 Now, we are ready to plot some good looking pictures of hydrogen atom wave functions that is orbitals. We are used to drawing s orbitals like spheres, but what we will see today is that actually s orbitals look like this. How? Let us see. This is what we have done so far. We have written down the hydrogen atom Schrodinger equation in spherical polar coordinates. And we have been able to separate the equation into 3 different parts. The radial equation, the theta equation and the phi equation. The solution of phi equation gave us a actually 1 by root 2 pi e to the power i m phi from where we got the magnetic quantum number 0 plus minus 1 plus minus 2 plus minus 3 so on and so forth. And from the theta dependent part we got the secondary quantum number 0, 1, 2, 3 so on and so forth. Also we have seen why it is that magnitude of m has to be lesser than or equal to L. We have learnt how that comes. That comes from the requirement of z component of angular momentum never being greater than the total angular momentum. Now, we have not solved the r and theta dependent parts, but we have told you what the wave functions look like. The theta dependent wave function is essentially a polynomial in cos theta. This is called a Legendre polynomial. And the r dependent part essentially is a constant multiplied by r to the power l e to the power minus zr by n0 multiplied by a Lagrangian function which is a polynomial. And when we look at the solutions of radial part, we have not done it in detail. We have just told you the final results. We get the principal quantum number n equal to 1, 2, 3 and so on so forth. We also get to learn that l less than n. Now, what is the information about the molecule that we about the atom that we get from these 3 different parts? From the radial part, we get to know the energy. And the expression for energy is the same as what you get from Bohr theory minus 13.6 electron volt by n square. From the angular part, the theta part we get to know the total angular momentum which is h cross multiplied by root over l into l plus 1. And from the phi dependent part we get the z component of angular momentum m h cross. And once again that is why mod m has to be less than or equal to l. Now, what we often do is that we take the angular part together and we write this kind of an expression. This is called spherical harmonics and that turns out to be your the theta part multiplied by the phi part. So, what we will do is we will look at radial and angular parts separately. We have already shown you the radial functions of hydrogen atom and we have reminded you that number of radial nodes is n minus l minus 1. We also remind ourselves that we have to talk about probability and not probability density. So, r square capital R square small r square dr essentially is radial probability distribution function for s orbital this becomes 4 pi r square it is not very difficult to understand I hope. So, taking into account volume element there is some radius non-zero radius where radial distribution function undergoes a maximum. And we have briefly said that you can figure out the most probable value of radius and average value of radius and they are usually not the same. Now, let us try plotting the orbitals in this kind of 3D pictures. So, in these 3 dimensions what is it y axis is the orbital the other 2 axis can be x and y, y and z x and z whatever we require. So, see actually this is a 4 dimensional picture psi is the 4th dimension the special dimensions are r theta phi or x y z whatever you want. But psi is a 4th dimension how will I draw 4 dimensional picture I cannot. So, I can only draw 3 dimensional sections and then we make contour plots of them. So, let us take the simplest case scenario 1 s orbital where the radial part is the only part that is there in the wave function exponential decay as we said. Now, when I plot it against r remember if this is x and this is y what is r r square is equal to x square plus y square is not it r is essentially equal to square root of x square plus y square just like that. So, for any value of x and y you have an value of r. So, this exponential decay that we draw we can keep on changing x and y and what we will generate is this kind of a conical shape. So, what you see here is sort of a 3 dimensional picture and you can see these lines here these lines join all points with the same value of psi you might remember that we had encountered this when talking about 2D and 3D box. Now, if I look down from the top what will I see I will only see these lines. So, this is a projection of this 3 dimensional object in 2 dimensional plane. So, these lines essentially join the all the points having same psi these are called contour diagrams. For 1 s orbital of course, the contours are all circular and one more thing to notice see here the spacing between contours is large here the spacing between contours is small why because the slope is more initially it is not a straight line right slope is more initially and gradually it falls off what is the meaning of slope if I draw like this this I called horizontal equivalent and here now I am using language that is used in your survey maps and this is called vertical interval but V i means vertical interval H e is horizontal equivalent. So, slope is V i divided by H e what will happen if slope is more then for same H e V i will be more something like this right or I can draw like this I will take the same vertical interval like this. So, you see when slope is more then these 2 points are close when slope is less these 2 points are far apart. So, wherever slope is less contour lines are far apart wherever slope is more contour lines are close together. This is your 1 s 3d picture as well as contour diagram what about 2 s 2 s has remember r to the power l multiplied by that Lagrange function l here is 0 so r to the power 0 so r to the power 0 is essentially 1 no problem with that but the Laguier polynomial not Lagrange function Laguier function is 2 minus z r by a where a is both radius where does the radial node occur then r equal to 2 a by z that is why the function falls off becomes 0 at r equal to 2 a by z changes sign so this is a nodal point remember node is a point where a wave function goes to 0 and changes sign if it does not change sign then it is not a node then again it increases and becomes 0 asymptotically. This is something that I plotted myself so do not take this number seriously these numbers are just relative and I encourage you to plot yourself. Alright so this is your 2 s orbital how does it look if I try to make a similar 3 dimensional picture all I have to do is I have to turn it around by 360 degrees with respect to this psi axis this is what I get. Now, see this is the diagram that you get for 2 s orbital initially very high value it falls crosses 0 and becomes negative do you see the base in here this is the negative base in and then it is slowly recovers and becomes 0 at infinite value of small r this these are the contour diagrams I have just taken this and turned it around I have used graphite to plot this in a MacBook so it is very easy to do these things there you can use whatever graph plotting software that you want turned around these are the contours and once again see these lines are far apart these lines keep on getting closer and closer and closer neglect this arrow head this is just an artifact from the program I will show you another view this is the top view well this is the side view this is the top view this is the bottom view if you look from the bottom you see this hole where does this hole come from well wave function has started from a certain value so at r equal to 0 value is very positive but the minimum value is actually negative that is what is determined by this rim this is the contour line where you have negative and do you see the radial node in the contour diagram this whitish circle that you see that is your radial node so this is how orbitals are usually depicted orbital remember is a one electron wave function and of course if I ask you to just draw it on a plane paper this is how you can draw it how do you show sign here either write explicitly or use different colors for different signs all right this is another way in which orbitals are often depicted lots of dots with different color color denotes sign and density of dots denotes probability so the way this is done is that you plot more dots where the probability density is more and when you look at the entire picture you get the probability distribution another way of drawing it this pac-mine kind of figure I do not know what these diagrams are called these diagrams are all drawn by my senior colleague Professor Vayu Shashidhar you see his name here so this is another way of drawing it you cut a section of the orbital you can show there is one sign outside one sign inside very nice depictions 3s orbital as we said has a polynomial of second order so naturally two roots so in for two values of r it becomes 0 and remember these are legwear functions and property of legwear functions dictates that the roots are both real okay now two nodes so what will it look like I do not have the 3d picture here but this scatter plot looks like this you can see there are three different regions you can try to make the 3d plot yourself now another thing that I want to stress even though we have said earlier see this here is the 3s orbital that we have plotted the outer small slope is the smallest when I multiply by small r r square by r square capital r square is your probability density actually you have to multiply it by 4 pi now see what has happened since you have multiplied by r square the outermost lobe which was the smallest has actually become the largest so where is the probability of finding 3s orbital 3s electron more outside in this major lobe okay so these are probability distribution function plots now let us talk about p orbitals and d orbitals so this diagram that you see is actually of 3p orbital as we are going to arrive at slowly but first let us talk about 2p orbitals we will start with the simplest one 2p z orbital here is the radial part radial part c now this time we have r by a multiplied by e to the power minus z r by 2 a r is an increasing function e to the power minus z r by 2 a is a decreasing function multiply them together you get a maximum and where does this maximum occur you can differentiate equal to 0 equal to 0 you can find out where the maximum radial maximum of the radial part of the wave function occurs remember the position of maximum of the radial part of the wave function will not be the same as the position of the maximum of r square multiplied by capital R square I encourage you to work out both and see for yourself whether it is same or whether they are different okay this is the radial part what about the angular part angular part has cos theta now cos theta remember is z by r so I instead of cos theta I can just write z and I can write z by r so r this r and that r will cancel will be left with e to the power minus z r by 2 a multiplied by z right interesting right that is why it is called a 2p z orbital so now see if I I know how to get radial nodes already equate the radial part to 0 if I equate the angular part to 0 cos theta equal to 0 what is that cos theta equal to 0 is z equal to 0 that is the x z x y plane so x y plane turns out to be an angular node of the 2p z orbital okay so that is why the 3d picture turns out to be like this you start from 0 you get a positive going function which then again decays to 0 negative going function that again decays to 0 but why is it positive why is it negative because see this y z plane must be sorry z x what am I saying x y plane must be a node I am showing you the x z plane here so in this projection remember here the third axis is wave function so if third axis is wave function where are you going to get this node here it will be z equal to 0 this line okay so this is the meaning so these are the 2 lobes now if you look down from the top what will you see the contour diagram will look like this remember the 2 lobes of p orbital these are your 2 lobes of 2p orbital when we go to 3p orbital we see the situation becomes even more interesting so this is what it is 1 plus low 1 minus low what is plus what is minus electrons do not become positively charged when they go to the plus low sine of the wave function is positive sine of the wave function is negative in the negative lobe okay so plus and minus on the lobes denote the sine of the wave function and these lobes arise out of angular part they are different signs because the angular plane the angular node is essentially the y x y plane the angular node essentially is x y plane okay and these are the constant probability surface how do you plot them you decide what psi psi star you want for join all the points having that same psi psi star now this is 3d space you join all the so for some x y z value I know that psi psi star is 0.002 let us say I join all those points and then I get this kind of a shape now I know where psi is plus where psi is minus so I can use different color or right plus or minus then what I do is I work out the volume inside this the volume inside that will be probability of finding the electron within that constant probability surface then that is how you generate these pictures of probability distribution and generally people confuse that with orbitals but hopefully after today we will never confuse we will remember that orbitals are wave function and these shapes of probability distribution are generated using the functional forms of the orbitals but not neglecting the spherical not detecting the volume element as well great. Now we come to an interesting situation 2 p z is something we could plot very easily what about this m equal to plus 1 m equal to minus 1 see here phi part was 1 because m equal to 0 phi part is 1 so it is a real orbital however for m equal to plus 1 and minus 1 we have orbitals that are imaginary and we cannot draw them in real space you can actually do whatever you want to do with them but you cannot plot them and in chemistry we like to plot things it is easier to understand so what we do is we remember a theorem what is the theorem that we remember a theorem that the quantum mechanical operators are linear so if quantum mechanical operators are linear then if I take a linear combination of wave functions then what happens I have poor memory so I do not remember whether I work this out earlier in any case we will do it once but I will write and I will delete also so see it takes some any operator a hat let us say it operates on c 1 phi 1 plus c 2 phi 2 where phi 1 and phi 2 are wave functions c 1 c 2 are coefficients and let us say also I think we did it that a hat operates on phi 1 to give me a 1 phi 1 a 2 a hat operates on phi 2 to give me a 2 phi 2 so now what is this since the linear operator I can write it as c 1 multiplied by a hat phi 1 plus c 2 multiplied by a hat phi 2 what is a 1 phi 1 we know what is a 2 phi 2 we know as well so I will write c 1 a 1 phi 1 plus c 2 a 2 phi 2 is this an eigenvalue equation in the general case no in a special case where a 1 is equal to a 2 that is same eigenvalues let us say both are equal to a then I can take it out right I can write a multiplied by c 1 phi 1 plus c 2 phi 2 okay now see look at this I will call them p plus p minus orbital they are eigenfunctions of Hamiltonian operator and they are eigenfunctions with the same eigenvalue right remember energy depends only on n only on the radial part so there is no problem I can take linear combination and whatever linear combination I take will have the same energy as these orbitals so I take two linear combinations first I add them what happens when I add e to the power i phi and e to the power minus i phi remember e to the power i phi is cos phi plus i sin phi and e to the power minus i phi is cos phi minus sin i sin phi okay so when we add that this is what happens you are left with cos phi this i sin phi terms cancel each other so I got sin theta cos phi what happens what is that that is actually psi 2 p x why because remember r sin theta cos phi is x so this 2 p x orbital now behaves like your p z orbital the only difference is for 2 p x orbital the angular node is the y z plane okay nice what happens if we take a minus combination the only difference here is that now the cos terms will vanish cos phi terms and sin phi terms will be there they have i in their coefficient so you have to divide by i also this is root 2 multiplied by i this is not root over i please do not this is not very clear i multiplied by root that is what it is so then I get sin theta sin phi r sin theta sin phi is y remember so this is your familiar psi 2 p y orbitals so remember that for 2 p x and 2 p y for p x and p y orbitals m values are not defined we generate them by taking linear combinations of m equal to plus 1 and m equal to minus 1 orbitals so if m value is not defined what is not defined is the z component of angular momentum right so remember the particle in a box wave function it was a linear sum of a wave function that denoted the linear motion in plus x direction and another one that denoted linear motion in minus x direction it is sort of like that okay plus m h cross and minus m h cross they are combined so z component of angular momentum is indeterminate if you perform a measurement then you will see either z equal to plus 1 or z equal to minus 1 okay but p x and p y orbitals they are not eigenfunctions of the L z operator they are eigenfunctions of your energy Hamiltonian operator and also angular momentum operator L square okay let us quickly talk about 3 p z in 3 p z the complicating factor is a radial node you have 6 minus z r by a in the radial part so what happens if I equate that to 0 that gives me a radial node that was not there for 2 p orbitals so now I will just show you how to draw an orbital if I give you the function the first thing to do is to draw the nodes this radial node is going to be a circle in this section and angular node is going to be a line okay now what I do is I draw any one of the lobes and you call it either plus or minus does not matter so what it means is that if you cross the node you cross this node sign will change so this is plus we will get minus if you cross this node then also sign will change so if this is plus it will become minus and once again the same thing will happen when you cross this node so this was plus now this is minus this is going to be plus and this is going to be minus so this is the contour diagram of 3 p z orbital remember contour diagram of 2 p z and 3 p z orbitals have this difference because of the radial node okay let me show you the 3d picture nice and here you can see the contours as well so you see you have a big hill a big trough followed by a small trough and a small hill in fact to get this diagram is very difficult these are so small but multiply them by r square we will take square of this and multiply them by r square this is going to blow up okay then similarly you can plot this 3 p z orbitals let us talk about d orbitals 3d x square minus y square orbital this here is the wave function it has sin square theta cos to phi how do I write cos to phi in terms of sin phi and cos phi I hope that is not very difficult for us when we do that we will see that this sin square theta cos to phi becomes x square minus y square by r square and when you quit that to 0 you get the angular node x equal to plus minus y okay these are the angular nodes now we can draw the loops this is minus and this will be plus this will be minus this will be plus what about d x y for d x y the angular part of the wave function is sin square theta sin to phi that turns out to be x y by r square okay what is sin to phi 2 sin phi cos phi right so 1 sin phi gets multiplied by sin theta and the other sin theta gets multiplied by cos phi that is how you get x y okay angular nodes become x y equal to 0 that is x equal to 0 y equal to 0 that is how you get these loops similarly for 3d z square no one more thing see remember this 3d x y y z z x square minus y square these are actually obtained by taking linear combinations one set is obtained by taking linear combinations of m equal to plus 1 m equal to minus 1 orbitals and the other one is generated by taking linear combinations of m equal to plus 2 and m equal to minus 2 orbitals which gives you which I leave that for you to figure out last orbital that I want to talk about is your 3d z square my favorite orbital because the angular part is 3 cos square theta minus 1 in fact if you equate this to 0 you will get theta equal to cos inverse 1 by root 3 which comes out to be 54.7 degrees this is called magic angle and this quantity keeps on coming back to haunt us in many many different areas but we will not talk about that anymore what I want to say is that 54.7 degrees is not the only solution there is another solution and that solution is I will be lazy and I say 180 degrees minus 54.7 degrees so remember this is one node conical nodes here this is another node and since the angle is 54.7 degrees more than 45 degrees that is why one loop is bigger the other loop is smaller and since it is conical this one turns out to be when you just turn it around the smaller loop turns out to be a belt but this here is really the 3d picture so plus plus minus minus this is what 3d z square orbital is you can generate surface of constant probability and then you get this familiar picture. Now you can go on and draw the nodes here you are going to get 2 conical nodes similarly we are not going to talk about f orbitals but I just show you the constant probability surfaces of the f0 orbital and these are the nodal surfaces this so this is what we wanted to say about hydrogen atom wave functions that are orbitals now the question is this hydrogen has only one electron so why do we need s v d a f so many orbitals n equal to 1 2 3 because first of all we want to access excited states we want to talk about spectra we want to talk about many electron atoms we are going to see how these orbitals are used to work out wave functions for a simple molecule molecular ion if you call it that is h 2 plus they are called molecular orbitals.