Okay let's get going this morning.
Welcome back everybody. 
So last time we were talking about how we can take symmetry operations and express them in
terms of matrix notation and what we
want to do today is to develop what are
called character tables for different
kinds of point groups.
The character table was front and center in understanding how symmetry can be used
to explain things like vibrational
properties of molecules and bonding and so forth.
So the character table, you'll
hear this term a lot.
 It's a kind of a big scary idea. 
We're going to try to break it down into its components and
understand how character tables are
actually formed. 
And then we're going to use them for the rest of the entire class.
So we ended here last time where we had used the transformation matrices
 
or developed the transformation matrices
for a simple point group C2h here.
And we showed how the identity C2 i and the reflection operation can be expressed in matrix notation.
And what we said is that using matrices you can do the same kinds of manipulations that you would do just intuitively and you're
and you are going to get the same answer. So if you
do a c2 operation followed by another c2
you get the identity operation and this works out through the matrix multiplication that we talked about last time.
What we ended with last time is that if we take the sum of the elements along the diagonal,
this is called the character of the individual
transformation matrix 
and this sum of numbers here, the character, gives us a
shorthand version of the matrix representation
and we call this
shorthand version big gamma
And so we have for E 1+1+1 = 3, so we can write that gamma for the operation E gives us a character of 3.
For c2 we have -1+(-1)+1= -1
and so forth for I we have -3 and for the reflection we have 1.
This collection of characters here for gamma gives us a reducible representation 
that in this particular case explains how x, y, and z coordinates at the center of the molecule
transform with the three different or the four
different operations
that we have in this particular point
group. 
This reducible representation is so called 
because it can be further
simplified into components
and we're going to generate what are called irreducible representations 
which are the basic building block for symmetry of this point group.
So let's do that.
How can we further simplify these 3x3 matrices
to something more simple in this case. 
Well we can do what's called block diagonalization. 
We realize that most of the elements, in fact all of the elements,
in this particular case that
are off diagonal are zero. 
They contain no information so we can just ignore them and we can block diagonalize each 3x3 matrix 
into individual 1x1 matrices - three 1x1 matrices for each case.
So we have three 1x1's here, we have
three 1x1's here, etc. 
This first row here indicates what
happens to the x coordinate 
the second row what happens to the y coordinate 
and the third row what happens to the z coordinate because we're talking about transformation matrices.
So x, y, and z, we can take these numbers now - the
individual 1x1 matrix characters
and we can express them in the form of a table - a convenient way to show the information
and this is beginning to show what a character table will look like, okay?
It is a table in which we have
columns up here 
each column corresponding to a symmetry operation in
this case.
Under the columns each symmetry operation we have the characters in this case along the diagonal,
so we have for x, y, and z, we have 1, 1, and 1, for C2 we have -1, -1, and 1,
so that's -1, -1, and 1 and so forth for I and so forth for the reflection operation.
At the bottom here we have the sum of these characters
so 3, -1, -3, and 1 that we had developed in the previous slide.
So this would be called the reducible representation.
Each of these three rows here are called
irreducible representations. 
They cannot be further simplified because they're
based on 
the simplest components of these matrices, which are 1x1 block diagonal matrices.
We also have the x, y, and z coordinates that correspond to each one of these three different rows here.
 I will explain of course this notation in a little bit, 
but a character table is essentially going to have the symmetry operations as columns 
They're actually going to be called the symmetry classes
It's a slight distinction that we will make
clear in a little bit.
And along the rows here we're going to have the irreducible representations for the point group.
The point group designation is always given
in the upper left corner of the table.
So these three rows as I mentioned in
this case Bu, Bu, and Au
are irreducible representations of the C2h point group
that's the language we would use.
They can't be further simplified because
they're just one by one matrices.
There's no way that we can get more
simple than that.
Alright. So who cares,
what are we going to do with these
things?
Well, these individual irreducible
representations here,
the characters for these individual irreducible
representations tell us
how these irreducible representations transform
for the different symmetry operations.
And in the table if you have a character of 1, that means that you're unchanged upon the symmetry operation.
In other words you're symmetric
So symmetric and unchanged are interchangeable words here.
If you have a -1 as your character, that means you're anti symmetric 
or you're inverted upon the symmetry operation.
And if you see a zero
that means it's something else.
so let's look and see how this portion
of the character table for C2h
gives us information about how our different
coordinates change with these symmetry operations.
Let's use an example - this is the same
 example that we used before
1,5 - dibromonaphthalene.
Okay so this is a C2h molecule that 
you should be able to 
convince yourself up very quickly 
We use the coordinate system with the convention that the z axis is along the principal diagonal of the molecule
the C2 axis in this case. And
that means the molecular plane is the xy plane
so let's look and see. For this
particular, let's just start with this one arbitrarily
so for the Au irreducible representation we see
that it transforms like the z-axis
transforms in the molecule so if you
look at the identity operation it's
unchanged under identity
like everything is going to be unchanged under identity
for C2 the z axis is unchanged, right?
The C2 is parallel to the z axis so
flipping 180 degrees around the z axis
does not change the coordinate so we
have a symmetric or unchanged character
Upon inversion the positive z goes to -z and so we have a character of -1.
 indicating
that it's anti-symmetric 
And for a horizontal reflection that also 
changes positive z into -z 
and so we expect then that upon the horizontal reflection we're going to have an inverted character -1,
 and so that's why z and Au basically have the same symmetry in this particular point group
 
we would say that the Au irreducible representation has the symmetry 
of the z axis in this particular case 
What about these others here? We have two identical
irreducible representations here.
Both are called Bu and we'll explain why, but
you can see why they're identical
it's because they have identical characters.
One is given x and the other is given Y.
We can say that the bu transforms
like x and y in this case, 
so again we can look at the different operations and
see how they transform under identity - 
of course no change.
Under C2 both the x and the y invert and so we have a minus 1.
Under inversion they both invert and
so we have a minus 1,
and under a horizontal reflection, a reflection in the plane of the molecule, 
neither x nor y change and so we have a character of positive 1 in this case.
And so we have the bu irreducible representation transforming like the x and the y axes in this particular case.
that's what this the character information here tells you
 how these individual irreducible representations transform under the different symmetry operation.
Because we have 2 Bu's and they're exactly the 
same, they have the same exact characters 
we just merge them together in the table.
So instead of having two rows we just have one row that's called Bu
and we indicate that both x and y transform this way by just putting X, y 
indicating that both the x and the y coordinates or the axes have the same symmetry in this particular point group
 This of course will not always be the case. 
 If we chose another point group example then the symmetry of the x and the y axis could be different.
In general they will be
different. 
Okay so we have, in this case, part of the character table - this is not the total character table for the C2h point group
but what we've done is shown
how you can get two irreducible representations.
It turns out there are
four altogether.
But we can get two of these by looking at the transformation matrices, 
How X, y and z for the molecule transform under the different symmetry operations 
What we want to do is show in a little bit how we can generate the full character table
by starting with the transformation matrices and then
using some of the properties of character tables 
to kind of finish off
the problem.
So this gets us into the the meat here. 
So character tables as the
central focus for the rest of this this lecture.
 Hopefully we'll get to the
first application of character tables
and symmetry at the end of the class today.
Okay, so we were working with C2h,
and we showed two irreducible representations, which is basically half of the character table for c2h.
This is the full character table for c2h. 
This is what you'll see if you look at the back
of your book in Appendix C I think
is where these character tables are, and
this is the same format that you'll see
in almost any reference guide that has
character tables.
It talks about molecular symmetry.
You can see that we've got basically four different sections of the character table, okay?
So the first section, the characters and then these other two columns here.
So we want to understand exactly what this is telling us, so we're going to go through it.
What a character table is? It's simply a list or table of the complete set of irreducible representations that's shown here
and the symmetry classes which
are shown in the columns.
So it's the characters for every individual symmetry
class for each individual irreducible representation.
Let's start by understanding this notation over here and what this tells us about the characters.
So these guys here are the irreducible representations and they're given a label that tells their symmetry
The label is called the Mulliken label
for the representation 
and it has the following flavor here.
So if you see A or B and, in this case, we see only A or B. 
What this tells you is you have 1x1 representation. 
Remember we had for C2h only 1x1
matrices along the block diagonal.
If it's A that means it's symmetric with
respect to the principal axis.
If it's B that means it's anti symmetric with
respect to the principal axis.
So it tells you right away what the character
is for the principal axis.
You can see that the A has a 1 and the B has a -1 - it's either symmetric or anti-symmetric
Sometimes you'll see an E instead of an A or B and sometimes you'll see a T instead of an A or B.
 if you see an E this means that you have a 2x2 matrix
and the character under the identity
operation is then going to be a 2 
And if you see a T that means you have a 3x3
representation 
and that means that the character under the identity operation is going to be a 3.
We'll see this with an example in a little bit and when we talk about a molecule with the C3 axis.
Okay, in addition to the A and the B or the E and the T we have these subscripts,
In this case we have g and u subscripts. 
For point groups that have inversion like this - C2h has the inversion operation
the representations are labeled with the subscript g which is gerade, German word, or ungerade for u.
  to denote whether there's symmetric or anti-symmetric with respect to inversion.
 So you are gerade or g if
you have a symmetric relationship with inversion.
And you are ungerade or u if
you have an anti-symmetric relationship with inversion.
So you can just look at the Mulliken label  and tell right away what the characters have to be in the character table.
Then, of course there are character
tables with more than just four rows and four columns.
There's much more complicated point groups with a lot more symmetry classes.
In those cases what you'll see are additional labels on these Mulliken symbols, 
maybe a one or two or a prime or double prime things like this are going to pop up.
And if they are present these numbers, these additional subscripts or superscripts typically indicate
the symmetry with respect to the next operation after the principal axis and after inversion if it's present.
So you might have a 1 if it's symmetric with respect to a reflection plane 
that's farther along in the character table.
2 if it's anti-symmetric with respect to
that reflection plane
The bottom line is that what we want is a
simple labeling scheme that tells us
kind of immediately what the symmetry is
of that particular irreducible representation
so just at a glance we can tell 
the basic flavor of that particular representation.
[STUDENT QUESTION]
So it's a mathematical definition
so under inversion does the value x y or z
invert to its negative value x y or z.
So you can see here there's a bunch
of stuff that we're going to get into in a second.
but you can see if you look at
the z or the x and the y axes
 those are all anti symmetric with respect to
inversion because as we would expect
if you invert x goes to -x, y goes
to -y, z goes to -z.
But some of these others that have these R
symbols, which we'll talk about in a second,
are symmetric with respect to
inversion. We'll get to that in a bit.
Okay, so we understand a little bit about
the Mulliken symbols that are used for
the irreducible representations what
about these last two columns here
These last two columns indicate
functions that have the same symmetry
as these respective irreducible
representations 
The first column are linear functions or rotational functions.
So linear functions like x, y and z
Rz, Rx, Ry mean rotations with respect
to those individual axes
so rotation around the z axis for example would be Rz.
And the second column here are the quadratic functions, those that combine x^2, y^2, z^2, xy, xz, and yz.
 Why do we care about these particular functions? 
It's because these functions here, for example the quadratics, 
these contain the elements of the d-orbitals.
All right so if you remember d orbitals you have dxy, dxz, dyz, dx^2-y^2, dz^2.
Those are exactly the same functions as shown here
 This is so that we can tell at a glance what
symmetry the d orbitals have, the
individual d orbitals have in this particular point group.
 Why do we care about x, y and z? for two reasons x, y and z tell us about translations - 
movements of the molecule through space.
They also tell us about the symmetry of the px py
and pz orbitals 
And why do we care about rotations? Because this tells us the symmetry of the three different possible
rotations for a three-dimensional
molecule.
So we can assign which of these
individual irreducible representations have the symmetry of a rotation for this particular object.
Okay, so let's look down here. The last two columns as I said are these functions.
 In this case because we have an inversion center the functions are at the origin.
Let's look at a couple of different molecular orbital examples here or atomic orbital examples.
So here is the x^2-y^2 d orbital, here is the dz^2 d orbital.
You can see that in the character table
here that were that we're trying to understand, 
x^2, y^2, and z^2 are all located in the same row, 
happens to be the Ag representation so
what we're saying is that
the x^2-y^2 and the dz^2 have the same symmetry in this particular point group.
So let's look at the individual symmetry operations. With respect to E of course nothing changes
With respect to C2 you can see that if you do a C2 along the z-axis here which is coming out at you
you get the same object back.
and if you get in this case the z-axis is up-and-down, 
if you do a C2 with respect to that
you also get the same object, right?
No inversion. So that's why we have a 
 character of positive 1.
Upon inversion operation what happens to this guy,
this particular x^2 - y^2
orbital is unchanged, right?
The two negative lobes just interchange and the
two positive lobes interchange
 so the thing is not inverted it's not reversed
and it's polarity. 
The same thing for the dz^2, the positive lobes
interchange and the negative ring 
kind of just dances around and doesn't change
And under the horizontal reflection operation, same deal.
Horizontal reflection is in this plane so that's unchanged
and it's in the xy plane here
so that's also unchanged. 
And the object is just multiplied by 1 in that sense 
So that's why those two orbitals transform as Ag and that's exactly what we can just read right off the table
 that's one of the powers of the table -
we don't have to think very deeply, it's all there for us to see, and it's all in plain sight
 What about the pz orbital?
well the pz orbital has different symmetry
we can tell what the pz orbital
is we just read it right off the table
the pz orbital must have the same
symmetry as z-axis and that's going to
be an Au representation. Let's prove to
ourselves that that's true.
Under E nothing changes.
Under C2 nothing changes because that's along the z-axis, which are parallel with the lobes here.
Under inversion what happens? Under inversion
the positive lobe let's just call that
this dark lobe here jumps up here and
then and the negative lobe jumps down here
 the thing has been multiplied by
-1 so it's anti-symmetric that's
why we have a -1 for the
character and under the horizontal
mirror plane right that mirror plane is
in the xy plane and so again we
interchange the positive and the
negative lobes and so the function has
been multiplied by -1 it's
anti-symmetric with respect to this
horizontal mirror plane. So you kind
of get a sense for how the character table
can be used
and how the characters come to be right
so it all should be self-consistent we
do the same thing with the PX and py orbitals
and we should be able to read these characters off from those different symmetry operations.
Okay so if there are any questions at this point
yeah go ahead.
[STUDENT QUESTION]
Yep, so the question is
why not just put x^2-y^2
 in these in this function the
the convention here and the idea is that
any linear combination of these
functions also has this entry so you
could have x^2+y^2, x^2-y^2, x^2-z ^2
and all of those linear
combinations would still have Ag symmetry
so it's just to make it more
general because there may be instances
that you care about other things just
than the real space versions of the d-orbitals
So any linear combinations of
these functions and linear combinations of those functions
are still good that's
why
[STUDENT QUESTION]
It is. Yeah, it is, and in some cases
it doesn't matter in some cases it does
we'll show how you can take a gamma and
reduce it to its irreducible parts.
That's coming up very soon.
Okay, other questions?
Ok. So there are some rules
about character tables that we need to understand 
 and these rules come out
of group theory again we're talking
about the results we're not talking
about the derivations.
So when you look at a character table
 this is the full character table in its full glory, 
the total number of symmetry operations as
we've already said is the order h. 
In this case, we have 4, so h = 4 for C2h. We already know that.  
 Operations are going to belong to the
same class the same operational class
if they are identical within coordinate
systems that are accessible by a symmetry operation
 that's present in this group of symmetry operations.
One class of symmetry operations is
listed per column. It turns out in this particular example
 that there's only one symmetry operation per class 
and so we have an order of four and we only have four classes
Okay this is a simple example.
We'll show in a second
when we have the number of classes
not equal to the number of symmetry
operations present in the point group.
But we always list the classes not just the individual symmetry operations in the character table
 so we'll come back to that point in a second.
The number of irreducible representations
for any point group is the same as the number of classes,
 so in other words we
have a square table the number of rows
and the number of columns is the same. In this case, we have four columns and we have four rows.
This is always true for character tables.
We always have one representation that's totally symmetric.
In other words all the characters are 
1,
so our totally symmetric representation is the Ag in this particular point group
you can see we have characters of 1 all the way across.
this is a helpful rule that allows us to
actually build the character table
because we can always know that at least
one of them is totally symmetric
Ok, and then we have a certain interrelationship
for the order with the characters in the table
and the following two
relationships hold
so the order is related to the characters the characters
are usually given the Greek letter Chi
in the following two ways so that the
order of the overall point group is
equal to the sum of the characters under the E operation squared
 for each individual irreducible representation i.
so i here refers to the irreducible representations Ag, Bg, Au, Bu
Chi is the character under E that's what this notation tells us
and we square each of the characters 
we add them all together
and we better get the order
Ok, so let's look at this for this particular example:
all we're doing is taking 1x1+1x1+1x1+1x1 = 4
And that is in fact the order for this particular
point group.
So that has to be true, 
and we also have a relationship that's
horizontal. The order has to equal
the sum of the characters for all of the
different symmetry operations squared 
for any individual irreducible representation.
So we can just choose a random irreducible representation and prove that this is true.
Let's just choose Au so we have 1*1+1*1+(-1)*(-1)+(-1)*(-1) = 4.
Pretty trivial, but this always has to be true for character table.
For the E and for any arbitrary irreducible representation, these equations must be met.
And then, finally, there's one other very important rule for character tables
and that is the individual irreducible representations are orthogonal to one another in a mathematical sense
they don't have any component of 1 in any of
the other irreducible representations.
Mathematically what that means is that if you take the sum over all of the different symmetry operations 
of the product of the character on that particular symmetry operation for irreducible representation of i
multiplied by the character for that operation of
irreducible representation j,
sum them all up you have to get 0.
That's the mathematical definition of orthogonality in this context.
So what does that mean? You can just choose any two random  irreducible representations in the table
and do the sum of the products and you better get 0.
Ok, so let's look at the first two maybe.
We've got 1*1+1*(-1)+1*1+1*(-1) = 0.
and that's true for any combination  of two irreducible representations.
These rules are as I mentioned coming out of group
theory, they're super handy of, course,
for building a character table because you
can just make sure that all these rules are satisfied.
Once you've done that you've got your character table.
So let's see how these rules can help us build
the C2h.
We've shown what the C2h result is but we haven't actually arrived at it ourselves in an honest way.
What we ended up with last time just based on
transformation matrices
is we had two irreducible representations 
that we were able to generate
but we know from the rules of
character tables there must be two more
because we have four different symmetry
operations or symmetry classes.
So we're missing two irreducible representations.
How do we find them?
Well, we know there must be four and we know that one is totally symmetric
so immediately what we find is one of them - the Ag - the
totally symmetric representation
that must have characters of all one.
And based on the characters here we can 
 assign its label.
It must be A because it's symmetric with respect to the principle rotational axis
and it must be g because it's symmetric with respect to inversion
We have another one in red here that we don't know yet.
And so by the process of elimination we can
figure this out using the rules.
Okay so which rule would we want to use?
Well, the fourth representation has to be orthogonal to all the other three
and the fourth representation has to have a character under E of one just like all
of them do.
And so we have an unknown, where we know that this is one
we don't know what these three characters are
but we do know that the whole thing must be orthogonal to all the other irreducible representations.
So how can we do that?
Well the quick way to tell is you can
look down here at these other two representations.
They both have two negatives and a positive, right?
Two negatives and a positive.
These first two are negative. These last two are negative.
So I bet you this one has to have a negative in the first
and a negative in the third and a positive in the second, 
and that's going to give us an totally
orthogonal set of irreducible representations.
So that's what's shown here - we have a character under C2 of -1, a character under i of 1,
a character under the reflection operation of -1, and
that's going to give us a Bg irreducible representation
that is totally orthogonal to all the others the other three in the in the character table.
How do we know it's Bg? It's -1 under C2 and it's symmetric with respect 
to inversion,
so that means B and g. 
And you can see that we've got different
characters for all four irreducible representations
we have a 4x4 table as we must have
and you can do those sum rules - vertical and horizontal
sum of products rules
and they'll all work out so that'show we can generate a fairly straightforward character table.
In this case just 4x4, it's not too burly
from the transformation matrices and then using our rules for character tables
Questions about how we did this? I'm sure there are many questions as to why we're doing this, 
but those will become clear I promise.
[STUDENT QUESTION]
Yeah so symmetric with respect to inversion means that when you perform the inversion operation,
the object is multiplied by positive one, it doesn't change.
This...the polarity or the positive and negative [?] parts of the function don't change.
So if you go back here for example.
These guys are good examples of symmetric with respect to inversion,
so you can see that the negatives stay negative and the
positives stay positive under inversion.
[STUDENT QUESTION]
No, that does not mean that. So it means
that it's when you do the inversion operation
the object becomes negative of
itself, that's what it means.
So you can see that here if you do inversion with respect to the PZ
the positive and the negative lobes interchange it's like you took the function,
the orbitals all have functions 
 corresponding to them and you just
multiply the function by -1
that's what you're doing.
So it's anti-symmetric. It's not asymmetric, it's anti-symmetric - anti itself.
[STUDENT QUESTION]
The easiest way to look at the symmetry of an orbital is to look at the positive and the negative lobes,
and so that's why they're shaded in the way that they typically are,
so the shading is to indicate positive and negative parts of the wave function.
Okay, the C2h - a nice character table to build ourselves - fairly straightforward.
Let's look at another fairly simple example of a character table that has a couple of different new features to it.
Let's look at the C3v character table. We should
be able to recognize a C3v molecule
and list all the symmetry operations
that correspond to C3v.
So such a molecule is ammonia, and ammonia here is
C3v because it has a C3 axis 
right down this line here.
There are two operations that correspond to [?] the C3 and the C3 squared.
And we also have vertical mirror planes, right?
We have three of them, each one passing through a nitrogen and a hydrogen, 120 degree rotations around.
There are no other symmetry operations so altogether there are just six symmetry operations.
You can see there's 1+2+3. In the character table they're grouped into classes of operations.
We don't list all of the individual symmetry operations.
We group them together according to their
similarity.
The 2 C3 operations are identical they have identical
characters,
and the three vertical mirror planes are also identical by symmetry.
So they get grouped into the same class
And so in the end what we have is
a 3x3 character table.
So one new feature is that we have classes here that have multiple symmetry operations in them
and that was not the case with the C2h.
The other thing that's different about this particular character table is this guy down here.
We have an E irreducible representation, and you can see that for the E irreducible representation
that means that it's a 2x2 matrix and that means that it's got a character of two under the identity operation.
So this is identity operation. This is irreducible representation E.
Unfortunately, this is the notation. It can be a little confusing because you've
got two E's.
Just remember that this is always the symmetry operation
Okay, if we wanted to build the c3v up using the same approach that we just used for the C2h, 
what we can do is find that we can get the  A1 representation and the E representation from the transformation matrices.
 We have three different kinds
of symmetry operations.
So we have three transformation matrices
E, C3, and our reflection operation.
For E, we have of course just the unit matrix, right? It's unchanged.
For C3 we have a matrix that looks a little bit strange, but it's not that surprising.
This is the general transformation matrix for a rotation.
And we've already used this transformation matrix in disguise when we did the C2 operation.
For a general rotation about angle theta, this is the
kind of transformation matrix you would have.
You just plug in the value for theta and you get the values for these four different matrix elements here.
This is a rotation around the z-axis and so that's why the z coordinate doesn't change.
There's a 1 there.
If we do a C3 operation, we have a 2/3 pi angle that we're changing by.
If we stick in 2/3 pi for cosine, 2/3 pi for -sin and etc, we get these numbers out.
And one of the most important things to recognize here is that when you have a 120 degree rotation
the x and the y coordinates after the rotation,
each of those coordinates depends on
each of the coordinates before the
rotation.
The x and the Y's are linked,
they're coupled together they cannot be
decoupled,
and that's why we have a 2x2 block matrix here.
In the C2 example that we had before, you rotate 180 degrees
and you just change x to
-x and y to -Y.
They don't have any coupling in a C2 but
they do have coupling in a C3.
So that's why we have these off diagonal elements
now, 
because the eventual x and y coordinates depend on both the original
x and y coordinates.
We'll come back to this in a second. Okay, so this is actually the c3 transformation matrix,
and then we have the reflection matrix, okay, and all
we're going to do in the reflection is
we're going to take one of the
coordinates and bounce it through the plane,
the other two coordinates that are
in the plane stay put.
And we'll just choose the y coordinate to change. It
doesn't matter which one we choose,
as long as it's not z, because z is along the principal axis here.
Ok, so what does this give us? Well, if we try to
block diagonalize these three transformation matrices
we have the situation that we can make a block that's 2x2 and a block that's 1x1
Okay, we can't make three individual 1x1
blocks anymore
because we have this coupled situation from the C3 rotation.
So we have a 2x2 and we have a 1x1.
The 2x2 corresponds to the x and the
y and the 1x1 corresponds to the z.
Just as before we can take the character
of the 2x2 and the 1x1,
the character of the 2x2 is, remember the
sum of the elements along the diagonal,
so the sum of these two is 2 and so
that's why we have a 2 under the E operation.
The sum of these two is -1 and so that goes here and the sum of these two is 0
and so that goes there. So we call this an E irreducible representation
because it's a 2x2 matrix and it always is going to have a character of two under E.
The other two characters corresponding to the other two symmetry classes are just read right off
of the summation along the diagonal.
So that immediately gives us transformation matrices E
and if we look at the z the 1x1 in the green here
we have of course a character of 1, a character of 1, and a character of 1.
that's the totally symmetric
representation.
That's the A1, it goes up here at the top of the table: 1, 1, and 1.
Why is it not called the Ag?
It's because there's no inversion center in this molecule.
So g and u labels only
come to be when you have inversion centers.
If you don't have inversion
centers there's no reason to use g and u
because there's no even and no odd.
So in this case we just call it a one because it's symmetric with respect to the principal
and it's symmetric with respect to the next symmetry operation over.
How do we get a 2 to complete the table?
Well, we can do that by using the
orthogonality rules
and the fact that A2 has to have a character of 1 under
the identity operation.
So that gives us E in the red, it gives us the A1, and the
green matrices there.
A2 comes about by using orthogonality and the fact that the character under E has to be equal to 1,
so we can run through that very quickly. The easiest thing to do is to look at the difference between A2 and A1.
Alright, remember that the sum of the products over all of the symmetry operations has to be equal to 0,
and so we have 1x1+2x1 = 3 so far.
plus 3x(-1) gives us 0. So 1+2+(-3) = 0.
And you could do the same thing
down here as well.
You can do 2 + (-2) + 0 = 0.
Okay so that's how we would build the A2 out of the other the other two irreducible representations
and generate the entire character table.
Okay, so just some notes about this and how
it's different than the C2h,
just to cement these points: The C3 and the C3^2 are identical.
After doing a C3 operation, you can do a C3 and you can convert those two into one another and
so they're in the same class.
And they're grouped up here in the table as 2 C3 operations.
The three mirror planes are identical after 120 degree rotations.
In other words, after C3.
And so they are also in the same class.
We call those the three vertical mirror
planes and we group them up here into one column.
We see the E representation here. This is immediately telling us it's a two dimensional representation.
In other words, in this case x and y are coupled together, so we have a 2x2 matrix
that gives us a character under E of 2. And for this particular point group it mixes x and y.
That's not necessary. It could mix x and z
could mix y and z.
If you see a T that means it's a 3x3 that
means the x, y, and z are all coupled together, 
and there are various rotations that you could imagine that they're going to cause that sort of situation.
And the way that we think about this is that 
when we see the full character table, when you see functions that are in parenthesis like this.
This is telling you that the x and y together transform like an E.
The x and y are
inextricably linked together in this case,
so there's no sense in thinking of
them as just x or Y,
but you think of them as x and y together.
Or Rx and Ry together, rotations around those two axes, at the same time give us an E symmetry.
When they're not in parentheses like this.
This indicates that you can see the x^2+y^2 or the z^2 and it's not linked together in the sense of
having a 2x2 or 3x3 matrix.
And you can just quickly look through this table and make sure that it obeys the rules of the character tables that we've already described
Orthogonality, we've already shown. But you can show that the
summation rules along the column and
along the rows are also met and they are. 
And they sum to the order of this
particular point group which is 6.
Any questions about this procedure?
[STUDENT QUESTION]
Yeah, sure the orthogonality, so remember the orthogonality is that
we're going to take the sum of the products of the individual characters and they better add to zero.
So let's do that for A1 and A2. We take the product of
these two which is just equal to 1.
We sum these two but we have to do
it twice because there are two operations
So that's 1+1+1 = 3 so far and then we have -1x3 so that's six terms all together on our sum.
Three of them positive one, three of them
-one, so there's this zero.
Other questions about this? It will be a bit
perplexing at first. It's totally normal.
When we start using them to look at
molecular vibrations for example
it'll become much clearer and more familiar, so
no worries at this stage.
Other questions, though?
Okay, so let's do a little summary
here and then we're going to look at our
first application of symmetry and that's
what we're going to finish up on today.
Each molecule has a point group as we
know
and the point group contains the full set of symmetry operations that describe the molecules' overall symmetry.
And we know how we can assign point
groups this at this stage we can use the decision tree
or we can use whatever you know logical decision making process we want.
The idea is we want to get from the
point of using the decision tree and
relying on that to just intuitively
seeing what the point group is of the molecule.
Okay and I think that the time was something like less than 10 seconds is what we're shooting for.
Can be done.
Character tables which we just showed examples of, we showed how to build two of them
Okay we kind of cheated because we went to the final answer then we backtracked
but this is just for teaching purposes so hopefully it's easier to understand.
Character tables show us the complete set of irreducible
representations and how those transform
under all of the different symmetry
classes of the group.
So it lists all the symmetry information in a convenient tabulated form,
and we're going to use the symmetry tables to understand bonding and spectroscopy.
Again if you want more information then this is the book that I used back in the day
and would be a good reference book even even
now.
This is one of the classics in the field.
Okay, let's use symmetry our first application of symmetry is pretty straightforward
and in fact we don't need the character table at all to do this.
We will need the character tables extensively in the next class to look at molecular vibrations.
The first application of symmetry is the incredibly powerful concept,
so we can immediately use symmetry to tell when a molecule is going to be chiral.
There's just a simple rule of when you can have a chiral molecule and when you can't.
To be chiral, a molecule cannot have an improper rotation axis.
If it lacks an improper rotation axis, it is chiral.
What does that mean for our point groups? What point groups lack improper rotations?
We have to be in the C1 or the Cn, n can be anything,
or the Dn where n can be
anything, the pure rotation point groups. 
We can't have any mirror planes because if you have a mirror plane
remember a mirror plane is the same thing as an S1
and you can't have an inversion center in your molecule to be chiral
because according to our rule
and we remember that inversion is the
same thing as 180 degree rotation
followed by perpendicular reflection, 
otherwise known as an S2. So you can't
have a mirror plane, you cannot have an inversion center
and you can't have any of the other possible S axes.
So let's look at a bonehead example of chirality here:
this is the mirror image of this particular
tetrasubstituted methane with itself.
We know already we've assigned the point
group of this molecule at C1. 
It has no symmetry asymmetric. It only has E operation.
And if you take its mirror image and you try to
superimpose them of course it doesn't work out
and so it's a chiral molecule because it's a C1.
Let's look at a second example.
So this one's a little bit harder, but
you could sit there, and you could convince yourself
That this is a D3 molecule, and it has no reflection planes, and no inversion center in it.
but it does have a three-fold axis of rotation that's coming out at you
and it does have perpendicular C2 axes of which there must be three
It's a D3 and that tells us right away because it has no S operations associated with it
it must be chiral. So here's the molecule and its reflection.
Try to superimpose them
and you cannot do it.
Okay, so there are many rules like this.
There are rules that pertain to chirality,
there are rules that pertain
to other optical properties like whether
a material can be second order nonlinear
material
whether molecules have certain kinds of awful activity - many different things can be immediately described
as absolute yes or absolute no based on just symmetry and that's why it's so powerful.
So next time what we're
going to do is we're going to pursue our
second application of symmetry where we actually use the character tables
in detail to understand vibrations
