1. A Practical Procedure for ab initio Determination of Vibrational Spectroscopic Constants, Resonances, and Polyads William F. Polik Hope College, Holland, MI June 2006

2. Chemical Reactions Occur via Excited Vibrational States

3. HFCO Pure Vibrational Spectrum 3 1 HFCO

4. Vibrational State Models HarmonicAnharmonicPolyad

5. Calculation Method 1.Compute equilibration geometry 2.Compute PES derivatives 3.Calculate spectroscopic constants 4.Identify important resonances 5.Compute excited vibrational states CCSD(T)/aug-cc-pVQZ ω i x ij K POLYAD program

6. 1. Compute Equilibration Geometry Geometry of energy minimum needed for Taylor expansion of PES Key points in calculation –Correlated theory and high quality basis, e.g., CCSD(T) and aug-cc-pVQZ –Tight convergence of SCF wavefunction and optimized structure Program used –Molpro (Werner & Knowles)

7. 2. Compute PES Derivatives Taylor-series derivatives are molecular force constants Key points in calculation: –Symmetrized internal coordinates –Numerical derivatives Programs used –FE/BE (Martin): list of displaced geometries; assemble derivatives –Intder (Allen): coordinate transformations –Molpro (Werner & Knowles): energy points

8. 3. Calculate Spectroscopic Constants Force field is defined in terms of displacements q i but vibrational energy levels are quantized by v i Second order perturbation theory relate  ijk and  ijkl to x ij Program used –Spectro (Handy)

9. Spectroscopic Constants Refs: Nielsen (1959), Papousek & Aliev (1982)

10. 4. Identify Important Resonances Perturbation theory breaks down at resonances For each resonant interaction –Modify calculation of x ij –Determine resonance constant K Program used –Spectro-modified (Handy, Martin, Polik)

11. Spectroscopic Constants Refs: Nielsen (1959), Papousek & Aliev (1982) resonance denominator when 2ω i ≈ω k resonance denominator when ω i ≈ω j + ω j, ω j ≈ω i + ω k, or ω k ≈ω i + ω j

12. 4. Identify Important Resonances Perturbation theory breaks down at resonances For each resonant interaction –Modify calculation of x ij –Determine resonance constant K Program used –Spectro-modified (Handy, Martin, Polik)

13. Modified Spectroscopic Constants Refs: Papousek & Aliev (1982), Martin & Taylor (1997) partial fraction expansion drop resonance term(s)

14. Resonance Constants Refs: Lehmann (1989), Martin & Taylor (1997)

15. 5. Compute Excited Vibrational States Define model parameters ( , x, K) Determine polyads H2O CCSD(T): aug-cc-pVQZ/cc-pVTZ w1o 1 0 0 1 0 0 3684.92372284 -1 w2o 0 1 0 0 1 0 1610.20330698 -1 w3o 0 0 1 0 0 1 3797.96450773 -1 x11 2 0 0 2 0 0 -43.64165166 -1 x12* 1 1 0 1 1 0 -37.99219452 -1 x13 1 0 1 1 0 1 -167.62218355 -1 x22* 0 2 0 0 2 0 -11.33265207 -1 x23 0 1 1 0 1 1 -19.05478905 -1 x33 0 0 2 0 0 2 -49.68139858 -1 K22,1 0 2 0 1 0 0 -154.23334812 -1 K11,33 2 0 0 0 0 2 -160.77598615 -1 21322132 K 11,33  12211221 K 1,22  11231123 K 1,22  2525

16. Calculate matrix elements Diagonalize matrices; report energy & wavefunction Program used –Polyad (Polik) Hamiltonian matrix: 8718.167 -188.897 0.000 -80.388 -188.897 8255.922 -243.864 0.000 0.000 -243.864 7767.700 0.000 -80.388 0.000 0.000 8957.964 Eigenvalues and vectors (columns): 7661.582 8286.153 8764.315 8987.704 0.071 -0.375 0.857 -0.345 0.398 -0.838 -0.361 0.095 0.915 0.394 0.088 -0.019 0.004 -0.045 0.356 0.933

17. Compare to Manual Method

18. Summary of Method 1.Compute equilibration geometry 2.Compute PES derivatives 3.Calculate vibrational spectroscopic constants 4.Identify important resonances; modify constants & calculate resonance constants 5.Compute excited vibrational states using polyad model

23. Interpretations and Conclusions Polyad model is useful and practical –Experimental fits are excellent for predicting excited vibrational states (± 10 cm -1 ) –Ab initio computation of excited states is relatively accurate (± 20 cm -1 ) Appropriate basis sets are –AVQZ for harmonic force field –VTZ for anharmonic force field Include resonances when K*HO/  E>0.1~0.3

24. Acknowledgements Ruud van Ommen (Netherlands) Ben Ellingson (Univ of Minnesota) John Davisson (Hope College) Bob Field (MIT) Peter Taylor (Univ of Warwick) Research Corporation, Dreyfus Foundation, NSF