The generalized correlated sampling approach: toward an exact - - PowerPoint PPT Presentation

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 Th´

eorique, CNRS-UMR 7616, Universit´ e Pierre et Marie Curie Paris VI, Case 137, 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 Universit´

e Paul Sabatier, 118 route de Narbonne 31062 Toulouse Cedex, France

Plan Introduction The generalized correlated sampling approach 9 avril 2008 2 de 44

General perspective

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

In principle adapted to the many body problem (weak limitation in system sizes) In Practice

Reference methods for groundstate energies of large systems. Less successful for other quantities.

Plan Introduction The generalized correlated sampling approach 9 avril 2008 3 de 44

Quantities of physical interest ?

Most of them can be obtained from total energies

Binding 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 . . .

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

Plan Introduction The generalized correlated sampling approach 9 avril 2008 4 de 44

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 The generalized correlated sampling approach 9 avril 2008 5 de 44

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 The generalized correlated sampling approach 9 avril 2008 6 de 44

Exploiting accurate QMC total energies to obtain accurate small differences 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 The generalized correlated sampling approach 9 avril 2008 6 de 44

Exploiting accurate QMC total energies to obtain accurate small differences 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 The generalized correlated sampling approach 9 avril 2008 6 de 44

Exploiting accurate QMC total energies to obtain accurate small differences 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 The generalized correlated sampling approach 9 avril 2008 7 de 44

Plan

1 QMC : A reference method for the total energy

Variational energy Diffusion Monte Carlo

2 Small energy differences in VMC

Observable in VMC

3 Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stabilty

4 Conclusion and perspectives

Plan Introduction The generalized correlated sampling approach 9 avril 2008 7 de 44

Plan

1 QMC : A reference method for the total energy

Variational energy Diffusion Monte Carlo

2 Small energy differences in VMC

Observable in VMC

3 Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stabilty

4 Conclusion and perspectives

Plan Introduction The generalized correlated sampling approach 9 avril 2008 7 de 44

Plan

1 QMC : A reference method for the total energy

Variational energy Diffusion Monte Carlo

2 Small energy differences in VMC

Observable in VMC

3 Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stabilty

4 Conclusion and perspectives

Plan Introduction The generalized correlated sampling approach 9 avril 2008 7 de 44

Plan

1 QMC : A reference method for the total energy

Variational energy Diffusion Monte Carlo

2 Small energy differences in VMC

Observable in VMC

3 Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stabilty

4 Conclusion and perspectives

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 8 de 44

Variational energy

EV ≡ Ψ| ˆ H |Ψ

Average on a probability distribution

Ψ| ˆ H |Ψ =

• dRΨ2(R) H Ψ

Ψ (R) =H Ψ

Ψ (R)Ψ2 =EL(R)Ψ2

Ev = 1 N

N

• k=1

EL(Rk)

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 8 de 44

Variational energy

EV ≡ Ψ| ˆ H |Ψ

Average on a probability distribution

Ψ| ˆ H |Ψ =

• dRΨ2(R) H Ψ

Ψ (R) =H Ψ

Ψ (R)Ψ2 =EL(R)Ψ2

Ev = 1 N

N

• k=1

EL(Rk)

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 9 de 44

Sampling of Ψ2

Dynamic over the configurations

R(t + dt) = R(t) + bdt + dW (2) b(t) ≡ ∇Ψ

Ψ (drift)

dW gaussian random numbers (diffusion). dWidWj = √ dtδij A trajectory R(t) <=> Sample of Ψ2.

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 10 de 44

Why VMC can be a very accurate method ?

Zero-variance - zero-bias property (ZVZB)

Bias EV − E0 |Ψ − Φ|2 Variance σ2(EL) |Ψ − Φ|2 φ Exact groundstate Important since Accuracy in QMC <=> EV − E0 systematic error + σ(EL)

√ N

statistical error.

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 11 de 44

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 space

• f functions having the same nodes as Ψ).

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 12 de 44

Conclusion

We can sample something better than Ψ, ΦFN , even if its analytic 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 Energie (Hartree) 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 derivatives

• n the VMC curve, on the DMC curve ?

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 13 de 44

Observable and energy on a similar footing in VMC

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

Hλ = H + λO => OΦ2 = dEλ dλ ≃ dEV [Ψλ] dλ Ψλ = Ψ + λ˜ Ψ dEV [Ψλ] dλ = ˜ OΨ2 O = ⇒ ˜ O[Ψ, ˜ Ψ] = O + (H − EL)˜ Ψ Ψ + 2(EL − EV ) ˜ Ψ Ψ

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 13 de 44

Observable and energy on a similar footing in VMC

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

Hλ = H + λO => OΦ2 = dEλ dλ ≃ dEV [Ψλ] dλ Ψλ = Ψ + λ˜ Ψ dEV [Ψλ] dλ = ˜ OΨ2 O = ⇒ ˜ O[Ψ, ˜ Ψ] = O + (H − EL)˜ Ψ Ψ + 2(EL − EV ) ˜ Ψ Ψ

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 14 de 44

Energy/observable in VMC

Bias Variance Choice of EV [Ψ] δΨ2 Idem Ψ EV ′

λ = ˜

O[Ψ, ˜ Ψ]Ψ2 δΨ2 + δΨδΨ′ Idem (Ψ, ˜ Ψ) OΨ2 δΨ O(1) (Ψ, ˜ Ψ = 0) δΨ = Ψ − Φ δΨ′ = ˜ Ψ − dΦλ

dλ

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 15 de 44

Electronic density

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

ˆ 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

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 16 de 44

Simplest improved estimator

δ(ri − r) = −1

4π ∆ 1 |ri−r| + Green formula

˜ ρZV (r) =

• − 1

4π

N

• i=1

1 |ri − r| ∇2

i (Ψ2)

Ψ2

• Ψ2

.

Used by P. Langfelder 1 et 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

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 17 de 44

Illustration : Helium atom

0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2 2.5 3 Usual Estimator Simple Improved Estimator Best Improved Estimator Exact

2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 0.02 0.04 0.06

• 0.001
• 0.0005

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

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 18 de 44

Application : water dimer

Fig.: Isodensity surfaces of the water dime

Bare regularized estimator Improved estimator

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 19 de 44

Pair correlation function

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

i<j δ(rij − u)

Probability density to find a pair of electrons at distance u

0.05 0.1 0.15 0.2 0.25 0.3 0.5 1 1.5 2 I(u) (a.u.) u (a.u.) histogram estimator with HF wave function ZV1 estimator with HF wave function ZV1ZB1 estimator with HF wave function accurate intracule He atom

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 20 de 44

Impact of the optimisation of Ψ

Illustration : Intracule calculations (collaboration with J. Toulouse et C. Umrigar)

• 0.4
• 0.2

0.2 0.4 0.6 1 2 3 4 5 4 π u2 [ I (u) - IHF (u) ] (a.u.) u (a.u.) Jastrow × HF Jastrow × SD Jastrow × CAS(10,8) N2 molecule VMC with ZVZB2 estimator

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

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 21 de 44

In summary

Derivatives (or small differences) of VMC energies : on the same footing as total VMC energies Strategy : Using improved estimators ˜ OZVZB[Ψ, ˜ Ψ]. => control of the statistical error and the systematic error, at a reasonable computational cost. Current work : Improving estimators for forces (geometry optimization). Application to a variety of other properties (dipole moments...).

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 22 de 44

Generalization to DMC

• R. Assaraf, A. Kollias, M. Caffarel

The DMC energy Trial function Groundstate of H EDMC = Ψ | H | Φ = ELΨΦ ΨΦ = Ψe−tH | Ψ =

• e−

R t

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

• [R(s)]

(2)

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

Ψ

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 22 de 44

Generalization to DMC

• R. Assaraf, A. Kollias, M. Caffarel

The DMC energy Trial function Groundstate of H EDMC = Ψ | H | Φ = ELΨΦ ΨΦ = Ψe−tH | Ψ =

• e−

R t

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

• [R(s)]

(2)

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

Ψ

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 22 de 44

Generalization to DMC

• R. Assaraf, A. Kollias, M. Caffarel

The DMC energy Trial function Groundstate of H EDMC = Ψ | H | Φ = ELΨΦ ΨΦ = Ψe−tH | Ψ =

• e−

R t

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

• [R(s)]

(2)

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

Ψ

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 23 de 44

Forward walking

Hλ = H + λO => HλΦλ = EλΦλ Computation of dEλ dλ = dELλΦλΨ dλ

Forward walking

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

• −

t dsO[R(s)]

• e− R t

0 EL(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

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 24 de 44

Approximations

Example : Hybrid improved estimator, in a FN approximation

˜ Oh ≡ 2 ˜ ODMC − ˜ OVMC Systematic error : O(|Ψ − Φ|2) + O(|˜ Ψ − Φ1|.|ΦFN − Φ|) O(|ΦFN − Φ|) for Oh Variance O(|Ψ − Φ|2) O(1) forOh Application to forces (R. Assaraf, M. Caffarel, 2003)

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 25 de 44

Generalized correlated sampling

First ingredient : two different guiding functions

Ψλ = Ψ + λ˜ Ψ => bλ = b + λ˜ b Eλ − E0 = ELλΨλΦλ − ELΨΦ

Second ingredient correlating the Overamped Langevin process

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

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 26 de 44

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

Eλ − E0 =

• ELλ(Rλ)e−

R t

0 dsELλ(Rλ) − EL(R)e−

R t

0 dsEL(R)

[R,Rλ]

Why a hope of improvement over usual forward walking ?

t

0 dsO[R(s)] −

→ t

0 ds ˜

O[R(s)]

˜ O = dELλ(Rλ) dλ = λ ˜ OZV (R)+

• T

. ∇EL(R)

• T

≡ Rλ − R λ ZV principle on ˜ O (choice of ˜ ψ) ! ! Small fluctuations if T 2 finite !

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 27 de 44

Why can we hope T 2 finite ?

Analysis in 1D U = Id

dT dt = ∂b ∂x T 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

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 28 de 44

Illustration in many dimensions

1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1 100 100 200 300 400 500 600 700

(R2-R1)2(t) Simulation time (au)

Convergence of the joint process, molecule Li8

Fig.: Quadratic deviation of two correlated simulations on the Li8 molecule

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 29 de 44

Stability for the H2 molecule

1e-11 1e-10 1e-09 50 100 150 200 250

<(R2-R1)2(t)> Simulation time (au)

H2 molecule at equilibrium Delta=1e-5

Fig.: Quadratic distance between the two different processes

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 30 de 44

DMC energy derivative in H2

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

• 0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 2 4 6 8 10 12 14 16

• 0.1
• 0.05

0.05 0.1 0.15

Energy (au) Energy derivative (au) Projecton time (au)

PDMC convergence for H2 (R=1.6 au, ∆R=1e-4) E(t)-Eexact dE/dR(t)-dEexact/dR

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 31 de 44

• 0.02

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]exact 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

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 32 de 44

Finite time step error

• 1.0205
• 1.02
• 1.0195
• 1.019
• 1.0185
• 1.018
• 1.0175
• 1.017
• 1.0165
• 1.016

0.005 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

0.005 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

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 33 de 44

Calculation on Li2

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

• 0.02

0.02 0.04 0.06 0.08 0.1 0.12 0.14 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 20 40 60 80 100 120 140

< dEL/dR > (au) <T2> (au) Simulation time (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

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 34 de 44

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 with large local kinetic energy) Note : Finite time step error O(τ).

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 35 de 44

Effect on T 2

20 40 60 80 100 120 140 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

< T2 > (au) Simulation time (au) Li2 molecule: R=5.051, λ=∆R = 1e-3 Fixed time step, unmodified detailed balance Adaptative time step, modified detailed balance

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 36 de 44

Effect on dEL

dR

• 0.02
• 0.01

0.01 0.02 0.03 0.04 0.05 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

< dEL/dR > (au) Simulation time (au) Li2 molecule: R=5.051, λ=∆R = 1e-3 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

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 37 de 44

Application : Force calculation in the dissociation of Li2

0.02 0.

• 0.02

8 7 6 5 4

Force (a.u.)

R (a.u.)

Li2

Exact Force RHF FN derivative

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 38 de 44

Impact of the unitary transform

0.01 0.1 1 10 100 1000 100 200 300 400 500 600 700 800

(R2-R1)2(t) Simulation time (au)

<(R2-R1)2>, U=Id / U = Pe F2H2 molecule, ∆ dFH-FH=0.1 U=Pe U=Id

Fig.: F2H2, 2 geometries ∆dFH−FH = 0.1, U = Id / U = Pe permutation in the space of electrons (the closest same spin electrons have the same random numbers).

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 39 de 44

0.05 0.1 0.15 0.2 0.25 0.3 100 200 300 400 500 600

Statistical error Simulation time (au)

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 electrons have the same random numbers).

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 40 de 44

Stable case

1e-11 1e-10 1e-09 50 100 150 200 250

<(R2-R1)2(t)> Simulation time (au)

H2 molecule at equilibrium Delta=1e-5

1e-25 1e-20 1e-15 1e-10 1e-05 1 50 100 150 200 250

<(R2-R1)2(t)> Simulation time (au)

H2 molecule at equilibrium Delta=0

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

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 41 de 44

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

<(R2-R1)2(t)> Simulation time (au)

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

<(R2-R1)2(t)> Simulation time (au)

CO molecule at equilibrium Delta=0

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

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 42 de 44

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 0.5e10 Li4 1e-5 eq 1e5 1e10 Li2 1e-5 5.051 (eq) 4e5 16e10 Li2 1e-5 3.0 1.7e5 2.9e10 Li2 0.1 3 42 1600 Li2 0.1 1 3.6 13 CO 0.1 2.175 B (eq) 3.9 15 CO 0.1 20 15 225 Li4 0.1 eq 14 196 F2H2 0.1 eq 10.3 106 Li4 1 eq 5.6 31 Li2 1 eq 5.25 26

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 43 de 44

Conclusion and perspectives

Small differences, derivatives of VMC energies

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

Generalized correlated sampling method for DMC small differences and derivatives

Different guiding functions for the two systems. Pseudo-random numbers (Wiener process) related by a unitary transform. Finite time step stabilization (joint acceptation probability, adaptative time step).

Plan

• I. Energy calculation

Variational energy Diffusion Monte Carlo

Small energy differences in VMC

Observable in VMC

Generalization to DMC

Forward walking Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability

Conclusion and perspectives The generalized correlated sampling approach 9 avril 2008 44 de 44

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 the Overdamped Langevin process. Stabilization techniques to be searched for.