Foundations of Theoretical Chemistry

  • H. Primas
Part of the NATO Advanced Study Institutes Series book series (volume 57)


The objective of these lectures is to present a unified approach to the modern theory of molecular matter. Careful attention is paid to the fundamentals and questions of interpretation, the emphasis is upon ideas, concepts and a realistic link between theory and experiment.


Measure Space Boolean Algebra Quantum Logic Orthomodular Lattice Theoretical Chemistry 
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  1. Aarnes, J.F., 1970, “Quasi-states on C*-algebras”. Trans.Amer, Math.Soc, 149, 601–625.MathSciNetMATHGoogle Scholar
  2. Abraham, R, 1967. “Foundations of mechanics”. Benjamin, New York; second edition (with J.E. Marsden), 1978.MATHGoogle Scholar
  3. Akcoglu, M.A., 1975. “Positive contractions of L 1-spaces”. Math.Z. 143, 5–13.MathSciNetCrossRefGoogle Scholar
  4. Alfsen, E.M. and Shultz, F.W., 1976. “Non-commutative spectral theory for affine function spaces on convex sets”. Memoirs of the Amer. Math. Soc., number 172.Google Scholar
  5. Amann, A., 1978. “Eine Verallgemeinerung des quantenmechanischen Elementaritätsbegriffs”. Diplomarbeit an der Abteilung für Chemie der ETH Zürich. Unpublished.Google Scholar
  6. Araki, H. and Woods, E.J., 1963. “Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas”. J.Math.Phys. (N.Y.) 4, 637–662.MathSciNetADSCrossRefGoogle Scholar
  7. Araki, H. and Woods, E.J., 1968. “A classification of factors”. Publ.Res.Inst.Math.Sci. (Kyoto) A 4, 51–130.MathSciNetCrossRefGoogle Scholar
  8. Arnold, V.I., 1962. “The classical theory of perturbations and the problem of stability of planetary systems”. Sovjet Math. Dokl. 3, 1008–1012. (Russian original: Dokl.Akad.Nauk SSSR 145, 487–490 (1962).)Google Scholar
  9. Arnold, V.I., 1963a. “Proof of a theorem of A.N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian”. Russian Math. Surveys 18, No.5, 9–36. (Russian original: Uspehi Mat.Nauk 18, No.5, 13–40 (1963).)ADSCrossRefGoogle Scholar
  10. Arnold, V.I., 1963b. “Small denominators and problems of stability of motion in classical and celestial mechanics”. Russian Math. Surveys 18, No.6, 85–191. (Russian original: Uspehi Mat.Nauk 18, No.6, 91–192 (1963).)ADSCrossRefGoogle Scholar
  11. Arnold, V.I. and Avez, A., 1968. “Ergodic problems of classical mechanics”. Benjamin, New York, 1968.Google Scholar
  12. Banach, S., 1932. “Théorie des operations linéaires”. Warsaw.Google Scholar
  13. Bargmann, V., 1954. “On unitary ray representations of continuous groups”. Annals of Mathematics 59, 1–46.MathSciNetMATHCrossRefGoogle Scholar
  14. Bartle, R.G., 1966. “The elements of integration”, Wiley, New York.MATHGoogle Scholar
  15. Birkhoff, G., 1940. “Lattice theory”. American Mathematical Society, Providence, Rhode Island; second revised edition, 1948; third new edition, 1967.Google Scholar
  16. Birkhoff, G. and Bartee, T.C., 1970. “Modern applied algebra”, McGraw-Hill, New York.MATHGoogle Scholar
  17. Bishop, D.M. and Cheung, L.M., 1979. “Natural orbital analysis of non-adiabatic H+ 2 wave functions”. Int.J. Quantum Chem. 15, 517–532.CrossRefGoogle Scholar
  18. Blau, U., 1973. “Zur 3-wertigen Logik ner natürlichen Sprache”. Papiere zur Linguistik 4, 20–96.Google Scholar
  19. Blau, U., 1978. “Die dreiwertige Logik der Sprache”. De Gruyter, Berlin.Google Scholar
  20. Bohr, N., 1928. “The quantum postulate and the recent development of atomic theory”. Nature (London) 121, 580–590.ADSMATHCrossRefGoogle Scholar
  21. Bohr, N., 1948. “On the notions of causality and complementarity”. Dialectica 2, 312–319. (Reprinted in: Science 111, 51–54, 1950).MATHCrossRefGoogle Scholar
  22. Bohr, N., 1949. “Discussion with Einstein on epistemological problems in atomic physics”. In: “Albert Einstein: Philosopher- Scientist”, ed. by P.A. Schilpp; Library of Living Philosophers, Evanston, Illinois, pp.199–241.Google Scholar
  23. Boole, G., 1847. “The mathematical analysis of logic”. Cambridge.Google Scholar
  24. Boole, G., 1854. “An investigation of the laws of thought”. Macmillan, London. Reprinted by Dover, New York, 1958.Google Scholar
  25. Borei, E., 1914. “Introduction géométrique à quelques théories physiques”. Gauthier-Villars, Paris.Google Scholar
  26. Bourbaki, N., 1963. “Eléments de mathématiques. Livre VI. Intégration”. Hermann, Paris.Google Scholar
  27. Brown, A., 1974. “A version of multiplicity theory”. In: “Topics in operator theory”; ed. by C. Pearcy; Mathematical surveys, number 13; American Mathematical Society, Providence, Rhode Island, pp.129–160.Google Scholar
  28. Brown, A. and Pearcy, C., 1977. “Introduction to operator theory I. V Elements of functional analysis”. Springer, New York.CrossRefGoogle Scholar
  29. Brown, J.R., 1976. “Ergodic theory and topological dynamics”. Academic Press, New York.MATHGoogle Scholar
  30. Bruns, H., 1884. “Bemerkungen zur Theorie der allgemeinen Störungen”. Astron. Nachr. 109, 215–222.ADSCrossRefGoogle Scholar
  31. Choi, M.D., 1972. “Positive linear maps on C*-algebras”. Canad.J. Math. 24, 520–529.MathSciNetMATHCrossRefGoogle Scholar
  32. Chong, K.M., 1976. “Doubly stochastic operators and rearrangement theorems”. J.Math.Analysis and Applications 56, 309–316.MathSciNetMATHCrossRefGoogle Scholar
  33. Clauser, J.F. and Shimony, A., 1978. “Bell’s theorem: experimental tests and implications”. Rep.Prog.Phys. 41, 1881–1927.ADSCrossRefGoogle Scholar
  34. Connes, A., 1973. “Une classification des facteurs de type III”. Annales Scientifiques de l’Ecole Normal Supérieur 6, 133–253.MathSciNetMATHGoogle Scholar
  35. Connes, A., 1976. “Classification of injective factors”. Annals of Mathematics 104, 73–115.MathSciNetMATHCrossRefGoogle Scholar
  36. Czelakowski, J., 1974. “Logics based on partial Boolean σ-algebras.1”. Studia Logica 33, 371–396.MathSciNetCrossRefGoogle Scholar
  37. Dell’Antonio, G.F., 1967. “On the limits of sequences of normal states”. Commun.Pure Appl.Math. 20, 413–429.MathSciNetMATHCrossRefGoogle Scholar
  38. Dixmier, J., 1957. “Les algèbres d’opérateurs dans l’espace Hilbertien. (Algèbres de von Neumann)”. Gauthier-Villars, Paris, premièr édition, 1957; deuxième édition, revue et augmentée, 1969.MATHGoogle Scholar
  39. Dixmier, J., 1964. “Les C*-algèbres et leurs représentations”. Gauthier-Villars, Paris; première édition, 1964; deuxième édition 1969. (English translation: “C*-algebras”, North-Holland, Amsterdam 1977).Google Scholar
  40. Domotor, Z., 1974. “The probability structure of quantum-mechanical systems”. Synthese 29, 155–185.MATHCrossRefGoogle Scholar
  41. Doob, J.L., 1953. “Stochastic processes”. Wiley, New York.MATHGoogle Scholar
  42. Douglas, R.G., 1972. “Banach algebra techniques in operator theory”. Academic Press, New York.MATHGoogle Scholar
  43. Dye, H.A., 1955. “On the geometry of projections in certain operator algebras”. Annals of Mathematics 61, 73–89.MathSciNetMATHCrossRefGoogle Scholar
  44. Einstein, A., Podolsky, B. and Rosen, N., 1935. “Can quantum-mechanical description of physical reality be considered complete?” Phys.Rev.47, 777–780.ADSMATHCrossRefGoogle Scholar
  45. Erber, T., Schweizer, B. and Sklar, A., 1973. “Mixing transformations on metric spaces”. Commun.Math.Phys. 29, 311–317.MathSciNetADSMATHCrossRefGoogle Scholar
  46. Evans, D.E. and Lewis, J.T., 1976. “Dilations of dynamical semigroups”. Commun.Math.Phys. 50, 219–227.MathSciNetADSMATHCrossRefGoogle Scholar
  47. Finch, P.D., 1969. “On the structure of quantum logic”. J.Symbolic Logic 34, 275–282.MATHCrossRefGoogle Scholar
  48. Golodec, V.Ja., 1977. “On automorphisms of von Neumann algebras”. Dokl.Akad.Nauk SSSR 237, 770–772. (English translation incorporating corrections made by the author: Sov.Math.Dokl. 18, 1477–1480 (1977).)MathSciNetGoogle Scholar
  49. Gorini, V., Kossakowski, A. and Sudarshan, E.C.G., 1976. “Completely positive dynamical semigroups of N-level systems”. J.Math.Phys. (N.Y.) 17, 821–825.MathSciNetADSCrossRefGoogle Scholar
  50. Greechie, R.J., 1971. “Orthomodular lattices admitting no states”. J.Combinatorial Theory 10, 119–132.MathSciNetMATHCrossRefGoogle Scholar
  51. Halmos, P.R., 1950. “Measure theory”. Van Nostrand Reinhold, New York.MATHGoogle Scholar
  52. Halmos, P.R. and Neumann, J. von, 1942. “Operator methods in classical mechanics, II”. Annals of Mathematics 43, 332–350.MathSciNetMATHCrossRefGoogle Scholar
  53. Holland, S.S., 1964. “Distributivity and perspectivity in orthomodular lattices”. Trans.Amer.Math.Soc. 112, 330–343.MathSciNetMATHCrossRefGoogle Scholar
  54. Holland, S.S., 1970a. “An m-orthocomplete orthomodular lattice is m-complete”. Proc.Amer.Math. Soc. 24, 716–718.MathSciNetMATHGoogle Scholar
  55. Holland, S.S., 1970b. “The current interest in orthomodular lattices”. In: “Trends in lattice theory”; ed. by J.C. Abbott; van Nostrand-Reinhold, New York, pp.41–126.Google Scholar
  56. Hooker, C.A., 1975. “The logico-algebraic approach to quantum mechanics. Volume 1. Historical evolution”. Reidel, Dordrecht-Holland.CrossRefGoogle Scholar
  57. Hopf, E., 1934. “On causality, statistics and probability”. J. Mathematics and Physics (Cambridge) 13, 51–102.Google Scholar
  58. Huntington, E.V., 1933. “New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell’s Principia Mathematica”. Trans.Amer.Math.Soc. 35, 274–304, 557–558, 971.MathSciNetGoogle Scholar
  59. Jammer, M., 1974. “The philosophy of quantum mechanics”. Wiley, New York.Google Scholar
  60. Jauch, J.M., 1968. “Foundations of quantum mechanics”. Addison-Wesley, Reading, Massachusetts.MATHGoogle Scholar
  61. Kadison, R.V., 1976. “Normal states and unitary equivalence of von Neumann algebras”. In “C*-algebras and their applications to statistical mechanics and quantum field theory”, ed. by D. Kastler; Proceedings of the international school of physics “Enrico Fermi”, Course 60, North-Holland, Amsterdam, pp.1–18.Google Scholar
  62. Kadison, R.V. and Singer, I.M., 1959. “Extensions of pure states”. Amer.J.Math. 81, 383–400.MathSciNetMATHCrossRefGoogle Scholar
  63. Kaplansky, I., 1951. “Projections in Banach algebras”. Annals of Mathematics 53, 235–249.MathSciNetMATHCrossRefGoogle Scholar
  64. Kochen, S. and Specker, E.P., 1965a. “Logical structures arising in quantum theory”. In: “The theory of models”; ed. by J. Addison, L. Henkin and A. Tarski; North-Holland, Amsterdam, pp.177–189.Google Scholar
  65. Kochen, S. and Specker, E.P., 1965b. “The calculus of partial propositional functions”. In: “Logic, methodology and philosophy of science”; ed. by Y. Bar-Hillel; North-Holland, Amsterdam; pp.45–57.Google Scholar
  66. Kochen, S. and Specker, E.P., 1967. “The problems of hidden variables in quantum mechanics”. J. Mathematics and Mechanics 17, 59–88.MathSciNetMATHGoogle Scholar
  67. Kolmogorov, A.N., 1933. “Grundbegriffe der Wahrscheinlichkeitsrechnung”. Springer, Berlin. (English translation: “Foundations of the theory of probability”; Chelsea, New York, 1950.)Google Scholar
  68. Kolmogorov, A.N., 1948. “Algèbres de Boole métriques complètes”. VI. Zjazd Matematykow Polskich. Appendix to Ann.Soc.Pol.Math. 20, 21–30.Google Scholar
  69. Kolmogorov, A.N., 1957. “General theory of dynamical systems and classical mechanics” (in Russian), Proc. 1954 Intern.Congr. Math.; North-Holland, Amsterdam; pp.315–333. (English translation in: R. Abraham, “Foundations of mechanics”; Benjamin, New York, 1967, pp.263–279.)Google Scholar
  70. Kolmogorov, A.N., 1958. “A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces”. (Russian). Dokl.Akad.Nauk SSSR 119, 861–864.MathSciNetMATHGoogle Scholar
  71. Koopmann, B.O., 1931. “Hamiltonian systems and linear transformations in Hilbert space”. Proc.Nat.Acad. Sci.U.S. 17, 315–318.ADSCrossRefGoogle Scholar
  72. Kraus, K., 1971. “General state changes in quantum theory”. Annals of Physics 64, 311–335.MathSciNetADSMATHCrossRefGoogle Scholar
  73. Lamperti, J., 1958. “On the isometries of certain function spaces”. Pacific J.Math. 8, 459–466.MathSciNetMATHCrossRefGoogle Scholar
  74. Loomis, L.H., 1947. “On the representation of σ-complete Boolean algebras”. Bull. Amer.Math. Soc. 53, 757–760.MathSciNetMATHCrossRefGoogle Scholar
  75. MacLane, S. and Birkhoff, G., 1967. “Algebra”. Macmillan, New York.MATHGoogle Scholar
  76. Maeda, F. and Maeda, S., 1970. “Theory of symmetric lattices”. Springer, Berlin.MATHCrossRefGoogle Scholar
  77. Misra, B., 1967. “When can hidden variables be excluded in quantum mechanics?” Nuovo Cimento A 47, 841–859.ADSMATHCrossRefGoogle Scholar
  78. Moser, J., 1962. “On invariant curves of area-preserving mappings of an annulus”. Nachr.Akad.Wiss. Göttingen, Math.-Phys.K1. 1962, 1–20.Google Scholar
  79. Moser, J., 1963. “Stable and random motions in dynamical systems”. Princeton University Press, Princeton, New Jersey.Google Scholar
  80. Murray, F.J. and Neumann, J. von, 1936. “On rings of operators”. Annals of Mathematics 37, 116–229.MathSciNetCrossRefGoogle Scholar
  81. Murray, F.J. and Neumann, J. von, 1943. “On rings of operators, IV”. Annals of Mathematics 44, 716–808.MathSciNetMATHCrossRefGoogle Scholar
  82. Neumann, J. von, 1929. “Zur Algebra der Funktionaloperatoren und Theorie der normalen Operatoren”. Mathematische Annalen 102, 370–427.MATHCrossRefGoogle Scholar
  83. Neumann, J. von, 1932a. “Mathematische Grundlagen der Quantenmechanik”. Springer, Berlin. (English translation: Mathematical foundations of quantum mechanics. Princeton University Press, Princeton, New Jersey, 1955.)MATHGoogle Scholar
  84. Neumann, J. von, 1932b. “Zur Operatorenmethode in der klassischen Mechanik”. Annals of Mathematics 33, 587–642, 789–791.MathSciNetCrossRefGoogle Scholar
  85. Ornstein, D.S., 1974. “Ergodic theory, randomness and dynamical systems”. Yale University Press, New Haven.MATHGoogle Scholar
  86. Parthasarathy, K.P., 1977. “Introduction to probability and measure”. Macmillan, Delhi and London.MATHGoogle Scholar
  87. Pfeifer, P., 1979. “Chiral molecules — a classical observable induced by the radiation field”. Thesis, ETH, Zürich. Preprint.Google Scholar
  88. Plymen, R.J., 1968. “Dispersion-free normal states”. Nuovo Cimento A 54, 862–870.ADSCrossRefGoogle Scholar
  89. Poincaré, H., 1892. “Les méthodes nouvelles de la mécanique céleste”. Gauthier-Villars, Paris.Google Scholar
  90. Primas, H., 1975. “Pattern recognition in molecular quantum mechanics”. Theoret.Chim.Acta 39, 127–148.CrossRefGoogle Scholar
  91. Primas, H., 1977. “Theory reduction and non-Boolean theories”. J.Math.Biology 4, 281–301.MathSciNetMATHCrossRefGoogle Scholar
  92. Primas, H., 1980. “Chemistry, quantum mechanics and reductionism. Perspectives in theoretical chemistry”. Lecture Notes in Chemistry, Springer, Berlin; in preparation.Google Scholar
  93. Raggio, G.A., 1978. “Dispersion-free states on the centre of C*-and W*-algebras”. Internal progress report, Lab.Phys.Chem. ETH-Zürich, October 1978. Unpublished.Google Scholar
  94. Royden, H.L., 1963. “Real analysis”. Macmillan, New York; second edition 1968.MATHGoogle Scholar
  95. Sakai, S., 1971. “C*-algebras and W*-algebras”. Springer, Berlin.CrossRefGoogle Scholar
  96. Sato, H. and Oka, Y., 1974. “A characterization of unitary operators induced by nonsingular transformations and its applications” Nagoya Math.J. 53, 189–198.MathSciNetMATHGoogle Scholar
  97. Segal, I.E., 1947a. Irreducible representations of operator algebras. Bull.Amer.Math.Soc. 53, 73–88.MathSciNetMATHCrossRefGoogle Scholar
  98. Segal, I.E., 1947b. “Postulates for general quantum mechanics”. Annals of Mathematics 48, 930–948.MathSciNetMATHCrossRefGoogle Scholar
  99. Sikorski, R., 1949. “On the inducing of homomorphisms by mappings”. Fund.Math. 36, 7–22.MathSciNetMATHGoogle Scholar
  100. Sikorski, R., 1960. “Boolean algebras”. Springer, Berlin; second revised edition, 1962; third edition (corrected reprint), 1968.MATHCrossRefGoogle Scholar
  101. Sinai, Ya., 1963. “On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics”. Sov.Math.Dokl. 4, 1818–1822. (Russian original: Dokl.Akad.Nauk SSSR 153, 1261–1264 (1963).)MathSciNetGoogle Scholar
  102. Sinai, Ya., 1976. “Introduction to ergodic theory”. Princeton University Press, Princeton.MATHGoogle Scholar
  103. Smoluchowski, M. von, 1918. “Ueber den Begriff des Zufalls und den Ursprung der Wahrscheinlichkeitsgesetze in der Physik”. Naturwissenschaften 6, 253–263.ADSMATHCrossRefGoogle Scholar
  104. Specker, E., 1960. “Die Logik nicht gleichzeitig entscheidbarer Aussagen”. Dialectica 14, 239–246.MathSciNetCrossRefGoogle Scholar
  105. Stinespring, W.F., 1955. “Positive functions of C*-algebras”. Proc.Amer.Math.Soc. 6, 211–216.MathSciNetMATHGoogle Scholar
  106. Stone, M.H., 1936. “The theory of representations for Boolean algebras”. Trans.Amer.Math.Soc. 40, 37–111.MathSciNetGoogle Scholar
  107. Størmer, E., 1968. “A characterization of pure states of C*-algebras” Proc.Amer.Math.Soc. 19, 1100–1102.MathSciNetGoogle Scholar
  108. Størmer, E, 1974. “Positive linear maps of C*-algebras”. In: “Foundations of quantum mechanics and ordered linear spaces”; ed. by A. Hartkämper and H. Neumann; Lecture Notes in Physics, vol.29; Springer, Berlin; pp.85–106.CrossRefGoogle Scholar
  109. Stroescu, E., 1973. “Isometric dilations of contractions on Banach spaces”. Pacific J.Math. 47, 257–262.MathSciNetMATHCrossRefGoogle Scholar
  110. Takesaki, M., 1970. “Tornita’s theory of modular Hilbert algebra and its applications”. Lecture Notes in Mathematics, vol.128, Springer, Berlin.Google Scholar
  111. Topping, D.M., 1967. “Asymptoticity and semimodularity in projection lattices”. Pacific J.Math. 20, 317–325.MathSciNetMATHCrossRefGoogle Scholar
  112. Varadarajan, V.S., 1968. “Geometry of quantum theory. Volume 1”. Van Nostrand, Princeton.CrossRefGoogle Scholar
  113. Watanabe, S., 1969. “Knowing and guessing”. Wiley, New York.MATHGoogle Scholar
  114. Whittaker, E.T., 1943. “Chance, freewill and necessity in the scientific conception of the universe”. Proc.Phys.Soc.55, 459–471.MathSciNetADSMATHCrossRefGoogle Scholar
  115. Woolley, R.G., 1978a. “Must a molecule have a shape?” J.Amer.Chem. Soc. 100, 1073–1078.CrossRefGoogle Scholar
  116. Woolley, R.G., 1978b. “Further remarks on molecular structure in quantum theory”. Chem.Phys.Lett. 55, 443–446.ADSCrossRefGoogle Scholar
  117. Yosida, K., 1965. “Functional analysis”. Springer, Berlin; fifth edition 1978.MATHGoogle Scholar

Copyright information

© Plenum Press, New York 1980

Authors and Affiliations

  • H. Primas
    • 1
  1. 1.Laboratory of Physical ChemistrySwiss Federal Institute of TechnologyZürichSwitzerland

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