1. 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, Japan and National Radio Astronomy of Japan, Mitaka, Japan
Jun. 21st 2016
71st International Symposium on Molecular Spectroscopy

2. 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

3. Motivation-trans-ethyl methyl ether
Dipole moment
　　　ma = D
　　　 mb = 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.

4. 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

5. Analysis-introduction of extended permutation-inversion group
Originally introduced by Hougen and DeKoven for 1-top molecule (J.Mol. Spectrosc. 98, (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.

6. 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.

7. 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).

8. 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:

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

10. Hamiltonian Operator

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

12. 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

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

14. 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

15. 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!