Good morning.
Welcome back.
I hope everybody had a great weekend.
So we started on Friday talking about the elements of symmetry. Beginnings of our discussion on symmetry of chapter 4,
we talked about symmetry elements and symmetry operations, and we built up the full list of symmetry operations for a non-molecular example, in that case, of snowflake,
right which had 6-fold symmetry. but what we want to do today is to start by using a molecular example,
and then we're going to jump into a discussion of point groups.
So point groups are  basically the complete set of symmetry operations describe a given molecule or object
and so all molecules have and fall into one of a variety of different kinds of point groups
And we want to understand how to assign point groups to different molecules
based on their symmetry operations - the symmetry operations that they possess. So let's get started.
Here our example is going to be staggered ethane.
Which is a nice example cuz we have several different symmetry operations that the molecule possesses.
So this is just standard notation.
Right?
If we just have a normal line.
It's in the plane.
If we have a solid wedge, it's coming out of the plane on the board; and if we have a dashed wedge
that atom is going into the plane on the board, cuz this is just like organic chemistry.
What we want to do is to identify.
First of all what its highest order rotational axes
this is often the easiest place to start in understanding the difference in symmetrical operations of a molecule
and so we can point the molecule this direction... we're looking perpendicular to the C-C Bond or we can look at it along the edge here.
so down the midth of the C-C bond looking from the left hand sid
And we can assign a,b,c,d,.. to these hydrogens
What we're going to see is if we specially we look at it in this particular orientation here.
It becomes fairly obvious that there's a C3 access to this molecule.
If we rotate by 120 degrees counterclockwise sense,
Then we are going to generate an equivalent configuration of that molecule
and we can't really tell that it's been changed at all - all we've done is we are gonna to cycle the labels of the hydrogens.
So Ha is going to go to Hc, Hb is going to go to Ha when we perform the C3 operation. And when after we perform the C3 operation,
we have an equivalent configuration that looks like this.
Because we have a C3 axis in this molecule.
We can perform the rotation twice we can do a 240 degree rotation.
That would be our second C3 operation that would drive Hb to Ha and so forth.
So these are the three hydrogens coming out of the plane of the board.
And then these are the three Hs would be going into the plane of the board
and you can see after the second C2 operation or the second C3 operation.
that we still have an equivalent configuration.
We have a good symmetry operation for this molecule.
If you do it a third time do the C3 the third time,
we were cover the identical configuration it we started with in other words.
We perform the operation E. So C3 cubed is equal to E.
And we can play other games we can do is see three Square directly and get this configuration.
So these particular operations, the C3 related operations are good operations for this particular staggered ethane molecule.
And so we would say that so far we can say that we have the E operation.
Of course,
we have a C3 and we have a C3 Square.
Those are the unique operations that come out of this C3 axis.
what other symmetry elements and operations pertain to this molecule. Well we can tell right away that there's going to be several more at least.
So let's look first at other axes along this this the same axis here.
Is there a C2 axis on that axis.
Is there a C4?
Is there a C6?
What do people think do we see other?
Rotation axis is along that direction.
What about just proper rotations - any other proper rotations?
Is there a C2?
I want that direction?
So if there were a C2 what would happen? Ha would rotate 180 degrees down.
It would be pointed directly in the plan of the board here.
That would not be the same configuration.
So there is no C2 axis.
There's no C4 axis is no C5.
The only thing this thing has is a C3 axis in that direction.
What about other rotational axis perpendicular to the C3?
So if you look at the molecule and look perpendicular to the principal axis,
we actually have three perpendicular see two axes in this molecule.
Remember that point group objects here.
We're going to have either zero or n perpendicular C2 axes in this particular example,
we have three perpendicular C2 axes, each one of which is perpendicular to a plane.
It's defined by two of the hydrogens and the two carbons.
So here's an axis coming out of the board that is C2 (on upper-right) if we rotate 180 degrees around its axis Ha goes to Hf, Hb goes to He, and Hc goes to Hd
So that is a good symmetry operation C2 perpendicular to this plane right here.
There are two more that are perpendicular to the other planes that contain the hydrogen. So those are inscribed here in this on the axis view of the molecule.
So we have our C2 here we have another C2 that's perpendicular to this plane and we have a third C2 perpendicular to this plane.
C2 rotation on any of those axes is going to give you equivalent configurations that look like this
we can describe these different C2s by differentiating them with primes
so we can just call one arbitrarily C2' and the other one C2'' just to tell that they're different axes.
It's just a total totally arbitrary designation.
But the basic point here is that in addition to our C3 axis and the operations that pertain to get
we have three additional operations C2 operations perpendicular to that C3 axis.
There are three of these perpendicular C2 axes.
[Student Question]
The point of the primes is just to differentiate the exact axes we are talking about.
So if we wanted to describe this one we could just call it C2 but to tell that we had another C2 axis.
It's a different from this original axis.
We give it some other designation in the convention is just to call it a prime or double prime or triple prime or whatever
Anyway,
we wanted to write but we have to use some kind of differentiation.
Okay.
So now we have six symmetry operations that we've built up for staggered ethane,
right?
We have the C3 and its variations and we have these perpendicular C2s.
That describes all of the proper rotations for this molecule may we can't find any additional proper rotations.
Let's move on to Reflections is the next symmetry operation at we care about what reflection planes are there - mirror planes in the molecule.
Well,
there are three mirror planes that are just like the C2 axis perpendicular to those C2 axes.
So there's one mirror plane in this plane, and you can see here that if you reflect across this plane Ha and Hf stay where they are, the carbon stay where they are.
But Hb and Hc are inter-related and Hd and Hc interrelated. This plane
Here is the plane that I've drawn right here.
We do the reflection operation.
You can see that Ha and Hf are the same,
but He Hd change, Hb and Hc change.
We generate an equivalent configuration.
The molecules unchanged except for our imaginary labels on the atoms.
Remember that these labels are imaginary right there just there to help us bookkeep
and tell what's going on when we do the operation,
of course,
they don't exist in reality.
In addition to this mirror plane.
There are more planes in that plane.
And also in this plane the other planes that contain the carbon atoms and two of the other hydrogen atoms.
And if we perform then your operation you can see that you get equivalent configurations.
So we've added to our six proper rotation.
We added three more symmetry operations that correspond to the mirror reflections
One thing important to know I guess is that  there is a sigma h a horizontal mirror plane in this molecule
the horizontal mirror plane remember is the one that is perpendicular to the principal axis.
And so that would have to be a plane that cuts right here perpendicular to the principal axis,
which is this axis - the C3 axis.
So is there a mirror plane here and the answer is no right?
Because if you reflect through the point Ha would go over here and there's no atom
so it is not a symmetry operation of this molecule.
So we have nine so far are there any more?
There are more.
The inversion operation is a is a symmetry operation of this molecule the inversion.
Remember you take coordinates XYZ and you change them to coords -X -Y -Z. Here' the center of the molecule
The inversion center in this particular case is at the center of the carbon-carbon bond. Right in the midpoint.
If we perform the inversion operation looking at the molecule in this orientation,
you can see that you're going to take Ha through the center.
It's going to pop off at Hf.
And so will Hf go to Ha.
So we have swapped them here. And the same thing for Hb and Hd.
This is above the plane that's below the plane.
So that passes through the center and goes below the plane
And the same for He and Hc and the carbons of course are also going to be interrelated by an inversion operation.
To generate an equivalent configuration.
We can see it perhaps more easily if we look and on on the molecule that takes the three hydrogen's that are popping out of the board and swaps them with their respective partners in the board.
So we have an inversion operation as well.
That's so far 10 total operations that we've tallied up.
We can only have 0 or 1 inversion.
And so we found that we have one.
So the game is over for inversion and we've looked at proper rotations.
We've looked at Reflections and we've looked at in the last symmetry operation class is the improper rotations.
So let's look and see if we have any improper rotations
and we probably do have an improper rotation...or two.
And the improper rotations in this case are parallel with the C3 principal axes and they are the S6 improper rotations.
So remember what S6 means?
Well,
first of all,
let's remember what an improper rotation is improper rotation is a rotation followed by a reflection perpendicular to the rotational axis.
An S6 means that we do a C6 and then we do a perpendicular reflection
and you can probably see right away how an S6 operation is going to be a good operation for this molecule.
What's look and on to describe how this is going to work out and proved ourselves that it generates an equivalent configuration.
So here's the original molecule before the operation.
Just hold on there and let's perform S6.
What's a 60 degree counterclockwise rotation?
So that sticks Hb here,
right 60°.
He goes to Ha
60° rotation and so forth. Notice that a C6 does not generate an equivalent configuration.
This is not the same as this because these items here or pointed out at us and they are pointed in the same orientation before so C6 is not cool.
But an S6 is because after we do our C6 operation,
we perform are perpendicular reflection that pops Ha, b, and c into the plane and brings Hd, e, and f out of the plane.
And gives us this orientation after that perpendicular reflection.
And this orientation - here is an equivalent configuration.
So the C6 is not a good operation,
but the S6 is a good operation.
I just like the the principal axis we can continue to do S6 operations.
And what we're interested in is what S6 is our unique that haven't already been described in another form.
If we perform a second S6,
then we rotate 120 degrees and we do a mere reflection once and then twice that's the same exact thing as just doing a C3 operation.
And so we can describe it as an S6 squared but we've already seen that that's C3.
We've already accounted for that particular operation in our list.
So that's not a unique one.
In fact,
there aren't that many unique S6 operations that haven't already been described in in other terminology.
Here are all the possible ones right at 1 2 3 4 5 6 times if you do it seven times at the same as doing it one time,
so we only have to account for 6.
Well in a 6 by itself is unique and there's no other operation that we've previously described that will that will accommodate or will achieve that same operation of 60° mirror reflection
While do it twice is the same as C3. Doing it three times?
Right?
An S6 means that we do a C6 and then we do a perpendicular reflection
Inversion can be understood by doing 180 degree rotation and a reflection that gives you the same exact result as doing the inversion in a so-called single step right bringing everything through that working.
So the cube is not original. What if you do it 4 times?
so that's the same thing is doing a C3 twice.
If you do it 5 times that's unique operation that you haven't previously described one that's going to do that.
We do 300 degree rotation and then we reflect reflect reflect reflect.
That's a unique result.
And if you do it six times that's just identity.
We already know that and so the unique additions to our list are just 2 of 6. S6 - only doing it once and doing a 5 times gives us unique new operation that we need to add to our list.
And so now we've done through all of the symmetry operations and we've tallied them all up.
What are we left with here?
This is our final table for staggered ethane.
Keep in mind that staggered ethane is going to be different than eclipse ethane or other conformations of ethane,
right?
Cuz you can imagine if you rotate along that.
Carbon-carbon Bond you going to change the Symmetry the molecule.
So here's our tally
We have an identity operation was always one for every object.
We have five individual rotation two of those are C3s -
the individual C3 and doing it twice.
And the other three are the perpendicular C2 axes that we described and so that's five total proper rotations that are independent.
We have three reflections of the so-called dihedral reflections.
They are parallel with the C3 axis. With an inversion that's 10.
With an inversion that's 10. And then we have to improper rotation is giving us 12 total
symmetry operations for this particular molecule in this confirmation.
This set of 12 is a unique and complete set of operations that describe all the symmetry properties of this molecule.
So these 12 operations describe completely and without redundancy the symmetry properties of any object that looks like staggered ethane in a mathematical sense.
The complete set of symmetry operations for molecule or an object.
We call this the point group of the molecule or the object.
And will describe how we can categorize point groups in a little bit just to give a prelude here - the point group of
staggered ethane is called D3d.
We would say it has D3d symmetry or it belongs to the D3d point group if we know that it's a D3d object.
Then we can use that information extraordinarily powerful e to the understand things like
molecular vibrations, the molecular orbital construction for the molecule, etc.
and that's kind of what we're doing over the next few chapters
taking this information and what we know about that particular point group
and making use of that to simplify calculations and to give an intuitive picture of
what it should have in terms of the number of molecular vibrations, etc.
It is quite beautiful how this all works out as you'll see.
The total number of operations in any point group is called the order of the point group.
We just add up all the operations.
So the order of the D3d is 12.
The symbol for order is usually a little h
and we'll use this quite a lot later.
So just remember that the order is simply the number of symmetry operations in the point group.
The order is always an integer multiple of n the principal axis.
And so in this case,
the principal axis is the C3 axis right is the largest value of n in the point group.
And so we have 12 operations in this particular case and multiple is for and so the order is equal to 4 times 3 or 12.
It's not always equal to 4 times then but it's always equal to an integer times n
All right,
are there any questions about this so far?
Who's our first real molecular example?
We have shown actually fairly complicated Point group as far as it goes cuz there's 12 operations.
That's a fairly reasonable number of operations.
What we want to do now is lay out the landscape for what all Point groups look like
categorize the point groups and describe a procedure by which we can assign point group, rapidly.
We don't want to have to and you won't end up needing to go through
all of the different operations in order to assign the point group.
Pretty soon you'll be able to assign point groups very rapidly, within 10 seconds.
Let's hope that's kind of the the goal.
I would say we're going to take this procedure that took us 20 minutes or whatever and we're going to narrow it down to 10 seconds.
Are we going to do that?
This was supposed to be the end of the first lecture here,
but I was a little slower than I expected.
So let me summarize what we know about symmetry elements and operations elements are geometric entities: planes, lines, and points.
The operations are movements with respect to these geometric entities.
And what we're going to do is we are interested in movements between equivalent configurations.
When we close our eyes we do the operation.
We open our eyes we can't tell the operation has been done.
That's what the equivalent configuration mean.
Another word indistinguishable for the original configuration.
However,
there is a discrimination here. Indistinguishable is not the same as identical.
If we have our little labels on our atoms like for staggered ethane,
there were lots of equivalent configurations that were indistinguishable if we took the the imaginary labels away.
but an identical configuration is one in which the labels are in exactly the same place as well.
Right?
So what if we perform the identity operation we have identical configurations.
So there's a slight distinction there that sometimes is important.
Equivalence or indistinguishable not necessarily identical.
Where is describing molecules in terms of their point symmetry?
What does symmetry mean? It means that at least one point in the molecule remains stationary when we do the operation.
And you'll notice that all of the operation that we've described that's true. Inversion...
there's a point that remains unchanged. Rotation is an axis. Reflection...
there's a plane. So we're not interested in molecules translating through space right now.
We're just talking about the object in some sort of mathematical vacuum,
and we're doing point operations to it.
And we've shown that many different there many different ways to describe the same sort of operation when that's true.
We always try to use the simpler description.
So in case of having an S6 squared we know that's the same as a C3 in all the tables and so forth.
It will see it's always listed as a C3 not an S6 squared just to keep it as simple as possible.
That's the convention.
Okay.
Let's now move into point groups and how we can order classify and designate a term to use in these different point groups.
Okay,
we start with the simplest Point groups.
These are the low symmetry Point groups.
This point groups have only one or two symmetry operations.
The point group labels in the following slides are going to be in these red numerals here.
So point group,
we have C1 point group, Cs point group, and the Ci Point group.
These are the three of simplest lowest symmetry point groups.
Directly below, the point group in red is this curly brackets notation.
This means the complete set with what is mathematical complete set of operations contained in this point group.
That's what I'm trying to say here.
And so in the C1 Point group,
you see that the only operation to see one point group objects have is the identity.
That's the complete set.
This is the lowest possible symmetry another which would say this has no symmetry at all - it's asymmetric.
Molecule that shows that possesses to see one point group or is in the C1 Point group.
Is this particular nothing like molecule,
you can see that there's no rotational axis of any sort.
There's no C3 axis, no C2 axis, no mirror planes, no inversion center. There's nothing.
Highly asymmetric molecule.
An object from everyday life that is got the same point group is a spring or Helix.
I matter how you try to do it.
There's no rotational axis.
There's no mirror planes.
There's no improper axis.
It's possible to have a point group that has an addition to Identity just a single mirror plane.
You're playing in the plane of the molecule.
This is a nice example of that.
You can see that there's a mirror plane in the plane of the molecule.
What does nothing else there's no C2 axis of any sort nothing else?
So we call this the Cs point group.
it's also possible to have just an inversion center.
That's the Ci Point group.
Here's an example of a molecule that just has an inversion center in addition to the trivial identity operation.
So you can see if we invert here, the Hs exchange, the Cls exchange, the Fs exchange, and the Cs exchange.
but there's nothing else. There's no C3 axis here because if you did a C3 the H would then occupy the F in position and that's not okay.
There's no C2 perpendicular, and nothing else.
Does everybody see that?
There's no C2 perpendicular like coming out of the coming out of the board.
Read if you do the C2 operation,
this Cl is above the plane that Cl is below the plane, so you don't have an equivalent configuration.
These are sometimes difficult to identify but they're the simplest
It sometimes it's hard to take an object and convince yourself that there's no operations other than a single mirror plane.
On the other side, so those are the low symmetry point groups, on the other side,
We have the highest symmetry point groups.
These are point groups that correspond to polyhedra.
Typically tetrahedron, octahedron, cubes, icosahedron - high symmetry objects.
There are three main types of high symmetry point groups: tetrahedral,
which is usually for the highest symmetry is called Td, octahedral Oh, and icosahedron Ih.
This point groups are derived from the platonic solids described and captured the imagination by Plato.
So these platonic solids are listed down here.
They're five regular platonic solids:  tetrahedron, octahedron, cube dodecahedron, icosahedron, and I have labeled their point groups next to it.
Let's look at the three different main point group here.
Tetrahedral group has all of these operations, 24 total operations in the complete set for the tetrahedral point group. For an example,
of course is methane, the most familiar example,
what distinguishes tetrahedral symmetry is that you have eight individuals C3 operations there
There are four C3 axes in the molecule: down the C-H bond, each individual C-H bond and there are two operations,
corresponding to each one of those axes.
So if you have eight different C3 operations in four different C3 axes than you have tetrahedral symmetry.
In addition to the C3 is C2s which split the H-C-H bond angle and as we describe last time you have S4 axes as well.
Octahedral symmetry here is SF6.
This is an octahedron.
48 operations in the full octahedral symmetry point group.
what's distinguishing here is that you have C4 axis and you have a ton of mirror planes in there and so if you have C4,
and a bunch of perpendicular mirror planes,
It's quite likely that you've got octahedral symmetry.
That's the defining characteristic of an octahedron.
Icosahedron has 120.
That's the most point of that's the most symmetry operations for any point group.
This is an example of an icosahedron symmetric object.
This is buckminsterfullerene,
which is C60 - 60 carbon atoms arranged in a cage that look like a soccer ball.
You have pentagonal symmetry so you can see this pentagon here.
There's also hexagons in this particular object,
but the characteristic feature of the symmetry is if you have C5 axis,
you have a ton of them.
Six different C5 axes.
So,
without being brutal we're not going to list all those different and visualize all those different operations.
But there you have it. There's the possibilities.
There are several different kinds of tetrahedral octahedral and icosahedral point groups.
They're actually seven altogether.
This is a table from book describing all seven different point groups.
We have two different kinds of icosahedron with two different kinds of octahedral and three different kinds of tetrahedral Point groups.
In addition to the three that I showed in the last slide Td, Oh, and Ih,
there are point groups that correspond to the same symmetry,
but without any near plans.
So that's just called T, O, and I so that's all I, O, and T and you can see in this list. Here are all the mirror planes.
And for the Os and the Is and Ts,
there aren't any mirror planes present.
They're just blank there.
So it's the same symmetry just without mirror planes, and it's possible to have molecules that have just O, I, or T symmetry.
Finally the 7th down here is called Th and that's generated by taking the T point group and adding an inversion center.
And then you get Th symmetry. That rounds out all the possibilities all the unique possibilities for point groups.
I show down here cuz I was interested for molecular example of Th symmetry and this is the example here.
This is hexapyridine iron(II), so you have these pyridines in units that are oriented like this.
There are canted relative to one another and so you don't have an octahedron here because there's no C4 axis.
But what you do have is a C3 axis.
It's coming right out at you.
There's a couple of others right there have to be four and you have an inversion center. You can see that if you invert this.
It comes right down here.
If you invert the pyridines
in through comes there and so forth so you can have tetrahedral symmetry with an inversion center,
which is not a full tetrahedral symmetry of like methane,
right?
Remember methane itself lacks an inversion Center.
Okay,
so we have the three simple low symmetry point groups, 7 high symmetry Point groups that you count encounter fairly commonly in molecular chemistry.
Let's look at the others.
There's not that many more. There are basically four more classes that we have to describe.
One is pretty simple: linear point groups for linear molecules. There are two point groups for linear molecules.
Here the linear molecule looks like this it's not symmetric in this plane.
What we have is C∞ axis in this direction the contains all the atoms in a molecule
Rotation by any angle gives us a symmetry operation, gives us an identical or an equivalent configuration.
So we label that as the C∞ axis and this little Greek letter there tells you that the angle is what is infinite.
In addition to that,
of course,
you also have an infinite number of mirror planes that run along the molecular axis with an infinite number of step sizes in the angle.
We call that point group the C∞v Point group.
Trivial to see, right? If you have a linear molecules it's going to be a linear point group.
So it is the C∞v or the D∞h
Let's talk about the D∞h got the stuff that the C∞v has
but in addition it has an inversion center and or a mirror plane that bisects the molecule like that.
And so we have inversion labeled there.
We have the infinite number of vertical mirror planes in this direction.
We also have in this case an S axis just because we have this mirror plane with this inversion center.
so if molecule symmetric when it's bisected and it is linear, then it's D∞h.
So all you really have to ask yourself when you encounter a linear example,
is is it symmetric or is it not symmetric?
That's the distinguishing characteristic.
Okay.
So, low, high, linear,
there are three more. We have the B.
We have the C and we have the S point groups.
Let's look at the D Point groups.
The D point groups are those that have n number of perpendicular C2 axis, perpendicular to the principal axis.
So anytime we have perpendicular C2 axes for molecule.
It's going to be a D point group.
The question then is just what the point group are we talking about.
We have Dn we have Dnh and we have Dnd.
Those are the three possibilities.
A Dn Point group does not have any mirror planes.
The complete set of symmetry operations depends on n.
So let's say that n = 3.
So we have a D3 for example,
which is shown here.
I'm going to have the identity.
We're going to have 2 C3 axes and we can have 3 C2 axes.
And that would give us the complete set of operations for a D3 point group.
Notice,
there are no mirror planes and no inversion centers, etc.
It is called a pure rotation group.
Cuz all you have is just simple rotations.
Okay.
This is a nice example of D3.
It basically looks like a propeller with three blades and if you rotate the propeller by hundred twenty degrees to get the same guy back.
You also have perpendicular C2 axes here,
which is why it's a D point group and not something else.
So here's a one of the three perpendicular C2 axis.
Alright,
let's see now that we have a flat molecule like this BF3.
We're obviously going to have now mirror planes.
they are going to pop up.
So we have in this particular case of horizontal mirror plane in the plane of the board that's perpendicular to the principal axis,
which is coming out of the C3 axis.
The number of symmetry operations is again going to depend on n. And this is how the order relates to n
Four times the order of the principal axis.
This molecule here has a C3 and has a perpendicular mirror plane, has three perpendicular C2 axes.
And those three directions there.
It has vertical mirror planes like there. This is a so-called D3h point group.
And this is the full list of symmetry operations,
of course bonded this particular D3h Point group.
So the differentiation between the n and the nh is we have a horizontal mirror plane and a bunch of other mirror planes popping up as a result.
And then finally we have the Dnd. Again the order is the same as it is for the Dnh.
Here's an example of a D2d Point group.
This is allene.
Here we have these two hydrogen's in the plane and these two hydrogen's perpendicular 90° coming out and in the plane and you can see what we've got here.
We have a C2 axis.
You can go along this direction here for the C2 axis.
There are perpendicular C2 axes as well.
They are located 45 degrees between this plane and this plane coming out at you.
So if you have one access 45° coming down and coming out of the board and another axe is 45° going from down into the board in that direction.
Those are your two perpendicular C2 axes that have to be there if it's going to be a D point group.
In addition we can count up all the other things that would be in this point group.
You're going to have an S4 axis and other things this is D2d with all of those symmetry operations present.
So, of course,
We can have a D3d we can have a D4d we can have a D5d, etc.
They're all going to have a different number of symmetry operations,
but in general they're going to look fairly fairly similar in terms of having their plans president having horizontal mirror plane for Dnh.
Not for Dnd.
Okay,
so we have the D point groups.
The last two are the C point groups and the S point groups. The C point groups are almost exactly the same except they don't have perpendicular C2 axis.
So here are the C point groups.
Here's a point groups that have a principal axis Cn,
but no perpendicular C2.
So that's the distinguishing characteristic differentiating D and C Point groups is the presence of perpendicular C2 axis.
Just like before we have we had a Dn.
We have a Cn, we have a Cnv now and we have a Cnh.
No mirror planes.
Here's an example of a C2 molecule here.
Rotational axis coming right out at you.
Cnv,
we have
vertical mirror planes in this case NH3 we have a threefold axis running this direction. The molecule is not planar so we don't have a horizontal mirror plane.
But we do have vertical mirror planes with three of them going through the N in one of the Hs
Those are the three different mirror planes.
And vertical so they contain the axis of the principal axis of molecule.
This is an example of C3v Point group.
And there's six total operations for that particular point group.
So Cnv is when we have vertical mirror planes. Cnh's when you have horizontal mirror plane present.
So an example of that is this molecule here.
This is a particular C2h with a twofold rotational axis coming out at you and you can see of course.
It has a horizontal mirror plane in the plane of molecule.
So C2h gives you identity, the C2, the inversion, seeing inversion center, and horizontal mirror plane.
That's the complete set of operations for this molecule.
There's only one more kind of point group.
And that is the S point groups.
So the S point group is as follows.
[Student Question]
On this one here coming right out at you.
So if you rotate a 180 degrees the hydrogens interchange,
the Cls are on the same side, and the Fs are on the same side of the molecule.
Are the questions about this so far?
We are kind of moving quickly through this so and it's going to take, depending on your ability to visualize three-dimensional space,
It's going to take some time to become familiar with these operations and comfortable with him.
The final point group are the S point groups,
and there's only one kind, typically label is S2n.
If an object has a principal axis and it has an S2n axis parallel to that principal axes,
but it has no perpendicular C2 and no mirror planes at all.
Then it's going to be an S2n point group.
These are much rarer than a lot of the other point groups.
So we won't we won't encounter this.
Maybe not at all other than this example we see.
so here's an S.
For example,
this is tetracobalt.
This is Cp - cyclopentadienyl.
So there are four cyclopentadienyl rings and a cluster of four Co. Here's the Co4 cluster here.
Now this is kind of a difficult one to visualize but there's a C2 axis coming out right there that goes right through the middle of those bonds.
If you rotate 180 degrees around that drives this cyclopentadienyl around and puts it here with the same orientation.
You can see this is slightly skewed right out of the plane.
This is slightly skewed in the appropriate way and these two which are pointed into the plane also rotate around and adopt an equivalent configuration after C2.
But there are no mirror plane in the molecule.
That's because these cyclopentadienyl rings are slightly canted relative to one another
and so there is no mirror plane say along this line here cuz that would move this atom slightly across in mirror plane to that position and there's no atom there.
But there is an S4 axis, it is parallel to the C2. 
We do an S4 operation.
We rotate this guy 90°, it would occupy that position in the plane,
and then we would do a reflection perpendicular to the plane and pop it out into this position.
And the same thing for the other for the other three Cp rings and the same thing,
of course with a Cobalt.
So we have in this case for an S4 just four symmetry operations identity, S4, C2, which is the same thing as doing this for twice
and yes for cubed.
You can find S4, S6, and S8.
It's very rare to see anything else in molecular symmetry.
Possibly, but very rare.
Okay,
how do we tell when we given a molecule just on a piece of paper?
What its point group is? Well, lucky for us
all there is a tried-and-true fail-safe decision tree that we can follow it to categorize point groups.
This is the decision tree page 81 in your book figure 4.7
It works every time.
What we do is we start here at the top of the decision tree and we make our way down in that you can see there's a split or two that happened.
First question to ask is that a group of low symmetry?
If it is then it's either the C1, Cs, or the Ci and we're done.
If it's not we go to second step here.
Is it a group of high symmetry?
Yes or no?
If it's not then we ask the question.
What is the highest order axis we can find?
What is the principal axis of the molecule and this set us up for proper determination of the point group.
Is there a C2, C3, or C5, can we identify what that principal axis is?
The next step is then to distinguish between D groups and C or S groups.
Are there perpendicular C2 axes in the molecule?
If there are then we go to the D decision tree at the branch if there are not we go to C and S branch.
Once we found that there are perpendicular C2 axis
So we know we have a D point group. Then the question is it D?
I can't read this...nh and nd or Dn of three different types,
right?
So the decision then comes down to is there a perpendicular mirror plane a horizontal mirror plane?
If yes,
it's Dnh. If not go down here.
Are there parallel mirror planes - parallel to the principal axis.
If yes,
it's Dnd.
If not,
it's just Dn with no mirror planes at all.
And that's the end of the that part of the decision tree.
What kind of business seemed so decisions here?
So if there are no perpendicular C2 axis,
it's either C or S.
We ask ourselves next.
Is there an H mirror plane? If there is, we have a Cnh,
If not, we come down here.
Are there vertical mirror planes?
If yes,
it's Cnv,
if not,
we come down here.
Is there an S2n axis. If yes,
it's S2n,
if no,
it's Cn with no mirror planes at all.
And so you play this little game here and typically for a lot of molecules it ends up practically being these questions here:
identify the principal axis and then find out if there are perpendicular C2 axis and that sets the stage for all the rest of it.
So perpendicular C2 axis, horizontal and vertical mirror plane,
and then you're done.
So let's look at a couple of examples in last 5 minutes.
Phosphorus pentafluoride is a nice example.
Is it low symmetry?
No. Is it linear?
No. Is it a high symmetry?
By symmetry I mean a tetrahedral, an octahedral, or an icosahedral.
It is not, right?
So tetrahedron has four triangular faces.
This one doesn't this has more so it's not low.
It's not high.
So we come to the decision.
What is the principal axis of molecule?
So what is the principal axis of this molecule?
There's a C3 axis. F-P-F right there rotate 120 degrees around that and you're going to interchange these Fs in the plane.
So we have a C3 axis. The next question:
Are there perpendicular C2 axes?
And there have to be either zero or three of them.
We know there are perpendicular C2 is right.
There they are.
So they go from P to F in each of the three individual F. Rotate 180 degrees around that axis.
and you flipped these Fs up there,
right and you flip the other two Fs in their place as well.
Okay.
So what does it mean now if we have a C3 and we have three perpendicular C2 is what is our point group? We're in the D branch of the decision tree.
So the next question is - is there a horizontal mirror plane?
Remember horizontal always means perpendicular to the principal axis.
Quick quick quick.
Yes.
There is a horizontal mirror plane,
right?
It's in the P-3F plane.
So the mirror plane would keep all the four atoms in the plane the same and [invert] the two Fs into one another.
And so yeah,
we've now finished we have determined that there is a horizontal mirror plane with perpendicular C2 and we have a C3 principal axes.
So that means that we're D3h.
That completes the decision tree in this case.
There's not that many decisions because we found we have a horizontal mirror plane.
D3h,
you could build up all of the different symmetry operations.
We will show later is that there's a table at the back of your book that has all these conveniently specified for you and the character tables in the back of your book.
I think it's appendix A or C or something.
So there's a list of all of the different character tables and D3h is one of them.
Okay, so PF5 has D3h symmetry and point group.
Looks like a diborane.
A cute little molecule.
We've got these hydrogen's 90° and we've got these hydrogens in the plane.
So there's two planes to this molecule.
Is it low is it high symmetry or is linear?
No,
no.
so let's find the principal axes.
No,
What is it?
Is there a C4 axis in this molecule?
You hope so big fantasy for symmetry.
There's only C2 in this one.
So there was one of the C2 axis.
There's a couple perpendicular C2 axis.
We chose this one doesn't really matter which one you choose but we chose this one because it contains the B-B Bond.
But you can see that there's also a C2 that goes out of the plane and a C2 that goes up and down like this.
So,
what does that tell you right away if there are two perpendicular see two axes?
We're down in the D Point group selection tree and see D2 D2h and D2d we have to ask ourselves about the horizontal mirror plane.
There is a horizontal mirror plane,
right that's perpendicular to the principal axis that we selected at the beginning between the Bs.
And so that again drives us to the D2h point group for this molecule.
Sorry that this one and the previous example are kind of similar.
I'm bored. One last example,
and then we'll finish up.
Let's look at this one.
This is a famous molecule 18-crown-6-ether.
It's called crown ether because it looks like a crown that you can put on your head.
This is the full ridiculous organic chemistry name to it.
But this is why everybody calls it 18-Crown-6.
There's six oxygens and there are 18 carbons.
That's why are 18 total molecules in the ring or total atoms of the ring.
So it's 18 and 6.
Is it high symmetry or is it low symmetry?
No.
Anyway,
find the principal axis.
What's the principal axis?
There's a C6 right there right. Here is the symbol for the C6 little red hexagon here right in the middle of molecule.
You rotate 60° and you interchange all the atoms.
Does it have perpendicular C2s?
It does not right and that's because the Os are high the Cs are low and you can't get perpendicular C2.
So that's what you would try to draw it.
But in fact,
it doesn't have one. If we have C2 axis that we rotate the Os down underneath that axis and doesn't work out so we know we're not D.
We know that we're either.
C or S,
right.
Is there a horizontal mirror plane?
So there is no horizontal mirror plane in a perpendicular to the C6 axis.
There are however other me airplanes in a molecule there are vertical mirror planes and dihedral mirror planes.
vertical plane passing through the atoms, dihedral passing between atoms
and so what do we have then? we have C6v Point group here and we ruled out the h we ruled out the D4.
We have parallel vertical mirror plane C6v in this case.
Practice assigning point groups and I'll see you on Wednesday.
