Appendix K: Magnetism in Transition Metal Complexes

Appendix J (Electronic Absorption Spectra of Metal Complexes) explains in detail the origin of color in complexes of the transition metals. We learn that 6 sigma bonding ligands disposed about a metal ion at the vertices of a regular octahedron cause a splitting of the d orbitals of the metal into two sets, one triply degenerate set, n, of non-bonding orbitals (d_xy,xz,yz), and one doubly degenerate set, s*, of antibonding orbitals (d_z2,x2-y2). If these orbitals are partially filled with electrons originally belonging to the metal, and if the ligand field splitting, D_o, between the two d orbital sets is such that visible light can cause the transition of an electron from the lower to the upper set, then the complex will be colored. We will now see that the very interesting and variable magnetic behavior of transition metals in complexes has its origin in the same basic structural features--split, partially filled d orbitals. With a couple of additional concepts, we can develop a satisfactory explanation of this magnetic behavior on the basis of the same theory that we used in discussing electronic spectra and color. We begin with an examination of the various types of magnetic behavior displayed by matter.

When a piece of matter is placed in a magnetic field, H (italic signifies a vector quantity), the matter becomes magnetically polarized; that is, a magnetic field is set up in the matter as a result of its presence within the external field, and it has been found experimentally that the strength of the internal field is proportional to the strength of the applied (external) field. We express this mathematically by equation (1) where M is the magnetic polarization of the matter and c is a

We are interested now in the phenomenon of paramagnetism, which arises whenever a molecule contains unpaired electrons. Oxygen is paramagnetic because it contains two unpaired electrons. NO₂ is paramagnetic since it contains an odd number of electrons. (Any substance containing an odd number of electrons must be paramagnetic. Nature usually avoids this situation in compounds of the representative elements, but makes up for it in transition and inner transition elements, where there can be as many as 7 unpaired electrons per molecule.) Organic free radicals are paramagnetic due to the presence of an odd electron (remember the methyl radical, CH₃, in free radical halogenation?). As it happens, complexes of most transition metals are paramagnetic as a result of the presence of unpaired electrons in the split d orbitals. We have already encountered an example of this in the complex Cr(NH₃)₆³⁺, which contains 3 unpaired electrons, as discussed in the Appendix J. Consider another example, this time involving Fe^III, whose electronic configuration is [Ar]3d⁵ in the absence of ligands. We would expect the free gaseous ion to contain 5 unpaired electrons, as indeed it does--the electrons are distributed in the five degenerate d orbitals, according to the Pauli Exclusion Principle and Hund's Rules, as shown below:

Hund's Rules state that electrons filling a set of degenerate orbitals will fill them so as to maintain their spins parallel for as long as possible. Similarly, the complex Fe(H₂O)₆³⁺, present in strongly acid aqueous solution, contains 5 unpaired electrons. However, the very stable complex Fe(CN)₆^3-, named hexacyanoferrate(-3), contains only one unpaired electron, even though it also contains Fe(III). We deduce from this that the nature of the ligands must in some way influence the pairing of electrons. But how? Let's call on what we have learned about the splitting of the d orbitals by the ligands. Figure 2 shows the situation in terms of an energy level diagram.

In the complex, 5 electrons must distribute themselves among the d orbitals, which as a result of the presence of 6 ligands are no longer all degenerate. There is no ambiguity about where the first three electrons will go--one electron will enter each of the three degenerate orbitals in the n set. The fourth electron, however, is faced with a choice: it can either pair up with one of the electrons in a lower orbital, requiring the expenditure of an amount of energy, P, called the pairing energy; or it can occupy one of the orbitals in the upper doubly degenerate set at the expense of the energy, D_o , since the upper set, s*, is less stable by than the lower set, n. The electron will take the option which requires the expenditure of the least energy. If D_o < P, it will occupy one of the s* orbitals. If D_o > P, the electron will pair up in n. The fifth electron will be faced with the same choice, and will make the same decision. Two situations are therefore possible in octahedral complexes of Fe^III:

In one case, five unpaired electrons are present. This is called the high spin (HS) case and must correspond to the situation in Fe(H₂O)₆³⁺. In the second case, called the low spin (LS) case, only one unpaired electron is present--the maximum possible amount of electron pairing has occurred. This is the situation which in Fe(CN)₆^3-. We conclude, then, that when H₂O is the ligand, D_o < P and that when CN- is the ligand, D_o > P (P remains essentially the same for a given metal ion, regardless of ligand, and can be obtained from spectroscopic measurements of the free metal ion, in this case Fe^III.) We call water a relatively "weak-field ligand" and cyanide a "strong-field ligand."

Any ion having as few as 4 or as many as 7 d electrons can exhibit either high- or low-spin behavior in an octahedral complex. You should work out the numbers of unpaired electrons expected in octahedral complexes of metal ions with configurations d1 to d9 to convince yourself of this. The theory that we have developed above explains beautifully most aspects of the magnetic behavior of transition metal complexes, the exceptions being the more subtle aspects that we will avoid for the time being. Let's turn now to the relationship between the number of unpaired electrons possessed by a molecule and its behavior in a magnetic field.

There are two sources of magnetic behavior in matter: 1) orbital motion of electrons; 2) spin of electrons. We will deal with spin motion only now, since it is the most important source of magnetic behavior in complexes of transition metals. Pictured below is an electron spinning (rotating) about its own axis.

This motion is analogous to the rotation of the earth about its axis. Since the electron is a charged particle, this rotational motion constitutes an electric current. To see this more clearly, mentally replace the electron with a loop of wire in which a current is flowing:

Recall from your study of electricity and magnetism in physics, that a flow of current always generates a magnetic field. When the current is flowing in a loop, the direction of the magnetic field is perpendicular to the plane of the loop. Thus

The spin of the electron about its axis, since it is an electric current, similarly generates a magnetic field parallel to the axis of rotation. This magnetic field is called the spin magnetic moment of the electron, and is given the symbol m_S. The arrow over the symbol indicates that the magnetic moment is a vector quantity. Now rotational motion of any kind always generates angular momentum, which is also a vector parallel to the axis of rotation. The magnetic moment and the angular momentum are therefore colinear vectors. Furthermore, their magnitudes are directly proportional. This is indicated in the following equation

where L = angular momentum and g is the proportionality constant relating magnetic moment and angular momentum. When a paramagnetic sample is placed in an external magnetic field, the individual spin magnetic moments of the unpaired electrons all line up with the field, just as iron filings line up along the magnetic lines of flux generated by a bar magnet. Their vector sum is the magnetic polarization defined by equation (1). Thus we see that magnetic polarization, which is a macroscopic property, is related to individual spin magnetic moments, which are microscopic properties of the system.

It can be shown that the magnetic moment of an individual ion in a sample is related to the number of unpaired electrons on the ion by the following equation.

The Bohr magneton is a convenient unit for expressing magnetic moments, and is defined in terms of fundamental constants as 1 BM = eh/4pmc. Here e = the charge on the electron, h = Planck's constant, m = the mass of the electron, and c = the speed of light, and, in equation (K-3), n = the number of unpaired electrons per ion. We can therefore calculate the number of unpaired electrons per molecule of sample if we can measure the magnetic moment. In practice, we can relatively easily measure the molar magnetic susceptibility of a material, which is then related to the magnetic moment by equation (4).