AcademyAtomic Quantum Structure
Academy
Many-Electron Atoms
Level 1 - Physics topic page in Atomic Quantum Structure.
Principle
Many-electron atoms cannot be described as independent hydrogen atoms because electrons repel each other and shield nuclear charge. Atomic structure is built by filling orbitals using quantum numbers and energy ordering.
The effective nuclear charge felt by an electron is less than the full nuclear charge because other electrons partially screen the nucleus.
Notation
\(Z\)
atomic number
1
\(Z_{\mathrm{eff}}\)
effective nuclear charge
1
\(n\)
principal quantum number
1
\(\ell\)
orbital angular momentum quantum number
1
\(m_\ell\)
magnetic quantum number
1
\(m_s\)
spin projection quantum number
1
Method
Derivation 1: Start with orbital labels
Orbitals are labelled by \(n\) and subshell letters set by \(\ell\).
Subshell labels
\[\ell=0,1,2,3\Rightarrow s,p,d,f\]
Orbitals in a subshell
\[N_{\mathrm{orbitals}}=2\ell+1\]
Derivation 2: Count electron capacity
Each orbital can hold two electrons with opposite spin projections.
Subshell capacity
\[N_e=2(2\ell+1)\]
Shell capacity
\[N_e=2n^2\]
Derivation 3: Fill lower-energy states first
The Aufbau pattern orders orbitals by energy, modified by shielding and penetration.
Effective charge idea
\[Z_{\mathrm{eff}}=Z-S\]
Hydrogen-like scaling
\[E_n\propto-\frac{Z_{\mathrm{eff}}^2}{n^2}\]
Rules
Subshell orbitals
\[N_{\mathrm{orbitals}}=2\ell+1\]
Subshell electron capacity
\[N_e=2(2\ell+1)\]
Shell electron capacity
\[N_e=2n^2\]
Effective nuclear charge
\[Z_{\mathrm{eff}}=Z-S\]
Examples
Question
How many electrons fit in a \(p\) subshell?
Answer
For a \(p\) subshell,
\[\ell=1\]
so \[2(2\ell+1)=2(3)=6\]
Checks
- Multi-electron energies depend on both \(n\) and \(\ell\).
- Shielding reduces the effective attraction felt by outer electrons.
- Each orbital holds at most two electrons.
- Electron configurations are built by filling lower-energy orbitals first.