Multiple-Electron Atoms

Multiple-Electron Atoms

All atoms except hydrogen are multiple-electron atoms. The physical and chemical properties of elements are directly related to the number of electrons a neutral atom has. The periodic table of the elements groups elements with similar properties into columns. This systematic organization is related to the number of electrons in a neutral atom, called the atomic number, \(Z\). We shall see in this section that the exclusion principle is key to the underlying explanations, and that it applies far beyond the realm of atomic physics.

In 1925, the Austrian physicist Wolfgang Pauli (see this figure) proposed the following rule: No two electrons can have the same set of quantum numbers. That is, no two electrons can be in the same state. This statement is known as the Pauli exclusion principle, because it excludes electrons from being in the same state. The Pauli exclusion principle is extremely powerful and very broadly applicable. It applies to any identical particles with half-integral intrinsic spin—that is, having \(s=1/2, 3/2, ...\) Thus no two electrons can have the same set of quantum numbers.

The Austrian physicist Wolfgang Pauli (1900–1958) played a major role in the development of quantum mechanics. He proposed the exclusion principle; hypothesized the existence of an important particle, called the neutrino, before it was directly observed; made fundamental contributions to several areas of theoretical physics; and influenced many students who went on to do important work of their own. (credit: Nobel Foundation, via Wikimedia Commons)

Let us examine how the exclusion principle applies to electrons in atoms. The quantum numbers involved were defined in Quantum Numbers and Rules as l\(\mathrm{n, l,}\phantom{\rule{0.25em}{0ex}}{m}_{l}\mathrm{, s}\), and \({m}_{s}\). Since \(s\) is always \(1/2\) for electrons, it is redundant to list \(s\), and so we omit it and specify the state of an electron by a set of four numbers \((n,\phantom{\rule{0.25em}{0ex}}l,\phantom{\rule{0.25em}{0ex}}{m}_{l},\phantom{\rule{0.25em}{0ex}}{m}_{s})\). For example, the quantum numbers \((2, 1, 0,\phantom{\rule{0.25em}{0ex}}-1/2)\) completely specify the state of an electron in an atom.

Since no two electrons can have the same set of quantum numbers, there are limits to how many of them can be in the same energy state. Note that \(n\) determines the energy state in the absence of a magnetic field. So we first choose \(n\), and then we see how many electrons can be in this energy state or energy level. Consider the \(n=1\) level, for example. The only value \(l\) can have is 0 (see this table for a list of possible values once \(n\) is known), and thus \({m}_{l}\) can only be 0. The spin projection \({m}_{s}\) can be either \(+1/2\) or \(-1/2\), and so there can be two electrons in the \(n=1\) state. One has quantum numbers \((1, 0, 0,\phantom{\rule{0.25em}{0ex}}+1/2)\), and the other has \((1, 0, 0,\phantom{\rule{0.25em}{0ex}}-1/2)\). This figure illustrates that there can be one or two electrons having \(n=1\), but not three.

The Pauli exclusion principle explains why some configurations of electrons are allowed while others are not. Since electrons cannot have the same set of quantum numbers, a maximum of two can be in the \(n=1\) level, and a third electron must reside in the higher-energy \(n=2\) level. If there are two electrons in the \(n=1\) level, their spins must be in opposite directions. (More precisely, their spin projections must differ.)