Linear Combination of Atomic Orbitals

Two 1s electrons in H-, hydride ion, react with H+ to give H2. The two electrons are now in a molecular orbital, ψ, like two 1s orbitals, φ, one for each H atom A and B, added together. 
Equ 1

Coefficients c1 and c2 are scaling-down factors, the squares of which define how much of each AO goes to make each MO. Squaring the combination equation: 

Equ 2

Since the square of the wavefunction is proportional to electron density, the first two terms represent scaled-down versions of the electron density around each atom separately, while the last term represents the electron density in the overlap region between the two atoms. If we integrate, 

Equ 3

the last term is the contribution of this MO to bonding between atom A and atom B. If it is positive, some of the electron is attracted by both nuclei at the same time, so the electron is more stable than if the atoms were separate. The MO is then called a bonding orbital. The integral is called the overlap integral between AOs φA and φB, and measures how much the atomic wavefunctions overlap in space.

An AO or a MO each represents one electron, i.e.  and  

The overlap integral is relatively small because most of the electron is around the separate nuclei, so for the purposes of examining the coefficients we could ignore the last term of the integral equation. Then

Equ 4

Once formed, H2 consists of two identical H atoms, so equal amounts of φA and φB must be taken, i.e.

Equ 5

Solving Equs. 4 and 5, each coefficient can be ±1/√2.   Both 1s orbitals are positive in the bonding region, so taking the same sign for both c1 and c2 (e.g. positive) gives a bonding MO.

Suppose we could push together two H- ions until their AOs overlapped. We would need to accomodate 4 electrons, i.e. we need 2 MOs. There is an unbreakable rule that the number of MOs made by LCAO equals the total number of AOs taken for the separate atoms. For 2 H- (or H2) we take two 1s AOs, so we must make 2 MOs. The only other, different, MO has opposite signs for c1 and c2. This MO changes sign at a nodal plane between the two atoms. There is less electron density between the nuclei than if the AOs did not interact, so the electron described by this MO is less stable than in the separate atoms, and the contribution to bonding is negative. This is called an antibonding MO. The energy level is higher than for the bonding MO.