Everybody, good morning!
Hope you're all well.
A couple of announcements to begin this morning.
The TA's are both going to hold their office hours on the 4th floor Lobby of Natural Sciences 1.
So this is kind of an open breezy space that sometimes serves the purpose of office hour room.
So for the time being that's where they're going to hold office hours.
In the future that may change because we're trying to get some dedicated rooms for the use of office hours.
We'll see if that pans out.
Okay,
so but for now Natural Sciences 1 fourth floor with the times that are already in the syllabus and on the website,
Okay.
So with that let's get started this morning.
So we were talking last time about point groups and I hope everybody has been practicing assigning their point groups.
I'm sure it's been that's been very much the case.
That's not good.
okay.
So what I thought we would do this morning is start with little self test of point group assignments.
I saw a show here four hypothetical molecules that serve as nice examples of assigning point group.
And of course the decision tree that helps us if we need it to make the decisions to assign the point groups rapidly and correctly,
but I'd like to do is to take about 2 minutes you guys try to assign a point groups for these four molecules,
and I'm going to walk through what we think the answer should be
We will have a couple of different examples. Molecular and non-molecular examples of assigning point groups.
So take 2 minutes now and try to work through these four examples here and then in 2 minutes will discuss them.
Okay,
everybody finished?
So some looks of confidence and looks of complete panic and in the audience. We'll go through the examples counter-clockwise.
So let's start with this one here.
What do we see about this molecules at high symmetry or is it very low symmetry?
That's kind of the first set of questions.
It's neither.
Right?
And so what we want to do then is a sign the principal axis the principal rotational axis of molecule.
So what do we think that is?
Cancel we see it's easy to axis along the let's call this the metal metal bond.
That is what M could could indicate so we cac2 along the metal metal Bond.
If we do a rotation 180 degrees we change As and Bs on the sides.
Okay,
next question.
Any perpendicular C2 axis? Are any perpendicular C2 axis present?
Okay.
So let's look at the vertical axis as possible.
Right if we had a vertical axis and we did a rotation of 180° B would go into A.
So that's not a good rotation waxes.
Are there any others that are possible perpendicular to the principal axis?
We have to kind of deep focus our eyes a little bit.
What if you had an axis it was pointed 45 degrees directly between these two atoms here coming out this way and going in that way.
A 45 degree angled axis would rotate B to B, A to A in the front, and A to A, B to B in the back.
And so there is one perpendicular C to axis that cuts right through the face of this plane here
that goes this direction and there's a second one that goes this direction - cuts right through the face of this plane here -
these four atoms and cut it down to the left.
So there is a principle C2 and two perpendicular C2 axes.
It's...yeah,
it's a diagonal C2 axis,
right?
It doesn't pass through any of the atoms.
It passes between all the atoms
But that immediately tells us that we have a what kind of point group?
A D point group and we have specifically a D2 something point group because we know the principal axis is two.
So then the question becomes one of your planes.
Is there a horizontal mirror plane in that molecule.
Remember horizontal mirror plane would give you one that bisects the metal metal Bond here.
And it's perpendicular to the C2 axis.
And there is not
Because if there were
we reflect B into A and that doesn't work out for example. And all the other atoms along the square don't work out.
What about horizontal...oh sorry
What about vertical mirror planes, those that are along the metal metal axis?
There are vertical mirror planes, so we can have a vertical mirror plane that's in this plane here.
That reflects A to A and B to B and all the other six atoms here stay unmoved.
And so what we have is a D2d point group in this particular case.
It's not D2h because there's no horizontal mirror plane,
but we do have vertical mirror plane.
So it's D2d.
So everybody see how that's done?
[Student Question]
Yes so in the D point group.
So you either have an h or a d possibility there is no v possibility.
So basically d and v are kind of the same idea in the D point groups.
If it's perpendicular,
if there's a mirror plane perpendicular to the principal axis is horizontal,
but if it's along the principal axis then it's either d or v.
Okay.
So D2d is the first one.
What about the second example here?
What do we have here?
So do we see a C2 axis in this molecule?
Where is it?
There's one that goes right through the same way that we showed last time right 45° comes right down.
Is there perpendicular C2 axes in this case?
Turns out that there is none, so this is just a C2h.
So in this case,
we have a horizontal mirror plane.
Let me see where it is.
The horizontal mirror plane is right between these two atoms
It's a plane that cuts right through here and it would reflect A to A, B to B,..., and it's perpendicular to the C2 axis.
It's coming 45° out of this plane of atoms here coming out this way and going in this way.
There's no perpendicular C2 cuz if there was a C2 axis that we tried to do in this direction,
we'd flip 180 degrees and put the Bs where As are.
And there's no perpendicular C2 along the metal-metal bond
because if we did the same thing this B would go to that A, this B would go to that A, and it doesn't work out.
So this case we only have a C2 with no perpendicular C2,
and we have a horizontal mirror plane.
So that takes us into the C point groups and it's a C2h in this case.
Makes sense so far?
So the C2 axis.
This plane here this made up of these four atoms or three atoms if you wanted to pick 3,
the axis is perpendicular to that plane.
So points 45 degrees in this direction cuts right through that point, right through the middle of the metal-metal bond.
It comes down to the right of the molecule.
From this way from the right down through the bottom this way.
Well,
if you look at that,
what do you think?
So if you do 180 degree rotation along that direction the Bs will swap with the As.
B rotate into the A position so you don't have C2 axis in that direction.
So the axis that you're talking about would be this one,
right?
If you rotate a 180 degrees, these four atoms here in this plane
perpendicular to the axes will rotate 180 degrees around. These Bs will be then located where these As are.
so maybe throw you off a little bit that you know,
we're not dealing with just X Y and Z, but we are also dealing with
coordinates for rotational axes that are between the main Cartesian axes.
So it's a little bit difficult to visualize maybe.
People see that there's only one C2 axis in this molecule.
So if it's in this direction here.
Can you have A-B-A-B-A-B-A-B if you rotate 180 degrees the B's and A's interchange?
Okay,
what about example number three?
What do we think about this one?
Are there any rotational axis in this molecule?
This one here.
So if we chose a C4 axis along the metal metal bond,
for example,
and did a 90 degree rotation.
This A goes there but this A goes to B's position
So that doesn't work.
So C1 is one possibility.
Okay,
so I agree with you that this is a low symmetry molecule.
Ci, Cs are the other two possibilities.
What do you think about an a Ci point group, with a Ci it has the identity and it has an inversion center.
Right?
Does this molecule have an inversion Center?
Does not. Does this molecule have a mirror plane?
Where is that mirror plane?
how about this plane right here?
Right? That keeps these six atoms in the plane unmoved,
but it's swaps the Bs and it swaps the As and we're good.
So this molecule is a Cs.
A low symmetry example where we only have one mirror plane present. That's it.
Other than the identity.
Everybody with me so far?
Okay,
what about the last example?
[Student Question]
Yeah, for this kind of example is definitely right in the center of the molecule.
Yes.
And so in general that's going to be true. It has to take all of the coordinates of the molecule and swap them,
you know XYZ.
It's a negative x negative y negative see it has to be in the geometric center of the object.
[Student question]
So so this doesn't make any sense to to call this an important point in the molecule
because if you try to invert through this you would displace the molecule actually,
and it wouldn't be a point group anymore.
So yeah,
the inversion center is going to be the center of the molecule.
What about the fourth example?
What what is the principal axis of this molecule?
So we have a C4 axis.
I'll agree with that.
The C4 axis is the metal metal axis.
You do rotations,
the A's interchange and the Bs interchange.
So then the question is horizontal mirror plane.
Yes or no?
Hearing equal mixtures of yes and no.
So remember the horizontal has to be perpendicular to the principal axis,
which would mean it's this playing right here that passes right through the metal metal bond and passes through all of these four bonds there too.
If we were reflected across that plane the Bs goes to the As.
That's not a good symmetry operation.
So there is no horizontal mirror plane in this molecule.
So we know now that we're either C4 or we could be in S something.
What about other kinds of mirror planes, vertical mirror planes?
Do we see a bunch of vertical mirror planes?
There's a kind of a ton of them right?
So there should be four vertical mirror planes.
So you have a mirror plane in this plane.
You have a mirror plane perpendicular to that plane.
And you also have mirror planes that are at the 45 degree bisecting this plane here and bisecting this plane here.
Right away, based on the decision tree,
we fall in here,
right?
We said there's no horizontal mirror plane.
We asked if there are vertical mirror planes. We say yes that must mean that we are C4v symmetry.
So the trickiest two things about these examples are first of all,
we have rotational axis that are not passing between the atoms in some cases.
So those can be difficult to see it first,
but you need to keep in mind that they can exist and they're just as important and relevant as any other rotational axes
and the other thing that I find particularly difficult is to verify that you have a low symmetry point group when you have a lot of atoms.
So you really have to stare at this for a little bit too, considering to convince yourself that you don't have any rotational axis,
you don't have an inversion center and so forth.
Questions about these four molecular examples?
[Student Question]
Horizontal mirror.
So so if you have mirror planes, you always reflect through the plane.
So the coordinates that is perpendicular to the plane goes from positive to negative coordinate.
If you have a rotational axis, then you're actually doing a rotation rather than of a reflection.
So the the mirror planes can be anywhere.
So here for example,
there's a mirror plane that's you know,
coming out 45 degrees and that reflects B to B, A to A, on the bottom here, B to B, and A to A.
Are the questions about this?
I promise with practice you will become experts. It will be like second nature to do this.
Let's look at another example.
What is the point group of a baseball?
So you can have a lot of fun just going through everyday life and assigning point groups to random objects.
It helps to have some symmetry to the object.
So,
you know signing a point group for the chair is not too interesting.
It's C1, right? But assigning point groups to other things can be a lot of fun.
So what about the point group of a baseball including the stitching?
So we're not just looking at the sphere but we're including the pattern that's made by the stitching
and if you have difficulty looking at a baseball and understanding what the symmetry could be for baseball,
you can look at a tennis ball instead.
It's brighter and it has exactly the same stitching pattern right, exactly the same symmetry.
So what do we think about the symmetry of a baseball or tennis ball?
So we see that there's a C2 axis in the baseball and the tennis ball.
Where is that C2 axis?
...closest point of stitches.
Yeah,
so it's may be easiest to see because of the in the in the tennis ball because of its rotational orientation,
but the rotational C2 axis is coming right through here, passing down that way, coming out this way.
So if you hold a tennis ball in your hand with the narrow part of the stitching close to you
the C2 axis is coming right at you. You rotate around that 180 degrees and you get the same object back.
Again,
that axis is popping out of the board kind of at a shallow angle in this case, going in and through the center of the tennis ball in this way.
Passing between the closest parts of the stitching here.
So we have a C2, I'll agree with that.
Do we have perpendicular C2s?
Does anybody have a tennis ball with him?
[Laughter]
Or baseball?
I should look this morning and I think my daughter has thrown all of our tennis balls over the fence.
[Laughters]
So it turns out that there are two there must be zero or two.
There are two perpendicular C2 rotational axis.
And you can kind of see it again with the tennis ball easiest and the rotational axis passes through this stitching and right through the stitching.
And also another one right through the stitching and right through that stitching.
If you get yourself a tennis ball and put your fingers one finger here and one finger here,
and use your other hand to rotate the tennis ball on your hand.
You'll see that after a hundred eighty degrees.
The tennis ball looks exactly the same stitching.
Remember the stitching kind of nut does a funny pattern on the back.
And when you rotate around you'll see that in factors the same object. C2 perpendicular is a good rotational operation.
I wanted to find also a video, but there were videos last year,
but I couldn't find the videos anymore.
Somebody taking them down of actually assigning the point group of a tennis ball.
So we have a D2.
The question is are there any mirror planes in the tennis ball?
They're definitely on your planes.
Right?
So there's a mirror plane that cuts right the tennis ball a half in this direction.
For example,
and a mirror plane to cut a tennis ball in half in that direction.
So what's the point group of a tennis ball or baseball?
D2?
is there an H?
D2d.
Tennis ball is D2d and here are the different symmetry operations that pertain to that particular point group.
The principal axis is the one that we chose to the beginning,
but it doesn't matter which one you choose.
The principal axis is this one that comes this direction and passes through the middle of tennis ball
kind of in that direction passes right between the narrow parts of the stitching.
And so then you have, you know,
mirror plane that is parallel to the axis in that direction and another mirror plane that is parallel to the axis in this direction.
That's why it's a D2d and not a D2h.
[Student Question]
Yeah,
this is the hardest part about assigning the point group of a tennis ball is finding a perpendicular c2s
and it's not totally satisfying to do it in a two dimensional projection,
but it goes from here to here and it's perpendicular to the principal axis.
And another one goes from here to here also perpendicular to the principal axis.
Okay tricky.
So you can assign point groups lots of things the Eiffel Tower, whatever you want, pyramids, ...
Lots of architecture has high symmetry.
Let's get back to the molecules.
What about in this, the final self test example,
what about the point groups of these orbitals?
This orbital is called what?
py, px, yep.
This is what?
That's a d orbital right?
And then do you know what this is?
That's the fx, y, z orbital.
Let's start with the easiest, the 2px orbital.
What's the symmetry of px orbital?
I'm hearing the right answer.
Yeah,
so it's linear, right?
This is a linear object.
This is C∞v.
It has an infinite rotational axis along the x-axis and it has an infinite number of vertical mirror planes,
but it does not have a horizontal mirror plane.
So it's not D∞H,
but it's C∞V.
What about the the D orbital here?
This is an object right with four lobes of alternating sign,
right this could be positive and that's a negative. So this is + - + - in the XY plane.
So what's our principal axis in this case?
So it has a C2 axis as the principal axis.
And in this case,
it's just the most convenient to pick the z-axis as the principal axis.
If you do a hundred eighty degrees this blue lobe rotate into that blue lobe, this white lobe rotates into that white lobe.
And we have perpendicular C2s.
We have a perpendicular C2 along the Y and a perpendicular C2 along the X.
Okay.
And we have mirror planes.
What mirror planes do we have?
Do we have a horizontal mirror plane?
We do. So in the plane of this object.
We have a horizontal mirror plane.
And so that immediately tells us that we have d2h.
So we can say that this particular the orbital has d2h symmetry
and this is one of the main reasons that we care about symmetry
because we can assign the symmetry of orbitals and that is an enormously powerful thing
because then we can understand how orbitals mix how they make molecular orbitals and so forth.
What about this fx, y, z orbital?
What does this remind you of?
Like if you just look at the blue lobes or you just look at the red lobes.
So it looks like a tetrahedral model.
that's just the red lobes.
You can consider these lobes that are pointing toward hydrogen for example, as in methane, right?
So these two and then you have these two red lobes low and 90 degrees rotated.
So now look at this and say: hmmm, this looks like a high symmetry object.
The first example of a high symmetry object we've seen.
if you wanted to tell if it were, for example, octahedral,
what would we look for for octahedral symmetry?
What's the defining characteristic of octahedral symmetry?
Octahedron. Icosahedron C5. C4 is octahedron.
So we have 90 degree rotation axis. Do we have any 90° rotation?
Axis here? Proper axis?
No,
we don't right? If you had an axis this direction here and you rotated 90 to blue and the red would interchange,
which is not cool.
So it's not octahedral symmetry.
You can't find a C4 axis in this object.
What about tetrahedral?
What's the defining characteristic of the tetrahedron? C3 and multiple C3s in different directions.
So what about C3 axis?
Do we see any of those?
So just imagine that you only have the blue lobes present and that each of those lobes is a C-H bond,
for example,
if you had an axis that goes through this CH Bond and you did 120 degree rotation,
you'd rotate these three blue lobes in [...] another. When you add the red lumps on if you'd like, and they do the same thing.
So you have one C3 axis through there, and one C3 axis through there.
You have a third one coming from the back of the plane of the board through this way,  and you have the fourth one this way.
So we have four C3 axes.
That is the defining characteristic of a tetrahedral point group.
You can go through all the rest if you want to do it turns out that this is Td.
Remember we have the three options T, Td, and Th.
Td is the one that's full tetrahedral symmetry with all the different mirror planes present
and you can see that you have a lot of mirror planes present in this object.
[Student Question]
Yes,
just there.
That's right.
That's right.
Yeah,
so this particular example here has mirror planes and axes that have points in common.
Okay,
So self test - successful...maybe.
[Laughter]
Or maybe it gives you an idea that it would pay to study up on how to assign point groups
What we want to do now is we want to move into some more of the mathematical machinery that we'll need
in order to make use of our symmetry classifications to describe molecules in different ways.
So, we're going to jump into
The mathematical machinery of point groups, and this comes out of an area branch of mathematics called group theory.
We're not going to derive group theory,
but we're going in and play with it in detail.
What we're going to do instead is use the results of group theory to apply to different chemical problems.
If you want more information, more in-depth description of how group theory applies to chemistry.
This book here is a great example of a great reference guide.
So this is Cotton's Chemical Applications of Group Theory
And I recommended very highly for anybody that wants to have more information about what we're going to talk about.
So in many cases,
what we're going to do is just present the results if you want the derivation or the proofs and so forth.
This is one reference book to look for.
There are other books that are mentioned at the back of the chapter 4 that are also very good references.
So all the point group that we've talked about so far are examples of an algebraic structure that's called a mathematical group.
And the definition of the group is fairly abstract.
It's a collection of elements that obey rules,  certain algebraic rules.
And the rules that these different elements obey give rise to the properties of mathematical groups
and it turns out that these properties are very important for chemistry.
So let's look at the rules that mathematical groups obey
And there are four main rules that we're going to describe.
It's just a copy out here.
This is going to be very abstract.
And it seems like it takes us away from chemistry totally but it really doesn't.
Just bear with me about a one class one full lecture of describing group Theory and how it applies to what we care about
We will come back then to chemical problems and it will become very clear.
I think that's why it's so powerful.
So the first rule is that every group, in our case every point group,
Has an identity operation.
That commutes with all other members of the group leaving them unchanged.
Remember our identity is just multiply by 1
What does commute mean? Commute means that if you take the identity and some other operation,
Let's call it A, and you
perform the math in this way.
So we do A first and then E, it's the same thing as doing E first and then A. The order doesn't matter.
And because its identity operations just multiplies by 1
so acting with the identity of operation just gives us the other operation out unchanged A, whatever A might be.
In this case A is a symmetry operations C2.
It's a reflection plane, it's an inversion,..., it's something like this.
So commutation means that the order doesn't matter, you get the same result.
So that's the first rule. The second rule of our point groups
Is that every operation in the set of operations in our point group
has an inverse operation that when combined with our operation yields the identity.
When we talk about combination we're talking about multiplication in the matrix sense.
So when we do combinations, when we act on a molecule, we are always going to act from right to left.
So in this case,
we do A first and then E, just like you would do in a matrix multiplication.
So the inverse is such that when you multiply it together with the operation you get the identity.
Let's look at an example of how this works out in C3v point group.
We have these different symmetry operations - 6 operations all together.
Every one of those operations has to have an inverse that gives us identity.
So what is the inverse of the vertical mirror plane? While it's just simply itself, if you do a reflection,
and then you do reflection back that gives you identity, gives you nothing changed.
So the inverse of the vertical mirror plane is itself
the inverse of the C3^2 operation, the 120 degree rotation, is just the C3 operation,
which is also in this set.
So we do 360 degree rotation and we get identity.
So we would say that the inverse of the C2^2 is the C3^2.
I'm sorry, it's the C3.
And all of these inverses have to be in the set of objects in our point group.
[Student Question]
I'll show you in just a second now.
So I'll show you how we multiply these operations together in a minute.
Okay,
so we have the identity and every operation has an inverse in this sense.
The product of any two operations, that is any binary product of the operations in the point group, must also be a member of the group.
So we will multiply any two of them together.
We get a third perhaps but it's also already in the set of objects.
So for example,
let's look at the C4v example,
which has these symmetry operations present.
We choose any binary product, that is, two things multiplied together, in this case C4,
Let's multiply that by C4.
So do C4 twice, that gives us of course C2.
And C2 is in the set already, as it must be a complete set.
What if we do a vertical reflection and then a C4?
Well,
it's not super easy to see just looking at it.
But if you write it down on a piece of paper,
you can see that that result gives you a dihedral reflection,
which is already in the set.
If you do a vertical mirror.
and then a dihedral reflection that turns out to give you a C4, which you can prove very easily by just jotting it down on a piece of paper.
And of course the C4 is in the set.
So this is the way that we can actually tell that we had a complete set of symmetry operations in our point group.
Is that all the binary products have to yield something that's already in the set.
This is the way that we can find all of the symmetry operations if we don't already know what they are.
So this is sort of a beautiful completeness to these two these sets of elements.
And the fourth rule,
is that the associative law of multiplication holds.
So if we have three operations A, B, and C it doesn't matter if we do C and B combined first and then A,
or if we do C first and then B and A combined.
As long as they're done in the same order, you get the same result.
It turns out that it's a very convenient and powerful to think of this math in terms of matrix multiplication
so we can assign basically a matrix to any individual symmetry operation.
So in order to really understand what we're doing here,
we have to review very quickly matrices and matrix multiplication, the very basics of matrix math.
If you are totally unfamiliar with this subject,
and there is a little primer that's on website: basic algebra for matrices, introduction to matrices, and matrix basic algebra.
And so we're just going to look at the very essentials here.
A matrix is just an array of numbers
and we're interested in mostly square matrices that have an equal number of rows and columns.
And these numbers could be anything.
I just threw up here or just totally random example of 3 by 3.
matrix is what we would call this a square matrix.
We're also going to come into contact with column matrices.
So call a matrix is just one in which you have only one column of numbers.
And a row matrix is one in which you have just one row of numbers.
We talked about multiplying symmetry operations.
We're talking about multiplying in terms of matrix multiplication.
Multiply two matrices together what we do is we add all of the products element by element.
Of each row of the first matrix with each column in the second matrix,
so it's row * column and adding up all of the different elements.
So let's show how this is done by just a two-by-two matrix multiplication.
So we're going to take this matrix,
which is a 2x2 multiplies this one which of the 2x2.
And what we want to do is we want to add up the products of each element by row and then by column.
So, for example, if we wanted to look at the element is going to be located in the 1-1 position in the product matrix we're going to have 1 * 1 + 2 * 3.
So we take the products and we add them together 1 * 1 + 2 * 3, first row, first column.
So that would give us a number here turns out to be 7.
What about the next one?
Let's just do this one here.
This is going to be the second row and the First Column so 3 * 1 + 4 * 3 is the value for that element in the product matrix.
And we can do the other two.
So this is going to be the first row second column.
That's this one here first row second column 1 * 2 + 2 * 4 gives us that number
and then finally the second row second column 3 * 2 + 4 * 4 is what we should expect there.
We can then simplify that and that's what you get for the product. A two-by-two matrix, in this case, and what you have,
you know.
the sum of the products.
it turns out that for almost all of the matrix multiplication that we're going to be doing
what we're interested in is a 3x3 * a column matrix.
This is almost always what we'll be doing.
And in this case if we have a 3x3 square matrix * column matrix, in many cases were have a very simple 3x3 matrix.
You do the same prescription, row by column.
There's only one column here.
And so we would have in this case, for the first elements of the product, we are going to have 1 * 1 + 0 * 2 + 0 * 3.
That's one.
the second element here is going to be 1 * 0 + -1 * 2 + 0 * 3. That's -2.
and then the third one down here is going to be 0 * 1 + 0 * 2 + 2 * 3, or 6.
That's how we would get the column the product column matrix from this.
Why am I talking about matrices here?
Let's show how this can really be powerful tool in understanding what the symmetry operations are doing.
Okay,
we can represent every symmetry operation that we've talked about.
As a 3 by 3 matrix that converts the original coordinates XYZ into the new coordinates after the transformation by the symmetry operation.
That's why we call this a transformation matrix.
It takes the original coordinates and transforms them into the new coordinates after the symmetry operations done.
To make it concrete,
Let's consider a specific example C2h point group. Here's a molecular example of that
We should be able to quickly tell that this is C2h.
The C2 axis is right there at the middle point of that bond, interchanges the Br, and it has a horizontal mirror plane that makes molecules flat.
Let's assign XYZ coordinates to this molecule.
The convention is always to put the principal axis has the Z axis
and then you can put the X and Y as the molecular plane and that's the convention that we will follow.
The z-axis is coming out at you.
Okay,
let's develop transformation matrices 3x3 matrices for each of the symmetry operations in this point group.
We have identity, C2, inversion, and the horizontal mirror plane.
What is the C2 operation due to the X Y and Z axis?
If we do a C2 operation the x rotates 180 degrees around it becomes -x.
The new coordinates for y, y rotates 180 degrees around and becomes negative y.
And z is unaffected by C2 operation because it's happening along the z-axis.
So the z just becomes itself.
No change.
We can represent this as a three-by-three transformation matrix like this.
We have elements only along what's called a diagonal of the matrix.
We have a negative one which tells us that the x is going to be * -1.
We have a negative one for y. We have a positive one for z.
Let's see how this works.
This is the transformation matrix for C2 in this particular point group.
This is how the matrix multiplication looks in order to get the new coordinates x', y', z'.
We take the old coordinates x, y, z and we act on the coordinates with the transformation matrix here.
We've got here and that's going to give us the new coordinates in terms of the old coordinates.
So what do we do for x here, for the first element in that column matrix?
-1 * x + 0 * y + 0 * z we get negative x
for the second 0 * x + -1 * y + 0 * z gives us -y?
and z is unchanged: 0 * x + 0 * y + 1 * z = z
so matrices are natural way to show how the coordinate system transforms under the symmetry operations.
So that's C2
what about inversion? Inversion takes x to -x, y to -y z the -z.
for the transformation matrix for an inversion operation is just negative ones along the principal diagonal
and nothing else for the other elements in the matrix.
And so if we do the same math here for the inversion, we act with the inversion operation,
We're going to convert the x, y, z into -x, -y, -z.
So we've done C2 we've done i, E is just trivial. It's just the unit matrix with just ones along the principal diagonal
and we could do the same thing for the horizontal mirror plane.
What would the transformation matrix look like for the horizontal mirror plane?
It would be 1, 1, -1 along the diagonal of the matrix.
So the only axis it's going to change in horizontal reflection is going to be the Z-axis,
the X and the Y coordinates don't change.
And that would be our four transformation matrices for this point group.
So here they are.
Here's the four transformation matrices and we're going to...yet we have about 5 minutes.
So we should be able to reasonably describe this.
So here's for the identity.
Here's for C2.
Here's for inversion.
And here's for the reflection.
Okay.
One of the things that we can immediately do is show that matrix multiplication gives us
the exact same answer that we know intuitively if we multiply two of these operations together.
We know that if we take a C2 and we then do it twice that we're going to generate the identity
that the molecule is reflected on or is rotate 180 degrees and then rotated 90° back.
So we've done nothing.
Matrix multiplication better give us an answer and it does
so if you take this matrix C2 and multiplies second matrix C2, you can convince yourself that you just going to have the unit matrix 1-1-1.
which is E, that's the matrix for identity.
So it really works in any arbitrary sequence.
What we want to do is to take the information that is buried in these matrices
and we want to tabulate in a very convenient way for reference.
We're going to develop what it's called a character table for this point group.
So what do I mean by that if you look at these transformation matrices,
you see there's only values among the principal diagonal.
If you take the sum of the values along the diagonal this is called character of the matrix
in math it's called the trace of the matrix. The trace is equal to the character.
So the sum of the numbers along each matrix diagonal is called a character in our description
and it gives you basically a shorthand version of The matrix representation.
You don't need to know all of the zeros in the 3x3 matrix in order to get the essential symmetry information out of what the identity operation does.
All you need to know is the sum of the diagonal.
Here's three that tells you in another way a simpler way what the symmetry is of that particular operation.
So we take the sum that's called the character and for E we would have the sum giving us 3.
So we would have a character of 3 for the identity operation.
That's how we would describe that in words.
If you look at C2 and we did the sum we're going to have negative one.
So the character for the C2 operations -1, character for the inversion operation is -3, the character for the reflection operation is 1,
and so what we've done here is we've condensed the transformation matrices, matrices all of these numbers.
That's just it redundant information into a simple 4 number set that contains all of the information up there of the transformation matrices.
This thing here this list of characters is called gamma.
This is called a reducible representation of the transformation matrices for this point group.
It's reducible in the sense that we can divide into simpler components as we'll see maybe next time
but what we would do is we would list the point group,
list the symmetry operations for that point group.
And in this case,
we would say that we have a reducible representation with the character of 3 for E, -1 for C2,
-3 for the inversion operation, and one for the reflection
will show next time how we can break the reducible representation into irreducible components
and build the entire character table for this particular point group.
