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

The generalized correlated sampling approach 9 avril 2008 1 de 44 Plan Introduction The generalized correlated sampling approach: toward an exact calculation of energy derivatives in Diffusion Monte Carlo Roland Assaraf a ,Alexander Kollias b and Michel Caffarel c 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

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

The generalized correlated sampling approach 9 avril 2008 3 de 44 Plan Quantities of physical interest ? Introduction 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)

The generalized correlated sampling approach 9 avril 2008 4 de 44 Plan Paradigm : Calculation of an observable O Introduction d λ ≃ E λ − E 0 O = dE λ H λ = H + λ O = > ¯ (1) λ Direct calculation E λ , E 0 computed independently. δ ( E λ − E 0 ) ∼ δ E 0 − → ∞ λ λ λ → 0

The generalized correlated sampling approach 9 avril 2008 5 de 44 Plan System size dependency in a direct calculation Introduction Example : a one particle gap ∆ = E ( N + 1) − E ( N ) δ E ∼ 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

The generalized correlated sampling approach 9 avril 2008 6 de 44 Plan Exploiting accurate QMC total energies to obtain accurate small Introduction 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 E λ − E 0 ∼ λ = ⇒ δ ( E λ − E 0 ) ∼ λ ⇐ ⇒ Finite statistical error on energy derivatives.

The generalized correlated sampling approach 9 avril 2008 6 de 44 Plan Exploiting accurate QMC total energies to obtain accurate small Introduction 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 E λ − E 0 ∼ λ = ⇒ δ ( E λ − E 0 ) ∼ λ ⇐ ⇒ Finite statistical error on energy derivatives.

The generalized correlated sampling approach 9 avril 2008 6 de 44 Plan Exploiting accurate QMC total energies to obtain accurate small Introduction 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 E λ − E 0 ∼ λ = ⇒ δ ( E λ − E 0 ) ∼ λ ⇐ ⇒ Finite statistical error on energy derivatives.

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

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

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

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

The generalized correlated sampling approach 9 avril 2008 8 de 44 Plan Variational energy I. Energy calculation Variational energy E V ≡ � Ψ | ˆ Diffusion Monte Carlo H | Ψ � Small energy differences in VMC Average on a probability distribution Observable in VMC Generalization to DMC � d R Ψ 2 ( R ) H Ψ Forward walking � Ψ | ˆ H | Ψ � = Ψ ( R ) Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other = � H Ψ Ψ ( R ) � Ψ 2 = � E L ( R ) � Ψ 2 molecules Criteria of stability Conclusion and perspectives N E v = 1 � E L ( R k ) N k =1

The generalized correlated sampling approach 9 avril 2008 8 de 44 Plan Variational energy I. Energy calculation Variational energy E V ≡ � Ψ | ˆ Diffusion Monte Carlo H | Ψ � Small energy differences in VMC Average on a probability distribution Observable in VMC Generalization to DMC � d R Ψ 2 ( R ) H Ψ Forward walking � Ψ | ˆ H | Ψ � = Ψ ( R ) Generalized correlated sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other = � H Ψ Ψ ( R ) � Ψ 2 = � E L ( R ) � Ψ 2 molecules Criteria of stability Conclusion and perspectives N E v = 1 � E L ( R k ) N k =1

The generalized correlated sampling approach 9 avril 2008 9 de 44 Plan Sampling of Ψ 2 I. Energy calculation Variational energy Diffusion Monte Carlo Dynamic over the configurations Small energy differences in VMC Observable in VMC R ( t + dt ) = R ( t ) + b dt + d W (2) Generalization to DMC Forward walking Generalized correlated sampling Calculation of the DMC energy b ( t ) ≡ ∇ Ψ Ψ (drift) derivative in H2 and Li2 Preliminary results on other molecules d W gaussian random numbers (diffusion). √ Criteria of stability � dW i dW j � = dt δ ij Conclusion and perspectives A trajectory R ( t ) < = > Sample of Ψ 2 .

The generalized correlated sampling approach 9 avril 2008 10 de 44 Plan Why VMC can be a very accurate method ? I. Energy calculation Variational energy Diffusion Monte Carlo Zero-variance - zero-bias property (ZVZB) Small energy differences in VMC Observable in VMC Generalization to DMC | Ψ − Φ | 2 φ Exact groundstate Bias E V − E 0 Forward walking Generalized correlated σ 2 ( E L ) | Ψ − Φ | 2 Variance sampling Calculation of the DMC energy derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability Important since Accuracy in QMC < = > E V − E 0 systematic error Conclusion and perspectives + σ ( E L ) √ statistical error. N

The generalized correlated sampling approach 9 avril 2008 11 de 44 Plan Diffusion Monte Carlo (DMC) I. Energy calculation Variational energy Sampling the exact groundstate Diffusion Monte Carlo Small energy differences in VMC e − tH | Ψ � = Φ (3) Observable in VMC Generalization to DMC Trotter formula : e − tH = e − δ tH e − δ tH . . . e − δ tH Forward walking Generalized correlated sampling Calculation of the DMC energy � R ′ | e − δ tH | R � = P ( R → R ′ ) W ( R ) derivative in H2 and Li2 Preliminary results on other molecules Criteria of stability Overdamped Langevin Weight Conclusion and perspectives In practice for fermions, Fixed node approximation : H − → H FN = ⇒ Φ − → Φ FN (variational solution in the space of functions having the same nodes as Ψ ).