1. FULL DIMENSIONAL VIBRATIONAL CALCULATIONS FOR METHANE USING AN ACCURATE NEW AB INITIO BASED POTENTIAL ENERGY SURFACE International Symposium on Molecular Spectroscopy, 2014 June 20, 2014 Moumita Majumder, Richard Dawes, Xiao-Gang Wang, Tucker Carrington Jr., Jun Li and Hua Guo Missouri University of Science and Technology Queen’s University University of New Mexico

2. 2 Introduction Methane (CH 4 ): Simplest saturated hydrocarbon. Astrophysics: Present in the atmosphere of many planetary systems, Titan, Triton, etc. Methane combines with nitrogen clusters and forms aerosol in Titan’s atmosphere. Large number of brown dwarf exhibit strong methane bands. Atmospheric chemistry: Second greenhouse gas (AGGI : ~ 0.5)* whose emission should be reduced. Combustion chemistry: *The NOAA Annual Greenhouse Gas Index, www.esrl.noaa.gov + ΔH ~ 890 kJ

3. Motivation 3 Construct a Permutation Invariant (PI) PES for CH 4 with negligible fitting error. 24 permutations Strategies to treat permutation symmetry Low permutation symmetry: Replicate the electronic energy for each permutation (expanding data set). High permutation symmetry: Use permutation invariant fitting functions. H4 H1 H2 H3 H4 H3 H1 H2 PES PES should be invariant with respect to all possible permutations !

4. 4 Single expansion: (analytically fit the PES using PI polynomials 1,2 ) Multiple expansions: Interpolating moving least square (IMLS) method. (interpolate the local fitting functions using a weight function) [works very well for van der Waals systems, N 2 O dimer 3, CO dimer 4 etc. Complex topography. Multi-well systems + 1 B. J. Braams and J. M. Bowman Int. Rev. Phys. Chem. 28, 577 (2009) 2 Z. Xie and J. M. Bowman, J. Chem Theory Comput. 6, 26, (2010) 3 R. Dawes, X-G Wang, A. W. Jasper and T. Carrington Jr., J. Chem. Phys. 133, 134304, (2010) Combining PES fitting strategies PIP-IMLS PIPs

5. 5 CO dimer PES 4 R. Dawes, X-G Wang, T. Carrington Jr., J. Phys. Chem. A, 117, 7612 (2013). AB φAφA φBφB C 3 Rotation Barriers: NH 3 dimer These van der Waals systems were successfully fit using IMLS. Ro-vibrational energy values are in accord with experimental values. van der Waals systems Interpolation of many expansions produces negligible fitting error

6. Coordinates: Each basis function is expressed as products of Morse variables of internuclear bond distances between atom pairs ij. no divergent or oscillatory behavior of the potential toward asymptote. 6 Potential energy function: 24 terms, including all permutations (PIP) This approach ensures that each basis function is permutation invariant Solving linear square fit Fitting using PIPs (review of Bowman approach)

7. 7 Choose one and treat rest of them as redundant. deg.Int. basis func(s)Reduc. basis func(s) 1113 26610 4100183 8437582327 9923784685 101847569041 No. of basis functions reduced drastically, especially for higher degree functions Computational cost for IMLS varies with square of the basis size

8. Details of ab initio data set and pruned fitting basis Ab initio data : –~10 6 symmetry unique ab initio points at (MRCI-F12(AE)/CVQZ- F12) level of theory extending to the dissociation of methane to methyl radical + H. –~ beyond 1.6 Å, multi-configuration dominates. CCSD(T) 5,6 failed! Fitting basis : all PIPs up to 6 th degree, i.e., 495 PIPs plus best determined higher degree PIPs. Rotating test sets were reserved to identify poorly determined functions for removal Fitting error of RMSD ~ 8.5 cm -1 was obtained for 88250 points, covering the energy range 0 – 30,000 cm -1 using a single expansion and an energy dependent bias. w i = 6000 cm -1 /(E i + 6000 cm -1 ) 8 5 Schwenke and Partridge, Spectrochim. Acta A, 57, 887(2001). 6 Cassam-Chenaї and Liévin, J. Chem. Phys., 136, 174309 (2012)

9. Energy Range (cm -1 ) RMSD (cm -1 ) 0-80001.83 0-150002.6 0-20,0003.6 0-26,0005.4 9

10. Vibrational levels of CH 4 : Comparison with Experiment Stateobs 5 obs-cal PIP-NNNNPIP-WLS (0000)A19708.279709.509709.38 (0001)F21310.76-2.00-2.01-2.26 (0100)E1533.33-0.96-0.46-2.10 (0002)A12587.04-4.11-4.04-4.42 (0002)F22614.26-4.26-3.97-4.46 (0002)E2624.61-3.68-4.13-4.53 (0101)F22830.31-2.85-2.68-4.07 (0101)F12846.07-3.20-2.55-4.28 (1000)A12916.48-0.96-1.78-1.18 (0010)F23019.490.09-0.060.39 (0200)A13063.64-2.10-1.26-2.51 10 Full-dimensional (9D) vibrational calculations were carried out on 3 single expansion fits to the data using an exact kinetic energy operator (Wang and Carrington) and a symmetry adapted Lanczos method to obtain eigenvalues and eigenvectors.

11. 11 StateObs 5 Obs-cal PP-NNNNPIP-WLS (0200)E3065.14-2.03-1.10-3.40 (0003)F23870.48-6.36-6.03-6.58 (0003)A13909.20-6.53-5.82-6.59 (0003)F13920.51-6.27-6.03-6.64 (0003)F23930.92-5.57-6.19-6.73 ReRe 1.0860511.0861331.081625 RMSD3.47---4.50 1.Computed vibrational energy levels are consistent with those calculated using two other single expansion fits to the same data. 2.The magnitude of errors in the vibrational levels is similar to the PES fitting error making it difficult to fully assess the quality of the ab initio data. 3.Removal of the fitting error using multiple expansions will allow the benefit of various further small corrections to the ab initio data to be determined. 5 E. Venuti, L. Halonen, R. G. Della Valle, J. Chem. Phys. 110, 7339, (1999)

12. PIP-IMLS 12 Potential energy function Interpolative weight functionLocal fitted functions

13. 13 Local fit: PIP-IMLS FITTING Interpolative weight function: {b j } = PIPs, { a j } = expansion coefficients M = total no. of PIPs PI distance metric Local data density tiny positive value, maintained continuity at q k = q i Parameter, p = 14 R. Dawes, X-G Wang, A. W. Jasper and T. Carrington Jr., J. Chem. Phys. 133, 134304, (2010) w i = f(d); local fitting function close to evaluation point get maximum weight Small portion of dataset is fitted (WLS method) with a negligible fitting error

14. 14 Permutation Invariant distance metric: If q k and q i are permutationally symmetric geometries of any ordered ten-tuples, d 2 (qk, qi) = 0 pd = 2. J. D. Bender, S. Doraiswamy, D. G. Truhlar and G. V. Candler, J. Chem. Phys. 140, 054302, (2014) d 2 : Minimum distance between one geometry and all possible permutations of other geometry

15. Quality of the fit 15 Energy Range (cm -1 )RMSD (cm -1 ) PIP- IMLS PIP-WLS 0-80000.21.8 0-150001.32.6 0-20,0002.13.6 208 local expansions at ab initio data points 3796 PIPs PES fitting error (RMSD) is reduced to sub wave number with ~ 200 expansions in the energy range 0 - 10,000 cm -1 ! Adding more expansions will reduce the error further.

16. Conclusions PIP-IMLS fitting is advantageous in cases with high permutation symmetry with challenging topographies (or when negligible fitting error is required). We will be able to produce a good quality PES for methane with negligible fitting error. 16 Future work Further validation of ab initio corrections to the surface by performing vibrational calculations on it. Apply the fitting strategy to other high permutation symmetry systems.

17. Acknowledgements Dr. Richard Dawes Missouri University of Science and Technology, USA Dr. Xiao-Gang Wang and Prof.Tucker Carrington Jr. Queen’s University, Canada Dr. Jun Li and Prof. Hua Guo University of New Mexico, USA Dr. Sergei Manzhos National University of Singapore, Singapore 17

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