the generalized correlated sampling approach: toward an...

Plan

Introduction

The generalized correlated sampling approach 9 avril 2008 1 de 44

The generalized correlated sampling approach:toward an exact calculation of energy

derivatives in Diffusion Monte Carlo

Roland Assarafa,Alexander Kolliasb and Michel Caffarelc

a) Laboratoire de Chimie Theorique, CNRS-UMR 7616, Universite Pierre et Marie Curie Paris VI, Case137, 4, place Jussieu 75252 PARIS Cedex 05, France

b) Carnegie Institution of Washington, Geophysical Laboratory 5251 Broad Branch Rd., N.W.,Washington, DC 20015, USA

c) Laboratoire de Chimie et Physique Quantiques, CNRS-UMR 5626, IRSAMC Universite PaulSabatier, 118 route de Narbonne 31062 Toulouse Cedex, France

Plan

Introduction


General perspective

Quantum Monte Carlo (QMC) : stochastic technics usedto solve the Schroedinger equation

In principle adapted to the many body problem (weaklimitation in system sizes)

In Practice

Reference methods for groundstate energies of largesystems.

Less successful for other quantities.

Plan

Introduction


Quantities of physical interest ?

Most of them can be obtained from total energies

Binding energies. Transition state energies. One, two particlegaps (electron affinities, ionization energies) . . .

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

Higher order derivatives : spectroscopic constants . . .

Differences of energies of very close systems(small energy differences)

Plan

Introduction


Paradigm : Calculation of an observable O

Hλ = H + λO => O =dEλ

dλ≃ Eλ − E0

λ(1)

Direct calculation

Eλ, E0 computed independently.

δ(Eλ −E0

λ) ∼ δE0

λ−→∞λ→ 0

Plan

Introduction


System size dependency in a direct calculation

Example : a one particle gap ∆ = E (N + 1)− E (N )

δE ∼ N and ∆ ∼ 1 (best case)

Independent calculation of energies :

=⇒ δ∆

∆∼ N

1/N plays the role of the small parameter λ

In practice limiting factor on system sizes

Plan

Introduction


Exploiting accurate QMC total energies to obtain accurate smalldifferences is not as simple as in a deterministic method.

=> At the heart of practical limitations regarding properties one can

compute and system sizes one can reach in QMC.

Objective

Eλ − E0 ∼ λ =⇒ δ(Eλ − E0) ∼ λ

⇐⇒Finite statistical error on energy derivatives.

Plan

Introduction


Plan

1 QMC : A reference method for the total energyVariational energyDiffusion Monte Carlo

2 Small energy differences in VMCObservable in VMC

3 Generalization to DMCForward walkingGeneralized correlated samplingCalculation of the DMC energy derivative in H2 and Li2Preliminary results on other moleculesCriteria of stabilty

4 Conclusion and perspectives

Plan

I. Energy calculation

Variational energy

Diffusion Monte Carlo

Small energydifferences in VMC

Observable in VMC

Generalization toDMC

Forward walking

Generalized correlatedsampling

Calculation of the DMC energy

derivative in H2 and Li2

Preliminary results on othermolecules

Criteria of stability

Conclusion andperspectives


Variational energy

EV ≡ 〈Ψ|H |Ψ〉

Average on a probability distribution

〈Ψ|H |Ψ〉 =

∫

dRΨ2(R)HΨ

Ψ(R)

= 〈HΨΨ (R)〉Ψ2 = 〈EL(R)〉Ψ2

Ev =1

N

N∑

k=1

EL(Rk )

Plan


Variational energy



Observable in VMC


Forward walking








Sampling of Ψ2

Dynamic over the configurations

R(t + dt) = R(t) + bdt + dW (2)

b(t) ≡ ∇ΨΨ (drift)

dW gaussian random numbers (diffusion).〈dWidWj 〉 =

√dtδij

A trajectory R(t) <=> Sample of Ψ2.

Plan


Variational energy



Observable in VMC


Forward walking








Why VMC can be a very accurate method ?

Zero-variance - zero-bias property (ZVZB)

Bias EV − E0 |Ψ− Φ|2Variance σ2(EL) |Ψ− Φ|2

φ Exact groundstate

Important sinceAccuracy in QMC <=> EV − E0 systematic error

+ σ(EL)√N

statistical error.

Plan


Variational energy



Observable in VMC


Forward walking








Diffusion Monte Carlo (DMC)

Sampling the exact groundstate

e−tH |Ψ〉 = Φ (3)

Trotter formula : e−tH = e−δtH e−δtH . . . e−δtH

〈R′|e−δtH|R〉 = P(R→ R′)W(R)

Overdamped Langevin Weight

In practice for fermions, Fixed node approximation :H −→ HFN =⇒ Φ −→ ΦFN (variational solution in the spaceof functions having the same nodes as Ψ).

Plan


Variational energy



Observable in VMC


Forward walking








Conclusion

We can sample something better than Ψ, ΦFN , even if itsanalytic form is unknown.

Illustration

-460.8

-460.6

-460.4

-460.2

-460

-459.8

1 2 3 4 5 6 7 8 9 10

En

ergie

(H

art

ree)

R (Bohr)

RHF

MCSCF CAS(8,5)

MRCISD CAS(8,5)

VMC J × CAS(8,5)

DMC J × CAS(8,5) Exact

Dissociation curve for HCl (collaboration J. Toulouse)

Can we compute accurate small differences or derivativeson the VMC curve, on the DMC curve ?

Plan


Variational energy



Observable in VMC


Forward walking








Observable and energy on a similar footing inVMC

ZVZB improved estimators (R. Assaraf, M. Caffarel, 2003)

Hλ = H + λO => 〈O〉Φ2 =dEλ

dλ≃ dEV [Ψλ]

dλ

Ψλ = Ψ + λΨ

dEV [Ψλ]

dλ= 〈O〉Ψ2

O =⇒ O [Ψ, Ψ] = O +(H − EL)Ψ

Ψ+ 2(EL − EV )

Ψ

Ψ

Plan


Variational energy



Observable in VMC


Forward walking








Energy/observable in VMC

Bias Variance Choice of

EV [Ψ] δΨ2 Idem Ψ

EV′λ = 〈O [Ψ, Ψ]〉Ψ2 δΨ2 + δΨδΨ′ Idem (Ψ, Ψ)

〈O〉Ψ2 δΨ O(1) (Ψ, Ψ = 0)

δΨ = Ψ−Φ

δΨ′ = Ψ− dΦλ

dλ

Plan


Variational energy



Observable in VMC


Forward walking








Electronic densityR. Assaraf, M. Caffarel, and A. Scemama, Phys. Rev. E. 75 035701, (2007)

ρ(r) =∑

i

δ(ri − r)

Regularized bare estimator :

Counting electrons in small boxes of size ∆R around r :=> Bias O(∆R)

=> Variance O( 1∆R

3).

Poorest, in regions rarely or never visited by electrons

Plan


Variational energy



Observable in VMC


Forward walking








Simplest improved estimator

δ(ri − r) = −14π

∆ 1|ri−r| + Green formula

ρZV (r) =

⟨

− 1

4π

N∑

i=1

1

|ri − r|∇2

i (Ψ2)

Ψ2

⟩

Ψ2

.

Used by P. Langfelder 1et al for the density at a nucleus, in

combination with a modified important sampling (to cure the still

infinite variance).

1P. Langfelder, S.M. Rothstein, J. Vrbik, J. Chem. Phys. 107 8525 (1997)

Plan


Variational energy



Observable in VMC


Forward walking








Illustration : Helium atom

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 3

Usual EstimatorSimple Improved Estimator

Best Improved EstimatorExact

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

0 0.02 0.04 0.06

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

2.5 2.6 2.7 2.8 2.9 3

ρ

r

(r)

Plan


Variational energy



Observable in VMC


Forward walking








Application : water dimer

Fig.: Isodensity surfaces of the water dime

Bare regularized estimator Improved estimator

Plan


Variational energy



Observable in VMC


Forward walking








Pair correlation function

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

I (u) =∑

i<j δ(rij − u)Probability density to find a pair of electrons at distance u

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

I(u

) (a

.u.)

u (a.u.)

histogram estimator with HF wave functionZV1 estimator with HF wave function

ZV1ZB1 estimator with HF wave functionaccurate intracule

He atom

Plan


Variational energy



Observable in VMC


Forward walking








Impact of the optimisation of Ψ

Illustration : Intracule calculations (collaboration with J.

Toulouse et C. Umrigar)

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5

4 π

u2 [

I (u

) - IH

F (u

) ]

(a.u

.)

u (a.u.)

Jastrow × HFJastrow × SD

Jastrow × CAS(10,8)

N2 molecule

VMC with ZVZB2 estimator

Fig.: Correlation part of the radial intracule in function of thedistance electron-electron u for the molecule N2.

Plan


Variational energy



Observable in VMC


Forward walking








In summary

Derivatives (or small differences) of VMC energies : onthe same footing as total VMC energies

Strategy : Using improved estimators OZVZB [Ψ, Ψ].

=> control of the statistical error and the systematic error, ata reasonable computational cost.

Current work :

Improving estimators for forces (geometry optimization).

Application to a variety of other properties (dipolemoments...).

Plan


Variational energy



Observable in VMC


Forward walking








Generalization to DMCR. Assaraf, A. Kollias, M. Caffarel

The DMC energy

Trial function Groundstate of H

EDMC = 〈Ψ | H | Φ〉 = 〈EL〉ΨΦ

ΨΦ = Ψe−tH | Ψ〉 =

⟨

e−

R

t

0EL(R(s))ds | R(t)〉

⟩

[R(s)](2)

[R(s)] Overdamped Langevin process with the drift b = ∇ΨΨ

Plan


Variational energy



Observable in VMC


Forward walking








Forward walking

Hλ = H + λO => HλΦλ = EλΦλ

Computation ofdEλ

dλ=

d〈ELλ〉ΦλΨ

dλ

Forward walking

Same trial function for H and Hλ (Ψ = Ψλ).

dΨΦλ

dλ=

⟨[

−∫

t

0

dsO[R(s)]

]

e−

R

t

0EL(R(s))ds | R(t)

⟩

[R(s)]

In principle exact, but tractable only if O has a small variance.Example for a force at the nucleus, variance infinite.

Plan


Variational energy



Observable in VMC


Forward walking








Approximations

Example : Hybrid improved estimator, in a FNapproximation

〈O〉h ≡ 2〈O〉DMC − 〈O〉VMC

Systematic error : O(|Ψ− Φ|2) + O(|Ψ− Φ1|.|ΦFN − Φ|)

O(|ΦFN − Φ|) for 〈O〉hVariance O(|Ψ− Φ|2)

O(1) for〈O〉hApplication to forces (R. Assaraf, M. Caffarel, 2003)

Plan


Variational energy



Observable in VMC


Forward walking








Generalized correlated sampling

First ingredient : two different guiding functions

Ψλ = Ψ + λΨ => bλ = b + λb

Eλ − E0 = 〈ELλ〉ΨλΦλ− 〈EL〉ΨΦ

Second ingredient correlating the Overamped Langevinprocess

R(t + dt) = R(t) + bdt + dW

Rλ(t + dt) = Rλ(t) + bλdt + dWλ

Wiener processes related by a unitary transform

dWλ = UλdW (3)

Plan


Variational energy



Observable in VMC


Forward walking








Difference of energies : average on the process [R,Rλ].

Eλ − E0 =⟨

ELλ(Rλ)e−R

t

0dsELλ(Rλ) − EL(R)e−

R

t

0dsEL(R)

⟩

[R,Rλ]

Why a hope of improvement over usual forward walking ?

∫ t

0dsO [R(s)] −→

∫ t

0dsO [R(s)]

O =dELλ(Rλ)

dλ= λOZV (R)+ ~T .~∇EL(R)

~T ≡ Rλ −R

λ

ZV principle on O (choice of ψ) ! !Small fluctuations if 〈~T 2〉 finite !

Plan


Variational energy



Observable in VMC


Forward walking








Why can we hope 〈T 2〉 finite ?

Analysis in 1D U = Id

dT

dt=

∂b

∂xT Growth rate term (4)

+ b Deviation term

If b turned off at time t0

The accumulated deviation T (t0) will evolve as

T (t) = e

R

t

t0

∂b

∂x T (t0)→ 0

Since 〈 ∂b∂x〉 < 0 (the system lies in a finite region).

Plan


Variational energy



Observable in VMC


Forward walking








Illustration in many dimensions

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

100

0 100 200 300 400 500 600 700

(R2-

R1)

2 (t)

Simulation time (au)

Convergence of the joint process, molecule Li8

Fig.: Quadratic deviation of two correlated simulations on the Li8molecule

Plan


Variational energy



Observable in VMC


Forward walking








Stability for the H2 molecule

1e-11

1e-10

1e-09

0 50 100 150 200 250

<(R

2-R

1)2

(t)>


H2 molecule at equilibrium Delta=1e-5

Fig.: Quadratic distance between the two different processes

Plan


Variational energy



Observable in VMC


Forward walking








DMC energy derivative in H2

Crude Ψ : Unoptimized CSF × Minimal jastrow (e-e cusp)

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 2 4 6 8 10 12 14 16-0.1

-0.05

0

0.05

0.1

0.15

Ene

rgy

(au)

Ene

rgy

deriv

ativ

e (a

u)

Projecton time (au)

PDMC convergence for H2 (R=1.6 au, ∆R=1e-4)

E(t)-Eexact

dE/dR(t)-dEexact/dR

Plan


Variational energy



Observable in VMC


Forward walking








-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

[dE

/dR

](t)

-[dE

/dR

] exa

ct

d(H-H) (au)

PDMC convergences for H2

t=0

t=0.05

t=2.5

t=5

t=1

t=2

t=4

t=8

Plan


Variational energy



Observable in VMC


Forward walking








Finite time step error

-1.0205

-1.02

-1.0195

-1.019

-1.0185

-1.018

-1.0175

-1.017

-1.0165

-1.016

0 0.005 0.01 0.015 0.02 0.025 0.03

E (

au)

Time step (au)

Time step error on the energy

Exact energy (Kolos)

VMC (Projection time t=0)

DMC (t=10)

-0.79

-0.785

-0.78

-0.775

-0.77

-0.765

0 0.005 0.01 0.015 0.02 0.025 0.03

∆E

/∆R

(au),

∆R

=1e-5

Time step (au)

Time step error on the energy derivative

Exact derivative (Kolos)

VMC (Projection time t=0)

DMC (t=10)

Fig.: Finite step error for the energy its derivative. H2 molecule,R = 0.8. Ψ = simple CSF × Full optimized jastrow

Plan


Variational energy



Observable in VMC


Forward walking








Calculation on Li2

2 VMC with same pseudo random numbers (U = Id)

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 0

20

40

60

80

100

120

140

< d

EL/

dR >

(au

)

<T

2 > (

au)


Naive calculation on Li2 (R=5.051, ∆R=1e-3), time step=0.05 au

Li2 molecule: R=5.051, λ=∆R = 1e-3

< dEL/dR >

<T2>

Plan


Variational energy



Observable in VMC


Forward walking








Origin of the instabilities for Li2 ?

Finite time is a destabilization factor

Proposition

Pacc(R,Rλ) = min[Pacc(R),Pacc(Rλ)]

Adaptative time step τ (smaller time steps in regions withlarge local kinetic energy)

Note : Finite time step error O(〈τ〉).

Plan


Variational energy



Observable in VMC


Forward walking








Effect on 〈T 2〉

0

20

40

60

80

100

120

140

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

< T

2 > (

au)



Fixed time step, unmodified detailed balance

Adaptative time step, modified detailed balance

Plan


Variational energy



Observable in VMC


Forward walking








Effect on dEL

dR

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

< d

EL/

dR >

(au

)



Fixed time step, unmodified detailed balance

Adaptative time step, modified detailed balance

σ(E ) = 0.16e−02

σ(∆E∆R

) = 0.43e−03 => σ(∆E ) = 0.43e−06 << σ(E ) ! !

Plan


Variational energy



Observable in VMC


Forward walking








Application : Force calculation in thedissociation of Li2

0.02

0.

-0.02

87654

For

ce (

a.u.

)

R (a.u.)

Li2

Exact

Force RHF

FN derivative

Plan


Variational energy



Observable in VMC


Forward walking








Impact of the unitary transform

0.01

0.1

1

10

100

1000

0 100 200 300 400 500 600 700 800

(R2-

R1)

2 (t)


<(R2-R1)2>, U=Id / U = Pe

F2H2 molecule, ∆ dFH-FH=0.1

U=PeU=Id

Fig.: F2H2, 2 geometries ∆dFH−FH = 0.1, U = Id / U = Pe

permutation in the space of electrons (the closest same spin electronshave the same random numbers).

Plan


Variational energy



Observable in VMC


Forward walking








0

0.05

0.1

0.15

0.2

0.25

0.3

100 200 300 400 500 600

Sta

tistic

al e

rror


Error on the energy difference U=Id / U = Pe

F2H2 molecule, ∆ dFH-FH=0.1

sqrt(2)σ(E1)

σ(E2-E1), U=Id

σ(E2-E1), U=Pe

Fig.: F2H2, 2 geometries ∆dFH−FH = 0.1, U = Id / U = Pe

permutation in the space of electrons (the closest same spin electronshave the same random numbers).

Plan


Variational energy



Observable in VMC


Forward walking








Stable case

1e-11

1e-10

1e-09

0 50 100 150 200 250

<(R

2-R

1)2

(t)>


H2 molecule at equilibrium Delta=1e-5

1e-25

1e-20

1e-15

1e-10

1e-05

1

0 50 100 150 200 250

<(R

2-R

1)2

(t)>


H2 molecule at equilibrium Delta=0

Fig.: Quadratic distance between the processes / sensitivity of theunperturbed process to initial conditions (H2 molecule).

Plan


Variational energy



Observable in VMC


Forward walking








Unstable case

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

50 100 150 200 250 300 350 400

<(R

2-R

1)2

(t)>


CO molecule at equilibrium Delta=1e-5

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

50 100 150 200 250 300 350 400

<(R

2-R

1)2

(t)>


CO molecule at equilibrium Delta=0

Fig.: Quadratic distance between the two different processes /sensitivity of the unperturbed process to initial conditions (COmolecule.

Plan


Variational energy



Observable in VMC


Forward walking








Some comparisons DMC energies differences correlated /independent

Calculation of ∆E = E (dA−B + ∆R)− E (dA−B)

Molecule ∆R (B) Geometry gain on σ(∆) cpu gain

Li8 1e-5 eq 7e4 0.5e10Li4 1e-5 eq 1e5 1e10Li2 1e-5 5.051 (eq) 4e5 16e10Li2 1e-5 3.0 1.7e5 2.9e10Li2 0.1 3 42 1600Li2 0.1 1 3.6 13CO 0.1 2.175 B (eq) 3.9 15CO 0.1 20 15 225Li4 0.1 eq 14 196F2H2 0.1 eq 10.3 106Li4 1 eq 5.6 31Li2 1 eq 5.25 26

Plan


Variational energy



Observable in VMC


Forward walking








Conclusion and perspectives

Small differences, derivatives of VMC energies

Basically a solved problem, though a lot of work has yet to bedone (improving estimators).

Generalized correlated sampling method for DMC smalldifferences and derivatives

Different guiding functions for the two systems.

Pseudo-random numbers (Wiener process) related by aunitary transform.

Finite time step stabilization (joint acceptation probability,adaptative time step).

Plan


Variational energy



Observable in VMC


Forward walking








Main results

Non trivial DMC energy derivatives obtained.

DMC derivatives ← chaotic properties of the dynamics.

Substantial gain for small but finite differences (∼ 0.1)even for chaotic systems.

Work still to be done to have 〈T 2〉 always finite !

Open questions

Better insight needed into the chaotic properties of theOverdamped Langevin process.

Stabilization techniques to be searched for.

the generalized correlated sampling approach: toward an...

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