemistry of the noble gas The chemistry of the noble gas neonneon is simple.is simple.

�� When atoms are brought together, their When atoms are brought together, their electron clouds overlap, giving a electron clouds overlap, giving a hardhard--core core repulsionrepulsion..

�� Atoms are weakly attracted to one another by Atoms are weakly attracted to one another by van van derder WaalsWaals forces.forces.

Two electrically neutral, closed-

shell atoms

Gives net

attraction

Temporary dipole resulting from quantum fluctuation

δ+δ+δ− δ−

Induced dipole, due to presence of other dipole

Solid NeonSolid NeonSolid NeonSolid NeonSolid NeonSolid NeonSolid NeonSolid Neon�� At low temperatures or high pressures, neon forms a At low temperatures or high pressures, neon forms a

crystalline solid with the facecrystalline solid with the face--centred cubic structure.centred cubic structure.

�� Highly accurate experimental data are available.Highly accurate experimental data are available.

�� Solid neon has long been used as a test system for Solid neon has long been used as a test system for theoretical manytheoretical many--body physics.body physics.

�� Accurate pressureAccurate pressure--volume data at high pressures volume data at high pressures may be of relevance to diamondmay be of relevance to diamond--anvil experiments.anvil experiments.

Diamond anvil

Metal gasket

Pressure-conducting

medium, e.g. neon

Sample

Equation of State of Solid NeonEquation of State of Solid NeonEquation of State of Solid NeonEquation of State of Solid NeonEquation of State of Solid NeonEquation of State of Solid NeonEquation of State of Solid NeonEquation of State of Solid Neon

Accuracy of DMC Equation Accuracy of DMC Equation Accuracy of DMC Equation Accuracy of DMC Equation Accuracy of DMC Equation Accuracy of DMC Equation Accuracy of DMC Equation Accuracy of DMC Equation

of Stateof Stateof Stateof Stateof Stateof Stateof Stateof State

Inhomogeneous Electron GasInhomogeneous Electron GasInhomogeneous Electron GasInhomogeneous Electron GasInhomogeneous Electron GasInhomogeneous Electron GasInhomogeneous Electron GasInhomogeneous Electron Gas

QMC electron-gas calculations provide data required by a less accurate (but faster) electronic-structure method called density-functional theory.

This plot shows the distribution of electrons about one fixed electron in an inhomogeneous electron gas.

The usual approximations in DFT use QMC data for a uniform electron gas. The corresponding distribution of electrons is shown on the right. The QMC-calculated distribution for the inhomogeneous system is on the left. Our results give insight into the nature of the approximations made in DFT.

SummarySummarySummarySummarySummarySummarySummarySummary

�� QMCQMC methods are highly methods are highly accurate techniques for accurate techniques for calculating material properties calculating material properties from first principles through from first principles through the use of random sampling.the use of random sampling.

�� QMC methods scale well with QMC methods scale well with system size and are easy to system size and are easy to implement on parallel implement on parallel computers.computers.

�� QMC methods can now be QMC methods can now be used to study nanometreused to study nanometre--sized systems.sized systems.

ReferencesReferencesReferencesReferencesReferencesReferencesReferencesReferences

�� CASINO web site: CASINO web site: www.tcm.phy.cam.ac.uk/~mdt26/casino2.htmlwww.tcm.phy.cam.ac.uk/~mdt26/casino2.html..

�� Good review article on VMC and DMC:Good review article on VMC and DMC: Foulkes Foulkes et alet al., Rev. Mod. Phys. ., Rev. Mod. Phys. 7373, 33, 33..

�� Studies of 3D Studies of 3D WignerWigner crystals:crystals: Drummond Drummond et et alal., Phys. Rev. B ., Phys. Rev. B 6969, 085116, 085116.

�� Studies of the stability of fullerenes:Studies of the stability of fullerenes: Kent Kent et alet al., ., Phys. Rev. B Phys. Rev. B 6262, 15394, 15394..

�� DiamondoidDiamondoid optical gap and electron affinities:optical gap and electron affinities:Drummond Drummond et alet al., Phys. Rev. ., Phys. Rev. LettLett. . 9595, 096801, 096801..

�� Equation of state of solid neon:Equation of state of solid neon: Drummond Drummond et et alal., Phys. Rev. B, ., Phys. Rev. B, 7373, 024107, 024107..