This animation shows how
the allowed wave functions for two atoms vary as the atoms, represented
here by finite wells, mover closer together and farther apart. (These wave
functions are obtained by numerical solution of the Schroedinger equation
for the potential energy shown.) Each atom has a fixed width and its energy
well is of depth 80 (in arbitrary units).
When the atoms are far apart, there are two pairs of states/wave functions, one pair at a low energy and one at a higher energy. The pair of lower-energy states differ only in that one, the blue one, is an even function (relative to the point between the atoms), while the red one is an odd function. Their probability densities, obtained by squaring the functions, are virtually identical. The functions essentially join the n = 1 ground-state wave functions of each individual atom, one in an even way, one in an odd way. The same can be said of the higher-energy pair, except that they join the n=2 wave functions for the two essentially isolated atoms. Since the energies of a given pair are virtually identical, an electron could occupy a state that is a linear combination of the two, and a sum or difference would be a state that is nonzero in only one atom or the other. When the atoms are far apart, electrons orbit one atom or the other.
When the atoms draw close, each pair of wave functions splits into a lower-energy even (blue) state and a higher energy odd (red) state. Electrons tend to occupy lowest-energy states, so if there were only one electron, it would occupy the lowest-energy blue state. As the animation shows, this state is lower in energy than either of the pair of states when the atoms were far apart, so the overall state would be lower in energy--a diatomic molecule has formed. Even if there were two electrons, as there would be in a hydrogen molecule, both could occupy this lowest-energy wave function (with opposite spins), resulting, again, in a lower energy than for isolated atoms. [It should be noted that as the atoms draw near, the repulsion the protons share increases, tending to raise the overall energy. The molecule forms at a separation where the decreasing of the lowest-energy wave function occupied by the electrons and the increasing of the protons' repulsive energy form a minimum.]
A final observation about the close-atom, or molecular, states: In the even/blue states of each pair, the probability density tends to be concentrated between the atoms and the energy is lower than for isolated atoms--electrons occupying these states would truly be shared by both of the molecule's atoms. In the odd/red states the probability density tends to form two separated regions and the energy is relatively high. For a given pair, we would refer to the blue function as a "bonding" state, and to the red function as an "antibonding" state.