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By Root 332 0

The lower portion of the figure shows what happens when the two atoms are brought together so they are separated by the bond length, r0 (see Figure 12.1). Wavefunctions have signs. In this case, both signs are positive. The probability amplitude waves add together to form a molecular orbital. We discussed waves in some detail in Chapters 3 and 5. In Chapter 3, we saw that waves could be combined to have constructive or destructive interference. In Chapter 5, the interference of photons was explained in terms of the Born interpretation of the wavefunction as a probability amplitude wave. Here, the two atomic orbital electron probability amplitude waves combine constructively to form a molecular orbital. The molecular orbital is a probability amplitude wave. The absolute value squared of the wave gives the probability of finding the electrons in some region of space.

FIGURE 12.2. The upper portion is a schematic of two hydrogen atom 1s orbitals. These are actually delocalized electron probability amplitude waves, represented here simply as circles. The lower proportion shows what happens when the two atoms are brought together to form the H2molecule. The two atomic orbitals combine to make a molecular orbital.

Figure 12.3 shows a one-dimensional plot of the probability distribution (square of the wavefunction) of the two atomic orbitals 1sa and 1sb, and the square of the sum of the atomic orbitals. The atomic orbitals are centered on the H atom nuclei, which are separated by the bond length, r0. The protons (nuclei) are positively charged and repel each other. However, the molecular orbital concentrates the negatively charged electron density between the nuclei and holds them together. The important feature of the molecular orbital is that the electrons no longer belong to one atom or the other. The molecular orbital describes a delocalized probability distribution for the two electrons. Both electrons are free to roam over the entire molecule. The two electrons, which belonged to different atoms when the atoms were far apart, belong to the entire molecule. They are shared by the atoms, which are no longer independent.

FIGURE 12.3. A one-dimensional plot of the square of the two 1s orbitals that belonged to H atoms a and b (solid curves), and the square of the sum of the atomic orbitals, which is the square of the molecular orbital. The electron density is concentrated between the two nuclei.

BONDING AND ANTIBONDING MOLECULAR ORBITALS

Something very important has been left out of the discussion so far. The Pauli Principle (Chapter 11) states that at most two electrons can be in any orbital. This is true of atomic orbitals or molecular orbitals. To form the hydrogen molecule, we began with two hydrogen atoms. Each hydrogen atom has a 1s orbital. According to the Pauli Principle, it would be possible to put four electrons in these two orbitals. The two hydrogen atoms only have two electrons, but it would not violate the Pauli Principle to add another electron to each of the atomic 1s orbitals. We added together the two 1s orbitals (constructive interference) to form one molecular orbital. The Pauli Principle tells us that we can put at most two electrons in this molecular orbital. We started with two atomic orbitals that could hold four electrons, but now we have one molecular orbital that can only hold two electrons. Something is missing. You never lose or gain orbitals or places for electrons when forming molecules. If you start with two atomic orbitals, then two molecular orbitals will be formed, which can hold four electrons.

In Figures 12.2 and 12.3, the two 1s hydrogen atomic orbitals were added with the same sign to produce a molecular orbital that concentrates the electron density between the two atomic nuclei. The 1s orbitals are probability amplitude waves and can also be added with opposite sign. When the atomic orbitals are added with opposite sign, they destructively interfere. The addition with opposite sign is shown in Figure 12.4. Because the signs of the two atomic orbitals are opposite,