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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
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Chem 104A, UC, Berkeley
Linear Combination of Atomic Orbitals (LCAO)
1. n atomic orbitals n molecular orbitals.
2. Like atomic orbitals, MOs are ortho-normal
)...(0
)...(1
jidv
jidv
ji
ji
nnk ccc ....2211
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
)11()(
)11()( **
BAbb
g
BAu
ssN
ssN
1])1()1)(1(2)1([)(
1])11([)]([
222
22
dvsdvssdvsN
dvssNdv
BBAAb
BAbb
g
1 SAB 1
Overlap integralDegree of spatial overlap between any two AOs From different atoms in a molecule.
AB
AB
b
SN
SN
22
1
22
1
*
Chem 104A, UC, Berkeley
Gerade and Ungerade MO symmetry
)1(2
11
)1(2
11
ab
ab
ab
ba
S
ss
S
ss
a, b, exchange position, MO does change, no inversionodd parity, ungerade: German word for “odd”.
)1(2
11
)1(2
11
ab
ab
ab
ba
S
ss
S
ss
a, b, exchange position, MO does not change, inversion Even parity, Gerade: German word for “even”.
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Chem 104A, UC, Berkeley
The energy of these two orbitals are obtained by applying Schrodinger Equation
Orbital Interaction Diagram1. Always draw axis
2. Fill in electron using Aufbau principle
Chem 104A, UC, Berkeley
*u
bg
*b
N*=1.4
Nb=0.53
-13.6 eV
* b>
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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
hv electron
Photoelectric effect (Einstein)
2
2
1eM mvIEh
eMhM
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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
5 4 . 7
X-raySource
Optics
Hemispherical Energy Analyzer
Position Sensitive Detector (PSD)
Magnetic ShieldOuter Sphere
Inner Sphere
Sample
Computer System
Analyzer Control
Multi-Channel Plate Electron Multiplier
Resistive Anode Encoder
Lenses for Energy Adjustment (Retardation)
Lenses for Analysis Area Definition
Position Address Converter
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Chem 104A, UC, Berkeley
IE(eV)
Electronemitted
12 16 20 24
13.6 eV
24.59 eV
15.45 eV
H
H2
He
12200 cm
Chem 104A, UC, Berkeley
Koopman’s Theorem
IE (n) = - En
The ionization energy of electron n is equal to the negative of itsOrbital energy.
We can obtain orbital energies via PES!
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Chem 104A, UC, Berkeley
*u
bg
*
b
N*=1.4
Nb=0.53
-13.6 eV
Chem 104A, UC, Berkeley
Quantum Harmonic Oscillator
k
2
1
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Chem 104A, UC, Berkeley
Energy
Internuclear Distance
H2
2H
Ground state molecule is typically in its lowest vibrational state.Upon ionization, H2
+ could be in a vibrational excited state.
ehv
eHhvHbg
bg
12
22
)()(
hvnEv )2
1(
Probability distributionOf the vibrational wavefunction
Chem 104A, UC, Berkeley
Energy
Internuclear Distance
H2
2H
Vibrational Probability Distribution:
For n=0: largest at equil. Distance
For higher levels:
Largest at the extreme of compression& expansion (KE=0)
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Chem 104A, UC, Berkeley
Wave functions, allowed energies, and corresponding probability densities for the harmonic oscillator
Chem 104A, UC, Berkeley
Energy
Internuclear Distance
H2
2H
Adiabaticionization
00 , nn
0n
0, n
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Chem 104A, UC, Berkeley
Energy
Internuclear Distance
H2
2H
Adiabaticionization
Vertical ionization
Relative Intensity
overlap of vibrationalWave function
Chem 104A, UC, Berkeley
Energy
Internuclear Distance
H2
2H
Adiabaticionization
Vertical ionization
Removing electron from bonding MO
12200 cm
k
2
1
12
Chem 104A, UC, Berkeley
IE(eV)
Electronemitted
12 16 20 24
13.6 eV
24.59 eV
15.45 eV
H
H2
He
12200 cm
Chem 104A, UC, Berkeley
Energy
Internuclear Distance
Adiabaticionization
Vertical ionization
Removing electron from antibonding MO
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Chem 104A, UC, Berkeley
Energy
Internuclear Distance
Adiabaticionization
Removing electron from nonbonding MO
Chem 104A, UC, Berkeley
)11()(
)11()( **
BAbb
g
BAu
ssN
ssN
Overlap integralDegree 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.
AB
AB
b
SN
SN
22
1
22
1
*
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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.
*
*
*
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley