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A Practical Procedure for ab initio Determination of Vibrational Spectroscopic Constants, Resonances, and Polyads William F. Polik Hope College, Holland, MI June 2006
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Chemical Reactions Occur via Excited Vibrational States
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HFCO Pure Vibrational Spectrum 3 1 HFCO
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Vibrational State Models HarmonicAnharmonicPolyad
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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
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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)
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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
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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)
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Spectroscopic Constants Refs: Nielsen (1959), Papousek & Aliev (1982)
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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)
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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
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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)
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Modified Spectroscopic Constants Refs: Papousek & Aliev (1982), Martin & Taylor (1997) partial fraction expansion drop resonance term(s)
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Resonance Constants Refs: Lehmann (1989), Martin & Taylor (1997)
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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
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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
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Compare to Manual Method
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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
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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
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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
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