 Now we are going to talk about molecular orbital theory for a very simple system dihydrogen molecular cation H2 plus why H2 plus because it has one electron. Now there are two approaches to bonding in valence bond theory what we do is we essentially extend Lewis electron dot model and we retain this concept of overlap of atomic orbitals and sharing of electron pairs is just that we write it in the language of quantum mechanics and it really works fine for many systems what it cannot handle is delocalization and it gives us very limited access to excited states and it can talk about singlet and triplet state of dihydrogen for example but there are many states that are not accessible. As we will see molecular orbital theory can give us access to many more things. So, in MOT what we do is we consider the electrons to move in the joint field of nuclei I will show you the Hamiltonian shortly. We try to find solutions for H2 plus and when you try to find solutions for this Hamiltonian for H2 plus we use some approximation method we use what is called linear combination of atomic orbitals. Good thing about MOT is that it is a general theory it can easily handle delocalization it does not necessarily have to be restricted to two centers but problem is it is a bit too general as we will see when we talk about H2 as simple as H2 it overdoes things and it over emphasizes the ionic structure. To start with let us write the Hamiltonian for H2 plus the one electron molecule of ours. So, this is a sort of an extension of hydrogen atom like hydrogen atom this also has one electron only the only additional thing here is that it has a second nucleus. So, we can write the Hamiltonian without much hassle first we write the kinetic energy term of the first nucleus HA then we write the kinetic energy term of the second nucleus H B then we write the kinetic energy term of the lone electron that is there we have this electron is attracted by both the nuclei. So, there will be two attractive potential energy terms and finally there is inter nuclear repulsion. Now before proceeding further inter nuclear repulsion can be a potential problem fortunately that is taken care of by this very elegant approximation proposed by Born and Oppenheimer. Now when you say Born and Oppenheimer if you are not someone who is studying science you think of something else you think of Manhattan project where atom bomb was made Oppenheimer was actually the overall in charge of this Manhattan project. But Born and Oppenheimer did many good things individually and together they proposed this approximation which says that this big fat nuclei cannot move as fast as this quick fast electron. So, nuclei are stationary with respect to electrons of course, I have put it in a very watered down almost scandalous manner, but this is what it essentially means it is okay if we understand that for now. We can consider the nuclei to be stationary with respect to the electrons. So, the moment we do that we get to ignore the kinetic energy terms in the Hamiltonian that is a great relief. Moreover, we can consider this last term to be constant. So, essentially we can work with these three terms kinetic energy of electron and potential energy for attraction of the electrons to with the two nuclei and I can take the last part as a constant. So, for different values of R I can set up this Hamiltonian and I can try to find solution and actually using elliptical polar coordinates it is possible to find solution. Those solutions are there in some specialized books not in the books that we study for this course and we are not interested in getting it in it also. You know why because what is the point you do it this is your Hamiltonian fine you solve it and you use spherical polar coordinates to solve it exactly. But what happens if you add one more electron it is no longer solvable because then you are going to have electron-electron repulsion which will mess things up completely. So, why get into all that trouble and people have got into all that trouble as I said they are actually available in some high level textbooks also on internet, but we are not going to get into that we are going to use a simpler approach utilizing certain things that we have generated already hydrogen atom wave functions orbitals. We are going to use linear sums of these orbitals to find approximate solutions. Our interest here is to find electron distribution and energy to do that one common way of doing it somehow I never correct typos that are there for a long time. So, it is not functional it is functions. So, if you want to generate some function a common approach is to take a linear combination of appropriate orthonormal functions it is not even required to take orthonormal functions, but it is better if you take them that makes life a little simpler and what is the meaning of appropriate we will elaborate in a minute. So, here to generate this polycentric one electron wave function I want to essentially take a linear combination of atomic orbitals. See atomic orbitals are they form an orthonormal set is not it and this sort of makes sense because well actually they were generated by considering the motion of electron in the field of at least one nucleus all we have here is that we have two nuclei together. So, in some parts or maybe in large parts the molecule orbital should resemble some atomic orbital or the other or a linear combination it sort of makes sense. And but then whenever I say these students are usually not happy and they say that you are giving some hand waving argument and all and I agree with that. So, let me just take this a little further this is the combination that we are going to use linear combination of two 1s one wave functions one for atom A one for atom B. Now, what I want to show you is this let us take a little break from there and talk about a general phenomenon what do you see what do you see here perhaps you see an elephant that is drawn by maybe a 3 year or 4 year old kid, but the shape of elephant is discernible you see an elephant. Now, this elephant was actually not drawn by any 3 or 4 year old kid. This is a figure taken from a legitimate research paper published as recently as 2010 in American Journal of Physics. How was this generated? So, it is a plot you can see you have x axis and y axis I have tried I tried my best to hide x and y I did hide x, but y was not hidden all that efficiently. So, what these guys did was that they expressed x and y in terms of some parameter t which they call time and so they decided some value of time t for that they worked out x and y. So, for that they got this function for more detail feel free to read this paper it is available freely on internet. So, what I am trying to show is that by using who would think that you can generate an elephant using some functions actually you can and the elephant does not have to be as rudimentary as this one you can draw better elephant also. And again if you go to this website I did not really check it out myself, but I think you get to plot different things try your hand at using different functions this is a more convincing elephant right. This elephant is drawn by maybe a 5 year or 6 year old kid again use generating functions. So, if you can generate an elephant using some mathematical function I should be able to generate the electron distribution also. To be honest I do not even need to use orbitals I can use Gaussian functions and exponential functions and stuff like that and that is what is done in higher level theories. But at least to start we are going to use wave functions. So, this is what you should be looking like at this point should be thing that does all this make sense all right I generate this function and maybe I can find a solution for energy using that Hamiltonian operator. But is that solution right does it make sense at all and here we have a saving race in the form of variation method variation theorem which says that if you calculate energy using some guess function guess function means a function that you have cooked up you call that epsilon 0 phi actually this is called functional epsilon 0 phi minus the actual energy E 0 is always going to be greater than equal to 0 or in other words you get an always get an upper bound to the actual energy if you use some function that you have cooked up. So, whatever function I get I will never do better than the best the energy I calculate will only approach the actual energy from higher energy side it will never cross it and become low. So, it is okay if I use a large number of functions the energy calculation will never be wrong the electron distribution can go wrong if you use 2 orbit functions that is why it makes sense to use atomic orbitals at least to start with. This is a very rudimentary introduction to this topic you might be wondering why it is called variation theorem that is because you actually this gives rise to this is also called the upper limit theorem. It is utilized in variation method where you play around with the contributions of these different functions and you see what is the best value of energy you can get and you have the assurance that it will never be less than the actual value of energy. So, this is where we are we have molecular orbitals we have given you some kind of an argument to say that this is not absolute nonsense we generate the molecular orbitals by linear combinations of these s orbitals. What do I do next? Let us see what your square of this molecular orbital is why am I calling it molecular orbital because it is a molecular wave function and this molecule has only one electron. So, I get c 1 square phi 1 s a square plus c 2 square phi 1 s b square plus 2 c 1 c 2 phi 1 s a phi 1 s b and since it is tiring to say phi phi phi so many times I am going to say just 1 s a 1 s b later on. So, I might even say s a s b or even I might say a and b they are all one and the same please bear with me. Now, see there is no logical reason to think that c 1 and c 2 should have different magnitudes in this expression for energy for probability density there is no reason why a should contribute more or less than b. So, c 1 square must be equal to c 2 square which implies that c 1 can be plus minus c 2. So, when c 1 equal to c 2 I call it c a when c 1 equal to minus c 2 I call it c b actually this is an ill conceived nomenclature it would have been better to call the first one c b it would have been better to call the second one c a you will see why I am saying this. So, this is the first wave function I take the plus combination. So, psi 1 is c a multiplied by 1 s a plus 1 s b the other one is 1 s a minus 1 s b multiplied by c b how do I proceed now the usual way I want to normalize I want to see what is the expectation value of energy and so on and so forth. But before that let me tell you this that this is called a bonding orbital 1 s a plus 1 s b and this is called an anti bonding orbital again I will take a range check on why this is called bonding why it is called anti bonding later on you can sort of guess but do you see why I said that this coefficient naming was ill conceived psi bonding is c a multiplied by 1 s a plus 1 s b psi anti bonding is c b multiplied by 1 s a plus 1 s b if I had only interchanged a and b then it would have been better bonding b anti bonding a would have been easier to remember but when then Ali Iyakhtay asked so we just leave with this. So, we have obtained two kinds of molecular orbitals bonding and anti bonding great this is where we are so far if I plot them this is what you get when they are at equilibrium bond distance there is some overlap of the orbitals. So, in this region you see they actually add in regions where a has some value of psi but b does not have too much then it is practically a here it is practically b. So, when you add them up for psi plus or psi 1 you get this kind of a profile these are the contour diagrams this is a profile like a circus tent and here you see this kind of a profile where you get a node. So, in bonding orbital the wave function has no node in anti bonding orbital you do have a node if you take psi square then you see you get this more well defined circus tent here as the profile here you have two separate circus tents it is not circus tents two separate tents two people who want their own individual tents that kind of a function is there. So, now the rudimentary discussion that is there in most textbooks is that here you end up increasing electron density between the two nuclei. So, they act as cement effectively here you end up decreasing electron density if you just plotted the wave functions they would have had some more electron density between the two nuclei that is why this is anti bonding energy associated with anti bonding should be more energy associated with bonding should be less this is okay but it is not the last word. So, for now we can live with it but there is more to it than what meets the eye. All right now let us try to do our favorite exercise let us try to normalize and here I want you to do the normalization yourself I am sure at this point you will be able to do it. So, please stop the video maybe after the next one this is what you want to get stop the video do the normalization yourself okay this is what you get Ca square multiplied by integral 1sa square plus integral 1sb square plus integral 1sa 1sb plus integral 1sb 1sa okay these as we know are one because the s orbitals are individually normalized what about these two these are not zero okay these are quantities that come up when we do valence bond theory treatment as well the only difference is that for valence bond theory we cannot work with H2 plus we have to work with H2 because it requires to center to electron so what is this integral this is called overlap integral they are the same first of all this sequence does not matter it is called overlap integral why is it called overlap integral this is something that we study when we study valence bond theory as well but here let us state it independently these are the contours of the two 1s orbitals okay what happens to the value of the integral when they are like this very far apart practically there is no overlap so there is no point in space where 1sa and 1sb simultaneously have some nonzero value so their product is zero when the distance is large and integral is also zero what happens when they move closer as overlap increases say in this region let us say that contours denote significant values of psi in this region of overlap both 1sa and 1sb will have significant value product will have significant value so when you integrate the value of the integral is going to increase and as they come closer and closer the value is also going to increase to a saturation something like this and this as is worked out using elliptical coordinates in Macquarie and Sammons book this can be written in terms of the inter nuclear distance capital R this is the functional form which we do not have to remember let we should understand the shape of the curve it is like this initially zero at very large R then it increases to a saturation value okay of course if this is zero that means what they will both be together saturation value what should it be you work out yourself these are normalized orbitals so if capital R is equal to zero this is your exercise please work out what is the value of the overlap integral when capital R is equal to zero and this is what I have shown for 2 1s orbitals you might as well work it out for say s and p orbitals p and p orbitals in different orientations so on and so forth this is your overlap integral which plays a very very important role in bonding now let us go a little further ahead this is what it is I have written the first two terms each of this is 1 add up to 2 second two terms each of these is overlap integrals add up to 2s so this is the condition I get from normalization c square is equal to well 1 equal to c square multiplied by 2 plus 2s now I do not know if you had stopped the video if you would have reached it here if you did not have a prior knowledge of s but well now you know what it is so ca turns out to be 1 by root over 2 plus 2s similarly you can work out yourself this time cb the coefficients for the anti-bonding orbital turns out to be 1 by root over 2 minus 2s so now we have the wave functions completely defined with these what we can try to do is that we can try to find the values of the energies and remember the best we can do now is we can try to find expectation values of the energies