new bonding lecture notes 2012
DESCRIPTION
chemical bondingTRANSCRIPT
Chemical Bonding
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Department of Chemical Engineering
& Biotechnology
Part IA. Convergence
CHEMICAL BONDING
Dr Mick Mantle
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Synopsis 1. Introduction
1.1 The Periodic Table
2. Valence bond theory
2.1. What is it?
2.2. Guidelines for drawing Lewis structures.
2.3. Valence shell electron pair repulsion (VSEPR) theory
2.4. Limitations of valence bond theory
3. Quantum mechanics in chemical bonding
3.1. Some general principles
3.1.1. Wave-particle duality
3.1.2. Heisenberg‘s uncertainty principle
3.1.3. Schrödinger equation
3.2. Atomic orbitals: one-electron atoms
3.2.1. Schrödinger equation
3.2.2. General features of wavefunctions
3.2.3. Energy level diagram for one-electron atoms
3.2.4. Shapes of atomic orbitals for one-electron atoms
3.3. Atomic orbitals: multi-electron atoms
3.4. Periodic trends in properties of atoms.
3.4.1. Example: 1st ionisation energy
3.4.2. Some comments on electronegativities
4. Molecular orbital theory
4.1. Linear Combination of Atomic Oritals (LCAO)
4.2. Formation of other molecular orbitals
4.3. Molecular orbital energy diagrams for homonuclear diatomic molecules
4.4. s-p mixing
4.5. Molecular orbital energy diagrams for heteronuclear diatomic molecules
4.6. Hybridisation of atomic orbitals
4.7. Three-centre two-electron bonds
4.8. Limitations of molecular orbital theory
5. Bonding in Solids
5.1. Types of solid
5.2. Molecular orbital theory for solids
5.3 Electrical Conductivity
5.3.1 Metals
5.3.2 Insulators
5.3.3 Intrinsic semi-conductors
5.3.4 Intrinsic semi-conductors
6. Transition Metal Chemistry
6.1. Oxidation numbers
6.2. Electronic configuration (18-electron rule)
6.3. Shape
6.4. Metal-Metal bonds
6.5. Cluster compounds
6.6. Metal-ligand bonding
6.7. Organometallic compounds
Recommended textbooks
James Keeler & Peter Wothers ―Chemical Structure and reactivity‖ OUP.
P.W. Atkins ―Physical Chemistry‖. OUP.
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1 Introduction
You probably already know quite a lot about chemical bonding from what you learnt at
school. You probably have some idea about:
The periodic table.
o Historically introduced as ―families‖ of elements with similar properties.
o Periodic trends in the table are useful in predicting the properties of elements
and their compounds (e.g. trends in ionisation energy, electronegativity,
melting points of compounds).
o BUT – have you ever asked yourself what is the theoretical basis for the
periodic table?
Atomic orbitals.
o Provides a basis for the periodic table, by saying electrons fill up “s
orbitals”, “p orbitals”, “d orbitals”, “f orbitals” within each atom.
o There is independent experimental evidence for the existence of atomic
orbitals in addition to the mere fact of existence of the periodic table.
o BUT – have you ever asked yourself about the theoretical basis for atomic
orbitals? Why do electrons around an atom arrange themselves in this way?
Chemical bonding.
o You should have some idea about ionic bonding, typified by Na+Cl
–.
o You should have some idea about covalent bonding, typified by Cl–Cl.
o This school-level bonding model is called ―valence bond theory‖.
o It is a simple theory, and satisfactorily explains the bonding in many
compounds (including NaCl, Cl2, diamond, SiO2 and virtually all organic
chemicals).
o BUT – the properties of quite a lot of compounds cannot be explained
completely by valence bond theory.
These include some simple molecules (such as CO and O2), metals, and many
inorganic molecules (e.g. organometallic complexes and cluster compounds).
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The purpose of this course is to:
Revise valence bond theory concepts (because they‘re useful in many, though not all,
cases).
Answer the question ―what is the theoretical basis for atomic orbitals and for the periodic
table‖.
Introduce you to the ―molecular orbital‖ theory of chemical bonding.
Discuss the various types of bonding in solids (drawing on molecular orbital theory where
appropriate).
Discuss the inorganic chemistry of transition-metal complexes (drawing on molecular
orbital theory where appropriate).
In addition to the formal lectures there will be an exercise sheet which seeks to provide
quantitative problems which illustrate the material covered.
Knowledge of the concepts taught in this course should be useful whenever you interact with
real chemists in the real world.
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1.1 THE PERIODIC TABLE
The Periodic Table
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2 Valence bond theory
2.1 What is it?
The key feature of valence bond theory is that it is a localised approach, in which
individual electrons can be recognised as being located either in a specific bond or
on a specific atom.
The number of Valence Electrons (VEs) in the ―outer shell‖ of an atom is termed the valency of that
element, and it can be determined by inspection of the ground state electronic configurations of the
elements of the periodic table. (We shall discuss what precisely ―outer shell‖ means when we
discuss quantum mechanics). As a rule of thumb to identify the number of valence electrons in an
atom we can use the following:
s-block : Use the superscripted number of last entry of the ground state electronic
configuration
Beryllium (Atomic number 4): VEs =
p and d-block : Use the summation of the TWO superscripted numbers of the ground
state electronic configuration ignoring any d10
or f14
Oxygen (Atomic number 8): [He] 2s22p
4 ` VEs =
Germanium (Atomic number 32): [Ar] 3d10
4s24p
2 VEs =
Iridium (Atomic Number 77): [Xe] 4f14
5d76s
2 VEs =
The square brackets around an individual element, i.e. [He] is a shorthand notation of the ground
state electronic configuration of that element.
There are generally two types of chemical bond:
(1) A covalent bond is formed when two atoms are held together by a pair of electrons which
are shared between the two atoms.
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(2) An ionic bond is formed when an electron is transferred from one atom to another. In
reality there never can be complete electron transfer, so a bond always has some covalent
character.
For both covalent and ionic bonding there is a propensity to attain the electronic configuration of the
nearest noble gas.
Atoms from the early rows of the periodic table have a tendency to have 8 electrons in their
outer valence shells. This is the so-called ―octet rule‖; however, it should be viewed as a
guideline rather than a rule.
Main-group atoms from latter rows of the periodic table have a tendency to have 18 electrons in
their outer valence shells if the d orbital electrons are counted (or 8 electrons if the d orbital
electrons are not counted).
Transition metal atoms sometimes have a tendency to have 18 electrons – we
shall consider this in more detail later in the course.
For covalent bonds, the number of bonding pairs of electrons is termed the bond order. Thus Cl2 has
a single bond, O2 a double bond, and N2 a triple bond. These bonds are commonly represented by:
Cl2 O2 N2
Covalent bonds will be polar if the bonded atoms have different electronegativities (a measure of
the power of an atom in a molecule to attract electrons). Polar bonds are expected to have some
degree of ionic character (e.g. δ+
H–Fδ–
).
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Pairs of electrons not involved in bonding are termed lone pairs (l.p.) or non-bonding pairs (n.b.p)
(e.g., water H2O and ammonia H3N).
Species with unpaired electrons are called radicals. They are often, but not always, reactive species.
There are only a few exceptions to the noble gas configuration rule for the elements in the early
rows of the periodic table.
Examples of ―electron deficient‖ species are BF3 and AlCl3. Electron deficient compounds are
sometimes termed Lewis acids as they can readily accept a pair of electrons from another atom
to form a dative bond, e.g. H3NBF3 (in which all atoms have now achieved noble gas
configurations). Note that, once formed, dative bonds are indistinguishable from normal
covalent bonds.
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For elements in the third row of the periodic table there are examples where the number of
outermost electrons around the elements exceeds that of the nearest noble gas, e.g. PCl5 and SF6.
These are sometimes termed ―hypervalent‖ species. They are traditionally explained as being
due to the involvement of d orbitals in the bonding (which can‘t occur for second row elements).
Some structures cannot be adequately represented by a single diagram showing the pattern of bonds
in the molecule, as they exist as a resonance hybrid of different contributing structures. Well-
known examples include benzene (in which all bond lengths are equivalent) and the CH3COO–
(acetate/ethanoate) anion. In some cases the contributing resonance structures can have quite
different electronic structures and energies, e.g. the cyanate anion:
NC–O– –N=C=O.
BUT BENZENE (C6H6)
in valence bond theory diagrams, each line is a PAIR of
electrons.
all electrons are indistinguishable from each other (i.e. it is impossible to state whether a
particular electron ―originated‖ from one atom or another).
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2.2 Guidelines for drawing Lewis Structures
Example: Carbonate Ion: CO32-
(i) Count total number of valence electrons in molecule to be formed. (For [XX]-
or [YY]+
ions
subtract / appropriate amount of electrons.
#ATOMS Additional
charge
on
molecule
Total Electron
count
(Σ #v.e.)
Carbon
Oxygen
Number of
valence
electrons
(# v.e)
(ii) Connect atoms with single bond initially using LEAST electronegative (see page 44) atom as
central atom
(iii) Complete OCTETS around MOST electronegative atom
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(iv) Place surplus electrons on central atom (even if it means more than OCTET..??hypervalent??).
(v) If central atom still not OCTET, donate non-bonding pair into bond.
(vi) If molecule has an overall negative charge put this on most electronegative atom(s), but
remember there may be resonance structures
ALSO, REMEMBER SYMMETRY, i.e. N2O (Nitrous oxide, laughing gas)
NNO VS. NON
LEWIS STRUCTURES:
Provides information about atom connectivities
Provides information about valence orbitals: bonding/non-bonding
Provides information about bond character, i.e. single, double, resonance structures.
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Examples:
Describe using valence bond theory the ―Lewis‖ structure of:
(a) Hydroxide Ion [OH]-
Guideline (i)
#ATOMS Additional
overall charge
On molecule
Total
Electron
count (Σ
#v.e.)
Hydrogen
Oxygen
#
(v.e)
Guidelines (ii & iii)
Guideline (vi)
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(b) Carbon Monoxide (CO)
Guideline (i)
#ATOMS Additional
Charge
On
molecule
Total
Electron
count
Carbon
Oxygen
# (v.e)
Guidelines (ii & iii)
Guideline (v)
Guideline (vi)
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(b) Trifluorochlorine ( ClF3 )
Guideline (i)
#ATOMS Additional
Charge
On
molecule
Total
Electron
count
Chlorine
Fluorine
# (v.e)
Guidelines (ii & iii)
Guideline (iv)
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2.3 Valence Shell Electron Pair Repulsion (VSEPR) theory
This is a method of predicting the shapes of molecules based on valence bond theory. It assumes
that the geometry of molecules is based on electron-electron repulsion. It works extremely well for
main-group elements. However, it isn‘t applicable to transition metal or lanthanide complexes (see
later in course). VSEPR theory is based on the following assumptions:
Bonds between atoms in a molecule consist of electron pairs.
Some atoms may possess lone pairs (i.e. non-bonded pairs).
Electron pairs adopt positions to minimise their mutual repulsion. The justification for this is
that areas of negative charge density repel other regions of negative charge density.
Lone pairs (lp) repel more than bonding electron pairs (bp),
i.e. repulsion is in order
If more than one possible structure exists we have to consider different geometrical
arrangements, BUT keep to the Basic geometric arrangement of electon pairs described in the
table below!. Make a table of interactions and ONLY CONSIDER ELECTRON PAIR
INTERACTIONS AT RIGHT ANGLES, i.e. 90 degrees to EACH OTHER
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Multiple bonds are treated as though they were a single electron pair for geometric purposes;
this means that the basic coordination geometry is dictated by the σ bonding framework only.
Multiple bonds will, however, occupy more space than single bonds.
The basic Geometric arrangement of electron pairs around the central atom will be
Electron
pairs
Geometry Molecule
2 Linear AX2
3 Trigonal planar AX3
4 Tetrahedral AX4
5 Trigonal bipyramid
(common)
Square-based
pyramid (rare)
AX5
6 Octahedral AX6
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When doing problems:
Count valence electrons + draw Lewis structure
Remember to bear in mind the charge on the complex if any
Determine the number of electron pairs present & remember the rule that repulsion is :
> >
Use Table of basic geometric arrangements as starting point for structure
If more than one possible arrangement of electron pairs exists, make a table of lp/lp, lp/bp and
bp/bp interactions and CONSIDER ONLY ELECTRON PAIR INTERACTIONS AT RIGHT
ANGLES, i.e. 90 degrees to EACH OTHER
Note that while π bonds don‘t directly affect the choice of molecular shape, they do need to be
considered for electron counting purposes!
Examples
H2O
Σ#v.e. Lewis
Structure
# e- pairs Basic
Structure
Repulsion Approx
Bond Angle
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NH3
Σ#v.e. Lewis
Structure
# e- pairs Basic
Structure
Repulsion Bond Angle
NH4+
Σ#v.e. Lewis
Structure
# e- pairs Basic
Structure
Repulsion Bond Angle
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ClF3
Σ#v.e. Lewis
Structure
# e- pairs Basic
Structure
Repulsion Bond Angle
BUT OTHER ARRANGEMENTS POSSIBLE
Structure lp/lp lp/bp bp/bp
(a)
(b)
(c)
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2.4 Limitations of valence bond theory
Valence bond theory, as described above, is simple and it is often an adequate description of
chemical bonding. However, it fails to explain the bonding in some molecules. For instance:
Why is O2 attracted to a magnet, implying that it must possess one or more
unpaired electrons?
Why does CO bind to metals through the carbon atom, i.e. behave as δ–
C≡Oδ+
in contrast to
what would be expected on simple electronegativity grounds?
What is the bonding like in the molecule B2H6, in which two of the hydrogen atoms bridge the
boron atoms?
What is the bonding like in Zeise‘s salt, K [Pt(C2H4)Cl3] .H2O? This was first made in 1827, but
it wasn‘t established unambiguously until the 1950‘s that ethene is bonded to the platinum atom.
What is the bonding like in ferrocene, Fe(C6H6)2, in which the iron atom is equidistant to all the
carbon atoms?
What is the bonding like in the [Re2Cl8]2–
anion, in which the Re–Re distance is only 2.24 Å (far
closer than the atoms are in rhenium metal), and the chlorine atoms adopt a configuration in
which they sterically hinder each other?
Valence bond theory also fails to provide any information on the ease with which an electron can be
removed from a molecule, or alternatively the energy change if an electron is added to a molecule.
For all these questions, we need another bonding theory that takes into account more features of
quantum mechanics.
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3 Quantum mechanics in chemical bonding
3.1 Some general principles
All of chemical bonding ultimately depend on quantum mechanics. The proof that quantum
mechanics is a good theory is that it works and has never yet failed (unlike classical mechanics
which only works for ―big‖ systems). We don‘t have time to go properly into quantum mechanics,
and so all we‘re going to do in this course is to use a few key points without any justification (sorry,
while it‘s a really interesting subject, it‘s not directly relevant to chemical engineering).
3.1.1 Wave-particle duality
Things that we normally associate as being particles have wave-like properties, while things we
normally associate as being waves have particle-like properties. Two important examples:
Light (electromagnetic radiation) of frequency ν consists of photons, each of
energy
E = hν (Planck equation)
where h = Planck‘s constant (6.626 x 10–34
J s).
For instance, light behaves as particles in the photoelectric effect.
When light behaves as a particle is has an associated kinetic energy T given by:
• As intensity of light , number of photoelectrons
• BUT the K.E. of photoelectrons does not change
• Red light does not cause electrons to be ejected whatever the intensity
• Only when light is of sufficient energy will it cause the ejection of electrons from the surface
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But de Broglie suggested that particles with momentum p have an associated
wavelength
This is demonstrated by experiments such as the diffraction of light (as electrons)
Using de Broglie’s relationship it is possible to express the Classical General
Wave Equation as:
The classic general wave equation tells us how a system evolves according to Newton‘s
Laws.
LIGHT SOURCELIGHT SOURCE
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3.1.2 Heisenberg‟s uncertainty principle
This states that it is not possible to know simultaneously the position and
momentum of a particle.
While classically unintuitive, it follows from the fact that a particle that is nominally at rest at a
single point must still be behaving like a wave (in some fashion) – it must have some oscillation in
position and/or momentum.
In quantitative terms, it can be shown that the product of the uncertainties always obeys:
Δp ˟ Δx ≥ ћ/2 where ћ = h/2π
An equivalent statement is ΔE ˟ Δt ≥ ћ/2, where Δt is the uncertainty in the time that the particle
spends in an energy level that has uncertainty ΔE.
Because of the Heisenberg uncertainty principle, we are no longer able to talk about the position of
a particle such as an electron.
Instead, we can only talk about the probability of finding the
particle in a specified volume.
3.1.3 Schrödinger Equation....(The equation of motion of waves!)
The Schrödinger equation is the quantum mechanical equivalent of the general wave equation given
back on the previous page. The Schrödinger equation describes how a quantum state, which we call
the WAVEFUNCTION {(x,y,z,t)}, evolves in time and space. The wavefunction, , is a
(quantum) mathematical description of where a particle is in space. The Schrödinger equation is
completely deterministic and may have many solutions for a given wavefunction (Schrödinger‘s
Cat!!). However in order to know something about the wavefunction we must somehow measure it.
Mathematically, we do this by collapsing the wavefunction and thus convert the wavefunction into a
PROBABILITY distribution of finding a particle in a volume element dV. The probability is given
by ψ2dV. (More rigorously, the probability function is ψψ
*dV where the asterisk denotes
complex conjugate). The time-independent Schrödinger wave equation for a particle of mass, m,
experiencing a potential energy, V(x, y, z) is:
EVm
2
2
2
2
2
22
2 zyx
where E is the energy of the system.
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For a system changing with time the time-dependent form of the equation is:
ti
zyx
V
m 2
2
2
2
2
22
2
where i is the square root of –1.
We can now define a system by specifying the potential energy function V, specifying the boundary
conditions and then attempt to solve this equation. It turns out that the time-independent equation
can be solved exactly only in very simple cases. One of these, usually termed the ―particle in a box‖,
is on your examples sheet. Another that can be solved exactly is the hydrogen atom – we shall
discuss this in a little while.
In those simple cases where we can solve the Schrödinger equation, we discover that:
The equation may only be solved if the energy E of the system takes certain
values – there are no solutions for other E values. Thus the concept of
quantisation of energy is in fact a direct consequence of nature obeying the
Schrödinger equation.
We characterise the solutions by “quantum numbers” – these are parameters that can
only take certain values. For instance, the solution to a particle in a box (see examples sheet)
implies that the only permitted values of E are n2π
2ћ
2/2ma
2 (where m is the particle mass and a
is the length of the box). Here n is a quantum number that takes values 1, 2, 3, 4, etc.
Each solution gives us an expression for the wavefunction ψ for the system
with that particular energy.
Each solution has a particular probability function for the location of the
particle in a given volume dV.
In order for the total probability to sum to unity, we must normalise the
wavefunction so that:
It turns out that these rules also hold for the more complicated cases in which we cannot solve the
Schrödinger equation exactly. In these cases, we can still obtain approximate solutions using
computing methods.
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Solving the Schrödinger wave equation for a free particle moving in one
dimension in a constant potential V0
(i) Write down the ―time independent‖ Hamiltonian operator
(ii) Write down the Eigenvalue (Scrodinger) equation and rearrange.
(iii) Consider the case where E > V0, i.e. (E-V0) is positive. Remember kinetic
energy is never negative in classical mechanics. To solve (1) we GUESS the
solution!
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(iv) Substitute the guess in (iii) into the equation you obtained in (ii) & then
evaluate
(v) Equate RHS of equation 1 with RHS of equation (2) in (iv)
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GRAPHICAL REPRESENTATION OF PREVIOUS EQUATION
Free particle moves/oscillates with increasing energy
BUT
This solution is NOT Quantised or Normalised
MUST impose Boundary Conditions to complete
quantisation
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3.2 Atomic orbitals: one-electron atoms
We shall initially discuss the case of atoms containing only one electron (i.e. H,
He+, Li
2+ etc.), as the Schrödinger equation can be solved exactly in this case.
3.2.1 Schrödinger equation
In this case the potential energy function is the Coulombic energy, and the time-independent
Schrödinger equation is:
E
Ze
2
0
2
2
2
2
2
2
22 1
42 rzyx
where Z is the atomic number, e the charge of an electron, r is the separation of electron and
nucleus, and μ is the reduced mass (given by 1/μ = 1/melectron + 1/mnucleus). Use of the reduced mass
allows for the fact that the nucleus as well as the electron will be moving. The Schrödinger equation
can then be transformed from Cartesian coordinates (x, y, z) into spherical polar coordinates (r, θ,
φ), and solved. The boundary conditions necessary to ensure that the wavefunction gives a
meaningful probability density are:
ψ must be single-valued in space
ψ must vary smoothly and
cannot suddenly jump from one value to another
The integral of ψ* ψ over all space
and must be finite
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The solutions for the wavefunction depend on three quantum numbers, normally designated n, l and
ml, and the solutions can be written in the form:
ψ = R(n, l, r) × Y(l, ml, θ, φ)
The solutions thus contain two separate parts:
a radial part (giving the r dependence) – this depends only on n and l quantum numbers
and an angular part (giving the θ, φ dependence) – this depends only on l and ml quantum
numbers.
It turns out that the functions R(n, l, r) and Y(l, ml, θ, φ) had been discovered before Schrödinger
was born as they are solutions to ―interesting differential equations‖ that had been previously
studied. The functions are known as ―associated Laguerre polynomials‖ and ―spherical harmonics‖,
respectively (you don‘t need to remember these names!).
The lowest energy solutions have the following form (no need to remember these expressions):
n l ml Wavefunction Atomic orbital
1 0 0 ψ = (1/√π) (1/ao)3/2
exp(–r/ao)
2 0 0 ψ = (1/4√2π) (1/ao)3/2
(2 – r/ao) exp(–r/2ao)
2 1 0 ψ = (1/4√2π) (1/ao)3/2
(r/ao) exp(–r/2ao) cos θ
2 1 ±1 ψ = (1/4√2π) (1/ao)3/2
(r/ao) exp(–r/2ao) sin θ cos φ
ψ = (1/4√2π) (1/ao)3/2
(r/ao) exp(–r/2ao) sin θ sin φ
3 0 0 ψ = (1/18√3π) (1/ao)3/2
(6 – 6r/ao+ r2/ao
2) exp(–r/3ao)
[ ao is a constant termed the ―Bohr radius‖ and is given by 4πεoћ2/mee
2 = 5.292 x10
–11 m ]
These solutions have different energies, and different probability functions for the
electron location.
The solutions are more commonly called atomic orbitals, and denoted 1s, 2s, 2p,
3s, 3p, 3d etc. Hence we have discovered that atomic orbitals are simply solutions
to the Schrödinger equation for one-electron atoms.
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We now need to consider the physical significance of the quantum numbers.
The „principle‟ quantum number 'n' (determines the ENERGY of the orbital)
o Has integral values of 1, 2, 3, etc.
o As n increases the electron density is further away from the nucleus
o As n increases the electron has a higher energy and is less tightly bound to the
nucleus:
The „orbital‟ (azimuthal ;second) quantum number 'l ' (determines the type of orbital)
o Has integral values from 0 to (n-1) for each value of n
o Instead of being listed as a numerical value, typically ' l ' is referred to by a letter
('s'=0, 'p'=1, 'd'=2, 'f'=3)
o Defines the shape of the orbital; shading indicates a different phase
o Magnitude is ħ √l(l+1)
22
00
22 1
42 nna
eZ
E
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The magnetic (third) quantum number 'ml' (determines orbital orientation)
o Has integral values between 'l' and -'l', including 0
o Describes the orientation of the orbital in space
The „spin‟ quantum number 'ms'
o Further detailed analysis reveals that a fourth quantum number becomes necessary
when relativity is taken into account as well as ―simple‖ quantum mechanics. This
quantum number is termed the electron spin quantum number ms, and it can take
values +½ and –½ only.
Thus we have discovered that for one-electron atoms there is:
One 1s orbital
One 2s orbital, and three 2p orbitals of the same energy
One 3s orbital, three 3p orbitals, and five 3d orbitals of the same energy
One 4s orbital, three 4p orbitals, five 4d orbitals, and seven 4f orbitals of the same energy
Atomic orbitals that have the same energy
as each other are termed “degenerate”.
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3.2.2 General features of WAVEFUNCTIONS, r,for one electron atom
The wavefunction (r, , ,) is a mathematical description of the electron in space
By calculating * × (remember * is the complex conjugate of ) at some point (r, ,
) in space we determine the entirely real and thus measurable PROBABILTIY DENSITY
for the electron AT THAT POINT.
(r) is a RADIAL wavefunction function; it is real and makes a contribution to the total
probability density function of 2(r). (its unit is 1/Volume and hence this is why it is
thought of as a density function…multiplication of this density function by a infinitesimal
volume of space, d=dxdydz, gives the occupation number in a small element of space).
Note: in general (r) does not give a true indication of the electron distribution in an
orbital since it only represents part of the wavefunction
• Integration of the occupation number, i.e. sum of various probabilities for each infinitesimal
volume element d over all space gives the total occupation of the electron in all space,
which must equal unity, i.e.
• Conversion of d from Cartesian to Spherical polar coordinates gives:
• The TOTAL probability density function is a product of both the square the RADIAL
FUNCTION, which having been converted to spherical polar coordinates, we now denote
R2(r), and the square of the ANGULAR FUNCTION Y*Y(l, ml, θ, f)
1,(* spaceall
dr
dddrrd sin2
ddYYdrrrR sin,,0 0
2
0
*22
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• NOTE: R2(r) is a probability (per unit volume) density function for the DISTANCE of the
electron from the nucleus EXCLUSIVE of any ANGULAR variation. It is more intuitive,
when thinking about the probability where electrons are in space, to think about the
probability of finding an electron in a thin shell of radius r and thickness dr.
• The reason this approach is particularly useful is that it adds up the probability of finding an
electron in all spatial directions and thus give us a measure of the probability of finding
an electron at a particular DISTANCE from the nucleus, regardless of direction.
• The surface density function(SDF) is defined by:
SDF = 4 r 2 R
2(r) ≡ P(r)
and takes into account explicit angular variation. The SDF is equivalent to P(r), the
RADIAL DISTRIBUTION FUNCTION (RDF).
• The product of the SDF/RDF with the thickness of the thin shell, i.e.
P(r) dr =4 r2 R
2(r) dr
is then defined as the probability of finding the electron in a shell of radius r and thickness
dr.
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r(a0)2 4 6 8 10
4R
2 (r)
×r2
, (1
/a0)
Graphical sketches of hydrogen radial wavefunctions(no inclusion of spherical harmonic, Y(l, ml, θ, ) )
1s surface density function, 4R2(r)×r2
Also known as the radial distribution/probability function RDF/RPF
1s radial function, R(r)
r(a0)2 4 6 8 10
R(r
), (
1/a
03/2
)
1s radial function, R(r)
r(a0)2 4 6 8 10
R(r
), (
1/a
03/2
)
1s radial probability
density function R2(r) for
a given set of coordinates
r(a0)2 4 6 8 10
R2 (
r),
(1/a
03 )
1s radial probability
density function R2(r) for
a given set of coordinates
r(a0)2 4 6 8 10
R2 (
r),
(1/a
03 )
ψ1s = R(r) = 2 (1/ao)3/2 exp(–r/ao)
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0
0 5 10 15
0
0 5 10 15
Other radial distribution functions
Points to note:
• 1s electrons are on average a lot closer to the nucleus than electrons in higher orbitals
• (you can show that the most likely distance = a0 for 1s case).
• There is a radial node in the probability distribution function of the 2s orbital. While the 2s and 2p orbitals are identical in energy for the hydrogen atom, there is a far higher chance of a 2s electron being very close to the nucleus than for a 2p electron.
• No. of Radial Nodes = (n - l - 1)
r(a0)
1s
2s
2p
Node
4
R2(r
) ×
r2,
(1/a
0)
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3.2.3 Energy level diagram for one-electron atoms
Energy
n = ∞
l = 0 l = 1 l = 2
n = 3
n = 2
n = 1
• Ground state is lowest energy configuration
– All systems try to adopt the lowest possible energy
– One e- atoms occupies 1s orbital
– Transistions (absorption/emission) to other orbitals possible (h ; kBT)
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3.2.4 Shapes of atomic orbitals for one-electron atoms
The combined radial ( R ( n , l , r ) ) and angular ( Y ( l , m l , θ , ) ) wave functions give rise to 3D electron ORBITALS. Below are isosurface representations of these for a particular
value of the wavefunction. !!NOTE the value of the iso-surface gives the apparent size of the
orbital, it doesn‘t reflect the electron density!!
• s - orbitals are spherically symmetric
• Higher s - orbitals have NODES where the electron density is zero
• NO ANGULAR ( , ) PART for s - orbitals
r
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• p - orbitals are dumbbell shaped
• Node at the nucleus.
• There are three distinct p - orbitals, they differ in their orientations
• There is no fixed correlation between the three orientations and the three magnetic quantum numbers (m
l )
• Combined radial (r) and angular ( , ) part
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• d - orbitals have following shape
UNDERSTANDING ORBITAL SHAPES IS KEY TO UNDERSTANDING
THE MOLECULES FORMED BY COMBINING ATOMS
r
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Some useful concepts
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3.3 Atomic orbitals: multi-electron atoms
For atoms containing more than one electron the Schrödinger equation is far more complicated as
interelectron repulsion needs to be included in the potential energy function. It turns out that even
with only two electrons present the equation becomes too complicated to solve exactly! Theoretical
chemists can, however, get very close to the solution using computing methods. Important results
from their work are:
• The approximate shape of the orbitals is unchanged use same labelling scheme as for the
hydrogen atom.
Energies now also depend on the l quantum number. Calculations show that:
E(2s) < E(2p) and E(3s) < E(3p) < E(3d) E(4s)
i.e. the {2s, 2p} and {3s, 3p, 3d}energy levels are no longer degenerate with each other.
This is precisely the situation that had been conjectured when explaining the form of the
periodic table, only now it has been given a sound theoretical basis.
This arrangement of atomic orbital energy levels is normally rationalised at a qualitative level by
introducing the concept of electron shielding. Electrons in outer orbitals are ―shielded‖ or
―screened‖ to a large extent from the full nuclear charge by the inner electrons due to interelectron
repulsion. Examination of the radial probability distribution plots for the hydrogen atom shows that
2s electrons are better able to penetrate close to the nucleus than 2p electrons, and thus that 2s
electrons are less effectively shielded from the nucleus by electrons in the 1s orbital. (see page 35)
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n l ml ms
2px
2py
2pz
n l ml ms
1s
Multi-electron atoms energy level diagram
RULES FOR PLACING ELECTRONS INTO ENERGY LEVELS
• Pauli Exclusion principle: no electron may have the same set of quantum numbers (n, l , ml , ms)
• Aufbau principle: lowest E levels fill first
• Hund ’s Rule: degenerate levels parallel spin-pair first, then once all orbitals have been used, they pair-up according to Pauli.
Energy
1s
2s
2px 2py 2pz
3s3px 3py 3pz
3d1 3d3 3d4 3d53d2
n = ∞
l = 0 l = 1 l = 2
n = 3
n = 2
n = 1
Energy
1s
2s
2px 2py 2pz2px 2py 2pz
3s3px 3py 3pz3px 3py 3pz
3d1 3d3 3d4 3d53d23d1 3d3 3d4 3d53d2
n = ∞
l = 0 l = 1 l = 2
n = 3
n = 2
n = 1
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This leads to the following ground state configuration of the elements:
1 H 1s1 19 K [Ar]4s
1
2 He 1s2 20 Ca [Ar]4s
2
3 Li [He]2s1 21 Sc [Ar]4s
23d
1
4 Be [He]2s2 22 Ti [Ar]4s
23d
2
5 B [He]2s22p
1 23 V [Ar]4s
23d
3
6 C [He]2s22p
2 24 Cr [Ar]4s
13d
5 (*)
7 N [He]2s22p
3 25 Mn [Ar]4s
23d
5
8 O [He]2s22p
4 26 Fe [Ar]4s
23d
6
9 F [He]2s22p
5 27 Co [Ar]4s
23d
7
10 Ne [He]2s22p
6 28 Ni [Ar]4s
23d
8
11 Na [Ne]3s1 29 Cu [Ar]4s
13d
10 (*)
12 Mg [Ne]3s2 30 Zn [Ar]4s
23d
10
13 Al [Ne]3s23p
1 31 Ga [Ar]4s
23d
104p
1
14 Si [Ne]3s23p
2 32 Ge [Ar]4s
23d
104p
2
15 P [Ne]3s23p
3 33 As [Ar]4s
23d
104p
3
16 S [Ne]3s23p
4 34 Se [Ar]4s
23d
104p
4
17 Cl [Ne]3s23p
5 35 Br [Ar]4s
23d
104p
5
18 Ar [Ne]3s23p
6 36 Kr [Ar]4s
23d
104p
6
This electronic structure is important as the outer (valence) electrons largely determine the
chemistry (in both valence bond theory and molecular orbital theory).
There is excellent direct experimental evidence for atomic orbitals and the ordering of their energy
levels from spectroscopy. If an atom is in an excited state, then it can fall down to a lower quantum
level by emitting a photon of frequency ν determined by the separation of the energy levels.
Measurements of the discrete frequencies emitted thus allows the separation of energy levels to be
determined experimentally.
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3.4 Periodic trends in properties of atoms
There are a number of periodic trends in the properties of elements across and down the periodic
table. These include ionisation energy, electron affinity, electronegativity and atomic radii. Most of
these can be easily rationalised using our knowledge of atomic orbitals.
3.4.1 Example: 1st ionisation energy
The 1st ionisation energy is the energy required to remove an electron from the neutral atom in the
gas phase:
0
10
20
30
0 10 20 30 40 50 60
Atomic number
1s
t io
nis
ati
on
en
erg
y (
eV
)
The electron removed will come from the highest occupied energy level. Thus ionisation energy will
depend on the highest filled atomic orbital, the atomic charge and the electron shielding.The
periodic trends in 1st ionisation energy can be rationalised on this basis.
1. The large increase in I.E. across period (n is the same) because effective nuclear charge is
increasing. Each additional nuclear charge is having a greater effect than the extra electron
shielding.
2. There is a decrease in I.E. down a group. The electron is being removed from an atomic
orbital with higher n quantum number.
3. Certain deviations in this overall pattern occur:
a) Be (2s2) has a higher I.E. than B (2s
22p
1): easier to remove 2p than from a full 2s level.
b) N (2s22p
3) has a higher I.E. than O (2s
22p
4): there is extra stability associated with a
half-filled energy level (or, equivalently, there is an extra instability associated with
having to pair up an electron).
4. There is only a small increase in ionisation energy across transition metals – effective
nuclear charge is increasing, but not outweighing the electron shielding term by as much as was the
He
Ne
Ar Kr
Xe
Li Na K Rb Cs
Zn
Ga In
Cd
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case for the main-group elements. This is because d (and f) orbitals are ―diffuse‖ – they do not
penetrate very close to the nucleus.
3.4.2 Some comments on electronegativity
Electronegativity is defined as the power of an atom IN A MOLECULE to attract
electrons. This means that electronegativities have no precise physical meaning. They cannot be
measured, and can vary for an element depending on which particular molecule it finds itself in.
Electronegativities predict that electrons are polarised towards electronegative atoms away from
electropositive atoms, and so are useful when discussing bond polarities. In most cases (but not all
cases), predictions based on electronegativities are correct.
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4 Molecular orbital theory
4.1 Linear Combination of Atomic Orbitals (LCAO) approach.
Solving the Schrödinger equation for one electron in a multi-atom system (i.e. a one-electron molecule) in principle gives us a set of wavefunctions ψ with energies E that correspond to molecular orbitals by analogy to atomic orbitals. However, this is too complicated to do rigorously. We thus have to make a simplifying assumption. Molecular orbitals (mo ‘s) arise from the overlap of atomic orbitals (ao ‘s).
I.e. mo is derived from the “LINEAR COMBINATION OFATOMIC ORBITALS” (LCAO),
mo= c1ao
1 + c2ao2 + c3ao
3 ...
Where ci represents the relative amounts contributed by the a.o. ‟s
The simplest case: the H2
+ molecule
Consider the H2
+ molecule (i.e. two H nuclei, which we shall call A and B, and one single electron).
In-phase linear combination of a.o wavefunctions
•CONSTRUCTIVE overlap of two A.O. ‘s give a single M.O. of lower energy, termed the
BONDING M.O.
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Anti - phase linear combination of a.o wavefunction
DESTRUCTIVE overlap of two A.O.‘s give a single M.O. of higher energy,
termed the ANTIBONDING M.O.
• Combine in - phase and anti - phase combinations into a single energy level
diagram for H 2 +
Energy
1s 1s Atomic orbital
Atomic orbital
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Note: the QM calculations show that the energy of the anti-bonding MO is raised slightly more in
energy, relative to the component AO ‘s, than bonding MO ‘s are lowered. However, this
behaviour is not always shown and often you will see no difference depicted in texts for simple
MO diagrams.
The way in which electrons are placed into MO‘ s is exactly the same as that for the single and
multi-electron atom model discussed earlier (Pauli, Aufbau and Hund)
•Some Examples
Energy
1s 1s
H2 molecule
Energy
1s 1s
He2 molecule?
He2 molecule…does not exist as it is unstable w.r.t He atoms
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The ―σ‖ indicates that the molecular orbital has the same symmetry as an ―s‖ atomic orbital - we use
greek letters for molecular orbitals. The subscript ―s‖ indicates that these molecular orbitals arise
from s atomic orbitals, and the asterisk is used to indicate an antibonding orbital.
and molecular orbitals are numbered according to symmetry. DO NOT GET CONFUSED
WITH the quantum number label ‗n‘
Electrons in bonding molecular orbitals have a lower energy as they are less constrained, i.e. more
delocalised, which means that is Kinetic Energy has decreased. The electron density is now shared
between two nuclei, which also results in a lowering in the energy of the system.
Electrons in anti-bonding orbitals have a higher energy due to the presence of a NODE/NODAL
PLANE
4.2 Formation of other molecular orbitals
The following general rules are obeyed:
1. Combining n atomic orbitals gives n molecular orbitals.
2. Only atomic orbitals of the correct symmetry will combine.
3. The energy of a molecular orbital relative to the atomic orbitals
from which it is derived depends on:
the relative energies of the atomic orbitals (close in energy
large interaction);
the degree of atomic orbital overlap between them (good overlap
large interaction).
4. Bond order = ½ (#bonding electrons – #antibonding electrons)
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There are two ways of combining p atomic orbitals, and these give different symmetry molecular
orbitals. Overlap of two pz orbitals ―end-on‖ forms a ζ bonding orbital. The sideways overlap of
two px orbitals, and the sideways overlap of two py orbitals form two π bonding orbitals. (Note that
π molecular orbitals have the same symmetry as p molecular orbitals). The resulting molecular
orbital shapes are shown below.
ψA(s) + ψB(s)
ψA(s) – ψB(s)
ψA(pz) + ψB(pz)
ψA(pz) - ψB(pz)
ψA(px) + ψB(px)
ψA(px) - ψB(px)
ψA(py) + ψB(py)
ψA(py) - ψB(py)
ψA(s) + ψB(s)
ψA(s) – ψB(s)
ψA(pz) + ψB(pz)
ψA(pz) - ψB(pz)
ψA(px) + ψB(px)
ψA(px) - ψB(px)
ψA(py) + ψB(py)
ψA(py) - ψB(py)
A.O.’sA.O.’s M.O.’sM.O.’s
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In the ―normal‖ case, the energy change on forming πp molecular orbitals is less than that on
forming σp molecular orbitals. We can thus draw an energy diagram showing the molecular orbitals
formed from the overlap of p atomic orbitals.
We are now in a position to construct a molecular orbital energy level diagram for simple diatomic
molecules.
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4.3 Molecular orbital energy diagrams for homonuclear diatomic molecules
The molecular orbital energy diagram for O2 and F2 (and Ne2 if it existed) can now be drawn, and is
shown below. Note that 1s – 1s overlap will be negligible because electrons in the inner shell are too
close to one of the nuclei to interact with the other one and thus on the energy level diagram below
they would be way below the 2s atomic orbitals and remain as individual atomic orbitals. However,
the naming system still takes into account that the 1s atomic orbitals are there.
2p 2p
Energy
2s 2s
Atomic
orbital
Atomic
orbital
Molecular
orbitals
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4.4 s-p mixing
The situation for molecules Li2, Be2, B2, C2 and N2 is a little more complicated. It turns out that the
2s and 2p atomic orbitals are sufficiently close in energy that the 2s – 2pz interaction is significant.
The 2s – 2pz is a σ symmetry interaction and this, along with their close energy match, means that
the resulting molecular orbitals also have the same symmetry. It is then possible to take linear
combinations of the subsequent molecular orbitals of the correct symmetry in exactly the same way
as for atomic orbitals resulting in a shifting of the MOLECULAR ORBITAL energy levels. The
next two diagrams illustrate this process starting with the linear combination of MOLECULAR
ORBITALS which were originally formed from the linear combination of ATOMIC ORBITALS:
2p 2p
3σp
3σp*
πp
πp*
2s 2s
2σs
2σs*
2p 2p3σp
3σp*
πp
πp*
2s 2s
2σs
2σs*
2p 2p
3σp
3σp*
πp
πp*
2s 2s
2σs
2σs*
2p 2p3σp
3σp*
πp
πp*
2s 2s
2σs
2σs*
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Energy level diagram for s-p mixing.
NOTE DESPITE THE APPEARANCE ON THE DIAGRAM THAT
THE 2s and 2p ATOMIC ORBITALS ARE FAR APART IN ENERGY
THIS IS NOT THE CASE FOR Li2, Be2, B2, C2 and N2 AND IN
REALITY THEY ARE CLOSE ENOUGH TO ALLOW OVERLAP.
NOTE: Whilst the 3p and 3p* have been raised in energy in the s-p
mixed case, this effect is outweighed by the lowering of the 2s and 2s*
orbitals
2p 2p
3σp
3σp*
πp
πp*
2s 2s
2σs
2σs*
2p 2p3σp
3σp*
πp
πp*
2s 2s
2σs
2σs*
2p 2p
3σp
3σp*
πp
πp*
2s 2s
2σs
2σs*
2p 2p3σp
3σp*
πp
πp*
2s 2s
2σs
2σs*
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We thus have the following energy level diagrams for the homonuclear diatomic molecules from Li2
to F2
We can now write down the ground-state electronic configurations of the following homonuclear molecules:
Li2 σs2
Be2 σs2 σs*
2
B2 σs2 σs*
2 πx1 πy
1
C2 σs2 σs*
2 πx2 πy
2
N2 σs2 σs*
2 πx2 πy
2 σp2
O2 σs2 σs*
2 σp2 πx
2 πy2 πx*
1 πy*1
F2 σs2 σs*
2 σp2 πx
2 πy2 πx*
2 πy*2
Ne2 σs2 σs*
2 σp2 πx
2 πy2 πx*
2 πy*2 σp*
2
It can be seen that molecular orbital theory can successfully predict bond orders predict magnetic properties. Molecules containing unpaired electrons are said to be paramagnetic; such molecules are (fairly strongly) attracted by magnetic fields. Molecules in which all the electrons are paired are termed diamagnetic; they are weakly repelled by magnetic fields.
Two key points:
A ―bond‖ in MO theory results from electrons occupying molecular orbitals formed from the favourable overlap of atomic orbitals of appropriate symmetry.
Much of the chemistry of molecules is determined by the properties of the Highest Occupied Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO), as these are the orbitals that will be involved in any electron transfer processes.
Li2 Be2 B2 C2 N2 O2 F2
Energ
y
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Naming system for molecular orbitals:
The naming system for molecular orbitals is not particularly logical and is usually based upon
symmetry operations/considerations. However with the exception of hydrogen, for homonuclear
diatomics of the second period row in the periodic table (which is all you‘ll ever be asked about)
the bonding starts with the overlap of the 2s atomic orbitals. The two molecular orbitals that are
produced (one bonding and one anti-bonding) are named the 2s and 2s* respectively. Often for
simplicity the ‗s‘ subscript is dropped.
NOTE: that despite the physical existence of the 1s atomic orbitals in the second row elements,
these orbitals are too close to the nucleus to overlap and do not take part in bonding. The next most
favourable overlap for homonuclear diatomics of the second period (row) of the periodic table is
the ―sigma symmetry‖ overlap of the 2pz atomic orbitals which gives rise to a bonding 3pz
molecular orbital and an anti-bonding 3pz* orbital. Note: the value of the number preceding the
molecular orbital does not have anything to do with the principle quantum number ‗n‘.
The situation is of course complicated for the second row hetero-nuclear diatomics and second row
molecules (that might contain hydrogen, e.g. H-F). Here the naming is different. First, despite the
fact that the non-hydrogen containing 1s atomic orbitals do not overlap or take part in any bonding,
we still have to acknowledge them to account for the numbering system. We count one of the 1s
atomic orbitals in the second row diatomics as being 1 and the next belongiong to the other atom
as 2. This then correctly predicts that the next molecular orbital (formed from the constructive
bonding overlap of either two 2s atomic orbitals or by the overlap of a single 2s atomic orbital with
a single 2pz atomic orbital) is then labelled the 3. The antibonding orbital that is also formed (by
the destructive antibonding overlap) is NOT a 3* and you must work through the molecular orbital
diagram to obtain the correct numbering. The following examples of carbon monoxide and
hydrogen fluoride will make the system clearer. Note also that for heteronuclear diatomics we do
not use the ‗s‘ or ‗p‘ subscripts at all.
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4.5 Molecular orbital energy diagrams for heteronuclear diatomic molecules
Similar energy diagrams can be drawn for heteronuclear diatomic molecules. Two examples will
illustrate the situation.
Carbon monoxide (CO)
Carbon monoxide is isoelectronic to N2 (i.e. it has the same number of electrons) and it has a
similar energy level diagram to N2. However, the atoms have different relative energies for their
atomic orbitals. The resulting mixing between orbitals of the same symmetry means that it is no
longer meaningful to describe the molecular orbitals using the subscript s and p we used in our
earlier diagrams. The dotted lines in the diagram show the principal atomic orbitals from which the
molecular orbital is derived. CO is correctly predicted to have a bond order of 3, and to have no
unpaired electrons.
Energ
y
A.O’s A.O’sM.O’s
Carbon OxygenCarbon monoxide
2s
2p
2s
2p
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Hydrogen Fluoride (HF)
In HF, there is a very large difference in the energies of the contributing atomic orbitals.
This means that the characteristics of the resulting molecular orbitals are often very close to the
contributing atomic orbitals. Indeed, the π molecular orbitals are derived exclusively from F, and are
termed ―non-bonding orbitals‖. There is no overlap of the hydrogen 1s atomic orbital and the
fluorine 2s atomic orbital due to the large difference in energy. Note that the occupied molecular
orbitals in HF are wavefunctions that are principally F in character (i.e. they have a high coefficient
for the F atomic orbitals in the linear combination expression). This accounts for the polarity of the
HF bond; this explanation of polarity doesn‘t require the nebulous property called electronegativity.
1s
2p
2s
1s
2p
Energy
2s
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4.6 Hybridisation of atomic orbitals
Hybridisation is actually a valence bond concept, but it is sometimes useful in MO theory as well.
Hybridisation may be conceived as the mixing of atomic orbitals on the same atom before
considering orbital overlap with other atoms. [While it turns out to be a useful concept, it does not
actually imply that this mixing occurs in real life!]
Consider methane, CH4. The ground state C atom (1s22s
22p
2) 2 unpaired e
-s for a covalent bond in
valence bond theory. It turns out that it is not too energetically unfavourable to promote one of the
2s electrons into the vacant 2p orbital. The four unpaired electrons can then be shared with the four
hydrogen atoms to form four equivalent covalent
It is thus convenient to describe the configuration of carbon as being derived from four equivalent
sp3 hybrid orbitals. The energy of a hybrid orbital is the weighted average of the contributing atomic
orbitals. It turns out that the shapes of sp3 hybrid orbitals point towards the corners of a tetrahedron.
The molecular orbitals of methane may thus be considered to arise from the overlap of the hydrogen
1s orbitals with the carbon sp3 hybrid orbitals:
Similarly hybrid sp2 orbitals can be invoked to explain the bonding and the planar geometry of
ethene, C2H4, with the electrons in the remaining 2p orbital on each carbon atom overlapping
sideways on to form a π bond
1sfour equivalent sp3
hyb. C
1sfour equivalent sp31sfour equivalent sp3
hyb. C
three equivalent 2p1s 2s
g.s. C
three equivalent 2p1s 2s three equivalent 2p1s 2s
g.s. C
1sthree equivalent sp2 2p1sthree equivalent sp2 2p
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For ethyne, C2H2, on the other hand, sp hybrid orbitals can be used to account for the linear
geometry and two π bonds formed.
More complicated hybridisation schemes are needed for the coordination geometries of other
elements. For instance, the six equivalent orbitals needed for an octahedral coordination (e.g. SF6)
can be treated as sp3d
2 hybrids.
1s two equivalent sp two 2p1s two equivalent sp two 2p
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4.7 Three-centre two-electron bonds
Earlier on, we mentioned that valence-bond theory couldn‘t easily explain the existence of species
such as diborane, B2H6, in which two of the hydrogen atoms are ―bridging‖ species. Diborane has
a total 12 valence electrons (B: 1s22s
22p
1 ; H = 1s
1) and has an approximate tetrahedral
arrangement of hydrogen atoms around each boron atom
For a B–Hterminal group (BHt) the bonding can be considered as two (out of four) normal sp3
hybridised covalent bonds giving a total count of EIGHT of the available twelve electrons in the
whole molecule. In order to understand the bonding of the bridging species B—H—B, in diborane
we need to consider that the B—H—B linkage as being formed from the overlap of 3 orbitals.
To obtain the MO diagram for a B-H-B fragment take linear combinations of two Boron sp3
―hybrid‖ atomic orbitals with a single hydrogen atomic orbital, i.e.
Boron A.O. Hydrogen A.O.
The net result is the 3-center-2-electron bridge bond
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4.8 Limitations of molecular orbital theory
Molecular orbital theory has two main disadvantages.
• Energy level diagrams are extremely difficult to construct for larger molecules, and they also can‘t
easily explain the shapes of any other than the simplest molecules.
We can get closer and closer to the ―true‖ wavefunction of a molecule by adding refinements
either to the valence bond approach or to the molecular orbital approach and so, in the limit,
both descriptions are equivalent. The following table gives an indication of the strengths of the
two main bonding theories.
.
explains “3-centre 2-electron”
bonds such as bridging hydrides
explains metallic bonding well (see
later)
Cannot fully explain bonding in
certain species
Difficult to get geometric
information
Gives information on molecular
geometry (VSEPR)
Gives electronic structure and
information on:
magnetic properties
photoionisation
electron attachement
Gives information on:
bond strengths
bond lengths
force constants
A “bond” results from electrons
occupying molecular orbitals
formed from the favourable
overlap of atomic orbitals of
appropriate symmetry
A bond is a shared pair of
electrons
Delocalised descriptionLocalised description
MO theoryVB theory
explains “3-centre 2-electron”
bonds such as bridging hydrides
explains metallic bonding well (see
later)
Cannot fully explain bonding in
certain species
Difficult to get geometric
information
Gives information on molecular
geometry (VSEPR)
Gives electronic structure and
information on:
magnetic properties
photoionisation
electron attachement
Gives information on:
bond strengths
bond lengths
force constants
A “bond” results from electrons
occupying molecular orbitals
formed from the favourable
overlap of atomic orbitals of
appropriate symmetry
A bond is a shared pair of
electrons
Delocalised descriptionLocalised description
MO theoryVB theory
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5 Bonding in Solids
5.1 Types of solid
Both valence bond and molecular orbital theory are useful descriptions of bonding in solids. Solids
may be classified as:
Ionic solids –
here an infinite structure is formed by bonds that have substantial ionic character (e.g. NaCl). The
coordination number around each atom can often be predicted by the ―radius ratio rule‖ based on the
ionic radii of the species involved:
o rsmaller/rlarger > 0.7 coordination number = 8 e.g. CsCl
o r smaller/rlarger = 0.4-0.7 coordination number = 6 e.g. NaCl
o r smaller /rlarger= 0.2-0.4 coordination number = 4 e.g. CuCl
(note that rsmaller is almost always that of the cation and rlarger that of the anion).
Covalent macromolecules
here an infinite structure is formed by bonds that are essentially covalent (e.g. diamond, SiO2).
Metals
metallic bonding is treated in more detail below, and is best explained by molecular orbital theory.
Virtually all metals exist in one of the three following forms:
o body-centred cubic (bcc): coordination number = 8
o face-centred cubic (fcc; sometimes called cubic close-packed, ccp): coordination
number = 12 (ABCABC packing)
o hexagonal close-packed (hcp): coordination number = 12 (ABAB packing)
HCP (e.g. Co,Zn)
FCC
(Ag, Ca)
BCC
(Ba, Cs)
HCP (e.g. Co,Zn)
FCC
(Ag, Ca)
BCC
(Ba, Cs)
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Molecular solids
these are solids in which there are only weak interactions between the molecules (the bonds within
the molecules themselves will be strong). These solids will normally have low melting points (some
of them will exist only at low temperature). The weak interactions may be due to:
o Hydrogen bonds – a hydrogen bond consists of an H atom between atoms of more
electronegative non-metallic elements (e.g. O–H…O in ice). Note bridging hydrides
in boranes B–H–B and metallic complexes M–H–M are not treated as hydrogen
bonds.
o Polar interactions – here the molecules have a permanent dipole moment, and so
there will be Coulombic interactions between them δ+
A–Bδ–
… δ+
A–Bδ–
o Van der Waals interactions – here the molecules don‘t have a permanent dipole
moment, but they can possess a temporary dipole moment which gives rise to a weak
attractive interaction. (e.g. P4, S8, C60)
Solids may be:
crystalline here there is long-range order and a regular ―unit cell‖ is periodically throughout the structure. Ionic
solids and metals are always crystalline. The other solids may or may not be crystalline.
Amorphous here there is no long-range order and thus no periodic repeating unit.
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5.2 Molecular orbital theory for solids
Molecular orbital theory can obviously be applied to solids, and it turns out to be particularly useful
in describing the properties of metals. In molecular orbital terms, we treat solids as being very large
molecules (typically containing ~1023
atoms). We need to consider overlap of the valence atomic
orbitals on each atom in order to generate the molecular orbitals. Thus it is possible for a valence
electron to be delocalised throughout the entire solid. (We can assume that overlap of the inner core
atomic orbitals is negligible).
Consider the molecular orbitals formed from the overlap of a linear chain for 2-11 Li 2s atomic
orbitals between 2 and 11.
-width of band represents strength of interaction/overlap between A.O.s
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-closer atoms become, stronger is interaction resulting in lowering, in energy of most bonding C.O.s
-and raising in energy of most anti -bonding C.O.s.
When the number n gets very large (as in a solid), we refer to the resulting molecular orbitals as
forming a band. The band will have finite energy width, but it consists of a very large number of
bonding and antibonding M.O.s that approximate a continuous distribution. The M.O.s that form
the band throughout a whole sample are also know as CRYSTAL ORBITALS (C.O.s). A more
detailed analysis shows a clustering of bonding M.O.s and antibonding M.O.s at the extrema of the
band, leading to the so called density of states. Hence for Lithium the band is half filled BUT not
evenly…the majority of electrons cluster in C.O.s at the lower energy end.
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5.3 Electrical Conductivity
5.3.1 Metals
Simply put, this occurs when electrons can move through bands following the application of a
voltage (or potential difference)! Consider the following diagram as snapshots of conduction:
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Overlap of other orbitals
We have seen that the overlap of Li 2s atomic orbitals produces a band whose C.O.s are half full of
electrons and how indeed this facilitates the conduction of electrons. What about Beryllium?
Beryllium has 2 valence electrons in its outer 2s shell and would thus produce a band that would be
completely filled and thus not allow conduction! However, we know that Beryllium is an excellent
conductor, so how can this be? Well, we need to consider so-called Band Overlap. For Beryllium:
The above diagram shows that the strength of the bonding can be increased if electrons from the top
(anti-bonding) C.O.s are transferred to the bottom (bonding) C.O.s of the p-band. Hence this
accounts for partially filled bands which intern gives Beryllium is electrical conductivity when a
voltage is applied.
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5.3.2 Insulators
For an insulator the valence band (i.e. the band occupied by the valence electrons) is completely
filled, and there is a large energy gap before the next band can be occupied. In this case, electrons
will be unable to move throughout the structure, as the Pauli exclusion principle stops the electron
changing molecular orbital.
For instance, in NaCl, if we take the number of chlorine atoms to be n. The n Cl– ions are very
close to each other. We have 7n valence electrons from chlorine, and 1n valence electrons from
sodium. This means that a continuous band/Crystal orbital will be formed by overlap of the 3s and
the three 3p atomic orbitals based on the chlorine atoms. This will create 4n energy levels where n is
the approximately Avagadro‘s number. These will be sufficient to fill completely the 4n energy
levels created by overlap of the chlorine atomic orbitals. There will also be a band due to sodium
atomic orbital overlap. However, this is far higher in energy (on the basis of ionisation energy,
electron affinity, and/or electronegativity).
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5.3.3 Intrinsic semiconductors
An intrinsic semiconductor would have a filled valence band at a temperature of absolute zero.
However, the band gap to the next available energy band is not very large, and thermal energy is
sufficient to promote some electrons up into it.
This means that at high temperatures, a small number of electrons will be able to move easily into
other molecular orbitals (this applies in both the conduction band and at the top of the valence
band), and so conductivity will be observed when a voltage is applied. Unlike metals the
conductivity of semiconductors increases with increasing temperature.
Small band gap (~1 eV)
Valence band
Conduction band
at T = 0 at T > 0
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5.3.4 Doped semiconductors
It is possible to add deliberately a very small amount of a dopant into an otherwise pure material.
For instance a small amount of arsenic can be substituted onto the silicon sites of pure silicon. In
this case each arsenic atom provides an extra valence electron that has to be accommodated. At
absolute zero, these extra electrons will be located on the arsenic atoms themselves (the arsenic
band will be very narrow, as As-As interactions will be negligible, i.e. they behave almost as As
atoms). However, at higher temperatures, the extra electrons may easily be promoted into the silicon
conduction band.
Even a doping level as low as 1 atom in 109 can cause
substantial changes in conductivity.
Doping arsenic into silicon produces what is termed an ―n-type‖ semiconductor (in which the
principle charge carriers are electrons).
Valence band (Si)
Conduction band (Si)
Donor band (As)
at T = 0 at T > 0
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Doping gallium into silicon on the other hand would produce a ―p-type‖ semiconductor. As gallium
has a lower valency than silicon, this creates a narrow acceptor band which has an energy just above
the valence band. Thermal excitation of electrons into the gallium band will thus enable electron
transport, effectively by having introduced positively charged ―holes‖ into the silicon valence band.
Valence band (Si)
Conduction band (Si)
Acceptor band (Ga)
at T = 0 at T > 0
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DIAMONDS!!
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6 Transition metal chemistry (a.k.a d-block chemistry)
Transition metals and their compounds are extremely important in nature. The key aspect to the
chemistry of transition metals is the properties of the d orbitals. d electrons are relatively easily lost
and gained, and so transition metals may show a wide range of different oxidation states in their
complexes. Such ―redox‖ behaviour is often very important in catalysis. [Note: reduction = gain of
electrons; oxidation = loss of electrons]. The presence of the d electrons also normally accounts for
the colour of transition-metal complexes.
6.1 Oxidation Number
The oxidation number is often defined as the apparent ionic charge a particular atom would have if
the compound was ionic. An alternative (and normally equivalent definition) is that it is the charge
remaining on the central metal atom if all the surrounding groups are removed in their closed shell
configuration. There are a range of metal halides, oxides and oxyanions in which the transition
metal has a high oxidation number. These include OsF6, CrO3, [VO4]3–
and [MnO4]–. In these cases,
the bonds around the metal are normal covalent bonds.
OXIDATION NUMBER = {Complex Ion Charge – Σ (ligand charge) }
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There are also a wide range of coordination complexes that involve the transition metal in a low
oxidation number. In these cases, the metal is acting as an acceptor of a pair of electrons (i.e. as a
Lewis acid) and the binding group is acting as a donor of electron pairs to form a dative bond. The
binding groups in these cases are commonly referred to as ligands.
Common ligands
neutral
C: CO C2H4 dienes/trienes
N: NH3 NR3 pyridine
P: PH3 PR3
O: OH2 OR2
S: SH2
Anionic:
H: H–
C: CN– CH3
– Ph
–
Si: SiR3–
N: NH2– NCS
–
O: OH– OR
– OCOR
– ONO
– OClO3
–
S: SH– SR
– SCN
–
Hal: F– Cl
– Br
– I
–
Cationic:
N: NO+
Some ligands are bidentate, which implies that they have two donor atoms that can bind to a metal
simultaneously. For example 1,2-diaminoethane (old name ethylene diamine, commonly
abbreviated en) may bind to the metal through both of its amine groups:
When a ligand binds through two or more donor atoms, the ligand is referred to as a chelate.
Examples of ligands that are tridentate and higher are also known.
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6.2 Electron configuration
Counting the number of electrons around a transition metal is less easy than main group elements.
In general:
Take the number of valence electrons for a neutral atom (from periodic table). Subtract the
oxidation number. Then add the number of electron pairs in bonds around the metal.
Example: [Co(III)Cl2(NH3)4]+
Transition metals do have a preference for adopting the electronic configuration of the nearest noble
gas in the same way as we discussed earlier, and this is sometimes termed the eighteen-electron rule.
For instance, the rule explains the stoichiometries of many transition-metal carbonyl complexes:
• Cr(CO)6
• Fe(CO)5
• Ni(CO)4
However, there are quite a large number of exceptions. For instance:
steric factors may prevent an 18-electron species being formed.
electronic factors (that are well understood, but too complicated to go into here) mean that some
complexes are stable 16-electron species (e.g. d8 species such as Ni/Pd/Pt in the +2 oxidation
state).
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6.3 Shape
VSEPR theory is not appropriate for transition-metal complexes as any d-orbital electrons not
involved in bonding do not form spatially defined lone pairs. The best guess of molecular
geometry is simply to place all the ligands as far apart as possible, with bulkier groups taking
up more space. Thus the geometry around the metal centre of [Cr(H2O)6]3+
is octahedral.
For transition-metal complexes there is a far greater likelihood of electronic factors affecting the
shape. For instance, there is an electronic driving force for 4-coordinate complexes of d8 metals
(sometimes for Ni, and almost always for Pd and Pt) to adopt a square planar conformation, even
though this is sterically less favoured than the tetrahedral arrangement.
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6.4 Metal-metal bonds
Some species have direct bonding between metal atoms. As well as overlap between s and p
orbitals, we now also have the possibility of overlap between d orbitals of the correct symmetry to
form energetically favourable bonds.
Co2(CO)8 contains a σ bond between the cobalt atoms and bridging CO species:
Note that Co(CO)4 would have 17 electrons and thus be unstable. Hence this is why Co(CO)4
dimerises so that each Cobalt atom has a share of eighteen electrons.
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[Re2Cl8]2–
contains one σ, two π and one δ bond between the rhenium atoms
Re
ReReRe
Re Re
ReRe
ReRe
z z
zz
x
y
x
y
x
y
x
y
++ -- ++ --
++ -- ++ --
+
+ -
- +
+ -
-
-
- +
+-
- +
+
+
+-
- +
+-
-
-
-+
+-
-+
+
+
+
+
+
+
+
+
+
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6.5 Cluster compounds
The existence of metal-metal bonding and bridging species leads to a whole variety of complexes
with clusters of metal atoms. Some of these are potential catalysts, while others can be considered
―models‖ of what happens at bulk metal surfaces
Example: Co3(CH)(CO)9 (using M to denote cobalt atoms)
M M
O≡C
O≡C
O≡C
C≡O
C≡OC≡OM
C
H
C≡
O
C≡O
C≡O
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6.6 Metal- ligand bonding (binding)
carbon monoxide
The binding of CO to transition metals is of particular importance both biologically (e.g. to
haemoglobin) and in catalysis (e.g. to the Cu/ZnO surface in the transformation of H2 + CO to
methanol). There are two contributions to the bonding:
There is a 5σ*-donor interaction from the carbon electron pair into an empty metal orbital. (Note
that the HOMO of CO has electron pair density on the carbon atom, see p. 56).
There is a π back-bonding from electrons in a filled d-orbital on the metal to the empty 2π*
LUMO orbital on the CO.
Thus CO acts as a σ donor and a π acceptor, with these two effects reinforcing each other.
Evidence: X-ray crystallography shows a shortening of the M–C bond and a lengthening of the
CO bond compared to that expected, while spectroscopy shows that the CO bond is weakened.
Similar considerations apply to the binding of CN– and N2 to transition metals (as these are
isoelectronic with CO).
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Ethene
Another important example of π back-bonding is the bonding of alkenes to transition metals. These
may be typified by Zeise‘s salt, [K] [PtCl3(H2C=CH2)].H2O, which was first prepared in 1827, but
the structure of which wasn‘t established until the 1950‘s.
There is σ donation from the π electrons in the C=C bond to the metal centre.
There is π back-bonding from a filled metal d orbital into the π* orbital of the alkene.
Evidence: increase in length of C=C bond.
Pt
C
C
z
x
y
++
Pt
C
C
z
x
y
+
-
+
-
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6.7 Organometallic compounds
The last example has moved us into the realm of organometallic chemistry, which is the study of
compounds that contain at least one M–C bond (though carbonyl complexes are sometimes
excluded).
One interesting class of molecules are ―sandwich complexes‖ such as Cr(C6H6)2.
Donation of electrons from filled π molecular orbitals on benzene into empty d orbitals on
chromium
Back-donation of electrons from filled d orbitals on chromium into empty π* molecular orbitals
on benzene.
As a result, the reactivity of the benzene ring in this complex will be different in this complex to
that of pure benzene.
A negatively charged cyclopentadienyl ring has aromatic properties like benzene and can bind to
transition metals in a similar way. For instance, the sandwich molecule ferrocene, Fe(C5H5)2, is
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particularly stable (it survives up to 500°C). Its discovery in 1951 caused the start of research in this
particular branch of chemistry.
It needs to be emphasised that organometallic compounds should not be viewed as ―outlandish‖ or
―unusual‖. Many are stable species, and some of them have commercial applications in the real
world. For instance the half-sandwich complex (C5H4CH3)Mn(CO)3 has been widely used as an
anti-knock additive in gasoline in North America. Metallocene complexes such as Ti(C5H5)2Cl2 are
widely used as catalysts in the preparation of polymers such as polyethylene.