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Topic #1: Bonding – What Holds Atoms Together?
Spring 2009
Prof. René Boeré
Some combinations of atoms lead to a minimum in the energy scale at a temperature-dependent average internuclear separation distance
Other combinations such a minimum: do not have example ?
Bonds are inherently quantummechanical
Our theory of bonding is based on the simplest case:
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What forces must we consider in this paradigm simplest molecule?
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nd
The material that we cover during the first one or two lectures is covered in the following section(s) of the course text book:
Chapter 11-5, pp. 438-441
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USEFUL REVIEW
Please review the material covered in Chemistry 1000 relating to (a) the quantum mechanical model of the atom, (b) Lewis theory
It would be highly recommended that you re-read all of Chapters 8, 9 and 10 during the first week of Chemistry 2000
Particularly important are:
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Sections 8.1 – 8.9
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Section 8.11
Sections 10.1 – 10.6
For a great web tool for visualizing H-atom atomic orbitals please visit: http://www.falstad.com/qmatom/ and explore the easy-to-use JAVA applet!
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2
+
Even the simplest case of two protons and one electron cannot be solved directly. What approximation must also be made?
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With these approximations in place, we solve the Schrödinger equation anew for H
2
+ and obtain an infinite series of solutions that have similarities and differences from the orbitals of the H atom.
The simplest of these is the ground state molecular orbital of the molecule. What does this ground state orbital look like?
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In H
2
+ , the electron doesn’t belong to either atom.
In H
2
+ , the electron is in an orbital which spans the molecule – a
molecular orbital!
Just as atoms have many atomic orbitals (1 can have many molecular orbitals. In H orbitals are all empty.
2
+ s , 2 s , 2 p , etc.), molecules
, the higher energy molecular
The energy of a molecular orbital depends in part on the relative positions of the nuclei.
The exact solutions of the H molecular orbitals can be used with only small modifications for any diatomic molecule.
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The orbitals of a hydrogen molecule (R = ~74 pm) must be somewhere between those two extremes. This thinking leads to a common way of implementing molecular orbitals by describing them as combinations of atomic orbitals. This is termed Linear Combination of Atomic
Orbitals (LCAO) and gives an LCAO-MO such as that below:
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By adding the two atomic orbitals, we obtain a sigma bonding
orbital ( σ ).
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Bonding: lots of electron density between the two nuclei
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Sigma symmetry: high electron density along the axis connecting the nuclei
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Recall from CHEM 1000 that it was possible to solve the
Schrödinger equation exactly for a hydrogen atom, but a helium atom had too many electrons. We encounter the same problem with H
2
. While H
2
+ can be solved, as soon as a second electron is introduced, there are too many moving bodies and the wavefunction cannot be solved exactly. This does not mean we’re finished with quantum mechanics! Instead, we make more approximations…
So, what’s a reasonable approximation? We know that, when two hydrogen atoms are far apart (i.e. R is large), they behave like two free hydrogen atoms:
If we were able to bring them together such that the nuclei overlapped (i.e. R = 0 pm), we would have _______________
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We can also subtract the two atomic orbitals equivalent to adding -5) :
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(equivalent to adding them after inverting the phase of one – just as subtracting 5 is
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This is a sigma antibonding orbital ( σ * ).
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Antibonding: depleted electron density between the two nuclei (look for a node perpendicular to the axis connecting the nuclei)
Sigma symmetry: high electron density along the axis connecting the nuclei
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LCAO
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We can draw an energy level diagram showing molecular orbitals and the atomic orbitals from which they were derived. This is referred to as a molecular orbital diagram ( LCAO MO diagram).
1
σ
*
Energy 1 s 1 s
1
σ
H H
2
H
Note that the energy difference is larger between the atomic orbitals and the antibonding orbital than between the atomic orbitals and the bonding orbital.
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2
+
1
σ
*
2
2
1
σ
*
Energy 1 s 1 s Energy 1 s 1 s
H H
2
1
σ
H He He
2
1
σ
He
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Thus, the orbital occupancy for H
2 and the orbital occupancy for He
2 in the ground state is in the ground state is
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We can calculate bond orders for these two “molecules” from their MO diagrams: bond order
=
1
2
(
# bonding e
− −
# antibondin g e
−
)
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LCAO
2
MO diagrams relate the energies of molecular orbitals to the atomic orbitals from which they were “derived”. If the total energy of the electrons is lower using molecular orbitals molecule forms. If the total energy of the electrons is lower using atomic orbitals (the two outside columns)
(the middle column) , the
, no molecule is formed.
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To fill a molecular orbital diagram with electrons, use the same rules as you would to fill in an atomic orbital diagram:
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Fill ___________________________________________ first.
Pauli’s exclusion principle still applies:
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Hund’s rule still applies:
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2
1
σ
* 1
σ
*
Energy 1 s 1 s Energy 1 s 1 s
1
σ
1
σ
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If a molecule of H
2 was irradiated with light, exciting an electron from 1 σ to 1 σ * , what would happen?
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Should it be possible for H
2
to exist?
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What about He
2
+ ?
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2
+
1
σ
* 1
σ
*
Energy 1 s 1 s Energy 1 s 1 s
1 σ 1 σ
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If a molecule of H
2 from 1 σ to 1 σ * was irradiated with light, exciting an electron
, what would happen?
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Should it be possible for H
2
to exist?
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What about He
2
+ ?
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2
+
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So far we have considered just two possible atomic orbitals derived for the simplest molecule H
2
+ , and have applied this model to several examples of diatomic molecules involving H and He
Diatomic molecules have bonds between two atoms: they are thus the key model for bonding in ones.
all molecules, even some very complicated
To do this requires an understanding of additional MO’s of H
2
+
We will now visualize these orbitals using the applet available from the website: http://www.falstad.com/qmmo/
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