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.

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Outline

• Motivation

• Introduction

• Results

• Conclusion & Future Directions

Motivation

• Highly accurate potentials are needed for spectroscopy and dynamics.

• Traditional high-accuracy quantum chemistry methods – Scaling with the number of electrons (n7 or worse)– Not yet efficiently parallelized on a large number of computing

cores

• Question: Can we improve the scaling and still produce accurate results?

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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.

1 R.J. Needs, M.D. Towler, N.D. Drummond and P. López Ríos, J. Phys.: Condensed Matter 22, 023201 (2010)

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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 n3 – Large pre-factor (i.e., expensive relative to traditional quantum

chemistry methods for small systems).

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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.

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N2: Single-reference breakdown 2

2 X. Li and J. Paldus, J. Chem. Phys. 129, 054104 (2008)

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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).

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Background of QMC

• DMC attempts to simultaneously create and sample the unknown exact ground state wave function. During the simulation, it evolves to the ground state.– Constrained above the exact solution by the fixed-node

approximation.

• Due to the statistical nature of the approach, there is uncertainty associated with a calculated energy – ΔE

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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.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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Trial Wave Function

• Multi-determinant expansion used to describe static electron correlation

• Includes Jastrow factor to capture dynamic correlation– Similar to geminal functions used for explicitly correlated F12

methods.• Backflow (coordinate) transformation

– Relaxes constraint of the fixed-node approximation

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Test Case 1: N2

• N(4S) + N(4S) N2

• Point group symmetry: D∞h

• Resolved into D2h symmetry 7,5,3,1Ag

• Ground state: X1Σg+

• aug-cc-pwCV5Z* with Jastrow – High angular momentum functions (l ≥ f) were removed

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N2 PECs

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Test Case 2: CO

• C(3P) + O(3P) CO

•Point group symmetry: C∞v

• Resolved into C2v symmetry as5,3,1(3A1 + 2B1 + 2B2 + 2A2)

• 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

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1-state MRCI calculation produces discontinuity…9-states are degenerate asymptotically

A DW-MRCI benchmark PEC for CO

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DW-MRCI/CBS with 9 states SO- and SR-Coupling

A DW-MRCI benchmark PEC for CO

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Accuracy of benchmark PEC for COCO J=0 Vibrational Levels

v Calculated Experiment Error0 1081.78 1081.59 0.191 3225.00 3224.86 0.132 5341.69 5341.65 0.043 7431.95 7432.03 -0.074 9495.86 9496.05 -0.195 11533.50 11533.76 -0.266 13544.98 13545.29 -0.317 15530.36 15530.64 -0.288 17489.76 17490.00 -0.249 19423.25 19423.50 -0.25

10 21330.93 21331.00 -0.0711 23212.88 23212.70 0.1812 25069.20 25068.60 0.6013 26899.97 26898.60 1.1714 28705.28 28703.40 1.88

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CO PECs

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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 O3 and HO3

• O + O2 O3

• HO + O2 HO3

• They often have spurious barriers or submerged reefs along the MEP.

• We are exploring fitting potential surfaces incorporating data that includes associated uncertainties.

Acknowledgements

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