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
Copyright © J. M. McBride 2011. Some rights reserved. Except for cited third-party materials, and those used by visiting speakers, all content is licensed under a Creative Commons License (Attribution-NonCommercial-ShareAlike 3.0).
Use of this content constitutes your acceptance of the noted license and the terms and conditions of use.
Materials from Wikimedia Commons are denoted by the symbol .
Third party materials may be subject to additional intellectual property notices, information, or restrictions.
The following attribution may be used when reusing material that is not identified as third-party content: J. M. McBride, Chem 125. License: Creative Commons BY-NC-SA 3.0