Crystal Field Theory

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At almost exactly the same time that chemists were developing the valence-bond model for coordination complexes, physicists such as Hans Bethe, John Van Vleck, and Leslie Orgel were developing an alternative known as crystal field theory. This theory tried to describe the effect of the electrical field of neighboring ions on the energies of the valence orbitals of an ion in a crystal. Crystal field theory was developed by considering two compounds: manganese(II) oxide, MnO, and copper(I) chloride, CuCl.

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Octahedral Crystal Fields

Each Mn²⁺ ion in manganese(II) oxide is surrounded by six O^2- ions arranged toward the corners of an octahedron, as shown in the figure below. MnO is therefore a model for an octahedral complex in which a transition-metal ion is coordinated to six ligands.

What happens to the energies of the 4s and 4p orbitals on an Mn²⁺ ion when this ion is buried in an MnO crystal? Repulsion between electrons that might be added to these orbitals and the electrons on the six O^2- ions that surround the metal ion in MnO increase the energies of these orbitals. The three 4p orbitals are still degenerate, however. These orbitals still have the same energy because each 4p orbital points toward two O^2- ions at the corners of the octahedron.

Repulsion between electrons on the O^2- ions and electrons in the 3d orbitals on the metal ion in MnO also increases the energy of these orbitals. But the five 3d orbitals on the Mn²⁺ ion are no longer degenerate. Let's assume that the six O^2- ions that surround each Mn²⁺ ion define an XYZ coordinate system. Two of the 3d orbitals (3d_x²_-y² and 3d_z²) on the Mn²⁺ ion point directly toward the six O^2- ions, as shown in the figure below. The other three orbitals (3d_xy, 3d_xz, and 3d_yz) lie between the O^2- ions.

The energy of the five 3d orbitals increases when the six O^2- ions are brought close to the Mn²⁺ ion. However, the energy of two of these orbitals (3d_x²_-y² and 3d_z²) increases much more than the energy of the other three (3d_xy, 3d_xz, and 3d_yz), as shown in the figure below. The crystal field of the six O^2- ions in MnO therefore splits the degeneracy of the five 3d orbitals. Three of these orbitals are now lower in energy than the other two.

By convention, the d_xy, d_xz, and d_yz orbitals in an octahedral complex are called the t_2g orbitals. The d_x²_-y² and d_z² orbitals, on the other hand, are called the e_g orbitals.

The easiest way to remember this convention is to note that there are three orbitals in the t_2g set.

The difference between the energies of the t_2g and e_g orbitals in an octahedral complex is represented by the symbol _o. This splitting of the energy of the d orbitals is not trivial; _o for the Ti(H₂O)₆³⁺ ion, for example, is 242 kJ/mol.

The magnitude of the splitting of the t_2g and e_g orbitals changes from one octahedral complex to another. It depends on the identity of the metal ion, the charge on this ion, and the nature of the ligands coordinated to the metal ion.

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Tetrahedral Crystal Fields

Each Cu⁺ ion in copper(I) chloride is surrounded by four Cl^- ions arranged toward the corners of a tetrahedron, as shown in the figure below. CuCl is therefore a model for a tetrahedral complex in which a transition-metal ion is coordinated to four ligands.

Once again, the negative ions in the crystal split the energy of the d atomic orbitals on the transition-metal ion. The tetrahedral crystal field splits these orbitals into the same t_2g and e_g sets of orbitals as does the octahedral crystal field.

But the two orbitals in the e_g set are now lower in energy than the three orbitals in the t_2g set, as shown in the figure below.

To understand the splitting of d orbitals in a tetrahedral crystal field, imagine four ligands lying at alternating corners of a cube to form a tetrahedral geometry, as shown in the figure below. The d_x²_-y² and d_z² orbitals on the metal ion at the center of the cube lie between the ligands, and the d_xy, d_xz, and d_yz orbitals point toward the ligands. As a result, the splitting observed in a tetrahedral crystal field is the opposite of the splitting in an octahedral complex.

Because a tetrahedral complex has fewer ligands, the magnitude of the splitting is smaller. The difference between the energies of the t_2g and e_g orbitals in a tetrahedral complex (_t) is slightly less than half as large as the splitting in analogous octahedral complexes (_o).

_t = ⁴/_{9 o}

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Square-Planar Complexes

The crystal field theory can be extended to square-planar complexes, such as Pt(NH₃)₂Cl₂. The splitting of the d orbitals in these compounds is shown in the figure below.

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The Spectrochemical Series

The splitting of d orbitals in the crystal field model not only depends on the geometry of the complex, it also depends on the nature of the metal ion, the charge on this ion, and the ligands that surround the metal. When the geometry and the ligands are held constant, this splitting decreases in the following order.

Metal ions at one end of this continuum are called strong-field ions, because the splitting due to the crystal field is unusually strong. Ions at the other end are known as weak-field ions.

When the geometry and the metal are held constant, the splitting of the d orbitals decreases in the following order.

Ligands that give rise to large differences between the energies of the t_2g and e_g orbitals are called strong-field ligands. Those at the opposite extreme are known as weak-field ligands.

Because they result from studies of the absorption spectra of transition-metal complexes, these generalizations are known as the spectrochemical series. The range of values of for a given geometry is remarkably large. The value of _o is 100 kJ/mol in the Ni(H₂O)₆²⁺ ion, for example, and 520 kJ/mol in the Rh(CN)₆^3- ion.

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High-Spin Versus Low-Spin Octahedral Complexes

Once we know the relative energies of the d orbitals in a transition-metal complex, we have to worry about how these orbitals are filled. Degenerate orbitals are filled according to Hund's rules.

• One electron is added to each of the degenerate orbitals in a subshell before a second electron is added to any orbital in the subshell.
• Electrons are added to a subshell with the same value of the spin quantum number until each orbital in the subshell has at least one electron.

Octahedral transition-metal ions with d¹, d², or d³ configurations can therefore be described by the following diagrams.

When we try to add a fourth electron, we are faced with a problem. This electron could be used to pair one of the electrons in the lower energy (t_2g) set of orbitals or it could be placed in one of the higher energy (e_g) orbitals. One of these configurations is called high-spin because it contains four unpaired electrons with the same spin. The other is called low-spin because it contains only two unpaired electrons. The same problem occurs with octahedral d⁵, d⁶, and d⁷ complexes.

For octahedral d⁸, d⁹, and d¹⁰ complexes, there is only one way to write satisfactory configurations.

As a result, we have to worry about high-spin versus low-spin octahedral complexes only when there are four, five, six, or seven electrons in the d orbitals.

The choice between high-spin and low-spin configurations for octahedral d⁴, d⁵, d⁶, or d⁷ complexes is easy. All we have to do is compare the energy it takes to pair electrons with the energy it takes to excite an electron to the higher energy (e_g) orbitals. If it takes less energy to pair the electrons, the complex is low-spin. If it takes less energy to excite the electron, the complex is high-spin.

The amount of energy required to pair electrons in the t_2g orbitals of an octahedral complex is more or less constant. The amount of energy needed to excite an electron into the higher energy (e_g) orbitals, however, depends on the value of _o for the complex. As a result, we expect to find low-spin complexes among metal ions and ligands that lie toward the high-field end of the spectrochemical series. High-spin complexes are expected among metal ions and ligands that lie toward the low-field end of these series.

Compounds in which all of the electrons are paired are diamagneticthey are repelled by both poles of a magnet. Compounds that contain one or more unpaired electrons are paramagneticthey are attracted to the poles of a magnet. The force of attraction between paramagnetic complexes and a magnetic field is proportional to the number of unpaired electrons in the complex. We can therefore determine whether a complex is high-spin or low-spin by measuring the strength of the interaction between the complex and a magnetic field.