Quantum Monte Carlo Simulations of Mixed 3 He/ 4 He Clusters [email protected] dario...

Quantum Monte Quantum Monte CarloCarlo Simulations Simulations of Mixedof Mixed 33He/He/44He He

ClustersClusters

[email protected]://www.unico.it/~dario

Dario BressaniniDario Bressanini

Universita’ degli Studi dell’Insubria Universita’ degli Studi dell’Insubria

© Dario Bressanini 2

OverviewOverview

Introduction to quantum monte carlo methodsIntroduction to quantum monte carlo methods

Mixed Mixed 33He/He/44He clustersHe clusters simulations simulations


Monte Carlo MethodsMonte Carlo Methods How to solve a How to solve a deterministicdeterministic problem using a Monte problem using a Monte

Carlo method?Carlo method?

Rephrase the problem using a Rephrase the problem using a probabilityprobability distributiondistribution

NdfPA RRRR )()( NdfPA RRRR )()(

““Measure” Measure” AA by sampling the probability distribution by sampling the probability distribution

)(~)(1

1

RRR PfN

A i

N

ii

)(~)(1

1

RRR PfN

A i

N

ii


Monte Carlo MethodsMonte Carlo Methods

The points The points RRii are generated using random numbers are generated using random numbers

We introduce noise into the problem!!We introduce noise into the problem!! Our results have error bars...Our results have error bars... ... Nevertheless it might be a good way to proceed... Nevertheless it might be a good way to proceed

This is why the methods are called Monte Carlo methods

Metropolis, Ulam, Fermi, Von Neumann (-1945)Metropolis, Ulam, Fermi, Von Neumann (-1945)


Quantum MechanicsQuantum Mechanics We wish to solve We wish to solve H H = E = E to high accuracy to high accuracy

The solution usually involves computing integrals in The solution usually involves computing integrals in

high dimensions: 3-30000high dimensions: 3-30000

The “classic” approach (from 1929):The “classic” approach (from 1929): Find approximate Find approximate ( ... but good ...)( ... but good ...) ... whose integrals are analitically computable (gaussians)... whose integrals are analitically computable (gaussians) Compute the approximate energyCompute the approximate energy

chemical accuracy chemical accuracy ~~ 0.001 hartree 0.001 hartree ~~ 0.027 eV 0.027 eVchemical accuracy chemical accuracy ~~ 0.001 hartree 0.001 hartree ~~ 0.027 eV 0.027 eV


VMC: Variational Monte VMC: Variational Monte CarloCarlo

02 )(

)()(E

d

dHH

RR

RRR02 )(

)()(E

d

dHH

RR

RRR

RR

RR

R

RR

RRR

dP

HE

dEPH

L

L

)(

)()(

)(

)()(

)()(

2

2

RR

RR

R

RR

RRR

dP

HE

dEPH

L

L

)(

)()(

)(

)()(

)()(

2

2

Start from the Variational PrincipleStart from the Variational Principle

Translate it into Monte Carlo languageTranslate it into Monte Carlo language



EE is a statistical average of the local energy is a statistical average of the local energy EELL over over PP((RR))

)(~)(1

1

RRR PEN

HE i

N

iiL

)(~)(1

1

RRR PEN

HE i

N

iiL

RRR dEPHE L )()( RRR dEPHE L )()(

Recipe:Recipe: take an appropriate trial wave functiontake an appropriate trial wave function distribute distribute NN points according to points according to PP((RR)) compute the average of the local energycompute the average of the local energy


The Metropolis AlgorithmThe Metropolis Algorithm

How do we sampleHow do we sample

RR

RR

dP

)(

)()(

2

2

RR

RR

dP

)(

)()(

2

2

Anyone who consider Anyone who consider arithmetical methods of arithmetical methods of producing random digitsproducing random digitsis, of course, in a state of sin.is, of course, in a state of sin.

John Von NeumannJohn Von Neumann

Use the Metropolis algorithm (M(RT)Use the Metropolis algorithm (M(RT)2 2 1953) ... 1953) ...

... and a powerful computer... and a powerful computer

??

The algorithm is a random The algorithm is a random

walk (markov chain) in walk (markov chain) in

configuration spaceconfiguration space


The Metropolis AlgorithmThe Metropolis Algorithm

movmovee

rejerejectct

acceacceptptRRii RRtrtr

yy

RRi+1i+1==RRii RRi+1i+1==RRtt

ryry

Call the OracleCall the Oracle

Compute Compute averagesaverages



No need to analytically compute integrals: No need to analytically compute integrals: completecomplete freedom in freedom in

the choice of the trial wave functionthe choice of the trial wave function..

r1

r2

r12

He atomHe atom

1221 rcrbrae 1221 rcrbrae

CCan use an use explicitly correlated explicitly correlated

wave functionswave functions

Can satisfy the cusp conditionsCan satisfy the cusp conditions


VMC advantagesVMC advantages

Can go beyond the Can go beyond the Born-OppenheimerBorn-Oppenheimer approximation approximation, ,

with with ANYANY potential, in potential, in ANYANY number of dimensions number of dimensions..

PsPs22 molecule (e molecule (e++ee++ee--ee--) in 2D and 3D) in 2D and 3DPsPs22 molecule (e molecule (e++ee++ee--ee--) in 2D and 3D) in 2D and 3D

MM++mm++MM--mm-- as a function of M/m as a function of M/mMM++mm++MM--mm-- as a function of M/m as a function of M/m

222 HH 222 HH

Can compute lower boundsCan compute lower bounds HEH 0 HEH 0


VMCVMC drawbacksdrawbacks Error bar goes down as NError bar goes down as N-1/2-1/2

It is computationally demandingIt is computationally demanding

The optimization of The optimization of becomes difficult as the becomes difficult as the

number of nonlinear parameters increasesnumber of nonlinear parameters increases

It depends critically on our skill to invent a good It depends critically on our skill to invent a good There exist exact, automatic ways to get better wave There exist exact, automatic ways to get better wave

functions.functions.

Let the computer do the work ...Let the computer do the work ...


Diffusion Monte CarloDiffusion Monte Carlo

Suggested by Fermi in 1945, but implemented only inSuggested by Fermi in 1945, but implemented only in

thethe 7 70’s0’s

Nature is not classical, dammit, and if you Nature is not classical, dammit, and if you want to make a simulation of nature, you'd want to make a simulation of nature, you'd better make it quantum mechanical, and by better make it quantum mechanical, and by golly it's a wonderful problem, because it golly it's a wonderful problem, because it doesn't look so easy.doesn't look so easy. Richard P. Feynman

VMC is a “classical” simulation methodVMC is a “classical” simulation method


The time dependent The time dependent SchrSchrödinger equation ödinger equation is is similarsimilar to a diffusion to a diffusion equationequation

Vmt

i 22

2

Vmt

i 22

2

kCCDt

C

2 kCCD

t

C

2

Time evolution

Diffusion Branch

The The diffusion diffusion equation can be equation can be “solved” by directly “solved” by directly simulating the systemsimulating the system

Can we simulate the Can we simulate the SchrSchrödinger equation?ödinger equation?

Diffusion Diffusion equation equation analogyanalogy


The analogy is only formalThe analogy is only formal is a complex quantity, while is a complex quantity, while CC is real and positive is real and positive

Imaginary Time Sch. Imaginary Time Sch. EquationEquation

)(),( / RR ntiEnet )(),( / RR ntiEnet

If we let the time If we let the time tt be imaginary, then be imaginary, then can be can be

real!real!

VD 2

VD 2

Imaginary time SchrImaginary time Schröödinger equationdinger equation


as a concentrationas a concentration is interpreted as a concentration of fictitious is interpreted as a concentration of fictitious

particles, called particles, called walkerswalkers

VD 2

VD 2

i

EEii

Riea )()(),( RR i

EEii

Riea )()(),( RR

The schrThe schröödinger equationdinger equationis simulated by a process is simulated by a process of diffusion, growth andof diffusion, growth anddisappearance of walkersdisappearance of walkers

)(0

0)(),( REEe RR )(0

0)(),( REEe RRGround State


Diffusion Monte CarloDiffusion Monte Carlo

SIMULATIONSIMULATION: discretize time: discretize time

•Kinetic process (branching)Kinetic process (branching)

2D

2D

De 4/)( 20),( RRR De 4/)( 20),( RRR

))(( REV R

))(( REV R

)0,(),( ))(( RR R REVe )0,(),( ))(( RR R REVe

•Diffusion processDiffusion process


The DMC algorithmThe DMC algorithm


QMC: a simple and useful QMC: a simple and useful tooltool

Yukawa potentialYukawa potential Plasma physics, solid-state physics, ...Plasma physics, solid-state physics, ...

re r / re r /

Stability of screened H, HStability of screened H, H22++ and H and H22 as a function of as a function of

, , without Born-Oppenheimer approximation without Born-Oppenheimer approximation (preliminary (preliminary

results)results)

=1.19=1.19 1.21.2BorromeanBorromean

H boundH boundH unboundH unbound

HH22++ bound bound

H unboundH unbound

HH22++ unbound unbound


The Fermion ProblemThe Fermion Problem Wave functions for fermions have nodes.Wave functions for fermions have nodes.

Diffusion equation analogy is lost. Need to introduce Diffusion equation analogy is lost. Need to introduce

positive positive andand negative negative walkers. walkers.

The The (In)(In)famous Sign Problemfamous Sign Problem

Restrict random walk to a positive region bounded by nodes. Restrict random walk to a positive region bounded by nodes.

Unfortunately, the Unfortunately, the exactexact nodes are unknown. nodes are unknown.

Use approximate nodes from Use approximate nodes from

a trial a trial . Kill the walkers if . Kill the walkers if

they cross a node.they cross a node.

++ --


HeliumHelium

A helium atom is an A helium atom is an

elementary particle. A weakly elementary particle. A weakly

interacting hard sphere.interacting hard sphere.

Interatomic potential is Interatomic potential is

known more accurately than known more accurately than

any other atom. any other atom.

Two isotopes: Two isotopes: • 33He (fermion: antisymmetric trial function, spin 1/2) He (fermion: antisymmetric trial function, spin 1/2) • 44He (boson: symmetric trial function, spin zero)He (boson: symmetric trial function, spin zero)• The interaction potential is the sameThe interaction potential is the same


Helium ClustersHelium Clusters

1.1. Small mass of helium atomSmall mass of helium atom

2.2. Very weak He-He interactionVery weak He-He interaction

0.02 Kcal/mol0.9 * 10-3 cm-1

0.4 * 10-8 hartree10-7 eV

0.02 Kcal/mol0.9 * 10-3 cm-1

0.4 * 10-8 hartree10-7 eV

Highly non-classical systems. No equilibrium structure.Highly non-classical systems. No equilibrium structure.ab-initio methods and normal mode ab-initio methods and normal mode analysisanalysis useless useless

SuperfluiditySuperfluidityHigh resolution High resolution spectroscopyspectroscopy

Low temperature Low temperature chemistrychemistry


The SimulationsThe Simulations

Both VMC and DMC simulationsBoth VMC and DMC simulations

StandardStandard

Potential = sum of two-body TTY pair-potentialPotential = sum of two-body TTY pair-potential

ji

ijHeHe rVV )()(R

ji

ijHeHe rVV )()(R

N

HeHe

N

HeHe)()( 3444

kji

rr

N

HeHe

N

HeHe)()( 3444

kji

rr


Pure Pure 44HeHenn ClustersClusters

2 3 4 5 6 7 8 9 10 11 12n

-8

-7

-6

-5

-4

-3

-2

-1

0

En

erg

y c

m-1

DMCVMC

2 3 4 5 6 7 8 9 10 11 12n

-8

-7

-6

-5

-4

-3

-2

-1

0

En

erg

y c

m-1

DMCVMC


Mixed Mixed 33He/He/44He He ClustersClusters

(0,3)(0,2)

(0,4)

(0,5)

(0,6)

(0,7)

(1,2)

(1,3)

(1,4)

(1,5)

(1,6)

(2,2)

1 2 3 4 5 6 7Number of atoms

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Ener

gy cm

-1

(0,3)(0,2)

(0,4)

(0,5)

(0,6)

(0,7)

(1,2)

(1,3)

(1,4)

(1,5)

(1,6)

(2,2)

1 2 3 4 5 6 7Number of atoms

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Ener

gy cm

-1

(m,n) = (m,n) = 33HeHemm44HeHenn (m,n) = (m,n) = 33HeHemm44HeHenn

Bressanini et. al.Bressanini et. al.J.Chem.Phys.J.Chem.Phys.112112, 717 (2000), 717 (2000)


Helium Helium Clusters: stabilityClusters: stability

44HeHeNN is destabilized by substituting a is destabilized by substituting a 44He with a He with a 33HeHe

The structure is only weakly perturbed.The structure is only weakly perturbed.

44HeHe44HeHeDimerDimerss

44HeHe33HeHe 33HeHe33HeHe

BoundBound UnboundUnbound UnboundUnbound

44HeHe33TrimeTrimersrs

44HeHe2233HeHe 44HeHe33HeHe22

BoundBound BoundBound UnboundUnbound

44HeHe44TetrameTetramersrs

44HeHe3333HeHe 44HeHe22

33HeHe22

BoundBound BoundBound BoundBound


Trimers and Tetramers Trimers and Tetramers StabilityStability44HeHe3 3 E = E = -0.08784(7)-0.08784(7) cm cm-1-1

44HeHe2233He He E = E = -0.00984(5)-0.00984(5) cm cm-1-1

Five out of six unbound pairs!Five out of six unbound pairs!

44HeHe4 4 E = E = -0.-0.33888866((11)) cm cm-1-1

44HeHe3333He He E = E = -0.-0.20622062((11)) cm cm-1-1

44HeHe2233HeHe22 E = E = -0.-0.071071((11)) cm cm-1-1

Bonding Bonding interactioninteractionNon-bonding Non-bonding interactioninteraction


33He/He/44He Distribution He Distribution FunctionsFunctions

0 10 20 30 40r (u.a.)

0.000

0.004

0.008

0.012

0.016

g(r

)

4He-4H e3He-4H e

0 10 20 30 40r (u.a.)

0.000

0.004

0.008

0.012

0.016

g(r

)

4He-4H e3He-4H e

33He(He(44He)He)55

Pair distribution functionsPair distribution functions


33He/He/44He Distribution He Distribution FunctionsFunctions

0 10 20 30 40r (u.a.)

0.000

0.004

0.008

0.012

0.016

g(r

)

4H e3H e

0 10 20 30 40r (u.a.)

0.000

0.004

0.008

0.012

0.016

g(r

)

4H e3H e

33He(He(44He)He)55

Distributions with respect to the center of massDistributions with respect to the center of mass

c.o.mc.o.m


Distribution Functions in Distribution Functions in 44HeHeNN

33HeHe

N=19

N=2

0 10 20 30r (bohr)

0.0

0.1

0.2

0.3

P(r

)

N=19

N=2

0 10 20 30r (bohr)

0.0

0.1

0.2

0.3

P(r

)

((44He-He-44He)He)

N=2

N=19

N=3

N=10

N=4

N=6

N=5

0 10 20 30

r (bohr)

0.000

0.005

0.010

0.015

P(r

) N=2

N=19

N=3

N=10

N=4

N=6

N=5

0 10 20 30

r (bohr)

0.000

0.005

0.010

0.015

P(r

)

((33He-He-44He)He)


N=19

N=3

0 10 20r (bohr)

0.000

0.005

0.010

0.015

(r)

(

boh

r-3)

3He4He2N=19

N=3

0 10 20r (bohr)

0.000

0.005

0.010

0.015

(r)

(

boh

r-3)

3He4He2

Distribution Functions in Distribution Functions in 44HeHeNN33HeHe

0 10 20 30r (bohr)

0.000

0.005

0.010

0.015

(r)

(

boh

r-3) 3He4He2

N=19

N=3

0 10 20 30r (bohr)

0.000

0.005

0.010

0.015

(r)

(

boh

r-3) 3He4He2

N=19

N=3

((44He-He-C.O.M.C.O.M.)) ((33He-He-C.O.M.C.O.M.))

c.o.m. = center of massc.o.m. = center of mass

Similar to pure Similar to pure clustersclusters

Fermion is pushed Fermion is pushed awayaway


44HeHe33 Angular Distributions Angular Distributions


NeNe33 Angular Distributions Angular Distributions

Ne trimerNe trimer


Helium Cluster StabilityHelium Cluster Stability

Is Is 33HeHemm44HeHenn stable ? stable ?

What is the smallest What is the smallest 33HeHemm stable cluster ? stable cluster ?

Liquid: stableLiquid: stable

Dimer: unboundDimer: unbound

33HeHemm

m = ?m = ? 20 < m < 35 20 < m < 35

critically boundcritically bound


44HeHenn

33HeHemm 0 1 2 3 4 5 6 70 1 2 3 4 5 6 700

11

22

33

44

55

BoundBound

UnboundUnbound

UnknownUnknown

3535

Probably Probably unboundunbound

Work in progress: Work in progress: 33HeHemm

44HeHenn


Work in ProgressWork in Progress

Various impurities embedded in a Helium clusterVarious impurities embedded in a Helium cluster

Different functional forms for Different functional forms for splines) splines)

Stability of Stability of 33HeHemm44HeHenn


ConclusionsConclusions

The substitution of a The substitution of a 44HeHe with a with a 33HeHe leads to an leads to an

energetic destabilization.energetic destabilization.

33HeHe weakly perturbes the weakly perturbes the 44HeHe atoms distribution. atoms distribution.

33HeHe moves on the surface of the cluster. moves on the surface of the cluster.

44HeHe2233He He bound, bound, 44HeHe33HeHe22 unbound.unbound.

44HeHe3333He He andand 44HeHe22

33HeHe22 bound.bound.

QMC gives accurate energies and structural informationQMC gives accurate energies and structural information


A reflection...A reflection...

A new method is initially not as well formulated or A new method is initially not as well formulated or understood as existing methodsunderstood as existing methods

It can seldom offer results of a comparable quality before It can seldom offer results of a comparable quality before a considerable amount of development has taken placea considerable amount of development has taken place

Only rarely do new methods differ in major ways from Only rarely do new methods differ in major ways from previous approachesprevious approaches

A new method for calculating properties in nuclei, atoms, A new method for calculating properties in nuclei, atoms, molecules, or solids automatically provokes three sorts of molecules, or solids automatically provokes three sorts of negative reactions:negative reactions:

Nonetheless, new methods need to be developed to Nonetheless, new methods need to be developed to handle problems that are vexing to or beyond the handle problems that are vexing to or beyond the scope of the current approachesscope of the current approaches

((Slightly modified fromSlightly modified from Steven R. White, John W. Wilkins and Kenneth G. Wilson) Steven R. White, John W. Wilkins and Kenneth G. Wilson)

Date post:	04-Jan-2016
Category:	Documents
Upload:	jasper-oconnor
View:	215 times
Download:	0 times

Quantum Monte Carlo Simulations of Mixed 3 He/ 4 He Clusters [email protected] dario...

Documents