Students often find the idea of linear combinations of atomic orbitals mysterious, even unfathomable. Let's consider a more familiar example, the combination of two sine waves as shown in Figure 1. The graph at the top of the left-hand panel shows the superposition of one half of a cycle of each of two sin waves, a and b. The linear combination a + b produces the lower graph in the left-hand panel. Notice how the amplitude of the new wave is greatest in the middle.
The graph at the top of the right-hand panel is the same as that at the top of the left-hand panel except that wave b has been multiplied by -1. The linear combination a + (-b), i.e. a-b, produces the wave at the bottom of the right-hand panel. Notice that this wave has both positive and negative values. The point at which the value of the wave is zero is called a node.
A sine wave is an example of a standing wave; although the amplitude of the wave changes continuously, the positions of the maxima and minima remain constant. Orbitals are 3-dimensional standing waves. The mathematical equations describing the orbitals are complex and we don't need to discuss them here. We are more interested in the pictorial representation of these combinations than in the mathematics that is involved. Figure 2 presents a color-coded convention for adding and subtracting atomic orbitals. The colors represent the phases of the orbitals; blue for positive, red for negative. The left hand panel shows the linear combinations of two s orbitals. When the orbitals are added, their colors remain unchanged. Subtraction causes the phase of one orbital to change from positive to negative.
Figure 3 offers a more pictorial representation of the MO diagram of dioxygen. Only the valence atomic orbitals, 2s and 2px,y,z, are shown. Note that these 8 atomic orbitals combine to form 8 molecular orbitals. The letters A and B are labels for the two oxygen atoms.
Admittedly, Figure 3 can send you into sensory overload in short order. So let's deconstruct it in order understand better how MO theory works. Figure 4 focuses on the combination of the 2s atomic orbital of oxygen A with the 2s atomic orbital of oxygen B. Compare this part of the MO diagram with the MO diagram of dihydrogen. The linear combinations 2sA + 2sB and 2sA - 2sB produce two new s molecular orbitals, ss and ss*, respectively. Since there are two electrons in the 2s orbital of each oxygen, the two MOs are both occupied.
Figure 5 diagrams the formation of two more s orbitals. This time they are generated by the head-to-head overlap of the 2pz orbitals on each oxygen.
Finally, Figure 6 shows the combinations of the 2px orbitals to produce a pi bonding and a pi anti-bonding MO. The overlap of the 2py AOs is left for you as an exercise.
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