In our discussion of valence bond theory, we saw
that chemical bonds are formed when two nuclei share a pair of
electrons between them. The sharing lowers the potential energy of
both electrons by exposing them to increased nuclear attractions. The
decrease in potential energy is greatest when the two electrons are
confined to a region between the two nuclei. This type of bond is
described as a __localized__ bond. For example, in methane,
CH_{4}, each pair of electrons is considered to be confined
to the region between the carbon nucleus and a hydrogen nucleus. One
of the limitations of valence bond theory is that it assumes all
bonds are localized bonds. As we will see, this is not a valid
assumption.

Another limitation of valence bond theory is that is sometimes makes incorrect predictions. The case of dioxygen provides a good example. Consider the two Lewis structures for dioxygen shown in Figure 1.

Whenever a theory makes incorrect predictions or is shown to be inconsistent with experimental fact, chemists have two choices:

- Modify the theory to accomodate the experimental facts.
- Create an alternative theory.

We will see how chemists have modified valence bond theory to deal the limitations of localized bonds when we discuss resonance theory. But first we will consider molecular orbital (MO) theory as an alternative to valence bond theory.

Chemists view molecules as combinations of atoms. They consider molecular orbitals as combinations of atomic orbitals, specifically as linear combinations of atomic orbitals. Don't let this term put you off. It simply means that molecular orbitals are formed by adding and subtracting atomic orbitals. Remember, electrons behave like waves, and their wavelike behavior may be described by mathematical functions similar to the sine or cosine function. These mathematical functions are what we call orbitals. The addition of atomic wave functions is analogous to constructive interference that occurs with sound waves. The subtraction corresponds to destructive interference.

Consider the simplest molecule, dihydrogen. In our discussion of
Lewis structures, we imagined a process in
which two hydrogen atoms came together to form a molecule of
dihydrogen. That process represents a linear combination of the two
hydrogen atoms' 1s atomic wave functions. Mathematically molecular
orbital theorists describe the process by the equation y =
1s_{A} + 1s_{B}, where y (psi) stands for the molecular orbital, while
1s_{A} and 1s_{B} represent the
atomic orbitals for H_{A} and H_{B}, respectively.
y is the
molecular orbital equivalent of a s bond in
valence bond theory.

A fundamental rule of molecular orbital theory is that the number
of molecular orbitals must be equal to the number of atomic orbitals.
For two hydrogen atoms, there are two atomic orbitals, which means
that there must be two molecular orbitals for dihydrogen. The second
molecular orbital is described by the equation y^{*} =
1s_{A} - 1s_{B}. There is no valence bond equivalent
of y^{*}.

Figure 2 illustrates the energy changes that accompany these linear combinations of atomic orbitals. The molecular orbital y corresponds to the minimum of the potential energy diagram we considered during our introductory discussion of valence bond theory.

Figure 3 offers an alternative description of the information shown in Figure 2. The colored spheres and elipses represent regions of electron density about the nuclei, which are shown as dots at the centers of the two 1s atomic orbitals.

There are several features of Figure 3 that deserve comment:

- The electron distribution about the separated hydrogen atoms is spherically symmetrical. In contrast, the electron distribution in the MO y is not. Rather, the electron density is concentrated in the inter-nuclear region where both electrons can experience the Coulombic attraction of both nuclei. (This is the result of constructive interference.)
- In the ground state there are no electrons in y
^{*}. However, if an electron is excited from y to y^{*}, the probability of the resulting electron density being in the inter-nuclear region of y^{* }is very low. (This is the result of destructive interference.) - As we've seen
previously, the
inter-nuclear distance in y is less
than the sum of the atomic radii of the two hydrogen atoms. The
inter-nuclear distance in y
^{*}is greater than the sum of the atomic radii of the two hydrogen atoms.

Before we turn our attention to the MO diagram of dioxygen, there
is one additional aspect of Figures 1 and 2 that you should know. The
MOs y and y^{*}are called the Highest Occupied Molecular Orbital (HOMO) and
the Lowest Unoccupied Molecular Orbital
(LUMO), respectively. These terms will be useful when we discuss
chemical reactivity and spectroscopy. Chemical reactions involve the
transfer of electron density from the HOMO of one reactant to the
LUMO of another. Spectroscopy is the interaction of light with
matter, an interaction which alters the populations of different
energy states.

Our discussion of the MO description of dihydrogen was a necessary interlude so that we could understand how MO theory accounts for the paramagnetic behavior of dioxygen. Figure 4 shows the MO diagram for dioxygen. Only the valence shell atomic orbitals are shown.

Don't worry about the details of this MO diagram. The important feature of the figure is that there are two HOMOs that have the same energy. Each one contains a single electron. According to Hund' Rule the energy of the system will be lower if the spins of these two electrons are unpaired than if they are paired. In other words, the most stable form of dioxygen should be paramagnetic, not diagmagnetic.