 So far we have discussed diatomic molecules. Now let us expand the scope of our discussion a little bit and now we want to talk about not necessarily diatomic, not necessarily linear molecules. We want to talk about little bigger molecules and while doing that we know that molecules have their own characteristic shape right. So this is something that we have studied Gillespie and Nyholm's approach of valence shell electron pair repulsion theory. There we know that we have an AB2 kind of molecule then it is going to be linear. If it is AB3 where A is one atom central atom and B is another atom you can say pendant atom then we will get a trigonal planar molecule AB4 tetrahedral AB5 trigonal bi-planar trigonal bipyramid sorry AB6 octahedral right. Of course these are all regular solids regular molecules and these are all molecules in which you have a central atom holds very nicely for metal complexes for example and the reason why molecules have these shapes please remember because very often by mistake we put the cart before the horse the reason why molecules have this shape is valence shell electron pair repulsion has to be minimized. So if you have a situation in which you have two bonds right this is one bond this is another bond then if they are like this and suppose there is no other lone pair or anything there will be some repulsion then what will happen they will try to open out open out open out when they are at 180 degrees to each other then this repulsion between what we can call bond pair of electrons is minimum that is why AB2 so two bonds means what have central A atom a B atom here and a B atom here this AB2 is going to be a linear molecule. What happens if I have a trigonal planar molecule unfortunately I do not have a third pen so I will just use this so trigonal planar again you have say three bonds you put them at some angle something like this all with like a pyramid they are going to repel repel repel repel and finally if these bonds are all equal equal if it remember we are talking about AB3 kind of molecules which means all the bond pairs are equivalent then the repulsion will be minimum when they are all in a plane and the angle between 2 is 120 degrees so generally when I go to class I do something which I have not been able to do today because well what I do in class is I take balloons I blow up balloons and I tie them up and I show that when you tie up two balloons hold like this you will always get a straight line right two balloons like this then if I tie two more balloons and wrap them together in front of your eyes you get a tetrahedron yeah balloons are balloons they are no hybridization or nothing so what holds for balloons also holds for electron clouds right in air in the balloon that wants to stay in a comfortable manner similarly electron clouds want to stay in a comfortable manner so it is important to understand that VACPR is essentially a steric effect okay loan pairs occupy more volume because they are not bound at one end so loan pair bond pair repulsion loan pair repulsion is maximum loan pair bond pair repulsion follows bond pair bond pair repulsion is minimum but whatever repulsion it is you want to minimize it and that is why the molecules take up whatever shape they take up okay so unfortunately right now we are in lockdown and I do not know where to buy balloons and then I have to blow all of them up by myself you do not have anybody to help so you do it yourself please blow up balloons you might have seen it in parties and all when they tie up balloons they always tie in pairs and then they roll them together and what is the shape you get is a tetrahedron why do you get a tetrahedron because a tetrahedron is the minimum bond pair bond pair repulsion geometry for an AB4 kind of molecule similarly if you roll 2 more balloons there you will get an octahedron okay and so on and so forth you burst one of the balloons in octahedron it will become trigonal bipyperamide tbp in front of your eyes okay unfortunately I cannot show that to you now but I encourage you to please do the experiment yourself and see is a lot of fun you can demonstrate to even it is 5 year old kid and they will be amazed and they will be amused and impressed and well it amuses me every time I do it so I hope it will amuse you as well right so this is the crux of the matter molecules have certain shapes because of VACPR they want to minimize the loan pair, loan pair, loan pair, bond pair, bond pair, bond pair, bond pair repulsions the problem is this if I want to build say a balance bond description of such a molecule how do I do it because I have to use orbitals that are there in the atoms right atoms that consider the molecules how do I do it because for example I want to make I want to talk about this AB3 kind of molecule BF3 let us say Boron has 2s orbital and 2p orbital it does not have orbitals which have angles of 120 degrees between each other so what do I do what I do is that I invoke hybridization which means I mix orbitals and I make new orbitals which gives us the suitable geometry required to minimize the loan pair, bond pair, bond pair, bond pair etc repulsions so please do not say that BF3 is trigonal planar because it uses sp2 hybridization it is the other way round BF3 uses sp2 hybrid orbitals of Boron because it has to be trigonal planar being an AB3 molecule and that is determined by VACPR right but perhaps I have also put the card before the horse a little bit because out of the blue I started talking about hybridization and hybrid orbitals without telling you what hybridization is and what am I talking about well this is the more formal discussion the handsome gentleman you see here is Linus Pauling who has been one of the founding fathers of the field of chemical bonding especially and Pauling as you might know got two normal prices one in chemistry one in peace very interesting life is not a lot of things he wrote this book on chemical bonding which for a long time was a textbook and now it has become a classic you can read it very easily I recommend that you read Pauling's book and Pauling also proposed this vitamin C theory by which he said that vitamin C is something that protects you from a lot of diseases he also tried to work in the structure of biomolecules very illustrious carrier extremely significant contribution to science as long as human civilization exists in this present form Pauling's name will not be forgotten. So hybridization was one of the concepts that was introduced by Linus Pauling so what I said is that we need more effective bonding more effective bonding means in this context we need to minimize these bond pair bond pair or whatever the pulsions to do that since we do not have suitable orbitals we have to produce suitable orbitals by taking linear combinations of atomic orbitals. So this is a schematic energy diagram of 2s and 2p orbitals remember in a multi electron atoms I do not remember whether I said it in as many words but what happens in multi electron atoms is that because of shielding 2s and 2p electrons now have different energies you might remember that sp these are all atomic orbitals and orbitals as you better remember by now me having said it so many times are one electron wave functions. So as long as it is a one electron system 2s and 2p orbitals actually have the same energy but in a multi electron atom many electron atom the extent of shielding of 2s and 2p electron is not same that is why 2p electrons have higher energy than 2s electron. So what Pauling said is that let us mix a required number of orbitals and we can mix them in different ways we can think that we have applied a field which induces mixing. So let us say we have mixed one orbital let us say 2p z orbital generally what we do is we take z to be a unique axis let us say we have done mixing of 2s and 2p orbital and we have made 2 orbitals one is gamma 2s plus beta 2p z the alpha 2s plus beta 2p z the other is gamma 2s plus delta 2p z. So now these are hybrid orbitals and the picture that I am showing you here is a general picture and sometimes people actually contest this picture because we are fixated upon thinking of equivalent hybrid orbitals I will come to that. But please believe me for now that this is this most general picture in which you can apply some kind of a field do something or require the molecule to undergo hybridization and make hybrid orbitals. These are coefficients alpha beta gamma delta depending on the relative magnitudes of alpha beta gamma delta the energies of the 2 orbitals are going to be more or less or equal to each other. So well more about that shortly. See this coefficients depend on field strength and square of a coefficient is contribution of the atomic orbital and hybrid orbital. So we have discussed linear combination already we are going to discuss linear combination in the later stage of this course as well. So what we know is that when we take linear combination and take mod square the contribution comes from mod square of the coefficient. So see in this orbital that I have drawn as lower in energy mod alpha square or alpha square if alpha is real that is going to give me the s contribution mod beta square or beta square if beta is real is going to give me the relative contribution of 2pz. Similarly here gamma square or mod gamma square will give me contribution of 2s mod delta square or delta square will give me contribution of 2pz in this hybrid orbital. Now suppose mod alpha square is equal to mod gamma square that means the orbitals are equivalent that means they have the same contribution from s orbital same contribution for p orbital. As you understand energy of the orbital will depend on since we are mixing now orbitals or wave functions that have little different energy from each other. If the contribution of 2s is more then the energy of the hybrid orbital will be closer to that of 2s that is lower in energy. If contribution of 2s is less then contribution of p is more we say p character is more or p contribution is more then that orbital will be closer in energy to p orbitals. So that would be non equivalent but if what we have shown here mod alpha square is equal to mod gamma square mod beta square is equal to mod delta square then you have equivalent hybrid orbitals. So the hybrid orbitals that we have used many times right from class 11 or class 12 sp2 sp3 sp these are all equivalent hybrid orbitals but it is not necessary that they always have to be equivalent. So in our discussion we will also talk about hybrid orbitals that are non equivalent we actually encountered them in say water orbitals used to used in bond pair you need to such orbitals are equivalent of one kind orbitals for lone pair are equivalent of another kind the 2 types are not equivalent to each other. So this is the most general picture there is no need to think that hybrid orbitals must necessarily always be equivalent if they are not equivalent then you cannot use notations like your sp sp2 sp3 and so on and so forth that is all. Now another important thing to remember is that hybrid orbitals are orthonormal to each other we make them mix them in such a way and we are actually going to do some calculations. So this will become clearer when we do that we mix them in such a way that they form an orthonormal set. So take sp orbitals the 2 sp orbitals are actually orthogonal to each other and they are normalized by themselves that is how we calculate the coefficients in the first place more about that when we come to it. Now let us start talking about the first kind of hybrid well equivalent hybridization that we want to discuss and that is sp or hybridization. How do I decide whether I want sp or sp2 or sp3 rule of thumb number of hybrid orbitals is equal to number of participating atomic orbitals and so if you need 2 equivalent bonds like you do in acetylene then I need 1 s orbital s will always be there I need only 1 p orbital to mix with s orbital so I use sp if you want 3 what did I say 1 if I have 2 bonds 2 equivalent bonds like in acetylene like in acetylene part was correct then we will do sp hybridization because 1 s orbital 1 p orbital makes to give you 2 sp hybrid orbitals and how they look will come to that if I want to talk about bf3 then I need 3 equivalent hybrid orbitals for the 3 sigma bonds. So then I need 3 orbitals 1 is s I need 2 of the piece if I want to talk about methane I need 4 orbitals for sigma bonds 1 s and all the 3 piece like that. Now one thing that we should mention here and this becomes most important because you are also going to talk about molecular orbital theory and we have already we already know about overlap integral from our discussion of your well you did not take the name overlap integral then discussion of many electron atoms please do not forget that s and p orbitals are of the same atom so overlap integral and all those things does not do not even arise. Now let us go ahead. So what we need is 2 equivalent hybrid orbitals of same energy and shape directions of course have to be at 180 degrees to each other because we want to handle a b2 kind of systems. What do I do this is what I need I have an s orbital and I have a p orbital what I am drawing here is your constant probability surfaces not really orbitals. So when you hybridize I want to hybridize in such a way that I will get one orbital pointing this way the other orbital pointing that way. So I want to measure low 1 1 side I want to minor low 1 the other side and this black dot is a nucleus please note that the nucleus is within the minor low is another mistake that we sometimes make we place the nucleus at the node it is at the minor low and I will convince you that the nucleus is in the minor low. So it is very simple I have only 2 orbitals here s and p and when I combine essentially I need linear combination. So when I have 2 functions I can take 2 kinds of combinations 1 with minus 1 with plus and then I want to normalize it. I said earlier that the hybrid orbitals have to form an orthonormal set. So I take psi s plus psi p psi s minus psi p 1 by root 2 is normalization constant please check for yourself that psi 2 is normalized psi 1 is normalized not very difficult to see because after all psi s and psi p by themselves are normalized. So what is your integral psi 2 square and should I write x no I will write psi 2 square d tau but I will write it in a little shorthand notation that will be equal to integral psi s square d tau plus integral psi p square d tau plus integral I am not even writing the constants all right oh no I think I should so 1 by 2 no this is 1 by 2 and this is going to be psi s psi p d tau this is 1 by 2 here and 2 I hope you understood what I did this is a plus b right so what is a plus b whole square that is a square plus 2 a b plus b square right. So this is a square this is 2 a b this is b square but now we are talking about not any a and b we are talking about orbitals. So now see this integral here what is this equal to integral psi s square that is equal to 1 because psi s by itself is normalized. So this thing becomes half second one again psi p is normalized this thing becomes half and here psi s and psi p remember are orthogonal to each other so this integral will be equal to 0 and this is what I said do not get confused with overlap integral we will get something like this later on when we talk about MO okay well even in BBT we have discussed a little bit. So we will do not think this is overlap integral overlap integral arises only when there are 2 nuclei here there is only one nucleus. So this is 0 this is 1 this is 1 this is half this is half so half plus half is equal to 1 so we have proven that psi 2 is normalized similarly you can show that psi 1 is normalized as well and what happens if I try to work out integral psi 1 psi 2 d tau psi s plus psi p this is psi s minus psi p and we have 1 by root 2 1 by root 2 that will be 1 by 2 integral psi s square d tau minus and take this outside the bracket integral psi p square d tau. So now we know this is 1 this is 1 1 minus 1 is 0 so we have proved that psi 1 and psi 2 actually are orthogonal to each other. Linear geometry with hybridized atom at the center this is what we achieve when we use sp hybridization but we have unfinished business I told you that the nucleus actually resides in the minor lobe which means that the way I have drawn it here the minor lobes are actually overlapping with each other I should actually prove it before we close this discussion. So now see contribution from s what is the contribution from s half contribution of p is also half so it is an equal mix contribution is 0.5 0.5 50 percent s character 50 percent p character how did I get half 1 by root 2 square remember coefficient square. Now let us think about the contours for the moment this is the contour of sp hybrid orbital and I show you how we get it this is where the nucleus is as I said the nucleus is actually engulfed by the minor lobe somehow this does not work on my computer but if you go to this link I am not checked it in a while I hope it is still there you can see very nicely how a 2s and a 2p orbital morph into giving you this sp hybrid orbital and the reason why you get a node here and the reason why you get this nucleus engulfed by the minor lobe is that you are working with 2s orbital and 2s orbital itself has a node right. So let us see if you understand that a little better from here now we go back to very early part not very early part sometime I do not remember module 15 16 where we have shown you the plots of orbitals against R as I said these are made by Professor Shashitar about 20 21 years ago. So this is the plot of your 2s orbital right 2s orbital has a radial node goes through goes through 0 changes sign and remains negative and becomes 0 asymptotically. So this is a radial node and I had shown you the 3d picture actually I shown you prettier 3d picture than what you show this black and white picture that I show you here and this is what it looks like and you also know that this is what the plot of this 2p z orbital would be 3d plot psi on one axis x and z on the other 2 axis right we have a hill we have a trough we have a valley. Now let us do something let us take a section let us take a razor blade and cut it from the top let us take a section of this 2s orbital let us also take a section of this 2p z orbital okay you understand what a section is I am sure you take a what should I say take a watermelon and cut it you cut it this way you see a cross section that is an oval you cut it this way you see a cross section that is about roughly a circle. Similarly we are going to take a cross section and this is what we see this here is your 2s right peak at the center falls off goes to the node becomes negative then goes to a minimum and becomes 0 asymptotically same thing on the other side what about 2p z plus slope here then at the nucleus we have a node that is a very important difference between 2s and 2p 2s has maximum at the nucleus 2p has maximum sorry 0 node at the nucleus and there it changes sign. Now do not forget the question we are trying to answer we are trying to answer the question why are we saying that the nucleus is engulfed by the minor lobe. So now look at the periphery of the nucleus and let me acknowledge Professor Sandeep Kar my colleague who actually drew this picture and showed us that this is a good way to explain this this picture I have not seen it any in any textbook all right. So now see near the nucleus what is the value of 2p z close to nucleus it is 0 what is the value of 2s near the nucleus it is very high. So I hope you will agree with me that in the immediate vicinity of the nucleus it is the value of the 2s orbital that is going to predominate as you go further away 2p z can take over in the immediate vicinity the value of 2p 2s orbital is going to predominate what happens if you go far away from nucleus on this side or that side if you go to that this side you see 2p z is plus and 2s is negative they cancel each other you have destructive interference so you eventually reach 0 if you go to this side then both are negative so there is constructive interference. So your minor lobe is here and I hope I have been able to convince you that all around the nucleus it is going to be plus sign which according to this is the sign of 2s orbital near the nucleus and that is the sign of the minor lobe usually minor lobe is showing with minus but do not forget that that is just convention we replace psi by minus psi nothing changes psi square mod psi square remains the same. So this is what gives you this kind of major lobe minor lobe and nucleus is within the minor lobe I hope you have been convinced that this is how it happens. So how does bonding take place you have 2s p orbitals overlapping like this and the remaining p orbitals are available for pi bonding that is why in acetylene you have one sigma orbital then one pi orbital like this one pi orbital like this say pz is used for sigma is for hybridization which is used for sigma bonding px and py or I should let us show like this px and py are used for pi bonding right that is what we have discussed about linear geometry next in the next class we will talk about trigonal geometry and sp2 hybrid orbitals.