Chapter 1 Fundamental Concepts - Yonsei

Chapter 1 Fundamental Concepts

1-4 Symmetry Point Groups

1-5 Point Groups of Molecules

1-6 Systematic Point Group Classification

1-7 Optical Activity and Symmetry

1-3 Combining Symmetry Operations

1-2 Defining the Coordinate System

1-1 Symmetry Operations and Elements

Molecular Symmetry and Group Theory

Focus in this text is the application of symmetry arguments to solve physical problems of chemical interest.

As a first step, we must identify and catalogue the complete symmetry.

Once this is done, we can employ the mathematics of groups to simplify the physical problem and subsequently to obtain chemically useful solutions.

Gas, liquid, solution, and solid

1.1 Symmetry Operations and Elements

Symmetry operations; it is a movement of an object about a symmetry element such that the object’s orientation and position before and after the operation are indistinguishable.

Symmetry elements; it is an imaginary geometrical entity such as a line, plane, or point about which a symmetry operation can be performed.

The object need not be in the identical position it had before the operations. It is only necessary that the position be indistinguishable and therefore equivalent.


The symmetry of a molecule or ion can be described in terms of the complete collection of symmetry operations it possesses.

Operation will be one of five types.-identity, rotation, reflection, inversion, rotation-reflection (or improper rotation)

Elements are (1) the object itself, (2) a line (rotation axis orproper axis), (3) a plane (reflection plane or mirror plane), (4) a point (inversion center), and (5) a line (improper axis).

All symmetry elements will pass through a common point at the center of the structure. For this reason, the symmetry of isolated molecules and ions is called point group symmetry.


Identity (E)-doing nothing, need it for mathematical requirements of group theory.

The object is said to be asymmetric, if it possesses only E.

Rotation (Cn)-rotation about an axis by 2π/n radians, the value n is the order of rotation


Rotation (Cn)-C4 indicates a fourfold rotation.Two successive C4 rotations (which we could designate C4 ) about the same axis has the same effect as a single C2 rotation.

C4 = E

Cn = E

Cn = Cn

This single element, a C4 axis, is associated with three unique operations: C4, C4 = C2, C4 =C4.

2

4

n-1 -1

n

2 3 -1


Rotation (Cn)-Rotations beyond full circle are expressed as the equivalent single rotation that is less than 2π.

C4 = C4

There also exists a C2 axis collinear with the C4 axis.

There are four other C2 axes.We can distinguish these by adding prime(’) and double prime (”)

5

Pass through more atoms than


Rotation (Cn)-

For square planar MX4, the rotational operations grouped by class are 2C4(C4 and C4), C2 (collinear with C4), 2C2, and 2C2.

The highest-order rotational axis an object possesses is called the principal axis of rotation.

3 ’ ”


Reflection (σ)-

For square planar MX4, three kinds of mirror plane.


The operation of σh (horizontal mirror plane) appears to do nothing to the molecule.-however, if atoms had a directional property perpendicular to the plane, the property will tune to negative.-defined as perpendicular to the principal axis of rotation-if no principal axis exists, defined as the plane of the molecule

Reflection (σ)-Performing two successive reflections about the same plane brings the object back into original configuration; σ σ = σ2 = E


A σv (vertical mirror plane) and σdplane (dihedral mirror plane) are defined as to contain a principal axis of rotation. -defining σv to contain the greater number of atoms

Reflection (σ)-


Inversion (i)-

Related to the central point within the molecule.-The central point (0, 0, 0) is called an inversion center or center of symmetry.-Molecules that have inversion symmetry are said to be centrosymmetric.-performing inversion twice in succession would bring every point back into itself; ii = i2 = E


Inversion (i)-

No inversion center for ethane in the eclipsed configuration


Rotation-reflection (Sn)-

-also called improper rotation-a proper rotation (Cn) followed by a reflection (σh) in a plane perpendicular to the axis of rotation.-if both Cn and σh exist, Sn must also exist.-however, the presence of both Cn and σh is not requirement for the existence of Sn



-like proper rotation, a series of improper rotations can be performed about the same axis.S4S4 = S4 = C2S4 = S4S4 = E-there are two S4 operation (S4 and S4)

2

3 -1

4

3



-there equivalent S4 axes-total six operations, which belong to a class designated 6S4

1.2 Defining the Coordination System

We will adopt a standard Cartesian coordination system with x, y, z defined by so-called “right-hand rule”.

1.3 Combining Symmetry Operations

Written in a right-to-left order-BA means “do A first, then B”.-S4S4 = C2, σhC4 = S4.-Commutation is nor generally observed does not mean that it is never observed.-Some cases AB = BA, however, some cases AB = BA


Symmetry elements of CBr2Cl2.-the complete set of symmetry operations for the molecule consists of identity (E), a twofold principal axis of rotation (C2), and two reflections about different mirror planes (σv and σv ).’

1 X 4 matrix to describe the locations of atoms

Represent operator matrices


EE = E, C2E = EC2 = C2, σvE = Eσv = σvSince E really does nothing to the molecule

C2σv

σv’C2σv σv’=


EE = E, C2E = EC2 = C2, σvE = Eσv = σvSince E really does nothing to the molecule

σvC2

σv’

C2σv σv’=

σvC2 σv’=

σvC2 =They are commutative


Multiplication table

Order of combination is row element (top) first, followed by column element (side)

1.4 Symmetry Point Groups

The complete set of symmetry operations exhibited by any molecule defines a symmetry point group.

The set must satisfy the four requirements of a mathematical group: closure, identity, associativity, and reciprocality.

Closure;If A and B are elements of the group G, and if AB =X, then X is also in the group.

In the case of symmetry groups, the group elements are the symmetry operations, not the symmetry elements.

The operations of CBr2Cl2 shows that all binary combinations equal either E, C2, σv or σv; this complete set of elements of a point group called C2v.The order of the group h = 4

’

Associativity;If A, B, C, and X are members of the group G and C(BA) =X, then (CB)A = X, too.C2(σvσv) = (C2σv)σv

Do not confuse to commutation.Not always allow this CBA = BAC.In the mathematics of groups, any group in which all combinations of elements commute is said to be Abelian.


Identity;Any group G must has an element E.

’’


Reciprocality;Every element A of the Group G has an inverse, A-1, such that AA-1 = A-1A = E.

SubgroupAmong the operations that constitute a point group, there generally exist smaller sets that also obey the four requirements of a group.

If g is the order of subgroup of a group whose order is h, then h/g = n, where n is an integer.

For C2v, where h = 4, only subgroups with orders g = 1 and g = 2 are possible.

The group C1 has no symmetry and is therefore asymmetric.

C1 C2 Cs Cs

1.5 Point Groups of Molecules

Schönflies notation;We have seen so far C2, Cs….

Four general categories

n is the order of the principal axis

Nonrotaional Groups

Single-Axis Groups

Dihedral Groups


Nonrotational Group

E E, σh E, i


Single-Axis Rotational Groups

E, C2,

Multiplication table for the point group C4

The recognition of the pattern makes it easy to construct the multiplication table for any cyclic group

Cn groups

Cyclic GroupsThey are Abelian



Cnv groups

2E, C3, C3, 3σvE, C4, C2, C4, 4σv

C∞ groups; all noncentrosymmetric linear molecules are belong to here (e.g., HCl, ClBeF)



Cnh groups

Since Cnσh = Sn and C2σh = S2 = i, these groups also have n-fold improper axes when n>2, and they are centrosymmetric when n is even.

E, C2, σh E, C3, C3, σh2



S2n groups

As the 2n notation implies, only groups of this type with even-order principal improper axes exist.An odd-order Sn axis is the same as that generated by the combination Cn and σh, which defines groups of the type Cnh.

3E, S4, S2= I, S4


Dihedral Groups

Dn groups

The dihedral groups have n twofold axes perpendicular to the principal n-fold axis

2E, C3, C3, 3C2


Dihedral Groups

Dnd groups

2E, C3, C3, 3C2, S6, S6, S6, S6, S6, 3σd2 3 4 5

out


Dihedral Groups

Dnh groups

2E, C3, C3, 3C2, σh, 3σv


Cubic Groups

24 operation, listed by classes: E, 8C3 (4C3, 4C3)3C2, 6S4 (3S4, 3S4), 6σd)

2

3

48 operation, listed by classes: E, 8C3 (4C3, 4C3)6C4 (3C4, 3C4), 6C2, 3C2 (3C4), i 6S4 (3S4, 3S4),8S6 (4S6, 4S6), 3σh (σxy, σyz, σxz) 6σd)

2

3

2 3

5

120 operation, listed by classes: E, 12C512C5, 20C3, 15C2, i 12S10, 12S10, 12S6, 15σ) 2

1.6 Systematic Point Group Classification

Identifying the point group of a molecule is a necessary first step for almost all applications of group theory in chemistry.

Check key symmetry elements in a prescribed sequence.

1.6 Systematic Point Group Classification

1.7 Optical Activity and Symmetry

EnantiomersCompounds that can exist as enantiomeric pairs are called chiral; they are said to be dissymmetric.Not always dissymmetric = asymmetric.

1.7 Optical Activity and Symmetry

A molecule is dissymmetric and may be chiral either if it is asymmetric or if it has no other symmetry than proper rotation.The possible chiral groups are C1, Cn, and Dn.

Belongs to C2 point group, but not be chiral.

Free rotation about O-O bond

Chapter 1 Fundamental Concepts - Yonsei

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