Explanation on Atomic Orbitals and Molecular Orbitals

We were reading about atomic structures and bond making and my teacher told me that when two atoms are fused or when they make bond, There are two orbitals formed. 1-Bonding Molecular Orbital & 2- Anti-Bonding Molecular Orbital. I can't figure it out why these two orbitals are created? I searched on internet and now stuck at the point that According to Pauli Exclusion Principal, No two electrons have same quantum properties in the orbital that's why two orbitals are formed, I thought two orbitals for each electron but I came to know that Anti-Bonding Molecular Orbital is completely empty having high energy, Means there is no electron or anything in it and now I am confused in it. Can anybody explain me this so I can explain to my other class mates.

~Thanks.

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In the theory of molecular orbitals (MO) you consider the movement of a single electron in the Coulomb field produced by the "naked" atomic nuclei of your molecule ("moiety"). In this sense the MO theory is a one-electron model and as such, it has nothing to do with the Pauli Principle. In the first step (i) you set up the possible MOs for your molecular moiety. You end up with a set of MOs ordered after their energy. In the second step (ii) you fill up these MOs with 2 elos each, in order of increasing energy, until you run out of elos. It is only in (ii) that the Pauli Principle is applied. –  Lupercus Dec 16 '12 at 22:26

Your teacher is referring to the LCAO approximation as a way of calculating molecular orbitals.

Suppose you bring two hydrogen atoms together i.e. create a hydrogen molecule. To calculate the electronic structure you need to solve the Schrodinger equation, but even for something as simple as the hydrogen molecule the Schrodinger equation is too complex to solve analytically. To make progress we need to use some approximate method.

When the hydrogen atoms are a long distance apart we know the electronic structure is simple the wavefunction of a hydrogen atom, $\psi_H$. So in the hydrogen molecule it's a reasonable guess that the H$_2$ wavefunction might look a bit like a combination of the two atomic wavefunctions. We could add or subtract the atomic orbitals to give:

$$\Psi_{H_2}^+ \approx \frac{1}{\sqrt{2}} \left( \psi_{H_a} + \psi_{H_b} \right)$$

or:

$$\Psi_{H_2}^- \approx \frac{1}{\sqrt{2}} \left( \psi_{H_a} - \psi_{H_b} \right)$$

If you have a look at the molecular orbital diagram for hydrogen:

the wavefunction $\Psi_{H_2}^+$ is lower in energy than the atomic wavefunction because it increases the electron density in between the protons where they are attracted to both protons. By contrast $\Psi_{H_2}^-$ is higher in energy because it reduces the electron density in between the protons. Hence it's said that the two atomic orbitals split as the hydrogen atoms approach each other to give bonding and anti-bonding molecular orbitals.

But I must emphasise that this is a hand waving approach. In many circumstances it can give you a rough idea of what's going on, but it's a crude model and only useful in simple cases.

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Can you tell me about the Anti-Bonding Molecular Orbital too? My teacher told me that its completely empty when two atoms make a molecule. It doesn't make sense for me, If two electrons in an orbital have different quantum numbers hence they spin in different direction in its own orbital. If the Anti-Bonding Orbital is empty, How is it formed and why? –  devWaleed Dec 17 '12 at 14:17
I am not in a big class so my teacher didn't tell me in this much detail about this, I don't know what is a wave-function of atom. But your answer was good.... –  devWaleed Dec 17 '12 at 14:21
Okay I understand few things now, But can you tell me which of the image for Geometry of CH4 bond is correct: bit.ly/SFrZUY OR bit.ly/V3eux4 –  devWaleed Dec 17 '12 at 16:02
In my example above $\Psi_{H_2}^+$ is the bonding orbital and $\Psi_{H_2}^-$ is the antibonding orbital. Each orbital holds two electrons, so if you join 2 hydrogen atoms, with one electron each, both electrons go into the bonding orbital and the atoms form an H$_2$ molecule. If you try to join two helium atoms you have a total of 4 electrons, so 2 go into the bonding orbital and 2 into the anti-bonding orbital. The result is that He atoms won't form an He$_2$ molecule. –  John Rennie Dec 17 '12 at 17:43
Re your question about methane, I would ask in the Chemistry SE - chemistry.stackexchange.com –  John Rennie Dec 17 '12 at 17:44