Post on 13-Jun-2022
4/30/18
1
Electronic Spectroscopy
Chapter 13Sections 6, 7, & 9
CHEM 4502: Introduction to Quantum Mechanics and Spectroscopy
Monday, April 30th
Mona Minkara
Electronic spectra contain electronic, vibrational, and rotational information
When a molecule absorbs certain types of radiation, they undergo certain types of electronic transitions
Microwave Radiation Γ Rotational
Infrared Γ Vibrational
Visible and Ultraviolet Γ Electronic
According to the Born-Oppenheimer approximation, electronic energy is independent of vibrational-rotational energy
πΈ"#$#%& = π£)*& + πΊ π£ + πΉ π½= π£)*& + π£)/ π + 1
2β π₯)/π£)/ π + 1
2
2+ π΅"π½ π½ + 1 β π·8π½2(π½ + 1)2
Trends in Molecular Energy Level Spacings
Electronic transitionsare accompanied by Vibrational and Rotational transitions.
Electronic > Vibrational > Rotational >> (Translational)
McQuarrie, p. 508 & 499
Electronic Spectroscopy
πΈ"#$#%& = π£)*& + πΊ π£ + πΉ π½= π£)*& + π£)/ π + 1
2β π₯)/π£)/ π + 1
2
2+ π΅"π½ π½ + 1 β π·8π½2(π½ + 1)2
Selection rule for vibronic transitions (vibrational transitions in electronic spectra)Γ Ξπ = integralvalue
Because vibrational energies >> rotational energies, we can ignore the rotational terms from the above equation.
Vibronic transitions usually originate from the π = 0 vibrational state (most populated at room temp), so the predicted frequencies of an electronic transition is:
π£)$HI = π"/ +12π£)/
L β14π₯)/
Lπ£)/L β12π£)/
LL β14π₯)/
LLπ£)/LL + π£)/LπL β π₯)/Lπ£)/LπL(πL + 1)
4/30/18
2
Electronic Spectroscopy
π£)N,N = π"/ +12π£)/
L β14π₯)/
Lπ£)/L β12π£)/
LL β14π₯)/
LLπ£)/LL
Harmonic oscillator approx.:
π·/ = π·N +12βπ£
Anharmonic oscillator approx.:
π·/ = π·N +12β π£/ β
12π₯/π£/
Zero-point energy of upper
state
Zero-point energy of lower
state
McQuarrie, p. 509
As πL, the vibrational quantum number of the upper state, has a higher value, the vibronic spacing becomes progressively smaller until the spectrum is essentially continuous
This electronic spectrum is due to πLL = 0 to πL = 0, 1, 2, β¦transitions. The set of transitions shown here is called an πL progression
π£)$HI = π£)N,N + π£)/LπL β π₯)/Lπ£)/LπL πL + 1 πL = 0, 1, 2, β¦
Electronic SpectroscopyMcQuarrie, p. 510
Example 13-6
Vibronic Transition π£)$HI/cm-1
0 β 0 39,699.10
0 β 1 40,786.80
0 β 2 41,858.90
Calculate π£)/L and π₯)/Lπ£)/L for the excited electron state of PN.
Analysis of electronic spectra yields data that would be difficult to find otherwise.
π£)$HI = π"/ +12π£)/
L β14π₯)/
Lπ£)/L β12π£)/
LL β14π₯)/
LLπ£)/LL + π£)/LπL β π₯)/Lπ£)/LπL(πL + 1)
π£)$HI = π£)N,N + π£)/LπL β π₯)/Lπ£)/LπL πL + 1 πL = 0, 1, 2, β¦
39,699.10 = π£)N,N40,786.80 = π£)N,N + π£)/L β 2π₯)/Lπ£)/L
41,858.90 = π£)N,N + 2π£)/L β 6π₯)/Lπ£)/L
πL = 0πL = 1πL = 2
π£)/L = 1103.3cm]1 π₯)/Lπ£)/L = 7.80cm]1
McQuarrie, p. 511 Electronic Spectroscopy: Franck-Condon Principle
What is the βselection ruleβ that governs which vibrational states will be observed in electronic absorption and emission spectra?
Franck-Condon Principle: The electronic excitation is much faster than nuclear motion, so the electronic transition will be βverticalβ (positions of nuclei initially unchanged).
As a result, if the 2 electronic states have similar equilibrium geometries, little vibrational excitation will be observed.
If their equilibrium geometries (bond lengths) are very different, a vibrational βprogressionβ will be observed, as in the I2 emission spectrum.
4/30/18
3
Franck-Condon Principle
An electronic transition can be depicted as vertical lines in the potential energy diagram because the nuclei do not move during this transition
After the electronic transition (which is accompanied by a vibrational transition), the nuclei relax so the R value of the minima shifts
McQuarrie, p. 512 & 513
Vibrational Normal ModesTo specify positions of N nuclei in a molecule requires 3Ncoordinates (called 3N βdegrees of freedomβ).
Vibrational Rotational Translational
Nuclear Motion: 3 Types
Translation of the molecule in space(motion of the center of mass)
Rotation of the molecule about its (fixed) center of mass (βRigid Rotatorβ)
Vibrations of the nuclei(moleculeβs center of mass fixed)
Degrees of Freedom (3N Total)
3
3 (nonlinear molecule)2 (linear molecule)
3Nβ6 (nonlinear)3Nβ5 (linear)
11
Vibrational Normal Modes of Polyatomic MoleculesH2O
CO2linear
nonlinearN = 3 (# atoms)3Nβ6 = 3normal modes
3Nβ5 = 4normal modes
The normal modes can be excited independently of each other.
So, the total vibrational energy is the sum: ( ) j
n
jjvib hvE
vib
nΓ₯=
+=1
21vj Β½
McQuarrie, p. 521
Review & Thank you!
Electronic > Vibrational >Rotational >> (Translational)
Degrees of Freedom
Molecule Vibrational Rotational Translational
Linear 3N β 5 2 3
Nonlinear 3N β 6 3 3