# Spectroscopic and Magnetic Properties

### Crystal Field Theory

Crystal field theory is a model used to predict bonding in octahedral and tetrahedral complexes considering a central cation surrounded by anionic (ligand) charge points in 3D space using simple VSEPR concepts.

Main-group elements form ionic or covalent compounds in order to satisfy the octet rule. However, this is not the case for transition metals. The bonding rules that govern coordination compounds are explained by an electrostatic model called crystal field theory that is used to describe and predict differences between $n$d orbital sets in transition metal complexes. This describes the structures of coordination compounds as well as their colors and magnetic properties.

In crystal field theory the focus is on the nonbonding electrons rather than the bonds between the central metal ion and the ligands. The ligands and the metal are considered as infinitesimally small point charges.

Because electrons all carry a negative charge, the electrons on the ligands repel those found on the central metal ion. In the tetrahedral configuration, the ligands form a triangular pyramid with the cation at the center of the pyramid. In the octahedral configuration, the ligand bonds are found along the Cartesian axes (x, y, z). The five d orbitals have lobes that lie along (x2-y2 and z2) and between (xz, yz, xy) on the Cartesian axes, respectively.

#### Orientation of d Orbitals

When the metal ion is isolated (no ligands are attached to it; gas phase), the five orbitals, $d_{{z}^2}$, $d_{x^2-y^2}$, $d_{xy}$, $d_{yz}$, and $d_{xz}$, all have the same energies. However, when ligands bond along the axes, this is no longer the case. The ${d_{{z^2}}}$and ${d_{{x^2}-{y^2}}}$ orbitals, collectively known as the $e_g$ orbitals, lie along the axes. The ${d_{xy}}$, ${d_{yz}}$, and ${d_{xz}}$ orbitals, collectively known as the ${t_{2g}}$ orbitals, lie between the axes. This means that electrons in the $e_g$ orbitals experience greater repulsion than those in the ${t_{2g}}$ orbitals since the electrons of the ligands are points directly at the electrons of the metal. Thus, the energy of electrons in the $e_g$ orbitals are higher than those in the ${t_{2g}}$ orbitals, that is, more energy is required to fill the $e_g$ orbitals than to fill the ${t_{2g}}$ orbitals. The difference between these energies is known as crystal field splitting energy ($\Delta$). The ${t_{2g}}$ and the $e_g$ change places when the coordination complex has tetrahedral geometry. The magnitude of $\Delta$ depends on the charge of the central metal ion, the orbitals the metal is using, and the nature of the ligands. The spectrochemical series is a list of ligands ordered according to their ability to induce crystal field splitting for a common metal center. Ligands on the left side of the series give small crystal field splittings while those on the right give greater energy separations between the ${t_{2g}}$ and $e_g$ orbital sets. A sample spectrochemical series can be formed using the following common ligands.
$\begin{gathered}{\rm{O}_2}^{2-}\lt\rm{I}^-\lt\rm{S}^{2-}\lt\rm{SCN}^-\rm{(S\rm{-}bonded)}\lt\rm{Cl}^-\lt\rm{OH}^-\lt\rm{C}_2{\rm{O}_4}^{2-}\lt\rm{H}_2\rm{O}\\\lt\rm{NCS}^-\rm{(N\rm{-}bonded)}\lt\rm{CH}_3\rm{CN}\lt\rm{en}{\lt\rm{NO}_2}^-\lt\rm{CN}^-\lt\rm{CO}\end{gathered}$
In this spectrochemical series, the ligand with the lowest $\Delta$ is peroxide ion (O22−), and the ligand with the highest $\Delta$ is carbon monoxide (CO). Ligands with low $\Delta$ (those on the left side of the series) are known as weak-field ligands, and ligands with high $\Delta$ (those on the right side of the series) are known as strong-field ligands. Metal ions can also be arranged according to their $\Delta$.
$\rm{Mn}^{2+}\lt\rm{Ni}^{2+}\lt\rm{Co}^{2+}\lt\rm{Fe}^{2+}\lt\rm{V}^{2+}\lt\rm{Cu}^{2+}\lt\rm{Fe}^{3+}\lt\rm{Cr}^{3+}\lt\rm{V}^{3+}\lt\rm{Co}^{3+}$
Similarly, in this spectrochemical series, the metal ion with the lowest $\Delta$ is manganese ion (Mn2+), and the one with the highest $\Delta$ is cobalt(III) ion (Co3+).

Generally, it is not possible to predict whether a particular ligand will have a strong or weak effect on a particular metal ion. However, $\Delta$ tends to increase with increasing oxidation state, and $\Delta$ tends to increase down a group on the periodic table.

The energy required for two electrons to occupy a single orbital is known as pairing energy (P). Because electrons in a single orbital repel each other, electrons will occupy orbitals singly, in the lowest energy levels possible, before they pair, according to Hund's rule. Orbitals that have the same energy level are degenerate orbitals. Electrons occupy each degenerate orbital singly before they pair.

For example, consider two different sets of ligands pairing with iron(II) ion (Fe2+ or FeII). When no ligands are bonded to the metal, the electrons occupy the lowest available orbital. When the weak-field ligand water (H2O) binds, $\Delta$ is less than P, so electrons fill the $e_g$ orbitals singly. A high-spin complex is a complex in which electrons fill higher orbitals singly and the pairing energy is large relative to the crystal field splitting energy. However, when a strong-field ligand, such as cyanide (CN), binds, P is less than $\Delta$, so electrons will pair in the ${t_{2g}}$ orbitals. A low-spin complex is a complex in which electrons preferentially pair in lower energy orbitals and the pairing energy is small relative to the crystal field splitting energy.

### Magnetic Properties

The magnetic properties of a coordination complex are affected by the arrangement of electrons, ligand field strength, electron-electron repulsion and pairing energies.
Molecules that contain unpaired electrons are paramagnetic, that is, attracted to magnetic fields. Molecules that contain no unpaired electrons are diamagnetic, that is, repelled by magnetic fields. Because it is possible to predict the way in which electrons will fill orbitals based on the energies required to pair electrons, it is possible to predict the magnetic moments, the strength and orientation of a magnetic field, of coordination complexes. The greater the number of unpaired electrons in an ion, complex, or molecule, the larger the magnetic moment. The magnetic moment is directly related to the number of unpaired electrons present. Therefore, high-spin complexes tend to be paramagnetic, and low-spin complexes tend to be diamagnetic. For example, high-spin and cationic [FeII(OH2)6]2+ complexes have four unpaired electrons (a ${t_{2g}}^4{e_g}^2$ electronic configuration), so it is paramagnetic. On the other hand, low-spin [FeII(CN)6]4- anions have no unpaired electrons (a ${t_{2g}}^6$ electronic configuration), so they are diamagnetic. Recall that water (aqua ligands) are weak-field (small $\Delta$) while cyanides (CN) are strong-field ligands (big $\Delta$).

### Spectroscopic Properties

The arrangement of electrons in a coordination complex or transition metal ion and their interaction with light results in the bright colors associated with them.

The ways in which electrons are distributed in orbitals is related to the energy of those electrons. Light energy is absorbed or reflected differently depending on the arrangement of electrons in the complex. For this reason, the distribution of electrons also affects the color of the coordination complex.

Most main-group elements are not brightly colored, because they absorb visible wavelengths of light and reflect wavelengths that are outside the range of visible light (e.g., ultraviolet region of the electromagnetic spectrum). In comparison, many transition complexes absorb visible light wavelengths, as electronic transitions between d orbitals are at lower energies—that is, in the visual range of our eyes. When one or a few wavelengths of visible light are absorbed and the rest are reflected, the eye perceives color based on the reflected light. In other words, an object appears red if light of other wavelengths is all absorbed and red frequency wavelengths are reflected to reach a person's eye. Transition metals often reflect some of the wavelengths of visible light, resulting in bright colors, whether the metal is in its elemental form or bonded to ligands in a coordination complex.

Coordination complexes fill orbitals according to the lowest energy requirements; that is, orbitals are filled such that the electrons are at their lowest energy state. When light strikes these electrons, it excites them to a higher state. Electrons move from the ${t_{2g}}$ orbital to the $e_g$ orbital, absorbing the energy from a specific wavelength of light (or a small range of wavelengths). Because some wavelengths of light have been absorbed, the complex appears brightly colored to human eyes. For example, [Fe(H2O)6]2+ has four unpaired electrons, two of which occupy the ${t_{2g}}$ orbitals. When white light strikes the complex, a photon excites one of these electrons to the $e_g$ orbital. The amount of energy required to do so is equal to $\Delta$ for the ligand. In this case, the complex absorbs red light and appears green.

Furthermore, different oxidation states of the metal ion can result in different colors. For example, vanadium is purple at +2, green at +3, blue at +4, and yellow at +5.