# Molecular Orbital Theory

Molecular orbitals form from bonding pairs of electrons and exist around the entire molecule instead of just one atom.

Neither VSEPR nor valence bond theory is able to explain all of the stable molecular structures. In addition, some molecules, such as oxygen (O2), have been shown experimentally to be paramagnetic (weakly attracted to an external magnetic field). A third model of covalent bonds can better rationalize these phenomena. Molecular orbital theory is the theory that atomic electron orbitals in covalent bonds are replaced by electron orbitals that belong to the entire molecule. A molecular orbital is a mathematical function that gives the probability of locating an electron in a localized volume of space. Similar to atomic orbitals and hybrid orbitals, molecular orbitals can hold up to two electrons with opposite spins.

Molecular orbitals are formed when atomic orbitals combine. Just like with atomic orbital hybridization, the number of molecular orbitals formed is equal to the total number of atomic orbitals combining to form them. Consider the two 1s atomic orbitals in each of the hydrogen atoms of the diatomic hydrogen (H2) molecule. Molecular orbital theory says that these two 1s orbitals combine to form two new molecular orbitals in the hydrogen molecule. One of these molecular orbitals is at a lower energy state than the 1s orbitals. This new molecular orbital is denoted by $\sigma_{1s}$. A molecular orbital that has a high electron density between two or more atoms due to overlap of multiple atomic orbitals is called a bonding molecular orbital. A molecular orbital that induces a low electron density between two or more atoms and is called an antibonding molecular orbital. It is at a higher energy state than the atomic orbitals it is formed from. When two 1s orbitals combine, the antibonding molecular orbital is denoted by ${\sigma_{1s}}^*$.
Molecular orbitals formed from p orbitals are more complicated than those formed from s orbitals because there are more of them. When oxygen (O2) forms, for example, the one 2s and three 2p orbitals from each oxygen atom combine to form eight molecular orbitals. The 2s orbitals are spherical and symmetric; they can combine in only one way because no matter what direction they approach each other from, the combination is the same. But because the p orbitals have x, y, and z orientations, they can combine either end to end like a $\sigma$ bond or side by side like a $\pi$ bond. Each combining pair of atomic orbitals produces a bonding and an antibonding molecular orbital. The bonding orbital has a high electron density between the atomic nuclei and exists at a lower energy state. The antibonding orbital has a low electron density between nuclei and exists at a higher energy state. Note that the x, y, and z subscripts are markers to describe the orientation of the p orbitals of an atom during bonding. These subscripts do not represent an inherent difference between the three p orbitals. During bonding, the p orbitals that lie along with the x-axis of a model are called px.

### Bonding and Antibonding Orbital Shapes

Atomic Orbitals Atomic Orbital Shape Bonding Orbital Name Bonding Orbital Shape Antibonding Orbital Name Antibonding Orbital Shape
$2s+2s$
$\sigma_{2s}$
${\sigma_{2s}}^*$
$2p_x+2p_x$
$\sigma_{2p}$
${\sigma_{2p}}^*$
$2p_y+2p_y$
$\pi_{2p}$
${\pi_{2p}}^*$
$2p_z+2p_z$
$\pi_{2p}$
${\pi_{2p}}^*$

Two atomic orbitals combine to form each molecular orbital. The bonding molecular orbitals are at a lower energy state than the antibonding orbitals, and the antibonding orbitals have nodes where there are no electrons, represented by a dotted line. The 2p orbitals can combine in any of the three Cartesian coordinate planes, which results in three sets of bonding and antibonding molecular orbitals.

It is important to keep track of the energy levels of the molecular orbitals, because just like Aufbau filling of atomic orbitals, electrons fill the lower-energy molecular orbitals before entering higher-energy ones. And just as with atomic orbitals, they fill all available orbitals at identical energy states singly before they pair up (Hund's rule). Only when orbitals at a low energy state are full of electron pairs do electrons begin to fill higher-energy molecular orbitals. With that said, remember that antibonding orbitals are at a higher energy state than the bonding orbitals and that $\sigma$ bonds are usually at a lower energy state than $\pi$ bonds because a greater fraction of the p orbitals can overlap with each other in end-to-end bonding compared to side-by-side bonding.