CHEM 2060 Lecture 22: VB Theory L22-1
PART FIVE: The Covalent Bond
We can model covalent bonding in molecules in essentially two ways:
1. Localized Bonds (retains electron pair concept of Lewis Structures)
⇒ Valence Bond (VB) model
⇒ Hybridization of atomic orbitals (e.g., sp3 hybridization)
Advantages:
• Concept of bond order applies (single, double, triple bonds, etc.)
• Can discuss a bond and ignore the rest of the molecule
Disadvantages:
• Hard to describe the properties of some species (e.g., O2)
2. Delocalized Molecular Orbitals.
⇒ Molecular Orbital (MO) model
Advantages:
• Easily describes spectroscopic properties of most species.
Disadvantages:
• Can be unnecessarily cumbersome.
CHEM 2060 Lecture 22: VB Theory L22-2
Valence Bond Theory … H2 as an example
In order to describe the bonding in H2, we use the atomic orbitals of the H atoms
and mix them.
Ha + Hb → H2
use the 1s orbitals of atoms Ha and Hb
“Combine” wavefunction of 1sa and 1sb
First guess: Multiply the functions.
ψI = 1sa(1) 1sb(2)
This puts electron 1 in the 1s orbital of Ha and electron 2 in the 1s orbital of Hb.
This is a lousy guess because it doesn’t take into account that we will not be able
to distinguish electrons 1 and 2 at short internuclear distances!
Better is:
ψII(+) = 1sa(1) 1sb(2) + 1sa(2) 1sb(1)
ψII(-) = 1sa (1) 1sb(2) - 1sa(2) 1sb(1)
CHEM 2060 Lecture 22: VB Theory L22-3
curves:
a) ψI – first guess (above)
b) ψII(+) – bonding combination
c) using Z* = 1.17
d) hybridize s orbital with 1% p
e) include ionic contributions
f) experimental energy
g) ψII(-) – antibonding combo.
Further improvements can be made.
3 −Zr
'2 a
0
1 $Z
ψ (1s) =
& ) e
π % a0 (
Change Z in atomic hydrogen wavefunction.
There are 2 nuclei in the H2 atom, Z* > Z.
Can lower the curve the most using Z* = 1.17
(i.e., fits experimental curve best with 1.17)
CHEM 2060 Lecture 22: VB Theory L22-4
Better Guess: Invoke Hybridization
Mixing atomic orbitals of the same atom is called HYBRIDIZATION.
This is quite reasonable. After all the orbitals are only wavefunctions.
Example: sp hybrid atomic orbitals are created from one s plus one p orbital
RULE: Mix n AOs ⇒ Get n HAOs
Mix an s orbital and a p orbital ⇒ Get 2 sp orbitals
CHEM 2060 Lecture 22: VB Theory L22-5
Why invoke hybridization for H2 ?
Conceptually, it seems unlikely that the 1s orbital should be able to mix with the
2p orbital because they are very far apart in energy.
This is a “bone of contention” between the VB advocates and MO advocates!
NEVERTHELESS, hybridization gives a lower energy.
We know the real measured energy, so this is clearly nearer the “truth”.
AND the bond is represented more accurately.
• Each H atom ought to distort the electron cloud on the other.
• Since 1s is spherically symmetric this does not represent the bond very well.
• 2pz orbital is not spherically symmetric, so we can add in a bit of this to
make the bond look better (more overlap, better bonding).
CHEM 2060 Lecture 22: VB Theory L22-6
Wavefunctions now are:
φa = N(1sa + γza)
φb = N(1sb + γzb)
N = Normalization constant
to make sure
∫ φ2 = 1
γ is the amount of 2pz orbital
that is mixed with the 1s
orbital
€ on each atom
• Best fit is for only 1%
contribution from 2pz.
• But 1% p-character
improves wavefunction
stability by 5%
CHEM 2060 Lecture 22: VB Theory L22-7
Ionic contribution
Hybridization has improved the wavefunction stability, but the wavefunction
from hybridization alone is still not a very good curve.
It is not that close to the experimental value.
Further improvements can be obtained by adding terms to the wavefunction
where both electrons are on one atom (i.e., charge separation)
Think of the Lewis dot resonance contributors:
We do this by adding a weighting factor λ
i.e.
ψ = φa(1) φb(2) + φa(2) φb(1) + λφa(1) φa(2) + λφb(1) φb(2)
For H2, the best value for λ ≈ 0.25
λ2 ~ 0.06, contribution is about 6%
The resulting improvement in wavefunction stability is only 2-3%.
CHEM 2060 Lecture 22: VB Theory L22-8
VB theory and Quantum Mechanics
In quantum mechanics, we use a trial wavefunction and improve it to better fit
the experimental values (e.g., bond energies, orbital energies, distances, etc.)
Better wavefunction = closer to experimental (minimum energy)
Valence bond theory is one quantum mechanical approach and it uses hybridized
atomic orbitals (hao’s) to improve the wavefunctions such that they are a closer
fit to experimental values.
For the H2 molecule, a 50-term wavefunction reproduces the experimental bond
energy to within 0.0001 of the experimental.
Obviously, this is done computationally. You will see this in CHEM 3860.
You will also learn that some approaches give very good energies and very poor
distances, and vice versa.
This is a huge field of theoretical research, limited only by processor speed.