Molecular Forces—A More Quantitative Look

It is commonly believed that quantum mechanics, which can be used to predict the detailed shape of electron clouds, is needed for any quantitative understanding of molecular forces. This is only partly true. We can get a fair understanding of molecular forces from Newtonian mechanics, as was demonstrated by the student Bob Piela in a project for an introductory physics course. This section will closely follow the approach presented in Piela’s project. In this section we will discuss only the simplest of all molecules, the hydrogen molecule ion consisting of two protons and one electron. We will use Newtonian mechanics to get a better picture of how the electron holds the molecule together, and to see why the lower, the more negative the energy of the electron, the more tightly the protons are bound together.
If you do a straightforward Newtonian mechanics calculation of the hydrogen molecule ion, letting all three particles move under the influence of the Coulomb forces between them, the system eventually flies apart. As a number of student projects using computer calculations have shown, eventually the electron gets captured by one of the protons and the other proton gets kicked out of the system. With Newtonian mechanics we cannot explain the stability of the hydrogen molecule ion, quantum mechanics is required for that.
Piela avoided the stability problem by assuming that the two protons were fixed at their experimentally known separation of 1.0 7 × 10^{– 8} cm (1.07 angstroms) as shown in Figure (1a), and let the computer calculate the orbit of the electron about the two fixed protons, as seen in Figure (1b). By letting the calculation run for a long time and plotting the position of the electron at equal intervals of time, as in a strobe photograph of the electron’s motion, one obtains the dot pattern seen in Figure (1c). This dot pattern can be thought of as the classical electron cloud pattern for the electron.

Figure 1: Orbit of an electron about two fixed protons.