Orbital hybridisation

In chemistry, hybridisation is the concept of mixing atomic orbitals to form new hybrid orbitals suitable for the qualitative description of atomic bonding properties. Hybridised orbitals are very useful in the explanation of the shape of molecular orbitals for molecules. It is an integral part of valence bond theory. Although sometimes taught together with the valence shell electron-pair repulsion theory, valence bond and hybridisation are in fact not related to the VSEPR model. 

History
Chemist Linus Pauling first used the hybridization theory to explain the structure of molecules such as methane. This concept was developed for such simple chemical systems. But the approach was later applied more widely. Today, chemists use it to explain the structures of organic compounds.

Orbitals represent how electrons behave within molecules. In the case of simple hybridization, this approximation is based on atomic orbitals. Chemists use the atomic orbitals of the hydrogen atom, which the only atom for which an exact analytic solution to its Schrödinger equation is known. In heavier atoms, like carbon, nitrogen, and oxygen, the atomic orbitals involved in bonding are the 2s and 2p orbitals. These orbitals can be occupied in a hydrogen atom, but only when the electron is in an excited state. Hybridized orbitals are assumed to be mixtures of these atomic orbitals, superimposed on each other in various proportions. The theory of hybridization is most applicable under these assumptions. It gives a simple orbital picture equivalent to Lewis structures. Hybridization is not required to describe molecules, but makes the description much easier.

Chemists use hybridization theory mainly in organic chemistry. Its explanation starts with the way the atoms in methane bond together. 

Hybridization of Atomic Orbitals and the Shape of Molecules
If the four hydrogen atoms in a methane molecule were bound to the three 2p orbitals and the 2s orbital of the carbon atom, the H-C-H bond angles would be 90o for 3 of the hydrogen atoms and the 4th hydrogen atom would be at 135o from the others. Experimental evidence has shown that the bond angles in methane are not arranged that way but are 109.5o giving the overall shape of a tetrahedron. The tetrahedral structure makes much more sense in that hydrogen atoms would naturally repel each other due to their negative electron clouds and form this shape. If you think electron-electron repulsion isn't significant, try walking through a wall! There is plenty of space for your nuclei to pass through the nuclei of the wall material but ouch, it just doesn't work that way.

Experimental evidence has also shown that the H-N-H bond angles in ammonia are 107o and the H-O-H bond angles in water are 105o. It is clear from these bond angles that the non-bonding pairs of electrons occupy a reasonable amount of space and are pushing the hydrogen atoms closer together compared to the angles found in methane.

The valence shell electron-pair repulsion model was devised to account for these molecular shapes. In this model, atoms and pairs of electrons will be arranged to minimize the repulsion of these atoms and pairs of electrons. Since the non-bonded electron pairs are held somewhat closer to the nucleus than the attached hydrogen atoms, they tend to crowd the hydrogen atoms. Thus ammonia exists as a distorted tetrahedron rather than a trigonal plane and water also exists as a distorted tetrahedron rather than a linear molecule with the hydrogen atoms at a 180o bond angle.

This concept proposes that since the attached groups are not at the angles of the p orbitals and their atomic orbitals would not have maximum overlap the s and p orbitals will be hybridized to match the bond angles of the attached groups.

The number of these new hybrid orbitals must be equal to the numbers of atoms and non-bonded electron pairs surrounding the central atom!

In the case of methane, the three 2p orbitals of the carbon atom are combined with its 2s orbital to form four new orbitals called "sp3" hybrid orbitals. The name is simply a tally of all the orbitals that were blended together to form these new hybrid orbitals. Four hybrid orbitals were required since there are four atoms attached to the central carbon atom. These new orbitals will have an energy slightly above the 2s orbital and below the 2p orbitals.

In the case of ammonia, the three 2p orbitals of the nitrogen atom are combined with the 2s orbital to form four sp3 hybrid orbitals. The non-bonded electron pair will occupy a hybrid orbital. Again we need a hybrid orbital for each atom and pair of non-bonding electrons. Ammonia has three hydrogen atoms and one non-bonded pair of electrons when we draw the electron-dot formula. In order to determine the hybridization of an atom, you must first draw the electron-dot formula.

In the case of water, the three 2p orbitals of the oxygen atom are combined with the 2s orbital to form four sp3 hybrid orbitals. The two non-bonded electron pairs will occupy hybrid orbitals. Again we need a hybrid orbital for each atom and each pair of non-bonding electrons. Water has two hydrogen atoms and two non-bonded pairs of electrons when we draw the electron-dot formula.

In the boron trifluoride molecule, only three groups are arranged around the central boron atom. In this case, the 2s orbital is combined with only two of the 2p orbitals forming three hybrid orbitals called sp2 hybrid orbitals. The other p-orbital remains unhybridized and is at right angles to the trigonal planar arrangement of the hybrid orbitals. The trigonal planar arrangement has bond angles of 120o. 

Hybrid Wave Equations
Hybrid orbitals are mathematically defined by linearly combining the wave equations of the orbitals involved. In sp, we combine the s orbital wave equation with one of the p orbital's. The sp2 and sp3 are simply the other two p orbitals mixed in. Intuitively, we can see that the hybrid orbitals are degenerate.