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Hamiltonians for Floppy Molecules (as needed for FIR astronomy) A broad overview of state-of-the-art successes and failures for molecules with large amplitude motions (LAMs) Jon T. Hougen Sensor Science Division, NIST Gaithersburg, MD

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What can experiment and Hamiltonians for floppy molecules “easily” provide, that FIR astronomers might want? Line lists for molecules of interest with: 1.Observed frequencies obs (exp.error) 2.Calculated frequencies calc (calc.error) 3.Calculated Intensities I 4.Lower state energies E” 5.Quantum number assignments 2

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Effective Hamiltonians (H eff ) are the only tools available at the moment for fitting and calculating (interpolating) line positions to experimental accuracy. In general ab initio methods cannot provide kHz or MHz accuracy. Effective Hamiltonians sacrifice physics “purity” by using “empirical” quantum mechanical interaction terms to get experimental accuracy in their H eff calculations. 3

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But effective Hamiltonians cannot be used at present for all molecules with large-amplitude motions (LAMs). We therefore now survey examples of: (i) classes of vibrational states and classes of molecules that can be accurately treated, and examples of: (ii) classes of states and molecules that cannot be accurately treated by H eff ’s. 4

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Classes of Vibrational States (= isolated versus not isolated): Rotational levels (& therefore transitions) in ground vibrational states can often be treated very well. Rotational levels (& therefore transitions) in excited vibrational states can only be treated “well” if the state is “isolated” from other vibrational states. 5

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2 broad classes of LAMs: Periodic motions = internal rotations Reciprocating motions (back and forth) = = inversions, proton exchange, etc. Periodic motions are easy to treat, using rapidly converging (short) Fourier series for V and . Reciprocating motions are hard to treat, using slowly converging (long) power series for V and . 6

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3 broad classes of molecules: 1. Only periodic motions (Fourier series in H = T + V and in wavefunctions). 2. Only reciprocating motions (power series in H = T + V and in wavefunctions). 3. Both periodic & reciprocating motions (high-barrier tunneling H = T + V). Make some general remarks in the next few slides on molecules in each class. 7

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Molecules with only one symmetric-top internal rotor LAM are fit to 1 to 20 kHz. Calc’d intensities are probably OK to 20%. Three-fold barrier (e.g., methanol) Six-fold barrier (e.g., toluene) Any point group symmetry at equilibrium My favorite computational method: Use Herbst’s procedure for any barrier height, but limited by computation time to low J values = low temperatures T < 100 K. 8

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9 Molecules with only one asymmetric-top internal rotor (e.g., CH 2 DOH or CD 2 HOH): Hilali, Coudert, Konov, and Klee report in JCP 135 (2011) a fit of 76 torsional subband centers in CH 2 DOH with rms = 0.09 cm -1 >> 20 kHz. This is good enough for low-resolution FIR astronomical observations, but probably not good enough for anticipated higher- resolution data.

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10 Molecules with only two symmetric-top internal rotors as LAMs are partly under control (to kHz), but no intensities. Programs now available (year or no) are: Point group symmetries C s (2010) and C 1 (no) for 2 inequivalent rotors, with PI group = G 18, e.g., methyl acetate. Point group symmetries C 2v (2012) and C 2h (no) for 2 equivalent rotors, with PI group = G 36, e.g., acetone, dimethyl ether.

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Molecules with only one reciprocating LAM and a low barrier (like NH 3 ) are much more difficult. For example, 1980’s papers by Š. Urban and coworkers report calculations of ground state energy levels to cm -1 = 30 MHz. Rumor has it that Urban has plans for to improve these calculations to 10 KHz using both new sub-mm measurements and a better H eff. 11

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Molecules with only one reciprocating LAM and one, two or three symmetric-top internal rotors (e.g., methyl amine with 1 CH 3 top & 1 NH 2 inversion; or e.g., methyl malonaldehyde with 1 CH 3 top & 1 H atom transfer C-O-H O=C). Only an empirical tunneling formalism is available now for calculating lines to kHz accuracy, but the formalism is applicable only if the barriers for all LAMs are high. 12

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13 As far as the high-barrier tunneling formalism is concerned: C 2 H 3 + ground state rotational levels are doable to kHz accuracy, because the barrier to H migration is high. CH 5 + ground state rotational levels are NOT doable to kHz accuracy, because the barriers to H roaming are low. C-H fundamental rotational levels are not doable because of dark state bath.

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14 What is needed in the next 20 years to produce floppy molecule line lists with 30 MHz = cm -1 accuracy for FIR astronomy, or alternatively with 30 kHz accuracy for THz astronomy? High-resolution molecular spectroscopists will need: 1. Better experimental and Int. data, 2. More powerful and accurate H eff ’s, 3. Faster computers.

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