Linear combinations of atomic orbitals

[Avardia]

So I've been looking at covalent bonds and come across the approx you can do of the molecular orbital for $H^+_2$ by just summing two 1s orbitals, the method is called the linear combinations of atomic orbitals, and you get what is below which I believe is exact in this case since the 1s orbital is all there is. And the orbital is just the wavefunction of electron.

So one is the bonding and the other is anti-bonding orbital. My question is why in the anti-bonding orbital do we have one of the summed 1s wavefunctions as negative?

 

[TeethWhitener]

My question is why in the anti-bonding orbital do we have one of the summed 1s wavefunctions as negative?

You don't have to. You're essentially doing a change of basis from the atomic orbitals to the molecular orbitals, so you can choose any two combinations of hydrogen 1s orbitals, as long as the combinations are linearly independent. It so happens that the fully symmetric and fully antisymmetric linear combinations of atomic orbitals closely approximate the bonding and antibonding molecular orbitals. You could choose a different basis, but you'd have much more complicated looking functions for your molecular orbitals.