H2 molecule by molecular orbital theory

According to molecular orbital (MO) theory, when two atoms come close, their atomic orbitals combine (Linear Combination of Atomic Orbitals, LCAO) to form molecular orbitals that belong to the whole molecule. The H₂ molecule forms because its two electrons can occupy a bonding orbital of lower energy than the separate 1s orbitals.

Molecular orbitals of H₂

Let $\psi_{1s}(r_A)$ and $\psi_{1s}(r_B)$ be the 1s orbitals of atoms A and B. They combine in two ways:

$$\psi_\sigma = \psi_{1s}(r_A) + \psi_{1s}(r_B) \qquad \text{(bonding)}$$ $$\psi_{\sigma^*} = \psi_{1s}(r_A) - \psi_{1s}(r_B) \qquad \text{(antibonding)}$$

 bonding  psi_s              antibonding psi_s*
      .--.   .--.               .--.
     /    \_/    \             /    \
    /      ^      \     ------/------\------- 0
   A  (overlap)    B          A       \    /B
                                       '--'
 |psi_s|^2 high between A,B   |psi_s*|^2 = 0 at mid-point

• In $\psi_\sigma$ the waves add between the nuclei, so electron density builds up there. This negative charge attracts both protons and holds them together. Energy is lowered.
• In $\psi_{\sigma^*}$ the waves cancel at the mid-point (a node), there is little charge between the nuclei, and the protons repel. Energy is raised.

Energy level diagram

 Energy
   |              sigma* (empty)
   |             /  ____  \
   |  1s  __ ---          --- __  1s
   |  (H_A) ^ \            / ^ (H_B)
   |           \  _^v_  /
   |              sigma (2 electrons,
   |                     opposite spins)

The two electrons (one from each H atom) go into $\sigma$ with opposite spins (Pauli principle). The antibonding orbital stays empty.

Energy versus separation

 E
 ^  \
 |   \  sigma* (repulsive, no minimum)
 |    \___
 |        ''----.....______________
0+-------------------------------------> R
 |    \              ____...----
 |     \        _---'   sigma (bonding)
 |      \    _-'
 |       '--'  <- minimum: R0 = 0.074 nm
 |              binding energy = 4.5 eV

The bonding curve has a minimum at $R_0 = 0.074$ nm. At this separation the H₂ molecule is about 4.5 eV lower in energy than two separate H atoms, so H₂ is stable. Bond order $= \frac{1}{2}(2 - 0) = 1$, a single covalent bond.