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Physical Background

This article provides an overview of the rudiments of quantum theory and its application to atomic and molecular states. Topics covered include the old quantum theory, mathematical apparatus, interpretation, hydrogen atom, many-electron systems, molecular states, and classification in chemistry.

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Physical Background

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  1. Physical Background

  2. Atomic and Molecular States • Rudiments of Quantum Theory • the old quantum theory • mathematical apparatus • interpretation • Atomic States • hydrogen • many-electron • Molecular states • classification • chemistry

  3. Rudiments of Quantum Theory

  4. Photons and Particles Maxwell equations Space and time change of phase Relativistic relation Ep • Electromagnetic field • The wave equation • Plane waves, The wave function • Planck’s hypothesis, dualism • Particle (E,p)(,k) de Broglie wave

  5. Bohr’s Model of the Hydrogen Atom • Classical atom • Unstable: accelerated motion, continuous radiation • Bohr’s rules • Quantized angular momentum • Only certain circular orbits allowed • Discrete set of stationary states • Discrete spectrum of energy • Discrete spectrum of radiation • Coulomb and centrifugal force • Bohr radius

  6. Mathematical Apparatus General operator Operator algebra • Philosophy: Act of observation • Interaction through which the quantity is ‘observed’ • Possible results of observation • Non-commuting observations • Mathematics: operators in Hilbert space • Eigenvalue equation • Commutation relations

  7. Mathematical Apparatus Mean value Complementarity principle Schrödinger representation • Interpretation postulates • Possible results of observation  are eigenvalues an • Observation  on a system in eigenstate n certainly leads to an • The mean value of the observable  on the ensemble of systems y • Physical postulates • The correspondence principle • In the limit of ‘large’ system quantum laws reduce to classical laws • Relation between classical quantities with no derivatives holds also for quantum operators • The principle of complementarity • The Heisenberg uncertainty principle “An experiment on one aspect of a system is supposed to destroy the possibility of learning about a 'complementary' aspect of the same system”.

  8. Angular Momentum • Space orientation of the orbit • Magnetic and electric moments • Internal and external interactions • Classical • Quantum • Spherical coordinates • Boundary condition +2n

  9. The Copenhagen Interpretation radioactive isotope cyanide capsule • Probabilistic approach • Probability density • Collapse of the wave function • Schrödinger’s Cat • Two-Slit Experiment Indeterminate quantum states “collapse” to definite values when they do, not when a human being catches them in the act

  10. Atomic States

  11. Hydrogen SO coupling – internal Zeeman • Particle in a central potential • Coulomb potential • Electron spin • Fine structure: relativistic corrections • Electron-nucleus, Kinetic energy, Spin-orbit interaction • Lamb shift • Hyperfine structure

  12. Hydrogen

  13. Many-Electron Atomic States • Ground state configuration • Pauli exclusion principle • Hund’s rules • e–with parallel s more separated • Lower repulsion, lower energy • Terms • LS coupling: small Z • L, S, J (M) • j-j coupling: large Z • J, M

  14. Molecular States

  15. Molecular Bonds • Ionic • Transfer of valence e–to produce a noble gas configuration • Coulomb force, long • Na+Cl-: re=0.24 nm, De=4.26 eV • Covalent • Shearing of pair of valence e–() • Quantum mechanical, short • H2: bonding S, anti-bonding A • Pauli principle  A(1,2)=S(1,2)(1,2) • H2: re=0.074 nm, De=4.75 eV • Metallic • Shared and delocalized valence e–- strong • Van der Waals • Dipole-dipole, weak, long • Hydrogen

  16. U U internuclear distance r Electronic States • Born-Oppenheimer Approximation • Separation of electronic and nuclear motion • Electronic motion – nuclei fixed

  17. Electronic potentials Mg2 Electronic States • Classification • Total orbital momentum along internuclear axis in the electric field • Internal Stark effect • Total spin along internuclear axis • magnetic coupling • Parity of el • Inversion about a plane through the axis: +/- • Inversion through the center of symmetry: g/u • Homonuclear molecules

  18. Harmonic (Hook) V(r) De Morse re Nuclear Motion • Rigid rotator • Harmonic oscillator • Anharmonic oscillator • Morse potential

  19. Nuclear Motion • Vibrating rotator • 100 vibrations during a revolution • Averaged rotational constant • Mean value (1/r2)v decreases as v increases • Centrifugal force • Coupling of electronic and nuclear motion • Hund’s cases • Coupling between various angular momentum vectors • Gyroscopic forces disturb orbital motion of electrons • Internal magnetic fields from the rotation of nuclei couple with the electron spin • Total angular momentum J

  20. 3d 4d 4p 3p 4s H2 separated atoms H bonding united atoms He + + + 2pu* + 3s 3d s s ssg 2ppg* antibonding 3p + 2pg + 2p + 3s s s 2su* ssu* 2ppu 2s 2sg bonding + + + 2p + pz pz 2s pzsg antibonding 1su*` + + + + + 1s pz pz pzsu* 1s 1sg energy bonding + + + + px px pxpu + + antibonding + Li2 N2 H2 + + internuclear distance re px px pxpg* Molecular Orbitals • LCAO (Linear Combination of Atomic Orbitals), perturbation theory • Homonuclear diatomics • Correlation diagram; surfaces of probability ||2, ||2

  21. sp3 Hybrid Orbitals • C • 2s2pz • Increase of Esp less than decrease of E due to 4 bonds instead of 2 • CH4: 3sps + 1sss ? • Hybridization • All bonds the same • Linear combination of atomic s and p orbitals in case of EsEp • Hybridization sp3CH4 • Each molecular orbital is combination of ¼ s and ¾ p • Tetrahedral geometry, 109.5°, strong directional s bonds 1s 2s 2px 2py 2pz     1s 2s 2px 2py 2pz     

  22. Hybrid Orbitals

  23. sp2 p sp s s H C C H s p H H C C H H s s s p s s Hybrid Orbitals • Hybridization sp2C2H4 • 3 sp2s bonds, ⅓s and ⅔p • 1 pp bond • sp2s approximately 120° • p perpendicular to axis • p out of the axis, more reactive • Hybridization spC2H2 • 2 hybrids ½ s + ½ p, s • 2 pure p, p

  24. Hybrid Orbitals

  25. resonance (A) (B) Hybrid Orbitals sp2s • Benzene (sp2)C6H6 • Valence bonding theory VB • Each C uses 3 sp2 orbitals to form s bonds with H and next C • Planar symmetrical hexagon, 120° • 6 e– in 6 p orbitals perpendicular to s bonds form 3 p bonds, 2 e– in each • 3 single and 3 double bonds • Shortcomings • Double bonds are not so stable • C–C 1.54 Ǻ; C=C 1.35 Ǻ • No isomeric compounds found • Resonance model • Resonance hybrid between structures (A) and (B) • 1.5 bonds between C atoms 1s p

  26. Hybrid Orbitals • Benzene (sp2)C6H6 • Molecular orbital theory MO • p system of delocalized e– • C bonds 1.40 Ǻ • Stability: “delocalization energy” • VSEPR • Valence Shell Electron Pair Repulsion Theory • Predicts the shapes of the molecules VB hybridizes the atomic orbitals first then overlaps the resulting hybrid orbitals by using LCAO. MO overlaps the atomic orbitals first by using LCAO followed by VSEPR concepts.

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