3.5 Electron states and box diagrams

So far, we have represented the electronic state of an atom as a collection of sub-shells. Now we turn to the states of the electrons within those sub-shells. Just as shells can be broken down into sub-shells, so sub-shells can be broken down into atomic orbitals. Each atomic orbital describes an allowed spatial distribution about the nucleus for an electron in the sub-shell. Here we shall only be concerned with their number.

Consider the formula for the sub-shell electron capacities, which is 2(2l + 1), l being the second quantum number. The factor (2l + 1) tells us the number of atomic orbitals in the sub-shell.

It turns out that each orbital in a sub-shell can contain up to two electrons. This is connected to a property of the electron called spin. This spin occurs in one of two senses, which are physically pictured as clockwise and anticlockwise (Figure 25).

Figure 25 The two senses of electron spin.

When an atomic orbital contains its maximum complement of two electrons, those two electrons must always have spins of opposite sense. This phenomenon of electron spin accounts for the '2' that precedes the bracket in the formula for sub-shell capacities, 2(2l + 1). For example, as we have seen, the factor (l + 1) tells us that there are three atomic orbitals in a p sub-shell. Each of the three orbitals can accommodate up to two electrons with opposed spins. So a p sub-shell can contain a maximum of 2 × 3 or 6 electrons.

A full atomic orbital, therefore, contains two electrons with spins of opposite sense. This is represented by writing one electron as an upward-pointing half-headed arrow, and the other as a downward-pointing half-headed arrow. For example, the helium atom has the electronic configuration 1s². The 1s sub-shell contains just one orbital, which can be represented by a single box containing the two electrons of opposite spin:

Two such electrons with opposed spins are said to be paired and the diagram that puts these electrons into boxes is referred to as a box diagram. Now consider the case of nitrogen, whose atom has the configuration 1s²2s²2p³. The 1s and 2s sub-shells contain one orbital each, and the 2p sub-shell contains three. The box diagram that must be filled therefore takes the form:

The final step is the assignment of the three 2p electrons to the three 2p boxes. There are several possibilities, but the one we want is the ground-state arrangement, the state of lowest energy. There is a simple rule, called Hund's rule, that tells us what this is:

Because electrons in the same box must have opposed spins, we must put the electrons, as far as possible, in different boxes with spins of the same sense, or, as it is usually termed, with parallel spins. In this case there are three 2p electrons and three boxes, so each box can take one electron with the same spin, and this is the preferred arrangement according to Hund's rule. The final result for the nitrogen atom is therefore: