CHEM 2060 Lecture 3: Symmetry L3-1

Benzene, C6H6 : σd mirror plane WARNING: Differentiating between a σv and a σd mirror plane can be a little tricky at first, but with practice it should become clear. Both σv and a σd mirror planes are collinear with the principle rotation axis… …BUT it is only possible to have a dihedral mirror plane in a molecule in which EITHER the principle axis Cn

OR an improper rotation axis Sn is 4-fold or larger (n ≥ 4)

In a molecule that has both σv and a σd mirror planes, the σv planes bisect as many atoms as possible and the σd planes bisect as many bonds as possible.

CHEM 2060 Lecture 3: Symmetry L3-2

σd mirror plane …cont’d NOTE: Some molecules have only σd planes!! These are molecules in which there is an S2n improper rotation collinear with the principle Cn axis (see next page). Molecules belonging to the point group Dnd (see ahead) have only σd planes. Molecules belonging to the point group Oh (see ahead) have σh & σd planes. NOTE: σd planes are always collinear with the principle axis. Okay, so what is an “improper rotation”? (Turn the page!)

CHEM 2060 Lecture 3: Symmetry L3-3

Improper rotations, Sn An improper rotation consists of a rotation followed by a reflection in the plane perpendicular to the rotation axis.

Methane, CH4 : S4

NOTE: Neither C4 nor the σ⊥ is a symmetry operator of methane but their product C4 × σ⊥ is (i.e. S4).

CHEM 2060 Lecture 3: Symmetry L3-4

NOTE: S1 = reflection S2 = i Pictorially So now we need to look at inversion i.

CHEM 2060 Lecture 3: Symmetry L3-5

Inversion Center, i A center of inversion is a point in the middle of the molecule (object). If you take each atom and move it through that point from its original position to the same distance on the other side of the center of inversion, the molecule appears unchanged. Hexafluoroantimonate, SbF6

- : i

Now you should check for yourself that S2 is indeed the same things as i.

CHEM 2060 Lecture 3: Symmetry L3-6

OK, That’s it: We can now identify molecules by their symmetry elements. E Cn σ(h, v, d) i Sn

Whether or not a molecule has all or some of these symmetry elements can easily be classified by assigning a label. This label identifies the POINT GROUP of the molecule. There are many, many Point Groups possible, but only a handful are common for molecular structures. Some of the most common points that you will be seeing include: C2v (which describes water, H2O) C3v (which describes ammonia, NH3) Td (tetrahedral, which describes methane CH4) D2h (which describes ethylene, H2CCH2) Oh (octahedral, which describes sulfur hexafluoride, SF6)

CHEM 2060 Lecture 3: Symmetry L3-7

Assigning Point Groups VERY IMPORTANT! Assigning a point group to a molecule depends on listing the symmetry elements of the molecule and then looking them up in a table. It is often easier (at first) to work through a YES/NO DECISION TREE.

Obviously, this can be tedious and eventually unnecessary as you become more comfortable with the concept. Eventually, you’ll be able to see the symmetry and assign the point group without a decision tree.

CHEM 2060 Lecture 3: Symmetry L3-8

Decision Tree For Assigning Point Groups

(b)

From (a)

N

NN

N

Y

Y

Y

Y

Y

N

CsCnv

N

Y

DnDndDnh

Cnh

σh?

S2n

Y

Cn

i ?

σ ? N

N

Y

Y

ClCi

Cn?

σh?

Select C n withhighest n i then, isnC2 Cn⊥ ?

nσv?nσd?

S2n?

Go to (b)

Two or moreCn n>2,

?

Molecule

Linear

i ?

NY

YY

Y

N

N

N

i ?

C5?Y N

IhD∞hC∞v

OhTd

(a)

Linear groups Cubic groups

⇓

CHEM 2060 Lecture 3: Symmetry L3-9

Let’s do an example: water, H2O Start at Top of Tree Linear → NO Two or more Cn’s, n > 2 → NO Cn → YES (H2O has a C2 rotation axis; n = 2) Select Highest Cn. Is nC2 ⊥ Cn? (In other words, are there 2 more C2 axes perpendicular to the C2 axis?) → NO σh → NO n σv’s → YES (H2O has two vertical mirror planes; 2 σv’s)

⇓ Cnv …because we’ve identified n = 2 for water, the answer is C2v

CHEM 2060 Lecture 3: Symmetry L3-10

Let’s try an example by identifying the symmetry elements: ammonia, NH3 Molecule possesses: Identity (E) C3 axis 3 σv planes ⇒ C3v or go through tree Linear ? – no, 2 or more Cn – no Cn ? – yes, C2 ⊥ C3 ? – no σh ?– no, nσv ?– yes ⇓ Cnv (n = 3) Point Group for NH3 is C3v

CHEM 2060 Lecture 3: Symmetry L3-11

Be careful with σh and σv ! In both examples so far we had to answer questions concerning the minor planes. Is it σh or σv? σh’s are ⊥ to the main rotation axis. σv’s are ⎜⎜ to the main rotation axis. H2O E, C2, 2σv

CHEM 2060 Lecture 3: Symmetry L3-12

Let’s try another example, boron trifluoride, BF3 : Linear ? – NO 2 or more Cn (n > 2)? – NO Cn ? – YES (C3 i.e., n = 3) Are there 3 C2 ⊥ C3 ? – YES σh ?– YES ⇓ Dnh

and because n = 3, the point group for BF3 is D3h HOMEWORK FOR TUTORIAL: Prove that BF3 and PCl5 have the same point group.

BF

FF

CHEM 2060 Lecture 3: Symmetry L3-13

More examples: C∞ν One infinite-fold rotation axis with infinite number of symmetry planes which include the rotation axis. D2 Three mutually perpendicular 2-fold rotation axes. D3 One 3-fold rotation axis and three 2-fold rotation axes perpendicular to the 3-fold axis. C∞ν D2 D3

CHEM 2060 Lecture 3: Symmetry L3-14

Summary of the truth table… There are only 2 linear groups, either D∞h or C∞v . (Question: Why no others?) If there are multiple Cn axes where n > 2, it is one of the cubic groups: Ih, Oh, Td Search for the highest order Cn axis… …if there are ⊥ C2 axes → D …if there are NO ⊥ C2 → C (with the exception of S2n) (This is the difference between D & C) For “D” groups

If D and σh → Dnh (See later σd’s bisect C2’s)

σd → Dnd (σd’s bisect C2’s) NO σ → Dn