Introduction. The purpose of this handout is to provide a simple model for the bonding in transition-metal complexes as a framework for understanding and interpreting the most important properties of these complexes. We shall deal mostly with the first row transition metals such as iron and copper as these are most important in industry and in living organisms. The most important question is the very existence of complexes and the nature of the coordinate covalent bond formed between a ligand and a transition metal species. Accordingly, the molecular-orbital (MO) model will be presented. The crystal-field model which is presented in the text will not be treated here because it does not address the origin of bonding in complexes.

Any successful bonding theory must address the following characteristics of complexes: 1) the nature of the coordinate covalent bond, 2) the dependence of the strength of the bond on the properties of the metal and the ligand, 3) the preference of complexes for octahedral coordination, 4) exceptions to octahedral coordination, 5) the color of complexes, and 5) the magnetic properties of complexes.

Model for Bonding. We shall present a very simple yet useful model for bonding in complexes. The origin of the bond between a ligand and a metal is the sharing of a pair of electrons on the ligand between the ligand and the metal species. The bond is predominantly covalent in nature as a pair of electrons is being shared. However, the shared electrons are provided exclusively by the ligand in our elementary model; hence the bond is often called a coordinate covalent bond.

With this idea in hand, we can address the basis for the preference of complexes for a coordination number of 6, i.e. for octahedral coordination. Each time a coordinate covalent bond is formed between a ligand and a metal, the sharing of the electrons results in a lowering of the electronic energy of the system. One might argue the more the merrier. However, once 6 ligands have formed bonds to a metal, the complex no longer has room for an additional ligand. That is, if a seventh coordinate covalent bond were formed, the energy of repulsion between the seventh ligand and the rest of the complex exceeds the stabilization resulting from the additional bond. The most common exceptions to this result are found with very large ions such as Pu+3 which has an ionic radius of 1.07 Å. By comparison, the ionic radius of Fe+3 is only 0.76 Å. Hence, the Pu+3 has additional space to accomodate more ligands and observed coordination numbers such as 9 are not surprising.

This simple model is powerful enough to handle the most important properties of complexes outlined above. The model can be extended to handle the subtleties and exceptions. However, we shall not deal with the extensions in general chemistry. This is in keeping with the first iteration approach in the course. Our model is close enough to the full story that it must be mastered in order to understand the extensions. Furthermore, the basic model provides the bulk of the complete picture. Those interested in a full discussion are encouraged to take a course in advanced inorganic chemistry after they have taken a course in quantum mechanics.

Relevant Features of Molecular Orbital Theory. It is worth reviewing at this time the important elements of the molecular orbital model as the properties of complexes are readily understood using them. These fundamental principles are:

Energy Level Diagram for Octahedral Complexes. If one applies the rules of symmetry, one obtains the molecular orbital energy diagram for an octahedral complex which is given in Figure 1. The metal orbitals and their relative energies are given on the left for large separation and hence no interaction. Similarly the ligand orbitals are given on the right. The molecular orbitals for the complex ML6 is given in the center. Each molecular orbital is labeled using the conventional notation. It is not necessary to memorize the names of the orbitals but the following features of the pattern are important:

The chemical and physical consequences of the simple model presented here are best illustrated with a simple example, hexacyanotitanate(III) or Ti(CN)6-3. The energy diagram for this complex is obtained by filling the diagram in Figure 1 with the correct number of electrons. Note the strategy. First construct the energy level diagram and then load in the electrons. Each ligand is monodentate and donates a pair of electrons. The electronic configuration of Ti is [Ar] 3d24s2 and hence that of Ti+3 is [Ar]3d. Note that the 4s electrons are removed first as they are shielded by the inner 3d electrons. A total of 13 electrons must be loaded and the result is shown in Figure 2. Only bonding and non-bonding orbitals are occupied and the complex is predicted to be stable which it is. Furthermore, there is one unpaired electron and the complex is expected to be paramagnetic.

The color or optical properties of the complex are determined by the electron in the non-bonding t2g molecular orbital. [I have used the symmetry designation of non-bonding orbital in this case for purposes of clarity in the discussion which follows. You don't need to memorize this nomenclature.] In this case, this is the highest molecular orbital which is occupied and is often called a HOMO for highest occupied molecular orbital. The orbital with the next highest energy is anti-bonding. It is called eg*. In this case, it is unoccupied and is called a LUMO for lowest unoccupied molecular orbital. The energy difference, D, between the t2g and eg* orbitals is fairly small and therefore a low energy photon has sufficient energy to excite the electron from the t2g orbital to the eg* orbital. For transition metal complexes, D is such that visible light is in the correct range. We therefore have a bonus from the model, an explanation for the color of transition metal complexes. The cyanide ligand interacts strongly with the metal and D is relatively large so a photon at the blue end of the visible spectrum is absorbed. The color of the complex is given by the transmitted light which is at the yellow end of the spectrum.

It is worth considering the effect of changing the metal ion or the ligand on the bonding and the magnitude of D. If the charge on the metal ion is increased and/or its size is decreased, the electric field around the ion is increased and the ion interacts more strongly with the ligand. In other words, the metal ion is a stronger Lewis acid. The resulting complex is more stable and the value of D increases. What consequences does this have for the color of the complex? Alternatively, one can vary the complexing power of the ligand. Roughly speaking, strong bases such as CN- are also strong ligands and very weak bases such as Cl- are weak ligands. I have provided a rule-of-thumb for classifying ligands which is based on acid-base theory. The same result can be explained by an extension of our simple model but the extension is unnecessarily complicated and will not be explored here as the payoff is not worth the effort.

Our simple model addresses one aspect of the optical properties of the complex, its color or at which wavelength is the light absorbed. A second parameter is the intensity of the color, how many photons are absorbed by the complex. The extinction coefficient, e, is a measure of this second effect. [Recall that the absorbance A of a sample equals A = elc where l is the cell length and c is the concentration of the absorbing species]. The value of e is largely determined by the symmetry of the complex. Recall that selection rules must be satisfied if a photon is to be absorbed. If we treat all ligands as equivalent, octahedral complexes possess a center of symmetry which has the quantum mechanical consequence that the propensity to absorb light measured by the value of e is greatly reduced. In other words, the complex is lightly colored irrespective of the nature of the ligand attached to the metal ion.

Consider the hexaämmine zinc(II) complex as a second example. Zinc is not considered a transition metal because all its d orbitals are filled but it does form complexes whose properties can be explained by our simple model. With zinc(II) as the metal ion, 10 metal electrons in addition to the 12 electrons from the ligands are available. The bonding molecular orbitals are completely filled as are the t2g and eg* orbitals. Antibonding orbitals are occupied and reduced stability is expected. However, the number of electrons in bonding molecular orbitals (12) still exceeds those in anti-bonding so the complex still possesses net bonding character. We can draw additional consequences. All the electrons are paired and the complex is diamagnetic. It is not possible to excite an electron from a t2g orbital to the eg* orbital as the eg* orbital is completely filled. One could excite the electron to one of vacant anti-bond orbitals which are mostly 4s and 4p in character. However, they are much higher in energy and a ultra-violet photon is required. Photons in the visible range do not have enough energy and cannot be absorbed at all. The zinc complex is expected to be colorless.

One can slightly modify the model to accomodate complexes of metals which are not transition metal ions. These would not have d valence electrons and only s and p electrons can be used in constructing the molecular orbital energy diagram. Fewer metal orbitals results in fewer interactions between the ligand and the metal ion. The bonding molecular orbitals will not be lowered as much and the complex will not be as stable. Consequently, one can form complexes with species such as Ca+2 but they will not be as stable as those formed with ions such as Fe+2. What role does ionic radius play? If one considers just this factor, which species will form the stronger complex, Ca+2 or Fe+2?

Energy Level Diagram of Tetrahedral Complexes. Some ions in combination with certain metal ions have a preference for tetrahedral coordination, e.g. tetrachlorocobaltate(II). There is no simple explanation for this result so it should be treated as an empirical fact whose basis may be very complicated. In this section, we shall accept the existence of tetrahedral complexes and apply the molecular orbital theory to their bonding. Since a tetrahedral complex has a radically different symmetry, the metal and ligand orbitals must necessarily combine differently. The molecular orbital diagram for a tetrahedral complex is given in Figure 3. The following features and consequences should be noted:

Energy Level Diagram of Square Planar Complexes. We have already noted that transition metals with a d8 configuration have a preference for square planar coordination. Octahedral complexes of these metal species can be formed but conditions such as high ligand concentration must be adjusted to force the issue. A combination of thermodynamics and molecular orbital theory provides an explanation for the formation of square planar complexes with d8 species. First entropy considerations alone would argue for a loss of ligands and a reduction in the coordination number. The energetics of this conversion from octahedral to square planar coordination must be examined to discover why this is most likely with d8 species and why square planar rather than tetrahedral coordination results.

An octahedral complex contains the elements of a square planar complex. Any two ligands which are trans to one another and the metal form an axis which might be compared to a line running through the north and south poles on the earth. The remaining 4 ligands lie in a plane on the equator and are called equatorial. The equatorial ligands and the metal constitute a square planar arrangement. Consider an octahedral complex of a d8 species such as Ni+2. Two of the 8 d electrons are present in the eg* anti-bonding molecular orbitals. If one examines the symmetry of these orbitals, one discovers that they involve the two axial metal-ligand bonds. These axial bonds are broken preferentially and the residual complex is a planar species.

For purposes of completeness, a molecular orbital energy diagram of a square planar complex is given in Figure 4. This figure like all the others is based on our simple model in which each ligand interacts with the metal only through its lone pair of electrons. There is no point in changing models at this point. If you compare Figure 4 with comparable figures in other texts, you may notice small differences as the figures in these texts make tacit use of extensions to our simple model. A square planar complex is not as symmetric as an octahedral complex and things are not as tidy when symmetry is reduced. Such is life but we should not let small subtleties obscure the big picture. The following elements of the diagram are the important ones. Some of these should be familiar by now.

Energy Level Diagram of Linear Complexes. Linear complexes with the formula ML2 are often observed with d10 species such as Cu(I), Ag(I), and Au(I). In this case, two ligand molecular orbitals interact with (n+1)s and ndz2 atomic orbitals and generate two bonding molecular orbitals and two antibonding molecular orbitals. The remaining four (n-1)d orbitals are non-bonding in our simplified model. The orbital diagram is given in Figure 5.


  1. Anhydrous manganese(III) chloride is very dark violet whereas the tetrahydrate is light red.

  2. 50 ml of aqueous 0.1 M cobalt(II) nitrate was prepared.

  3. A large excess of concentrated hydrochloric acid is added to the cobalt(II) nitrate solution.

  4. In the synthesis of hexamminenickel(II) chloride, concentrated aqueous ammonia is added to an aqueous solution of nickel(II) chloride. The sample is washed with small portions of ether and is stored in a closed vial.

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25 Feb. 1997