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Symmetry and group theoryThe symmetry of molecules and solids is a very powerful tool fordeveloping an understanding of bonding and physical properties– Used to predict the nature of molecular orbitals– Used to predict if electronic and vibration spectroscopic transitions can be

observed We will cover the following material:– Identification/classification of symmetry elements and symmetry operations– Assignment of point groups

» The point group of a molecule uniquely and fully describes the molecules symmetry– Identifying polarity and chirality using point groups– Introduction to what a “Character Table” is– Assigning symmetry labels to “Symmetry adapted linear combination or

orbitals”– Assigning symmetry labels to of vibration modes– Determining the IR and Raman activity of vibrational modes

What is symmetry?

An object displays symmetry if you can “do something to it” (rotate it, reflect it, invert it etc.) without changing its appearanceThe process of doing the rotation, reflection etc. is referred to as a SYMMETRY OPERATION if it does not change the appearance of the molecule

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Symmetry elementsA symmetry element is the line about which a rotation occurs or the plane through which a reflection takes place etc.Symmetry elements are labeled– Rotation axis, Cn

» The n indicates the angle through which the rotation takes place. Angle = 360/n– Mirror plane, σh, σv, σd

» n, v or d indicate orientation of plane with respect to any rotation axes that may be present

– Inversion center, i– Improper rotation axis, Sn

» This involves a combination of rotation and reflection. The n indicates the angle through which the rotation takes place. Angle = 360/n

– Identity, E» E is always a symmetry element as doing nothing to an object always leaves it

looking just the same as it originally did.

Mirror planes

σv includes the principle rotation axis

σh is perpendicular to the principle rotation axis

σd includes the principle rotation axis, but lies between C2 axes that are perpendicular to the principle axis

σh

σvBorazine

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Rotation axes

Rotation axes examples 1

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Rotation axes examples 2

Inversion center

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Note inversion (i) and C2 are not equivalent

Inversion center examples

Note (a) and (b) do not have inversion centers

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Improper rotation axesAn improper rotation involves a combination of rotation and reflection– The rotation and the reflection do not have to be

symmetry operations of the molecule in their own right

Improper rotation axes

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S1 and S2 operations

S1 is equivalent to σS2 is equivalent to i

What symmetry elements do you have?

F BCl

ClBr Pt Cl

Cl

Cl

Cl Pt Cl

Cl

Cl

C CBr

Br

Cl

Cl

Cl

CClCl

Br

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Chirality and polarity

For a molecule to be chiral it must not have a Sn symmetry element– this includes mirror planes (S1) and inversion

centers (S2)For a molecule to be polar it must not have an inversion center or more than one rotation axis or a mirror plane that does not include the principle rotation axis

Point groups

The full set of point symmetry operations that a particular molecule has is called a point group (group of symmetry operations) There are symbols for each of the possible point groupsThese symbols are often used to describe the symmetry of a moleculeRather than saying water is bent, you can say that water has C2v point symmetry

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Assignment of point groups

Look for symmetry elements of molecule– highest order rotation axis is usually most

important for assigning point groupPoint group may be obvious if the molecule has very low symmetry or very high symmetryOtherwise use systematic method

Point group flow chart

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Icosohedra

The icosohedral point group Ih has 5 fold symmetry

Examples 1

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Examples 2

Examples 3

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Classifying functions by symmetry

It is possible to classify functions according to how they behave when the symmetry operations of a point group are performed upon them– for example, what happens to the 2pz orbital in

ammonia when the symmetry operations for ammonia’s point group (C3v) are applied to the orbital

– the behavior of a function when the symmetry operations are applied to it is called its character

Character tablesAll possible sets of characters (irreducible representations) for a given point group are shown in a character table

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Irreducible representations

There are only a certain number of distinct ways in which a function can behave when subjected to the symmetry operations of a particular point group– each of these possibilities is called an irreducible

representation– the character of each possible irreducible

representation is listed in the character table for that point group

Water

Consider how the following orbitals behave when subjected to the symmetry operations of the point group C2v– O 2s, O 2px 2pz and 2py

– H1s + H2s and H1s - H2s

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Symmetry elements for H2O

Transformation of H1s orbitals in H2OWe can classify the combinations 1s(A) + 1s(B) and 1s(A) -1s(B) by how they transform when the symmetry operations of the point group for the molecule (C2v) are applied

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Transformation of Opz orbital in H2O

Transformation of the other O orbitals in H2O

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MO diagram for H2O

Symmetry adapted linear combinationsIn molecules like H2O, NH3, CH4 where more than one atom is symmetry equivalent, combinations of the symmetry equivalent orbitals transform according to a irreducible representations of the molecules point group– These combinations of orbitals are referred to as SALCS

(Symmetry Adapted Linear Combinations).

These combinations can be calculated using group theory– However, for this class we will just show which

combinations are appropriate for each point group

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SALCS – 2 and 3 fold symm

SALCS – 4 and 5 fold symm

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SALCS – 6 fold symm

SALCS – oct and tet symm

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SALCS for NH3Transforms as A1

Transform as E

Symmetry of N orbitals in NH3

The N 2pz orbital and the N 2s orbital transform as A1 in the point group C3v

N 2px and 2pytransform as E

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MO diagram for NH3

dsd

IR and Raman spectroscopyIn IR spectroscopy a photon is completely absorbed to excite a molecular vibrationIn Raman spectroscopy quite high energy photons are used and some energy is lost from the photon to excite a vibration. The intensities and energies of the photons that have lost energy to excite the vibration are measured in order to get a spectrum

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Raman spectroscopyLaser excitation source is used. Intensity of inelastically scattered photons is measured

Vibrational spectroscopy and symmetry

For molecules with some symmetry, group theory can be used to predict how many vibrational modes should appear in the infrared and Raman vibrational spectra of the molecule– Not all modes are both IR and Raman active

» For centrosymmetric molecules anything that is IR active is not Raman active an visa versa

– However, in neutron vibrational spectroscopy all modes can be observed although modes involving H motion tend to give the strongest signals

As different molecular symmetries lead to different numbers of IR and Raman active bands this is a useful approach for determining the shape (symmetry) of molecules where there is a possible ambiguity– Often used in matrix isolation studies where some very reactive

species has been trapped in a noble gas solid at low T and there are no easy alternative ways of determining symmetry

– Useful for establishing which isomer you are looking at. For example, cis and trans isomers will typically give different vibrational modes because the molecules have different symmetries

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Why does symmetry mater?For a transition involving the absorption of a photon (IR, optical etc.) there must be a change in dipole moment on going from the groundto the excited state. This occurs when the following integral has a none zero value:

τµψψ d21∫Ψ1 and ψ2 are the ground and excited state wavefunctions, µ is the dipole moment operator and the integral is over all space

This integral can only be none zero if the function being integrated has a component with symmetry like that of the highest symmetry irreducible representation of the molecules point group

As µ transforms like the functions x, y and z, this in practice means that you only observe an IR transition when the excited state wavefunction belongs to the same irreducible representation as either x, y or z. So we need to be able to determine the symmetry properties of the vibrationally excited state to predict if we will see a transition to it.

Raman transitionsRaman excitation occurs only when there is a change in polarizabilityon going from the ground to the excited state. This occurs when the following integral has a none zero value:

ταψψ d21∫ Ψ1 and ψ2 are the ground and excited state wavefunctions, α is the operator for polarizability and the integral is over all space

As α transforms like the functions x2, y2, z2, xy, xz etc. this in practice means that you only observe a Raman transition when the excited state wavefunction belongs to the same irreducible representation as either x2, y2, z2, xy, xz etc. So we need to be able to determine the symmetry properties of the vibrationally excited state to predict if we will see a transition to it.

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Examples of vibrational selection rulesA1, B1 and B2 symmetry vibrations will be IR active.

A1, A2, B1 and B2 symmetry vibrations will be Raman active.

A1 and E symmetry vibrations will be IR active.

A1, and E symmetry vibrations will be Raman active.

A1, and E symmetry vibrations will be IR active.

A1, B1, B2 and E symmetry vibrations will be Raman active.

How many vibrational modes?For a molecule with N atoms there are a total of 3N degrees of freedom (x, y and z coordinates for each atom)– Three of these degrees of freedom are

used to describe the position of the molecules center of mass

– Three are used in determining the orientation of the molecule in space (assuming it is non-linear)

– The remaining 3N-6 degrees of freedom correspond to distinct vibrational modes of the molecule (stretching and bending)

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Determining the symmetry of stretching modesSymmetry for a normal (vibrational) mode can be determined by group theoretical analysisConsider what happens to the displacement vectors for each mode when you apply the symmetry operations of the point group

C2v

D2h

Both IR active

Only B2uIR activeA1g is Ramanactive

B21-1-11χanti

A11111χsymm

σv(yz)σv(xz)C2EC2V

1

1σ(yz)

-1

1

i

-1

1

C2(x)

1

1

C2(y)

B2u-11-11χanti

A1g1111χsymm

σ(xz)σ(xy)C2(z)ED2h

y

x

Vibrational spectra for Pd(NH3)2Cl2

Two IR active Pd-Clstretches

Only one IR active Pd-Cl stretch