Transcript 06ColBiacet.ppt

Analysis and fit of the high-resolution spectrum
of the two-equivalent-top molecule biacetyl
Nobukimi Ohashi1, Jon T. Hougen2,
Cheng-Liang Huang3, Chen-Lin Liu4, Chi-Kung Ni4
1Kanazawa
University, Kanazawa, Japan
2National Institute of Standards and Technology, Gaithersburg, MD,
U.S.A.
3National Chiayi University, Chiayi, Taiwan
4Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei,
Taiwan
Biacetyl
CH3-C(=O)-C(=O)-CH3
2
3
9
C
12
O
8
C
C
7
4
11
C
O
10
5
6
1
C2h
Torsional energy levels of the Ã1Au state of biacetyl
Polyad
V=1
V=1
100
100
A2
A2
G
G
E4
E1
E4
E1
E1
95
v1v2
10
1 0
90
90
-1
Energy (cm )
Energy (cm-1)
A2
0 1
01
85
A
A
E
E
AE+EA
AA+AA
AE+EA
AA+AA
EE+EE
EA+AE
EE+EE
EA+AE
v1v2
00
0 0
-5
-5
G
A1
E
A
E
A
0
E3
E2
E1
E3
V=0
0
G
G
G
E3
E3
V=0
0
A3
E1 E2
E1
E3
G
A1
A1 E2
EE
EA+AE
EE
EA+AE
AA
AA
5
Two non-interacting local modes
0 splitting
1 splitting
2 splittings
Local-Mode Treatment
10
Exact calcn
Allowed
absorption
Exp.
spectrum
Exact Allowed Exp
V v 1v 2
4
spectra Components in G36
(in fit, forbidden at 1K, not analyzed, attempted)
22
Low
3113 Low
4004 High
A1 G A4 E3 E1 G E2 E4
3
2112 High
3003 Low
A3 G A2 E2 E3 G E4 E1
E2 E4 G E3 E1 A3 A2 G
2
11
High
2002 High
E3 E1 G A1
A1 G A4 E3 E1 G E2 E4
1
1001
High
E2 E3 G A3 E1 E4 G A2
0
00
High
A1 G E3 E1
Transitions used in the earlier torsional-rotational
analysis of the Ã1Au – X 1Ag transition of biacetyl
-
Ã1Au
G*
(v1, v2)**
G
E1
E3
G
E1
G
A2
(0, 0)
(0. 0)
(1, 0)
(1, 0)
(1, 0)
(1, 0)
(1, 0)
X 1Ag
#
G*
0 J  4, 0 Ka 3
0 J  3, 0 Ka 2
0 J  3, 0 Ka 2
0 J  4, 0 Ka 2
0 J  4, 0 Ka 2
0 J  3, 0 Ka 2
0 J  3, 0 Ka 2
G
E1
E3
G
E1
G
A1
v=0
v=0
v=0
v=0
v=0
v=0
v=0
0 J  3, 0 Ka 3
0 J  3, 0 Ka 2
0 J  3, 0 Ka 2
0 J  3, 0 Ka 2
0 J  3, 0 Ka 2
0 J  3, 0 Ka 1
0 J  2, 0 Ka 1
37
17
25
29
36
22
13
*G: symmetry species of torsional state. [G(Ã1Au) = A3, G(X1Ag) = A1]
**v1, v2 : local mode quantum numbers.
Diagonalization of Hamiltonian matrix for the Ã1Au state :
max. of m1 = max. of m2 = 24 & one-step diagonalization
Character Table* of Permutation-Inversion group G36
ammonia dimer, ethane (D3d), dimethylether (C2v), biacetyl (C2h)
E (123)
(456)
#
1
2
A1
A2
A3
A4
E1
E2
E3
E4
G
1
1
1
1
2
2
2
2
4
1
1
1
1
2
2
-1
-1
-2
(14)(26)(35)
(78)(9,10)
(11)(12)*
3
1
1
-1
-1
2
-2
0
0
0
(123) (123)
(465)
3
1
1
1
1
-1
-1
2
2
-2
4
1
1
1
1
-1
-1
-1
-1
1
(142635)
(78)(9,10)
(11,12)*
6
1
1
-1
-1
-1
1
0
0
0
(14)(25)(36)
(78)(9,10)
(11,12)
3
1
-1
1
-1
0
0
2
-2
0
(142536)
(78)(9,10)
(11,12)
6
1
-1
1
-1
0
0
-1
1
0
(23)
(56)*
9
1
-1
-1
1
0
0
0
0
0
* Taken from D. D. Nelson, W. Klemperer, J. Chem. Phys. 87 (1987) 139-149.
Generating operations for G36
Ri = R + S-1(c,q,f)ai(q1, q2)
Molecular variable transformations
PI operations
c-of-m Euler angles
Torsional angles
E
R c, q, f
q1
(123)
R c, q, f
q1+(2p/3) q1
(456)
R c, q, f
q2
(23)(56)*
-R p-c, p-q, p+f -q1
(14)(26)(35)(78) -R c, q, f
(9,10)(11,12)*
-q2
q2
q2+(2p/3)
-q2
-q1
Hamiltonian (Principal Axis Method)
(i) for the X1Ag ground electronic state
H = AJz2 + BJx2 + CJy2 + TG (G = A, E or G)
(ii) for the Ã1Au excited electronic state (S1)
HPAM = Te + AJz2 + BJx2 + CJy2
(rotation)
+ F(p12 + p22)
(torsion)
+ A01CC[cos(3q1) + cos(3q2) ]
+ 2A11SSsin(3q1)sin(3q2)
+ 2A11CCcos(3q1)cos(3q2)
+ q(p1 – p2)Jz + s (p1 – p2)Jx (Coriolis)
+ higher-order terms
Basis-set functions
A-species (A1  A2  A3  A4)
|A; m1, m2; J,K = | m1, m2  | J,K 
where | m1, m2  = (1/2p)exp(i m1q1) exp(i m2q2)
m1 = 0 mod 3 (… -3 , 0 , +3, …), m2 = 0 mod 3 and -J  K  +J
E1-species (E1  E2)
|E1; m1, m2; J,K  = | m1, m2  | J,K 
-J  K  +J
m1 = 1 mod 3 (..., -2, +1, +4, ….),
m2 = 2 mod 3 (..., -1 +2, +5,
)  m1+m2 = 0 mod 3
E3-species ( E3  E4)
|E3; m1, m2; J,K  = | m1, m2  | J,K 
m1 = 1 mod 3, m2 = 1 mod 3 
G-species
|G; m1, m2; J,K  = | m1, m2  | J,K 
m1 = 1 mod 3, m2 = 0 mod 3
-J  K  +J
m1-m2 = 0 mod 3
-J  K  +J
5. Molecular parameters (cm-1) obtained in the global fit
Ã1Au state
A
B
C
F (p12+p22)
F12 (2p1p2)
2A01CC (= V3)
2A03CC (= V9)
A11CC c3q1c3q2
A11SS s3q1s3q2
A12CC c3q1c6q2
A12SS s3q1s6q2
X1Ag state
0.18038(13)
0.11604(13)
0.07009(20)
5.28229(98)
-0.0939(33)
246.80(25)
4.23(14)
14.87(23)
-12.52(14)
2.982(68)
1.140(23)
A
B
C
T0(E) – T0(A)
T0(G) – T0(A)
_____________
0.17678(16)
0.11236(14)
0.07038(20)
0.0441(25)
0.0224(18)
_____________
q (p1-p2)Jz
s (p1-p2)Jx
Te
-0.2469(20)
0.1631(26)
22316.698(73)
One standard deviation in parentheses
310 lines. 19 parameters. RMS of fit = 0.0040 cm-1.
Conclusions and Future Work
The present fit is quite good, indicating that
the present Hamiltonian formalism captures
the necessary physics and has the necessary
numerical accuracy for the problem.
We plan further efforts to include in the fit
rotationally assigned bands from polyads with
V  v1+v2  3.