Molecular Properties and Spectroscopy

Almost all of the information we have about the Universe comes from the

study of electromagnetic radiation (light) Spectroscopy

Isaac Newton

Newton used the term Spectrum in context

with his experiments on the color of light:

The Beginning: Newton (1675)

Fraunhofer lines (1814)

Joseph von Fraunhofer

Gustav Robert Kirchhoff

18.03.1824-17.10.1887

Kirchhoff and Bunsen

Robert Wilhelm Bunsen

31.03.1811-16.08.1899

Spectroscopy pioneers in Heidelberg

The Birth of Spectral Analysis: Kirchoff (1860)sodium

Electromagnetic Spectrum

400 nm 800 nm

photon energy

1021 1018 1016 1014 1012 1010 108 106 104

Frequency (Hz)

Unit Conversion

Energy E of a photon: E = h (in eV or J)

Wave length: = c/ = hc/E (in nm)

h = 6.626010-34 Jsh = 4.1357x10-15 eVs

J eV cm-1 K

1 J 1 6.24146 x 1018 5.03404x1022 7.24290 x 1022

1 eV 1.60219 x 10-19 1 8.06548x103 1.16045 x 104

1 cm-1 1.98648 x 10-23 1.23985 x 10-4 1 1.43879

1 K 1.38066 x 10-23 8.61735 x 10-5 6.95030 x 10-1 1

Example:

Level spacing: 3.1 eV = 4.9667 x 10-19 J

Photon frequency: 7.4957 x 1014 s-1

Wavelength: = 4 x 10-7 m = 400 nm

Wavenumbers: = 1/ = 25000 cm-1

Wave numbers: = 1/ (cm-1)

c= 2.99792 x 108 m/s

Basics: Atomic Hydrogen

with

2

kin. energy pot. energy

Schrdinger Equation

1

2

0

Hamiltonian

Eigenvalue equation:

Discrete solutions with quantum numbers:

n : principle (energy) quantum number 1, 2, 3 .

: angular momentum 0, 1, 2, , (n-1)m : magnetic quantum number - , 1 , . ,

Atomic Hydrogen: Transitions

O eV

-13.6 eV n=1

n=2

n=3n=4 n=5

n=6n

= - R

Energy levels:

Discrete transitions:

=

L-

12

1.7

nm

H

6

56

.3 n

m

n=3 n=2 @ 656.3 nm

responsible for red glow in

H II regions containing

ionized H atoms

Lym

an

Ba

lme

r

Pa

sch

en

-3.4 eV

-1.9 eV

H (656nm) emission (red glow) in HII regions

Horsehead nebula

Atoms with more than one Electron

L2S+1

L= l1+l2

J

J=L+S

P3

L= 0+1=1

1

J=1+1=1

Notation

Atomic Hydrogen Observations

Emission, V630 Saggitari,Schmidtobreick A&A 432, 199 (2005)

Danish telescope, La Cilla (Chile)

Far Ultraviolet

Spectroscopic Explorer (FUSE)

operated 1999-2007

G 231-40 (white dwarf) Hebrad A&A 394, 647 (2002)

Lym

an

Lym

an

Balmer Series

Hydrogen 21cm line

Very Large Array (VLA)Arecibo Effelsberg

Doppler shift due to relative motion: v/c = /

Ly absorption, observed

with Hubble Space Telescope

at z =2.5

Burles and Tytler, ApJ 507,

732 (1998)

Example:

Cosmological Redshift

The redshift z is defined as

z = / =obs- emitemit

The observed wavelength is shifted by: obs = emit + z emit

Hydrogen

Ly shifted

to 426 nm

Molecular Properties

Diatomics (2 nuclei) Polyatomics (>2 nuclei)

Homonuclear (e.g. H2)

Heteronuclear (e.g. OH, CO)

Molecular binding energies are relatively small (1-5 eV),

smaller than ionization energies (>10 eV)

Molecules are easily destroyed and are found in cooler,

less ionized environments.

Present in objects with T < 8000 K Present in objects with T < 4000 K

The Born-Oppenheimer Approximation

The Born Oppenheimer Approximation assumes that electrons in a molecule

move much faster than the nuclei, and adapt instantaneously, finding the

lowest potential energy for each nuclear configuration. Therefore, it is

possible to calculate an electronic energy for each nuclear configuration,

considering the nuclei frozen.

The nuclear Hamiltonian is:

nuclear masses electronic potential

Coulomb potential

between the nuclei

H2 Molecular Orbitals

Electron orbitals of H atoms

Antibonding orbital

Energy [eV]

R Internuclear

Distance [a0]

Bonding orbital

H2 potential energy curves

-4.5 eV

Re = 1.4 a0

1

2

Molecular Vibration: Harmonic Oscillator Potential

Anharmonicity

leads to lowering

of energy levels

1

2

Harmonic Oscillator Potentail

! " 1

2#

Energy levels

transitions

Zero point energy ! 1

2#

H2 electronic potential curves

Internuclear distance Ra [a0]

Electronically

excited states

Bonding orbital

Antibonding

orbital

Notation

2S+1 (+/-)

S: spin quantum number

: projection of orbital angular momentum along

internuclear axis

(+/-): reflection symmetryu/g: parity

(g/u)

Energy [eV]

R Internuclear

Distance [a0]

Electronic and Vibrational Excitation

-4.5 eV

Pure

electronic

transition

Transition

With vibronic

coupling

v=0v=1

v=2

v=0v=1

v=2

Polyatomic Molecules: Example H3+

Simplest Polyatomic molecule

Consists of 3 protons and 2 electrons

3 internuclear distances

3 vibrational degrees of freedom

Electronic Potential curve is a Hyper-Surface of 3 coordinates,

can not be plotted!

Normal Modes of Vibration

Normal modes:

independent modes of vibration

diagonalize the Molecular Hamiltonian

linear combinations of the internuclear distances

Finding Normal Modes: Use Molecular Symmetry / Group Theory

The Normal Modes of H3+

Degenerate (the same frequency)

bending mode 2breathing modebending mode 1

+90 phase -90 phase

The Normal Modes of H3+

H3+ vibrational level scheme

Translational degrees of freedom

Rotational degrees of freedom

Vibrational degrees of freedom

Linear Non-linear

3 3

2 3

In general molecules have 3N degrees of freedom

3N -5 3N -6

Vibrations bear a lot of Potential For Complexity

But: Symmetry helps!

H3+ : 3 vibrational degrees of freedom 2 distinct normal modes

C60: 174 vibrational degrees of freedom 46 normal modes

Rotations: the Rigid Rotor

m1

m2center of mass

Energy levels

rotational constant

$

2%

& = $'(' + 1)

Diatomic or Linear Polyatomic Molecules

rotational

quantum

number

Moment of inertia

% =*+,,,

r1

r2

Large and heavy molecules have

small rotational constants!

Rigid Rotor Levels and Transitions

Energy levels

& $'(' 1)

Selection rule:

' .1

& ' 1 & ' $ ' 1 ' 2 '(' 1)

Transitions:

2$ ' 1

frequency2B 4B 6B 8B

10 21 32 43

P and R branch

P-Branch J=-1 R-Branch J=+1

P and R branch

transmission

100

0

Rotation-Vibration Spectrum of HBr

R-BranchP-Branch

General Rotational Structure of Polyatomic Molecules

A

B

C

Any arbitraryily shaped solid body has three principal axes of

rotation (A

Energy Level Diagram of H2O

Overview

Electronic

Transitions:

E = 1-15 eV

Visible-UV

Vibrational

Transitions:

E = 0.1-1 eV

Infrared

Rotational

Transitions:

E = 0.01-0.1 eV

(sub)-Millimeter

Einstein Coefficients

E1

E2h

B1

2

B2

1

A2

1

: energy density of the radiation field

AbsorptionSpontaneous

emission

Induced

emission

Equilibrium:

absorbed photons = emitted photons

$6 3 8 $86

6 = /9

19:19

Solving for (v):

Using Boltzmann radiation law:

9

9= ;:

?@

AB 8= 8;=

;

?@

AB

6 = /9

19;=;?@AB:19

CD

;=;?@AB:

(1)

(2)

(3)

Insert (3) into (2):

(4)

Einstein Coefficients

E1

E2h

B1

2

B2

1

A2

1

: energy density of the radiation field

AbsorptionSpontaneous

emission

Induced

emission6

3$

;=;?EAB 1

(4)

Plancks thermal radiation law

6 = 8G

HI

>?EAB 1 (5)

8GHI

>?EAB 1

=3$

;=;< $$ >

?EAB 1

Equating (4) and (5):

(6)

$= KK$ 3= LMNEOPO $ (7)

Lifetimes and Einstein A:

The lifetime T of an initial state i

Is given by the sum of the Einstein

coefficients summed over all final

states

1

AiffT =

Einstein A21 coefficient

Einstein B12 coefficient

Einstein B12 coefficient

Absorption cross section

Line strength

Transition dipole moment

Oscilator strength

Hillborn,

Am. J. Phys.

50, 982 (1982)

Level 2 (up)

Level 1 (low)

What makes tr