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Analysis of Multidimensional Potential Energy Surfaces — Stationary and Critical Points —

  • D. Heidrich
  • W. Kliesch
  • W. Quapp
Chapter
Part of the Lecture Notes in Chemistry book series (LNC, volume 56)

Abstract

As outlined in the introducing chapter, the minimizers and the saddle points of index one of the energy functional are corner-stones of most reaction theories in chemistry. Thus the applicability of these theories strongly depends on the availability of mathematical methods, which compute such points in an effective manner. Therefore the current chapter is engaged in the presentation of methods, which enable to compute minima and/or saddles of a PES.

Keywords

Saddle Point Stationary Point Hessian Matrix Descent Method Zero Eigenvalue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References (Chapter 2)

  1. 1.
    Kolman B, Trench WF (1971) Elementary Multivariable Calculus, Academic Press, New YorkGoogle Scholar
  2. 2.
    Leichtweiss K, Profke L (1972) Analytische Geometrie, Teubner, StuttgartGoogle Scholar
  3. 3.
    Hirsch W, Smale S (1974) Differential Equations,Dynamical Systems, and Linear Algebra, Academic Press, New YorkGoogle Scholar
  4. 4.
    Zurmühl R (1964) Matrizen und ihre technischen Anwendungen, Springer, BerlinGoogle Scholar
  5. 5.
    Morê JJ, Sorensen DC (1984) in: Golub GH (ed) Studies in Numerical Analysis, Mathem. Ass. America, p 29Google Scholar
  6. 6.
    Dennis Jr WC, Schnabel RB (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood CliffsGoogle Scholar
  7. 7.
    Heidrich D, Quapp W (1986) Theor Chim Acta (Berl) 70: 89CrossRefGoogle Scholar
  8. 8.
    Bell S, Crighton JS (1984) J Chem Phys 80: 2464CrossRefGoogle Scholar
  9. 9.
    Head JD, Zerner MC (1986) in: Jorgensen P, Simons J (eds) Geometrical Derivatives of Energy Surfaces and Molecular Properties, Reidel, DordrechtGoogle Scholar
  10. 10.
    Schlegel B (1987) in: Lawley KP (ed) Ab Initio Methods in Quantum Chemistry, Wiley, New York, p 249Google Scholar
  11. 11.
    a) Incerti S, Parisi V, Zirilli F (1979) SIAM J Numer Anal 16:779 (b) Zirilli F (1982) SIAM J Numer Anal 19: 800Google Scholar
  12. 12.
    Ackermann S, Schenk K (1990) The invariance problem in geometry optimization and vibrational analysis. A systematic theoretical study of non-redundant coordinate (in preparation)Google Scholar
  13. 13.
    Mclver JW, Komornicki A (1972) J Am Chem Soc 94: 2625CrossRefGoogle Scholar
  14. 14.
    Simons J, Jorgensen P, Taylor H, Ozment J (1983) J Phys Chem 87: 2745CrossRefGoogle Scholar
  15. 15.
    Ortega J, Rheinboldt W (1970) Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New YorkGoogle Scholar
  16. 16.
    Fletcher R (1987) Practical Methods for Optimization, 2 edn, Wiley, ChichesterGoogle Scholar
  17. 17.
    Sorensen DC (1982) SIAM J Numer Anal 19: 409CrossRefGoogle Scholar
  18. 18.
    Shu It z GA, Schnabel RB, Byrd RH (1985) SIAM J Numer Anal 22: 47Google Scholar
  19. 19.
    Dembo RS, lisenstat SC, Steinhaug T (1982) SIAM J Numer Anal 19: 400Google Scholar
  20. 20.
    Dembo RS, Steinhaug T (1983) Math Programming 26: 190CrossRefGoogle Scholar
  21. 21.
    Schlick T, Overton M (1987) J Comput Chem 8: 1025CrossRefGoogle Scholar
  22. 22.
    Dennis Jr JE, Moré JJ (1977) SIAM Rev 19: 46CrossRefGoogle Scholar
  23. 23.
    Broyden CG (1965) Math Comp 19: 577CrossRefGoogle Scholar
  24. 24.
    Davidon WC (1959) Variable metric methods for minimization, Rep ANL-5990 Rev, Argonne National Laboratory, Argonne, 111CrossRefGoogle Scholar
  25. 25.
    Broyden CG (1967) Math Comp 21: 368CrossRefGoogle Scholar
  26. 26.
    Murtagh BA, Sargent RWH (1972) Comput J 13: 185CrossRefGoogle Scholar
  27. 27.
    Dennis, jr JE, Schnabel RB (1979) SIAM Rev 21: 443Google Scholar
  28. 28.
    Powell MJD (1970) in: Rosen JB, Mangasarian OL, Ritter K (eds) Nonlinear Programming, Academic Press, New YorkGoogle Scholar
  29. 29.
    Fletcher R, Powell MJD (1963) Comput J 6: 163Google Scholar
  30. 30.
    Schwetlick H (1979) Numerische Lösung nichtlinearer Gleichungen, Deutscher Verlag der Wissenschaften, BerlinGoogle Scholar
  31. 31.
    Broyden CG (1971) J Int Math Appi 6: 76, 222Google Scholar
  32. 32.
    Fletcher R (1970) Comput J 13: 317CrossRefGoogle Scholar
  33. 33.
    Goldfarb D (1970) Math Comput 24: 23CrossRefGoogle Scholar
  34. 34.
    Shanno DF (1970) Math Comput 24: 647CrossRefGoogle Scholar
  35. 35.
    Brodlie KW, Gourlay AR, Greenstadt J (1973) J Inst Math Appi 11: 73CrossRefGoogle Scholar
  36. 36.
    Davidon WC (1975) Math Programming 9: 1CrossRefGoogle Scholar
  37. 37.
    Schnabel RB (1978) Math Programming 15: 247CrossRefGoogle Scholar
  38. 38.
    Dennis Jr JE, Moré JJ (1974) Math Comput 28: 549CrossRefGoogle Scholar
  39. 39.
    Rheinboldt WC, Vandercraft JS (1974) SIAM J Numer Anal 11: 1069CrossRefGoogle Scholar
  40. 40.
    Broyden CG, Dennis JE, Moré JJ (1973) J Inst Math 12: 223CrossRefGoogle Scholar
  41. 41.
    Rheinboldt WC (1974) Methods for Solving Systems of Nonlinear Equations, Society for Industrial and Applied Mathematics, PhiladelphiaGoogle Scholar
  42. 42.
    Head JD, Zerner MC (1985) Chem Phys Lett 122: 264CrossRefGoogle Scholar
  43. 43.
    Agrafiotis DK, Rzepa HS (1988) J Chem Research(S) 1988 (3): 100Google Scholar
  44. 44.
    Frisch MJ, Binkley JS, Schlegel HB, Raghavachari K, Melius CF, Martin RL, Stewart JJP, Bobrowicz FW, Rohlfing CM, Kahn LR, Defrees DJ, Seeger R, Whiteside RA, Fox DJ, Flunder EM, Pople JA “Gaussian 86”, Gaussian Ine, Pittsburgh, PAGoogle Scholar
  45. 45.
    Amos RD, Rice JE (1988) CADPAC: The Cambridge Analytic Derivatives Package, University of Cambridge, Cambridge, EnglandGoogle Scholar
  46. 46.
    Cerjan CJ, Miller WH (1981) J Chem Phys 75: 2800CrossRefGoogle Scholar
  47. 47.
    Banerjee A, Adams N, Simons J, Shepard R (1985) J Phys Chem 89:52; Nichols J, Taylor H, Schmidt P, Simons J (1990) J Chem Phys 92: 340Google Scholar
  48. 48.
    Baker J (1986) J Comput Chem 7:385; Baker J (1987) J Comput Chem 8: 563Google Scholar
  49. 49.
    Biegler-König F (1985) J Optimiz Theory amp; Appl 47: 393CrossRefGoogle Scholar
  50. 50.
    Thiel W (1988) J Mol Struct 163: 415Google Scholar
  51. 51.
    Komornicki A (1985) GRADSCF: An ab initio Gradient Program System, Version 9. 2, Polyatomics Research Institute, Mounta in ViewGoogle Scholar
  52. 52.
    Dixon LCW (1972) J Optimiz Theory amp; Appl 10: 34CrossRefGoogle Scholar
  53. 53.
    Wolfe P (1971) SIAM Rev 13: 185CrossRefGoogle Scholar
  54. 54.
    Byrd RH, Nocedal J, Yuan Y (1987) SIAM J Numer Anal 24: 1171CrossRefGoogle Scholar
  55. 55.
    Kliesch W, Schenk K, Heidrich D, Dachsei H (1988) J Comput Chem 9: 810CrossRefGoogle Scholar
  56. 56.
    Smith CM (1988) Theor Chim Acta (Beri) 74: 85CrossRefGoogle Scholar
  57. 57.
    Snyman JA (1982) Appi Math Modelling 6: 449CrossRefGoogle Scholar
  58. 58.
    Brauer F, Nohel, JA (1969) The Qualitative Theory of Ordinary Differential Equations, W Benjamin, New YorkGoogle Scholar
  59. 59.
    Aluffi-Pentini F, Parisi V, Zirilli F (1984) ACM Trans Math Software 16: 299CrossRefGoogle Scholar
  60. 60.
    Brown AA, Bartholomew-Biggs MC (1989) J Optimiz Theory amp; Appl 62: 211CrossRefGoogle Scholar
  61. 61.
    Crowder HP, Dembo RS, Mulvey JM (1978) Math Programming 15: 316CrossRefGoogle Scholar
  62. 62.
    Schenk K (1990) Dissertation A, Karl-Marx-Universität, LeipzigGoogle Scholar
  63. 63.
    Pechukas P (1976) J Chem Phys 64: 1516–21CrossRefGoogle Scholar
  64. 64.
    Beatty MF (1977) J Appl Mechanics 44, No3: 501Google Scholar
  65. 65.
    Miller WH, Handy NC, Adams JE (1980) J Chem Phys 72: 99CrossRefGoogle Scholar
  66. 66.
    Frederik JH, McClelland GM, Brumer P (1985) J Chem Phys 83: 190CrossRefGoogle Scholar
  67. 67.
    Kielbasinski A, Schwetlick H (1988) Numerische lineare Algebra, VEB Deutscher Verl.Wiss., BerlinGoogle Scholar
  68. 68.
    Guichardet A (1984) Ann Inst Henri Poincaré 40: 329Google Scholar
  69. 69.
    Iwai T (1987) J Math Phys 28:964,1315, (1988) 29: 1325Google Scholar
  70. 70.
    Mezey PG (1985) Potential Energy Hypersurfaces, Elsevier, AmsterdamGoogle Scholar
  71. 71.
    Randell R (1988) Stud Phys Theor Chem 54: 125Google Scholar
  72. 72.
    Iwai T (1987) Ann Inst Henri Poincaré 47: 199Google Scholar
  73. 73.
    Guichardet A (1986) ibid. 44:91Google Scholar
  74. 74.
    Tachibana A, Iwai T (1986) Phys Rev A 33: 2262CrossRefGoogle Scholar
  75. 75.
    Collard K, Hall GG (1977) Intl J Quant Chem 12: 623CrossRefGoogle Scholar
  76. 76.
    Connor JNL (1976) Mol Phys 31: 33CrossRefGoogle Scholar
  77. 77.
    Poston T, Steward I (1978) Catastrophe Theory and its Application, P itman, LondonGoogle Scholar
  78. 78.
    Thorn R (1977) Stabilite Structurelle et Morphogénèse, second edition, Inter Editions, ParisGoogle Scholar
  79. 79.
    Murrell JN, Laidler KJ (1968) Trans Faraday Soc 64: 371CrossRefGoogle Scholar
  80. 80.
    Stanton RE, Mclver JW (1975) J Am Chem Soc 97: 3632CrossRefGoogle Scholar
  81. 81.
    Hénon M, Heiles C (1964) Astron J 69: 73CrossRefGoogle Scholar
  82. 82.
    Noid DW, Koszykowski ML, Marcus RA (1981) Ann Rev Phys Chem 32: 267CrossRefGoogle Scholar
  83. 83.
    Aquilanti V, Cavalli S (1987) Chem Phys Lett 141: 309CrossRefGoogle Scholar
  84. 84.
    Aquilanti V, Cavalli S, Grossi G (1989) Theor Chim Acta 75: 33CrossRefGoogle Scholar
  85. 85.
    Eisenberg JM, Greiner W (1987) Nuclear Theory, Vol 1, 3rd edn, North Holland, AmsterdamGoogle Scholar
  86. 86.
    Waite BA, Miller WH (1981) J Chem Phys 74:3910, (1982) 76: 2412Google Scholar
  87. 87.
    Gilmore R (1981) Catastrophe Theory for Scientists and Engineers, Wiley, NYGoogle Scholar
  88. 88.
    Golubitzky M, Schaeffer D (1979) Corom Pure Appl Math 32: 21CrossRefGoogle Scholar
  89. 89.
    Quapp W (1989) Theor Chim Acta 75: 447CrossRefGoogle Scholar
  90. 90.
    Valtazano P, Ruedenberg K (1986) Theor Chim Acta 69: 281CrossRefGoogle Scholar
  91. 91.
    Gilmore R, Kais S, Levine RD (1986) Phys Rev A 34: 2442CrossRefGoogle Scholar
  92. 92.
    Della Valle RG, Brillante A (1987) Chem Phys 114: 141CrossRefGoogle Scholar
  93. 93.
    Schleyer PvR, Sawaryn A, Read AE, Hobza P (1986) J Comput Chem 7: 666CrossRefGoogle Scholar
  94. 94.
    Clementi E, Kistenmacher H, Popkie H (1973) J Chem Phys 58: 2460CrossRefGoogle Scholar
  95. 95.
    Vaals JJv, Meerts WL, Dymanus A (1983) Chem Phys 82: 385CrossRefGoogle Scholar
  96. 96.
    Somorjan RL, Hornig DF (1961) J Chem Phys 36: 1980CrossRefGoogle Scholar
  97. 97.
    Chan SI, Stelman D, Thompson LE (1964) J Chem Phys 41: 2828CrossRefGoogle Scholar
  98. 98.
    Janke W, Kleinert H (1987) Chem Phys Lett 137: 162CrossRefGoogle Scholar
  99. 99.
    Ali MK, Wood WR, Dewitt JS (1986) J Math Phys 27: 1806CrossRefGoogle Scholar
  100. 100.
    Tipping RH, Ogilvie JF (1983) Phys Rev A 27: 95CrossRefGoogle Scholar
  101. 101.
    Alvarez G, Graffi S, Silverstone HJ (1988) Phys Rev A 38: 1687CrossRefGoogle Scholar
  102. 102.
    Feix G (1983) Z phys Chem Leipzig 264: 369Google Scholar
  103. 103.
    Laane J (1970) Appl Spectrosc 24,Noi:73Google Scholar
  104. 104.
    Miller FA (1983) Proc Indian Acad Sci 92: 563Google Scholar
  105. 105.
    Winnewisser BP (1985) in: Narahari Rao K (ed) Molecular Spectroscopy — Modern Research, Vol I II, Academic Press, OrlandoGoogle Scholar
  106. 106.
    Quapp W, Winnewisser BP (1991) paper in preparationGoogle Scholar
  107. 107.
    Monkenbusch M (1987) Acta Cryst A 43: 198CrossRefGoogle Scholar
  108. 108.
    Wilson EB, Decius JC, Cross PC (1955) Molecular Vibrations, McGraw-Hill, New YorkGoogle Scholar
  109. 109.
    Dachsei H, Quapp W (1991) J Math Chem 6: 77CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • D. Heidrich
    • 1
  • W. Kliesch
    • 2
  • W. Quapp
    • 2
  1. 1.Sektion ChemieUniversität LeipzigLeipzigDeutschland
  2. 2.Sektion MathematikUniversität LeipzigLeipzigDeutschland

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