1. Outline 1. Introduction 2. User community and accuracy needs 3. Which large-amplitude motions 4. Which tools = which sym. operations 5. Example (in progress) C 2 H 3 + High-Barrier Multi-Dimensional Tunneling Hamiltonians

2. Acknowledgements Main collaborators on theoretical development of tunneling Hamiltonians 1980-2009 Nobukimi Ohashi (Japan) since 1984 Laurent H. Coudert (France) since 1986 Yung-Ching Chou (Taiwan) since 2003 Melanie Schnell (Germany) since 2006

3. High-Barrier Multi-Dimensional Tunneling Formalism Goal To parameterize high-resolution spectra of floppy molecules to experimental accuracy without knowledge of the potential surface Diatomic Example: Parameterize rotational levels as BJ(J+1) – DJ 2 (J+1) 2 instead of starting from an ab initio potential curve V(r)

4. High-Barrier Multi-Dimensional Tunneling Formalism  The Anti-Quantum-Chemist The Anti-Quantum-Chemist is: A great antagonist expected to fill the world with wickedness but to be conquered forever by the Quantum Chemist at his second coming. (from a reliable dictionary)

5. User Community Hierarchy High-barrier tunneling Hamiltonians Spectrum Analyzers  v J K a K c  Radio Astronomers , E , Int, molecule Physical Effects People  splittings =  E = E(vJ K a K c  ) - E(v  J  K a  K c   )  torsion, inversion, H-transfer, BO failure Primary Users Secondary Users

6. Parameterize to measurement accuracy Modern microwave measurements: 20 000.002 MHz or 200 000.02 MHz 1 part in 10 7 = 8 significant figures Requires energy calculations: with 8 + 1 + 1 = 10 significant figures = E'  E" and no computational error

7. Molecule List + How to Classify LAMs Use Vocabulary from Reciprocating Engine: (converts) to-and-fro of piston (to) circular motion of crankshaft Piston = N atom inversion, H atom transfer, H-bond exchange Crankshaft = CH 3 top, OH 2 top, H migration If Point Group, Permutation-Inversion Group, and types of motion are the same, then the Tunneling Hamiltonian is the same.

8. Molecule Year Groups LAMs H 2 N-NH 2 1981 C 2 G 16 (2) 1 torsion + 2 inversion HO-OH 1984 C 2 G 4 (2) 1 torsion HF-HF 1985 C s G 4 (2) 1 H-bond exchange H 2 O-H 2 O 1985 C s G 16 2 tors + 1 H-b exch CH 3 -NH 2 1987 C s G 12 (m) 1 torsion + 1 inversion C 2 H 3 + 1987 C 2v G 12 1 H-migration (  tor) (HCCH) 2 1988 C 2v G 16 1 torsion CH 3 NHD 1990 C 1 G 6 1 torsion + 1 inversion (CH 3 OH) 2 1993 C 1 G 36 2 torsion + 1 H-bond + 1 lone pair exchange (H 2 O) 2 H + 1994 C 2 G 16 (2) 1 tor + 2 inversions Na 3 1997 C 2v G 12 1 pseudorotation (tor)

9. (CH 3 O) 2 P(=O)CH 3 2002 C 1 G 18 2 torsion + 1 “inversion” CH 3 CHO S 1 2004 C 1 G 6 1 torsion + 1 inversion CH=O CH 3 -C H 2006 C s G 12 (m) CH O 1 torsion + 1 H transfer (CH 3 ) 3 SnCl 2008 C 3v G 162 3 torsions cis/trans HCCH 2010 C 2h G 4 (8) 1 torsion + 2 LAM bends

10. Tools 1. Point group at equilibrium  the number of frameworks 2. Tunneling Hamiltonian matrix itself 3. Permutation-Inversion group  symmetry as transformation of variables 4. Time reversal  complex conjugation 5. Extended Permutation-Inversion groups

11. Framework (E.B.Wilson 1935): Ball-and-stick model with atoms labeled Different frameworks cannot be oriented to make all atom labels match. H a OH b has 1 framework NH a H b H c has 2 frameworks (inversion) H a C b  C c H d has 2 frameworks (break CH bonds) CH 3 NH 2 has “6” frameworks (no bond breaking)

12. 6 Benzene out-of-plane  orbitals E W 0 0 0 W W E W 0 0 0 0 W E W 0 0 0 0 W E W 0 0 0 0 W E W W 0 0 0 W E H = Tunneling matrix has three kinds of elements: non-tunneling E, tunneling splitting W, and 0 |1;p z  |2;p z  |3;p z  |4;p z  |5;p z  |6;p z 

13. Some LAM-Rotation Tunneling Concepts: Use  for LAM angular (periodic) variables and  for LAM to-and-fro (  ) motions Framework basis functions = v,K set of fcns |  v (  s,  s)  n |KJM  n for each framework n H tun = Tunneling Hamiltonian =  i V i R i = f 0 (  s,  s) + f 1x (  s,  s)J x + f 1y (  s,  s)J y + f 1z (  s,  s)J z + f 2x (  s,  s)J x 2 + f 2xy (  s,  s)(J x J y +J y J x ) + … Tunneling matrix elements = T n =  |  v (  s,  s)  1 |KJM  1 |H tun ||  v (  s,  s)  n |KJM  n 

14. 1 2 3 4 …… n H rot T 2 T 3 T 4 …… T n T 2 † H rot T i T j …… T k T 3 † T i † H rot T p …… T q …………………………… T n † T k † T q † …….… H rot Framework # H rot dimension (2J+1)  (2J+1) T n = 1  n tunneling matrix (2J+1)  (2J+1) We want some T n = 0 Multi-Dimensional Tunneling Hamiltonian Matrix of dimension: (2J+1)n  (2J+1)n

15. P puts 1JK'  nJK" element also in another position in H (with possible phase change)  1;J,K'|H tun |n;J,K"  = P  1;J,K'|H tun |n;J,K"  = =  P1;J,K'|PH tun |Pn;J,K"  = = (-1) K'+K"  p;J,-K'|H tun |q;J,-K"  Permutation-Inversion symmetry operations as coordinate transformations Example for PI operation P: P  (  1,  2,  1,  2, , ,  ) = =  (  1 +2  /3,  2 -2  /3,-  2,+  1,-  + , ,  )

16. Molecule Year Lines/Parameters/rms H 2 N-NH 2 1981 HO-OH 1984 HF-HF 1985 H 2 O-H 2 O 1990 173 / 22 / 90 kHz (47) CH 3 -NH 2 2004 1523 IR+MW / 53 /18 kHz(346 MW ) C 2 H 3 + 2009 322 / 11 / 0.05 cm -1 (HCCH) 2 1988 CH 3 NHD 1990 (CH 3 OH) 2 1999 (H 2 O) 2 H + 1994 Na 3 1997

17. Year Lines/parameters/rms (CH 3 O) 2 P(=O)CH 3 2003 609 / 54 / 8 kHz CH 3 CHO S 1 2004 136 / 11 / 0.002 cm -1 CH=O CH 3 -C H 2008 2578 / 37 / 15 kHz CH O (CH 3 ) 3 SnCl 2008 cis/trans HCCH 2010

18. H 2   H 1  C a  C b  H 3  H 1  H 3  C a  C b  H 2  H 3  H 2  C a  C b  H 1 H 1  C a  C b  H 2  H 3 H 3  C a  C b  H 1   H 2 H 2  C a  C b  H 3  H 1  H migration in C 2 H 3 +

19. Parameterization of H-migration tunneling splittings is similar to the parameterization of methyl-top internal rotation splittings Int. rot. splittings Fa 1 cos(2  /3)(  K-  ) H-migration split. h 2  2 amplitude periodicity These two parameters and their higher-order J and K corrections occur in the off-diagonal matrix elements of the tunneling Hamiltonian

20. Six frameworks Nearest neighbor tunneling only 1 2 3 4 5 6 1 H rot T 2 0 0 0 T 2 2 T 2 † H rot T 2 0 0 0 3 0 T 2 † H rot T 2 0 0 4 0 0 T 2 † H rot T 2 0 5 0 0 0 T 2 † H rot T 2 6 T 2 † 0 0 0 T 2 † H rot C 2 H 3 + Tunneling Matrix (water dimer formalism) size of H rot T 2 (2J+1) x(2J+1)

21. C 2 H 3 + Data IR: Crofton, Jagod, Rehfuss, Oka, J. Chem. Phys. 91 (1989) 5139 MW: Bogey, Cordonnier, Demuynck, Destombes, Ap.J. 399 (1992) L103 C 2 H 3 + Tunneling Theory  K = 0: Hougen, JMS 123 (1987) 197  K = 0,  1: Cordonnier, Coudert JMS 178 (1996) 59

22. Laurent Coudert’s tunneling treatment uses h 2,  2 parameters to fit 3 splittings: Cordonnier, Coudert JMS 178 (1996) 59 Transition Obs. Splitting Calc. Splitting 9 09 – 8 18 -0.350(30) -0.341 MHz 1 10 – 1 01 -0.480(20) -0.492 MHz 3 12 – 3 03 -0.350(20) -0.351 MHz 5 lines < 0.3 < 0.26 MHz Obs. MW splittings from Lille Bogey, Cordonnier, Demuynck, Destombes Ap.J. 399 (1992) L103-L105

23. Can we use Coudert’s existing program to fit Oka et al.’s 500 assigned lines in the 3  C-H stretching region ??? Gabrys, Uy, Jagod, Oka, Amano on C 2 H 3 + in J. Phys. Chem. 99 (1995) 15611, say: “We hope that the observed splittings shown in Table 4 and Figure 3 will be quantitatively explained some day.”

24. Unknown factor: What is the magnitude of random vibrational perturbations at 3  ? Tunneling splittings  0.1 to 0.2 cm -1 If perturbations < 0.05 cm -1, try to fit. If perturbations > 0.2 cm -1, it’s hopeless. Next slide shows attempt to fit K = 0 splittings. K = 1 and 2 fits are similar, but with different amplitudes, damping, and periods.

25. C 2 H 3 + K = 0 A-E Splittings Calculated Observed

26. Conclusion (as of 20 June 2009): We have in hand a theoretical high-barrier tunneling-formalism fitting program for Oka’s C 2 H 3 + H-migration rotational energy levels. Before this Columbus meeting, there were some problems with the fit. At this Columbus meeting Laurent Coudert, Takeshi Oka, and I (a) were able to … (b) were not able to …