# 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

Example

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