Determining the Ground State Term for $[Ni(H_2O)_6]^{2+}$

To find the ground state term symbol for the complex ion $[Ni(H_2O)_6]^{2+}$, we need to consider the electronic configuration of the central metal ion, Nickel (Ni), and the geometry of the complex.

Central Metal Ion Configuration

• Nickel (Ni) has an atomic number of 28. Its electronic configuration is $[Ar] 3d^8 4s^2$.
• The complex is formed by Nickel in the +2 oxidation state, $Ni^{2+}$.
• Therefore, the electronic configuration of the central ion is $Ni^{2+} = [Ar] 3d^8$.

Complex Geometry and Ligand Field

• The complex $[Ni(H_2O)_6]^{2+}$ has an octahedral geometry ($O_h$).
• Water ($H_2O$) is a weak field ligand.
• We need to determine the ground state term symbol for a $d^8$ configuration in an octahedral field.

Deriving the Ground State Term Symbol

• For a free $d^8$ ion, the ground state term symbol is $^3F$ (derived from the Russell-Saunders coupling scheme).
• In an octahedral ($O_h$) ligand field, the free ion terms split. The $^3F$ term splits into the following terms, listed in order of increasing energy for a typical $d^8$ octahedral complex:
  • $^3A_{2g}$ (Lowest energy)
  • $^3T_{2g}$
  • $^3T_{1g}$(F)
• The $^3P$ term from the free ion splits into $^3T_{1g}$(P), which is generally higher in energy.
• For a weak field ligand like water, the energy ordering determined by ligand field theory dictates that the $^3A_{2g}$ term lies lowest in energy.

Conclusion

The ground state term symbol for the $d^8$ configuration of $Ni^{2+}$ in the octahedral complex $[Ni(H_2O)_6]^{2+}$ is $^3A_{2g}$.

This corresponds to Option 3.