Dario Bressanini UNAM, Mexico City, 2007 Universita’ dell’Insubria, Como, Italy Introduction to Quantum Monte.

Dario Dario BressaniniBressanini

UNAM, Mexico CityUNAM, Mexico City, , 2007 2007

http://scienze-como.uninsubria.it/http://scienze-como.uninsubria.it/bressaninibressanini

Universita’ dell’Insubria, Como, ItalyUniversita’ dell’Insubria, Como, Italy

Introduction toIntroduction to Quantum Monte Carlo Quantum Monte Carlo

EH

2

Why do simulations?Why do simulations?

• Simulations are a general method for Simulations are a general method for ““solvingsolving” many-body problems. Other ” many-body problems. Other methods usually involve approximations.methods usually involve approximations.

• Experiment is limited and expensive. Experiment is limited and expensive. Simulations can complement the Simulations can complement the experiment.experiment.

• Simulations are easy even for complex Simulations are easy even for complex systems.systems.

• They scale up with the computer power.They scale up with the computer power.

3

Buffon needle experiment, Buffon needle experiment, AD 1777AD 1777

d

d

Lp

2

L

4

SimulationsSimulations

• ““The general theory of quantum The general theory of quantum mechanics is now almost complete. The mechanics is now almost complete. The underlying physical laws necessary for underlying physical laws necessary for the mathematical theory of a large part the mathematical theory of a large part of physics and the whole of chemistry are of physics and the whole of chemistry are thus completely known, and the difficulty thus completely known, and the difficulty is only that the is only that the exact application of these exact application of these laws leads to equations much too laws leads to equations much too complicated to be solublecomplicated to be soluble.”.”

Dirac, 1929Dirac, 1929

5

General strategyGeneral strategy How to solve a How to solve a deterministicdeterministic problem problem

using a Monte Carlo method?using a Monte Carlo method? Rephrase the problem using a Rephrase the problem using a probabilityprobability

distributiondistributionNdfPA 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

6

Monte Carlo MethodsMonte Carlo Methods

The points The points RRii are generated using are generated using

random numbersrandom numbers

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

to proceedto proceed

This is why the methods are called Monte Carlo methods

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

7

Stanislaw Ulam (1909-Stanislaw Ulam (1909-1984)1984)

S. Ulam is credited as the inventor of Monte Carlo method in 1940s, which solves mathematical problems using statistical sampling.

8

Why Monte Carlo?Why Monte Carlo?

• We can approximate the numerical value We can approximate the numerical value of a definite integral by the definition:of a definite integral by the definition:

b

a

L

ii xxfdxxf

1

)()(

• where we use where we use LL points points xxii uniformly spaced. uniformly spaced.

9

Error in QuadratureError in Quadrature

• Consider an integral in Consider an integral in DD dimensions: dimensions:

• N= N= LLD D uniformly spaceduniformly spaced points,points, to CPU time to CPU time

• The error with The error with NN sampling points is sampling points is

DD

V

xfdxdxdxf )()( 21 RR

DD Nxfdf /1)()( RRR

10

Monte Carlo Estimates of Monte Carlo Estimates of IntegralsIntegrals

• If we sample the points not on regular If we sample the points not on regular grids, but grids, but randomlyrandomly (uniformly (uniformly distributed), thendistributed), then

1

f ( ) f ( )N

iiV

VX dX X

N

Where we assume the integration domain is a regular box of V=LD.

11

Monte Carlo ErrorMonte Carlo Error

• From probability theory one can show From probability theory one can show that the Monte Carlo error decreases that the Monte Carlo error decreases with sample size with sample size NN as as

• Independent of dimension Independent of dimension DD ( (goodgood).).

• To get another decimal place takes 100 To get another decimal place takes 100 times longer! (times longer! (badbad))

N1

12

MC is advantageous for large MC is advantageous for large dimensionsdimensions

•Error by simple quadrature Error by simple quadrature NN-1/D-1/D

•Using smarter quadrature Using smarter quadrature NN-A/D-A/D

•Error by Monte Carlo always Error by Monte Carlo always NN-1/2-1/2

•Monte Carlo is always more efficient Monte Carlo is always more efficient for large D (usually D > 4 - 6)for large D (usually D > 4 - 6)

13

Monte Carlo Estimates Monte Carlo Estimates of of ππ

(1,0)

We can estimate π using Monte Carlo

1

1

212

dxx

14

Monte Carlo Monte Carlo IntegrationIntegration

•Note thatNote that Can Can automatically estimate the errorautomatically estimate the error

by computing the standard deviation by computing the standard deviation of the sampled function valuesof the sampled function values

All points generated are All points generated are independentindependent

All points generated are All points generated are usedused

15

Inefficient?Inefficient?

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

N

ii

b

a

xfN

abdxxf1

)(1

)()(

• If the function is If the function is strongly peaked, strongly peaked, the process is the process is inefficientinefficient

• We should We should generate more generate more points where the points where the function is largefunction is large

• Use a non-Use a non-uniform uniform distribution!distribution!

16

General Monte CarloGeneral Monte Carlo

• If the samples are not drawn uniformly If the samples are not drawn uniformly but with some probability distribution but with some probability distribution pp(R),(R), we can compute by Monte Carlo: we can compute by Monte Carlo:

Where p(R) is normalized,

)(~)(1

)()(1

RRRRRR pfN

dpf i

N

ii

1)( RR dp

17

Monte CarloMonte Carlo

• soso

RR

RRRR d

p

fpdfI

)(

)()()(

Convergence guaranteed by the Central Limit Theorem

•The statistical error0 if p(R) f(R), convergence is faster

i i

i

p

fNp

fI

)(

)(1)(

)(

R

R

R

R

18

Warning!Warning!

• Beware of Monte Carlo integration Beware of Monte Carlo integration routines in libraries: they usually cannot routines in libraries: they usually cannot assume anything about your functions assume anything about your functions since they must be general.since they must be general.

• Can be quite inefficientsCan be quite inefficients

• Also beware of standard compiler Also beware of standard compiler supplied Random Number Generators supplied Random Number Generators (they are known to be bad!!) (they are known to be bad!!)

19

Equation of state of a fluidEquation of state of a fluid

The problem: The problem: compute the compute the equation of state (equation of state (pp as function of as function of particle density particle density N/VN/V ) of a fluid in a box ) of a fluid in a box given some given some interaction potential interaction potential between the between the particlesparticles

Assume for every position of particles we can Assume for every position of particles we can compute the potential energy V(compute the potential energy V(R)R)

20

The Statistical Mechanics The Statistical Mechanics ProblemProblem

For equilibrium properties we can just For equilibrium properties we can just compute the Boltzmann multi-dimensional compute the Boltzmann multi-dimensional integralsintegrals

R

RRR

R

de

deAA

Tk

E

Tk

E

B

B

)(

)(

)(

R

RRR

R

de

deAA

Tk

E

Tk

E

B

B

)(

)(

)(

Where the energy usually is a sumWhere the energy usually is a sum

ji

ijdVE )()(R

ji

ijdVE )()(R

21

An inefficient recipeAn inefficient recipe

For 100 particles (not really the For 100 particles (not really the thermodynamic limit), integrals are in 300 thermodynamic limit), integrals are in 300 dimensions.dimensions.

The The naïvenaïve MC procedure would be to MC procedure would be to uniformly distribute the particles in the uniformly distribute the particles in the box, throwing them randomly.box, throwing them randomly.

If the density is high, throwing particles If the density is high, throwing particles at random will put them some of them too at random will put them some of them too close to each other.close to each other.

almost all such generated points will give almost all such generated points will give negligible contribution, due to the negligible contribution, due to the boltzmann factorboltzmann factor

22

An inefficient recipeAn inefficient recipe

E(E(RR) becomes very large and positive) becomes very large and positive We should try to generate more points We should try to generate more points

where E(where E(RR) is close to the minima) is close to the minima

R

RRR

R

de

deAA

Tk

E

Tk

E

B

B

)(

)(

)(

R

RRR

R

de

deAA

Tk

E

Tk

E

B

B

)(

)(

)(

23

The Metropolis AlgorithmThe Metropolis Algorithm

How do we do it?How do we do it?

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 (walk (markov chainmarkov chain) in ) in

configuration space. Points configuration space. Points

are are notnot independent independent

24

25

26

Importance SamplingImportance Sampling

The idea is to use The idea is to use Importance Importance SamplingSampling, that is sampling more where , that is sampling more where the function is largethe function is large

“…“…, instead of choosing , instead of choosing configurations randomly, …, configurations randomly, …, we choose we choose configuration with a probability exp(-configuration with a probability exp(-EE//kkBBTT)) and weight them evenly.” and weight them evenly.”

- - from M(RT)from M(RT)22 paper paper

27

The key IdeasThe key Ideas

Points are Points are no longer independentno longer independent!! We consider a point (a We consider a point (a WalkerWalker) that ) that

moves in configuration space according moves in configuration space according to some rulesto some rules

28

A Markov ChainA Markov Chain

A Markov chain is a random walk through A Markov chain is a random walk through configuration space: configuration space:

RR11RR2 2 RR3 3 RR4 4 ……

Given Given RRn n there is a there is a transition probabilitytransition probability

to go to the next point to go to the next point RRn+1n+1 : : p(Rp(RnnRRn+1n+1)) stochastic matrixstochastic matrix

In a Markov chain, the distribution of RIn a Markov chain, the distribution of Rn+1n+1

depends only on depends only on RRnn. There is no memory. There is no memory

We must use an We must use an ergodicergodic markov chain markov chain

29

The key IdeasThe key Ideas

Choose an appropriate Choose an appropriate p(Rp(RnnRRn+1n+1)) so that so that

at equilibrium we sample a distribution at equilibrium we sample a distribution ππ((RR) ) (for this problem is just (for this problem is just ππ = = exp(-exp(-EE//kkBBTT)) ) )

A A sufficientsufficient condition is to apply condition is to apply detailed detailed balancebalance. .

Consider an infinite number of walkers, Consider an infinite number of walkers, and two positions R, and R’and two positions R, and R’

At equilibrium, the #of walkers that go At equilibrium, the #of walkers that go from Rfrom RR’ is equal to the #of walkers R’ is equal to the #of walkers R’R’R R

pp((RRR’R’) ) ≠ ≠ pp((R’R’RR))

30

The Detailed BalanceThe Detailed Balance

)()()()( RRRRRR pp )()()()( RRRRRR pp ππ((RR) is the distribution we want to sample) is the distribution we want to sample We have the freedom to choose We have the freedom to choose pp(R(RR’)R’)

)(

)(

)(

)(

R

R

RR

RR

p

p)(

)(

)(

)(

R

R

RR

RR

p

p

31

Rejecting pointsRejecting points

The third key idea is to use The third key idea is to use rejectionrejection to to enforce detailed balanceenforce detailed balance

pp(R(RR’)R’) is split into a is split into a TransitionTransition step and step and an an Acceptance/Rejection Acceptance/Rejection stepstep

)()()( RRRRRR ATp )()()( RRRRRR ATp TT((RRR’R’) generate the next “candidate” ) generate the next “candidate”

pointpoint AA((RRR’R’) will decide to accept or reject this ) will decide to accept or reject this

pointpoint

32

The Acceptance The Acceptance probabilityprobability

Given some Given some TT, a possible choice for , a possible choice for AA is is

)(

)(

)()(

)()(

R

R

RRRR

RRRR

AT

AT)(

)(

)()(

)()(

R

R

RRRR

RRRR

AT

AT

)()(

)()(,1min)(

RRR

RRRRR

T

TA

)()(

)()(,1min)(

RRR

RRRRR

T

TA

For symmetric For symmetric TT

)(

)(,1min)(

R

RRR

A

)(

)(,1min)(

R

RRR

A

33

What it doesWhat it does

Suppose Suppose ππ((R’R’) ≥ ) ≥ ππ((RR)) move is move is alwaysalways accepted accepted

Suppose Suppose ππ((R’R’) < ) < ππ((RR)) move is accepted with probability move is accepted with probability

ππ((R’R’)/)/ππ((RR)) Flip a coinFlip a coin

The algorithm samples regions of The algorithm samples regions of large large ππ((RR))

Convergence is guaranteed but the rate Convergence is guaranteed but the rate

is not!!is not!!

)(

)(,1min)(

R

RRR

A

)(

)(,1min)(

R

RRR

A

34

IMPORTANT!IMPORTANT!

Accepted and rejected states count the Accepted and rejected states count the same!same!

When a point is rejected, you add the When a point is rejected, you add the previous one to the averagesprevious one to the averages

Measure acceptance ratio. Set to roughly Measure acceptance ratio. Set to roughly 1/2 by varying the “step size”1/2 by varying the “step size”

ExactExact: no time step error, no ergodic : no time step error, no ergodic problems problems in principlein principle (but no dynamics). (but no dynamics).

35

Quantum MechanicsQuantum Mechanics We wish to solve We wish to solve HH = E = E to high to high

accuracyaccuracy The solution usually involves computing The solution usually involves computing

integrals in high dimensions: 3-30000integrals in high 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 ... whose integrals are analitically computable

(gaussians)(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

36

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

37

VMC: Variational Monte VMC: Variational Monte CarloCarlo

EE is a statistical average of the local energy is a statistical average of the local energy 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

38

Error bars estimationError bars estimation In Monte Carlo it is easy to estimate the In Monte Carlo it is easy to estimate the

statistical errorstatistical errorNdfPA RRRR )()( NdfPA RRRR )()(

ifif

)(~)(1

1

RRR PfN

A i

N

ii

)(~)(1

1

RRR PfN

A i

N

ii

The associated statistical error isThe associated statistical error is

N

ii Af

NAAA

1

2222 )(1

)( R

N

ii Af

NAAA

1

2222 )(1

)( RN

A)(N

A)(

39

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

40

if p if p ≥≥ 1 1 /* accept always *//* accept always */ accept moveaccept move

If 0 If 0 ≤≤ p p << 1 1 /* accept with probability p *//* accept with probability p */

if p if p >> rnd() rnd()accept moveaccept move

elseelsereject movereject move

The Metropolis AlgorithmThe Metropolis Algorithm

The OracleThe Oracle2

)(

)(

old

newp

2

)(

)(

old

newp

41

VMC: Variational Monte VMC: Variational Monte CarloCarlo

No need to analytically compute integrals: No need to analytically compute integrals: completecomplete freedom in the choice of the trial freedom in the choice of the trial wave functionwave function..

r1

r2

r12

He atomHe atom

1221 rcrbrae 1221 rcrbrae

CCan use an use explicitly explicitly

correlated wave functionscorrelated wave functions

Can satisfy the cusp Can satisfy the cusp

conditionsconditions

42

VMC advantagesVMC advantages

Can go beyond the Can go beyond the Born-Oppenheimer Born-Oppenheimer approximationapproximation, with , with anyany potential, in potential, in anyany number of dimensionsnumber 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

43

Properties of the Local Properties of the Local energyenergy

For an exact eigenstate For an exact eigenstate EELL is a constant is a constant

At particles coalescence the divergence of At particles coalescence the divergence of V must be cancelled by the divergence of V must be cancelled by the divergence of the kinetic termthe kinetic term

For an approximate trial function, For an approximate trial function, EELL is not is not

constantconstant

)()(

)(

2

1

)(

)()(

2

RR

R

R

RR V

HEL

)()(

)(

2

1

)(

)()(

2

RR

R

R

RR V

HEL

44

Reducing ErrorsReducing Errors

For a trial function, if For a trial function, if EELL can diverge, the can diverge, the

statistical error will be largestatistical error will be large To eliminate the divergence we impose the To eliminate the divergence we impose the

Kato’s cusp conditionsKato’s cusp conditions

N

iiLE

NHE

1

)(1

R

N

iiLE

NHE

1

)(1

R

N

iiLL HE

NHHE

1

2222 )(1

)( R

N

iiLL HE

NHHE

1

2222 )(1

)( R

N

EL )(N

EL )(

45

Kato’s cusps conditions on Kato’s cusps conditions on

We can include the correct analytical We can include the correct analytical structurestructure

electron – electron cusps:electron – electron cusps:2

1)0( 1212

rr

21)0( 12

12

rr

Zrr 1)0( Zrr 1)0(electron – nucleus cusps:electron – nucleus cusps:

46

Optimization of Optimization of

Suppose we have variational parameters in Suppose we have variational parameters in the trial wave function that we want to the trial wave function that we want to optimizeoptimize

The straigthforward optimization of The straigthforward optimization of EE is is numerically unstable, because numerically unstable, because EELL can can

diverge diverge

N

iiLT E

NE

1

),(1

)( cRc

N

iiLT E

NE

1

),(1

)( cRc);( cR );( cR

For a finite For a finite N can be unboundN can be unbound Also, our energies have error bars. Can be Also, our energies have error bars. Can be

difficult to comparedifficult to compare

47

Optimization of Optimization of

It is better to It is better to optimizeoptimize

0)(),(1

))((1

22

N

iiL HE

NH ccRc 0)(),(

1))((

1

22

N

iiL HE

NH ccRc

Even for finite Even for finite NN is numerically stable. is numerically stable. The lowest The lowest will not have the lowest will not have the lowest EE but but

it is usually closeit is usually close

222 )( HHH 222 )( HHH

0)(2 H 0)(2 H

It is a measure of the quality of the trial It is a measure of the quality of the trial functionfunction

48

Optimization of Optimization of

Meaning of optimization of Meaning of optimization of

We want We want VV’ to be “close” to the real ’ to be “close” to the real VV

For which potential For which potential V’V’ is is TT an an

eigenfunction?eigenfunction?

HEH 0 HEH 0

Trying to reduce the Trying to reduce the distance between distance between upper and lower upper and lower boundbound

TT

T EV

2

2

1T

T

T EV

2

2

1

)(min)(min 222 HdVVT R )(min)(min 222 HdVVT R

49

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 becomes difficult

as the number of nonlinear parameters as the number of nonlinear parameters increasesincreases

It depends critically on our skill to invent It depends critically on our skill to invent a good a good

There exist exact, automatic ways to get There exist exact, automatic ways to get better wave functions. better wave functions. Let the computer Let the computer do the work ...do the work ...To be continued...To be continued...

In the last episode: In the last episode: VMCVMC

Today: DMCToday: DMC

51

First Major VMC First Major VMC CalculationCalculation

W. McMillan Thesis in 1964W. McMillan Thesis in 1964 VMC calculation of ground state of liquid VMC calculation of ground state of liquid

helium 4.helium 4. Applied MC techniques from classical liquid Applied MC techniques from classical liquid

theory.theory.

52

VMCVMC advantages and advantages and drawbacksdrawbacks

Simple, easy to implementSimple, easy to implement Intrinsic error barsIntrinsic error bars Usually obtains 60-90% of correlation energyUsually obtains 60-90% of correlation energy 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 becomes difficult as

the number of nonlinear parameters the number of nonlinear parameters increasesincreases

It depends critically on our skill to invent a It depends critically on our skill to invent a good good

53

Diffusion Monte CarloDiffusion Monte Carlo

Suggested by Fermi in 1945, but Suggested by Fermi in 1945, but implemented only inimplemented only in the the 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

54

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 Can we simulatesimulate the the SchrSchrödinger equation?ödinger equation?

Diffusion Diffusion equation equation analogyanalogy

55

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

positivepositive

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

56

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

fictitious particles, called fictitious 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

57

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

58

First QMC calculation in First QMC calculation in chemistrychemistry

77 lines of Fortran code!77 lines of Fortran code!

59

Formal developmentFormal development

Formally, in imaginary timeFormally, in imaginary time

)0()(ˆ ˆ HitetHt

i )0()(ˆ ˆ HitetHt

i

)0()(ˆ Het )0()(ˆ Het

In coordinate In coordinate representationrepresentation

RRRR

RR

de

et

H

H

)0(

)0()(ˆ

ˆ

RRRR

RR

de

et

H

H

)0(

)0()(ˆ

ˆ

60

Schrödinger Equation in integral Schrödinger Equation in integral formform

Monte Carlo is good at integrals...Monte Carlo is good at integrals...

RRRRR dG )0,(),'(),'( RRRRR dG )0,(),'(),'(

RRRR HeGˆ

'),'( RRRR HeGˆ

'),'(

We interpret G as a probability to move We interpret G as a probability to move from R to R’ in an time step from R to R’ in an time step . . We iterate We iterate this equationthis equation

61

Iteration of Schrödinger Iteration of Schrödinger EquationEquation

We can iterate this equationWe can iterate this equation

')0,(),'(),'''()2,''( RRRRRRRR ddGG ')0,(),'(),'''()2,''( RRRRRRRR ddGG

62

Zassenhaus formulaZassenhaus formula

We must use a small time step We must use a small time step , but at , but at the same time we must let the same time we must let

2/],[)(ˆ 2 VTVTVTH eeeee 2/],[)(ˆ 2 VTVTVTH eeeee

)( 2ˆ Oeee VTH )( 2ˆ Oeee VTH

In general we do not have the exact GIn general we do not have the exact G

63

Trotter theoremTrotter theorem

A and B do not commute, uA and B do not commute, use Trotter se Trotter TheoremTheorem

nnBnA

n

BA eee //lim

nnBnA

n

BA eee //lim

Figure out what each operator does Figure out what each operator does independently and then alternate their independently and then alternate their effect. This is rigorous in the limit as effect. This is rigorous in the limit as nn

In DMC A is diffusion operator, B is a In DMC A is diffusion operator, B is a branching operatorbranching operator

64

Short Time approximationShort Time approximation

Diffusion + branchingDiffusion + branching

At equilibrium the algorithm will sample At equilibrium the algorithm will sample 00

The energy can be estimated asThe energy can be estimated as

RRRRR dG )0,(),'(),'( RRRRR dG )0,(),'(),'(

2/)'()'( 2

),'( RRRRR eeG V 2/)'()'( 2

),'( RRRRR eeG V

N

iiT

N

iiT

T

TH

d

dHE

1

1

0

0

0

)(

)(

R

R

R

R

N

iiT

N

iiT

T

TH

d

dHE

1

1

0

0

0

)(

)(

R

R

R

R

65

The DMC algorithmThe DMC algorithm

66

A picture for HA picture for H22++

67

Short Time approximationShort Time approximation

68

Importance samplingImportance sampling

2/)'()'( 2

),'( RRRRR eeG V 2/)'()'( 2

),'( RRRRR eeG V

),()(),( RRR Tf ),()(),( RRR Tf

),()())(ln(2

1),( 2

RRR

RfEff

fLT

),()())(ln(2

1),( 2

RRR

RfEff

fLT

V can diverge, so branching can be V can diverge, so branching can be inefficientinefficient

We can transform the SchrWe can transform the Schröödinger dinger equation, by multiplying by equation, by multiplying by TT

69

Importance samplingImportance sampling

),()())(ln(2

1),( 2

RRR

RfEff

fLT

),()())(ln(2

1),( 2

RRR

RfEff

fLT

Similar to a Fokker-Plank equationSimilar to a Fokker-Plank equation Simulated by diffusion+drift+branchingSimulated by diffusion+drift+branching To the pure diffusion algorithm we added a To the pure diffusion algorithm we added a

drift drift step that pushes the random walk in step that pushes the random walk in directions of increasing trial functiondirections of increasing trial function

)(ln' RRR T )(ln' RRR T

70

Importance samplingImportance sampling

),()())(ln(2

1),( 2

RRR

RfEff

fLT

),()())(ln(2

1),( 2

RRR

RfEff

fLT

The branching term now isThe branching term now is )(RELe )(RELe

Fluctuations are controlledFluctuations are controlled At equilibrium it samples:At equilibrium it samples:

)()(),( 0 RRR Tf )()(),( 0 RRR Tf

71

DMC AlgorithmDMC Algorithm

• Initialize a population of walkers {RInitialize a population of walkers {Rii}}

• For each walkerFor each walker

)(ln' RRR T )(ln' RRR T

RR

DriftDrift

DiffusionDiffusionR’R’

72

DMC AlgorithmDMC Algorithm

• Compute branchingCompute branching

))'(( refL EREew ))'(( refL EREew

• Duplicate R’ to M copies: M = int( Duplicate R’ to M copies: M = int( ξξ + + w )w )

• Compute statisticsCompute statistics

• Adjust EAdjust Erefref to make average population to make average population

constant.constant.

• Iterate….Iterate….

73

Good for Helium studiesGood for Helium studies

ThousandsThousands of theoretical and experimental of theoretical and experimental paperspapers

)()(ˆ RR nnn EH )()(ˆ RR nnn EH

have been published on Helium, in its various forms:have been published on Helium, in its various forms:

AtomAtom Small ClustersSmall Clusters DropletsDroplets BulkBulk

74

33HeHemm44HeHenn Stability Chart Stability Chart

3232

44HeHenn 33HeHemm 0 1 2 3 4 5 6 70 1 2 3 4 5 6 7 8 9 10 11 8 9 10 11

00

11

22

33

44

55

33HeHe3344HeHe88 L=0 S=1/2 L=0 S=1/2

33HeHe2244HeHe44 L=1 S=1 L=1 S=1

33HeHe2244HeHe22 L=0 S=0 L=0 S=0

33HeHe3344HeHe44 L=1 S=1/2 L=1 S=1/2

Terra IncognitaTerra IncognitaTerra IncognitaTerra Incognita

Bound L=0Bound L=0

UnboundUnbound

UnknownUnknown

L=1 S=1/2L=1 S=1/2

L=1 S=1L=1 S=1

BoundBound

75

Good for vibrational Good for vibrational problemsproblems

76

For electronic structure?For electronic structure?

77

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

Diffusion equation analogy is lost. Need to Diffusion equation analogy is lost. Need to introduce introduce positive positive andand negative negative walkers. walkers.

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

If we knew the If we knew the exact nodesexact nodes of of , we could , we could exactly exactly simulatesimulate the system by QMC methods, the system by QMC methods, restricting restricting

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

Unfortunately, the Unfortunately, the exactexact nodes nodes

are unknown. Use approximate are unknown. Use approximate

nodes from a nodes from a trial trial . Kill the . Kill the

walkers if they cross a node.walkers if they cross a node.

++ --

78

Common misconception Common misconception on nodeson nodes

• Nodes are Nodes are notnot fixed by antisymmetry fixed by antisymmetry alone, only a 3N-3 sub-dimensional alone, only a 3N-3 sub-dimensional subsetsubset

79

Common misconception Common misconception on nodeson nodes

•They have They have (almost)(almost) nothing to do with nothing to do with Orbital Nodes.Orbital Nodes. It is It is (sometimes)(sometimes) possible to use nodeless possible to use nodeless

orbitalsorbitals

80

Common misconceptions Common misconceptions on on nodesnodes

• A common misconception is that A common misconception is that on a on a nodenode, two like-electrons are always , two like-electrons are always close. This is not trueclose. This is not true

22 11

0

0

0

11 22

81

Common misconceptions on Common misconceptions on nodesnodes

• Nodal theorem is Nodal theorem is NOT VALID in N-DimensionsNOT VALID in N-Dimensions Higher energy states Higher energy states does notdoes not mean more nodes mean more nodes ((Courant and Courant and

Hilbert Hilbert )) It is only an upper boundIt is only an upper bound

82

Common misconceptions on Common misconceptions on nodesnodes

• Not even for the same symmetry speciesNot even for the same symmetry species

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3

Courant counterexampleCourant counterexample

83

Tiling Theorem Tiling Theorem (Ceperley)(Ceperley)

Impossible for Impossible for ground stateground state

The Tiling Theorem does not say how The Tiling Theorem does not say how many nodal domains we should expect!many nodal domains we should expect!

Nodal domains must have the same shapeNodal domains must have the same shape

84

Nodes are relevantNodes are relevant

• Levinson Theorem:Levinson Theorem: the number of nodes of the zero-energy the number of nodes of the zero-energy

scattering wave function gives the number of scattering wave function gives the number of bound statesbound states

• Fractional quantum Hall effectFractional quantum Hall effect

• Quantum Chaos (billiards)Quantum Chaos (billiards)

Integrable systemIntegrable system Chaotic systemChaotic system

87

The Fixed Node The Fixed Node approximationapproximation

Since in general we do not know the exact Since in general we do not know the exact nodes, we resort to approximate nodesnodes, we resort to approximate nodes

We use the nodes of some trial functionWe use the nodes of some trial function

The energy is an upper bound to EThe energy is an upper bound to E00

The energy depends The energy depends onlyonly on the nodes, the on the nodes, the rest of rest of affects the statistical error affects the statistical error

Usually very good results! Even poor Usually very good results! Even poor

usually have good nodesusually have good nodes

88

Trial Wave functionsTrial Wave functions For small systems (N<7)For small systems (N<7)

Specialized forms (linear expansions, hylleraas, Specialized forms (linear expansions, hylleraas, ...)...)

For larger systems (up to For larger systems (up to ~ ~ 200)200) Slater-Jastrow FormSlater-Jastrow Form

J

iii eDc

J

iii eDc

A sum of Slater DeterminantsA sum of Slater Determinants Jastrow factor: a polynomial parametrized Jastrow factor: a polynomial parametrized

in interparticle distancesin interparticle distances

A little A little intermezzointermezzo

Be atom nodal structureBe atom nodal structure

94

Be Nodal StructureBe Nodal Structure

0HF 0HF

r3-r4

r1-r2

r1+r2

0CI 0CI

r1-r2

r1+r2

r3-r4

2222 2121 pscss 2222 2121 pscss

95

Be nodal structureBe nodal structure

Now there are only Now there are only twotwo nodal domains nodal domains

It can be proved that It can be proved that the the exactexact Be wave Be wave function has exactly function has exactly two regionstwo regions

See See Bressanini, Ceperley and ReynoldsBressanini, Ceperley and Reynoldshttp://scienze-como.uninsubria.it/bressanini/http://scienze-como.uninsubria.it/bressanini/

Node isNode is 0...)())(( 224

223

214

2134321 rrrrcrrrr

96

Be nodal structureBe nodal structure

A physicist proof...(A physicist proof...(David CeperleyDavid Ceperley)) 4 electrons: 1 and 2 spin up, 3 and 4 spin 4 electrons: 1 and 2 spin up, 3 and 4 spin

downdown Tiling Theorem applies. There are at most Tiling Theorem applies. There are at most

4 nodal domains4 nodal domains

++

12P12P

--

34P34P--

3412ˆˆ PP 3412ˆˆ PP

++

RR

97

Be nodal structureBe nodal structure

We need to find a point We need to find a point RR and a path and a path RR(t) (t) that connects that connects RR to to PP1212PP3434R so that R so that (R(t)) ≠ (R(t)) ≠

00

Consider the point R = (Consider the point R = (rr11,-,-rr11,,rr33,-,-rr33))

rr

11

rr

22

rr

33

rr

44

is invariant w.r.t. is invariant w.r.t. rotationsrotations

Path: Rotating by Path: Rotating by along along rr11x rx r33 , , is constant is constant

But But ((RR) ≠ 0:) ≠ 0: exactexact= = HF HF + higher terms+ higher terms

HFHF((RR) = 0) = 0

higher terms ≠ 0 higher terms ≠ 0

99

An exampleAn example

High precision total energy calculations of High precision total energy calculations of moleculesmolecules

An example: what is the most stable fullerene?An example: what is the most stable fullerene?

C24

QMC could make QMC could make consistentconsistent predictions of the predictions of the lowest structurelowest structure

Other methods are not capable of making consistent predictions about the stability of fullerenes

100

DMCDMC advantages and advantages and drawbacksdrawbacks

Correlation between particles is Correlation between particles is automaticallyautomatically taken into account.taken into account.

ExactExact for boson systems for boson systems Fixed node for electrons obtains 85-95% of Fixed node for electrons obtains 85-95% of

correlation energy. Very good results in many correlation energy. Very good results in many different fieldsdifferent fields

Works for T=0. For T > 0 must use Path Works for T=0. For T > 0 must use Path Integral MCIntegral MC

Not a “black box”Not a “black box” It is computationally demanding for large It is computationally demanding for large

systemssystems Derivatives of Derivatives of are very hard. Not good are very hard. Not good

enoughenough

101

Current researchCurrent research

Current research focusses onCurrent research focusses on Applications: nanoscience, solid state, Applications: nanoscience, solid state,

condensed matter, nuclear physics, geometry condensed matter, nuclear physics, geometry for molecules,...for molecules,...

Estimating derivatives of wave functionEstimating derivatives of wave function Solving the sign problem (Solving the sign problem (very hardvery hard!!)!!) Make it O(N) method (currently is O(N^3)) to Make it O(N) method (currently is O(N^3)) to

treat bigger systems (currently about 200 treat bigger systems (currently about 200 particles)particles)

Better wave functionsBetter wave functions Better optimization methodsBetter optimization methods

102

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)

THE ENDTHE END