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Effective Hamiltonians

What are the Resonance and Overlap Integrals?

Lots of algebra but the gist is:

From our determinant, we have a way to think about what the values mean

\[ \begin{vmatrix} H_{11}-ES_{11} & H_{12}-ES_{12} & \cdots & H_{1N}-ES_{1N} \\ H_{21}-ES_{21} & H_{22}-ES_{22} & \cdots & H_{2N}-ES_{2N} \\ \vdots & \vdots & \ddots & \vdots\\ H_{N1}-ES_{N1} & H_{N2}-ES_{N2} & \cdots & H_{NN}-ES_{NN} \end{vmatrix}=0 \]

  • The overlap integrals are normalised values (\(-1\) to \(1\)) that measure the nearness and phase relationships between the orbitals
  • The resonance integrals on the diagonal give us the energy of each of the MO
  • The resonance integrals on the off-diagonal allow for a mixing of orbitals that improve the energy of one orbital at the expense of another


For a 2 basis function system, with zero overlap integrals, our determinant looks like this:

\[ \begin{vmatrix} H_{11}-E& H_{12}\\ H_{21} & H_{22}-E\\ \end{vmatrix}=0 \]

When we solve this we get:

\[ E=\frac{(H_{11}+H_{22})\pm\sqrt{(H_{11}-H_{22})^2+4H_{12}^2}}{2} \]

Which we can think of as:

  • \(H_{11}/H_{22}=\) The initial MO energy
  • \(\sqrt{(H_{11}-H_{22})^2}=\) The difference between the two MOs
  • \(\frac{(H_{11}+H_{22})}{2}=\) The middle point between the two MOs
  • \(4H_{12}^2=\) A corrective term that allows for the basis functions to mix and provide resonance stabilisation/destabilisaiton