Tip:
Highlight text to annotate it
X
[ Silence ]
>> Good morning everybody.
Let's go ahead and get started.
It's about that time.
So, if you didn't already get a character table, raise your hand
because we have a bunch right here.
All right.
It looks like almost everybody did.
In case you haven't met our TAs yet, this is Jerry.
I think John Mark and Upali [assumed spelling],
you know from last quarter, yes?
OK.
[ Pause ]
All right.
So, we have a lot to do today.
We're going to do a few more examples
of assigning molecules to point groups.
So, I hope everybody has your flowcharts hopefully corrected.
There's a corrected one posted online.
There were a couple mistakes.
So, sorry about that.
Please do check out the corrected version.
Also, so you should have your flowchart
and your character table that we're handing out in order
to be able to go through the examples.
And so, hopefully everybody tried the practice problems
on assigning things to point groups.
And we're going to go through a few of those this morning.
And then, we're going to continue to talking
about matrix representations.
So, somebody also asked me
when the lectures are going to be posted online.
And I actually don't know.
But the person who does know is Shawn
who is doing the filming in the back.
Can you tell us when that's going to be?
>> Yes. So, well actually,
the first lecture is this demo module search on YouTube--
[ Inaudible Remark ]
For UCI OpenCourseWare that's on department.
You'll be able to find the professor's name
and the class name.
[ Inaudible Remark ]
>> Great, thanks.
So, in general, how long do you think it's going to take for--
yeah, so after each lecture, how long--
[ Inaudible Remark ]
Great. Thank you very much.
So, that's where you'll be.
All right.
Any more questions about logistics, things like that?
One thing I should mention is
that my Wednesday office hours are 3 to 4.
I'm going to have that-- a bunch of people showed
up at office hours yesterday.
We had a really good discussion.
So I encourage you to do that.
The Tuesday office hours I think are going to be moved to 11
to 12 from now on to avoid conflicting
with our discussions.
So, is there some class that every one has to take Tuesday
from 11 to 12 that I don't know about?
OK, good. That sounds like a winner.
OK. So, let's talk about some point group examples.
[ Pause ]
OK. Can everybody see?
OK. S o unfortunately, I don't have very much control
over the lights.
Our choices are like that and like that.
So, raise your hand if you like it better darker.
Raise your hand if you like it better brighter.
OK, darker it is.
All right.
So, if we have a molecule like this and we want to assign it
to a point group, what do we want to do?
We need to get out our flowcharts and look at--
does this thing belong to any of the special groups?
So we have low symmetry, high symmetry, and linear.
So, if we added one more substituent here,
then that would be, you know,
something like sulfur hexafluoride
or xenon hexafluoride.
Then, that would be an octahedral molecule.
But we don't.
We have this.
So, what do you think?
I think I'm going to get somebody to volunteer
to assign this thing to a point group.
>> Is that high symmetry?
>> You just volunteered.
>> Oh no.
[ Laughter ]
>> OK. So, just to-- so everyone can see what it looks like here.
So we've got this thing that has a square on the bottom,
and then it's got one substituent sticking up.
And the thing that keeps it
from being high symmetry is it doesn't have another one
on the bottom.
>> So, it's not high symmetry?
>> So it's not high symmetry.
>> OK.
>> So now what?
Now, we need to find the principal axis.
>> Would it be C4?
>> That's right.
It has a C4 access.
OK, so that's good.
So now, the next question is do we have some C2 axes
that are perpendicular to that C4 axis?
>> Some what?
>> C2 axes.
Can we do some 180 degree rotations perpendicular
to the C4?
>> I'd say no.
>> Go on. Grab the model and pick it up.
It will be easier to see.
>> So--
>> OK. So try to hold it under the camera so everybody can see.
There you go.
>> OK. So, if that's the C4--
>> Yeah.
>> And so, it goes 90 degrees for the C4, right?
>> Also, that's your C4 operation.
>> Right. But-- So, perpendicular to that.
>> But perpendicular to that can be--
>> So, perpendicular to that, it would be running through there.
So, if you rotate 180 degrees, it's not the same, right?
>> If you rotated 180 degrees, it's not the same like that, no.
>> Yeah.
>> So what do you think, everybody agree?
>> Yeah.
>> Yeah. So, there's no C2 axes.
OK. So, now we've got-- we know that it's got
to be a C or S2N group.
So, what do you think?
Does it have a horizontal plane perpendicular to that axis?
>> Yes-- No--
>> So remember, here's our principal axis.
>> Yeah, I know there's one parallel.
>> Yeah. There's-- Well, there's--
>> There's two parallel.
>> There's two parallel.
>> Is that octahedral?
>> It's-- Well, it's vertical, right,
because it contains the principal axis.
>> So, two planes.
>> To be dihedral, it would also have to bisect some C2 axes
and we don't have any C2 axes.
So, they're vertical planes.
So you're--
>> Write the note that like this?
>> Well, you're on the right track.
But the question if we're following the flowchart is just,
does it have a horizontal plane?
>> No.
[ Inaudible Remark ]
>> OK, sure.
So the question is how do we know
which symmetry planes are vertical
and which ones are horizontal?
And the definition of that in this context is
that if your symmetry plane contains the principal axis,
it's vertical.
We also have dihedral planes.
Those also contain the principal axis.
Not every molecule has them.
And they bisect the C2 axes
that are perpendicular if there are any.
In this case, there aren't.
A horizontal plane would be one that cuts perpendicular
to the principal axis.
So, what do you think?
Does this molecule have one?
No, right, because if we flip it over,
then this substituent would be down instead of up
and it wouldn't be the same thing.
>> So, is it C2V?
>> So, it is-- you are very close.
It would be C2V--
>> C4V-- C4V, right?
>> Right. It would be C2V if its principal axis was a C2 axis.
>> Yeah, yeah, yeah.
>> But instead, it's a C4.
So, great job and thanks for--
>> I never seen this stuff before.
>> You did a good job.
[ Applause ]
Yes?
[ Inaudible Remarks ]
A dihedral plane is also-- it's all--
so it's like a vertical plane
and that it contains the principal axis.
But, it also bisects your C2 planes
that are perpendicular to it.
All right.
We need to have one discussion going on and not many.
I'm really happy to answer everybody's question.
But, we need to do them in series and not in parallel.
OK. So the question is what's a dihedral plane?
So let's find a molecule that has some--
I made a whole bunch of models which is really nice except
that I can't find anything.
OK. So, here's benzene.
So some rules for looking at symmetry, in general,
we're going to say that, you know, resonant structures
when we're assigning things to a point group,
we're going to assume
that resonant structures are fluctuating back and forth
so quickly that we can't see the individual structures
which of course is that's why we have resonant structures anyway.
It's an-- You have an average of these bond links.
So, this molecule which you can go through the point group,
it belongs to the D6H group.
So it belongs to one of the D groups
because it has a horizontal plane.
It has-- Its principal axis is a C6 axis.
So, benzene is D6H.
And so, and we were talking about just now,
do we have C2 axis perpendicular to the principal axis?
In this case, we do.
We can flip it over all kinds of different ways.
>> I stole your pen.
>> Oh, thank you.
So, we can flip it over.
It looks like three different ways perpendicular to that axis.
And so then, we also have some dihedral planes
which are vertical planes that bisect those C2 axes.
And one thing that's really nice about the character tables is
that some of the symmetry elements are really hard
to visualize or at least when you go through and try
to count all of them and make sure you didn't miss any.
And the good news is that you don't really have to do
that because once you assign the molecule to a point group,
if you open up your point group table
and look at the D6H group--
[ Pause ]
The first thing you notice is that benzene has a lot
of symmetry operations.
But it lists for you what they all are.
[ Pause ]
And so, you don't necessarily have to go through and find all
of them yourself once you get it into a point group.
Then, you can go back after the fact and check
out what all the symmetry elements are.
The character table gives you a lot
of other information about the molecule.
And we're going to talk about a lot of that today.
But before we go on, I do want to talk a little bit more
about assigning things to point groups
because this is a really important skill
that if you have a hard time with it,
it's going to be challenging to keep up later on.
So let's make sure that everyone gets it.
And again, if you don't and you need more practice,
stay after class.
I'll be here answering questions.
Come to office hours.
Ask the TAs in discussion.
You know, it is something that once you get it you do.
But it can take a little bit of practice.
OK. So--
[ Pause ]
Let's look at this molecule.
Can I have another victim?
I mean, volunteer?
[ Pause ]
So that molecule has some interesting things going on.
[ Pause ]
OK. So here's your flowchart.
So the first thing we want
to know is does it-- it's not linear.
We can tell that.
Is it low symmetry or high symmetry?
[ Inaudible Remarks ]
OK. This is a controversial molecule.
Some people are saying it's high symmetry
and other people are saying it's low symmetry.
OK. So, I don't think it's high symmetry, right,
because it's not icosahedral and it's not tetrahedral.
That would be like this.
And octahedral, we saw a little while ago.
So, well, let's see if it's low symmetry.
So, to give examples of some of the low symmetry point groups,
C1 is the one that doesn't have any symmetry elements.
That's like that.
Do you think it's like that?
Does it have no symmetry elements?
I can see one, right?
It looks like it has an inversion center.
If I turned inside out, this carbonyl would go over here,
and the CH2 would go over here.
And this methyl group would go over there.
So, another assumption that we make, you know,
I said that we assumed
that resonant structures are their average structure.
We also assume that there's free rotation about single bonds.
So those methyl groups are just spinning around.
If we're going to talk about it in terms where that's not going
to happen, I'll tell you that the molecule is really,
really cold, so it's not moving.
Otherwise, we're going to assume that it does.
OK. So, it's not low symmetry or at least it's not C1.
So, what about something like CS?
That's where-- It just has the identity in a mirror plane.
>> Yeah.
>> We already know it's not
that because we said it has an inversion center.
One of the other low symmetry groups is CI
which means it only has an inversion center.
So what do you think?
Does that thing have anything going on other
than its inversion center?
So, if we cut it like this, we would have this carbonyl
over here and that one over there.
And that wouldn't be the same, right?
Is there any way we can rotate it?
What do you think?
Yeah, I think she's right.
It doesn't have anything else going on.
So, that is CI.
Thank you.
OK. So that's-- Those are some examples of--
you know, now you see what the low symmetry groups look like.
And I think we're going to stop there for examples.
Although, you know, I'm happy to do huge numbers more if you come
to office hours or stay after class and things like that.
I guess another thing that I want to point out is
that until you get good at doing it really fast in just looking
at them, it's best to go through the flowchart and assign them.
So, one thing that people get confused about is looking
at the symmetry operations
versus the names of the point groups.
So, for instance, I noticed that one mistake
that people make sometimes is looking at something
and saying it has an improper rotation axis.
And so then, they think it has to belong
to one of the S2N groups.
And not necessarily lots
of things can have an improper rotation axis
without belonging those to those groups.
So, you know, it's good if you just go through systematically
and look at the flowchart.
OK. So I think that's it for playing
with the tinkertoys today.
Let's go on and do some other things.
So, I'm going to switch back to PowerPoint.
Yes?
[ Inaudible Remark ]
I'm sorry, can you speak up, please?
[ Inaudible Remark ]
Sure. So, in improper rotation, again is when you rotate
by 360 degrees over N, and then reflect about--
reflect through a plane that's perpendicular to that axis.
So, if it were in S3 axis, we would rotate
by 120 degrees, and then reflect.
[ Inaudible Remark ]
I can show that, sure.
So-- All right.
I still have that up there.
So, if I were going to do an improper rotation,
I would rotate by a third of a turn, and then reflect.
You know, flip it.
So, it's-- So rotate it by a third of a turn,
and then reflect through this plane.
So, you see what I mean?
I can't quite do it to the model but that's what it is.
OK. So, let's-- One more question.
[ Inaudible Remark ]
That's a good question.
So the question is for ethane, if you're looking
for the point group, would it be staggered or eclipsed?
So, remember we said that by default
if I don't tell you anything about it, we're going to assume
that there's free rotation about single bonds.
So, you can just assume
that those methyl groups are rotating.
If I wanted you to do it for staggered or eclipsed ethane,
I would have to tell you that specifically.
Otherwise, you wouldn't know.
And, you know, of course those configurations do exist
at low temperatures.
It's just, you know, otherwise we're assuming things
like methyl groups are just rotating all
around all the time.
[ Inaudible Remark ]
In that case-- I mean, yeah.
You just treat it as one big substituent.
Yeah. It's just the methyl group is just freely rotating.
You know, that said, we might see problems where we say
that something is staggered or eclipsed.
And you just have to pay attention
to the description of the molecule.
OK. So now, let's talk about all the information that you get
in the character table.
So, so far, we've done examples where we look at how
to put things into a particular point group.
And that leaves aside the question
of why do we want to do this?
So, the reason we want to do this is that once we do,
we get all kinds of information about the molecule for free.
Somebody already collected it and put it
in this character table and we can use it.
[ Pause ]
OK. Now, I would really like just to show my slides now.
[ Pause ]
OK, good, there we go.
All right.
So there's our flow chart.
OK. So now, let's talk about the character table.
So, everybody has this in front of them.
Let's look at the information that it gives you.
OK. So I have put some examples here
on this sheet just to show some.
And we're going to be using this a lot.
So please bring it to class
to follow along with the discussion.
And these are the same character tables
that you'll be given on the exam.
So-- OK. So, if we have the C2V character table,
so that's a familiar molecule that belongs to that point group
as water just to visualize it.
And let's look at the information that we have here.
So in the top left, we have the name of the point group.
And then, going along the top, we have the names
of the symmetry operations that belong to that point group.
So, E is the identity.
So that's do nothing.
C2 is 180 degree rotation.
And then, we have these two planes, sigma V, XZ,
and sigma V prime, YZ.
So, they're called sigma and sigma prime just to distinguish
that they're not equivalent to each other.
Because if we have a water molecule
and this is a little bit hard to see because it's small but,
you know, everybody knows what water looks
like so it should be OK.
We have our two planes.
One is we can slice through the molecule like this
so that one hydrogen ends up on either side.
And the other one is we can cut through the whole thing
so that we're slicing through the hydrogen oxygen
hydrogen bonds.
And those two planes are not equivalent to each other.
So that's why they get separate entries in the table.
And then, the next question is what are the XY--
how are the X, Y and Z axes defined?
The principal axis is always the Z axis
and then you just use the right hand rule.
OK. So that tells us the total number of operations.
And the number of operations that exists in the group is kind
of a measure of the symmetry of the molecule.
And so for C2V, there are only four of them.
We have the identity.
We have the 180 degree rotation.
And then, we've got these two reflection planes.
And so, that's kind of all there is.
All right.
We're going to come back to what all the rest of the stuff is.
But let's look at C3V now.
So, that's a molecule like ammonia.
Question over here?
[ Inaudible Remark ]
We haven't gotten to that yet.
We're going to come back to it.
>> OK.
>> OK. So, right now, we're just talking
about the symmetry operations in the group.
OK. So, if we think about ammonia that has the identity
in this group on which everything does.
And then, if we look at the next entry, we have two C3.
And so, what that means is
that there are two C3 operations that you can do.
So I have my ammonia molecule.
And I have one of the hydrogen sticking out toward you
and the other ones are pointing off to the sides.
And what the two C3 designation means is
that I can rotate this once
and that gives us an equivalent state as far as symmetry
but it's not identical to where it started out.
And then, I can rotate it again.
And, you know, again, it's symmetrically equivalent.
But if we could tag all of these hydrogens, so, I mean,
imagine that we can isotopically label them.
So that one is a proton and one is tritium,
and the other one is deuterium, so we can tell them apart.
We have to go around the third time before we get back
to the initial configuration.
And this is an important thing.
We have to be able to make a distinction in between things
that are valid symmetry operations which this is
and being able to tell the difference between that
and the original configuration.
So that's why we have two C3 operations because we have one,
two before we get back to the original configuration.
It doesn't mean that it has two separate C3 axes.
Now, don't get confused because in some other point groups,
it might mean that something has multiple axes that are the same.
But the important lesson here is
that when you have something listed as, you know,
that operation selects sigma and sigma prime,
that means they're not equivalent.
But if it's called two sigma, then those--
it's describing two operations that are equivalent.
So then similarly, we have three sigma V. So remember,
a vertical plane contains the principal axis and we have three
of them because we can cut through any of these bonds
and that gives us symmetry operation
and they're equivalent to each other.
So, yeah, a question over there?
[ Inaudible Remark ]
The molecule that had the square on the bottom
and then something sticking up, it was in the C4V.
Yeah, that's an interesting question.
So, you do have-- so you have C3, C4, but only two C2, right,
because you can go, you know, one way and then the other way.
OK. So, tetrahedral, we're not going to go through all of them
but notice it has a lot of symmetry operations
and that should fit with your intuition
that a tetrahedral molecule like methane is more symmetric
than these other things.
So, another important characteristic is the number
that you get when you add up all of these symmetry operations.
That's called H. Some point group tables give it to you.
This one doesn't so you have to add it up yourself.
But that's something you can do.
All right.
Question?
[ Inaudible Remark ]
Before you get back to the original molecule, yeah.
>> But you said it doesn't matter which way you rotated?
>> You know, by convention, we usually do it counterclockwise.
But, you know, no.
If you did everything the other way,
as long as you're consistent, you get the same answers.
But for purposes of doing stuff in class,
the convention is usually we do it counterclockwise.
Yes?
[ Inaudible Remark ]
Well, also if you rotate about the principal axis,
you can do the C4 three times before you get back
to the first--
[ Inaudible Remark ]
Because in that case, if you rotated it like this
or like this, you would get-- there were two ways to do it.
>> There's two ways to go through that one?
>> So, my point is just, you know,
be careful because there are--
these operations with coefficients in front
of them indicating that you have multiple ways
to do the same operation.
And sometimes it means just
that you can do the operation a couple
of times before you get back to the original state.
And other times, it means that you have different axes
or different planes that are equivalent.
We'll see more examples of this as it comes up.
I don't want to spend a whole bunch of time talking
about every case because it gets a little bit abstract.
Let's wait and see examples.
OK. So now, what is all the rest of this stuff
on the character table?
That's a lot of what we're going
to spend time on today and Friday.
OK. So these As and Es and Ts,
those are the irreducible representations
or the symmetry species of the group.
And what those are, it's a complete description of objects
that can behave in certain ways
under these particular symmetry operations.
And we're going to talk about that
with some concrete examples a little bit later on.
So, some things to know about them, the ones that are called A
and B are singly degenerate.
The ones that are called E are doubly degenerate.
And the ones that are called T are triply degenerate.
And then, let's look at the other information that you get
in this table which starts to give you some hints
about how you might be able to use this information.
And that is we have things like X, Y, and Z. We have XY, XZ, YZ.
These are linear and quadratic terms, you know,
in terms of the Cartesian coordinates.
So, for X, Y, and Z right now, you can think about that
as either a little unit vector directed along the
appropriate axis.
Or you can think about it as a PX, PY, or a PZ orbital in terms
of how it transforms related to symmetry.
Those are very intuitive concepts for chemists
and chemical engineers.
So it helps if you visualize it as an orbital.
The XY, XZ, et cetera, X squared minus Y squared, you can think
about those as D orbitals.
They're going to have other interpretations when we get
into talking about infrared
and raman spectroscopy later in the course.
But for now, you can think about these just in terms of orbitals.
OK. So you can start to see what's useful about this table.
So once you assign something to a point group, for one thing,
there's a limited number of objects
that can behave a certain way under these symmetry operations.
We have a complete set
of symmetry operations to work with.
And we can already see that we learn some information about how
at least orbitals behave with respect to this symmetry.
And this is already written down for you in the table.
OK. So, having gone over that a little bit, we are going
to switch gears and talk about matrices and how
to make matrix representations of operators.
And we're going to do a little review of how
to deal with matrices.
Hopefully, this is a review for everybody.
If not, we're going to go over what you didn't know
about it so don't worry.
If you need a little bit of extra practice or background,
please check out the Wikipedia page and/or the Wolfram site
on matrices and matrix multiplications,
rotation operators, things like that.
OK. So, if we have a matrix which we're going to call A,
these entries are its matrix elements.
And we can call those AIJ.
We'll see that kind of terminology a lot.
So, in this case, A11 is minus 3.
A12 is 6, et cetera.
That's just how we label them.
And we're just going to go through a quick review of how
to deal with matrices.
So, you can add them if they have the same number
of rows and columns.
And if you can do it, it's pretty easy.
You just add up the individual matrix elements.
And so, here's what you get.
In this case, we just add the individual matrix elements
and get these cells.
[ Pause ]
So, I know everybody's probably seen this stuff before
but it doesn't hurt to have a little bit of a review,
especially since if you didn't really talk
about matrix representations of operators last quarter.
It actually makes your life quite a bit easier, I think.
I think it's much easier to deal
with operators in that formalism.
OK. So that's how we add them.
That actually doesn't come up terribly often in the kind
of things that we're going to do.
Here's something that does.
If you want to find the trace of a matrix,
you just add the elements on the diagonal.
And ignore everything else that might be in the matrix.
It doesn't matter.
We're just going to add the elements
that are along the diagonal.
The trace is also often called the character
which gives you a hint as to what the character table is
about and why we're talking about this right now.
So, all of those 1s and minus 1s and 0s and 2s,
et cetera on the character tables,
each one represents the character of the matrix
that corresponds to that particular operation
for a particular symmetry species.
And we're going to learn how to make our own,
if not by the end of-- if not today, by Friday.
OK. So the character is a lot of times given the symbol chi.
In this case, it's 7.
So that's a really important matrix operation.
Fortunately, it's easy.
We can also multiply them by scalers.
In order to do that, we just multiply each element
in the matrix by the scaler.
And we can take the trace of that one too.
So again, pretty straightforward stuff but it's good
to go over it just in case.
All right.
Let's talk about matrix multiplication also
in case you haven't seen it in a little while.
So, when we go to multiple the matrices,
I'm going to write this all out once.
So, we go through and multiply the row of--
the first row of this one by the first column of that one.
So we get 1 times 5 plus 2 times 8
as the first matrix element in the new matrix.
And then, we just go across.
So now, we have 1 times 6 plus 2 times 9, et cetera.
And we build up our new matrix like that.
So, pretty simple but you have to double check
because it's easy to make a mistake.
How many people have taken Chem 5
or otherwise known as mathematica?
It's a lot easier if you use mathematica.
So, most of the examples that we'll do
in class will be relatively simple.
And, you know, you'll be able to do it in your head fine enough
but if you have to do these for matrices
of any size, use mathematica.
It makes it a lot easier.
OK. So here's what we get for this particular one.
And it's also worth pointing
out that matrix multiplications don't necessarily commute.
So, if we multiply these two things together
and then we do it in the other order,
you don't get the same answer.
And, you know, of course this relates to stuff
that you learned last quarter in quantum mechanics a lot
of operators don't necessarily commute
and they can be represented as matrices.
And we'll also see that in some point groups,
symmetry operations may around commute.
All right.
So, other things to look at.
If the product of two matrices equals zero,
that doesn't necessarily imply that either
of the matrices has all zeros in it.
There are different ways to get that.
So that's our little review of matrix properties.
Again, if you need more review than that,
check out the Wolfram site and/or Wikipedia.
Wikipedia is a really great resource on things like this
that are non-controversial.
Of course, things where there's--
where there are differences of opinion.
People can change it all the time and troll each other.
Nobody really does that on sort of the basic math
and chemistry, and physics topic.
So, it's a good thing to use it as a resource.
OK. So now that we've talked about properties of matrices,
let's start looking at how
to construct transformation matrices for actual operations
that we might want to do.
And we're going to do it in two dimensional space to start
with just to make things easier.
OK. So the way we're going to do this is we're going
to think about, you know,
I want to accomplish some transformation.
And I'm going to apply it to a test vector
which I'm just calling alpha beta.
And we need to think about what do we want alpha beta
to transform into, and then what matrix do we have to multiply
by it to get that result.
So, if we want a reflection about the Y axis, remember,
we're in a two dimensional plane.
So we need to think about what do we multiple by alpha beta
in order to get it reflected about the Y axis?
So, of course if we reflected about the Y axis,
beta isn't going to change and alpha is going to change sign.
And so, working backwards, we have to think
about what do we need to multiply by that vector in order
to accomplish our transformation.
And as we're going to see, the matrix you get depends
on what you're trying to do,
what object you're applying it to.
But we're going to talk about the cases of just doing this
in two dimensional and three dimensional space.
OK. So what if we want to do a projection in the X axis
so we only want to see the X component?
So, what do we have to multiply by alpha beta
to get just the projection on the X axis?
Yeah. So I hear people following along.
So everyone gets it.
That's cool.
All right.
What if we want to scale it by 3?
So we just need something that has 3s on the diagonal.
So, this is why I like group theory in these kind
of geometric transformations
because it really gives intuition into how we can set
up matrix representations of different operators.
The quantum mechanical operators
of course are all linear operators
as you learned last quarter.
So they can be represented this way.
But doing this with these geometrical things helps gives
us an intuition for how to use it before we have to get
into more complicated concepts.
OK. So in general, if we have some vector and we want
to rotate it, so we have our first vector, R1,
and now we move it into this position, R2.
If we just set up how we want to do this rotation,
if we look at X2 and Y2, we have R cosine alphas plus theta,
and R sine alpha plus theta.
And we can expand this out.
And that gives us the rotation matrix that we need to be able
to perform this particular transformation.
Rotation matrices are something that we're going to see a lot.
We're going to use them now when we talk about group theory.
So hopefully it's clear how that's going to work
and how we're going to use that quite a bit.
We're also going to use them when we talk
about NMR spectroscopy and look
at how spins behave in a magnetic field.
And really they come up in all kinds of different areas
of chemistry and physics.
It's a useful thing to know how to do.
OK. So having gotten this far, you have enough information
to definitely do the practice problems which are posted online
so don't try to write them all down right now.
I just wanted to point out that that's there.
So, do go ahead and check these out online and try
to do them for Friday.
Having looked at that, let's move on to three dimensions.
So, we talked about our little two dimensional rotation matrix.
Now, let's look at this in three dimensions.
And our basis is little unit vectors pointing
in the X, Y, and Z dimensions.
And notice I'm going to try to be really careful
about telling you what basis I'm using and if I don't,
you should ask me because it's a really important question.
That affects everything about the problem.
So, right now, it's just our unit vectors.
OK. So, what if we want to do a C2 rotation, so 180 degrees?
So, if we have our X, Y, and Z unit vectors, that's going
to flip the signs of X and Y and leave Z alone.
And so, this is going to tell us what our matrix is.
Yeah?
[ Inaudible Remark ]
Well--
>> 3 by 1?
>> This is in three dimensions now.
We were doing it in two dimensions before.
[ Inaudible Remark ]
Yeah. So that-- So that you raised a really important point
which is why I have said that I have to be very careful
to always tell you what the basis is that we're using
because that changes everything about the problem.
So, you know, if we're starting with--
so before we were starting with a-- you know, a two--
we're starting with a two by two
because we had a two dimensional vector.
Now, we have a three dimensional vector.
>> OK.
>> And so--
[ Inaudible Remark ]
I'm going to check it and write up something about it.
Sorry about that.
It's just-- I want to get through a little bit more
of this before class and, you know,
whatever is confusing we can go over later.
OK. So, let's talk about our rotation matrix for C4.
So, this one is a little bit more complicated
because we flipped the position of X and Y. We made X negative.
And again, Z stays the same because we're rotating
about the principal axis.
Oops. And so, that's the rotation matrix
that we ended up with for C4.
[ Pause ]
And so, what I want to point out is that here's what we get
for in general rotation matrix about any angle.
We need to put in the sines and cosines.
And so, in Cartesian coordinates,
here are the general rotation matrices for some angle
about the X, Y, and Z axis.
And these are things that are going to come up over
and over again and we're going to use them.
So, again, you don't have to write it down right now.
This is-- You know, it's available.
You can look it up.
But they're going to come up and it's important.
I also want to point out that the inverse
of a matrix is the matrix
that if you multiply a matrix times its inverse, you're going
to get an identity matrix which has just 1s on the diagonal
and 0s everywhere else.
Sometimes it's called I if we're talking
about the identify operation and in terms
of the character tables, we call it E.
And if A represents some transformation, then its inverse
which is called A to minus 1 undoes it and returns it
to its original state.
And here, that is written out.
So, OK, that's pretty good as far
as where I wanted to get this time.
Next time, we're going to tie it all together and see how
to use this in terms of group theory.
Yes?
[ Inaudible Discussions ] ------------------------------8a65dbcf519b--