MO Diagram "greatest stabilisation when orbitals are close in energy" I was studying MO diagrams in Chemistry and we were given a diagram explaining this:



I understood the diagram furthest to the left describing the situation where the Electrophile (E) and Nucleophile (Nu) result in the largest energy gain when both are equal in energy. 

However, the lecturer later said that when there is a large difference in energy between E and Nu (far right), the energy gain is negligible. He didn't go further to explaining why this was true. 

I tried looking on the internet for an answer but was led to s-p mixing and content I am not familiar with nor find relevant to the question. 

I'm confused here. What explains this phenomena? Or, is it just an effect observed without explanation - like a Postulate?
Fundamentally it's mathematics. The energies of the MOs are the eigenvalues of the matrix $$\begin{pmatrix}\alpha_1 &amp; \beta \\ \beta &amp; \alpha_2 \end{pmatrix},$$ where $(\alpha_1,\alpha_2)$ are the energies of the two constituent orbitals and $\beta$ is (loosely speaking) the overlap between them.

For the same size of $\beta$, if $\alpha_1 \approx \alpha_2$, then the eigenvalues can differ quite a lot from $(\alpha_1,\alpha_2)$. On the other hand, if $\alpha_1$ is very different from $\alpha_2$, then the eigenvalues will be closer to $(\alpha_1,\alpha_2)$.

As an illustration consider the matrix

$$\begin{pmatrix}1.1 &amp; 0.5 \\ 0.5 &amp; 1 \end{pmatrix}$$

This has eigenvalues of $1.55$ and $0.55$, which are relatively distant from the "original energies" of $1.1$ and $1$. On the other hand, the matrix

$$\begin{pmatrix}3 &amp; 0.5 \\ 0.5 &amp; 1 \end{pmatrix}$$

has eigenvalues $3.12$ and $0.88$, which are closer to $3$ and $1$.

Taking a more abstract perspective, the eigenvalues of $$\begin{pmatrix}\alpha_1 &amp; \beta \\ \beta &amp; \alpha_2 \end{pmatrix}$$ are

$$\lambda_\pm = \frac{\alpha_1 + \alpha_2}{2} \pm \frac{\sqrt{(\alpha_1 - \alpha_2)^2 + 4\beta^2}}{2}$$

and you can play around with this expression to gain some insight. For example, in the limit where  $(\alpha_1 - \alpha_2)^2 \gg 4\beta^2$ (corresponding to a large energy difference between the original interacting orbitals), the eigenvalues reduce to

$$\lambda_\pm \to \frac{\alpha_1 + \alpha_2}{2} \pm \frac{\alpha_1 - \alpha_2}{2}$$

which are simply $\alpha_1$ and $\alpha_2$.