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1. COMPARISON OF VBT AND MOT
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2. Let us compare the molecular-orbital and valence-bond
treatments of the H2 ground state.
𝝋ₐ
1s
𝝋b
1s
Consider 𝝋a symbolize atomic orbital of nucleus a and 𝝋b atomic
orbital of nucleus b
Then the MO wavefunction will be,
𝝋(MO)=[𝝋ₐ(1)+ 𝝋b (1)] [𝝋b(2)+ 𝝋ₐ (2)]
𝝋(MO) = 𝝋ₐ (𝟏) 𝝋ₐ(𝟐) + 𝝋b (𝟏) 𝝋b(𝟐) + 𝝋b (𝟏) 𝝋ₐ(𝟐)
+𝝋ₐ (𝟏) 𝝋b(𝟐)
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3.  Last two terms represent equal sharing of electrons. So Covalent
𝝋b (𝟏) 𝝋ₐ(𝟐) + 𝝋ₐ (𝟏) 𝝋b(𝟐)
H2  H+ H (Covalent)
 First two terms represent both the electrons present in a single AO. So
Ionic .
𝝋ₐ (𝟏) 𝝋ₐ(𝟐) + 𝝋b (𝟏) 𝝋b(𝟐)
H2  Ha
+ + Hb
- (Ionic)
 Molecular wave-function gives a combination of ionic and covalent
contributions.
 50% ionic and 50% covalent.
Actually, the H2 ground state dissociates to two neutral H atoms. Thus
the MO function gives the wrong limiting value of the energy.
So in order to correct this omit the ionic terms,
𝝋b(𝟏) 𝝋ₐ(𝟐) +𝝋ₐ (𝟏) 𝝋b (𝟐)
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4. In case of VB theory, the wavefunction will be,
𝝋b (𝟏) 𝝋ₐ(𝟐) +𝝋ₐ (𝟏) 𝝋b(𝟐)
 This equation tells that if one electron of H2 in orbital
A and the other must be in orbital B.
 No chance of both the electron being in the same
orbital.
 Probability of finding both electrons near the same
nucleus.
So the improved equation will be,
𝝋(VBimp) = 𝝋ₐ (𝟏) 𝝋b(𝟐) + 𝝋b (𝟏) 𝝋ₐ (𝟐) +
δ [𝝋ₐ (𝟏) 𝝋ₐ(𝟐) + 𝝋b (𝟏) 𝝋b (𝟐)]
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5.  When δ equal to zero - VB function
When δ equal to 1 - MO function
 The Exp value of δ is closer to zero than to 1, and also VB
function gives a better dissociation energy than the MO function.
Dissociation
Energy
Bond Length
MO 2.65 eV 0.83 Ao
VB 3.16 eV 0.80 Ao
Imp VB 4.00 eV 0.77 Ao
Exp 4.72 eV 0.74 Ao
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6. Compare the improved valence-bond trial function with improved
MO function.
We can improve the MO function by configuration interaction.
𝝋(MOimp) = [𝝋ₐ (𝟏) + 𝝋b(1)] [𝝋b (2) + 𝝋ₐ (𝟐)] +
γ [𝝋ₐ (𝟏) - 𝝋b (1)] [𝝋ₐ (2) - 𝝋b (𝟐)]
Multiplying by
If we take δ = , then Imp VB function is equal to Imp MO function
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7.  MO function says that structures with both electrons on the same
atom are just as likely as structures with each electron on a different
atom.
 VB function says that it has no contribution from structures with
both electrons on the same atom.
 In MO theory, wavefunction can be improved by configuration
interaction.
𝝋(MOimp) = [𝝋ₐ (𝟏) + 𝝋b(1)] [𝝋b (2) + 𝝋ₐ (𝟐)] +
γ [𝝋ₐ (𝟏) - 𝝋b (1)] [𝝋ₐ (2) - 𝝋b (𝟐)]
 In VB theory, electron correlation is improved by ionic–covalent
resonance.
𝝋(VBimp) = 𝝋ₐ (𝟏) 𝝋b(𝟐) + 𝝋b (𝟏) 𝝋ₐ (𝟐) +
δ [𝝋ₐ (𝟏) 𝝋ₐ(𝟐) + 𝝋b (𝟏) 𝝋b (𝟐)]
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8. Comparing MO and VB based on Hamiltonian operation,
For MO method,
Here the unperturbed Hamiltonian consists of the bracketed terms. In MO
theory the unperturbed Hamiltonian for H2 is the sum of two H2
+
Hamiltonians, one for each electron.
Accordingly, the zeroth-order MO wave function is a product of two H2
+
like wave functions, one for each electron.
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9. For VB method,
The unperturbed system is two hydrogen atoms. We have two
zeroth-order functions consisting of products of hydrogen-atom
wave functions.
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10.  The MO method is used far more often than the VB method, because
it is computationally much simpler than the VB method.
 The MO method was developed by Hund, Mulliken, and Lennard-
Jones in the late 1920s.
Originally, it was used largely for qualitative descriptions of
molecules, but the electronic digital computer has made possible the
calculation of accurate MO functions.
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11. Reference:
 Quantum Chemistry - Ira N. Levine.
 Quantum Chemistry – Donald A. McQuarrie
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