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3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3.012 Fund of Mat Sci: Bonding – Lecture 12
PolymerspOlymerspoLymerSPOLYmerSPOLYMerSPOLYMErspolymeRpolymerS
Image of a DNA strand removed for copyright reasons.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Homework for Fri Oct 21
• Study:25.7 (Huckel model), 18.1 (quantum oscillator),
• Read 18.6 (classical harmonic oscillator)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Last time: 1. Hartree and Hartree-Fock equations2. Slater determinants3. H2 solution4. Symmetries5. Homonuclear diatomic levels6. Bond order
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Fluorine dimer F2
Correction: Nodal plane containing molecular axis → πFigure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Matrix Formulation (I)
H Eψ ψ=
{ }1,
orthogonal n n nn k
cψ ϕ ϕ=
= ∑
ψϕψϕ mm EH =ˆ
mnkn
mn EcHc =∑=
ϕϕ ˆ,1
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Matrix Formulation (II)
mkn
nmn EccH =∑= ,1
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
⋅
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
kkkkk
k
c
c
E
c
c
HH
HH
.
.
.
.
.
.
............
...... 11
1
111
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Matrix Formulation (III)
11 1
22
1
....... .
det . . 0. .
......
k
k kk
H E HH E
H H E
−⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟ =⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Empirical tight binding andHückel approach
• TB: The matrix elements of the Hamiltonian are “universal empirical parameters”
• Hückel: Planar / quasi-planar systems with delocalized π bonding: two parameters– α: matrix element between same orbital – β: matrix element between neighboring orbitals– Hamiltonian between further neighbors is 0
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Example: Benzene (C6H6)
Diagram of the sigma bond network of benzene removed for copyright reasons.
See page 462, Figure 12-26 in Petrucci, R. H., W. S. Harwood, and F. G. Herring. General Chemistry: Principles and Modern Applications. 8th ed. Upper Saddle River, NJ: Prentice Hall, 2002.
Benzene – energy levels
0 0 00 0 0
0 0 0det 0
0 0 00 0 0
0 0 0
EE
EE
EE
α β ββ α β
β α ββ α β
β α ββ β α
−⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−
=⎜ ⎟−⎜ ⎟
⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
H
CC C
CCC
H
H
H
H
H
π
π
ππ
π
π bonding
π
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Benzene – molecular orbitals
π6
π5π4
π2 π3
α − 2β
α − β
α + β
α + 2βπ1
Energy
π5 π4
π2
π1
π6
π3
Figure by MIT OCW. Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
HOMO-LUMO in naphthalene
Images of orbitals in naphthalene removed for copyright reasons.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The Quantization of Vibrations
• Electrons are much lighter than nuclei (mproton/melectron~1800)
• Electronic wave-functions always rearrange themselves to be in the ground state (lowest energy possible for the electrons), even if the ions are moving around
• Born-Oppenheimer approximation: electrons in the instantaneous potential of the ions (so, electrons can not be excited – FALSE in general)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Nuclei have some quantum action…• Go back to Lecture 1 – remember the
harmonic oscillator
Stationaryobject
Spring
Mass
Equilibriumposition of
mass
z
z0
A mass on a spring. This system can berepresented by a harmonic oscillator.
6
4
2
0
-2
-4
-6
8
1 2 3 4
σ*1s energyu
σg1s energy
Born-Oppenheimer energyof exact orbitals
E BO
/eV
R/10-10m
uOrbital Energies for the σg 1s and σ* 1s LCAO Molecular Orbitals.
Figure by MIT OCW.
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The quantum harmonic oscillator (I)2 2
22
1 ( ) ( )2 2
d kz z E zM dz
ϕ ϕ⎛ ⎞− + =⎜ ⎟
⎝ ⎠
h
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The quantum harmonic oscillator (I)2 2
22
1 ( ) ( )2 2
d kz z E zM dz
ϕ ϕ⎛ ⎞− + =⎜ ⎟
⎝ ⎠
h
km
ω =kma =h
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The quantum harmonic oscillator (II)
Graph of harmonic oscillator wave functions removed for copyright reasons.
See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 532, figure 14.21.
12
E nω⎛ ⎞= +⎜ ⎟⎝ ⎠
h
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Quantized atomic vibrations
Image removed for copyright reasons.See http://w3.rz-berlin.mpg.de/%7Ehermann/hermann/Phono1.gif.
Figure by MIT OCW.