Coordinate reduction for exploring chemical reaction paths

  • Adam B. Birkholz
  • H. Bernhard SchlegelEmail author
Regular Article


The potential energy surface for the reaction of a typical molecular system composed of N atoms is defined uniquely by 3N-6 coordinates. These coordinates can be defined by the Cartesian coordinates of the atomic centers (minus overall translation and rotation), or a set of internally defined coordinates such as bond stretches, angle bends, and torsions. By applying principal component analysis to the geometries along a reaction path, a reduced set of coordinates, d ≪ 3N-6, can be obtained. This reduced set of coordinates can reproduce the changes in geometry along the reaction path with chemical accuracy and may help improve the efficiency of reaction path optimization algorithms.


Reaction path PCA Potential energy surface Optimization 



This work was supported by a grant from the National Science Foundation (CHE0910858). Wayne State University’s computing grid provided computational support.


  1. 1.
    Wales D (2004) Energy landscapes: applications to clusters, biomolecules and glasses (Cambridge Molecular Science). Cambridge University Press, CambridgeGoogle Scholar
  2. 2.
    Schlegel HB (2011) Geometry optimization. WIREs Comput Mol Sci 1:790–809. doi: 10.1002/wcms.34 CrossRefGoogle Scholar
  3. 3.
    Heidrich D (1995) The reaction path in chemistry: current approaches and perspectives. Understanding chemical reactivity, vol 16. Kluwer Academic Publishers, DordrechtGoogle Scholar
  4. 4.
    Elber R, Karplus M (1987) A method for determining reaction paths in large molecules: Application to myoglobin. Chem Phys Lett 139:375–380CrossRefGoogle Scholar
  5. 5.
    Sheppard D, Terrell R, Henkelman G (2008) Optimization methods for finding minimum energy paths. J Chem Phys 128:134106. doi: 10.1063/1.2841941 CrossRefGoogle Scholar
  6. 6.
    Burger SK, Yang WT (2006) Quadratic string method for determining the minimum-energy path based on multiobjective optimization. J Chem Phys 124:054109CrossRefGoogle Scholar
  7. 7.
    Ischtwan J, Collins MA (1994) Molecular potential energy surfaces by interpolation. J Chem Phys 100:8080–8088. doi: 10.1063/1.466801 CrossRefGoogle Scholar
  8. 8.
    Schatz GC (1990) The analytical representation of potential energy surfaces for chemical reactions. Advances in molecular electronic structure theory. JAI Press, LondonGoogle Scholar
  9. 9.
    Bolton K, Hase WL, Peslherbe GH (1998) Direct dynamics of reactive systems. Modern methods for multidimensional dynamics computation in chemistry. World Scientific, SingaporeGoogle Scholar
  10. 10.
    Baker J, Kessi A, Delley B (1996) The generation and use of delocalized internal coordinates in geometry optimization. J Chem Phys 105:192–212. doi: 10.1063/1.471864 CrossRefGoogle Scholar
  11. 11.
    Kato S, Morokuma K (1980) Potential-energy characteristics and energy partitioning in chemical-reactions—ab initio MO study of 4-centered elimination-reaction CH3CH2F–CH2 = CH2 + HF. J Chem Phys 73:3900–3914. doi: 10.1063/1.440576 CrossRefGoogle Scholar
  12. 12.
    Konkoli Z, Kraka E, Cremer D (1997) Unified reaction valley approach mechanism of the reaction CH3 + H-2- > CH4 + H. J Phys Chem A 101:1742–1757. doi: 10.1021/jp962877j CrossRefGoogle Scholar
  13. 13.
    Jolliffe IT (2002) Principal component analysis, vol XXIX, 2nd edn. Springer Series in Statistics, NYGoogle Scholar
  14. 14.
    Jensen F, Palmer DS (2011) Harmonic vibrational analysis in delocalized internal coordinates. J Chem Theor Comput 7:223–230. doi: 10.1021/ct100463a CrossRefGoogle Scholar
  15. 15.
    Balsera MA, Wriggers W, Oono Y, Schulten K (1996) Principal component analysis and long time protein dynamics. J Phys Chem 100:2567–2572. doi: 10.1021/jp9536920 CrossRefGoogle Scholar
  16. 16.
    Palmer DS, Jensen F (2011) Predicting large-scale conformational changes in proteins using energy-weighted normal modes. Proteins 79:2778–2793. doi: 10.1002/prot.23105 CrossRefGoogle Scholar
  17. 17.
    Fukui K (1981) The path of chemical-reactions—the IRC approach. Acc Chem Res 14:363–368. doi: 10.1021/ar00072a001 CrossRefGoogle Scholar
  18. 18.
    Gonzalez C, Schlegel HB (1990) Reaction-path following in mass-weighted internal coordinates. J Phys Chem 94:5523–5527. doi: 10.1021/j100377a021 CrossRefGoogle Scholar
  19. 19.
    Gonzalez C, Schlegel HB (1989) An improved algorithm for reaction-path following. J Chem Phys 90:2154–2161. doi: 10.1063/1.456010 CrossRefGoogle Scholar
  20. 20.
    Pulay P, Fogarasi G (1992) Geometry optimization in redundant internal coordinates. J Chem Phys 96:2856–2860CrossRefGoogle Scholar
  21. 21.
    Hratchian HP, Schlegel HB (2005) Using Hessian updating to increase the efficiency of a Hessian based predictor-corrector reaction path following method. J Chem Theory Comput 1:61–69. doi: 10.1021/ct0499783 CrossRefGoogle Scholar
  22. 22.
    Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Mennucci B, Petersson GA, Nakatsuji H, Caricato M, Li X, Hratchian HP, Izmaylov AF, Bloino J, Zheng G, Sonnenberg JL, Liang W, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Vreven T, Montgomery JA, Peralta JE, Ogliaro F, Bearpark M, Heyd JJ, Brothers E, Kudin KN, Staroverov VN, Keith T, Kobayashi R, Normand J, Raghavachari K, Rendell A, Burant JC, Iyengar SS, Tomasi J, Cossi M, Rega N, Millam JM, Klene M, Knox JE, Cross JB, Bakken V, Adamo C, Jaramillo J, Gomperts R, Stratman RE, Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochierski JW, Martin RL, Morokuma K, Zakrzewski VG, Voth GA, Salvador P, Dannenberg JJ, Dapprich S, Mayhall NJ, Daniels AD, Farkas O, Foresman JB, Ortiz JV, Cioslowski J, Fox DJ (2010) Gaussian 09 Version B.01; Gaussian, Inc., Wallingford, CTGoogle Scholar
  23. 23.
    Mathematica Version 7.0 (2008) Wolfram Research, Inc. Champaign, ILGoogle Scholar
  24. 24.
    Heathcock CH, Mahaim C, Schlecht MF, Utawanit T (1984) A synthetic approach to the quassinoids. J Org Chem 49:3264–3274. doi: 10.1021/jo00192a004 CrossRefGoogle Scholar
  25. 25.
    Headgordon T, Headgordon M, Frisch MJ, Brooks CL, Pople JA (1991) Theoretical-study of blocked glycine and alanine peptide analogs. J Am Chem Soc 113:5989–5997CrossRefGoogle Scholar
  26. 26.
    Stewart J (2007) Optimization of parameters for semiempirical methods V: modification of NDDO approximations and application to 70 elements. J Mol Model 13:1173–1213. doi: 10.1007/s00894-007-0233-4 CrossRefGoogle Scholar
  27. 27.
    Sonnenberg JL, Wong KF, Voth GA, Schlegel HB (2009) Distributed gaussian valence bond surface derived from ab initio calculations. J Chem Theory Comput 5:949–961. doi: 10.1021/ct800477y CrossRefGoogle Scholar
  28. 28.
    Niu SQ, Hall MB (2000) Theoretical studies on reactions of transition-metal complexes. Chem Rev 100:353–405CrossRefGoogle Scholar
  29. 29.
    Zhou J, Tao P, Fisher JF, Shi Q, Mobashery S, Schlegel HB (2010) QM/MM studies of the matrix metalloproteinase 2 (MMP2) inhibition mechanism of (S)-SB-3CT and its oxirane analogue. J Chem Theory Comput 6:3580–3587. doi: 10.1021/ct100382k CrossRefGoogle Scholar
  30. 30.
    Becke AD (1993) Density-functional thermochemistry. 3. The role of exact exchange. J Chem Phys 98:5648–5652CrossRefGoogle Scholar
  31. 31.
    Perdew JP, Chevary JA, Vosko SH, Jackson KA, Pederson MR, Singh DJ, Fiolhais C (1992) Atoms, molecules, solids, and surfaces—applications of the generalized gradient approximation for exchange and correlation. Phys Rev B 46:6671–6687CrossRefGoogle Scholar
  32. 32.
    Perdew JP (1991) Generalized gradient approximations for exchange and correlation—a look backward and forward. Phys B 172:1–6CrossRefGoogle Scholar
  33. 33.
    Perdew JP (1991) Electronic structure of solids. Akadernie Verlag, BerlinGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of ChemistryWayne State UniversityDetroitUSA

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