molecular many electron systems: electronic & nuclear movement … · 2017. 5. 11. · optical...

IPC Friedrich-Schiller-Universität Jena 1

Hamiltonian for a polyatomic molecule treated as Coulomb system with N nuclei

(coordinates {R}) and n electrons (coordinates {ri}) :

In atomic units i.e. ~ = qe = me = 1

Kinetic energy operator for nuclei

Kinetic energy operator for electrons

Nuclei-electron interaction operator

Electron-electron interaction operator

Nuclei-nuclei interaction operator

Molecular many electron systems: electronic & nuclear movement


(3N + 3n)-dimensional problem:

Born-Oppenheimer Approximation: separate treatment of electronic and nuclear

motion allows the total wavefunction of a molecule to be broken into its electronic

and nuclear components:

Decomposition of Hamiltonian:

= adiabatic potential energy surfaces

Schrödinger equation for complete problem:


Does not depend on {ri} =

constant for given nuclear

geometry


Multiplikation with and integration over electron coordinates

Schrödinger equation for nuclear motion:

C describe coupling between nuclear and electron motion thus the resulting

coupling of electronic states (non-adiabatic coupling)



Born-Oppenheimer approximation neglects coupling between nuclear and electron

motion

C = 0

Electrons adjust immediately or adiabatic to any nuclear motion:

displays the potential for nuclear motion

Within the Born-Oppenheimer approximation the nuclear dynamic is treated

in a way that the nuclear motion is described on adiabatic potential energy

surfaces



Description of quantized molecular electronic energy states by many-electron

wavefunctions:

Approximation of many electron wavefunctions as Slater determinant

(antisymmetrized product) of one electron wavefunctions called molecular orbitals

(MOs):

Electronic states can be approximated by a single electronic configuration which is

commonly displayed by a MO diagram

MO

UV-Vis-Absorption


In a MO diagram (representing a single electronic configuration) the highest occupied

MO is called HOMO and the lowest unoccupied MO is called LUMO

MOs are represented as linear combination of atomic orbitals (AOs)

Bonding MOs result when AOs enhance each other in the nuclei region

Antibonding MOs are formed when AOs cancel each other in the nuclei region

Classification of MOs:

s-orbitals: bonding orbitals which are symmetric with respect to rotation around

the molecular axis

s*-orbitals: antibonding orbitals with nodal plane within molecular axis

p-orbital: results from overlap of two lobes of one AO with the two lobes of

another AO

Nonbonding MOs contain lone pairs of electrons which do not participate in bonding

atoms together since they are unshared

UV-Vis-Absorption


Molecular electronic transitions

UV-Vis-Absorption

Molecular electronic transitions take place when valence electrons in a molecule

are excited from one energy level to a higher energy level.

Electrons residing in the HOMO of a sigma bond can get excited to the LUMO of

that bond. This process is written down as a σ → σ* transition.

Likewise promotion of an electron from a π-bonding orbital to an antibonding π

orbital* is denoted as a π → π* transition.

Auxochromes with free electron pairs denoted as n have their own transitions, as

do aromatic pi bond transitions.

The following molecular electronic transitions exist:

σ → σ* π → π* n → σ* n → π* aromatic π → aromatic π*

p,p* np* ns*

(C=C, C=O) (C=O, C=N, C=S) (–Hal, -S-, -Se- etc.)


Molecular orbital or electronic configuration (z.B. Formaldehyd)

Energetic order of transitions:

p* ← n < p* ← p < s* ← n < p* ← s < s* ← s

UV-Vis-Absorption


spin multiplicity

• Total spin quantum number S = ∑ si

with si = +½ or - ½

• Multiplicity M = 2S + 1

• M = 1: Singulet

• M = 2: Dublet

• M = 3: Triplet

HOMO

LUMO

UV-Vis-Absorption


Molecular

orbital

Electronic

configuration

Electronic

states

UV/Vis-absorption spectrum

UV-Vis-Absorption


J = 0

J = 1

J = 2

J = 3

J = 4

Ro

tatio

na

l le

ve

ls

v = 0

v = 1

v = 0

v = 1

v = 3

v = 4

Vib

ratio

na

l le

ve

ls

Excitation [

10

-15 s

]

Internal conversion

[10-14 s]

Fluorescence

[10-9 s]

Intersystem crossing

Phosphorescence

[10-3 s]

S0

S1

S2

S3

S4

T1

Tn

IR- & NIR-

spectroscopy

UV-VIS-spectroscopy Microwave-

spectroscopy


Jablonski-Scheme


Principle to interpret electronic absorption spectra based on the probability distrubtion

||2 of the vibrational levels within the electronic states.

The basis of this principle is that

electronic transitions happen on a

timescale (~10-16s) that is

significantly smaller than the

vibrational period (~10-13s) of a

given molecule and therefore the

distance at which they happen can

be assumed to be fixed during the

transition.

Franck-Condon principle

UV-Vis-Absorption


Transition dipole moment for a transition between the states |i and |f:

For excitation follows:

Electronic transition dipole moment is developed in a rapidly converging Taylor

expansion about nuclear displacements from the equilibrium position

Condon approximation neglects higher order terms i.e. electronic transition dipole

moment is assumed to be constant i.e. nuclear coordinates correspond to

equilibrium geometry

Condon approximation:

Transition dipole moment:

B.O.-approximation


UV-Vis-Absorption


= degree of redistribution of electron density during transition

= degree of similarity of nuclear configuration between vibrational

wavefunctions of initial and final states.

Intensity of a vibronic transition is direct proportional to the square modulus of the

overlap integral between vibrational wavefunctions of the two electronic states =

Franck-Condon-Factor:


UV-Vis-Absorption


|i

|f |i

|f


UV-Vis-Absorption


Transition metal complexes

A biologically very important group of metal complex bonds are the porphyrin

pigments such as:

Hemoglobin (pigment of the blood, central ion Fe2+)

Cytochromes of respiratory chain

Chlorophyll (green molecules in leaves, central atom Mg)

UV-Vis-Absorption

In these molecules the octahedron

structure with a central atom is

incorporated into particular proteins

The four ligand positions of the base of

the pyramid are occupied by the lone

electron pairs of nitrogen atoms of the

plane porphyrin ring system

The two corners of the pyramid are

occupied by specific amino acids

(histidine) and/or by an oxygen molecule

(hemoglobin)

Heme-group


Cytochrome c:

Pyramid corners of heme unit are occupied by N-atom of a histidine residue and

S-atom of a mezhionine residue

Redox change of cytochromes predominatly occurs at the central iron atom

[(Fe2+) ↔ (Fe3+)]

-Peaks

= sensitive for redox change

(analysis of mitochondria)


UV-Vis-Absorption



Hemoglobin (iron is always found as Fe2+)

Arterial oxygen-loaded blood = light red

Blood in veines free of oxygen = deep red

Desoxy Hemoglobin

(Fe2+ / 92 pm / high spin)

End-on coordination of O2 (Fe2+ / 75 pm / low spin)

0,4 A °

B

UV-Vis-Absorption


Fundamental terms:

Polarimetry, optical rotation, circular birefringence:

turning of the plane of linearly polarized light

Optically active molecules exhibit different refractive indices for right nR and

left nL polarized light nR ≠ nL

Optical rotatory dispersion (ORD):

Wavelength dependency of rotation

Allows determination of absolute configuration of chiral molecules

Circular dichroism:

linearly polarized light is transformed into elliptically polarized light upon traveling

through matter

Different absorption coefficients for left and right circular polarized light

(eR ≠ eL ).

Polarimetry & Optical rotatory dispersion & Circular dichroism

UV-Vis-Absorption


Polarimetry

What happens if light interacts with chiral molecules?

Enantiomeric molecules interact differently with circular polarized light.

Polarizability depends on direction of rotation of incoming circular polarized light

Optically active substances exhibit different refractive indices for right nR and

left nL polarized light nR ≠ nL

UV-Vis-Absorption



Polarimetry:

Incoming linear polarized light beam experiences different refractive indices for its

left and right circular components.

Phasing of left and right rotating component of exiting light beam is shifted while

the absolute E-field vectors do not change

Vector addition leads to linear polarized light with rotated polarization plane

UV-Vis-Absorption



Polarimetry:

For follows:

Na-D line l = 589 nm

2-Butanol = 11.2° (Messwert)

T = 20°C

l = 1dm

Difference is rather small!

Due to the different refractive indices a phase difference d = jL –jR builds up in the active medium which is proportional to the path

length l.

When exiting the medium linear polarized light where the oscillation

plane is rotated by d/2 arises

It follows:

l

UV-Vis-Absorption



Optical rotatory dispersion(ORD)

ORD measures molar rotation [F] as function of the wavelength!

If the substance to be investigated has no

electronic absorption within the

investigated spectral region the following

ORD spectra are obtained

Reason:

refractive indices for left and right

polarized light change differently with

wavelength (rotatory dispersion is

proportional to refractive index

difference).

ORD-spectra of 17ß- and 17-

hydroxy-5-androstan

UV-Vis-Absorption



Optical rotatory dispersion(ORD)

Refractive indices for left and right polarized light exhibit anomalous dispersion in

the range of an absorption band

Cotton effect

Positiv negativ Cotton effect

UV-Vis-Absorption



Circular Dichroism (CD)

Enantiomeric molecules exhibit besides different refractive indices for left and right

circular polarized light also different absorption coefficients:

It follows:

For pure ORD bands left and right circular polarized components of linear polarized

light experience only different retardation when passing through the sample while for a

CD band also one component gets more absorbed than the other

Exiting light is elliptically polarized.

Circular Dichroism

UV-Vis-Absorption




[10-1 × deg × cm2 × g-1]

[10 × deg × cm2 × mol-1]

The ratio between short and the long elliptical axis is defined as tangent of an

angle , the so called ellipticity (tan = b/a):

a = ER + EL

b = ER - EL

The specific ellipticity is defined as:

where 0bs is the experimentally determined

ellipticity.

The molar ellipticity is defined as:

UV-Vis-Absorption



Circular Dichroismus (CD) Simple model:

For an electronic transition to be CD active the following must be true:

µe is the electronic transition dipole moment (corresponds to a linear displacement

of electrons upon transition into an excited state)

µm is the magnetic transition moment (corresponds a radial displacement of

electrons upon excited state transition)

Scalar product is characterized by a helical electron displacement.

Depending on the chirality of the helix preferably more right or left circular

polarized light will be absorbed, respectively.

Electronic transition Magnetic transition Optical activity

UV-Vis-Absorption




Application field:

b-sheet

random coil

-helix

UV-Vis-Absorption




Application field:

Typical reference CD spectra:

Poly-L-Lysine in different conformations:

-Helix, b-sheet and random coil.

Temperature

dependent CD

spectra of insuline:

For increasing

temperature the

molecule changes

form -helix into the

denaturated random

coil form with ß-sheet

contributions.

UV-Vis-Absorption



Vibrational microspectroscopy

Polyatomic molecules – Normal modes

Number of vibrational degrees of freedom:

For a non-linear molecule consisting of N atoms

there are 3N – 6 vibrational degrees of freedom =

Normal modes.

Normal modes can be excited independently (they

are decoupled).

Every normal mode q is a harmonic oscillator

independent of the rest of the molecule:

q = effective mass of the vibration = measure of the mass,

which is moved during the vibration. amplitude, x

Possib

le e

nerg

y levels

, E

qv Potential energy

V

0



Polyatomic molecules – Normal modes

Example Water: 3 normal modes

d (1595 cm-1) as (3756 cm-1) s (3652 cm

-1)

Adenine

ar ring breathing

Thymine

(C=O) ar



Polyatomic molecules – band assignment


Raman vs. IR and NIR Absorption spectroscopy

v = 1

v = 0

a(R-R eq )

IR

V

v = 1

v = 0

a(R-R eq )

V

NIR

IR Absorption NIR Absorption

v = 1

v = 0

a(R-R eq )

V

Raman-Signal

(Stokes)

Raman scattering



Raman microspectroscopy


Classical description

Incident electromagnetic field: (1)

Induced dipole moment: (2)

(1) in (2): (3)

Oscillating molecule: (4)

Expansion of around q = 0:

(5)


(5) in (3): (6)

Anti-Stokes-Raman-Streeung

Rayleigh-Streuung Stokes-Raman-Streuung



Classical description

Trigonometric transformation:

(7)



Raman microspectroscopy – quantum mechanical description

v = 0

v = 1

laser – vib laser + vib

virtual states

v = 0

v = 1

v = 2

v = 3

vibrational

states

Rayleigh scattering

Stokes-Raman

scattering

Anti-Stokes-Raman scattering




3500 3000 2500 2000 1500 1000 5000

5000

10000

15000

20000

25000

30000

wavenumber / cm-1

Ra

ma

n in

ten

sity / a

rb. u

.

(C=C)

(=C-H)

s(CH2)

s(CH3)

as(CH2)

s(CH)

as(CH)

as(CH3)

d(CH3/CH2)

Pyridin – ring breathing

N

N

C H 3

H

H

H H

H

H

H

H H

H

H

Pyrrolidin – ring breathing

Raman spectrum of nicotine



Raman spectra of cells

3000 2500 2000 1500 1000

(C

-H)

(C

-C)

(C

-O)

(C

H2)

(P

O2

- )sym

d(C

H2,C

H3)

d(C

H2,C

H3)

d(C

H2,C

H3)

s(O

-H)

(C

=O

)

d(O

-H)

water

protein

lipide

polysaccharide

Ra

ma

n In

ten

sity

Wavenumber / cm-1

dna

T A

, G

A. C

, U

C, U

A, G

A, G

T

Phe, T

rp

am

ide III

Tyr,

Trp

, H

is

Trp

, am

ide II

Trpam

ide I

(C

-C) s

cele

tal

(C

-C) s

cele

tal

(C

=C

) cis

d(=

CH

) in p

lane

(C

H2)

(C

-H)

(C

-H)

(C

-H)

endoplasmic

reticulum

nucleus

ribosome

mitochondria

DNA/RNA

polysaccharides

lipids

proteins

water



Raman spectra of cells

endoplasmic

reticulum

nucleus

ribosome

mitochondria

DNA/RNA

polysaccharides

lipids

proteins

water

3000 2500 2000 1500 1000 500

Experimentell bacterial spectrum

Simulated bacterial spectrum

Linearcombination of the main components

Ra

ma

n In

ten

sity

Wavenumber / cm-1

* *

Quartz *

Raman microscopy



CCD spectrometer

laser

microscope

L

BS1

BS2

M

MO

S

N IF

C

W

High specificity

High spatial resolution ( < 1 µm)

Minimal sample preparation

All solvents can be applied

(inclusive water)

44


Requirement of statistical models

Biological samples

Biopolymers (proteins, DNA, …)

Similar vibrations

Similar spectra

Classification model

Calculating a marker for groups

0

Classification models:

Artificial neural networks

Linear discriminant analysis

Support vector machines



788

DNA

1005

phe

1575

G, A

1800 1600 1400 1200 1000 800 600

Ra

ma

n-I

nte

nsity

Wavenumber / cm-1

Breast cancer cell lines (MCF-7, MCF-10A) as model system

finger print region with chemical

information

Univariate analysis show the chemical

information



Multivariate Analysis Strategy for Cancer Diagnosis

In order to compare two eukaryotic cells it is necessary to focus on one compartment.

The nucleus was chosen because major differences arise while tumorgenesis.

All spectra which exhibit a higher DNA/RNA content were

used to build a classification model for MCF-7/-10A.

The decision malignant cell (MCF-7) / benign cell (MCF-

10A) was possible with a accuracy of 99,11%.

Raman microscopy

Raman spectroscopy of a colon tissue section. A, representative Raman spectra of connective tissue (1),

muscle layer (2) and epithelial tissue (3). B, 79 x 79 Raman map with a step size of 2,5 µm. The colors of

the Raman represent cluster memberships. C, the arrow in the photograph shows the position of a

ganglion. E, enlarged ganglion from C.

47


Biomedical diagnostics

Special properties of Raman spectroscopy:

Stokes ( = 0 – R) Raman scattering intensity:

Increase of the

laser intensity

or by

SERS

Use of higher

excitation frequency

(shorter wavelengths)

Excitation of the electronic

resonance

Resonance Raman scattering

48



inte

nsity

400 450 500 550 600 650 700 750

wavelength / nm

Fluorescence Raman

Resonance

Raman

Resonance Raman spectroscopy



Advantages of resonance Raman spectroscopy

Resonance enhancement of the scattering intensities by a factor 106 – 108

Improved S/N ratio

Allows detection of low concentrated substances in solution

Resonance-Raman spectra are dominated by modes characteristic for the

geometrical changes of the molecules during the electronic transition

Selective excitation (e.g. of chromophores pivotal for the biological activity of the

molecule) possible due to variation of the excitation wavelength

simplified spectrum, since less vibrations contribute to the spectrum; only

Franck-Condon active modes are enhanced




Excitation of different chromophors within a molecule

Guanosin-5‘-Mono-Phosphat






3000 2500 2000 1500 1000

wavenumber / cm-1

AG

-Vort

rag

NIR

-bandassig

nm

ent

grating: 300 l/mm

Hole: 1000 m

central spectral

position: 2000 cm-1

wavebumbercorrection:

+ 2cm-1

IR spectrum

Raman spectrum (532 nm)

UV resonance Raman spectrum (244 nm)

phospho

die

ste

r

(C

C/C

N)

d(C

H)

Am

id II

Am

id I

satu

rate

d e

ste

rscarbohydrates

proteins

lipids

Tyr

A +

G +

Tyr

T +

AA

+ G

G +

AC

G +

AT

yr+

Trp

T

S. pseudovenezuela DSM 40212

(C

H)

G+A

Am

id III

DNA – components

&

Proteins

Contribution from

whole cell

5 µm

10

20

30

40

50

20 30 40 50 60 70

bulk

single cell

Applied vibrational spectroscopic methods Retrieved information


SERS = Surface Enhanced Raman Scattering


Stokes ( = 0 – R) Raman scattering intensity:

Increase of the

laser intensity

or by

SERS

Use of higher

excitation frequency

(shorter wavelengths)

Excitation of the electronic

resonance

Resonance Raman scattering


Electromagnetic enhancement

Localized surface plasmon resonance (LSPR)

Plasmons can be described in the classical picture as an oscillation of free electron

density against the fixed positive ions in a metal. Surface plasmons are surface

electromagnetic waves (evanescent wave) that propagate in a direction parallel to

the metal/dielectric (or metal/vacuum) interface.

+

-

+

-

Electric field

Electron cloud

Gold sphere




ELM

EDIP

ESC

R

Molecule = Etot

mit Etot = E0 + ELM

Observator

EDIP

EO

Raman intensity IR

IR ER2

ER = EDIP + ESC



Electromagnetic enhancement

Localized surface plasmon resonance (LSPR)


SERS increases detection limit!




L

L - S = R

R

Excitation of coherent molecular vibrations

within common focus of two laser beam whose

difference frequency L - S matches a

molecular vibration.

Non-linear Raman microspectroscopy

Coherent Raman spectroscopy

Fundamental concepts of coherent anti-Stokes Raman scattering (CARS)


Coherent molecular

vibrations excited by

the two laser fields L

and S modulate a

third laser field (L)

and thus generate a

coherent light beam at

anti-Stokes frequency

aS = 2L - S



IPC Friedrich-Schiller-Universität Jena 72 Stokes pulse

Pump pulse 1

Pump pulse 2

CARS-Signal

Energy conservation

Momentum

conservation


Fundamental concepts of coherent anti-Stokes

Raman scattering (CARS)


CARS intensity

starting point for derivation of CARS intensity is non-linear wave equation for anti-

Stokes amplitude EaS

for k = 0 CARS intensity increase for L2

for k 0 build up of CARS signal over limited coherence

length lPh p/k; thereafter spatial-time-periodic intensity

modulation due constructive and destructive interference.

CARS signal oscillates as function of L with periodicity

laS/2n



L

IaS

lPh


CARS line profiles

Besides resonant susceptibility due

to molecular vibrations also a non-

resonant contribution NR due to

electronic structure of medium must

be considered. NR is a real value

and does not show a strong

frequency dependency i.e. can be

considered as constant in vicinity of

vibrational ferquency.

resonant

susceptibility

Non-resonant

susceptibility




Collinear CARS microscopy:

Phase-matching for strong focusing with microscopy objective

Phase-matching uncritical because:

For every wave vector component of

pump beam the corresponding

component of Stokes beam needed

to fulfill phase-matching can be found

in focus.

CARS signal is only generated for

short interaction length. Focusing with

objective goes below coherence

length. Interaction length is too short

in order for a large phase mismatch

to occur.

Wavevector mismatch in water for collinear CARS geometry

as a function of NA. Solid line: pump wavelength: 600 nm,

Raman shift: 4167 cm-1; dashed line: pump wavelength:

500 nm, Raman-Shift: 7500 cm-1; L is defined as FWHM of

the focal excitation intensity along optical axis

CARS microscopy




Beam geometries in CARS microscopy

F-CARS

E-CARS

p

S aS

aS F-CARS = Forward CARS

E-CARS = Epi CARS

E-CARS signal for scattering

objects smaller than half anti-

Stokes wavelength

Magnitude of phase mismatch

acts as filter for scattering

objects of certain size.


CARS microscopy


CARS signal dependence on the size of

a sphere placed in vacuum at the origin

of tightly focused excitation beams for

different experimental geometries.

F-CARS signal increases quadratically with D

until it reaches a constant value where the

diameter of the sphere exceeds the longitudinal

dimension of the focal volume.

E-CARS signal has the same amplitude as F-

CARS signal for sphere diameters smaller than

the pump wavelength and reaches a maximum

for~ 0.3lp. E-CARS amplitude decreases quickly

for increasing sphere diameters and oscillates as

function of D with periodicity laS/2n as a result of

interference effects due to a large phase

mismatch . For large D (= bulk) no E-CARS

signal can be observed anymore.


Beam geometries in CARS microscopy


CARS microscopy



CARS microscopy is not background free

Contrast reduction due to:

Non-resonant background signals.

Water as solvent leads due to ist broad

Raman bands strong resonant

background signals

CARS field of object in F-CARS geometry is superimposed on much stronger CARS signal of medium

(due to larger interaction length).

Low contrast of CARS image of object

E-CARS detection allows imaging with much better contrast because E-CARS signal of medium

disappears due to destructive interference. However, this is not true for E-CARS signal of object if its size

is comparable to anti-Stokes wavelength where only the object leads to image contrast.


CARS microscopy


Comparison of contrast in F-CARS

and E-CARS images of unstained

live epithelial cells taken with 5 ps

pulse trains at a repetition rate of

400 kHz. (a) F-CARS image

recorded with parallel-polarized

pump and Stokes beams with

average powers of 0.4mW and

0.2mW, respectively, at a Raman

shift of 1579 cm−1 (protein vibration).

(b) E-CARS image recorded with

average pump and Stokes powers

of 2.0mW and 1.0mW, respectively,

at a Raman shift of 1570 cm−1.(DNA

vibration).

Larger water background Background supression! Small

features (~ 100 nm) become visible.


CARS microscopy


CARS microscopy is not background free

Raman images 1659cm-1; 32 x 32 data points; Measuring time 9 h

Bright field

CARS image

CARS images at 1660cm-1; 512 x 512 data points; Measuring time 5.4s,

A comparative study on colon tissue by means of CARS and Raman imaging


CARS microscopy


Multimodal Optical Imaging

Future trends in non-linear microscopy


Multimodal image:

Composite of

CARS, SHG and TPEF

Cerebellum of a domestic pig:

Multimodal image retrieves simlar information as

H&E

In-vivo detection of tumor

H&E CARS CH SHG (collagen)

Purkinje cells

White matter

arachnoidea

Granule layer

TPEF

Grey matter





Brain metastasis of lung carcinoma:

Detection of tumor by CARS

Detection of tumor boundary by spectral profile

Subcellular details (cell cores)

Multimodal information of the tissue morphochemistry

H&E




Multi-modal imaging

Various methods are necessary in order to achieve molecular contrast

as well as multivariate information (absorption, fluorescence based

methods, SHG, THG, Raman and FTIR, CARS, etc.)

Implementation of in-vivo multimodal imaging for clinical detection and

ultimately to the diagnosis of disease

Multi-modal imaging


molecular many electron systems: electronic & nuclear movement … · 2017. 5. 11. · optical...

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