Calculating Potential Energy Curves With Quantum Monte Carlo Andrew D Powell, Richard Dawes Department of Chemistry, Missouri University of Science and Technology, Rolla, MO, USA.

Outline Motivation Introduction Results Conclusion & Future Directions 2

Motivation Highly accurate potentials are needed for spectroscopy and dynamics. Traditional high-accuracy quantum chemistry methods – Scaling with the number of electrons (n 7 or worse) – Not yet efficiently parallelized on a large number of computing cores Question: Can we improve the scaling and still produce accurate results? 3

Introduction Quantum Monte Carlo (QMC) is an alternative method to solve the Schrödinger equation. – The CASINO 1 QMC package is used to solve the electronic Schrödinger equation. It has demonstrated the capability of capturing large fractions of the correlation energy. 4 1 R.J. Needs, M.D. Towler, N.D. Drummond and P. López Ríos, J. Phys.: Condensed Matter 22, (2010)

Scaling of QMC Scales almost linearly with the number of cores. – Has been tested with ≥ 500,000 cores. – Well-suited for next-generation computer architectures with millions of cores Scales well with the number of electrons – Scales as n 3 – Large pre-factor (i.e., expensive relative to traditional quantum chemistry methods for small systems). 5

Approach To Generate Global Potential Surfaces Generally, multi-reference methods (such as MRCI) are required. Limitations of traditional high accuracy multi-configurational quantum chemistry (e.g. MRCI) – Usually lacks high order dynamic electron correlation – Some error introduced by internal contraction (ic-MRCI) – Scaling with the number of electrons is poor, especially with large active spaces. – Not yet efficient for large scale parallelization. 6

N 2 : Single-reference breakdown 2 2 X. Li and J. Paldus, J. Chem. Phys. 129, (2008) 7

Background of QMC QMC methods use random sampling – Two types of QMC, Variational Monte Carlo (VMC) and Diffusion Monte Carlo (DMC). VMC is designed to sample a wave function and to calculate the expectation value of the Hamiltonian using Monte Carlo numerical integration. – VMC is also used to optimize parameters for dynamic electron correlation (Jastrow and backflow). 8

Background of QMC 9

Preparation of trial wave function A trial wave function is used as an initial reference for the method It can be prepared by methods such as DFT, HF, CASSCF, etc. For global potentials, a multi-configurational method is necessary. We use CASSCF trial wave functions from GAMESS 3. – Molecular orbitals and configuration coefficients are prepared by scripts for use in the CASINO program M.W.Schmidt, K.K.Baldridge, J.A.Boatz, S.T.Elbert, M.S.Gordon, J.H.Jensen, S.Koseki N.Matsunaga, K.A.Nguyen, S.J.Su, T.L.Windus, M.Dupuis, J.A.Montgomery J.Comput.Chem. 14, 1347 (1993).

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Test Case 1: N 2 N( 4 S) + N( 4 S)  N 2 Point group symmetry: D ∞h Resolved into D 2h symmetry 7,5,3,1 A g Ground state: X 1 Σ g + aug-cc-pwCV5Z* with Jastrow – High angular momentum functions (l ≥ f) were removed 12

N 2 PECs 13

Test Case 2: CO C( 3 P) + O( 3 P)  CO Point group symmetry: C ∞v Resolved into C 2v symmetry as 5,3,1 (3A 1 + 2B 1 + 2B 2 + 2A 2 ) DW-SA-CASSCF for the nine singlet states – aug-cc-pwCVTZ* basis with dynamic weighting *Angular momentum functions (l ≥ f) were removed – With Jastrow and backflow 14

1-state MRCI calculation produces discontinuity… 9-states are degenerate asymptotically A DW-MRCI benchmark PEC for CO 15

DW-MRCI/CBS with 9 statesSO- and SR-Coupling A DW-MRCI benchmark PEC for CO 16

Accuracy of benchmark PEC for CO 17 CO J=0 Vibrational Levels vCalculatedExperimentError

CO PECs 18

Conclusion & Future Directions QMC methods show promising accuracy It is of interest to benchmark systems which have proven difficult for traditional quantum chemistry methods. – e.g. MEP for formation of species such as O 3 and HO 3 O + O 2  O 3 HO + O 2  HO 3 They often have spurious barriers or submerged reefs along the MEP. We are exploring fitting potential surfaces incorporating data that includes associated uncertainties. 19

Acknowledgements 20