Physical Background

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

Rudiments of Quantum Theory

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

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

General operator Operator algebra Mathematical Apparatus 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

Mathematical Apparatus Interpretation postulates –Possible results of observation  are eigenvalues a n –Observation  on a system in eigenstate  n certainly leads to a n –The mean value of the observable  on the ensemble of systems  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 Mean value Complementarity principle Schrödinger representation “An experiment on one aspect of a system is supposed to destroy the possibility of learning about a 'complementary' aspect of the same system”.

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

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

Atomic States

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

Hydrogen

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

Molecular States

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

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

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 Electronic potentials Mg 2

Nuclear Motion Rigid rotator Harmonic oscillator Anharmonic oscillator –Morse potential V(r) rere DeDe Harmonic (Hook) Morse

Nuclear Motion Vibrating rotator –100 vibrations during a revolution –Averaged rotational constant –Mean value (1/r 2 ) 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

Molecular Orbitals LCAO (Linear Combination of Atomic Orbitals), perturbation theory Homonuclear diatomics –Correlation diagram; surfaces of probability |  | 2, |  | 2 1s 2s 2p 3s 3p 3d 4s 4p 4d 1s 2s 2p 3s 3p 3d 1s  g 1s  u *` 2s  g 2s  u * 2p  u 2p  g * 2p  g 2p  u * internuclear distance r e H2H2 Li 2 N2N2 energy united atoms He separated atoms HH2H s s s s sgsg su*su* pzpz pzpz pzpz pzpz pxpx pxpx pxpx pxpx pzu*pzu* pzgpzg pxupxu pxg*pxg* bonding antibonding bonding antibonding bonding antibonding

C –2s  2p z Increase of E s  p less than decrease of E due to 4 bonds instead of 2 CH 4 : 3sp  + 1ss  ? –Hybridization All bonds the same Linear combination of atomic s and p orbitals in case of E s  E p Hybridization sp 3 CH 4 –Each molecular orbital is combination of ¼ s and ¾ p –Tetrahedral geometry, 109.5°, strong directional  bonds 1s 2s 2p x 2p y 2p z     1s 2s 2p x 2p y 2p z      sp 3 Hybrid Orbitals

sp Hybrid Orbitals Hybridization sp 2 C 2 H 4 –3 sp 2  bonds, ⅓ s and ⅔ p –1 p  bond –sp 2  approximately 120° –p perpendicular to axis –  out of the axis, more reactive Hybridization sp C 2 H 2 –2 hybrids ½ s + ½ p,  –2 pure p,  sp 2 H C C H       H C C H     

Hybrid Orbitals

Benzene (sp 2 ) C 6 H 6 Valence bonding theory VB –Each C uses 3 sp 2 orbitals to form  bonds with H and next C –Planar symmetrical hexagon, 120° –6 e – in 6 p orbitals perpendicular to  bonds form 3  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 sp 2  1s p (A) (B) resonance

Hybrid Orbitals Benzene (sp 2 ) C 6 H 6 Molecular orbital theory MO –  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.