1. Theoretical Study of the Ethyl Radical Daniel Tabor and Edwin L. Sibert III June 20, 2014

2. Outline Background Construction of Hamiltonian Van Vleck perturbation theory CH stretch analysis Torsion-inversion modes Concluding remarks 2

3. Background Raston et al. J. Chem. Phys. 138, 194303 (2013). He droplet spectra S. Davis, D. Uy, D.J. Nesbitt J. Chem. Phys. 112, 1823 (2000). 3

4. Background R.S Bhatta, A. Gao, D. S. Perry, THEOCHEM 941, 22-29 (2010). Dominant torsion-inversion coupling term is where for CCSD(T)/6-311++G(d,p) Coupling involving large-amplitude degrees of freedom 4

5. Main Calculation Steps Construct a Hamiltonian Use Van Vleck perturbation theory on the torsion-averaged Hamiltonian Use symmetry, diagonalize Apply transformation to all torsion-dependent terms 5

6. Constructing Hamiltonian Potential Energy – Polynomial fit to a functional form (CCSD(T)/cc-pVTZ energies) Kinetic Energy – Expand the G-matrix in each coordinate 6 Symmetry Coordinates (further transformed)

7. About 47000 Basis Functions G 12 PI Symmetry 6 Different Symmetry Blocks a states e states Diagonalization—Symmetry 7 a1a1 4271 a2a2 3922 b1b1 3999 b2b2 4099 e1e1 7882 e2e2 7795 Product basis set: easy to evaluate matrix elements TorsionProduct of harmonic oscillators But only trivially separates e and a states Switch to a sin/cos-type basis... Also reduces torsional basis size for E states

8. Vibration-Torsion Basis Set and Symmetry Symmetrymod(l-m,3)n6+n7n6+n7 n 6 +n 7 +n 8 +n 12 a1a1 zeroeven b1b1 zeroevenodd a2a2 zerooddeven b2b2 zeroodd e1e1 nonzeroN/Aeven e2e2 nonzeroN/Aodd Vibrational Angular Momentum Use “old” basis to determine the symmetry of each basis function Apply the ideas of the new basis upon matrix element evaluation 8

9. Evaluating Matrix Elements Symmetry (l value)Ac1c1 c2c2 c3c3 c4c4 e (all)1/21111 a 1 (both zero)11000 a 1 (bra zero)1/Sqrt(2)1010 a 1 (ket zero)1/Sqrt(2)1100 a 1 (neither zero)11010 a 2 (both zero)11000 a 2 (bra zero)1/Sqrt(2)1010 a 2 (ket zero)1/Sqrt(2)100 a 2 (neither zero)1100 9 Four possible terms combine to form new matrix elements in new basis Matrix Element =

10. Effect of Coupling All CH 3 group terms included Energy (cm -1 ) 10 CH stretch region

11. Zooming In All CH 3 group terms included Still not much happening? VV Dressed H! Energy (cm -1 ) 11

12. Torsion-Inversion: Complexities of the Potential For coordinates s 5 and s 8, a quartic potential is not sufficient 12

13. Torsion-Inversion Potential In Normal Coordinates 13

14. Coupling in VV Wavefunctions a1e1 e2 a2 b2b1 14

15. Lowest VV-Transformed Wavefunctions of Each Symmetry a1a2 b1 b2 e1 e2 15

16. Conclusions/Future Work Ethyl radical has been treated with full consideration of the torsion Application of Van Vleck perturbation theory produces rapidly converging eigenvalues w.r.t. basis set size Next: incorporate more degrees of freedom in torsion-inversion analysis 16

17. Acknowledgements Sibert Group – Britta Johnson – Amber Jain – Ned 17