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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 is defined as "=c/!#

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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 Plancks 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 be appreciated

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Electromagnetic spectrum

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Electromagnetic spectrum and Molecules

Electromagnetic Spectrum %#

Associated Spectroscopy %#

(IR and Rot Raman) XR UV/Vis IR Rotational NMR

Effect on Molecules %#

diffraction electron transitions

nuclear vibration

molecular rotation

nuclear spin

Information Content %#

Atom position +

Basic on structure +

functional groups +

bond length +

atom connections

+

UV IR MW r.f. X-Ray V I S

9 12 15 17

Frequency 10n HZ

n

+ MS % Structure % Chemical-Physical properties Final Target %

The arrangement of electrons gives rise

to the Electron Energy, EE

The rotation of the whole molecule gives

rise to the Rotational Energy, ER

The vibrations of the nuclei give rise to the Vibrational Energy, EV

The spin of nuclei gives rise to the

Nuclear Spin Energy, ES

H &E,V,R,S = EE,V,R,S&E,V,R,S

Molecular Energy Levels

Schrdinger Equation

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 #

If &=&a ! &b!&c ! then Ea+Eb+Ec+

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

Separation of molecular energies

EE EV ER

E

ES

UV IR MW r.f. X-Ray

V

I

S

9 12 15 17 Frequency 10n HZ

n

Rotational Spectroscopy

Vibrational Spectroscopy

Electronic Spectroscopy

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Population of energy levels

E

'E=h!#

j#

i#

When dealing with a collection of molecules (ensamble) we need to figure out how many molecules are actually in each energy state

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#$E!kT

'E=10cm-1 i.e. in the MW region

'E=1000cm-1 i.e. in the IR region

'E=10000cm-1 i.e. in the UV region

gj ! gi ! 1

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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 (), 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

The exchange of energy between and radiation is

maximized if they oscillate at the same frequency

QuantumTheory

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:

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 Einsteins coefficient: A=(16$3!3||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. is the transition moment integral and )0 is the vacuum permittivity

E

'E=h!#

j#

i#

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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||2)/(3)0hc3) (s-1) is the Einsteins 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 a rate: I=NjBji*(!)h! (J s-1)

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Spontaneous emission ~A

Induced absorption ~B

Induced emission ~B

j

i

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:

Iem ! nj"h!A $ B ""#$ Iab ! ni"h B "!"

I ! !ni " nj"#h B ##$

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Spectroscopic selection rules

E

'E=h!#

&j#

&i#

The intensity (I) of a transition from a state i (described by &i) and a state j (described by &j) is proportional to:

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

I ! !nj # ni" # # % $ &i'% &j'(

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To calculate the integral of the transition moment we need to know better the quantities involved

" ! #i$" #j$%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. z, thus

" ! #i,el$#i,vib$#i,rot$z #j,el$#j,vib$#j,rot$%rot$%vib$%el

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now it is possible to change the reference system in the one rotating with the molecule

where "z( is the director cosine which involve the Euler angles between the two frames.

! "z(!s are function of molecular rotational coordinates ! ! depends only on electronic and nuclear coordinates

It is actually not possible to factorize rigorously the second integral into an electronic and a vibrational part but an approximation can be used

z " !"x,y,z

z%

" !"x,y,z

" $i,rot%z $j,rot%'rot%" $i,el%$i,vib% $j,el%$j,vib%'vib%'el

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The ( can be expanded in series with respect to the vibrational coordinates Qi

Thus:

# e $ !

i#1

3%N&6

Qi e%Qi $

1

2% !i,j#1

3%N&6 2%

Qi%Qj e%Qi%Qj $ ...

" !"x,y,z

" $i,rot%z $j,rot%'rot ) # Pure rotational selection rules ! !i,el"e"!j,el"%el"! !i,vib"!j,vib"%vib Electronic selection rules

! !i"1

3#N$6" %i,el# Qi e

#%j,el#)el#" %i,vib#Qi#%j,vib#)vib# Vibrational selection rules

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For pure rotational transitions:

So, the 2nd and 3rd integral reduces to z " 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)

!i,el " !j,el and !i,vib " !j,vib

For vibrational transitions:

the 1st is non-zero only if " 'J=1 (etc)

the 2nd is always 0 as vibrational functions are ortho-normal the 3rd is non-zero if the vibration creates a dipole and if

" 'v=1

" !"x,y,z

" $i,rot%z $j,rot%'rot%" $i,el%$i,vib% $j,el%$j,vib%'vib%'el

! !i,el"e"!j,el"%el"! !i,vib"!j,vib"%vib

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" !"x,y,z

" $i,rot%z $j,rot%'rot ) #" $i,el%e%$j,el%'el%" $i,vib%$j,vib%'vib * !i"1

3%N+6" $i,el% Qi e

%$j,el%'el%" $i,vib%Qi%$j,vib%'vib$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 is called Franck-Condon factor and scales the intensity of the transition

! !i,vib"!j,vib"#vibFinally, even if the 2nd term is 0 for symmetry here can be vibronic transition allowed when the 3rd integral is non-zero

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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 Einsteins coefficients

! Selection rules