An introduction to Quantum Monte Carlo - TDDFT. · PDF fileAn introduction to Quantum Monte...

An introduction toQuantum Monte Carlo

Jose Guilherme VilhenaLaboratoire de Physique de la Matière Condensée et Nanostructures,

Université Claude Bernard Lyon 1

1) Quantum many body problem;

2) Metropolis Algorithm;

3) Statistical foundations of Monte Carlo methods;

4) Quantum Monte Carlo methods ; 4.1) Overview; 4.2) Variational Quantum Monte Carlo; 4.3) Diffusion Quantum Monte Carlo; 4.4) Typical QMC calculation;

Outline

Consider of N electrons. Since me<<M

N , to a good

approximation, the electronic dynamics is governed by :

which is the Born Oppenheimer Hamiltonian.

We want to find the eigenvalues and eigenvectos of this hamiltonian, i.e. :

QMC allow us to solve numerically this, thus providing E0

and Ψ0 .

1 - Quantum many body problem

H=−12∑i

∇ i2−∑

Z∣r i−d∣

12∑i∑j≠i

Z∣ri−r j∣

Hr1,r 2,. .. ,rN =Er 1,r2,. .. , rN

Consider of N electrons. Since me<<M

N , to a good

approximation, the elotonic dynamics is governed by :

which is the Born Oppenheimer Hamiltonian.

We want to find the eigenvalues and eigenvectos of this hamiltonian, i.e. :

QMC allow us to solve numerically this, thus providing E0

and Ψ0 .

H=−12∑i

∇ i2−∑

Z∣r i−d∣

12∑i∑j≠i

Z∣ri−r j∣

Hr1,r 2,. .. ,rN =Er 1,r2,. .. , rN

Varionational principleFor any normalized wfn Ψ:

Zero Varience Property The variance of the “local energy” must vanish for Ψ

⟨T R∣ H∣T R⟩E0

∣T∣2

∫ dR∣T∣2EL−⟨EL⟩

2=0 ;where EL=

Example :

Harmonic Oscilator

EL x =HT x

T xthen ...∂ xEL x =0 ;if T=0

≠0 ; if T≠ 0

∣T∣2

2=0 ;where EL=

Example :

Harmonic Oscilator

EL x =HT x

≠0 ; if T≠ 0

∣T∣2

2=0 ;where EL=

Example :

Harmonic Oscilator

EL x =HT x

≠0 ; if T≠ 0

4) Quantum Monte Carlo methods ; 4.1) Overview; 4.2) Variational Quantum Monte Carlo; 4.3) Diffusion Quantum Monte Carlo; 4.4) Typical QMC calculation

Outline

Markov Chain stocastic process (no memory).

Metropolis-Hastings algorith (Markov chain) randomly samples a “problematic” probability distribuition function;

Some definitions ...

> P(R): normalized probability distribuition function> R=(r

1,..., r

N) : configuration or walker (can be vector with

the positions of all electrons)> T(R'←R): Proposal density (can be 1, a gaussian, ...)> q=rand(0,1): random number in [0,1]

2 - Metropolis Algorithm

We seek a set {R1, R

2, ..., R

M} of sample points of P(R).

How can we get them ?!? Metropolis Algorithm:

1) Start in a point Rt ;2) Propose a point R', with a probability of T(R'←R) ;3) Calculate the “acceptance rate” : 4) if A > 1 then : Rt+1=R' else if A > q then : Rt+1=R' else : Rt+1=Rt

5) Repeat all from setp 2;6) stop at k>M , and take away the first k-M points.

The sampled points match P(R). The mixing is controled by T(R'←R).

A R 'R=T R 'RPRT RR ' PR '

2, ..., R

Outline

Monte Carlo methods: solve problems using random numbers.

Evaluate integrals:- expectation value of a random variable is just the integral over its probability distribution - generate a bunch of random numbers and average to get the integral;

Simulate random processes with random walkers.

Number of dimensions doesn’t matter.- Simpsons rule (error ~ O (M-4/d))- Monte Carlo method (error ~ O (M-1/2))

3 – Statistical foundations of MC methods

Condider a well behaved f(R) such that:

where P(R) is any normed probability density function.

Central limit theorem:

For any p.d.f. , and a large enough ramdom sample {R

2,...R

M} of it :

i.e., F(R) is normally distributed with F=μf and σ

fM-1/2

F R=1M∑k=0

f Rk ≈f ; limM∞

F R=∫ d R f RP R

F2M =⟨F R ;M −f

2⟩= f

f=∫d R f R PR ; f=∫ d R [ f R−f ]2PR

Condider a well behaved f(R) such that:

where P(R) is any normed probability density function.

Central limit theorem:

For any p.d.f. , and a large enough ramdom sample {R

2,...R

M} of it :

i.e., F(R) is normally distributed with F=μf and σ

fM-1/2

F R=1M∑k=0

f Rk ≈f ; limM∞

F R=∫ d R f RP R

F2M =⟨F R ;M −f

2⟩= f

f=∫d R f R PR ; f=∫ d R [ f R−f ]2PR

Consider the integral:

P(R) is a normed p.d.f. and f(R)=g(R)/P(R)

i) Metropolis algorithm samples P(R) to get {R1,R

2,...R

ii) Monte Carlo estimative of I is

with a statistical error of σIMC

=σfM-1/2

iii) We can estimate σIMC

using MC again, i.e. :

I=∫ g Rd R=∫P R f RdR

IMC=1M∑k=1

f Rk ≈I

f=1M∑i=1

[ f Ri−1M∑j=1

f R j]2

P(R) is any normed p.d.f. and f(R)=g(R)/P(R)

2,...R

=σfM-1/2

iii) The σf can be estimated using MC again, i.e. :

IMC=1M∑k=1

f R k ≈I

f=1M∑i=1

[ f Ri−1M∑j=1

f R j]2

2,...R

=σfM-1/2

IMC=1M∑k=1

f R k ≈I

f=1M∑i=1

[ f Ri−1M∑j=1

f R j]2

2,...R

=σfM-1/2

IMC=1M∑k=1

f R k ≈I

f=1M∑i=1

[ f Ri−1M∑j=1

f R j]2

Outline

Quantum Monte Carlo simulate the quantum MB problem;(Variational MC; Difusion MC; Path integral MC; Auxiliary field MC; Reptation MC ; Gaussian quantum MC; etc ...)

MC handles directly the many dimention electronic wave function ( R=[r

2,…,r

We’ll cover two main flavors: Variational Monte Carlo (integration over (3N)D space); Diffusion Monte Carlo (projector approach)

4 – Quantum Monte Carlo methods 4.1 - Overview

On 55 molecules, mean absolute deviation of atomization energy is 2.9 kcal/mol (MPPT and CI ~ 2kcal/mol ;O(N4-N6) )

Same accuraccy and better scaling than post Hartree-Fock methods (~ N3). (Non Fixed node DMC scales as eN)

Successfully applied to organic molecules, transition metal oxides, solid state silicon, systems up to ~1000 electrons

Can calculate accurate atomization energies, phase energy differences, excitation energies, one particle densities, correlation functions, etc..

No free lunch!! Statistical error ~ (Computational time)-1/2

4 – Quantum Monte Carlo methods 4.1 - Overview

Outline

Consider: R=(r1, r

2, ...r

i position of the ith e- ; and a trial wfn Ψ

such that:i) Ψ

T(R) and Ψ∇

T(R) exist and are continous in all* R;

ii) and exist;

The <H> with respect to ΨT(R) is:

where EL={HΨ

T} and P(R)=|Ψ

T|2 |Ψ

Using metropolis MC method we sample P(R) ( {R1,..., R

M} ) and

average the EL to get:

Optimizing T to get acceptance ~ 50% we improve the mixing of the

sample;

4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo

EV= ⟨T∣ H∣T ⟩=∫∣T∣

∫d R∣T∣2 EL Rd R

EV= ⟨ H ⟩= 1M∑k=1

∫∣T∣2d R ∫T

∗ HT d R

Consider: R=(r1, r

2, ...r

such that:i) Ψ

T(R) and Ψ∇

ii) and exist;

where EL={HΨ

T} and P(R)=|Ψ

T|2 |Ψ

M} ) and

sample;

EV= ⟨T∣ H∣T ⟩=∫∣T∣

EV= ⟨ H ⟩= 1M∑k=1

∫∣T∣2d R ∫T

∗ HT d R

Consider: R=(r1, r

2, ...r

such that:i) Ψ

T(R) and Ψ∇

ii) and exist;

where EL={HΨ

T} and P(R)=|Ψ

T|2 |Ψ

M} ) and

sample;

EV= ⟨T∣ H∣T ⟩=∫∣T∣

EV= ⟨ H ⟩= 1M∑k=1

∫∣T∣2d R ∫T

∗ HT d R

Consider: R=(r1, r

2, ...r

such that:i) Ψ

T(R) and Ψ∇

ii) and exist;

where EL={HΨ

T} and P(R)=|Ψ

T|2 |Ψ

M} ) and

sample;

EV= ⟨T∣ H∣T ⟩=∫∣T∣

EV= ⟨ H ⟩= 1M∑k=1

∫∣T∣2d R ∫T

∗ HT d R

Evaluate Probability ratio

Propose a move

Initial Setup

Update electrons positions

Metropolis accept/reject

Calculate local energy

Output the result

acceptreject

" How many graduate students lives have been lost optimizing wavefuctions ... " D. Ceperley

Good news !!! T can have any functional form !!!

Most commonly used T is a Slater-Jastrow wavefunction:

X =eJ X ; 1,. ..,i {∑j=1

jD jX }> X=x1, ... ,X N , with x i= {ri ,i }> j i , coefficients to be optimized ;

> DnX =∣ 1x1 ⋯ 1xN ⋮ ⋱ ⋮

N x1 ⋯ N xN ∣ hxg are single particle Slater orbs.

> eJ X ;1,. ..,i , is the Jastrow correlation function;> J=F d ,r i , r ij , the Jastrow factor: usually sums of Chebyshev polynomials

How do we acctually perform the optimization of the wfn ? (minimize

E() or E

v or a mixing of both )

The most used is the minimization of E() , because:

i) We know it's exact value for the g.s.ii) Better numerical stability;

The variance of the energy is given by:

where: parameters to optimize ; Ev variational energy, E

local energy.

E2 =∫

∣∣2

∫∣∣2d R

[EL −EV ]2d R

Correlated sampling method1) Start form a guessed set of paramters {

2) Sample the P(R,) using MMC method;

3) With this sampling minimize E() like this :

Calculate EL ; E

E() , for a different set of parameters {

(choosen in a way to minimize E()), like this:

4) Once E reaches a minimum, set {

5) Repeat all the steps 2-5 until E()~0

EL Ri ,N =HRi ,N Ri ,N

; N =N

EV=∫∣T ∣

∫∣T ∣2N d R

E LN d R≈1

∑k=1

Rk ,N ∑i=1

Ri ,N EL Ri ,N

E2N =∫

∣T ∣2N

∫∣T ∣2N d R

[EL N −EV N ]2d R≈

∑k=1

R k ,N ∑i=1

Ri ,N [EL Ri ,N −EV ]2

PR= ∣R,T ∣2

∫∣ R,T ∣2d R

Calculate EL ; E

; N =N

EV=∫∣T ∣

∫∣T ∣2N d R

E LN d R≈1

∑k=1

Rk ,N ∑i=1

Ri ,N EL Ri ,N

E2N =∫

∣T ∣2N

∫∣T ∣2N d R

∑k=1

R k ,N ∑i=1

PR= ∣R,T ∣2

∫∣ R,T ∣2d R

Calculate EL ; E

; N =N

EV=∫∣T ∣

∫∣T ∣2N d R

E LN d R≈1

∑k=1

Rk ,N ∑i=1

Ri ,N EL Ri ,N

E2N =∫

∣T ∣2N

∫∣T ∣2N d R

∑k=1

R k ,N ∑i=1

PR= ∣R,T ∣2

∫∣ R,T ∣2d R

Calculate EL ; E

; N =N

EV=∫∣T ∣

∫∣T ∣2N d R

E LN d R≈1

∑k=1

Rk ,N ∑i=1

Ri ,N EL Ri ,N

E2N =∫

∣T ∣2N

∫∣T ∣2N d R

∑k=1

R k ,N ∑i=1

PR= ∣R,T ∣2

∫∣ R,T ∣2d R

Calculate EL ; E

; N =N

EV=∫∣T ∣

∫∣T ∣2N d R

E LN d R≈1

∑k=1

Rk ,N ∑i=1

Ri ,N EL Ri ,N

E2N =∫

∣T ∣2N

∫∣T ∣2N d R

∑k=1

R k ,N ∑i=1

PR= ∣R,T ∣2

∫∣ R,T ∣2d R

Calculate EL ; E

; N =N

EV=∫∣T ∣

∫∣T ∣2N d R

E LN d R≈1

∑k=1

Rk ,N ∑i=1

Ri ,N EL Ri ,N

E2N =∫

∣T ∣2N

∫∣T ∣2N d R

∑k=1

R k ,N ∑i=1

PR= ∣R,T ∣2

∫∣ R,T ∣2d R

Outline

4 – Quantum Monte Carlo methods 4.3 – Diffusion Quantum Monte Carlo

General strategy: stochastically simulate a differential equation that converges to the eigenstate

Equation:

Must propagate an entire function forward in time <=> distribution of walkers

−dR , t d t

= H−E R , t

We want to find a wave function so

Our differential equation

Suppose that

decreases

Kinetic energy (curvature)decreases, potential energystays the same

Time derivative is zero when

t=infinity

−12∇ 2V R E

−dR , t d t

= H−E R , t

Generate walkers with a guess distribution Each time step:

Take a random step (diffuse)

A walker can either die, give birth, or just keep going

Keep following rules, and we find the ground state!

Works in an arbitrary number of dimensions

−dR , t d t

=−12∇

2R V R−E R , t

Diffusion Birth/death

Initial

−dR , t d t

=−12∇

2R V R−E R , t

Initial

−dR , t d t

=−12∇

2R V R−E R , t

Initial

−dR , t d t

=−12∇

2R V R−E R , t

Initial

Diffusion Birth/deathInitial

−dR , t d t

=−12∇

2R V R−E R , t

Importance sampling: multiply the differential equation by a trial wave function Converges to instead of The better the trial function, the faster DMC is-- feed it a

wave function from VMC

Fixed node approximation: for fermions, ground state has negative and positive parts Not a pdf--can’t sample it Approximation:

Importance sampling: multiply the differential equation by a trial wave function Converges to instead of The better the trial function, the faster DMC is-- feed it a

wave function from VMC

Fixed node approximation: for fermions, ground state has negative and positive parts Not a pdf--can’t sample it Approximation:

Outline

Choose system and get one-particle orbitals

Optimize wave function using VMC, evaluate energy and properties of wave function

Use optimized wave function in DMC for most accurate, lowest energy calculations

4 – Quantum Monte Carlo methods 4.4 – Typical QMC calculation

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