3.012 Fund of Mat Sci: Bonding – Lecture 12
P O L Y
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M E R S
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Image of a DNA strand removed for copyright reasons.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
polymer
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.
2.
3.
4.
5.
6.
Hartree and Hartree-Fock equations
Slater determinants
H2 solution
Symmetries
Homonuclear diatomic levels
Bond order
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Fluorine dimer F2
Figure by MIT OCW.
Correction: Nodal plane containing molecular axis → π
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Matrix Formulation (I)
Ĥ ψ = E ψ
ψ =
∑c
n =1, k
n
ϕn
{ ϕ } orthogonal
n
ϕ m Hˆ ψ = E ϕ m ψ
ˆ
c
ϕ
H
∑ n m ϕn = Ecm
n =1, k
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Matrix Formulation (II)
∑H
n =1, k
⎛ H11
⎜
⎜ .
⎜ .
⎜
⎜ .
⎜H
⎝ k1
......
......
c = Ecm
mn n
H1k ⎞ ⎛ c1 ⎞
⎛ c1 ⎞
⎟ ⎜ ⎟
⎜ ⎟
. ⎟ ⎜ .⎟
⎜ .⎟
. ⎟⋅⎜ . ⎟ = E ⎜ . ⎟
⎟ ⎜ ⎟
⎜ ⎟
. ⎟ ⎜ .⎟
⎜ .⎟
⎟
⎜
⎟
⎜
⎟
H kk ⎠ ⎝ ck ⎠
⎝ ck ⎠
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Matrix Formulation (III)
⎛ H11 − E
⎜
.
⎜
det ⎜
.
⎜
.
⎜
⎜ H
k1
⎝
......
H 22 − E
......
⎞
⎟
.
⎟
⎟=0
.
⎟
.
⎟
H kk − E ⎟⎠
H1k
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Empirical tight binding and
Hückel 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.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
π
π
π
H
H
C
C
H
π
H
C
C
C
C
π
H
H
Benzene – energy levels
π
π bonding
Figure by MIT OCW.
β
0
0
0
β ⎞
⎛α − E
⎜
⎟
α −E
β
0
0
0 ⎟
⎜ β
⎜ 0
β
α −E
β
0
0 ⎟
det ⎜
⎟=0
β
α −E
β
0
0 ⎟
⎜ 0
⎜ 0
β
α −E
β ⎟
0
0
⎜⎜
⎟⎟
β
α −E⎠
0
0
0
⎝ β
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Benzene – molecular orbitals
α − 2β
π6
π4
π5
α−β
π6
π5
π4
Energy
π2
π3
α+β
π2
π1
α + 2β
π1
Figure by MIT OCW.
π3
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
8
Stationary
object
σ*1s
u energy
6
4
2
EBO/eV
Spring
z
Mass
0
Born-Oppenheimer energy
of exact orbitals
σg1s energy
1
2
3
4
-2
Equilibrium
position of
mass
0
z
-4
-6
A mass on a spring. This system can be
represented by a harmonic oscillator.
R/10-10m
Orbital Energies for the σg 1s and σ*u 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)
⎛ h2 d 2 1 2 ⎞
+ kz ⎟ ϕ ( z ) = E ϕ ( z )
⎜−
2
2
⎝ 2 M dz
⎠
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The quantum harmonic oscillator (I)
⎛ h2 d 2 1 2 ⎞
+ kz ⎟ ϕ ( z ) = E ϕ ( z )
⎜−
2
2
⎝ 2 M dz
⎠
k
ω=
m
km
a=
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.
1⎞
⎛
E = hω ⎜ n + ⎟
2⎠
⎝
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Quantized atomic vibrations
Figure by MIT OCW.
Image removed for copyright reasons.
See http://w3.rz-berlin.mpg.de/%7Ehermann/hermann/Phono1.gif.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Specific Heat of Graphite (Dulong and Petit)
2500
CP (J.K-1.kg-1)
2000
1500
1000
500
0
0
500
1000
1500
2000
2500
Temperature (K)
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)