Electromagnatic radiation :- Vibrational Spectroscopy

 Vibrational Spectroscopy:

Vibrational spectroscopy can be thought of by starting with a simple harmonic 

oscillator (SHO) model. In this model, we consider two atoms joined by a bond to be 

equivalent to two masses joined by a spring. The spring can be compressed, forcing 

the spheres close to each other - stretched, moving them apart - or allowed to freely 

come to rest in the spheres' equilibrium positions. This can be shown in a potential 

energy curve:

However, not all energies are possible. The possible vibrational states are given by the 

vibration quantum number, v, and vibrational selection rule Δv = ±1. 

The energy of each level, E

, is given by

h

E 











 

2

1

 

(the second ν is the Greek character nu, the fundamental frequency). The 

fundamental frequency is given by the equation ν = (1/2π)×(k/μ)1/2, where μ is the 

reduced mass of the molecule, and k is thebond force constant. Notice that this k is 

similar to the spring force constant from Hooke's Law F = -kx. This leads to a potential 

energy surface that looks like this:

As you can see, the energy levels are equally spaced, and the equilibrium bond length 

is constant for all energy levels. However, this model is imperfect - it does not account 

for the posibility of bond dissociation (under this model, the bond would never break, 

no matter the magnitude of the vibrational energy). It also does not account for extra 

repulsive effects at very small bond lengths caused by the electroweak force. This is 

the force which prevents atoms for being forced together as the distance between them 

gets very small (the reason nuclear fusion only occurs at very high temperatures, for 

example). A model which takes into account these factors, and which more accurately 

models a vibration diatomic molecule, is the Anharmonic Oscillator (AHO), and the 

corresponding potential energy surface called the Morse potential. The potential 

energy curve now looks like this:

Notice how the expectation (or average) bond length, rexp is no longer the same for all 

energy levels. One effect of the anharmonicity is this deformation as the energy of the 

vibrations increases. Another is that now, the vibrational energy levels are no longer 

equally spaced, but instead get closer together as the vibrational quantum number 

increases. This model also illustrates that there aren't an infinite number of vibrational 

energy levels - above some energy, the bond breaks and the molecule dissociates.

Vibrational energies occur roughly in the 100 - 4000 cm-1 (about 1 - 50 kJ mol-1), 

or infra-red (IR) region of the electromagnetic spectrum. The fundamental principle 

for obtaining a vibrational spectra is that the electric dipole moment of the molecule 

must change during the vibration. If there is no change in dipole moment, then this 

particular vibration will not give rise to any absorption in the IR region. An example 

of this can be seen below for the symmetric stretch (Σg+) mode of carbon dioxide. 

Because both C=O bond lengths change exactly in phase, there is never a net dipole 

moment on the molecule. A homonuclear diatomic molecule such as dioxygen O2

has 

zero dipole moment, so it has no IR spectrum.

The number of modes (types) of vibration can be predicted for a molecule, containing 

N atoms, using the following general expressions.

Electromagnatic radiation :- Rotational energy level

 Rotational energy level

Molecular rotations require little energy to excite them. Pure rotation spectra occur in 

the microwave region of the spectrum (~1 - 200 cm-1

). It is important to note that a molecule cannot rotate about some arbitrary axis - the principle of conservation of 

angular momentum dictates that only a few rotations are possible. In general, rotation 

must be about the centre of mass of a molecule, and the axis must allow for 

conservation of angular momentum. In simple cases, this can often be recognised 

intuitively through symmetry - such as with the water molecule.

A pure rotation spectrum can only arise when the molecule possesses a permanent 

electric dipole moment. Like with vibrational spectroscopy, the physical effect that 

couples to photons is a changing dipole moment. Since molecular bond lengths remain 

constant in pure rotation, the magnitude of a molecule's dipole cannot change. 

However, since electric dipole is a vector quantity (it has both size and direction) 

rotation can cause a permanent dipole to change direction, and hence we observe its 

spectra. Since homonuclear molecules such as dinitrogen N2

have no dipole moment 

they have no rotation spectrum. Highly symmetric polyatomic molecules, such as 

carbon dioxide, also have no net dipole moment - the dipoles along the C-O bonds are 

always equal and opposite and cancel each other out. It is important to recognise also 

that if a molecule has a permanent dipole, but this dipole lies along the main rotation 

axis, then the molecule will not have a rotational spectrum - such as for a water 

molecule.

In pure rotational spectroscopy for a simple diatomic molecule, the energy levels - as 

displayed below - are given by EJ = BJ (J+1), where J is the rotational quantum 

number, B is the rotational constant for the particular molecule given by

I

h

B 2

2

8



with the unit of Joules, where I is the moment of inertia, given by I = μr

2

- where r is 

the bond length of this particular diatomic molecule and μ is the reduced mass, given 

by μ = m1m2

/ m1 + m2

.

Most energy level transitions in spectroscopy come withselection rules. These rules 

restrict certain transitions from occuring - though often they can be broken. In pure 

rotational spectroscopy, the selection rule is ΔJ = ±1.

A vibrational spectrum would have the following appearence. Each line corresponds to 

a transition between energy levels, as shown. Notice that there are no lines for, for 

example, J = 0 to J = 2 etc. This is because the pure rotation spectrum obeys the

selection rule ΔJ = ±1. The energy gap between each level increases by 2B as the 

energy levels we consider increase by J = 1. This leads to the line spacing of 2B in the 

spectrum. Each transition has an energy value of 2B more than the previous transition.

Electromagnatic radiation :- Born-Oppenheimer approximation

 Born-Oppenheimer approximation 

A molecule is an assembly of positively charged nuclei and negatively charged 

electrons that form a stable entity through the electrostatic forces which hold it all 

together. Since all the particles which make up the molecule are moving relative to 

each other, a full quantum mechanical description of the molecule is very complicated 

and can only be obtained approximately. Fortunately, the overall motion of the 

molecule can be broken down into various types of motion, namely, translational, 

rotational, vibrational, and electronic. To a good approximation, the so called Born-

Oppenheimer approximation, each of these motions can be considered on its own 

(Born & Oppenheimer 1927). 

Their main objective was the separation of electronic and nuclear motions in a 

molecule. The physical basis of this separation is quite simple. Both electrons and 

nuclei experience similar forces in a molecular system, since they arise from a mutual 

electrostatic interaction. However, the mass of the electron, is about four orders of

magnitude smaller than the mass of the nucleus. Consequently, the electrons are 

accelerated at a much greater rate and move much more quickly than the nuclei. 

Therefore, as an approximation, we can regard the dynamics of the electrons and 

nuclei as largely independent. When describing the electrons, the nuclei can be 

considered as being fixed in space. On longer timescales, the electrons can 

immediately follow the much slower nuclear motions. To a very good approximation, 

the total energy of a molecule is given as the sum of three components, corresponding 

to the three modes of motion mentioned above, namely

Where the three terms on the right hand side represent the electronic, vibrational, and 

rotational energy. We disregard overall translational energy of the molecule because it 

leads mainly to collisions and, thus, has only an indirect influence on the internal 

structure of molecules. Furthermore, we have neglected in Eq. (3.1) the interaction 

between the three different types of motion. These interaction terms have to be 

considered in a consistent treatment, and we will come back to them in the course of 

this chapter, but it is more illustrative to neglect them as a first approximation. In 

molecular spectroscopy, Eq. (3.1) is often expressed in units of the wave number ύ = 

v/c , using the notation

 (3.2)

with

 (3.3)

The energy difference between two states can then be denoted by 

 ( 3.4)

where the units of ύ are cm

–1

Since electrons move more rapidly than the nuclei, and the vibrations of nuclei are 

more rapid than the rotation of molecules, the following relation holds:

 (3.5)

We have already argued that the electronic energy is largest. The relative energies of 

vibrations and rotations follow from simple order of magnitude estimates. The valence 

electrons spread over the whole molecule (of size a ≈ 1 Å) and thus have typical

energies

 (3.6)

In this estimate we have used the Rydberg energy but with a instead of the Bohr 

radius. The energy scale of vibrations about the equilibrium separation req of two 

nuclei in a molecule is defined by ℏω . To find ℏω we consider the potential energy 

of the vibrational modes given by Mnω2(r – req) 2 , cf. Eqs. (3.33) and (3.34). We 

have req ~ a. For vibrations with (r – req) ~ a the electron configuration would be 

modified strongly and “costs” roughly one electronic energy Ee. From Mnω 2 (r –

req)2 ~ 2 2 e ℏ m a we find

(3.7)

Rotational energies are estimated by J 2 /I (cf. Eq. (3.13)) where J is the angular 

momentum and I is the moment of inertia. With J 2 ~ 2 2 ℏ ∼ ℏ j j ( ) +1 and I ~ 

Mna 2 we get

 (3.8)

The corresponding length scales are λe ~ 100–1000 nm (i.e. in the order of visible 

wavelengths), λvib ~ 10 µm (i.e. in the infrared region), and λ rot ~ 1 mm. Therefore, 

spectral lines of molecules are observed in the visible (actually near UV to about 1 

µm) in the case of electronic transitions, in the infrared for vibrational transitions, and 

in the sub-mm to about 1 mm region (microwaves) for rotational transitions.

Electromagnatic Rotation :- Rotational Selection rules

 Rotational Selection rules:

1. Transitions with ΔJ=±1 are allowed;

Photons do not have any mass, but they have angular momentum. The conservation of 

angular momentum is the fundamental criteria for spectroscopic transitions. As a 

result, the total angular momentum has to be conserved after a molecule absorbs or 

emits a photon. The rotational selection rule relies on the fact that photon has one unit 

of quantized angular momentum. During the photon emission and absorption process, 

the angular moment J cannot change by more than one unit.

Electromagnatic radiation :- Vibrational Selection rules

 Vibrational Selection rules

1 Transitions with Δv=±1, ±2, ... are all allowed for anharmonic potential, but the 

intensity of the peaks become weaker as Δv increases.

2 v=0 to v=1 transition is normally called the fundamental vibration, while those 

with larger Δv are calledovertones.

3 Δv=0 transition is allowed between the lower and upper electronic states with 

energy E1 and E2 are involved, i.e. (E1, v''=n) → (E2, v'=n), where the double 

prime and single prime indicate the lower and upper quantum state.

4 The geometry of vibrational wavefunctions plays an important role in vibrational 

selection rules. For diatomic molecules, the vibrational wavefunction is symmetric 

with respect to all the electronic states. Therefore, the Franck-Condon integral is 

always totally symmetric for diatomic molecules. The vibrational selection rule 

does not exist for diatomic molecules.

For polyatomic molecules, the nonlinear molecules possess 3N-6 normal vibrational 

modes, while linear molecules possess 3N-5 vibrational modes.

Let's consider a single photon transition process for a diatomic molecule. The 

rotational selection rule requires that transitions with ΔJ=±1 are allowed. Transitions 

with ΔJ=1 are defined as R branch transitions, while those with ΔJ=-1 are defined as 

P branch transitions. Rotational transitions are conventional labeled as P or R with the 

rotational quantum number J of the lower electronic state in the parentheses. For 

example, R(2) specifies the rotational transition from J=2 in the lower electronic state 

to J=3 in the upper electronic state.

2. ΔJ=0 transitions are allowed when two different electronic or vibrational states are 

involved.

Electronic Selection Rules :- Electronic transitions in atoms

Electronic Selection Rules 

Electronic transitions in atoms

Atoms are described by the primary quantum number n, angular momentum quantum 

number L, spin quantum number S, and total angular momentum quantum number J. 

Based on Russell-Saunders approximation of electron coupling, the atomic term 

symbol can be represented as (2S+1) LJ

.

1. The total spin cannot change, ΔS=0;

2. The change in total orbital angular momentum can be ΔL=0, ±1, but 

L=0 ↔ L=0 transition is not allowed;

3. The change in the total angular momentum can be ΔJ=0, ±1, but J=0 ↔ J=0 

transition is not allowed;

4. The initial and final wave functions must change in parity. Parity is related to the 

orbital angular momentum summation over all elections Σ li, which can be even 

or odd; only even ↔ odd transitions are allowed.

1.6.2 Electronic transitions in molecules:

The electronic-state configurations for molecules can be described by the primary 

quantum number n, the angular momentum quantum number Λ, the spin quantum 

number S, which remains a good quantum number, the quantum number Σ (S, S-1, ..., 

-S), and the projection of the total angular momentum quantum number onto the 

molecular symmetry axis Ω, which can be derived as Ω=Λ+Σ. The term symbol for 

the electronic states can be represented as

Group theory makes great contributions to the prediction of the electronic selection 

rules for many molecules. An example is used to illustrate the possibility of electronic 

transitions via group theory.

1. The total spin cannot change, ΔS=0; the rule ΣΔ=0holds for multiplets;

If the spin-orbit coupling is not large, the electronic spin wavefunction can be 

separated from the electronic wavefunctions. Since the electron spin is a magnetic

effect, electronic dipole transitions will not alter the electron spin. As a result, the spin 

multiplicity should not change during the electronic dipole transition.

2. The total orbital angular momentum change should be ΔΛ=0, ±1;

3. Parity conditions are related to the symmetry of the molecular wavefunction 

reflecting against its symmetry axis. For homonuclear molecules, the g ↔ u transition 

is allowed.

Electromagnatic radiation iv :- Scattering

  Scattering :-

When electromagnetic radiation passes through matter, most of the radiation continues 

in its original direction but a small fraction is scattered in other directions. Light that is 

scattered at the same wavelength as the incoming light is called Rayleigh scattering. 

Light that is scattered in transparent solids due to vibrations (phonons) is called 

Brillouin scattering. Brillouin scattering is typically shifted by 0.1 to 1 cm-1

from the 

incident light. Light that is scattered due to vibrations in molecules or optical phonons 

in solids is called Raman scattering. Raman scattered light is shifted by as much as 

4000 cm-1 from the incident light.

Electromagnetic radiation part iii :- Diffused reflection

Diffused reflection: 

it is produced when light reaches a surface or object which has 

texture such as, for example, a wall with texture.

A diffused reflection will produce a softer light than that produced by direct reflection. 

It will also generate less contrast in the scene, clearer shades and a smoother transition 

between highlights and shadows.

A direct reflection will produce a more intense light, with higher contrast and darker 

and well-defined shadows.

As we mentioned previously, white reflects (theoretically) every form of light. A 

coloured surface will reflect its own colour while absorbing the rest. For example, a 

green object will reflect green and it will absorb red and blue.

Electromagnetic radiation part 2

  Interaction of electromagnetic radiation with matter

It is well known that all matter is comprised of atoms. But subatomically, matter is 

made up of mostly empty space. For example, consider the hydrogen atom with its one 

proton, and one electron. The diameter of a single proton has been measured to be 

about 10-15 meters. The diameter of a single hydrogen atom has been determined to be 

10-10 meters; therefore the ratio of the size of a hydrogen atom to the size of the proton 

is 100,000:1. Consider this in terms of something more easily pictured in your mind. If 

the nucleus of the atom could be enlarged to the size of a softball (about 10 cm), its 

electron would be approximately 10 kilometers away. Therefore, when 

electromagnetic waves pass through a material, they primarily move through free 

space, but may have a chance to encounter with the nucleus or an electron of an atom.

Because the encounters of photons with sub atomic particles are by chance, a given 

photon has a finite probability of passing completely through the medium it is 

traversing. The probability that a photon will pass completely through a medium 

depends on numerous factors including the photon’s energy and the composition and 

thickness of the medium. The more densely packed a medium’s atoms, the more likely 

the photon will encounter an atomic particle. In other words, the more subatomic 

particles in a material (higher Z number), the greater the likelihood that interactions 

will occur Similarly, the more material a photon must cross through, the more likely 

the chance of an encounter.

When a photon does encounter an atomic particle, it transfers energy to the particle. 

The energy may be reemitted back the way it came (reflected), scattered in a different 

direction or transmitted forward into the material. Let us first consider the interaction 

of visible light. Reflection and transmission of light waves occur because the light 

waves transfer energy to the electrons of the material and cause them to vibrate. If the

material is transparent, then the vibrations of the electrons are passed on to 

neighboring atoms through the bulk of the material and reemitted on the opposite side 

of the object. If the material is opaque, then the vibrations of the electrons are not 

passed from atom to atom through the bulk of the material, but rather the electrons 

vibrate for short periods of time and then reemit the energy as a reflected light wave. 

The light may be reemitted from the surface of the material at a different wavelength, 

thus changing its color.

The interactions between electromagnetic radiation and matter cause changes in the 

energy states of the electrons in matter.

Electrons can be transferred from one energy level to another, while absorbing or 

emitting a certain amount of energy. This amount of energy is equal to the energy 

difference between these two energy levels (E2

-E1

).

When this energy is absorbed or emitted in a form of electromagnetic radiation, the 

energy difference between these two energy levels (E2

-E1

) determines uniquely the 

frequency (λ) of the electromagnetic radiation:

(  E) = E2

-E1 = h

1.4.1 Absorption

Absorption of electromagnetic radiation is the way in which the energy of a photon is 

taken up by matter, typically the electrons of an atom. Atoms or molecules absorb 

light, the incoming energy excites a quantized structure to a higher energy level. The 

type of excitation depends on the wavelength of the light. Electrons are promoted to 

higher orbitals by ultraviolet or visible light, vibrations are excited by infrared light, 

and rotations are excited by microwaves.

An absorption spectrum is the absorption of light as a function of wavelength. The 

spectrum of an atom or molecule depends on its energy level structure, and absorption 

spectra are useful for identifying of compounds.

Emission

The emission spectrum of a chemical element or chemical compound is the spectrum 

of frequencies of electromagnetic radiation emitted due to an atom's electrons making 

a transition from a high energy state to a lower energy state. Atoms or molecules that 

are excited to high energy levels can decay to lower levels by emitting radiation 

(emission or luminescence). For atoms excited by a high-temperature energy source 

this light emission is commonly called atomic or optical emission, and for atoms 

excited with light it is called atomic fluorescence. For molecules it is called 

fluorescence if the transition is between states of the same spin (singlet to singlet 

transition) and phosphorescence if the transition occurs between states of different spin

(triplet to singlet transition).

The emission intensity of an emitting substance is linearly proportional to analyte 

concentration at low concentrations, and is useful for quantitating emitting species.

1.4.3 Transmission

Transmission happens when light goes through a surface or object. There are 3 types 

of transmission: direct, diffuse or selective.

1. Direct transmission: it is when light goes through an object and no change in 

direction or quality takes place. For example, through glass or air.

2. Diffuse transmission: it is produced when light goes through a transparent or semi-

transparent object with texture. For example, frosted glass or drafting paper. Light, 

instead of going in one direction, is redirected to other directions. Light which is 

transmitted in a diffused manner tends to be softer; it will have less contrast and 

less intensity; it will generate clearer shades; and it will have a smoother transition 

between highlights and shadows than direct light.

Selective transmission: it is produced when light goes through a coloured object. A 

portion of light will be absorbed and another portion will be transmitted through this 

object. In the example below, white light (red, green and blue) goes through a red 

surface. The green and blue are absorbed and only red is transmitted. As a result, we 

will only see red light on the other side of this surface.

Filters or gels, which we mentioned in the lesson about colour temperature, work 

through selective transmission. Colour filters will only allow one colour to go through 

(a blue filter allows only blue light go through) and it will absorb the rest of the 

colours. A blue filter lets blue wave lengths through and absorbs red and green wave 

lengths.

1.4.4. Reflection

Reflection happens when light reaches an object and it bounces or is reflected, 

partially or totally, from this object. Light can be reflected directly or in a diffused 

manner.

1. Direct reflection: it is produced when light is reflected from a flat or smooth

surface such as, for example, a mirror. Light will be reflected in the same angle as it 

reached this surface (law of reflection).

Electromagnetic Radiation

 Introduction

Transmission of energy through a vacuum or using no medium is accomplished by 

electromagnetic waves, caused by the oscillation of electric and magnetic fields. They 

move at a constant speed of 3x108 m/s. often, they are called electromagnetic 

radiation, light, or photons. An electromagnetic radiation, it has both electric and 

magnetic field components, which oscillate in a fixed relationship to one another, 

perpendicular to each other and perpendicular to the direction of propagation.

The two components making up an electromagnetic radiation are the:

a) Electric field B) Magnetic field

The two fields are always perpendicular to each other and both are perpendicular to the 

direction of propagation.

 

Fig. Electromagnetic wave

The notion that electromagnetic radiation contains a quantifiable amount of energy can 

perhaps be better understood if we talk about light as a stream ofparticles, 

called photons, rather than as a wave. (Recall the concept known as ‘wave-particle 

duality’: at the quantum level, wave behavior and particle behavior become 

indistinguishable, and very small particles have an observable ‘wavelength’). If we

describe light as a stream of photons, the energy of a particular wavelength can be 

expressed as:

E = hc / λ

where E is energy in kcal/mol, λ (the Greek letter lambda) is wavelength in 

meters, c is 3.00 x 108 m/s (the speed of light), and h is 9.537 x 10

-14 kcal/s/mol-1

, a 

number known as Planck’s constant.

Because electromagnetic radiation travels at a constant speed, each wavelength 

corresponds to a given frequency, which is the number of times per second that a crest 

passes a given point. Longer waves have lower frequencies, and shorter waves have 

higher frequencies. Frequency is commonly reported in hertz (Hz), meaning ‘cycles 

per second’, or ‘waves per second’. The standard unit for frequency is s-1

.

When talking about electromagnetic waves, we can refer either to wavelength or to 

frequency - the two values are interconverted using the simple expression:

 C 

where ν (the Greek letter ‘nu’) is frequency in s-1

. Visible red light with a wavelength 

of 700 nm, for example, has a frequency of 4.29 x 1014Hz, and an energy of 40.9 kcal 

per mole of photons.

The full range of electromagnetic radiation wavelengths is referred to as 

the electromagnetic spectrum.

1.3 Properties of Electromagnetic radiation 

The radiated EM radiation has certain properties: 

• EM waves travel at the speed of light c

• The electric and magnetic fields are perpendicular to each other. 

• The electric and magnetic fields are in phase (both reach a maximum and minimum 

at the same time). 

• The electric and magnetic fields are perpendicular to the direction of travel 

(transverse waves).

Terms Used 

Wavelength - Is the distance between any two equivalent points on successive waves.

Wavenumber - Is the reciprocal of the wavelength in centimeters.

Frequency - Is the number of oscillations of the field which occur each second.

Velocity- In a vacuum, the velocity of electromagnetic radiation is 2.9979 x 108 m/s

Amplitude - The height of the wave.

Their wavelengths and the corresponding differences in their energies: shorter 

wavelengths correspond to higher energy.

PROBLEM: Calculate wavelength and frequency of waves. 

(a) A local radio station broadcasts at a frequency of 91.7 MHz (91.7 x 106 Hz). What is the 

Wavelength of these radio waves?

b) What is the frequency of blue light with a wavelength of 435 nm?

SOLUTION: 

You are asked to calculate the wavelength or frequency of electromagnetic radiation. 

You are given the frequency or wavelength of the radiation. 

(a) First rearrange Equation 6.1 to solve for wavelength (λ). Then substitute the known values 

into the equation and solve for wavelength.

ʋ = c / λ or λ =c / ʋ

(b) First rearrange Equation 6.1 to solve for frequency (ʋ). Then substitute the known values 

into the equation and solve for frequency. Notice that wavelength must be converted to units 

of meters before using it in Equation (c = λʋ)

435nm43510 9m

= 4.35 x 10-7 m