A New Hybrid Program For Fitting Rotationally Resolved Spectra Of methylamine-like Molecules: Application to 2-Methylmalonaldehyde Isabelle Kleiner a and Jon T. Hougen b a LISA, Université de Paris Est and CNRS, Créteil, F-94010, France b Sensor Science Division, NIST, Gaithersburg, MD 20899, USA Outline 1. General need for a new fitting program. 2. Theoretical approach of the “hybrid” program: Partitioned H matrix, Basis set, Interaction terms, 3. Specific problem in 2-methyl malonaldehyde. 4. Successful solution of this specific problem. 5. Future directions. 1

Need for a hybrid program for methylamine-type molecules. What is a “methylamine-like molecule”? = A molecule with 2 Large Amplitude Motions: 1 internal rotation motion (rotatory) 1 back-and-forth motion (oscillatory) In CH 3 -NH 2 : Internal rotation = methyl-group rotation Back-and-forth motion = amino-group inversion In 2-methyl malonaldehyde (see next slide): Internal rotation = methyl-group rotation Back-and-forth motion = hydrogen-atom transfer 2

Two large-amplitude motions in methyl malonaldehyde: Intramolecular hydrogen transfer Internal rotation of a methyl rotor Intramolecular hydrogen transfer induces a tautomerization in the ring, which then triggers a 60 degree internal rotation of the methyl rotor. C4C4 C6C6 C5C5 O7O7 H9H9 H 11 H 10 C 12 H3H3 H2H2 H1H1 C4C4 C6C6 C5C5 O7O7 O8O8 H9H9 H 11 H 10 C 12 H1H1 H3H3 H2H2 (123)(45)(78)(9,10) O8O8

Need for a hybrid program Up to now, the rotational levels of methylamine-like molecules have been fit nearly to measurement error by a pure tunneling Hamiltonian formalism *. Its two main deficiencies (which the hybrid program is supposed to fix) are: -It cannot treat vibrational states near or above the top of the barrier to any tunneling motion. -It cannot treat the tunneling components of two different vibrational states at the same time. 4 * N. Ohashi, J. T. Hougen, J. Mol. Spectrosc. 121 (1987)

Theoretical approach of the “hybrid” program For internal rotation use the RAM Hamiltonian of Herbst et al (1984): F(P   J z ) 2 + ½V 3 (1  cos3  ), + higher order torsion-rotation interaction terms as found in the BELGI code. For the motion in a double-well potential For the motion in a double-well potential (-NH 2 inversion or H transfer motion), use a tunneling formalism, where H = T + V is replaced with one tunneling splitting parameter and its many higher- order torsion-rotation corrections. 5

THEORETICAL MODEL: THE GLOBAL APPROACH H RAM = H rot + H tor + H int + H d.c. RAM = Rho Axis Method (axis system) for a C s (plane) frame Constant s 1 1-cos3  p2p2 JapJap 1-cos6  p4p4 Jap3Jap3 1V 3 /2F  V 6 /2k4k4 k3k3 J2J2 (B+C)/2*FvFv GvGv LvLv NvNv MvMv k 3J Ja2Ja2 A-(B+C)/2*k5k5 k2k2 k1k1 K2K2 K1K1 k 3K J b 2 - J c 2 (B-C)/2*c2c2 c1c1 c4c4 c 11 c3c3 c 12 J a J b +J b J a D ab or E ab d ab  ab  ab d ab6  ab  ab Torsional operators and potential function V(  ) Rotational Operators Hougen, Kleiner, Godefroid JMS 1994  = angle of torsion,  = couples internal rotation and global rotation, ratio of the moment of inertia of the top and the moment of inertia of the whole molecule Kirtman et al 1962 Lees and Baker, 1968 Herbst et al 1986

7 Double-well potential for the back-and-forth motion, i.e., for the –NH 2 inversion or the H transfer motion L(eft)R(ight) L()L() R()R()

8 Theoretical approach of the “hybrid” program Hamiltonian matrix is partitioned into Left-well and Right-well blocks = LL, LR, RL, RR Basis set: exp[+i(3k+  )  ]  L (  ) |K,J,M  exp[+i(3k+  )  ]  R (  ) |K,J,M  BELGI Tunneling L 7x(2J+1) R 7x(2J+1) L 7x(2J+1) R 7x(2J+1) H =

Theoretical approach of the “hybrid” program Interaction terms include all G 12 group-theoretically allowed products of powers of the basic operators: Torsional motion: P  k, cos3m , sin3n , Back-and-forth motion: P ,  Rotational motion: J x p, J y q, J z r e.g., Operators Occur in blocks P  2, cos6 , J x 2, J y 2, J z 2 LL, RR, LR, RL  cos3 ,  (J x J z +J z J x ) LL, RR P  J y LR, RL 9

Specific problem in 2-methyl malonaldehyde Pure tunneling formalism ( Ilyushin et al. JMS 251 (2008) 56 ) : 3-fold increase in torsional splitting when OH  OD, but CH 3 remains CH 3

Specific problem in 2-methyl malonaldehyde Use the tunneling splittings for 2MMA-d0 and -d1 from Ilyushin et al to determine a V 3 barrier. Lin and Swalen, Rev. Mod. Phys. 31 (1959) 841. H = F P  2 + ½ V 3 (1-cos3  ) Calculate F from ab initio structure and fix it. Determine V 3 from A-E tunneling splitting for both –OH and –OD isotopologs of 2-MMA. Check for consistency: V 3 (OH) = 399 cm -1 V 3 (OD) = 311 cm -1

12 Successful solution of the specific 2-methylmalonaldehyde problem Overview of the 2-MMA-d0 and -d1 fits in Ilyushin et al. JMS 251 (2008) 56 and in the present work 2-MMA-d0 2-MMA-d1 Ref. [8] This work Ref. [8] This work Total lines Total parameters  fit (unitless) _______________________________________________________________ A-species lines wrms of A lines E-species lines wrms of E lines ___________________________________________________________________________________________________________ # of 2 kHz lines   rms of 2 kHz lines   # of 10 kHz lines rms of 10 kHz lines # of 50 kHz lines rms of 50 kHz lines

13 Successful solution of the specific 2-methylmalonaldehyde problem V 3 values from the old pure tunneling and the new hybrid formalism 2-MMA-OH 2-MMA-OD Diff. Pure tunneling formalism: V 3 = 399 cm -1 V 3 = 311 cm cm -1 Hybrid formalism: V 3 = cm -1 V 3 = cm cm -1 N.B. The hybrid fit numbers are not yet final, but the -OH vs -OD discrepancy has been greatly reduced.

Future directions Fit more than one vibrational state simultaneously Try to get a global fit of CH 3 NH 2 rotational levels in the v torsion = 0 and 1 states with v inversion = 0. Much of this MW and FIR data is already in the literature. Fit torsional levels near or above the top of a low barrier Try to fit the rotational levels of the hydrogen-bonded complexes H 3 N-HOH and CH 3 CN-HOH. Both of these have a C 3v rotor with a low barrier (~10.5 cm-1) and a double minimum for the H exchange motion: H 3 N-H a OH b  H 3 N-H b OH a. Some data for these species are already in the literature. (Stockman, Bumgarner, Suzuki, and Blake, JCP 1992) 14

15 Stockman et al 1992

16 Diagram of Frameworks for the Pure Tunneling Formalism Diagram of Frameworks for the Hybrid Formalism