Fast computation of analytical second derivatives with effective core potentials: Application to Si₈C₁₂, Ge₈C₁₂, and Sn₈C₁₂

An improved method is described for the computation of integrals involving effective core potentials. The improved method provides better scalability to higher angular momenta as well as improved speed. The new method is also applied to the determination of the minimum energy structures of Si₈C¹², Ge₈C¹², and Sn₈C₁₂, main group analogs of the Ti₈C₁₂ compounds (known as metcars). Relative energies, geometries, and vibrational frequencies are reported for several novel structures.