Molecular Structure

Bonding mechanisms:

Atoms interact primarily via electrostatic forces.

Electrons in different atoms repel each other but are attracted to the

protons in the nucleus.

At long range, the force is primarily attractive.

At very short range, the force is primarily repulsive.

The net potential energy is due to a combination of attractive and

repulsive potentials:

U = !A

rn+B

rm

Molecular Structure

Bonding mechanisms:

Ionic bonds:

In an ionic bond electrons are transferred from 1 atom to the other.

Example: Sodium Chloride : NaCl

Electronic structure of sodium: 1s22s22p63s1

Energy input required to ionize sodium (remove 3s electron): 5.1eV

Electronic structure of chlorine: 1s22s22p5.

Adding an electron completes the 2p shell and lowers the energy

Energy released by adding an electron (electron affinity) = 3.7eV

Net energy of the system: 5.1eV-3.7eV = 1.4eV

Molecular StructureBonding mechanisms:

Ionic bonds:

Example: Sodium Chloride : NaCl

At large separations the total energy is 5.1eV-3.7eV = 1.4eV

At smaller separations, the energy decreases due to electrostatic

attraction.

At the equilibrium separation energy is minimized: -4.2eV.

This is the dissociation energy: energy required to break the ionic bond

At very small separations, repulsive forces dominate and energy

increases

Molecular StructureBonding mechanisms:

Covalent bonds:

In a covalent bond, electrons are shared between atoms

Example: Hydrogen molecule: H2.

Each hydrogen atom shares its electron with the other atom.

The shared electrons are most likely to be found between the nuclei.

The probability distribution is concentrated here.

The spins of the the two electrons are oppositely aligned to satisfy the

exclusion principle.

CH4 molecule:

Molecular StructureBonding mechanisms:

van der Waal’s bonds:

Very weak electrostatic bonds called van der Waals bonds can occur

between molecules or neutral atoms with a dipole moment.

Neutral molecules or atoms in which the charge distribution is not

uniform (polarized) have a permanent dipole moment.

These molecules or atoms can interact with other molecules or atoms

with a dipole moment.

Hydrogen bonds: Bond created between molecules by a hydrogen atom.

Example: Water (ice)

HF2 molecule:

Molecular StructureMolecular energies:

Molecular rotation: Consider a diatomic molecule

The speeds of the rotating atoms are v1 and v2 with

Erot

=1

2m1v1

2+1

2m2v2

2=1

2m1r1

2+ m

2r2

2( )! 2

Total angular momentum:

L = m1v1r1+ m

2v2r2

= m1r1

2+ m

2r2

2( )!

Total kinetic energy:

Erot

=1

2

L2

m1r1

2+ m

2r2

2( )

v1=!r

1v2=!r

2

Molecular StructureMolecular energies: Molecular rotation

Icm

= m1r1

2+ m

2r2

2= µR

0

2

Erot

=1

2

L2

m1r1

2+ m

2r2

2( )

Erot

=L2

2Icm

L is quantized,

L2= l(l +1)!, l = 0,1,2...

Rotational energy is quantized,

Erot

=!2

2Icm

l(l +1), l = 0,1,2...

Molecular StructureMolecular energies: Molecular vibration

U =1

2k(!x)

2

p1= ! p

2

In the centre of mass frame,

Bonds in diatomic molecules act like springs.

If the bond is stretch or compressed the atoms vibrate:

Total kinetic energy,

KE =p1

2

2m1

+p2

2

2m2

=1

2

1

m1

+1

m2

!"#

$%&p1

2=1

2

m1+ m

2

m1m2

!"#

$%&p1

2=p1

2

2µ

Molecular StructureMolecular energies: Molecular vibration

U =1

2k(!x)

2

Bonds in diatomic molecules act like springs.

If the bond is stretch or compressed the atoms vibrate:

Total energy,

KE =p1

2

2µ

Evib = KE +U =p1

2

2µ+1

2k(!x)

2

This describes the familiar harmonic oscillator with quantized energies,

Evib

= ! +1

2

"#$

%&'!( , ! = 0,1,2... ! =

k

µOscillation frequency:

Molecular StructureMolecular energies: Molecular vibration

The ground vibrational energy of a hydrogen (H2) molecule is 0.273eV.

What is the spring constant ?

Evib

= ! +1

2

"#$

%&'!( , ! = 0,1,2...

For !=0,

Evib

=1

2!! =

1

2!

k

µ= 0.273eV

! =k

µ

k =4(0.273eV )

2µ

!2

=4(0.273eV )

2

!2

mH

2

2mH

!"#

$%&= 576N / m

Molecular Spectra

Total energy:

A molecule can simultaneously rotate and vibrate.

These can be treated as independent motions, so the total energy is a

sum of the two energies:

E!l =h2

2ICM

l(l +1) + ! +1

2

"#$

%&'!( , l = 0,1,2.....,! = 0,1,2....

Separation of rotational energies are

much smaller than the separation of

vibrational energies.

For each vibrational level l there is a

whole set of closely spaced rotational

levels

Molecular SpectraTotal energy:

E!l =h2

2ICM

l(l +1) + ! +1

2

"#$

%&'!( , l = 0,1,2.....,! = 0,1,2....

!E = Ef " Ei = hf

All rules have to be satisfied simultaneously. Hence we cannot get a

pure rotational or pure vibrational transitions since !l or !! cannot be

0. In general we get a mixed rotational-vibrational spectrum.

!" = " f #" i = 1!l = l f " li = 1

HCl spectrum

Molecular SpectraTotal energy:

E!l =h2

2ICM

l(l +1) + ! +1

2

"#$

%&'!( , l = 0,1,2.....,! = 0,1,2....

!E = Ef " Ei = hf

Absorption spectra: If a molecule is excited so

that !! = 1, then there is a sequence of transitions

for which !l =1 and another sequence with

!l=-1:

!" = " f #" i = 1!l = l f " li = 1

!E = !" +!2

ICM

(l +1)

!E = !" #!2

ICM

l

l is the quantum number of the initial state. This

explains the two lobes seen in molecular spectra

Molecular Spectra

The l=5 to l=6 rotational absorption line of a diatomic molecule occurs

at a wavelength of "1 whereas the l=1 to l=0 rotational absorption line

of a diatomic molecule occurs at a wavelength of "2. What is the

moment of inertia of the molecule ?

!l =+1:

!E = !" +!2

ICM

(l +1)

!E = !" #!2

ICM

l!l=-1:

For l=5:

hc

!1

= !" +6!

2

ICM

For l=1:

hc

!2

= !" #!2

ICM

hc

!1

"hc

!2

= !# +6!

2

ICM

" !# +!2

ICM

=7!

2

ICM

ICM

=7!

2

hc

!1

"hc

!2

=7!

2#c!1!2

!2" !

1

$

%&'

()

Molecular SpectraTotal energy:

E!l =h2

2ICM

l(l +1) + ! +1

2

"#$

%&'!( , l = 0,1,2.....,! = 0,1,2....

Global warming:

In CO2 (carbon dioxide) molecules, absorption lines lie in the infrared

rather than visible part of the spectrum.

Hence incoming visible light from the Sun is not absorbed.

However, infrared radiation (heat) emitted by the earth is absorbed by

the CO2 molecules.

Thus CO2 molecules act like a one-way valve to trap heat and help to

raise the temperature of the surface.

This greenhouse effect is one of the factors leading to global warming.