Electron Configurations — Theory & Guide
This guide summarizes the rules used to write ground-state electron configurations for neutral atoms and
how the valence shell is determined in s/p-block (A-group), d-block (transition), and f-block (inner-transition) elements.
1) Three governing rules
- Aufbau principle (building up): Fill orbitals in the empirical order of increasing energy.
Order used up to 7p: 1s, 2s, 2p, 3s, 3p, 4s,
3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d,
6p, 7s, 5f, 6d, 7p.
- Pauli exclusion principle: Each orbital holds at most two electrons with opposite spins
\( (m_s=+\tfrac12 \text{ and } -\tfrac12) \).
- Hund’s rule: For a given subshell, place electrons singly in different orbitals with parallel spins before pairing.
2) Subshell capacities and quantum numbers
A subshell is labeled by principal quantum number \(n=1,2,\dots\) and orbital quantum number \( \ell \in \{0,1,2,3\} \)
for \(s,p,d,f\) respectively. Each subshell contains \(2\ell+1\) orbitals, each orbital holds 2 electrons,
so the maximum population is \(2(2\ell+1)\).
| Subshell | \(\ell\) | # orbitals \(2\ell+1\) | Max e⁻ \(2(2\ell+1)\) | Allowed \(m_\ell\) |
|---|
| s | 0 | 1 | 2 | 0 |
| p | 1 | 3 | 6 | \(-1, 0, +1\) |
| d | 2 | 5 | 10 | \(-2, -1, 0, +1, +2\) |
| f | 3 | 7 | 14 | \(-3, -2, -1, 0, +1, +2, +3\) |
3) Writing configurations
Electrons are distributed across subshells in the aufbau order. Configurations are written as
\( n\ell^{\,x} \) blocks (e.g., \(3p^4\)). A noble-gas core may be used for brevity,
e.g., [Ar] 4s2 3d10 4p5 for bromine.
4) What counts as the valence shell?
- Main-group (A-group, s/p-block):
Valence = subshells with the highest principal quantum number \(n_\text{max}\), i.e.,
nmaxs and
nmaxp only. Any filled \((n_\text{max}-1)d\) belongs to the core.
Example (Br, Z=35): valence is 4s2 4p5; the 3d10 is core.
- Transition metals (d-block):
Valence =
nmaxs plus occupied (nmax−1)d.
Example (Fe, Z=26): valence is 4s2 3d6.
- Lanthanides/actinides (f-block):
Valence =
nmaxs plus occupied (nmax−2)f (and sometimes (nmax−1)d if populated).
5) Common stability exceptions
Some d-block elements are more stable when a d subshell is half-filled (\(d^5\)) or filled (\(d^{10}\)).
Neutral-atom ground states frequently observed:
- Cr: [Ar] 3d5 4s1
- Cu: [Ar] 3d10 4s1
- Nb: [Kr] 4d4 5s1
- Mo: [Kr] 4d5 5s1
- Ru: [Kr] 4d7 5s1
- Rh: [Kr] 4d8 5s1
- Pd: [Kr] 4d10 5s0
- Ag: [Kr] 4d10 5s1
- Pt: [Xe] 4f14 5d9 6s1
- Au: [Xe] 4f14 5d10 6s1
These patterns are captured in the calculator when constructing configurations.
6) Orbital diagram method (boxes and arrows)
A subshell with \(2\ell+1\) orbitals is drawn as that many boxes. Following Hund’s rule, place one
\( \uparrow \) in each box before adding \( \downarrow \) to pair. For example, \(2p^4\) is
7) Quantum numbers for valence electrons
For each valence electron, quantum numbers are assigned as:
- \(n\): the shell index of its subshell (e.g., \(n=4\) for 4p).
- \(\ell\): 0, 1, 2, 3 for s, p, d, f.
- \(m_\ell\): one of the \(2\ell+1\) values from \(-\ell\) to \(+\ell\); when applying Hund’s rule,
singly occupied orbitals receive distinct \(m_\ell\) values.
- \(m_s\): \(+\tfrac12\) for the first pass (parallel spins), then \(-\tfrac12\) for paired electrons.
The calculator enumerates \(m_\ell\) values in increasing order and assigns spins \(+\tfrac12\) first, then \(-\tfrac12\), to match Hund’s rule.
8) Examples
- Phosphorus (Z=15):
[Ne] 3s2 3p3.
Valence (A-group): 3s, 3p. Orbital diagram for 3p shows three unpaired \( \uparrow \).
- Iron (Z=26):
[Ar] 4s2 3d6.
Valence (d-block): 4s and 3d together.
- Bromine (Z=35):
[Ar] 4s2 3d10 4p5.
Valence (A-group): 4s, 4p only; 3d10 is core.
- Cerium (Z=58): begins the lanthanides; valence includes 6s and occupied 4f (and sometimes 5d).
9) How this tool decides the valence set
- Build the full configuration using the aufbau order with Pauli + Hund.
- Apply known d-block stability exceptions (Cr, Cu, Nb, Mo, Ru, Rh, Pd, Ag, Pt, Au).
- Detect the block from the last filled subshell in the aufbau sequence.
-
Valence selection:
- s/p-block (A-group): include only \(n_\text{max}s\) and \(n_\text{max}p\).
- d-block: include \(n_\text{max}s\) and \((n_\text{max}-1)d\).
- f-block: include \(n_\text{max}s\) and \((n_\text{max}-2)f\) (and any occupied \((n_\text{max}-1)d\)).
Note: Real energy ordering can vary in ions or excited states; this guide targets neutral-atom ground states used in general chemistry.
