1. Calculation of rovibrational H 3 + lines. New level of accuracy Slides of invited talk at Royal Society conference on H 3 + Oleg L. Polyansky 1,2 1 Institute of Applied Physics, Russian Academy of Sciences, Uljanov Street 46, Nizhnii Novgorod, Russia 603950 2Department of Physics and Astronomy, University College London, London WC1E 6BT, UK. 9 th February, 2012

2. Calculation of rovibrational H 3 + lines. New level of accuracy Oleg Polyansky, Alex Alijah Kolya Zobov, Irina Mizus, Roman Ovsyannikov Lorenzo Lodi, Jonathan Tennyson, Attila Csaszar,

3. Analytical PES from the ab initio points

4. RV- Schroedinger equation with exact kinetic energy and PES V(r 1,r 2,  ) H  =E 

5. The highest H 3 + line. -3.0 and +8.5 cm -1 – previous predictions

7. Quotation...H 3 + spectroscopy which now entered the visible region with transitions up to 13676 cm -1. For such energies the deviations from theory are often more than 1 cm -1 and it gives further challenges to theorists... Morong, Gottfried and Oka, JMS, v.255, p.13, (2009)

8. Major goal of this talk is to demonstrate and prove 3 basic points 1.Before: 0.1 cm -1 up to 10000 cm -1 1 cm -1 between 10 000cm -1 and 13 000 cm -1 2.Now: 1 cm -1 => 0.1cm -1 Up to 17000 cm -1 3. Future: Opens the way to further progress 0.1cm -1 up to 20,25,30,35 000 cm-1 0.1 cm -1 => 0.01cm -1 Because some aspects of calculations – BO PES, adiabatic correction and relativistic correction are already 0.01 cm -1

9. Structure of this talk 1.Motivation (helps to appreciate the basic goal) 2. Global Analytical PES ( accurate to 0.1 cm -1 ) and comparison with previous PES 3. Accuracy of previous RV calcs (0.1 cm -1 up to 10 000 cm -1 and 1 cm -1 up to 13000 cm -1 ) 4. Our RV calcs (variational calculations and nBO models) 5. Comparison with experiment (0.1 cm -1 up to 17000 cm -1 ) 6. Conclusions and Future work

10. H3+H3+ Motivation Will help to appreciate the major goal Many honorary titles Simplest unsolved QM problem Smallest large QM system Smallest polyatomic molecule Smallest poly-electronic system

11. Ab initio predictions of water levels Isotopologue N levels J max  H 2 16 O 9426 20 1.17 H 2 17 O 1083 12 0.56 H 2 18 O 2460 12 0.65 D 2 16 O 2807 12 0.71 HD 16 O 1976 12 0.47 All water 17338 20 0.95 O. L. Polyansky, A. G. Csaszar, S. V. Shirin, N. F. Zobov, P. Barletta, J. Tennyson, D. W. Schwenke, P. J. Knowles, High-accuracy ab initio rotation-vibration transitions for water, SCIENCE, vol. 299, p. 539-542, 2003.

12. How it should be and how it is H 2 + H 2 H 3 + H 2 O Below barrier of 10 000 cm -1 10 -5 cm -1 3x10 -5 cm -1 10 -2 cm -1 1 cm -1 Above barrier 10 -5 cm -1 3x10 -5 cm -1 1 cm -1 1 cm -1

13. Water spectrum above disociation. The density of lines 1000 times lower than in Carrington-Kennedy predissociation spectrum of H 3 +

14. EXPERIMENTAL AND CALCULATED SPECTRUM OF WATER ABOVE DISSOCIATION Zobov,Shirin,Lodi,Siva,Tennyson,Csaszar,and Polyansky, Chem.phys.Lett.v.507,p.48,(2011)

15. Our Starting point (from previous talk) 10 -8 E h accuracy 42 000 points Dense and global grid Now I’ll show that all these aspects are important for our purposes

16. First 9 MBB-geometry points for various ab initio PES N n x n y n z E h E this work - E x (cm -1 ) CRJK RKJK LF MBB 1 -4 0 0 -1.255924 - 0.012 -1.206 -2.6 -40.3 2 -3 0 0 -1.296828 - 0.013 -1.127 -1.8 -23.1 3 -2 0 0 -1.323893 - 0.013 -0.960 -1.2 -10.9 4 -1 0 0 -1.339057 - 0.014 -0.724 -1.1 - 3.4 5 0 0 0 -1.343835 - 0.018 0.000 0.0 0.0 6 1 0 0 -1.339388 - 0.015 -0.037 -2.1 - 1.7 7 2 0 0 -1.326560 - 0.015 0.528 -6.5 1.8 8 3 0 0 -1.305893 - 0.021 1.323 -15.0 6.3 9 4 0 0 -1.277607 - 0.022 2.473 -49.1 11.0 CRJK – Cencek,Rychlewski, Jaquet,Kutzelnigg, JCP, v.108, 2831 (1998) RKJK - Roese, Kutzelnigg, Jaquet, Klopper, JCP, v.101, 2231 (1994) LF - Lie and Frye, JCP, v.96, 6784 (1992) MBB - Meyer, Botschwina, Burton, JCP, v.84, 891 (1986)

17. N n x n y n z E h E this work - E x (cm -1 ) CRJK RKJK LF MBB 10 5 0 0 -1.241529 - 0.021 4.276 -48.0 14.2 11 0 -4 0 -1.240043 - 0.017 -2.539 -4.1 - 7.2 12 0 -3 0 -1.287329 - 0.019 -1.849 -0.2 - 3.5 13 0 -2 0 -1.319311 - 0.019 -1.186 -1.1 - 1.3 14 0 -1 0 -1.337797 - 0.018 -0.661 -1.1 - 0.4 15 0 1 0 -1.337839 - 0.019 -0.741 -1.1 - 0.6 16 0 2 0 -1.319646 - 0.018 -1.338 -1.1 5.6 17 0 3 0 -1.288451 - 0.014 -1.384 -5.8 20.4 18 -4 -1 0 -1.244891 - 0.015 -1.286 -113.0 -31.9 19 -4 1 0 -1.245426 - 0.014 -1.275 -102.1 -40.4 20 -3 -2 0 -1.257927 - 0.015 -1.416 -87.0 -26.2 21 -3 -1 0 -1.287408 - 0.014 -1.205 -55.0 -23.8 22 -3 1 0 -1.287788 - 0.014 -1.202 -52.2 -23.5 CRJK – Cencek,Rychlewski Jaquet Kutzelnigg, JCP, v.108, 2831 (1998) RKJK - Roese Kutzelnigg Jaquet Klopper, JCP, v.101, 2231 (1994) LF - Lie and Frye, JCP, v.96, 6784 (1992) MBB - Meyer, Botschwina, Burton, JCP, v.84, 891 (1986) Next 12 MBB-geometry points for various ab initio PES

18. Analytical PES from the ab initio points

19. Functional form of the fitted PES Viegas, Alijah, Varandas, JCP,126,074309(2007)

20. Number of PES points and their sd in different energy regions of the GLH3P

23. Comparison of some ab intio points with Bachorz et. Al, JCP,v.131,24105(2009)   in cm -1 10.18 10.27 10.39

24. Rovibrational energy levels from Schroedinger equation Vibrational Energy Rotational Energy Potential Energy

25. Vibrational KE Non-orthogonal coordinates only Rotational & Coriolis terms Non-orthogonal coordinates only Reduced masses (g 1,g 2 ) define coordinates H  =E 

26. mode E obs / cm  1 MBB CP RKJK PT(BO) PT(nBO) (uncorr) 01 1 2521.409 2.5 5 0.3 -0.11 +0.056 10 0 3178.290 0.1 7 0.09 -1.3 +0.025 02 0 4778.350 5.4 21 1.1 0.0 +0.020 02 2 4998.045 5.0 6 0.8 -0.3 +0.010 11 1 5554.155 3.2 14 0.07 -1.4 0.000 1 2992.505 0.0 0.5 -1.46  0.020 2 2205.869 1.4 0.04 -0.47  0.050 3 2335.449 2.6 0.9 +0.47 +0.090 1 2736.981 0.2 0.2 -1.04 +0.001 2 1968.169 2.0 0.8 +0.58 +0.023 3 2078.430 1.2 0.4 -0.74  0.004 Ab initio vibrational band origins H2D+H2D+ H3+H3+ D2H+D2H+ MBB - - Meyer, Botschwina, Burton, JCP, v.84, 891 (1986) CP – Carney, Porter, JCP, v.65,3547(1976) RKJK- - Roese, Kutzelnigg, Jaquet, Klopper, JCP, v.101, 2231 (1994) PT - Polyansky and Tennyson, J. Chem. Phys., 110, 5056 (1999) – based on the points of CRJK - Cencek,Rychlewski, Jaquet, Kutzelnigg, JCP, v.108, 2831 (1998)

27. Correction to potential Adiabatic effects in H 3 + The Born-Handy approximation

28. Bunker and Moss, JMS, v.80, p.217 (1980) Correction to kinetic energy Nonadiabatic correction

29. mode E obs / cm  1 MBB CP RKJK PT(BO) PT(nBO) (uncorr) 01 1 2521.409 2.5 5 0.3 -0.11 +0.056 10 0 3178.290 0.1 7 0.09 -1.3 +0.025 02 0 4778.350 5.4 21 1.1 0.0 +0.020 02 2 4998.045 5.0 6 0.8 -0.3 +0.010 11 1 5554.155 3.2 14 0.07 -1.4 0.000 1 2992.505 0.0 0.5 -1.46  0.020 2 2205.869 1.4 0.04 -0.47  0.050 3 2335.449 2.6 0.9 +0.47 +0.090 1 2736.981 0.2 0.2 -1.04 +0.001 2 1968.169 2.0 0.8 +0.58 +0.023 3 2078.430 1.2 0.4 -0.74  0.004 Ab initio vibrational band origins H2D+H2D+ H3+H3+ D2H+D2H+ MBB - - Meyer, Botschwina, Burton, JCP, v.84, 891 (1986) CP – Carney, Porter, JCP, v.65,3547(1976) RKJK- - Roese, Kutzelnigg, Jaquet, Klopper, JCP, v.101, 2231 (1994) PT - Polyansky and Tennyson, J. Chem. Phys., 110, 5056 (1999) – based on the points of CRJK - Cencek,Rychlewski, Jaquet, Kutzelnigg, JCP, v.108, 2831 (1998)

30. mode E obs / cm  1 MBB CP RKJK PT(BO) PT(nBO) (uncorr) 01 1 2521.409 2.5 5 0.3 -0.11 +0.056 10 0 3178.290 0.1 7 0.09 -1.3 +0.025 02 0 4778.350 5.4 21 1.1 0.0 +0.020 02 2 4998.045 5.0 6 0.8 -0.3 +0.010 11 1 5554.155 3.2 14 0.07 -1.4 0.000 7870.020 -0.81 9113.080 +0.93 11323.100 +0.55 11658.400 +7.58 Ab initio vibrational band origins MBB - - Meyer, Botschwina, Burton, JCP, v.84, 891 (1986) CP – Carney, Porter, JCP, v.65,3547(1976) RKJK- - Roese, Kutzelnigg, Jaquet, Klopper, JCP, v.101, 2231 (1994) PT - Polyansky and Tennyson, J. Chem. Phys., 110, 5056 (1999) – based on the points of CRJK - Cencek,Rychlewski, Jaquet, Kutzelnigg, JCP, v.108, 2831 (1998)

31. 2521.512521.460.05 3179.593179.560.02 4778.334778.160.17 4998.314998.170.14 5555.415555.47-0.05 6264.446264.57-0.14 7006.087005.810.27 7285.487285.440.05 7770.187770.55-0.36 7870.837871.34-0.51 8489.368490.23-0.87 9001.009000.890.10 9112.159112.21-0.06 9254.759255.41-0.66 9653.299653.96-0.67 Difference between energy levels of H3+ for BO on (PT99 and GLH3P) Difference as it should be for levels below 10000 cm -1

32. 9966.779969.20-2.43 9997.479998.37-0.90 10592.7310595.27-2.55 10643.4210647.00-3.58 10856.9010857.99-1.09 10918.0310919.75-1.72 11322.5511326.12-3.57 11650.8211655.10-4.28 11809.7011814.75-5.05 13283.3513293.69-10.34 13306.9113319.92-13.01 13388.5213398.25-9.72 13432.9313443.71-10.78 13579.2913590.81-11.53 13705.0113715.01-10.00 14044.6114057.69-13.08 14139.8714157.99-18.11 Difference between energy levels of H3+ for BO only (PT99 and GLH3P) Ab initio points differ no more than 0.1 cm -1 in 69 MBB geometries Why this big difference in energies?

33. GLH3P – PT99 PT99 – Polyansky and Tennyson, JCP,v.110,5056 (1999) – 69 MBB geometries, sd – 4.5 cm -1

34. GLH3P-PPKT PPKT – Polyansky, Prosmiti, Klopper and Tennyson, Mol.Phys,v.98,261(2000) – 200 geometries, sd – 1.0 cm -1

35. mode E obs / cm  1 MBB CP RKJK PT(BO) PT(nBO) 01 1 2521.409 2.5 5 0.3 -0.11 +0.056 10 0 3178.290 0.1 7 0.09 -1.3 +0.025 02 0 4778.350 5.4 21 1.1 0.0 +0.020 02 2 4998.045 5.0 6 0.8 -0.3 +0.010 11 1 5554.155 3.2 14 0.07 -1.4 0.000 7870.020 -0.81 9113.080 +0.93 11323.100 +0.55 11658.400 +7.58 THUS the reason for this large discrepancy – BO PES used in PT Ab initio vibrational band origins MBB - - Meyer, Botschwina, Burton, JCP, v.84, 891 (1986) CP – Carney, Porter, JCP, v.65,3547(1976) RKJK- - Roese, Kutzelnigg, Jaquet, Klopper, JCP, v.101, 2231 (1994) PT - Polyansky and Tennyson, J. Chem. Phys., 110, 5056 (1999) – based on the points of CRJK - Cencek,Rychlewski, Jaquet, Kutzelnigg, JCP, v.108, 2831 (1998)

36. Thus, we proved that better BO PES is needed. Now we can use this global GLH3P BO PES, which is now extremely accurate and dense For rovibrational calculations

37. Relative contribution of BO-PES, adiabatic, nonadiabatic and relativistic corrections to the accuracy of optical lines calculations

38. Obs-calc. BO+adiabatic –grey, full model – red and yellow

40. The highest H 3 + line. -3.0 and +8.5 cm -1 – previous predictions

41. Part of a table from Bachorz et. al, JCP, v131, 024105 (2009) Last column – our calculations

42. H3+H3+

43. H2D+H2D+

44. Quotation...Our measurements include high rotational lines up to J=6. Such high J lines have high deviations from theory and are particularly challenging to theorists... Morong, Gottfried and Oka, JMS, v.255, p.13, (2009)

45. Part of the table of Morong, Gottfried and Oka, JMS, v.255, p.13, (2009) with the mentioned high J lines

46. We fitted 4250 dipole moments with the standard deviation 0.001 to DMS. Using our PES and DMS calculated the intensities of McKellar, Watson JMS, 191, 215(1998)

47. Table of intensities. Comparison with Watson and McKellar, JMS, v.191, 215 (1998)

48. Table of intensities. Comparison with Watson and McKellar, JMS, v.191, 215 (1998),continued

49. Intensity calcs Strong lines on average 2%, for all lines -4% Need more accurate intensity measurements to be able to demonstrate the full potential of our DMS, but even now we can state that our linelists can provide not only 0.1 cm-1 line positions, but few % lineintensity

50. CONCLUSIONS Accurate ab initio calculations 10 -8 E h (previous talk) Dense grid and 42000 points Accurate fit to analytical surface 0.097 cm -1 Globally accurate PES GLH3P 0.1 cm -1 observed – calculated Hopefully 0.01 cm -1 of BO, adiabatic, relativistic

51. CONCLUSIONS 0.1 cm -1 observed – calculated up to 17000 cm the work done. Future work: 0.1 cm for 20,25,30,35 000 cm -1 could demonstrate only if experiment could be done 0.1 cm -1 => 0.01 cm -1 – improvement of non- adiabatic models and QED calculations