 In this module and the next we are going to have a lot of fun. We have done plenty of mathematics which might have been taxing for some of us. But now what we are going to do is we have reached the stage where we can talk about the results of all that mathematics that we have done. Well as you know we have actually skipped a lot of tedious steps. But still now we are in a position to draw these beautiful pictures like the one that is shown here. But better still we get to understand what we are talking about. So we will discuss the acceptable hydrogen atom wave functions as we said earlier these are called orbitals. So once again let me repeat an orbital is an acceptable solution of Schrodinger equation of a hydrogenic atom a one electron system. That is what an orbital is nothing more, nothing less. Let us get this very very clear. Hearing this I would expect you to start asking questions right away. Because if these orbitals are one electron wave functions how is it that we talk about electron configuration of many electron atoms? How is it that we use them to generate molecular orbitals in molecules? They are not one electron systems. Well we will cross those bridges when we come to them. Right now let us just see what kind of pictures we can draw when we consider the functions that the orbitals are. So we already know what kind of functions they are. We have formulated the hydrogen atom problem in spherical polar coordinates and we have got three equations. We have got three equations one in phi, one in theta and the third in R. We have explicitly solved the phi dependent part and we have got a very simple solution 1 by root over 2 pi for whatever reason I have forgotten to write the normalization constant everywhere e to the power plus minus im phi. You could just say e to the power im phi because plus minus is incorporated in the values of M themselves. The theta dependent part we have given you the solution and it is worked out from commutativity of angular momentum component and total angular momentum square. We are actually going to perhaps record those lectures a little later but they might come before the lecture I am presenting now. So at the moment I do not know whether we have done the solution, whether you have seen the solution but if you have not it will come sometime later. We do plan to do a complete discussion of angular momentum but for now it is sufficient to know that the solution of the theta part is essentially some polynomials in cos theta and solution well they are characterized by the quantum numbers L, L ranges from 0, 1, 2, 3 so on and so forth and magnitude of M has to be less than L. When we solve the R part we get another series of polynomials the Laguier polynomials and then by applying the boundary conditions to these wave functions the R dependent wave functions we get another quantum number the third quantum number the principal quantum number N that ranges from 1, 2, 3 so on and so forth and also as has become a pattern by now from this part we get a limit to the value of L and L always has to be less than N again this result is something that we have known since perhaps class 11 or class 12. We have said this several times already but before going on to start drawing them I thought I will just present them once and also what you have learnt is that from these wave functions we can find out the energy from the R dependent part and you can find out total angular momentum from the theta dependent part you can find out the z component of angular momentum from the phi dependent part. These are the informations contained in each of these parts of the wave function. So here it is R R multiplied by capital theta multiplied by capital phi and you can also put theta and phi dependent parts together and you can write this as a spherical harmonics because after all in spherical polar coordinates you have one radius which is a length and you have two angles so theta and phi are coordinates of a kind. So usually they are put together and they are called spherical harmonics and you can write it as the radial part multiplied by the spherical harmonics. What we are going to do in the next couple of modules is that we are going to try to plot them one by one and we are going to try to see what they look like. Of course, naturally a question that will arise is that what happens when m is non-zero then you will have an imaginary function how do you plot an imaginary function we will see. We have already talked about these radial functions I will not repeat them once again number of radial nodes as we know is n minus l minus 1 and a very important concept that we have discussed is that of radial probability density. We have said that it is important to consider the volume of the volume element written in spherical polar coordinates if you want to talk about probability distribution. So from there we have realized that it is not enough to talk just about capital R square you must talk about capital R square multiplied by small r square if you are going to understand what kind of probability density is there across well along a radius they will be modulated by the theta and phi dependent parts we will come to them one by one. In this module we only want to talk about s orbitals which do not really have any contribution from the theta and phi dependent parts so it is enough if you consider this part to start with. When we talk about p orbitals then of course we cannot not consider the angular part as well. So this is the radial distribution function we will actually plot them one by one we have already shown you the result today we will plot them in front of you and see how we get this okay and as you said the information you get from radial part are average value of radius and most probable value of radius in addition of course the most important energy and when we go to the angular part we will talk about angular distribution function as well. Now the way we represent this is by using 3 dimensions 3 dimensional representation and since we cannot represent I mean unless we draw a model if you want to draw on paper the most convenient way of doing it is to use contours. Those who have studied geography to some extent or those who are interested in maps will have seen maps like this what you see here is the map of an island right. So here the island is shown in relief looking at the color coding you can see that this is the peak and then this is a ridge and as you go down well as you go in this direction or this direction or this direction radially out the height decreases and the small lines that are there these are the contour lines contour lines essentially join all the parts all the points that have the same height as far as an island is concerned okay. So maybe we will go to the next slide and show it because it is more of a sketch all this is from internet I have not drawn these so let us say this is the island you take a section of the island this is what it looks like this side is steep this slide has a more gentle slope. So what you do is first of all you look at all the points connect all the points that have the same height. So one way one analogy we can use is that suppose the sea level keeps rising and let us say the sea level rises to this side let us say this is 10 meter or something then if you look from the top from a helicopter what will you see you will see this out right now what you see is this zero line that is the outline of the island that is the 0th contour line suppose the sea rose 25 meters then you would see this line that is marked 25 is not it then if it rose to 50 meters you would see this line this would be the outline of the island okay and that is what we draw here okay so essentially what you do is you take a section and you drop perpendicular and then you join all the points that you get such these are the contour lines contour lines denote the same height we are going to demonstrate with wave functions shortly. Now see one thing that is very important to remember is that wherever the lines are close together on this side for example the slope is steeper wherever the lines are far apart the slope is gentler because the slope is essentially the vertical displacement between 2 contour lines divided by the horizontal distance. Now generally contour lines are drawn in such a way that the vertical displacement between successive lines is same so 0, 25, 50, 75 and so on and so forth. But then if the horizontal separation is more between 2 successive contour lines of course they will be far apart from each other if the horizontal separation is not much like here they will be close to each other this is how one reads contour maps of course one problem that we face once again is that in our case if you want to draw contour maps for wave functions wave functions do not always have plus sign when we talk about islands they only have plus heights there is no minus so it is a little simpler in case of wave functions we will have troughs as well so how do you designate troughs well you can write minus 25, minus 50 and so on and so forth or you can use contour lines of different color. Generally as we are going to show contour lines and color shading together gives us very beautiful pictures. So let us go ahead first picture that I show and I drop a bombshell on you right away is 1s orbital this is my depiction of 1s orbital but it is important to understand what I have drawn here to understand that let us have a look at the function first is the simplest possible function one can get some constant multiplied by an exponential decay in r e to the power minus r by a 0. So if you draw in 2 dimensions what will you get we will just get an exponential decay isn't it so why do not I just draw it let me draw the function and let us see what we get I have given you the preview of an even better looking function but we will come to that eventually. So let me just plot it I want to plot e to the power minus x that is all so I will plot x into e to the power minus x and I strongly encourage you to do this yourself it is a lot of fun I am using I have a MacBook so I am using grapher but you can use any graph plotting software of your choice and you can play around like I am doing right now. Here goes so exponential decay one thing to remember is that this function for you when you plot it is going to have values for plus x and minus x but in case of wave function remember x axis is really r so there is no minus r r if you remember goes from 0 to infinity. So we will just neglect since I do not know how to restrict my picture to your only positive values of x we will just neglect the negative one. So let me change this frame limits minus 0.1 to 10 is fine it is just that the maximum will definitely go beyond 1 so we will go to 1 so I will make it 1.1 please neglect the one that goes this side this is a simple exponential decay. Now this is a very drab one-dimensional plot is not it we want to make it a little more interesting so we can think like this that we have r and this r is independent of phi. So for all values of phi we can draw plot like this what is the picture you get take this exponential decay and turn it around by 360 degrees whatever you get is your 1s orbital but remember when we plot this 1s orbital 1 axis is r fine second axis what we are saying is phi third axis is psi itself or r itself in case of 1s orbital well in case of s orbitals r and psi are one and the same so but we are not considering theta theta does not have an effect here but these three dimensions are really not the three physical dimensions two are physical dimensions r and phi the third one is psi so how to go from there to drawing a picture of an orbital in real space we will come to that eventually but for now let me show this nice 3d picture also maybe instead of destroying that let me just open a new graph I want to plot a 3d plot I want to make a 3d plot and here z we have to write is sorry e to the power minus now I want r right and here it is written as x and y z axis is psi and x and y so I have to convert x and y to r so in two dimensions r square equal to x square plus y square so instead of r I am going to write e to the power minus square root of x square did I say minus then I was wrong x square plus y square actually isn't it x square plus y square let me have a look for a minute just in case I am wrong z equal to e to the power minus square root of x square plus y square this is the plot that I get I will make it a little more good looking so that we see it better frame limits I think we went to 10 up to there so this I will make minus 10 to plus 10 so y will also be minus 10 to plus 10 these are all arbitrary values please do not worry about the actual values here look at the picture I do not have anything in the negative side so I will just make it 0.2 or something positive side I do not have anything beyond 1 actually so I will just make it 1.1 okay so this is your 1s orbital it started looking like it but I will make it look better and the reason why I am doing all this is that you should have an idea of how these pictures are generated so see what I am doing I am coloring this picture of mine according to height so this color unfortunately I am color blind so I will not take the risk of saying that this is red blue and this is red but I hope that is correct you can figure out which color it is all I can say is that it is going from this end to that end of the spectrum as height increases and also to make us understand what contours are I will turn contours on you see this this circle here this is one contour line it joins all the points that are say 0 this one joins all the points that are at some particular height same for all the contours and fortunately I can play around with this I will just increase the number of lines yes this is what I have so this here is my depiction of 1s orbital in a three-dimensional plane that is the diagram that we have in these slides and the good thing about this software is that you can play around with it you can turn it around and look inside it it is basically it looks like a cone now I will turn it around like this do you see what you get now if you look from the top remember when we talked about the island example we talked about a helicopter on top of the island looking down on the hill and then you see a picture like this so it is like a two-dimensional projection of a three-dimensional object that is what we have drawn this here is the contour lines this allows us to represent a three-dimensional feature in two-dimensional paper or screen okay so this is essentially what you have and then of course there is no node here the wave function is always positive or always negative depending on how you define it okay so this is 1s orbital for you we will go back to the presentation so this is the picture that I showed you perhaps with different limits and these are the contour lines well I have created this picture but the contour lines were created many many years ago by my senior colleague Professor Bayou Shashigar I am immensely grateful to him for having created these pictures before when I had means to actually plot them like this so you will see his name on many slides in this presentation and the next okay so these are the contour lines that represent 1s orbital and for good measure we usually write a plus sign here okay so remember these are contour lines and remember the z axis that is pointing towards you is psi it is not the Cartesian z axis it is very important to understand this it is a common source of confusion amongst students right so this is 1s orbital for you let us go over to the 2s orbital in 2s orbital what you have is you have r to the power 0 it is fine 2 minus z r by a a polynomial of first order multiplied by the exponential term the difference between this exponential term and the earlier one is that here in the denominator you have a 2 right so the falloff is supposed to be faster so what do we have here we have a falloff and then this function as we discussed earlier can become 0 at some value at which value r equal to 2 a by z that is where we will get a node and that is where the wave function will change sign right so once again we will just plot it for you and we will make it a little interesting so let us do that we will go to the 2d plot here we have the exponential function already but now I want to divide it by 2 I hope you understand what I am doing here I am setting z to 1 and I am drawing these functions in terms of a board radius a 0 so a 0 is also essentially set to 1 and I am not really considering the normalization constants all the normalization constant would do is just multiply everything it will not change the shape right now I am not interested in values my interest is in shape remember what we had earlier we had a factor a polynomial that polynomial was 2 minus r by a or something like that so I will just make it 2 minus x so x is going to be 2 x equal to 2 is going to be the node right so here goes let us see what the function looks like we will have to change the frame limits this one is fine x 1 maybe I will make it a little bigger maybe I will make it 16 remember arbitrary units do not forget this one will need a little more this one will be I think 4 or something like that let us see that is too much and this is too little this is 2 actually so I will make it 0.4 and I will make it 2.1 now we see it this here is your 2s wave function so the 2s wave function does have a radial node we will go back and we will show it once again on the slides but here you can see the node clearly wave function goes from positive values to negative values through a value of 0 here it is showing 2 actually it is going to be 2 a 0 remember I have plotted this in terms of your a 0 we will come back to this and we will see what the radial probability distribution function turns out to be from here but before that let me show you the more beautiful picture let me show you the 3D picture here so what I need to do is this is divided by 2 that seems to be correct and this has to be multiplied by if you remember 2 minus r once again r is essentially root over x square plus y square if you are talking about a 2 dimensional situation the third dimension is psi so let us see what we get this is what we get we have to change the limits once again remember what we had done we had said this to 16 if I remember correctly in the 2D plot this will be plus 16 y has to be the same thing otherwise it will be distorted and this one I think we had made it 0.4 and this one was 2.1 or something no harm if I say 2.2 this is your 2s orbital in 3D I think you can already see the depression but as usual we are going to color it according to height and we are also going to get the contours now we got the picture that should be easy to understand I will turn it around what do we have in the immediate vicinity of the nucleus is a large positive value or large negative value if you choose to be later on when we talk about hybridization we will actually make this negative it does not matter because for the sine of wave function is only relative and then do you see the depression this is the region where it has gone below the x y plane so it has negative values and then it is gradually recovers and becomes 0 at infinity asymptotically it never crosses 0 so there is we do not have a second node we do do have that in 3s orbital we will see but if we turn around this is the view that I really like do you see do you see the negative part and the positive part this is plus inside outside this is the negative basin and again you can look and inside and you can see do you see these contour lines these are all negative this one at best is 0 these are all minus whatever you have inside is plus so a sharp peak in the middle followed by a basin that is negative and it becomes 0 asymptotically at r equal to infinity this is the 2s wave function for you really like it that we can make this rotate and it can sort of entertain us while we learn something beautiful so this is 2s orbital for you as promised let us go back once to the 2d picture and let us see what r square multiplied by capital R square is you already know the answer but I think if I plot it and if you see it evolving in front of your eyes I think you will understand better so I want r square so I can be a little lazy and just copy paste the same function once again and I have to multiply it by do not forget by r square in this case x is r this is what we get see this is what this is a picture that we have shown you earlier is not it this is your radial probability distribution function and in case you miss something I will just repeat the show for you for now I will remove x square or r square remember where small r square came from it came from the volume element so if I just remove it what I have is we have psi 2s square psi 2s multiplied by psi 2s square of the wave function this is what the square of the wave function looks like where is the maximum where is the minimum I have to change the frame limit once again earlier it was 2.1 so now it should be 4.2 or something see this is psi square for the 2s orbital near the nucleus the wave function and therefore square of the wave function is huge remember square of wave function is probability density not probability the outer loop that we have here is actually very very small and this is positive because I have taken square but the moment I multiply it by the volume element though and behold this small apparently small probability density region turns out to be the region with much greater probability so this is what we have been trying to emphasize probability density is not the be all and end all probability really depends on the volume element as well of course it depends upon probability density but not just probability density okay so let us entertain you a little bit and repeat the same thing here let us see what it looks like in three dimensions please do not forget here the third dimension is actually the wave function itself here I have to write well here I have to write simply x square plus y square why because in two dimensions radius of a circle is given by what r square equal to x square plus y square this radius is the coordinate do not forget here we go looks good does not it go back to our usual way of doing things more contour lines maybe all right so see what is this this is the r value for which the probability of finding the 2s electron is maximum okay you get it by differentiating r square multiplied by capital R square with respect to r and equating it to 0 what do we have inside it is a little difficult to see looks like the crater of a fall canal does not it and our vision is a little messed up by this arrow head that is part of the software let us not bother about it but inside inside the crater like that lava down below do you see the second peak this is the inner peak that you have for r square multiplied by capital R square and that is much smaller than the outer maximum that we have this is a maximum of the inner one at r equal to 0 much smaller compared to this even though the wave function itself if you just plot the wave function itself the square of a function here are the contour lines do you see it well that was bad I do not know if that helped maybe I just stick to this do you see it now if we just take psi psi star or psi square in this case you see the peak inside the one that is close to the nucleus is so much bigger you do not even see the other outer one you even see the outer one this is the outer one very very small compared to this peak but just multiplied by r square the entire shape changes so it is important am I doing multiplied by r square the entire shape changes and the maximum the small band of psi psi star actually turns out to be associated with a greater probability okay so we will close this module here and we will come back with the next one continuing from here