There are two main approaches to the description of bonding in molecules, valence bond (VB) theory and molecular orbital (MO) theory. Though molecular orbital orbital theory is considered a more satisfactory explanation and has largely superseded valence bond theory, it is still worth briefly considering VB theory, as it provides a reasonable explanation of the structure of many molecules. From there, we can develop MO theory, see how it reduces to the same results as VB theory for simple molecules, and how it explains those cases that VB theory cannot.

Before we begin, it is appropriate to mention an approximation that is made in both theories to allow solution of the Schrodinger equation. The Schrodinger equation for any molecule (or indeed any chemical bond) contains at least three bodies (two atomic nuclei and one electron), and a three body problem cannot be solved analytically (though it may be possible to use an iterative procedure to obtain a solution for a specific set of conditions). We thus make the Born-Oppenheimer approximation, which is based upon the fact that the nuclei are much more massive than an electron and move far more slowly - it is thus reasonable to consider the nuclei as fixed, stationary objects about which the electron(s) move. This simplifies the problem considerably.

Valence Bond theory is most easily explained by way of an example, and the simplest example to choose is the H₂ molecule, as it contains only two electrons and two nuclei.
The wavefunction for an electron on each of two widely separated hydrogen atoms is written as:

This notation indicates that the overall wavefunction for the situation is the product of two other wavefunctions. The first is the wavefunction for an electron in a hydrogen 1s orbital on atom A. This is a function solely of r₁, the coordinates of electron 1. The second wavefunction is that for an electron in a hydrogen 1s orbital on atom B, and is a function solely of r₂, the coordinates of the second electron. To simplify the notation, we shall represent the above overall wavefunction as follows:

which indicates the atom that each electron is on.
However, when the two atoms are close to each other, it is not possible to know which of the two electrons is on which atom (since the electrons are indistinguishable from one another), and so an equally valid wavefunction for the system would be the following:

Now, in quantum mechanics, when there are two equally acceptable wavefunctions for a system, the true wavefunction is a superposition of the two possibilities, so the best wavefunction for the system when the atoms are close is:

Note either the addition or the subtraction gives an acceptable wavefunction for the system, so there are two different states that the system may be in. It can be shown that the additive combination lies at a lower energy than the subtractive.

The additive wavefunction thus corresponds to two hydrogen atoms close to each other, lying at lower energy than two hydrogen atoms separated in space. i.e. two hydrogen atoms bonded together:

We can represent this wavefunction pictorially as follows. In these diagrams, the nuclei of the two atoms A and B are marked with the appropriate letter. The horizontal axis represents distance, and the green and blue lines show how the probability density for electrons 1 and 2 respectively changes with distance from the nuclei:

The black line for the final picture represents the total electron density associated with both electrons for the bonding wavefunction - note the increased electron density between the two nuclei relative to the two other wavefunctions. This arises from the superposition of the two wavefunctions on the left of the diagram, which interfere with each other constructively to give rise to the wavefunction on the right of the diagram..

The formation of a bond is considered to be due to the enhanced probability of finding both electrons between the two nuclei, where they can bind the two nuclei together by Coulombic attraction.

(Compare this to the elementary idea of a covalent bond, in which two electrons pair up between two nuclei, bonding occurring because the positive nuclei are attracted to the negative electrons, and it is obvious that the two are almost exactly the same. The main difference is the inclusion of the wavefunctions for the system to provide some justification for the model.)

The electron distribution described by the wavefunction on the right of the above diagram is called a σ bond. A σ bond has cylindrical symmetry around the internuclear axis. Classically, it is thought of as a pair of electrons located between the two nuclei. (Note that in the valence bond theory, it is still possible for the electrons to be found outside the two nuclei, a situation which is not considered in the elementary picture.)

Thus far, we have not referred to the spin of the electrons at any point, though a covalent bond is normally thought of as forming when two atomic orbitals overlap and the spins of the electrons in the orbitals pair up. The bonding wavefunction written out above is concerned purely with the spatial distribution of the electrons and is thus incomplete, as to describe the system completely it should include a part that is dependent upon the spins of the electrons. This has been omitted for the sake of simplicity. 

In a more detailed treatment of the quantum mechanics of this situation, it may be shown that the spins of the two electrons must be opposite to form the bonding wavefunction. Though there is no intrinsically favourable interaction between electrons of opposite spin, they must have opposite spins if they are to form a low energy wavefunction that corresponds to bonding.