18.04.30 Electronic Spectroscopy Lecture

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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 à Rotational

Infrared à Vibrational

Visible and Ultraviolet à Electronic

According to the Born-Oppenheimer approximation, electronic energy is independent of vibrational-rotational energy

𝐸"#$#%& = 𝑣)*& + 𝐺 𝑣 + 𝐹 𝐽= 𝑣)*& + 𝑣)/ 𝜐 + 1

2− 𝑥)/𝑣)/ 𝜐 + 1

2

2+ 𝐵"𝐽 𝐽 + 1 − 𝐷8𝐽2(𝐽 + 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

𝐸"#$#%& = 𝑣)*& + 𝐺 𝑣 + 𝐹 𝐽= 𝑣)*& + 𝑣)/ 𝜐 + 1

2− 𝑥)/𝑣)/ 𝜐 + 1

2

2+ 𝐵"𝐽 𝐽 + 1 − 𝐷8𝐽2(𝐽 + 1)2

Selection rule for vibronic transitions (vibrational transitions in electronic spectra)à Δ𝜐 = integralvalue

Because vibrational energies >> rotational energies, we can ignore the rotational terms from the above equation.

Vibronic transitions usually originate from the 𝜐 = 0 vibrational state (most populated at room temp), so the predicted frequencies of an electronic transition is:

𝑣)$HI = 𝑇"/ +12𝑣)/

L −14𝑥)/

L𝑣)/L −12𝑣)/

LL −14𝑥)/

LL𝑣)/LL + 𝑣)/L𝜐L − 𝑥)/L𝑣)/L𝜐L(𝜐L + 1)

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Electronic Spectroscopy

𝑣)N,N = 𝑇"/ +12𝑣)/

L −14𝑥)/

L𝑣)/L −12𝑣)/

LL −14𝑥)/

LL𝑣)/LL

Harmonic oscillator approx.:

𝐷/ = 𝐷N +12ℎ𝑣

Anharmonic oscillator approx.:

𝐷/ = 𝐷N +12ℎ 𝑣/ −

12𝑥/𝑣/

Zero-point energy of upper

state

Zero-point energy of lower

state

McQuarrie, p. 509

As 𝜐L, 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 𝜐LL = 0 to 𝜐L = 0, 1, 2, …transitions. The set of transitions shown here is called an 𝜐L progression

𝑣)$HI = 𝑣)N,N + 𝑣)/L𝜐L − 𝑥)/L𝑣)/L𝜐L 𝜐L + 1 𝜐L = 0, 1, 2, …

Electronic SpectroscopyMcQuarrie, p. 510

Example 13-6

Vibronic Transition 𝑣)$HI/cm-1

0 → 0 39,699.10

0 → 1 40,786.80

0 → 2 41,858.90

Calculate 𝑣)/L and 𝑥)/L𝑣)/L for the excited electron state of PN.

Analysis of electronic spectra yields data that would be difficult to find otherwise.

𝑣)$HI = 𝑇"/ +12𝑣)/

L −14𝑥)/

L𝑣)/L −12𝑣)/

LL −14𝑥)/

LL𝑣)/LL + 𝑣)/L𝜐L − 𝑥)/L𝑣)/L𝜐L(𝜐L + 1)

𝑣)$HI = 𝑣)N,N + 𝑣)/L𝜐L − 𝑥)/L𝑣)/L𝜐L 𝜐L + 1 𝜐L = 0, 1, 2, …

39,699.10 = 𝑣)N,N40,786.80 = 𝑣)N,N + 𝑣)/L − 2𝑥)/L𝑣)/L

41,858.90 = 𝑣)N,N + 2𝑣)/L − 6𝑥)/L𝑣)/L

𝜐L = 0𝜐L = 1𝜐L = 2

𝑣)/L = 1103.3cm]1 𝑥)/L𝑣)/L = 7.80cm]1

McQuarrie, p. 511 Electronic Spectroscopy: Franck-Condon Principle

What is the “selection rule” 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 “vertical” (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 “progression” will be observed, as in the I2 emission spectrum.

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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 “degrees of freedom”).

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 (“Rigid Rotator”)

Vibrations of the nuclei(molecule’s center of mass fixed)

Degrees of Freedom (3N Total)

3

3 (nonlinear molecule)2 (linear molecule)

3N−6 (nonlinear)3N−5 (linear)

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Vibrational Normal Modes of Polyatomic MoleculesH2O

CO2linear

nonlinearN = 3 (# atoms)3N−6 = 3normal modes

3N−5 = 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

nå=

+=1

21vj ½

McQuarrie, p. 521

Review & Thank you!

Electronic > Vibrational >Rotational >> (Translational)

Degrees of Freedom

Molecule Vibrational Rotational Translational

Linear 3N – 5 2 3

Nonlinear 3N – 6 3 3