The splitting of the d-orbitals into different energy levels in transition metal complexes has important consequences for their stability, reactivity, and magnetic properties. Let us first consider the simple case of the octahedral complexes [M(H2O)6]3+, where M = Ti, V, Cr. Because the complexes are octahedral, they all have the same energy level diagram:

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The Ti3+, V3+, and Cr3+ complexes have one, two and three d-electrons respectively, which fill the degenerate t2g orbitals singly. The spins align parallel according to Hund's rule, which states that the lowest energy state has the highest spin angular momentum.

For each of these complexes we can calculate a crystal field stabilization energy, CFSE, which is the energy difference between the complex in its ground state and in a hypothetical state in which all five d-orbitals are at the energy barycenter.

For Ti3+, there is one electron stabilized by 2/5 ΔO, so CFSE=−(1)(25)(ΔO)=−25ΔO
Similarly, CFSE = -4/5 ΔO and -6/5 ΔO for V3+ and Cr3+, respectively.

For Cr2+ complexes, which have four d-electrons, the situation is more complicated. Now we can have a high spin configuration (t2g)3(eg)1, or a low spin configuration (t2g)4(eg)0 in which two of the electrons are paired. What are the energies of these two states?

High spin: CFSE=(−3)(25)ΔO+(1)(35)ΔO=−35ΔO
Low spin: CFSE=(−4)(25)ΔO+P=−85ΔO+P , where P is the pairing energy

Energy difference = -ΔO + P

The pairing energy P is the energy penalty for putting two electrons in the same orbital, resulting from the electrostatic repulsion between electrons. For 3d elements, a typical value of P is about 15,000 cm-1.

The important result here is that a complex will be low spin if ΔO > P, and high spin if ΔO < P.

Because ΔO depends on both the metals and the ligands, it determines the spin state of the complex.

Rules of thumb:

3d complexes are high spin with weak field ligands and low spin with strong field ligands.

High valent 3d complexes (e.g., Co3+ complexes) tend to be low spin (large ΔO)

4d and 5d complexes are always low spin (large ΔO)

Note that high and low spin states occur only for 3d metal complexes with between 4 and 7 d-electrons. Complexes with 1 to 3 d-electrons can accommodate all electrons in individual orbitals in the t2g set. Complexes with 8, 9, or 10 d-electrons will always have completely filled t2g orbitals and 2-4 electrons in the eg set.

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d-orbital energy diagrams for high and low spin Co2+ complexes, d7

Examples of high and low spin complexes:

[Co(H2O)62+] contains a d7 metal ion with a weak field ligand. This complex is known to be high spin from magnetic susceptibility measurements, which detect three unpaired electrons per molecule. Its orbital occupancy is (t2g)5(eg)2.

We can calculate the CFSE as −(5)(25)ΔO+(2)(35)ΔO=−45ΔO
[Co(CN)64-] is also an octahedral d7 complex but it contains CN-, a strong field ligand. Its orbital occupancy is (t2g)6(eg)1 and it therefore has one unpaired electron.

In this case the CFSE is −(6)(25)ΔO+(1)(35)ΔO+P=−95ΔO+P.
Magnetism of transition metal complexes
Compounds with unpaired electrons have an inherent magnetic moment that arises from the electron spin. Such compounds interact strongly with applied magnetic fields. Their magnetic susceptibility provides a simple way to measure the number of unpaired electrons in a transition metal complex.

If a transition metal complex has no unpaired electrons, it is diamagnetic and is weakly repelled from the high field region of an inhomogeneous magnetic field. Complexes with unpaired electrons are typically paramagnetic. The spins in paramagnets align independently in an applied magnetic field but do not align spontaneously in the absence of a field. Such compounds are attracted to a magnet, i.e., they are drawn into the high field region of an inhomogeneous field. The attractive force, which can be measured with a Guoy balance or a SQUID magnetometer, is proportional to the magnetic susceptibility (χ) of the complex.

The effective magnetic moment of an ion (µeff), in the absence of spin-orbit coupling, is given by the sum of its spin and orbital moments:

μeff=μspin+μorbital=μs+μL(5.11.1)
In octahedral 3d metal complexes, the orbital angular momentum is largely "quenched" by symmetry, so we can approximate:

μeff≈μs(5.11.2)
We can calculate µs from the number of unpaired electrons (n) using:

μeff=n(n+2)−−−−−−−√μB(5.11.3)
Here µB is the Bohr magneton (= eh/4πme) = 9.3 x 10-24 J/T. This spin-only formula is a good approximation for first-row transition metal complexes, especially high spin complexes. The table below compares calculated and experimentally measured values of µeff for octahedral complexes with 1-5 unpaired electrons.