Why do bonds form?
Chemical bonds...
... result from the sharing of electrons between atoms.
electrons
Lennard-Jones potential energy
diagram for the hydrogen molecule.
represented
in a Lewis
structure as:
H
H
protons
Energy of two
separate H atoms
r
E
Forces involved:
We understand that the electrons are shared
between the to H nuclei. How do we
describe their positions? How do we
describe electron position in atoms?
The Schrödinger equation – the solution for the H atom
o
o
Austrian physicist Erwin Schrödinger used de Broglie’s insight to calculate what electron
waves might act like, using a branch of mathematics known as wave mechanics
The result is encapsulated in a complex formula known as the Schrödinger equation:
 2  2  2
8 2m


  2 ( E  V )
2
2
2
x
y
z
b
o
o
o
o
o
o
This equation was know to belong to a special class known as an eigenvector equation: an
operator acts on a function (ψ) and generates a scalar times the same function
Ψ is known as the wavefunction of the electron: there are an infinite number of such
wavefunctions, each of which is characterized by a precise energy En, where n is an integer
from 1,2,3,4….∞. V in the equation is a constant value: the potential energy from attraction to
the nucleus. The remaining terms are all fundamental constants
A wave function that satisfies the Schrödinger equation is often called an orbital. Orbitals are
named for the orbits of the Bohr theory, but are fundamentally different entities
An orbital is a wave function
An orbital is a region of space in which an electron is most likely to be found
The square of the amplitude of the wavefunction, ψ2, expresses the probability of finding the
electron within a given region of space, which is called the electron density. Wavefunctions
do not have a precise size, since they represent a distribution of possible locations of the
electron, but like most distributions, they do have a maximum value.
Atomic Orbitals
We have a picture of where electrons are found around atoms: in atomic orbitals of
various types and energies.
2p
The size and energy of orbitals depend on …
2s
E
1s
Orbitals are wavefunctions and have wave properties:
These 2s orbitals are the same in all respects
except that they have opposite phase.
This is a single 2p orbital. It
has two lobes of different
phase and a node (an angular
node) at the nucleus.
Atoms to Molecules
• Recall that the Schroedinger solution works for the H atom but becomes
extremely complex for multi-electron atoms.
• We begin then by examining the simplest possible molecule: H2+
Forces involved:
• This is a three-body problem which can be simplified by assuming that the
motion of the nuclei is small compared to the motion of the electron (a fixed
internuclear distance).
• The result is an approximation of what one obtains by overlapping the atomic
orbitals at the optimal bonding distance. This is the LCAO approach: Linear
Combination of Atomic Orbitals.
What happens when two standing waves interact?
Out of phase overlap:
In phase overlap:
=
=
In orbital terms...
r221s
two overlapping 1s H atomic orbitals
r221s
• In H2+, the electron is found in an orbital that is the sum of the overlap of the
atomic orbitals
• The electron is distributed equally between both nuclei
• This is, therefore, a molecular orbital (MO) – an orbital for the H2+
molecule
• This type of MO has radial symmetry and electron density directly between
the two nuclei. These are classified as σ bonding orbitals.
MOs
• are orbitals just like atomic orbitals
• can span the entire molecule
• can house up to two electrons
σ1s bonding MO
But if you can form MOs by combining AOs, then electron occupancy must be
conserved. Two AOs can hold four electrons, so their combination must also
hold four electrons. There must be a second MO… In all cases, the number of
MOs that exist for a molecule must equal the number of AOs in its constituent
atoms.
Out of phase:
=
This orbital has low electron density
between the two nuclei and destabilizes
the attraction between them. This is
referred to as …
The MO diagram for the H2+ cation…
We use Correlation Diagrams (MO Diagrams) to show the MO electron configuration
and show from which AOs the MOs are derived.
AOs appear on the outer parts of the diagram at their corresponding energy level…
… and MOs in the middle.
Electron energies…
Orbital occupancy…
Orbital labels…
MO electron configuration…
Bond order = ½(#bonding electrons - #antibonding electrons)
The Hydrogen Molecule – all that changes is the number of electrons…
We construct a picture of the electronic structure of molecules using
correlation diagrams, which indicate the energy and type of molecular
orbitals that are occupied and vacant.
300
pm
1s
200
pm
150
pm
E
100
pm
1s
1s
250
pm
220
pm
73
pm
1s
MO electron configuration: (1σ1s)2
The AOs appear on the left (and for diatomics, right) side of the diagram at their
corresponding energy level and the MOs that result from the AOs are in the
middle at their corresponding energy level.
An appropriate no. of valence electrons are added to the diagram following Hund's Rule and
the Pauli exclusion principle to determine which orbitals are occupied.
Irradiation of H2
1s
1s
1s
1s
Ground state BO:
1s
hυ
1s
1s
1s
Excited state BO:
First Row Diatomics
1s
1s
1s
1s
1s
1s
1s
1s
1s
1s
1s
1s
1s
1s
1s
1s
Second Row Homonuclear Diatomics: Li2 – Ne2
We now have access to four new orbitals: 2s, 2px, 2py and 2pz. These are the valence
AOs.
The 1s orbitals are too small to overlap at the optimal bond distances so they can be
ignored.
Li
Li
1s
1s
2s
Constructing MO Diagrams
… for the second row diatomics.
Principles:
1) We have four valence AOs per atom, therefore there must be ___________ MOs.
2) Only orbitals of compatible symmetry (both σ or both π) can be combined.
OK
Not OK
3) The energy of an MO is closely related to the energy of the AOs from which it is derived.
4) The best (i.e. most energetically favourable) MOs are formed from AOs of similar energy.
The 2s AOs are lowest in energy and equal in energy…
2σ*
2σ
2s
2s
2s
2s
2s
2s
The 2pz AOs overlap to form a σ bond…
Using two 2pz orbitals…
2p
Out of phase
2pz
2p
In phase
2p
2p
2pz
2s
2s
2s
2s
The remaining 2p orbitals?
Once the 2pz AO has been used to form a σ bond, the remaining 2p orbitals cannot
overlap head on because these AOs are at right angles to the 2pz and are therefore…
The  Bond
Out of phase
=
2px
In phase
=
2px
The 2py AOs overlap in exactly the same way to form 2py and π2py orbitals with
the same shape and same energy but different…
As easy as…
Where do these fit?
Correlation diagram for homonuclear diatomics, Z = 8 and above
(O2-Ne2)
2pz
Rules for filling MO diagram:
• The number of electrons
added to the
MO diagram must equal the
total number of valence
electrons
• Orbitals of lowest energy areE
filled first.
• The Pauli exclusion
principle applies.
• Hund’s Rule must be
obeyed.
O2
Bond order
BDE 498 kJ
MOEC:
2px 2py
2p
2p
2px 2py
2pz
2s
2s
2s
2s
F2
Bond order
BDE 155 kJ
2pz
2px 2py
2p
2p
2px 2py
E
2pz
2s
2s
2s
MOEC:
2s
Ne2
Bond order
BDE
2pz
2px 2py
2p
2p
2px 2py
E
2pz
2s
2s
2s
MOEC:
2s
MO Diagrams for Z = 3-7 differ from 8-10

Which p orbital combines with which other p orbital is symmetrydetermined. Since orthogonal orbitals on different atoms don’t combine,
you won’t see combination of a px orbital on one atom and a py orbital
on its neighbour, for example.

s orbitals *can* combine
with p orbitals when making
 MOs *if* the orbitals are
close enough in energy.
The figure at the right shows
2s and 2p atomic orbital
energies for the elements in
period 2. We can see that
there will be little mixing
between 2s and 2p orbitals
for the heavier elements in period 2.
Figure courtesy of Prof. Marc Roussel
21
Because Li, Be, B, C and N have smaller
energy gaps between their 2s and 2p orbitals,
some mixing is observed when forming the σ
and σ* orbitals, primarily σ*2s and σ2pz:
Mixing in some “p character” lowers the
energy of the σ*2s MO
instead of
Mixing in some “s character” raises the
energy of the σ2pz MO
instead of
If this effect is strong enough, the 3σ orbital
can end up higher in energy than the 1π orbital,
giving the MO diagram on the next page. This
is the case in Li2, Be2, B2, C2 and N2.
The MO Diagram for Li2-N2
2p
2p 2p
2p
2p
3σ2pz
2p
2p
2s
2s
2s
2s
Li2 and Be2
Li2
Be2
BO:
Bond energy: 106 kJ/mol
BO:
MOEC:
2s
2s
2s
2s
2s
2s
2s
2s
Correlation diagram for homonuclear diatomics, Z up to 7 (Li2-N2)
2p
B2?
Bond order =
2p 2p
BDE = 290 kJ
2p
2p
2p
2p
2s
2s
2s
MOEC:
2s
C2
Bond order
BDE 620 kJ
2p
2p 2p
2p
2p
2p
2p
2s
2s
2s
MOEC:
2s
2p
N2
Bond order
BDE 945 kJ
2p 2p
2p
2p
2p
2p
2s
2s
2s
MOEC:
2s
Recap