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.

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

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,

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

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).

C3 Vibrational Levels vibrational Energy assignment Botschwina et.al. Jensen et.al. This work Exp. (0,2,0) 133.267 154.3 130.438 132.7993(19) (0,4,0) 283.414 333.2 281.496 286.11(10) (0,6,0) 450.111 527.8 450.665 461.09(17) (0,8,0) 632.479 735.5 636.098 647.59(18) (0,10,0) 829.317 955.2 836.300 848.40(20) (0,12,0) 1050.193 1062.4(25) (1,0,0) 1218.950 1191.9 1231.189 1224.49 (0,14,0) 1277.175 1287.6(22) (1,2,0) 1403.064 1378.3 1407.964 1404.39(13) (1,4,0) 1590.100 1586.8 1517.331 1525.57(24) (0,16,0) 1590.824 1590.34(24) (1,6,0) 1782.723 1780.5 1771.823 1773.37(25) (0,18,0) 1782.912 (1,8,0) 1983.924 2032.4 1985.748 1992.9 (0,0,1) 2040.589 2007.3 2023.554 2040.0192(6) (0,2,1) 2138.538 2130.7 2043.139 2133.8876(19) (1,10,0) 2195.196 2267.8 2118.267 2210.5 (0,4,1) 2262.474 2283.2 2199.907 C3 Vibrational Levels

(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.

C3Ar_gs1000, R=7.329 ,lr = 2.457 ,sr = 3.61,α=0.32 ,β= 0.1 ,γ=3.2 E = 1756.771 cm-1 z φ = 88.979° ρ = 152.133° R = 3.878Å lr = 1.3Å X = 1.3Å C θ = 3.33° Ar w = 75.228°

C3Ar ( ) Vibrational Levels Assignment Energy (cm-1) Spectroscopic constants observed calculated V=0 0.0 A 0.45427(111) 0.45919 (1.1%) B 0.061077(822) 0.05654 (-8.0%) C 0.053818(700) 0.05032 (- 6.5%) 1.719 2.607 Ar above the C3 plane, R3.93 Å 8.059 8.486 Vb=1 9.10 Vs=1 28.841

C3Ne_gs1000, R=6.8 ,lr = 2.46 ,sr = 3.6 ,α=0.3 ,β= 0.08 ,γ=3.5 E = 1737.147 cm-1 z φ = 89.517° ρ = 153.931° R = 3.598Å lr =1.302Å X = 1.304Å C θ = 20.466° Ne w = 76.579°

C3Ne ( ) Vibrational Levels Assignment Energy (cm-1) Spectroscopic constants observed calculated V=0 0.0 A - 0.46689 B - 0.09235 C - 0.07694 (B+C)/2 0.087(1) 0.08465 Vb=1 8.43

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.