The Aufbau Principle
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You should be comfortable with the four quantum numbers (\(n, l, m_l, m_s\)) and understand that the Pauli exclusion principle limits each orbital to two electrons with opposite spins.
Why this matters
The Aufbau principle provides the algorithmic logic for predicting the electron configuration of any element.
Without a systematic method for assigning electrons to orbitals, the periodic table would appear as an arbitrary collection of chemical behaviours. By establishing a clear hierarchy of orbital energies, the Aufbau principle allows chemists to predict valence electron counts, magnetic properties, and chemical reactivity directly from atomic number. This lecture bridges the gap between the abstract quantum mechanical model and the practical utility of the periodic table.
The Logic of Filling
Imagine a multi-storey building where rooms (orbitals) are available for tenants (electrons). The building has a strict hierarchy of desirability based on the room's location and size. Tenants do not choose randomly; they follow a specific protocol. First, they occupy the lowest energy rooms available. Second, if two rooms are identical in energy, they prefer to sit in separate rooms rather than sharing one, maintaining a certain distance. Third, no room can hold more than two tenants, and if two do share, they must be of opposite orientation. The Aufbau principle is simply the codification of this protocol for atomic orbitals. It dictates that we start with the 1s orbital and proceed upwards, filling each orbital to its capacity before moving to the next higher energy level. This sequential filling explains why the first two elements (Hydrogen and Helium) have similar configurations, while Lithium starts a new pattern, and so on.
The Madelung Rule
The order in which orbitals are filled is not strictly by principal quantum number \(n\). Instead, it follows the Madelung rule (or \(n+l\) rule). Orbitals are ranked by the sum of their principal quantum number \(n\) and azimuthal quantum number \(l\). Orbitals with a lower \(n+l\) value are filled first. If two orbitals have the same \(n+l\) value, the one with the lower \(n\) value is filled first.
Let us apply this to the first few orbitals. The 1s orbital has \(n=1, l=0\), so \(n+l=1\). The 2s orbital has \(n=2, l=0\), so \(n+l=2\). The 2p orbital has \(n=2, l=1\), so \(n+l=3\). The 3s orbital has \(n=3, l=0\), so \(n+l=3\). The 3p orbital has \(n=3, l=1\), so \(n+l=4\). The 4s orbital has \(n=4, l=0\), so \(n+l=4\). The 3d orbital has \(n=3, l=2\), so \(n+l=5\). The 4p orbital has \(n=4, l=1\), so \(n+l=5\).
This sequence explains the "diagonal" filling pattern often seen in the periodic table, where the 4s orbital fills before the 3d orbital, despite 3d having a lower principal quantum number.
Applying Pauli and Hund
Once the order of orbitals is established, we apply the Pauli exclusion principle and Hund's rule to determine the specific arrangement of electrons within those orbitals. The Pauli principle states that no two electrons in an atom can have the same set of four quantum numbers. Practically, this means each orbital (defined by \(n, l, m_l\)) can hold a maximum of two electrons, and these two must have opposite spins (\(m_s = +\tfrac{1}{2}\) and \(m_s = -\tfrac{1}{2}\)).
Hund's rule addresses the distribution of electrons among degenerate orbitals (orbitals with the same \(n\) and \(l\), such as the three 2p orbitals). It states that electrons will occupy empty degenerate orbitals singly with parallel spins before any orbital is doubly occupied. This minimizes electron-electron repulsion and maximizes total spin, leading to a lower energy state. For example, in a carbon atom (6 electrons), the configuration is \(1s^2 2s^2 2p^2\). The two 2p electrons will occupy two different 2p orbitals with parallel spins, rather than pairing up in a single 2p orbital.
Writing Configurations
To write the ground-state electron configuration of an atom, we sum the number of electrons required to fill the orbitals in the Madelung order until we reach the total electron count of the atom. For instance, Sodium (Na, \(Z=11\)) has 11 electrons. We fill 1s (2), 2s (2), 2p (6), which totals 10 electrons. The 11th electron goes into the next available orbital, 3s. Thus, the configuration is \(1s^2 2s^2 2p^6 3s^1\). We can often use noble gas shorthand to simplify this, writing \([\mathrm{Ne}] 3s^1\), where \([\mathrm{Ne}]\) represents the core electrons of Neon.
It is crucial to note that the "valence electrons" are those in the outermost shell (highest \(n\)). For Sodium, the valence electron is the 3s electron. This single valence electron explains Sodium's high reactivity and its tendency to form \(\mathrm{Na^+}\) ions. The periodic table is essentially a map of these valence electron configurations, grouped by their chemical similarity.
The result
The building-up principle
The ground-state electron configuration of an atom is obtained by filling the atomic orbitals of the previous result one electron at a time, subject to three rules: (1) the Madelung rule: orbitals fill in order of increasing \(n+l\), and for orbitals sharing the same \(n+l\), in order of increasing \(n\); (2) the Pauli exclusion principle: no orbital holds more than two electrons, and those two must have opposite spin (\(m_s=+\tfrac12\) and \(m_s=-\tfrac12\)); (3) Hund's rule: within a set of degenerate orbitals (equal \(n\) and \(l\), differing only in \(m_l\)), electrons occupy separate orbitals with parallel spin before any orbital is doubly occupied.
A common misconception is that the Madelung rule is a fundamental law of nature. In reality, it is an empirical approximation that works well for many elements but fails for some transition metals and lanthanides due to subtle energy differences between orbitals (e.g., Cr and Cu have anomalous configurations where an electron is promoted to a d-orbital to achieve a half-filled or fully-filled subshell, which is more stable). Always verify configurations against experimental data for these elements. C
Worked example
Problem. Determine the ground-state electron configuration of Phosphorus (P, \(Z=15\)) and identify the number of unpaired electrons.
- Count the total number of electrons: \(Z=15\), so there are 15 electrons.
- List orbitals in Madelung order: 1s, 2s, 2p, 3s, 3p.
- Fill orbitals sequentially: 1s holds 2 (total 2), 2s holds 2 (total 4), 2p holds 6 (total 10).
- Remaining electrons: \(15 - 10 = 5\). The next orbital is 3s, which holds 2 (total 12).
- Remaining electrons: \(15 - 12 = 3\). The next orbital is 3p, which holds 3.
- Apply Hund's rule to the 3p subshell: The three electrons occupy the three separate 3p orbitals with parallel spins.
- Write the full configuration: \(1s^2 2s^2 2p^6 3s^2 3p^3\).
- Count unpaired electrons: The 3p subshell has 3 unpaired electrons. All other subshells are full.
Units check. The number of electrons is dimensionless, and the configuration notation is standard.
Three quick tests
- What is the Madelung rule order for the 4s and 3d orbitals?
Answer
4s fills before 3d because \(4+0=4\) and \(3+2=5\). - How many unpaired electrons are in a neutral Iron atom (Fe, \(Z=26\))?
Answer
4 unpaired electrons (configuration \([\mathrm{Ar}] 3d^6 4s^2\), with 4 unpaired in 3d). - Why does the 2p subshell hold a maximum of 6 electrons?
Answer
There are 3 orbitals in the 2p subshell, and each holds 2 electrons (Pauli principle).
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Every lecture and every result in this unit, with prerequisites and threads.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.