1. 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.

2. Outline Motivation Introduction Results Conclusion & Future Directions 2

3. 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

4. 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, 023201 (2010)

5. 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

6. 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

7. N 2 : Single-reference breakdown 2 2 X. Li and J. Paldus, J. Chem. Phys. 129, 054104 (2008) 7

8. 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

9. Background of QMC 9

10. 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. 10 3 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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12. 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

13. N 2 PECs 13

14. 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

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

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

17. Accuracy of benchmark PEC for CO 17 CO J=0 Vibrational Levels vCalculatedExperimentError 01081.781081.590.19 13225.003224.860.13 25341.695341.650.04 37431.957432.03-0.07 49495.869496.05-0.19 511533.5011533.76-0.26 613544.9813545.29-0.31 715530.3615530.64-0.28 817489.7617490.00-0.24 919423.2519423.50-0.25 1021330.9321331.00-0.07 1123212.8823212.700.18 1225069.2025068.600.60 1326899.9726898.601.17 1428705.2828703.401.88

18. CO PECs 18

19. 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

20. Acknowledgements 20