Magnetochemistry

MAGNETOCHEMISTRY

SPIN AND ORBITS

Where does magnetism come from? Magnetism is a complex
subject and the
Electrons are charged particles and are therefore magnetic. The description of electrons as
electrons generate a magnetic field by moving in one of two spinning and orbiting the
ways: nucleus is a gross
simplification but useful
• The electron spins around its own axis – represented by for a first order analysis of
the spin quantum number S. magnetic behaviour.

• The electron also orbits the nucleus – represented by the The spin from individual
orbital angular momentum quantum number L. electrons can be added
using a simple equation:
We can combine these two moments to produce the total
angular momentum, given the symbol, J. S =  ms

J=S+L As an example if you have
2 unpaired electrons, the
calculation would be:

S=½+½=1

The value of L depends, as
we shall see, strongly on
the nature of the orbital
the electron is in and the
electron configuration. For
instance, the d4
configuration is a D term
and thus L = 2, whereas
the d7 configuration is
described by an F term,
and therefore L = 3.

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PARAMAGNETISM & Magnetism in Transition Metal compounds
DIAMGANETISM
Transition metal compounds often have unpaired electrons and if
Paramagnetism occurs in they do the complexes are paramagnetic. However, some
any system where there transition metal complexes have all their electrons paired (i.e. S =
are unpaired electrons. 0) and these compounds are termed diamagnetic. An example
The magnetic moment in which illustrates this is shown below.
such systems aligns itself
with an applied magnetic For the d6 electron configuration if the complex is high spin (e.g.
field and is attracted into [Fe(H2O)6]2+) there are four unpaired electrons and so the system
the magnetic field. It is is paramagnetic but if the complex is low spin (e.g. [Fe(CN)6]4-) all
important to realize that the electrons are paired and the complex is diamagnetic. It follows
paramagnetic materials do that measuring their magnetism is an easy way to distinguish
not retain any between these two complexes.
magnetization outside the For first row transition metals we can often neglect the
magnetic field. contribution from the orbital angular momentum as L is generally
close to zero. To calculate the magnetism of transition metal
Diamagnetism occurs in complexes we use the spin-only formula:
systems where all the
electrons are paired. In eff = 2√ ( + 1) = √ ( + 2)
these systems the sample
is repelled by the magnetic where eff is the effective magnetic moment reported in the units
field. It is important to of Bohr magnetons represented by the symbol, B or sometimes
note that all materials BM and n = no. of unpaired electrons. Unfortunately, the effective
exhibit diamagnetism but magnetic moment cannot be measured directly but it is related to
it is very weak compared a measureable quantity called the molar magnetic susceptibility,
to paramagnetism, M as shown below:
typically 100 times weaker.
In first principle eff = 2.828√ (if Gaussian units, cm3mol-1, are used)
calculations diamagnetism It’s important to note that the above equation doesn’t work with
can be ignored but for a SI units but as these are generally not used in magnetochemistry
more thorough treatment this is not usually a problem.
it must be considered. The effective magnetic moment of a wide range of transition metal
complexes have been measured and a summary of the findings are
shown in Table 1 below.

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Table 1 Spin-only values of eff compared with the observed values.

Metal Ion dn configuration S eff(spin-only)/ B Observed eff / B

Sc3+, Ti4+ d0 0 0 0
Ti3+ d1 1 1.73 1.7-1.8

V3+ 2 2.8-3.1
V2+, Cr3+ 3.7-3.9
d2 1 2.83
Cr2+, Mn3+ d3 3 3.87 4.8-4.9
Mn2+, Fe3+ 5.7-6.0
2
Fe2+, Co3+ 5.0-5.6
Co2+ d4 2 4.90 4.3-5.2
d5 5 5.92
Ni2+ 2.9-3.9
Cu2+ 2 1.9-2.1

Zn2+ d6 2 4.90 0
d7 3 3.87

2

d8 1 2.83
d9 1 1.73

2

d10 0 0

Examining Table 1 it is clear that the spin-only formula is GEOMETRY &
remarkably successful at predicting the observed magnetic MAGNETISM
moment most of the time. Let us look at a real example:
In using the spin-only
Worked example formula, once you know
the dn configuration it is
At 298 K, eff for [Cr(NH3)6]Cl2 is 4.85 B. Is the complex high spin crucial to think about the
or low spin? geometry of the complex.
For instance, four
Answer coordinate Ni2+ (d8)
1. The metal is Cr and as there are two Cl- anions it must be complexes such as
Cr2+, therefore the dn configuration is d4 (remember the dn [NiX2(PR3)2] can be square
configuration = group no. – oxidation state = 6 - 2 = 4). planar or tetrahedral. If
2. As there are six ligands the complex will be octahedral. the complex is tetrahedral
Applying an octahedral crystal field gives us these two there are two unpaired
configurations: electrons (e4t24) and eff =
2.83 B, while if the
3. We know calculate eff using the spin only formula. complex is square planar
eff = √ ( + 2) eff = 0 B. Remember that
eff = √2(2 + 2) = √8 = 2.83 B (low spin) the high spin or low spin
eff = √4(4 + 2) = √24 = 4.90 B (high spin) problem only applies to 1st
row octahedral transition
4. It is clear that the observed value is closer to 4.90 than metal complexes;
2.83 B, therefore the complex is high spin. tetrahedral complexes are
invariably high spin.

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Beyond the spin-only formula

Although the spin-only formula is successful for many 1st row transition metal complexes there are
certain systems where it fails. Take the example of Co2+ from Table 1, the spin-only formula predicts
that eff = 3.87 B but the actual values are between 4.30-5.20 B, higher even than that expected for
four unpaired electrons! One of the main reasons that the spin-only formula fails is that the orbital
angular momentum, L is not zero. It follows that we have to consider both the spin (S) and orbital
angular (L) momenta. In the case of d-block ions we assume that the spin and orbital angular momenta
operate independently, as we shall see later this is not true for all metal complexes. But assuming that
this is true gives us the van Vleck formula and although strictly only applicable to metal ions works
quite well for many transition metal complexes.

eff = √4 ( + 1) + ( + 1)

Table 2 Values of eff calculated with the van Vleck formula.

Metal Ion Ground term Calculated eff / B Although, this formula looks complex it’s
actually quite simple to use. Let’s look at
Ti3+ 2D 3.01 Co2+ complexes for which the spin-only
V3+ 3F 4.49 formula failed so spectacularly.
V2+, Cr3+ 4F 5.21
Cr2+, Mn3+ 5D 5.50 • In an octahedral Co2+ complex the dn
Mn2+, Fe3+ 6S 5.92
Fe2+, Co3+ 5D 5.50 configuration is d7 and the ground term
Co2+ 4F 5.21
Ni2+ 3F 4.49 symbol is 4F. It follows that S = 3 and L =
Cu2+ 2D 3.01 2

3, therefore:

eff = √4 3/2(3/2 + 1) + 3(3 + 1) =
√27 = 5.21 B

As we can see from Table 1 some of the observed values for Co2+ complexes are close to 5.21 B and
this implies that they have a large amount orbital angular momentum. However, many other
complexes have values that are much lower than that predicted by the van Vleck formula and this
suggests that in most complexes there is some orbital angular momentum but it is not as much as that
predicted by the van Vleck formula. To understand why we must consider how the application of a
crystal field effects the orbital angular momentum.

Applying a crystal field

We begin by asking how electrons acquire orbital angular momentum. The easiest way to visualise
this in a metal complex is that electrons will have orbital angular momentum if they are in an orbital
that can be transformed into an equivalent and degenerate (i.e. same energy) orbital by rotation. For
instance, in an octahedral crystal field, the t2g set of orbitals all have the same energy and can be
interconverted by rotation:

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This means that an electron in a t2g orbital has orbital angular momentum, i.e. L  0. It’s important to
note that there must be space in the orbital that the electron is moving into. In the case of low spin d4

shown on the
left, the ‘spin
down’ electron
can be moved
from a dxz to dyz
to dxy orbital.
However, for the
d3 configuration
each orbital has one electron, therefore none of the electrons can be moved and in this case L = 0; this
is why the spin-only formula works well for d3.

SELF-STUDY EXERCISE If the electrons are in
the eg set these
• Which other dn configurations do you think will have orbital orbitals, dx2-y2 and dz2,
angular momentum? have different shapes
and thus electrons
• Will the geometry of complex make any difference?

cannot be transferred between them. Consequently, these SPLITTING OF TERMS
electrons never have orbital angular momentum. It’s important to
note that different geometries alter the relative energies of the d- In an octahedral crystal
orbitals and different orbitals may be degenerate. field terms are split:

Worked examples • P terms give a T1g
term.
Consider a free-ion 3F term. Here L = 3 and S = 1, using the van
Vleck equation: • D terms split into T2g
and Eg terms.
eff = √4 1(1 + 1) + 3(3 + 1) = √20 = 4.49 B
• F terms split into T1g,
In an octahedral field the 3F term splits into A2g + T2g + T1g and in T2g and A2g.terms.
the case of Ni(II) the ground term is 3A2g. As there are two unpaired
electrons S = 1 and the A term has a single way of arranging the • Remember also that
electrons effectively Leff = 0 and using the van Vleck equation: for every F term there
will be one P term of
eff = √4 1(1 + 1) + 0(0 + 1) = √8 = 2.83 B the same multiplicity
above it.
In essence the octahedral crystal field completely removes the
orbital angular momentum.

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Now let’s consider an octahedral V3+ complex. The electron configuration here is d2 and the ground
term is 3T1g. The spatial degeneracy of a T term is three-fold and we describe this with Leff = 1. It follows
that:

eff = √4 1(1 + 1) + 1(1 + 1) = √10 = 3.16 B

We now see that application of the crystal field has removed some but not all of the angular
momentum. This is true for many transition metal complexes and this is why the effective magnetic
moment is often in between that predicted by the van Vleck and spin-only formulas.

Spin-orbit coupling

A final complication we must consider when discussing magnetism is a phenomenon called spin-orbit
coupling. As the name suggests this involves a coupling of spin (S) and orbital angular (L) momenta
which causes mixing of magnetic states of the same spin. When spin-orbit coupling is strong we can’t
treat S and L separately and the magnetism can only be described using the total angular momentum,
J. The strength of spin-orbit coupling is determined by , the spin-orbit coupling constant; simply put
the larger  is the stronger spin-orbit coupling. Spin-orbit coupling constants vary from a few cm-1 for
light atoms to thousands of cm-1 for heavy elements such as lanthanides and actinides.

Although spin-orbital coupling is generally quite weak for transition metal complexes, it is sometimes
present and for Cu2+ complexes cannot be ignored (see Table 3). Note that  is positive for less than
half-filled shells and negative for shells that are more than half-filled.

Table 4 Spin-orbit coupling constant for the 1st row transition metals.

Metal Ion Ti3+ V3+ Cr3+ Mn3+ Fe2+ Co2+ Ni2+ Cu2+

dn configuration d1 d2 d3 d4 d6 d7 d8 d9
 / cm-1 155 105 90 88 -102 -177 -315 -830

Detailed theory shows that the spin-only formula may be modified to take into account spin-orbit
coupling and while dependent upon oct also applies to tetrahedral complexes,

eff = eff(spin-only)  (1 −   )

where  is a constant that depends on the ground term: 4 for an A ground state, and 2 for an E ground
state. This simple approach is unfortunately not applicable to ions with a T ground state.

Measuring magnetic susceptibility

Having discussed the various types of magnetism exhibited by metal complexes we now consider how
the magnetism of metal complexes is measured.

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• Gouy balance. This measures the change in
force as the sample (as a solid) is placed
between the poles of a permanent magnet.
The raw data is converted into the mass
susceptibility, g and by multiplying by the
molecular weight we calculate the molar
magnetic susceptibility, M which is then
related to eff.
M = g  MW

• SQUID magnetometer. SQUID stands for Superconducting
QUantum Interference Device and involves the use of two
superconductors separated by an insulating layer forming
two parallel Josephson junctions. This is the most sensitive
technique for measuring magnetism as it can detect very
small changes. As with the Guoy balance the sample used is a
solid. The data produced is processed to give the molar
magnetic susceptibility, M which can then be related to the
number of unpaired electrons via eff.

• Evan’s method. This method uses 1H NMR spectroscopy and
involves dissolving the complex in a suitable solvent, e.g. CDCl3 and then adding a capillary of
the pure solvent (CHCl3) into the sample solution. The shift in the solvent peak of the solution
compared with the pure solvent is related to the degree of paramagnetism in the complex.
Processing of the data in a similar way to the Gouy method gives g and ultimately, eff.

Exercises

1. Comment on the observations that octahedral Ni(II) complexes have magnetic moments in
the range 2.9-3.4 B, tetrahedral Ni(II) complexes have moments up to  4.1 B, and square
planar Ni(II) complexes are diamagnetic.

2. Find x in the formulae of the following complexes by determining the oxidation state of the
metal from the experimental values of eff: a) [VClx(bpy)], 1.77 B; b) Kx[V(ox)3], 2.80 B; c)
[Mn(CN)6]x-, 3.94 B. What assumption have you made and how valid is it?

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Further reading

1. Inorganic Chemistry, Weller, Overton, Rourke, Armstrong and Atkins, 6th edition, 2014, Oxford
University Press. An excellent textbook which provides a nice introduction to
magnetochemistry and covers many of the topics in this course. The bulk magnetism and
SQUID magnetometer pictures are taken from this textbook.

2. Inorganic Chemistry, Housecroft & Sharpe, 4th edition, 2012, Pearson. Another very good
textbook which covers all of the topics in this course. This text goes a little deeper than the
above OUP textbook. The Gouy balance picture comes from this textbook.

3. Coordination Chemistry, Ribas Gispert, 2008, Wiley-VCH. Chapter 10 on molecular magnetism
provides a very thorough treatment of the magnetism of both mono- and polynuclear
coordination compounds and is recommended for those interested in more subtle aspects of
magnetochemistry.

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