Exam 2 Set 1 - Symmetry and MO of Complex Molecules

group theory

the mathematical method of describing the symmetry of an object based on its geometry

• symmetry element - geometric entity like a plane, axis, point, ect that a symmetry operation is centered on (around which the operation is defined) - referred to as an object

  • e.g. rotating around an AXIS, reflecting across a PLANE, etc.

• symmetry operation - transformations of an object into a configuration that is INDISTINGUISHABLE from the original - changing the original in a way that the transformed object looks identical to the original post-transformation

• any 2 atoms interchanged by a symmetry operation are equivalent

symmetry elements

identity - E

proper rotation - Cn

reflection - sigma (σ)

improper rotation - Sn

inversion - i

identity - E

basically just a full 360 deg rotaion

trivial

every molecule has an E operation

proper rotation axis - Cn

rotation by 360/n deg, with n being an integer and replaced by the chosen integer when writing C

• e.g. C2, C5, etc.

• C2 = 180, C3 = 120, C4 = 90, and so on

• molecules have different Cn axes of different n values at different locations

principle rotation axis (PRA) = highest order n of a molecule

• z axis by convention

primes (like the ‘ applied to a Cn) - sometimes primes are used to rank and group different Cn axes

• coincident (lie exactly on top of each other) with the principle rotation axis - ?

• passes through more atoms - ?

mirror plane - σ

reflection across a mirror plane

3 types:

• σn - horizontal, perpendicular to the PRA

• σv - vertical, parallel to the PRA and actually contains it (as every axis of symmetry must cross the center of the atom) as well as a perpendicular C2 or whatever axis - makes sense, sig v always drawn in line with molecule spokes

• σd - dihedral, contains PRA and cuts the space between perpendicular C2 or whatever axes

• which is the “contains” in the notes doc referring to ?

molecules can have more than one σv or σd, but only ONE σh

usually going to have mirror planes between each spoke formed by an outer atom bonded to a central atom

• that operation represented an Nσv, with N representing how many of this type of mirror plane there are

improper rotation axis - Sn

rotation by 350/n followed up by a reflection through a plane perpendicular to the rotation axis

works best with stuff coming off of 2 central atoms, like 2 Cs, bonded together (staggered ethane, kind of dumbell shape)

Sn of CH4 ?

inversion center

best for stuff like a cyclobutane (square ring) with alternating groups off the corners variably sticking up or down

more examples in the notes

point group

summary of all symmetry operations for a particular object

represented by a point group symbol, a shorthand name

can be used to derive all possible operations by using a character table

point group symbol ID

1. Does the molecule have a unique PRA, and what is the n?

2. Does the molecule have any C2 perpendicular to the PRA?

   1. yes = symbol starts with D

   2. no = symbol starts with C

3. Does the molecule have σh?

   1. yes = symbol is Dnh or Cnh

   2. no = proceed to step 4

4. Does the molecule have any other mirror planes?

   1. yes = symbol is Dnd or Cnv

   2. no = symbol is Dn or Cn

a good portion of molecules end up being C2v

high symmetries

some molecules have very high symmetries

tend to lack a unique PRA, but their point group symbols are usually easy to find

• Td, Oh, Ih, Cinfv, Dinfv

• inf = infinity

Td - tetrahedral, the 4 C3s running along each bond to the central atom

Oh - octahedral - C3s along each bond ?

Cinfv - linear between 2 atoms, axis is along the bond, can spin around at any angle and still look the same

• no perpendicular C2

Dinfh - linear between 3 atoms

• perpendicular C2

low symmetries

some molecules have very low symmetries

either have just E or E with a mirror plane or inversion center

C1 - asymmetric - e.g. tetra with all diff groups

Cs - E and σ - e.g. tetra with all but 2 diff groups

Ci - E and i, e.g. weird square ring thing

cyclic groups

C (as in the point group symbol)

do not have perpendicular C2 axes relative to the PRA

• can be rotated but appear different if flipped over

• dissymmetric

dihedral groups

D

have perpendicular C2 axes relative to the PRA

molecular properties

chiral molecules - only have one proper axis of rotation

• possible point groups = Cn, Dn

polar molecules - only have one Cn axis and no σn

• possible point groups = Cn, Cnv

anatomy of a character table

yeah I ain’t writing all dat

read it in the notes, page 7

some highlights

• value of +1 indicates no change

• value -1 = exact opposite

• p and d orbitals assigned to one of the symmetry patterns that the molecule has

irreducible representations and their names

a basic pattern of symmetry transformations

most basic representation - cannot be further broken down

?

describes how an object/set of objects, like an orbital, changes under each of the symmetry operations in that point group

need to memorize naming rules ?

sigma

h has to be perp to a C2

v parallel to C2

don’t put a sigma where another C2 could go

octahedral

C2’ and “

sig v and d?

tetrahedral

S4

also C3 and C2 and sigv

i = S2

i is the same as rotating 180 and then flipping

more EN atoms

get the axial regions

need less room on trig bi

very high symmetry

probably no principal rotation axis

Cinfv

if the ends of the linear molecule are DIFFERENT

if onyl have E

is a C1

asymmetric

chiral with symmetry

Cn or Dn

dissymmetric

S operations

look for alternating up/down parts

complex MO diagrams solving steps

1. find the molecular structure - if it has a resonance form then use that one. Use FC to confirm that the form is right.

2. Find the symmetry of the structure and its corresponding character table

3. Prep MO diagram. Put central atom to the left and terminal atoms to the right. Put the combo in the middle.

4. Use the periodic table to find the valence orbitals (whichever ones are on the lowest row). Position those of the more EN atom closer to the bottom. Any nonbonding S has to be the lowest of all

5. Use the character table to find the letter designation for each central orbital.

6. Calc number code for terminal atom designation by testing each symmetry operation (+1 if don’t move, 0 if they do). Find the letter designations with the codes that add to the calculated number code.

7. Note down how many of each central bond MO there should be and tally those.

8. Write in the a1s first them go from there. All sets should be spaced equally relative to the AOs that form them. The bonding should be below the lowest AO that forms it and the anti above the highest, with the nobondings in between. Nonbonding should be straight across their corresponding AO. Draw lines between corresponding MO and AOs

9. Fill in the e-

10. tally up bonding (first of any MO set), antibonding (last of any set) and nonbonding (the rest and any s that were nonbonding to begin with).

11. Compare results to lewis structure from before.

reducible and irreducible representations

the number code you calculate for the terminal atom AO designations is a reducible representation

the codes you add up to figure out the corresponding letter designations are all IRreducible

the act of identifying the irreducible representations is called decomposing

the combo of letters you get from the calculation defining the terminal AO types is called a SALC, or symmetry adopted linear combinations

ions

ionic molecules are going to add or take away from your e- total

don’t forget this!

degeneracy

a and b are single

e is doubly degenerate - each MO has 2 orbital slots

t is triply degenerate

nonbonding effect

the orbital ends up spread across the molecule and doesn’t strongly contribute 1 way or another to bonding

discounted in BO calc

H terminal atoms

usually its 1s AO orbital is midway between the 2p and 2s AOs of the central atom

repeat MO designations

multiple AOs may call for the same MO designation

just put it once - they’re just both contributing to the same MO

if the terminal atom valence is 3p, for example

4s is likely too high energy to interact with 3p

will still be sandwiching the terminal 3p between the central 3p and 3s

e orbital position

usually listed above a1

bs also usually higher e than a1

nonbonding tally

any terminal s orbitals that are too low E to interact with the rest of the AO are also considered nonbonding

1e

probably above the lower central s AO

gonna be placed higher than any AOs not contributing to the e MO

for every bonding orbital

there must be an anti

picture of bond shading drawings

4/5/2025

number of bonding orbitals and BO

the number of bonding orbitals will only correctly predict the number of molecular bonds if the BO is a whole number!

this will also screw up the lone pair count

diatomics

since there’s only 2 atoms involved, you can get the MO letter designations of both from the table, since there isn’t the same “central-terminal” relationship going on