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Some of you will recognize this
molecule that I have rotating up

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on the screen because it is one
of the coordination complexes of

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Werner that we talked about last
time.

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And Werner, of course,
was well-known.

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And some of you probably read
is biography over the last few

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days on the Nobel Prize website.
Well-known for his theory that

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broke down the preconceived
notions that had been prevailing

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at that time concerning the
structure of systems that have

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unusual salt-like behavior in
some cases, but that contain the

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3D elements.
I am focusing on 3D,

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but this is also true of 4D and
5D elements, the transition

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metals.
Elements like titanium,

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vanadium, chromium,
iron, cobalt,

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nickel, etc.
This molecule here,

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that you see spinning up on the
screen, is being represented

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here on the screen in one of
these kind of arbitrary forms,

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wherein the cobalt center at
the middle of the molecule is

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just a round ball of an
arbitrary size.

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And then, this particular
molecule that contains a cobalt

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three ion is
surrounded by a compliment of

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six ligands.
We have three chloride ligands,

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that because this program chose
to do so colored those chlorides

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the same as the cobalt,
not what I would do,

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and put the three ammonia
ligands, colored the ammonia

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nitrogens in dark blue,
and the ammonia hydrogens in

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white.
And we talked about the fact

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that these transition elements,
in fact, these 3+ ions of

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metals like cobalt,
can behave simultaneously as

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Lewis acids toward a multitude
of Lewis bases,

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here six.
This is a times six Lewis acid

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in the middle and six Lewis
bases, which are three chloride

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ions and three ammonia molecules
oriented around it,

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in an arrangement that is
quasi-octahedral.

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Because the positions of the
three nitrogens of the ammonias

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that are interacting with the
cobalt center and the three

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chloride ions are located near
the vertices of a regular

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octahedron.
So we will be taking use of the

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geometry of molecules like this
in discussing electronic

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structure properties of these
molecules today.

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We will be doing that beginning
today.

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And that will be important to
understanding the magnetism and

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the color and also the reactions
of molecules like this.

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Now, one other arbitrary thing
that this program did is it

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chose not to draw lines between
the cobalt and any of the six

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ligands.
But normally,

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when you see these molecules
drawn in textbooks,

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you will see that the lines
drawn are the same as the lines

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between the nitrogens and these
hydrogens.

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So we would have to add six
more lines to this drawing to

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get the typical textbook
representation of a molecule

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like this.
But, still, that would be a

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somewhat arbitrary
representation.

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And so, I would like to show
you a less arbitrary

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representation.
And we will do that forthwith.

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And this one has to do with
looking at the same molecule,

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but represented as an electron
density isosurface.

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And that isosurface will be
colored according to this

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function, that tells us about
the propensity to pair electrons

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in 3D space.
And so you will recall that

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when we talk about coloring
electron density isosurfaces in

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this way, so this now is a
physically important kind of

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representation of this
coordination complex,

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this color scale will run from
red all the way to blue.

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And at blue is where you find
regions in space where you are

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most likely to find pairs of
electrons mapped onto the value

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here, colored mapped onto the
value of an electron density

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isosurface.
Here the value of the electron

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density that is present for
every point represented on this

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surface is 0.11 electrons per
unit volume.

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And so what you see here is
that, in fact,

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the electron density does
become low as you move from the

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cobalt in the direction of any
one of the ligands.

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But the lone pair on each
ammonia is certainly polarized

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in the direction of the cobalt
center.

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And the cobalt center is not
uniform in terms of the way that

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electron density is organized
around it.

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You will see that right on the
metal center,

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we see red.
And then sort of at the corners

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of a cube, you see this yellow
or green color on the cobalt,

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and that is very significant.
And what we are going to be

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today is we are going to try to
understand what happens when you

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put a set of ligands into an
octahedral array around a

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central metal ion that has
d-electrons and d orbitals to

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play with.
And, in fact,

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what you will see is that the
oxidation state of the cobalt

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center here is +3.
That is why I am referring to

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it as a cobalt plus three ion.

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And because cobalt is in group
nine of the periodic table,

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you then know that there are
six valance electrons on the

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cobalt center that you have to
put into orbitals.

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And so what we are really
seeking to know is how can we

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get an energy-level diagram for
a system like this,

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so we will know how to put
those six electrons into that

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diagram, the ones that are
mostly localized on the cobalt,

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and make predictions on whether
the electrons should be paired

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up or not, for example.
We saw with homonuclear

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diatomics, like the dioxygen
molecule, that electrons are not

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all paired up.
Two of the electrons are

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unpaired.
And, when we have six electrons

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to put into an energy-level
diagram for the cobalt ion,

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we are going to wonder just
what the case is.

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How many energy levels are
there, and where do we put the

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electrons?
And so we are going to need to

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very briefly review what I did
at end of lecture last time.

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And that has to do with the
properties of the five d

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orbitals --
-- because, in order to answer

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the kinds of questions that we
are posing about energy levels

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of metal ions that are
transition metals,

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we are going to really need to
know very well the nodal

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properties of these d orbitals.
And so, remember the m quantum

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number here can be zero for the
d z squared.

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It can be plus one for the
d(xz).

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And plus two for the d x
squared minus y squared.

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Minus one for the d(yz)

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orbital.
And minus two for the d(xy)

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orbital.
And let's draw those.

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d z squared is an
interesting one.

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And, because it is distinct
from the other four d orbitals,

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we are going to be spending
more time on it today than the

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others.
What it has,

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these are the x,
let's say y and z axes here,

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is it has a positive lobe along
both plus and minus z.

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So it looks like a p orbital so
far, except that we have the

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same sign both in plus or minus
z.

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And then what we have is a very
interesting toroidal shape that

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goes around in the x,y-plane all
the way around in a

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cylindrically symmetric manner.
And so, if you were to look

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down z onto this orbital,
it might look something like

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this.

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Like that.
It would be cylindrically

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symmetric about z.
And that was another feature

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that I didn't get to add to this
diagram at the end of last hour,

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which is that because of this
property, this m equals zero

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that means this is a sigma
orbital with respect to the

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z-axis.
Cylindrically symmetric about

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z.
And then, let's go to the xz,

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since I am making this x,
y, and z over here.

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The d(xz) has four lobes,
and they are in between the x

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and z axes, as I am trying to
represent here.

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These two are kind of coming
out in front here,

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and these two back behind.
And then, the phases go as

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follows.
Unshaded is,

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again, plus.
And then we have the x squared

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minus y squared
orbital, which has four lobes.

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In fact, the shape of xz,
x squared minus y squared,

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yz, and xy are all the
same.

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00:10:06,000 --> 00:10:10,000
It's just that they point in
different directions in space

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with respect to the Cartesian
coordinate axes.

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x squared minus y squared,
like d z squared,

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is an orbital whose
lobes point along the coordinate

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axes, like this.
And it is minus along y,

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as the name suggests,
and plus along x,

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as the name suggests.
That is our d x squared minus y

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squared orbital.

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And then yz is like xz,
but let me finish this part

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here.
If d z squared is

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sigma with respect to z,
and you imagine looking down z

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onto d(xz), what would that be
with respect to z?

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00:11:02,000 --> 00:11:02,000
Anyone.
Pi.
Thank you.
That would be pi with respect

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to z, because the yz plane is a
nodal surface for this d

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orbital, as is the x,y-plane.
And that is what we are trying

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to do here, is become familiar
with these orbital surfaces.

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And, accordingly,
now, if you look down z onto

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the d x squared minus y squared
orbital,

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you are now going to see two
nodal surfaces when you look

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down it that way.
Because you are going to see

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that there is one located over
here.

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That is a plane that contains
the z-axis, but it bisects the x

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and y axes.
And then there is another one

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over here, but that is 90
degrees to the first one.

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And that makes this one delta
with respect to z.

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Delta is when you have two
nodes that contain the z-axis

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00:12:06,000 --> 00:12:10,000
and if we are looking down that
z-axis.

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And then, over here,
the d(yz) orbital has its four

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lobes between y and z,
like that.

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And it is positive between y
and z, and negative over here,

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as I am shading.
And this one,

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like d(xz), is pi with respect
to z.

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And then, over here,
we have a d(xy) orbital as our

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final orbital.
And what you might be able to

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guess, we have a sigma with
respect to z,

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we have a pair of pi with
respect to z for m equals plus

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and minus one.
We also must have a pair of

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delta with respect to z for m
equals plus and minus two.

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And that means that the d(xy)
orbital, like d x squared minus

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00:13:03,000 --> 00:13:08,000
y squared,
must lie in the x,y-plane.

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00:13:08,000 --> 00:13:13,000
And, in order to be orthogonal
to d x squared minus y squared,

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00:13:13,000 --> 00:13:17,000
we are going to have to rotate
it such that its four lobes

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point now between the x and y
Cartesian coordinate axes like

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this, although I am trying to
improve on that with my

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coloration.
That is like that.

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Again, this is an orbital
perpendicular to z and which has

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delta symmetry.
And now, the two nodes are,

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in fact, the xz and yz
coordinate axes.

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Indicating those nodal planes
there.

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This is certainly the basics
for what you need to know about

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the d orbitals.
And just briefly,

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I would like to switch to
Athena terminal,

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00:13:57,000 --> 00:14:02,000
here, to show you that there
are ways for you to go ahead and

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00:14:02,000 --> 00:14:08,000
visualize the orbitals.
And I am going to make this

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00:14:08,000 --> 00:14:12,000
information available to you,
so that you can go ahead and do

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00:14:12,000 --> 00:14:16,000
this yourself in order to
visualize these in a way that

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00:14:16,000 --> 00:14:21,000
will take you right from the
equations for the orbitals to

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00:14:21,000 --> 00:14:25,000
their graphical representation.
I think that is really

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00:14:25,000 --> 00:14:30,000
important to get a good
understanding of orbitals.

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00:14:30,000 --> 00:14:38,000
Let's zoom this a little bit,
so that you can begin to see

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00:14:38,000 --> 00:14:39,000
it.

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00:14:46,000 --> 00:14:49,000
This is a worksheet put
together that,

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00:14:49,000 --> 00:14:54,000
in fact, contains all the
functional forms for the d

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00:14:54,000 --> 00:14:56,000
orbitals.
Let me see.

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00:14:56,000 --> 00:15:00,000
Where is that?
Here we go.

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00:15:00,000 --> 00:15:02,000
And you can look at this with
Maple on Athena.

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00:15:02,000 --> 00:15:06,000
And then, you can look at the
equation that represents the

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00:15:06,000 --> 00:15:09,000
angular part of the wave
function for the orbital that

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00:15:09,000 --> 00:15:12,000
you are interested in.
And then, you can go ahead and

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00:15:12,000 --> 00:15:14,000
plot it.
And you can plot it in such a

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00:15:14,000 --> 00:15:16,000
way that the function is
animated.

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00:15:16,000 --> 00:15:20,000
And, rather than just seeing it
projected on a board as well as

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00:15:20,000 --> 00:15:25,000
I can draw, you will be able to
see it drawn up graphically.

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00:15:25,000 --> 00:15:30,000
In fact, these come from the
solution for the Schrˆdinger

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00:15:30,000 --> 00:15:36,000
equation for the hydrogen atom.
And the angular part of these

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00:15:36,000 --> 00:15:42,000
wave functions that is going to
be oh so important to us is

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00:15:42,000 --> 00:15:48,000
something known as the set of
spherical harmonic equations.

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00:15:48,000 --> 00:15:54,000
And that should reference you
to this issue of standing waves,

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00:15:54,000 --> 00:16:00,000
that we have discussed.
Let's just see what we can do,

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00:16:00,000 --> 00:16:03,000
here.
Sometimes, I am not so good at

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00:16:03,000 --> 00:16:06,000
using Maple up in front of the
class.

232
00:16:11,000 --> 00:16:14,000
What you are seeing is we are
getting representations for d z

233
00:16:14,000 --> 00:16:17,000
squared.
Right here is a way of writing

234
00:16:17,000 --> 00:16:19,000
d z squared.
You are going to see an

235
00:16:19,000 --> 00:16:22,000
important term here,
three cosine squared theta

236
00:16:22,000 --> 00:16:25,000
minus one.
And we will come back to that

237
00:16:25,000 --> 00:16:27,000
in a moment.
That is the angular part of the

238
00:16:27,000 --> 00:16:31,000
d z squared wave
function.

239
00:16:31,000 --> 00:16:34,000
And then we can look at some of
these other ones.

240
00:16:34,000 --> 00:16:39,000
You will see that some of the d
orbitals come as combinations of

241
00:16:39,000 --> 00:16:43,000
real and imaginary functions
that are the solutions to the

242
00:16:43,000 --> 00:16:46,000
differential form of the
Schrˆdinger equation.

243
00:16:46,000 --> 00:16:50,000
And then we take linear
combinations of these to get

244
00:16:50,000 --> 00:16:54,000
real forms, so that we can get
plots that we can look at.

245
00:16:54,000 --> 00:16:58,000
And let's see if we can get d z
squared,

246
00:16:58,000 --> 00:17:00,000
here.
There it is.

247
00:17:00,000 --> 00:17:04,000
There is a picture of d z
squared.

248
00:17:04,000 --> 00:17:07,000
And you see,
if you are using this Maple

249
00:17:07,000 --> 00:17:12,000
worksheet, that you can actually
rotate that around and animate

250
00:17:12,000 --> 00:17:16,000
it a little bit.
You see that we have this torus

251
00:17:16,000 --> 00:17:20,000
that is in the x,y-plane.
And you have the two large

252
00:17:20,000 --> 00:17:24,000
lobes that extend up along plus
and minus z.

253
00:17:24,000 --> 00:17:29,000
And so I am going to encourage
you to go ahead and look at that

254
00:17:29,000 --> 00:17:35,000
worksheet, which will be
available from our website.

255
00:17:35,000 --> 00:17:37,000
Go ahead and look at some of
the functions.

256
00:17:37,000 --> 00:17:41,000
And if some of you are
interested in higher orbitals,

257
00:17:41,000 --> 00:17:44,000
the f orbitals are also
available in this worksheet.

258
00:17:44,000 --> 00:17:48,000
So you can visualize the f
orbitals, that are important for

259
00:17:48,000 --> 00:17:51,000
understanding the chemistry of
elements like,

260
00:17:51,000 --> 00:17:54,000
for example,
uranium, which is a little bit

261
00:17:54,000 --> 00:17:57,000
beyond the scope of 5.112.
Now let's switch to the

262
00:17:57,000 --> 00:18:01,000
document camera.
And, if you could,

263
00:18:01,000 --> 00:18:06,000
I would like you to make this
part big.

264
00:18:06,000 --> 00:18:11,000
This is a table in your
textbook that has the angular

265
00:18:11,000 --> 00:18:18,000
part of the wave functions for
various hydrogen-like orbitals.

266
00:18:18,000 --> 00:18:23,000
And this is the part that I am
most interested in,

267
00:18:23,000 --> 00:18:28,000
over here.
If you could just focus in on

268
00:18:28,000 --> 00:18:34,000
the d orbitals over here.
When I was talking about the d

269
00:18:34,000 --> 00:18:38,000
z squared orbital a
moment ago, I was focusing on

270
00:18:38,000 --> 00:18:41,000
this term here,
this cosine squared theta minus

271
00:18:41,000 --> 00:18:43,000
one term.
And here are the other

272
00:18:43,000 --> 00:18:46,000
d-orbitals.
These are the descriptors for

273
00:18:46,000 --> 00:18:49,000
the d-orbitals,
zy, yz, xz, x squared minus y

274
00:18:49,000 --> 00:18:52,000
squared,
and z squared.

275
00:18:52,000 --> 00:18:55,000
And I will need to refer to
this in a moment,

276
00:18:55,000 --> 00:18:59,000
so we will leave this up.
The reason why I have this

277
00:18:59,000 --> 00:19:03,000
arrow written into my book here
is because these are backwards

278
00:19:03,000 --> 00:19:06,000
in the text.
This one is actually x squared

279
00:19:06,000 --> 00:19:10,000
minus y squared,
and this one is actually the xy

280
00:19:10,000 --> 00:19:13,000
orbital.
We figured that out last year

281
00:19:13,000 --> 00:19:17,000
when we were doing this lecture.
And so your book isn't always

282
00:19:17,000 --> 00:19:19,000
right.
You should make sure you check.

283
00:19:19,000 --> 00:19:22,000
The same is certainly true of
your instructor,

284
00:19:22,000 --> 00:19:24,000
but we will try not to mislead
you.

285
00:19:24,000 --> 00:19:30,000
And so now here is the approach
that we are going to take.

286
00:19:40,000 --> 00:19:42,000
Let's say that we have our
coordinate system,

287
00:19:42,000 --> 00:19:46,000
and we want to know how to
evaluate one of these d orbital

288
00:19:46,000 --> 00:19:49,000
wave functions for a particular
point in space.

289
00:19:49,000 --> 00:19:53,000
Actually, we are going to want
to evaluate the square of the

290
00:19:53,000 --> 00:19:56,000
wave function.
And so we are going to make use

291
00:19:56,000 --> 00:19:58,000
of a Cartesian coordinate
system.

292
00:19:58,000 --> 00:20:02,000
And we are going to express
things in terms of polar

293
00:20:02,000 --> 00:20:04,000
coordinates.

294
00:20:10,000 --> 00:20:15,000
Here is a sphere projected onto
our Cartesian coordinate system.

295
00:20:15,000 --> 00:20:20,000
And some of you will be very
familiar with this.

296
00:20:20,000 --> 00:20:25,000
We are going to say if you are
at a point here in space,

297
00:20:25,000 --> 00:20:31,000
then we can describe that point
in space by a set of variables

298
00:20:31,000 --> 00:20:35,000
which will be r,
theta, and phi.

299
00:20:35,000 --> 00:20:42,000
And here is our angle theta.
And if we drop down to a

300
00:20:42,000 --> 00:20:49,000
perpendicular on the x,y-plane
from our point,

301
00:20:49,000 --> 00:20:58,000
then we are going to define phi
as being from the x-axis and

302
00:20:58,000 --> 00:21:04,000
going over in the direction of
y.

303
00:21:04,000 --> 00:21:08,000
And so, when you look at the d
orbital wave functions,

304
00:21:08,000 --> 00:21:13,000
you see that they are all
written here in terms of just

305
00:21:13,000 --> 00:21:16,000
theta and phi,
and not in terms of r,

306
00:21:16,000 --> 00:21:20,000
which is this distance here
from the nucleus,

307
00:21:20,000 --> 00:21:26,000
from the center of this metal
ion that we are talking about.

308
00:21:26,000 --> 00:21:30,000
This would be the r.
And what we are going to do,

309
00:21:30,000 --> 00:21:35,000
is if this point on the surface
of our sphere represents one of

310
00:21:35,000 --> 00:21:38,000
our ligand atoms,
so think back to that complex

311
00:21:38,000 --> 00:21:43,000
we were describing a few moments
ago, cobalt with three ammonia

312
00:21:43,000 --> 00:21:46,000
ligands and three chloride
ligands, we are going to

313
00:21:46,000 --> 00:21:50,000
approximate each of those six
ligands by a point on the

314
00:21:50,000 --> 00:21:54,000
surface of our sphere.
And then, we are going to say,

315
00:21:54,000 --> 00:21:57,000
if there is a metal at the
center, that cobalt ion in

316
00:21:57,000 --> 00:22:01,000
particular, what is the
probability of finding a d

317
00:22:01,000 --> 00:22:05,000
electron where that ligand atom
is?

318
00:22:05,000 --> 00:22:08,000
And, in order to do that,
we are going to need to be able

319
00:22:08,000 --> 00:22:11,000
to evaluate the wave function.
And we are making the

320
00:22:11,000 --> 00:22:14,000
assumption, for simplicity,
that this is a perfect sphere,

321
00:22:14,000 --> 00:22:17,000
and also that our coordination
geometry is a perfect

322
00:22:17,000 --> 00:22:19,000
octahedron.
And so we are not going to

323
00:22:19,000 --> 00:22:22,000
consider r, because r is going
to be the same everywhere.

324
00:22:22,000 --> 00:22:25,000
It is just a perfect sphere
with one value of r,

325
00:22:25,000 --> 00:22:29,000
no matter which ligand position
we are looking at.

326
00:22:29,000 --> 00:22:33,000
And so that means we can focus
in just on these which are the

327
00:22:33,000 --> 00:22:36,000
angular part of the wave
function for the molecule in

328
00:22:36,000 --> 00:22:39,000
question.
And so let's see what this

329
00:22:39,000 --> 00:22:43,000
means with respect to d z
squared.

330
00:22:54,000 --> 00:22:57,000
The probability of finding an
electron at some point in space

331
00:22:57,000 --> 00:23:01,000
in a particular atomic orbital
is proportional to the square of

332
00:23:01,000 --> 00:23:05,000
that atomic orbital at that
point in space.

333
00:23:05,000 --> 00:23:07,000
That is our probability
density.

334
00:23:07,000 --> 00:23:10,000
And we encountered that very
early in the semester.

335
00:23:10,000 --> 00:23:14,000
Now, we are going to make use
of that to make a plot.

336
00:23:14,000 --> 00:23:18,000
We are going to plot this
probability density of finding

337
00:23:18,000 --> 00:23:22,000
electron in d z squared
as a function of

338
00:23:22,000 --> 00:23:25,000
theta.
Why can I do that ignoring phi?

339
00:23:25,000 --> 00:23:29,000
Well, that is because if you
look at d z squared,

340
00:23:29,000 --> 00:23:33,000
there is no phi in that
equation.

341
00:23:33,000 --> 00:23:35,000
Why is that?
That is because d z squared

342
00:23:35,000 --> 00:23:39,000
is cylindrically
symmetric about z.

343
00:23:39,000 --> 00:23:42,000
And look at how we defined phi.
It is as you go in the

344
00:23:42,000 --> 00:23:45,000
x,y-plane around starting from
x.

345
00:23:45,000 --> 00:23:49,000
Because of the sigma symmetry
of d z squared with respect to

346
00:23:49,000 --> 00:23:53,000
z, there is no phi dependence of
this wave function.

347
00:23:53,000 --> 00:23:57,000
And that should appear here in
the angular form of the

348
00:23:57,000 --> 00:24:01,000
description of the d z squared
orbital.

349
00:24:01,000 --> 00:24:06,000
But we do know that d z squared
does depend on theta,

350
00:24:06,000 --> 00:24:10,000
because theta starts out
somewhere here along,

351
00:24:10,000 --> 00:24:15,000
let's say initially theta
equals zero would be right on

352
00:24:15,000 --> 00:24:19,000
the positive z-axis.
And then, as we keep a constant

353
00:24:19,000 --> 00:24:23,000
r and we sweep down here toward
the x,y-plane,

354
00:24:23,000 --> 00:24:28,000
the d z squared
probability density is dropping

355
00:24:28,000 --> 00:24:32,000
off.
And then at some point,

356
00:24:32,000 --> 00:24:36,000
when we get to the node here,
and we are going to be

357
00:24:36,000 --> 00:24:39,000
interested in that,
it goes to zero,

358
00:24:39,000 --> 00:24:44,000
because that is what happens on
nodes, as we continue down

359
00:24:44,000 --> 00:24:50,000
toward the x,y-plane past that
node, it is going to be nonzero

360
00:24:50,000 --> 00:24:55,000
again and rise up as we approach
this smaller torus in the

361
00:24:55,000 --> 00:24:57,000
x,y-plane.
Smaller, that is,

362
00:24:57,000 --> 00:25:03,000
than the big lobes that extend
up along z and down along minus

363
00:25:03,000 --> 00:25:05,000
z.
Let's represent that

364
00:25:05,000 --> 00:25:08,000
graphically.

365
00:25:14,000 --> 00:25:20,000
Coming down in theta from theta
equals zero to some value here,

366
00:25:20,000 --> 00:25:26,000
we are going to be interested
in just what that value is.

367
00:25:26,000 --> 00:25:32,000
And then, rising up again to
theta is equal to pi over two.

368
00:25:32,000 --> 00:25:36,000
Do you see that?
This is another way of

369
00:25:36,000 --> 00:25:43,000
displaying this property.
This is at constant r --

370
00:25:48,000 --> 00:25:50,000
-- and varying theta.

371
00:25:56,000 --> 00:25:59,000
First of all,
how can we find out at what

372
00:25:59,000 --> 00:26:04,000
value theta goes to zero?
Well, we look up here at the

373
00:26:04,000 --> 00:26:08,000
functional form of the d z
squared.

374
00:26:08,000 --> 00:26:12,000
We get nodal properties of d z
squared.

375
00:26:22,000 --> 00:26:25,000
We can pretty quickly see that
in each of these,

376
00:26:25,000 --> 00:26:30,000
we have a factor leading out in
front, which is a normalization

377
00:26:30,000 --> 00:26:35,000
factor that assures us that the
sum integrated over all space of

378
00:26:35,000 --> 00:26:39,000
this wave function will come out
to be one.

379
00:26:39,000 --> 00:26:42,000
If there is an electron in that
orbital somewhere,

380
00:26:42,000 --> 00:26:45,000
the probability of finding that
electron somewhere in space will

381
00:26:45,000 --> 00:26:47,000
be one.
We have these normalizing

382
00:26:47,000 --> 00:26:51,000
constants out in front that
allow for that and ensure that

383
00:26:51,000 --> 00:26:54,000
that is the case.
But where we actually find the

384
00:26:54,000 --> 00:26:56,000
angular dependence is in that
second term.

385
00:26:56,000 --> 00:26:59,000
Here it is three cosine squared
theta minus one.

386
00:27:10,000 --> 00:27:14,000
And what we want to do is say,
when does this function go to

387
00:27:14,000 --> 00:27:17,000
zero?
Because when that goes to zero,

388
00:27:17,000 --> 00:27:22,000
we will have this angle here.
If we say, let this equal zero,

389
00:27:22,000 --> 00:27:26,000
we are looking for the value of
the angle theta which

390
00:27:26,000 --> 00:27:32,000
corresponds to the node of the d
z squared orbital.

391
00:27:32,000 --> 00:27:35,000
Remember, we mentioned this
last time, because of this

392
00:27:35,000 --> 00:27:39,000
cylindrical symmetry of the d z
squared orbital,

393
00:27:39,000 --> 00:27:44,000
this really is a conical nodal
surface that is above and below

394
00:27:44,000 --> 00:27:47,000
the plane here.
If you are anywhere on that

395
00:27:47,000 --> 00:27:51,000
cone, either in plus or minus z,
the value of that d z squared

396
00:27:51,000 --> 00:27:55,000
orbital is zero.
And so we are setting it equal

397
00:27:55,000 --> 00:28:02,000
to zero to find the angle theta.
And we can rearrange this and

398
00:28:02,000 --> 00:28:07,000
say that cosine squared theta is
equal to one over three.

399
00:28:07,000 --> 00:28:14,000
And, if we go ahead
and solve that,

400
00:28:14,000 --> 00:28:19,000
this comes out to the arccosine
of root three over three.

401
00:28:19,000 --> 00:28:25,000
And the other possibility that

402
00:28:25,000 --> 00:28:31,000
satisfies that relation is pi
minus the arccosine of root

403
00:28:31,000 --> 00:28:37,000
three over three

404
00:28:37,000 --> 00:28:41,000
And so what that means is that
this relation,

405
00:28:41,000 --> 00:28:48,000
here, gives us the angle for
that cone in the plus z axis.

406
00:28:48,000 --> 00:28:54,000
And then this one down here,
pi minus arccosine root three

407
00:28:54,000 --> 00:29:01,000
over three,
gives us the angle for

408
00:29:01,000 --> 00:29:05,000
that cone down in the minus
z-axis.

409
00:29:05,000 --> 00:29:09,000
And what is this?
This is, in degrees,

410
00:29:09,000 --> 00:29:14,000
something like 54.476 dot,
dot, dot degrees,

411
00:29:14,000 --> 00:29:18,000
approximately.
That is just how far down you

412
00:29:18,000 --> 00:29:27,000
are from the z-axis when you hit
that nodal surface of z.

413
00:29:27,000 --> 00:29:29,000
And that number,
you are going to see,

414
00:29:29,000 --> 00:29:32,000
is kind of a magical number in
chemistry.

415
00:29:32,000 --> 00:29:37,000
And it will hearken back to
some of the things that we have

416
00:29:37,000 --> 00:29:39,000
been taking about,
recently.

417
00:29:39,000 --> 00:29:44,000
And, in order to get to that
point, I am going to need to now

418
00:29:44,000 --> 00:29:48,000
talk about our ligands again
with respect to d z squared.

419
00:30:07,000 --> 00:30:10,000
We are considering,
now, an octahedral metal

420
00:30:10,000 --> 00:30:11,000
complex.

421
00:30:20,000 --> 00:30:24,000
This is the type of
Werner-esque complex that we

422
00:30:24,000 --> 00:30:29,000
talked about last time.
You are going to see that we

423
00:30:29,000 --> 00:30:35,000
are going to have ligands.
We are not really specifying

424
00:30:35,000 --> 00:30:38,000
them.
We are numbering them and

425
00:30:38,000 --> 00:30:43,000
locating them at positions one,
two, three, four,

426
00:30:43,000 --> 00:30:50,000
five, and six relative to our
metal center at the middle.

427
00:30:50,000 --> 00:30:54,000
And what you might begin to
realize is that in order to find

428
00:30:54,000 --> 00:30:58,000
out what our energy-level
diagram will be that refers only

429
00:30:58,000 --> 00:31:02,000
to the five d orbitals on the
cobalt center,

430
00:31:02,000 --> 00:31:06,000
or whatever metal center is at
the middle of this ion,

431
00:31:06,000 --> 00:31:10,000
what we are going to have to do
is evaluate the square of the

432
00:31:10,000 --> 00:31:15,000
wave function at each of these
ligand positions.

433
00:31:15,000 --> 00:31:19,000
And we are assuming that all
the ligands are equivalent.

434
00:31:19,000 --> 00:31:25,000
If you think of the ligand as
an electron or as a point charge

435
00:31:25,000 --> 00:31:30,000
in space, then you can imagine
that if we have an electron in d

436
00:31:30,000 --> 00:31:35,000
z squared up here,
that it is going to interact

437
00:31:35,000 --> 00:31:40,000
strongly and very repulsively
with a point charge that would

438
00:31:40,000 --> 00:31:45,000
be located at position one.
Whereas, if we instead had a

439
00:31:45,000 --> 00:31:49,000
point charge located at that
theta angle of 54 point whatever

440
00:31:49,000 --> 00:31:53,000
over there that we solved for,
since that is on the node of d

441
00:31:53,000 --> 00:31:56,000
z squared,
that would be the least

442
00:31:56,000 --> 00:32:00,000
possible repulsive interaction
that you could get between an

443
00:32:00,000 --> 00:32:04,000
electron and an electron in d z
squared because it is

444
00:32:04,000 --> 00:32:07,000
on the node.
And so what we would like to do

445
00:32:07,000 --> 00:32:10,000
is to go ahead and solve for
some of these things.

446
00:32:10,000 --> 00:32:13,000
Essentially,
what we seek --

447
00:32:25,000 --> 00:32:28,000
-- is an energy-level diagram.

448
00:32:33,000 --> 00:32:35,000
And so let's write this,
up here.

449
00:32:35,000 --> 00:32:39,000
This is five over 16pi.
This is d z squared

450
00:32:39,000 --> 00:32:43,000
that I am writing up.
Three cosine squared theta

451
00:32:43,000 --> 00:32:45,000
minus one.

452
00:32:45,000 --> 00:32:50,000
And because we are talking
about the probability density of

453
00:32:50,000 --> 00:32:54,000
finding an electron at a
particular point in space,

454
00:32:54,000 --> 00:32:58,000
which will mean a particular
theta value in our case for d z

455
00:32:58,000 --> 00:33:02,000
squared,
we are talking about that wave

456
00:33:02,000 --> 00:33:07,000
function squared.
What we need to do is evaluate

457
00:33:07,000 --> 00:33:10,000
it.
We have already evaluated it

458
00:33:10,000 --> 00:33:14,000
where the node is,
but we would like to evaluate

459
00:33:14,000 --> 00:33:18,000
it at position one.
Because we have a ligand at

460
00:33:18,000 --> 00:33:21,000
position one,
and we need to know what the

461
00:33:21,000 --> 00:33:26,000
relative response will be of d z
squared to a ligand

462
00:33:26,000 --> 00:33:32,000
along position one versus the
other five positions.

463
00:33:32,000 --> 00:33:35,000
We can say something about that
by symmetry, already.

464
00:33:35,000 --> 00:33:38,000
And so we are going to find out
that the value that you get for

465
00:33:38,000 --> 00:33:42,000
evaluating d z squared
at position one is

466
00:33:42,000 --> 00:33:45,000
the same as you would get for
evaluating it at position six.

467
00:33:45,000 --> 00:33:49,000
And that is because the big
lobes of d z squared

468
00:33:49,000 --> 00:33:52,000
are along plus and minus z.
That is where ligands one and

469
00:33:52,000 --> 00:33:54,000
six are.
And then, because the torus is

470
00:33:54,000 --> 00:33:58,000
also cylindrically symmetric,
two, three, four and five have

471
00:33:58,000 --> 00:34:02,000
the same value when you evaluate
d z squared at those

472
00:34:02,000 --> 00:34:06,000
positions.
But that value is smaller than

473
00:34:06,000 --> 00:34:10,000
along z, as illustrated by this
graph over here.

474
00:34:10,000 --> 00:34:14,000
But we just want to know,
how much smaller.

475
00:34:30,000 --> 00:34:33,000
And so this is position one.
We get a value of

476
00:34:33,000 --> 00:34:36,000
five-quarters.
And down here,

477
00:34:36,000 --> 00:34:39,000
position two,
a value of five-sixteenths.

478
00:34:39,000 --> 00:34:42,000
I am leaving off a factor of
pi.

479
00:34:42,000 --> 00:34:45,000
Please don't be concerned by
that.

480
00:34:45,000 --> 00:34:50,000
But these are the relative
values that you get when you

481
00:34:50,000 --> 00:34:54,000
evaluate this function at ligand
positions one and two,

482
00:34:54,000 --> 00:34:59,000
which is all we need to do
because of the symmetry of this

483
00:34:59,000 --> 00:35:05,000
because that value for position
one is also true for position

484
00:35:05,000 --> 00:35:09,000
six.
So we have positions one and

485
00:35:09,000 --> 00:35:12,000
six.
And then, this is also three,

486
00:35:12,000 --> 00:35:17,000
four, and five.
By doing two quick evaluations

487
00:35:17,000 --> 00:35:21,000
at two different theta
positions, position one,

488
00:35:21,000 --> 00:35:25,000
of course, theta is equal to
zero, up here.

489
00:35:25,000 --> 00:35:32,000
And we evaluate that squared
function for theta equals zero.

490
00:35:32,000 --> 00:35:36,000
And we get five-fourths pi,
but I am leaving off the pi.

491
00:35:36,000 --> 00:35:38,000
And down here,
at position two,

492
00:35:38,000 --> 00:35:42,000
we evaluate this for theta
equals pi over two because two,

493
00:35:42,000 --> 00:35:45,000
three, four,
and five are all in the

494
00:35:45,000 --> 00:35:49,000
x,y-plane at 90 degrees to z.
So the value of theta anywhere

495
00:35:49,000 --> 00:35:53,000
for those four ligands would be
pi over two.

496
00:35:53,000 --> 00:35:57,000
You evaluate this function for
pi over two, and you will get

497
00:35:57,000 --> 00:36:04,000
five-sixteenths pi.
And I have just left off the

498
00:36:04,000 --> 00:36:06,000
pi.
That is useful.

499
00:36:06,000 --> 00:36:12,000
Let's take this one up to the
top.

500
00:36:23,000 --> 00:36:27,000
Here we have done one of the d
orbitals.

501
00:36:27,000 --> 00:36:35,000
This is d z squared
for all six ligand positions.

502
00:36:40,000 --> 00:36:49,000
Now, let's do d x squared minus
y squared for

503
00:36:49,000 --> 00:36:54,000
all six ligand positions.

504
00:36:58,000 --> 00:37:00,000
First of all,
d x squared minus y squared

505
00:37:00,000 --> 00:37:05,000
has a pretty
interesting relationship with

506
00:37:05,000 --> 00:37:09,000
ligands one and six.
What is that relationship?

507
00:37:17,000 --> 00:37:19,000
Zero.
Because x squared minus y

508
00:37:19,000 --> 00:37:23,000
squared has two
nodal surfaces that intersect

509
00:37:23,000 --> 00:37:26,000
along the z-axis.
And so those ligands,

510
00:37:26,000 --> 00:37:30,000
four and six,
lie on a nodal surface.

511
00:37:30,000 --> 00:37:35,000
And so, we know that that is
going to be equal to zero.

512
00:37:35,000 --> 00:37:38,000
If, however,
we go ahead and evaluate the

513
00:37:38,000 --> 00:37:44,000
square of d x squared minus y
squared, let me just write it

514
00:37:44,000 --> 00:37:46,000
up.
And I have to switch it,

515
00:37:46,000 --> 00:37:49,000
since they are wrong in the
book.

516
00:37:49,000 --> 00:37:55,000
15 over 16pi square root sine
squared theta cosine two phi.

517
00:38:01,000 --> 00:38:03,000
We have that.
Now, just as we do,

518
00:38:03,000 --> 00:38:07,000
we are converting an orbital
angular property into a

519
00:38:07,000 --> 00:38:09,000
probability density by squaring
it.

520
00:38:09,000 --> 00:38:13,000
This is what we normally do.
When you see pictures of

521
00:38:13,000 --> 00:38:18,000
orbitals, they are representing
the square of the wave function

522
00:38:18,000 --> 00:38:21,000
in space.
We need to evaluate this as a

523
00:38:21,000 --> 00:38:26,000
function of theta and phi in
order to find out what --

524
00:38:26,000 --> 00:38:28,000
The first two,
one and six,

525
00:38:28,000 --> 00:38:31,000
we did by inspection,
but what about positions two,

526
00:38:31,000 --> 00:38:33,000
three, four,
and five?

527
00:38:33,000 --> 00:38:37,000
In the case of those four,
you can see that where they lie

528
00:38:37,000 --> 00:38:42,000
in the x,y-plane with respect to
the x squared minus y squared

529
00:38:42,000 --> 00:38:46,000
orbital is all
identical to each other by

530
00:38:46,000 --> 00:38:49,000
symmetry.
Because x squared minus y

531
00:38:49,000 --> 00:38:53,000
squared has lobes that extend
along x and plus x and y and

532
00:38:53,000 --> 00:38:56,000
minus y.
And that is where all these

533
00:38:56,000 --> 00:38:59,000
ligands lie, at positions two,
three, four,

534
00:38:59,000 --> 00:39:06,000
and five.
We know that theta is equal to

535
00:39:06,000 --> 00:39:11,000
what?
It is going to be pi over two.

536
00:39:11,000 --> 00:39:17,000
And phi, of course,
can be zero pi over two pi,

537
00:39:17,000 --> 00:39:24,000
and three pi over two for any
of those positions.

538
00:39:24,000 --> 00:39:32,000
And what we will find is that
this evaluates --

539
00:39:32,000 --> 00:39:37,000
For any of those,
let's just use phi equals zero.

540
00:39:37,000 --> 00:39:43,000
This evaluates as 15 over 16pi.
And I am dropping the pi.

541
00:39:43,000 --> 00:39:49,000
Now we have d z squared
and d x squared minus

542
00:39:49,000 --> 00:39:54,000
y squared
evaluated for all six ligand

543
00:39:54,000 --> 00:39:59,000
positions.
And then, let's make a table of

544
00:39:59,000 --> 00:40:01,000
this.

545
00:40:23,000 --> 00:40:27,000
z squared.
x squared minus y squared.

546
00:40:27,000 --> 00:40:29,000
Let's do xz,

547
00:40:29,000 --> 00:40:32,000
yz, and xy.
And here are our ligand

548
00:40:32,000 --> 00:40:35,000
positions.
And in the table here,

549
00:40:35,000 --> 00:40:41,000
we are going to put what these
relative evaluated squared wave

550
00:40:41,000 --> 00:40:45,000
functions are.
And the biggest one of all is

551
00:40:45,000 --> 00:40:51,000
this one here that we got at
position one for d z squared,

552
00:40:51,000 --> 00:40:56,000
which is
five-fourths.

553
00:40:56,000 --> 00:41:00,000
I am going to divide everybody
through by five-fourths,

554
00:41:00,000 --> 00:41:03,000
so that I can make our biggest
value equal to one for

555
00:41:03,000 --> 00:41:07,000
simplicity here.
And then, when we do that,

556
00:41:07,000 --> 00:41:11,000
you are going to see that two
gives a value of a quarter.

557
00:41:11,000 --> 00:41:14,000
That is position two,
down in the torus.

558
00:41:14,000 --> 00:41:18,000
Three, down in the torus,
is one-quarter relative to that

559
00:41:18,000 --> 00:41:20,000
one.
Four is one-quarter.

560
00:41:20,000 --> 00:41:23,000
Five is one-quarter.
And six, down in minus z,

561
00:41:23,000 --> 00:41:26,000
is one.
So we have evaluated the d z

562
00:41:26,000 --> 00:41:30,000
squared orbital
squared at the ligand positions

563
00:41:30,000 --> 00:41:35,000
one through six.
And these are the relative

564
00:41:35,000 --> 00:41:40,000
values that we got for the
probability of finding an

565
00:41:40,000 --> 00:41:46,000
electron at that point in space,
given a constant value of r.

566
00:41:46,000 --> 00:41:51,000
And x squared minus y squared,

567
00:41:51,000 --> 00:41:56,000
we got for positions one and
six zero, we just said that.

568
00:41:56,000 --> 00:42:02,000
And then, on the same scale
here, we get three-quarter,

569
00:42:02,000 --> 00:42:04,000
three-quarter,
three-quarter,

570
00:42:04,000 --> 00:42:09,000
and three-quarter.
And then for xz,

571
00:42:09,000 --> 00:42:15,000
yz, and xy, we have gone
through these two steps to do z

572
00:42:15,000 --> 00:42:21,000
squared and x squared
minus y squared explicitly.

573
00:42:21,000 --> 00:42:25,000
Now, we have to look at where

574
00:42:25,000 --> 00:42:31,000
ligands one through six are
relative to the nodes of xz or

575
00:42:31,000 --> 00:42:37,000
yz and xy.
And what we will find is that,

576
00:42:37,000 --> 00:42:44,000
in each case,
these ligands lie on nodal

577
00:42:44,000 --> 00:42:48,000
surfaces of xz,
yz, and xy.

578
00:42:48,000 --> 00:42:53,000
This is zero,
zero, zero, zero,

579
00:42:53,000 --> 00:42:57,000
zero, zero, zero,
and so on.

580
00:42:57,000 --> 00:43:01,000
Okay?
All the ligands,

581
00:43:01,000 --> 00:43:04,000
one through six,
lie on nodal planes of xz,

582
00:43:04,000 --> 00:43:06,000
yz, and xy.
Only four of the ligands

583
00:43:06,000 --> 00:43:10,000
interact with x squared minus y
squared,

584
00:43:10,000 --> 00:43:14,000
ligands two through five,
because those are the ones that

585
00:43:14,000 --> 00:43:17,000
lie in the x,y-plane.
And they interact strongly,

586
00:43:17,000 --> 00:43:20,000
this relative value of
three-quarters,

587
00:43:20,000 --> 00:43:25,000
but not as strongly as the two
ligands in positions one and six

588
00:43:25,000 --> 00:43:30,000
interact with the big lobes of d
z squared.

589
00:43:30,000 --> 00:43:32,000
And then, also,
ligands two through five,

590
00:43:32,000 --> 00:43:36,000
which lie in the x,y-plane,
interact with the torus of d z

591
00:43:36,000 --> 00:43:40,000
squared,
but to a much smaller extent

592
00:43:40,000 --> 00:43:44,000
because the torus does not have
as great a radial extent.

593
00:43:44,000 --> 00:43:48,000
And these ligands are all at
the same radius out from the

594
00:43:48,000 --> 00:43:51,000
metal center.
And so what this corresponds to

595
00:43:51,000 --> 00:43:54,000
is now an energy-level diagram
as follows.

596
00:43:54,000 --> 00:43:58,000
And this is for an octahedral
complex, --

597
00:44:03,000 --> 00:44:07,000
-- where we have relative
energy units of zero,

598
00:44:07,000 --> 00:44:11,000
one, two, and three.
And what we have to do is say

599
00:44:11,000 --> 00:44:17,000
that the amount an electron in d
z squared would be

600
00:44:17,000 --> 00:44:22,000
repelled simultaneously by
electrons in positions one

601
00:44:22,000 --> 00:44:27,000
through six would be the sum of
these values.

602
00:44:27,000 --> 00:44:31,000
And so we add that up and get,
in fact for d z squared,

603
00:44:31,000 --> 00:44:34,000
up here,
a three.

604
00:44:34,000 --> 00:44:38,000
And, interestingly,
for d x squared minus y

605
00:44:38,000 --> 00:44:42,000
squared,
if we take a sum of the four

606
00:44:42,000 --> 00:44:46,000
interactions that we found that
are non-zero,

607
00:44:46,000 --> 00:44:50,000
we also get a three for d x
squared minus y squared.

608
00:44:50,000 --> 00:44:55,000
So then the net of all their
interactions is the same for x

609
00:44:55,000 --> 00:45:00,000
squared minus y squared and z
squared.

610
00:45:00,000 --> 00:45:03,000
And then down here,
we found that these three

611
00:45:03,000 --> 00:45:07,000
orbitals, xz,
yz, and xy, where the ligands

612
00:45:07,000 --> 00:45:11,000
at positions one through six lie
on their nodal surfaces,

613
00:45:11,000 --> 00:45:14,000
lie right on their nodal
surfaces.

614
00:45:14,000 --> 00:45:17,000
And so the number evaluates to
zero.

615
00:45:17,000 --> 00:45:22,000
And so a wave function squared
will evaluate to zero any time

616
00:45:22,000 --> 00:45:26,000
you are looking at a position
that is on one of its nodal

617
00:45:26,000 --> 00:45:29,000
surfaces.
It evaluates to zero.

618
00:45:29,000 --> 00:45:34,000
And so we have d(xz),
d(yz), and d(xy).

619
00:45:34,000 --> 00:45:39,000
And then, a diagram like this,
which is a d-orbital splitting

620
00:45:39,000 --> 00:45:40,000
diagram --

621
00:45:53,000 --> 00:45:56,000
-- is associated with a couple
of different parameters.

622
00:45:56,000 --> 00:46:00,000
One is, down here we have a
triply degenerate energy level.

623
00:46:00,000 --> 00:46:04,000
Previously, we had only seen
doubly degenerate energy levels.

624
00:46:04,000 --> 00:46:08,000
Now we have a triply degenerate
on composed of xz,

625
00:46:08,000 --> 00:46:11,000
yz, and xy.
And that level we are going to

626
00:46:11,000 --> 00:46:14,000
be calling t(2g).
And then, up here,

627
00:46:14,000 --> 00:46:19,000
we have a doubly degenerate
level that will get the label

628
00:46:19,000 --> 00:46:22,000
e(g).
And then, whatever the value of

629
00:46:22,000 --> 00:46:25,000
the splitting of these two
energy levels,

630
00:46:25,000 --> 00:46:29,000
one triply and one doubly
degenerate, we are going to give

631
00:46:29,000 --> 00:46:34,000
that the label delta O for
octahedral.

632
00:46:34,000 --> 00:46:38,000
And at the beginning of next
hour, I will say more about

633
00:46:38,000 --> 00:46:42,000
tables like this.
And I will show you how you can

634
00:46:42,000 --> 00:46:46,000
actually figure out d-orbital
splitting diagrams for other

635
00:46:46,000 --> 00:46:51,000
coordination geometries and how
they compare for a couple of the

636
00:46:51,000 --> 00:46:54,000
most popular coordination
geometries.