Search for the ground-state electronic configurations of correlated organometallic metallocenes from constraint density functional theory

The ground-state electronic configurations of the correlated organometallic metallocenes, $M{\mathrm{Cp}}_2,M=\mathrm{V}$, Cr, Mn, Fe, Co, and Ni, are investigated using constraint density functional theory combined with nonempirical $U_{\mathrm{eff}}$ parameters determined from linear-response theory. The relative stability of the various $d$-orbital electronic configurations of these organometallic molecules is found to be sensitive to the amount of correlation. Using nonempirical values of $U_{\mathrm{eff}}$, the calculated electronic configurations are in agreement with the experiments: ${}^4{A}_{2g},{}^3{E}_{2g},{}^6{A}_{1g},{}^1{A}_{1g},{}^2{E}_{1g}$, and ${}^3{A}_{2g}$ for the ${\mathrm{VCp}}_2,{\mathrm{CrCp}}_2,{\mathrm{MnCp}}_2,{\mathrm{FeCp}}_2,{\mathrm{CoCp}}_2$, and ${\mathrm{NiCp}}_2$, respectively.

Figure 1

(a) Structure of a metallocene where large (red), middle (gray), and small (white) circles indicate transition-metal, carbon, and hydrogen atoms, respectively. (b) Schematic of the energy diagrams of the crystal-field splitting of transition-metal ($M$) $d$ orbitals for $D_{5d}$ symmetry, the molecular orbitals in the two cyclopentadienyl rings ${\mathrm{Cp}}_2$, and the hybridized orbitals in the $M{\mathrm{Cp}}_2$. (c) and (d) Schematics of the molecular $e_{1g}^{*}$ (antibonding $M{d}_{xz,yz}-{\mathrm{Cp}}_2{e}_{1g}$) and $e_{2g}$ (bonding $M{d}_{x^2-y^2,xy}-{\mathrm{Cp}}_2{e}_{2g}$) states in $M{\mathrm{Cp}}_2$.

Figure 2

Energy diagram of the $d$ orbitals in (a) ${\mathrm{VCp}}_2\left({}^4{A}_{2g}\right)$, (b) ${\mathrm{CrCp}}_2\left({}^3{E}_{2g}\right)$, (c) ${\mathrm{MnCp}}_2\left({}^2{E}_{2g}\right)$, (d) ${\mathrm{FeCp}}_2\left({}^1{A}_{1g}\right)$, (e) ${\mathrm{CoCp}}_2\left({}^2{E}_{1g}\right)$, and (f) ${\mathrm{NiCp}}_2\left({}^3{A}_{2g}\right)$, calculated using GGA. Up and down arrows indicate the electron occupations of the majority- and minority-spin states, respectively. The reference energy (0 eV) is set to the vacuum level.

Figure 3

Energy diagram of the $d$ orbitals in (a) ${\mathrm{VCp}}_2\left({}^4{A}_{2g}\right)$, (b) ${\mathrm{CrCp}}_2\left({}^3{E}_{2g}\right)$, (c) ${\mathrm{MnCp}}_2\left({}^6{A}_{1g}\right)$, (d) ${\mathrm{FeCp}}_2\left({}^1{A}_{1g}\right)$, (e) ${\mathrm{CoCp}}_2\left({}^2{E}_{1g}\right)$, and (f) ${\mathrm{NiCp}}_2\left({}^3{A}_{2g}\right)$, calculated using the $\mathrm{GGA}+U$ with the $U_{\mathrm{eff}}^{\mathrm{LRT}}$ values in Table 1. Notation is the same as in Fig. 2.

Figure 4

Total energy differences among the electronic configurations $\mathrm{\Delta }E$ as a function of $U_{\mathrm{eff}}$ for (a) ${\mathrm{MnCp}}_2$ and (b) ${\mathrm{CrCp}}_2$. The diamonds (blue) and triangles (red) in (a) represent the ${}^2{E}_{2g}$ and ${}^6{A}_{1g}$ states, respectively, and the diamonds (blue), triangles (red), and squares (green) in (b) represent the ${}^3{E}_{2g},{}^3{A}_{1g}$, and ${}^5{E}_{1g}$ states, respectively. $U_{\mathrm{eff}}=0\mathrm{eV}$ corresponds to the GGA results.

Figure 5

Charge-density difference $\mathrm{\Delta }\rho ={\rho }_{M{\mathrm{Cp}}_2}-({\rho }_{{\mathrm{Cp}}_2}+{\rho }_M)$ for (a) ${}^6{A}_{1g}{\mathrm{MnCp}}_2$ and (b) ${}^3{E}_{2g}{\mathrm{CrCp}}_2$ for $\mathrm{GGA}+U$ with $U_{\mathrm{eff}}^{\mathrm{LRT}}$. The blue and red regions indicate positive (accumulation) and negative (depletion) differences, respectively.