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Mathematical Problems of Nuclear Configuration Averaging

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Abstract

The concept of the average of a family of related nuclear configurations, for example, the average of those configurations which are slightly distorted versions of a given stable conformer of a molecule, has a role as both interpretative tool and also as a reference configuration in practical, computational use. However, depending on the actual coordinates used along which the average is defined, the average of nuclear configurations is not necessarily a physically viable arrangement, a fact that has to be taken into account when generating the corresponding electron density averages. Some of the associated mathematical and computational problems are described and the validity of a macroscopically motivated approach to conformation averaging is discussed.

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References

  1. P.G. Mezey, Optimization and analysis of energy hypersurfaces, in: Computational Theoretical Organic Chemistry (Reidel, New York, 1981) pp. 101–128.

  2. P.G. Mezey, Catchment region partitioning of energy hypersurfaces, I, Theor. Chim. Acta 58 (1981) 309–330.

    Google Scholar 

  3. P.G. Mezey, The isoelectronic and isoprotonic energy hypersurface and the topology of the nuclear charge space, Internat. J. Quantum Chem. Symp. 15 (1981) 279–285.

    Google Scholar 

  4. P.G. Mezey, Manifold theory of multidimensional potential surfaces, Internat. J. Quantum Chem., Quant. Biol. Symp. 8 (1981) 185–196.

    Google Scholar 

  5. P.G. Mezey, Critical level topology of energy hypersurfaces, Theor. Chim. Acta 60 (1981) 97–110.

    Google Scholar 

  6. P.G. Mezey, Lower and upper bounds for the number of critical points on energy hypersurfaces, Chem. Phys. Letters 82 (1981) 100–104; 86 (1982) 562.

    Google Scholar 

  7. P.G. Mezey, The symmetry of electronic energy level sets and total energy relations in the abstract nuclear charge space, Mol. Phys. 47 (1982) 121–126.

    Google Scholar 

  8. P.G. Mezey, Topology of energy hypersurfaces, Theor. Chim. Acta 62 (1982) 133–161.

    Google Scholar 

  9. P.G. Mezey, Level set topologies and convexity relations for Hamiltonians with linear parameters, Chem. Phys. Letters 87 (1982) 277–279.

    Google Scholar 

  10. P.G. Mezey, Level set topology of the nuclear charge space and the electronic energy functional, Internat. J. Quantum Chem. 22 (1982) 101–114.

    Google Scholar 

  11. P.G. Mezey, Quantum chemical reaction networks, reaction graphs and structure of potential energy hypersurfaces, Theor. Chim. Acta 60 (1982) 409–428.

    Google Scholar 

  12. P.G. Mezey, Reaction topology: manifold theory of potential surfaces and quantum chemical synthesis design, in: Chemical Applications of Topology and Graph Theory, ed. R.B. King (Elsevier, Amsterdam, 1983) pp. 75–98.

    Google Scholar 

  13. P.G. Mezey, An approach to conformation analysis on multidimensional potential surfaces, Internat. J. Quantum Chem., Quant. Biol. Symp. 10 (1983) 153–160.

    Google Scholar 

  14. P.G. Mezey, Inequalities and homotopy relations in reaction topology, Internat. J. Quantum Chem., Quant. Chem. Symp. 17 (1983) 453–460.

    Google Scholar 

  15. P.G. Mezey, The differentiable manifold model of quantum chemical reaction networks, Internat. J. Quantum Chem., Quant. Chem. Symp. 17 (1983) 137–152.

    Google Scholar 

  16. P.G. Mezey, Classification schemes of nuclear geometries and the concept of chemical structure. Metric spaces of chemical structure sets over potential energy hypersurfaces, J. Chem. Phys. 78 (1983) 6182–6186.

    Google Scholar 

  17. P.G. Mezey, A molecular geometry invariant property of energy level set boundaries in Z-space, Internat. J. Quantum Chem. 24 (1983) 523–526.

    Google Scholar 

  18. P.G. Mezey, The topology of energy hypersurfaces. II. Reaction topology in Euclidean spaces, Theor. Chim. Acta 63 (1983) 9–33.

    Google Scholar 

  19. P.G. Mezey, Molecular structure and reaction mechanism: a topological approach to quantum chemistry, J. Mol. Struct. Theochem 103 (1983) 81–99 (volume dedicated to Nobel Laureate Prof. K. Fukui).

    Google Scholar 

  20. P.G. Mezey, Reaction topology of excited state potential energy hypersurfaces, Canad. J. Chem. 61 (1983) 956–961 (volume dedicated to Prof. H. Gunning).

    Google Scholar 

  21. P.G. Mezey, The topological model of non-rigid molecules and reaction mechanisms, in: Symmetries and Properties of Non-Rigid Molecules: A Comprehensive Survey, eds. J. Maruani and J. Serre (Elsevier, Amsterdam, 1983) pp. 335–353.

    Google Scholar 

  22. P.G. Mezey, Topology of doublet potential surfaces, Bull. Soc. Chim. Belg. 92 (1983) 555.

    Google Scholar 

  23. P.G. Mezey, A general formulation of the "quantum chemical Le Chatelier principle", Internat. J. Quant. Chem. 25 (1984) 853–861.

    Google Scholar 

  24. P.G. Mezey, The hyperspherical coordinate representation of potential surfaces of large molecules, Internat. J. Quantum Chem., Quant. Biol. Symp. 11 (1984) 267–272.

    Google Scholar 

  25. P.G. Mezey, The algebraic structure of quantum-chemical reaction mechanisms, Internat. J. Quantum Chem. Symp. 18 (1984) 77–85.

    Google Scholar 

  26. P.G. Mezey, Network relations on potential surfaces as aids to computer-based quantum chemical synthesis planning, Internat. J. Quantum Chem. Symp. 18 (1984) 675–681.

    Google Scholar 

  27. P.G. Mezey, The metric properties of the reduced nuclear configuration space, Internat. J. Quantum Chem. 26 (1984) 983–985.

    Google Scholar 

  28. P.G. Mezey, Constraints on electronic energy hypersurfaces of higher multiplicities, J. Chem. Phys. 80 (1984) 5055–5057.

    Google Scholar 

  29. P.G. Mezey, Simple lower and upper bounds for isomerization energies, Canad. J. Chem. 62 (1984) 1356–1357.

    Google Scholar 

  30. P.G. Mezey, The reaction polyhedron and group theory of reaction mechanisms, Internat. J. Quantum Chem., Quant. Chem. Symp. 19 (1985) 93–105.

    Google Scholar 

  31. P.G. Mezey, A simple relation between nuclear charges and potential surfaces, J. Amer. Chem. Soc. 107 (1985) 3100–3105.

    Google Scholar 

  32. P.G. Mezey, New global constraints on electronic energy hypersurfaces, Internat. J. Quantum Chem. 29 (1986) 85–99.

    Google Scholar 

  33. P.G. Mezey, Cluster topology and bounds for the electronic energy, Surface Science 156 (1985) 597–604.

    Google Scholar 

  34. P.G. Mezey, A comparison of two group theoretical models of reaction mechanisms on potential surfaces, Internat. J. Quantum Chem. 28 (1985) 387–398.

    Google Scholar 

  35. P.G. Mezey, Group theory of constrained reaction mechanisms, Canad. J. Chem. 63 (1985) 1972–1975 (volume dedicated to Prof. C. Sandorfy).

    Google Scholar 

  36. P.G. Mezey, Catchment regions as "molecular loges" on potential energy hypersurfaces, J.Mol. Structure, Theochem 123 (1985) 171–177 (volume dedicated to Prof. R. Daudel).

    Google Scholar 

  37. P.G. Mezey, Topological model of reaction mechanisms, in: Structure and Dynamics of Molecular Systems, Vol. I., eds. R. Daudel, J.-P. Korb, J.-P. Lemaistre and J. Maruani (Reidel, Dordrecht, 1985) pp. 57–70.

    Google Scholar 

  38. P.G. Mezey, Topological theory of molecular conformations, in: Structure and Dynamics ofMolecular Systems, Vol. I., eds. R. Daudel, J.-P. Korb, J.-P. Lemaistre and J. Maruani (Reidel, Dordrecht, 1985) pp. 41–56.

    Google Scholar 

  39. P.G. Mezey, The topology of energy hypersurfaces. III. The fundamental group of reaction mechanisms on potential energy hypersurfaces, Theor. Chim. Acta 67 (1985) 43–61.

    Google Scholar 

  40. P.G. Mezey, The topology of energy hypersurfaces. IV. Generator sets for the fundamental group of reaction mechanism and the complete set of reaction paths, Theor. Chim. Acta 67 (1985) 91–113.

    Google Scholar 

  41. P.G. Mezey, The topology of energy hypersurfaces. V. Potential defying chemical species: a global analysis of vibrational stabilization and destabilization on potential energy hypersurfaces, Theor. Chim. Acta 67 (1985) 115–136.

    Google Scholar 

  42. P.G. Mezey, Reaction topology, in: Applied Quantum Chemistry, Proceedings of the Hawaii 1985 Nobel Laureate Symposium on Applied Quantum Chemistry, eds. V.H. Smith, Jr., H.F. Schaefer III and K. Morokuma, (Reidel, Dordrecht, 1986) pp. 53–74.

    Google Scholar 

  43. P.G. Mezey, Nuclear charges and molecular total energies: a rule on nested reaction globes, Internat. J. Quantum Chem. 29 (1986) 333–343.

    Google Scholar 

  44. P.G. Mezey, Differential and algebraic topology of chemical potential surfaces, in: Mathematics and Computational Concepts in Chemistry, ed. N. Trinajstic (Ellis Horwood, Chichester, UK, 1986) chapter 19, pp. 208–221.

    Google Scholar 

  45. P.G. Mezey, Theory of reaction mechanisms and molecular design, J. Mol. Struct. Theochem 138 (l986) 13–21.

    Google Scholar 

  46. P.G. Mezey, Reflection properties of reaction paths in the reduced nuclear configuration space, Internat. J. Quantum Chem. Symp. 21 (1987) 191–198.

    Google Scholar 

  47. G.A. Arteca and P.G. Mezey, A method for the characterization of molecular conformations, Internat. J. Quantum Chem., Quant. Biol. Symp. 14 (1987) 133–147.

    Google Scholar 

  48. R.K. Gosavi, O.P. Strausz, F. Bernardi, A. Kapur and P.G. Mezey, A molecular orbital study of triplet state [Be•C2H4] exciplexes and their reaction hypersurfaces, J. Phys. Chem. 91 (1987) 283–288.

    Google Scholar 

  49. P.G. Mezey, Global analysis and group theory of reaction mechanisms, J. Mol. Struct. Theochem 149 (1987) 57–66 (volume dedicated to Nobel Laureate Prof. G. Herzberg).

    Google Scholar 

  50. P.G. Mezey, Potential Energy Hypersurfaces (Elsevier, Amsterdam, 1987).

    Google Scholar 

  51. P.G. Mezey, Symmetry and periodicity of potential surfaces: a test for multicenter interactions, Theor. Chim. Acta 73 (1988) 221–228.

    Google Scholar 

  52. F. Harary and P.G. Mezey, Embedding and characterization of quantum chemical reaction graphs on two-dimensional orientable surfaces, Discrete Appl. Math. 19 (1988) 205–214.

    Google Scholar 

  53. G.A. Arteca and P.G. Mezey, Shape description of conformationally flexible molecules: application to two-dimensional conformational problems, Internat. J. Quantum Chem., Quant. Biol. Symp. 15 (1988) 33–54.

    Google Scholar 

  54. G.A. Arteca and P.G. Mezey, Validity of the Hammond postulate and constraints on general onedimensional barriers, J. Comput. Chem. 9 (1988) 728–744.

    Google Scholar 

  55. P.G. Mezey, From geometrical molecules to topological molecules: a quantum mechanical view, in: Molecules in Physics, Chemistry and Biology, Vol. II, ed. J. Maruani (Reidel, Dordrecht, 1988) chapter 2, pp. 61–81.

    Google Scholar 

  56. P.G. Mezey, Reaction topology and quantum chemical molecular design on potential energy surfaces, in: New Theoretical Concepts for Understanding Organic Reactions, eds. J. Bertran and I.G. Csizmadia, Nato ASI Series (Kluwer Academic, Dordrecht, 1989) pp. 55–76.

    Google Scholar 

  57. P.G. Mezey and H. Flakus, Rotation-independent conjugation of sulfur-nitrogen bonds, J.Mol. Struct. Theochem 186 (1989) 117–129.

    Google Scholar 

  58. G.A. Arteca and P.G.Mezey, Molecular similarity and molecular shape changes along reaction paths: a topological analysis and consequences on the Hammond postulate, J. Phys. Chem. 93 (1989) 4746–4751.

    Google Scholar 

  59. P.G. Mezey, Molecular point symmetry and the phase of the electronic wavefunction; Tools for the prediction of critical points of potential energy surfaces, Internat. J. Quantum Chem. 38 (1990) 699–711.

    Google Scholar 

  60. P.G. Mezey, Point symmetry groups of all distorted configurations of a molecule form a lattice, J. Math. Chem. 4 (1990) 377–381.

    Google Scholar 

  61. G.A. Arteca and P.G.Mezey, A quantitative approach to structural similarity from molecular topology of reaction paths, Internat. J. Quantum Chem. Symp. 24 (1990) 1–13.

    Google Scholar 

  62. G.A. Arteca, G.A. Heal and P.G. Mezey, Comparison of potential energy maps and molecular shape invariance maps for two-dimensional conformational problems, Theor. Chim. Acta 76 (1990) 377–390.

    Google Scholar 

  63. G.A. Arteca and P.G. Mezey, Analysis of molecular shape changes along reaction paths, Internat. J. Quantum Chem. 38 (1990) 713–726.

    Google Scholar 

  64. P.G. Mezey, Non-visual molecular shape analysis: shape changes in electronic excitations and chemical reactions, in: Computational Advances in Organic Chemistry (Molecular Structure and Reactivity), eds. C. Ogretir and I.G. Csizmadia, Nato ASI Series (Kluwer Academic, Dordrecht, 1991) pp. 261–288.

    Google Scholar 

  65. G.A. Arteca and P.G. Mezey, Energy and shape analysis along reaction paths of chemical reactions. The case of hydrogen-deuterium exchange, J. Mol. Structure Theochem 230 (1991) 323–338.

    Google Scholar 

  66. X. Luo, G.A. Arteca and P.G. Mezey, Shape analysis along reaction paths of ring opening reactions, Internat. J. Quantum Chem. Symp. 25 (1991) 335–345.

    Google Scholar 

  67. G.A. Arteca and P.G. Mezey, Configurational dependence of molecular shape, J. Math. Chem 10 (1992) 329–371.

    Google Scholar 

  68. X. Luo and P.G. Mezey, A global characterization and similarity analysis of two-dimensional potential energy surfaces, Internat. J. Quantum Chem. 41 (1992) 557–579.

    Google Scholar 

  69. G.A. Arteca and P.G. Mezey, Similarities between the effects of configurational changes and applied electric fields on the shape of electron densities, J. Mol. Struct. Theochem 256 (1992) 125–134 (special volume on "Electrostatics in Molecules", eds. G. Náray-Szabó and W.J. Orwille Thomas).

  70. X. Luo, G.A. Arteca and P.G. Mezey, Shape similarity and shape stability along reaction paths. The case of the PPO?OPP isomerization, Internat. J. Quantum Chem. 42 (1992) 459–474.

    Google Scholar 

  71. P.G.Mezey, On the allowed symmetries of all distorted forms of conformers, molecules, and transition structures, Canad. J. Chem. 70 (1992) 343–347 (special issue dedicated to Prof. S. Huzinaga).

    Google Scholar 

  72. P.G. Mezey, Dynamic shape analysis of biomolecules using topological shape codes, in: The Role of Computational Models and Theories in Biotechnology, ed. J. Bertran (Kluwer Academic, Dordrecht, 1992) pp. 83–104.

    Google Scholar 

  73. J.-E. Dubois and P.G. Mezey, Relations among functional groups within a stoichiometry: a nuclear configuration space approach, Internat. J. Quantum Chem. 43 (1992) 647–658.

    Google Scholar 

  74. P.G. Mezey and J. Maruani, The fundamental syntopy of quasi-symmetric systems: geometric criteria and the underlying syntopy of a nuclear configuration space, Internat. J. Quantum Chem. 45 (1993) 177–187.

    Google Scholar 

  75. P.G. Mezey, Dynamic shape analysis of molecules in restricted domains of a configuration space, J. Math. Chem. 13 (1993) 59–70.

    Google Scholar 

  76. P.G. Mezey, New rules on potential surface topology and critical point search, J. Math. Chem. 14 (1993) 79–90.

    Google Scholar 

  77. P.G. Mezey, Discrete representations of three-dimensional molecular bodies and their shape changes in chemical reactions, in: Graph Theoretical Approaches to Chemical Reactivity, eds. D. Bonchev and O. Mekenyan (Kluwer Academic, Dordrecht, The Netherlands, l994) pp. 181–208.

    Google Scholar 

  78. P.G. Mezey, From reaction path to reaction mechanism: fundamental groups and symmetry rules, in: Reaction Path in Chemistry, ed. D. Heidrich (Kluwer Academic, Dordrecht, 1995) pp. 11–38.

    Google Scholar 

  79. P.D. Walker, P.G. Mezey, G.M. Maggiora, M.A. Johnson and J.D. Petke, Application of the shape group method to conformational processes: shape and conjugation changes in the conformers of 2-phenyl pyrimidine, J. Comput. Chem. 16 (1995) 1474–1482.

    Google Scholar 

  80. P.G. Mezey, Two symmetry constraints on the identity and deformations of chemical species, J. Phys. Chem. 99 (1995) 4947–4954.

    Google Scholar 

  81. P.G. Mezey, Molecular similarity measures of conformational changes and electron density deformations, Advances in Molecular Similarity 1 (1996) 89–120.

    Google Scholar 

  82. P.G. Mezey, Averaged electron densities for averaged conformations, J. Comput. Chem. 19 (1998) 1337–1344.

    Google Scholar 

  83. P.G. Mezey, The topology of catchment regions of potential energy hypersurfaces, Theor. Chem. Acc. 102 (1999) 279–284.

    Google Scholar 

  84. P.G. Mezey, K. Fukui and S. Arimoto, A treatment of small deformations of polyhedral shapes of functional group distributions in biomolecules, Internat. J. Quant. Chem. 76 (2000) 756–761.

    Google Scholar 

  85. P.G. Mezey, Distributions and averages of molecular conformations, Comput. Chem. 25 (2001) 69–75.

    PubMed  Google Scholar 

  86. P.G. Mezey, Computational aspects of combinatorial quantum chemistry, J. Comput. Methods Sci. Engrg. (JCMSE), in press.

  87. P.G. Mezey, Molecular informatics and topology in chemistry, in: Topology in Chemistry, eds. R.B. King and D.H. Rouvray (Ellis Horwood, UK) in press.

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Authors and Affiliations

  1. Institute of Chemistry, University of Budapest, Pázmány Péter Sétány 1, Budapest, Hungary

    Zsolt Szekeres & Paul G. Mezey

  2. Mathematical Chemistry Research Unit, Department of Chemistry and Department of Mathematics and Statistics, University of Saskatchewan, 110 Science Place, Saskatoon, SK, Canada, S7N 5C9

    Zsolt Szekeres & Paul G. Mezey

  3. nstitute for Advanced Study, Collegium Budapest, Szentháromság u. 2, H 1014, Budapest, Hungary

    Paul G. Mezey

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Szekeres, Z., Mezey, P.G. Mathematical Problems of Nuclear Configuration Averaging. Journal of Mathematical Chemistry 30, 315–324 (2001). https://doi.org/10.1023/A:1015127811798

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