Molecular Spectroscopy Symposium June 2013 Modeling the Spectrum of the 2 2 and 4 States of Ammonia to Experimental Accuracy John C. Pearson & Shanshan Yu Jet Propulsion Laboratory, California Institute of Technology, Pasadena CA 91109

2 Molecular Spectroscopy Symposium June 2013 Motivation In 2010 we asked if the spectrum of 2 2 / 4 could be modeled to experimental accuracy The answer is NO with the previous formulation of the Hamiltonian The same applies to all other ammonia bands lying above 2 The perils of ammonia have lead to widespread belief that we cannot fit the LAM-inversion and small amplitude vibration A huge effort has gone into spectroscopic potential to explain ammonia It is important in hot exoplanets While this has extended assignments, assignments are still quite limited in J There is lot of high quality data for example: – Sasada et al., 1992, JMS 151, 33 – Cottaz et al., 2000, JMS 203, 285 NH 3 inversion-normal vibration quantum mechanics worked out in detail – Full contact transformations performed – Simpler than 3-fold internal rotation – See Urban, 1988, JMS 131, and references therein

3 Molecular Spectroscopy Symposium June 2013 History Hamiltonian formulation fits the ground state and 2 to experimental accuracy Yu et al., 2010, JCP 133, Extrapolation and physical meaning of constants are known challenges, but it does model all experimental line positions correctly Predicted intensities of  K=3 transitions do not agree with experimental data Up to a factor of 100 differences! 2 2 / 4 Hamiltonian accounts of l-doubling and Coriolis but otherwise it is exactly the same formulation used in the ground state This applies to higher lying states as well. Two possibilities Something is fundamentally wrong with our understanding of NH 3 Something is wrong with the Hamiltonian as applied or as formulated Given the amount of theory the second possibility was the starting point

4 Molecular Spectroscopy Symposium June 2013 Quantum Mechanics 1/3 A relatively simple to read review of NH 3 quantum mechanics was presented by Sarka & Schrötter, JMS 179, (1996) Here H diag is diagonal in the basis of symmetric (s) and antisymmetric (a) wavefunctions and is the standard symmetric top Hamiltonian modified with an energy for the origin of (s) and (a) and an on-diagonal  K=6 term Here the subscript number is the power of the operators Where [A,B] + denotes the usual anticommutator AB+BA

5 Molecular Spectroscopy Symposium June 2013 Quantum Mechanics 2/3 Parameters with ,  j,  k are represented by nondiagonal symmetric matrices Parameters with ,  j,  k are represented by nondiagonal antisymmetric matrices Ground state and 2 Hamiltonians have only  parameters i.e. only symmetric operators The 4 state is l-doubled (K-l is the good quantum number) resulting in 4 states l=1,-1 (a) and l=1, -1 (s) Sasada et al and Cottaz et al Hamiltonian: –Diagonal part of Hamiltonian (  K=6 term) results in “K-type interactions” in 4 –Coriolis interactions between 2 2 and 4  K=  l=+/-1 –“l-type” interaction  (K-l)=0 between l=1 and l=-1 in 4 –And  (K-l)=3 interactions i.e. where <a| and <b| have different overall a/s symmetry

6 Molecular Spectroscopy Symposium June 2013 Quantum Mechanics 3/3 The part of the Hamiltonian: All matrix elements previously used are symmetric i.e.  terms –Same as ground state and 2 –Should we expect this to work for 2 2 and 4 ?

7 Molecular Spectroscopy Symposium June and 4 The splitting in 2 2 is much larger than in 4 in addition the primary interaction Involves the 2 2 (s) and the 4 l=1 (a) other interactions are more isolated. Numbers correspond to SPFIT V labels

8 Molecular Spectroscopy Symposium June 2013 Do we need antisymmetric  terms? Only one known attempt to include  terms (Belov et al. JMS, 84, (1988)) –Uncertainty was twice value ( 2 state) Validity of contact transformations was the subject of Sarka & Schötter JMS 179, (1996) Paper. –Contact transformation to remove  and  possible in very large inversion splitting i.e. isolated state limit –Intermediate inversion splitting contact transformation can remove  (i.e. ground state and 2 ) –Small inversion splitting contact transformation can remove  –Contact transformations such as these must be applied to the dipole  Not in the intensity calculations for ground state and 2 Contact transformations generally require an isolated state –The  4 & 2 2 states are degenerate and have two orders of magnitude difference in inversion splitting. –Unreasonable that a single transformation would work for both states –Perturbations can also amplify small effects The  terms should not be ignored in 4 & 2 2

9 Molecular Spectroscopy Symposium June 2013 Additional  (K-l)=3 Terms 1/2 1. Between (a) and (s) in Between (a) and (s) in 4 l=1 and l=-1 3. Between (a) and (s) in 4 l=1/-1 and l=1/-1 4. Between (s/a) and (s/a) 2 2 and 4  l=1  K=4 5. Between (s/a) and (s/a) 2 2 and 4  l=-1  K=2 The 4 mode is antisymmetric so interactions with 2 2 are s-s and a-a Within the 2 2 or 4 the symmetric terms Results in a 23JKVV (#1 & 3) for the same l and 21JKVV (#2) for  l=2 in SPFIT With in 2 2 or 4 the antisymmetric terms Results in a 22JKVV (#1 & 3) for the same l and 20JKVV (#2) for  l=2 in SPFIT

10 Molecular Spectroscopy Symposium June 2013 Additional  (K-l)=3 Terms 2/2 The symmetric term between 2 2 and 4  l=1  K=4 Results in a 64JKVV term (#4) The antisymmetric term between 2 2 and 4  l=1  K=4 Result is a 63JKVV term (#4) The symmetric term between 2 2 and 4  l=-1  K=2 Results in a 62JKVV term (#5) The antisymmetric term between 2 2 and 4  l=-1  K=2 Results in a 61JKVV term (#5)

11 Molecular Spectroscopy Symposium June 2013 Previous Data Sets Historical 2 2 and 4 band data –Cottaz et al., 2000, JMS 203, 285 –Sasada et al., 1992, JMS 151, 33 (inc microwave lines) –Lellouch et al., 1987, JMS 124, 333 (inc new assignments) –Papoušek et al., 1986, J. Mol. Struct. 141, 361 –Urban et al., 1984, Can. J. Phys 62, 1775 –Weber et al., 1984, JMS 107, 405 Historical Hot band data –D’Cunha, 1987, JMS 122, 130 –Hermanussen, Bizzarri & Baldacchini, 1986, JMS 119, 291 (inc new assignments) –Sasada et al., 1986, JMS 117, 317 Historical laser measurements –Chu, Li, & Cheo, 1994, JSQRT 51, 591 –Hillman, Jennings, & Faris, 1979, Appl. Opt. 18, 1808 –Kostiuk et al., 1977, IR Phys. 17, 431 –Sattler et al., 1981, JMS 88, 347 & JMS 90, 297 –Nereson, 1978, JMS 69, 489 –Bischel, Kelly, & Rhodes, 1976, Phys. Rev. A 13, 1829

12 Molecular Spectroscopy Symposium June 2013 New Data AC Discharge Emission Spectrum cm -1 –Pirali & Vervloet, 2006, Chem. Phys. Lett. 423, 376 Long path low pressure Synchrotron absorption Spectrum cm -1 ~2000 new lines assigned –Rotation-Inversion , , –Hot bands & A few hundred additional submillimeter frequency measurements S- 4 S l = A-S l = S l =1-2 2 S S- 4 S l = S- 4 S l = S l =1-2 2 S S- 4 S l = S-A

13 Molecular Spectroscopy Symposium June 2013 Fitting results Fits all but a few microwave transitions to <<1 MHz –A few of the outliers have been confirmed to be bad Fits the high J forbidden  (K-l)=3 band data –Urban et al., 1984, Can. J. Phys 62, 1775 –Only a few lines are problematic Allowed numerous additional assignment in –Hermanussen, Bizzarri & Baldacchini, 1986, JMS 119, 291 –Lellouch et al., 1987, JMS 124, 333 High J lines from discharge and Hot spectrum are being added –Does require more constants but looks like it will fit all

14 Molecular Spectroscopy Symposium June 2013 Preliminary Fitting Results (a)Calculated values from ground state force field  =2.93 MHz,  =0.45 MHz (b)Effectively a Coriolis interaction Most important is all of previous data (a few bad lines excluded) fits to a reduced RMS of 1.4 Including ~400 microwave transitions Should fit all with a few more higher order constants.

15 Molecular Spectroscopy Symposium June 2013 Conclusions The antisymmetric  K=3 terms cannot be excluded in NH 3 when there are interacting states –The interactions preclude the contact transformations required to re-formulate the Hamiltonian in a way that eliminates  in both states. The dipole moment in the ground and 2 states needs to account for the contact transformation made to eliminate the  terms in the analysis. –This probably accounts for the observed problem in the  K=3 transitions

16 Molecular Spectroscopy Symposium June 2013 Acknowledgement This work was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration