Exam 2 Refined 1 - Symmetry and MO

group theory

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

symmetry element

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

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

Cn

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

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

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

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

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

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

improper rotation axis - Sn

rotation by 360/n followed up by a reflection through a plane perpendicular to the rotation axis

look for alternating up down bits

inversion center

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

point group

summary of all symmetry operations for a particular object

point group ID

1. find PRA and n of it

2. perpendicular C2s? - Y then D, N then C

3. σh? Y then nh, N then continue

4. mirror planes? Y then Dnd or Cnv, N then n

high symmetries

tend to lack a unique PRA

• Td, Oh, Ih, Cinfv, Dinfv

• inf = infinity

Cinfv - linear between 2 atoms or 2 diff 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 or 2 same atoms

• perpendicular C2

low symmetries

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

point group Cx

no perpendicular C2

dissymmetric

dihedral groups

Dx

like cyclic except they DO have perp D2 and can be flipped while retaining symmetry

assymetric v dissymetric

both are chiral

assymetric - can only be C1 symmetry, only has operation E

dissymetric - Cn or Dn where n isn’t 1, can have perp C2s IF D

chiral and polar

chiral - only 1 PRA

• Cn and Dn

• chiral - not all Cs

polar - only 1 Cn axis and no sigma

• Cn or Cnv

sigma placement

don’t use where another C2 could go (try rotating first over reflecting)

tetrahedral

don’t forget it can do S4

i = S2

same as rotating 180 and flipping

more EN atoms

axial regions (right angle from main plane)

reducible and irreducible representations

calced number code for terminal atom AO via scoring if each symmetry operation gets it to move or not is reducible

the codes you add up to get this same score are irreducible

codes are SALC (symmetry adopted linear combos)

degeneracy

a and b are single

e is double

t is triple

MO e- total

don’t forget to factor in molecular charge!

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

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

bonding and nonbonding MOs

bonding orbitals are formed from the constructive overlap of AOs

anti-bonding are formed from destructive overlap

significance of vertical arrangement of MOs

only significant when compared to the AOs that formed the MO - used to determine B, N, A

does not matter between MOs