Symmetry  Contents -  The Equations for the orbitals

Valence-Bond Theory

The Valence-Bond (Heitler-London) wavefunction for H₂ is

$$[{\rm 1s}_A(1){\rm 1s}_B(2)+{\rm 1s}_B(1){\rm 1s}_A(2)][\alpha(1)\beta(2)-\alpha(2)\beta(1)]$$

$$\equiv\{{\rm 1s}_A{\rm 1s}_B\}$$ (38)

 Compare it to the molecular orbital wavefunction

$$({\rm 1s}_A(1)+{\rm 1s}_B(1))({\rm 1s}_A(2)+{\rm 1s}_B(2))[\alpha(1)\beta(2)-\alpha(2)\beta(1)]$$ (39)

 We see that the VB wavefunction likes to have one electron near A and one electron near B, whereas the MO wavefunction allows the possibility for 2 electrons on A. The VB wavefunction will therefore describe the dissociation of H₂ into H+H, unlike the MO wavefunction. We say that the VB wavefunction has covalent structures and the MO wavefunction has covelent and ionic structures.
We also introduce the concept of hybrid orbitals, particularly relevant for the C atom where the 2s and 2p orbitals lie close in energy. In particular sp³ orbitals, defined as

$$\chi_1=\frac{1}{2}({\rm 2s}+{\rm 2p}_x+{\rm 2p}_y+{\rm 2p}_z)$$ (40)

$$\chi_2=\frac{1}{2}({\rm 2s}+{\rm 2p}_x-{\rm 2p}_y-{\rm 2p}_z)$$ (41)

$$\chi_3=\frac{1}{2}({\rm 2s}-{\rm 2p}_x+{\rm 2p}_y-{\rm 2p}_z)$$ (42)

$$\chi_4=\frac{1}{2}({\rm 2s}-{\rm 2p}_x-{\rm 2p}_y+{\rm 2p}_z)$$ (43)

 Each hybrid orbital points towards one of the corners of the cube.
   
 
Hence one can write down a VB wavefunction for CH₄:

$$\{{\rm 1s}_C^2\}\{\chi_1{\rm 1s}_A\}\{\chi_2{\rm 1s}_B\}\{\chi_3{\rm 1s}_C\}\{\chi_4{\rm 1s}_D\}$$ (44)

 and even for H₂O

$$\{{\rm 1s}_O^2\}\{\chi_1{\rm 1s}_A\}\{\chi_2{\rm 1s}_B\}\{\chi_3^2\}\{\chi_4^2\}$$ (45)

 In this case there are two electrons in each of the sp³ orbitals $\chi_3,\\chi_4$, which constitute the lone pairs, or `rabbit ears'.
VB theory predates MO theory. It gives a more pleasing description because it maintains the atomic nature of the atoms in a molecule. However it is much more difficult to work with, because atomic orbitals on different atoms are not orthogonal, whereas molecular orbitals being eigenfunctions, are orthogonal. The essential difference between VB and MO is that the former is a localised picture (ie atomic) and the latter is a delocalised picture (electrons shared between atoms).

But it is important to realise that although the MO picture gives delocalised orbitals it is possible to construct linear combinations of them (eq $\cos(\theta)\phi_a+\sin(\theta)\phi_b,-\sin(\theta)\phi_a+\cos(\theta)\phi_b$) which do not change the electron density or the wavefunction, and become localised. In H₂CO, instead of having the a₁ and b₂ bonding CH₂ orbitals, doing the above constructs two localised orbitals, one along each CH bond. Of course they no longer carry a symmetry label, but symmetry has been used to construct them. The organic chemistry interpretations use such localised orbitals, and computer programs can construct localised orbitals at the end of calculations..

However, because the MO wavefunction is a determinant, it is possible to add and subtract rows (ie orbitals) to construct functions which are localised. 

  
 Symmetry  Contents -  The Equations for the orbitals

1998-09-23