Symmetry adapted basis functions for the transition metal complex tetracyanonickelate using the eigenfunction method

In this work we obtain the symmetrised basis functions using the eigenfunction method by taking an example of transition metal complex tetracyanonickelate [Ni(CN)₄]²⁻ having square planar structure with D_4h symmetry. We consider a subgroup D₄ for simplicity. For this symmetry group, class operators are obtained and a linear combination of class operators is taken such that it has distinct eigenvalues. A representation of this class operator is obtained by making use of the p and d orbitals of the central Ni atom. The reduction of this representation is obtained by making use of a canonical subgroup chain for the symmetric group of the Hamiltonian. This yields the required symmetrised basis functions using simple matrix algebra, thus illustrating the merits and elegance of the eigenfunction method over the conventional methods.