Consider the following molecular orbitals formed from the combination of two hydrogen \(1 s\) orbitals: a. Which is the bonding molecular orbital and which is the antibonding molecular orbital? Explain how you can tell by looking at their shapes. b. Which of the two molecular orbitals is lower in energy? Why is this true?

Molecular orbitals are the result of the combination of atomic orbitals from different atoms in a molecule. When atomic orbitals combine, they can form either a bonding molecular orbital, which has lower energy than the original atomic orbitals, or an antibonding molecular orbital, which has higher energy than the atomic orbitals. For the hydrogen 1s orbitals, let us consider two orbitals A and B combining:

When two hydrogen 1s orbitals combine in-phase (i.e., the positive phase of one orbital combines with the positive phase of the other, and the negative phase of one orbital combines with the negative phase of the other), it results in constructive interference that creates a bonding molecular orbital. The electron density in the bonding molecular orbital will lie between the two atomic nuclei, providing a stabilizing force that reduces the energy of the system. In this case, the molecular orbital formed is the bonding molecular orbital. - Shape: The bonding orbital appears as if it is formed by two overlapping spheres with electron densities concentrated in the middle region between the two hydrogen atoms.

When two hydrogen 1s orbitals combine out-of-phase (i.e., the positive phase of one orbital combines with the negative phase of the other, and the negative phase of one orbital combines with the positive phase of the other), it results in destructive interference that creates an antibonding molecular orbital. The electron density in the antibonding molecular orbital will be found less between the two atomic nuclei, creating a node or area of low electron density, and destabilizes the system. In this case, the molecular orbital formed is the antibonding molecular orbital. - Shape: The antibonding orbital appears as if it is formed by two non-overlapping spheres with electron densities concentrated on opposites sides, and a node is formed between the two hydrogen atoms.

As discussed earlier, the bonding molecular orbital formed from the in-phase combination of hydrogen 1s orbitals is lower in energy than the original atomic orbitals. This is true because the electron density is concentrated between the atomic nuclei, creating a stabilizing force that reduces the energy of the system. On the other hand, the antibonding molecular orbital formed from the out-of-phase combination of hydrogen 1s orbitals is higher in energy than the original atomic orbitals. This is because there is a node between the atomic nuclei, resulting in less electron density between the nuclei and no stabilizing force. This leads to an increase in the energy of the system. So, the bonding molecular orbital is lower in energy than the antibonding molecular orbital.