Download PG510 Symmetry and Molecular Spectroscopy

PG510
Symmetry and
Molecular Spectroscopy
Lecture no. 6
Molecular Spectroscopy:
General Concepts
Giuseppe Pileio
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Learning Outcomes
By the end of this lecture you will be able to:
!! Understand the basics of spectroscopy
!! Link the electromagnetic spectrum to molecular
spectroscopy
!! Link the different kind of molecular spectroscopy with
molecular motions
!! Understand how to calculate spectral intensity
!! Understand selection rules and their link with group
theory
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Spectroscopy
Spectroscopy refers to the study of the interaction
between an electromagnetic radiation and the matter
The term spectrum refers to a plot of intensities versus
radiation frequency (or related quantities)
The term spectrometer refers to a device that is able to
record such a spectrum
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Electromagnetic Radiation
Classical view: transverse waveform
Radiation consists of a magnetic and an electric field
oscillating on perpendicular planes which are perpendicular
to the direction of propagation of the wave
The two fields oscillate at the same frequency (!) and the
wave propagates through vacuum at the speed of light
c=2.99 108 m/s
The distance between two crests is called wavelength and
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is defined as "=c/!#
Quantum view: photons
Radiation consists of a stream of particles (photons) each
of which has no mass and carries an energy E=h! (where h is
Planck’s constant, h=6.6 10-34 Js)#
Photons have linear momentum p=h!/c
Photons have angular momentum ±h/2$
Unification: De Broglie Hypothesis
Linear momentum and wavelength are inversely proportional
"=h/p so that:
•! Photons: "=c/!#
•! Electrons: "=h/mv
•! macroscopic objects: "=h/p=~10-34 " too small to
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be appreciated
Electromagnetic spectrum
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Electromagnetic spectrum and Molecules
+ MS % Structure % Chemical-Physical properties
Final Target %
Information
Content %#
atom
bond length connections
+…
+…
functional
Atom
Basic on
groups
+…
position + … structure + …
Effect on
Molecules %#
diffraction
Associated
Spectroscopy %#
electron
transitions
XR
Electromagnetic
Spectrum %#
UV/Vis
X-Ray
UV
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n
V
I
S
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nuclear
vibration
molecular
rotation
nuclear
spin
(IR and Rot Raman)
IR
Rotational
IR
NMR
MW
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Frequency 10n HZ
r.f.
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Molecular Energy Levels
The arrangement
of electrons gives rise
to the Electron Energy, EE
The vibrations of the
nuclei give rise to the
Vibrational Energy, EV
The rotation of the whole
molecule gives
rise to the
Rotational Energy, ER
The spin of nuclei gives
rise to the
Nuclear Spin Energy, ES
Schrödinger Equation
H &E,V,R,S = EE,V,R,S&E,V,R,S
Separation of molecular energies
Quantum Mechanics: If a wavefunction can be written as
the product of different ones the energy of the system is
the sum of relative energies
If &=&a ! &b!&c ! …
then
Ea+Eb+Ec+…
Born-Oppenheimer: Nuclei are ~4 order of magnitude
more massive than electrons. Thus, since the same
forces act on both of them, nuclei are, approximately,
fixed when the electron transitions happen
H&E&V,R,S = (EE+EV,R,S) &E&V,R,S #
Empirical observations show that separation between
vibrational levels is larger than the one between rotational
levels which is larger than the one between spin levels
H &E&V &R& S = (EE+EV+ER+ES) &E&V&R& S #
Electronic
Spectroscopy
E
Vibrational
Spectroscopy
Rotational
Spectroscopy
EE
X-Ray
n
UV
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V
I
S
15
EV
ER
IR
MW
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Frequency 10n HZ
ES
r.f.
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Population of energy levels
When dealing with a collection of
molecules (ensamble) we need to figure
out how many molecules are actually in
each energy state
j#
E
'E=h!#
i#
This problem was solved by Boltzmann in the so called
Boltzmann distribution law:
nj
ni
!
gj
gi
"e#$E!kT
where nj and ni is the number of molecules which are
actually in the state j and i, respectively, while gj and gi is
the respective degeneracy of those states (i.e. how many
energy levels have the same energy value)
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nj
ni
!
gj
gi
'E=10cm-1 i.e. in
the MW region
"e#$E!kT
gj ! gi ! 1
'E=1000cm-1 i.e.
in the IR region
'E=10000cm-1 i.e.
in the UV region
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Spectroscopic Transitions
A spectroscopic transition is the change of a molecule from
one quantum state to another
The energy for the transition to happen is provided by the
electromagnetic radiation
Transitions may involve the electric dipole moment (µ), the
magnetic dipole moment (m) or the polarizability tensor (().
Those quantities vary as a result of molecular or electronic
motions (rotations, vibrations, electron motions)
Classical Theory
QuantumTheory
The exchange of energy
between µ and radiation is
maximized if they oscillate at
the same frequency
The exchange of energy is
maximized if the frequency of
the radiation (!) and the energy
difference between two levels
('E) satisfy: 'E=h!#
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Transition modes and Einstein coefficients
Considering two isolated energy levels of
a single molecule, there are only three
ways in which the molecule can move
between those two levels:
E
j#
'E=h!#
i#
1.! Spontaneous emission:
- If a molecule is in the state j it tends to loose energy
to go to state i emitting a photon of frequency !
- The probability for this to occur is given by Einstein’s
coefficient: A=(16$3!3|µ|2)/(3)0hc3) (s-1)
- If we have Nj molecules in j then energy is emitted at
a rate: I=NjAh! (J s-1)
The process goes with v4 i.e. becomes increasingly important
at higher frequencies. µ is the transition moment integral
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and )0 is the vacuum permittivity
2.! Induced absorption:
- If a molecule in the state i is irradiated by a radiation
of frequency ! it is promoted in the state j
- The probability for this to occur is: P = *(!)Bij where
Bij=(2$2|µ|2)/(3)0hc3) (s-1) is the Einstein’s coefficient
and *(!) is the radiation density at ! in Jm-3
- If we have Ni molecules in i then energy is emitted at
a rate: I=NiBij*(!)h! (J s-1)
3.! Induced emission:
- If a molecule in the state j is irradiated by a radiation
of frequency ! it can go to state i by emitting at !
- The probability for this to occur is: P = *(!)Bji with
Bji=Bij
- If we have Nj molecules in j then energy is emitted at
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a rate: I=NjBji*(!)h! (J s-1)
j
Spontaneous
emission ~A
Induced
absorption ~B
Induced
emission ~B
i
Iem ! nj "hΝ!A $ B Ρ""Ν#$
Iab ! ni "hΝ B Ρ"!Ν"
If nj is significant (j level significantly populated) at a
certain T then it means that 'E is not big and so the
frequency is small enough to neglect spontaneous
emissions, so irradiating the sample we have an overall
intensity due to both induced absorption and emission:
I ! !ni " nj "#hΝ B Ρ##Ν$
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Spectroscopic selection rules
The intensity (I) of a transition from a
state i (described by &i) and a state j
(described by &j) is proportional to:
I ! Ν !nj # ni " # Μ #
E
&j#
'E=h!#
%
Μ % $ &i 'Μ &j '( Τ
&i#
µ is the transition moment operator. It is usually different
for various kind of spectroscopy: electric dipole (MW, IR,
UV/Vis), polarizability tensor (Raman), magnetic dipole
(NMR)
The rules by which the integral of the transition moment is
identically zero are called selection rules
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To calculate the integral of the transition moment we need
to know better the quantities involved
"
Μ " ! #i $Μ #j $% Τ
First of all we already said that the wavefunction that
describe the system in the two levels i and j can be
factorized in the electronic (el), vibrational (vib) and
rotational (rot) so:
! " !el #!vib #!rot
Furthermore, we can think to have a light polarized along
z so that will interact with the z component of the dipole
moment i.e. µz, thus
Μ " ! #i,el $#i,vib $#i,rot $Μz #j,el $#j,vib $#j,rot $% Τrot $% Τvib $% Τel
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now it is possible to change the reference system in the
one rotating with the molecule
!
Μz "
ΛzΑ %ΜΑ
Α"x,y,z
where "z( is the director cosine which involve the Euler
angles between the two frames.
•! "z(!s are function of molecular rotational coordinates
•! µ! depends only on electronic and nuclear coordinates
!
Μ"
" $i,rot %ΛzΑ $j,rot %' Τrot %" $i,el %$i,vib %ΜΑ $j,el %$j,vib %' Τvib %' Τel
Α"x,y,z
It is actually not possible to factorize rigorously the
second integral into an electronic and a vibrational part
but an approximation can be used
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The µ( can be expanded in series with respect to the
vibrational coordinates Qi
ΜΑ # ΜeΑ $ !
3%N&6
i#1
∆ΜΑ
∆Qi
%Qi $
e
1
2
% !
3%N&6
i,j#1
∆2 %ΜΑ
∆Qi %∆Qj
%Qi %Qj $ ...
e
Thus:
Μ"
!
" $i,rot %ΛzΑ $j,rot %' Τrot ) #
Α"x,y,z
e
! !i,el "ΜΑ "!j,el "% Τel "! !i,vib "!j,vib "% Τvib
! ! " %i,el #
3#N$6
i"1
Electronic selection rules
#%j,el #) Τel #" %i,vib #Qi #%j,vib #) Τvib #
∆ΜΑ
∆Qi
Pure rotational selection rules
e
Vibrational selection
rules
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For pure rotational transitions:
Μ"
!
!i,el " !j,el
and
!i,vib " !j,vib
" $i,rot %ΛzΑ $j,rot %' Τrot %" $i,el %$i,vib %ΜΑ $j,el %$j,vib %' Τvib %' Τel
Α"x,y,z
So, the 2nd and 3rd integral reduces to µz
" the electric dipole moment must be ! 0
while the 1st is non-zero only if
" 'J=±1 (linear)
" 'J=0,±1 (asymmetric)
" 'J=0,±1 (symmetric) & 'K=0 (k!0, K=±1,…,±J)
#
# 'J=±1 (symmetric) & 'K=0 (k=0)
For vibrational transitions:
e
! !i,el "ΜΑ "!j,el "% Τel "! !i,vib "!j,vib "% Τvib
the 1st is non-zero only if
" 'J=±1 (etc…)
nd
the 2 is always 0 as vibrational functions are ortho-normal
the 3rd is non-zero if the vibration creates a dipole and if
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" 'v=±1
Μ"
!
e
" $i,rot %ΛzΑ $j,rot %' Τrot ) #" $i,el %ΜΑ %$j,el %' Τel %" $i,vib %$j,vib %' Τvib * ! " $i,el %
Α"x,y,z
3%N+6
i"1
%$j,el %' Τel %" $i,vib %Qi %$j,vib %' Τvib $
∆ΜΑ
∆Qi
e
For electronic transitions:
the 1st is non-zero only if
" 'J=±1 (etc…)
the 2nd is non-zero only if
" the direct product &i,el"&j,el transforms like
x, y or z
The term ! !i,vib"!j,vib"# Τvib is called Franck-Condon factor
and scales the intensity of the transition
Finally, even if the 2nd term is 0 for symmetry here can be
vibronic transition allowed when the 3rd integral is nonzero
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What did we learn in this lecture?
•! The concept of spectroscopy
•! The classical and quantum description of electromagnetic
radiation
•! The link between radiofrequency, spectroscopy and
energy levels
•! The population of energy levels
•! Transition probabilities and Einstein’s coefficients
•! Selection rules
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