Ground State Term Symbol of \(\text{Mn}^{4+}\) Ion:
WBSET 2024
Electron Configuration:
Mn: \([\text{Ar}]3d^{5}4s^{2}\).
\(\text{Mn}^{4+}\) ion: \([\text{Ar}]3d^{3}\).
Total Spin Angular Momentum (S) and Multiplicity:
For the \(3d^{3}\) configuration, the three electrons occupy different d-orbitals with parallel spins to maximize spin multiplicity (Hund's first rule).
Each electron has a spin quantum number \(m_{s}=+1/2\).
Total spin \(S=1/2+1/2+1/2=3/2\).
Spin multiplicity is \(2S+1=2(3/2)+1=4\).
Total Orbital Angular Momentum (L):
The electrons occupy the \(d\) orbitals (where \(l=2\)) with magnetic quantum numbers \(m_{l}\) of +2, +1, 0, -1, -2.
To maximize total orbital angular momentum (Hund's second rule, after maximizing spin), the three electrons are placed in the \(m_{l}=+2\), \(m_{l}=+1\), and \(m_{l}=0\) orbitals.
Total orbital angular momentum \(L=(+2)+(+1)+(0)=3\).
Spectroscopic symbol = \(L\)
Total Angular Momentum (J):
The total angular momentum \(J\) is determined by Hund's third rule.
J = \(|L-S|\) to \(|L+S|\)
For less than half-filled orbitals, the ground state has the minimum possible \(J\) value.
The minimum value is \(J=3/2\).
Term Symbol:
Combining these values, we get the ground state term symbol is \({}^{2S+1}\text{L}_{J}\).
For \(\text{Mn}^{4+}\), this is \({}^{4}\text{F}_{3/2}\).