Lecture 12 - Electronic Spectroscopy - Molecular Spectroscopy Notes | EduRev

: Lecture 12 - Electronic Spectroscopy - Molecular Spectroscopy Notes | EduRev

 Page 1


 Module 3 : Molecular Spectroscopy
 Lecture 12 : Electronic Spectroscopy
  
 
Objectives
 
After studying this lecture, you will be able to
Qualitatively order the molecular energy levels into electronic, vibrational, rotational and other energy levels.
Calculate the relative population of these energy levels.
Identify the regions of the electromagnetic spectrum corresponding to different molecular transitions.
Calculate the energy differences in various units that are commonly used in molecular spectroscopy.
Understand the physical basis of selection rules in general and for electronic spectroscopy in particular.
Identify different parts of an absorption spectrometer.
Label electronic energy levels using the total angular momentum quantum number.
Rationalize the multiplet structure of electronic spectra
  
12.1 Introduction
  
 
We have already studied various aspects of atomic and molecular structure in terms of molecular energy
levels and charge densities. In addition to electronic energy levels in molecules, there are other energy
levels in a molecule such as vibrational and rotational, just to name a few. All these levels are quantized or
discrete. The quantization is a consequence of the boundary conditions (such as finiteness, single
valuedness, square integrability) on the wave functions. What this means in physical terms is that in
molecular vibrations, the molecules undergo oscillatory motion in a small region of space around the
minimum energy structure. In rotational motion, molecules look identical after a rotation by 360 
0
 with
respect to any axis.
We can not see molecular structures with either the naked eye or even by microscopic cameras. We can
only study the transitions between the energy levels of molecules resulting from the absorption and emission
of light. To infer from the spectroscopic transition data the correct molecular structure is almost like the
work of a detective.
Molecular spectroscopy is a vast and growing subject and we shall qualitatively explore some aspects the
principles of spectroscopy and of UV-visible spectroscopy in the present lecture. Vibrational and rotational
spectroscopy will be taken up in the next lecture; magnetic resonance methods in lecture 14 and some of
the remaining methods in lecture 15.
  
Page 2


 Module 3 : Molecular Spectroscopy
 Lecture 12 : Electronic Spectroscopy
  
 
Objectives
 
After studying this lecture, you will be able to
Qualitatively order the molecular energy levels into electronic, vibrational, rotational and other energy levels.
Calculate the relative population of these energy levels.
Identify the regions of the electromagnetic spectrum corresponding to different molecular transitions.
Calculate the energy differences in various units that are commonly used in molecular spectroscopy.
Understand the physical basis of selection rules in general and for electronic spectroscopy in particular.
Identify different parts of an absorption spectrometer.
Label electronic energy levels using the total angular momentum quantum number.
Rationalize the multiplet structure of electronic spectra
  
12.1 Introduction
  
 
We have already studied various aspects of atomic and molecular structure in terms of molecular energy
levels and charge densities. In addition to electronic energy levels in molecules, there are other energy
levels in a molecule such as vibrational and rotational, just to name a few. All these levels are quantized or
discrete. The quantization is a consequence of the boundary conditions (such as finiteness, single
valuedness, square integrability) on the wave functions. What this means in physical terms is that in
molecular vibrations, the molecules undergo oscillatory motion in a small region of space around the
minimum energy structure. In rotational motion, molecules look identical after a rotation by 360 
0
 with
respect to any axis.
We can not see molecular structures with either the naked eye or even by microscopic cameras. We can
only study the transitions between the energy levels of molecules resulting from the absorption and emission
of light. To infer from the spectroscopic transition data the correct molecular structure is almost like the
work of a detective.
Molecular spectroscopy is a vast and growing subject and we shall qualitatively explore some aspects the
principles of spectroscopy and of UV-visible spectroscopy in the present lecture. Vibrational and rotational
spectroscopy will be taken up in the next lecture; magnetic resonance methods in lecture 14 and some of
the remaining methods in lecture 15.
  
Molecular Energy Levels
 
A schematic energy level diagram of a polyatomic molecule is shown in Fig. 12.1  
5 
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0 ..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__.. 0  
4 
   3………………………………………………………………………………….
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0.. __..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__.... 0
3 
   _ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 1
   3………………………………………………………………………………….
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0...__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__... 0
2 
   2……………………………………………………………………………........
   1………………………………………………………………………………….
   0 __..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__. 2
   2…………………………………………………………………………………
   1…………………………………………………………………………………
   _ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 1
   3………………………………………………………………………………….
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0 ..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__.. 0
1 
Fig. 12.1 Molecular energy levels. Electronic energy levels are shown by thick lines, vibrational energy levels
are shown by dashed lines and rotational energy levels are shown by dotted lines. The quantum numbers
corresponding to electronic and rotational levels are shown on the left and the quantum numbers for the
vibrational levels are shown on the right.
  
In Fig. 12.1, the thick lines represent electronic energy levels which are separated from one another by
energies in the range of 0.5 to a few electron volts. “Within” (or between two) each electronic energy levels
there are several vibrational energy levels and within each vibrational level, there are several rotational
energy levels. These energy levels are not equally placed and the spacings between higher levels decrease.
The range of energy levels and the region of the electromagnetic radiation that causes transitions between
these levels is shown in Table 12.1
Page 3


 Module 3 : Molecular Spectroscopy
 Lecture 12 : Electronic Spectroscopy
  
 
Objectives
 
After studying this lecture, you will be able to
Qualitatively order the molecular energy levels into electronic, vibrational, rotational and other energy levels.
Calculate the relative population of these energy levels.
Identify the regions of the electromagnetic spectrum corresponding to different molecular transitions.
Calculate the energy differences in various units that are commonly used in molecular spectroscopy.
Understand the physical basis of selection rules in general and for electronic spectroscopy in particular.
Identify different parts of an absorption spectrometer.
Label electronic energy levels using the total angular momentum quantum number.
Rationalize the multiplet structure of electronic spectra
  
12.1 Introduction
  
 
We have already studied various aspects of atomic and molecular structure in terms of molecular energy
levels and charge densities. In addition to electronic energy levels in molecules, there are other energy
levels in a molecule such as vibrational and rotational, just to name a few. All these levels are quantized or
discrete. The quantization is a consequence of the boundary conditions (such as finiteness, single
valuedness, square integrability) on the wave functions. What this means in physical terms is that in
molecular vibrations, the molecules undergo oscillatory motion in a small region of space around the
minimum energy structure. In rotational motion, molecules look identical after a rotation by 360 
0
 with
respect to any axis.
We can not see molecular structures with either the naked eye or even by microscopic cameras. We can
only study the transitions between the energy levels of molecules resulting from the absorption and emission
of light. To infer from the spectroscopic transition data the correct molecular structure is almost like the
work of a detective.
Molecular spectroscopy is a vast and growing subject and we shall qualitatively explore some aspects the
principles of spectroscopy and of UV-visible spectroscopy in the present lecture. Vibrational and rotational
spectroscopy will be taken up in the next lecture; magnetic resonance methods in lecture 14 and some of
the remaining methods in lecture 15.
  
Molecular Energy Levels
 
A schematic energy level diagram of a polyatomic molecule is shown in Fig. 12.1  
5 
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0 ..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__.. 0  
4 
   3………………………………………………………………………………….
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0.. __..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__.... 0
3 
   _ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 1
   3………………………………………………………………………………….
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0...__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__... 0
2 
   2……………………………………………………………………………........
   1………………………………………………………………………………….
   0 __..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__. 2
   2…………………………………………………………………………………
   1…………………………………………………………………………………
   _ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 1
   3………………………………………………………………………………….
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0 ..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__.. 0
1 
Fig. 12.1 Molecular energy levels. Electronic energy levels are shown by thick lines, vibrational energy levels
are shown by dashed lines and rotational energy levels are shown by dotted lines. The quantum numbers
corresponding to electronic and rotational levels are shown on the left and the quantum numbers for the
vibrational levels are shown on the right.
  
In Fig. 12.1, the thick lines represent electronic energy levels which are separated from one another by
energies in the range of 0.5 to a few electron volts. “Within” (or between two) each electronic energy levels
there are several vibrational energy levels and within each vibrational level, there are several rotational
energy levels. These energy levels are not equally placed and the spacings between higher levels decrease.
The range of energy levels and the region of the electromagnetic radiation that causes transitions between
these levels is shown in Table 12.1
 
Table 12.1 Molecular Energy levels and the Regions of Electromagnetic Radiation
Region of the 
Spectrum
Frequency 
Range (Hz)
Wavelength Range Energy range Kind of 
Spectroscopy
X-rays
3 x 10 
16
 to 3x10 
18 10nm to 100pm
~10 
4
 kJ/mol
X-ray photo-electron
Spectroscopy (inner
electrons)
Visible and UV
3x10
14
 to 3x10
16 1µm to 10nm
~10 
2
 kJ/mol
Electronic
Spectroscopy
Infrared
3x10 
12
 to 3x10 
14 100µm to 1µm ~10 kJ/mol Vibrational
Spectroscopy
Microwave
3x10 
10
 to 3x10 
12 1cm to 100µm 100J/mol Rotational
Spectroscopy
Radio frequency
3x10 
6
 to 3x10 
10 10m to 1 cm 0.001 to 10J/mol Magnetic Resonance
Spectroscopy
 
It is a very convenient approximation to treat the total energy of a molecule as a sum of electronic,
vibrational, rotational, nuclear spin, electron spin and other energy levels.
E 
total
 = E 
electronic
 + E 
Vibrational
 + E 
rotational
 + ……                    (12.1)
The above equation follows from the following approximation
H 
total
 = H 
electronic
 + H 
vibrational
 + H 
rotational
 +……                     (12.2)
On solving the equation (12.2), the following solutions emerge
H 
total
  
total
 = E 
total
 
total
                                                       (12.3a)
H 
electronic
  
electronic
 = E
electronic
  
electronic
                            (12.3b)
H 
vibrational
 
vibrational
 = E 
vibrational
  
vibrational
                      (12.3c)
H 
rotational
 
rotational
 = E 
rotational
 
rotational
                            (12.3d)
For simplicity we will denote rotational vibrational and electronic by subscripts ‘r', ‘v' and ‘e' respectively. The
nature of E
e
 and 
e
 has already been discussed in the first few lectures by using atomic and molecular
orbitals. In the present lecture, we are considering transitions between electronic energy levels. The energy
levels have to be calculated by solving the approximate Schrodinger equation (12.3b).
  
12.3 Selection Rules
Although there are a very large number of energy levels in molecules, transitions are allowed only between
specific levels. Several conditions have to met in addition to the primary transition energy requirement, ie,
?E = nh ?                                                                                  (12.4)
Normally n = 1 and only one photon is absorbed. However, with advances in laser techniques, multiphoton
absorption has become possible. In addition to energy conservation in an absorption or emission process,
angular momentum has to be conserved during a transition. Photons (particles of light) possess an angular
Page 4


 Module 3 : Molecular Spectroscopy
 Lecture 12 : Electronic Spectroscopy
  
 
Objectives
 
After studying this lecture, you will be able to
Qualitatively order the molecular energy levels into electronic, vibrational, rotational and other energy levels.
Calculate the relative population of these energy levels.
Identify the regions of the electromagnetic spectrum corresponding to different molecular transitions.
Calculate the energy differences in various units that are commonly used in molecular spectroscopy.
Understand the physical basis of selection rules in general and for electronic spectroscopy in particular.
Identify different parts of an absorption spectrometer.
Label electronic energy levels using the total angular momentum quantum number.
Rationalize the multiplet structure of electronic spectra
  
12.1 Introduction
  
 
We have already studied various aspects of atomic and molecular structure in terms of molecular energy
levels and charge densities. In addition to electronic energy levels in molecules, there are other energy
levels in a molecule such as vibrational and rotational, just to name a few. All these levels are quantized or
discrete. The quantization is a consequence of the boundary conditions (such as finiteness, single
valuedness, square integrability) on the wave functions. What this means in physical terms is that in
molecular vibrations, the molecules undergo oscillatory motion in a small region of space around the
minimum energy structure. In rotational motion, molecules look identical after a rotation by 360 
0
 with
respect to any axis.
We can not see molecular structures with either the naked eye or even by microscopic cameras. We can
only study the transitions between the energy levels of molecules resulting from the absorption and emission
of light. To infer from the spectroscopic transition data the correct molecular structure is almost like the
work of a detective.
Molecular spectroscopy is a vast and growing subject and we shall qualitatively explore some aspects the
principles of spectroscopy and of UV-visible spectroscopy in the present lecture. Vibrational and rotational
spectroscopy will be taken up in the next lecture; magnetic resonance methods in lecture 14 and some of
the remaining methods in lecture 15.
  
Molecular Energy Levels
 
A schematic energy level diagram of a polyatomic molecule is shown in Fig. 12.1  
5 
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0 ..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__.. 0  
4 
   3………………………………………………………………………………….
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0.. __..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__.... 0
3 
   _ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 1
   3………………………………………………………………………………….
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0...__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__... 0
2 
   2……………………………………………………………………………........
   1………………………………………………………………………………….
   0 __..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__. 2
   2…………………………………………………………………………………
   1…………………………………………………………………………………
   _ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 1
   3………………………………………………………………………………….
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0 ..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__.. 0
1 
Fig. 12.1 Molecular energy levels. Electronic energy levels are shown by thick lines, vibrational energy levels
are shown by dashed lines and rotational energy levels are shown by dotted lines. The quantum numbers
corresponding to electronic and rotational levels are shown on the left and the quantum numbers for the
vibrational levels are shown on the right.
  
In Fig. 12.1, the thick lines represent electronic energy levels which are separated from one another by
energies in the range of 0.5 to a few electron volts. “Within” (or between two) each electronic energy levels
there are several vibrational energy levels and within each vibrational level, there are several rotational
energy levels. These energy levels are not equally placed and the spacings between higher levels decrease.
The range of energy levels and the region of the electromagnetic radiation that causes transitions between
these levels is shown in Table 12.1
 
Table 12.1 Molecular Energy levels and the Regions of Electromagnetic Radiation
Region of the 
Spectrum
Frequency 
Range (Hz)
Wavelength Range Energy range Kind of 
Spectroscopy
X-rays
3 x 10 
16
 to 3x10 
18 10nm to 100pm
~10 
4
 kJ/mol
X-ray photo-electron
Spectroscopy (inner
electrons)
Visible and UV
3x10
14
 to 3x10
16 1µm to 10nm
~10 
2
 kJ/mol
Electronic
Spectroscopy
Infrared
3x10 
12
 to 3x10 
14 100µm to 1µm ~10 kJ/mol Vibrational
Spectroscopy
Microwave
3x10 
10
 to 3x10 
12 1cm to 100µm 100J/mol Rotational
Spectroscopy
Radio frequency
3x10 
6
 to 3x10 
10 10m to 1 cm 0.001 to 10J/mol Magnetic Resonance
Spectroscopy
 
It is a very convenient approximation to treat the total energy of a molecule as a sum of electronic,
vibrational, rotational, nuclear spin, electron spin and other energy levels.
E 
total
 = E 
electronic
 + E 
Vibrational
 + E 
rotational
 + ……                    (12.1)
The above equation follows from the following approximation
H 
total
 = H 
electronic
 + H 
vibrational
 + H 
rotational
 +……                     (12.2)
On solving the equation (12.2), the following solutions emerge
H 
total
  
total
 = E 
total
 
total
                                                       (12.3a)
H 
electronic
  
electronic
 = E
electronic
  
electronic
                            (12.3b)
H 
vibrational
 
vibrational
 = E 
vibrational
  
vibrational
                      (12.3c)
H 
rotational
 
rotational
 = E 
rotational
 
rotational
                            (12.3d)
For simplicity we will denote rotational vibrational and electronic by subscripts ‘r', ‘v' and ‘e' respectively. The
nature of E
e
 and 
e
 has already been discussed in the first few lectures by using atomic and molecular
orbitals. In the present lecture, we are considering transitions between electronic energy levels. The energy
levels have to be calculated by solving the approximate Schrodinger equation (12.3b).
  
12.3 Selection Rules
Although there are a very large number of energy levels in molecules, transitions are allowed only between
specific levels. Several conditions have to met in addition to the primary transition energy requirement, ie,
?E = nh ?                                                                                  (12.4)
Normally n = 1 and only one photon is absorbed. However, with advances in laser techniques, multiphoton
absorption has become possible. In addition to energy conservation in an absorption or emission process,
angular momentum has to be conserved during a transition. Photons (particles of light) possess an angular
 
momentum of ?. During an electronic transition, for example, a molecule may get exited from an S state to
a P state. In the S state, angular momentum is zero and in the P state, angular momentum is 1 (in units of
? ). The transition probability P
i? f
   from a state  ?
i
 to a state  is given by  ?
f  
  is given
P
i?f
 a <?
i
¦µ ¦?
f
>¦
2                                                             
 (12.5)
Where µ is the transition dipole operator. Fluctuations in molecular charge densities, dipole moment,
polarizability and so on are responsible for this “transition moment” operator.
Another feature that contributes to the intensity of a spectral line is the population of the energy levels. The
lower levels are usually more populated than the higher levels and the intensities of absorption lines from
lower levels are usually higher. The relative population of energy levels is governed by the Boltzmann
distribution at equilibrium.
N
upper
/N
lower
 a exp (-?E/k
B
T)                                                   (12.6)
Here,  ?E = E 
upper
 – E
 lower,
 k
B
 = Boltzmann constant = 1.38 x 10 
-16
 erg/K and T = absolute temperature. 
When ?E is large, the number of molecules in the upper level, N
 upper
 is very small.
Most of spectroscopy deals with absorption of light.  The amount of light absorbed depends on the path
length l of the sample (ie the linear extent through which the light travels through the sample), the
concentration of the absorbing molecules, c, contained in the sample and the molar absorption coefficient ?.  
The Beer-Lambert law which relates the absorbed intensity of light, I, to the incident intensity I
o
 is given by
I/I
o
 =  10
 – ?cl
                                                            (12.7)
The quantity I/I
o 
is called transmittance.  This law is very useful in determining unknown concentrations of
molecules in samples when  ? is already determined by previous experiments.  A block diagram of an
absorption spectrometer is shown in Fig. 12.2.
  
                            Fig. 12.2 Block diagram of an absorption spectrometer
The modulator is a device that interrupts the radiation several times in a second and converts the signal to
an alternating current which is then used for recording. This is useful for amplification of signals and also to
reduce the noise levels in the spectrum.
Page 5


 Module 3 : Molecular Spectroscopy
 Lecture 12 : Electronic Spectroscopy
  
 
Objectives
 
After studying this lecture, you will be able to
Qualitatively order the molecular energy levels into electronic, vibrational, rotational and other energy levels.
Calculate the relative population of these energy levels.
Identify the regions of the electromagnetic spectrum corresponding to different molecular transitions.
Calculate the energy differences in various units that are commonly used in molecular spectroscopy.
Understand the physical basis of selection rules in general and for electronic spectroscopy in particular.
Identify different parts of an absorption spectrometer.
Label electronic energy levels using the total angular momentum quantum number.
Rationalize the multiplet structure of electronic spectra
  
12.1 Introduction
  
 
We have already studied various aspects of atomic and molecular structure in terms of molecular energy
levels and charge densities. In addition to electronic energy levels in molecules, there are other energy
levels in a molecule such as vibrational and rotational, just to name a few. All these levels are quantized or
discrete. The quantization is a consequence of the boundary conditions (such as finiteness, single
valuedness, square integrability) on the wave functions. What this means in physical terms is that in
molecular vibrations, the molecules undergo oscillatory motion in a small region of space around the
minimum energy structure. In rotational motion, molecules look identical after a rotation by 360 
0
 with
respect to any axis.
We can not see molecular structures with either the naked eye or even by microscopic cameras. We can
only study the transitions between the energy levels of molecules resulting from the absorption and emission
of light. To infer from the spectroscopic transition data the correct molecular structure is almost like the
work of a detective.
Molecular spectroscopy is a vast and growing subject and we shall qualitatively explore some aspects the
principles of spectroscopy and of UV-visible spectroscopy in the present lecture. Vibrational and rotational
spectroscopy will be taken up in the next lecture; magnetic resonance methods in lecture 14 and some of
the remaining methods in lecture 15.
  
Molecular Energy Levels
 
A schematic energy level diagram of a polyatomic molecule is shown in Fig. 12.1  
5 
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0 ..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__.. 0  
4 
   3………………………………………………………………………………….
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0.. __..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__.... 0
3 
   _ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 1
   3………………………………………………………………………………….
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0...__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__... 0
2 
   2……………………………………………………………………………........
   1………………………………………………………………………………….
   0 __..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__. 2
   2…………………………………………………………………………………
   1…………………………………………………………………………………
   _ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 1
   3………………………………………………………………………………….
   2………………………………………………………………………………….
   1 …………………………………………………………………………………
   0 ..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__..__.. 0
1 
Fig. 12.1 Molecular energy levels. Electronic energy levels are shown by thick lines, vibrational energy levels
are shown by dashed lines and rotational energy levels are shown by dotted lines. The quantum numbers
corresponding to electronic and rotational levels are shown on the left and the quantum numbers for the
vibrational levels are shown on the right.
  
In Fig. 12.1, the thick lines represent electronic energy levels which are separated from one another by
energies in the range of 0.5 to a few electron volts. “Within” (or between two) each electronic energy levels
there are several vibrational energy levels and within each vibrational level, there are several rotational
energy levels. These energy levels are not equally placed and the spacings between higher levels decrease.
The range of energy levels and the region of the electromagnetic radiation that causes transitions between
these levels is shown in Table 12.1
 
Table 12.1 Molecular Energy levels and the Regions of Electromagnetic Radiation
Region of the 
Spectrum
Frequency 
Range (Hz)
Wavelength Range Energy range Kind of 
Spectroscopy
X-rays
3 x 10 
16
 to 3x10 
18 10nm to 100pm
~10 
4
 kJ/mol
X-ray photo-electron
Spectroscopy (inner
electrons)
Visible and UV
3x10
14
 to 3x10
16 1µm to 10nm
~10 
2
 kJ/mol
Electronic
Spectroscopy
Infrared
3x10 
12
 to 3x10 
14 100µm to 1µm ~10 kJ/mol Vibrational
Spectroscopy
Microwave
3x10 
10
 to 3x10 
12 1cm to 100µm 100J/mol Rotational
Spectroscopy
Radio frequency
3x10 
6
 to 3x10 
10 10m to 1 cm 0.001 to 10J/mol Magnetic Resonance
Spectroscopy
 
It is a very convenient approximation to treat the total energy of a molecule as a sum of electronic,
vibrational, rotational, nuclear spin, electron spin and other energy levels.
E 
total
 = E 
electronic
 + E 
Vibrational
 + E 
rotational
 + ……                    (12.1)
The above equation follows from the following approximation
H 
total
 = H 
electronic
 + H 
vibrational
 + H 
rotational
 +……                     (12.2)
On solving the equation (12.2), the following solutions emerge
H 
total
  
total
 = E 
total
 
total
                                                       (12.3a)
H 
electronic
  
electronic
 = E
electronic
  
electronic
                            (12.3b)
H 
vibrational
 
vibrational
 = E 
vibrational
  
vibrational
                      (12.3c)
H 
rotational
 
rotational
 = E 
rotational
 
rotational
                            (12.3d)
For simplicity we will denote rotational vibrational and electronic by subscripts ‘r', ‘v' and ‘e' respectively. The
nature of E
e
 and 
e
 has already been discussed in the first few lectures by using atomic and molecular
orbitals. In the present lecture, we are considering transitions between electronic energy levels. The energy
levels have to be calculated by solving the approximate Schrodinger equation (12.3b).
  
12.3 Selection Rules
Although there are a very large number of energy levels in molecules, transitions are allowed only between
specific levels. Several conditions have to met in addition to the primary transition energy requirement, ie,
?E = nh ?                                                                                  (12.4)
Normally n = 1 and only one photon is absorbed. However, with advances in laser techniques, multiphoton
absorption has become possible. In addition to energy conservation in an absorption or emission process,
angular momentum has to be conserved during a transition. Photons (particles of light) possess an angular
 
momentum of ?. During an electronic transition, for example, a molecule may get exited from an S state to
a P state. In the S state, angular momentum is zero and in the P state, angular momentum is 1 (in units of
? ). The transition probability P
i? f
   from a state  ?
i
 to a state  is given by  ?
f  
  is given
P
i?f
 a <?
i
¦µ ¦?
f
>¦
2                                                             
 (12.5)
Where µ is the transition dipole operator. Fluctuations in molecular charge densities, dipole moment,
polarizability and so on are responsible for this “transition moment” operator.
Another feature that contributes to the intensity of a spectral line is the population of the energy levels. The
lower levels are usually more populated than the higher levels and the intensities of absorption lines from
lower levels are usually higher. The relative population of energy levels is governed by the Boltzmann
distribution at equilibrium.
N
upper
/N
lower
 a exp (-?E/k
B
T)                                                   (12.6)
Here,  ?E = E 
upper
 – E
 lower,
 k
B
 = Boltzmann constant = 1.38 x 10 
-16
 erg/K and T = absolute temperature. 
When ?E is large, the number of molecules in the upper level, N
 upper
 is very small.
Most of spectroscopy deals with absorption of light.  The amount of light absorbed depends on the path
length l of the sample (ie the linear extent through which the light travels through the sample), the
concentration of the absorbing molecules, c, contained in the sample and the molar absorption coefficient ?.  
The Beer-Lambert law which relates the absorbed intensity of light, I, to the incident intensity I
o
 is given by
I/I
o
 =  10
 – ?cl
                                                            (12.7)
The quantity I/I
o 
is called transmittance.  This law is very useful in determining unknown concentrations of
molecules in samples when  ? is already determined by previous experiments.  A block diagram of an
absorption spectrometer is shown in Fig. 12.2.
  
                            Fig. 12.2 Block diagram of an absorption spectrometer
The modulator is a device that interrupts the radiation several times in a second and converts the signal to
an alternating current which is then used for recording. This is useful for amplification of signals and also to
reduce the noise levels in the spectrum.
                                              Fig. 12.3 A typical absorption spectrum
Three absorption regions are shown. The one at high frequency is very sharp, while the other two are broad.
The width of spectral lines is governed by the uncertainty principle.
?E .   ?t   = ?/2                          (12.8)
 
where ?E  is the uncertainty in energy of a state and ?t is the duration in which the state exists.
  
12.4 Electronic Spectroscopy of Atoms
For hydrogen atom, the selection rules (the rules that govern whether a transition is allowed or not) are
?n   =  any integer
?l   =   ± 1                                                                                                                  (12.9)
The consequences of these conditions are that all transitions from 1s ? np  (n = 2)  are allowed.  Similarly
2p  ? 3s, 2p  ? 3d,  2p  ? 1s, 3s  ? 4p, ….. are all allowed transitions.  In Equation (12.9), n  is the
principal quantum number.  The well known Lyman series arises from the transions from the n = 1  state to
n´ = 2, 3,4,5…  The Balmer series arise from the transitions from the n = 2 state to n´ = 3,4,5…. The other
series named after Paschen (n = 3), Brackett (n = 4) and Pfund (n = 5) can be readily rationalized.  The
formula for ?E is the well known Bohr’s formula.
                                                (12.10)
Where n is the initial quantum number and n’, is the number or state resulting from the absorption of
radiation.  The Rydberg constant R  =  109677.581  cm
-1
We have already studied that for each angular momentum L, there are (2 l + 1)  values of m
l
.  The values
of m
l
  
correspond to the projection of the angular momentum vector L   in different directions.  In the same
manner, the spin angular momentum vector  s   can also be projected in (2s + 1)  = 2 directions.  The
projections are pictorially shown in Fig. 12.4.
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