A multi-dimensional Smolyak collocation method in curvilinear coordinates for computing vibrational spectra

Journal of Chemical Physics

Volumn 143, Issue 21, 2015, Pages

  Avila, Gustavo ^a   Carrington, Tucker ^a  

^a QUEEN S UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

ITERATIVE METHODS; KINETICS; MATHEMATICAL OPERATORS; VIBRATIONAL SPECTRA;

COLLOCATION METHOD; CURVILINEAR COORDINATE; DIFFERENTIAL OPERATORS; EIGENSOLVERS; ENERGY MATRIX; KINETIC-ENERGY OPERATOR; MULTI DIMENSIONAL;

KINETIC ENERGY;

EID: 84949033213     PISSN: 00219606     EISSN: 10897690     Source Type: Journal    
DOI: 10.1063/1.4936294     Document Type: Article

References (89)

1. W. Yang and A. C. Peet, " The collocation method for bound solutions of the Schrödinger equation," Chem. Phys. Lett. 153 (1), 98-104 (1988). 10.1016/0009-2614(88)80139-8
2. W. Yang and A. C. Peet, " A method for calculating vibrational bound states: Iterative solution of the collocation equations constructed from localized basis sets," J. Chem. Phys. 92 (1), 522-526 (1990). 10.1063/1.458454
3. S. Carter and N. C. Handy, " The variational method for the calculation of ro-vibrational energy levels," Comput. Phys. Rep. 5 (3), 117-171 (1986). 10.1016/0167-7977(86)90006-7
4. J. Tennyson, " The calculation of the vibration-rotation energies of triatomic molecules using scattering coordinates," Comput. Phys. Rep. 4 (1), 1-36 (1986). 10.1016/0167-7977(86)90005-5
5. J. C. Light and T. Carrington, Jr., " Discrete-variable representations and their utilization," Adv. Chem. Phys. 114, 263-310 (2000). 10.1002/9780470141731.ch4
6. J. M. Bowman, T. Carrington, Jr., and H.-D. Meyer, " Variational quantum approaches for computing vibrational energies of polyatomic molecules," Mol. Phys. 106 (16-18), 2145-2182 (2008). 10.1080/00268970802258609
7. M. H. Beck, A. Jäckle, G. A. Worth, and H. D. Meyer, " The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets," Phys. Rep. 324 (1), 1-105 (2000). 10.1016/S0370-1573(99)00047-2
8. S. Carter, S. J. Culik, and J. M. Bowman, " Vibrational self-consistent field method for many-mode systems: A new approach and application to the vibrations of CO adsorbed on Cu (100)," J. Chem. Phys. 107 (24), 10458-10469 (1997). 10.1063/1.474210
9. + isotopologues," J. Chem. Phys. 131 (3), 034308 (2009). 10.1063/1.3183166
10. A. Zoccante, P. Seidler, M. Bo Hansen, and O. Christiansen, J. Chem. Phys. 136, 204118 (2012). 10.1063/1.4721626
11. X.-G. Wang and T. Carrington, Jr., " A contracted basis-Lanczos calculation of vibrational levels of methane: Solving the Schrödinger equation in nine dimensions," J. Chem. Phys. 119 (1), 101-117 (2003). 10.1063/1.1574016
12. G. Avila and T. Carrington, Jr., " Nonproduct quadrature grids for solving the vibrational Schrödinger equation," J. Chem. Phys. 131 (17), 174103 (2009). 10.1063/1.3246593
13. D. Lauvergnat and A. Nauts, " Quantum dynamics with sparse grids: A combination of Smolyak scheme and cubature. Application to methanol in full dimensionality," Spectrochim. Acta, Part A 119, 18-25 (2014). 10.1016/j.saa.2013.05.068
14. G. Avila and T. Carrington, Jr., J. Chem. Phys. 139, 134114 (2013). 10.1063/1.4821348
15. A. C. Peet, W. Yang, A. C. Peet, and W. Yang, " The collocation method for calculating vibrational bound states of molecular systems with application to ArHCl," J. Chem. Phys. 90 (3), 1746-1751 (1989). 10.1063/1.456068
16. S. Manzhos and T. Carrington, Jr., " An improved neural network method for solving the Schrödinger equation 1," Can. J. Chem. 87 (7), 864-871 (2009). 10.1139/V09-025
17. S. Manzhos, K. Yamashita, and T. Carrington, Jr., " On the advantages of a rectangular matrix collocation equation for computing vibrational spectra from small basis sets," Chem. Phys. Lett. 511 (4), 434-439 (2011). 10.1016/j.cplett.2011.06.040
18. 2: Implications for adsorption mode assignment in dye-sensitized solar cells," Phys. Chem. Chem. Phys. 15 (25), 10028-10034 (2013). 10.1039/c3cp00065f
19. S. Manzhos, M. Chan, and T. Carrington, Jr., " Communication: Favorable dimensionality scaling of rectangular collocation with adaptable basis functions up to 7 dimensions," J. Chem. Phys. 139 (5), 051101 (2013). 10.1063/1.4817182
20. U. Manthe and H. Koeppel, J. Chem. Phys. 93, 345 (1990). 10.1063/1.459606
21. M. J. Bramley and T. Carrington, Jr., " A general discrete variable method to calculate vibrational energy levels of three and four atom molecules," J. Chem. Phys. 99 (11), 8519-8541 (1993). 10.1063/1.465576
22. J. C. Tremblay and T. Carrington, Jr., " Calculating vibrational energies and wave functions of vinylidene using a contracted basis with a locally reorthogonalized coupled two-term Lanczos eigensolver," J. Chem. Phys. 125 (9), 094311 (2006). 10.1063/1.2234480
23. P. Sarkar, N. Poulin, and T. Carrington, Jr., " Calculating rovibrational energy levels of a triatomic molecule with a simple Lanczos method," J. Chem. Phys. 110 (21), 10269-10274 (1999). 10.1063/1.478960
24. +," J. Chem. Phys. 129 (23), 234102 (2008). 10.1063/1.3027825
25. R. Dawes, X. G. Wang, A. W. Jasper, and T. Carrington, Jr., " Nitrous oxide dimer: A new potential energy surface and rovibrational spectrum of the nonpolar isomer," J. Chem. Phys. 133 (13), 134304 (2010). 10.1063/1.3494542
26. X.-G. Wang and T. Carrington, Jr., " The utility of constraining basis function indices when using the Lanczos algorithm to calculate vibrational energy levels," J. Phys. Chem. A 105 (12), 2575-2581 (2001). 10.1021/jp003792s
27. G. Avila and T. Carrington, Jr., " Using nonproduct quadrature grids to solve the vibrational Schrödinger equation in 12D," J. Chem. Phys. 134 (5), 054126 (2011). 10.1063/1.3549817
28. 4," J. Chem. Phys. 135 (6), 064101 (2011). 10.1063/1.3617249
29. G. Avila and T. Carrington, Jr., " Solving the vibrational Schrödinger equation using bases pruned to include strongly coupled functions and compatible quadratures," J. Chem. Phys. 137 (17), 174108 (2012). 10.1063/1.4764099
30. E. L. Sibert, J. T. Hynes, and W. P. Reinhardt, J. Chem. Phys. 77, 3595-3604 (1982). 10.1063/1.444261
31. S. F. Boys, " Some bilinear convergence characteristics of the solutions of dissymmetric secular equations," Proc. R. Soc. London, Ser. A 309, 195-208 (1969). 10.1098/rspa.1969.0037
32. J.-P. Berrut and L. N. Trefethen, " Barycentric Lagrange interpolation," SIAM Rev. 46 (3), 501-517 (2004). 10.1137/S0036144502417715
33. G. Mastroianni and D. Occorsio, " Lagrange interpolation at Laguerre zeros in some weighted uniform spaces," Acta Math. Hung. 91 (1-2), 27-52 (2001). 10.1023/A:1010678709857
34. H.-J. Bungartz and M. Griebel, " Sparse grids," Acta Numer. 13, 147-269 (2004). 10.1017/S0962492904000182
35. V. Barthelmann, E. Novak, and K. Ritter, " High dimensional polynomial interpolation on sparse grids," Adv. Comput. Math. 12 (4), 273-288 (2000). 10.1023/A:1018977404843
36. J. Shen and Y. Haijun, " Efficient spectral sparse grid methods and applications to high-dimensional elliptic problems," SIAM J. Sci. Comput. 32 (6), 3228-3250 (2010). 10.1137/100787842
37. B. A. Malin, D. Krueger, and F. Kubler, " Solving the multi-country real business cycle model using a Smolyak-collocation method," J. Econ. Dyn. Control 35 (2), 229-239 (2011). 10.1016/j.jedc.2010.09.015
38. B. Ganapathysubramanian and N. Zabaras, " Sparse grid collocation schemes for stochastic natural convection problems," J. Comput. Phys. 225 (1), 652-685 (2007). 10.1016/j.jcp.2006.12.014
39. F. Nobile, R. Tempone, and C. G. Webster, " A sparse grid stochastic collocation method for partial differential equations with random input data," SIAM J. Numer. Anal. 46 (5), 2309-2345 (2008). 10.1137/060663660
40. G. Avila and T. Carrington, Jr., " Using multi-dimensional Smolyak interpolation to make a sum-of-products potential," J. Chem. Phys. 143 (4), 044106 (2015). 10.1063/1.4926651
41. K. Hallatschek, " Fourier-transform on sparse grids with hierarchical bases," Numer. Math. 63 (1), 83-97 (1992). 10.1007/BF01385849
42. V. Gradinaru, " Fourier transform on sparse grids: Code design and the time dependent Schrödinger equation," Computing 80 (1), 1-22 (2007). 10.1007/s00607-007-0225-3
43. X. Ma and N. Zabaras, " An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations," J. Comput. Phys. 228 (8), 3084-3113 (2009). 10.1016/j.jcp.2009.01.006
44. Sparse Grids and Applications, edited by J. Garcke and M. Griebel (Springer Science Business Media, 2012), Vol. 88.
45. A. Klimke and B. Wohlmuth, " Algorithm 847: Spinterp. Piecewise multilinear hierarchical sparse grid interpolation in MATLAB," ACM Trans. Math. Software 31 (4), 561-579 (2005). 10.1145/1114268.1114275
46. M. Griebel and J. Hamaekers, " Fast discrete Fourier transform on generalized sparse grids," in Sparse Grids and Applications (Springer International Publishing, 2014), pp. 75-107.
47. K. Petras, " Smolyak cubature of given polynomial degree with few nodes for increasing dimension," Numer. Math. 93 (4), 729-753 (2003). 10.1007/s002110200401
48. J. Szabados, " Weighted Lagrange and Hermite-Fejér interpolation on the real line," J. Inequalities Appl. 1997, 481267, pdf available at http://www.emis.ams.org/journals/HOA/JIA/1/299.pdf.
49. M. Griebel and S. Knapek, " Optimized tensor-product approximation spaces," Constr. Approximation 16 (4), 525-540 (2000). 10.1007/s003650010010
50. F. Richter, M. Hochlaf, P. Rosmus, F. Gatti, and H.-D. Meyer, " A study of the mode-selective transcis isomerization in HONO using ab initio methodology," J. Chem. Phys. 120 (3), 1306-1317 (2004). 10.1063/1.1632471
51. A. McNichols and T. Carrington, Jr., " Vibrational energy levels of formaldehyde calculated from an internal coordinate Hamiltonian using the Lanczos algorithm," Chem. Phys. Lett. 202 (6), 464-470 (1993). 10.1016/0009-2614(93)90031-U
52. H.-G. Yu and J. T. Muckerman, J. Mol. Spectrosc. 214, 11-20 (2002). 10.1006/jmsp.2002.8569
53. D. Neuhauser, " Bound state eigenfunctions from wave packets: Time energy resolution," J. Chem. Phys. 93 (4), 2611-2616 (1990). 10.1063/1.458900
54. R. B. Lehoucq, S. K. Gray, D.-H. Zhang, and J. C. Light, " Vibrational eigenstates of four-atom molecules: A parallel strategy employing the implicitly restarted Lanczos method," Comput. Phy. Commun. 109 (1), 15-25 (1998). 10.1016/S0010-4655(98)00002-2
55. F. Ribeiro, C. Iung, and C. Leforestier, " A Jacobi-Wilson description coupled to a block-Davidson algorithm: An efficient scheme to calculate highly excited vibrational levels," J. Chem. Phys. 123 (5), 054106 (2005). 10.1063/1.1997129
56. 3H," J. Chem. Phys. 90 (6), 3198-3203 (1989). 10.1063/1.455871
57. S.-W. Huang and T. Carrington, Jr., " A comparison of filter diagonalisation methods with the Lanczos method for calculating vibrational energy levels," Chem. Phys. Lett. 312 (2), 311-318 (1999). 10.1016/S0009-2614(99)00889-1
58. G. Balint-Kurti and P. Pulay, " A new grid-based method for the direct computation of excited molecular vibrational states: Test application to formaldehyde," J. Mol. Struct.: THEOCHEM 341 (1), 1-11 (1995). 10.1016/0166-1280(95)04240-7
59. D. Xu, R. Chen, and H. Guo, " Probing highly excited vibrational eigenfunctions using a modified single Lanczos propagation method: Application to acetylene (HCCH)," J. Chem. Phys. 118 (16), 7273-7282 (2003). 10.1063/1.1562626
60. R. Chen, G. Ma, and H. Guo, " Six-dimensional quantum calculations of highly excited vibrational energy levels of hydrogen peroxide and its deuterated isotopomers," J. Chem. Phys. 114 (11), 4763-4774 (2001). 10.1063/1.1348274
61. 3 with geometric phase in hyperspherical coordinates," Chem. Phys. Lett. 281 (4), 312-318 (1997). 10.1016/S0009-2614(97)01253-0
62. G. H. Golub and C. F. Van Loan, Matrix Computations (JHU Press, 2012), Vol. 3.
63. R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods (SIAM, Philadelphia, PA, 1998), available at http://www.caam.rice.edu/software/ARPACK.
64. G. Avila and T. Carrington, Jr., " Non-product quadrature grids: Solving the vibrational Schroedinger equation in 12D," in Quantum Dynamic Imaging: Theoretical and Numerical Methods, edited by A. D. Bandrauk and M. Y. Ivanov (Springer, 2010). 10.1021/jp003792s
65. B. Brumm, " A fast matrix-free algorithm for spectral approximations to the Schrödinger equation," J. Scien. Comput. 37, A2003-A2025 (2015). 10.1137/140981022
66. N. C. Handy, " The derivation of vibration-rotation kinetic energy operators, in internal coordinates," Mol. Phys. 61 (1), 207-223 (1987). 10.1080/00268978700101081
67. E.-W. Knapp and D. J. Diestler, " Quantum-mechanical treatment of collision induced dissociation," J. Chem. Phys. 67 (11), 4969-4975 (1977). 10.1063/1.434680
68. 2Ne as examples," J. Chem. Phys. 77 (8), 4061-4072 (1982). 10.1063/1.444316
69. H. Wei and T. Carrington, Jr., " The discrete variable representation for a triatomic Hamiltonian in bond length-bond angle coordinates," J. Chem. Phys. 97, 3029-3037 (1992). 10.1063/1.463044
70. H. Wei and T. Carrington, Jr., " Discrete variable representations of complicated kinetic energy operators," J. Chem. Phys. 101, 1343-1360 (1994). 10.1063/1.467827
71. D. W. Schwenke, " On the computation of ro-vibrational energy levels of triatomic molecules," Comput. Phys. Commun. 70 (1), 1-14 (1992). 10.1016/0010-4655(92)90086-E
72. See supplementary material at http://dx.doi.org/10.1063/1.4936294 E-JCPSA6-143-019545 for a table of all the assignable HONO vibrational levels.
73. J. M. Hutson, " Intermolecular forces from the spectroscopy of van der Waals complexes," Ann. Rev. Phys. Chem. 41, 123-154 (1990). 10.1146/annurev.pc.41.100190.001011
74. E. L. Sibert, Int. Rev. Phys. Chem. 9, 1 (1990). 10.1080/01442359009353236
75. H. Koeppel, W. Domcke, and L. S. Cederbaum, Adv. Chem. Phys. 57, 59 (1984).
76. C. Leforestier, K. Szalewicz, and A. van der Avoird, " Spectra of water dimer from a new ab initio potential with flexible monomers," J. Chem. Phys. 137, 014305 (2012). 10.1063/1.4722338
77. H. Y. Mussa and J. Tennyson, " Calculation of the rotation-vibration states of water up to dissociation," J. Chem. Phys. 109, 10885-10892 (1998). 10.1063/1.476519
78. V. A. Mandelshtam and H. S. Taylor, " Spectral projection approach to the quantum scattering calculations," J. Chem. Phys. 102, 7390-7399 (1995). 10.1063/1.469051
79. H.-G. Yu and S. C. Smith, " The calculation of vibrational eigenstates by MINRES filter diagonalization," Ber. Bunsenges. Phys. Chem. 101, 400-406 (1997). 10.1002/bbpc.19971010312
80. R. Chen and H. Guo, " Discrete energy representation and generalized propagation of physical systems," J. Chem. Phys. 108, 6068-6077 (1998). 10.1063/1.476017
81. P.-N. Roy and T. Carrington, Jr., " An evaluation of methods designed to calculate energy levels in a selected range and application to a (one-dimensional) Morse oscillator and (three-dimensional) HCN/HNC," J. Chem. Phys. 103, 5600-5612 (1995). 10.1063/1.470543
82. -1," Chem. Phys. Lett. 365, 480-486 (2002). 10.1016/s0009-2614(02)01503-8
83. A. Jaeckle and H.-D. Meyer, " Product representation of potential energy surfaces," J. Chem. Phys. 104 (20), 7974-7984 (1996). 10.1063/1.471513
84. S. Manzhos and T. Carrington, Jr., " Using redundant coordinates to represent potential energy surfaces with lower-dimensional functions," J. Chem. Phys. 127 (1), 014103 (2007). 10.1063/1.2746846
85. H.-G. Yu and J. T. Muckerman, " A general variational algorithm to calculate vibrational energy levels of tetraatomic molecules," J. Mol. Spectrosc. 214, 11 (2002). 10.1006/jmsp.2002.8569
86. -1," J. Chem. Phys. 100 (9), 6175-6194 (1994). 10.1063/1.467273
87. J. C. Tremblay and T. Carrington, Jr., " A refined unsymmetric Lanczos eigensolver for computing accurate eigentriplets of a real unsymmetric matrix," Electron. Trans. Numer. Anal. 28, 95-113 (2007), pdf available at http://www.emis.ams.org/journals/ETNA/vol.28.2007/pp95-113.dir/pp95-113.pdf.
88. F. Heiss and V. Winschel, " Likelihood approximation by numerical integration on sparse grids," J. Econ. 144 (1), 62-80 (2008). 10.1016/j.jeconom.2007.12.004
89. R. Bourquin, " Exhaustive search for higher-order Kronrod-Patterson extensions " (2015), pdf available at http://www.sam.math.ethz.ch/~raoulb/research/preprints/kes.pdf.