Assessment of density functional theory for iron(II) molecules across the spin-crossover transition

Octahedral Fe$^{2+}$ molecules are particularly interesting as they often exhibit a spin-crossover transition. In spite of the many efforts aimed at assessing the performances of density functional theory for such systems, an exchange-correlation functional able to account accurately for the energetic of the various possible spin-states has not been identified yet. Here we critically discuss the issues related to the theoretical description of this class of molecules from first principles. In particular we present a comparison between different density functionals for four ions, namely [Fe(H$_2$O)$_6$]$^{2+}$, [Fe(NH$_3$)$_6$]$^{2+}$, [Fe(NCH)$_6$]$^{2+}$ and [Fe(CO)$_6$]$^{2+}$. These are characterized by different ligand-field splittings and ground state spin multiplicities. Since no experimental data are available for the gas phase, the density functional theory results are benchmarked against those obtained with diffusion Monte Carlo, one of the most accurate methods available to compute ground state total energies of quantum systems. On the one hand, we show that most of the functionals considered provide a good description of the geometry and of the shape of the potential energy surfaces. On the other hand, the same functionals fail badly in predicting the energy differences between the various spin states. In the case of [Fe(H$_2$O)$_6$]$^{2+}$, [Fe(NH$_3$)$_6$]$^{2+}$, [Fe(NCH)$_6$]$^{2+}$, this failure is related to the drastic underestimation of the exchange energy. Therefore quite accurate results can be achieved with hybrid functionals including about 50% of Hartree-Fock exchange. In contrast, in the case of [Fe(CO)$_6$]$^{2+}$, the failure is likely to be caused by the multiconfigurational character of the ground state wave-function and no suitable exchange and correlation functional has been identified.