Lecture 6Lecture 6

Molecular Symmetry

Reading: Shriver & Atkins, 6.1-6.3g ,

Intro to symmetrySymmetry, from Merriam-Webster

1: balanced proportions; also : beauty of form arising from balanced proportions 2: the property of remaining invariant under certain changes (as of

i i i f h i f h l i h f i f h orientation in space, of the sign of the electric charge, of parity, or of the direction of time flow) —used of physical phenomena and of equations describing them

Ceiling of Lotfollah mosque (I ) h i l

Taj Mahal, an example of (Iran) has rotational symmetry

of order 32 and 32 lines of reflection. A Kolam, the first ritual daily act of a

Hindu household

bilateral symmetry

CPT Symmetry

The implication of CPT symmetry is that a "mirror-image" of our universe — with The implication of CPT symmetry is that a mirror image of our universe with all objects having their positions reflected by an imaginary plane (corresponding to a parityinversion), all momenta reversed (corresponding to a time inversion) and with all matter replaced by antimatter (corresponding to a charge inversion)—

ld l d l h l l

1980 Noble Prize: CP breaking in m mesons (kaons)

would evolve under exactly our physical laws.

in m-mesons (kaons)

Cronin & Fitch

The importance of symmetry

To predict the translational, rotation, and vibrational motions of a molecule, as well as the transformation of orbitals

Which in turn tells us:

•Allowed electronic and vibrational energies•Optical spectra

O ti l ti it & i l di h i•Optical activity & circular dichroism•Chemical reactions involving breaking and forming bonds

•Spectroscopic selection rules that determine which transitions p pare allowed can be expressed in terms of molecular symmetry of

energy states

Symmetry Operations

• Symmetry operation: an action that leaves the molecule unchanged• Symmetry operation: an action that leaves the molecule unchanged.

• Symmetry element: a point, line, or plane with respect to which the symmetry operation is performedoperation is performed

n-fold Rotations: Cn

• n an integer g• rotation by 360°/n about a particular axis defined as the n-fold rotation axis.

• C2 = 180° rotation• C3 = 120° rotation• C4 = 90° rotation

• C5 = 72° rotation, C6 = 60° rotation, etc.

xnnx02sin2cos

'

zyx

nn

nn

zyx

100

02cos2sin'

'

zz 100

Transformation matrix

n-fold Rotations: Cn

• n an integer g• rotation by 360°/n about a particular axis defined as the n-fold rotation axis.

• C2 = 180° rotation• C3 = 120° rotation• C4 = 90° rotation

• C5 = 72° rotation, C6 = 60° rotation, etc.

Rotation of H2O(note that clockwise and anti-clockwise rotations by 180o are identical)

n-fold Rotations: Cn

R i f NHRotation of NH3• Trigonal pyrimidal molecule

• Three-fold rotation axis, C3, but two operations associated with this axis: l k i i b 120o d h i l k i i b 120oa clockwise rotation by 120o and another anticlockwise rotation by 120o.

C32C3

n-fold Rotations: Cn

R i f X FRotation of XeF4 • Square planar molecule

• Four-fold rotation axis, C4; also two C2 rotation axesB ti th hi h t d t ti l i (th i i l i ) d fi • By convention, the highest order rotational axis (the principal axis) defines z.

Labeling convention:• A single prime (C2’)

indicates axis passes through several atoms

• A double prime (C2’’) i di t th t it indicates that it passes between outer atoms.

Mirror plane, σ•Invariance to reflection, possessed either by a single object or by a set of objects, p y g j y j

N

H HF

001

010001

yz 100

Mirror plane, σ•Invariance to reflection, possessed either by a single object or by a set of objects, p d by g bj by bj

•when perpendicular to principal axis (σh)•when parallel to principal axis:

σvvσd (‘dihedral’) if bisects C2’ axes

XeF4

H OH2O

Center of Symmetry (Inversion), i

•Takes a point on a line through the origin (the inversion center) to an equal distance on the other side

• Transforms a point with coordinates (x,y,z) to one with the coordinates (-x, -y, -z)

SF6

001

SF6

100010i

Center of Symmetry (Inversion), i

•There need not be an atom at the center of inversion (i.e., N2)• While an inversion and two-fold rotation may sometimes achieve the

same result, they must be distinguished• no tetrahedral molecule has a center of inversion

i C2i 2

Example

Whi h f h f ll i Which of the following isomers of

(gly)2CO(OH)2CO(gly)2

have a center of symmetry?have a center of symmetry?

(a) (b) (c) (d)

(e) (f) (g)

Example

Which have mirror planes?

(a) (b) (c) (d)

(e) (f) (g)

Improper Rotation (Sn)

•A t ti fl ti•A rotation-reflection•A rotation Cn followed by a reflection in the plane perpendicular to the

Cn axis•To obtain the produce of two symmetry operations [σ (C ) ] multiply To obtain the produce of two symmetry operations [σxy(Cn)z], multiply

the transformation matrices in sequence from right to left

22

02sin2cos

22

02sin2cos001 nnnn

100

02cos2sin

100

02cos2sin100010)(

nnnnS zn

Improper Rotation (Sn)

•A t ti fl ti•A rotation-reflection•A rotation Cn followed by a reflection in the plane perpendicular to the

Cn axis•To obtain the product of two symmetry operations [σ (C ) ] multiply To obtain the product of two symmetry operations [σxy(Cn)z], multiply

the transformation matrices in sequence from right to left

C4 σh

Improper Rotation (Sn)

(a) An S1 axis is equivalent to a mirror plane(b) A l f(b) An S2 axis is equivalent to a center of inversion

Identity Operation E

• Does noting to the moleculeE l l h hi i• Every molecule has this operation

• Some molecules have only this operation• E has the same importance as the identity matrix in mathp y

001

100010E 100

Example: Eclipsed Ethane

C CC C

H

What are the symmetry elements?

Point Groups

Properties of a Group:Properties of a Group:

• The product of any two operations must be an operation of the group (closed under multiplication)(closed under multiplication)

• Every operation must have an inverse (an operation that will undo the effect of the first operation)

E t t i th id tit E• Every group must contain the identity E

• All operations of the group are associative: ABC=(AB)C=A(BC)

• The product of any two operations or elements in defined. Groups for which all elements commute are Abelian.

Point Group:p

• The set of symmetry operations that describe a molecule’s overall symmetry

Point Groups

Generated by repetition of Cn operation

The C1h group contains only a horizontal mirror plane and is termed “Cs”

Mirror plane contains the rotational axis: Cnv

Linear molecule no CLinear molecule, no C2

axis perpendicular to C ∞

More point groupsDihedral group D : addition of C2Dihedral group Dn: addition of C2

axis perpendicular to a Cn axis

Adding σh to a D group generates a D hAdding σh to a Dn group generates a Dnh

group, with 4n symmetry operations

Linear molecule

Tetr hedron groupTetrahedron group

Octahedral group

Assigning Molecules to Point Groups

Assigning Molecules to Point Groups

H2O

Assigning Molecules to Point Groups

C60

Identifying symmetry species

Let’s identify the symmetry species of the oxygen

l h ll i valence-shell atomic orbitals in an H2O molecule (C2v symmetry)( 2v y y)

O: 2s2 2p4

E C2 σv σv’

2s

2pz

2pxpx

2py

Identifying symmetry species

Let’s identify the symmetry species of the oxygen

l h ll i valence-shell atomic orbitals in an H2O molecule (C2v symmetry)( 2v y y)

O: 2s2 2p4

E C2 σv σv’

2s 1 1 1 1

2pz

2pxpx

2py

Identifying symmetry species

Let’s identify the symmetry species of the oxygen

l h ll i valence-shell atomic orbitals in an H2O molecule (C2v symmetry)( 2v y y)

O: 2s2 2p4

E C2 σv σv’

2s 1 1 1 1

2pz 1

2px 1px

2py 1

Identifying symmetry species

Let’s identify the symmetry species of the oxygen

l h ll i valence-shell atomic orbitals in an H2O molecule (C2v symmetry)( 2v y y)

O: 2s2 2p4

E C2 σv σv’

2s 1 1 1 1

2pz 1 1 1 1

2px 1px

2py 1

Identifying symmetry species

Let’s identify the symmetry species of the oxygen

l h ll i valence-shell atomic orbitals in an H2O molecule (C2v symmetry)( 2v y y)

O: 2s2 2p4

E C2 σv σv’

2s 1 1 1 1

2pz 1 1 1 1

2px 1 ‐1 1 ‐1px

2py 1

Identifying symmetry species

Let’s identify the symmetry species of the oxygen

l h ll i valence-shell atomic orbitals in an H2O molecule (C2v symmetry)( 2v y y)

O: 2s2 2p4

E C2 σv σv’

2s 1 1 1 1

2pz 1 1 1 1

2px 1 ‐1 1 ‐1px

2py 1 ‐1 ‐1 1

Identifying symmetry species

Let’s identify the symmetry species of the oxygen

l h ll i valence-shell atomic orbitals in an H2O molecule (C2v symmetry)( 2v y y)

O: 2s2 2p4

E C2 σv σv’

2s 1 1 1 1 A1

symmetric

2pz 1 1 1 1

2px 1 ‐1 1 ‐1

1

A1

Bantisymm.

px

2py 1 ‐1 ‐1 1

B1

B2

Identifying symmetry species

Ch t t bl Di l ll th t l t f th i t • Character table: Displays all the symmetry elements of the point group , together with a description of how objects (such as an

atomic orbital) transform under the symmetry operation

• the order of a group, h, is the total number of symmetry operations that can be carried outoperations that can be carried out

• The point group of a molecule and its associated character table b d t di t th t l ti l t ti d ib ti lcan be used to predict the translational, rotation, and vibrational

motions of a molecule, as well as its dipole moment, chirality, and optical spectra.

• Also, in molecular orbital theory, all components of the MO must behave identically under transformationbehave identically under transformation