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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 Robert B. Woodward Theoritical Model for Concerted Reactions

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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

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Classical Mechanics Sir Isaac Newton 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

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

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

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Hydrogen Atom Niels Bohr Electron on a Stable Orbit Hydrogen Atom Equilibrium Electric Force Centrifugal Force Total Energy of the Electron Continuum 1922

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

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

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Wave-Particle Duality Electron Case Louis de Broglie Destructive Electron Wave-Particle Duality Angular momentum 1929

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

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

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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

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

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

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Schrödinger Equation Erwin Schrödinger Stationary Wave Equation Kinetic Energy Total Energy Schrödinger Equation 1933

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Free Electron Schrödinger Equation Stationary Wave Function Free Electron Wavefunction

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Particle in a Box 2 Conditions Particle in a Box Wavefunction

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Particle in a Box

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Example: -Carotene 22 electrons

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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.

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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.

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

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

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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

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Quantum Mechanics Erwin Schrödinger Paul Dirac « for the discovery of new productive forms of atomic theory » The Nobel Prize in Physics 1933 Werner Heisenberg « for the creation of quantum mechanics… »

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Molecular Orbital Theory of Conjugated Systems Hückel Molecular Orbitals Erich Hückel

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

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

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Hückel Theory Planar/symmetric systems Secular Equations 4 Approximations for planar and symmetrical polyenes Erich Hückel

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Hückel Theory Planar/symmetric systems Secular Equations Approximation 1

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

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

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Hückel Theory Planar/symmetric systems Approximation 3 Secular Equations

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Hückel Theory Planar/symmetric systems Approximation 3 Secular Equations

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

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

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Hückel Theory Planar/symmetric systems Secular Equations Secular Determinant

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

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Hückel Theory Planar/symmetric systems Highest Energy Lowest Energy

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

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Hückel Theory Planar/symmetric systems Highest Energy Lowest Energy

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

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Secular Equations Example: Butadiene Secular Equations

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Secular Equations Example: Butadiene Secular Equations Symmetrical Anti-Symmetrical

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Secular Equations Example: Butadiene Secular Equations Symmetrical

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Secular Equations Example: Butadiene Secular Equations Anti-Symmetrical

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Secular Equations Example: Butadiene Highest Energy Lowest Energy

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

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Secular Equations Example: Butadiene Highest Energy Lowest Energy

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

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Diels-Alder Cycloaddition

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Diels-Alder Cycloaddition Conservation of Orbital Symmetry Robert B. Woodward Roald Hoffman Elias J. Corey ?

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Diels-Alder Cycloaddition Symmetry of Orbitals Robert B. Woodward Roald Hoffman Ethylene Butadiene

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

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

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Diels-Alder Reaction Correlation Diagrams

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

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[2+2] Cycloaddition Correlation Diagrams

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

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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

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