Vlasta Mohaček Grošev

Theory of Raman spectroscopy on molecules

and crystals

Training School on Raman Spectroscopy, COST Action MP 1302 “Nanospectroscopy”,

Zagreb 23.-25. 09. 2015.

John Tyndall studied light scattering in 1860s

• differing types of molecules have differing absorptions of infrared radiation because their molecular structures give them differing oscillating resonances

1863. book “Heat Considered as a Mode of Motion”

I ∝ ω4

Gustav Mie (1869-1957)

• used Maxwell’s theory of radiation to explain plasmon resonances in colloid suspensions of metal particles

scattering on particles d ≳ λ

Tcherniak et al.: Probing a Century Old Prediction One Plasmonic Particle at a Time. Nano Letters 10 (4) (2010) 1398–1404.

Why is the sky blue?

John William Strutt, 3rd Baron Rayleigh

scattering on particles d << λ

I ∝ ω4

Adolf Smekal (1895 -1959)theoretically predicted what we

call Raman effectA. Smekal: Zur Quantentheorie der Dispersion.In: Die Naturwissenschaften. 11, Nr. 43, 1923,

S. 873-875

Sir Chandrasekhara Raman

• searched for “optical analogue of the Compton effect”

• 1924. met Compton at a meeting of British Association in Toronto

• performed measurements on liquids usig focused sunlight

On 7th July 1928. published the article in Nature with K. S. Krishnan in which explicitly they attributed the newly found spectral lines to molecular vibrations.

K. S. Krishnan

Independently, beginning from 1926, L.I. Mandelstam and G.S. Landsberg initiated experimental studies on vibrational scattering of light in crystals (quartz and calcite) at the Moscow State University.

They discovered the “combinatorial scattering” on 28th February 1928., but published the work with attributions to vibrations on July 9 and 31st 1928.

NOBEL PRIZE FOR PHYSICS FOR 1930. WENT TO RAMAN.

laser light: hν0scattered light: hν

wavenumber is defined as 1/ λ measured in cm-1

Scattering of light:

H. J. Galla, Spectroscopische Methoden in der Biochemie, Georg Thieme Verlag 1988.

A prototype of Raman spectrum: CCl4

H. J. Galla, Spectroscopische Methoden in der Biochemie, Georg Thieme Verlag 1988.

H. J. Galla, Spectroscopische Methoden in der Biochemie, Georg Thieme Verlag 1988.

G. Placzek 1934, Handb. Radiol., 6, 205.D. A. Long: The Raman Effect, J. Wiley 2002.

Intensity of the oscillating dipole p0νS frequency of the dipole oscillation

is the wavenumber of the oscillations

What is the nature of the induced electric dipole moment of the molecule and the field that is producing it?

linear Raman effect

second hyperpolarizability tensor is of the fourth order γρσξ

hyperpolarizability tensor is of the third order: βρστ

polarizability tensor is of the second order: α ρσ

Quantum mechanically:

Time dependent wave functions of the initial and final states are approximated with the unperturbed functions and corrections:

When interaction involves only electric dipole term corrections can be approximated as

and the summation is over all possible states of the system.

while the transition dipole moment corresponds to the matrix element between unperturbed wave functions of the initial and final states:

For linear Raman effect first order of the induced electric dipole moment because of external electric field

It is important for absorption and emission transitions, such as those measured with infrared or UV-VIS spectroscopy.

For Rayleigh scattering

For Raman scattering :

generally when r is not a stationary state - it decays in time

denoting the difference

real electric dipole moment is

for stationary initial and final states

Indices ρ and σ run over x, y, z

A prototype of Raman spectrum: CCl4

H. J. Galla, Spectroscopische Methoden in der Biochemie, Georg Thieme Verlag 1988.

H. J. Galla, Spectroscopische Methoden in der Biochemie, Georg Thieme Verlag 1988.

Stokes scattering

Rayleigh scattering

ωfi = ωf – ωi < 0ωfi = ωf – ωi = 0ωfi = ωf – ωi > 0

antiStokes scattering

Polarizability of the molecule changes with the vibrational state

where Q denotes a normal coordinate.

The selection rules for the i ⤍ f transition requirethat both the and the

be different from zero.

• symmetry of the tensor αρσ is given for each point group

• symmetry of Qk and the nature of Qk one finds via normal coordinate analysis

How to find normal coordinates {Qk} ? For molecules with the stable configuration Born-Oppenheimer approximation is valid:

ξ represents all nuclear coordinatesx represents all electronic coordinates

For diatomic molecules there is one vibrational coordinate: Δr (change of the internuclear separation distance from the equilibrium value)

Eigenvalues of energy are found by approximating the equipotential energy surface with a parabolic function of Δr.

Schrődinger equation for a free diatomic molecule is:

Vibrational-rotational energies of a free molecule in the 1st approximation:

From Pauling&Wilson: “Introduction to Quantum Mechanics”, Dover 1963.

Full line - probability density of finding a particle in the oscillator quantum state with n =1 (left) and n =

10 (right). Dashed line – classical probability.

pure rotational Raman spectrum of N2

vibrational-rotational Raman spectrum of N2

vibrational-rotational Raman spectrum of N2

Molecular vibrations

in such a way so that kinetic and potential energy have DIAGONAL form:

Normal coordinates Qk are defined as

Atoms vibrate in normal modes in such a way that each atom reaches maximum amplitude at the same time.

Quantum mechanically:

Each normal mode has quantized energy levels

and the total vibrational energy of a molecule with N atoms is:

There are 3N-6 vibrational degrees of freedom.

Water

Derivation of symmetry coordinates:

1. find the character of the reducible representation containing all 9 coordinates of water atoms

2. subtract translations and rotations of the molecule as a whole

Derivation of symmetry coordinates:

Molecules with many atoms – symmetry

gaucheC2

transC2h

Technische Universitaet Darmstadt, dr.Stefan Immel

Tutorials Symmetry Point groups

(COOH)2 oxalic acid tTt conformer

V. Mohaček Grošev et al. J. Raman Spectrosc. J. Raman Spectrosc. 2009, 40, 1605–1614.

Point group symmetry is C2h

Point group symmetry is C2h

Γvib = 7 Ag + 2 Bg + 3 Au + 6 Bu

Ag and Bg modes are RAMAN active

Vibrations are calculated numerically,e.g. via Gaussian.

In the crystal, symmetry elements are not necessarily on the molecule

Removing the acoustic modes

Matrix of the polarisability tensor

in the crystal has components of

Ag and Bg

symmetry.

Symmetry elements are now CRYSTAL SYMMETRY

elements.

XXI International Conference on Horizons in Hydrogen Bond Research

T64000 HORIBA Jobin YvonRaman spectrometer514.5 nm (green) laser line

CCS 350 Janis Research cryostat

XXI International Conference on Horizons in Hydrogen Bond Research

single crystal was placed in the capillary and mounted inside the cryostat

figure by K. Furić, V. Volovšek: J. Mol. Struct. 976 (2010) 174–180.

Spectra of powder sample exhibit the same behaviour found in z(yy)x +z(yz)x spectra:

z(yy)x + z(yz)x

Low frequency Raman spectra do not show any change that would indicate phase change:

However, when the polarization of the electric field is along a crystal axis (here z), significant increase of hydrogen bond modes is observed:

Thank you for your attention!