Electronic Structure of Molecules

AST380E

Yancy L. Shirley

Where do we find molecules ?

Molecular Clouds &Star Formation Regions

Orion in CO J=2-1

VLT, Japan

CH4 Absorption

Brown Dwarfs:Gliese 229B

Geballe, et al. 1995

Planetary Atmospheres

ISO Spectrum of Jupiter

CH4

CH3D

PH3

NH3

C2H6

C2H2

Voyager, NASA

Comets:Hale-Bopp

HCN J=4-3

D. Jewitt, U. Hawaii

Evolved Stars:IRC10216

HCN BIMA Image C4H BIMA Image

Why do molecules form?

Classical Argument

20

2

2/42Re

F attractive

20

2

4 Re

F repulsive

FF repulsiveattractive

Stable molecule has e- with HIGH PROBABILITY of being found between nuclei

+ +

R

e-

The Born-Oppenheimer Approximation

R

e

r

e

r

e

mmH

BAe

eBA

p 044422

ˆ2

0

2

0

22

222

2

The H2+ Hamiltonian

Since the mass(p+) >> mass(e-) , we can ignore nuclear motion (the 1st term) and solve Schroedinger’s Eq. assuming a fixed nucleus.

+ +

R

e-

rA rB

ignore

p+p+

Quantum Calculations of H2 Structure

ear

sa 0/2/101

sBsAsym 11

sBsAasym 11

Molecular Orbitals - LCAO

+

+ + + +

BONDING ANTI-BONDING

e- binding Energy

Bonding

Anti-bonding

-13.6 eV

-54.4 eV

p+ repulsion

Symmetric

Anti-symmetric

En

erg

y

Distance (nm) K. Krane, Modern Physics

L

Lz

E

Electronic State in a molecule

lllm

ml

l

lz

,...,1,

Electronic orbital angular momenta precess about internuclear axis due to strong electric field.

Symbol: , ,

2,1,0: lm

Molecular Orbitals

+ +

+ -

AO MO

+

g1s

u1s

1s

-+

1s

Z

Z

Molecular Orbitals

+- + - +- -

g2pz

+- +-

u2pz

+ +- -

2pz

2pz

AO MO

+

-

+

-

+

-

+

- +

-

Z

Z

+

+

-

-

Molecular Orbitals

u2px

g2px

AO MO

2px

Correlation of Molecular Orbitals - Homonuclear

G. Herzberg, Molecular Spectra & Molecular Strucutre , vol I.

Unified Atom Separated Atom

Total Electronic Term of a Linear Molecule

O

LS

i

ilm

i

ismS

12STotal electronic term:

Symmetries of electronic state

x

y

z

Inversion

-x

-y

-z

g

- u

ONLY Applies to homonuclear linear molecules

x

y

z

Reflection

x

-y

z

+

- -

ONLY Applies to linear molecules in states

E

g1s

H2+

= 0

S = ½ 2g

+

E

g1s

H2

= 0

S = 0 1g

+

E

g1s

He2+

= 0

S = ½ 2u

+

u1s

E

g1s

C2

= 0

S = 0 1g

+u1s

g2s

u2s

u2p

E

g1s

N2

= 0

S = 0 1g

+u1s

g2s

u2s

u2p

g2p

E

g1s

O2

= 0

S = 1 3g

-u1s

g2s

u2s

u2p

g2p

= 2

S = 0 1g

= 0S = 0 1g

+

Ground State

g2p

O2 (g2p)2 Configurations

1 1 + - 2 0

1 1 - + 2 0

-1 -1 + - -2 0

-1 -1 - + -2 0

1 -1 + + 0 1

-1 1 + + 0 1

1 -1 - - 0 -1

-1 1 - - 0 -1

1 -1 + - 0 0

1 -1 - + 0 0

1 2 s1 s2

1

3

1

D. McQuarrie, Quantum Chemistry

Bond Order

Bond Energy

Bond Length

B2 C2 N2 O2 F2 Ne2

G. Herzberg, Molecular Spectra & Molecular Strucutre , vol I.

Correlation of Molecular Orbitals - Heteronuclear

Unified AtomSeparated Atom

E

s

OH

= 1

S = ½ 2

s

p

p

E

1sA

CO

= 0

S = 0 1g

+1sB

2sA

2sB

2pA

2pA

Photoelectron Spectrum CO

s

s

p

p

# e- e

ject

ed

Binding Energy D. McQuarrie, Quantum Chemistry

MO Notation

D. McQuarrie, Quantum Chemistry

Ground Electronic State of a few Linear Molecules

2 H2+, CN, CO+, CCH

3 O2, NH, SO, CCS, C2O

2 CH, OH, NO

1 H2, CH+, CO, CS, SiO, HCN, HCO+, N2H+, CO2, HCNH+, HC3N

E

g1s

Excited States of H2

= 0

S = 0 X1g

+

u1s

g2s

u2s

u2p

g2p

= 0S = 1 b3u

+

= 0S = 0 B1u

+

= 0S = 0 C1u

Lyman Band

Werner Band

UV Electronic Transitions:

H2

Shu, Radiation

Electronic Transitions:

C2Phillips Band