 So, we are discussing the hydrogen problem and we said that if you have a sigma g square which is sigma g1, sigma g2, then the special part of the hydrogen molecule, so this is H2 molecule, the special part contains covalent plus ionic in equal measure, so they are actually same and this leads to the problem that at the large dissociation limit or large H hydrogen distance limit, the results become bad and it actually does not go over to H plus H. So, this is been realized and people have tried to correct this by using UHF, I already told you about UHF. The reason is that if you analyze the problem, one of the problem that I discussed yesterday was that both these are in the same orbitals with the paired spin, whenever you write UHF, the problem is that the electrons are spared as up and down and when I try to use this function for large distance, they remain alpha and beta when it has dissociated. But you know that when it has dissociated, this electron should not know what the spin of the other electron is, they are completely independent, so that independence is lost because of the RHA, so that is one of the problem that happens, particularly when closed shell is breaking up into open plus open, if they break up into close plus close, then it remains spin paired, both, so then this problem does not occur, so that is what we discussed yesterday that whenever closed shell system breaks into open plus open, this problem happens. The other way to look at it is that the covalent character is lost because you have a too much of ionic character, there are many different ways of looking at it, so one remedy is of course to use UHF, because UHF at least removes this spin paired because now I am just saying that they are not in the same orbital, they are in different orbitals with opposite spins, so when they go back, they may go back to up and down spin because here because of the Fermi rule, it is essential that if this is up, this has to be down but in the UHF, this can be up and this can be down but when I take it back, it is possible that this can go up, flip up because they are not constrained by Pauli's tuition principle because they are in different orbitals and I told you UHF has, UHF essentially means the up spins and down spins need not have the same space orbitals, so one way to correct this is UHF and actually UHF gives more or less correct dissociation results but UHF is also single determinant picture, this is also single determinant picture and the single determinant picture gives more or less correct dissociation except that now there is a confusion that what state I am getting because it is no longer spin adopted. So I already discussed yesterday that if I do unrestricted Hartree-Fock, it is no longer spin adopted, so we are not sure what state we are getting, so this is not really a very good way of doing it, so a better way to do it is the following, let me go back to the problem first again, so you had the two state problem, one is 1 SA, one is 1 SB, you have sigma G, you have sigma U and the normal UHF is to two electrons stuck in sigma G, so this is the standard problem and this determinant which I now call sigma G square is not good for dissociation of H2 or any molecule which breaks into open plus open, but it is a singlet state, so can I ensure that I keep the wave function singlet and yet try to dissociate correctly, so let us see if I can do that by forming some other determinants from these two orbitals, so you can see as I discussed in configuration interaction I can have many other determinants, so one of them is that lift one of the electrons to sigma U, so I can have a configuration determinant which is sigma G1, sigma U1, note if I have this configuration how many spin states I can have, how many spin states I can have, 4, the 4 can be written as one triplet and one singlet, is it clear, the two electrons state, so sigma G1, sigma U1 can be written as a triplet state with a symmetric spin part, I think we have done this in the 425 also and a singlet state with an anti-symmetric spin part, correct, note that all singlet state must have anti-symmetric spin part, so it is very easy to form this sigma G1 into sigma U2 plus sigma U1 into sigma G2 that is a symmetric space part into alpha 1, beta 2 minus beta 1, alpha 2, so that will become your anti-symmetric spin part, however this cannot be written as single determinant, that is a problem that we have discussed very elaborately, the triplet of course one of two of them which are sigma G1, sigma U2, alpha 1, alpha 2, beta 1, beta 2 can be written as one determinant and the other one cannot be, MS equal to 0 cannot be written, but this determinant again is not allowed because if you see what is this surface that we are looking at, what is sigma G2 symmetry, what is the symmetry of sigma G2, singlet, capital sigma G, right, I hope all of you know how to write molecular terms involved, right, so if I have sigma G2, I have sigma into sigma that gives you capital sigma state, I have two electrons in G, G into G that gives you grad A and of course it is a singlet, so it is called singlet sigma G state, so I am actually interested in a singlet sigma G state, on the other hand if I make one G and one U, I can get a singlet, but this is no longer a G state, correct, so this state is a singlet sigma U state, this symbol you have, none of you have done this symbol, spectroscopy is not done, this is what is called molecular, like atomic term symbol you have done, right, this is called molecular term symbol, alright, it is very easy to understand, this is a singlet, these term symbols are written as 2s plus 1 on the superscript, then total l value and then j value, okay, then yeah, I mean you can write j value or whatever, you can write the j value, but here I am not writing the j, what is the total spring 2s plus 1 is 1, what is the l value, it is 0, right, so that is capital sigma, just as for atom l value 0 is s, for molecule it is called capital sigma, okay, so that is just the nomenclature and then this grad A and ungrad A comes because both of them are symmetric orbitals, if I put both of them in the symmetric orbital, the result is also symmetric, symmetric essentially means from the origin if I invert, it should remain exactly identical, the phase, so U and G have a opposite phase, so the point that I am trying to say that if I put one of them as U, then obviously G is plus, U is minus, so plus into minus becomes minus and it gets a singlet sigma, so I am not really interested in this, even if I can make a singlet, tripled anyway I am not interested, I am interested in states of the symmetry singlet sigma, so how can I form this, I can form this if I put both the electrons in sigma U, so analyze the determinant sigma U square, now if you see sigma U square they will be again paired, right, both the electrons will go here, so it is a singlet state, it is also capital sigma, right, but now we have a U into U that becomes G, so it is a singlet sigma, so this is a determinant, if I form the determinant that is not the restricted heart rate for determinant incidentally, that is a W-excited determinant in the CI language, but if I form the W-excited determinant by pushing these two electrons from here to here, I get a determinant of the same symmetry, so what I am going to do is to analyze this determinant, so how will it look like, sigma U1, sigma U2 and of course alpha 1 beta 2 minus beta 1 alpha 2, that I am not going to write, that is a singlet formula. Now what is sigma U, it is 1sA minus 1sB, correct, times some constant k1 square into 1sA minus 1sB, just as I did this sigma G square analysis in valence bond picture, I can analyze sigma U square, 1sA minus 1sB, but remember sigma U is minus, sigma G was plus, so expand this now, exactly as we do it, so here k1 square, then you had 1sA1, 1sA2, remember this is 1, this is 2, 1sA2, then you have minus 1sA1, 1sB2, minus 1sB1, 1sA2, plus 1s, 1sB1, 1sB2, okay, so just exactly do the same thing that we did, in the case of sigma G square, they were plus, so everything was plus, all the four terms were plus, right, now because this is minus, two terms are minus, two terms are plus, which are plus the ionic terms, the covalent terms are minus, but again plus and minus is relative, so what it means that the ionic and covalent have different phase, so I can write this as an ionic term minus covalent, whereas sigma G square can be written as ionic term plus covalent term, the sigma U square can be written as ionic term minus covalent term, so while each of them does not give correct result, I can make a combination of these two and I can get purely covalent term or I can get purely ionic terms, in fact what is interesting is that I can make a linear combination of C1 sigma G square plus C2 sigma U square to get anything I want, right, it need not be pure covalent pure ionic, I can multiply this by 0.1, 0.2, this can be 0.8, whatever, so I can keep on mixing the sigma G square and sigma U square such that the entire wave function is normalized, okay and I can get any amount of ionicity and covalency that is required and that is very important to understand because when I have the potential energy surface, remember this is the surface, it goes to hydrogen plus hydrogen, right, V and R, here it is purely covalent, here there is some amount of covalent plus some 0.1, let us say 0.1 times ionic, whatever, so this ionicity and covalency is changing as R in changes, so in fact there is not a fixed value of covalency and ionicity, it keeps on changing and by simply mixing these two I can keep on changing, okay, so this is a very nice way to analyze the chemistry because sigma G square by valence bond is ionic and covalent, but plus sigma U square is ionic and covalent, but with the minus sign and then by different combination of sigma C1 and C2 I can generate different levels of ionicity and covalence, right, I just gave a two extreme example where I can have a purely covalent term, where can I have a purely ionic term and you can see now there is no problem of symmetry because each of them is singlet sigma G square, each of them has a right symmetry, so you have the right symmetry potential surface and that is how a surface can be generated, while doing that I am not going to discuss in detail giving an overview, while doing this now I have already realized that I have gone beyond single determinant, right, this is no longer single determinant, this was single determinant, this is a W-excited determinant, I have already mixed it, so now you see the inadequacy of Hartree-Fock, where Hartree-Fock we said we will try to get best single determinant, but you can only go this far, so that is one example to show why a single determinant does not work, in this case of course UHF you will argue in works, but UHF is not, there is no spin, so if you want to project it you will again get a linear combination of determinants, so the essential story is that as you go forward restricted Hartree-Fock or generally single determinant is good up to a point, it recovers 95 to 97% of total energy which is very nice, yet that 3 to 5% of the total energy that is missing is very crucial, in addition there are problems of dissociation which simply cannot be handled by single determinant, so some of the issues that I want to raise here and so let me write them down what is the problem of Hartree-Fock, so whenever we mean Hartree-Fock we actually mean a single determinant here, so we have a single determinant Hartree-Fock recovers 95 to 97% of total energy, however most of the chemistry deals with difference energies, chemistry deals with the difference energies which are very very small, very small difference energies, so these difference energies cannot be adequately described at this level because they are much less than 3 to 5% of the total energy, so typical example is that I have two quantities E1 and another quantity E2, so I am subtracting these two quantities E1 and E2, these are very large numbers but the difference here delta is very very small and this delta is usually much less than the 3 to 5% of E1 or E2, so this is the typical scenario that you get in chemistry like spectroscopy, you have a ground and excited states of a large system each of them is a huge number but the difference may be you know few electron volt, few kilocalories but this may be million kilocalories, million kilojoules at least, this may be also the order of million kilojoules, so if you have 3% error and you are trying this may be 15, 20 kilojoules, so it is absolutely ridiculous to describe such a small difference by 3 to 5% error of a very large number, this is very very large E1 and E2, so this is a typical problem that happens in chemistry, when you do structure optimization it is exactly the same problem because optimization of structure is also essentially dealing with energy of two different geometries, so that is also difference of two energies, so that they are also the same problem comes up, this is spectroscopy, all spectra essentially deals with difference of energies, optimization of structure has the same problem, in fact binding energy that is another important, so binding energy is also again difference of two such large numbers, in all these cases you can imagine that the 3 to 5% of very large number is not sufficient, the error bar of 3 to 5% is not sufficient, so one needs to go to a much more, much higher accuracy, maybe less than 1%, maybe 0.4, 0.5% of the total energy, so we need to have methods which go much beyond Hartree-Fock single determinant, so that is one problem, the second problem is which you have already recognized is that particularly the RHS restricted Hartree-Fock, restricted Hartree-Fock allows the electrons to be paired and this can actually create problem and one of the things that we saw is the dissociation of hydrogen, we will come back to this problem, this is a much more serious problem, so I do not want to discuss today, the spin pairing of RHA is a problem that we saw in dissociation but this in terms of going beyond has much more serious issues, so we will come back to this problem but since RHA particularly gets electrons to be paired it is usually not good okay and this has a serious problem in dissociation of hydrogen, one of the earliest example where the Hartree-Fock was failed was actually to analyze the nitrogen N2 molecule, the photoelectron spectra, so you know that in nitrogen if you do MO theory you have sigma g and pi u right, at the last two orbitals which are very close line, so the issue is that which is higher energy, so most of the Hartree-Fock theory predicts actually wrong result and that is one of the problems that was actually recognized, even the fluorine binding energy, F2 binding energy was also not predicted correctly, so these are some of the earlier examples where people did Hartree-Fock and did not get correct results alright, so I mean essentially we can keep on giving why Hartree-Fock is not good but I just thought at this point I want to set the stage that we have to do something better and I already told you that if you do for the dissociation of hydrogen mixing of sigma g square and sigma u square you can already go do much better, you can do at least solve the dissociation problem, the covalency and ionicity that I discussed but I think the final message is that we have to do better than the Hartree-Fock and that brings us to the next part of the course which is basically I just call it post-Hartree-Fock theory, so I think all this is just for introduction to this, many people actually find it very demanding but as you will see that it is not very difficult to go beyond single determinant, the only problem is when you go beyond single determinant the chemistry is lost because you had nice n spin orbitals, n electrons that is lost because they have two determinants where orbitals are not different, simply in the case of sigma g square and sigma u square now you have a problem, are the electrons in sigma g or the electrons in sigma u, sometimes they are in sigma g, sometimes they are in sigma u just as you know quantum mechanics superposition that there is a probability, so now the probability comes in in terms of where the electrons are, so the chemistry becomes very difficult but at the same time the situation warrants that we must go beyond single determinant that is something that I want to first tell you and at this point I must tell you that there are lots of literature of course of the post-Hartree-Fock theory, in fact the post-Hartree-Fock theory started almost in late 40s when it was first realized that Hartree-Fock is not good, there was a binding energy of alkali halide crystals you know Wigner was actually you know Wigner was very important, how all of you heard the name of Wigner, he has a very important solid state, Wigner you have not heard, E Wigner is a very good physicist actually, very important physics person, so Wigner student was parol of Lafdine, have you heard of Lafdine, I have already mentioned his name, Lafdine was Wigner student, so in 1948 when Lafdine was doing his PhD thesis in Uppsala, Wigner gave him a problem that find out the cohesive energy of an alkali halide crystal, you understand what is cohesive energy like binding energy okay, some alkali halide crystal and then he tried to calculate the Hartree-Fock and he found that the binding energy as the crystal is increasing, the binding energy is slowly decreasing and going towards 0 within what is called the Hartree-Fock model, then there is a first realization that there is something seriously wrong with the Hartree-Fock and Wigner did not do much beyond Hartree-Fock you know 1940s people did not even know how to go beyond Hartree-Fock, the Lafdine was the one who actually owned up the subject and the subject is of the post Hartree-Fock theory particularly in chemistry started with parallel of Lafdine and I must mention his name historically there are many, many figures that I will mention but Lafdine is a very important figure who said that the energy is not exact, of course everybody knew that the Hartree-Fock is not exact but nobody bothered about it because they thought the approximation is good enough. But Lafdine realized that this approximation is not good enough, he was the first to realize and he defined a very important term which is called the correlation energy, it is important to realize that when he defined the correlation energy the Hamiltonian was did not have any terms which exactly what we are discussing of relativistic origin, so there is no electron spin interaction and so on. So basically what we are discussing is also for a non-relativistic Hamiltonian, so this is where he discussed and he said that the difference of please remember that the Hartree-Fock energy is of course less than or sorry greater than or equal to the exact energy right that all of you realize that it is a single determinant, so CI will give you exact energy. So it is only one determinant, so obviously by variation method this should only be greater than or equal to E0, the Lafdine defined this difference of E0 and E Hartree-Fock, now depending on the sign you can write E0-E Hartree-Fock or E Hartree-Fock-E0 it is a sign problem, so this he said is correlation energy. So he defined for the first time this word correlation that whatever I could not account in the Hartree-Fock energy that difference is called the correlation energy, so this was the first definition of correlation energy which is the exact energy-E Hartree-Fock, if I define like this then correlation energy is negative, if you define E Hartree-Fock-E0 the correlation energy is positive but that is only a question of you know success. So depending on how you define the correlation energy can be positive or negative okay but basically you know this was the first definition of correlation energy and the Lafdine started discussing how to you know how to include this correlation, so this was the first very important attempt and Lafdine wrote several reviews in the early 50s if you look at Lafdine you know that was probably he was the only guy who was talking in quantum chemistry of going beyond Hartree-Fock and even people did not believe him but now of course there is a huge bandwagon and you can imagine for 65, 66 years you know there is so much of all of quantum chemistry is actually this which you normally do not teach alright unfortunately. So I am going to teach half of the course on that at least it demands much more than even a full course at least at the PSD level but you know we will do as much as possible. So the techniques of mathematics were however developed how to go beyond was already developed and one of these has been already in told to you in the form of configuration interaction I have already told you that if you do CI then you can actually approach the exact states right this is what is called CI which is basically in any basis take all possible determinants and make a linear combination right we have already seen this for a two electron problem you remember okay that you expanded then you saw that it is a combination of determinants I will show again. So this is one technique that became very important I am just giving a overview in the next 5, 10 minutes I am not going to do anything today the second technique that came very important was a perturbation theory. So this was already known in fact perturbation theory you have already done even for single particle problem like anharmonic oscillator etc. etc. you can do perturbation you have already done in even 4 to 5 but the same perturbation theory can be applied the question is how? How do I apply because remember in perturbation theory I have to define H0 and V correct you do you remember perturbation theory I will do it all over again you have to define H0 and V how do I define H0 and V so that is a really important problem so that I can address the improvement of the Hartree form these two theories became very very important much later in the 60s and the 70s there are theories which came which were really not variational remember the CI is variational right CI is variational and this is perturbation. So neither variational nor perturbation non perturbation so there are theories which came which were neither variational nor perturbation and these are very very interesting theories so we will also discuss these theories and one of the important theory which has come up today and I discussed yesterday also I just told you is the couple cluster theory which is actually more standard in the range of neither variational nor perturbation. The perturbation theories are known as Moyler-Plazette MP perturbation variations are known as CI so these are the three major class of theories which today handle the correlation. So we will try to discuss one by one since I have already discussed CI little bit I will come back to CI in much greater detail for this many particle problem I think I will start with the perturbation tomorrow I mean it is just choice but if you feel that we should complete CI all over again I can do that but I think we have not done perturbation in this class so probably I will start with the perturbation theory tomorrow. So as you know that when you say perturbation theory first question to ask what is H what is H0 what is V that is the first question to ask and I just want to remind you in the perturbation theory that one very important condition is that the H0 must be such an operator whose solution should be completely known I hope you remember and V is an operator which is significantly smaller than H0 because there are two very important conditions that I have to fulfill. So now that knowing Hartree-Fock how do I do how do I devise my perturbation theory to improve the Hartree-Fock energy so I am going to come to that and this will be basically MP class of theories Moyler-Plazette in fact the perturbation theory itself you will realize there are various shades there are various different types of perturbation theory because there are various different ways I can write H equal to H0 plus V remember in the perturbation theory most important part is to write this split H0 plus V depending on how I choose my H0 and V the perturbation theory will change alright so I have different types of perturbation theory but one that I am going to discuss is called the Moyler-Plazette I hope all of you know how to write Moyler spelling does anybody know MO well they write like this actually okay ELLER Moyler and the Plazette PLE SSE DTE so Moyler please plus try to at least spell correctly Moyler and Plazette they did this perturbation theory which I am going to start tomorrow okay it is called the Moyler-Plazette all of you know MP2 MP2 is basically is this so we will do at least MP2 and tell you how to go beyond we are not going to do MP3 MP4 in this course because that will be too much to expect but at least MP2 level of perturbation theory we will do and I think after that I am going to come back to the CI which is basically variation method to solve in between I am going to discuss the nature of electron correlation and that is a very important part what are the physical things that contributes to this change in energy and Lafdine actually did very good job in fact I will give you some review articles tomorrow of Lafdine maybe since I am in the job I can still finish it advances in chemical physics those who are interested because this is something that I am doing today a very overview I am not going to do it all the time so just please note down those who are interested I think volume 2 page 207 1959 if my memory is still right I have not yet gone to a state where I have forgotten but I think it is a volume 2 1959 it is a very good review article in advances in chemical physics you can see it is a very old article but much of what he learned in 10 years from the time that he did his PhD with Wigner he actually has put down in this review article but I mean I am just requesting you some of you not immediately but little bit later after I have discussed can go because everything that is there in this thing I am not going to discuss obviously because it is just too much outside the syllabus and also the time is not there you will not understand but it is also the time alright I think I close here