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Transcript
PG510 Symmetry and Molecular Spectroscopy Lecture no. 6 Molecular Spectroscopy: General Concepts Giuseppe Pileio 1 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 2 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 3 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 4 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 5 be appreciated Electromagnetic spectrum 6 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 17 n V I S 15 nuclear vibration molecular rotation nuclear spin (IR and Rot Raman) IR Rotational IR NMR MW 12 Frequency 10n HZ r.f. 9 7 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 17 V I S 15 EV ER IR MW 12 Frequency 10n HZ ES r.f. 9 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#$E!kT where nj and ni is the number of molecules which are actually in the state j and i, respectively, while gj and gi is the respective degeneracy of those states (i.e. how many energy levels have the same energy value) 11 nj ni ! gj gi 'E=10cm-1 i.e. in the MW region "e#$E!kT gj ! gi ! 1 'E=1000cm-1 i.e. in the IR region 'E=10000cm-1 i.e. in the UV region 12 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!# 13 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 14 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 15 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$ 16 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 17 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 18 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 19 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 20 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 21 " '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 22 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 23