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Moderne Methoden der numerischen Quantenchemie

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Mathematik in der Praxis

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Zusammenfassung

Die Quantenmechanik ist der mathematische Schlüssel zur Berechnung von physikalischen und chemischen Eigenschaften von Atomen, Molekülen und Festkörpern. Nach der von Heisenberg und Schrödinger 1926 entwickelten Theorie, die die Gesetze der klassischen Mechanik auf atomarer Ebene ablöst, wird die Elektronenstruktur eines Systems beschrieben durch eine Funktion von mehreren Variablen, die einer gewissen partiellen Differentialgleichung, der Schrödinger-Gleichung, gehorchen muß. Mit Kenntnis dieser „Wellenfunktion“ lassen sich im Prinzip alle physikalisch oder chemisch meßbaren Größen des untersuchten Systems bestimmen — dazu weiter unten Genaueres. Da die Schrödinger-Gleichung für Moleküle nicht mehr analytisch lösbar ist, blieben Anwendungen quantenmechanischer Prinzipien auf chemische Fragestellungen, insbesondere auf das Phänomen der chemischen Bindung, lange Zeit entweder rein qualitativ oder benutzten sehr grobe Näherungen (wie z.B. die Hückel-Theorie 1930–33). Die ersten numerischen Berechnungen an Molekülen, die auf rein quantenmechanischer Grundlage („ab initio“) ohne zusätzliche Approximationen durchgeführt wurden, fallen in den Zeitraum 1955–60, in dem auch die ersten Digitalrechner aufkamen. Ende der 60er Jahre setzte dann eine stürmische Entwicklung von neuen Näherungsmodellen und Algorithmen ein, die in den 70er und 80er Jahren zu einigen entscheidenden Durchbrüchen führte. Inzwischen steht eine Reihe von bewährten und effizienten Methoden zur Verfügung, die teilweise sehr genaue Voraussagen ermöglichen. Insbesondere die sog. direkten Methoden erlauben heute Berechnungen, von denen man vor 30 Jahren nur träumen konnte. Mittlerweile sind ab-initio-Rechnungen an Systemen mit über hundert Atomen ohne weiteres möglich, und dank ihrer Durchführbarkeit und Vorhersagekraft für Moleküle von industriellem Interesse haben quantenchemische Verfahren inzwischen auch Einzug in die Forschungsabteilungen der chemischen, petrochemischen und pharmazeutischen Unternehmen gehalten.

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Literatur

  1. Almlöf, J., Taylor, P.R.: Computational Aspects of Direct SCF and MCSCF methods. In: C. E. Dykstra (ed.) Advanced Theories and Computational Approaches to the Electronic Structure of Molecules. Reidel, Dordrecht 1984

    Google Scholar 

  2. Bauschlicher Jr., C.W., Langhoff, S.W.: Quantum Mechanical Calculations to Chemical Accuracy. Science 254 (1991) 394

    Article  MathSciNet  Google Scholar 

  3. Becke, A.D.: Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 38 (1988) 3098

    Article  Google Scholar 

  4. Becke, A. D.: A multicenter numerical integration scheme for polyatomic molecules. J. Chem. Phys. 88 (1988) 2547

    Google Scholar 

  5. Bickelhaupt, F.M., Baerends, E.J., Nibbering, N.M.M., Ziegler, T.: Theoretical Investigation on Base-Induced 1,2-Eliminations in the Model System F- + CH3CH2F. The Role of the Base as a Catalyst. J. Am. Chem. Soc. 115 (1993) 9160

    Article  Google Scholar 

  6. Böhm, H.-J., Brode, S.: Ab Initio SCF Calculations on Low-Energy Conformers of N-Acetyl-N’-methylalaninamide and N-Acetyl-N’- methylglycinamide. J. Am. Chem. Soc. 113 (1991) 7129

    Article  Google Scholar 

  7. BČF94] Bonačić-Koutecký, V., Češpiva, L., Fantucci, P., Fuchs, C., Koutecký, J., Pittner, J.: Quantum Chemical Interpretation of Optical Response of Small Metal Clusters. Comm. Atomic & Mol. Phys. Im Druck

    Google Scholar 

  8. Brown, F.K., Chandra Singh, U., Kollman, P.A., Raimondi, L., Houk, K.N., Bock, C. W.: A Theoretical Study of Intramolecular Diels-Alder and 1,3-Dipolar Cycloaddition Stereoselectivity Using ab Initio Methods, Semiempirical Methods, and a Tandem Quantum Mechanic-Molecular Mechanic Method. J. Org. Chem. 57 (1992) 4862

    Article  Google Scholar 

  9. Buenker, R. J., Peyerimhoff, S.D., Butscher, W.: Applicability of the multi-reference double-excitation CI (MRD-CI) method to the calculation of electronic wavefunctions and comparison with related techniques. Mol. Phys. 35 (1978) 771

    Article  Google Scholar 

  10. Curl, R.F., Smalley, R.E.: Fullerenes, Scientific American 10 (1991) 54

    Google Scholar 

  11. Davidson, E. R.: The Iterative Calculation of a Few of the Lowest Eigenvalues and Corresponding Eigenvectors of Large Real-Symmetric Matrices. J. Comp. Phys. 17 (1975) 87

    Article  MATH  Google Scholar 

  12. Diercksen, G. H. F.: Optimized Transformation of Four Center Integrals. Theoret. Chim. Acta 33 (1974) 1

    Article  Google Scholar 

  13. Dunning Jr., T. H., Hay, P. J.: Gaussian basis sets for molecular calculations. In: H. F. Schaefer III (ed.) Methods of Electronic Structure Theory. Plenum Press, New York 1977

    Google Scholar 

  14. Fan, L., Ziegler, T.: Optimization of molecular structures by self-consistent and nonlocal density-functional theory. J. Chem. Phys. 95 (1991) 7401

    Article  Google Scholar 

  15. Flurry, Jr., R. L.: Symmetry Groups. Prentice-Hall, Englewood Cliffs 1980

    Google Scholar 

  16. Fuchs, C., Bonačić-Koutecký, V., Koutecký, J.: Compact Formulation of Multiconfigurational Response Theory. Applications to Small Alkali Metal Clusters. J. Chem. Phys. 98 (1993) 3121

    Article  Google Scholar 

  17. Geertsen, J., Rittby, M., Bartlett, R.J.: The Equation-of-Motion Coupled-Cluster Method: Excitation Energies of Be and CO. Chem. Phys. Lett. 164 (1989) 57

    Article  Google Scholar 

  18. Gill, P. M. W., Head-Gordon, M., Pople, J. A.: An Efficient Algorithm for the Generation of Two-Electron Repulsion Integrals over Gaussian Basis Functions. Int. J. Quantum Chem. Symp. 23 (1989) 269

    Google Scholar 

  19. Guest, M. F., Sherwood, P., van Lenthe, J. H.: Parallelism in computational chemistry. I. Hypercube-connected multicomputers. Theor. Chim. Acta 84 (1993) 423

    Article  Google Scholar 

  20. Harrison, R. J., Zarrabian, S.: An efficient implementation of the Full-CI method using an (n- 2)-electron projection space. Chem. Phys. Lett. 158 (1989) 393

    Article  Google Scholar 

  21. Häser, M., Almlöf, J., Scuseria, G.E.: The equilibrium geometry of Cöo as predicted by second-order (MP2) perturbation theory Chem. Phys. Lett. 181 (1991) 497

    Google Scholar 

  22. Hedberg, K., Hedberg, L., Bethune, D.S., Brown, C.A., Dorn, H.C., Johnson, R. D., de Vries, M.: Bond Lengths in Free Molecules of Buckminsterfullerene, C60, from Gas-Phase Electron Diffraction. Science 254 (1991) 410

    Article  Google Scholar 

  23. Hinze, J. (ed.): The Unitary Group for the Evaluation of Electronic Energy Matrix Elements, Springer, Berlin Heidelberg New York 1981 (Lecture Notes In Chemistry 22)

    Google Scholar 

  24. Hohenberg, P., Kohn, W.: Inhomogeneous electron gas, Phys. Rev. 136 (1964) 864

    Article  MathSciNet  Google Scholar 

  25. Illas, F., Rubio, J., Ricart, J.M., Bagus, P.S.: Selected versus complete configuration interaction expansions. J. Chem. Phys. 95 (1991) 1877

    Google Scholar 

  26. Jeziorski, B., Monkhorst, H. J.: Coupled-cluster method for multideter-minantal reference states. Phys. Rev. A 24 (1981) 1668

    Article  Google Scholar 

  27. Jørgensen, P., Simons, J.: Second Quantization-Based Methods in Quantum Chemistry. Academic Press, New York 1981

    Google Scholar 

  28. Jørgensen, P., Simons, J. (eds.): Geometrical Derivatives of Energy Surfaces and Molecular Properties, Reidel, Dordrecht 1986

    Google Scholar 

  29. Kato, T.: On the Eigenfunctions of Many-Particle Systems in Quantum Mechanics. Commun. Pure Appl. Math. 10 (1957) 151

    MATH  Google Scholar 

  30. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin Heidelberg New York 1980

    MATH  Google Scholar 

  31. Knowles, D.B., Alvarez-Collado, J.R., Hirsch, G., Buenker, R. J.: Comparison of perturbatively corrected energy results from multiple reference double-excitation configuration-interaction method calculations with exact full configuration-interaction benchmark values. J. Chem. Phys. 92 (1990) 585

    Article  Google Scholar 

  32. Knowles, P. J., Handy, N. C.: A new determinant-based full configuration interaction method, Chem. Phys. Lett. 111 (1984) 315

    Article  Google Scholar 

  33. Kutzelnigg, W., Klopper, W.: Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. I. General theory. J. Chem. Phys. 94 (1991) 1985

    Article  Google Scholar 

  34. Lieb, E.H.: Density Functional for Coulomb Systems, In: A. Shimony, H. Feshbach (eds.) Physics as Natural Philosophy. MIT Press, Cambridge (Mass.) 1982

    Google Scholar 

  35. Lieb, E. H., Simon, B.: The Hartree-Fock Theory for Coulomb Systems. Commun. Math. Phys. 53 (1977) 185

    Article  MathSciNet  Google Scholar 

  36. Lundqvist, S., March, N.H. (eds.): Theory of the Inhomogeneous Electron Gas. Plenum Press, New York London 1983

    Google Scholar 

  37. Matsen, F. A., Pauncz, R.: The Unitary Group In Quantum Chemistry. Elsevier, Amsterdam 1986 (Studies in physical and theoretical chemistry 44)

    Google Scholar 

  38. Oliphant, N., Bartlett, R. J.: A systematic comparison of molecular properties obtained using Hartree-Fock, a hybrid Hartree-Fock density-functional-theory, and coupled-cluster methods. J. Chem. Phys. 100 (1994) 6550

    Article  Google Scholar 

  39. Olsen, J., Jensen, H. J. Aa., Jørgensen, P.: Solution of the Large Matrix Equations Which Occur in Response Theory. J. Comp. Phys. 74 (1988) 265

    Article  MATH  Google Scholar 

  40. Olsen, J., Roos, B. O., Jørgensen, P., Jensen, H. J. Aa.: Determinant based configuration interaction algorithms for complete and restricted configuration interaction spaces. J. Chem. Phys. 89 (1989) 2185

    Article  Google Scholar 

  41. Parr, R. G., Yang, W.: Density Functional Theory of Atoms and Molecules. Oxford University Press, New York 1989

    Google Scholar 

  42. Primas, H., Müller-Herold, U.: Elementare Quantenchemie. Teubner, Stuttgart 1984

    Google Scholar 

  43. Pulay, P.: Convergence acceleration of iterative sequences: The case of SCF iteration. Chem. Phys. Lett. 73 (1980) 393

    Article  Google Scholar 

  44. Rettrup, S.: An Iterative Method for Calculating Several of the Extreme Eigensolutions of Large Real Non-symmetric Matrices. J. Comp. Phys. 45 (1982) 100

    Article  MathSciNet  MATH  Google Scholar 

  45. Roos, B.O., Siegbahn, P. E. M.: The Direct Configuration Interaction Method from Molecular Integrals. In: H. F. Schaefer III (ed.) Methods of Electronic Structure Theory. Plenum Press, New York 1977

    Google Scholar 

  46. Roos, B.O.: The Multiconfigurational (MC) SCF Method. In: G.H.F. Diercksen, S. Wilson (eds.) Methods in Computational Molecular Physics. Reidel, Dordrecht 1983

    Google Scholar 

  47. Roos, B.O.: The Complete Active Space Self-Consistent Field Method and its Application in Electronic Structure Calculations. In: K. P. Lawley (ed.) Ab Initio Methods in Quantum Chemistry, Bd. II. John Wiley & Sons, Chichester 1987 (Adv. Chem. Phys. 69)

    Google Scholar 

  48. Roos, B. O., Andersson, K., Flilscher, M. P.: Towards an accurate molecular orbital theory for excited states: the benzene molecule. Chem. Phys. Lett. 192 (1992) 5

    Article  Google Scholar 

  49. Roothaan, C. C. J.: New developments in molecular orbital theory. Rev. Mod. Phys. 23 (1951) 69

    Article  MATH  Google Scholar 

  50. Saunders, V.R.: Molecular Integrals for Gaussian Type Functions. In: G.H.F. Diercksen, S. Wilson (eds.) Methods in Computational Molecular Physics. Reidel, Dordrecht 1983

    Google Scholar 

  51. Saunders, V. R., Hillier, I. H.: A „Level-Shifting“Method for Converging Closed Shell Hartree-Fock Wave Functions, Int. J. Quantum Chem. 7 (1973) 699

    Article  Google Scholar 

  52. Sch87] Schlegel, H.B.: Optimization of equilibrium geometries and transition structures. In: K. P. Lawley (ed.) Ab Initio Methods in Quantum Chemistry, Bd. I. John Wiley & Sons, Chichester 1987 (Adv. Chem. Phys. 68)

    Google Scholar 

  53. Schrödinger, E.: Quantisierung als Eigenwertproblem, Ann. d. Phys. 80 (1926) 437

    Article  Google Scholar 

  54. Serre, J.-P.: Linear Representations of Finite Groups. Springer, Berlin Heidelberg New York 1977

    MATH  Google Scholar 

  55. Shavitt, I.: The Method of Configuration Interaction. In: H.F. Schaefer III (ed.) Methods of Electronic Structure Theory. Plenum Press, New York 1977

    Google Scholar 

  56. Shavitt, I.: The unitary group and the electron correlation problem. In: P.-O. Löwdin, B. Pullman (eds.) New Horizons of Quantum Chemistry. Reidel, Dordrecht 1983

    Google Scholar 

  57. Shepard, R.: The Multiconfiguration Self-Consistent Field Method. In: K. P. Lawley (ed.) Ab Initio Methods in Quantum Chemistry, Bd. II. John Wiley & Sons, Chichester 1987 (Adv. Chem. Phys. 69)

    Google Scholar 

  58. Siegbahn, P.E.M., Almlöf, J., Heiberg, A., Roos, B.O.: The complete active space SCF (CASSCF)method in a Newton-Raphson formulation with application to the HNO molecule. J. Chem. Phys. 74 (1981) 2384

    Google Scholar 

  59. Siegbahn, P.E.M.: The Direct CI Method. In: G.H.F. Diercksen, S. Wilson (eds.) Methods in Computational Molecular Physics. Reidel, Dordrecht 1983

    Google Scholar 

  60. Siegbahn, P. E. M., Pettersson, L. G. M., Wahlgren, U.: A theoretical study of atomic fluorine chemisorption on the Ni(100) surface. J. Chem. Phys. 94 (1991) 4024

    Google Scholar 

  61. Sutcliffe, B.T.: Fundamentals of computational quantum chemistry. In: G.H.F. Diercksen, A. Veillard (eds.) Computational Techniques in Quantum Chemistry. Reidel, Boston 1975

    Google Scholar 

  62. Szabo, A., Ostlund, N.: Modern Quantum Chemistry. MacMillan, New York 1982

    Google Scholar 

  63. We87] Werner, H.-J.: Matrix-Formulated Direct Multiconfiguration Self- Consistent Field and Multiconfiguration Reference Configuration Interaction Methods. In: K.P. Lawley (ed.) Ab Initio Methods in Quantum Chemistry, Bd. II. John Wiley k Sons, Chichester 1987 (Adv. Chem. Phys. 69)

    Google Scholar 

  64. Werner, H.-J., Knowles, P. J.: A second order multiconfiguration SCF procedure with optimum convergence. J. Chem. Phys. 82 (1985) 5053

    Article  Google Scholar 

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  1. Fachbereich Chemie, Freie Universität Berlin, Germany

    Carsten Fuchs

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  1. Carsten Fuchs
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  1. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931, Köln, Germany

    Achim Bachem

  2. Institut für Informatik, Universität zu Köln, Pohligstraße 1, D-50969, Köln, Germany

    Michael Jünger  & Rainer Schrader  & 

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Fuchs, C. (1995). Moderne Methoden der numerischen Quantenchemie. In: Bachem, A., Jünger, M., Schrader, R. (eds) Mathematik in der Praxis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-79763-7_5

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  • DOI: https://doi.org/10.1007/978-3-642-79763-7_5

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