ki2141-2011-molecular_symmetry_2013_11_29

Molecular Symmetry Achmad Rochliadi, MS., PhD.

and Dr. Veinardi suendo

Molecular Symmetry 2

Contents

What and Why?

The symmetry elements and operations

The symmetry classification

Consequences of symmetry

Linear algebra in symmetry

Group representation and character tables


What is symmetry ?

According to Webster Dictionary

Correspondence in size, shape and relative

position of parts that are on opposite sides of a

dividing line or median plane or that are

distributed about a center of axis.

Molecular symmetry

If a molecule has two or more orientation that

are indistinguishable then the molecule

possesses symmetry.


Why symmetry important in chemistry

The symmetry of the molecule tells us whether

the molecule is chiral, and whether it has a

dipole moment.

Symmetry will allow us to interpret

spectroscopic measurements on molecules. It is

particularly important when we come to

interpreting the infrared (vibrational) spectra of

molecules.


Symmetry is important in interpreting the crystal

structures of molecules. Modern X-ray diffraction

methods use symmetry in order to interpret the spectra

obtained and determine the absolute position of atoms

within a crystalline solid, and hence its structure.

Symmetry is crucial both in understanding the

electronic structure of molecules (Molecular orbital, or

MO theory). It is crucial in simplifying the otherwise

computationally intensive calculations that need to be

carried out in order to find the energies of molecules

and hence predict their structure and the chemical

reactions that can be carried out on them


Symmetry Elements and Operations

Symmetry Elements

A point, line or plane in the molecule about

which the symmetry operation take out. There is

only 5 symmetry elements related to molecule

symmetry.

Symmetry Operations

Some transformations of the molecule such as a

rotation or reflection which leaves the molecule

in a configuration in space that is

indistinguishable from its initial configuration.


Symmetry Elements

Elements Symbol Operation

Identity E Leaves each particles in its

original position

N-fold proper

axis

Cn Rotation about the axis by

3600/n (or by multiply)

Plane Reflexion in plane

Inversion

center

i Inversion through center

N-fold

improper axis

Sn Rotation by 3600/n followed

by reflexion in a plane

perpendicular to the axis


1. The identity operators, E

The simplest symmetry operation is known as the

identity operator, given the symbol E (E from the

German word Einheit, meaning unity). The E operator

basically means do nothing to the molecule.

Evidently, if you do nothing to the molecule it will look

the same as when you started. The identity operation

will thus work on all molecules.

Many non-symmetrical molecules,

such as the amino acid alanine

shown here, contain only the

E operator.


2. The n-fold rotation operators, Cn

Rotation axes have the nomenclature Cn which means rotate the molecule around the specified axis through an angle of 360/n. Thus, a C2 axis means rotate by 180, C3 by 120, C4 by 90 and so on.

An easy way to remember this is that n is the number of times you would have to rotate the molecule before you would get back to the beginning.

Note that some molecules can contain more than one rotation axis. When this occurs we refer to the axis with the highest degree of rotational symmetry as the principle axis.

Operation C1 is a rotation through 360, and it is equivalent to E Operator.

Operation Cnm

, corresponding to m successive Cn rotations.


Some of the rotation symmetry elements of a cube. The twofold, threefold and fourfold axes are labeled with the conventional symbols.


(a) An NH3 molecule has a threefold (C3) axis and (b) H2O molecule has a twofold (C2) axis.

Symmetry modeling : http://www.ch.ic.ac.uk/local/symmetry/


Successive rotations


Benzene have C6, C2, the principal axis is the sixfold

axis that perpendicular to the hexagonal ring.


3. The reflections operators,

A mirror plane (symbol ) is a symmetry

element that results in the reflections of

the molecule through a mirror plane.

If the plane is parallel to the principal

axis, it is called vertical and denoted v.

If the plane is perpendicular to the

principal axis, it is called horizontal and

denoted h

A vertical mirror plane that bisect

(divide) the angle between two C2 axes is

called a dihedral plane and denoted d.


An H2O molecule has two mirror planes. They are both

vertical so denoted v and v.


Dihedral mirror planes (d) bisect the C2 axes

perpendicular to the principal axis.


3. The inversion operators, i

Imagine taking each point in a molecule, moving it to the centre of the molecule, and then moving it out the same distance on the other side.

Move every atom at position (x,y,z) to position (-x,-y,-z). If the molecule still looks the same, then it contains a centre of inversion.


A Regular octahedron has a centre of inversion


Inversion of benzene, notice that the three of the C-H

groups have been color coded. When the inversion is

performed, these groups move to the mirrored side.


5. The n-fold improper rotation, Sn

An improper is the most complex symmetry element to

understand. An improper rotation consist of TWO steps

and neither operation alone needs to be a symmetry

operation:

The rotation like Cn, where the molecule is rotate around the

axis.

Reflection through a plane perpendicular to the axis of that

rotation,


(a) A CH4 molecule has a fourfold improper rotation axis (S4); the

molecule is indistinguishable after a 90 rotation followed by a

reflection across the horizontal plane, (b) The staggered form of

ethane has an S6 axis composed of a 60 rotation followed by a

reflections.


To classify the molecules according to their

symmetries, the molecule symmetry elements is

listed and collect together the molecules with

the same list of elements.

The name of the group is determined by the

symmetry elements it possesses.

Two system of notation

The Schoenflies system, more common.

The Hermann-Mauguin system/International

system, exclusively used in crystal symmetry.

The symmetry classification (Group Theory)


What is point group

A point group is a collection of symmetry

operations that together are specific to a wide

number of different molecules. These

molecules are from a symmetry viewpoint,

equivalent. For example, both water and cis-

dichloroethene are members of the C2v point

group. Once you have learned about the various

symmetry operations, go to the link on point

groups to find out more about this concept.


Group Theory

In essence, group theory is a set of

mathematical relationships that allow us to

study symmetry. An in depth and rigorous

study of group theory requires an extensive

knowledge of matrix algebra. As chemists, we

can usually concern ourselves less with the

details of the math, and more on visualizing

how symmetry operations transform molecules

in three dimensional space.


The diagram for

determining the point

group of a molecule


Summary of shapes corresponding to

different point groups.


Group C1 Ci Cs

Molecule belong to C1 if has no other element

than the identity (1). Ci if has identity and

inversion (3), and Cs if it has identity and a

mirror plane alone (4).


Group Cn Cnv Cnh

Cn : possess n-fold axis (5)

Cnv : possess n-fold + v (H2O; NH3)

Cnh : possess n-fold + h (6) - (7)


Group Dn Dnv Dnh

Dn : possess n-fold axis ntwofold axes

perpendicular to Cn

Dnh : possess n-fold axis n-twofold axes

perpendicular to Cn + h (8, 9, 10, 11)

Dnd : possess Dn + n dihedral mirror planes d

(12. 13)


Group S2n

Its the molecule that has not classified into one

of the group above but possess one S2n axis.


The cubic groups

Molecule that possess more than one principal

axis belong to the cubic groups.

Tetrahedral groups : Td (a), T (a), and Th (a)

Octahedral groups : Oh (b) and O (b)

Icosahedral group : Ih (c) and I (c)


(a) T (b) O

(c) I (a) Th


The full rotation group

The molecule rotational group, R3, consists of

infinite number of rotation axes with all

possible values of n.

Sphere and an atom belong to R3

C C


Cv and Dh point groups

Asymmetrical diatomics (e.g. HF, CO and [CN]-) and linear polyatomics that do not possess a centre of symmetry (e.g. OCS and HCN) possess an infinite number of sv planes but no sh plane or inversion centre. These species belong to the Cv point group.

Symmetrical diatomics (e.g. H2, [O2]2-) and linear polyatomics that

contain a centre of symmetry (e.g. [N3]-, CO2, HCCH) possess a sh

plane in addition to a C axis and an infinite number of sv planes. These species belong to the Dh point group.


Groups of high symmetry


D5h D5d

Exercise 1



Polarity

If the molecule belongs to group Cn with n > 1,

it cannot possess a charge distribution with a

moment dipole perpedicular to the symmetry

axis but it may have one parallel to the axis.

Molecule belong to Cn, Cnv, Cs may be polar.

All other group such C3h, D, ets there are

symmetry operations that take one end ot

molecule into the other.


(a) A molecule with a Cn

axis cannot have a dipole

perpendicular to the axis,

but (b) it may have one

parallel to the axis.



Chirality

A chiral molecule is a molecule that cannot be superimposed on its mirror image. A chiral molecule is an optic active molecule.

A molecule may be chiral only if it does not posses an axis of improper rotation, Sn.

Take notice that Sn operation could be present under different symmetry element. Example: molecules belonging to the groups Cnh posses an Sn axis implicity because the possess both Cn and h, which are the two components of an improper rotation axis. So as the molecule have i elements.


Some symmetry elements are implied by the other symmetry elements in a group. Any molecule containing an inversion also possesses at least an S2 element because i and S2 are equivalent


S4


Exercise 2


Linear algebra in symmetry: Dictionary


The effect on a matrix of a change in coordinate system


Traces and determinants

The trace of a matrix is defined as the sum of the

diagonal elements.

The determinant of a matrix is a value associated

with a square matrix that can be computed from

the entries of the matrix by a specific arithmetic

expression:


Group representation and character tables


Example: C2v point group

C2 operation

sv(xz) operation


Example 1: C2v point group (H2O)

C2 sv(xz) operation

Proof the following statements:

C2 C2 = E

sv(xz) sv(yz) = C2

sv(yz) sv(yz) = E


Characters

The character, defined only for a square matrix, is the trace of the matrix, or the sum of the numbers on the diagonal from upper left to lower right. For the C2v point group, we can obtained the following characters:

We can say that this set of characters also forms a representation, which is an alternate shorthand version of the matrix representation.

Whether in matrix or character format, this is called a reducible representation, a combination of more fundamental irreducible representations.

Reducible representations are designated with a capital gamma (G).


Reducible and irreducible representations


Character tables


Properties of characters of irreducible

representations in point groups

Also

i

i Eh )(


Properties of characters of irreducible

representations in point groups


Example 2: C3v point group (NH3)


Transformation matrices


In the C3v point group (C32) = (C3), which means that

they are in the same class and described as 2C3 in character table.

In addition, the three reflections have identical characters and are in the same class, as described as 3sv.

The transformation matrices for C3 and C32cannot be

block diagonalized into 1 1 matrices because the C3 matrix has off-diagonal entries. However, they can be block diagonalized into 2 2 and 1 1 matrices, with all other matrix elements equal to zero.

Transformation matrices


Character tables of C3v point group

The C3 matrix must be blocked this way because the (x,y) combination is needed for the new x and y, while the other matrices must follow the same pattern for consistency across the representation.

The set 2 2 matrices has the characters corresponding to the E representation, while the set of 1 1 matrices matches the A1 representation.

The A2 representation can be found using the defining properties of a mathematical group as in previous example.


Character tables of C3v point group: A1


Character tables of C3v point group: E

Antisymmetric Symmetric disymmetric


Properties of the characters for C3v point group


Addition feature of character tables

The expression listed to the right of the characters indicate the symmetry of mathematical functions of the coordinates x, y and z and of rotation about the axes (Rx, Ry, Rz).

This can be used to find the orbitals that match the representation. For example: x with (+) and (-) direction matches the px orbital with (+) and (-) lobes in the quadrants in the xy plane; the product xy with alternating signs on the quadrants matches lobes of the dxy orbital.



In all cases, the totally symmetric s orbital matches the first representation of in the group, one of the A set.

The rotational functions are used to describe the rotational motion of the molecule.

In the C3v example, the x and y coordinates appeared together in the E irreducible representation with notation (x,y). This means that x and y together have the same symmetry properties as the E irreducible representation. Consequently, the px and py orbitals together have the same symmetry as the E irreducible representation in this point group.



Matching the symmetry operations of a molecule with those listed in the top row of the character table will confirm any point group assignment.

Irreducible representations are assigned labels according to the following rules, in which symmetric means a character of 1 and antisymmetric a character of -1.

Letter are assigned according to the dimension of the irreducible representation.



This might also give us information about degeneracies as follows:

A and B (or a and b) indicate non-degenerate

E (or e) refers to doubly degenerate

T (or t) means triply degenerate

Subscript 1 designates a representation symmetric to a C2 rotation perpendicular to the principal axis, and subscript 2 designates a representation of antisymmetric to the C2. If there are no perpendicular C2 axes, 1 designates a representation symmetric to vertical plane, and 2 designates a representation antisymmetric to a vertical plane.

Subscript g (gerade) designates symmetric to inversion, and subscript u (ungerade) designates antisymmetric to inversion.

Single prime () are symmetric to sh and double prime () are antisymmetric to sh when a distinction between representations is needed (C3h, C5h, D3h, D5h).


Molecular vibration: H2O (C2v)

Degree of freedom



Full C2 operation of H2O

The Ha and Hb entries are not on the principal diagonal

because Ha and Hb exchange each other in a C2 rotation, and x(Ha) = -x(Hb), y(Ha) = -y(Hb) and z(Ha) = z(Hb).

Only oxygen contribute to the character for this operation, for total of -1.


Reducing representations to irreducible representations


Symmetry molecular motion of water


IR Spectra of H2O

Experimental values are 3756, 3657 and 1595 cm1

Calculated IR spectrum of gaseous H2O


A bent triatomic: H2O (C2v)



2s atomic orbital of the O (a1)



2px atomic orbital of the O (b1)



2py atomic orbital of the O (b2)



2pz atomic orbital of the O (a1)



1s atomic orbital of the H (a1 and b2)

and

a1 b2




summation




Normalization


The MO Diagram of Water with the 2a1 Showing Bonding

and Antibonding Character

The MO Diagram of Water with the 2a1 Labeled as

Non-bonding


The MO Diagram of Water with the 1a1 Labeled as Non-

bonding


Further on character tables



References and further readings

P. Atkins and J. de Paula, Atkins Physical Chemistry, 8th Edition, Oxford University Press, Oxford, 2006.

S.F.A. Kettle, Symmetry and Structure: Readable Group Theory for Chemists, 3rd Edition, John Wiley & Sons, Chichester, 2007.

A.M. Lesk, Introduction to Symmetry and Group Theory for Chemists, Kluwer Academic Publishers, Dordrecht, 2004.

C.E. Housecroft and A.G. Sharpe, Inorganic Chemistry, 3rd Edition, Pearson Education Limited, Harlow, 2008.

G.L. Miessler and D.A. Tarr, Inorganic Chemistry, 4th Edition, Pearson Education Limited, Harlow, 2010.

ki2141-2011-molecular_symmetry_2013_11_29

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