Many-electron states

The one-electron picture is modified by the strong Coulomb interactions among the electrons. There are only three distinct types of energy-level diagrams for 3d ions, corresponding to the D,F or S ground-state terms. The D-states map directly onto the one-electron levels because there is just one electron or hole in an otherwise empty, half-filled or filled d-shell. These are the d¹, d⁴, d⁶ and d⁹ configurations, with L = 2 according to Hund’s second rule. The F term is found for the d², d³ , d⁷ and d8 configurations, with L = 3. The S term is for d⁵, where L = 0. Orgel diagrams (Fig. 1) show the splitting of the ground-state D or F level in a cubic crystal field. The S term does not split, on account of the spherical symmetry of the half-filled shell.

Figure 1: Orgel diagrams for D and F terms. The corresponding dn ion configuration is shown.

The hierarchy of interactions H₀ > H_cf > H_so is not always maintained for 3d ions. Some ligands create such intense electrostatic fields that they manage to overturn not only Hund’s second rule, but also his first rule as well.
The crystal field can drive the ion into a low-spin state. The on-site Coulomb interaction has the effect of raising the energy of doubly occupied orbitals by U, compared with singly occupied orbitals. This is the electrostatic penalty for double occupancy. When U exceeds the crystal-field splitting of the oneelectron levels, Hund’s first rule applies as advertised, but when U<  the t_2g- orbitals for an octahedral site, for example, will tend to be doubly occupied. Figure 2 illustrates the high-spin and low-spin states for Fe²⁺.

Figure 2: Comparison of the one-electron energy levels for Fe²⁺ (3d⁶) in FeCl₂, where there is a high-spin state with S = 2, and FeS₂ where the crystal field stabilizes a low-spin state with S = 0.

In some materials, where the low-spin state lies only slightly lower in energy than the high-spin state, a spin crossover may occur as a function of temperature at a phase transition, driven by magnetic entropy R ln(2S + 1).
Tanabe–Sugano diagrams show the splitting of both ground-state and higher terms in the crystal field. They are drawn with the ground-state energy set to zero. Some diagrams for 3d ions in octahedral coordination are shown in Fig. 3. The stabilization of low-spin states in strong crystal fields is evident from these diagrams. The crystal-field parameters, and especially the cubic crystal-field parameter   (also known as 10Dq) can be deduced from the wavelengths of optical transitions from the ground state to different excited states.

Figure 3: Tanabe–Sugano diagrams for: (a) a 3d⁶ ion and (b) a 3d⁷ ion. The high-spin–low-spin crossover is indicated by the vertical line.