 Let me go ahead little bit around and discuss something more on Hartree-Fock, so some more on Hartree-Fock. So, let us see what more we want to do. And we have particularly discussed RHF, I want to again tell you. We have particularly discussed restricted Hartree-Fock determinant and we have mentioned that this is a singlet state. What is a singlet state or a triplet state means it is an eigenfunction of S square and one of the S operator conveniently it is taken as Sz just like angular momentum. So you have for eigenfunction of sorry, S square, for eigenfunction of S square you have square root S into S plus 1 over h cross and for Sz you have just MS h cross, so these are the two eigenvalues that you get and when you do this S square the capital S or S that comes in the S into S plus 1 is 0. So that is why it is called singlet, okay. Now there is an important theorem for a determinant I might have already mentioned but again I should mention this, when is a determinant spin adopted? So this is called spin adopted, so what is a spin adopted state? The spin adopted determinant or a state is an eigenfunction of S square and is one of the S operators, so that is conveniently Sz operator, okay. So now what we want to say ask the question when is a determinant spin adopted? I hope the question is clear, when is it an eigenfunction of S square and Sz operator? Again this proof is very elaborate but the result is very important. I want to make the result very clear. So when you discuss spin adaptation remember we have to worry about what kind of spin it is, so how many of them are alpha spin, how many of them are beta spins, so that is very important. So obviously we have to write not in terms of general spin orbitals but a space part and alpha and beta attached. So let us assume that in a determinant we have n alpha spin orbitals and this is a very general statement that we are making n alpha spin orbitals of alpha spin. I hope the point is very clear okay and n beta number of spin orbitals which are beta spin such that of course n is equal to n alpha plus n beta that is very clear because you have only two spins. So you have n spin orbitals, I am categorizing the n spin orbitals as sum with alpha spin sum with beta spin okay that is what is that is the only possibility that you have right. And let us say that n alpha number is alpha spin, n beta number is beta spin such that the total n is n alpha plus n beta is it clear? So this is a very general statement I am making in a determinant or a single determinant when I have got chi 1, chi 2 to chi n if I analyze them for RHS of course it is very clear that n alpha is equal to n beta correct and each of them is n by 2 half of n so it n is equal to n alpha plus n beta. So both the number two numbers are identical for RHS but in general I can have n alpha number of alpha spin and n beta number of beta spin then there is a theorem which states which is the following theorem then let us assume one of them is maybe larger than the other so n alpha is greater than equal to n beta let us say I mean it could be the other way round of course equality equal to beta is also assumed so it is either greater or equal to n beta then the theorem says that the determinant will be spin adopted again I told you what is spin adopted. Spin adopted means it is an eigen function of s square and s z so that is spin adaptation okay for singlet the value of s is 0 but n is triplet is also spin adopted so determinant will be spin adopted if and only if this is very important if and only if the lower number so let us say here n beta number of space orbitals or n beta space orbitals with beta spin which means those spin or space orbitals which have beta spin clearly those numbers are n beta or subset or fully subsumed of subsumed in or of whatever of the n alpha space orbitals with alpha spin which means the lower number of spin orbitals have this space part these space parts are complete subset of the larger n alpha space parts I hope that point is clear now what you mean by subset I hope you understand so for example let me give a given example so 1s alpha a determinant not necessarily R H F 1s alpha sorry 2s alpha 1s beta 3 electron does this follow this sorry 2s alpha yeah does this follow this yes because your number of space orbital for beta part is 1s which is completely contained here for alpha you have 1s 2s you have 1s so this ticks this box so this is a subset of this that means these number of space orbitals have to be already contained in the n alpha space orbitals least so they are common in addition of course n alpha will have more because n alpha is greater than equal to n beta in which case determinant will be spin adopted and the value of s and the value of s will then be n alpha minus n beta tell me the value like for example here by 2 and of course ms value will exactly same thing with h cross whatever so this is the strict theorem so I can look at any determinant there are space and spin parts which is alpha which is beta and then as then make a comment if it is been adopted or not if it is been adopted this will be the final s state so then I can say it is a single a double a triple a everything for a determinant so for example I give you a determinant 1s alpha 2s alpha and some 1s prime alpha 1s beta 1s prime alpha or even 2s alpha does not matter is this been adopted why yes answer is yes because this please check n beta 1s is contained here so I wrote something else initially so I wrote 1s alpha 2s beta 1s prime alpha is this been adopted no because so hope you will be able to find out from this spin from the space part and the spins one which is been adopted one which is not spin adopted so if I have in general if I have in general orbitals a determinant of this kind phi 1 alpha phi 2 alpha phi 3 beta phi 4 alpha let us say phi 4 beta where phi 1 phi 2 phi 3 phi 4 are different then it is not spin adopted for spin adaptation in this case it is essential that phi 3 and because there are 2 beta orbitals there are 2 alpha orbital they have to be identical there is no other option so phi 3 and phi 4 must be same as phi 1 and phi 2 of course in this case we know it is a beryllium like system so 1s alpha 1s beta 2s alpha 2s beta so obviously it is spin adopted I have just written it juxtaposedly does not matter so that is a beryllium system so it will be singlet or nothing here okay RHF by construction is been adopted now you know that we have made sure that the alpha spin orbitals are identical as the beta spin orbital so the space part of the alpha spin orbitals is identical to the beta so it is by construction spin adopted and of course S is 0 which means it is a singlet state is it clear alright with this I now present to you two different types of hot reform which are very useful particularly when you go to the open shell systems or sometimes a more general closed shell systems or open shell which you have not discussed please remember our discussion has been based on strictly on closed shell system which is restricted hot reform but there are some closed shell system which does not satisfy this where I cannot make this satisfied so in which case I have to worry about more general systems and also open shell systems I am not going to discuss more about it but before I leave hot refog I must say that there are other kinds of hot refogs so one of them is called the restricted open shell hot refog again many of you might have heard this restricted open shell hot refog so this is called ROHF and the second is unrestricted hot refog which is UHF and of course what we have done so far is restricted hot reform so far what we have done is only RHF but there are two important hot refogs which exist one is restricted but can be applied for open shell ROHF another is completely unrestricted so what is the difference between these two is what I will explain now obviously by very name you can see restricted open shell means it is applied to open shells if it is applied to open shells the first important thing is that N alpha is not equal to N beta that is the first thing that we have learnt because closed shell is N alpha equal to N beta of course open shell can have N alpha number of alpha and beta electron same but then they will have different other different properties so let us say now for a particular case that we are discussing N alpha is greater than N beta so then the restricted open shell hot refog is precisely one which fulfills this condition for spin adaptation that means the number of N beta space orbital is or constitute a subset of N alpha space orbit so this is your ROHF so I will give examples and hence it is of course spin adapted by this theorem now that I have already stated the theorem it is very clear that it is spin adapted so N alpha may be greater than N beta it is greater than N beta but then the N beta the lower ones form a subset of this and of course if you write N alpha greater than equal to N beta then ROHF rather ROHF subsumes ROHF which means ROHF is a special case of ROHF so there is nothing great about what we have done in fact if you write this as N alpha greater than or equal to N beta and write this same statement then the ROHF ROHF is a special example of ROHF because N alpha equal to N beta is ROHF is it clear but in general when you write ROHF we assume that the N alpha and N beta different so I am specifically writing that this N beta constitute a subset of N alpha space orbit so let us assume I have four electrons three of them alpha one of them beta so how do I write this so four electron so I must have one as alpha I can have two as alpha I can have three as alpha but the beta must be one of these three so it lets say one as beta that is ROHF I have four electrons so alphas must have different space orbitals by the Pauli principle of course but the beta space orbital must be one of these three so if I write in a very simple chemist notation so this is 1s this is 2s this is 3s then 1s is w occupied this is singly occupied so that is your ROHF and clearly you can see that is a triplet if I write the diagram you can immediately see the triplet and it shows here because N alpha is 3 N beta is 1 so s equal to 1 and it is a triplet state okay and that is the diagram that it shows so it means that whatever is the lower number must be contained in the space orbitals higher numbers then it is called ROHF and I can use now this for open cell system because I can do triplet for example I can do doublet so three electron that is a doublet right it is ROHF yes yes no problem it will be an excited state yeah so it will have it will have a structure like this if it is 2s beta it is still a triplet but it is not an excited it is not a ground among the triplet this is not a ground state that is a different matter we are not talking about which is the lowest energy yes why we need spin adapted because Hamiltonian it commutes to the square very simple so our Schrodinger equation solution must also be eigenfunctional square very often sometimes we cannot enforce but that is because of the approximation we are not exactly solving but in doing approximations if we can enforce it it is obviously more desirable okay just like we like to enforce all symmetry in a problem so spin is another symmetry note that the actual Hamiltonian physical Hamiltonian commutes to square so all exact wave functions are spin adapted is just that because we are doing approximations we are not taking making attention on these so that is the reason we have to bring back and we have to remind ourselves okay on the other hand unrestricted Hartree form is of course by definition has no such restriction so your n beta space orbital need not be a subset of n alpha one of them can be contained others may not so it can be anything so here if it is let us say 1s prime beta which means your alpha and beta are not exactly sitting here but beta is sitting somewhere here call it 1s prime which is slightly different in energy and obviously this is also not there this is a 4 electron system so in this case it is not spin adapted it is not spin adapted and you can actually see if I write this determinant so if I write this 4 you can immediately tell that it is not spin adapted right because you will not call it triplet but if your triplet is just count out number of alpha minus number of beta then it is a triplet but that is not right you count them out when the lower numbers are already contained which means there is a maximum double occupancy so what does ROHF mean ROHF mean do as much double occupancy as possible whatever is open shell whatever you cannot doubly occupy leave them all in the same spin so for example again I come back to ROHF so let us say I have 3 I said 4 1 if I decide that there are 3 alpha spins and 1 beta spins then what is the maximum double occupancy I can do 1 because I have only 1 beta spin so 1 space orbital I can doubly occupy leave the rest 2 with the same spin whatever is not doubly occupied has to have the same spin so I can redefine ROHF in a different way whatever is not doubly occupied must have same spin that is another way of looking at ROHF I decide to write it in this manner because this is strictly more right but the consequence of this that if there are no doubly occupied orbitals then the electron sitting in all those orbitals must have the same spin then it is called ROHF so in a way ROHF ROHF is maximum double occupancy maximum and quite clearly in this case ROHF is a special case of ROHF because I am doing real maximum double occupancy I have 2 alpha 2 beta okay. UHF I do not put any such attention so 2 alpha can be in 2 special orbitals 2 beta can be in another special orbitals and hence UHF is not spin adapted so that is important to realize that the UHF the unrestricted heart rate for determinant is not spin adapted and this is a special problem of UHF at the same time we must recognize that in terms of energy the same problem if I do with UHF and ROHF or ROHF whatever I am again writing ROHF and ROHF in the same box because it is a special case then this energy will be lower than this energy why because in ROHF and ROHF I am making a condition that the beta space part is same as the alpha space part subset in UHF I have no such condition so what does the variational theorem say if you have more flexibility energy will be lower so that is interesting if you want a lower energy and that is my objective then you should better do UHF right but the problem with UHF is that you don't get the spin so you have to worry about what do you want to do okay each of them is approximation of course there is a way to get spin adaptation after doing UHF and that is called projected UHF I will again not discuss all this but there is a method which is called PUHF which means I first do UHF and from the UHF wave function I project singlet, triplet, doublet so by projection operators so this is a highly mathematical in group theory also we do the same thing so this is called PUHF which also exists in the literature so I just thought you should know the name so PUHF in a way is been adopted without projected so this is been adopted so PUHF is against been adopted but moment you do PUHF you lose that energy variational thing because you can't have both because I am now not taking the full wave function I am taking a part of it so I have lost the original part of the wave function okay so I think this these are some of the Hartree fog that you should know again no more details ROHF, UHF equations are actually very easy to derive because we have the general equations the during the spin integration you should have to just take care with alpha and beta not same so you can easily get an alpha so but there will be two equations one for the alpha spin orbital one for the beta and do that ROHF is somewhat more difficult to understand actually UHF is actually much easier ROHF is more difficult and ROHF is understood in many many ways if you analyze the ROHF determinant there is actually two parts to it one is a close part another is open part if you notice all ROHF one is a close part one is open part because close part is the number of beta which is doubly occupied and the rest is open part so many times ROHF determinant is actually said to have a close part and an open part so if I have a determinant form which is close so there is a doubly occupied all of them are doubly occupied and then you have an open part which have open part essentially means all have same spin either alpha or beta all have same spin so this is one characteristic of all ROHF determinant how many how much is the close part how much is the open part that will determine your spin and that depends on also number of electron because together it must be total number of electrons so there is a very complicated Hartree-Fock calculation for the close part open part they are also coupled the Fock operator for the close part Fock operator for the open part they are coupled so ROHF solutions are much more complex than UHF UHF is actually easy however I lose this spin adaptation