International Symposium on Molecular Spectroscopy 62 nd Meeting

  • Slides: 22
Download presentation
International Symposium on Molecular Spectroscopy 62 nd Meeting - - June 18 -22, 2007

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

Background • CO 2 as a Dopant Molecule in Helium Cluster Microscopic understanding of

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(2 D) by Park Russell T. Pack, J. Chem. Phys. 61, 2091 (1974)

 Experiment • Infrared spectrum of CO 2 -He in the υ3 region M.

Experiment • Infrared spectrum of CO 2 -He in the υ3 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. Mc. Kellar et. al, Phys. Rev. Lett. 92, 145503 (2004) J. Tang and A. R. M. Mc. Kellar, J. Chem. Phys. 121, 181 (2004) Portion of observed infrared absorption Comparison of the Observed Vibrational shifts for CO 2 -He. N, OCS-He. N, N 2 O-He. N spectrum and prediction for CO 2 -He

Problems • All previous predicted spectra based on 2 -D PES CO 2 fixed

Problems • All previous predicted spectra based on 2 -D PES CO 2 fixed at equilibrium (2 D) 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, -1 gave RMS residual 1. 0 -8. 0 cm. • 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

Motivation • The spectra of clusters are more sensitive to the long range potential.

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) • Recently, Le Roy et al. introduced a new “Morse / Long-Range” (MLR) potential form. 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).

Computational Details • CO 2 -He in Jacobi coordinates (Q 3, θ, R) •

Computational Details • CO 2 -He in Jacobi coordinates (Q 3, θ, R) • Ab initio calculation of 3450 points at CCSD(T)/aug-cc-p. VQZ level of theory. • Bond functions 3 s 3 p 2 d 1 f 1 g incorporated in the basis. • Intermolecular potential produced by the ‘supermolecular’ approach. • Full counterpoise procedure used to correct for basis set superposition error.

3 -Dimenstional Analytic Potential Form The ab intio points fitted to the generalized MLR

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 De(Q 3, θ), Re(Q 3, θ) and φi (Q 3, θ) represented by polynomials in Q 3 and Legendre expansions in the angle.

C 6 and C 8 Coefficients The coefficients C 6(Q 3, θ) are expanded

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

Fitted Potential Energy Surface Ab initio points vs. fitted curves PES Depend on Q

Fitted Potential Energy Surface Ab initio points vs. fitted curves PES Depend on Q 3 -1 • RMS residual of 0. 032 cm is obtained on this fitting • the 2832 ab initio • a generalized MLR potential defined by only 55 fitting parameters. -1 points at energies below 1000 cm to

Features of Potential Energy Surface T Shape minimum: E(0, 90°, 3. 062) -1 =-49.

Features of Potential Energy Surface T Shape minimum: E(0, 90°, 3. 062) -1 =-49. 562 cm Transition state: E(0. 0, 40. 77°, 3. 977) -1 =-24. 408 cm Linear minimum: E(0. 0, 180°, 4. 263) -1 =-26. 693 cm Linear minimum: E(0. 0, 0°, 4. 263) -1 =-26. 693 cm Linear minimum: E(-0. 2, 180°, 4. 251) -1 =-29. 389 cm Linear minimum: E(-0. 2, 0°, 4. 268) -1 =-24. 581 cm Transition state 2: E(-0. 2, 130. 91°, 3. 895) -1 =-23. 996 cm T Shape minimum: E(-0. 2, 86°, 3. 063) -1 =-49. 758 cm Transition state 1: E(-0. 2, 28. 58°, 4. 263) -1 =-24. 164 cm

The energy and position of potential minimum depend on θ and Q 3 Previous

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

Ro-vibrational Energy Levels • 3 -D Hamiltonian in Jacobi coordinates (Q 3, R, )

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

Vibrational Energy Levels and Wave Function This work a YYX b BCVVBLS c KMTLHBW

Vibrational Energy Levels and Wave Function This work a YYX b BCVVBLS c KMTLHBW Bending vibrations of CO 2 in complex for J=0 Complex formed from CO 2 (υ3=0) n 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 (υ3=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 ΔE 0. 081 -0. 012 Exp. 0. 095 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) Even Odd But, 35, 000 Lanczos steps to convergence for υ3=1 complex, only 1, 000 for υ3=0

Vibrationally Averaged PES • Total ro-vibrational wave function written as direct product • vibrational

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

Question: How is the accuracy of calculated vibrational levels affected by method for averaging

Question: How is the accuracy of calculated vibrational levels affected by method for averaging potential energy surface Test: For J=0, compare vibrational levels obtained using 3 -D PES and the two 2 -D averaged PESs 3 -D n 1 2 3 4 5 Exp. -17. 045 -8. 752 -7. 647 -4. 036 -1. 280 -16. 964 -8. 755 -7. 659 -4. 035 -1. 276 ΔE 2 -D ΔE Diff. (3 D-2 D) Complex formed from CO 2 (υ3=0) -17. 045 0. 000 -8. 749 -0. 002 -7. 646 -0. 001 -4. 036 0. 000 -1. 282 0. 002 Complex formed from CO 2 (υ3=1) 0. 081 -16. 964 0. 081 0. 000 -0. 003 -8. 752 -0. 003 -0. 012 -7. 657 -0. 011 -0. 002 0. 001 -4. 033 0. 003 -0. 002 0. 004 -1. 276 0. 000 0. 095 ΔE: vibration shift from ground state to corresponding He-CO 2(υ3=1)

Infrared transition frequencies of CO 2 -He (in J'Ka. Kc-J"Ka. Kc Calc. Cal. -Obs.

Infrared transition frequencies of CO 2 -He (in J'Ka. Kc-J"Ka. Kc Calc. Cal. -Obs. J'Ka. Kc-J"Ka. Kc -1 cm ) Calc. Cal. -Obs. 616 -707 2346. 410 0. 031 331 -422 2346. 746 0. 025 515 -606 414 -505 2346. 771 2347. 138 0. 010 -0. 001 330 -321 331 -322 2349. 512 2349. 945 0. 003 0. 005 313 -404 2347. 502 -0. 012 431 -422 2349. 637 -0. 015 212 -303 2347. 867 -0. 020 432 -423 2350. 159 0. 001 111 -202 2348. 257 -0. 024 533 -524 2350. 462 -0. 002 331 -220 2351. 473 -0. 002 110 -101 2349. 459 -0. 028 330 -221 2351. 792 -0. 019 211 -202 2349. 838 -0. 032 432 -321 2351. 726 0. 002 312 -303 2350. 318 -0. 021 533 -422 2351. 915 -0. 002 413 -404 2350. 727 -0. 007 331 -202 2352. 337 -0. 003 514 -505 2351. 117 0. 003 432 -303 2353. 349 -0. 012 Continue RMS discrepancy for 49 transitions is 0. 018 -1 cm

Conclusions • A 3 -D analytic potential energy surface for CO 2 -He that

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(υ3=0) or CO 2(υ3=1). • The calculated IR spectra are in excellent agreement with recent experimental results validating the quality of PES. -1 • The predicted band origin shift of 0. 081 cm in good agreement with -1 experiment 0. 095 cm , 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.

Acknowledgments • Professor M. Nooijen (University of Waterloo) • Research supported by the Natural

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

 DVR and Lanczos algorithm • A direct product discrete variable representation (DVR) grid

DVR and Lanczos algorithm • 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 • Lanczos recursion used to diagonalize the Hamiltonian 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. 001 cm

Remove Spurious Eigenvalues Methods • The Cullum-Willoughby (CW) method compare the eigenvalues of its

Remove Spurious Eigenvalues Methods • 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. • Different Recursion Steps 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”. Since the “spurious” eigenvalues are shared by both the Lanczos matrix and its submatrix, they have small z 1 i and can be considered as copies generated from the round -off errors. J. Cullum and R. A. Willoughby, J. Comput. Phys. 44, 329 (1981) R. Chen and H. Guo J. Chem. Phys. 111, 9944 (1999)

The energy and position of potential minimum depend on θ and Qe Previous This

The energy and position of potential minimum depend on θ and Qe Previous This work De Empirical Ab initio de Empirical Re Ab initio (Rm, θ, De) T-shaped minima a YYX (3. 1 , 90, -44. 41) (3. 1 , 90, -45. 98) (3. 07, 90, -50. 38) (3. 062, 90, -49. 56) (3. 063, 86, -49. 76) b NAGF c KMTLHBW d =0. 0 Q 3 d Q 3=-0. 2 Saddle point (4. 1, 39, -19. 81) (3. 95, 45, -16. 01) (3. 98, 40. 8, -24. 41) (4. 26, 28. 6, -24. 16) (3. 90, 130. 9, -24. 00) Linear minima (4. 27, 0, -27. 69) (4. 3, 0, -26. 31) (4. 25, 0, -28. 94) (4. 26, 0, -26. 69) (4. 27, 0, -24. 58) (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

Question: How is the accuracy of calculated vibrational levels affected by method for averaging

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 R / Å 3. 30 3. 90 4. 20 v 01 v 02 859. 9855198 1. 5003777 -26. 3005472 θ=0. 0° -0. 0598981 -0. 0002655 -0. 0003883 -0. 0000350 -0. 0000065 -0. 0000011 5. 00 -12. 4926762 -0. 0000620 0. 0000000 7. 00 -1. 1232185 -0. 0000005 0. 0000000 θ=85° 2. 20 837. 5047253 -0. 0276864 -0. 0008079 2. 50 146. 7624441 -0. 0098472 -0. 0000857 2. 70 6. 7833468 -0. 0036810 -0. 0000271 3. 10 -47. 7360964 -0. 0003041 -0. 0000034 3. 60 -30. 4316809 -0. 0000057 -0. 0000002 10. 00 -0. 0711728 0. 0000000