1. Vibrational Level Structures of the Ground Electronic States of the C3Ar and C3Ne Complexes
Yi-Ren Chen and Yen-Chu Hsu
Institute of Atomic and Molecular Sciences
Academia Sinica
Taiwan, R. O. C.
National Science Council, R. O. C.

2. Steps in the Calculations,
1. Converting the PE function obtained from ab initio calculations from internal coordinates to Jacobi coordinates.
Interpolating to a finer grid for DVR calculations. C
C.M. of C3
C.M. of C2 Ar R ℛ r Ө θ φ

3. 2. The Kinetic energy operators
ref: Yonggang Yang and Oliver Kühn, Mol. Phys. 106, 2445(2008).
Writing vectors, (C2 unit), (the third carbon to the C. M. of C2) and
(the rare gas atom to the C. M. of C3) in spherical coordinates,
To minimize the numerical effort,  and β were replaced by uα(=cos ) and
u β(=cos β). The kinetic operators for J=0 are,

4. where 1, 2, and 3 refer to the reduced mass of C2, C3, and C3Rg,
respectively.

5. Refs: 3. Heidelberg MCTDH program
U. Manthe, H.-D. Meyer, and L. S. Cederbaum, J. Chem. Phys.
97, 3199(1992).
M. H. Beck, A. Jäckle, G. A. Worth, and H.-D. Meyer, Phys. Rep.
324, 1(2000).
H.-D. Meyer and G. A. Worth, Theor. Chem. Acc. 109, 251(2003).

6. C3 Vibrational Levels vibrational Energy
assignment Botschwina et.al Jensen et.al This work Exp (0,2,0) (19) (0,4,0) (10) (0,6,0) (17)
(0,8,0) (18)
(0,10,0) (20)
(0,12,0) (25)
(1,0,0)
(0,14,0) (22)
(1,2,0) (13)
(1,4,0) (24)
(0,16,0) (24)
(1,6,0) (25)
(0,18,0)
(1,8,0)
(0,0,1) (6)
(0,2,1) (19)
(1,10,0)
(0,4,1)
C3 Vibrational Levels

7. (0,14,0) (1,2,0) Θ- (a.u.) Θ- (a.u.) ℛ(a.u.) ℛ(a.u.)
Ψ Ψ*
Ψ Ψ*
Θ- (a.u.)
Θ- (a.u.)
ℛ(a.u.)
ℛ(a.u.)
The plot of Ψ Ψ* of C3 versus Jacobi coordinates, Θ-  and ℛ at given r=1.298Å.

8. C3Ar_gs1000, R=7.329 ,lr = 2.457 ,sr = 3.61,α=0.32 ,β= 0.1 ,γ=3.2
E = cm-1 z
φ = °
ρ = °
R = 3.878Å
lr = 1.3Å
X = 1.3Å C
θ = 3.33° Ar
w = °

9. C3Ar ( ) Vibrational Levels
Assignment Energy (cm-1) Spectroscopic constants
observed calculated
V= A (111) (1.1%)
B (822) (-8.0%)
C (700) (- 6.5%)
1.719
Ar above the C3 plane, R3.93 Å
8.059
8.486
Vb=
Vs=

10. C3Ne_gs1000, R=6.8 ,lr = 2.46 ,sr = 3.6 ,α=0.3 ,β= 0.08 ,γ=3.5
E = cm-1 z
φ = °
ρ = °
R = 3.598Å
lr =1.302Å
X = 1.304Å C
θ = ° Ne
w = °

11. C3Ne ( ) Vibrational Levels
Assignment Energy (cm-1) Spectroscopic constants
observed calculated
V= A B C
(B+C)/ (1)
Vb=

12. Conclusions
Both Ar and Ne change the C3-bending potential considerably. Although the equilibrium geometry of free C3 is linear, the wavefunction of the zero-point energy level of the ground electronic state of C3Ar shows that the C-C-C bond angle is bent, to around 152. This bent C3 conformation explains why the A constant of the T-shaped C3Ar-complex is larger than the B constant of the C3 monomer. There are four highly-coupled large-amplitude vibrational degrees of freedom: these are the in- and out-of-plane C-C-C bends, the vdW stretch and the vdW bend. Their amplitudes are large even at the energy of the zero-point level.
Our calculated B and C constants of the Ar complex are both smaller than the experimental values, suggesting that our ab initio potential underestimates the vdW bond length slightly. This seems to be true also for the Ne complex.
One isomer was identified at about 3 cm-1 above the zero-point energy of C3Ar. The Ar atom is above the bent C3 molecular plane.en
Even before convergence to 10-4 cm-1 was reached, it was found that the vdW bending level was pushed upwards to higher energy due to interaction with two low-lying levels. The identities of these two lying levels remain to be assigned.