IAM(-LIKE) Tunneling Matrix Formalism for One- and Two-Methyl-Top Molecules Based on the Extended Permutation-Inversion Group Idea and Its Application.

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Presentation transcript:

IAM(-LIKE) Tunneling Matrix Formalism for One- and Two-Methyl-Top Molecules Based on the Extended Permutation-Inversion Group Idea and Its Application to the Analyses of The Methyl-Torsional Rotational Spectra NOBUKIMI OHASHI1, KAORI KOBAYASHI2, and MASAHARU FUJITAKE1 1. Kanazawa University, Japan 2. Department of Physics, University of Toyama, 3190 Gofuku,Toyama, 930-8555 Japan and National Radio Astronomy of Japan, Mitaka, Japan Jun. 21st 2016 71st International Symposium on Molecular Spectroscopy

Outline Motivation Coordinate systems for internal rotation problems IAM-like tunneling matrix formalism for inequivalent two-top molecules Comparison with Groner’s ERHAM theory Future possibility- application to equivalent two-top molecules

Motivation-trans-ethyl methyl ether Dipole moment 　　　ma = 0.146 D 　　　 mb = 1.165 D Two internal rotors barrier height ⋍　850 cm-1 (OCH3 internal rotation) barrier height ⋍　1150 cm-1 (CCH3 internal rotation) O C C C Tunneling Matrix Formalism with PAM Hamiltonian was good but not good enough to extend higher J, K.

Coordinate systems for internal rotation problems Hougen, Kleiner, & Godefroid, J. Mol. Spectrosc., 163, 559 (1994) Principal axis method Rho axis method Internal axis method PAM RAM IAM Elimination of the Jx terms Elimination of the Jx and Jz terms

Analysis-introduction of extended permutation-inversion group Originally introduced by Hougen and DeKoven for 1-top molecule (J.Mol. Spectrosc. 98, 373-391 (1983)) Useful to avoid the inconvenient problem that appears in the IAM coordinate system. As a result, periodic behavior in K explicitly appears in the matrix elements. Useful to treat each vibrational state independently.

Transformations of variables under the generating operations of the PI group G6 (123) (23)* This is inconvenient. As a method for avoiding this inconvenience, putting the m-extended PI group G6m was introduced by Hougen and DeKoven.

Essence of the present formalism for the two internal rotor problem Coordinate system (IAM-like) where we neglect small amplitude vibrations. a1 : OCH3 internal rotation angle, a2: CCH3 internal rotation angle, Why is this coordinate system adopted ? (1) dominant Coriolis terms in PAM are cancelled because (2) both the r1 and r2 axes are nearly parallel with the PAM z-axis (= a axis).

Introduction of an extended permutation-inversion group Generating operations a, b and c construct an extended PI group G18mn having 18×mn group elements. Torsional framework functions based on the tunneling formalism We have the following 9mn (= 3m×3n) framework functions:

Nonzero-overlap integral b = 0 Nonzero-overlap integral Orthonormalized at the later step not shown in this talk

Hamiltonian Operator

Construction of Hamiltonian matrix Hamiltonian operator From phenomenological and symmetry consideration, we have + higher order terms From comparison with HIAM-like

Hamiltonian matrix elements This is an ERHAM!! : for the terms other than : for the terms (We assumed m, n to be odd .) : for Vi other than

ERHAM Matrix Elements n1 / n1 = n2 =3 s1, s2 = 0 or 1

Possible application to two equivalent rotors - a future plan Reconsideration of symmetry of molecules Still similar K-dependence appears Should be possible with this IAM-like formalism

Thank you for your attention! Acknoledgment M. Fujitake (Kanazawa U.) N. Ohashi National Astronomical Observatory of Japan (NAOJ) ALMA-J Grant-in- Aid from the Ministry of Education, Science, Sports and Culture of Japan Thank you for your attention!