Chem 104A, UC, Berkeley Molecular Orbital Theory Reading: DeKock and Gray, Chap 4 (but not 4-8) Chap 5 (through 5-8) Miessler and Tarr, Chap 5 Chem 104A, UC, Berkeley 1 Chem 104A, UC, Berkeley Linear Combination of Atomic Orbitals (LCAO) k c11 c22 .... cnn 1. n atomic orbitals n molecular orbitals. 2. Like atomic orbitals, MOs are ortho-normal dv 1...(i j) i j dv 0...(i j) i j Chem 104A, UC, Berkeley 2 Chem 104A, UC, Berkeley * * ( u ) N (1sA 1sB ) b b ( g ) N (1sA 1sB ) [ ( b )]2dv [N b (1s 1s )]2 dv 1 g A B (N b )2[ (1s )2 dv 2 (1s )(1s )dv (1s )2 dv] 1 A A B B 1 SAB 1 1 N b 2 2S AB Overlap integral * 1 Degree of spatial overlap between any two AOs N From different atoms in a molecule. 2 2S AB Chem 104A, UC, Berkeley Gerade and Ungerade MO symmetry 1s 1s 1s 1s a b b a 2(1 Sab ) 2(1 Sab ) a, b, exchange position, MO does not change, inversion Even parity, Gerade: German word for “even”. 1s 1s 1s 1s a b b a 2(1 Sab ) 2(1 Sab ) a, b, exchange position, MO does change, no inversion odd parity, ungerade: German word for “odd”. 3 Chem 104A, UC, Berkeley The energy of these two orbitals are obtained by applying Schrodinger Equation Orbital Interaction Diagram 1. Always draw axis 2. Fill in electron using Aufbau principle Chem 104A, UC, Berkeley * N*=1.4 * u -13.6 eV b Nb=0.53 * b > b g 4 Chem 104A, UC, Berkeley •Bond order = ½ (#bonding e-s - #antibonding e-s) •Bond order = 0 indicates the species will not exist. Chem 104A, UC, Berkeley Photoelectron Spectroscopy M h M e hv electron Photoelectric effect (Einstein) 1 h IE mv2 M 2 e 5 Chem 104A, UC, Berkeley PES instruments consist of an X-ray or UV source, an energy analyzer for the photoelectrons, and an electron detector. Chem 104A, UC, Berkeley Computer System Hemispherical Energy Analyzer Outer Sphere Magnetic Shield Analyzer Control Inner Sphere Multi-Channel Plate Electron Lenses for Energy Multiplier Adjustment Optics (Retardation) Resistive Anode X-ray Encoder Source Lenses for Analysis Position Address Area Definition Converter Position Sensitive Detector (PSD) Sample 54.7 6 Chem 104A, UC, Berkeley 13.6 eV H Electron emitted 1 2200cm H2 15.45 eV 24.59 eV He 12 16 20 IE(eV) 24 Chem 104A, UC, Berkeley Koopman’s Theorem IE (n) = - En The ionization energy of electron n is equal to the negative of its Orbital energy. We can obtain orbital energies via PES! 7 Chem 104A, UC, Berkeley N*=1.4 * * u -13.6 eV b Nb=0.53 b g Chem 104A, UC, Berkeley 1 k 2 Quantum Harmonic Oscillator 8 Chem 104A, UC, Berkeley Energy H 2 1 Ev (n )hv Probability distribution 2 Of the vibrational wave function H 2 hv H 2 e b 2 b 1 ( g ) hv ( g ) e H2 Internuclear Distance Ground state molecule is typically in its lowest vibrational state. + Upon ionization, H2 could be in a vibrational excited state. Chem 104A, UC, Berkeley H 2 Energy Vibrational Probability Distribution: For n=0: largest at equil. Distance For higher levels: 2 H Largest at the extreme of compression & expansion (KE=0) Internuclear Distance 9 Chem 104A, UC, Berkeley Wave functions, allowed energies, and corresponding probability densities for the harmonic oscillator Chem 104A, UC, Berkeley Energy H 2 n 0 n, 0 , n 0 Adiabatic ionization H2 n 0 Internuclear Distance 10 Chem 104A, UC,Vertical Berkeley Energy ionization H 2 Adiabatic ionization H2 Relative Intensity overlap of vibrational Wave function Internuclear Distance Chem 104A, UC, Berkeley Vertical Energy ionization H 2 2200cm1 Adiabatic ionization H2 1 k 2 Internuclear Distance Removing electron from bonding MO 11 Chem 104A, UC, Berkeley 13.6 eV H Electron emitted 1 2200cm H2 15.45 eV 24.59 eV He 12 16 20 IE(eV) 24 Chem 104A, UC, Berkeley Vertical Energy ionization Adiabatic ionization Internuclear Distance Removing electron from antibonding MO 12 Chem 104A, UC, Berkeley Energy Adiabatic ionization Internuclear Distance Removing electron from nonbonding MO Chem 104A, UC, Berkeley * * ( u ) N (1sA 1sB ) b b ( g ) N (1sA 1sB ) 1 N b 2 2S AB 1 N * 2 2S AB Overlap integral Degree of spatial overlap between any two AOs From different atoms in a molecule. Degree of AO interaction is closely related to their overlap Functions of distance, symmetry. 13 Chem 104A, UC, Berkeley SAB >0, bonding interaction, E stabilized Chem 104A, UC, Berkeley SAB <0, antibonding interaction, E destabilized * * * SAB =0, nonbonding, no orbital interaction. 14 Chem 104A, UC, Berkeley Chem 104A, UC, Berkeley 15.