Molecules Slide 1

1
Atomic Binding - Molecular Bonds

• Atoms are known to come together to form
  molecules which can be relatively simple (e.g.
  the hydrogen molecule H2) or very complex (e.g.
  the DNA molecule).
• In this part of the course we first identify the
  interatomic forces which bind atoms to form
  molecules and, subsequently, study the resulting
  energy levels (in particular, how they differ
  from those of the constituent atoms) and spectra
  of simple molecules.
• Finally, we investigate how atoms come together
  to form solids, the properties of these solids
  arising from the allowed energy levels and some
  practical applications of these properties.

2
Molecular Bonds - Ionic Bonding

• There are two main types of strong molecular
  bonds namely ionic and covalent bonds.
• The sodium chloride (NaCl) molecule provides an
  example of ionic bonding. Recall that the
  electronic configuration of the sodium atom is
  1s22s22p63s1 and that of chlorine is
  1s22s22p63s23p5.
• The outermost 3s electron of the sodium atom is
  weakly bound because it is partly shielded from
  the nucleus by the inner closed shells (the
  figure on the next slide shows that the
  ionisation energy is only 5.1eV).
• The 3s electron essentially experiences the
  effect of a net charge of only 1e because the
  nucleus has a charge of 11e and there are 10
  electrons (charge of -10e) in the inner closed
  shells.

3
Molecular Bonds - Ionic Bonding (ctd.)

• The ionisation energy of an atom is the energy
  required to remove an electron from the atom when
  it is in the ground state, resulting in the
  formation of a positive ion.

4
Molecular Bonds - Ionic Bonding (ctd.)

• The chlorine atom has 17 electrons, 12 of which
  are in closed shells or sub-shells
  (1s22s22p63s2). These form a spherically
  symmetric distribution. The remaining five
  electrons are in 3p states which are not
  spherically symmetric.
• Four of the five 3p electrons are in states whose
  distributions are in the shape of a doughnut.
  Since these states are filled, they cannot
  accommodate the fifth electron. That electron is
  in another state (in the shape of a dumbbell)
  which is half filled since it can accept two
  electrons of opposite spins (refer to the figure
  on the next slide).
• If the weakly bound 1s electron from a sodium
  atom is in the vicinity of the chlorine atom, it
  can fill the state. The electron would experience
  an attraction of 5e because the charge of 17e
  of the chlorine nucleus is partly shielded by the
  charge of -12e of electrons in the inner filled
  shells and sub-shells.

5
Molecular Bonds - Ionic Bonding (ctd.)
Can accommodate 3s electron from sodium atom
3s electron
Doughnut shaped states
Chlorine
Sodium

• The electron is therefore more strongly attracted
  by the 5e charge of the chlorine atom than the
  1e charge of the sodium atom.
• It spends most of its time near the chlorine atom
  which, therefore, acquires a net negative charge
  in comparison with the sodium atom which acquires
  a net positive charge.

6
Molecular Bonds - Ionic Bonding (ctd.)

• Since the two atoms are oppositely charged, the
  electrostatic attraction between them forms a
  bond which holds them together. Such a bond is
  called an ionic bond because it is due to the
  electrostatic attraction between two ions of
  opposite charges.

• Many compounds are ionic in nature. Examples are
  compounds formed from elements in group I (H, Li,
  Na, K, ) and group (VII) (F, Cl, Br, I, ) of
  the periodic table.

• Ionic bonds that involve more than one outermost
  electron (valence electron) are also formed from
  elements of group (II) (Be, Mg, Ca, ) and group
  (VII) of the periodic table. One example is
  calcium chloride (CaCl2).

7
Molecular Bonds - Covalent Bonding

• To gain an insight into the formation of covalent
  bonds, we consider the formation of a hydrogen
  molecule (H2) from two isolated hydrogen atoms in
  the ground state.
• When the atoms come close to each other, the
  electron distributions for the two atoms overlap
  and the two available electrons are shared by the
  atoms.
• This results in two possibilities for the total
  spin S.
• Either the spins of the two electrons are
  parallel in which case S 1/2 1/2 1
• or
• The spins are opposite resulting in S 1/2
  (-1/2) 0

8
Molecular Bonds - Covalent Bonding (ctd.)

• If the spins are the same (i.e. S 1), the two
  electrons cannot both be in the lowest energy
  state for one of the atoms, since this would
  violate the Pauli exclusion principle (the two
  electrons would have the same set of quantum
  numbers).
• Therefore when the atoms approach each other, the
  electron distributions do not overlap. This is
  shown by the probability distributions in the
  diagram.
• Consequently, the atomic nuclei repel each other
  and bonding of the two hydrogen atoms does not
  occur.

Electron clouds
Probability distribution
9
Molecular Bonds - Covalent Bonding (ctd.)

• If the spins are opposite ( i.e. S 0), the two
  electrons have different sets of quantum numbers
  and can come together (i.e. the electron
  distributions can overlap).
• The electrons, therefore, spend much of their
  time between the two nuclei.
• The resulting attraction of the two nuclei by the
  electron cloud predominates over electrostatic
  repulsion of the positive nuclei.
• The net attraction of the two atoms holds them
  together, forming a covalent bond.
• Note that in this case the electrons are shared
  by the two atoms.

Electron cloud
Probability distribution
10
Molecular Bonds - Partially-ionic Nature of
Covalent Bonds

• Oxygen has 8 electrons in the configuration
  1s22s22p4. Of the 8 electrons, 4 are in closed
  shells or subshells.

• Electrons from the hydrogen atoms are more
  attracted to the oxygen nucleus (effective charge
  of 4e due to shielding effects) than to their
  own nucleus.
• The two H atoms are therefore more positive than
  the Oxygen atom, which is more negative, due to
  the imbalance in the electron probability
  distribution.
• This imbalance causes the molecule to be polar
  and is the basis of its ability to solvate atoms
  and other molecules, particularly ionic species.

• The properties of molecules may be traced to the
  quantum mechanical nature of their electron
  distributions.

11
Molecular Bonds - Other Types of Bonds

• In addition to the strong ionic and covalent
  bond, there are other types of bonds such as the
  van der Waals bond and the hydrogen bond which
  are weak bonds.
• The van der Waals bond is due to weak
  electrostatic attractions between molecules. It
  is important in liquids and solids at room
  temperature, when thermal excitations are
  negligible.
• The hydrogen bond plays an important part in
  holding many organic molecules together.

12
Molecular Bonds - Binding Energy

• From the discussion so far, it is clear that two
  atoms come together to form a molecule because
  of a net attractive force between the atoms or
  ions formed.
• For the molecule to be stable, the total energy
  of the molecule must be less than the total
  energy of the atoms when the are separated.
• Energy must, therefore, be supplied to overcome
  the bond and separate the atoms to infinity.
• The required energy is known as the Binding
  Energy. It is typically 2 to 5eV for ionic and
  covalent bonds.

13
Molecular Spectra - Potential Energy Diagrams

• Solution of the Schrodinger equation requires
  knowledge of the potential energy of a system of
  atoms as a function of the internuclear
  separations.
• For two point charges q1(C) and q2(C), the
  potential energy U(r) is given by

Potential energy term (function of x only in this
case)

• As the charge separation decreases, U(r)
  increases for like charges (figure a) because of
  the repulsive force between them, but decreases
  for unlike charges (figure b) because the force
  is attractive.

14
Molecular Spectra - Potential Energy Diagrams

• The figure shows the potential energy of a system
  of two atoms (e.g. two hydrogen atoms) as a
  function of the internuclear distance r.

• At infinite separation, the force between the
  atoms is zero, and so is the potential energy.

• The potential energy decreases as the atoms
  approach each other, but for very small
  separations, the force is repulsive. These are
  related to changes in the electron distribution
  between the atomic nuclei.

• At a particular separation ro, called the
  equilibrium separation, the potential energy is a
  minimum.
• The binding energy is roughly equal to the depth
  of the potential well. They are not equal
  because the ground state energy may not lie
  exactly at the minimum of the potential energy
  curve.

15
Molecular Spectra - Potential Energy Diagrams
(ctd.)

• The total potential energy can be approximated by
  an expression of the form
• where A and B are constants related to the
  attractive and repulsive potentials respectively,
  and m and n are small integers.

U(r)
(repulsive)
r
Ground state energy
(attractive)
16
Molecular Spectra

• The energy states of a molecule arise from
• (a) rotation of the molecule as a whole
• (b) vibration of its atoms relative to one
  another
• (c) changes in its electronic configuration
• The spectra of polyatomic molecules can be very
  complicated. In this course, the treatment will
  be confined to diatomic molecules for simplicity.
  However, the main conclusions will also apply to
  more complicated molecules.

17
Molecular Spectra

• The spectrum of a molecule carries information
  about the structure of the molecule. Parameters
  such as the bond lengths, bond angles and force
  constants can often be determined from measured
  spectra.
• Different types of molecules have characteristic
  spectra which can be used for identification.
  This is done by comparing unknown spectra with
  fingerprint spectra of known molecules.
• Solution of the Schrodinger equation for
  molecules shows that the rotational and
  vibrational energies of a molecule are quantised.
• Transitions between rotational and vibrational
  energy levels are subject to selection rules (as
  is the case for atomic transitions), and result
  in molecular spectra which consist of closely
  spaced spectral lines. These are known as band
  spectra.

18
Molecular Spectra - Rotational States

• Consider a diatomic molecule that is rotating
  bout an axis through its centre of mass (CM)
  perpendicular to the line joining the atoms.
• Its rotational energy Erot is given by

where I is the moment of inertia and w is the
angular velocity.

• The rotational angular momentum of the molecule
  is equal to Iw.
• Solution of the Schrodinger equation shows that
  the rotational angular momentum Iw is quantised
  and given by

19
Molecular Spectra - Rotational States

• Consequently, the rotational energy Erot is also
  quantised and given by

L 0, 1, 2, 3, .....

• Transitions between rotational energy levels are
  not all allowed. The selection rule is DL ?1

• For a transition between two rotational energy
  levels with quantum numbers L and L - 1 (remember
  that DL ?1), the energy DErot of the photon
  that is emitted or absorbed is

L
Emission
Absorption
L - 1
In the above expression, L is the quantum number
of the upper state.
20
Molecular Spectra - Rotational Spectra
Erot

• If f is the frequency of emitted or absorbed
  radiation,

Emission
Absorption
Or
Frequency f
Rotational spectrum
21
Molecular Spectra - Rotational States

• The moment of inertia I of the diatomic molecule
  shown in the figure is given by
• Since the molecule rotates about an axis through
  its centre of mass

(1)
(2)

• From equations (1) and (2), noting that r
  r1r2, the moment of inertia can be written as

22
Rotational Spectra- Example

• Example
• The rotational transition from L 0 to L1 of
  the CO (carbon monoxide) molecule has an
  absorption frequency of 1.15x1011Hz. The mass of
  12C is 1.99x10-26 kg and that of 16O is
  2.66x1-26 kg.
• (a) What is the moment of inertia of the CO
  molecule?
• (b) What is the CO bond length?
• (c) What is the wavelength of the emitted photon
  for the transition from L 4 to L 3?

23
Rotational Spectra- Example

• The frequency f is given by

f 1.15x1011 Hz, h 6.626x10-34 Js, L 1 (for
upper state)
(a)
(b)
(c)
Emission is in the microwave spectral region (l
0.1mm - 10mm)
24
Molecular Spectra - Vibrational States

• We have seen that molecular energy states can
  result from rotation of a molecule as a whole.
• Molecular energy states can also result from
  vibrations of the atoms in a molecule relative
  to one another.

• Near the equilibrium position ro, the potential
  energy U can be approximated by that of a
  harmonic oscillator.

Displacement
Uo

• The restoring force F is given by

(Hookes law)

• For small displacements about ro, the restoring
  force is proportional to the displacement and
  the molecule undergoes simple harmonic motion
  about ro.

25
Molecular Spectra - Vibrational States

• For a simple harmonic oscillator, the classical
  frequency f of oscillation is given by

k is the stiffness constant m is the reduced mass

• Solution of the Schrodinger equation for the
  simple harmonic oscillator potential shows that
  the oscillator energy Evib is quantised.

Vibrational quantum number v 0, 1, 2, 3, ....
f is the frequency

• Note that the lowest vibrational energy ( for v
  0) is not zero (as is the case for rotation) but
  hf/2. This is known as the zero point energy.

• Also note that the energy levels are equally
  spaced. Energy spacing is hf.

26
Molecular Spectra - Vibrational States

• Vibrational transitions are subject to the
  following selection rule

Dv ?1

• The selection rule shows that allowed vibrational
  transitions can only occur between adjacent
  vibrational energy levels.
• In the simple harmonic approximation, the energy
  DE of emitted or absorbed photon is given by

• Transition energies are 10 to 100 times those for
  rotational and wavelengths are in the infrared
  spectral region (l 1mm to 100mm)

27
Molecular Spectra - Vibrational States

• Example
• The hydrogen molecule emits infrared radiation
  of wavelength of approximately 2.3mm.
• (a) What is the energy separation (in eV) of
  adjacent vibrational levels?
• (b) What is the energy (in eV) of the lowest
  vibrational state?

(a)
(b)
For lowest vibrational state, v 0
28
Molecular Spectra - Anharmonic Oscillators

• For large displacements ( larger than 10 of
  bond length), molecular vibrations are not simple
  harmonic in nature.
• The shape of the potential energy curve is more
  complicated as shown by the solid line in the top
  figure (the dotted line shows the simple harmonic
  oscillator approximation).

Displacement
v

• The vibrational energy levels are then not
  equally spaced, and transitions frequencies and
  wavelengths are different from those of the
  simple harmonic oscillator approximation.

5
4
Energy
3
Energy levels for large displacements
2
1
0
Internuclear distance
29
Molecular Spectra - Rotational and Vibrational
transitions

• Molecules can rotate and vibrate simultaneously.
  Transitions between rotational and vibrational
  energy levels of a molecule are subject to the
  following selection rule

DL 1
DL -1
DL ?1
and
Dv ?1
L

• The figure shows some transitions between the
  rotational-vibrational energy levels of a
  diatomic molecule and the resulting spectrum.
• Note that there is a gap in the spectrum
  corresponding to DL 0.

v 1
v 0
Energy
30
Molecular Spectra - Rotational and Vibrational
transitions

• The figure shows the absorption spectrum of the
  HCl molecule.
• Each line is made up of two peaks because
  chlorine has two isotopes which have different
  masses and, therefore, different moments of
  inertia.