lecture_8

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PG510

Symmetry and Molecular Spectroscopy

Lecture no. 8

Molecular Spectroscopy:

Vibrational Spectroscopy of Polyatomic Molecules

Giuseppe Pileio

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Learning Outcomes

By the end of this lecture you will be able to:

!! Understand the concept of vibrational modes

!! Calculate the number, the symmetry and the form of normal modes of vibrations

!! Use group theory to predict how many of these modes will contribute to the spectrum and their polarization

!! Extract molecular information from vibrational spectra

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Normal modes of vibration

A molecular vibration is a movement that involves the nuclei without changing the position of the center of mass

Although the vibration of a molecule appears a very complex motion it can be decomposed into a finite set of normal modes of vibrations

A normal mode is a collective motion of all the atoms. Each atom moves in phase with each other at a characteristic frequency. The blue vector are called displacement vectors

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Three are the most important features of normal modes of vibrations:

1.! Their number is finite and depends on the number of atoms

2.! Each normal mode oscillates at a frequency which is characteristic

3.! Each normal mode transforms as one of the irreducible representations of the group the molecule belongs to

Group theory: given a molecule the group theory is able to predict how many normal modes there are, their symmetry and how many of them are IR active (will contribute to the vibrational spectrum)

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The number of normal modes

To count the number of normal modes few facts should be noted:

2.! However, if all the N atoms move along the x (y or z) direction then the center of mass translates. Then three of these 3N modes are in fact molecular translations

x y

z

x y

z

x y

z 1.! In a molecule of N atoms each atom can move in the three direction of the Cartesian space, so there are 3N degrees of freedom

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3.! Moreover, the movement in which all the atoms moves in circular paths about one of the three axes corresponds to molecular rotations, i.e. another 3* of the 3N total modes are not vibrations

* linear molecules have only 2 rotational modes (the rotation along the principal axis does not, in fact, rotate any atom)

Then, a molecule of N atoms has:

! 3 translational modes ! 3 or 2 rotational modes (non-linear/linear) ! 3N-6 or 3N-5 normal vibrational modes (non-linear/linear)

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The symmetry of normal modes

Each normal mode transforms as one of the irreducible representations of the group the molecule belongs to.

Thus, the entire set of 3N Cartesian displacement vectors can be used as a base for a reducible representation of the group

Also, translations and rotations transform as one of the IRREP of the group.

x y

z

x y

z

x y

z

Group theory: Since we have 3N vectors the dimension of the reducible representation will be 3N

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In order to find out the symmetry of the normal modes using group theory only we need:

1.! Assign the molecular point group

Group Theory: for H2O, the principal axis is a C2, there are no other perpendicular C2 but there are planes of symmetry along the C2 i.e. the point group is C2v

2.! Find out the table of characters and assign the axis frame correctly according to the table

C2v E C2 !v (xz) !v (yz)

"1# 1 1 1 1 z x2, y2, z2

"2# 1 1 -1 -1 Rz xy

$1# 1 -1 1 -1 x, Ry xz

$2# 1 -1 -1 1 y, Rx yz

%xyz 3 -1 1 1

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3.! Write the 3N-order reducible representation of the Cartesian displacements, %tot

Group Theory: This step is immediate, just multiply the representation of the Cartesian axes (!xyz bottom line of character table or sum of the representation which x, y and z belong to) by the number of unmoved atoms

E ! leaves all unmoved C2 ! leaves only O unmoved "(xz) ! leaves all unmoved "(yz) ! leaves anly O unmoved


%xyz 3 -1 1 1

unmoved atoms

3 1 3 1

%tot 9 -1 3 1

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!tot = 3A1 + A2 + 3B1 + 2B2

4.! Decompose the reducible representation %tot in the sum of irreps

Group Theory: aj = 1/h (!R #(R) #j(R) )

5.! Remove from the decomposition the 3 IRREP which translations and the 3 (or 2 if linear molecules) which rotations belongs to

Group Theory: rotations (Rx, Ry, Rz) and translations (x, y, z) transformations are reported in the table of characters

!rot = A2 + B1 + B2 !tran = A1 + B1 + B2 $

!vib = !tot !rot !tran = 2A1 + B1

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Separation of modes

Polyatomic molecules have a bunch of different way to vibrate: stretching of bond, in-plane bending of bonds, out-of-plane bending, twisting, rocking, torsions etc

It is actually possible to isolate them and look for their symmetries and properties

This is done using the same technique but substituting the reducible representation of the Cartesian displacement with the reducible representation of all the stretching or bending etc

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Stretching analysis:


%str 2 0 2 0

Group Theory: it is possible to calculate the character of a representative of a symmetry operation simply using the following rules:

1.! each object moved to another place contribute as 0

2.! each object inverted contributes as -1 3.! each object unmoved contributes as 1

1.! Write the reducible representation of stretching modes

!str = A1 + B1

2.! Decompose the reducible representation %str in the sum of irreps

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Bending analysis:


%str 1 1 1 1

1.! Write the reducible representation of bending modes

2.! Decompose the reducible representation %str in the sum of irreps

Note that this is one object only

Note that in this case the representation of the bending is already an irrep

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Symmetry Coordinates

Symmetry coordinates are pictures of normal modes, i.e. it is possible to write down the picture of the normal mode of a given symmetry

Notes:

! The symmetry coordinates are approximations of the normal modes

! In molecules where the principal axis is a Cn the sum can be restricted to the Cn pure rotational group

! In the case in which the energies of two modes are close (and the two modes have the same symmetry) then the symmetry coordinates are no more a good approximation to the real normal modes

S!j!!

R

j#"R##R#s$

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choosing the vector s to be the stretching of OH bonds:

for the mode with A1 symmetry:

E 1

C2 1

And the mode with B1 symmetry

E 1

C2 -1

symmetric stretch

asymmetric stretch

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Selection rules

Vibrational selection rules for polyatomic molecules can be rationalized recalling the form of the transition moment operator:

" !"x,y,z

" $i,rot%z $j,rot%'rot ) #" $i,el%e%$j,el%'el%" $i,vib%$j,vib%'vib * !i"1

3%N+6" $i,el% Qi e

%$j,el%'el%" $i,vib%Qi%$j,vib%'vib$

! !i,vib""!j,vib"%vibPure vibrational transition Group Theory: The integral is different from zero only if %(&i,vib)x%(')x%(&j,vib) does contain the total symmetric of the group the molecule belongs to

The problem is now to find the symmetry of the three objects involved

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The dipole moment transforms as x,y and z being expressed as:

Group Theory: the symmetry of x,y and z is reported in the table of characters

So that the excitation of its normal mode no. 1 from v=0 to v=1 ((v=1) can be written as:

! " e.r!At room temperature most of the vibrations occur with their zero point (v=0) energy and since a molecule has 3N-6 (3N-5 if linear) it is possible to write:

!i,vib " !1#!v1"#!2#!v2" ...#!3#N$6#!v3#N$6"!1"!0""!2"!0" ..."!3"N#6"!0" $ !1"!1""!2"!0" ..."!3"N#6"!0"

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Now it is possible to demonstrate that:

Moreover, it is also possible to demonstrate that:

!i"!0" # total symmetric!i"!1"#ith normal mode

And, by the way, that for v=even &i(v) is total symmetric while transforms as the i-th normal mode for v=odd (only if the mode is not degenerate otherwise more complicate equations occur)

Thus, basically, the only thing to calculate is !&f(v) and:

Group Theory: the representation of the direct product AxB (!AB) will contain the totally symmetric only if the IRREP !A is equal to the IRREP !B

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Overtones, Combination bands and Hot bands

An IR spectrum of a polyatomic molecule shows more peaks than expected this is because there are:

1.! Overtones: they appear nearly to a multiple of a fundamental transition and occur when a mode is excited beyond v=1

2.! Combination bands: they occur when more than one mode is excited and appear at the sum of the frequency of all the modes involved

3.! Hot bands: they occur when an already excited vibration is further excited and appear at a slightly less frequency of the fundamental of that mode

!1"!0""!2"!0" ..."!3"N#6"!0" $ !1"!2""!2"!0" ..."!3"N#6"!0"!1"!0""!2"!0" ..."!3"N#6"!0" $ !1"!1""!2"!1" ..."!3"N#6"!0"!1"!1""!2"!0" ..."!3"N#6"!0" $ !1"!2""!2"!0" ..."!3"N#6"!0"

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Vibrational Spectra of polyatomic molecules

Ethyl ethanoate

Functional Group Fingerprint

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Information Content

An IR spectrum contains a lot of information that with the help of group theory give important hints on the molecule:

The spectrum contains the frequency of normal modes whose number, symmetry and form can be derived by group theory

The 400-1200 cm-1 region (called fingerprint region) is unique for each molecule

The 1200-4000 cm-1 region contains the fundamental transition characteristic of different functional group

identification of unknown samples

identification of functional groups

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Another very interesting information provided by the combination of IR spectroscopy and group theory is the assignment of structure. Suppose we have:

M

L

L

L

C=O O=C

C=O

M

C=O

L

L

C=O L

C=O mer fac


mer %str 3 1 1 3

C3v E 2C3 3!v

fac %str 3 0 1

!str = 2A1 + B2 all IR active !str = A1 + E all IR active

3 bands expected in the C=O region

2 bands expected in the C=O region

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What did we learn in this lecture?

! What is a normal mode

! Number and symmetry of normal modes through group theory

! How to draw a picture as a good approximation of the normal mode

! How to calculate how many modes will appear in the vibrational spectrum and their polarization

! The info encoded in vibrational spectra

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