Crystal Field Theory
Crystal field theory (CFT) describes the breaking of orbital degeneracy in transition metal complexes due to the presence of ligands. CFT qualitatively describes the strength of the metal-ligand bonds. Based on the strength of the metal-ligand bonds, the energy of the system is altered. This may lead to a change in magnetic properties as well as color. This theory was developed by Hans Bethe and John Hasbrouck van Vleck .
When examining a single transition metal ion, the five d-orbitals have the same energy. When ligands approach the metal ion, some experience more opposition from the d-orbital electrons than others based on the geometric structure of the molecule. Since ligands approach from different directions, not all d-orbitals interact directly. These interactions, however, create a splitting due to the electrostatic environment.
For example, consider a molecule with octahedral geometry. Ligands approach the metal ion along the x, y, and z axes. Therefore, the electrons in the dz² and dx²-y² orbitals (which lie along these axes) experience greater repulsion. It requires more energy to have an electron in these orbitals than it would to put an electron in one of the other orbitals. This causes a splitting in the energy levels of the d-orbitals. This is known as crystal field splitting. For octahedral complexes, crystal field splitting is denoted by ∆o (or ∆oct).The energies of the dz² and dx²-y² orbitals increase due to greater interaction with the ligands. The dxy, dxz, and dyz orbitals decrease with respect to this normal energy level and become more stable.
d-Orbitals Splitting Different Energy Levels
Electrons in Orbitals
According to the Aufbau principle, electrons are filled from lower to higher energy orbitals. For the octahedral case above, this corresponds to the dxy, dxz, and dyz orbitals. Following Hund's rule, electrons are filled in order to have the highest number of unpaired electrons. For example, if one had a d3 complex, there would be three unpaired electrons. If one were to add an electron, however, it has the ability to fill a higher energy orbital ( dz² or dx²-y²) or pair with an electron residing in the dxy, dxz, or dyz orbitals. This pairing of the electrons requires energy (spin pairing energy). If the pairing energy is less than the crystal field splitting energy, ∆₀, then the next electron will go into the dxy, dxz, or dyz orbitals due to stability. This situation allows for the least amount of unpaired electrons, and is known as low spin. If the pairing energy is greater than ∆₀, then the next electron will go into the dz² or dx²-y² orbitals as an unpaired electron. This situation allows for the most number of unpaired electrons, and is known as high spin. Ligands that cause a transition metal to have a small crystal field splitting, which leads to high spin, are called weak-field ligands. Ligands that produce a large crystal field splitting, which leads to low spin, are called strong field ligands.
Low Spin, Strong Field (∆o˃P) High Spin, Weak Field (∆o˂P)
As mentioned above, CFT is based primarily on symmetry of ligands around a central metal/ion and how this anisotropic (properties depending on direction) ligand field affects the metal's atomic orbitals; the energies of which may increase, decrease or not be affected at all. Once the ligands' electrons interact with the electrons of the d-orbitals, the electrostatic interactions cause the energy levels of the d-orbital to fluctuate depending on the orientation and the nature of the ligands. For example, the oxidation state and the strength of the ligands determine splitting; the higher the oxidation state or the stronger the ligand, the larger the splitting. Ligands are classified as strong or weak based on the spectrochemical series:
I- < Br- < Cl- < SCN- < F- < OH- < ox2-< ONO- < H2O < SCN- < EDTA4- < NH3 < en < NO2- < CN-
In addition to octahedral complexes, two common geometries observed are that of tetrahedral and square planar. These complexes differ from the octahedral complexes in that the orbital levels are raised in energy due to the interference with electrons from ligands. For the tetrahedral complex, the dxy, dxz, and dyz orbitals are raised in energy while the dz², dx²-y² orbitals are lowered. For the square planar complexes, there is greatest interaction with the dx²-y² orbital and therefore it has higher energy. The next orbital with the greatest interaction is dxy, followed below by dz². The orbitals with the lowest energy are the dxz and dyz orbitals. There is a large energy separation between the dz² orbital and the dxz and dyz orbitals, meaning that the crystal field splitting energy is large. We find that the square planar complexes have the greatest crystal field splitting energy compared to all the other complexes. This means that most square planar complexes are low spin, strong field ligands.
Description of d-Orbitals
In order to understand CFT, one must understand the description of the lobes:
In an octahedral complex, there are six ligands attached to the central transition metal. The d-orbital splits into two different levels. The bottom three energy levels are named dxy, dxz, and dyz (also referred to as t2g). The two upper energy levels are named dx²-y², and dz² (also referred to as eg). The reason they split is because of the electrostatic interactions between the electrons of the ligand and the lobes of the d-orbital. In an octahedral, the electrons are attracted to the axes. Any orbital that has a lobe on the axes moves to a higher energy level. This means that in an octahedral, the energy levels of eg are higher (0.6∆o) while t2g is lower (0.4∆o). The distance that the electrons have to move from t2g from eg and it dictates the energy that the complex will absorb from white light, which will determine the color. Whether the complex is paramagnetic or diamagnetic will be determined by the spin state. If there are unpaired electrons, the complex is paramagnetic; if all electrons are paired, the complex is diamagnetic.
In a tetrahedral complex, there are four ligands attached to the central metal. The d orbitals also split into two different energy levels. The top three consist of the dxy, dxz, and dyz orbitals. The bottom two consist of the dx²-y² and dz² orbitals. The reason for this is due to poor orbital overlap between the metal and the ligand orbitals. The orbitals are directed on the axes, while the ligands are not. This corresponds
The difference in height of the splitting energy is called delta tetrahedral (Δt). The size of Δt tends to be smaller than the spin pairing energy, so it is usually high spin. Δt = 0.44Δo.
Square Planar Complexes
In a square planar, there are four ligands as well. However, the difference is that the electrons of the ligands are only attracted to the xy plane. Any orbital in the xy plane has a higher energy level. There are four different energy levels for the square planar (from the highest energy level to the lowest energy level): dx2-y2, dxy, dz2, and both dxz and dyz.
The difference in height of the splitting energy (from highest orbital to lowest orbital) is called delta square planar (Δsp). The size of Δsp tends to be larger then the pairing energy, so the ligands are usually low spin. Δsp = 1.74Δo.
Question: For the complex ion [Fe(Cl)6]3- determine the number of d electrons for Fe, sketch the d-orbital energy levels and the distribution of d electrons among them, list the number of lone electrons, and label whether they are paramagnetic or dimagnetic.
Step 1: Determine the oxidation state of Fe. Here it is Fe3+. Based on its electron configuration, Fe3+ has 5 d-electrons.
Step 2: Determine the geometry of the ion. Here it is an octahedral which means the energy splitting should look like:
Step 3: Determine whether the ion is low or high spin by looking at the spectrochemical series. Cl- is high spin. Therefore, electrons fill all orbitals before being paired.
Step four: Count the number of lone electrons. Here, there are 5 electrons.
Step five: lone pairs are paramagnetic. This ion is paramagnetic.
Question: A tetrahedral metal complex absorbs 545 nm. What is the crystal field splitting ∆o? What is the color of the metal?
Answer: Δt = hc/λ, Δt = 0.44Δo
∆t = (6.626 x 10-34 Js)(3 x 108 m/s)/545 x 10-9 m=3.65 x 10-19J
∆o = ∆t/0.44 = 3.65 x 10-19J/0.44 = 8.30 x 10-14J, RED (via color wheel)
For each of the following, sketch the d-orbital energy levels and the distribution of d electrons among them, state the geometry, list the number of d-electrons, list the number of lone electrons, and label whether they are paramagnetic or dimagnetic. :
1. octahedral, 2, 2, paramagnetic
2. tetrahedral, 8, 2, paramagnetic
3. octahedral, 6, 4, paramagnetic, high spin
4. octahedral, 6, 0, diamagnetic, low spin
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