Advanced Theoretical Chemistry (Simons)

1.1: Why Quantum Mechanics is Necessary

The field of theoretical chemistry deals with the structures, bonding, reactivity, and physical properties of atoms, molecules, radicals, and ions all of whose sizes range from ca. 1 Å for atoms and small molecules to a few hundred Å for polymers and biological molecules such as DNA and proteins. Sometimes these building blocks combine to form nanoscopic materials (e.g., quantum dots, graphene sheets) whose dimensions span up to thousands of Å, making them amenable to detection using specialized microscopic tools. However, description of the motions and properties of the particles comprising such small systems has been found to not be amenable to treatment using classical mechanics. Their structures, energies, and other properties have only been successfully described within the framework of quantum mechanics. This is why quantum mechanics has to be mastered as part of learning theoretical chemistry.

We know that all molecules are made of atoms that, in turn, contain nuclei and electrons. As I discuss in this Chapter, the equations that govern the motions of electrons and of nuclei are not the familiar Newton equations but a new set of equations called Schrödinger equations.

When scientists first studied the behavior of electrons and nuclei, they tried to interpret their experimental findings in terms of classical Newtonian motions, but such attempts eventually failed. They found that such small light particles behaved in a way that simply is not consistent with the Newton equations. Let me now illustrate some of the experimental data that gave rise to these paradoxes and show you how the scientists of those early times then used these data to suggest new equations that these particles might obey. I want to stress that the Schrödinger equation was not derived but postulated by these scientists. In fact, to date, to the best of my knowledge, no one has been able to derive the Schrödinger equation. From the pioneering work of Bragg on diffraction of x-rays from planes of atoms or ions in crystals, it was known that peaks in the intensity of diffracted x-rays having wavelength l would occur at scattering angles q determined by the famous Bragg equation:

where d is the spacing between neighboring planes of atoms or ions. These quantities are illustrated in Figure 1.1 shown below. There are may such diffraction peaks, each labeled by a different value of the integer ( ). The Bragg formula can be derived by considering when two photons, one scattering from the second plane in the figure and the second scattering from the third plane, will undergo constructive interference. This condition is met when the extra path length covered by the second photon (i.e., the length from points to to ) is an integer multiple of the wavelength of the photons.