3.012 Fund of Mat Sci: Bonding – Lecture 12 P O L Y

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.


Homework for Fri Oct 21

• Study:25.7 (Huckel model), 18.1 (quantum oscillator),

• Read 18.6 (classical harmonic oscillator)


Last time: 1. Hartree and Hartree-Fock equations2. Slater determinants3. H2 solution4. Symmetries5. Homonuclear diatomic levels6. Bond order


Fluorine dimer F2

Correction: Nodal plane containing molecular axis → πFigure by MIT OCW.


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


Matrix Formulation (II)

mkn

nmn EccH =∑= ,1

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⋅

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

kkkkk

k

c

c

E

c

c

HH

HH

.

.

.

.

.

.

............

...... 11

1

111


Matrix Formulation (III)

11 1

22

1

....... .

det . . 0. .

......

k

k kk

H E HH E

H H E

−⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟ =⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠


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


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

−⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−

=⎜ ⎟−⎜ ⎟

⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠


H

CC C

CCC

H

H

H

H

H

π

π

ππ

π

π bonding

π

Figure by MIT OCW.


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.


HOMO-LUMO in naphthalene

Images of orbitals in naphthalene removed for copyright reasons.


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)


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.


The quantum harmonic oscillator (I)2 2

22

1 ( ) ( )2 2

d kz z E zM dz

ϕ ϕ⎛ ⎞− + =⎜ ⎟

⎝ ⎠

h


The quantum harmonic oscillator (I)2 2

22

1 ( ) ( )2 2

d kz z E zM dz

ϕ ϕ⎛ ⎞− + =⎜ ⎟

⎝ ⎠

h

km

ω =kma =h


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


Quantized atomic vibrations

Image removed for copyright reasons.See http://w3.rz-berlin.mpg.de/%7Ehermann/hermann/Phono1.gif.

Figure by MIT OCW.


Specific Heat of Graphite (Dulong and Petit)

00

500

500

1000

1000

1500

1500

2000

2000

2500

2500

Temperature (K)

CP

(J.K

-1.k

g-1)

Figure by MIT OCW.

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3.012 Fund of Mat Sci: Bonding – Lecture 12 P O L Y