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

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1 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 Théorique, Université Pierre et Marie Curie - CNRS, Paris, France. 2 Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York, USA. julien.toulouse@upmc.fr Web page: March 2009

2 1 Optimization of wave functions 2 Calculation of observables

3 1 Optimization of wave functions 2 Calculation of observables

4 Trial wave function Jastrow-Slater wave function Ψ(p) = Ĵ(α) N CSF i=1 c i C i Ĵ(α) = Jastrow factor (with e-e, e-n, e-e-n terms) C i = Configuration state function (CSF) = linear combination of Slater determinants of given symmetry.

5 Trial wave function Jastrow-Slater wave function Ψ(p) = Ĵ(α) N CSF i=1 c i C i Ĵ(α) = Jastrow factor (with e-e, e-n, e-e-n terms) C i = Configuration state function (CSF) = linear combination of Slater determinants of given symmetry. The Slater determinants are made of orbitals expanded on a Slater basis: φ k (r) = N basis µ=1 λ kµ χ µ (r) χ(r) = N(ζ)r n 1 e ζr S l,m (θ, φ)

6 Trial wave function Jastrow-Slater wave function Ψ(p) = Ĵ(α) N CSF i=1 c i C i Ĵ(α) = Jastrow factor (with e-e, e-n, e-e-n terms) C i = Configuration state function (CSF) = linear combination of Slater determinants of given symmetry. The Slater determinants are made of orbitals expanded on a Slater basis: φ k (r) = N basis µ=1 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

7 Wave function optimization: why and how? Important for both VMC and DMC in order to reduce the systematic error reduce the statistical uncertainty

8 Wave function optimization: why and how? Important for both VMC and DMC in order to reduce the systematic error reduce the statistical uncertainty How to optimize? Until recently: minimization of the variance of the energy OK for the few Jastrow parameters but does not work well for the many CSF and orbital parameters Since recently: minimization of the energy (+ possibly a small fraction of variance) works well for all the parameters the energy is a better criterion

9 Optimization method: principle Expansion of the wave function around p 0 to linear order 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(p0 )) p j.

10 Optimization method: principle Expansion of the wave function around p 0 to linear order 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(p0 )) p j. Normalization of wave function chosen so that the derivatives Ψ j are orthogonal to Ψ 0.

11 Optimization method: principle Expansion of the wave function around p 0 to linear order 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(p0 )) p j. Normalization of wave function chosen so that the derivatives Ψ j are orthogonal to Ψ 0. Minimization of the energy = generalized eigenvalue equation: ( E0 g T )( ) ( )( ) / T 1 = E g/2 H p lin 0 S p where E 0 = Ψ 0 Ĥ Ψ 0, g i = E(p0 ) p i, H ij = Ψ i Ĥ Ψ j, S ij = Ψ i Ψ j.

12 Optimization method: principle Expansion of the wave function around p 0 to linear order 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(p0 )) p j. Normalization of wave function chosen so that the derivatives Ψ j are orthogonal to Ψ 0. Minimization of the energy = generalized eigenvalue equation: ( E0 g T )( ) ( )( ) / T 1 = E g/2 H p lin 0 S p where E 0 = Ψ 0 Ĥ Ψ 0, g i = E(p0 ) p i, H ij = Ψ i Ĥ Ψ j, S ij = Ψ i Ψ j. Update of the parameters: p 0 p 0 + p.

13 Optimization method: robustness The linear method is equivalent to a stabilized Newton method: ( E0 g T )( ) ( )( ) / T 1 = E g/2 H p lin 0 S p { (h + 2 E S) p = g 2 E = g T p where h = 2(H E 0 S) is an approximate Hessian, and E = E 0 E lin > 0 is the energy stabilization. = more robust than Newton method

14 Optimization method: robustness The linear method is equivalent to a stabilized Newton method: ( E0 g T )( ) ( )( ) / T 1 = E g/2 H p lin 0 S p { (h + 2 E S) p = g 2 E = g T p where h = 2(H E 0 S) is an approximate Hessian, and E = E 0 E lin > 0 is the energy stabilization. = more robust than Newton method In quantum chemistry, it is known as super-ci method or augmented Hessian method.

15 Optimization method: robustness The linear method is equivalent to a stabilized Newton method: ( E0 g T )( ) ( )( ) / T 1 = E g/2 H p lin 0 S p { (h + 2 E S) p = g 2 E = g T p where h = 2(H E 0 S) is an approximate Hessian, and E = E 0 E lin > 0 is the energy stabilization. = more robust than Newton method In quantum chemistry, it is known as super-ci method or augmented Hessian method. Additional stabilization: H ij H ij + a δ ij where a 0.

16 Optimization method: on a finite VMC sample The generalized eigenvalue equation is estimated as ( E0 gr T/2 )( ) ( ) ( T 1 = E g L /2 H p lin 0 S p with Ψi (R) g L,i /2 = Ψ 0 (R) Ψi (R) H ij = Ψ 0 (R) H(R)Ψ 0 (R) Ψ 0 (R) H(R)Ψ j (R) Ψ 0 (R) non-symmetric! Ψ 2 0 Ψ 2 0 Ψ0 (R) H(R)Ψ j (R) and g R,j /2 = Ψ 0 (R) Ψ 0 (R) Ψi (R) Ψ j (R) and S ij = Ψ 0 (R) Ψ 0 (R) ) Ψ 2 0 Ψ 2 0

17 Optimization method: on a finite VMC sample The generalized eigenvalue equation is estimated as ( E0 gr T/2 )( ) ( ) ( T 1 = E g L /2 H p lin 0 S p with Ψi (R) g L,i /2 = Ψ 0 (R) Ψi (R) H ij = Ψ 0 (R) H(R)Ψ 0 (R) Ψ 0 (R) H(R)Ψ j (R) Ψ 0 (R) non-symmetric! Ψ 2 0 Ψ 2 0 Ψ0 (R) H(R)Ψ j (R) and g R,j /2 = Ψ 0 (R) Ψ 0 (R) Ψi (R) Ψ j (R) and S ij = Ψ 0 (R) Ψ 0 (R) = Zero-variance principle of Nightingale et al. (PRL 2001): If there is some p so that Ψ 0 (R) + j p j Ψ j (R) = Ψ exact (R) then p is found with zero variance. In practice, these non-symmetric estimators reduce the fluctuations on p by 1 or 2 orders of magnitude. ) Ψ 2 0 Ψ 2 0

18 Optimization method: mixing a fraction of variance How to minimize the energy variance with the linear method? { V = min V 0 + gv T p + 1 } p 2 pt h V p

19 Optimization method: mixing a fraction of variance How to minimize the energy variance with the linear method? { V = min V 0 + gv T p + 1 } p 2 pt h V p V = min p ( 1 p T ) ( V 0 gv T/2 )( 1 g V /2 h V /2 + V 0 S p ( 1 p T ) ( 1 0 T )( ) 1 0 S p )

20 Optimization method: mixing a fraction of variance How to minimize the energy variance with the linear method? { V = min V 0 + gv T p + 1 } p 2 pt h V p V = min p ( 1 p T ) ( V 0 gv T/2 )( 1 g V /2 h V /2 + V 0 S p ( 1 p T ) ( 1 0 T )( ) 1 0 S p ) ( V0 g T V /2 g V /2 h V /2 + V 0 S ) ( 1 p ) ( 1 0 T = V 0 S )( 1 p )

21 Optimization method: mixing a fraction of variance How to minimize the energy variance with the linear method? { V = min V 0 + gv T p + 1 } p 2 pt h V p V = min p ( 1 p T ) ( V 0 gv T/2 )( 1 g V /2 h V /2 + V 0 S p ( 1 p T ) ( 1 0 T )( ) 1 0 S p ) ( V0 g T V /2 g V /2 h V /2 + V 0 S ) ( 1 p matrix to add to the energy matrix ) ( 1 0 T = V 0 S )( 1 p )

22 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 up to 1 mhartree in a few iterations

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

24 Systematic improvement in QMC For C 2 molecule: total energies for a series of fully optimized Jastrow-Slater wave functions: Energy (Hartree) VMC DMC CCSD(T)/cc-pVQZ Exact 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!

25 Potential energy curve of C 2 molecule ( 1 Σ + g ) Jastrow single determinant wave function VMC J SD : Energy (Hartree) Morse potential size-consistency error Interatomic distance (Bohr)

26 Potential energy curve of C 2 molecule ( 1 Σ + g ) Jastrow single determinant wave function VMC J SD : Energy (Hartree) DMC J SD Morse potential size-consistency error Interatomic distance (Bohr) = Single-determinant DMC is size-consistent but with broken spin symmetry at dissociation, Ψ DMC Ŝ 2 Ψ DMC = 2

27 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) Morse potential Interatomic distance (Bohr)

28 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) Morse potential Interatomic distance (Bohr) = DMC gives dissociation energy with chemical accuracy (1 kcal/mol 0.04 ev): D DMC = 6.482(3) vs D exact = 6.44(2) ev

29 Dissociation energies of diatomic molecules Jastrow multideterminant (full valence CAS) wave functions: Errors on dissociation energy (ev) DMC J CAS VMC J CAS MCSCF CAS Li 2 Be 2 B 2 C 2 N 2 O 2 F 2 Ne 2 Molecules = Near chemical accuracy in DMC

30 Example of application Binding energy of 2 NO 2 to a fragment of carbon nanotube: Estimates for the full nanotube (9,0): B3LYP calculations: no binding QMC calculations: weak binding ( 10 kcal/mol) Lawson, Bauschlicher, Toulouse, Filippi, Umrigar, Chem. Phys. Lett., 466, 170 (2008)

31 1 Optimization of wave functions 2 Calculation of observables

32 Calculation of an observable in VMC Energy Estimator: E L (R) = H(R)Ψ(R) Ψ(R) Systematic error: δe = O(δΨ 2 ) Variance: σ 2 (E L ) = O(δΨ 2 ) } Quadratic Zero-Variance Zero-Bias property

33 Calculation of an observable in VMC Energy Estimator: E L (R) = H(R)Ψ(R) Ψ(R) Systematic error: δe = O(δΨ 2 ) Variance: σ 2 (E L ) = O(δΨ 2 ) } Quadratic Zero-Variance Zero-Bias property Arbitrary observable Ô (which does not commute with Ĥ) Estimator: O L (R) = O(R)Ψ(R) Ψ(R) } Systematic error: δo = O(δΨ) Quadratic Zero-Variance Variance: σ 2 (O L ) = O(1) Zero-Bias property

34 Zero-Variance Zero-Bias estimators (Assaraf & Caffarel) Based on the Hellmann-Feynman theorem ( ) de λ Ô = dλ λ=0 where E λ = Ψ λ Ĥ + λô Ψ λ

35 Zero-Variance Zero-Bias estimators (Assaraf & Caffarel) Based on the Hellmann-Feynman theorem ( ) de λ Ô = dλ λ=0 one can define an improved estimator: O improved (R) = O(R)Ψ(R) Ψ(R) where E λ = Ψ λ Ĥ + λô Ψ λ + O ZV (R) + O ZB (R) [ H(R)Ψ ] (R) Ψ (R) with the ZV term: O ZV (R) = Ψ E L (R) (R) Ψ(R) and the ZB term: O ZB (R) = 2 [E L (R) E] Ψ (R) Ψ(R)

36 Zero-Variance Zero-Bias estimators (Assaraf & Caffarel) Based on the Hellmann-Feynman theorem ( ) de λ Ô = dλ λ=0 one can define an improved estimator: O improved (R) = O(R)Ψ(R) Ψ(R) where E λ = Ψ λ Ĥ + λô Ψ λ + O ZV (R) + O ZB (R) [ H(R)Ψ ] (R) Ψ (R) with the ZV term: O ZV (R) = Ψ E L (R) (R) Ψ(R) and the ZB term: O ZB (R) = 2 [E L (R) E] Ψ (R) Ψ(R) Quadratic Zero-Variance Zero-Bias property Systematic error: δo improved = O(δΨ 2 + δψ δψ ) Variance: σ 2 (O improved ) = O(δΨ 2 + δψ 2 + δψ δψ )

37 Example of improved QMC estimators Dipole moment of CH molecule ( 2 Π) in VMC: Dipole moment (Debye) 1.9 usual estimator Interatomic distance (Bohr)

38 Example of improved QMC estimators Dipole moment of CH molecule ( 2 Π) in VMC: Dipole moment (Debye) usual estimator improved estimator Interatomic distance (Bohr) = Reduction of statistical uncertainty!

39 Example of improved QMC estimators Correlation hole of C 2 molecule in VMC: 0.5 usual histogram estimator Correlation hole (Bohr -1 ) Interelectronic distance (Bohr)

40 Example of improved QMC estimators Correlation hole of C 2 molecule in VMC: Correlation hole (Bohr -1 ) usual histogram estimator improved estimator Interelectronic distance (Bohr) = Reduction of statistical uncertainty!

41 Summary and perspectives Summary efficient wave function optimization method by minimization of VMC energy near chemical accuracy with compact wave functions improved estimators for observables in QMC Toulouse, Umrigar, JCP 126, (2007) Umrigar, Toulouse, Filippi, Sorella, Hennig, PRL 98, (2007) Toulouse, Assaraf, Umrigar, JCP 26, (2007) Perspectives Toulouse, Umrigar, JCP 128, (2008) optimization by minimization of DMC energy optimization of molecular geometry excited states