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Ligand Field Theory

The ligand field theory (LFT) describes the interaction of the ligand’s electron pairs with the d-orbitals of the central transition metal. A bunch of physical and chemical properties like magnetic behavior, absorption spectra and preferred occurrence of oxidation numbers and coordinations can be explained by this theory.[1]

A transition metal (ion) possesses five d-orbitals. The five d-orbitals are the dz2, dx2y2, dxy, dxz, dyz orbitals. Bringing a metal atom or ion into a (hypothetically) spherically symmetrical ligand field causes an elevation of the energy of the d-orbitals. Because interactions are identically in all spatial directions, all d-orbitals have the same energy and the orbitals are still degenerated.

Real interactions between ligands and central metal atoms do not show a spherically symmetrical ligand field. Consequently, the energetic levels of the orbitals differ depending on the geometrical position of the ligands as well as the properties of both, the ligand and the central atom. In the following sections, the characteristics of octahedral and tetrahedral complexes will be discussed as examples. For information on other geometrical coordinations the reader should refer to textbooks of inorganic chemistry.[1;2]


Figure 1: Octahedral coordination of d-orbitals.

Octahedral Complexes In an octahedral complex, the transition metal ion in the center is coordinated by 6 ligands, one ligand on each position of the axes of coordinates. As shown in figure 1, the interaction of the d-orbitals with the ligand orbitals differs depending on the orbital under consideration. Interaction of the orbitals dxy, dxz, dyz, that are lying between the axes of coordinates, is less pronounced as for the orbitals dz2 and dx2y2, that are lying on the axes. This results in a separation of the d-orbitals in two groups with different energy levels as shown in Figure 0.2. The dxy, dxz, dyz orbitals, referred to as t2g for symmetry reasons, have a lower energy level as the dz2 and dx2y2 orbitals, referred to as eg states.

The energy difference between the t2g and the eg state equals ∆O = 10 Dq. Because the sum of the orbital energies has to be equal to the energy of the spherically symmetrical ligand field, the energy of the of the t2g orbitals is decreased by 2/5∆O and the energy of the eg is increased by 3/5∆O. This leads to a stabilization of the t2g orbitals and to a destabilization of the eg orbitals. In total, the barycenter of the orbital energies is equal to the energy for a spherically symmetrical ligand field. The overall energy increase of the d-orbitals in an octahedral complex compared to the free ion depends on the ligands and metal ions and is around 20 eV to 40 eV, ∆O is in the range of 1 eV to 4 eV.[1] LFT2

Figure 2: Ligand field scheme of an octahedral complex.

Tetrahedaral Complexes For complexes showing a tetrahedral coordination the situation is different to that of an octahedral coordinated one. Interaction between the orbitals lying between the axes of coordinates (dxy, dxz, dyz) is pronounced, while the interaction with the orbitals lying on the axes (dz2 an dx2y2) is weaker (see Figure 3). With this, the splitting of the orbitals is inverse to that in an octahedral ligand field. Figure 4 illustrates the resulting ligand field.


Figure 3: Tetrahedral coordination of d-orbitals.

Again, the dxy, dxz, dyz orbitals (t2 states) and the dz2 and dx2y2 (e states) are degenerated. Because only four ligands interact with the metal center, the ligand field splitting between these two states is lower as observed for octahedral complexes (6 ligands). The ligand field spliting energy is approximately ∆t = 4/9|∆O| = 4.45 Dq and splits between the two states in such a way, that the barycenter of the orbital energies is equal to the energy for a spherically symmetrical ligand field.

Spin States In some transition metal complexes, the electronic configuration of the transition metal can basically possess two different spin states for the same oxidation state (i.e. number of d-electrons): the high-spin and the low-spin configurations. These configurations differ in the kind of occupation of the different orbitals. Ont the one hand, it is possible to occupy all orbitals by unpaired electrons only (high-spin), or by occupying certain orbitals with paired electrons (low-spin). Because energy has to be spend for pairing electrons, it is a question of the energy difference between the t2g and the eg state, whether it is energetically more beneficial to occupy the orbitals with paired or unpaired electrons. For instance, in the case of octahedral coordination, population of one dxy, dxz, dyz orbital results in a stabilzation by 4 Dq.


Figure 4: Ligand field scheme of a tetrahedral complex.

  Addition of a second electron will gain further 4 Dq and for d3 configuration the stabilization energy is −12 Dq. For the d4 configuration two cases occur. On the one hand, by population of a further t2g orbital the stabilization energy would be −16 Dq. But for this energy is required for spin pairing, which reduces the overall energy gain. Therefore, occupation of the eg states may result in larger energy gains, although the ligand field stabilization energy is only −6 Dq (-12 + 6). Thus, if the ligand field splitting energy (∆ = 10 Dq) exceeds the spin pairing energy, the formation of low spin complexes is preferred. For an octahedral ligand field, the high-spin and low-spin configurations are shown for the d5 configuration of the transition metal in figure 5. These two different spin states exist in the octahedral coordination only for d4, d5, d6, d7 electron configurations of the transition metal. In d1 - d3 only the low energy orbitals are occupied and for d8 - d10 the low energy orbitals (t2g orbitals) are always occupied by six electrons. Which of the two spin state is preferred depends on the ligands and the center metal ion. The absolute values for the ligand field splitting energy ∆O depends on the nature of the metal center and the ligands involved. The order of the ligand field splitting energy and with this the preferred spin state can be derived from the spectrochemical series of the ligands and metals/ions. The spectrochemical series for the ligands is:


A lower value in this series means a lower ligand field splitting energy (∆ in figure 5) and thus more likely a high-spin complex.

The spectrochemical series for the metal ions is:



The order of the metal ions follows the same order as for the ligands. A position on the left equals to a smaller ligand field splitting energy. The 3d-metals are found on the left side, while the 4d- and 5d-metals are on the right side. As a result, high spin complexes are rarely known for 4d- and 5d-metal complexes.


Figure 5: Low-spin (left) and high-spin configuration (right) of an octahedral complex.

The ligand field splitting also depends on the coordination environment. For example the ligand field splitting in tetrahedral complexes is lower than in octahedral complexes. Hence, due to the smaller energy difference between the t2 and the e state, low-spin complexes are not known for tetrahedral complexes till now.


  1. RIEDEL, E.: Anorganische Chemie (German Edition), 2004, Walter de Gruyter & Co, ISBN 978–3–11–018168–5.
  2. HOLLEMAN, A. F.; WIBERG, E.; WIBERG, N.: Lehrbuch der Anorganischen Chemie., 1995, Gruyter, ISBN 978–3–11–012641–9.

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