Numerical Methods

  • Stavros C. Farantos
Part of the SpringerBriefs in Molecular Science book series (BRIEFSMOLECULAR)


The theories developed in the previous chapters, classical and quantum mechanical, are put in action by discretizing the corresponding differential equations. The variable order finite difference approximation to the unknown solutions and their derivatives is the preferred method, not only because of their well understood convergence properties and the relatively easy way of their programming, but also, finite differences provide a uniform approach to different type of equations, especially when we work in a Cartesian coordinate system. With respect to Schrödinger equation several grids are examined and comparisons with the more popular pseudospectral methods is made. For the location of periodic orbits the multiple shooting method is developed as it has been thoroughly tested. Finally, computer codes for studying classical nonlinear molecular dynamics and solving the Schrödinger equation are described.


Periodic Orbit Pseudospectral Method Finite Difference Approximation Sinc Function Cardinal Function 
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  1. 1.
    Aguiar MAM, Malta CP, Baranger M, Davies KTR (1987) Bifurcations of periodic trajectories in non-integrable Hamiltonian systems with two degrees of freedom: numerical and analytical results. Annals Phys 180:167CrossRefGoogle Scholar
  2. 2.
    Allqower EL, Georg K (1990) Numerical continuation methods., Springer series in computational mathematics, Springer, BerlinGoogle Scholar
  3. 3.
    Balay S, Gropp WD, McInnes LC, Smith BF (1997) Efficient management of parallelism in object oriented numerical software libraries. In: Arge E, Bruaset AM, Langtangen HP (eds) Modern software tools in scientific computing, Birkhäuser Press, pp 163–202Google Scholar
  4. 4.
    Balay S, Buschelman K, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Smith BF, Zhang H (2001) PETSc Web page.
  5. 5.
    Balay S, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Smith BF, Zhang H (2004) PETSc users manual. Technical Report ANL-95/11— Revision 2.1.5, Argonne national laboratory.Google Scholar
  6. 6.
    Baranger M, Davies KTR, Mahoney JH (1988) The calculation of periodic trajectories. Annals Phys 186:95–110CrossRefGoogle Scholar
  7. 7.
    Baye D, Heenen PH (1986) Generalised meshes for quantum mechanical problems. J Phys A: Math Gen 19:2041–2059CrossRefGoogle Scholar
  8. 8.
    Boyd JP (1992) A fast algorithm for Chebyshev, Fourier, and Sinc interpolation onto an irregular grid. J Comp Phys 103:243–257CrossRefGoogle Scholar
  9. 9.
    Boyd JP (1994) Sum-accelerated pseudospectral methods: finite differences and sech-weighted differences. Comp Methods Appl Mech Engrg 116:1–11CrossRefGoogle Scholar
  10. 10.
    Carrington T Jr (2004) Methods for calculating vibrational energy levels. Can J Chem 82:900–914CrossRefGoogle Scholar
  11. 11.
    Chen R, Guo H (1996) A general and efficient filter-diagonalization method without time propagation. J Chem Phys 105(4):1311–1317CrossRefGoogle Scholar
  12. 12.
    Colbert DT, Miller WH (1992) A novel discrete variable representation for quantum mechanical reactive scattering via the S-matrix Kohn method. J Chem Phys 96:1982–1991CrossRefGoogle Scholar
  13. 13.
    Collins P, Burbanks A, Wiggins S, Waalkens H, Schubert R (2008) Background and documentation of software for computing Hamiltonian normal forms. School of mathematics, University of Bristol, University Walk, Bristol BS8 1TW, 1st ednGoogle Scholar
  14. 14.
    Davies KTR, Huston TE, Baranger M (1992) Calculations of periodic trajectories for the Hénon-Heiles Hamiltonian using the monodromy method. Chaos 2:215–224CrossRefGoogle Scholar
  15. 15.
    Deuflhard P (1974) A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting. Numer Math 22:189–315CrossRefGoogle Scholar
  16. 16.
    Deuflhard P (1979) A stepsize control for continuation methods and its special application to multiple shooting techniques. Numer Math 33:115–146CrossRefGoogle Scholar
  17. 17.
    Farantos SC (1998) POMULT: a program for computing periodic orbits in Hamiltonian systems based on multiple shooting algorithms. Comp Phys Comm 108:240–258CrossRefGoogle Scholar
  18. 18.
    Feudel U, Jansen W (1992) CANDYS/QA—a software system for the qualitative analysis of nonlinear dynamical systems. Int J Bifurc and Chaos 2:773–794CrossRefGoogle Scholar
  19. 19.
    Fornberg B (1998) A practical guide to pseudospectral methods. Cambridge monographs on applied and computational mathematics, Cambridge University Press, CambridgeGoogle Scholar
  20. 20.
    Fornberg B, Sloan DM (1994) A review of pseudospectral methods for solving partial differential equations. Acta Numerica 3:203–267CrossRefGoogle Scholar
  21. 21.
    Gradshteyn IS, Ryzhik IM (1994) Table of integrals, series and products. Academic Press, New YorkGoogle Scholar
  22. 22.
    Guantes R, Farantos SC (1999) High order finite difference algorithms for solving the Schrödinger equation in molecular dynamics. J Chem Phys 111:10,827–10,835CrossRefGoogle Scholar
  23. 23.
    Guantes R, Farantos SC (2000) High order finite difference algorithms for solving the Schrödinger equation in molecular dynamics II: periodic variables. J Chem Phys 113:10,429–10,437CrossRefGoogle Scholar
  24. 24.
    Guantes R, Nezis A, Farantos SC (1999) Periodic orbit—quantum mechanical investigation of the inversion mechanism of Ar\(_3\). J Chem Phys 111:10,836–10,842CrossRefGoogle Scholar
  25. 25.
    Guo H (2007) Recursive solutions to large eigenproblems in molecular spectroscopy and reaction dynamics. Rev Comput Chem 25:285–347Google Scholar
  26. 26.
    Hénon M (1982) On the numerical computation of Poincaré maps. Physica D 5:412–414CrossRefGoogle Scholar
  27. 27.
    Karabulut H, Sibert EL III (1997) Trigonometric discrete variable representations. J Phys B: At Mol Opt Phys 30:L513–L516CrossRefGoogle Scholar
  28. 28.
    Keller HB (1976) Regional Conf Ser in Appl Math Numerical solution of two point boundary value problems SIAM. 24:61Google Scholar
  29. 29.
    Kosloff R (1994) Propagation methods for quantum molecular dynamics. Annu Rev Phys Chem 45:145–178CrossRefGoogle Scholar
  30. 30.
    Kosloff R (1996) Quantum molecular dynamics on grids. In: Zhang JZH Wyatt RE (eds) Marcel Dekker Inc, Dynamics of molecules and chemical reactions, pp 185–230Google Scholar
  31. 31.
    Lanczos C (1950) An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J Res Natl Bur Stand 45:255–282CrossRefGoogle Scholar
  32. 32.
    Light JC, Hamilton IP, Lill JV (1985) Generalized discrete variable approximation in quantum mechanics. J Chem Phys 82:1400–1409CrossRefGoogle Scholar
  33. 33.
    Mandelshtam VA, Taylor HS (1995) Spectral projection approach to the quantum scattering calculations. J Chem Phys 102(19):7390–7399CrossRefGoogle Scholar
  34. 34.
    Mandelshtam VA, Taylor HS (1997) The quantum resonance spectrum of the H\(^{3+}\) molecular ion for J\(=\)0. An accurate calculation using filter-diagonalization. J Chem Soc, Faraday Trans 93:847–860CrossRefGoogle Scholar
  35. 35.
    Mandelshtam VA, Grozdanov TP, Taylor HS (1995) Bound states and resonances of the hydroperoxyl radical HO\(_2\): an accurate quantum mechanical calculation using filter-diagonalization. J Chem Phys 103(23):10,074–10,084CrossRefGoogle Scholar
  36. 36.
    Mathews J, Walker RL (1970) Mathematical methods of physics. Addison-Wesley, ReadingGoogle Scholar
  37. 37.
    Muckerman JT (1990) Some useful discrete variable representations for problems in time-dependent and time-independent quantum mechanics. Chem Phys Lett 173:200–205CrossRefGoogle Scholar
  38. 38.
    Ponder JW (2014) Tinker molecular modelling
  39. 39.
    Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipies. Cambridge University Press, CambridgeGoogle Scholar
  40. 40.
    Reithmeier E (1991) Periodic solutions of nonlinear dynamical systems. Lecture notes in mathematics, SpringerGoogle Scholar
  41. 41.
    Seydel R (1988) From equilibrium to chaos: practical bifurcation and stability analysis. Elsevier, New YorkGoogle Scholar
  42. 42.
    Stamatiadis S, Farantos SC (2010) AUTO_DERIV: tool for automatic differentiation of a FORTRAN code (New Version). Comp Phys Comm 181(10):1818–1819CrossRefGoogle Scholar
  43. 43.
    Stamatiadis S, Prosmiti R, Farantos SC (2000) AUTO_DERIV: tool for automatic differentiation of a FORTRAN code. Comp Phys Comm 127:343–355CrossRefGoogle Scholar
  44. 44.
    Stenger F (1981) Numerical methods based on Whittaker cardinal, or Sinc functions. SIAM Rev 23:165–224CrossRefGoogle Scholar
  45. 45.
    Stoer J, Bulirsch R (1980) Introduction to numerical analysis. Springer, New YorkCrossRefGoogle Scholar
  46. 46.
    Suarez J, Farantos SC, Stamatiadis S, Lathouwers L (2009) A method for solving the molecular Schrödinger equation in Cartesian coordinates via angular momentum projection operators. Comp Phys Comm 180:2025–2033CrossRefGoogle Scholar
  47. 47.
    Szegö G (1948) Orthogonal polynomials. Am Math Soc Colloq Publ 23:42Google Scholar
  48. 48.
    Tal-Ezer H, Kosloff R (1984) An accurate and efficient scheme for propagating the time dependent Schrödinger equation. J Chem Phys 81(9):3967–3971CrossRefGoogle Scholar
  49. 49.
    Wall MR, Neuhauser D (1995) Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. I. Theory and application to a quantum-dynamics model. J Chem Phys 102(20):8011–8022CrossRefGoogle Scholar
  50. 50.
    Zhang DH, Zhang JZH (1996) Time-dependent quantum dynamics for gas-phase and gas-surface reactions. In: Zhang JZH Wyatt RE (eds) Dynamics of molecules and chemical reactions, Marcel Dekker Inc, pp 231–276Google Scholar
  51. 51.
    Zhu W, Zhao X, Tang Y (1996) Numerical methods with high order of accuracy applied in the quantum system. J Chem Phys 104(6):2275–2286CrossRefGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Department of ChemistryUniversity of CreteIraklionGreece
  2. 2.Institute of Electronic Structure and LaserFoundation for Research and Technology-HellasIraklionGreece

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