# SYMMETRY ELEMENTS AND POINT GROUPS

**SYMMETRY ELEMENTS**

The use of symmetry elements has enabled shape of objects to be classified in categories for identification. This however is used in chemistry to classify the shape of the structure of chemical compounds into different groups based on the symmetry elements that can be found in the structure of the compound. The elements include, **E, C _{n}, I, S_{n} and σ**

**.**

**IDENTITY ELEMENT (E)**

**This** is the symmetry element obtained by viewing the shape of an object or the structure of compound when **nothing** is done to it. This implies doing nothing or viewing the structure of the compound through several **angles or axis** in space and arriving back at the same **original **position as if nothing has happened before. This is like turning an object through 360-degree angle. Every object has E symmetry element.

**AXIS OF ROTATION (C**_{n})

_{n})

**C _{n}** is a symmetry element that refers to the

**axis or direction**by which we can

**turn an object**to obtain a view of the shape that cannot be differentiated from the first view. This can occur along multiple axis or viewpoint on the object. Each turn or rotation in a particular direction or plane is

**C**is (C

_{n}axis_{n})

**where n is the maximum number of turns needed to obtain an**

**indistinguishable view in the particular plane or direction**. Turning the object to get an indistinguishable view is a C

_{n}(n=

**number of turns by which the original shape can be obtained**). A turn of one 180 degree to get an indistinguishable view is a C

_{2}axis. If only one turn of 360 degrees can give the indistinguishable view, then we have a C

_{1}or E.

**Principal C _{n} axis**

**is**

**maximum number of turns needed to obtain indistinguishable view of the shape of an object.**

**Subsidiary C _{n} axes**:

Any Cn axis less than the principal Cn axis is a subsidiary axis. Some objects have other C_{n} axes in addition to the main principal C_{n} axes, for example a square has a **C _{4} principal axes**, but it also has four C

_{2}axes. An equilateral triangle has C

_{3}principal axes and three subsidiary C

_{2}axes.

**PLANE OF SYMMETRY ( σ )**

**σ** is the symmetry element obtained by viewing the object along a plane that divides the object into two equal parts. This can be a **vertical or horizontal plane.**

**Horizontal plane** (σ_{h}) is the plane that cuts through along the plane of the page **but is perpendicular to the principal axis.**

**A vertical plane** (σ_{v}) cuts through the object out and above the shape into two equal parts **but it contains or coincides with the principal axis.**

The 2-d shape of some objects can have **multiples vertical planes** of symmetry (**σ _{v}**) but only one horizontal plane of symmetry (

**σ**) if possible while other objects have no plane of symmetry.

_{h} **σ _{h (plane of the page)}**

**CENTER OF INVERSION (***i *)

*i*)

**I** or **center of inversion is a symmetry element that is present if **all the point locations on one half of an object connects along lines that crosses the center to reach points of the same kind at the opposite half equal distance from the center. This cannot be really done on a real object or model structure. [Trying to look simultaneously the front and back of a ruler will end up breaking the ruler.]

**IMPROPER ROTATION (***S*_{n})

*S*)

_{n}**S _{n} is the symmetry** element

**ob**tained by turning the object clockwise along a principal

**C**axis and then viewing the object reflected image through a plane right-angled to the

_{n}**C**axis. This is called

_{n}**improper rotation**of the object.

Example of illustrations

__Dihedral plane (__*d*) or σ_{d}

__Dihedral plane (__

*d*) or σ

_{d}This is a vertical plane that **bisect two C _{2} axis but **

**only passes through principal C**such that the shape is divided into two equal parts. Examples given below have vertical planes along the C

_{n}axis and no other point or atom_{2}axes. Other examples involve objects with shape or structure made up of two layers in

**staggered**conformations. Best view of this is obtained by using a model kit.

**Multiplication of Symmetry elements**

**Symmetry Operations: ****Effect of symmetry elements on others **

** **This is what is obtained by turning a model or shape of an object through an axis or a plane of reflection followed by another symmetry operation. For example, turning a C_{1} symmetry element through 360 degrees (C_{1}) gives you an E symmetry, that is **C _{1} x C_{1} = E. **

**Symmetry element with even number of axes raised to exponent power equal to the number of axes gives the identity element. Example C**_{2}^{2}**=E, or also E= C**_{4}^{4 }

** Symmetry element with odd number of axes raised to exponent power equal to double number of axes gives the identity element E. Example, C _{3}^{6}=E .^{ }**

In the example below, the object has a **C _{2}** axis and two perpendicular vertical planes of symmetry (

*) .*

**σ**_{v}and σ_{h }Reflection through the **vertical plane (σ _{v1})** and then rotation through

**C**is the same as reflecting the original object directly through the vertical plane

_{2}axis**σ**This is written as

_{v2}.**C**

_{2 }x σ_{v }= σ_{v2}TABLE OF SYMMETRY OPERATIONS

You will also see that **σ _{v2}_{ }x σ_{v1 }= C**

_{2 }This means reflecting the shape through the

**vertical plane (σ**and then reflecting again through the perpendicular vertical

_{v1})**plane (σ**

_{v2}

**)**gives the same result as a

**C**.

_{2}symmetry elementPractice with the shape or structure of **WATER (H _{2}O)** molecule.

P.S. In water structure the **principal axis C _{2}** lies with the

**planes of symmetry σ**and is

_{v1}and σ_{v2}**not**perpendicular to any of the planes so there is

**no**

**horizontal plane of symmetry**.

Other molecules that fit this description include SO_{2} , or three membered ring compounds containing one different heteroatom but two atoms of the same kind. [epoxides]

Try and practice with an equilateral triangle and write a table of the results of the symmetry operations.

Examples of compounds that fit this description include **cyclopropane. **The example below shows the results of rotation of a **rectangle** through a **C _{2}** axis followed by reflection in a

**vertical plane σ**. This gives several results. The black arrow is nothing but just showing the orientation of the object after rotation or reflection.

_{v}We can use this to make multiple symmetry operations and use the results to write equations. However, the equations are only true for this particular object or structure under consideration. It may not be true for another object.

We can write several equations based on the combination of symmetry operations on the **rectangle** shown above. A few examples can be written as follows.

**TABLE OF SYMMETRY OPERATIONS**

Example of compound that fits this description is 1,3-**cyclobutadiene.**

**Symmetry** **Commutations**

Two symmetry operations **commutes** if the inverse is equal to the same result. For example, **C**_{2 }**x σ**_{v2 }** commute if σ**_{v}_{2 }** _{x }C_{2}** gives the same result. Looking at the results in the diagram above and the equations, we can see that the two symmetry elements

**C**actually commute because the reverse operation on the rectangle gives the same result. The two symmetry elements are perpendicular to each other in the rectangle. What is the name of this symmetry element that is generated as a result of this operation.

_{2 }and σ_{v2 }**[IMPROPER ROTATION S**in this case it will be

_{n}]**S**. However, in this particular case, other symmetry elements such as inv

_{2}**ersion**(

**) or combination of**

*i***pC**and

_{2}**σ**will also give the same result. [

_{h}**S**and

_{2}*gives same results in the table above. This shows that they are the same].*

**i****POINT GROUPS**

These are sets of alphanumeric characters used to designate the state of the shape or structure of an object in two-dimensional view. There are rules needed to correctly classify the point group of the shape of a structure. Examples compounds and their point groups include Water is C_{2v}, Ammonia is C_{3v}, HF is C_{αv}, HCCH, C_{αh}, Cyclopropane D_{3h }etc.

The names can be seen as a Capital upper case alphabet followed by a subscript number and or an alphabet. There are many of these, but the most common first upper-case letter is usually a **D or C**.

**D** is used if there are two or more subsidiary a C_{2} axis that crosses a perpendicular principal axis. Example is an equilateral triangle or Cyclopropane structure. It has three **C _{2}** axis that crosses at a perpendicular

**C**axis; therefore, we have a

_{3}principal**D**

_{3}.**The horizontal plane of symmetry supersedes the vertical plane, so we use h instead of v. Therefore, the point group is D_{3h}**

The same results will be obtained for BCl_{3}, B(OH)_{3}, H_{3}O^{+} and other compounds of that shape.

What is the point group of a rectangle, an isosceles triangle and a scalene triangle?

Other examples of point groups are given below using water and ammonia.

**All the vertical planes are perpendicular to each other, however all of them contain or are colinear with the Principal C _{n} axis and not perpendicular to principal C_{n} axis so there is no horizontal plane of symmetry in water and ammonia**.

**Special Cases**

**HIGH SYMMETRY:**

Certain objects have special names notations for the rotational axis because of very high symmetry number of rotational axis.

Examples of these point groups are listed below;

Tetrahedral shapes (T_{d}),

Octahedral (O_{h})

Dodecahedral or Icosahedral (I_{h}).

Infinite rotational axis:

Other objects have infinite number of rotational axis (C** _{α}**) e.g. a circle or a cylinder or a line or a dot.

C** _{α}** is perpendicular to a C

_{2}=> D

_{α}All the four objects have infinite number of C_{2 }axes and perpendicular principal axis **C _{α}**

There are infinite number of vertical planes of symmetry (**σ**_{v}) along the C_{α} axis.

The plane of symmetry (**σ**_{h}) that cuts the cylinder into two parts is **perpendicular to the C _{α} axis** =>

**horizontal plane (h).**This is in the plane of the page for the circle or dot. Therefore,

**Point group = D**

_{αh}Molecules with straight line or cylindrical shapes that fits this model are symmetrical linear molecules. e.g., HCCH, N_{2}, F_{2}, Cl_{2}, CO_{2} etc.

However, a slight variation gives a different point group in this example below.

In this example, the C** _{α}** has no perpendicular C

_{2}axis => we only have C

**principal axis.**

_{α}There is infinite number of vertical planes of symmetry (**σ**_{v}).

There is **no** horizontal plane of symmetry perpendicular C** _{α}** axis => we have ONLY VERTICAL planes of symmetry (

**σ**

_{v}) lying along the C

**axis. Therefore,**

_{α}**Point group = C**

_{αv}Examples of molecules that fit this model are non-symmetrical linear molecules, including HCl, HBr, CO etc.

**LOW SYMMETRY:**

Some objects have zero or very low number of symmetry elements.

Objects with only C_{n} rotational axis (n=1 or more) but **no plane of symmetry** =>Point group= C_{n}

Object with C_{n} =1 or E rotational axis but only **One plane of symmetry** =>Point group= C_{s}

Object with C_{n}=1 or E and Only (*i*) center of inversion =>Point group = C_{i}

Object with C_{n}=1 or E and Only S_{n} =>Point group= S_{n} or S_{2n} is related to Ci or i since S_{2} the same as C*i*

S_{n} can have several values based on the value of n or principal C_{n} axis if there is no plane of symmetry.

Example an object with principal axis of C_{2} (with no plane of symmetry) and two points of inversion exchange will have S_{2} point group. If the object that is C_{1} is passed through a C_{2} operation and then reflected to get the same object and orientation, then the point group is S_{2. }If the object gives the same shape and orientation after the C_{n} operation and there is no plane of symmetry, then the point group is C_{n} or D_{n} (if perpendicular C_{2} crosses the principal C_{n} Axis).

Examples of chemical structures with their symmetry and point groups are listed below.

**Try and identify which compound is chiral or achiral.**

;

OTHER EXAMPLES WITH LOW SYMMETRY

More difficult examples.

Other examples;

Other structures and their point groups;

More low symmetry compounds.

Structural derivatives of Methane and their point groups.

**S _{3} Cyclopropane derivative:**

This structure below has no plane of symmetry by careful examination. It can be rotated and reflected six times but the point group vertex corner as one rotation reflection(S_{2n}). since each vertex point has both rotation reflection substituents. Therefore, there each count as one S_{n} where n=3. Point group is S_{3}. All the structures are the same, but they are just drawn in three different ways.

**Organometallic complexes (hypothetical**):

These compounds shown below have no vertical plane of symmetry. however, the eclipsed conformation has horizontal plane of symmetry since the substituents are the same and aligned above and below. Point group is **C _{3h}**. The other compound has no horizontal plane of symmetry since it does not have the same number of substituents above and below.

.

**Organometallic complexes below are in staggered conformations**. They have no vertical or horizontal plane of symmetry. There is a principal **C _{3}** axis at the center of each triangular structure. There is also a

**C**axis through both structures above and below perpendicular to the

_{2}**C**axis. The point group is

_{3}**D**.

_{6}