 That brings us to the final question of basis sets which we have been trying to avoid because it is too technical. So basis sets are usually atomic orbitals that we have discussed but of course you can use any basis set. Before I say basis set there is nothing sacrosanct about basis set, chemists like to use atomic orbitals. So I should mention and that is why we have taken a general non-orthanormal basis set but your A mu, A nu could have been anything. So normally we would like to use atomic orbitals and the orbitals which are eigenfunctions of atoms are called the slater orbitals. So it is basically called slater orbitals. So one example is hydrogen atom, you have already done slater orbital. So slater orbital let us say 1s function, let us take a 1s function, a slater function. We need not even worry about 1s. It can be 2s, 1s, they will have a similar structure. So let me call it 1s though. So there is some constant, so let us say some k, it is a proportionality constant, exponential minus some alpha and then you have r minus rA which is the coordinate. So this is let us say 1s function centered on A. So this would be a typical form of a slater function. So this is basically some constant, again I do not want to write the details but there are some constant. If you look at the hydrogen atom it is very similar, you have exponential minus r, exponential minus r. So in general this slater function can be like this. I have written it in terms of a vector again for generality, for hydrogen atom for example you just have r which is a scalar quantity number and then you can write down the integrals. The problem with the slater functions is that they are not very easy to manipulate and that is the reason people go to what is called the Gaussian functions. The Gaussian functions are easy to manipulate. So let me write down the form of a Gaussian function, again barring, so phi Gaussian function S type, so S type centered on atom A again. So again there is some proportionality constant, I will not bother, let us say some k1 and you have some exponential minus alpha, this can change, this is not the same alpha, so let me call this let us say some gamma or whatever, some alpha and then you have modulus r minus rs square, note that this modulus is important to make it scalar. So just as here I made it scalar by taking modulus, of course for hydrogen atom we do not write rA explicitly because my origin is 0, 0, 0, so I do not write so we just get exponential minus r. So the difference between Gaussian and scalar is just this, that you have a square here and you do not have a square here and this makes a lot of difference but we have to understand why should I use a Gaussian because obviously slater functions are better. The problem with the slater function is the following, that when I calculate let us say mu, nu, lambda, sigma integrals, so mu can be centered on A, nu can be centered on B, lambda can be centered on C, sigma can be centered on D, in general when a polyatomic molecule, so these are integrals and each of them can be slater although I have written only S type, you can write P type, D type and all that, P type, D type will have angular parts and so on, it is not important for the understanding. So these kind of integrals are called force center two electron integrals, I hope it is clear why it is called force center because each of the four atomic orbitals are centered on four different atoms, A, B, C, D, so they are called force center two electron integral. These integrals are considerably complex to evaluate because of the fact that each of them has different R A, R B, R C, R D, so they become computationally extremely expensive and that is the reason people did not want to use slater function. On the other hand if I use instead of slater function Gaussian functions, then these integrals should be very easy to evaluate for a very simple reason because Gaussian functions follow a very interesting theorem, so which I will write the theorem. So let us see that, let us say that I have two Gaussians, so let me first state the theorem, so let us say I am only talking of S type Gaussian with a center alpha, with a exponent alpha and a center R A. Now let me write away tell some of the nomenclatures that we are going to use, this is the center, I hope it is clear that is the coordinate, this is what is called the exponent, this is a very important name because it is exponential, under exponential, so this is generally called exponent. So let us say that I have two one Gaussian, S type Gaussian with exponent alpha and a center R A, so I am not going to write the full thing, so that is clear. I multiply by another S type Gaussian which has an exponent beta and center R B, so just as mu and nu, you have a center A, center B, assume instead of slater they are Gaussians, so then what I am saying that the product of this is also a Gaussian and exactly S type with some exponent P and a center R C, some C, R C. Now this is an excellent theorem because this essentially tells that as I keep multiplying, let us say coordinate one, so I am going to multiply mu and lambda, then what will happen is that this into this will become one Gaussian on a particular center, this will become another Gaussian on a particular center because they have coordinates and difference, so I cannot multiply one and two and the entire thing will then become two center, two electron integral. I hope you understand except the centers have changed, so for example what is the center here, so first of all P will become alpha plus beta and that is very easy if you do the multiplication you will see P becomes alpha plus beta and R C becomes alpha, it is a alpha times R A plus beta times R B, it is actually weighted average of R A and R B divided by alpha plus beta, so note that this is weighted by alpha, the other one is weighted by beta, it can be shown that the new Gaussian has an exponent alpha plus beta and it is centered on alpha R A plus beta R B times alpha divided by alpha plus beta, so this property of the Gaussian makes these computations extremely easy because now you have only two center integral, so you have to just multiply your coordinate one A and C, coordinate B and D, so between A and C you will get some center, it does not matter A prime, between B and D you will get some B prime and the entire integral will be a two electron integral on two Gaussians, so there will be no longer four Gaussians because the product is one Gaussian, so I have already simplified, so there will be only two Gaussian one by R 1 2 integrated, so this makes it much simpler to evaluate, much faster, computationally much simpler to evaluate because of this theorem of the Gaussian functions, so that is the reason atomic orbitals have to be used as Gaussians, but how? The question is atomic orbitals are not Gaussians, actual atomic orbitals are slater but you can always say that no, I need a basis, so I do not care, but then the chemistry will say is it a good basis, so we must go back to the atomic orbitals and see what can I do to express atomic orbitals in terms of Gaussians, can I do anything? First of all there are two important differences between a slater and Gaussian functions, so that is something to be understood, so these differences take place at r equal to 0, so typically at the nucleus of that center, r equal to 0 is essentially nucleus here or at r equal to rA and of course at r tends to infinity, so this I can actually say 0 or rA, I mean depending on what is the atom center and when r becomes infinity, so the important property, the slater function derivative with respect to r is actually non-zero, whereas if I do Gaussian function derivative at r equal to 0 with respect to r at r equal to 0, I should write that, at r equal to 0 or rA whatever is actually 0, so this means my Gaussian cannot represent the slater function properly, if this is the physics, this is missing a very important physics at r equal to 0, further the Gaussian function decays more rapidly than the slater function and this has a connotation at r equal to infinity, in the limit r equal to infinity it has a problem, what do you mean by decays? Decays essentially means because it is square, r square, here it is r, so all of you know exponential minus r square will decay much faster than exponential minus r, so if slater decays like this Gaussian will start to decay something like this, so it does not represent the physics of the slater orbital neither at the origin of the atom, so when I say r equal to 0 it is actually rA or r minus rA equal to 0 whatever, nor at r tending to infinity, of course the intermediates are also different, but these two differences become extremely difficult to handle, if I just write MU as a Gaussian then I will miss the physics, so essentially I am not using atomic orbitals and that was the purpose of using the atomic orbitals as the basis, I am not using of course you can argue that does not matter, I just have a basis, but then I do not know how good is the basis and how many functions I have to use, I wanted to use slater only to get a good basis, then of course the chemistry are clever, they said okay let me map a slater, a slater cannot be a Gaussian, but let me map a slater as a linear combination of Gaussian, a linear combination of Gaussian function, so let us say I have a S type slater, can I write this S type slater as a linear combination of several S type Gaussians, now in principle it is possible because they are two different functions, but the idea is again mathematics, I am representing one function as a linear combination of another set, I do not want to bring that linear combination inside the Hartree-Fock, so what we do is that do an atomic calculation beforehand and find out this linear combination and fix it, so this linear combination is pre-decided before I start the calculation, so that today has become part of the basis set, so I will explain that little bit you know without going into details that this becomes a part of the basis set, for each atom I know what is hydrogen 1S or what is carbon 2S, carbon 2P, how to expand that into a combination of Gaussian I first decide even before I go for a calculation, let us say I want to do a methane calculation, I should find out the expansion of carbon 1S, carbon 2S, carbon 2P etc in terms of the Gaussians before I start the calculation, so I will do a separate calculation for atoms, same thing for the hydrogen and these are already pre-decided, people have actually worked out lots of calculations for atoms and have pre-decided, so they then become part of the basis, so then we say that these integrals are slater but we will not use the form of this slater, we will write them now as this combination of Gaussians, each of them, so I will of course have a little bit more problem, the integral will not be so simple but will be linear combination of several Gaussian integrals and each of the Gaussian integrals can be done fast, so the whole thing can still be done quite quickly, so that is the whole idea of the computation, so I will explain that how do I write this as a linear combination, of course the linear combination by itself brings in a new dimension to the basis set, so one dimension was what is MU, atomic orbitals, what atomic orbitals will come to that but each atomic orbital also now I do not know how to represent, how many linear combinations, so that brings a new dimension to the basis set, so then the basis set itself now has two parameters, one is the number of MU, number of atomic orbitals that I want to use and for each atomic orbitals how many Gaussians, these two together becomes my basis set, so what I will do now I will explain in a simple manner by taking examples, so I hope I just wanted to tell you why Gaussians, one Gaussian per se is not good because they have totally different property from the slater but a linear combination is very good, so let us see that I want to do a calculations, I want to go back to my methane calculation as an example and I first decided my basis sets, atomic basis sets are carbon 1s, carbon 2s, carbon 2p, let us say I just stop here, please note however that there is no reason to stop here because when I did a molecular orbital as a linear combination of atomic orbitals it must be as large as possible, so I would in principle like to have carbon 3s, carbon 3p, carbon 3d and so on but let us assume for the time being that I stop here and I must say that you must go at least up to here, otherwise results will be very bad for a very simple chemistry reason that this is the balance, you know you can, you need not worry, so I must at least take up to 2p and then you can take more, so let us assume that I am taking up to 3p and then I would also take in the same spirit hydrogen 1s and then I have hydrogen 2s, 2p etc. I can take more if I want but let us concentrate on this set, so this is basically just the balance, balance only whatever is actually participating in the bond, so let us assume I take this but here itself I have first a choice, I can go further, okay, then my choice is how do I expand carbon 1s, carbon 2s, carbon 2p in terms of Gaussians, how many Gaussians? Similarly hydrogen 1s, how many Gaussians? How do I expand? When I expand in terms of Gaussians remember the Gaussian has an exponent and a coefficient of expansion, so there are 2 parameters, so for example carbon 1s can be written as a linear combination of Gaussians gf, sorry Gaussian function s type with different exponent alpha i, so that is one way to write it, so this is my s type Gaussian with different exponent alpha i, for each alpha i I have a linear combination of di, this i can be let us say 1 to k, so what this essentially says that I have decided to expand the atomic orbital of carbon 1s as a linear combination of k Gaussian functions, k number of Gaussian functions, each of them is of course s type with alpha 1, alpha 2, alpha 3, alpha k as exponents which again I choose, of course center is same, yeah center will be always r a, whatever is the center of this carbon 1s Gaussians will be centered exactly on that, so that is needless to say because obviously otherwise they will not even represent, so I can make k number of Gaussians to represent carbon 1s, then I can represent carbon 2s, for carbon 2s I can include more number of Gaussians or less number of Gaussians, it need not be same number of k, so I can use some other number, so let us say some l whatever or k1, k2 whatever d equal to i equal to 1 to l again di, so Gaussian function s type with some other alpha i and I can keep doing it, so you can now imagine why there are so many basic sets, because first of all I have a choice here and then I have a plethora of choice here how to expand, but these are pre decided, remember these things will not come in the SCF calculation the d and alphas, they are pre decided, so I am going to simply use them in calculation of the two electron integrals or one electron integrals, s type, that is of course by symmetry, I am not going to expand in s types later by p type Gaussian, that is meaningless, it will become actually 0 the coefficient, expansion coefficient, so I am always going to, so if I have p type I am going to use p type Gaussian, more interestingly even before I do that the valence set itself is modified saying that I can use more number of valence orbitals and that is very strange, how can you have more number of valence orbitals, what they do is that for example and this is very typical basis I use a carbon another valence carbon I call it 2s prime and another 2p prime, this is a very typical valence basis set and each of them would be expanded 2s and 2p are normally expanded by the same number of functions, 2s prime and 2p prime are also expanded by the same number of, this is one type of basis set, again it does not matter you know I am just telling you one basis set which I will explain tomorrow, these particular basis sets, so hydrogen also you have on a prime for example, these particular basis sets are called double zeta valence basis set, I think by the name it is very clear, you know most of the names are obvious d z v which means double zeta means I have doubled them, this zeta is just this zeta or I can call it zeta whatever to make sure, slatter is zeta you know it is just an exponent name, I mean these are, so I have 2 zeta functions, each of them will be expanded by any way I want, now normally this is a basis that John Pope, no this is a basis that yeah John Pope and many others actually did, Huzinaga and Dunning was the first to do it, what they normally do they have found that this set of 2s and 2p are expanded by in one particular way number of functions and this is done by another particular way and I will explain that with a specific basis and so on, similarly hydrogen 1s, 1s prime can be different but s and prime and unprime are different, so I will particularly give you one basis set tomorrow, just the number I am not the exact values of alpha that is not important, so to show that how the basis sets have been constructed, yes, yeah so even before I expand them in the Gaussian functions, the valence set itself is doubled simply because I need more functions, so I also said that atom in the basis set more the merrier, remember the simple philosophy, more you have better you are, I can even have carbon 2s double prime another set, carbon 2s double prime, 2p double prime, quite clearly this will become triple zeta, TGV, you have heard all these nomenclatures, so I am just kind of telling you, the POPOLU is somewhat different nomenclature, this was actually done by Huzinaga and Dunning and POPOLU is a different nomenclature, yes, by atomic calculation that is separately done, so atomic hot refog they fit it, basically it is fitting, yes so this is a lot of trial and error have gone on fitting, so parameterization, so first they see that after how many numbers is not important to add more, so I am not going to go, it is a very dirty thing and whole of 40s, 50s people have done so many paper just fitting, how many Gaussians fit into a slater well as well as possible, but again well is a relative term, to you what is well may not be well to me, I will add more, so that is how 631g, 431g, stars all this will come, star star is completely different when you add this, then you become what is called polarize functions, so I will talk about the later, okay, so I think what I am going to do that I have just given a general philosophy, I will take up couple of basis sets, one is Huzinaga Dunning, this is Huzinaga Dunning basis sets, another is POPOLU basis set, they are actually identical in terms of numbers, but this exponents and the coefficients may be different, see 2s and 2s prime have different Gaussians, 2p and 2p prime have different Gaussians, but number of Gaussians between 2s and 2p are same, I mean this is not important, I mean this is what Dunning and Huzinaga use, you can use a different, everybody can have their own Gauss basis set, you can say I do not follow it, but they found that the results are nearly equal, these are all trial and error outcomes, so what I am going to tell there is no rational, there is a computational results which are people have done and there is an outcome which actually tells some famous basis sets which are reasonably good, so I am going to only discuss this, so one is this DGV and TGV type of basis sets which is Huzinaga Dunning, other is this 321G, 431G, 631G kind of basis set which are POPOLU basis set, so what I am going to just discuss is these two types of basis sets, okay, but you can keep on having many, many basis sets and then I will discuss what happens when you add this which are basically polarization functions which you can add to either of these basis sets, okay, so these are the and that should come, because if you want to discuss variety of basis sets it just continues, I mean it is not of interest, but as long as you know that they are by trial and error, people have actually found out and the, there is a basis set library, so you can go and pick up those exponents, yes, yes, yes, yes, yes, yes, yes, yes, no, you can have more, something else, yes, no, accurate is always more, more number, more number and different, why you, yes, you want to make them equal, no, no, obviously they are different, variationally they are more flexible, so energy will be always minimum if you give more and more variation flexibility, moment you are constraining it is not good, but the people constrained simply because if they do not want to have too many Gaussians and it all depends on what is the accuracy of the results, so this is a trial and error, but if you ask me a philosophical question which is better, I would of course say that everything should be different, yeah, ideally everything should be different, yeah, construct what, no, everything can be written in terms of some of the Gaussians, yeah, once I calculate the integrals, these are now my integrals which is, which will be written in sum over all blah, blah, blah Gaussians which I will write, I will expand and those these are known, alphas are known, so they are once and for all construct, constructed, so when I said calculate one and two electron integrals, this means I am actually calculating this, it is linear combination and all that are put at the back end of the computations, so they are all done, first Gaussians are done, multiplied, then I compute this and that's it, finally I use them all the time, I don't you go back to Gaussian again, I don't go back to Gaussian, I just have to do it once, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, so now I don't go back, I have to calculate only once the integral evaluation, so when I gave you a step, calculate one and two electron integral, in that step all this is done, after it is all Slater, oh Slater or whatever, I am not using Slater, okay, in fact there is a name for it I will tell you, contracted Gaussian, these mu nu are no longer called Slater, they are called Cgf and these these functions which are actual Gaussians, they are called Pgf, primitive Gaussians, primitive means raw Gaussians, they are contracted in linear combination, so we finally our basis sets are no longer actual basis of atomic orbital, they are actually basis of Cgf, contracted Gaussians and we assume this later, that's it.