A Science of Aesthetics

Transformations You Cannot See.

Topics

Point Symmetry Operations, E C i S• Classes

Point Groups• Identification Scheme

Uses and Consequences

Symmetry Operations on Points

The nuclei of molecules are the points. Symmetry operations transform identical

nuclei into themselves.• After such operations, the molecule looks

absolutely unchanged.• Nature finds all these ways to fool you, so

symmetry has Entropy consequences!• We must be able to identify the operations too.

The Identity Operation, E

One way of finding a molecule is after nothing has been done to it! That counts.

No matter how asymmetrical a molecule is, it must have an identity operation, E.• The symbol “E” comes from the German,

“eigen,” meaning “the same.”• CBrClFI, bromochlorofluoroiodomethane, has

E as its only symmetry operation, for example.

The Rotation Element, Cn

Axes can often be found in molecules around which rotation leaves the atoms identical.

An n-fold rotation, if present, is symbolized by the element Cn., and represents n–1 rotational operations about the axis.• Each operation is a rotation through yet another

360°/n, but the operation C1 is merely E.

Finding Rotations

Symmetrical molecules often have rotation axes through their atoms.• SF6 is octahedral and has fourfold axes through

the atoms F–S–F that invisibly cycle the remaining 4 fluorine atoms.

But the axes need not pierce any atom!• The crown-shaped S8 molecule also has a C4,

but it goes through the empty center.

Principal Axis

Molecules may have many rotation axes.• E.g., S8 has four C2

axes, one through each pair of bonds opposing across the middle.

• Only one of these is shown in the molecular model above.

But the axis with the highest n is designated as the principal axis.• It is used to find other

operations, and often lends its name to the symmetry of the molecule.

• In S8, that would be the C4 axis.

Inobvious Axes

Remember the trick for drawing tetrahedra like that for CCl4?

The chlorines occupy opposite corners on opposite faces of the cube.

Like this …

That purple C3 axis is one of 4 diagonals of the cube on which you could spin the molecular top.

But what about the 3 C2 axes straight through each face?• Only 1 shown here.

Mirror Planes,

Reflection in a mirror leaves some molecules looking identical to themselves.

What distinguishes these operations is the physical placement of the mirror so that the image coincides with the original molecule!

There are 3 types of mirror planes:• Vertical, horizontal, and dihedral?!?

Vertical Mirrors, v

If the reflection plane contains the Principle Axis, it is called a “vertical mirror plane.”

Just as rotation axes need not pierce atoms, neither do v, but they often do.

E.g., in SF4, the principle axis is C2.

The two reflection planes are both“vertical” and happen to contain allof the atoms in the molecule.

Horizontal Mirror, h

Horizontal is to vertical so you can infer that h is to the Principle Axis.

While a molecule might have several vertical mirrors, it can have only one h.In PCl5, the horizontal plane is obvious. It’s what we’ve been calling the equatorial plane that contains the three 120° separated chlorines. (The polar axis is C3.)

Dihedral Mirror, d

Greek “two-sided” doesn’t help.• All planes are two-sided!

In symmetry it means vertical planes that lie between the C2 axes the Principal Axis.

• E.g., in S8 above, one of S8’s four dihedral planes contains the C4 Principal Axis and bisects the adjacent C2 axes.

• The plane contains opposite sulfur atoms,

The Magic of Mirrors

Rotations and the identity operation do not rearrange a molecule in any way.

But mirrors (and inverse and improper rotations) do. Mirror planes reflect a mirror image whose

“handedness” has changed. Left right.• “Chiral” molecules have mirror images that

cannot be superimposed on the original! So they cannot have a . You’d see the change.

Chiral Molecules

Caraway flavor agent

• S-Carvone

Spearmint flavor agent

• R-Carvone

These molecules cannot be aligned.

SF6

Inversion, i

Mirrors merely transpose along one axis (their axis), but inversion transposes atoms along all 3 axes at once.• i is like xyz so points (x,y,z)(–x,–y,–z)

• Therefore, there must be identical atoms opposite one another through the center of the molecule.

Improper Rotation, Sn

Proper rotations (Cn) do not rearrange the molecule, but improper rotations (Sn) do; they are rotations by 360°/n followed by reflection in a plane to the Sn axis.

• For Sn to be, neither Cn nor the need exist!

• To see that, consider the molecule S8 above; believe it or not, it has an S8 axis coincident with the C4 Principle Axis.

Clearly there are seven S8 operations.

Point Groups

“Point” refers to atoms, and “Group” refers to the collection of symmetry operations a molecule obeys.• The group is complete because no sequences of

operations ever generates one not in the group!

E.g., H2O, besides E,has v

v

and v’

v’

and C2.

C2

It’s clear that C2•C2 = E, but

C2•v = v’ needs a little thought.

Motivational Factors

The reason we want to know the Point Group of a molecule is that all symmetry consequences are encoded in the Group.• The nature and degeneracies of vibrations.• The legitimate AO combinations for MOs.• The appearances and absences of lines in a

molecule’s spectrum.• The polarity and chirality of a molecule.

Common Point Groups Cs molecules have only E and one .

• flat and asymmetrical.

Cnv besides E have Cn and n v planes.• NH3, for example.

Dnh has E, Cn, n C2 axes lying in a h.• like BF3.

Td, Oh, and I are the Groups for• tetrahedral, octahedral, and icosahedral molecules

• like CH4, SF6, and C60, for example.

The Grand Scheme, Part I Is the molecule linear?

• If so, does it have inversion, i?• If so, then it is in the group Dh.

• If no i, then it’s Ch.

• If not linear, has it 2 or more Cn with n>2?• If so, does it have inversion, i?

– If no i, then it is Td.

– If it has i, then is there a C5?

• That C5 means it’s the icosahedral group, I.

• But if C5 is absent, it’s the octahedral group, Oh.

• If n<2 or there aren’t 2 or more Cn, go to Part II.

The Grand Scheme, Part II

OK, does it have any Cn?

• No? How about any ?• Has a , thank Lewis, it’s a Cs molecule.

• If no , then has it an inversion, i?– If an i, then it’s Ci.

– But without the i, it’s only C1 (and it has only E left)!

• There is a Cn? OK, pick the highest n and proceed to Part III.

The Grand Scheme, Part III to the (highest n) Cn, are there n C2 axes?

• If so, a h guarantees it’s Dnh.

• Without h, are there n d planes?

• The dihedrals identify Dnd, without them it’s Dn.

If no C2,

• A h makes it Cnh, but without h,

• n v would make it Cnv, but if n h are missing,

– Is there a 2n-fold improper rotation, S2n?

• If so, it’s S2n, but if not, it’s just Cn.

In fewer than 8 questions, we have it!

What we’ve bought.

Cn or Cnv (n>1) means no dipole to axis.

• If no polarity along axis, molecule isn’t polar!

No Cnh or D group molecule can be polar.

Molecule can’t be chiral with an Sn!

• But inversion, i is S2; so i counts.

• Also a Cn and its h is also Sn; so they both count too. If present, they deny chirality.

Anatomy of a Character Table

C3v E 2C3 3v h=6

A1 1 1 1 z z2, x2+y2

A2 1 1 –1 Rz

E 2 –1 0 (x,y) (xy,x2–y2),(xz,yz) (Rx,Ry)

Group Name

Identity

Pair of C3 operations

Trio of v

Order of the Group(# of operations)

SymmetrySpecies

DoubleDegeneracy Functions transforming as the

symmetry species (e.g. orbitals)

Motion in NH3, a C3v Molecule

x

y

x

zx

z

x

zx

yx

y

x

y

What part of a coordinate survives each symmetry operation?E leaves all 12 coordinates alone. Therefore 12 survive.C3 leaves only Nz and –½ each of Nx and Ny. So 1–½–½=0.

v leaves only Nx, Nz, H1x, & H1z but makes –1 each for Ny and H1y. So 1+1+1+1–1–1=2.

ammonia motion = 12 0 2

C3v E 2C3 3v h=6

A1 1 1 1 z z2, x2+y2

A2 1 1 –1 Rz

E 2 –1 0 (x,y) (xy,x2–y2),(xz,yz) (Rx,Ry)

12 0 2 x+y+z+Rx+Ry+Rz+vibrations

A1 = 1(12) + 2(1)(0) + 3(1)(2) = 18/h = 3A1

A2 = 1(12) + 2(1)(0) + 3(–1)(2) = 6/h = 1A2

E = 2(12) + 2(–1)(0) + 3(0)(2) = 24/h = 4 E

Ammonia Motion = 3A1+A2+4E Since E is doubly degenerate, that means

3+1+8 = 12 motions (3 coords for 4 atoms). Since x+y+z+Rx+Ry+Rz = A1+A2+2E, the

six vibrations in NH3 must be 2A1+2E.

C3v E 2C3 3v h=6

A1 1 1 1 z z2, x2+y2

A2 1 1 –1 Rz

E 2 –1 0 (x,y) (xy,x2–y2),(xz,yz) (Rx,Ry)