Absolutely Small - Michael D. Fayer [75]
It is possible to solve the Schrödinger Equation for this problem. It can’t be done exactly, but it can be done very accurately. What does the solution of the Schrödinger Equation give you? It gives the energies of the system, and it gives the wavefunctions. When we solved the particle in a box problem, we obtained the wavefunction for a single particle in a hypothetical one-dimensional box with infinite walls. When we solve the Schrödinger Equation for the hydrogen atom or other atoms, we get the energy levels and the atomic wavefunctions, the atomic orbitals. When we solve a molecular problem, we get the quantized energies of the molecular energy levels and the molecular wavefunctions. The molecular wavefunctions are usually called molecular orbitals. So for atoms we get atomic orbitals that describe the electron’s probability distribution about the atomic nucleus. This is a probability amplitude wave. The molecular orbital describes the probability distribution for the electrons in the molecule relative to the atomic nuclei of the atoms that make up the molecule. For the hydrogen molecule there are two electrons and two atomic nuclei, the two protons.
THE BORN-OPPENHEIMER APPROXIMATION
A very useful way to understand bonding of hydrogen atoms as they come together to form the hydrogen molecule uses a concept called the Born-Oppenheimer Approximation. As discussed in Chapter 5, Born won the Nobel Prize in 1954 for the Born probability interpretation of the wavefunction. Oppenheimer made major contributions to the field of physics. He is probably best known as the physicist who led the Manhattan Project during World War II that developed and tested the first atomic bomb. The Born-Oppenheimer Approximation works by placing the two hydrogen atom nuclei (the two protons) a fixed distance apart. Start with a distance that you suspect to be so far apart that the hydrogen atoms don’t feel each other. A quantum mechanical calculation of the energy is performed. If the atoms are far apart, then the energy will be just twice the energy of the 1s state of the hydrogen atom because you have two hydrogen atoms. Then you make the distance a little closer and do the calculation again. Then you make the distance even closer and do the calculation yet again. When the distance between the nuclei in the calculation becomes small enough, the atoms feel each other. If a chemical bond is going to form, that is, if the two hydrogen atoms are going to combine into a hydrogen molecule, then the energy must decrease. To form a bond, the energy of the molecule must be lower than the energy of the atoms when they are separated far apart.
Figure 12.1 is a plot of the energy of two hydrogen atoms as they are brought closer and closer together. As mentioned, when the two H atoms are very far apart, they don’t interact with each other. Each is just a hydrogen atom, and each has the energy of the hydrogen 1s atomic orbital. We are going to take this as the zero of energy. The hydrogen atom itself has a negative energy as described in Chapter 10. That energy represents the binding of the electron to the proton (nucleus). Here we are interested in the change in energy when the two H atoms interact. We want to understand the energy associated with chemical bond, so the zero of energy is the energy when there is no chemical bond. In Figure 12.1, the zero of energy is indicated in the diagram by the dashed line. This is the energy when the atoms are completely separated. The horizontal axis is the separation, r, of the two H atoms. As the H atoms are brought together the energy begins to decrease, then decreases more rapidly. The energy reaches a minimum at a separation, r0 (see Figure 12.1). As the atoms are brought even closer together the energy increases very rapidly, that is, when the atoms are too close, they repel each other. Because the energy goes down when the atoms are brought together, a chemical bond is formed between the two atoms.
FIGURE 12.1. A plot of the energy of two hydrogen atoms as they are brought close