# Molecular Physics

```PRS Question
For a molecule to be stable, the following must
hold



Its energy should exceed the sum of the
energy of its constituents
Its energy should exactly equal the sum of
the energy of its constituents
Its energy should be less than the sum of
the energy of its constituents
1
Molecular Structure
2
2
Topics

Recap

Exclusion Principle Repulsion


The Covalent Bond
 H2
 He2
Summary
3
Recap
A molecule, such as KCl, forms only if the
following holds
Energy(molecule) < ∑i Energy(atomi)
that is, the total energy of the molecule is
less than the total energy of its
constituent atoms when they are widely
separated
4
Recap
For example, the net total energy of the K+
and Cl- ions, after the transfer of the 4s
electron from K to the unoccupied 3p state
in Cl is 0.72 eV.
This is not low enough to cause the ions to
bind. However, if they are brought within
2.8 nm of each other their total energy falls
below zero and the ions bind
K+
2.8 nm
Cl5
Exclusion Principle Repulsion



As atoms are brought closer and closer
together, their wavefunctions overlap more
and more
But, since only one electron can occupy a given
state, some electrons will be forced to occupy
higher (empty) energy levels
Another way to say this is that electrons in
the same spin state tend to repel each other
because of the Pauli exclusion principle
6
Exclusion Principle Repulsion


This tendency of electrons to stay away from
each other increases the energy of a system
of atoms as they are brought together, while
the electrostatic energy decreases
Consequently, there will be some separation
between the atoms, called the equilibrium
position, r0, for which the energy of the
molecule is a minimum
7
Exclusion Principle Repulsion
Example: For molecules like KCl, the potential
energy is a sum of the electrostatic potential,
the exclusion principle energy and the net
ionization energy:
2
e
U (r )  k  Eex  Eion
r
The energy needed for the reaction
K+Cl- -> K + Cl is called the
dissociation energy, Ed
8
Exclusion Principle Repulsion
r0 = 0.27 nm, Ed = 4.4 eV
9
10
The Covalent Bond




To make an H2 molecule, we could first try to
transfer an electron and create the ions H+
and H- and then bring them together
The net ionization energy is 12 eV
It turns out there is no distance at which the
loss in potential energy is greater than 12 eV
So the H2 molecule cannot form this way
11
The Covalent Bond


The binding of two hydrogen atoms to form
H2 is a pure quantum effect: the sharing of
the two electrons by the hydrogen atoms
To see how this works, we consider the
quantum mechanics of two 1-D square wells of
finite depth and width L
12
The Covalent Bond
Wells far
apart, wave
functions do
not overlap
Energies are
identical for
both states
13
The Covalent Bond
When close,
the wave
functions
look as shown
Then
Energy(yA)
is now
greater than
Energy(yS)
14
The Covalent Bond – H2
Symmetric
Anti-symmetric
When far apart the total energy of the two
hydrogen atoms, each in their 1s (ground)
state, is -13.6 eV + -13.6 eV = -27.2 eV.
Unstable configuration
Stable configuration
In other words, the energies of the
symmetric and anti-symmetric states are the
same. The ground state with energy -27.2 is
said to be doubly degenerate
15
The Covalent Bond – H2
When the atoms are brought close together,
however, the energies of the symmetric and
Symmetric
anti-symmetric
states areAnti-symmetric
no longer identical
Stable configuration
Unstable configuration
16
The Covalent Bond – H2
The symmetric state, in which electrons have
a high probability of being between the protons,
Symmetric
has
lower energy than theAnti-symmetric
anti-symmetric state.
The symmetric state is called a bonding orbital
The other state is called an anti-bonding orbital
Stable configuration
Unstable configuration
17
The Covalent Bond – H2
However, for H2 to form the symmetric state
must be able to accommodate both electrons.
Anti-symmetric
Symmetric
This can happen because the electron spins
are anti-parallel, that is, have total spin S = 0.
H2 is said to be s-bonded
Stable configuration
Unstable configuration
18
The Covalent Bond – H2
The energy diagram of
H2 as a function of separation r
This kind of bonding is called a covalent bond
19
The Covalent Bond – He2
He has no valence electrons that can be shared.
Consequently, 2 electrons will occupy the 1s
bonding orbital, which is then a saturated bond,
while the other 2 electrons are forced to
occupy the higher energy anti-bonding orbital
Since the combined energy of the bonding and
anti-bonding orbitals exceeds that of 2 widely
separated He atoms, the He2 molecule does not
form
20
Summary


The exclusion principle induces an additional
repulsive force between electrons, which can
balance the electrostatic forces yielding a
separation between atoms for which the total
energy is a minimum
As two atoms are brought together, bonding
and anti-bonding orbitals form with the
former having the lower energy. Whether or
not a molecule forms depends on the total
energy of the occupied orbitals.
21
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