1. Guillaume Barbe (1979- ) Université de Montréal November 11 th 2008 From Newton to Woodward Complete Construction of the Diels-Alder Correlation Diagram Sir Isaac Newton 1642-1727 Robert B. Woodward 1917-1979 Theoritical Model for Concerted Reactions

2. Outlines Construction of Schrödinger equation from classical mechanics and routine mathematics Hückel Model of molecular orbitals will provide a quantification of the energies and orbital coefficients for polyenes Quick excursion in the rational of the symmetry- allowed Diels-Alder Cycloaddition

3. Classical Mechanics Sir Isaac Newton 1642-1727 First Law Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. Second Law The relationship between an object's mass m, its acceleration a, and the applied force F Potential Energy Kinetic Energy Continuum

4. Plum-pudding (1904) Joseph J. Thomson 1856-1940 1906 His son George Paget Thomson 1856-1940 Nobel Prize Physics 1937 Disovery of the Particlelike property of Electron 7 of his students won the Nobel Prize

5. Planetary Model (1909) Ernest Rutherford 1871-1937 By emitting radiation, the electron should lose energy and collapse into the nucleous Atom is not stable ! 1908

6. Hydrogen Atom Niels Bohr 1885-1962 Electron on a Stable Orbit Hydrogen Atom Equilibrium Electric Force Centrifugal Force Total Energy of the Electron Continuum 1922

7. Hydrogen Spectra Photoelectric effect (1905) Albert Einstein 1879-1955 h = Planck Constant photon = frequency of the incident photon h 0 =  = Work function = Energy needed to remove an electron 1921

8. Wave-Particle Duality Photon Case Albert Einstein 1879-1955 Special Theory of Relativity Limit Case Speed of object is low Photon 1921

9. Wave-Particle Duality Electron Case Louis de Broglie 1892-1987 Destructive Electron Wave-Particle Duality Angular momentum 1929

10. Ultraviolet Catastrophe Beginning of quantum theory (1900) Max Planck 1858-1947 Black-body Radiation Density of Energy Rayleigh-Jeans Law Planck Distribution Planck Suggestion 1918

11. Bohr Model (1913) Niels Bohr 1885-1962 Electron on a Stable Orbit Equilibrium Electric Force Centrifugal Force Hydrogen Radius Continuum Quantification Quantum Mechanic 1922

12. Wave Equation Lieou, C. K. C. Eur. J. Phys. 2007, 28, N17-N19 Wave Function Classical Mechanic Non-absortive and Non-dispersive medium Continuously Differentiable Wave Equation

13. Stationary Wave Lieou, C. K. C. Eur. J. Phys. 2007, 28, N17-N19 Wave Equation Variable Separation Stationary Wave Function Trigonomeric Identity Wave Function

14. Stationary Wave Erwin Schrödinger 1887-1961 Lieou, C. K. C. Eur. J. Phys. 2007, 28, N17-N19 Wave Equation Wave Function Stationary Wave Equation 1933

15. Schrödinger Equation Erwin Schrödinger 1887-1961 Stationary Wave Equation Kinetic Energy Total Energy Schrödinger Equation 1933

16. Free Electron Schrödinger Equation Stationary Wave Function Free Electron Wavefunction

17. Particle in a Box 2 Conditions Particle in a Box Wavefunction

18. Particle in a Box

19. Example:  -Carotene 22  electrons

20. Quantum Mechanics Postulates Postulate 1 Schrödinger Equation 1. Associated with any particle moving in a conservative field of force is a wave function which determines everything that can be known about the system.

21. Quantum Mechanics Postulates Postulate 2 Schrödinger Equation 2. With every physical observable q there is associated an operator Q, which when operating upon the wavefunction associated with a definite value of that observable will yield that value times the wavefunction.

22. Quantum Mechanics Postulates Postulate 3 Schrödinger Equation 3. Any operator Q associated with a physically measurable property q will be Hermitian.

23. Quantum Mechanics Postulates Postulate 4 Schrödinger Equation 4. The set of eigenfunctions of operator Q will form a complete set of linearly independent functions.

24. Quantum Mechanics Postulates Postulate 5 Schrödinger Equation 5. For a system described by a given wavefunction, the expectation value of any property q can be found by performing the expectation value integral with respect to that wavefunction. Hermetian Operator Expectation Value

25. Quantum Mechanics Erwin Schrödinger 1887-1961 Paul Dirac 1902-1984 « for the discovery of new productive forms of atomic theory » The Nobel Prize in Physics 1933 Werner Heisenberg 1901-1976 1932 « for the creation of quantum mechanics… »

26. Molecular Orbital Theory of Conjugated Systems Hückel Molecular Orbitals Erich Hückel 1896-1980

27. Secular Equations Schrödinger EquationPostulate 4 Determination of c a and E Overlap Integral

28. Secular Equations We want to determine the value and sign of c a and E

29. Hückel Theory Planar/symmetric systems Secular Equations 4 Approximations for planar and symmetrical polyenes Erich Hückel 1896-1980

30. Hückel Theory Planar/symmetric systems Secular Equations Approximation 1

31. Hückel Theory Planar/symmetric systems Approximation 2 Secular Equations = Coulomb Integral = Energy of bound electron = Constant

32. Hückel Theory Planar/symmetric systems Approximation 2 Secular Equations = Coulomb Integral = Energy of bound electron = Constant

33. Hückel Theory Planar/symmetric systems Approximation 3 Secular Equations

34. Hückel Theory Planar/symmetric systems Approximation 3 Secular Equations

35. Hückel Theory Planar/symmetric systems Approximation 4 Kronecker Symbol Overlap Integral Secular Equations

36. Hückel Theory Planar/symmetric systems Approximation 4 Secular Equations Kronecker Symbol Overlap Integral

37. Hückel Theory Planar/symmetric systems Secular Equations Secular Determinant

38. Hückel Theory Planar/symmetric systems Secular Determinant Highest Energy Lowest Energy 3 Molecular Orbitals  is negative

39. Hückel Theory Planar/symmetric systems Highest Energy Lowest Energy

40. Hückel Theory Planar/symmetric systems Secular Equations Highest Energy Lowest Energy Normalization

41. Hückel Theory Planar/symmetric systems Highest Energy Lowest Energy

42. Secular Equations Example: Butadiene Secular Equations We want to determine the value and sign of c a and E

43. Secular Equations Example: Butadiene Secular Equations

44. Secular Equations Example: Butadiene Secular Equations Symmetrical Anti-Symmetrical

45. Secular Equations Example: Butadiene Secular Equations Symmetrical

46. Secular Equations Example: Butadiene Secular Equations Anti-Symmetrical

47. Secular Equations Example: Butadiene Highest Energy Lowest Energy

48. Secular Equations Example: Butadiene Secular Equations Normalization Highest Energy Lowest Energy

49. Secular Equations Example: Butadiene Highest Energy Lowest Energy

50. Sinusoidal Lobe Alternance Ethene Allyle Butadiene « Electron in a Box »

51. Diels-Alder Cycloaddition

52. Diels-Alder Cycloaddition Conservation of Orbital Symmetry Robert B. Woodward 1917-1979 Roald Hoffman 1937- Elias J. Corey 1928- ? 1965 1981 1990

53. Diels-Alder Cycloaddition Symmetry of Orbitals Robert B. Woodward 1917-1979 Roald Hoffman 1937- Ethylene Butadiene 1965 1981

54. Diels-Alder Cycloaddition Symmetry of Orbitals Butadiene Ethylene , C 2

55. Diels-Alder Reaction Reaction Path: Plan Symmetry Cyclobutene + EthelyneCyclohexene Only a  plan symmetry along the reaction path

56. Diels-Alder Reaction Correlation Diagrams

58. [2+2] Cycloaddition Correlation Diagrams Ethylene + EthelyneCyclobutane Two  plan symmetry along the reaction path

59. [2+2] Cycloaddition Correlation Diagrams

61. Diels-Alder Cycloaddition Frontier Molecular Orbitals Butadiene Ethylene FMO Fukui Acc. Chem. Res. 1971, 4, 57. HOMO LUMO HOMO LUMO Kenichi Fukui 1918-1988 1981 Spino et al. Angew. Chem., Int. Ed. 1998, 37, 3262.

62. Conclusions Schrödinger equation can be easily obtained from classical mechanics through routine mathematical procedures Application of Hückel Model to polyenes provides an approximate but reliable quantification of energies and orbital coefficients Conservation orbital symmetry and FMO are useful in predicting the course of concerted reactions