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
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