4. Molecular spectroscopy

Basel, 2008

4.4.5 Fluorescence radiation

CLSM micrographof convolaria

Fluorescence microscopy

The excited molecule:

- is subject to collisions with the surrounding molecules and gives up energy by decreasing the vibrational levels (vibrational relaxation) –radiationless decay.

- the rest of energy is released by emission of a photon > fluorescence occur at a lower frequency that of the incident radiation!

Fluorescence correlation spectroscopyA fluorescent labeled molecule in solution emits photons as long as it moves through a laser spot, those number depends on:

1. number of molecules (concentration)

2. diffusion time (mass of molecule)

3. spot size (instrumental parameter)

4. quantum yield (property of dye)

5. bleaching, triplet states (laser power and dye)

What can we learn by using FCS?- Diffusion time of fluorescent particles size

- Concentration of fluorescent molecules

- Molecular brightness number of dye molecules/diffusing particle

- Binding constant (8x diff. in MW, knowledge of brightness necessary) for protein-ligand interactions

FCS: example

1 10 100 1000 10000 1000001.0

1.5

2.0

G (t

)Time [µs]

SOD+Dye SOD+Dye+Polymer

Example: encapsulation of superoxide dismutase in nanocontainers

Diffusion times of fluorescent particles passing through the confocal volume are proportinal with the molecular mass of these particles:

G(τ)

Increasing diffusion time - mass

SOD - τD = 120SOD+ vesicles –

τD = 2.3 ± 0.5 ms

rkTDπηω

τ 64

21=

τD = diffusion time of the molecule

ω1 = radius of laser focus

η= viscosity of the solution

r = hydrodynamic radius of the molecule

T = absolute temperature

k = Boltzmann constant

NA = Avogadro number

ρ= mean density of the molecule

m = molecular mass of the moleculeANmr

πρ433 =

(4.17)

(4.18)

4.5 MW spectroscopy

x

E->

Domain:

- λ = 10-2 - 10-4 m

- ν = 1010 - 1012 Hz

- = 104 –106 cm-1ν~

Hypothesis: the molecule is a collection of mass points (for each atom one mass point), and can rotate as a whole (rigid rotor).

The moment of inertia around the centre of the mass:

2i

iixx xmI ∑= xi - coordinate x of the particle of mass mi

from the centre of the mass.

(4.19)

Rotor types

The molecule is considered as a rigid rotor:

• Spheric rotor: CH4

• Symmetric rotor: NH3

• Linear rotor: CO2, HCl

• Asymmetirc rotor: H2O

4.5.1 Moments of inertiaDiatomic molecules:

Triatomic linear molecules:

Spherical rotors:

2RmmmI BA=

( ) ( )222 '' RmRmmmRR

mmmI CA

ACA +++=

22 RmI A=If mA=mC, R=R‘ :

2

38 RmI A=

22 RmI A=

(4.20)

(4.21)

(4.22)

(4.23)

(4.24)

Symmetric rotors( )θcos12 2

// −= RmI A

( ) ( ) ( )

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦⎤

⎢⎣⎡ +++

++++−=⊥

21

22

cos21316'3'

cos21cos1

θ

θθ

RmRmmRmm

RmmmmRmI

ABAC

CBA

A

( )θcos12 2// −= RmI A

( ) ( )θθ cos21cos1 22 ++−=⊥ RmmmRmI BA

A

2// 4 RmI A=

22 '22 RmRmI CA +=⊥

I⏐⏐

I⊥ I⊥

(4.25)

(4.26)

(4.27)

(4.28)

(4.29)

(4.30)

4.5.2. Rotational energy levels

I = mR2

m = mAmB/(mA+mB)

EJ = B J(J+1)

B = h2/(8π2I)( )1

8 2

2

+= JJI

hEJ π

(4.32)(4.31)

(4.34)

(4.33)

(4.35)

Rotational energy levels:

Example: CCl4 has I = 4.85x10-45 kgm2 , and thus a rotation constant B = 1.04x10-22 J = 5.24 cm-1

Rotational energy levels of a diatomic rigid rotor are:

J = 0, 1, 2, 3...J = rotational quantum number

B - rotational constant of the molecule

4.5.3 Rotational transitions

J = 0

J = 1

J = 2

J = 3

J = 4

E /BJ

0

2

6

12

20

4

6

8 ∆J 0 -> 1 1 -> 2 2 -> 3 3 -> 4 4 -> 5

2B 4B 6B 8B 10Bcm-1

´ 2

The transition energy between two rotational levels (spherical rotors, orlinear rotors), when J > J – 1 is:

( ) ( ) BJJBJJBJEJ 211 =−−+=∆ (4.36)

For large molecules, the inertia momentul increases > B decreases > ∆EJ decreases !

Rotational transitions for linear rotors

Linear rotors: HCl, CO2, C2H2...

EJ = B J(J+1)

Rotational energy levels of a linear rigid rotor are:

J = 0, 1, 2, 3...

Centrifugal distortion: when the molecules is rotating, the atmos are subject to centrifugal forces which induce distortions of the molecular geometry change in I !

Rotational energy levels for a diatomic molecule (centrifugal distortion induces a stretch of the bond):

EJ = B J(J+1) - DJ2(J+1)2 D - centrifugal distortion constant

A high D value indicates a high stretch of the bond

(4.37)

4.5.4 Selection rules: rotational transitions

1. The molecule must have a permanent electric dipole moment.

Inactive molecules (in normal conditions do not have a pure rotationalspectrum):

• Homonuclear diatomic molecules

• Symmetric linear molecules

• Spherical rotors if they are not significantly centrifugal distorted.

A molecule gives a pure rotational spectrum only if it is polar (it possess a fluctuating dipole momentul when rotating)

Active molecules: OCS, H2O, NO, NO2...

2. Specific rules for linear molecules: ∆J = ± 1 and ∆MJ = o, ±1

MJ – quantum number of the projection of J on Oz axis.

Rotational spectra

Rotation spectrum of the Orion nebula, showing the fingerprint of diatomic and poliatomic molecules present in the interstellar cloud.

Wavenumbers of the allowed J > J+1 absorptions for a linear rotor are:

( ) ( )121~ +=←+ JBJJν J = 0, 1, 2, 3...

Pure rotational spectrum: a serie of lines with wavenumbers 2B, 4B, 6B, ... and of separation 2B. > Determine the moment of inertia perpendicular to the principal axis of the molecule.

Example:

(4.38)

x

E->

4.6 IR spectroscopyMolecules vibrate in a a large number of different modes:

- benzen has 30 different vibration modes (swelling or shrinking of the ring, and buckling in distorted shapes)

- one protein has thousends of different ways of vibration (twisting, stretching, buckling in different regions).

λ ≈ 10-6 – 10-5 m

ν ≈ 1013 - 1014 Hz

E ≈10-20 – 10-19 J

Electromagnetic waves can induce a change in the dipole moment of molecules (IR waves can resonate with the molecules vibrations, as theyhave comparable frequency).

ν~ ≈ 300 – 3000 cm-1

4.6.1. Vibrations of diatomic moleculesAs vibrations in a large molecule can be considered as the summ of motions of each two bonded atoms, we describe first the diatomic molecules.

a. Harmonic potential energy curve (see chapter 3.2.3)

( )221)( errkrV −=

k - force constant of the bond

r - internuclear distance

re – internuclear distance at equilibrium

Solutions of the Schrödinger equation for a harmonic oscillator:

νhE ⎟⎠⎞

⎜⎝⎛ +=

21vv

v – vibration quantum number v = 0,1,2,3... µπ

k21v =

µ - efective mass, or reduced mass

BA

BA

mmmm+

=µ

(4.39)

(4.40)

(4.42)

(4.41)

4.6.2 Low vibrational states

Ev = hν (v+1/2) v = 0, 1, 2, …

v = 0

v = 1

v = 2

v = 3

v = 4

Rre

V(R)

1/2hν

3/2hν

5/2hν

7/2hν

9/2hν

Vibration energy levels in the harmonic aproximation (in the regions close to the equilibrium)

Low vibrational states

Isotopic effect: substitution of an atom in a bond by different isotopes induces a change in vibrational frequency > change in vibrational energy, Ev

Heavier isotopes > decrease in vibrational frequency (see reduced mass formula 4.42)

v = 0 > Ev = hν

zero point energy

(4.43)

4.6.3 IR- Selection rules

Selection rules for IR transitions:

• Change of dipole moment during the transition (∆µ ≠ 0)

• ∆v = ± 1

During the vibration of the molecule the dipole momentum is oscillating, too, and if it changes, will interact with the electromagnetic radiation (IR domain).

ννν hhhE =⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ +=∆

21v

23vv

At T = 293K, the most important vibration transition: v = 0 to v = 1 energy levels.

Examples:

• CO group of a peptide link has k = 1.2 kN m-1. Thus it absorbes at = 1700cm-1

• HCl has k = 0.516 kN m-1. Thus it absorbes at = 2990cm-1

ν~

(4.44) (4.45)

ν~

IR transitions

Clasification of molecules with respect of their IR spectra:

1. Homonuclear molecules are IR inactive (the stretching motion does not change the dipole momentum).

2. Heteronuclear molecules are IR active.

Transitions frequencies:

- bending modes are less stiff that the stretching modes:

νbend. < νstr.

- high ν appear for large k (stiff bonds), and small reduced mass.

4.6.4 High vibrational states

Anharmonic oscillator (the force is not proportional with the displacement) > potential energy is not a parabola!

For high vibrational states (high v) the energy levels are not equally spaced > harmonic oscillator approximation is not appropriate!

re

νχν hhE e

2

v 21v

21v ⎟

⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ +=

χe- anharmonicity constant

Transition energy:

( ) νχν hhE e1v2 +−=∆

Wavenumber of the transition:

( ) ( ) 00~1v2~v1v~ νχνν e+−=←+

(4.47)

(4.46)

(4.48)

High vibrational states

The number of vibrational levels of an anharmonic ascillator is finite because the second term in 4.31 is negative and induces the convergence of the levels at high vibrational quantum numbers. > vmax

In the IR spectra appear overtones (additional weak absorption lines) with ∆v = 2, 3, ...

∆Ev - not equal for high vibrational states

Explanation of the overtones: the selection rules are derived for harmonic oscillator, thus a slight anharmonicity will induce weak overtones (allow forbidden transitions, but with weak intensity).

3.2.1 Internal and external degrees of freedom of a molecule3.2.1 Internal and external degrees of freedom of a molecule

Aim: determine the number of independent coordinates (=degrees of freedom, dof) to describe the motions of a molecule.

However, the motions of the atoms inside a molecule are not independent from one another:

Analyse the possible types of motion:

1. An atom can move in all three dimensions in 3D space -> 3 dof

2. A molecule consisting of N atoms can be regarded as a cluster of the

constituent atoms -> 3N dof

• The molecule can move as a hole: Translation -> 3 dof

• The molecule can rotate as a whole: Rotation: linear molecule -> 2 dof

non-linear molecule -> 3 dof

• The remaining 3N-3-2=3N-5 (linear mol.) or 3N-6 (non-linear mol.) dof

account for the internal vibrations of the molecule.

3.2 The intramolecular potential

4.6.5 Normal modes of vibration

O

H H

O

H H

O

H H

Definition: Normal mode of vibration represents an independent, synchronous motion of atoms or groups of atoms which may be excited without leading to the excitation of any other normal mode.

Normal modes of vibration = combination of vibrational displacement of atoms

The description of the vibration motion of a polyatomic molecule is simpler if we consider combinations of the stretching and bending motions of individual bonds. > normal modes of vibration

Example 1: Normal modes for H2O molecule:

ν~

Symm. stretching Asymm. stretchingbending

Normal modes of vibration - examplesExample 2 : Normal modes for CO2 molecule:

ν~

Each normal mode of vibration = an independent harmonic oscillator

How many normal modes of vibration are in a protein of 6000 atoms?

Number of normal modes of vibration = number of vibration modes of the molecule

4.6.6 Selection rules for normal modesEnergy levels in the harmonic aproximation (similar as formula to 4.25 and 4.26):

νhE ⎟⎠⎞

⎜⎝⎛ +=

21vv

µπk

21v =

(4.49)

(4.50)

BUT:

k = the extent to which bonds bend and stretch during the vibration

µ = extent to which each atom contributes to the vibration

Observation: atoms which do not move during the vibration, do not contribute to µ!

Selection rules:

- the motion of a normal mode of vibration should induce a change in the dipole momentul of the molecule (∆µ ≠ 0)

- ∆v = ± 1

Active Normal modes: CO2

Example CO2:

• Symmetric stretching mode – inactive IR

• Antisymmetric stretching mode – active IR

• Bending modes – active IR

4.6.7 IR spectraIR spectra have two regions:

- fingerprint region (some of the normal modes of vibration of organic molecules can be regarded as collective motions of the molecule as a whole, thus they are characteristic for that molecule) ν < 1500 cm-1

-stretching normal modes with ν > νfingerprint region

Confirm the presence of a molecule in a mixture (usually the bending normal modes belong to this region)

Identify an unknown compound (Tables of stretching ν).

Stretching frequencies: examples

IR spectrum: example

Example: thioacetic acid

CO in gas phase

2 peaks (with double maxima) :

- fundamental absorption - Transitionv=0 > v=1, at 2143 cm-1 ~4.256x10-20 J

- first overtone - Transition v=0 > v=2, at 4260 cm-1 ~ 8.462x10-20 J

Energy of the overton is double as that of the fundamental transition. Ist intensity is weaker as it is allowed due to the anharmonicity of the oscillator.

In gas phase a vibration spectrum of a heteronuclear diatomic molecule can be analysed at high resolution > each line consists of a large number of closely spaced components.

The structure of components for each vibrational line is due to the rotational transitions which accompagne each vibration transition.

4.6.8 Rotation structure of a vibration transition

Energy levels of a diatomic molecule taking into acount the vibration and the rotation movements (in the harmonic approximation):

( )121v,v ++⎟⎠⎞

⎜⎝⎛ += JBJhE J ν

Selection rule: ∆J = ±1, 0

P branch(∆J = -1) BJ2~~0 −=νν

Q branch (∆J = 0) 0~~ νν =

R branch (∆J = +1) ( )12~~0 ++= JBνν

Rotational structure of vibrational transition

v > v+1

P RQ

(4.51)

(4.54)

(4.53)

(4.52)

Rotation structure of a vibration transition

Transition v > v+1 :

P branch (∆J = -1) R branch (∆J = +1)Q branch (∆J = 0)

4.7 Magnetic resonance

EPR (Electron Paramagnetic Resonance) or ESR (Electron Spin Resonance) are synonymous terms which describe the resonant absorption of microwave radiation (1010 Hz) by paramagnetic ions or molecules when they are located in a static magnetic field.

Magnetic resonance: resonant absorption of radiation (microwave or radio) by molecules which contains magnetic spins (electronic or nuclear), when they are placed in a magnetic field.

NMR (Nuclear Magnetic Resonance) describes the resonant absorption of radiofrequency radiation (∼500 MHz) by molecules which contain magnetic nuclei (I ≠ 0), when they are located in a static magnetic field.

4.7.1 EPR systems

EPR spectra are obtained from paramagnetic transition ions in crystals, chemical complexes, biomolecules, defect centers in semiconductors and the samples can be:

•Single crystals

•Solutions

•Frozen solutions

•Powders

In paramagnetic substances magnetic moments are weekly coupled they can be considered as isolated from one-another.

iµr

4.7.2 Two energy levels system

A two-level system is characteristic for a paramagnetic centre with an electron spin ½ (such as free radicals, conduction electrons in metals, F-centers trapped in alkali halides, transition metal ions, etc)

When the paramagnetic system is placed in a static magnetic field (B), the energy of interaction between the paramagnetic ion and the magnetic field is:

Brr

⋅−= µE

B

µr

- magnetic momentµr

(4.55)

Two energy levels system

The relation between magnetic moment and the spin S associated to electron is:

Sgrr βµ −=

β - Bohr magneton

g - electronic splitting factor

The potential energy of the dipol in magnetic field is:

Brr

⋅= SgV β BMgE sβ=±

21

±=sMwhere:

MS quantum number associated to the projection of the spin to Oz axis.

(4.56)

(4.57) (4.58)

4.7.3 Resonance condition

When the energy of the radiofrequency source (hνc) is equal with the difference between the energy levels at a resonance value ofmagnetic field B0, an absorption takes places.

0Bβν gh c =

( ) ( ) gGHzmT c1448.71B 0 ν×=

For a free electron (g=2.0023) in a magnetic field B=330mT, the νc=9.248 GHz (so called X-band frequency)

Resonance condition(4.59)

(4.60)

g – gyromagnetic factor

νc – microwave frequency

EPR for two energy levels system

EPR line

4.7.4 Selection rules - EPR

EPR spectra are based on magnetic dipolar transitions, while electronic spectra are based on electric-dipole transitions.

Selection rule: the magnetic dipolar transitions take place when the magnetic quantum number Ms (which has as values: S, S-1, … -S) changes according to:

1=∆ sM

In some cases the “forbidden transitions” can also be observed in the spectra.

4.7.5 Paramagnetic transition ions

List of most used paramagnetic transition ions.

Biologically important elements: V, Fe, Mn, Co, Cu, Mo and Ni.

4.7.6 EPR spectrometer

Microwave source: klistron, of gun diode

Detector: detector sensitive for phase detection

Sample cavity: microwave resonant cavity

Magnet system

Sample types:

- solid, single crystals

- fluid (frozen solutions, or fluid solutions)

- gas phase