sym 5 Flashcards
in the D4h,, operations mix x and y but leave z
independent
so we can do a seperate x and y matrix,, and treat them separately to the z matrix
bc zā normally = z bc its the principal so rotating it kinda just makes it stay in its place.
for D4h
so by splitting the xy from the z matrix we go from a 3x3 matrix to a
2x2 + 1x1 matrix
2x2 being xy
1x1 being z
for D4h
3x3 matrix can be simplified due to zā being z,, and so it called a
reducible representation
for D4h
the 2x2 + 1x1 representation is therefore called the
irreducible representation
bc u cannot simplify the relationship between them any more.
for D4h
in Dh4,, the x and y vectors are
degenerate
theyre both in the same chemical environment
operations cause them to interchange
the px and py orbitals have the same energy
a vibration changing the x dipole moment has the same frequerncy as the one changing the y dipole moment
Eu meaning
ungerade
inversion // i does reverse the vectors!!!
Eg meaning
gerade// inversion // i doesnt change the vectors
the x and y orbitals in D4h belong to XXXX bc what
they belong to Eu
an irreducible representation
Eu bc inversion reverses both of them!!
from x ā> -x
from y ā> -y
what do matrixes help us do
they help us realise how much of the original vector is present in the new one
when the character chnages its 0
but then like the one where it says -1 is the new vector.
how do we find gamma // r // the reproducible representation
r for reproducible representation
we draw an arrow at each atom around the central atom and see how these chnage with the operation:
for E in ammonia,, its 3,, bc when u do the identity operation all the arrows stay in the same place (they dont change so 1 + 1 + 1 bc theres 3 arrows)
for the 2C3 its 0,, bc they all change,, so 0 + 0 + 0 = 0
for 3sigmav (plane running through each NH bond) when u do it only one think stays the same (the bond ur going through) so u get 0 + 0 + 1 = 1
remmeber if it chnages its 0 and if it doesnt change its 1
how to remmeber when we use 0 and when we use 1
1 will stay where it is bc its standing up and straight
0 will roll away bc its curved so it will move
how do we know if what weve foun usung ārā is reducble
bc it doesnt match any of the standard irriducible representations
group theory and reducible representation exppp
group theory says that any reducible representation can be broken down into a sum of standard irreducible representations
these will give the labels for the representations of the fundamental vibrations we need to predict the lines that should be seen in ir and raman spec.
h =
order of the group
total number of operations in the group (add the big numbers of the clases up)
what is the gc of a class
the big number of the classss
what is the gc of the operation 2C3
the gc would be 2
the large number
finding the order of a column,, the order of A1
sum of (gc) (xi ^2)
(u do big number x character^2) + (big number x character^2) across the whole row (each class) then add the numbers together and this gives u the order,, h
rows are the
bits on the lhs,, Xi
columns are the ones that
fall down,, like groups in a periodic table
C
2 different irreducible representations are
orthogonal (at right angles to eachother)
sum of: gc X xi X xj = 0
h =
sum of : gc x xi^2
xi = the characters for the irreducible representations
gc = the stoichiometry of the class/column
we only square the xi value when
were looking for the order!!
we then multiply this by gc and do this for every one.
when making the multiplication matrix,, where do u write the 0s and 1s
u put the 1 where the think moved to
aka u put a 1 in the atoms new position!!
then u put 0 where the atom didnt go
what do u do to the matrix characters to get a character
u add them together!!
if the matrix is:
|0 1|
|-1 0|
the character would be
1+(-1) = 0
Xi means we need to use the characters in what row
we use the characters in the E irreducible row š
Ei
Xj means we use characters in which row
the A1 irreducible row
AJ
to see if irreducible reps are orthogonal,, what equation do we do and what do we do
we use the sum: gc Xi Xj formula
u should get 0 if theyre orthogonal.
aka u use the big stoichiometry number (gc) X the character for the A1 X character for E
and then add the answers uppp!!!
if theyre orthogonal,, youll get a 0 š
what do we use the reduction formula for
to go from the reproducible rep to the irreducible rep
what is the reduction formula
its given in the booklet and in exams
what is h
the order
gc X xi^2
remember xi = E
what is gc
gc is the large number,, the stoichiometry if the class
what is xr
r is gamma
this is where we use the arrows to see if they change (0) or stay the same (1) or are reversed (-1) when the operation its under is done.
what is Xi
Xi is the E!!
how can x,y,z be split
x,y = doubly degenerate bc they kind just move into each others area!! theyre Eu
z is A2u
it tells u this in the table tho!!
when we have r, A2u and Eu,, whats special about them
the irreducible representation characters sum to those for the reducible ones for every class.
r = Eu + A2u
for r,, u do the operation for
for every axis
then u see where the x, y and z went etccc
then u see what numbers they give: 1 for no change,, 0 for change,, -1 for reverse
what else does it mean when things are ungerade
they reverse when theyre inverted
when analysing vibrational modes,, what does the irreducible representations tell us
tell us the symmetry of each fundamental mode,, the table then identifies which modes are Ir and raman active
helps us predict the number of bands seen in spectra for a given set of vibrations.
if character tables are square,, this means
the number of irreducible representations = number of classes!!
how many operations are there in a 3C2 class
3!!
will h and the sum thing always be the same
im guessing nope
bc its 1/h x sum
so i fear not,, or else it would always be 1.
when do we do the thing where we add an arrow to the atom
planar molecules!! to see if the arrow changes direction during the operation.
