correlated sampling without reweighting, computing …roland assaraf laboratoire de chimie...

IntroductionCorrelated sampling with reweighting

Correlated sampling with no reweightingConclusion and perspectives

Correlated sampling without reweighting,computing properties with size-independent

variances

Roland Assaraf

Laboratoire de Chimie Théorique, CNRS-UMR 7616, Université Pierre et MarieCurie Paris VI, Case 137, 4, place Jussieu 75252 PARIS Cedex 05, France

BIRS 2012

Roland Assaraf Correlated sampling without reweighting



Some perspective on Quantum Monte Carlo (QMC)

Many problem in Quantum physics at zero temperature

The Schroedinger equation,

HΦ = (−N∑

i=1

∆i + V(r1, r2 . . . rN))Φ = EΦ (1)

N number of particles.

ri, 3 spatial coordinates of particle i.

E lowest eigenvalue, the groundstate energy.

Φ(r1 . . . rN) the lowest eigen vector, the groundstate

Φ antisymetric for electrons (fermions).




Stochastic technics in principle adapted for solving theSchroedinger equation :

Solving the many problem in Quantum Physics

Computing integrals in large dimensions.




Example : variational energy

Variational energy

EV ≡ 〈Ψ|Ĥ|Ψ〉

Average on a probability distribution

〈Ψ|Ĥ|Ψ〉 =∫

dRΨ2(R)HΨΨ

(R)

= 〈HΨΨ (R)〉Ψ2 = 〈e(R)〉Ψ2R : 3N coordinates of the N interacting particles

Ev =1N

N∑k=1

e(Rk)




In general

More generally

EQMC = 〈e(R)〉ΠDepending on the QMC method, the nature of R might change :

3N particle coordinates (VMC,DMC..).

Trajectories in the space of 3N particle coordinates(PDMC, PIMC, reptation...)




Accurate energies

No analytical integration

Flexibility (choice of ψ in VMC).

Weak limitation in system sizes.

Possibility to improve “arbitrarily” the accuracy(“zero-variance zero-bias principle”, choice of ψ in VMC..).

In practice, reference methods for total energies on largesystems (large N)




Quantites of physical interest

They are energy differences.

Exploiting accurate total energies

More tricky in QMC than in a deterministic method.

Energy differences are usually very small.

Statistical uncertainties.

Small statistical uncertainty on a total energy might be huge ona difference if energies are computed independently.




Why energy differences are usually small ?

ExamplesBinding energies, transition state energies. One, two particle gaps (electron

affinities, ionization energies) . . .

First order derivatives of the energy : Any observable (force, dipole, moment,

densities...).

Higher order derivatives : spectroscopic constants . . .

They are groundstate energies of similar systems




Paradigm : Calculation of an observable O

Hλ = H + λO => Ō =dEλdλ

≃ Eλ − E0λ

=∆λλ

(2)

∆λ = Eλ − E0 ∝ λ small

Behavior as a function of the system size

limN→∞ ∆λ(N) = K finite.

The perturbation λO depends usually on a few degrees offreedom.

∆λ has a locality property




In summary

Small λ and large N

∆λ(N) ∝ λ

Accuracy on ∆λ in an independent energy calculation

δ∆λ∆λ

∝ δE0λ

∝√

Nλ

No locality property for the statistical uncertainty.

Comparison to total energy

δ∆λ∆λ

∝ N32

λ

δEE




Overview

1 Introduction

2 Correlated sampling with reweightingThe methodStatistical uncertaintiesNumerical illustration

3 Correlated sampling with no reweightingThe methodNumerical illustration

4 Conclusion and perspectives




The methodStatistical uncertaintiesNumerical illustration

correlated sampling with reweighting

We have to compute the difference

Eλ − E0 = 〈eλ(R)〉πλ − 〈e(R)〉π

Sampling the same distribution for the two energies

Eλ − E0 =〈eλ πλπ 〉π〈πλπ〉π

− 〈e〉π. (3)weight wλ

Different contexts

Variational Monte Carlo eλ(R) =Hλψλψλ

(R), wλ(R) =ψ2

λ

ψ2(R)

Forward walking method inc ontext of DMC algorithms.

...





General expression

Compact expression

∆λ = Eλ − E0 = 〈eλ − e〉π +cov(eλ,wλ)

〈wλ〉π(4)

λ dependence

Eλ − E0 = λ∂Eλdλ |λ=0 + o(λ).

E′λ = 〈e′λ〉π + cov(eλ,w′λ)

Zero-Variance (ZV) estimator Pulay correction

Finite statistical uncertainty on E′λ =⇒ δ∆λ∆λ = K + o(λ)





Pair correlation function

[J. Toulouse, R. Assaraf, C. J. Umrigar. J. Chem. Phys. 126 244112 (2007)]Ou =

∑i




N-dependence

R. Assaraf, D. Domin, W. Lester.

Model of two separated (non interacting) subsystems

Particles coordinates Rl and Ru. Hλ = Hlλ + Hu

Variational Monte Carlo

R = (Rl,Ru)

Ψλ(R) = Ψλ(Rl,Ru) = Ψlλ(Rl)Ψu(Ru)

Local energy eλ(R) = elλ(Rl) + eu(Ru)

Eλ − E = 〈elλ − el0〉 +cov(eλ,wl)

〈wl〉 (5)





First term (ZV)

〈elλ − el0〉 depends only on Rl

=⇒ Locality property of its variance





Pulay term

cov(eλ,wl)〈wl〉 =

cov(elλ,wl)

〈wl〉 +cov(eu,wl)

〈wl〉 (6)

Local Non local

The non local contribution is 0 (eu and wl independant) !

Its variance on a finite sample (eu(Rui ),wl(Rli))i∈[1..M] :

∝ V(eu) ∝ N.=⇒ δ∆λ(N) ∝

√N for large N.

Non locality property of the Pulay term.





Conclusion

δ∆λ∆λ

∝√

N (7)

Correlated sampling with reweighting solves the small λdifficulty but not the large N one





Illustration on Hn chains

Is the analysis for non interacting subsystems holds forinteracting systems ?

Hydrogen chains, metallic and insulating

Calculation of the force on the first nucleus : derivative ofthe energy with respect to the position of the first nucleus

Variational calculation

ψ is a single determinant (Restricted Hartree Fock)





Metallic hydrogen chains

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 10 20 30 40 50

For

ce

Number of H atoms

Metallic chain ZV [~ψmin,0]

ZV [ψ´λ,0]

ZV [ψ´λ,v]

ZV [ψ´λ,v]+P[ψ´λ,v]

ZV [ψ´λ,0]+P[ψ´λ,0]

ZV [~ψmin,0]+P[ψ´λ,0]

RHF

FIG.: Energy derivative, different estimators





Insulating hydrogen chains

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 10 20 30 40 50

For

ce

Number of H atoms

Insulating chain

ZV [~ψmin,0]

ZV [ψ´λ,0]

ZV [ψ´λ,v]


ZV [ψ´λ,0]+P[ψ´λ,0]


RHF

FIG.: Energy derivative, different estimators





Histogram of the ZV term, metallic chain

0

200

400

600

800

1000

1200

1400

-3 -2 -1 0 1 2 3

Fre

quen

cies

eλ´-

H2H4H8

H24H48

FIG.: Histogram of the energy derivative in the Hn chain

ZV contribution has the local property ! !Roland Assaraf Correlated sampling without reweighting




Histogram of the local energy, metallic chain

0

200

400

600

800

1000

1200

1400

1600

-10 -8 -6 -4 -2 0 2 4 6 8 10

Fre

quen

cies

eL-

Local energy histogram

Metallic chain

H2H4H8

H24H48

FIG.: Histogram of the local in the Hn chain

The local energy has not the local propertyRoland Assaraf Correlated sampling without reweighting




Statistical uncertainties

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0 10 20 30 40 50 60 70 80 90

Sta

tistic

al u

ncer

tain

ty

Number of H atoms

ZV [~ψmin,0]

ZV [ψ´λ,0]

ZV [ψ´λ,v]


ZV [ψ´λ,0]+P[ψ´λ,0]


P[ψ´λ,0]

P[ψ´λ,v]

FIG.: Statistical uncertainties in the insulating Hn chain





Statistical uncertainties

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 10 20 30 40 50 60 70 80 90

Sta

tistic

al u

ncer

tain

ty

Number of H atoms

ZV [~ψmin,0]

ZV [ψ´λ,0]

ZV [ψ´λ,v]


ZV [ψ´λ,0]+P[ψ´λ,0]


P[ψ´λ,0]

P[ψ´λ,v]

FIG.: Statistical uncertainties in the metallic Hn chain




The methodNumerical illustration

The methodAssaraf, Caffarel, Kollias 2011

Basic idea

〈eλ(R)〉πλ − 〈e(R)〉π = 〈eλ(Rλ) − e(R)〉Π(R,Rλ)

Marginal distributions of Π(R,Rλ) must be π(R), πλ(Rλ).

differences of the order of λ, 〈(Rλ) − R)2〉 = Kλ2





How to build such a process

Choosing close stochastic processe, L, Lλ having π and πλas stationary states.

Stability versus chaos. Two trajectories with the differentinitial conditions and same pseudo random numbers meetexponentially fast.

Insures that close processes will produce closetrajectories.





For example, with the overdamped Langevin process onewould have

R(t + dt) = R(t) + b [R(t)] dt + dW (8)

Rλ(t + dt) = Rλ(t) + bλ [Rλ(t)] dt + dW (9)





Stability of the process versus chaos

Chain of 120 Hydrogens (120 electrons).Same process but different initial conditions.Perturbed system one atom displaced of λ = 10−4a.u (finitedifference derivative).

1e-20

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0 2 4 6 8 10 12 14

<(R

λ=0-

R)2

>

Time (au)

(t)

(average on 100 walkers at time t)

Synchronization H120 molecule

FIG.: Synchronization of trajectories in H120Roland Assaraf Correlated sampling without reweighting




Independance of the uncertainties on λ

Exact derivative

H120 finite differences

〈(Rλ−Rλ

)2〉

〈ELλ−ELλ〉

Finite difference parameterλ (au)

〈(Rλ−

Rλ

)2〉

〈E

Lλ−

EL

λ〉

2

1.8

1.6

1.4

1.2

1

10.10.010.0010.0001

-0.05

-0.1

-0.15

-0.2

-0.25

FIG.: Quadratic distances betwen the two processes





Locality of the algorithm

1e-16

1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

0 10 20 30 40 50 60 70 80

<r i(

λ)-r

i)2>

Distance from the first atom (a.u.)

Metallic chain

H2H4H6H8

H16H24H48

1e-26

1e-24

1e-22

1e-20

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

0 10 20 30 40 50 60 70 80<

r i(λ)

-ri)2

>Distance from the first atom (a.u.)

Insulating chainH2H4H8

H16H24H48

FIG.: Square average of the inter electron distance at a givendistance from the first atom





-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 10 20 30 40 50

For

ce

Number of H atoms

Metallic / correlated sampling with no reweighting

Metallic / RHF

Insulating / correlated sampling with no reweighting

Insulating / RHF

FIG.: Energy derivative with the correlated sampling with noreweighting





0

0.0005

0.001

0.0015

0.002

0 10 20 30 40 50

Sta

tistic

al u

ncer

tain

ty

Number of H atoms

Metallic / correlated sampling with no reweighting

Insulating / correlated sampling with no reweighting

FIG.: Uncertainty as a function of N, metallic chains





0

200

400

600

800

1000

1200

-3 -2 -1 0 1 2 3

Fre

quen

cies

(eλ(Rλ)-e(R))/λ

Exact derivative histogram

Metallic chain

H2H4H8

H24H48

FIG.: Histogram of the correlated difference metallic chain




Reweighting introduces statistitical fluctuations difficult tocontrol

Solves the small perturbation problem (λ small).

Sometimes large prefactors in the variance.

Same large N behavior as independent energycalculations.

Correlated sampling with no reweighting

Solves the small λ and large N undesirable behavior.

Perspective to obtain small energy differences withcomparable accuracy to the energy.

Relies on some particular dynamics (stability with respectto the chaos).




Possible to build such stable dynamics

At the core of perfect sampling (criteria of timeconvergence, see Fahy, Krauth...).

Building such dynamics for general molecules is underway.

Vast subject (numerically, mathematically). Collaborationsare welcome...


IntroductionCorrelated sampling with reweightingThe methodStatistical uncertaintiesNumerical illustration

Correlated sampling with no reweightingThe methodNumerical illustration

Conclusion and perspectives

correlated sampling without reweighting, computing …roland assaraf laboratoire de chimie...

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