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.