A unifying property of the transition metals is their ability to form what are known as complexes: species in which ligands (neutral or anionic molecules with a lone pair of electrons) surround a metal ion at the centre and bind to it through donation of non-bonding electrons into the metal ion's vacant d-orbitals. A simple example is the hexaaquatitanium(III) ion, [Ti(H2O)6]3+, in which six molecules of water, a neutral ligand, bind to a titanium(III) ion through donation of the lone pair on oxygen, forming an octahedral geometry around the titanium centre.

Transition metal, or d-metal complexes are an extremely important phenomenon of inorganic chemistry and have many interesting physical properties, such as magnetism and deep colour. A simple but often powerful method of describing and explaining the properties of these complexes is given by crystal field theory.

Assumptions

Crystal field theory is based upon a simple model of the electronic structure of d-metal complexes. The ligands, whether neutral or anionic, are treated as point charges: that is, the negative charge on the donated electrons is treated as if it occupied a single point in space. This is a common approach in the physics and chemistry of electronic charge, and it resembles the model of a crystal as an array of point charges, hence the name crystal field theory.

The significance of this assumption is that the incoming electrons from the ligands interact differently with the five different d-orbitals on the transition metal ion. All d-block elements have five d-orbitals: electron orbitals whose value of l, the angular momentum quantum number, is 2. These are not all filled in any transition metal cation, but they exist whether filled or not, and receive the incoming electrons from ligands in a complex.

Orbital splitting

The five d-orbitals are given the symbols dxy, dzx, dyz, dx2-y2 and dz2. In a complex they are all differently aligned relative to the incoming charge. Depending on the geometry of the complex, some of the d-orbitals will point directly towards the ligands, while some will point between them. Those which point at the ligands will experience more repulsion between their own electrons and those of the incoming ligands, than will those which do not point directly at them. Thus the orbitals pointing at the ligands will be less stable and higher in energy. Now all the d-orbitals are no longer equivalent, giving rise to the phenomenon of orbital splitting, and the difference in energy between the more and less repelled orbitals is called the crystal field splitting parameter.

Although the overall effect of electron donation is added stability, due to the attractive interaction between the ligands and the positive metal ion, the difference in repulsive interaction experience by the d-orbitals has important consequences for d-metal complex chemistry.

Octahedral complexes as a typical example

As an example, take the commonest d-metal complex geometry, octahedral, in which six ligands surround each metal ion and describe the shape of an octahedron. In such a complex, the orbitals dxy, dzx and dyz point between the ligands and are less repelled than the dx2-y2 and dz2 orbitals, which point at the ligands. The crystal field splitting parameter, the difference in energy, for an octahedral complex is given the symbol ΔO. Each of the first three, less repelled orbitals is lower than the mean energy by two-fifths ΔO. Each of the second two, more repelled orbitals is higher than the mean energy by three-fifths ΔO. (The mean energy itself is unchanged from that of the mean energy of the orbitals in a free atom.)

All this talk concerns orbitals. How do the metal's d-electrons now occupy those orbitals? (All orbitals can hold up to two electrons.) If all five d-orbitals were equivalent in energy, as in a free atom, electrons would obey Hund's rule and occupy, as far as possible, different orbitals, and have equivalent spins. However, three of the orbitals are now lower in energy than the other two. If there are no more than three d-electrons, there is no problem: they will each occupy one of the three lower-energy orbitals. If there are more than three, the subsequent electrons might reoccupy the lower energy orbitals, or they might go into the higher-energy ones.

If the former, they experience a repulsion called the pairing energy arising from the double-occupation of an electron orbital. If the latter, they must overcome the increase in energy, ΔO, between the less and more repelled d-orbitals. Which of these happens depends on the value of ΔO. If it is greater than the pairing energy, electrons will first doubly occupy the lower energy orbitals before singly occupying the higher energy ones. This is called the strong-field case, and complexes where it happens are called low-spin. If it is less than the pairing energy, electrons will instead occupy the higher energy orbitals before doubly occupying the lower energy ones. This is called the weak-field case, and complexes in which it happens are called high-spin.

The value of ΔO depends on the identity both of the ligands and of the metal. As far as metals are concerned, ΔO tends to increase with oxidation number, and to increase down a group: both are due to increased metal-ligand interaction. As for the ligands, they form a series, from those tending to lower values of ΔO, to those which give rise to higher values. In general, values of ΔO increase from halide ligands, to oxygen-based ligands, to nitrogen-based ligands, to carbon-based ligands.

For example, consider the octahedral complex [Cr(H2O)6]2+. Here we have chromium, which is in the first row of the d-block, in a low oxidation state (+2), and with an oxygen-based ligand, water. All these factors tend towards a low value of ΔO, hence the complex is high-spin: chromium(II) has four d-electrons, and the fourth of these occupies one of the higher-energy d-orbitals.

Conversely, consider [Ru(CN)6]2-. Here we have ruthenium, which is in the second row of the d-block, has the oxidation state +4 and is complexed to cyanide, a ligand consisting of carbon and nitrogen. Hence it has a high value of ΔO and is low-spin: its fourth and final d-electron occupies one of the lower-energy orbitals, alongside the first three.

Other geometries

The same basic principles apply to the other common geometries of d-metal complexes, although each geometry must be taken separately. In tetrahedal complexes, for example, four ligands surround each metal ion and describe the shape of a tetrahedron. In this case, because of the relative alignments of the ligands and the d-orbitals, the dx2-y2 and dz2 orbitals are lower in energy than the other three, but the same principle of strong- and weak-field ligands applies. The crystal field splitting parameter for tetrahedral complexes, ΔT, is lower than ΔO and such complexes are usually high-spin.

Meanwhile in square planar complexes, where four equiplanar ligands describe a square shape around the metal, the situation is slightly more complicated: the five d-orbitals are split into four different energy levels. Here, the crystal field splitting parameter, ΔSP, is higher than ΔO.

Crystal field stabilization energy

In many complexes the phenomenon of orbital splitting affords a slight extra stabilization of the metal ion, further to that of complex formation itself. This decrease in energy is called the crystal field stabilization energy and for octahedral complexes it is given by the formula CFSE = (-0.4x + 0.6y) ΔO, where x is the number of electrons in the lower energy-orbitals and y the number in the higher-energy orbitals.

For example, consider a low-spin iron(II) octahedral complex. In free iron(II), six electrons occupy five degenerate (equal energy) d-orbitals. In a low-spin octahedral complex, all six electrons occupy the three lower-energy orbitals. Since each of these orbitals is lower than the mean energy by two-fifths ΔO, there is a reduction in energy of twelve-fifths ΔO. However, there are now also three electron-pairings (one in each orbital), whereas before there was only one. This increase the total energy by 2P, where P is the pairing energy. Hence the total energy reduction - the crystal field stabilization energy - is given by 12/5ΔO - 2P.

Physical properties of d-metal complexes

As I said earlier, crystal field theory can explain many of the important properties of such complexes, such as colour and magnetism.

Colour arises when substances are able to absorb certain frequencies of visible light in order to promote electrons to higher energy levels. The remaining, non-absorbed frequencies are reflected back as coloured light. In many transition metal complexes, the crystal field splitting parameter, e.g. ΔO, is equal to the energy of a certain frequency of light, so this light can be absorbed to promote electrons from lower-energy to higher-energy d-orbitals: this is d-d transition. For example, copper(II) ions are well-known for their striking blue colour in aqueous solution: that is, when they form a complex with water.

The magnetism of a substance is closely related to the presence and behaviour of unpaired electrons, that is, electrons which singly occupy an orbital. The two most common forms of magnetism are paramagnetism, in which a substance is attracted into a magnetic field, and diamagnetism, in which it is repelled by one. Among transition metal complexes, the former is found when the metal ions have unpaired electrons, and the latter arises when they do not.

For paramagnetic substances, the magnitude of the paramagnetism can be estimated from the spin-only formula: μ = √ (n2+2n), where n is the number of unpaired electrons. μ is the paramagnetism and is given in units of Bohr magnetons. High-spin complexes have more unpaired electrons, so they tend to be more paramagnetic.

Other less obvious physical properties of d-metal complexes can also be explained by crystal field theory. One example is metal ion hydration energy, which varies across the top row of the d-block in a way related to the crystal field stabilization energies of these metals with water.

Crystal field theory is, however, far from perfect. Its treatment of ligands as point charges is not always valid, and sometimes a more sophisticated approach must be used. This approach, which is outside the scope of this node, is known as ligand field theory and models the interactions between metal and ligand orbitals as based on molecular orbital theory.


Reference: Shriver, D.F. and Atkins, P.W. Inorganic Chemistry (third edition). 2001, Oxford University Press.

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