Chemistry 125: Lecture 57March 2, 2011

Spectroscopy Electronic & IR Spectroscopy

Normal Modes:Mixing and Independence

This

For copyright notice see final page of this file

Spectroscopy forStructure and Dynamics

“Sunbeams..passing through a Glass Prism to the opposite Wall, exhibited there a Spectrum of divers colours”

Newton (1674)

“Specters or straunge Sights, Visions and Apparitions” (1605)

O.E.D.

Electronic (Visible/UV) e.g. F&J sec. 12.7-12.8 pp. 533

Vibrational (Infrared) e.g. F&J sec. 15.4, pp. 707-713

NMR (Radio) e.g. F&J sec. 15.5-15.9, pp. 713-749

“Atom in a Box”can be used to show:

(1) Spectral transitions for H atom (levels, energy, wavelength)

(2) Static shift of e-density from mixing 2s with 2p (same energy)

(3) Oscillation of e-density from mixing orbitals with different energy because of change in relative phase* with time (add, then subtract).

(b) “Breathing” from mixing 1s with 2s. (no interaction with light)

(a) Oscillating “dipole” from mixing 1s with 2p. (makes or interacts with light)

* This is a feature of time-dependent quantum mechanics, where the (complex) phase of a wavefunction changes at a rate proportional to its energy. When

energies of the components differ, their relative phases vary in time.

++

+

1s 2p

(1s + 2p)2

superposition e-density

time-dependent

Oscillation frequency given by the energy difference between

1s and 2p

Time-Dependence Footnote A time-dependent wavefunction looks just like the spatial s we have been talking about, except that it is multiplied by eit = cos(wt) + i sin(wt), where i = (-1), is the energy (in

frequency units) of the spatial wavefunction and t is time. In many cases this makes no difference, because when you “square” the wave function you get eit e-it = 2. BUT when a problem involves actually mixing two states of different energy, one considers a wavefunction of the form eit + eit . If 1 and 2 are different, this means that the two spatial functions cycle in- and out-of-phase with one another. If at a certain time they add, at a time 0.5/(1-2) later they will subtract. e.g. (1s+2pz) will become (1s-2pz).

* This is different from the mixing involved in forming hybrids or LCAO-MOs, where we just try to guess the best shape for an orbital of one particular energy for a molecule by analogy with known solutions for a simpler situation (atoms).

*

This is the source of the oscillation we observe when superimposing functions of different n using Atom-in-a-Box.

time

cos

++

+

1s 2p

Oscillating dipole has“oscillator strength”

interacts with /generates / absorbs

light

(1s + 2p)2

superposition e-density

time-dependent

1s - 2p transitionis “allowed”

Oscillation frequency given by the energy difference between

1s and 2p

+

+

+

1s 2s

(1s + 2s)2

superposition e-density

time-dependent

Symmetrical “breathing”e-density deformation has

no “oscillator strength”does not interact

with light’s E-field.

1s - 2s transitionis “forbidden”

Pulsing frequency given by the energy difference between

1s and 2s

:

n-*

n-* Transitions ofOrganic “Chromophores”

:C X+

+ -

- Oscillating electric field wags electrons

up and down by mixing n with *.

:

n+*

The large energy gap between n and * makes this transition occur at high

frequency (in the ultraviolet).

:

n-*

R

n-* Transitions ofOrganic “Chromophores”

:C X+

+ -

- Oscillating electric field wags electrons

up and down by mixing n with *.

With sufficient “conjugation” the * LUMO energy shifts close enough to n

that the transition is at visible wavelength.e.g. the retinaldehyde imine

of rhodopsin, which is the visual pigment in our eyes.

+

+ -

-+

+ -

-

* mix approaches energy of 2p orbital

During work on the synthesis of Vitamin-A a Palladium-Lead catalyst was developed, with which one can hydrogenate a triple bond without attacking double bonds already present in the starting material or those created by the hydrogenation.

Helvetica Chimica Acta, 35, 447 (1952)

OPP OPPOPP

PPO

-Carotenyne

-CaroteneCCH

RR

HH2 Pd/Pb

hn D

CCH R

R H

Autumn Scarlet(?) Tanager

©Birdwatchers Digest

Early Fall

isozeaxanthin-caroteneretinal

O

canthaxanthin

O: isolated

O: conjugated

Summer Scarlet(!) Tanager

with

kin

d p

erm

issi

on

of

Llo

yd S

pita

lnik

Graph of a Spectrum (IR of Paxil)

(1) Color (wavelength)

(2) Molecular Energy Gap

(3) Molecular Vibration Frequency

(1) LightIntensity

(2) Light-InducedOverlap

(3) Light’s“Handle”

(changing dipole)

(1) Experiment

(2) Quantum Mechanics

(3) Classical Mechanics

Meaning of Axes :

Infrared Spectroscopy

Using Light to Fingerprint molecules, to identify Functional Groups,

Infrared Spectroscopy

.Using Light to Fingerprint molecules , ,,,,,,

and to use molecular dynamics to study Bonding and whether Atoms are linked by “Springs”

€

x = h sin(ωt)

€

dx

dt= hω cos(ωt)

€

d2x

dt 2= a = −hω2 sin(ωt)

€

a

x= −ω2

€

ω 2 =f

m

What Makes Vibration Sinusoidal?

€

a

x=

− f

m

€

a = F mNewton

Hooke F = - fx

€

ω =f

m

FrequencyConstant!independent of amplitude 2h

€

=−ω 2

displacement frequency

velocity

acceleration

(Text Fig12.6)

-fx

(half) amplitude

FrequencyConstant!independent of amplitude 2h

Hooke, of Spring (1678)

© National Maritime Museum, Greenwich, LondonHarrison’s Marine Chronometer (1761)

Frequency

For atoms f should be

Bond Stiffness.(1, 2, 3?)

m is mass

or, for free diatomic, “reduced” mass

is dominated by the smaller mass!

When Hooke’s Law Applies:

H-C = 1 12

1 + 12= 0.9

12 12

12 + 12C-C = = 6.0

C-Cl = 12 35

12 + 35= 8.9

H-X stands apart

C-O = 12 16

12 + 16= 6.9 m1 m2

m1 + m2 =

√ fm

C-H sqrt (1/0.9)

C-O sqrt (1/6.9)

C=O sqrt (2/6.9)

~3000/cm ; 1014Hz

~1100 ~3 x 1013

~1500

~1900

(Cf. Eyring)

C=N sqrt (3/6.5)

Quartz Crystal Microbalance can weigh a monolayer of adhering molecules

(e.g. H2 + H2C=CH2 / Pt)

Possibility of effective independence

Coupled Oscillatorsillustrate:

Complexity

“Normal” mode analysis

Phase of mixing

Coupled Oscillators

Simple2 = f/m

Coupled Oscillators

Coupled to Frozen Partner2 = (f +s)/m

Simple2 = f/m

Coupled Oscillators

In-Phase Coupling

2 = 2f/2m = f/m

Simple

Coupled to Frozen Partner2 = (f +s)/m

2 = f/m

Coupled Oscillators

2 = 2(f +2s)/2m = (f +2s)/m

Out-of-Phase Coupling

Simple

Coupled to Frozen Partner

In-Phase Coupling

2 = (f +s)/m

2 = f/m

2 = 2f/2m = f/m

•ip

oopcoupled

isolated

In such “Normal” Modes all atoms

oscillate at the same frequency

oopip oop + ip

Coupled Oscillators

Superposition of Two Normal Modes of

different frequencyVibration switches between

oscillators as the two modes beat in- and out-of-

phase

Out-of-Phase Coupling

In-Phase CouplingIn such

“Normal” Modes all atoms

oscillate at the same frequency

2 = 2(f +2s)/2m = (f +2s)/m

2 = 2f/2m = f/m

ipoop

Very Different Oscillators are ~Independent

Vibration remains localized when coupling is weak

compared to -mismatch

ip

oopcoupled

low

high

oop + ip

A General Molecule of N Atoms has 3N Independent Geometric Parameters.

(e.g. as Cartesian Coordinates)

or3 to Fix Center of Mass

3 to Fix Orientation

3N-6 for Internal Vibrations(Normal Modes)

3N-6 Mixed-up Normal Modes sounds hopelessly complex.

(though good for “fingerprint”)

(Cf. Energy-match / Overlap)

butmixing requires:

Frequency Match&

Coupling Mechanism ip

oopcoupled

isolated

Butane C4H10

3 x (4 + 10) = 42 degrees of freedom

- 3 (translation) - 3 (rotation) = 36 vibrations

C4 : 3 stretch, 2 bend, 1 twist

10 C-H : 10 stretch, 20 bend or twist

Mixed (according to frequency-match / coupling) into 36 normal modes.

C8 Straight Chain Hydrocarbons

OctaneC8H18

C-H stretch

C-CH3 umbrella+ C-C stretch

CH2

rockCH2 wag

CH2 scissors

26 atoms 72 normal modes (not all IR active)

C-H stretch

“Breathing” gives no net dipole change - no IR peak

Half ofC4H10’s tenC-H stretch

normal modes have no “handle”

E(t) helps push 8 H in and out

E(t) helps push 4 Hs up and down

Timing has been disabled on this slide so you can step back and forth with the arrow keys to study vibrational modes.

End of Lecture 57March 2, 2011

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