correlated sampling without reweighting, computing …roland assaraf laboratoire de chimie...
TRANSCRIPT
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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
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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).
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
Stochastic technics in principle adapted for solving theSchroedinger equation :
Solving the many problem in Quantum Physics
Computing integrals in large dimensions.
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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)
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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)
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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)
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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...)
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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)
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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.
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
Overview
1 Introduction
2 Correlated sampling with reweightingThe methodStatistical uncertaintiesNumerical illustration
3 Correlated sampling with no reweightingThe methodNumerical illustration
4 Conclusion and perspectives
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion 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.
...
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion 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.
...
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion 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.
...
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
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)
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
Pair correlation function
[J. Toulouse, R. Assaraf, C. J. Umrigar. J. Chem. Phys. 126 244112 (2007)]Ou =
∑i
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
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)
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
First term (ZV)
〈elλ − el0〉 depends only on Rl
=⇒ Locality property of its variance
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
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.
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
Conclusion
δ∆λ∆λ
∝√
N (7)
Correlated sampling with reweighting solves the small λdifficulty but not the large N one
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
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)
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
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
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
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,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
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
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
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
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
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
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,v]+P[ψ´λ,v]
ZV [ψ´λ,0]+P[ψ´λ,0]
ZV [~ψmin,0]+P[ψ´λ,0]
P[ψ´λ,0]
P[ψ´λ,v]
FIG.: Statistical uncertainties in the insulating Hn chain
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodStatistical uncertaintiesNumerical illustration
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,v]+P[ψ´λ,v]
ZV [ψ´λ,0]+P[ψ´λ,0]
ZV [~ψmin,0]+P[ψ´λ,0]
P[ψ´λ,0]
P[ψ´λ,v]
FIG.: Statistical uncertainties in the metallic Hn chain
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodNumerical illustration
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.
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodNumerical illustration
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)
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodNumerical illustration
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
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodNumerical illustration
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
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodNumerical illustration
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
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodNumerical illustration
-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
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodNumerical illustration
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
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodNumerical illustration
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
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
The methodNumerical illustration
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
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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).
Roland Assaraf Correlated sampling without reweighting
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IntroductionCorrelated sampling with reweighting
Correlated sampling with no reweightingConclusion and perspectives
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...
Roland Assaraf Correlated sampling without reweighting
IntroductionCorrelated sampling with reweightingThe methodStatistical uncertaintiesNumerical illustration
Correlated sampling with no reweightingThe methodNumerical illustration
Conclusion and perspectives