 We are in a very interesting juncture of discussing Hartree-Fock equations and how they are solved using self-consistent fields. We have learned what Hartree-Fock equations are. We have written the Hartree-Fock equation for Helium atom right and while doing that we have invoked this effective potential energy of electron 1 at its position r 1 due to the presence of electron and we have written it as integral phi r 2 phi of r 2 star multiplied by phi of r 2 divided by r 1 2 d tau okay. Well I should say dr 2 not d tau okay. What does this mean? It means that phi star phi that is essentially charge density. What is the charge density that is given by phi star phi and the distance is r 1 2. So, that is the interaction between a charge cloud and a point charge right that is the electron-electron repulsion okay. Electron number 1 is taken as a point at r 1. Electron number 2 is taken as a cloud okay. You get similar expressions if you think of say ionic atmosphere in solutions of electrolytes okay very similar kind of thing okay. And then using it we wrote this effective 1 electron Hamiltonian for electron number 1. The change that we see in this 1 electron Hamiltonian is the incorporation of this effective potential energy due to electron-electron repulsion that we have discussed just now and with that we wrote down Schrodinger equation as usual we said this Schrodinger equation is called Hartree-Fock equation. We also said that Hartree-Fock equation is obtained from variation principle by defining energy like this take the wave function as a product of two orbitals work out the expectation value of energy using the Hamiltonian for Helium atom. So, we have not explicitly used that effective potential here okay eventually we will reach the same place okay but this is just what we know for a long long time okay. Then we expand it and some of the double integrals become a single integral one remains a double integral that is called J12 the Coulomb integral the other ones which essentially give your 1 electron energies for electron number 1 and electron number 2 they are called I1 and I2. So, what does I stand for if I write a general index I for 1 or 2 that would give me a energy of a 1 electron system and J12 is the electron-electron repulsion term okay that is what we have that is how far we got in the last class. Now we learn how to invoke what is called self-consistent fields and try to solve these equations okay. This is the beginning of a very what should I say ubiquitous technique that keeps on coming back in the field of computational chemistry but we will talk about that in a more appropriate time for now what we have is we have this orbital well product of orbitals that is the wave function that we have said. So, we are working under orbital approximation and so if I remember you the way Atkins puts it is very nice Atkins says that the meaning of this is that each electron is in its own orbital right. So, electron number 1 its coordinates are defined by its coordinates only electron number 2 well sorry what am I saying its wave function is defined by its own coordinates only phi is only a function of R 1 nothing to do with R 2 phi of R 2 is only a function of R 2 nothing to do with R 1 right. So, we are pretending as if the electrons move around in whatever way they move around that we cannot really talk about trajectories but their movement of course there is a repulsion and the wave function gets corrected and all that but then they move around as if the other one is not even there when they come close together they repel each other when they go far away they repel each other to a lesser extent all right but there is no correlation in the movements of the 2 electrons that is what orbital approximation essentially suggests whether that model holds or not let us hope we will get there by the end of this class all right and we have talked about the Hamiltonian already make this Hamiltonian operate on this wave function and we get this equation and the first question that we will answer actually is what is epsilon 1 all right and this is the effective potential that we talked about earlier great. Now, let us think a little bit about what the functional form of this R dependent wave function should be well not R dependent R remember is a combination of R theta and phi. So, remembering what we had done for hydrogen atom wave functions we can write it as a product of an R dependent part and a theta phi dependent part capital R multiplied by y. So, capital R would lead to a quantum number n the only thing is that eventually n becomes a variational parameter and the theta phi dependent part is associated with the quantum numbers l and m. So, the moment I write it like this I have to do something to the Hamiltonian right I should actually write the Hamiltonian in terms of spherical polar coordinates also and if you remember what we did in case of hydrogen atom we do not we had showed you 13 will we rush through 13 slides which we did not discuss and that tells us how to generate the Hamiltonian in terms of R theta and phi and then we did separation of variables. So, when you do all that I am not going through the steps again they are quite mundane and you do not have to remember the steps you do not even have to remember the final answer. But I hope you will not have problems in believe me if I write that the equation that we end up with for the R dependent part and we do not bother about the theta phi dependent part because we are discussing helium in case of helium ground state only 1s orbitals are involved they are theta phi independent. So, who cares about theta phi right now. So, we get this equation in R dependent part the first term and the second term the second term remember what this L into L plus 1 is that is beta if you remember the separation of variables in case of hydrogen atom that we have performed this beta turned out to be that L into L plus 1. So, exactly similar treatment but for helium not for hydrogen this is what we have got then what have we got here this R of R 1 is a wave function in terms of one coordinate that is R 1 then we have something like energy epsilon or epsilon 1 that we have written here I here I have forgotten to write that subscript 1 and we are an operator that operator essentially what is the operator some differential d dr kind of thing is there. So, it is a differential equation. So, this is a time when we should be really happy and say hey we got a differential equation we have solved so many or well at least we know the solutions of so many this can be solved as well look up a book it can be solved this is a problem. The problem is that in this differential equation in the operator part itself you have a contribution from the wave function and that sort of landser landsers in a little bit of a fix because how are we supposed to get the wave function by solving the equation. But then how do you solve the equation if the wave function is sitting in the form of the operator itself. So, this is sort of a egg and chicken problem which came first egg or chicken or is the egg inside the chicken or what. So, we got a problem we cannot really solve it as easily as we solve the differential equation for the hydrogen atom problem. But we have come a long way from hydrogen atom problem already and we know many ways in which we can at least have an intelligent guess of wave functions. So, since we do not know what the wave function would be and that knowledge is required in order to write the Hamiltonian itself we can make an intelligent guess and we have already discussed at length how many different kinds of trial functions are used and what we discussed are actually very classical kind of trial functions there are many more that came later. So, we make an intelligent guess of an appropriate wave function and then we evaluate this U effective with it that now becomes a number. Now, you can solve this particular differential equation for the trial wave function. What do you get now you get a new set of wave functions will they be exactly the same as a trial wave function not necessarily this is the beauty of this technique. So, you guess some function. So, what you do essentially is that you kind of come up with a rough model of the potential energy with that solve this Schrodinger equation kind of thing and get what the wave function is that wave function will be for the system that you have modeled using that rough potential energy and mostly that wave function that you generate is going to be slightly different usually not very different but slightly different from the trial wave function. So, solve the equation and to get what I call improved values of wave functions wave functions that are closer to reality what do you do then we will do it again. Now, using those wave functions newly generated wave functions plug them into the expression for the effective potential write the equation once again the Hamiltonian will change solve it the solution will also change a little bit most likely and you are going to go towards better and better and better match to the reality of wave function remember building that elephant perhaps you start with a mouse the wave function that you choose and slowly as you add terms and all that mouse becomes fatter and fatter and fatter starts looking like a pig and then like a tapir or an anteater which has a small snout and finally it looks something that is close to the elephant maybe a little thin elephant since you started from a mouse so that is the idea. So, when there is not much of change in successive iteration then you have what is called self consistent field the field that you have generated is now self consistent and that is as good as you can get using the wave functions that you have chosen to work with. So, the orbitals that you thus generate are called Hartree-Fock orbitals and usually the guess function that you use are linear combinations of slater orbitals. Remember construction drawing of elephant problem there is no restriction on how many functions we want to use how many coefficients we want to use you just play around with them until we get a suitable match. So, you see you may not end up getting a proper basis set to start with suppose you start with once later orbital and you might get what looks like convergence but that may be far away from reality. So, you want to increase the number of terms. So, it is not just iteration with one trial function you want to play around with trial functions as well that is what makes it a numerical computational problem. Analytically if you want to do it by hand it will take ages and you get frustrated and not do it that is why you want computers to do it for you and that is what leads to this very sophisticated software that is written and is now available some for a charge some in free domain in which with which you can do this computational chemistry calculations and their choice of basis is of utmost important importance not all basis work equally well for all problems. So, here at all level for now we talk about linear combinations of slater orbitals as the initial guess and then we keep on using this self consistent field method until we get convergence. This is called the Hartree-Fock-Ruthan method which we might mention in the next class I am actually in two minds because we are running out of time we want to talk about molecules not much time is left but we have to finish what is started let us see. That being established we want to now focus on this chap epsilon 1, epsilon 1 looks like energy and we get it by as an eigen function of an equation that looks like Schrodinger equation one electron Schrodinger equation is it energy of the atom is it energy of the orbital or is it something else what is it to know that well it is called orbital energy its expectation value is called the orbital energy and this is how you define it of course. Now how do you evaluate this again you can expand so this will be integral phi of r 1 star multiplying minus half del 1 square operating on phi of r 1 plus integral phi of r 1 multiplying this u 1 effective r 1 and so on and so forth. So, what you get is sorry I forgot to say that in the first integral you include this minus z by r 1 as well since I forgot I will write. So, this becomes integral phi of r 1 star please do not forget that when you write a wave function in the bra vector you are writing its complex conjugate it is minus half del 1 square minus z by r 1. So, this is a Hamiltonian that you would get for one electron system pretending as if there is no other electron that operates on phi of r 1 that is my first integral the second integral is integral phi of r 1 u 1 effective of r 1 we do not need to write that bracket phi of r 1 something like that. So, essentially you get j 1 i 1 plus j 1 2 I hope it is not very difficult for you to see that this u 1 effective actually contains this phi r 2 star 1 by r 1 you might be wondering the way I have written it here in my bad handwriting how am I equating this to j 1 2 that is because you have to plug this in there as well u 1 effective of r 1 so all this will go in there that is how you will get it this is the energy of the helium atom actually it is not because if you remember what we discussed earlier energy of helium atom turned out to be i 1 plus i 2 plus j 1 2 right remember that variational treatment that we did before this that is what we get so it is not the same as epsilon 1 so epsilon 1 is not really the energy of helium atom what is it subtract one from the other epsilon 1 turns out to be e minus i 2 or I can write e equal to epsilon 1 plus i 2 something like that whatever way you want to write what is i 2 remember i 2 is i 2 is the expectation value of energy for helium right helium no expectation value of a one electron system with the same atomic number as helium what would that be helium has two electrons so if I remove one of those electrons I get a one electron system with the same atomic number as helium what is that is helium plus ion so this i 2 essentially is the energy of helium plus same way you can define i 1 also so but there is a catch here it is energy of helium plus but using a Hartree-Fock orbital not using your hydrogen atom orbital using a Hartree-Fock orbital that we have discussed so if we just plug that value what does it turn out to be so ionization energy right this minus epsilon 1 turns out to be the ionization energy of helium what is ionization energy energy of H e plus minus energy of your helium atom so essentially it is going to be minus of right hand side so minus epsilon 1 so we see that this epsilon 1 that we got well the negative of that gives us a good measure of the ionization energy of helium and that is the celebrated Koopman's theorem do not forget that the approximation that we use here is that the same set of orbitals can be used for neutral atom and the ion which may or may not be correct or necessary but then when you do a calculation this is from Clementi's work of 1965 the value of ionization energy that we get is 0.919796 atomic unit and from experiment it is 0.904 atomic units so not very bad actually so Koopman's theorem is something that forms a cornerstone in discussion of quantum mechanics of many electron systems we end the discussion today with another very very interesting perhaps intriguing concept and that is of correlation energy remember we had touched upon this a little while ago we said that the moment we use orbital approximation we essentially are saying that the electrons are uncorrelated the motion is uncorrelated their expressions might get modified a little bit of course they depend each other but they move in the same way as they were moving when the other one was not present so all this Hartree-Fock calculation that we do is for uncorrelated electrons so the energy that we get during the calculation is minus 2.8617 atomic unit when we do a more exact calculation when I say exact calculation of course you cannot solve it like that but when you go back to what we had discussed a couple of classes ago that perturbation theory using 13th order perturbation or that variation method using a large number of terms like 1000 something terms remember then the value that we had reported or not reported the value that we had seen at that time was minus 2.9037 atomic units so this is as good as it gets so using Hartree-Fock method the energy that we get is more than that minus 2.8617 atomic unit okay but then see it is more not because we have not tried to account for these repulsions and all you actually work very hard right we have tried to modify the wave function we have tried to play around with the field right we have done SCF calculation so the only thing that we have not done is that we have not considered the correlation between electrons we have not considered that motion of one electron can affect the motion of the other what does it mean you know sometimes celebrities do not like each other right so in a party if the moment one celebrity enters another celebrity leaves by the other door that is correlation two electrons behave like that in an atom so otherwise they would move in whatever way but now if they happen to come close together if they see that they are going to come close together maybe they avoid each other that would be correlation what we are saying is that this difference in energy what is the difference in energy correlation energy turns out to be minus 0.0420 atomic unit you think that is a small number it is not it is like 1.14 when minus 1.14 electron volt significant for the systems that we are talking about so what we learn from here is very interesting electrons are the celebrities of atomic world they find a way of avoiding each other when they are in the same atom and the energy that they save by avoiding each other is about 1.14 electron volt get what I am saying if you do not account for correlation then the value that I get is minus 2.8617 when we account for correlation unknowingly when we do those exact calculations we are not really explicitly saying that okay electrons are correlated all we are doing is that we are trying to minimize the energy and your upper limit theorem tells us that you can get as close to the actual energy as possible and actual energy would involve correlation even though we cannot calculate it right so the difference gives us the correlation energy so electrons to this two electrons of helium end up saving 1.14 electron volt energy for the atom by moving in an intelligent manner and avoiding each other I strongly suggest that you also read this we have followed McQuarrie's approach but please also read this from Piller's book their notation is a little different so you have to go back and read a little more and in fact they have used something that we thought we will discuss but then for the want of time we are not getting into they use something called Virial theorem okay using Virial theorem they have reached the same conclusion and they have used some very nice language which are now forgotten but it essentially says electrons in that atom find a way of avoiding each other smart particles right that is the in my opinion most interesting thing that we have learned in our discussion today.