Section 3: The Quantum Atom
Figure 11: A simple classical model fails to explain the stability of the helium atom.
Source: © William P. Reinhardt, 2010. More info
As we learned in Unit 5, quantum theory replaces Bohr's orbits with standing, or stationary state, wavefunctions. Just as in the Bohr picture, each wavefunction corresponds to the possibility of having the system, such as an electron in an atom, in a specific and definite energy level. As to what would happen when an atom contained more than one electron, Bohr was mute. Astronomers do not need to consider the interactions between the different planets in their orbits around the Sun in setting up a first approximation to the dynamics of the solar system, as their gravitational interactions with each other are far weaker than with the Sun itself. The Sun is simply so massive that to a good first approximation it controls the motions of all the planets. Bohr recognized that the same does not apply to the classical motion of electrons in the next most complex atom: Helium, with a charge of +2 on its nucleus and two moving electrons. The electrons interact almost as strongly with each other as with the nucleus that holds the whole system together. In fact, most two-electron classical orbits are chaotically unstable, causing theoretical difficulties in reconciling the classical and quantum dynamics of helium that physicists have overcome only recently and with great cleverness.
Perhaps surprisingly, the energy levels described by Schrödinger's standing wave picture very quickly allowed the development of an approximate and qualitative understanding of not only helium, but most of the periodic table of chemical elements.
Energy levels for electrons in atoms
What is the origin of the Lewis filled shell and electron pair bonding pictures? It took the full quantum revolution described in Unit 5 to find the explanation—an explanation that not only qualitatively explains the periodic table and the pair bond, but also gives an actual theory that allows us to make quantitative computations and predictions.
When an atom contains more than one electron, it has different energies than the simple hydrogen atom; we must take both the quantum numbers n (from Unit 5) and l (which describes a particle's quantized angular momentum), into account. This is because the electrons do not move independently in a many-electron atom: They notice the presence of one another. They not only affect each other through their electrical repulsion, but also via a surprising and novel property of the electron, its spin, which appeared in Unit 5 as a quantum number with the values 1/2, and which controls the hyperfine energies used in the construction of phenomenally accurate and precise atomic clocks.
The effect of electron spin on the hyperfine energy is tiny, as the magnetic moment of the electron is small. On the other hand, when two spin-1/2 electrons interact, something truly incredible happens if the two electrons try to occupy the same quantum state: In that case, one might say that their interaction becomes infinitely strong, as they simply cannot do it. So, if we like, we can think of the exclusion principle mentioned in the introduction to this unit as an extremely strong interaction between identical fermions.
Figure 12: The ground state of helium: energy levels and electron probability distribution.
We are now ready to try building the periodic table using a simple recipe: To get the lowest energy, or ground state of an atom, place the electrons needed to make the appropriate atomic or ionic system in their lowest possible energy levels, noting that two parallel spins can never occupy the same quantum state. This is called the Pauli exclusion principle. Tradition has us write spin +1/2 as and spin -1/2 as , and these are pronounced "spin up" and "spin down." In this notation, the ground state of the He atom would be represented as n = 1 and , meaning that both electrons have the lowest energy principal quantum number n = 1, as in Unit 5, and must be put into that quantum state with opposite spin projections.
Quantum mechanics also allows us to understand the size of atoms, and how seemingly tiny electrons take up so much space. The probability density of an electron in a helium atom is a balance of three things: its electrical attraction to the nucleus, its electrical repulsion from the other electron, and the fact that the kinetic energy of an electron gets too large if its wavelength gets too small, as we learned in Unit 5. This is actually the same balance between confinement and kinetic energy that allowed Bohr, with his first circular orbit, to also correctly estimate the size of the hydrogen atom as being 105 times larger than the nucleus which confines it.
Assigning electrons to energy levels past H and He
Now, what happens if we have three electrons, as in the lithium (Li) atom? Lewis would write Li, simply not showing the inert inner shell electrons. Where does this come from in the quantum mechanics of the aufbau? Examining the energy levels occupied by the two electrons in the He atom shown in Figure 12 and thinking about the Pauli principle make it clear that we cannot simply put the third electron in the n = 1 state. If we did, its spin would be parallel to the spin of one of the other two electrons, which is not allowed by the exclusion principle. Thus the ground state for the third electron must go into the next lowest unoccupied energy level, in this case n = 2, l = 0.
Figure 13: Electrons in atomic energy levels for Li, Na, and K
Using the exclusion principle to determine electron occupancy of energy levels up through the lithium (Li), sodium (Na), and potassium (K) atoms, vindicates the empirical shell structures implicit in the Mendeleev table and explicit in Lewis's dot diagrams. Namely Li, Na, and K all have a single unpaired electron outside of a filled (and thus inert and non-magnetic) shell. It is these single unpaired electrons that allow these alkali atoms to be great candidates for making atomic clocks, and for trapping and making ultra-cold gases, as the magnetic traps grab that magnetic moment of that unpaired electron.
Where did this magical Pauli exclusion principle come from? Here, as it turns out, we need an entirely new, unexpected, and not at all intuitive fundamental principle. With it, we will have our first go-around at distinguishing between fermions and bosons.