19_01fig_PChem.jpg

Spectroscopy

18_12afig_PChem.jpg

Rotational Motion

Center of Mass

Translational Motion

r1

r2

2

ˆ ˆ ˆ2

Hm

2

ˆ ˆ ˆ2

HI

L L

22ˆ ˆ ˆ ( )

2 2 eq

kH

r r

Motion of Two Bodies

Each type of motion is best represented in its own coordinate system best suited to solving the equations involved

k

2 22 2

1 2 1 21 2

ˆ ˆ ˆ ˆ ( , )2 2

H Vm m

r r

RcInternal coordinates

Cartesian

Internal motion (w.r.t CM)

Motion of the C.M.

2r

1r

Origin

rVibrational Motion

18_11fig_PChem.jpg

( ) ( )E K r V r r

Simple Harmonic Motion

( ) ( )0

dE dK r dV r

dt dt dt r

2ˆ( ) ( )2 e

kV r r r

k

Hooks Law

Max

Min

Max

Min

K

V

Still

fast

Still

Conservation of Energy

22ˆ ( )

2K r

Kinetic Energy

Potential Energy

18_01fig_PChem.jpg

22ˆ ˆ( ) ( )

2H r V r

k

2 30 1 2 3

ˆ( ) ( ) ( ) ( ) ( )ne e e n eV r c c r r c r r c r r c r r

( ) ( )ˆ( )!

n ne e

nn

d V r r rV r

dr n

2

2

( )ed V rk

dr

Odd OddEven

SymmetricCan be approximated by a quadraticHarmonic Approx.

Hamiltonian of a Diatomic

22 2ˆ ( ) ( ) ( )

2 2e e

kH r V r r r

r

Vibrational Wavefunctions2 2

22

( ) ( ) ( )2 2 e vib

d kr r r E r

dr

2

24( ) ( )

2 !

r

nn

er H r

n

2

3/2 3

2 4

1 0

2 1

4 2 2

8 12 3

16 48 12 4

n

r n

rH

r

r r

r r

Hermite polynomials

k

Gaussian Tunneling

Oscillation

Highly excited state n=12

19_02tbl_PChem.jpg

Vibrational Spectroscopy

D(t)

r(t)

Band structure

19_10fig_PChem.jpg

3 6N

Polyatomic VibrationsFor an N atom molecule:

3 CM Coordinates (X,Y,Z)

3 Axes of Rotation

Remaining coordinates are Vibrational modes

Normal modes have a characteristic frequency, i,determined by

the motion they represent, and are independent of each other3 6 1

2

N

vib i ii

E n

Total of 3N Coordinates (x,y,z)

19_04tbl_PChem.jpg

Vibrational Spectra of Molecules

Modes of Vibration

Correlation Tables

Vibrational Spectroscopy

E(n)

0

1

234

5

2

3

2

5

2

7

2

9

2

11

2

1

2nE n

n1n

1n n nE E E

Selection Rule

1 11

2 2n n

For perfect HarmonicBehaviour

2n n nE E E 2n 1st Overtone

2

2n n nE E E 3n 2nd Overtone

3

not exactly 2x due to anharmonicity

19_p08_PChem.jpg

Selection Rules and Line Intensities

Boltzmann Distribution

0

1

0

b

E

k TNe

N

At ambient T, most are in the ground state:

ex) k = 250 N/m, = 2x10-26 kg and

E(n)

0

1

234

5

2

3

2

5

2

7

2

9

2

11

2

94 %

5.4 %

0.3%

0.06%

0.003%

3n

5n

*( ) ( )n i i nP x x x dx

x

1

0

0.054N

N

1n

19_02tbl_PChem.jpg

Coupled ModesMode i

Mode j

Combination Mode 1in 1jn

i jn n i jE E E

Difference Mode 1in 1jn

i jn n i jE E E

Fermi Resonance in p 1jn

0i jn n i jE E E p

Causes linebroadening, and splitting

19_18fig_PChem.jpg

InstrumentationScanning

Grating Orientation ()

Abs

orpt

ion

0

( )log

( )

IA

I

I0

I

19_18fig_PChem.jpg

Instrumentation

FTSpectrum Inteferogram

The reference and sample beams are coherent, therefore they can interfere with each other. The phase of the reference beam can be modulated by changing the mirror position

Mirror Displacement