Crystal Field Theory

Crystal Field Theory

To explain the observed behavior of transition metal complexes (such as how colors arise), a model involving electrostatic interactions between the electrons from the ligands and the electrons in the unhybridized d orbitals of the central metal atom has been developed. This electrostatic model is crystal field theory (CFT). It allows us to understand, interpret, and predict the colors, magnetic behavior, and some structures of coordination compounds of transition metals.

CFT focuses on the nonbonding electrons on the central metal ion in coordination complexes not on the metal-ligand bonds. Like valence bond theory, CFT tells only part of the story of the behavior of complexes. However, it tells the part that valence bond theory does not. In its pure form, CFT ignores any covalent bonding between ligands and metal ions. Both the ligand and the metal are treated as infinitesimally small point charges.

All electrons are negative, so the electrons donated from the ligands will repel the electrons of the central metal. Let us consider the behavior of the electrons in the unhybridized d orbitals in an octahedral complex. The five d orbitals consist of lobe-shaped regions and are arranged in space, as shown in the figure below. In an octahedral complex, the six ligands coordinate along the axes.

The directional characteristics of the five d orbitals are shown here. The shaded portions indicate the phase of the orbitals. The ligands (L) coordinate along the axes. For clarity, the ligands have been omitted from the \({d}_{{x}^{2}\text{−}{y}^{2}}\) orbital so that the axis labels could be shown.

In an uncomplexed metal ion in the gas phase, the electrons are distributed among the five d orbitals in accord with Hund’s rule because the orbitals all have the same energy. However, when ligands coordinate to a metal ion, the energies of the d orbitals are no longer the same.

In octahedral complexes, the lobes in two of the five d orbitals, the \({d}_{{z}^{2}}\) and \({d}_{{x}^{2}\text{−}{y}^{2}}\) orbitals, point toward the ligands (see the figure above). These two orbitals are called the e_g orbitals (the symbol actually refers to the symmetry of the orbitals, but we will use it as a convenient name for these two orbitals in an octahedral complex).

The other three orbitals, the d_xy, d_xz, and d_yz orbitals, have lobes that point between the ligands and are called the t_2g orbitals (again, the symbol really refers to the symmetry of the orbitals). As six ligands approach the metal ion along the axes of the octahedron, their point charges repel the electrons in the d orbitals of the metal ion.

However, the repulsions between the electrons in the e_g orbitals (the \({d}_{{z}^{2}}\) and \({d}_{{x}^{2}\text{−}{y}^{2}}\) orbitals) and the ligands are greater than the repulsions between the electrons in the t_2g orbitals (the d_zy, d_xz, and d_yz orbitals) and the ligands. This is because the lobes of the e_g orbitals point directly at the ligands, whereas the lobes of the t_2g orbitals point between them. Thus, electrons in the e_g orbitals of the metal ion in an octahedral complex have higher potential energies than those of electrons in the t_2g orbitals. The difference in energy may be represented as shown in the figure below.

In octahedral complexes, the e_g orbitals are destabilized (higher in energy) compared to the t_2g orbitals because the ligands interact more strongly with the d orbitals at which they are pointed directly.

The difference in energy between the e_g and the t_2g orbitals is called the crystal field splitting and is symbolized by Δoct, where oct stands for octahedral.

The magnitude of Δ_oct depends on many factors, including the nature of the six ligands located around the central metal ion, the charge on the metal, and whether the metal is using 3d, 4d, or 5d orbitals. Different ligands produce different crystal field splittings. The increasing crystal field splitting produced by ligands is expressed in the spectrochemical series, a short version of which is given here:

\(\underset{\text{a few ligands of the spectrochemical series, in order of increasing field strength of the ligand}}{\xrightarrow{\text{I}^{-}<\text{Br}^{-}\text{Cl}^{-}<\text{F}^{-}<\text{H}_2\text{O}<\text{C}_2\text{O}_4^{\;\;2-}<\text{NH}_3<en<\text{NO}_2^{\;\;-}<\text{CN}^{-}}}\)

In this series, ligands on the left cause small crystal field splittings and are weak-field ligands, whereas those on the right cause larger splittings and are strong-field ligands. Thus, the Δ_oct value for an octahedral complex with iodide ligands (I⁻) is much smaller than the Δ_oct value for the same metal with cyanide ligands (CN⁻).

Electrons in the d orbitals follow the aufbau (“filling up”) principle, which says that the orbitals will be filled to give the lowest total energy, just as in main group chemistry. When two electrons occupy the same orbital, the like charges repel each other. The energy needed to pair up two electrons in a single orbital is called the pairing energy (P). Electrons will always singly occupy each orbital in a degenerate set before pairing. P is similar in magnitude to Δ_oct. When electrons fill the d orbitals, the relative magnitudes of Δ_oct and P determine which orbitals will be occupied.

In [Fe(CN)₆]⁴⁻, the strong field of six cyanide ligands produces a large Δ_oct. Under these conditions, the electrons require less energy to pair than they require to be excited to the e_g orbitals (Δ_oct > P). The six 3d electrons of the Fe²⁺ ion pair in the three t_2g orbitals (see the figure below). Complexes in which the electrons are paired because of the large crystal field splitting are called low-spin complexes because the number of unpaired electrons (spins) is minimized.

Iron(II) complexes have six electrons in the 5d orbitals. In the absence of a crystal field, the orbitals are degenerate. For coordination complexes with strong-field ligands such as [Fe(CN)₆]⁴⁻, Δ_oct is greater than P, and the electrons pair in the lower energy t_2g orbitals before occupying the eg orbitals. With weak-field ligands such as H₂O, the ligand field splitting is less than the pairing energy, Δ_oct less than P, so the electrons occupy all d orbitals singly before any pairing occurs.

In [Fe(H₂O)₆]²⁺, on the other hand, the weak field of the water molecules produces only a small crystal field splitting (Δ_oct < P). Because it requires less energy for the electrons to occupy the e_g orbitals than to pair together, there will be an electron in each of the five 3d orbitals before pairing occurs. For the six d electrons on the iron(II) center in [Fe(H₂O)₆]²⁺, there will be one pair of electrons and four unpaired electrons (see the figure above).

Complexes such as the [Fe(H₂O)₆]²⁺ ion, in which the electrons are unpaired because the crystal field splitting is not large enough to cause them to pair, are called high-spin complexes because the number of unpaired electrons (spins) is maximized.

A similar line of reasoning shows why the [Fe(CN)₆]³⁻ ion is a low-spin complex with only one unpaired electron, whereas both the [Fe(H₂O)₆]³⁺ and [FeF₆]³⁻ ions are high-spin complexes with five unpaired electrons.

Example

CFT for Other Geometries

CFT is applicable to molecules in geometries other than octahedral. In octahedral complexes, remember that the lobes of the e_g set point directly at the ligands. For tetrahedral complexes, the d orbitals remain in place, but now we have only four ligands located between the axes (see the figure below). None of the orbitals points directly at the tetrahedral ligands. However, the e_g set (along the Cartesian axes) overlaps with the ligands less than does the t_2g set. By analogy with the octahedral case, predict the energy diagram for the d orbitals in a tetrahedral crystal field. To avoid confusion, the octahedral e_g set becomes a tetrahedral e set, and the octahedral t_2g set becomes a t₂ set.

This diagram shows the orientation of the tetrahedral ligands with respect to the axis system for the orbitals.

Solution

Since CFT is based on electrostatic repulsion, the orbitals closer to the ligands will be destabilized and raised in energy relative to the other set of orbitals. The splitting is less than for octahedral complexes because the overlap is less, so Δ_tet is usually small \(({\text{Δ}}_{\text{tet}}=\;\cfrac{4}{9}\;{\text{Δ}}_{\text{oct}}):\)

The other common geometry is square planar. It is possible to consider a square planar geometry as an octahedral structure with a pair of trans ligands removed. The removed ligands are assumed to be on the z-axis. This changes the distribution of the d orbitals, as orbitals on or near the z-axis become more stable, and those on or near the x- or y-axes become less stable. This results in the octahedral t_2g and the e_g sets splitting and gives a more complicated pattern with no simple Δ_oct. The basic pattern is: