 We are almost done with our discussion of orbitals. We have seen these beautiful pretty pictures of orbitals that come up when you try to make 3D plots or contour plots of them. We have discussed s orbitals p orbitals. This is how far we have got so far. Now let us conclude this part of the discussion by talking about d orbitals. Where does pz orbital get its name from? Why is it z? Well spdf come from some ancient spectroscopic nomenclature. But why is it z? Because it is along z orbital sorry it is along z axis. Same symmetry as z. Or another way of thinking is that if I go back where is the angular node? The angular node is at cos theta equal to 0 that is z is equal to 0. So the left hand side of the equation of the node is a subscript that it comes and that is how d orbitals also get their names. Now before talking about d orbitals let me remind you something that we said in the previous module. Only one d orbital comes out to be real when we solve Schrodinger equation in the way that we have discussed and it is not 3D x square minus y square orbital. It is 3D z square orbital. For 3D z square orbital this m is equal to 0 so it is a real orbital. These others that we are talking about now are actually generated by appropriate linear combination of m equal to plus minus 1 orbitals and m equal to plus minus 2 orbitals. I leave it to you to figure out which ones are generated by taking linear combination of m equal to plus 1 and minus 1 orbitals which ones are generated from m equal to plus and minus 2 orbitals. With that background let us come back and see what a plot of 3D x square minus y square would look like. This is what we have we have a constant multiplied by r square multiplied by e to the power minus zr by 3a multiplied by sin square theta cos 2 phi. This is the angular part. Let us worry about the radial part to start with. The radial part again goes through a maximum. When r equal to 0 it is 0 r equal to infinity it is again 0 and in the middle r square keeps increasing and e to the power minus zr which has a maximum value at r equal to 0 keeps decreasing. So the product of course would go through a maximum that is simple. Let us have a look at the angular part of 3D x square minus y square that will lead us to this label x square minus y square in the first place. So what we have there is sin square theta multiplied by cos 2 phi. So what is cos 2 phi? Cos 2 phi is let me write now cos 2 phi is essentially cos square phi minus sin square phi. This is multiplied by sin square theta. So I can write like this sin square theta here as well. So what do we have here? Do you remember what sin theta cos phi is? r sin theta cos phi is essentially x. So sin square theta cos square phi is x square divided by r square is not it? Minus we have sin square theta sin square phi. Sin theta sin phi is y is it y? Actually r sin theta sin phi is equal to y. So sin theta sin phi is equal to y by r and sin square theta sin square phi is equal to y square by r square. So we get y square divided by r square. So you can take r square common and you get this x square minus y square divided by r square. So that is what gives a name and then tell me now what is the node? How do I get the angular node? I get the angular node by equating x square minus y square by r square to 0. So what I get essentially is angular node is x equal to plus minus y. So as discussed earlier what we do is we draw the nodes first x equal to plus minus y and we draw a loop we cross a node wave function changes sin cross the node again it changes sin once again cross the node once again once again you see there is a change in sin. So this is 3D x square minus y square for you. Now we come to 3D xy sin square theta sin 2 phi again using the relationship between sin 2 phi and sin phi cos phi what is sin 2 phi? Sin 2 phi is essentially 2 sin phi cos phi. So sin square theta multiplied by sin phi cos phi becomes sin theta sin phi multiplied by sin theta cos phi. So you get x y and of course it has to be divided by r multiplied by r so you get x y by r square. So x y equal to 0 x y equal to 0 means x equal to 0 is a node and y equal to 0 is a node. So once again the moment you cross wherever you cross the node wave function has to change sin. So this is 3D xy. Now we come to 3D z square the most interesting d orbital as far as I am concerned. Here the angular part is 3 cos square theta minus 1 and I believe we have worked out the polar plot of 3 cos square theta minus 1 some time ago. There I told you that when this is equated to 0 you get magic angle which has applications in many different fields. So 3 cos square theta minus 1 what will it be is going to be 3 z square by r square minus 1. So you get 2 angular nodes where will the angular nodes be here it is easier for me to think in terms of theta. Angular nodes will be at theta equal to 54.7 degrees and 180 minus 54.7 degrees we had worked this out earlier. So these are the 2 angular nodes remember these are the 2 angular nodes theta equal to 54.7 degrees and theta equal to 54 well 180 minus 54.7 degrees please do not think that this is one angular node straight line this is another angular node straight line they are not they are actually 2 funnel shape funnel shaped angular nodes. And now since theta is 54.7 degrees this angular space available is smaller that is why this slope is larger this slope is smaller and every time you cross a node sign has to change. So this is 3 z d z square let me show you the 3d plot along with contours I hope you can relate this with the one that we showed earlier these are the 2 nodes once again this is one angular node remember this is z axis so this is theta this way as well as that way this is another value of theta. So this is one node this is another node here the way I have drawn it the bigger lobes have plus sign of psi the smaller lobes have minus sign of psi. So this is your 3d z square please have a look at this diagram and look at the good thing here is that the 3d plots have contours shown in them I hope it is not very difficult for you to correlate this with this one. And now I think you understand the shape of d orbitals and also why d orbitals are called why d orbitals have got the subscripts that they have got. So once again you can work out the surface of constant probability these are the angular nodes shown with the surface and this is how you can show probability distribution as dots. To conclude this discussion let me say that 3d square 3d z square is actually a nickname why because the way we are written it here we have not really converted completely to Cartesian coordinates. So z square by r square is there so if we want to convert completely to Cartesian coordinates we should remember that r square is x square plus y square plus z square. So when you substitute that and equate to 0 we get 2 z square minus x square minus y square equal to 0. So the full level of 3d z square is really 3d 2 z square minus x square minus y square. This comes handy when you try to discuss things like d orbital splitting in an octahedral field or whatever field it is in using symmetry using what are called character tables. So it is important to understand what these names mean. We are not going to discuss f orbitals we will stop at d but let me for the sake of completeness show you some constant probability surface for f orbitals along with nodal planes as well. That brings us to an end of this part of the discussion we have something more to say about orbitals and then we will see how they are used at least as a first approximation in multi-electron systems and that is where we can answer that question how is it that we can use 1s 2s 2p in selithium or beryllium or other multi-electron atoms even though we have emphasized so much that orbitals are one electron functions.