1. An Analytic 3-Dimensional Potential Energy Surface for CO 2 -He and Its Predicted Infrared Spectrum Hui Li, Robert J. Le Roy υ International Symposium on Molecular Spectroscopy 62nd Meeting - - June 18-22, 2007

2. Background CO 2 as a Dopant Molecule in Helium Cluster Microscopic understanding of superfluidity, a collective bulk property An Essential Starting Point Is an accurate potential function for the He-CO 2 pair depending on CO 2 stretching Test Case Empirical ab initio First Quantitative Work on He-CO 2 (2D) by Park Russell T. Pack, J. Chem. Phys. 61, 2091 (1974)

3. Experiment Infrared spectrum of CO 2 -He in the υ 3 region M. J. Weida et.al, J. Chem. Phys. 101, 8351 (1994) Observed vibration shifts for CO 2 -He N J. Tang, A. R. M. McKellar et. al, Phys. Rev. Lett. 92, 145503 (2004) J. Tang and A. R. M. McKellar, J. Chem. Phys. 121, 181 (2004) Comparison of the Observed Vibrational shifts for CO 2 -He N, OCS-He N, N 2 O-He N Portion of observed observed infrared absorption spectrum and prediction for CO 2 -He

4. Problems All previous predicted spectra based on 2-D PES CO 2 fixed at equilibrium (2D) 1) Adequate approximation for MW spectra of ground state, but not for IR spectra involving vibrational excitation of CO 2 component 2) Want to predict shift of CO 2 vibrational frequency on forming the complex 3) Most PES fit to an ESMMSV or HFD forms, gave RMS residual 1.0-8.0 cm -1. Vibrational shift simulation needs a PES involving vibrational stretching of the CO 2 in the complex No existing work reports realistic estimate of these shifts

5. Motivation The spectra of clusters are more sensitive to the long range potential. Computational spectroscopy of helium-solvated molecules. S. Paolini, et al. J. Chem. Phys. 123, 114306 (2005) Advantage: 1) Incorporates theoretically known long-range inverse-power behaviour 2) Potential is a single smooth analytic function, not one made up of joined segments R. J. Le Roy and R. D. E. Henderson, Mol. Phys. 105, 691 (2007). Recently, Le Roy et al. introduced a new “Morse / Long-Range” (MLR) potential form.

6. Computational Details CO 2 -He in Jacobi coordinates (Q 3, θ, R) Ab initio calculation of 3450 points at CCSD(T)/aug-cc-pVQZ level of theory. Bond functions 3s3p2d1f1g incorporated in the basis. Intermolecular potential produced by the ‘supermolecular’ approach. Full counterpoise procedure used to correct for basis set superposition error.

7. 3-Dimenstional Analytic Potential Form The ab intio points fitted to the generalized MLR potential function form Radial behaviour expressed in terms of the dimensionless radial variable Exponent coefficient function φ(Q 3, θ, R) is a constrained polynomial in this variable. u LR (Q 3, θ, R) defines the limiting attractive long-range behaviour of the potential energy function, and has the form D e (Q 3, θ ), R e (Q 3, θ) and φ i (Q 3, θ ) represented by polynomials in Q 3 and Legendre expansions in the angle.

8. C 6 and C 8 Coefficients The coefficients C 6 (Q 3, θ ) are expanded as from He and CO 2 pseudo-dipole oscillator strength distributions of Meath et al. 1 The stretching-dependent come from recent theoretical result. 3 1 A.Kumar and W. J. Meath, Mol. Phys. 54, 823 (1985); B. L. Jhanwar and W. J. Meath, Chem. Phys. 67, 185 (1982). 2 Russell T. Pack, J. Chem. Phys.. 64, 1659 (1976). 3 A.Haskopoulos and G. Maroulis Chem. Phys. Lett. 417. 235 (2006). Where, is calculated from the ratio are calculated from the ratio C 8 /C 6 reported by Russell T Pack. 2

9. Fitted Potential Energy Surface RMS residual of 0.032 cm -1 is obtained on this fitting the 2832 ab initio points at energies below 1000 cm -1 to a generalized MLR potential defined by only 55 fitting parameters. Ab initio points vs. fitted curves PES Depend on Q 3

10. Features of Potential Energy Surface Linear minimum: E(0.0, 0°, 4.263) =-26.693 cm -1 Linear minimum: E(0.0, 180°, 4.263) =-26.693 cm -1 T Shape minimum: E(0, 90°, 3.062) =-49.562 cm -1 Transition state: E(0.0, 40.77°, 3.977) =-24.408 cm -1 T Shape minimum: E(-0.2, 86°, 3.063) =-49.758 cm -1 Linear minimum: E(-0.2, 180°, 4.251) =-29.389 cm -1 Linear minimum: E(-0.2, 0°, 4.268) =-24.581 cm -1 Transition state 1: E(-0.2, 28.58°, 4.263) =-24.164 cm -1 Transition state 2: E(-0.2, 130.91°, 3.895) =-23.996 cm -1

11. The energy and position of potential minimum depend on θ and Q 3 a G. Yan et al., J. Chem. Phys. 109, 10284 (1998) b M. Keil and G. A. Parker, J. Chem. Phys. 82, 1947 (1985) c L. Beneventi, et al., J. Chem. Phys. 89,4671 (1988) Empirical de -D e (θ) R e (θ) Ab initio Empirical Ab initio Previous This work

12. Ro-vibrational Energy Levels Lanczos recursion used to diagonalize the Hamiltonian 3-D Hamiltonian in Jacobi coordinates (Q 3, R,  ) is Use direct product discrete variable representation (DVR) µ is reduce mass of the CO 2 -He dimer, M=m c m O /(2 m O +m C ) is the reduce mass for Q 3 mode of CO 2. J x, J y, J z are the components of the total angular momentum J The z axis of the body-fixed frame lies along the R radial vector

13. Vibrational Energy Levels and Wave Function This workYYX a BCVVBLS b KMTLHBW c nComplex formed from CO 2 ( υ 3 =0) 1-17.045-15.806-15.689-18.052 2-8.749-7.143-9.756-9.247 3-7.646-5.771-6.968-8.154 4-4.036-3.035 5-1.282-0.576 Complex formed from CO 2 ( υ 3 =1) 1-16.964-15.818 2-8.752-7.155 3-7.657-5.781 4-4.033-3.068 5-1.276-0.596 ΔEΔE 0.081-0.012 Exp. 0.095 Even Odd ΔE is the band origin shift. a G. Yan et al., J. Chem. Phys. 109, 10284 (1998) b M. Weida et al., J. Chem. Phys. 101, 8351 (1994) c T. Korona et al., J. Chem. Phys. 115, 3074 (2001) But, 35,000 Lanczos steps to convergence for υ 3 =1 complex, only 1,000 for υ 3 =0 Bending vibrations of CO 2 in complex for J=0

14. Vibrationally Averaged PES Vibrational averaged intermolecular PES Total ro-vibrational wave function written as direct product vibrational wave function of CO 2 obtained from

15. Question: How is the accuracy of calculated vibrational levels affected by method for averaging potential energy surface ΔE : vibration shift from ground state to corresponding He-CO 2 ( υ 3 =1) Test: For J=0, compare vibrational levels obtained using 3-D PES and the two 2-D averaged PESs 3-DΔEΔE2-DΔEΔEDiff.(3D-2D) nComplex formed from CO 2 ( υ 3 =0) 1-17.045 0.000 2 -8.752-8.749-0.002 3-7.647-7.646-0.001 4-4.036 0.000 5-1.280-1.2820.002 Complex formed from CO 2 ( υ 3 =1) 1-16.9640.081-16.9640.0810.000 2-8.755-0.003-8.752-0.003 3-7.659-0.012-7.657-0.011-0.002 4-4.0350.001-4.0330.003-0.002 5-1.2760.004-1.2760.0060.000 Exp.0.095

16. Infrared transition frequencies of CO 2 -He (in cm -1 ) J' KaKc -J" KaKc Calc.Cal.-Obs.J' KaKc -J" KaKc Calc.Cal.-Obs. 6 16 -7 07 2346.4100.0313 31 -4 22 2346.7460.025 5 15 -6 06 2346.7710.0103 30 -3 21 2349.5120.003 4 14 -5 05 2347.138-0.0013 31 -3 22 2349.9450.005 3 13 -4 04 2347.502-0.0124 31 -4 22 2349.637-0.015 2 12 -3 03 2347.867-0.0204 32 -4 23 2350.1590.001 1 11 -2 02 2348.257-0.0245 33 -5 24 2350.462-0.002 3 31 -2 20 2351.473-0.002 1 10 -1 01 2349.459-0.0283 30 -2 21 2351.792-0.019 2 11 -2 02 2349.838-0.0324 32 -3 21 2351.7260.002 3 12 -3 03 2350.318-0.0215 33 -4 22 2351.915-0.002 4 13 -4 04 2350.727-0.0073 31 -2 02 2352.337-0.003 5 14 -5 05 2351.1170.0034 32 -3 03 2353.349-0.012 Continue RMS discrepancy for 49 transitions is 0.018 cm -1

17. Conclusions A 3-D analytic potential energy surface for CO 2 -He that explicitly incorporates its dependence on the Q 3 normal-mode of the CO 2, has been obtained by least-squares fitting new ab initio interaction energies to a new “Morse /Long-Range” potential form. Obtained separate high accuracy, vibrational-averaged 2-D PESs He with CO 2 ( υ 3 =0) or CO 2 ( υ 3 =1). The calculated IR spectra are in excellent agreement with recent experimental results validating the quality of PES. The predicted band origin shift of 0.081 cm -1 in good agreement with experiment 0.095 cm -1, and should provide a reliable starting point for future CO 2 -He N clusters simulation. Future work: Quantum Monte Carlo simulations to predict CO 2 -He N vibrational shifts.

18. Acknowledgments Professor M. Nooijen (University of Waterloo) Research supported by the Natural Sciences and Engineering Research Council of Canada

19. Lanczos recursion used to diagonalize the Hamiltonian A direct product discrete variable representation (DVR) grid was used in vibrational energy calculation Each stretching coordinate was represented by 70 PODVR grid, with 200 equidistant sine-DVR grid on interval [1.6, 5.0] 80 Gauss-Legendre grid points on the interval [60-180] were used for the angular variable DVR and Lanczos algorithm When eigenfunctions are needed, for selected eigenvalue, using inverse iteration method get the eigenvector and repeat Lanczos recursion get the wavefunction 10000 Lanczos iterations were found adequate to converge the levels within 9000 cm 0.001cm

20. Different Recursion Steps The Cullum-Willoughby (CW) method compare the eigenvalues of its sub-matrix obtained by deleting the first row and first column. If a Lanczos eigenvalue appears for both matrices, it is regarded “spurious” and deleted. Remove Spurious Eigenvalues Methods Compare the Lanczos eigenvalues in different recursion steps: a true eigenvalue should not depend on recursion steps. With earliest appearance in the recursion is regarded as “good”. J. Cullum and R. A. Willoughby, J. Comput. Phys. 44, 329 (1981) R. Chen and H. Guo J. Chem. Phys. 111, 9944 (1999) Since the “spurious” eigenvalues are shared by both the Lanczos matrix and its sub- matrix, they have small z 1i and can be considered as copies generated from the round-off errors.

21. The energy and position of potential minimum depend on θ and Q e (R m,θ, D e )T-shaped minimaSaddle pointLinear minima YYX a (3.1, 90, -44.41)(4.1, 39, -19.81)(4.27, 0, -27.69) NAGF b (3.1, 90, -45.98)(3.95, 45,-16.01)(4.3, 0, -26.31) KMTLHBW c (3.07, 90, -50.38)(4.25, 0, -28.94) Q 3 =0.0 d (3.062, 90, -49.56)(3.98, 40.8, -24.41)(4.26, 0, -26.69) Q 3 =-0.2 d (3.063, 86, -49.76)(4.26, 28.6, -24.16)(4.27, 0, -24.58) Q 3 =-0.2 d (3.063, 86, -49.76)(3.90,130.9, -24.00)(4.25, 180, -29.39) a G. Yan et al., J. Chem. Phys. 109, 10284 (1998) b F. Legri et al., J. Chem. Phys. 111, 6439 (1999) c T. Korona et al., J. Chem. Phys. 115, 3074 (2001) d Present work Empirical de DeDe ReRe Ab initio Empirical Ab initio Previous This work

22. Question: How is the accuracy of calculated vibrational levels affected by method for averaging potential energy surface Test 1: Compare the diagonal and off-diagonal averaged energies v 00 v 01 v 02 R / Åθ=0.0° 3.30 859.9855198-0.0598981-0.0000350 3.901.5003777-0.0002655-0.0000065 4.20-26.3005472-0.0003883-0.0000011 5.00-12.4926762-0.00006200.0000000 7.00-1.1232185-0.00000050.0000000 θ=85° 2.20837.5047253-0.0276864-0.0008079 2.50146.7624441-0.0098472-0.0000857 2.706.7833468-0.0036810-0.0000271 3.10-47.7360964-0.0003041-0.0000034 3.60-30.4316809-0.0000057-0.0000002 10.00 -0.07117280.0000000