Chemistry 3211 – Coordination Chemistry Part 3
Ligand Field and Molecular Orbital Theory
Electronic Structure of Six and Four-Coordinate Complexes
Using Crystal Field Theory, we can generate energy level diagrams describing the relative energies of
the different metal-based d-orbitals in an octahedral crystal field. This assumes only electrostatic
interactions and does not consider any covalent bonding (σ, π or δ) interaction or orbital overlap. This
gives rise to two sets of orbital energy levels. The doubly degenerate eg set consists of the axial orbitals
dx2-y2 and dz2 and is higher in energy than the triply degenerate t2g set consisting of the interaxial dxy, dxz
and dyz orbitals. The energy difference is Δo.
Distortions from the ideal octahedral environment cause variations in the orbital energy levels. For
example, if we start pulling away one of the ligands along the z-axis, ultimately removing it to form a
five-coordinate square pyramidal geometry, this partly removes an unfavourable (according to CF
theory) interaction in the z-direction. All orbitals with a z-component (dz2, dxz and dyz) fall in energy.
The others go up in energy. This gives rise to the energy level splitting diagram shown below.
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Removing the other ligand on the z-axis has the same effect, leading to the energy level diagram for a
square planar complex. Note the large energy gap between the dxy and dx2-y2 orbitals. As a result, square
planar complexes are almost exclusively those of d8 metals (all metal orbitals are filled except the dx2nd
Ed. illustrates
y2) and so have 16 electrons rather than 18. Figure 20-10 in Housecroft and Sharpe 2
the crystal field splitting parameters for common geometries.
Four coordinate complexes are the second most abundant and the same kinds of arguments can be
applied to these species, but we have to take into account the different order of the d-orbital energies. A
tetrahedral geometry can be described by superposition within a cube. None of the metal d orbitals
points exactly at the ligands, but the dxy, dxz and dyz orbitals approach the ligand location more closely
than do the dz2 and dx2-y2 orbitals. Therefore, for a regular tetrahedron, the splitting of the d orbitals is
inverted compared to that of an octahedron, and the crystal field splitting, ΔT, is smaller than Δo (by a
crude approximation, ΔT is generally 4/9 the magnitude of Δo).
t2
Z
ΔT
X
Y
0.4ΔT
e
0.6ΔT
Because there is no centre of inversion in a tetrahedron, the orbital sets are labelled t2 for the triply
degenerate set and e for the lower energy doubly degenerate set (the orbital designations do not include
the parity label g or u). Ligand field stabilization energies for tetrahedral complexes can be calculate in
exactly the same way as for octahedral complexes, the only differences being the order of occupation (e
before t2) and the contribution of each orbital to the total energy (-3/5 ΔT for an e electron and +2/5 ΔT
for a t2 electron).
Tetragonal Distortions and the Jahn-Teller Effect
As described above, distortion from the ideal octahedral geometry results in a new distribution of d
orbital energy levels. These distortions may arise as a result of steric interactions among the ligands, or
indeed from the nature of the metal ion itself. For example, copper(II), which has nine d electrons,
forms six-coordinate complexes that usually depart considerably from true Oh symmetry and often
show significant tetragonal distortions. Low-spin d7 and high-spin d4 six-coordinate complexes may
show a similar distortion. Extension of the z-axis and compression of the remaining axes reduces the
energy of the eg(dz2) orbital and increased the energy of the eg(dx2-y2) orbital. Therefore, if the eg set is
occupied by one, two or three electrons (as in l.s. d7, d8, d9 and h.s. d4 complexes) a tetragonal
distortion may be energetically advantageous. As a special case, the distortion of a h.s. d8 complex may
be large enough to encourage the pairing of two electrons in a dz2 orbital. Effectively, this may lead to
the loss of two ligands on the z-axis and the formation of d8 square planar complexes, such as those
found for Rh(I), Ir(I), Pt(II), Pd(II) and Au(III). The common occurrence of square planar 4d8 and 5d8
complexes relates to the high value of the ligand field splitting parameter for these heavier transition
metals, which gives rise to large ligand field stabilization by going to low spin square planar
complexes. On the other hand, 3d8 metal complexes, such as [NiX4]2- where X is a halogen are
generally tetrahedral because the ligand field splitting is generally smaller for the first row transition
metals. Only when coordinated to strong field ligands such as CO or CN- does a large enough ligand
field splitting occur, resulting in the formation of a square planar complex, such as [Ni(CN)4]2-.
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For six-coordinate systems possessing an odd number of electrons in the eg orbitals, that is, for d9, high
spin d4 and low spin d7 electron configurations, these tetragonal distortions can be attributed
electronically to the Jahn-Teller effect. It states that if the ground state electronic configuration of a
non-linear complex is orbitally degenerate, the complex will distort so as to remove the degeneracy
and achieve a lower energy. An octahedral d9 complex has three electrons in its eg set of orbitals. The
unpaired electron can occupy either of the dz2 or dx2-y2 orbital because they are energetically degenerate.
Essentially, it resonates between the two orbitals and this destabilizes the complex. However, if the
geometry is physically distorted and we allow one metal-ligand bond axis to lengthen, this will remove
the degeneracy of the eg set and one orbital will become lower in energy than the other. Therefore, the
paired electrons will preferentially occupy the lower energy orbital and the unpaired electron will
populate the highest energy orbital. This results in a lowering of the overall ligand field stabilization
energy for the complex. The distortion to give a square planar complex, where two of the ligands are
removed entirely, is an extreme case of tetragonal distortion, but should not be confused with the JahnTeller distortion because d8 electron configurations are not orbitally degenerate. The Jahn-Teller effect
can only identify an unstable electron configuration; it does not predict the preferred distortion.
dx2-y2
dz2 dx2-y2
dx2-y2
dz2 dx2-y2
z-axis
elongation
dz2
z-axis
elongation
dz2
Barycentre
Barycentre
dxy
dyz
dxy
dxz
d9
dyz
dxy
dyz
dxy
dxz
l.s. d7
dxz
dyz
dxz
dx2-y2
dz2 dx2-y2
dxy
dyz
h.s. d4
dxz
z-axis
elongation
dz2
Barycentre
dxy
dyz
dxz
Ligand Field Theory
Although crystal field theory is useful in approximating energy levels in transition metal complexes,
the fact that it ignores any possible covalent bonding interactions between the metal and its surrounding
ligands is rubbish. We have already noted that the ligand field splitting parameters, Δ, are strongly
dependant upon both the metal AND the ligand (thus the spectrochemical series of ligands). So then,
why does the spectrochemical series arrange the ligands the way it does? What is common with the
strong field ligands that allows them to increase Δ? Does the electron configuration in the atomic and
molecular orbitals in these ligands have something to do with it? Of course it does!
You are recommended to revise your understanding of basic Molecular Orbital theory for simple
diatomic molecules (covered in Sections 2.8 and 2.9 of S & A 4Ed). Recall that in constructing
molecular orbitals from atomic orbitals, the number of orbitals must remain constant (i.e. number of
MOs equals the number of AOs). Typically, we place the more electronegative element on the right
hand side and that its AOs are lower in energy. Remember also that for most heteronuclear diatomic
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complexes, orbital mixing occurs (i.e. there is a hybridization of the atomic s and p orbitals because of
similar energies).
Below are molecular orbital diagrams for three simple diatomics, which will be useful in explaining
various aspects of Ligand Field/Molecular Orbital theory for metal complexes.
Note the difference in the arrangement of the σ2p and π2p orbitals between O2 and N2. Recall that for O2
(and F2 and Ne2) the energy difference between the 2s and 2p orbitals is large and therefore no s-p
orbital mixing is observed. The highest occupied molecular orbital (HOMO) in N2 is of σ symmetry (a
lone pair which can act as a Lewis base to form a coordinate covalent bond) whereas the HOMO in O2
is of π* symmetry. The bonding modes exhibited by N2 and O2 should therefore be slightly different,
and indeed they are, but more on that later!
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Molecular Orbital Diagram for CO
Carbon monoxide, or carbonyl, ligands are found throughout coordination chemistry and metal
carbonyl compounds are particularly important as catalysts in organic synthesis. CO is isoelectronic
with N2 (C has one less electron than N, O has one more), therefore the MO diagram is very similar to
that of N2. It is diamagnetic, with a bond order of 3 (C≡O) and has a HOMO of σ2p (weakly
antibonding) symmetry and a lowest unoccupied molecular orbital (LUMO) of π*2p (strongly
antibonding) symmetry.
Since O is far more electronegative than C, we would expect a large dipole moment with δ- on O.
However, CO actually has only a small dipole moment (0.1 Debye) with the δ- charge on carbon.
Usually electronegativities are a good indication of the direction and magnitude of the dipole on a
molecule, but especially in molecules where orbitals with antibonding character are occupied, things
are not so straightforward. As a result, CO ligands are good Lewis bases and are σ donors at the carbon
atom. Also, the vacant π* orbital LUMO can act as an electron pair acceptor capable of forming metal
ligand π bonds. These types of bonding in metal complexes are described by Ligand Field theory,
which combines aspects of Crystal Field theory with the more rigorous treatment of bonding given by
MO theory. It uses symmetry-adapted linear combinations (SALCs) of metal-ligand interactions
and, as the name suggests, relies on some application of symmetry elements and group theory to
produce a viable bonding model.
Sigma (σ) Bonding
Consider an octahedral complex in which each ligand has a single valence orbital directed toward the
central metal atom. Each of these orbitals has local σ symmetry with respect to the M-L axis. Such
ligands are termed “sigma donor ligands” and can be neutral, such as ammonia, NH3, or charged, such
as fluoride, F-. In a strictly octahedral environment, the metal orbitals divide by symmetry into four
sets, which are given labels that describe their symmetry (subscripts g or u, 1 or 2), and their
multiplicity (a or b, e, t). The resulting Symmetry Labels (also called Mulliken Symbols) are used to
now describe sets of orbitals in a complex, rather than just the atomic orbital. Therefore, orbitals or
groups of metal orbitals that are labelled by a particular Mulliken Symbol can only interact with a
ligand set of orbitals that is described by the same label.
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Metal Orbital
Symmetry Label
Degeneracy
s
a1g
1
px, py, pz
t1u
3
dxy, dyx, dxz
t2g
3
dx2-y2, dz2
eg
2
Six symmetry-adapted linear combinations of the six ligand σ symmetry orbitals can also be formed.
These combinations are shown below, where we are only considering ligand s orbitals to keep things
simple. Bear in mind that these are not specifically s or p atomic orbitals any more, but only the
orbitals, atomic or molecular, which exhibit σ symmetry with respect to the metal-ligand bond axis. We
are now treating all six ligands as a group. Among the metal orbitals, only the t2g set (dxy, dxz, dyz) does
not possess suitable symmetry to undergo σ-bonding interactions with the ligand SALCs. However, it
does possess π symmetry, which will be discussed later.
Molecular orbitals for the resulting complex are formed by combining SALCs and metal orbitals of the
same symmetry type. Since we have six metal orbitals that can be broken down into three groups
according to their symmetry (a1g, tiu and eg) and six ligand SALCs, these combine to form 6 bonding
and 6 antibonding MOs. The metal t2g set remains non-bonding in this scenario.
The resulting MO diagram can now be populated with electrons according to the Aufbau process, Pauli
principle and Hund’s rule. For the resulting energy levels, the greatest contribution to the molecular
orbital of lowest energy is from atomic orbitals of lowest energy. For most ligands, particularly those of
period 2 and 3 (n quantum number ≤ 3) in the main group, the ligand σ orbitals are derived from AOs
with energies much lower than those of the metal d-orbitals. Thus, the six bonding MOs of the complex
are mostly ligand-orbital in character. These six bonding orbitals can accommodate the 12 electrons
provided by the six ligand lone pairs. Therefore, the electrons provided by the ligands are largely
confined to the ligands in the complex. However, this is not to say that there is no metal character in
these MOs, or that there is no ‘ligand electron’ density present near the central metal atom.
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The d electrons of the metal centre, if any, occupy the lower energy, non-bonding metal t2g set and the
higher energy, antibonding eg set. Therefore, the metal based d electrons tend to reside largely on the
metal atom. In summary, the frontier orbitals of the complex are the non-bonding, entirely metal based
t2g orbitals and the antibonding eg orbitals, which are mainly metal in character anyway. Therefore,
because most bonding (and therefore “chemistry”) occurs in the valence (or “frontier”) orbitals of a
compound, most reactions of metal complexes occur at the metal centre. The octahedral ligand field
splitting parameter, Δo, in this approach is called the HOMO-LUMO separation, or HOMO-LUMO
gap. It is approximately the splitting of the metal d orbitals caused by the ligands, whereas in CFT Δo is
purely a metal d orbital separation.
The diagram below shows how the SALCs of the ligand orbital sets combine with the metal orbitals to
produce molecular orbital energy levels (the MO diagram) of an octahedral ML6 complex. The t2g and
eg* (which is now of antibonding symmetry with respect to the M-L bond) are empty in this diagram,
suggesting the metal ion, M, has a d0 electron configuration. LFT explains very well the problem that
Werner faced regarding the ability of a metal to bind numerous ligands (secondary valence). With
several d orbitals (as well as s and p orbitals) available in the valence shell, enough delocalized MOs
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can be constructed to accommodate all the electrons needed to bind the ligands. Also, since lanthanides
have f orbitals as well, they can accommodate even more ligand electrons and, therefore, can easily
have coordination numbers higher than 6.
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Pi (π) Bonding
The above diagram ignores any potential π bonding that may be present (the t2g set is non-bonding). If
the ligands in a complex have orbitals with local π symmetry with respect to the M-L axis (a t2g ligand
SALC), then they may form π MOs with the t2g set of metal orbitals. These ligand π orbitals may be
populated or vacant and, therefore, may act as electron donors or acceptors as shown in the diagram
below. The effect of this bonding on Δo depends on where the electron density is going. For π-acceptor
ligands, the bonding is SYNERGIC: σ-donation to the metal strengthens π-backbonding to the ligand,
and π-donation from the metal to the ligand strengthens the σ-donor component of bonding. This is
because σ-donation leads to increased electron density on the metal, which allows increased πbackdonation. Conversely, π-backdonation reduces the amount of electron density on the metal, which
allows more σ-donation from the ligand to the metal.
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A π donor ligand has filled orbitals that have π symmetry around the M-L axis. The energies of these
orbitals are similar to those of the metal d orbitals and the ligand has no vacant low energy π orbitals
(strictly speaking these are of π* symmetry). Examples of π donor ligands include Cl-, OH-, NR2- and
H2O. These all turn out to be low in the spectrochemical series (weak field ligands) that lower Δo. The
full π orbitals of the π donor ligands lie lower in energy than the partially filled d orbitals of the metal.
This means that when they form molecular orbitals with the metal t2g orbitals, the bonding combination
lies lower than the ligand orbitals and the antibonding combination lies above the energy of the d
orbitals of the free metal (figure below on the left). The result is that ligand electrons occupy the
bonding t2g MO, while the metal electrons now occupy the antibonding t2g MO set, which has been
raised in energy.
Ligands that are π donors decrease Δo.
Ligands that are π acceptors increase Δo.
A π-acceptor ligand is a ligand that has filled π orbitals, which are much lower in energy than the metal
t2g orbitals. It also has empty π∗ orbitals which are close enough in energy to the metal d orbitals to
allow population. Examples of ligands that possess orbitals of this appropriate energy are CO and N2,
which their MO diagrams we have described previously. The bonding π orbital of MO is populated and
is largely localized on the O atom because O is more electronegative than C, hence the lower energy π
bonding orbital is mostly O in character. The antibonding π* orbital of CO has its largest amplitude on
C and has appropriate symmetry and energy to overlap the t2g orbitals of the metal. Because of the
resulting energy correlation, the π donor ability of CO is very small, and its bonding to metals is
dominated by its π acceptor character along with its σ donor ability (see diagram below). Because these
π acceptor ligands are typically of antibonding character with respect to the ligand atoms, they lie
higher in energy than the metal d orbitals, they form molecular orbitals of t2g symmetry which are
largely of metal d orbital in nature. These bonding combinations lie slightly lower in energy than the
metal d orbitals themselves. As a result, Δo is increased due to the “pushing down” in energy of the
bonding t2g symmetry molecular orbitals.
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The spectrochemical series can now be put in perspective on account of the σ and π binding ability of
the ligands. Strong σ donors, like H- and CH3-, are high in the series. When π bonding is taken into
consideration, strong π acceptors are high in the series due to their ability to increase Δo. Strong π
donors are low in the spectrochemical series due to their lowering effect on Δo.
Below is the resulting molecular orbital (MO) diagram for an octahedral complex with strong πacceptor ligands (e.g. Cr(CO)6). The Cr0 centre is a 3d6 configuration. The 6 CO ligands each donate
two electrons to the metal centre via σ bonding. The π bonding symmetry orbitals of the ligands are
also populated, but cannot effectively act as π donors to the metal centre, partially because of their
energies being much lower than the appropriate symmetry metal t2g set, but also because the metal d6
centre has its t2g set populated by metal based electrons; electron-electron repulsion further hinders any
π donor ability from the CO ligand.
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The antibonding π* orbitals of the CO ligand are vacant and are low enough in energy to undergo
orbital overlap with the populated metal t2g set. Therefore, the metal “back-donates” this electron
density into the CO π* orbitals. In total, this complex has 18 electrons that participate in bonding. In
transition metal complexes, this “18 electron rule” is similar to the “octet rule” used in main group
chemistry. Complexes possessing 18 bonding electrons are particularly stable. Werner studied Co3+
ammine complexes because of their stability. The Co3+ ion has a 3d6 configuration. Each ammine
ligand is a two electron donor, hence [Co(NH3)6]3+ possess 6 + (6×2) = 18 electrons. Although a large
number of exceptions to this rule are found in classical coordination chemistry, the stability of
organometallic compounds (those possessing metal-carbon bonds) is strongly attributed to this effect.
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