Quantum Monte Carlo wave functions and their optimization for quantum chemistry

Size: px

Start display at page:

Download "Quantum Monte Carlo wave functions and their optimization for quantum chemistry"

Francis Manning
5 years ago
Views:

1 Quantum Monte Carlo wave functions and their optimization for quantum chemistry Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France CEA Saclay, SPhN Orme des Merisiers April 2015

2 Outline 2/26 1 Quantum Monte Carlo (QMC) in a nutshell 2 Wave function optimization 3 Calculation of excited states 4 Symmetry breaking 5 New forms of wave functions

3 Outline 3/26 1 Quantum Monte Carlo (QMC) in a nutshell 2 Wave function optimization 3 Calculation of excited states 4 Symmetry breaking 5 New forms of wave functions

4 Variational Monte Carlo (VMC) 4/26 a method for calculating multidimensional integrals e.g., the energy Ψ Ĥ Ψ = ( ) H(R)Ψ(R) dr Ψ(R) 2 Ψ(R)

5 Variational Monte Carlo (VMC) 4/26 a method for calculating multidimensional integrals e.g., the energy Ψ Ĥ Ψ = ( ) H(R)Ψ(R) dr Ψ(R) 2 1 Ψ(R) M M k=1 H(R k )Ψ(R k ) Ψ(R k ) Metropolis sampling: M points R k

6 Variational Monte Carlo (VMC) 4/26 a method for calculating multidimensional integrals e.g., the energy Ψ Ĥ Ψ = ( ) H(R)Ψ(R) dr Ψ(R) 2 1 Ψ(R) M M k=1 H(R k )Ψ(R k ) Ψ(R k ) Metropolis sampling: M points R k Advantage: can use a flexible explicitly correlated Ψ(R) In practice, 2 types of error: unknown systematic error due to approximate wave function known statistical uncertainty (finite sampling) 1/ M

7 Diffusion Monte Carlo (DMC) 5/26 Basic idea: diffusion in imaginary time τ = i t Ψ exact (R) e τĥ (τ ) Ψ(R) sampling

8 Diffusion Monte Carlo (DMC) 5/26 Basic idea: diffusion in imaginary time τ = i t Ψ exact (R) e τĥ (τ ) Ψ(R) sampling but: sign problem: converges to the bosonic ground-state!

9 Diffusion Monte Carlo (DMC) 5/26 Basic idea: diffusion in imaginary time τ = i t Ψ exact (R) e τĥ (τ ) Ψ(R) sampling but: sign problem: converges to the bosonic ground-state! In practice: fixed-node (FN) approximation Ψ FN (R) e τĥ FN (τ ) Ψ(R) E DMC E fermionic gs diffusion with nodes of Ψ(R) fixed

10 Standard Jastrow-Slater wave functions 6/26 Ψ(R,p) = J(R,α) m c m Φ m (R) J(R,α) : Jastrow factor = exponential of a function depending explicitly on e-n and e-e distances = short-range weak/dynamic correlation m c mφ m (R) : linear combination of Slater determinants or CSFs of given spatial and spin symmetry = long-range strong/static correlation

11 Standard Jastrow-Slater wave functions 6/26 Ψ(R,p) = J(R,α) m c m Φ m (R) J(R,α) : Jastrow factor = exponential of a function depending explicitly on e-n and e-e distances = short-range weak/dynamic correlation m c mφ m (R) : linear combination of Slater determinants or CSFs of given spatial and spin symmetry = long-range strong/static correlation Slater determinants made of orbitals expanded on a Slater basis: φ k (r) = µ λ kµ χ µ (r) χ(r) = N(ζ)r n 1 e ζr S l,m (θ,φ)

12 Standard Jastrow-Slater wave functions Ψ(R,p) = J(R,α) m c m Φ m (R) J(R,α) : Jastrow factor = exponential of a function depending explicitly on e-n and e-e distances = short-range weak/dynamic correlation m c mφ m (R) : linear combination of Slater determinants or CSFs of given spatial and spin symmetry = long-range strong/static correlation Slater determinants made of orbitals expanded on a Slater basis: φ k (r) = µ λ kµ χ µ (r) χ(r) = N(ζ)r n 1 e ζr S l,m (θ,φ) Parameters to optimize p = {α,c,λ,ζ}: Jastrow parameters α, CSF coefficients c, orbital coefficients λ and basis exponents ζ 6/26

13 Outline 7/26 1 Quantum Monte Carlo (QMC) in a nutshell 2 Wave function optimization 3 Calculation of excited states 4 Symmetry breaking 5 New forms of wave functions

14 Linear optimization method Toulouse, Umrigar, JCP, 2007 Umrigar, Toulouse, Filippi, Sorella, Hennig, PRL, 2007 Toulouse, Umrigar, JCP, /26 Wave function is linearly expanded in p = p p 0 : Ψ (1) (p) = Ψ 0 + j p j Ψ j where Ψ 0 = Ψ(p 0 ) and Ψ j = Ψ(p) p j p=p 0

15 Linear optimization method Toulouse, Umrigar, JCP, 2007 Umrigar, Toulouse, Filippi, Sorella, Hennig, PRL, 2007 Toulouse, Umrigar, JCP, /26 Wave function is linearly expanded in p = p p 0 : Ψ (1) (p) = Ψ 0 + j p j Ψ j where Ψ 0 = Ψ(p 0 ) and Ψ j = Ψ(p) p j p=p 0 Minimization of energy min p Ψ (1) Ĥ Ψ (1) Ψ (1) Ψ (1) = where H ij = Ψ i Ĥ Ψ j and S ij = Ψ i Ψ j generalized eigenvalue equation = H p = E S p

16 Linear optimization method Toulouse, Umrigar, JCP, 2007 Umrigar, Toulouse, Filippi, Sorella, Hennig, PRL, 2007 Toulouse, Umrigar, JCP, /26 Wave function is linearly expanded in p = p p 0 : Ψ (1) (p) = Ψ 0 + j p j Ψ j where Ψ 0 = Ψ(p 0 ) and Ψ j = Ψ(p) p j p=p 0 Minimization of energy min p Ψ (1) Ĥ Ψ (1) Ψ (1) Ψ (1) = where H ij = Ψ i Ĥ Ψ j and S ij = Ψ i Ψ j Update of parameters: p 0 p 0 + p generalized eigenvalue equation = H p = E S p

17 Linear optimization method in VMC Estimators on a finite sample: H ij = 1 M Ψ i (R k ) H(R k )Ψ j (R k ), S ij = 1 M Ψ 0 (R k ) Ψ 0 (R k ) M k=1 non-symmetric! M Ψ i (R k ) Ψ j (R k ) Ψ 0 (R k ) Ψ 0 (R k ) k=1 Toulouse, Umrigar, JCP, 2007 Umrigar, Toulouse, Filippi, Sorella, Hennig, PRL, 2007 Toulouse, Umrigar, JCP, /26

18 Linear optimization method in VMC Estimators on a finite sample: H ij = 1 M Ψ i (R k ) H(R k )Ψ j (R k ), S ij = 1 M Ψ 0 (R k ) Ψ 0 (R k ) M k=1 non-symmetric! M Ψ i (R k ) Ψ j (R k ) Ψ 0 (R k ) Ψ 0 (R k ) k=1 = Zero-variance principle: If there is some p so that Ψ 0 + j p j Ψ j = Ψ exact p is found from H p = E S p with zero variance In practice, this non-symmetric estimator greatly reduces the fluctuations on p then Toulouse, Umrigar, JCP, 2007 Umrigar, Toulouse, Filippi, Sorella, Hennig, PRL, 2007 Toulouse, Umrigar, JCP, /26

19 Simultaneous optimization of all parameters Optimization of 149 parameters = 24 (Jastrow) + 49 (CSF) + 64 (orbitals) + 12 (exponents) for C 2 molecule : Energy (Hartree) Energy (Hartree) Iterations Iterations = Energy converges within error bars in a few iterations Toulouse, Umrigar, JCP, /26

20 Systematic improvement in QMC For C 2 molecule: total energies for a series of fully optimized Jastrow-Slater wave functions: Energy (Hartree) Exact VMC J*SD J*CAS(8,5) J*CAS(8,7) J*CAS(8,8)J*RAS(8,26) Wave function = Systematic improvement in VMC Toulouse, Umrigar, JCP, /26

21 Systematic improvement in QMC For C 2 molecule: total energies for a series of fully optimized Jastrow-Slater wave functions: Energy (Hartree) Exact VMC DMC J*SD J*CAS(8,5) J*CAS(8,7) J*CAS(8,8)J*RAS(8,26) Wave function = Systematic improvement in VMC and DMC Toulouse, Umrigar, JCP, /26

22 Dissociation energies of diatomic molecules Single-determinant (SD) and multideterminant (full valence CAS) wave functions: Error on dissociation energy (ev) VMC J SD Li 2 Be 2 B 2 C 2 N 2 O 2 F 2 Ne 2 Molecules Toulouse, Umrigar, JCP, /26

23 Dissociation energies of diatomic molecules Single-determinant (SD) and multideterminant (full valence CAS) wave functions: Error on dissociation energy (ev) DMC J SD VMC J SD Li 2 Be 2 B 2 C 2 N 2 O 2 F 2 Ne 2 Molecules Toulouse, Umrigar, JCP, /26

24 Dissociation energies of diatomic molecules Single-determinant (SD) and multideterminant (full valence CAS) wave functions: Error on dissociation energy (ev) VMC J CAS DMC J SD VMC J SD Li 2 Be 2 B 2 C 2 N 2 O 2 F 2 Ne 2 Molecules Toulouse, Umrigar, JCP, /26

25 Dissociation energies of diatomic molecules Single-determinant (SD) and multideterminant (full valence CAS) wave functions: Error on dissociation energy (ev) DMC J CAS VMC J CAS DMC J SD VMC J SD Li 2 Be 2 B 2 C 2 N 2 O 2 F 2 Ne 2 Molecules = Near chemical accuracy in DMC with Jastrow CAS Toulouse, Umrigar, JCP, /26

26 Outline 13/26 1 Quantum Monte Carlo (QMC) in a nutshell 2 Wave function optimization 3 Calculation of excited states 4 Symmetry breaking 5 New forms of wave functions

27 Calculation of excited states in QMC 14/26 VMC excited-state calculation with wave-function optimization For lowest energy state of a given symmetry: same as ground state calculation

28 Calculation of excited states in QMC 14/26 VMC excited-state calculation with wave-function optimization For lowest energy state of a given symmetry: same as ground state calculation For a state that is not the lowest one in a given symmetry, two strategies: - state-average approach: minimization of a weighted average of the energies of n states (C. Filippi et al.) - state-specific approach: minimization of the energy of the targeted state by selecting the n th eigenvector p in the linear optimization method (Zimmerman, Toulouse, Zhang, Musgrave, Umrigar, JCP, 2009)

29 Calculation of excited states in QMC 14/26 VMC excited-state calculation with wave-function optimization For lowest energy state of a given symmetry: same as ground state calculation For a state that is not the lowest one in a given symmetry, two strategies: - state-average approach: minimization of a weighted average of the energies of n states (C. Filippi et al.) - state-specific approach: minimization of the energy of the targeted state by selecting the n th eigenvector p in the linear optimization method (Zimmerman, Toulouse, Zhang, Musgrave, Umrigar, JCP, 2009) DMC excited-state calculation with VMC-optimized wave function Fixed-node approximation prevents collapse on the ground state

30 Example of methylene CH 2 Adiabatic excitation energies (ev) with full-valence CAS wave functions: 2.550(8) 1 A 1 VMC DMC 2.524(4) CR-CC Exp. 1 B (8) 1.416(4) A (8) 0.406(4) B Zimmerman, Toulouse, Zhang, Musgrave, Umrigar, JCP, 2009 Gour, Piecuch,Wloch, MP, /26

31 Outline 16/26 1 Quantum Monte Carlo (QMC) in a nutshell 2 Wave function optimization 3 Calculation of excited states 4 Symmetry breaking 5 New forms of wave functions

32 Ground-state potential energy curve of C 2 molecule ( 1 Σ + g) Jastrow single determinant wave function VMC J SD : Energy (hartree) Accurate size-consistency error Internuclear distance (bohr) Toulouse, Umrigar, JCP, /26

33 Ground-state potential energy curve of C 2 molecule ( 1 Σ + g) Jastrow single determinant wave function VMC J SD : Energy (hartree) DMC J SD Accurate size-consistency error Internuclear distance (bohr) = Single-determinant DMC is size consistent but with broken spin symmetry at dissociation, Ψ FN Ŝ 2 Ψ FN = 2 Toulouse, Umrigar, JCP, /26

34 Ground-state potential energy curve of C 2 molecule ( 1 Σ + g) Jastrow multideterminant wave function: Energy (hartree) VMC J CAS(8,8) DMC J CAS(8,8) Accurate Internuclear distance (bohr) = VMC and DMC are now size consistent without symmetry breaking Toulouse, Umrigar, JCP, /26

35 Spatial symmetry breaking in hydrogen rings 19/26 For large enough rings, three Hartree-Fock solutions: symmetry adapted (SA) symmetry broken atom-centered (SB-AC) symmetry broken bond-centered (SB-BC) Which Hartree-Fock wave function should we use in DMC? Reinhardt, Toulouse, Assaraf, Umrigar, Hoggan, ACS Proceedings, 2012

36 Spatial symmetry breaking in hydrogen rings DMC total energies with symmetry-adapted (SA) or symmetry-broken (SB) Hartree-Fock wave functions: Total energy / H 2 (mhartree) SB-BC SB-AC SA Number of H atoms = Symmetry-adapted wave function has better nodes Reinhardt, Toulouse, Assaraf, Umrigar, Hoggan, ACS Proceedings, /26

37 Outline 21/26 1 Quantum Monte Carlo (QMC) in a nutshell 2 Wave function optimization 3 Calculation of excited states 4 Symmetry breaking 5 New forms of wave functions

38 Jastrow-Valence-Bond wave functions Ψ J VB = Ĵ c m Φ VB,m m where Φ VB,m are VB structures: Φ VB,m = inactive p â p â p active pairs (ij) ( â i â j â i â j ) unpaired active with nonorthogonal active orbitals localized on single atom q â q vac = Compact wave functions from chemical picture Braïda, Toulouse, Caffarel, Umrigar, JCP, /26

39 Jastrow-Valence-Bond wave functions Ψ J VB = Ĵ c m Φ VB,m m where Φ VB,m are VB structures: Φ VB,m = inactive p â p â p active pairs (ij) ( â i â j â i â j ) unpaired active with nonorthogonal active orbitals localized on single atom Example: N 2 q â q vac = Compact wave functions from chemical picture Braïda, Toulouse, Caffarel, Umrigar, JCP, /26

40 Jastrow-Valence-Bond wave functions Ψ J VB = Ĵ c m Φ VB,m m where Φ VB,m are VB structures: Φ VB,m = inactive p â p â p active pairs (ij) ( â i â j â i â j ) unpaired active with nonorthogonal active orbitals localized on single atom Example: N 2 q â q vac = Compact wave functions from chemical picture Braïda, Toulouse, Caffarel, Umrigar, JCP, /26

41 Jastrow-Valence-Bond wave functions Ψ J VB = Ĵ c m Φ VB,m m where Φ VB,m are VB structures: Φ VB,m = inactive p â p â p active pairs (ij) ( â i â j â i â j ) unpaired active with nonorthogonal active orbitals localized on single atom Example: N 2 q â q vac = Compact wave functions from chemical picture Braïda, Toulouse, Caffarel, Umrigar, JCP, /26

42 Jastrow-Valence-Bond wave functions Ψ J VB = Ĵ c m Φ VB,m m where Φ VB,m are VB structures: Φ VB,m = inactive p â p â p active pairs (ij) ( â i â j â i â j ) unpaired active with nonorthogonal active orbitals localized on single atom Example: N 2 q â q vac = Compact wave functions from chemical picture Braïda, Toulouse, Caffarel, Umrigar, JCP, /26

43 Jastrow-Valence-Bond wave functions Ψ J VB = Ĵ c m Φ VB,m m where Φ VB,m are VB structures: Φ VB,m = inactive p â p â p active pairs (ij) ( â i â j â i â j ) unpaired active with nonorthogonal active orbitals localized on single atom Example: N 2 q â q vac = Compact wave functions from chemical picture Braïda, Toulouse, Caffarel, Umrigar, JCP, /26

Jastrow-Valence-Bond wave functions Ψ J VB = Ĵ c m Φ VB,m m where Φ VB,m are VB structures: Φ VB,m = inactive p â p â p active pairs (ij) ( â i â j â i â j ) unpaired active with nonorthogonal active

44 Jastrow-Valence-Bond wave functions Ψ J VB = Ĵ c m Φ VB,m m where Φ VB,m are VB structures: Φ VB,m = inactive p â p â p active pairs (ij) ( â i â j â i â j ) unpaired active with nonorthogonal active orbitals localized on single atom Example: N 2 q â q vac = Compact wave functions from chemical picture VB: 27 structures, 16 determinants CAS: 112 determinants Braïda, Toulouse, Caffarel, Umrigar, JCP, /26

45 Dissociation energies of diatomic molecules with J VB Comparison with single-determinant (SD) and full-valence CAS wave functions: Error on dissociation energy (ev) DMC J*VB 1 Lewis DMC J*CAS DMC J*SD C 2 N 2 O 2 F 2 Molecule = Compromise between compactness and accuracy Braïda, Toulouse, Caffarel, Umrigar, JCP, /26

46 Dissociation energies of diatomic molecules with J VB Comparison with single-determinant (SD) and full-valence CAS wave functions: Error on dissociation energy (ev) or 3 Lewis DMC J*VB 1 Lewis DMC J*CAS DMC J*SD C 2 N 2 O 2 F 2 Molecule = Compromise between compactness and accuracy Braïda, Toulouse, Caffarel, Umrigar, JCP, /26

47 New basis function form for nondivergent pseudopotentials Gauss-Slater basis function χ nlm (r,ζ) = N n (ζ) r n 1 2 e (ζr) 1+ζr S l,m (θ,φ) For r 1, its reduces to a Gaussian function χ nlm (r,ζ) N n (ζ) r n 1 e (ζr)2 S l,m (θ,φ) which is appropriate since no e-n cusp with nondivergent pseudopotentials For r 1, its reduces to a Slater function χ nlm (r,ζ) N n (ζ) r n 1 e ζr S l,m (θ,φ) which is the correct asymptotic behavior in finite systems Petruzielo, Toulouse, Umrigar, JCP, 2011 Petruzielo, Toulouse, Umrigar, JCP, /26

48 Atomization energies of 55 molecules (G2 set) 25/26 DMC calculations with reoptimized truncated multideterminant CAS wave functions with pseudopotentials and Gauss-Slater basis: = Mean absolute deviation in DMC = 1.2 kcal/mol Petruzielo, Toulouse, Umrigar, JCP, 2012

49 Summary and acknowledgements 26/26 Summary QMC methods can handle weak and strong correlation efficient wave function optimization method by minimization of VMC energy near chemical accuracy on energy differences calculation of excited states possible must be careful with symmetry breaking in DMC exploration of new forms of compact wave functions Acknowledgements R. Assaraf (Paris), B. Braïda (Paris), M. Caffarel (France), C. Filippi (Netherlands), R. Hennig (USA), P. Hoggan (France), C. Musgrave (USA), B. Mussard (Paris), F. Petruzielo (USA), P. Reinhardt (Paris), S. Sorella (Italy), C. Umrigar (USA), P. Zimmerman (USA), Z. Zhang (USA)

Recent advances in quantum Monte Carlo for quantum chemistry: optimization of wave functions and calculation of observables

Recent advances in quantum Monte Carlo for quantum chemistry: optimization of wave functions and calculation of observables Julien Toulouse 1, Cyrus J. Umrigar 2, Roland Assaraf 1 1 Laboratoire de Chimie