QMC dissociation energy of the water dimer: Time step errors and backflow calculations

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QMC dissociation energy of the water dimer: Time step errors and backflow calculations Idoia G. de Gurtubay and Richard J. Needs TCM group. Cavendish Laboratory University of Cambridge Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 1/19

Introduction Water Physical and chemical properties: strong polar hydrogen bonds Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 2/19

Introduction Water Physical and chemical properties: strong polar hydrogen bonds Monomer Dimer Trimer Bulk water Binding energy of only a few kcal/mol Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 2/19

Introduction Water Physical and chemical properties: strong polar hydrogen bonds Monomer Dimer Trimer Bulk water Binding energy of only a few kcal/mol DIMER: prototype of all hydrogen-bonded systems Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 2/19

Previous calculations on H 2 O and (H 2 O) 2 MP2, CCSD(T), CI + correlation effects quite accurate - basis set truncation errors - basis set superposition errors - N 5, N 7 Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 3/19

Previous calculations on H 2 O and (H 2 O) 2 MP2, CCSD(T), CI + correlation effects quite accurate - basis set truncation errors - basis set superposition errors - N 5, N 7 Density functional theory (DFT) + More favorable scaling - Strong dependence on the XC functional Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 3/19

This work on H 2 O and (H 2 O) 2 Quantum Monte Carlo + electronic correlation explicitly + Scales as N 3 Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 4/19

This work on H 2 O and (H 2 O) 2 Quantum Monte Carlo + electronic correlation explicitly + Scales as N 3 THIS WORK = VMC and DMC Energies of H 2 O and (H 2 O) 2 All-electron (AE) and Pseudopotential (PP) calculations Slater-Jastrow (SJ) and Slater-Jastrow-Backflow (BF) wave functions = Electronic dissociation energy of (H 2 O) 2 Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 4/19

VMC, DMC VMC Accuracy determined by the trial wave function biased energy differences for optmizing parameters of the wave function DMC Project out the ground state component of the trial wave function Fermionic symmetry: fixed node approximation CASINO code Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 5/19

Trial wave function Slater-Jastrow (SJ) wave function R={r i } Ψ SJ (R) = e J(R) Ψ S (R) Jastrow factor: e J(R) Slater determinant: Ψ S = D D Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 6/19

Trial wave function Slater-Jastrow (SJ) wave function R={r i } Ψ SJ (R) = e J(R) Ψ S (R) Jastrow factor: e J(R) Slater determinant: Ψ S = D D Slater-Jastrow-Backflow (BF) wave function Ψ BF (X) = e J(R) Ψ S (X) x i = r i + ξ i (R) Backflow displacement: ξ i (R) [PLR: 24/07, 9:30am] Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 6/19

Single particle orbitals CRYSTAL98 code Basis set: Roos augmented double zeta ANO (s, p, d) B3LYP orbitals seem to give better nodes than Hartree-Fock orbitals use parameters in XC functional to optimise orbitals Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 7/19

Single particle orbitals CRYSTAL98 code Basis set: Roos augmented double zeta ANO (s, p, d) B3LYP orbitals seem to give better nodes than Hartree-Fock orbitals use parameters in XC functional to optimise orbitals E xc = (1 A)(E LDA x A : Fock exchange B : non-local exchange C : non-local correlation +B Ex Becke )+A Ex HF +(1 C) Ec VWN +C Ec LYP True B3LYP A = 0.2, B = 0.9 and C = 0.81 Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 7/19

B3LYP orbitals E xc = (1 A)(E LDA x + B Ex Becke ) + A Ex HF + (1 C) E VWN c A = 1 = E DMC = 76.4218(1) Ha HF = E DMC = 76.42205(8) Ha + C E LYP c Exchange is the most important contribution = Use A as optimisable parameter keeping B and C constant Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 8/19

B3LYP-25 orbitals DMC Energy (Ha) -76.4216-76.4218-76.4220-76.4222-76.4224-76.4226-76.4228-76.4230-76.4232-76.4234-76.4236-76.4238-76.4240 DFT Energy (Ha) -76.44-76.45-76.46-76.47 0 20 40 60 80 100 % Fock Exchange -76.4242 0 10 20 30 40 50 60 70 80 90 100 % Fock Exchange in XC functional Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 9/19

Summary of details SJ and BF trial wave functions J/BF parameters: minimize variance of local energy B3LYP-25 single-particle orbitals AE: single-particle orbitals corrected obey cusp conditions PP: From Hartree-Fock theory work well with QMC avoid short-range variations of the wavefunction near the nuclei = larger time steps Various time steps + extrapolation to zero time Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 10/19

Geometries Monomer: experimental equilibrium geometry r OH = r OH = 0.9572 Å = 104.52 o Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 11/19

Geometries Monomer: experimental equilibrium geometry r OH = r OH = 0.9572 Å = 104.52 o Dimer: CCSD(T) geometry (Klopper et. al) O a acceptor donor H a H H f d H a Od r OO = 2.912 Å (Deformations neglected) Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 11/19

Geometries Monomer: experimental equilibrium geometry r OH = r OH = 0.9572 Å = 104.52 o Dimer: CCSD(T) geometry (Klopper et. al) O a acceptor donor H a H H f d H a Od Electronic dissociation energy r OO = 2.912 Å (Deformations neglected) (H 2 O) 2 2 H 2 O D e = D o ZPE Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 11/19

Water monomer total energy (Ha) 97.4% B3LYP-like -76.4361 Exact -76.438 Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 12/19

Water monomer total energy (Ha) VMC-SJ -76.3773(2) DMC-SJ B3LYP-like Exact -76.4230(1) -76.42371(6) -76.4361-76.438 93.4% Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 12/19

Water monomer total energy (Ha) VMC-SJ -76.3773(2) DMC-SJ (HF) DMC-SJ B3LYP-like Exact -76.42102(4) -76.4219(1) -76.4230(1) -76.42371(6) -76.4361-76.438 97.4% Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 12/19

Water monomer total energy (Ha) VMC-SJ -76.3773(2) 84% VMC-BF -76.4010(1) 90% DMC-SJ DMC-BF -76.42371(6) -76.42792(15) 96.2% 97.4% Exact -76.438 Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 12/19

Water monomer total energy (Ha) VMC-SJ -76.3773(2) VMC-BF -76.4010(1) DMC-SJ DMC-BF DMC-PNO-CI -76.42371(6) -76.42792(15) -76.429(1) 97.4% Exact -76.438 Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 12/19

Water monomer total energy (Ha) VMC-SJ -76.3773(2) VMC-BF -76.4010(1) DMC-SJ DMC-BF DMC-PNO-CI CCSD(T)-R12 Exact -76.42371(6) -76.42792(15) -76.429(1) -76.4373-76.438 97.4% Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 12/19

H 2 O, extrapolation to zero time DMC Energy (Ha) -76.4230-76.4235-76.4240-76.4245-76.4250-76.4255-76.4260-76.4265-76.4270-76.4275-76.4280-76.4285-76.4290-76.4295-76.4300-76.4305-76.4310 0 0.002 0.004 0.006 0.008 0.01 0.012 Time step (au) SJ BF Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 13/19

Water dimer acceptor donor O a H H f d H a Od H a Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 14/19

Pseudopotential calculations DMC Energy (Ha) -34.4140-34.4160-34.4180-34.4200-34.4220-34.4240-34.4260-34.4280-34.4300-34.4320-34.4340-34.4360-34.4380 (H 2 O) 2 (SJ) (H 2 O) 2 (BF) 0 0.01 0.02 0.03 0.04 0.05 0.06 Time step (au) Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 15/19

Pseudopotential calculations DMC Energy (Ha) -34.4140-34.4160-34.4180-34.4200-34.4220-34.4240-34.4260-34.4280-34.4300-34.4320-34.4340-34.4360-34.4380 2 x H 2 O (SJ) 2 x H 2 O (BF) 0 0.01 0.02 0.03 0.04 0.05 0.06 Time step (au) Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 15/19

Pseudopotential calculations DMC Energy (Ha) -34.4140-34.4160-34.4180-34.4200-34.4220-34.4240-34.4260-34.4280-34.4300-34.4320-34.4340-34.4360-34.4380 2 x H 2 O (SJ) (H 2 O) 2 (SJ) 2 x H 2 O (BF) (H 2 O) 2 (BF) 0 0.01 0.02 0.03 0.04 0.05 0.06 Time step (au) Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 15/19

All-electron calculations DMC Energy (Ha) -152.8460-152.8480-152.8500-152.8520-152.8540-152.8560-152.8580-152.8600-152.8620-152.8640-152.8660-152.8680-152.8700 2 x H 2 O (SJ) (H 2 O) 2 (SJ) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time step (au) Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 16/19

All-electron calculations DMC Energy (Ha) -152.8460-152.8480-152.8500-152.8520-152.8540-152.8560-152.8580-152.8600-152.8620-152.8640-152.8660-152.8680-152.8700 2 x H 2 O (SJ) (H 2 O) 2 (SJ) 2 x H 2 O (BF) (H 2 O) 2 (BF) 0 0.002 0.004 0.006 0.008 0.01 0.012 Time step (au) Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 16/19

Dissociation energy (D e ) D e (kcal/mol) 5.7 5.6 5.5 5.4 5.3 5.2 5.1 4.9 5 4.8 AE-SJ PP-BF PP-SJ 0 0.01 0.02 0.03 0.04 0.05 Time step (au) Method D e (kcal/mol) AE-SJ 5.23(11) PP-SJ 5.03(7) PP-BF 5.43(10) Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 17/19

Dissociation energy (D e ) D e (kcal/mol) 5.7 5.6 5.5 5.4 5.3 5.2 5.1 4.9 5 4.8 AE-SJ PP-BF PP-SJ 0 0.01 0.02 0.03 0.04 0.05 Time step (au) Method D e (kcal/mol) AE-SJ 5.23(11) PP-SJ 5.03(7) PP-BF 5.43(10) DMC-HF 5.02(18) DMC-B3LYP 5.21(18) CCSD(T) 5.04(5) Exp 5.44(70) Exp 5.00(70) Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 17/19

Conclusions AE and PP total energies of H 2 O and (H 2 O) 2 SJ and BF wave functions with B3LYP-like single-particle orbitals Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 18/19

Conclusions AE and PP total energies of H 2 O and (H 2 O) 2 SJ and BF wave functions with B3LYP-like single-particle orbitals Water monomer B3LYP-like orbitals lower the H 2 O DMC energy 1.5 mha compared to HF orbitals better nodal surface BF correlations reduce the DMC energy by an additional 4-5 mha E = -76.42792(15) Ha (10 mha above exact value) Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 18/19

Conclusions AE and PP total energies of H 2 O and (H 2 O) 2 SJ and BF wave functions with B3LYP-like single-particle orbitals Water monomer B3LYP-like orbitals lower the H 2 O DMC energy 1.5 mha compared to HF orbitals better nodal surface BF correlations reduce the DMC energy by an additional 4-5 mha E = -76.42792(15) Ha (10 mha above exact value) D e of water dimer Time step errors cancel Extrapolated D e for AE and PP with SJ and BF wave functions within error bars of exact value Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 18/19

Acknowledgments... Financial support from the Basque Government through Research Fellowship BFI 04/183 Calculations were performed at the Cambridge-Cranfield High Performance Computing Facility Idoia G. de Gurtubay. Quantum Monte Carlo in the Apuan Alps II. July 23, 2006 QMC dissociation energy of the water dimer - p. 19/19

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