Applying Quantum Monte Carlo to the Electronic Structure Problem: Potential Energy Curves of N2 and CO with Diffusion Monte Carlo Andrew D Powell, Richard Dawes Department of Chemistry, Missouri University of Science and Technology, Rolla, MO, USA. 1 Outline Motivation Scaling and Accuracy Introduction Background of QMC methodology Results

Comparison with spectroscopically accurate PECs Cost analysis Conclusion 2 Motivation Scaling Quantum Monte Carlo (QMC) methods are efficiently parallelized with a large number of cores n3 with the number of electrons as n7 or worse with traditional quantum methods With its cost prefactor, how practical is it for routine application? Accuracy Capable of capturing large fractions of dynamic correlation energy How does it compare with MRCI or CCSD(T) for small molecules? Can QMC produce all data for global PESs?

3 Introduction Quantum Monte Carlo (QMC) is an alternative method to solve the Schrdinger equation. CASINO1 QMCPACK2 For traditional quantum chemistry methods, there are many codes available. MOLPRO Gaussian

GAMESS CFOUR R.J. Needs, M.D. Towler, N.D. Drummond and P. Lpez Ros, J. Phys.: Condensed Matter 22, 023201 (2010). 2 J. Kim, K.P. Esler, J. McMinis, M.A. Morales, B.K. Clark, L. Shulenburger, and D.M. Ceperley, J. Phys.: Conf. Ser. 402, 012008 (2012). 1 4 Introduction Test systems: diatomic PECs for N2 and CO CASINO code Background of QMC Variational Monte Carlo (VMC) and Diffusion Monte Carlo (DMC)

5 Background of VMC VMC is designed to sample a trial wave function and to calculate the expectation value of the Hamiltonian using Monte Carlo numerical integration. VMC is primarily used to optimize parameters for dynamic electron correlation. 6 Background of DMC The DMC method attempts to simultaneously create and sample the unknown exact ground state wave function.

Constrained from above the exact solution by the fixed-node approximation Fixed-node approximation: the nodal surface of the wave function is constrained to be that of the trial wave function. (1) (2) (3) Due to the statistical nature of the approach, there is uncertainty associated with a calculated energy. E E 7 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. We use MCSCF trial wave functions from GAMESS.3

Molecular orbitals and configuration coefficients are prepared by scripts for use in the CASINO program. aug-cc-pwCV5Z* basis set High angular momentum functions (l f) were removed 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 8 Trial Wave Function Multi-determinant expansion is used to describe static electron correlation Includes a Jastrow factor to capture dynamic correlation

Makes the trial wave function depend explicitly on the interparticle separations Maintains nodal structure of the trial wave function. Enforces the cusp conditions. Tightness of VMC optimization Procedure 1: 20,000 CPU-hrs. Procedure 2: 2,000 CPU-hrs. 9 Single-reference breakdown Divergence of CCSD(T) occurs for rNN > 2 .4 4 X. Li and J. Paldus, J. Chem. Phys. 129, 054104 (2008) 10

N(4S) + N(4S) N2 (X1g+) ~80 cm-1 uncertainties after 105 samples (1500-2000 cpu-hrs for DMC) 11 A DW-MRCI benchmark PEC for CO 1-state MRCI calculation produces discontinuity 9-states are degenerate asymptotically 12 A DW-MRCI benchmark PEC for CO DW-MRCI/CBS with 9 states SO- and SR-Coupling

13 Accuracy of benchmark PEC for CO CO J=0 Vibrational Levels v Calculated Experiment Error 0 1 2 3 4

5 6 7 8 9 10 11 12 13 14 1081.78 3225.00 5341.69 7431.95 9495.86 11533.50 13544.98

15530.36 17489.76 19423.25 21330.93 23212.88 25069.20 26899.97 28705.28 1081.59 3224.86 5341.65 7432.03 9496.05 11533.76 13545.29 15530.64 17490.00

19423.50 21331.00 23212.70 25068.60 26898.60 28703.40 0.19 0.13 0.04 -0.07 -0.19 -0.26 -0.31 -0.28 -0.24 -0.25 -0.07

0.18 0.60 1.17 1.88 14 C(3P) + O(3P) CO(X1+) ~80 cm-1 uncertainties after 105 samples (1500-2000 cpu-hrs for DMC) 15 Conclusions Cost of QMC Tightening of the VMC optimization gave no benefit in the subsequent DMC. Not competitive with traditional chemistry (N2 and CO take ~5 min per point with MRCI/CBS. Unlikely to be used to generate all points of a PES. n3 scaling ensures crossover point.

Accuracy Comparable to the best ab initio or empirical PECs. Future Arbiter in difficult cases? Testing reaction barriers for F + H2O, O + O2, and O2 + propargyl. Using QMCPACK (more efficient for MCSCF trial wave functions, capable of open-shell systems, allows spdfg functions) Testing FCIQMC in same applications 16 17 Dynamically-weighted state-averaged multireference electronic structure theory E SA MCSCF

n w i EiMCSCF i 0 W n W wi i 0 HCO Dynamic weighting1 2 wi cosh (Ei 0 / )

Theta Ei0 = Ei E0 rCH Generalized dynamic weighting2 2 wi cosh (Eij / ) HO2 Eij = |Ei Ej| rOH Theta

M.P. Deskevich, D.J. Nesbitt and H-J. Werner, JCP 120 7281 (2004) 6 R. Dawes, A.W. JasperS.A. Reid et al. J. Phys. Chem. Lett. 1 641 (2010) 5 Importance of Dynamic Weighting: BeOBe BeO + Be 1 state 23 states equal weights In single state calc (at left), avoided crossing with higher lying state (blue oval, at right) causes discontinuity. In fixed weights calc (at right) crossing with higher lying state (red

oval) causes discontinuity cascade affecting ground state. Six lowest A1 states (co-linear asymmetric stretch C2V) Importance of Dynamic Weighting: BeOBe BeO + Be 23 states equal weights 23 states dynamic weights With dynamic weights (at right), crossing still occurs (blue oval), but due to negligible weight, no disruption results.