Atoms to Molecules
Single electron atom or ion (arbitrary Z)
orbitals known exactly
 Ze 2
V (r ) 
4 o r
ψ(r , ,  )  R(r ) ( ) ( ) solutions characterized by QNs: n, l, ml
R(rn) 
 13.6Z 2
En 
n2
n2
rn 
Z
Consider 1s orbital:
 1 0 0 (r, ,  )  Ae Zr / ao
n l ml

1
π ao3 / 2
single electron (neutral)
atom is hydrogen, Z = 1
Electrons in Molecules
H2+ ionic molecule: 2 protons and 1 electron
Take new molecular*symmetry
orbital about
to bex =a0linear
atomic
orbitals
requirescombination
coefficients to be of
equal
in magnitude
Orbitals*
 1 0 0 (r, ,  )  Ae Zr / ao

~ S  ψ a  ψb
ψ
~ A  ψ a  ψb
ψ
a
b
- ½R
½R
a
b
position
Probability densities: * = ||2
||2
Bonding
Antibonding
electron between nuclei
electron on either side of nuclei
a
b
- ½R
½R
a
position
intuitively: this orbital has lower coulombic energy
b
“node” high energy
Types of MOs from LCAOs
s
s
+
pz
pz
+
s bonds
s
pz
+
px (py)
+
 bond
Types of MOs from LCAOs
s
s
+
Showed: symmetric = bonding
s bonds
Symmetric
pz

+
Anti-bonding
pz
+
+

Antisymmetric
+

+
+

+

Bonding

+
+

+
+
The H2 Ion
Vary inter-nucleus distance, R, between the two protons
R0
anti-bonding
E
b
a
R
~ A  ψ a  ψb
ψ
electron sees only one
proton, no interaction
-13.6 eV
bonding
E
 0  R0
R
R

a
b
To find equilibrium bond distance, R0
~ S  ψ a  ψb
ψ
The H2 Molecule
• Introduce 2nd electron
• Solution has perturbations due to electron-electron interactions
• Ignore these and place 2nd electron in ‘same’ bonding orbital but
with opposite spin
alternative
depictions
1s
R
E

H  H
2 anti-bonding states
-13.6 eV
2 protons

2 bonding states
• Result: H2 covalent bond
• Directional; typical of molecules
two 1s states each
 4 states total
1s
The N-atom Hydrogen Solid
1s1 N-atom solid  N electrons
Chemical Bonding  Continuous Bands
R0
E
N states
unoccupied
R
overlap of states
discrete  continuous
N anti-bonding states
1s
N states
occupied
N bonding states
2N states
H2 molecule: N = 2
1s orbital is close to nucleus, nuclear interaction prevents strong overlap
Lithium: A Simple Metal
Li #3
1s22s1 N-atom solid 2s orbitals overlap without nuclear repulsion
N 2s electrons, 2N states
R0
R
E
N states
unoccupied
N states
occupied
anti-bonding
overlap of states
discrete  continuous
2s
2N states
1s
2N states
bonding
E
 0  R0
R
All states occupied, independent
Silicon: A Semiconductor
N-atom solid  4N relevant electrons
Si: #14 1s22s22p6 3s23p2
[3(sp3)4] hybrid orbital composed 3s: 2N states
[Ne]
of 3s and all 3p orbitals:
Hybridization:
consider just 2 atoms
R0
anti-bonding
3
3p
3s
1
6 states
2 states
3p
3s
4N states
unoccupied
Eg
1 3
bonding
4 states
(+ 4)
N-atom Solid
 Continuous Bands
8N states
R
overlap of states
discrete  continuous
E
4 states
(+ 4)
3p: 6N states
4N states
occupied
E
 0  R0
R
4N anti-bonding states
“sp3”
8N states
4N bonding states
Magnesium: A Metal?
Mg: #12 1s22s22p6 3s2
N atom solid, 2N electrons
[Ne]
R0
R
3s and 3p overlap to create a
band with 8N states; only 2N
states occupied  yes, a metal
E
3p
2N states
occupied
6N states
anti-bonding
3s
2N states
bonding
E
 0  R0
R
metal requires a partially
occupied band??
LiF: An Ionic Solid


Li: #3 1s22s1
F: #9 1s22s22p5
1s2
2p6
 Li+1 + e F + e-  F-1
Energy of bonding for a hypothetical ion pair
E = Eionization + Ecoulombic
Eion ( Li) 
Eion ( F ) 
13.6

Li
Z eff
2
13.6


Eion
13.6Z 2

n2
2

5.4 eV
> 0 requires energy

-3.7 eV
< 0 releases energy

-7.2 eV
2
F
Z eff
22

2
+1 -1
ECoul
Z1Z 2 e 2

4 o R
Zeff < Zact due to shielding
Epair = 5.4 - 3.7 - 7.2 eV = 5.5 eV
N-atom Pair Solid of LiF
Li: #3 1s22s1
F: #9 1s22s22p5


e: Li(2s)  F(2p) Li+1 + e-
1s2
2p6
 E(F2p) < E(Li2s)
6N electrons
R0
R
LiF is a non-metal
E
Li 2s 2N states
Eg
6N states
occupied
E
 0  R0
R
F 2p
6N states
Thoughts on how to
transform it a metal?
Summary: MO/LCAO Approach
• N atom solid
– bonding and anti-bonding states
– isolated states  bands due to exclusion principle
• Metal
– no energy gap between occupied and unoccupied states
– many need to consider orbitals of slightly higher energy
• Semi-conductor
– hybrid orbitals  bands
– ‘small’ bandgap between occupied and unoccupied
• Ionic
qualitative distinction
– electron transfer from electropositive to electronegative ion
– orbitals  bands
– ‘large’ bandgap between occupied and unoccupied states