So our first topic is symmetry
 
and when we talk about symmetry, we have an intuitive
understanding of what symmetry is.
There are symmetric objects all around us, right?
And in nature, there are very beautiful examples of symmetry.
Also in the human-made universe landscape, there are great examples of symmetry.
In buildings and bridges and things like that.
This is just a montage here of some objects from nature that have various symmetries.
So we have things like ice crystals that have six fold symmetry.
You have starfish, or seastars that have pentagonal symmetry.
Lots of animals have bilateral symmetry. There's a mirror plane in the middle of the animal.
It's Alright, so this bird... I like to picture it, because this bird looks like he has a frog riding on his back.
It's a very strange bird, I don't know what species that is.
And plants have a variety of different symmetries.
There's spiral... different kinds of features.The way the leaves are shaped.
The way the stems branch out...
Alright, there.. They have various kinds of symmetries that we're going to understand how to describe.
Here's another example of bilateral symmetry. Another example of a honeycomb of hexagonal symmetry.. Flowers..
Here's a shell that has a spiral.
And of course we have chiral kinds of symmetries as well, right? With your two hands for example.
... as an example of a good chiral object or good pair of objects.
So if you look around, you may find that there's symmetric beings next to you.
You might see somebody that looks like this.
And if you ignore the Medusa-like hair...
And the kind of crooked nose...
This individual here has rough bilateral symmetry.
Right? There's a mirror plane that runs down, splits down the middle of him.
(Less audible) Try finding somebody like this sitting next to you, which has a different set of symmetry, watch, it's the same kind of symmetry.
So humans have a kind of bilateral symmetry.. *inaudible*
So, if we want to talk about symmetry more mathematically, and concretely, we need to understand
the basic concepts of symmetry elements and symmetry operations.
This is the language that we're going to use to actually
quantify and rigorously describe the symmetries of objects, including molecules.
So everything we've talked about can be applied to objects, and a subset of objects or molecules.
So let's introduce first what a symmetry element is.
A symmetry element is just a geometric object, like a plane, a 2-dimensional plane.
Or a 1-dimensional line (which is also called an axis).
Or a point. (A zero-dimensional point)
Symmetry operation is a movement that we do with respect to one of these symmetry elements.
So movement might be that we reflect across a plane.
Might be that we rotate around an axis.
Or that we invert the object through a point.
So we're performing a motion (a movement) with respect to a symmetry element.
Now what we are most interested in here is which objects, so called possess or have given symmetry operations.
When we say that an object has or possesses a symmetry operation, what we mean is that
when we do the symmetry operation to the object..
...before and after, the object looks indistinguishable.
To us.
So if we, for example, have an object.
We close our eyes, we do the symmetry operation, we open our eyes, we can't tell that the operation has been done.
If that's true of a symmetry operation for a given object, then that object possesses that symmetry operation.
That's the idea.
There are basically four different elements (symmetry elements).
And each of the elements has an operation that is associated with it.
So the elements that we have, we have planes as I mentioned up here.
We call these mirror planes.
We have two different kinds of axes or lines.
We have a proper axis, and we have something called an improper axis.
So a mirror plane, you simply do a reflection through the plane or in the plane, that's kind of the language.
If you have a proper axis, that's just a simple rotation about an axis.
You could rotate 180°, you could rotate 120°, you could rotate 60° for example.
An improper axis is a combination of these two.
It's a rotation, followed by reflection perpendicular to the rotational axis.
So we do a rotation, and then we do a reflection.
So the improper axis, basically is with respect to two symmetry elements (an axis and a plane).
And then, the center of inversion is the element that corresponds to a point of symmetry.
So a center of inversion is when we invert all atoms through the center of the object.
And we will describe exactly what that means, but you can kind of intuitively understand.
You're going to take a point that's here and you're going to go through the origin, through the center, and pop out the other side.
And you're going to do that for all the points.
That's what's called inversion.
So, we have four symmetry elements. There are only five classes of symmetry operations.
We have the reflection, this is usually given the symbol σ ("sigma").
Okay, so when we see σ, think reflection.
Our proper rotation here is Cn.
So C tells us it's a proper rotation. n tells us what amount of rotation we have.
The improper rotation is given the label Sn.
An inversion center is little "i". Little italic "i".
And then the fifth one that corresponds to no symmetry element is the so called identity operation. Big "E".
And the identity operation is the simplest of all operations and we'll start with that one.
What we want to do is understand with examples here, what these symmetry operations are all about.
How to identify them in objects.
And then how to build up the full list of symmetry elements and symmetry operations that a given molecule might have.
Okay, so this is all so that we can find the so-called "point group" of a given molecule.
And if we know the point group, then we know a ton about
the molecular vibrations of that molecule, the molecular orbital structure of that molecule, etc.
So let's start with identity.
So the identity operation, big "E", is the so-called "do nothing" operation.
That's why it's the easiest.
Mathematically, it just means multiply by one.
And it's included in the operations for completeness, for mathematical completeness.
So almost everything that we'll talk about over the next three or four lectures
is coming out of a mathematical theory called "group theory".
That was originally developed in pure mathematics and then it was applied very productively to chemistry and
other kinds of disclipines.
We're not going to get into the "meat" of group theory, but this is one of the results of group theory..
..That we have this mathematical "multiply by one" identity operation.
All objects have the identity operation.
And so if we take an object
And we apply the identity operation, we get back exactly the same thing.
So that's the trivial identity operation.
Okay, let's go now through proper rotations.
And we're going to go through mirror planes, the inversion center, and then improper rotations.
To understand and to follow kind of what's going on with some of these other symmetry operations,
we're going to use a snowflake.
So snowflakes are nice, they often have hexagonal symmetry. This particular one has six long arms.
And that's what we're going to focus on here.
These arms are all labeled with just imaginary labels "a", "b", "c", "d", "e", and "f".
That allow us to keep track of what happens to the arms of the snowflake when we do the symmetry operations.
So the proper rotation (Cn) is just a counter-clockwise rotation of 2π radians divided by n.
Or if you like to think of it in degrees, 360° divided by n.
About some axis.
And you can see with the snowflake that we have six-fold rotational symmetry.
There's an axis coming out of the board, here, which is a six-fold axis.
*gum smack*
And if we do a C6 operation.. So 6 here means that we take 360°, we divide by 6.
That gives us a 60° rotation.
In a counter-clockwise direction, that is the convention.
Is that we're talking about counter-clockwise rotations here.
And what we're going to do is we're going to move a to f, f to e, e to d, and so forth.
And after the operation is complete..
We have the same snowflake indistinguishable.
We have, just labeled here, the different arms, so these are just fake labels.
If we took the labels away, we couldn't tell that the snowflake had been rotated by 60°.
And so this C6 operation is a symmetry operation of this snowflake.
Crowd member: "I've heard of it being a six-fold rotation or six-fold axis, is it just referring to the number of arms
...that are coming off from the center of the axis?"
Dr. Law: "It's referring to this number here actually"
So it's the same thing. The number of arms is six, it's a hexagonal symmetry, and that's described by having a C6 axis.
So you have six-fold rotational axis.
Now they're all different ways of saying the same thing.
Okay, so we can do one C6 operation, we could continue to spin it around ,we could do another C6 operation.
And if we do that second C6 operation, we move point "c" to point "b", and point "b" to point "a", etc.
Again, it's a valid symmetry operation for the snowflake.
One way of writing this is that we do two consecutive C6 operations.
So C6 done consecutively is "C6 squared".
Okay, but that's the same thing as just doing 120° rotation.
Which is called a C3 operation, because if we take 360°, divide it by 3, that gives us 120° rotation.
And if we just go from here, and jump directly to to there, we can get from here to there by 120° rotation.
So two C6's in series is the same thing as a C3.
Let's do it again, let's continue to rotate. We do a third C6 operation.
Still a valid operation.
We get our snowflake back.
This is the same thing as doing three consecutive C6's. This is a "C6 cubed".
And this is the same thing as doing 180° rotation, which is also known as a C2.
Alright, 360° divided by 2.. 180°
And you can see that we just put "f" where "c" is, and "c" where "f" is (that way).
That's a 180° flip.
We can continue!
We do four C6's... That's fine.
Four C6's, as you might guess, four C6's in a row is a "C6 to the fourth".
Which is the same thing as doing the C3 operation twice.
240° rotation.
Okay, so..
It's getting a little bit redundant.
Interestingly, we can also think about clockwise rotations. These are usually called inverse rotations.
And so if we add a "C3 squared", which is a 240° rotation counter-clockwise, that's the same thing as an inverse C3, an inverse 120° rotation.
So we can think of it that way too if we wanted.
Not so common to use these in the rest of the class, but it's just so that you understand
how to describe clockwise and counter-clockwise rotation if you wanted to.
Crowd Member: "So for n, it always has to be a whole number, right?"
Dr. Law: "Yes, that's right. Yep. Always a whole number."
Okay, let's continue, we do five.
Okay.
Five C6's does 300° rotation, it doesn't correspond to anything else in particular.
We could do a negative, or an inverse C6. That's the same thing as a "C6 5" ("see six five").
And then, if we completed the loop, and we did six C6 operations, then we've gone 360°
And all the labels are going to be exactly where they were to start and so six C6's is the same thing as a C1.
Alright, if we put one here, a 360° rotation doesn't change anything, and a C1 is the same thing as identity.
So a C6 to the sixth is the same as E.
And that closes the loop.
So we have a number of different ways we can specify these operations.
Normally what we would say is that this molecule has a C2, it has a couple of C3's operations, a couple of C6's, and the E operation.
And that would round out and complete all of the six different C6 rotations that we can do.
Okay, are there any questions about the proper rotation example here for this snowflake?
We will apply it in many times to different objects so it will become clearer.
Crowd Member: "So power just signifies the amount of rotations? Cause when we say like C6 squared, do we not like input in 60 squared *inaudible*?"
Dr. Law: "It's just the number of consecutive rotations that you do."
Crowd Member: "And, uh, the subscript 6, is it just how many arms a molecule has based on its axis of rotation?"
Dr. Law: "So it's related to the symmetry of an object, right? So you can have a C4 for example."
Which would be an object that has 90° or square-like for example, is one object that would have that kind of rotational symmetry.
You can have a C3 kind of rotational symmetry, like a turbine blade with 3 arms for example.
Or BF3, the molecule. Right? Trigonal planar molecule.
Yep. That's right, so this is just chosen.
The C6 is chosen so that we can see many different rotations and what they would correspond to.
But you can have C2, C3, C4, you could have a C10.
In principle, any are possible. In practice, there is just a small number of very common ones.
And higher, very high order rotational axes are fairly uncommon.
Other questions about the proper axes?
*silence this time*
Okay, let's look at additional proper rotations here.
So we've picked out a C6 axis that's perpendicular to the snowflake.
The question is are there any other axes that are rotational axes in the snowflake.
And the answer is definitely yes!
Right, so there are perpendicular axes to this one, that are also rotational axes for the snowflake.
So here's an example of one of these perpendicular axes, it's in the plane of the board now.
And this is a C2 axis, alright?
So there's a C2 axis here along this arm, there's a C2 axis here, and a third on here .
If we rotate this around 180°, so this notation says perpendicular C2 along the length ad here, along this line.
What we get is we're going to get "a" staying where "a" is, "d" staying where "d" is, but "f" and "b" are going to switch.
And "e" and "c" are going to switch.
And that's what's shown here, so "f" and "b" swap places, and "c" and "e" swap places after the symmetry operation is performed.
So that's a valid symmetry operation for the snowflake, a perpendicular C2 axis.
If you do the C2 operation twice, you get E.
Right? You get the identity operation.
So we have these three axes that go through the snowflakes arms, but there are in addition to those three
three more C2 axes that pass directly between the arms.
You can see that.
And so if we line up, write up all the different C2 axes on the snowflake here, we can see that we have three red ones that pass through the arms
and three blue ones which pass between the arms. In other words, there are six perpendicular C2 axes for this snowflake.
And this is a nice rule.
*presenter clicks*
Any object that has a Cn axis must have either zero or n perpendicular C2 axes.
Here what we have is an object with a C6 axes, so it has to have either zero or six perpendicular C2 axes.
In this case it has six.
The C6 axis that comes out of the board here is called the highest order or the principal axis of this object.
The principal axis is the axis with the largest number of n.
So in this case C6 is the highest order axis or the principal axis for the snowflake.
And so if we count up all of the different proper rotation axes in this particular object
We have a C6. Parallel to that C6 is a C3 and a C2, and then perpendicular to this principal axis are six perpendicular C2 axes.
And if you try to find additional axes, you'll find that there are none.
So that completes all of the different proper rotation axes and operations for this particular object.
*click*
Okay.
Any questions about that?
Yes in the back.
Crowd Member: "Uh, what was the C2 axis? Because there are six perpendicular axes? What does the 'just' C2 axis in the middle *inaudible*"
Dr. Law: "Uh, where, I'm sorry, this one?"
Crowd Member: "Uh, no, the last bullet point."
Dr. Law: "Uh huh/Mmhmm"
Crowd Member: "That C2.."
Dr. Law: "Where is 'this' C2? (That the crowd member is asking about)"
Crowd Member: "In the plane"
Dr. Law: "Yup. So this C2 is parallel to the C6, so we looked here.."
*presenter clicks*
And we said back here a couple steps.
*many presenter clicks* Oops, this one.
Three C6's consecutively gives us a C2.
And so one way you can think about this is that the C6 and the C2 are parallel.
And there's a C2 operation in addition to C6 operations and C3 operations, all along the same rotational axis.
Yep
Crowd Member: "Just sort of looking at the math there, would you then take the um, the subscript and divide it by the exponent?"
Dr. Law: "Yep. That's right. That's a quick way to do it. Yep, exactly."
Mmhmm.
*presenter click barrage*
Okay
Reflections are the next symmetry operation that we're going to apply to this snowflake.
The reflection operation, remember, is σ.
And when we talk about reflection here, we're talking about internal reflection.
We're talking about reflection through a plane of symmetry that is within the object.
And there are many reflection planes, mirror planes in this snowflake.
Perhaps the most obvious is the plane of the board, right?
The plane of the screen here.
That's a reflection plane.
And this one is a mirror plane that is called σh ("sigma sub h").
Or a horizontal mirror plane, because it is perpendicular to the principal axis of the object.
So the principal axis is coming out at us, the perpendicular mirror plane is the one in the board,
and so it's given a horizontal designation, a horizontal mirror plane.
So that's one of the many particular mirror planes in this object.
Okay, so let's look at some other mirror planes.
We'll come back to this in a second here.
Here's the snowflake written with additional mirror planes inscribed.
These mirror planes are parallel with the principal axis.
And there are six total mirror planes. That is, three that run through the arms.
Right, each individual pair of arms.
And there's three that run between the arms. Dihedral mirror planes.
Those that run along the arms and are parallel with the principal axis are called vertical mirror planes.
And those that run between the arms are called dihedral mirror planes.
Okay, and they are given the designations "v" and "d"  (respectively).
All of these are parallel to the principal axis of the object.
And just like we had a certain number of perpendicular C2 axes that are possible, we have a certain number of
vertical and horizontal mirror planes that are possible for an object.
It's either "zero" or "n".
And in this case, it's "n". It's six. We have three vertical and three dihedral.
And so all together, the number of mirror planes that this object has is seven.
We have the horizontal mirror plane in the plane of the board.
We have these three verticals, and we have these three dihedrals.
So we'd say there are seven different mirror plane possibilities in this object.
We go back a step here,
just for completeness.
If you perform a mirror plane, "n times". A mirror operation n times, where you have n is an even number, then you just get the identity.
For example, if n is 2, now if I reflect once and I reflect back, then I have done nothing.
And if I do that four times, then I have also done nothing.
*lol*
If you have n odd, then no matter what the value of n is, that's just equal to sigma itself.
Right, I reflect, I reflect back, I reflect a third time, it's the same thing as just reflecting once.
So this is just for completeness. This may be useful in the future.
Okay *click*
So we've done proper axes, and we've done mirror planes, reflections.
The next symmetry operation that we want to talk about for this object is the inversion operation.
This is little italic "i".
What does inversion mean?
It means the following:
Each point of the object is moved along a straight line
through the center of the object
which we call the inversion center.
And it's moved to a point an equal distance from the origin on the other side.
In other words, all coordinates x, y, z become negative x, negative y, negative z.
You're flipping the coordinates through the center of the object.
It's possible for an object to have either one or zero inversion centers.
You can't have two inversion centers.
And the snowflake, by almost by inspection, just looking at it, you can tell that it has an inversion center.
Right, if we take this arm, and we go through the center and we pop out the other side and we are good.
That arm goes there, that arm goes there, and so forth.
All of the points of the object, this little arm here, if it was drawn to be perfect, would go end up there, going right through the origin
and pop out and would give us an indistinguishable orientation of that snowflake.
So the snowflake, ignoring fine detail on the snowflake, has inversion symmetry.
And it has an inversion center.
So we now have a fairly large number of symmetry operations that this snowflake has.
But we're not totally finished yet.
Here's the inversion, if we do the inversion, all of the places just swap places through the origin, through the center.
So inversion is quite an interesting operation.
Similarly to the reflection operation, if you perform it n times, where n is even, you get the identity.
If you perform it n times, where n is odd, you just get the inversion.
So if I invert, and I invert back, n would be equal to two, that's the same as doing nothing.
If I invert, invert, and invert, that's the same thing as just doing it once.
*click*
So what objects have inversion centers?
.. or inversion symmetry?
These objects here have inversion centers.
These objects here, which are of some chemical importance have no inversion centers.
So octahedra, this guy has an inversion center. Boxes, squares, rectangles and parallelograms all have inversion centers.
You can find a center of that object whereby if you exchange all the coordinates, you will get the same object.
Objects that have no inversion centers are (include) triangles, so if you find the middle of the triangle
and let's say you invert this vertex through the center, it'll pop out here.
But you'll have flipped the triangle over during the inversion operation, it's not an inversion operation.
Right, there's no inversion symmetry.
Same thing with a tetrahedron.
Okay, this is one of the most important examples of course for Chemistry.
So CH4 does not have a center of inversion.
If you do the inversion operation, what you do is you move this fourth Hydrogen here over here, the third Hydrogen down here to these two places
and you swap these guys through the carbon and over on this side, what you see is that this does not
is not the same thing as that.
Alright, it's related by rotation, but it's not an inversion operation.
And pentagons also don't have inversion symmetry.
Crowd Member: "Um, so looking at the tetrahedron there, I can't help but think back to Organic Chemistry
Crowd Member: "..when we had stereogenic centers, you normally didn't see inversion of those
Crowd Member: "...is C(H)4 not applicable, just because the Hydrogen atoms are identical
Crowd Member: "..around that center there, and so you don't have a stereogenic center? Or um..."
Dr. Law: "So you don't have a stereogenic center in this case, right? This is not a chiral center."
Crowd Member: "But would you still inversion even if it could be considered..."
Dr. Law: "So, we have to be careful about what we mean by inversion."
Here what we're talking about is a mathematical operation, a geometric operation.
What you may be referring to is actually movement of one of the atoms to the other side of the molecule for example.
Or, uh, you know a trigonal pyramid or something like that flopping back and forth.
That's inversion in a sense, but that's an Organic Chemistry definition version.
This is different.
So what we're talking about here is a mathematical description of taking all the coordinates and giving them their negative values.
And seeing if the object is indistinguishable after that operation or not.
Crowd Member: "So from your inversion image of the snowflake, it looked the same as a C6 cubed, is that always the case?"
Dr. Law: "A C6 cubed... Um, 180°, yeah it is exactly the same thing, right. For that 'particular' object."
For that hexagonally symmetric object, yes.
Crowd Member: "Are there objects that don't have..."
Dr. Law: "Yes, yeah. So the um.."
For hexagonally symmetric objects, that is going to be true, then for C4, it's going to be different.
It would be the same thing as a C2 operation if it's a flat object like that.
If it's a 3-dimensional object like an octahedron, it's not going to have that kind of relationship
because then you've got to worry about the below-plane and above-plane arms of the octahedron.
And those will not be reproduced in the same way like a C4 squared, an inversion operation.
It'd be different.
Okay
*click*
Okay, and then finally, and most confusingly in some ways, this is the most complex operation.
The improper rotations.
So the improper rotation operation, remember is Sn.
And this is a rotation, followed by a reflection perpendicular to the rotation axis.
So this is also called a roto-reflection, it's a combined operation.
Rotation. Reflection. Roto-reflection.
So one object that has a improper rotation axis is methane.
And it has an S4 axis.
Actually a few.
So if we look at the S4 operation, what are we gonna do?
So here's methane that's inscribed in a cube.
You may know that you can draw a tetrahedron by painting in opposite corners of the cube.
with the object.
So we're going to label this H1, H2, H3, and H4 so that we can just keep track of where the Hydrogens go.
So an S4 operation means that we do a C4 rotation followed by a perpendicular reflection.
So here's our S4, we first do a C4. That moves H2 to this position here, and moves H1 to this position here
and the same thing for the Hydrogens on the bottom.
So we get this kind of orientation.
Clearly, C4 rotation in itself is not a symmetry operation of a tetrahedron.
But, if we follow the C4 to complete the S4 with a perpendicular reflection
right, here's the reflection plane. It's inscribed here, it's through the carbon of course, through the center.
Then what happens is that H1 goes down here, the H2 goes down here, the H3 and H4 pop up top, and you end up with this orientation.
And this is indistinguishable from this.
Alright, but with our labels, of course, we can see what has happened to the individual Hydrogens.
This Hydrogen here has moved down here and so forth.
But that is a good symmetry operation for methane.
We could do it a second time.
Okay, so if we do a second S4 operation starting from this position here.
Okay, we bring that down here.
We do a C4 followed by a perpendicular reflection.
The C4 is gonna move H4 here to there. H3 to here, and so forth.
And then we reflect, and we end up with this particular orientation.
It's the same thing, indistinguishable from the starting object.
It's also the same thing as if we just did a C2 operation.
And so we can go from here to here directly with the C2 that moves H2 here and H1 there.
You can see that's what's happened.
Or you can do two sequential S4's.
So we can build up a little table.
If we just do one S4 operation we just call that an S4.
If we do two, we call that a C2, rather than an S4 squared.
Because it's simpler.
The C2 operation is simpler, so that's what we go with.
If we did 270°, three sequential S4's, we call that a "S4 cubed".
That's a unique symmetry operation for this particular object.
And if we did 360°, that's four S4 operations consecutively, we just call that E, rather than S4 *inaudible* to the fourth.
Again, because it's simpler.
So the unique operations in this particular example are just the S4 and the S4 cubed.
The other two possible S4 operations are more easily described as a C2 and an E.
Crowd Member: *inaudible question*
Dr. Law: "Um... Oh the H is horizontal"
So this is horizontal in the sense that it is perpendicular the rotational axis.
Now anytime you see a σh ("sigma h"), think
"I have a rotational axis, and now I'm going to do a reflection perpendicular to that axis"
Okay.
The snowflake has S operations. It has S3 and S6 operations but they're not too insightful,
so instead, what we're going to do, and it looks like we're going to tackle this mostly next time
is describe the Sn operations with the real molecular example.
*click*
So we'll, we'll begin that next time, and let me just finish by saying that if we do an S2 operation
it's the same thing as inversion.
An S2 operation will be a C2 followed by a mirror reflection perpendicular
and anAnd that's gonna go C2 and then here this would be the n position for an S2 operation
and you can see that this is just an inversion of the original object.So
So an S2 is equal to "i".
And an S1 is simply a 360° rotation followed by a reflection.
So all that is, is just a reflection.
So S1 is just equal to σ.
It's the same thing as σ.
So it's important to be able to recognize, um, when an operation that looks complicated is really just a simpler operation in disguise.
So we will pick this up with our molecular example of staggered ethane on Monday. *sound of desks*
