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PG510

Symmetry and Molecular Spectroscopy

Lecture no. 2

Group Theory:

Molecular Symmetry

Giuseppe Pileio

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Learning Outcomes

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

!! Understand the concepts of symmetry element and symmetry operation

!! Classify and recognize the symmetry elements and their associated operations as required to specify molecular symmetry

!! Understand some important properties of symmetry operations and how those operations combines

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Symmetry Elements and Operations

What is a Symmetry Element?

It is a geometrical entity (point, line or plane) with respect to which a symmetry operations can be carried out

What is a Symmetry Operation?

It is a movement of a body such that, once it has been done, every point of the body is in an orientation which is indistinguishable from the original one

There are only 3 symmetry elements but infinite symmetry operations

A symmetry plane (!) is a geometrical plane that passing through the body allows a symmetry operation called reflection to be performed.

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Symmetry Planes and Reflections

The operation has to be done in this way:

i.! drop a perpendicular from each atom to the plane ii.! extend that line at equal distance on the opposite side iii.! move the atom to this other end of the plane

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•! The atoms on the plane make a special case since the operations let them unmoved.

" Any planar molecule has at least a plane (molecular plane)

" If there is only one atom of a given species it must be in each and every symmetry plane

" Atoms that do not lie on the plane must occur even number

The product of two reflections (!!, i.e. two successive application of the operation !) produces a configuration which is identical to the original. This operation is called identity and denoted by E.

! !n = E for n = even " !n = ! "for n = odd

A symmetry point of inversion (i) is a geometrical point that passing through the body allows a symmetry operation called inversion to be performed.

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Symmetry points and Inversions

The operation has to be done in this way:

i.! drop a line from each atom to the point ii.! extend that line at equal distance on the opposite side iii.! move the atom to this other end of the point

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•! If there is an atom at the inversion point, that atom has to be unique in the molecule

•! Atoms that do not lie on the point must occur in pairs

•! If a molecule contains an odd number of atoms of more than a species then the molecule does not have a point of inversion

! in = E for n = even " in = i "for n = odd

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Symmetry Proper Axes and Proper Rotations

A symmetry Proper Axis (Cn ) is a geometrical line that passing through the body allows a symmetry operation called proper rotation to be performed.

The operation has to be done in this way:

i.! take every atom and perform a rotation about the axis ii.! draw the arc of the circle described iii.! move the atom to the end of the arc

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•! n denotes the order of the axis and is the largest number that give an equivalent configuration upon rotation of 2#/n

•! A proper axis of order n generates n rotation operations denoted by Cn

m. They are performed by repeating a 2#/n rotation m times

•! There can be any number (even or odd) of each species lying on the axis (unless other elements impose restrictions)

•! If one atom of a certain species lies off a Cn axis then there must be n-1 more atoms of that species

•! If the fraction m/n can be reduced to m’, n’ then Cnm

coincides with the reduced version Cn’m’

! Cnn= E Cn

n+1 = Cn Cnn+2 = Cn

2 and so on "

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Symmetry Improper Axes and improper Rotations

A symmetry improper Axis (Sn ) is a geometrical line that passing through the body allows a symmetry operation called improper rotation to be performed.

The operation has to be done in this way:

i.! take every atom and perform a rotation about the axis ii.! drop a line to a plane normal to the axis iii.! extend that line at equal distance on the opposite side iv.! move the atom to this other end of the plane

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•! n denotes the order of the axis and is the largest number that give an equivalent configuration upon rotation of 2#/n

•! If Cn and ! exist independently then also Sn exists. The opposite is not true

•! The order in which Cn and ! are applied is unimportant

•! The existence of Sn with n even implies the existence of Cn/2

•! The existence of Sn with n odd implies the existence of both Cn and !

! Snn= E Sn

m = can be rewritten (i.e. S2=i, S62=C3 …)

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More on Symmetry elements and operations

The product of two symmetry operations X and Y:

Z=XY

means the application of the two operations consecutively from right to left i.e. first Y and then X.

C2(z)

•!The product of two proper rotations must be a proper rotation

[x,y,z] !C2(x) ! [x,-y,-z] !C2(y) ! [-x,-y,z]

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•! The product of two reflections with respect to planes intersecting at an angle $ is a rotation by 2$, i.e. a Cn=#/$ "

•!The presence of a Cn and a plane containing it assures that n of those planes exists (separated by $=#/n)

$

Cn

!

!

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•! The product of two C2 about axes intersecting at an angle $ is a rotation by 2$ about an axis perpendicular to them, i.e. a Cn=#/$"

•! The presence of a Cn and a perpendicular C2 assures that n of those C2 axes exist

$

C2

C2

C#/$"

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•! The product of a proper rotation of even order and a perpendicular plane is an inversion lying on the point of intersection "

" C2nn ! = ! C2n

n = C2 ! = ! C2 = i " C2n

n i = i C2nn = C2 i = i C2 = !

Cn2n

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•! If XY=Z=YX then X and Y are said to commute

These pairs of operations always commute

o! Two rotations about the same axis o! Reflections to perpendicular planes o! Inversion and any reflection or rotation o! Two C2 about perpendicular axes o! Rotation and reflection in a plane perpendicular

Group Theory: A molecule that has no Improper Rotation Axes must be dyssimmetric"

•! Molecules not superimposable to their mirror image (enantiomer) are called dyssimmetric

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What did we learn in this lecture?

•! The concept and the difference between symmetry elements and operations

•! The four symmetry elements and the operations they can generate

•! Some properties of symmetry operations

•! Product and commutation rules of symmetry operations

•! Optical isomerism and group theory