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 = hν (in eV or J)

Wave length: λ = c/ν = hc/E (in nm)

h = 6.6260×10-34 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: λ = 4 x 10-7 m = 400 nm

Wavenumbers: ω = 1/λ = 25000 cm-1

Wave numbers: ω = 1/λ (cm-1)

c= 2.99792 x 108 m/s

Basics: Atomic Hydrogen

with � �

���

2��� �

�

kin. energy pot. energy

�Ψ � Ψ

Schrödinger Equation

�1

2��� �

�� Ψ �� � 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=6

n ∞

� = - R��

��Energy levels:

Discrete transitions:

�

�= ��

�

��� �

�

���

Lα-

12

1.7

nm

Hα

–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

Hα (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

α

Lym

an

β

Balmer Series

Hydrogen 21cm line

Very Large Array (VLA)Arecibo Effelsberg

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

Lyα 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 = ∆λ/λ =λobs- λemit

λemit

The observed wavelength is shifted by: λobs = λemit + z λemit

Hydrogen

Lyα 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

Λ2S+1 (+/-)

S: spin quantum number

Λ: projection of orbital

angular momentum along

internuclear axis

(+/-): reflection symmetry

u/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

ν frequency2B 4B 6B 8B

1→0 2→1 3→2 4→3

P and R branch

P-Branch ΔJ=-1 R-Branch ΔJ=+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<B<C), the principle axes are orthogonal and they

diagonalize the moment of inertia matrix

%/ ≤ %1 ≤ %2

Rotational constants:

$ � ��

2%13 �

��

2%24 �

��

2%2

Class 1: linear molecules (IA=0, IB=IC)

H2, C2, C4, …

& � $'(' � 1)

Class 2: spherical molecules (IA=IB=IC) & � $'(' � 1)CH4, C60, …

Class 3: symmetric top molecules (IB=IC or IA=IB)

H3+, NH3, …

& � $' ' � 1 − 3 − $ 52 & = $' ' + 1 − (4 − $)5�

Class 4: asymmetric top molecules (IA≠IB≠IC)

Water, H2CO, etc …No simple general formula

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

E2

hν

B1

2ρ

B2

1ρ

A2

1

ρ: energy density of the radiation field

AbsorptionSpontaneous

emission

Induced

emission

Equilibrium:

absorbed photons = emitted photons

$��6 ν � 3�� 8�� $��8�6 ν

6 ν = /��9�

1��9�:1��9�

Solving for ρ(v):

Using Boltzmann radiation law:

9�

9�= ;<

;=>:

?@

AB 8�= 8�;=

;<>

?@

AB

6 ν = /��9�

1��9�;=;<>?@AB:1��9�

�

C��D��

;=;<D��D��

>?@AB:�

(1)

(2)

(3)

Insert (3) into (2):

(4)

Einstein Coefficients

E1

E2

hν

B1

2ρ

B2

1ρ

A2

1

ρ: energy density of the radiation field

AbsorptionSpontaneous

emission

Induced

emission6 ν �

3��$��

;=;<

$��$��

>?EAB � 1

(4)

Planck’s thermal radiation law

6 ν = 8Gν�

HIℎν

>?EAB − 1 (5)

8Gν�HI

ℎν>?EAB − 1

=3��$��

;=;< $��$�� >

?EAB − 1

Equating (4) and (5):

(6)

$��= K�K�$�� 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

Σ Aiff

T =

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 transitions strong:

Transition dipole moment

A21 = µ212

H2

H2O

3ε0 hc3

2ω213

Electric Dipole transitions: µ21 ≈ e a0 A21 ≈ e2a02

Dipole moments are measured in Debye, molecules with permanent dipole moments typically have 1-3D

Electric Quadrupole transitions are weaker by a factor (e a0 λ)2/(e a0)4 ≈ 108

Magnetic dipole transitions scale with the Bohr magneton (eh/4πmc),

They are wekaer by α2 ≈ 10-5

Selection rules

Vibrations: vvvv1 1 1 1 , v, v, v, v2 2 2 2 ,,,, l l l l ; ; ; ; Nuclear Spin I

Rotation: total motional angular momentum Jprojection of J onto the symmetry axis K

|vvvv1111, v, v, v, v2222,l,l,l,l> |J,K>

Dipole selection rules

∆I =0,∆J = 0, +1,-1; J=0 J=0,

∆K = 2n+1,

∆(K-l ) ) ) ) = 3n, (∆G=3n).

Example: H3+ Selection rules

Example transition strengths

Type A21 (s-1) Example λ A21 (s-1)

Electric dipole

UV 109 Lyα 121.6 nm 2.4 x 108

Visible 107 Hα 656 nm 6 x 106

Vibrational 102 CO 4.67 µm 34.0

Rotational 10-6 CS 6.1 mm 1.7 x 10-6

Forbidden

Opt. (el. Quadrupole) 1 [OIII] 436.3 nm 1.7

Opt. (magn. Dipole) 102 [OIII] 500.7 nm 2 x 102

Hyperfine HI 21 cm 2.9 x 10-15

Some remarkable transitions in Hydrogen

A = 2.9 x 10-15 s-121 cm line

Atomic H / electron spin flip

Ortho-para transiton in molecular hydrogen

Nuclear spin flip

A = 6.2 x 10-14 yr-1

Some remarkable transitions in Hydrogen

Molecular Hydrogen J=2 → J=0

Rotational Quadrupole transition

ground state

Stat.

weight

A ≈ 2.94 10-11 s-1

Space is big. You just won't believe how vastly, hugely,

mind- bogglingly big it is. I mean, you may think it's a

long way down the road to the chemist's, but that's

just peanuts to space.

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Literature

Molecules: Haken/ Wolf

“Molecular Physics and Elements of Quantum

Chemistry:Introduction to Experiments and Theory”

Springer

Herzberg, Gerald“Molecular Spectra and Molecular Structure: Spectra of Diatomic

Molecules”

Krieger

Khristenko, Maslov, Shevelko“Molecules and their spectroscopic properties”

Springer

Molecular

Astrophysics:

Tennyson, Jonathan“Astronomical spectroscopy”

World Scientific

Tielens, A.G.G.M.“The physics and chemistry of the interstellar medium”

Cambridge University Press