1.6: Rules Governing Ground State Electron Configurations

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Introduction

In addition to having different energy levels, orbitals also have different shapes and orientations, and each can be occupied by two electrons. For each principal quantum number, n, there is up to one s orbital, three p orbitals, five d orbitals and seven f orbitals. Each orbital can hold up to two electrons of different spin (m_s = +1/2, -1/2); therefore, the s subshell can hold two electrons, the p subshell can hold six electrons, the d subshell can hold ten electrons, and the f subshell can hold 14 electrons.

The ground state electron configuration specifies the lowest-energy arrangement of electrons into subshells for a given element. Ground state electron configurations are the foundation for understanding molecular bonding, properties, and structures. The following rules explain how to determine the lowest energy arrangement of electrons into the orbitals. The ground state electron configurations contribute to a fundamental understanding of the periodic table and will later be used to understand different atomic properties.

The Aufbau Principle

The Aufbau Principle (also called the building-up principle or the Aufbau rule) states that, in the ground state of an atom or ion, electrons fill atomic orbitals of the lowest available energy level before occupying higher-energy levels Figure \(\PageIndex{1}\). In general, an electron will occupy an atomic orbital with the lowest value of \(n, l, m_l\), in that order of priority.

Figure \(\PageIndex{1}\): The order of filling of ground state electron orbitals is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, etc... (CC-BY-NC-SA; Kathryn Haas)

Electrons will occupy the lowest energy orbitals in order to minimize the total energy. The two quantum numbers that are related to energy in multi-electron atoms are \(n\), and \(l\). Thus, orbitals with the lowest values of \(n\) and \(l\) will fill first.

Hund's Rule of Maximum Multiplicity

When electron fill into orbitals, the multiplicity refers to the number of unpaired electrons. Multiplicity is calculated as n + 1, where n is the number of unpaired electrons. Hund's Rule of Maximum Multiplicity states that for a given electron configuration, the lowest energy arrangement of electrons in degenerate orbitals is the arrangement with the greatest multiplicity (or the greatest number of unpaired electrons). This rule is used to predict the ground state of an atom or molecule with one or more open electronic shells.

Hund's rule is based on empirical observation of atomic spectra, and it is a consequence of the energy required to pair two electrons in the same orbital. This energy of repulsion between two electrons in the same orbitals is a Coulombic energy of repulsion, \(\Pi_c\), caused by two electrons with like charge sharing the same area of space (an orbital). When more than one electron occupies a set of degenerate orbitals, the most favorable arrangement is one where the number of paired electrons is minimized.

A simplified definition of Hund's rule is that the lowest energy arrangement is the one with the greatest number of unpaired electrons. This implies that if two or more orbitals of equal energy are available, electrons will occupy them singly before filling them in pairs. Examples of ground state arrangements of electrons in three degenerate p-orbitals is given in Figure \(\PageIndex{2}\).

Figure \(\PageIndex{2}\): Hund's rule is that the number of unpaired electrons (multiplicity) must be maximized in the ground state. (CC-BY-NC-SA; Libretexts)

The Pauli Exclusion Principle

The Pauli exclusion principle states that it is impossible for two electrons of a multi-electron atom to have the same set of values for all four quantum numbers. Two electrons in different orbitals will have a different set of \(n, l\), and \(m_l\) values. When two electrons reside in the same orbital, they posses the same \(n, l\), and \(m_l\) values, therefore their m_s must be different. Thus, two electrons in the same orbital must have opposite spin values of \(+\frac{1}{2}\) and \(-\frac{1}{2}\). The value of \(m_s\) for an unpaired electron is conventionally assigned a value of \(+\frac{1}{2}\).

Figure \(\PageIndex{3}\): Pauli's principle tells us that paired electrons must have opposite spin. CC-BY-NC-SA; Libretexts)

References

1. Miessler, Gary L., and Donald A. Tarr. Inorganic Chemistry. Upper Saddle River, NJ: Pearson Prentice Hall, 2010. Print.
2. Brown, Ian David. The Chemical Bond in Inorganic Chemistry the Bond Valence Model. Oxford: Oxford UP, 2006. Print.