Quantum mechanics successfully explains the structure of the periodic table. We need three basic ideas to see how this works:

1. The energy levels of an atom are found by solving the Schrödinger equation for multielectron atoms. Figure 41.14, a very important figure for understanding the periodic table, showed that the energy depends on the quantum numbers $n$ and $l$.

2. For each value $l$ of the orbital quantum number, there are $2l+1$ possible values of the magnetic quantum number $m$ and, for each of these, two possible values of the spin quantum number $m_s$. Consequently, each energy level in Figure 41.14 is actually $2(2l+1)$ different states. Each of these states has the same energy.

3. The ground state of the atom is the lowest-energy electron configuration that is consistent with the Pauli exclusion principle.

We used these ideas in the last section to look at the elements helium $(Z=2)$ and lithium $(Z=3)$. Four-electron beryllium $(Z=4)$ comes next. The first two electrons go into $1s$ states, forming a closed shell, and the third goes into 2s. There is room in the $2s$ level for a second electron as long as its spin is opposite that of the first $2s$ electron. Thus the third and fourth electrons occupy states $(2\text{,}0\text{,}0\text{,}+\frac{1}{2})$ and $(2\text{,}0\text{,}0\text{,}-\frac{1}{2})$. These are the only two possible $2s$ states. All the states with the same values of $n$ and $l$ are called a subshell, so the fourth electron closes the $2s$ subshell. (The outer two electrons are called a subshell, rather than a shell, because they complete only the $2s$ possibilities. There are still spaces for $2p$ electrons.) The ground state of beryllium, shown in FIGURE 41.20, is $1s^{2}2s^2$.

FIGURE 41.20 The ground state of beryllium $(Z=4)$.

These principles can continue to be applied as we work our way through the elements. There are $2l+1$ values of $m$ associated with each value of l, and each of these can have $m_s=\pm \frac{1}{2}$. This gives, altogether, $2(2l+1)$ distinct quantum states in each nl subshell. TABLE 41.2 lists the number of states in each subshell.

Boron $\left(1s^{2}2s^{2}2p\right)$ opens the $2p$ subshell. The remaining possible $2p$ states are filled as we continue across the second row of the periodic table. These elements are shown in FIGURE 41.21. With neon $\left(1s^{2}2s^{2}2p^6\right)$, which has six $2p$ electrons, the $n=2$ shell is complete, and we have another closed shell. The second row of the periodic table is eight elements wide because of the two $2s$ electrons plus the six $2p$ electrons needed to fill the $n=2$ shell.

FIGURE 41.21 Filling the $2p$ subshell with the elements boron through neon.