# Molecular Orbitals

## Main Question or Discussion Point

I have a few questions on molecular orbital theory which I hope you guys can help me settle!

So I understand orthogonality meaning that the molecular orbitals have zero overlap, due to the Pauli Exclusion Principle.

How do a 2s and 3s molecular orbital achieve orthogonality? Is it due to a node? Does the 3s electron density penetrate the inner 2s at all?

And how do 2p and 2s orbitals achieve orthogonality?

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So I understand orthogonality meaning that the molecular orbitals have zero overlap, due to the Pauli Exclusion Principle.
It doesn't mean zero overlap, but rather that for two orbitals, $$\psi_1, \psi_2$$ the integral $$\int_{-\infty}^{\infty}\psi_1^\ast\psi_2 dx = 0$$
So they can overlap as much as they want, as long as the overall integral becomes zero.

This isn't due to the exclusion principle, but due to the fact that the orbitals are eigenfunctions of the Hamiltonian and form an orthonormal basis of a Hilbert space.

How do a 2s and 3s molecular orbital achieve orthogonality? Is it due to a node? Does the 3s electron density penetrate the inner 2s at all?
2s and 3s are atomic orbitals. But just look at the hydrogen case (just to simplify, I'll take 1s and 2s):
$$\psi_{1s} = e^{-r}\quad\psi_{2s}=(1-\frac{r}{2})e^{-r/2}$$

Obviously the 2s orbital has a node, it must change sign at r=2 given the (1-r/2). Integrate $$\int_0^{\infty}r^2\psi_1\psi_2 dr$$ and see what you get.
(the r^2 comes in because you're integrating the radial wave function spherically)

And how do 2p and 2s orbitals achieve orthogonality?
The same way.