SymmetryLecture by Inorganic Laboratory

CHEM4115 Inorganic Chemistry Laboratory II

Symmetry of Inorganic Molecules and Applications

January 29, 2009

Why do we need to study molecular symmetry? "Symmetry, unity, regularity, and simplicity are essential elementary qualities of beauty in both complex and simple things." ‐

Plato

Does the six‐fold symmetry axis of a flower make it more beautiful than the single reflection plane in a cockroach?

Such questions are beyond the scope of inorganic chemistry. Instead, we have a more modest goal, that of classifying the symmetry point groups of the molecules.

Why do we need to study molecular symmetry? Chemistry is the study of molecules and transformations between molecules.

Quantum chemistry provides prior knowledge of many molecular properties without experimentation.

Symmetry plays important part in: Molecular structures Molecular vibrations Electronic structures Many molecules have symmetry. These symmetries can be classified, and then used to predict molecular properties. Group theory simplifies quantum chemical calculations.

Group theory provides the link between molecular symmetry and molecular properties and changes.

Applications of Molecular Symmetry We can predict the number of signals in the NMR spectrum based on the symmetry of a molecule (or vice versa). For example, the 19F NMR spectrum of a series of hexa‐ haloantimonate anions.

[SbF6]‐ : Oh

trans‐[SbBr2F4]‐ : D4h

mer‐[SbBr3F3]‐ : C2v

1 signal

2 signals (int. = 2:1)

[SbBrF5]‐ : C4v

cis‐[SbBr2F4]‐ : C2v

fac‐[SbBr3F3]‐ : C3v

2 signals (int. = 4:1)

2 signals (int. = 1:1)

1 signal

Applications of Molecular Symmetry We can predict number of vibrational modes in the IR and Raman spectra of the molecule. Different molecular symmetries lead to different numbers of IR and Raman active bands. For example, cis‐ and trans‐isomers will typically give different vibrational modes because the molecules have different symmetries.

What is Molecular Symmetry? Symmetry operations Symmetry elements Point group

Symmetry Elements Imaginary lines or planes or points about which when certain operations can be performed, so that the molecule appears same before and after operation. * * Rotation

* *

Inversion

SymmetryÂ Elements

Symmetry Operations • The action that moves a molecule into a new orientation equivalent to the initial one. • Lines: Rotations about lines by 360o/n. • Planes: Reflection of the molecule once. • Points: changing the coordinates of all atoms in a molecule to negative values.

Symmetry Elements and Operations Element

Symbol

Operation

Rotation axis

n‐fold rotation

Plane of symmetry

Reflection in plane of symmetry

Rotation‐ reflection axis

n‐fold rotation followed by reflection in plane perpendicular to the axis

Center of symmetry

Projection through center of inversion to equal distance the other side from center

Identity element

Leaves the object unchanged

The table describes several symmetry elements and operations. Symmetry operations are actions that can be performed on a molecule and result in an equivalent structure. The simplest operation is to do nothing, the identity element (E) – the one element that all structures will have. A molecule in a specific geometry, or conformation, can be characterized by the set of all symmetry operations that will result in equivalent structures.

1. Rotation Axis, Cn • Rotation of the molecule about an axis by an angle θ where θ = 360o/n. • The axis is known as n‐fold axis.

180°

H(2)

H(1)

H(2)

H(1)

2‐fold axis, C2, n = 2, θ = 180o

1. Rotation Axis, Cn H(3)

H(1)

H(2)

H(1)

120°

C31

C32 H(3)

H(1)

H(2)

H(3)

3‐fold axis, C3, n = 3, θ = 120o Notes about rotation operations: ‐Rotations are considered positive in the counter‐clockwise direction. ‐Each possible rotation operation is assigned using a superscript integer m of the form Cnm. ‐The rotation Cnn is equivalent to the identity operation (nothing is moved).

1. Rotation Axis, Cn Linear molecules have an infinite number of rotation axes C because any rotation on the molecular axis will give the same arrangement. C

C∞

O(1)

C∞ C

N(1)

N(2)

O(2)

C , n = , θ = any angle

1. Rotation Axis, Cn The Principal axis in an object is the highest order rotation axis and typically assigned to the z‐axis if we are using Cartesian coordinates.

2‐fold axes, C2 3‐fold axis, C3 The principal axis is the 3‐fold axis through the center of the central atom.

2. Plane of Symmetry, σ  Reflection across a plane of symmetry (mirror plane).  A reflection operation exchanges one half of the object with the reflection of the other half.  Two successive reflections are equivalent to the identity operation (nothing is moved).  Reflection planes may be vertical, horizontal or dihedral.

2. Plane of Symmetry, σ • A vertical mirror planes, v, contains the principal axis. • A horizontal mirror plane, h, is perpendicular to the principal axis. This must be the xy‐plane if the z‐axis is the principal axis. • A dihedral mirror plane, d, generally bisects two vertical mirror planes.

2. Plane of Symmetry, σ σv

σv Cl

σh σd

Pt Cl

σd

2. Plane of Symmetry, σ

3. Rotation‐Reflection Axis, Sn •

Rotation of a molecule about an axis followed by reflection in a plane perpendicular to the axis.

•

It is a single operation. Neither the axis nor the plane is symmetry element, but their combination is. Also known as improper rotation.

•

4. Center of symmetry, i In this operation, every part of the object is reflected through the inversion center, which must be at the center of mass of the [x, y, z] object. 2 2

1 1

11 1 1 1 1

[‐x, ‐y, ‐z]

2 2

1 1 22

1 1

2 2

4. Center of symmetry, i

5. Identity, E • The identity operation does nothing to the object. • It is necessary for mathematical completeness, as we will see later.

Combination of Operations C44

C41

+90°

C44 = E +90°

C43

C42 = C2 C2

+180°

C42 +90°

Combination of Operations

Symmetry and Point Groups • Molecules do not have any random set of symmetry elements.

• Only certain specific set of symmetry elements are possible for a molecule.

• The collection of all the symmetry elements present in a molecule, along with an identity element, which does nothing, is known to form a GROUP.

• Because all of the symmetry elements of such groups intersect at a single point in the molecule, such groups are called point groups.

What is Molecular Symmetry?

Symmetry operations Symmetry elements Point group

To Study the Molecular Symmetry Two ways to study it: 1. Graphical approach – use picture 2. Mathematical approach – use matrix Limitations: 1. Free molecule 2. No influence from neighboring molecules

Identifying Point Groups We can use a flow chart such as this one to determine the point group of any object. The steps in this process are: 1. First determine shape using Lewis Structure and VSEPR Theory. 2. Determine the symmetry is special (e.g. octahedral, tetrahedral). 3. Determine if there is a principal rotation axis. 4. Determine if there are n 2‐fold axes perpendicular to the principal axis. 5. Determine if there are mirror planes. 6. Assign point group.

Are there n vertical planes, σv, containing Cn?

Special cases: Perfect tetrahedral (Td) e.g. P4, CH4

Perfect octahedral (Oh) e.g. SF6, [B6H6]2‐

Perfect icosahedral (Ih) e.g. [B12H12]2‐, C60

Are there n vertical planes, σv, containing Cn?

H2O O(1)

H(3)

H(2)

NH3

BH3

trans‐[SbF4ClBr]‐

Sb Br

CHFCl2

CHFClBr

C2F2Cl2Br2

2 1

1 1

H3C‐CH3 (staggered conformation)

CO2

Ni(CH2)4

BenzeneÂ

AcetoneÂ

Identifying Point Groups Group labels Symmetry elements*

Examples

One plane

NOCl, HCOOH

One n‐fold axis

Neither staggered nor eclipsed CCl3‐CH3

One centre of symmetry

Perfectly staggered (CHClBr)2

Cnv

One n fold axis and n vertical planes

H2O (C2v), CH3Cl (C3v), O3 (C2v), Co(NH3)4ClBr (C4v)

Cnh

One n‐fold axis, one plane perpendicular to the axis

trans D2‐Ethylene (C2h), B(OH)3 (C3h)

One n fold axis and n C2 axes perpendicular to the n‐fold axis

Neither staggered nor eclipsed ethane (D3)

Dnh

One n‐fold axis, one plane perpendicular to the axis and n C2 axes perpendicular to Cn

Ethylene (D2h), Benzene (D6h), [AuCl4]‐ anion (D4h)

*All groups have identity element as a natural member.

Keep practicing • You will be able to indentify the point group within eyes blinking. • There will be questions about point group in every quizzes along this semester. • http://symmetry.otterbein.edu/index.html

Ni(en)3 , en = H2NCH2CH2NH2

Propeller Square antiprism

Mg(5‐Cp)2 , eclipsed conformation)

Pentagonal prism

F2 F1 H

F3 O

H Cl(5)

H H

Cl(2)

Ni(1)

Cl(3)

H H

H O

Cl(4)

H N

H O

B H O

H H

O S

F Ph Me

C4v

Cl N

B Cl

F F

Br O

N F

C3h

C2h

D5d

F Co

chiral!

F F

F B

D5h

H C3v

F Fe

D3h

It is quite likely that you would forget everything about today but remember that there are

real pictures in group theory too, besides imaginary lines, planes and points!

Consciously or subconsciously, we all may consider issues of symmetry in determining beauty. Research suggests that animals and humans are influenced by symmetry in choosing mates.

Basics Properties of Groups 1. The combination of any two elements of the group must produce an element of the group. – AB = C, C must be member of the group

2. There must be an identity element (E). – AE = EA = A for all symmetry elements

3. Every element A must have a unique inverse denoted by A‐1 such that A‐1 must be a member of the group. – AA‐1 = A‐1A = E

4. The combination of elements in the group must be associative. – A(BC) = (AB)C = ABC

Basics Properties of Groups C41 +90°

+180°

C42

1. The combination of any two elements of the group must produce an element of the group.

+90°

C4 × C4 = C2

Basics Properties of Groups 2. There must be an identity element. The numbers {1, i, ‐1, ‐i} form a group under multiplication as the combining operation.

The elements {E, C2, σv , σ’v} form a group under multiplication as the combining operation.

1 × i = i 1 × ‐i = ‐i 1 × ‐1 = ‐1

E × C2 = C2 E × σv = σv E × σ’v = σ’v

Identity element for this group is 1

Identity element for this group is E

Basics Properties of Groups C4 +90°

Exactly the same

C4‐1 C 3 4

‐90°

+270°

3. Every element A must have a unique inverse denoted by A‐1 such that A‐1 must be a member of the group.

C4 × C43 = E

Basics Properties of Groups 4. The combination of elements in the group must be associative. C4

+90°

+180°

(C4 × C4) × C2 = C4 × (C4 × C2) C4

+90°

+180°

+90°