1. Vibration-rotation spectra from first principles Lecture 2: Calculations of spectroscopic accuracy Jonathan Tennyson Department of Physics and Astronomy University College London OSU, February 2002

2. “Experiments (are) measured to tenths of wave numbers… this level of accuracy in a calculation is meaningless” Freisner, Bentley, Menou and Leforestier, J. Chem. Phys. 99, 324 (1993)

3. For triatomics accuracy determined by: The potential energy surface The validity of a potential (ie the Born-Oppenheimer approximation) Potentials: from electronic structure calculations spectroscopically determined

4. Potentials: Ab initio or Spectroscopically determined

5. Using spectra to improve a potential? 1.Guess form eg V(r 1,r 2,  ) =  c i f i (r 1,r 2,  ) 2.Compute obs  calc and standard deviation 3.Compute derivatives. Hellman-Feynman theorem d /dc = gives d /dc i = 4. Repeat calculation with improved V(r 1,r 2,  ) Guesses improved using specialist techniques eg I-NoLLS: a program for interactive nonlinear least-squares fitting of the parameters of physical models, M.M. Law & J. M. Hutson, Comp. Phys. Commun., 102, 252 (1997).

6. Fitting to spectroscopic data Best start: high quality ab initio calculation (starting point usually determines quality of fit). Final fit usual in 2 – 3 iterations (But many tests first!) Usually fit energy levels rather than spectra Fit vibrational and rotational data simultaneously (Essential for light molecules) Born-Oppenheimer approximation !? Fit 20 – 30 parameters (only).

7. Spectroscopically determined water potentials ReferenceYear  vib /cm -1 N vib E max /cm -1 Hoy, Mills & Strey19722142513000 Carter & Handy1987 2.422513000 Halonen & Carrington1988 5.355418000 Jensen1989 3.225518000 Polyansky et al (PJT1)1994 0.64018000 Polyansky et al (PJT2)1996 0.946325000 Partridge & Schwenke1997 0.334218000 Shirin et al2002 0.109925000  mportant to treat vibrations and rotations

8. Spectroscopically determined potential Polyansky, Jenson & Tennyson (PJT1), J. Chem. Phys., 101, 7561 (1994) Fit: 1600 term values with J up to 14 a New experimental value: 10899.64 cm -1

9. Ab initio accuracy better than 1cm  1 Adiabatic or Born-Oppenheimer Diagonal Correction (BODC) Non-adiabatic corrections for vibration and rotation Electronic (kinetic) relativistic effect Relativistic Coulomb potential (Breit effect) Radiative correction (Lamb shift or qed) Can BO electronic structure calculations be done this accurately? Variational rotation-vibration calculations with exact kinetic energy operator accurate to better than 0.001 cm  1

10. H 3 + H 2 O H 2 S HCN/HNC Molecule considered at high accuracy

11. Ab initio Born-Oppenheimer potentials for H 3 + Year Authors E min / E h  E / cm 1975 Carney & Porter  1.33519 1900 1980 Schinke et al  1.34023 790 1985 Burton et al  1.34188 430 1986 Meyer et al  1.34309 160 1990/2 Frye et al  1.343828 9 1994 Rohse et al  1.3438336 1 1998 Cencek et al a  1.3438355 0.04 For spectroscopy shape is more important than magnitude a Also electronic relativistic correction, ~ 3 cm  1

12. Adiabatic effects in H 3 + The Born-Handy approximation

13. mode E obs / cm -1 BO +  V ad 01 1 2521.409  0.11  0.24 10 0 3178.290  1.30  0.40 02 0 4778.350 0.00  0.50 02 2 4998.045  0.30  0.64 11 1 5554.155  1.40  0.50 1 2992.505  1.46  0.36 2 2205.869  0.47  0.25 3 2335.449 +0.47  0.14 1 2736.981  1.04  0.28 2 1968.169 +0.58  0.11 3 2078.430  0.74  0.18 Ab initio vibrational band origins H2D+H2D+ H3+H3+ D2H+D2H+

14. Non-adiabatic effects in diatomics P.R. Bunker and R.E. Moss, Mol. Phys., 33, 417 (1977)

15. Vibrational KE Non-orthogonal coordinates only Rotational & Coriolis terms Non-orthogonal coordinates only Effective Hamiltonian after intergration over angular and rotational coordinates. Case where z is along r 1 Reduced masses (g 1,g 2 ) define coordinates

16. Non-adiabatic effects in the ST Hamiltonian

17. mode E obs / cm  1 BO +  V ad  v   nuc 01 1 2521.409  0.11  0.24 +0.056 10 0 3178.290  1.30  0.40 +0.025 02 0 4778.350 0.00  0.50 +0.020 02 2 4998.045  0.30  0.64 +0.010 11 1 5554.155  1.40  0.50 0.000 1 2992.505  1.46  0.36  0.020 2 2205.869  0.47  0.25  0.050 3 2335.449 +0.47  0.14 +0.090 1 2736.981  1.04  0.28 +0.001 2 1968.169 +0.58  0.11 +0.023 3 2078.430  0.74  0.18  0.004 Ab initio vibrational band origins H2D+H2D+ H3+H3+ D2H+D2H+ O.L. Polyansky and J. Tennyson, J. Chem. Phys., 110, 5056 (1999).

18. J K a K c J K a K c E obs / cm -1 BO +  V ad  v   nuc + K NBO 3 2 1 3 2 2 2225.501  0.385  0.245  0.062  0.044 3 2 1 2 0 2 2448.627  0.521  0.259  0.011  0.076 2 2 0 2 2 1 2208.417  0.435  0.242  0.050  0.068 2 2 1 2 0 2 2283.810  0.521  0.239 +0.030  0.059 2 2 0 1 0 1 2381.367  0.573  0.250 +0.008  0.060 3 3 1 2 1 2 2512.598  0.647  0.250 +0.075  0.099 2 0 2 3 1 3 2223.706  0.418  0.163 +0.050 +0.068 2 2 1 3 1 2 2242.303  0.753  0.151 +0.140 +0.095 2 1 2 2 2 1 2272.395  0.420  0.168 +0.035 +0.099 2 2 0 2 1 1 2393.633  0.320  0.162 +0.140 +0.087 3 3 1 3 2 2 2466.041  0.224  0.164 +0.190 +0.080 3 3 1 2 2 0 2596.960  0.185  0.177 +0.167 +0.077 3 3 0 2 2 1 2602.146  0.203  0.172 +0.167 +0.080 2 3 H 2 D + : ab initio spectra

19. Rotational non-adiabatic effects Use  nuc given by nuclear mass Explicit inclusion of effect via rotational g-factors PR Bunker & RE Moss, J. Mol. Spectrosc., 80, 217 (1980) Preliminary results for H 3 + Calculations for all observed levels, J up to 15 Reproduces observations to better than 0.001 x J 2 cm  1 for vibrational ground state OL Polyansky, MA Kostin, J Tennyson, BT Sutcliffe, I Paidarova & SPA Sauer, to be published

20. Ab initio predictions of water vibrational fundamentals

21. Reference Year Barrier height/cm  1 Comment Carter and Handy 1987 11493 Spectroscopic Empirical Jensen 1989 11246 Spectroscopic Empirical Polyansky et al (PJT2) 1994 10966 Spectroscopic Empirical Lanquetin et al 1999 11154 Effective Hamiltonian Partridge & Schwenke (PS) 1997 11155 Ab initio Partridge & Schwenke 1997 11128 Spectroscopic Empirical PS + adiabatic + relativistic 1998 11192 Ab initio Csaszar et al 1998 11046  70 Extrapolated ab initio Tarczay et al 1999 11127  35 High accuracy ab initio Kain et al 2000 11105  5 Corrected ab initio Valeev et al 2001 11119  12 Ab initio (MP2 – R12) Water: Barrier to linearity

22. Achieving a “perfect” ab initio potential Need to consider: (for water) SCF at full basis set limit (done) Valence CI to full basis set limit (by extrapolating from large basis calculation) Extension of CI to full CI limit (only possible with v. small, eg DZP, basis set) Core – valence correlation New high accuracy extrapolated ab initio calculations in progress Polyansky, Csaszar, Tennyson, Barletta, Shirin, Zobov & Schwenke The future: explicit inclusion of r 12 into the wavefunction

23. Ab initio: vibrational errors

24. Ab initio + Adiabatic: vib. errors

25. Ab initio + adiabatic + relativistic MVD1 Csaszar, Kain, Polyansky, Zobov and Tennyson, Chem. Phys. Lett., 293, 317 (1998).

26. Ab initio +Gaunt 1 +Breit 2 Obs / cm  1 (010) 1598.19 +0.10 +0.04 1594.75 (020) 3158.49 +0.18 +0.09 3151.63 (030) 4677.22 +0.21 +0.10 4666.79 (040) 6148.29 +0.20 +0.05 6134.01 (050) 7561.09 +0.10  0.10 7542.44 (060) 8894.52  0.16  0.35 8869.95 (101) 7249.52 +1.60 +1.32 7249.82 (201) 10612.70 +2.34 +1.94 10613.35 (301) 13829.31 +3.07 +2.54 13830.94 (401) 16896.50 +3.87 +3.20 16898.84 (501) 19776.00 +4.44 +4.04 19781.10 Relativistic electronic potential effects in water 1 Gaunt correction: 1 electron approximation 2 Breit correction: full calculation H.M. Quiney, P. Barletta, G. Tarczay, A.G. Csaszar, O.L. Polyansky and J. Tennyson, Chem. Phys. Lett., 344, 413 (2001). (also D2)

27. 2s, 2p 2p 3/2 2p 1/2 2s 1/2 The hydrogen atom: n = 2 levels Fine structure Non- relativistic 0.365 cm -1 2p 1/2 0.035 cm -1 Lamb shift 2p 3/2 SchrodingerDiracQED

28. Ab initio + Lamb Obs / cm  1 (010) 1598.19  0.09 1594.75 (020) 3158.49  0.18 3151.63 (030) 4677.22  0.29 4666.79 (040) 6148.29  0.43 6134.01 (050) 7561.09  0.60 7542.44 (060) 8894.52  0.86 8869.95 (101) 7249.52 +0.37 7249.82 (201) 10612.70 +0.54 10613.35 (301) 13829.31 +0.71 13830.94 (401) 16896.50 +0.83 16898.84 (501) 19776.00 +1.01 19781.10 (601) 22519.69 +1.19 22529.44 (701) 25105.51 +1.29 25120.28 One-electron Lamb shift effects in water P. Pyykko, K.G. Dyall, A.G. Csaszar, G. Tarczay, O.L. Polyansky and J. Tennyson, Phys. Rev. A, 63, 024502 (2001)

29. BO / cm  1 +BODC 1 + Non-adiabatic  v   nuc 2 diag 3 full 4 (010) 1597.60  0.46  0.19  0.06  0.07 (020) 3157.14  0.94  0.38  0.12  0.15 (100) 3661.00  0.55  0.46  0.72  0.70 (030) 4674.88  1.43  0.55  0.18  0.23 (110) 5241.83  0.16  0.65  0.77  0.76 (040) 6144.64  2.00  0.71  0.23  0.30 (120) 6784.56  0.23  0.83  0.83  0.84 (200) 7208.80  1.25  0.88  1.39  1.37 (002) 7450.86  1.47  0.90  1.47  1.57 (050) 7555.62  2.71  0.84  0.28  0.32 Born-Oppenheimer corrections for water 1 Born-Oppenheimer diagonal correction using CASSCF wavefunction 2 Non-adiabatic correction by scaling vibrational mass,  V 3 Two parameter diagonal correction 4 Full treatment by Schwenke (J. Phys. Chem. A, 105, 2352 (2001).) J. Tennyson, P. Barletta, M.A. Kostin, N.F.Zobov, and O.L. Polyansky, Spectrachimica Acta A (in press).

30. Assignments using branches Ab initio potential J Error / cm -1 Determined potential Spectroscopically Variational calculations:

32. Polyad structure in water absorption spectrum Long pathlength Fourier Transform spectrum recorded by R Schmeraul

33. R. Schermaul, R.C.M. Learner, J.W. Brault, A.A.D. Canas, O.L. Polyansky, D. Belmiloud, N.F. Zobov and J. Tennyson J. Molec. Spectrosc. (in press) Weak lines

34. New experimental measurements REIMS data Carleer et al. Bruker F.T.S Range :13200  25020 cm -1 T : 291 K p(H 2 O) : 18.5 hPa pathlength ~ 602.32 m Number of new lines : 2286 IMPERIAL data (R.A.L) Schermaul et al. Bruker F.T.S Range :13350  14750 cm -1 T : 294.4 K p(H 2 O) : 23.02 hPa pathlength ~ 800.75 m Number of lines : 3179 Number of new lines : 963 Weak water linesVery difficult to record Only a few weak lines in HITRAN Also Kitt Peak archive dataAlso spectra 8000 – 13500 cm 

35. Water vapour spectrum: new assignments in the blue Long pathlength FTS M Carleer et al, J. Chem. Phys., 111, 2444 (1999)

36. Vibrational mode Previous work a This work b band origin Local Normal lines levels lines levels cm  1 (4,2)  1 (115) 10 5 22513. (7,0) + 0 (700) 5 2 90 39 22529.296 (7,0)  0 (601) 42 20 57 15 22529.445 (6,0)  2 (521) 16 10 22630. (7,0)  1 (611) 16 10 23947. (8,0) + 0 (800) 24 20 25120. (8,0)  0 (701) 12 6 23 18 25120.278 Water: Rotation-Vibration spectra in the near ultra violet a C. Camy-Peyret et al, J. Mol. Spectrosc., 113, 208 (1985). b N.F. Zobov et al, J. Chem. Phys., 113, 1546 (2000).

37. Intensity data compared to HITRAN-96 by polyad for spectral region 8500 – 15800 cm -1 HITRAN underestimates intensity of strong lines! D Belmiloud et al, Geophys. Res. Lett., 27, 3703 (2000). Numbers are ratio of total intensity to HITRAN PolyadIntegrated absorbance Spectral linefits Ab Initio calculation Correction Giver et al.  1.261.310.92  1.191.211.041.14  1.261.25 1.09  1.061.040.96

38. Frequency / cm -1 Water absorption by the atmosphere: Standard Model W Zhong, JD Haigh, D Belmiloud, R Schermaul & J Tennyson, Quart. J. Roy. Metr. Soc., 127, 1615 (2001)

39. Frequency / cm -1 Water absorption by the atmosphere: correction of Giver et al (2000)

40. Frequency / cm -1 Water absorption by the atmosphere: Effect of weak water lines

41. Frequency / cm -1 Water absorption by the atmosphere: Effect of ESA-WVR linelist

42. Missing absorption due to water: First estimates Theory Experiment Radiative Transfer Model Atmospheric absorption  In the red and visible :  Unobserved weak lines have a significant effect : ~ 3 Wm -2  Estimated additional 2.5-3 % absorption in the near I.R/Red.  Estimated additional 8-11 % absorption in the ‘Blue’ ?  Underestimate of strong lines even more important : ~ 8 Wm -2  Estimated additional 8 % absorption in the near I.R/Red.

43. Missing absorption due to water: Outstanding issues  In the near infrared and red:  Contributions due to H 2 18 O, H 2 17 O and HDO.  Possible role of water dimer (H 2 O) 2.  In the blue and ultraviolet:  Are H 2 16 O line intensities also underestimated?  Contribution due to weak lines

44. Sensitivity of vibrational band origins EffectContribution / cm -1 H 2 O H 2 S H 3 + BO convergence+ 30 +/  0.03 Relativistic correction (1e)  19  +/  0.03 Darwin term (2e)  0.8  a Gaunt correction+ 5  a Breit correction+ 6+ 0.03 a QED+1.3+ 1.5 a Adiabatic correction (BODC)+ 5+ 2 +/  1.5 Non-adiabatic correction  4~  3  0.5 a Unknown, assumed negligible

45. Water assignments using variational calculations Long pathlength absoption (T = 296K) 11000 - 25000 cm -1 Fourier Transform and Cavity Ring Down Laboratory emisson spectra (T =1300  1800K) 400 – 6000 cm -1 Absorption in sunspots (T = 3200 K) N band, L band, K band 10-12  m 3  m 2  m  25000 new lines assigned Dataset of 12000 measured H 2 16 O energy levels J. Tennyson, N.F. Zobov, R. Williamson, O.L. Polyansky & P.F. Bernath, J. Phys. Chem. Ref. Data, 30, 735 (2001).

46. Oleg Polyansky Nikolai Zobov Maxim Kostin Paolo Barletta Mizuho Tanaka Roman Tolchenov