In a metal complex, the \( t_{2g}^6 e_g^1 \) configuration is typically observed when the central metal ion has an electronic configuration that favours a particular distribution of electrons between the lower \( t_{2g} \) and higher \( e_g \) orbitals.
- When a weak field ligand is present, the splitting of the \( d \)-orbitals is small, and this allows the electrons to occupy higher energy \( e_g \) orbitals, leading to a low spin complex. The \( e_g^1 \) electron is placed in the higher energy \( e_g \) orbital, while the remaining \( t_{2g} \) orbitals are fully occupied.
Thus, the correct conditions for the \( t_{2g}^6 e_g^1 \) configuration are a weak field ligand and a low spin complex.