Constraint density functional calculations for multiplets in ligand-fields: Applications to Fe-phthalocyanine and Al$_2$O$_3$:Cr$^{3+}$

In transition-metal-based complexes and molecules, multiplet structures are essential in understanding the electronic structure. However, it is often difficult to evaluate a {\it true} ground state or the {\it lowest} state within a given ligand (or crystal) symmetry from first principles calculations based on density-functional theory (DFT). Here, we propose a simple DFT approach, implemented into the FLAPW method\footnote{Wimmer, Krakauer, Weinert, Freeman, PRB{\bf 24}, 864; Weinert, Wimmer, Freeman, PRB{\bf 26}, 4571}, to treat multiplets in ligand-fields, by imposing a density matrix constraint on the $d$-orbital occupation numbers. We demonstrate the utility of this approach for the case of an isolated single Fe phthalocyanine (FePc) and a Cr impurity in a corundum Al$_2$O$_3$. For the FePc, results predict that there are three stationary states of $^3E_g$, $^3B_2$, and $^3A_2$ in the Fe$^{2+}$ ion, and our total energy calculations clearly demonstrate that the ground state is $^3A_{2g}$. In the case of the Al$_2$O$_3$:Cr$^{3+}$, where an on-site Coulomb correlation correction (+$U$) is incorporated, the ground state is $^4A_2$ and the total energy difference between the ground state and the excited state $^4T_2$, 2.9~eV, roughly agrees with an experimental value of 2.23~eV.